Characterization, Modeling, and Control of the Nonlinear
Actuation Response of Ionic Polymer Transducers
by
Curt S. Kothera
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Mechanical Engineering
Donald J. Leo, Chair
Daniel J. Inman
Seth L. Lacy
Harry H. Robertshaw
William R. Saunders
Craig A. Woolsey
September 2005
Blacksburg, Virginia
Keywords: ionic polymer transducer, nonlinear, characterization, modeling, control, IPMC
Copyright by Curt S. Kothera, 2005
Abstract
Characterization, Modeling, and Control of the Nonlinear Actuation Response of Ionic Polymer Transducers
Curt S. Kothera
Ionic polymer transducers are a class of electroactive polymer materials that exhibit coupling
between the electrical, chemical, and mechanical domains. With the ability for use as both
sensors and actuators, these compliant, light weight, low voltage materials have the potential
to benefit diverse application areas. Since the transduction properties of these materials
were recently discovered, full understanding of their dynamic characteristics has not yet
been achieved.
This research has the goal of better understanding the actuation response of ionic
polymers. A specific emphasis has been placed on investigating the observed nonlinear
behavior because the existing proposed models do not account for these characteristics.
Employing the Volterra representation, harmonic ratio analysis, and multisine excitations,
characterization results for cantilever samples showed that the nonlinearity is dynamic and
input-dependent, dominant at low frequencies, and that its influence varies depending on the
solvent. It was determined that lower viscosity solvents trigger the nonlinear mechanisms
at higher frequencies. Additionally, the primary components of the harmonic distortion
appear to result from quadratic and cubic nonlinearities.
Using knowledge gained from the characterization study, the utility of different candidate system structures was explored to model these nonlinear response characteristics.
The ideal structure for modeling the current-controlled voltage and tip velocity was shown
to consist of an underlying linear system with a dynamic input nonlinearity. The input
nonlinearity is composed of a parallel connection of linear and nonlinear terms, where each
nonlinear element has the form of a Hammerstein system. This system structure was validated against data from measured time and frequency responses.
As a potential application, and consequently further validation of the chosen model
structure, a square-plate polymer actuator was considered. In this study, the plate was
clamped at the four corners where a uniform input was applied, measuring the centerpoint displacement. Characterization and modeling were performed on this system, with
results similar to the cantilever sample. Applying output feedback control, in the form of
proportional-integral compensation, showed that accurate tracking performance could be
achieved in the presence of nonlinear distortions. Special attention was extended here to
the potential application in deformable mirror systems.
iii
Acknowledgments
I first would like to acknowledge my academic advisor Dr. Donald J. Leo for his invaluable
support and guidance in performing this research, editing draft versions of this document,
and making himself available to me when he was busy with several other things. The
opportunity he has provided me has positively affected my life and will continue to do so in
the future. I would also like to thank Dr. Daniel J. Inman, Dr. Harry H. Robertshaw, Dr.
William R. Saunders, Dr. Craig A. Woolsey, and Dr. Nakhiah C. Goulbourne for serving
as my committee members and offering their advice.
I must also acknowledge the direction provided to me from Dr. Seth L. Lacy and
Dr. R. Scott Erwin of the Air Force Research Laboratory’s Space Vehicles Directorate.
Spending a summer working with them early on in my doctorate studies opened new doors
to my research and formulated the basis for much of the research that was performed.
Additionally, I would like to thank Dr. Lacy for his continued influence on my work and
for serving as a part of my committee from outside the university.
As for my colleagues in the Center for Intelligent Material Systems and Structures
at Virginia Tech, I need to express my gratitude to all of them for the assistance they
provided me with various problems I ran into along the way and the help they gave me in
the laboratory. Most specifically from the polymer group, I would like to acknowledge the
newly appointed Dr. Barbar J. Akle, Matthew D. Bennett, Miles A. Buechler, and Kevin
M. Farinholt for their daily support and many conversations.
Since funding was critical in allowing me to carry out my graduate research, I need
to thank the National Science Foundation. Also for supplemental support, I would like to
thank the Virginia Space Grant Consortium.
And last, but not least, I would like to thank my family and friends for putting up
with me through my extended academic career. I have seen my older sister, Connie, and
iv
my younger sister, Carla, both get married and start their “real” lives during my tenure at
Virginia Tech, and now I can say that I am finally finished with school. Mom and Dad, you
should be especially relieved because I will no longer need to use your house as a storage
facility for all my stuff. Thanks everyone!
v
To my parents,
Sharyn R. Kothera and Clark S. Kothera
Contents
List of Tables
xi
List of Figures
xii
Nomenclature
xvii
Chapter 1 Introduction
1.1
1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
History of Ionic Polymers . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.2
Actuation Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Proposed Ionic Polymer Actuator Models . . . . . . . . . . . . . . . . . . .
7
1.2.1
Black-Box Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2.2
Gray-Box Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.3
White-Box Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.2.4
Modeling Comments . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Improvements to Transducer Materials . . . . . . . . . . . . . . . . . . . . .
23
1.3.1
Electrode Improvements . . . . . . . . . . . . . . . . . . . . . . . . .
24
1.3.2
Counter-ion Research . . . . . . . . . . . . . . . . . . . . . . . . . .
24
1.3.3
Solvent Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
1.3.4
Ionomer Research
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
1.3.5
Other Fabrication Improvements . . . . . . . . . . . . . . . . . . . .
27
1.3.6
Control Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
1.4
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
1.5
Motivating Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
1.6
System Identification and Modeling Procedure
30
1.2
1.3
vii
. . . . . . . . . . . . . . . .
1.7
Technical Objectives and Contributions . . . . . . . . . . . . . . . . . . . .
32
1.8
Document Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
Chapter 2 Motivation and Volterra Analysis
35
2.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.2
Nonlinear Identification Technique . . . . . . . . . . . . . . . . . . . . . . .
40
2.2.1
Volterra Series Fundamentals . . . . . . . . . . . . . . . . . . . . . .
40
2.2.2
Comparison of Linear Identification Methods . . . . . . . . . . . . .
42
2.2.3
Related Volterra Research . . . . . . . . . . . . . . . . . . . . . . . .
44
Analysis Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2.3.1
Linear System Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.3.2
Nonlinear System Analysis . . . . . . . . . . . . . . . . . . . . . . .
48
2.3.3
Methodology Comments . . . . . . . . . . . . . . . . . . . . . . . . .
58
Experimental Identification using the Volterra Series . . . . . . . . . . . . .
59
2.4.1
Actuator Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
2.4.2
Laboratory Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
2.4.3
Experimental Identification Results . . . . . . . . . . . . . . . . . . .
62
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
2.3
2.4
2.5
Chapter 3 Characterization of the Nonlinearity
74
3.1
System Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
3.2
Single-Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.2.1
Laboratory Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.2.2
Experimental Volterra Results . . . . . . . . . . . . . . . . . . . . .
78
3.2.3
Harmonic Distortion Analysis . . . . . . . . . . . . . . . . . . . . . .
81
Multiple-Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
3.3.1
Multisine Excitations . . . . . . . . . . . . . . . . . . . . . . . . . .
84
3.3.2
Illustration of Multisine Analysis . . . . . . . . . . . . . . . . . . . .
85
3.3.3
Laboratory Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.3.4
Experimental Multisine Results . . . . . . . . . . . . . . . . . . . . .
89
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
3.3
3.4
viii
Chapter 4 Nonlinear Identification Techniques
94
4.1
Nonlinear System Structures . . . . . . . . . . . . . . . . . . . . . . . . . .
94
4.2
Multi-Input/Single-Output Technique (MI/SO) . . . . . . . . . . . . . . . .
95
4.2.1
MI/SO System Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.2.2
Example MI/SO Identification . . . . . . . . . . . . . . . . . . . . .
103
Dynamic Input Nonlinearity Technique (DIN) . . . . . . . . . . . . . . . . .
107
4.3.1
DIN System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
108
4.3.2
Example DIN Identification . . . . . . . . . . . . . . . . . . . . . . .
111
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
4.3
4.4
Chapter 5 Dynamic System Modeling
116
5.1
Data Collection and Modeling Preliminaries . . . . . . . . . . . . . . . . . .
116
5.2
Modeling the Electrical Response . . . . . . . . . . . . . . . . . . . . . . . .
117
5.2.1
Impedance – MI/SO Identification . . . . . . . . . . . . . . . . . . .
118
5.2.2
Impedance – DIN Identification . . . . . . . . . . . . . . . . . . . . .
124
5.2.3
Impedance Model Improvements . . . . . . . . . . . . . . . . . . . .
130
Modeling the Mechanical Response . . . . . . . . . . . . . . . . . . . . . . .
134
5.3.1
Deformation – MI/SO Identification . . . . . . . . . . . . . . . . . .
135
5.3.2
Deformation – DIN Identification . . . . . . . . . . . . . . . . . . . .
138
5.3.3
Deformation Model Improvements . . . . . . . . . . . . . . . . . . .
145
5.4
Modeling Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
149
5.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
151
5.3
Chapter 6 Position Control of a Square-Plate Actuator
6.1
6.2
6.3
152
System Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . .
152
6.1.1
Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
152
6.1.2
Nonlinear Characterization . . . . . . . . . . . . . . . . . . . . . . .
155
6.1.3
Deflection Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . .
156
Output Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . .
164
6.2.1
Linear Control Simulations . . . . . . . . . . . . . . . . . . . . . . .
164
6.2.2
Experimental PI-Control . . . . . . . . . . . . . . . . . . . . . . . .
168
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
174
ix
Chapter 7 Summary and Conclusions
176
7.1
Research Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
176
7.2
Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
178
7.3
Future Work Recommendations . . . . . . . . . . . . . . . . . . . . . . . . .
179
Bibliography
181
Vita
192
x
List of Tables
2.1
Summary of Actuator Compositions. . . . . . . . . . . . . . . . . . . . . . .
60
2.2
Nonlinear Volterra model error for Sample 1 (values in percent, %). . . . . .
66
2.3
Linear Volterra model error for Sample 1 (values in percent, %). . . . . . .
66
2.4
Nonlinear Volterra model error for Sample 2 (values in percent, %). . . . . .
70
2.5
Linear Volterra model error for Sample 2 (values in percent, %). . . . . . .
71
3.1
Summary of Actuator Compositions (viscosity values at 25o C). . . . . . . .
77
3.2
Volterra model errors for Sample 1 (values in percent, %). . . . . . . . . . .
79
3.3
Volterra model errors for Sample 2 (values in percent, %). . . . . . . . . . .
79
3.4
Volterra model errors for Sample 3 (values in percent, %). . . . . . . . . . .
80
3.5
Multisine excitation design parameters. . . . . . . . . . . . . . . . . . . . .
88
5.1
MI/SO Voltage Model Parameters (Hz). . . . . . . . . . . . . . . . . . . . .
120
5.2
DIN Voltage Model Parameters for H(f ). . . . . . . . . . . . . . . . . . . .
125
5.3
DIN Voltage Model Parameters for A(f ). . . . . . . . . . . . . . . . . . . .
130
5.4
DIN Velocity Model Parameters for H(f ). . . . . . . . . . . . . . . . . . . .
139
5.5
DIN Velocity Model Parameters for Ai (f ) (Hz). . . . . . . . . . . . . . . . .
143
5.6
DIN Improved Velocity Model Parameters (Hz).
. . . . . . . . . . . . . . .
146
6.1
Plate Model Parameters for H(f ). . . . . . . . . . . . . . . . . . . . . . . .
159
6.2
Plate Model Parameters for input system (Hz). . . . . . . . . . . . . . . . .
160
6.3
PI-Compensator design summary. . . . . . . . . . . . . . . . . . . . . . . . .
166
6.4
Controlled step response results. . . . . . . . . . . . . . . . . . . . . . . . .
172
6.5
Controlled smoothed pulse response results. . . . . . . . . . . . . . . . . . .
174
xi
List of Figures
1.1
r
Chemical structure of the ionomer Nafion
. . . . . . . . . . . . . . . . . . .
2
1.2
Drawing of polymer transducer components. . . . . . . . . . . . . . . . . . .
4
1.3
Large bending deflection of an ionic polymer actuator in the cantilever configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
6
Linear model responses: (a) step response model; (b) frequency response
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.2
Combined model predictions: (a) step response; (b) sinusoidal response. . .
37
2.3
Nonlinear response characteristics of ionic polymer actuator: (a) harmonic
distortion; (b) amplitude scaling. . . . . . . . . . . . . . . . . . . . . . . . .
2.4
38
Step response comparison of different solvents: (a) deionized water; (b) ionic
liquid (EMI-Tf). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
2.5
Comparison of some linear identification techniques. . . . . . . . . . . . . .
43
2.6
Example prediction and validation data sets used for model identification. .
46
2.7
Average validation error versus model order (M ) and degree (N ). . . . . . .
47
2.8
System structures used in the nonlinear analysis: (a) total system; (b) subsystem 1 input and outputs; (c) subsystem 2 inputs and outputs. . . . . . .
49
Input and output signals for the two subsystems. . . . . . . . . . . . . . . .
50
2.10 Average validation error for subsystem 1 (voltage-to-current) simulations. .
51
2.9
2.11 Volterra results for subsystem 1: (a) prediction and validation; (b) power
spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
2.12 Average validation error for subsystem 2 (current-to-velocity) simulations. .
55
2.13 Volterra results for subsystem 2: (a) prediction and validation; (b) power
spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
56
2.14 Forward running sum RMS of each subsystem. . . . . . . . . . . . . . . . .
57
2.15 Experimental setup for ionic polymer actuator testing. . . . . . . . . . . . .
61
2.16 Error of models for subsystem 1 at 1.5 V, 5.0 Hz (Sample 1): a) prediction;
b) validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
2.17 Volterra model h32 results for subsystem 1 at 1.5 V, 5.0 Hz (Sample 1): a)
times series; b) power spectra. . . . . . . . . . . . . . . . . . . . . . . . . . .
65
2.18 Actuation response at different frequencies for water-based sample. . . . . .
67
2.19 Error of models for subsystem 1 at 1.5V, 5.0Hz (Sample 2): a) prediction; b)
validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
2.20 Volterra model h34 results for subsystem 1 at 1.5V, 5.0Hz (Sample 2): a) time
series; b) power spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
69
Polymer model arrangements: (a) full 2-input, 2-output system; (b) modified
1-input, 2-output system under no mechanical load. . . . . . . . . . . . . .
75
3.2
Experimental setup for ionic polymer actuator testing. . . . . . . . . . . . .
78
3.3
Linear Volterra model error: (a) electrical output; (b) mechanical output. .
81
3.4
Harmonic ratio analysis: (a) quadratic terms – 2f ; (b) cubic terms – 3f . . .
82
3.5
Example of multisine excitation design. . . . . . . . . . . . . . . . . . . . .
85
3.6
Multisine example: (a) representative system structure; (b) frequency-dependence
of system components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
3.7
Multisine response of system: (a) time series (quarter-period); (b) autospectra. 87
3.8
Experimental multisine response characteristics (3.5 mA-rms): (a) voltage
response; (b) velocity response. . . . . . . . . . . . . . . . . . . . . . . . . .
3.9
Odd multisine components for increasing amplitudes: (a) voltage response;
(b) velocity response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.4
95
System descriptions: (a) general system; (b) decomposed system of MI/SO
nonlinear analysis technique. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
92
System arrangements used for impedance modeling: (a) MI/SO-Direct; (b)
MI/SO-Reverse; (c) DIN. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
90
96
Equivalent MI/SO representations: (a) initial (correlated) system; (b) revised
(uncorrelated) system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
Representative system for MI/SO identification example. . . . . . . . . . . .
103
xiii
4.5
Frequency response plots of system components for MI/SO example. . . . .
4.6
MI/SO example – Identified underlying linear system: (a) correct assumption; (b) incorrect assumption. . . . . . . . . . . . . . . . . . . . . . . . . .
4.7
105
MI/SO example – Identified frequency dependence of nonlinearity: (a) correct assumption; (b) incorrect assumption. . . . . . . . . . . . . . . . . . . .
4.8
104
106
MI/SO example – Autospectral functions: (a) correct assumption; (b) incorrect assumption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
System arrangements for low amplitude data. . . . . . . . . . . . . . . . . .
108
4.10 Nonlinear system structures: (a) Hammerstein model; (b) Wiener model. .
110
4.11 Representative system for DIN modeling example. . . . . . . . . . . . . . .
111
4.12 Frequency response plots of system components for DIN example. . . . . . .
111
4.9
4.13 DIN example: (a) identification of linear system for two input amplitudes;
(b) model fit to low amplitude, identified system. . . . . . . . . . . . . . . .
112
4.14 DIN example – inverse system simulations: (a) low amplitude; (b) high amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
4.15 DIN example – Identified frequency-dependence of nonlinearity: (a) correct
assumption; (b) incorrect assumption. . . . . . . . . . . . . . . . . . . . . .
5.1
114
Evidence of voltage nonlinearity in EMI-Tf sample: (a) frequency response;
(b) time response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
5.2
MI/SO voltage results with cubic nonlinearity: (a) direct; (b) reverse. . . .
119
5.3
MI/SO voltage modeling results: (a) underlying linear system H(f ); (b)
frequency-dependence of nonlinearity A(f ). . . . . . . . . . . . . . . . . . .
5.4
MI/SO voltage modeling simulation results: (a) frequency response; (b) time
response.
5.5
121
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
Average percentage error for sinusoidal response of MI/SO simulation model
(voltage). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
5.6
DIN voltage modeling results: underlying linear system H(f ). . . . . . . . .
124
5.7
DIN voltage modeling inverse simulations: (a) low amplitude; (b) high amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8
126
DIN voltage identification and modeling of nonlinear system: (a) Hammerstein assumption; (b) Wiener assumption. . . . . . . . . . . . . . . . . . . .
xiv
127
5.9
DIN voltage model simulation results of frequency response: (a) Hammerstein assumption; (b) Wiener assumption. . . . . . . . . . . . . . . . . . . .
128
5.10 DIN voltage model simulation results of time response. . . . . . . . . . . . .
129
5.11 DIN voltage model improvements: (a) frequency-dependence of nonlinearity;
(b) linear system in direct input path. . . . . . . . . . . . . . . . . . . . . .
131
5.12 DIN voltage model improvements: (a) frequency response; (b) sinusoidal
response.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132
5.13 DIN voltage model improvements – sinusoidal response error: (a) overall
percent error; (b) amplitude and phase metrics. . . . . . . . . . . . . . . . .
133
5.14 Evidence of velocity nonlinearity in EMI-Tf sample: (a) frequency response;
(b) time response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135
5.15 MI/SO velocity results with quadratic and cubic nonlinearity: (a) direct; (b)
reverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
136
5.16 MI/SO autospectral velocity results with quadratic and cubic nonlinearity:
(a) direct; (b) reverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137
5.17 DIN velocity modeling results: underlying linear system H(f ). . . . . . . .
138
5.18 Example of DIN inverse system augmentation using Fourier transform components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
140
5.19 DIN velocity identification and modeling of nonlinear systems: (a) quadratic
frequency-dependence; (b) cubic frequency-dependence. . . . . . . . . . . .
142
5.20 DIN velocity modeling simulation results: (a) frequency response; (b) sinusoidal response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
143
5.21 DIN velocity model error. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
5.22 DIN velocity model improvements: (a) frequency-dependence of quadratic
nonlinearity; (b) frequency-dependence of cubic nonlinearity. . . . . . . . .
146
5.23 DIN velocity model improvements: (a) frequency response; (b) sinusoidal
response.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
147
5.24 DIN velocity model improvements – sinusoidal response error: (a) overall
percent error; (b) amplitude and phase metrics. . . . . . . . . . . . . . . . .
148
5.25 Final nonlinear model forms: (a) electrical system; (b) mechanical system. .
150
6.1
153
Laboratory setup for plate actuator experiments. . . . . . . . . . . . . . . .
xv
6.2
Close-up of polymer plate actuator. . . . . . . . . . . . . . . . . . . . . . . .
154
6.3
Harmonic ratio analysis for plate deflection. . . . . . . . . . . . . . . . . . .
155
6.4
Evidence of deflection nonlinearity in plate actuator: (a) frequency response;
(b) time response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
6.5
Plate modeling result for underlying linear system H(f ). . . . . . . . . . . .
158
6.6
Plate modeling results – frequency-dependence of nonlinear components: (a)
quadratic term; (b) cubic term. . . . . . . . . . . . . . . . . . . . . . . . . .
160
6.7
Plate modeling results – linear system in direct input path. . . . . . . . . .
161
6.8
Plate modeling results: (a) frequency response; (b) time response.
162
6.9
Plate modeling results – sinusoidal response error: (a) overall percent error;
. . . . .
(b) amplitude and phase metrics. . . . . . . . . . . . . . . . . . . . . . . . .
163
6.10 Simulated transfer functions of square-plate actuator system with PI-control
design: (a) system response; (b) control effort. . . . . . . . . . . . . . . . .
166
6.11 Simulation results for open- and closed-loop sine response using PI-control
in the nonlinear operating range. . . . . . . . . . . . . . . . . . . . . . . . .
167
6.12 Simulation results for open- and closed-loop step response using PI-control.
168
6.13 Block diagram of experimental control investigation. . . . . . . . . . . . . .
169
6.14 Experimental results of PI-controller tracking a sine wave in the nonlinear
operating range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
170
6.15 Experimental comparison of open- and closed-loop tracking: (a) step input;
(b) smoothed pulse input. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
171
6.16 Experimental PI-control results for a step input: (a) constant gains; (b)
variable gains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
172
6.17 Experimental PI-control results for a smoothed pulse: (a) constant gains; (b)
variable gains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvi
173
Nomenclature
x(t)
– input vector
y(t)
– output vector
n(t)
– noise vector
t
– time variable
s
– Laplace variable
f
– frequency [Hz]
∆t
– time spacing
∆f
– frequency spacing
ω
– frequency [rad/s]
z
– system zero
p
– system pole
k
– system gain
ε
– error
ρ
– correlation coefficient
kp
– proportional gain
ki
– integral gain
GM
– gain margin
PM
– phase margin
Mp
– overshoot
tr
– rise time
ts
– settling time
xvii
hN
M (·)
– Volterra kernel of degree N and order M
M
– Volterra model order
N
– Volterra model degree
I(·)
– current signal (simulation)
U (·)
– displacement, velocity signal (simulation)
V (·)
– voltage signal (simulation)
IL (·)
– linear portion of current
IN (·)
– nonlinear portion of current
˜
I(·)
– current signal (data)
A(·)
– frequency-dependence of nonlinearity
B(·)
– augmentation of linear system
C(·)
– linear system through parallel nonlinear component
D(·)
– linear control system
G(·)
– input-output system description
H(·)
– underlying linear system
Ho (·)
– optimal linear system
L(·)
– linear system through nonlinear function
g(·)
– nonlinear function, zero-memory
v(t)
– output of nonlinear system (correlated with x(t))
u(t)
– output of nonlinear system (uncorrelated with x(t))
yv (t)
– output of parallel nonlinear component (correlated input)
yu (t)
– output of parallel nonlinear component (uncorrelated input)
yo (t)
– output of optimal linear system
yh (t)
– output of underlying linear system
X(f )
– Fourier transform of input, x(t)
Y (f )
– Fourier transform of output, y(t)
Sxx (f )
– autospectrum of x(t)
Sxy (f )
– cross-spectrum of x(t) and y(t)
2
(f )
γxy
– coherence function between x(t) and y(t)
2 (f )
qxy
– linear coherence of nonlinear component
xviii
X b(·)
– low amplitude input
X B (·)
– high amplitude input
XnB (·)
– nonlinear component of input
Y b (·)
– low amplitude output
Y B (·)
– high amplitude output
Yhb (·)
– low amplitude linear output component
Ynb (·)
– low amplitude nonlinear output component
xix
Chapter 1
Introduction
Ionic polymer transducers exhibit coupling between three physical domains: electrical,
chemical, and mechanical. This coupling enables their use as both sensors and actuators. Because this phenomenon was only recently discovered, the fundamental mechanisms
governing the actuation and sensing response are still open for debate. However, since ionic
polymers are very compliant, low voltage transducers, they have the potential for wide
applicability in active material systems. The present work aims to gain a better understanding of the physical mechanisms involved in the actuation response with an emphasis
on analyzing the nonlinear behavior of the material.
This chapter will begin with some general background information of ionic polymer
materials. Then a review of some leading models will be given. This will be followed by a
discussion of performance enhancing research for the actuators. A brief overview of some
key applications is discussed next, followed by some motivation for continuing work in this
area. Then the specific objectives of the research will be highlighted, leading into the close
of the chapter with an outline of the remainder of this document.
1.1
Background
As an introduction to ionic polymer materials, this first section will provide some general
background information. One section will discuss the development of ionic polymers in a
historical context, while another section is aimed at detailing the basic concept of actuation.
1
r
Figure 1.1: Chemical structure of the ionomer Nafion
.
1.1.1
History of Ionic Polymers
Ionic polymer materials have been around for decades in fuel cell technology, but only
recently have their parent class of materials, ionomers, begun receiving considerable attention. The exact definition of ionomers has had some historical uncertainty associated
with it (Eisenberg and Yeager, 1982). However, it can be said that ionomers are a class of
ion-containing copolymers that contain non-ionic repeat units, in addition to a small number of ionic repeat units, whose bulk properties result from ionic interactions occuring in
discrete regions of the material called ionic aggregates (Eisenberg and Rinaudo, 1990). The
confusion involved with this definition is that ionomers with high ionic content behave like
polyelectrolytes in solvents of high dielectric constant. The properties of these materials in
high dielectric constant solvents are determined by electrostatic interactions that occur over
relatively large distances (Eisenberg and Kim, 1998), as opposed to the ionic interactions
that dominate this behavior of ionomers. One of the most commonly known ionomers,
r
, is manufactured by E.I. DuPont de Nemours. This perfluorosulfonate ionomer is
Nafion
r
. It was
a copolymer of another commercially available DuPont product known as Teflon
invented in the 1960s by Walther Grot during the early years of fuel cell development (Grot,
1986), and was the first synthetic polymer developed with ionic properties, giving birth to
r
is
the field of ionomers (Rees and Vaughn, 1965). The chemical composition of Nafion
given in Figure 1.1. In this figure, the mobile cation, M + can take a variety of forms and
is therefore not specified.
r
have motivated its research in various areas since its
The properties of Nafion
development in fuel cell technology (DuPont, 2004). Some of these include high working
temperatures (Kinder and Meador, 2005), high resistance to chemical attack (Star et al.,
2
2004), high permeability to water (Jena and Gupta, 2002), and its ability to act as a superacid catalyst (Fujiwara et al., 2000). These properties have helped the material find its
way into several application areas, such as fine chemicals production, water electrolysis
and purification, selective gas humidifiers or dehumidifiers, fuel cells, and eventually smart
materials (Perma Pure Inc, 2000).
Although ionic polymers have been in use for some time, it was only recently that
the transduction properties were discovered. In the early 1990s, different research groups
independently demonstrated the sensing and actuation capabilities of the materials. A
research group in Japan (Oguro et al., 1992) presented actuation results with ionic polymer
materials, while another group in the United States (Sadeghipour et al., 1992) showed
the sensing capability in the material. These discoveries showed that when a voltage was
applied across the thickness of the material, it would mechanically deform, and conversely,
by mechanically deforming the material, a measurable charge flow was produced. Hence,
ionic polymers joined the field of smart materials.
The establishment of an entirely new branch of smart materials resulted, which has
come to be named electroactive polymers (EAPs). This branch now consists of two separate
groups, commonly categorized by their respective driving mechanisms. Electric EAPs are
materials driven solely by an applied electric field or Coulomb forces, whereas in ionic
EAPs, the motion involves mobility or diffusion of ions. The ionic EAP category consists
of ionic polymer gels, ionomeric polymer-metal composites (IPMCs), conductive polymers,
carbon nanotubes, and electro-rheological fluids (Bar-Cohen, 2001b). Ionic polymer-metal
composites are the specific type of transducer used in this research. Another common
grouping that several of these electroactive polymers fall into is that of “artificial muscles.”
This distinction comes about because of their similar properties to human muscle, namely
high compliance, high attainable strain, and light weight. A recent extensive review of
artificial muscle technology has been given by Madden et al. (2004).
1.1.2
Actuation Concepts
While the electromechanical coupling in ionic polymers was discovered just over a decade
ago, there is still no universally accepted set of equations that govern their response (deGennes et al., 2000; Nemat-Nasser and Li, 2000; Tadokoro et al., 2000). It may seem
strange that, with so many applications employing this material, there is an ongoing inves3
Figure 1.2: Drawing of polymer transducer components.
tigation into the fundamental transduction mechanisms. The electromechanical coupling
r
in these
complicates and changes the physical phenomena that occur when using Nafion
other “non-smart material” applications, however. The processes occuring in these particular applications are well understood, which is why such commercialization has taken
place. As an example, the reader can refer to Bernardi and Verbrugge (1991) for a model
of the mechanisms in fuel cell power generation using the material as a proton exchange
membrane.
One of the key reasons why there is such a difference between fuel cells and an ionic
polymer actuator is that the ionomer material must be plated with conductive electrodes
in order to apply an electric field that will induce deformation. This chemical deposition
of dendritic metal electrodes onto the surfaces of the actuator creates diffuse, non-uniform
interface layers, where the primary actuation phenomena are believed to take place (Asaka
and Oguro, 2000; Xiao and Bhattacharya, 2001; Nemat-Nasser, 2002). Related work studying the response in a polymeric gel (Wallmersperger et al., 2004) and varying the ratio of
active to inactive layers in a piezoelectric bimorph actuator (Leo et al., 2003) also draw
attention to the importance of this activation layer for actuation.
Some of the widely accepted basic concepts of actuation will now be reviewed. An
ionic polymer transducer consists of two basic components: a polymer membrane and metal
electrodes. Figure 1.2 has been included to help in this brief discussion. In this figure the top
and bottom surfaces represent the electrodes, where the outer surfaces are relatively flat and
the interface with the membrane (portion in between) is non-uniform. The base material of
the membrane is an ionomer that contains a hydrophobic fluorocarbon polymer backbone
with fixed, hydrophilic anion groups, mobile cations, and a solvent. The backbone in Figure
4
1.2 appears as long lines that make up the network of polymer chains. The circles attached
to the chains represent the fixed anions, while the cations are shown as the unconnected
circles. Additionally, the solvent is shown filling in the voids, where the solvent would be
spread throughout in the actual membrane material, not in localized sections like in the
drawing.
The cations are initially bound to the anionic groups in the neutral state, forming
clusters with the solvent molecules, and can easily be exchanged with other cation forms
(Lopez et al., 1977). This solvent, most commonly deionized water, serves as a transport
mechanism within the material. The metal electrodes are chemically deposited on each side
of the membrane, forming flexible electrode surfaces. After manufacturing is complete, the
final product is a compliant, miniaturizable transducer.
One note-worthy detriment to using water as the solvent is that water evaporates
in air. This means that when an ionic polymer transducer is operated in an open-air
environment, the material will suffer from dehydration. Concerning this adverse effect, it
has been demonstrated that the dynamic response characteristics can vary by up to 20%
in a 10 minute time interval (Kothera et al., 2003) and that the peak deflection of a step
response diminishes near 100,00 cycles (Bennett and Leo, 2003).
An ionic polymer transducer deforms when an electric field is applied across the
electrodes. The cantilever actuator is the most widely discussed configuration because
of the large attainable deflections. Figure 1.3 shows an example of a (30 x 5 x 0.2)mm
actuator bending under an applied 3.0 V step. The step response refers to a sudden change
in command voltage from 0.0 V to the designated level, in this case 3.0 V, where it is held
constant. The field is applied across the thickness of the actuator through a clamp coated
with gold foil electrodes. Prior to the application of the field, the cations and anions are
bonded together and form into clusters with the water.
Applying a field causes these weak ionic bonds to break apart as the positively
charged cations are attracted to the negatively charged surface (Lakshminarayanaiah, 1969).
This is where modeling theories split. Some derivations propose that the cations drag water
molecules with them as they move toward the cathode, making the dominant mechanism
out to be hydraulic (deGennes et al., 2000). Others propose the dominant mechanism to be
the electrostatic forces resulting from redistributed cations (Nemat-Nasser, 2002). However,
it could be that both forces actually play a role.
5
Figure 1.3: Large bending deflection of an ionic polymer actuator in the cantilever configuration.
Several theories have been presented, which will be discussed in the next section.
But back to the basic characteristics, a fast initial motion toward the anode arises when
a step voltage is applied. Then, while the voltage remains constant, the actuator slowly
relaxes back toward the cathode. This indicates that there are two fundamental mechanisms
governing the response of some ionic polymer actuators since there are two time scales in
the step response. It could be that the fast initial rise is more related to electrostatics,
while the relaxation is associated more with hydraulics as internal pressures equilibrate.
This relaxation that is seen in water samples does not always occur when other solvent
materials are used in fabrication, however, but more to this end will be mentioned later.
Having discussed some of the basic concepts involving the actuation of ionic polymers, the
next section will review the models proposed in the literature.
6
1.2
Proposed Ionic Polymer Actuator Models
This section will be broken up into three categories of different model types: black-box (no
prior system knowledge), gray-box (some system knowledge or structure), and white-box
(physical system derivation). Though the models are separated and grouped accordingly,
this is somewhat subjective because of the discrepancy of whether truly black or truly white
models exist and the unclear boundary of where one type ends and the next begins. There
are no model forms presently proposed for ionic polymer actuators that do not depend on
experiments for at least one parameter. On the other hand, most models developed do use
knowledge of the system response for decisions about model structure. Because of this, the
presentation of each category will be relative to ionic polymer technology. Additionally, due
to the growing amount of research in the area, this section does not attempt to summarize
every model of ionic polymers that has been developed. Rather, it gives examples of models
in each category. Also, due to the vast number of parameters and variables involved, they
do not all necessarily coincide with any standard nomenclature list, though they are defined
within the context of each model. The purpose of this section is to show that a wide variety
of different model structures exist.
1.2.1
Black-Box Models
Black-box models are widely considered to be models developed by a designer or modeler
that has no prior knowledge of the system at all. They are therefore referred to as “black”
since the box surrounding the structure and mechanisms of the actual physical system
cannot be seen. Only the input and output signals from the system are known. A model
form is then blindly applied to the signals, and a curve-fitting routine is used to fit the
model to the data. An advantage to this type of model is the relatively small amount of
time required for its derivation, but these models are rarely scalable and have limited use.
Some common forms of black-box modeling approaches include basis function expansions,
neural networks, and various auto-regressive routines (Eykhoff, 1974; Rugh, 1981; Sjoberg
et al., 1995). Two black-box modeling techniques will be discussed here.
7
Kanno
The first black-box approach is a model that was proposed by Kanno et al. (1994) only
two years after the electromechanical coupling was discovered in the material. This is no
doubt why a black-box model was proposed, with little known of the material at the time.
r
cantilever actuator electroded with platinum was considered in this work. The
A Nafion
bending displacement in response to a 1.5 V step input was modeled as a linear combination
of exponentials as
Y = Ae−αt + Be−βt + Ce−γt + De−δt + E
(1.1)
where Y is the tip displacement, A, B, C, D, E are coefficients, and α, β, γ, δ are time
constants. The model parameters were all determined experimentally by curve-fitting data
with the least-squares method. Putting equation 1.1 into transfer function form, its quality
was assessed by comparing simulation results to experimental results with a square wave
input. It makes sense that a square wave input would verify this model form since it is a
sequence of applied steps, which the model parameters were adjusted to fit. However, it is
not scalable.
Mallavarapu
The fact that the previous model did perform well in the step response led to its incorporation into a more control-friendly form, the state-space representation, to improve the poor
step response characteristics of the actuators (Mallavarapu, 2001). This particular model
was also adjustable to include resonance terms and has the form
⎡
−α
⎤
0
...
0
0
⎢
⎢
0
0
⎢ 0 −β . . .
⎢
⎢
.
..
ẋ(t) = ⎢ 0
0
0
0
⎢
⎢
⎢ 0
0
0
0
1
⎣
0
0
0 −ωn2 −2ζωn
y(t) =
1 1 . . . 1 0 x(t)
⎡
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎥ x(t) + ⎢
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
⎤
a
b
..
.
0
Kr ωn2
⎥
⎥
⎥
⎥
⎥
⎥ u(t)
⎥
⎥
⎥
⎦
(1.2)
where ωn are natural frequencies, ζ are damping ratios, and Kr are the resonant gain
coefficients. Simulation with this model was able to match measured data in the open-loop
step response and the large overshoot was reduced using an observer-based linear quadratic
8
Gaussian (LQG) control law. Control experiments also verified the effectiveness of the
controller, but this model suffered from the same downfalls of equation 1.1, those being the
lack of scalability and that little was learned from this model. These downfalls resulted in
the consideration of more elaborate model structures, which leads to the next category of
model types that will introduce more a-priori knowledge of the system and its structure to
give better results.
1.2.2
Gray-Box Models
This subsection discusses ionic polymer actuator models that have more structure or knowledge of physical processes than a black-box model. Some aspects of the overall system can
be seen by the model designer here, while others rely on experimental determination. Often, the case may be that the physical mechanisms of the system are not modeled, and only
the macroscopic behavior of the response is known. An example of this could be a system
where the order is known from physical interpretation, but the model parameters are determined experimentally (Ljung, 1999). Common types of gray-box models include various
auto-regressive and equivalent circuit forms. Any model between “blind” curve-fitting and
complete physical derivation falls into this broad category.
Jung
This model presents an equivalent circuit representation of an ionic polymer actuator (Jung
et al., 2003). The purpose of this work was to study some of the overall actuation characteristics of the polymer, so developing a simple model was all that was needed. Examining
the frequency response of a square actuator constrained not to move, an equivalent circuit
network of a damped high-pass filter was chosen to model the relationship between input
r
actuator with platinum electrodes was expressed as a
and output voltage. The Nafion
resistor, R1 , in series with a parallel connection of another resistor, R2 , and a capacitor,
C. Another resistor, R3 , was included in the model due to the experimental setup. The
transfer function for this system can be written as
G(s) =
R3 (1 + sT1 )
(R1 + R2 + R3 )(1 + sT2 )
(1.3)
where Ri are fixed resistance values and Ti are time constants determined from the resistances and the varying capacitance. The variability in capacitance was stated to be the
9
result of several factors, but is mostly due to the swelling and migrative motion within the
membrane, which can lead to changes in the electrical characteristics. Simulation with this
model shows that the maximum current rapidly increases with the frequency of the driving
voltage, and it saturates because of the resistance values. Experimental time responses are
seen to be similar to the simulated system.
With this model, an analysis of power consumption of the actuator was performed.
By varying the input waveform between a square, a triangle, and a sinusoid, and monitoring
the current, it was shown that waveforms containing high frequency components, such as
the square, draw much more current than smoother waveforms like the sine wave. This is
due to the high-pass filter design of the model, which amplifies the high frequency content
of the input signal. Comparing the three waveforms in equal peak-to-peak input voltages,
results showed that while the square wave produced the most displacement, the triangular
wave used the least overall power.
Kanno
This model, coming two years after the previously mentioned black-box model, has developed much more structure (Kanno et al., 1996). This model is composed of three different
stages that combine to cause the bending response of a cantilever actuator. First is an
electrical stage where voltage is the input and current is the output, followed by a stress
generation stage where current is the input and stress is the output. The final stage is a
mechanical stage that converts stress into deformation.
The actuator is divided into ten elements consisting of resistors and capacitors,
including surface resistances, and was represented as a first-order transfer function summation. The electrical stage is represented by a series of circuit networks that can be described
by
G(s) =
n
k=1
1 + sCRn
1 + sCRn
+ αn
Rd2,k + sCRd1,k
Rx + sCRx Rc
(1.4)
where Ra are different resistor combinations, C is the capacitance, n is the number of circuit
network elements, and α is the ratio of free to total electrode length. The resistances account
for effects of the surface, polymer, and characteristic response, along with the capacitance.
With voltage as the input and current as the output, this stage is essentially the actuator
impedance. The output current from this stage, with good matching between simulation
10
and experiment, then goes on to produce stress in the material.
It was proposed that compressive and tensile stresses are developed in the surface
layers on each side of the actuator, and are in proportion to the time-derivative of current.
Once the internal stress is developed, volumetric strain is induced. This relationship was
given as
σ = D(s)
− e
ωn2 s
s2 + 2ζωn s + ωn2
I
(1.5)
where σ is a stress vector, D is a mechanical characteristic matrix, is a strain vector, e is a
stress generation tensor, I is the current through the material, and ωn and ζ are parameters
of the second-order delay. The delay is used to account for the energy transformation.
For validation, a finite element model was created that replicates the response of the
above equations. Using Rayleigh damping and adjusting parameters to match experimental
response data, the total model response could closely predict the actual response.
Newbury
Of the models discussed so far, this model by Newbury and Leo (2002) is the most designoriented. It is a linear two-port model that can be used for both actuation and sensing for
r
bender with gold electrodes. The two-port form comes from modeling
a cantilever Nafion
the electromechanical system as an equivalent circuit. This dynamic transducer model has
the form
⎫
⎧
⎫ ⎡
⎤⎧
⎨
⎨ v(ω) ⎬
i(ω) ⎬
Z11 Z12
⎦
=⎣
⎩ f (ω) ⎭
Z21 Z22 ⎩ u̇(ω) ⎭
(1.6)
where v is voltage, f is force, i is current, u̇ is velocity, and Zij are different input-output
relationships affected by the electrochemical and chemomechanical processes within the material. These relationships are determined from optimizing the fit to measurements taken
with imposed electrical and mechanical boundary conditions on the system. The conditions
used were short-circuit, open-circuit, blocked motion, and free motion. Simulating this system with empirically determined parameter values gives good agreement with experimental
results. One important discovery in this work was that the short-circuit current is linearly
proportional to tip velocity of the actuator over a low frequency band, both from the model
equations and experimental investigation. Interesting to note here is that this finding contradicts the previous Kanno model, where it was assumed that the stress was in proportion
to the time-derivative of current. Also, a noted disadvantage of this model is that it does
11
not scale with actuator geometry, meaning that if the model were identified for a transducer
with dimensions A, it would not be valid for a transducer with dimensions B.
To correct this problem of scalability, the formulation was later modified to include
actuator geometry and viscoelasticity (Newbury and Leo, 2003a). This improved twoport equivalent circuit model establishes a macroscopic response model that can easily
interchange between strain and charge distribution. There are two mechanical terms in
this model (mechanical impedance due to stiffness and an inertial term) and two electrical
terms (DC resistance and charge storage). The mechanical impedance is determined from
Euler-Bernoulli beam theory in a quasi-static condition, assuming small deflections. The
modulus is then augmented to include viscoelastic effects using the Golla-Hughes-McTavish
(GHM) method. The inertial term was added to improve accuracy near resonance using
transverse beam equations under the cantilever boundary conditions. The DC resistance is
expressed in terms of the resistivity of the assumed homogeneous material with perfectly
conductive electrodes, and the electrical impedance is modeled as parallel branches of RC
circuits connected in series. Finally, electromechanical coupling in the system is determined
by the turns ratio of a linear transformer, serving as a link between the electrical and
mechanical components, where it was noted that the actual mechanisms for the coupling
likely occur at the electrode-polymer interface. In this work, the turns ratio was calculated
using mechanics of materials and an equation common to piezoelectric bimorph transducers.
It should also be noted that all of the parameters, Zij , in this form of the model are functions
of transducer dimensions, therefore, the model is scalable for different geometries, and the
model parameters have the freedom to be frequency-dependent (Zij → Zij (ω)).
Further experimentation and simulation with this system revealed that very little
viscoelastic behavior was present in the material in the frequency range of interest, the
u̇
f (ω)
4Ld
has a right half-plane zero),
strain coefficient is non-minimum phase (d(ω) = 3twY
v(ω)
and the highest charge densities occur at very low frequencies (Newbury and Leo, 2003b).
Model form and scalability were also verified experimentally. In examining the parameters
more closely, there was some evidence that they may be dependent on the input level,
however. It should be noted here that later studies of this particular model form, extending
the frequency range, showed that viscoelasticity was present in the material (Franklin, 2003;
Buechler and Leo, 2005).
12
de Gennes
This model takes a different approach than the others mentioned in this category, while
maintaining a functionally similar form to that of the Newbury model. Rather than presenting an equivalent circuit to describe the actuator response, this work presents a description
of coupled equations drawing on basic physical concepts (deGennes et al., 2000), and is
essentially a simplification of the work by Segalman et al. (1992). It is for this reason that
it is discussed as the last model in the gray-box category, which is also the lightest shade of
gray thus far. Although this model was developed for ionic gels and not specifically ionic
polymers, it has received considerable attention in ionic polymer literature.
Studying the linear actuation response in static conditions, this model uses irreversible thermodynamic principles. The model highlights two driving forces (the applied
electric field and the water pressure gradient) and two fluxes (electric current and water
current) as the fundamental mechanisms involved, and presents the model in a compact,
linear form. The charge transport is represented in the current density, J, and the solvent
(water) transport is represented by a flux, Q. These two phenomena are coupled in the
material and are activated directly by the electric field, E, and the pressure gradient, ∇P .
This system of equations is written as
⎫
⎧
⎫ ⎡
⎤⎧
⎨ J ⎬
σ −L12 ⎨ E ⎬
⎦
=⎣
⎩ ∇P ⎭
⎩ Q ⎭
L21 −K
(1.7)
where σ is the membrane conductance, L12 = L21 = L is the coupling term, and K is
the Darcy permeability. Using this set of equations, the assumption that the electrodes
are impermeable to water is made, giving a proportionality between pressure gradient and
electric field. This leads to an equation that directly relates the applied electric field to the
bending moment. The pressure gradient is then used to balance the induced bending stress,
which can then be related to the curvature.
The physical concepts explained by this model are that cations drift through the
thickness of the gel when an electric field is applied, and these cations also carry water
molecules with them. It is this accumulation of cations and water molecules on the cathode
side of the actuator, and subsequent depletion from the anode that generates local pressure
differentials in the material. Bending deformation then ensues. Two limitations that are
brought out in this model are that all electrode effects are ignored, as well as any contribu13
tion due to osmotic pressure. There were also no experimental verification results for this
model form.
1.2.3
White-Box Models
Models that are derived from first-principles and a comprehensive knowledge of the physics
involved in the system are called white-box models. These have the most structure and
scalability of all model forms, but also take the longest to derive and characterize. While
data is not necessarily needed during the development of the model, it is used for verification of the final form. This type of model does not generally include any of the previously
mentioned common forms of black- and gray-box models. Although this section is titled
“white-box models,” the models discussed may not necessarily be “white” when compared
to other engineering fields. However, they have the highest degree of structure and understanding of all the others discussed in the previous two subsections, making them the
closest thing to white-box models for ionic polymer actuators.
Xiao
This work presents the bending response of an ionic polymer actuator using an elastoelectro-chemical formulation (Xiao and Bhattacharya, 2001). While a detailed finite deformation model is presented at the start, it is later simplified to a one-dimensional form that
is more characteristic of the typical bending actuator. The basic derivation method will be
briefly discussed here, but only the simplified equations will be presented.
The starting point of this model uses micromechanical work by Nemat-Nasser and
Li (2000). Treating the polymer as an insulating, elastic medium, it is assumed that deformation is induced by a changing ionic concentration. Cations are free to move within
the membrane, while anions are fixed to the backbone. It should also be noted that this
model considers a quasi-static step response, where the excitation voltage is constant. An
expression for the total free energy is given to start. Next, a stress-free transformation
is introduced. Then a variational principle is applied, stating that the system evolves in
such a way that the ion concentrations, electric field, and deformation reduce the total free
energy, out of which the governing equations come. The proposed phenomena that can
be extracted from these equations are described by the Nernst-Einstein diffusion equation
with an elasticity term, Maxwell’s equations for electrostatics, and mechanical equilibrium.
14
Simplified to one-dimensional form, the two governing equations are
+
+d
Ct+ = dCxx
zF +
z2F 2 +
Cx φx − d
(C − C − )C +
RT
RT
φxx = −
zF +
(C − C − )
(1.8)
(1.9)
where C + and C − are the concentrations of cations and anions, d is the diffusion coefficient,
z is valence of ions, F is the Faraday constant, is the dielectric constant of the membrane,
φ is the electric potential, T is temperature, and R is the universal gas constant. The
subscripts denote differentiation with respect to either space, x, or time t. From looking
at equation 1.8, it can be seen that the third term is nonlinear, indicating that the main
nonlinearity, according to this model, arises in the electrochemistry. It is also stated that
this nonlinear term is essential in determining the activation layers at the boundaries of the
actuator.
Although simulation results for this model were not compared to experiments, a
study of the boundary layer phenomenon was carried out. From this study, the authors
determined that the actuation response is the consequence of cation accumulation and
depletion into thin layers near the cathode and anode of the actuator, respectively.
Asaka
The premise of this model is that electrokinetically-induced pressure gradients cause mechanical deformation in a polymer bender (Asaka and Oguro, 2000). This model differs
from the others mentioned in that it assumes a step current as input, rather than voltage. Recall that the gray-box model by Kanno et al. (1996) also assumed the deformation
resulted from a current input. Operating at voltages below the electrolysis level, capacitive current will only flow due to cation migration, where electro-osmosis also occurs. The
modeling begins by stating that the water flux in the membrane is a function of the fluid
pressure gradient and the electro-osmotic flow. Next, linear elasticity in the swelling stress,
hydration-dependent strain, and conservation of momentum are used to relate the hydration
dynamics to the water flux. Once the hydration state is determined, the total curvature
of the composite actuator is related to the stresses in the membrane and the plated layers.
Then by applying mechanical equilibrium conditions to differential cross-section elements of
the composite, the curvature is determined to be a function of one characteristic time, which
15
is different than most other models that have two time scales. The stress in the membraneplating interface region is then considered to arrive at a more general deformation equation
that includes dependence on the acidity of the aqueous solution surrounding the actuator.
The final form of the equation for the bending response of the actuator, assuming no initial
curvature, is given by
ς=
φj
2d3
1
9 1+Heq F Dm Em
2
1
π Dm t + M
1 − exp − 2d
Q
(1.10)
where ς is the induced curvature, d is half the thickness of the membrane, Heq is the
equilibrium hydration, φ is the water molar volume, is the water transference coefficient,
j is the current density, F is Faraday’s constant, Dm is the diffusion coefficient, Em is the
bulk modulus of the material, π is the osmotic pressure, M is the bending moment from
interfacial stress, and Q is the composite stiffness. For the sake of simplification, several
quantities in equation 1.10 were grouped together so the system could be represented with
ς = 2k1 Iτ0 [1 − exp(−t/τ0 )] + k2 It
(1.11)
where k1 and k2 are model parameters, τ0 is the characteristic time, and I is the current. The
first term in this equation represents the electrokinetics, while the second term addresses
the influence of the aqueous solution surrounding the actuator.
r
actuators with
Using this simpler equation, experiments were performed on Nafion
platinum electrodes. By comparing the model to measured results, it was also determined
that the parameters k1 and k2 are linearly proportional to the current density of the material.
The proportionalities of the model parameters to membrane thickness and cation form were
also discussed. Regarding the characteristic time, it was found to increase with actuator
thickness. Also, by observing the response at various sweep rates, it was concluded that
at low frequencies, the interfacial stresses resulting from the electro-osmotic effect are most
dominant in causing a bending deformation in the material, while at high frequencies, the
electrokinetic effects are more responsible for the pressure gradient that causes deformation.
Tadokoro
Tadokoro et al. (2000) propose a model based on six major phenomena: ionic motion induced
by electric field, water motion induced by ion-drag, membrane swelling and contraction,
momentum effects, electrostatic forces, and a change in conformation. A force balance of the
16
mobile cation gives an expression relating the electrostatic force to the viscous resistance, a
diffusion force from the deviation of cations, and a diffusion force from the deviation of water.
The flux of concentration satisfying continuity and the total charge are then incorporated
into the force balance, resulting in a partial differential equation for the charge of cations,
Q, as
η
t
∂ 2 Q(x, t)
e
∂lnw(x, t)
∂Q(x, t)
∂Q(x, t)
= kT
−
+
kT
i(τ
)dτ
+
Q(x,
t)
−
Q(x,
0)
∂t
∂x2
∂x
Sx
∂x
0
(1.12)
where η is a viscous resistance coefficient, k is Boltzmann’s constant, T is temperature,
w is the water concentration, e is the electric charge of a single cation, is the dielectric
constant of the hydrated membrane, Sx is the surface area over which the current acts, i is
the current, and x is the spatial coordinate.
Water travel is modeled beginning with an equilibrium equation of its diffusion force
and its resistance due to viscosity. Applying continuity then gives another partial differential
equation for the number of water molecules, W , that can be solved simultaneously with the
charge equation using appropriate initial and boundary conditions. This equation has the
form
η
∂ 2 W (x, t)
∂W (x, t)
= kT
.
∂t
∂x2
(1.13)
It was noted here that electrostatic forces dominate the migration of cations since they are
much larger than the diffusion forces. Since strain and water content are assumed to be
linearly proportional, based on experiments, the swelling and contraction was modeled as
such
ε = αC
(1.14)
where ε is the swell strain of the membrane, C is the water content, and α is the constant
of proportionality, determined from experiments. Also generating stress is the electrostatic
force by a fixed electric charge. This is because the anions are fixed to the polymer backbone,
while the cations can migrate. This results in electrostatic forces producing internal stress
in the membrane, near the anode. This is modeled as
σ(y, t) =
Fb (y, t)
Sy
(1.15)
where σ is the internal stress due to electrostatic force, Fb is the total electrostatic force
on the fixed charge, and Sy is the surface area acted upon. Finally, the conservation of
17
momentum is applied to the hydrated cations and membrane, resulting in an equation for
the reaction force of the actuator
Fa (t) = M
dU (t)
dt
(1.16)
where Fa is the reaction force, U is the velocity of the membrane, and M is the mass of the
hydrated membrane. It was also noted here that the internal stress caused by conformation
change of the polymer was not formulated because the mechanisms are not understood well
enough.
Using this set of model equations, simulations showed that electrostatic forces cause
cations to migrate toward the cathode and water to travel according to ionic migration,
but because of the diffusion force, the gradient is smaller than that for the cations. Also,
it was shown that the dominant initial force for bending is the result of water swelling and
contracting at the cathode and anode. The electrostatic force was shown to dominate the
internal stress during back relaxation. Lastly, nonlinearity between step voltage level and
peak displacement is demonstrated in this model to match well with experimental results.
Nemat-Nasser
Saving the most elaborate model for last, the work of Nemat-Nasser (2002) will now be
discussed. The initial form of this model was developed as a linear micromechanical formulation describing ion transport, electric field, and elastic deformation of an ionic polymermetal composite subject to a step voltage, where ion migration is responsible for the fast
initial response and water transport causes the relaxation (Nemat-Nasser and Li, 2000).
This formulation begins by separating the stress field within the membrane into an elastic
component related to the polymer backbone and an eigenstress related to the imbalance of
the charge density. Field equations describing the charge distribution are then combined
with the stresses, counter-ion continuity, and Darcy’s law for free water velocity to arrive
at a dynamic equilibrium equation
∇σ T = ρ0
dV
dt
(1.17)
where σ T is the total stress in the membrane, ρ0 is the mass density, and V is the velocity of
the actuator. Under a time-independent field and with no external loading, an equilibrium
18
state is reached in the membrane, where the following equation applies
∂ 2 Ex C − F 2
Ex = 0.
−
∂x2
ϑe RT
(1.18)
Here E is the electric field, C − is anion density, F is Faraday’s constant, ϑe is the effective
dielectric constant, R is the universal gas constant, T is temperature, and x is the thickness
coordinate. This equation is essentially the linear version of equation 1.8, which makes sense
because the work is related up to this point. Linear elasticity is assumed in the material,
as is the spherical shape of clusters with water on the inside and cation-anion pairs on the
surface. Later studies showed that the spherical assumption of the cluster shape proved to
be correct with no applied field, but has more of a spheroidal shape with an electric field
(Li and Nemat-Nasser, 2000), and that surface cluster effects play a significant role in the
equilibrium cluster state (Weiland and Leo, 2005). However, cluster interaction was ignored
in this model.
It was also noted that by the solution suggesting that the charge density is nearly
zero everywhere in the membrane, thin boundary layers exist on each face where the charge
imbalances occur that lead to deformation. Simulation and experimental results are in good
agreement with this initial model for both a step and sinusoidal input. This model does
not include the capacitive properties of the electrodes, however.
Getting back now to the elaborate model, the actuator was considered in a more
general case, where different backbone structures and cation forms could be incorporated.
This formulation begins with an analysis of the actuator stiffness as a function of hydration
by deriving expressions for stress in the polymer and pressure within the clusters, both
osmotic and electrostatic. An expression for the composite modulus as a function of water
uptake is the result:
ȲIP M C =
YM YB
.
BAB YM + (1 − BAB )YB
(1.19)
Here, ȲIP M C is the composite modulus, YB is the modulus of the bare membrane, YM is the
modulus of the metal electrodes, AB is a concentration factor for stress in the membrane,
and B is a function of water uptake and volume fraction of the metal plating. Simulations
with this equation match experimental results well for various composite forms.
In studying the actuation step response, measurements were taken of the current flow
that showed a continued accumulation of cations into the cathode boundary layer, even after
the back-relaxation occurred. This find led to the conclusion that electrostatic forces are
19
dominant over osmotic and hydraulic pressures during this phase of the response. In this
model, the assumption is made that an electric field redistributes the cations, causing volume
changes in the clusters within each boundary layer, which ultimately causes deformation of
the actuator. The derivation begins by developing a relationship between the eigenstrain
rate and the actuator tip velocity as a function of hydration. Since the application of an
electric field causes volumetric changes in the boundary layers and affects the hydration
state of the actuator, the volumetric strain rate due to cation transport is discussed next.
Phenomena occurring in the boundary layer regions were then considered by examining the
pressures, electrostatic interaction forces, charge distribution, and hydration in each. It
was noted here that the anode boundary layer is thicker than that of the cathode during
actuation. The resulting differential equation describing the bending motion of the actuator
can be expressed as
YBL
hLLA
LA
u̇
=
DA tA − t̄C
L
LC
3ȲIP M C − 2YB 4H 3
(1.20)
where u̇ is the tip velocity of the actuator, L its length, YBL is the effective modulus of
the boundary layer, h is half the thickness of the membrane, H is half the thickness of
the composite actuator, Lα is the effective length of the boundary layer, tα is the effective
pressure acting on the water within the boundary layer, and DA is a hydraulic permeability
coefficient including anode boundary layer thickness. The subscripts α denote the side
of the actuator under consideration, meaning the cathode, C, or the anode, A. The Lα
and DA terms are primarily functions of applied potential and temperature, while the tα
terms depend primarily on hydration since they consist of cluster pressure and elastic stress.
The cluster pressure itself considers dipole-dipole interaction, osmotic pressure, and other
interactions between either anions or cations. Comparing simulations of equation 1.20 with
experiments, the results were shown to be in good agreement.
A recent study inspired by this model has been conducted in contrast to one aspect,
that being the cause of the initial fast response. Weiland and Leo (2004) constructed a
computational model employing a polarization effect as the dominating mechanism, instead
of counter-ion migration. As a result, simulation studies showed that polarization does,
in fact, have significance in the actuation response. Indeed, this finding adds more to the
debate over the fundamental mechanisms in these materials.
20
1.2.4
Modeling Comments
Looking back at this section in review, it can be said that most black- and gray-box models were developed to study certain response characteristics or phenomena in the material.
The white-box models, on the other hand, attempt to model physical processes taking place
within the actuator. For the most part, these white-box models were discussed in chronological order, which makes sense that as time progresses and more is known of the materials,
better physical models were developed. It should be noted, too, that work by Segalman
et al. (1991) and Shahinpoor (1992) is also often credited in the actuation discovery of these
materials, but their work deals with ionic gels, which are not the same as the ionic polymers
used in the above studies. Thus, they laid the groundwork for the model by deGennes et al.
(2000) discussed as the last gray-box model. A modeling review covering a broader range
of active material systems, such as these, can be found in Shahinpoor and Kim (2004).
One modeling topic that was not discussed in this section has to do with sensing. It
was stated that the model by Newbury and Leo (2002) is valid at low frequencies for both
sensing and actuation, but the majority of modeling work with ionic polymer transducers
has been for actuation. Nemat-Nasser and Li (2000) briefly discuss some sensing models
based on an effective dipole resulting from an imposed deformation. This work states that,
given a deformation, the voltage induced is two orders of magnitude less than that required
to induce the same deformation. Recent work by Farinholt and Leo (2004b) began with
these electrostatic equations and focused entirely on developing a charge sensing model for
a step input position. Beginning with a partial differential equation governing the charge
density, boundary and initial conditions are imposed, and the stress is assumed to vary
linearly with charge. The final equation relating charge to displacement was given as
L b/2
D(h, t)dydz
1
3Y bh cosh(βh)
−
e−λt
= w(L)
2βψsL sinh(βh) βh
q(t) =
0
−b/2
(1.21)
where q(t) is the charge, D(h, t) is the electric displacement, w(L) is the imposed tip deflection, Y is the modulus, β is the length scale inverse, ψs is the stress-charge proportionality
constant, λ is the inverse time constant, and L, b, 2h are sensor length, width, and thickness,
respectively. Results of this sensing model showed that the material sensitivity increases
with transducer width and thickness, and decreases with increasing free length of a can-
21
tilever.
As mentioned in the beginning of this section, the primary reason for including
so much about the present models of ionic polymer actuators is to show that they are
derived using a variety of principles and different structural forms by researchers from
diverse fields of expertise, and they can all match their simulation results to experimental
data. However, no model exists that can account for all experimental results under all
experimental conditions.
This is why the modeling debate still exists. Putting this aside, some general remarks
will now be made about the modeling efforts discussed above. These comments pertain
mostly to observations and propositions made from the white-box models since they tend
to draw more on physical processes than either the black- or gray-box models. To this
end, it is commonly agreed upon that there exist two primary transport mechanisms in
the actuation response of an ionic polymer transducer. These are charge, from the applied
electric potential, and solvent, usually water in the models discussed. It is also widely
accepted that the major actuation mechanisms occur in thin activation layers near the
surfaces of the electrodes. These are often referred to as boundary layers that form near
the anode and cathode.
While these concepts are common to the models, discrepancies come about when
considering the dominant mechanisms.
Some models emphasize electrostatic forces as
dominating the response, but others owe the basis of actuation to hydraulic forces. In
the electrostatic model, it is stated that actuation is due to interaction between the ions
(Nemat-Nasser, 2002). This occurs after an electric field is applied across the thickness of
the polymer, causing additional cations to accumulate at the negatively charged surface.
Here, repulsive forces are generated that cause the polymer to bend. The hydraulic model
claims that ion migration resulting from an applied electric field causes water transport
through the thickness of the material (Shahinpoor and Kim, 2002). In this theory, water
displaced by motion of the ions causes local pressure gradients that produce strain in the
material.
The modeling debate is still ongoing because all of the proposed models have some
supporting experimental evidence, while no model has been developed that can predict
all of the observed behavior. It is also mentioned in the literature that polymer sample
preparation, actuator properties, and testing repeatability often vary widely from specimen
22
to specimen, making it difficult to compare the results of different research and development.
Mentioning now some of the limitations of the proposed models, most are very specific with regard to the derivation assumptions, boundary conditions, and material properties. Also, only two of the nine models reviewed include any nonlinear effects in the material
response. The others all present linear approximate models, which is a good place to begin
to understand the dominating mechanisms. However, the maximum deflection was shown
to scale nonlinearly with input voltage level in the model from Tadokoro et al. (2000), and
different nonlinearities have been reported by several other researchers. So before a comprehensive model is developed, nonlinear response characteristics will need to be considered
in the modeling. This leads to another common limiting factor in the proposed models,
which is how the quasi-static, or step response, is considered almost exclusively, neglecting
the response to a general input. Recent evidence of nonlinearities in the sinusoidal response
of current and velocity, for example, have been reported by Kothera et al. (2004). Overall, these factors provide evidence that there is much remaining to be developed before a
complete model is derived.
As a comprehensive actuation model is developed, some researchers have been experimenting with methods to improve the performance of ionic polymer actuators. This is
the topic of the next section.
1.3
Improvements to Transducer Materials
This section is dedicated to addressing some of the research that has enhanced the response
characteristics of ionic polymer actuators. Improvements to both open-loop response and
closed-loop response will be discussed, but the majority of this work has to do with various
manufacturing procedures or different material types. There are four main variables that
can be altered to modify the response of an ionic polymer actuator. These include the
electrodes, the mobile cation form, the solvent type, and the base ionomer material used.
Each will be discussed in turn, followed by some closed-loop control studies. Studying
the response of various actuator materials also provides valuable insight into the physical
processes occurring in the material.
23
1.3.1
Electrode Improvements
The first work done in this area involved experimenting with the chemically deposited electrodes. As noted in the modeling discussion, all of the actuators had electrodes made of
a single pure metal, typically gold or platinum. Shahinpoor and Kim (2000a) first experimented with applying a thin silver layer over a platinum electrode that reduced the electrical
surface resistance. This reduction in resistance signifies an increased level of conductivity,
which was confirmed experimentally as the mechanical force output was increased 10-20%.
While these metals work well with the actuators, precious metals are rather expensive, limiting commercialization. Work has been conducted by Bennett and Leo (2003) addressing
this potential drawback. In this work, a co-reduction process employing non-precious metal
electrodes was developed. Instead of pure gold or platinum electrodes, this research showed
that depositing a copper/platinum alloy, with a thin over-layer of gold to increase conductivity and prevent copper oxidation, created an actuator with comparable performance to
actuators manufactured with pure, precious metal electrodes by the previously mentioned
authors.
Also regarding electrodes, several researchers have reported that increasing the number of metal platings also increases conductivity and capacitance, and hence, actuator performance. Although a saturation limit will be reached, the number of platings can be
optimized (Oguro et al., 1999). An even more recent development in this area is the direct
assembly process, where carbon nanotubes and conducting polymers are incorporated into
the electrode plating (Akle and Leo, 2005). By fabricating composite electrodes with active materials, results showed increased performance in both response time and attainable
strain.
1.3.2
Counter-ion Research
Moving now to the counter-ion form, experiments have been performed using a variety
of cations by Shahinpoor and Kim (2000b). This work showed that actuators containing
lithium ions, Li+ , possessed a 40% higher efficiency (force output divided by power input)
than the nearest competitor sodium, Na+ . Others tested include potassium, K+ , hydrogen,
H+ , calcium, Ca++ , magnesium, Mg++ , and barium, Ba++ . This was stated to be the
expected result since it was assumed that cations with higher hydration numbers should
24
produce more force and displacement. Asaka et al. (2001) also experimented with a variety
of cation forms including alkali and alkaline Earth metals, like the previously mentioned
work, and alkylammonium ions like tetramethylammonium, TMA, and tetrabutylammonium, TBA, for example. The difference in these types of cations lies in their partiality
to water. The alkali and alkaline Earth metals are smaller and hydrophilic, meaning that
they like water, whereas alkylammonium ions are larger, hydrophobic, and therefore do not
want to bond with water. This study showed that the water drag phenomenon consists
of hydration and pumping effects. The hydration effects were reflected by the smaller, hydrophilic cations that were carried by water through the polymer matrix, but the pumping
effects were shown by the larger, hydrophobic ions as they pushed their way through the
membrane. As far as actuation results are concerned, cesium, Cs+ , and Li+ were the best
of the hydrophilic ions, while TMA and TEA were the best hydrophobic ions. It is difficult to determine an overall best performer because the responses changed differently with
frequency. A similar study was carried out by Nemat-Nasser and Wu (2003), in which an
added result was how TBA may suppress water electrolysis at relatively high voltages, but
r
is poor.
its actuation response in Nafion
1.3.3
Solvent Research
Solvent type is another variable in the manufacture of ionic polymer actuators. It has
been noted in several studies that the conductivity of the actuator is strongly dependent on
hydration level. Clearly, using water as the solvent presents limitations due to its volatility.
Another limitation with water is that electrolysis occurs at around 1.23 Volts. This is
the chemical breakdown of water into oxygen and hydrogen gas. Ionic liquids are a solvent
offering greater electrochemical stability than water and they are non-volatile, meaning that
they will not evaporate from the membrane. Work by Lu et al. (2002) first experimented
with using ionic liquids in conducting polymers, which are another type of artificial muscle.
Extending their favorable results to ionic polymers, ionic liquids were employed, specifically
1-ethyl-3-methylimidazolium trifluoromethanesulfonate (EMI-Tf), and results showed that
an electrochemical stability window over 4 Volts is achievable (Bennett and Leo, 2005a).
Experiments with this actuator indicate very consistent strain results when compared to
an actuator using water over several days of testing. The primary limitation of this ionic
liquid is the diminished speed of response in a step response. It displays the character of an
25
overdamped system, while with water the response is similar to an underdamped system.
This investigation does offer promise with ionic liquids as solvents, however.
A more in-depth look into using ionic liquids as solvents in ionic polymer actuators
was performed by Bennett and Leo (2004). This study examines the use of five different
ionic liquids and concludes that viscosity and degree of membrane swelling are important
parameters, where low viscosity and high swelling percentage provides for more strain in
the actuator. And when normalizing with respect to charge, actuators with ionic liquids
can have up to an order of magnitude increase in strain compared to a similar water-based
actuator in proton form, and nearly equivalent strain for the lithium form. Additionally,
ionic liquid actuators do not exhibit the back relaxation seen with water samples. In another
study, a wider variety of solvents were used, including glycerol and butyl acetate, where
evidence of a new low frequency mechanism (peak in the frequency response) was found
with the butyl acetate samples (Farinholt and Leo, 2004a).
1.3.4
Ionomer Research
The remaining manufacture variable to be mentioned is the ionomer material, which will
be discussed now. Experiments have shown that using a perfluorosulfonic acid membrane
r
can produce a back relaxation in the opposite direction of the initial step
like Nafion
response motion. On the contrary, actuators made of a perfluorocarboxylic acid membrane
r
demonstrated a relaxation in the same direction as the initial motion (Asaka
like Flemion
et al., 2001; Nemat-Nasser, 2002). More recently, synthetic ionomer materials have been
synthesized from the direct polymerization of sulfonated monomers, allowing precise control
on the placement of ionic groups on the polymer backbone (Akle et al., 2003a). The new
classes of ionomers that were developed are called sulfonated poly(arylene ether sulfone)s
(BPSH) and poly(arylene thioether sulfone)s (PATS), which both contain higher concentrar
. Of these two new
tion of ionic groups and have higher stiffnesses than the typical Nafion
materials, PATS is more compliant. As expected from the enhanced material properties,
actuators with these materials were capable of producing much more force and similar strain
r
. A further analysis with these new ionomer materials showed
levels compared to Nafion
the existence of a linear relationship between strain and capacitance, regardless of polymer
composition or plating method (Akle and Leo, 2004). More work in this area has led to the
conclusion that the actual driving mechanism for mechanical deformation is not the applied
26
voltage, but rather the charge accumulation at the electrodes (Akle et al., 2005). Evidence
of this was given in results showing that the strain to voltage response is nearly constant for
a variety of different actuator samples when normalizing with respect to charge, or electric
displacement, while a comparison of the original strain to voltages responses ranges over an
order of magnitude.
1.3.5
Other Fabrication Improvements
Some new concepts for actuator enhancement that do not fit into any of the aforementioned
categories will now be summarized. The first is that of tailoring the base polymer by adding
a clay-based nano-composite (Paquette et al., 2003). This investigation concludes that
power consumption can be reduced by decreasing the voltage-current hysteresis behavior,
which may be due to a relative increase in elastic forces. Second is a method to increase
the force output. Akle and Leo (2004) showed that by stacking multilayer transducers
connected in parallel, the output force increases in proportion to the number of layers for
actuation, while the sensitivity also increases with the number of layers when using the
stacked transducer as a sensor.
1.3.6
Control Research
The actuator will now be assessed on a precision performance basis. Here, some closedloop results of different control schemes will be presented. As noted before, a typical ionic
polymer cantilever exhibits a large initial displacement, followed by slow relaxation to different steady-state positions. These response characteristics translate into large overshoot,
long settling time, and steady-state error, in systems and controls terminology. In order
to show that ionic polymers are capable of being used as more precise actuators, different
types of control algorithms have been explored. The first work in the area of feedback was
presented by Mallavarapu and Leo (2001). This was an observer-based design where the
initial model form took after that of Kanno et al. (1994). Using a cost function analysis,
experiments showed that the overshoot could be significantly reduced, but settling time and
steady-state error remained somewhat of a problem. This is where the work by Kothera
and Leo (2005) began. Using integral control drastically reduced the undesirable open-loop
actuation properties, including back-relaxation, the large overshoot, and steady-state error.
It was shown that the polymer actuator could hold a commanded position to within ±2%
27
for nearly 30 seconds before going unstable. This instability was attributed to material
dehydration causing changes in system parameters. It was also noted that the closed-loop
bandwidth was approximately 10% of the first bending mode frequency, using this particular control scheme. To account for the changing hydration state of the actuators, Lavu
et al. (2005) recently developed a model that includes dehydration and applied an adaptive
control scheme to achieve the desired tracking performance.
Considering both displacement and force control, Richardson et al. (2003) have successfully incorporated an impedance control strategy based on a PID scheme to track displacement and force. As far as device design goes, Newbury (2002) constructed a small
rotary motor that implemented five polymer transducers (4 actuators, 1 sensor). This innovative system was driven by the four polymer actuators and was controlled to track a
reference input signal using a feedback signal from the single polymer sensor. Also, Bhat
and Kim (2004) demonstrated the novelty of a hybrid control scheme switching between position and force control schemes using ionic polymer actuators and sensors. This particular
work was done with application to a robotic micro-gripper in mind.
With improved performance characteristics, ionic polymer transducers are presently
being considered for a variety of different application areas. These are discussed in the
following section.
1.4
Applications
This section will be brief, only mentioning a few examples from some of the major fields
that are considering ionic polymers, as well as the most recent developments. The reader
is referred to Kothera (2002) for a more extensive review of ionic polymer applications
or Shahinpoor and Kim (2005) for specific industrial and medical applications. An overall
review of how the materials are nearing commercialization is also provided by Ashley (2003).
Three of the most promising areas are robotics, biomedicine, and space systems, all of
which express interest in ionic polymer transducers for their favorable properties, such as
compliance, low operation voltage, light weight, and ease of miniaturization.
The robotics field is eager to apply ionic polymer to biomimetics. This is the study
of artificial mechanisms intended to mimic biologically inspired motions. Examples of these
include studying the motion of fins (Guo et al., 2000; Laurent and Piat, 2001), wings (Akle
28
and Leo, 2003), and snakes or worms (Arena et al., 2002) that had previously been difficult
to reproduce using conventional stiff actuators. Also serving as a link between robotics and
biomedicine is the development of artificial muscles. Ionic polymers have similar properties
to human muscles, which has excited many researchers and inspired the publication of a
book (Bar-Cohen, 2001a) and establishment of an annual conference symposium at the SPIE
Smart Materials Conference to help promote and spread the technological advancements.
One application in this area is the development of a finger-like micro-gripper (Shahinpoor
et al., 1998). Other biomedical applications favor ionic polymers because the base material is
r
, which is biocompatible. This means that ionic polymer transducers
a copolymer of Teflon
have the potential to be used inside the human body, such as in an active catheter device
(Onishi et al., 2000), sensing blood pressure and pulse rate (Keshavarzi et al., 1999), sensing
vascular stenosis (Akle et al., 2003b), or monitoring bone healing progress (Mudarri et al.,
2004). Moving outside the body and into outer space, ionic polymers are also being consider
in space systems. The primary reason for this is that the development of next-generation
satellites and telescopes are considering the use of inflatable and membrane structures,
where the large displacements and membrane nature of ionic polymer materials have an
advantage over other more common transducers like PZT (Witherspoon and Tung, 2002;
Leo et al., 2001).
A couple recent applications that do not fall into the above-mentioned categories have
to do with sensing. The first is a shear sensing application. Etebari et al. (2004) have shown
that ionic polymers are a viable solution to obtaining dynamic skin friction measurements
under a broad range of operating conditions and environments. Results showed that ionic
polymer transducers are ideal candidates because they have high compliance, sensitivity as
high as PZT (two orders of magnitude greater than PVDF), and can be constructed in many
configurations. The other has more to do with environmental sensing. This application is for
a humidity measuring device in chlorine-air atmospheres (Tailoka et al., 2003). Monitoring
impedance measurements, moisture diffusion across the electrode-electrolyte interface can
be observed, and the sensor has been calibrated over a wide frequency range.
Considering the range of applications for ionic polymer transducers, they look to
have great promise. And while the majority of modeling efforts are focused on actuation,
it appears that there are at least as many, if not more, active sensing applications as there
are actuation applications. These materials are still in their early years, however. As ionic
29
polymer materials become more fully characterized, commercialized, and optimized to specific applications, they will undoubtedly have an impact on a wide assortment of disciplines.
Gaining a better understanding of the material is paramount for the applicability of ionic
polymer to be completely realized. This leads into the motivation for the present research.
1.5
Motivating Statement
The fact that the fundamental mechanisms of ionic polymer actuation are still under investigation and the great promise ionic polymer transducers hold in diverse applications are
the primary motivating factors concerning the present work. Also, nearly all of the proposed models in the field are linear and input-specific. This motivates the development of a
nonlinear dynamic model that is capable of better describing the behavior of ionic polymer
actuation. Characterizing the nonlinear response to obtain a phenomenological model will
aid in better understanding the physical interactions that take place within the polymer
material.
1.6
System Identification and Modeling Procedure
While the aim of this work is to develop a dynamic nonlinear model for the actuation
response of ionic polymer transducers, the model will not introduce new physical laws related
to electrochemical and electromechanical phenomena. Instead, a more control-oriented
model will be developed using selected identification techniques. As a brief introduction
to system identification and modeling, the basic procedure will be mentioned here. While
different texts vary in the name and number of steps to this process, the overall explanations
are very similar. Some recent publications that give detailed accounts of all the aspects
of system identification (linear and nonlinear) are Ljung (1999), Pintelon and Schoukens
(2001), and Nelles (2001).
The first step is to obtain information about the system. This means that data must
be collected first, and that the data collected should be relevant for the purpose of the model.
For instance, if the response over only a specific band of high frequencies is of interest, then
there is no need to conduct static tests. Once information has been collected for the system,
the next step in identification is to choose a model structure for the system. Though this can
be very time consuming, the process can be expedited by having some a-priori knowledge
30
of the system. This could be anything from the order of the system, to constraints on
the system response, to deciding between a linear or nonlinear architecture. With a model
structure selected, parameterization occurs next. In parameterizing a model, some form
of optimization is generally used, such as a least-squares rule. This is the stage where
parameters are adjusted so that the model matches, or approximates, the measured data.
Considering a measured frequency response function as the system data, parameterizing
the model will minimize the error between the selected model’s frequency response and the
frequency response data. Then with the identified and parameterized model, validation
tests can be performed. In this stage the actual performance of the model is assessed and
the need to iterate on any step in the process will be determined. A different set of data is
used here for validation than what was used to derive the identified model. Using different
data will give a better idea of the robustness of the model.
Since the different basic types of models (black-box, gray-box, white-box) were discussed in the polymer modeling overview, they will not be commented on further here.
Instead, different model types more related to system identification will be summarized.
These include parametric and non-parametric models. The primary difference between
these two types of models is that parametric models have a (known) finite number of parameters, corresponding to known order, whereas non-parametric models are not limited in
this regard. An example of a parametric model would be a transfer function with known order and relative degree, where the modeling just tuned the coefficients to minimize the error
with the data. A non-parametric model type is the Volterra series, which is an infinite series
expansion. Though in all practical instances, the summation is truncated, it is not limited
to a number of parameters and the model reduction is left to the discretion of the system
modeler. Another way these modeling methods could be looked at is that a model structure
is assumed at the outset of parametric model development, while non-parametric modeling looks directly at the input-output measurements and later determines which model is
ideal. System identification techniques can also be applied in different domains (time and
frequency). Time-domain identification techniques generally create a system model from
regression techniques and frequency-domain techniques typically make use of the Fourier
transform and spectral methods to develop a model.
This methodology will be employed in the present research to help model the actuation response of ionic polymer transducers. Here, a non-parametric approach will be
31
taken to see what knowledge can be gained from the modeling efforts without imposing any
restrictions up front. The identification techniques chosen also employ frequency-domain
analysis, as it will be shown. Having completed the introductory information, the next
section will highlight the key goals and contributions of this work.
1.7
Technical Objectives and Contributions
The specific objectives of this research will be detailed in this section, along with how they
contribute to the field of ionic polymer transducers. These goals are stated below, followed
by brief descriptions of each.
• Characterization of Solvent-induced Nonlinear Response. In an effort to improve the
response characteristics and applicability of ionic polymer actuators, research using
different solvent materials is under way. While this research has led to broadened electrochemical stability windows and increased life spans, the effect on various response
properties has not been investigated. In this work, the action of nonlinear response
mechanisms will be characterized and related to the solvent viscosity. This study
will aid in better understanding the inner mechanisms of actuation in ionic polymer
transducers and possibly lend support to some proposed physical processes.
• Development of Nonlinear Dynamic Actuation Model. Though several model forms
have been proposed for ionic polymer actuation, none are universally accepted because of competing dominant mechanisms. Additionally, most of these models are
linear approximations of input-dependent systems, where nonlinearities have been observed. While not developing new physical equations of state or motion, the present
research will explore various nonlinear identification and modeling techniques based
on input and output measurements. The goal of this work is to obtain a phenomenological dynamic model that accounts for the nonlinear behavior observed experimentally. Important for application, this effort will place an emphasis on identifying a
block-oriented model structure that will enable implementation of these actuators into
system design.
• Proof of Viability for Control of Structures. Since the ultimate objective of studying
ionic polymer materials is to help fully realize their potential as an active material
32
system, their ability to solve existing design obstacles is paramount. To show that ionic
polymer actuators are a viable solution to control the shape of flexible structures, a
practical application study will be performed. The specific target in this study will be
controlling a square-plate polymer actuator, which could mimic an optical membrane
device and the reference commands could simulate adjustments in the focus.
The work performed to achieve these goals will follow in the subsequent chapters of
this document. An outline for how the work will be presented is provided in the next, and
final, section of this opening chapter.
1.8
Document Overview
This introductory chapter was meant to provide a broad overview of the current state-ofthe-art in ionic polymer transducer technology, having discussed background information,
a modeling review, performance enhancing research, and some application areas. More
specific topics related to the present research work were also introduced, including the basic
motivation to study nonlinearity in ionic polymers, an introduction to the type of modeling
that will be performed, and how this work contributes to the field. The next chapter
will focus more on a motivating example and a time-domain identification technique that
is used to characterize nonlinear distortion to sinusoidal inputs. This technique, Volterra
series representation, is then applied to an example system and some initial data. Chapter 3
then takes a more in-depth look at characterizing the nonlinearity in the actuation response
by looking at both this time-domain technique for single-frequency inputs, and a multiplefrequency technique (multisine excitations) for broadband evaluation.
Once a better understanding of the nonlinear form is obtained from the characterization results, the utility of specific nonlinear model structures can be explored. In this
stage, knowledge of the nonlinearity can significantly aid in predicting the experimentally
observed response. To this end, two nonlinear identification techniques for different model
structures are detailed in Chapter 4. Having performed simulation studies on example systems of these methodologies, Chapter 5 will apply and compare the results for the different
model structures with experimental actuation data. Here, models for both the electrical
and mechanical responses will be given. Then in Chapter 6, the identification and modeling
techniques will be applied to a more complex system, a square-plate polymer actuator. A
33
model will be developed for this system and control experiments will be conducted, showing
that ionic polymers could serve as a viable solution to controlling the shape of membrane
structures. Following this control work, the document will be summarized and concluded
in Chapter 7.
34
Chapter 2
Motivation and Volterra Analysis
This chapter provides a more elaborate account of the motivation for studying the nonlinear
response of ionic polymer actuators. Following this, a brief introduction to a nonlinear
identification technique will be discussed in application to harmonic distortion analysis.
After this description will be a section dedicated to the distortion analysis of both a linear
and nonlinear illustrative system. Once results of this system are discussed, experimental
results for the actuation response of two ionic polymer transducers will be presented, along
with a discussion of conclusions that were found.
2.1
Motivation
Because of the linear assumption in existing modeling efforts, several empirical models
have been developed to study specific types of response characteristics, some of which were
mentioned in the modeling overview section in Chapter 1. Two primary cases are studied
most often: the step response and the higher frequency (1–20 Hz) sinusoidal response. The
step response can be characterized with two time scales, one being a fast initial response
and the other a slow relaxation to steady-state. For a water sample, the fast time scale
is on the order of 10 milli-seconds, while the slow time scale is three orders of magnitude
longer, at more than 10 seconds. The sinusoidal response begins with a transient, normally
prevailing for a few cycles before the phase shifted and amplitude scaled steady-state is
reached. Interesting to the sinusoidal response is that the output waveform often differs
from that of the input. The excitation frequency is dominant, but higher harmonics usually
add some noticeable distortion to the signal, indicating nonlinear dynamics.
35
Figure 2.1: Linear model responses: (a) step response model; (b) frequency
response model.
As a motivating example, linear models have been developed using empirical data,
but they were made to fit either step response data or a frequency response function. The
step response model investigated was taken after the form proposed by Kanno et al. (1994)
and later modified by Mallavarapu and Leo (2001). Without resonance terms, the simplified
model is composed of three exponential terms with different coefficients and time constants.
In state-space form, the equations can be written as
⎡
⎤
⎡ ⎤
−α 0
0
a
⎢
⎢ ⎥
⎥
⎢
⎥
⎢ ⎥
ẋ(t) = ⎢ 0 −β 0 ⎥ x(t) + ⎢ b ⎥ u(t)
⎣
⎦
⎣ ⎦
0
0 −γ
c
y(t) =
1 1 1 x(t)
(2.1)
where the time constants α, β, γ and the coefficients a, b, c are determined experimentally by
curve-fitting the displacement resulting from a sudden imposition of an electric field to the
actuator. The frequency response model used in this study was created using the frequencydomain observability range space extraction (FORSE) algorithm, refined to represent the
data as an 8th-order state-space system using Dynamod 4.01. This data was collected
using a Gaussian input voltage, Hanning window, and 5 averages by a Tektronix 2630
Fourier Analyzer. Figure 2.1a shows the step response data set (dotted) nearly overlaying
36
Figure 2.2: Combined model predictions: (a) step response; (b) sinusoidal response.
the step response model output (solid) and Figure 2.1b shows the frequency response data
(dotted) overlaying the frequency response model output (solid). While the step model does
not overlay the data perfectly, the relaxation rate is predicted well, so this was assumed
reasonable for this exercise. Since each of these two model forms was parameterized over a
more specific frequency range, they are unable to predict the response of the other. More
precisely, the step response model is unable to predict the response at a given frequency,
and the frequency response model cannot predict the measured step response.
In an attempt to exploit these different regions of frequency dominance, the two
models were combined by convolving the polynomials to form one, hopefully comprehensive,
model that would be able to predict both the step response and the response to a given
frequency. To this effect, the combined model was simulated to a step response and two
different sine waves. Figure 2.2a shows that the combined model matches fairly well with
step response data (similar response to step model alone). The combined model also matches
well with sinusoidal data near and above the first resonance (just over 20 Hz), but does a
poor job predicting the amplitude and phase at frequencies below the resonant peak, as
can be seen in Figure 2.2b for 4 Hz (top) and 16 Hz (bottom). In each of these figures,
the dotted line represents the data, the dash-dot line shows the response of the step model,
the dashed line shows the response for the frequency response model, and the solid line is
the combined model response. Because ionic polymers have such a low frequency response
37
Figure 2.3: Nonlinear response characteristics of ionic polymer actuator: (a)
harmonic distortion; (b) amplitude scaling.
compared to other smart materials used as transducers, the inability to accurately predict
the response in this frequency range would be detrimental to system implementation. There
is also the nonlinear distortion problem that this linear model is unable to predict, so these
shortcomings serve as motivation for studying the ionic polymer actuator as a nonlinear
system.
In addition to nonlinear distortion that has been seen in the sinusoidal response of
ionic polymer actuators, a nonlinear amplitude scaling has also been observed experimentally in the voltage to current response for broadband excitation. Figure 2.3 shows some
recent measurements collected displaying these nonlinear response characteristics. In Figure 2.3a, a single-tone 7 Hz, 1.0 V input is applied to a cantilever sample and distortion is
apparent in both the current and velocity response (normalized to peak). The amplitude
scaling is shown in Figure 2.3b. The figure shows that the magnitude increases by 25% with
an order of magnitude increase in the input voltage level, which is a sign of nonlinearity.
38
Figure 2.4: Step response comparison of different solvents: (a) deionized water;
(b) ionic liquid (EMI-Tf ).
Had the system been linear, the voltage-to-current frequency response would have been
constant at each frequency and amplitude since the magnitude is merely an input-output
ratio, however.
There are also marked characteristic changes in the quasi-static response resulting
from changing the solvent in the actuator. Recalling from the performance enhancing
section that using ionic liquids as the solvent eliminates the back-relaxation seen in the step
response of a sample employing deionized water as the solvent. This is a clear indication that
the solvent in the actuator has a large influence on the physical processes in the material
that ultimately lead to the actuation response. Figure 2.4 has been included to depict these
drastically different step responses. Water is the solvent in the actuator producing Figure
2.4a from a 300 mV input, while the actuator in Figure 2.4b uses the ionic liquid EMI-Tf
and responds to a 2.0 V step. Although the input step amplitudes vary from each sample,
they do show the common response properties of each type of actuator. Namely, the water
sample has a large overshoot and slow relaxation, while the ionic liquid sample has a small
overshoot (no overshoot is present at low voltages with EMI-Tf) and virtually no relaxation.
The results shown in Figures 2.3 and 2.4 provide the motivation for investigating how
nonlinearities enter into the response of ionic polymer actuators. More precisely, whether
39
the relationship between an electrical input and an electrical output or a mechanical output is more dominantly nonlinear, if either, and what role different solvents play in this
nonlinearity need to be further explored. As mentioned before, researchers are actively
working in the areas of electrode morphology, base ionomer material, and various cation
forms to gain more understanding of the physical mechanisms involved in the response of
ionic polymer materials. There has also been work employing different solvents. This work
has served as an introduction to the potential benefits of varying this transducer component
and the possibility of a new low frequency mechanism. However, studying the nonlinear
effects introduced by the solvent and their resulting part in the overall actuation response
has received little attention. It is for this reason that investigating the cause of these nonlinear trends will add another piece to the puzzle of attaining a comprehensive model. To
get an initial idea of what type(s) of nonlinearities are present in the actuation response, a
technique used to characterize this nonlinearity will be detailed next.
2.2
Nonlinear Identification Technique
This section will discuss the basics of the chosen identification technique for this case of
distortion analysis. The method chosen for this investigation is the Volterra series because
it has the form of a power series expansion, which should be capable of well representing
the higher order harmonics that have been observed. Next will be an example section of a
full linear identification comparison, followed by some relevant studies.
2.2.1
Volterra Series Fundamentals
The Italian mathematician and physicist Vito Volterra (1860–1940) is whom the series
expansion has come to be named after. He, along with Fredholm, Hilbert, and Schmidt
were arguably the founders of theory on integral equations (Tricomi, 1957). Volterra’s
work with functions that depend on values of other functions, beginning in the late 1800s
(Volterra, 1887), contributed much to the early development of functional analysis, which
eventually included a book-length publication on the subject (Volterra, 1930). His most
notable contributions were the set of integral equations with his name and the nonlinear
Lotka-Volterra differential equations for the population dynamics in biological systems,
which he studied later in his career.
40
The Volterra series in its theoretical form is a functional power series relating the
input and output of a system. It is assumed, like in linear convolution, that the input
x(t) = 0 for t < 0, and that the system has finite memory. The system description can be
written in the form
y(t) = y0 +
N n=1
t
0
hn (τ1 , . . . , τn )x(t − τ1 ) . . .x(t − τn )dτ1 . . . dτn
(2.2)
where y(t) is the output, x(t) is the input, and hn (·) are the input/output relations for each
degree n, called the Volterra kernels. This description is a direct generalization of describing
a linear system using a convolution integral, where each Volterra kernel hn (·) can be thought
of as an impulse response of the nth degree (Eykhoff, 1974), and identification using the
Volterra representation requires measurement of these kernels (Billings, 1980).
Using a digital computer in practice requires that an approximate discrete-time
form be implemented and the model be truncated. Therefore, the more practical form of
the Volterra representation of a system is
y(k) = y0 +
M
h1 (i)x(k − i)∆t +
i=0
M
M h2 (i, j)x(k − i)x(k − j)∆t∆t + . . .
(2.3)
i=0 j=0
where M is the order, or number of memory terms, used in the expansion. This corresponds
to the number of previous input samples used in calculation of the current model output.
It should also be noted that the initial condition y0 is often taken to be zero. This form
of equation 2.3 is the form that was used in the present work. For identification applications, the Volterra series is considered a non-parametric, black-box approach, where there
is no prior knowledge of the system. Additionally, solving for the kernel parameters in
equation 2.3 is carried out in the least-squares sense after constructing an expansion matrix
with the prediction input sequence. An advantage to this formulation is that the system
response under consideration can be decomposed into individual components of various
degrees (Eykhoff, 1974)
y(t) = ylinear (t) + yquadratic (t) + ycubic (t) + . . .
(2.4)
where the output y(t) is composed of terms raising the input to different powers, or degrees;
stated more precisely y(t) = f (x, x2 , x3 , . . . , xN ). This concept of nonlinear component
separation will prove useful in the identification procedure to be discussed next.
41
In looking at the above expressions, it is noticed that a convenient consequence of
using the Volterra series to describe a system is that when only the linear case is considered,
equation 2.2 is the same as the convolution integral for a linear system, while equation 2.3
is its approximation. This corresponds to the case where N = 1 in equation 2.2 and only
h1 (·) is considered in equation 2.3.
There are also certain restrictions involving the use of Volterra kernels in studying
a given system. These are assumptions and results of using the truncated Volterra series to
describe a system. Two examples are that the system must have finite memory and that
the response cannot exhibit chaotic behavior. Another restriction that becomes apparent
when considering the power series form of the equations is that the system exhibits only
harmonic generation, not sub-harmonic generation, when excited with a periodic input.
Finally, for a given steady-state input signal, the system must not have more than one
steady-state response. In accordance with the present study, ionic polymer actuators satisfy
these requirements.
2.2.2
Comparison of Linear Identification Methods
As a point of reference, a quick comparison will be made between two linear identification
techniques and the linear Volterra series. The arbitrary system used in this example is a
simple second-order system of the form
H(s) = k
ω2
s2 + 2ζωs + ω 2
(2.5)
where k = 0.4, ω = 5, and ζ = 0.01. The output of this system was simulated from a
normally distributed, random input sequence with a variance of 1.0 at a sample rate of 20
Hz. Then using this simulated input and output data, a few different linear methods were
r
functions ARX and N4SID, and the
used to identify the system. These include the Matlab
linear Volterra series. The auto-regression with exogenous variables (ARX) gives a transfer
function representation of the data, while the other method, N4SID, is a subspace method
for calculating a state-space model of the system. Using a fourth-order approximation,
the ARX model was able to accurately predict the simulated output, while the N4SID
model required only a second-order to achieve the same output prediction. However, it was
found that the linear Volterra expansion required an order of 150 to be able to match the
simulation output and the output from the models identified using the other two methods.
42
Figure 2.5: Comparison of some linear identification techniques.
In terms of equation 2.3, this expansion has N = 1 and M = 150. Figure 2.5 shows these
results. Although it can be seen that all the responses shown in the output plot overlay
each other, the simulated data is the solid line, the dashed line is the ARX output, the
dotted line is the N4SID output, and the dash-dot line is the Volterra output.
The order required for the Volterra series to identify this simple linear system does
seem high, and it could be thought that the Volterra series is inferior to the other two
methods shown. This is true for the case of linear identification, for which the ARX and
N4SID methods were developed, but it is the extension to nonlinear systems that shows the
strength of the Volterra expansion. This does not come without cost, however. While this
correlation to the impulse response function is an advantage of the Volterra series, when
nonlinearities of higher degree are considered, the number of kernel parameters increases
exponentially. This creates a disadvantage in computation time and parametric complexity.
For this reason, pruning or order reduction methods are often applied to Volterra models.
Examples of these methods include constraining kernels of certain degree to zero prior to
solving the system representation (model pruning), and setting all kernel parameters below
a desired threshold value equal to zero after the system has been solved (model reduction).
Another detriment when using high order and degree with experimental results is that, as
with all polynomial model types, the Volterra series has a tendency to fit the noise in the
43
measurements, causing the model to be too sensitive (Nelles, 2001). In the case of distortion
analysis where specific inputs with known spectral characteristics are used, the concept of
a “minimum-parameter” model can also be employed to keep the order and degree number
low and making the model less sensitive to noise. This will be discussed in the system
analysis sections to follow.
2.2.3
Related Volterra Research
Before proceeding with the analysis methodology, it may be worthwhile to briefly mention
a few related examples of how the Volterra series has been used in research. For the most
part, Volterra expansions of only a few degrees (up to 3) are used due to increased complexity, but there have been some cases where up to degree five has been used. Fifth-degree
Volterra kernels have been used for studying intermodulation distortion in a nonlinear,
broadband amplifier (McRory and Johnson, 1993) and approximating wing oscillations in
aircraft at high angles of attack (Stalford et al., 1987). Getting into the smart material
work, a 4th-degree Volterra model has been used in the distortion reduction of a Terfenol-D
actuator (D’Annunzio et al., 1996). Aside from Volterra modeling, nonlinear identification
using a Hammerstein model has led to successful model development with ionic polymer
materials in application to walking robots (Yamakita et al., 2003). Coincidentally, a booklength treatment of Volterra analysis has been written with several case studies involving
polymerization reactions by Doyle et al. (2002).
2.3
Analysis Methodology
The analysis procedure to be discussed will be applied to study the actuation response of
ionic polymer transducers, where the polymer system is assumed to be composed of two
single-input, single-output subsystems connected in series. Assuming this structure of the
system will facilitate the investigation into where nonlinearities enter into the response.
The first subsystem is described by the voltage and current signals. This is essentially the
electrical impedance of the system, and using voltage as the input and current as the output
will give insight into the electrochemical response of the polymer actuator. The second
subsystem is described with current as the input signal and velocity as the output. Since
this system includes the motion of the actuator, it provides information on the mechanical
44
response. An illustrative system will be used to explain the analysis methodology. For
the sake of comparison, this system also consists of two subsystems in series. Each of
the subsystems is arbitrarily composed of one pole and one zero, and the nonlinearity is
blindly introduced with the input. By “blindly,” it is meant that there is no measurement
of how this nonlinearity affects the original input, except that the output contains nonlinear
distortion. Only the applied voltage, output current, and output velocity are used in this
procedure.
2.3.1
Linear System Analysis
As a prelude to the nonlinear study, the Volterra series identification method was applied
to a purely linear system to see how it performed. In this study, an input waveform was
passed through a linear system to produce an amplitude-scaled, phase-shifted version of
that input. The linear system used consisted of one pole and one zero, and is defined as
H1 (s) =
s + 50
.
s + 15
(2.6)
These system parameters were arbitrarily chosen. Various sinusoids were also arbitrarily
chosen as the input signals used for the simulations. Both single-frequency sinusoids and
multiple-frequency sinusoids were used.
Figure 2.6 shows a sample data set used in one of the analyses. The dotted input
signal is the result of adding a 5 Hz and a 13 Hz sine wave. These frequencies, as with
all other parameters in this illustrative study, were chosen arbitrarily, but it was made
certain that they were not integer multiples of one another to help show the utility of the
approach. The solid line is the output of the linear system. Being that this system is
linear, the principle of superposition holds, where the output signal in the figure is the
same as it would be if the system was simulated with each of the two frequencies as single
sinusoids and adding their results together. An abstraction from this concept, namely signal
decomposition, also forms the basis for the nonlinear analysis methodology, so it will be
discussed more to this end in the following section. The figure also shows the prediction
data set, which was used to identify the Volterra model, and the validation data set. This
validation data set is different from that used in determining the model parameters, and is
therefore used to assess the quality of the model.
In applying the Volterra expansion to the two signals, models were identified for the
45
Figure 2.6: Example prediction and validation data sets used for model identification.
various input signals used. A sinusoid with a single frequency, two frequencies, and three
frequencies were used in three different simulations. In each of these simulations, several
Volterra models were calculated by varying the order (M ), or number of memory terms
in the model, and the degree (N ), or the highest power to be used in the power series
expansion. The model error, defined as the average of the absolute value of the difference
between the simulation output and the output of the Volterra models, was used to determine
which order and degree produced the best model of the system. This equation has the form
εV = E[|ysim(t) − yV olterra (t)|],
(2.7)
where E[·] is the expected value operator, or the mean. Using the average of the absolute
prediction and absolute validation error, the performance of the models was compared. It
was seen that the error converged to zero for models of all degrees calculated at the same
order. And this convergence order depended on the type of input waveform used.
In the three input waveforms tested (composed of one, two, and three frequencies),
the error converged to zero in an order of twice the number of frequencies contained in the
input. For example, the single-frequency signal required an order of 2 for convergence, while
the three-frequency signal converged at an order of 6. A plot of average validation error
versus order and degree for the two-frequency input (shown in Figure 2.6) can be seen in
46
Figure 2.7: Average validation error versus model order (M ) and degree (N ).
Figure 2.7, where the error converges at an order of 4. For sinusoids, this order of convergence happens to be the same as the order of persistence of excitation. A single-frequency
sine wave has a persistence of excitation order of 2 based on the two-sided spectrum of the
signal, with a peak at −f and f . What this indicates is that when using sinusoidal inputs
to a system that produces harmonic nonlinear distortion, the minimum-parameter, linear
Volterra model and the persistence of excitation of the input signal should have the same
order. Again, by using the terminology of a “minimum-parameter” model, it is meant that
the order, degree, and therefore the number of model parameters are kept to a minimum
in fully describing the system response. This concept will also be applied in the nonlinear
analysis by minimizing the number of possible model parameters. The figure also shows
that the results from all three degrees overlay each other at nearly all orders, which means
that the system is best approximated as the lowest degree. In this case, the system should
be linear, which by design, is a correct assessment.
A comment on the purpose of identifying the minimum-parameter model will briefly
be made here. In the simulations of the linear system here and nonlinear system to follow,
the “measured” input and output signals used to describe the system do not contain any
noise. However, actual experimental measurements will likely be affected by some level of
noise. This noise can create problems when parameterizing a Volterra model as the order
47
and degree increase. The calculation of the model parameters will work to actually fit the
noise, requiring a high order and degree to get the prediction error to converge to zero, or
near zero. If noise is in one or both measurements, the error will converge to some nonzero
value that is proportional to the amount of noise in the signals. This noise-fitting that occurs
in the prediction model will lead to poor validation results, assuming the noise is random.
Since the model would match the noise in the prediction data, if the noise in the validation
data differed at all, as it would, the validation error could be very large. This is why the
concept of finding the minimum-parameter model comes up. A minimum-parameter model
will fit only the general trend of a signal, leaving the error convergence value to reflect the
noise level.
Using the concepts of signal decomposition and minimum order and degree of the
Volterra expansion, the analysis procedure will now be extended to a system containing
nonlinearity. The next section will explain how these linear system characteristics can be
applied to a series of nonlinear systems.
2.3.2
Nonlinear System Analysis
This section will discuss how some principles of the Volterra representation can be applied
to gain understanding of a nonlinear system. Since this analysis procedure will be applied to
an ionic polymer actuator, the illustrative nonlinear system was designed to have the same
basic structure as some of the proposed models: two single-input, single-output subsystems
connected in series. Without an accepted mathematical set of equations governing the
actuator dynamics, the validity of this system decomposition can only be assumed correct.
In accordance with the theories discussed in the modeling review that it is the current
resulting from an applied voltage that actually causes mechanical deformation, this system
structure is deemed a reasonable representation of the system, and will aid in yielding
insight into whether the electrical or mechanical characteristics of the material cause more
nonlinearity.
As an analog to the experiment, the applied input to the first subsystem and hence,
overall system, will be referred to as the voltage (V ), the output of the first subsystem and
also the input to the second subsystem will be called the current (I), and the output of
the second subsystem and the overall system will be the velocity (U ). This is shown in
Figure 2.8a. The V-I subsystem is composed of a nonlinear distortion of the voltage signal,
48
Figure 2.8: System structures used in the nonlinear analysis: (a) total system;
(b) subsystem 1 input and outputs; (c) subsystem 2 inputs and outputs.
followed by the linear system of equation 2.6. A portion of the input signal is affected by a
cubic nonlinearity, which is then added with the applied signal before passing through the
linear system. This can be written as
I(t) = H1 (s)[V (t) + η1 V 3 (t)]
(2.8)
where η1 was arbitrarily chosen as 1.2. It should be noted here that this equation contains
both time- and Laplace-domain variables, t and s, respectively. The I-U subsystem is
structurally similar, but the nonlinear distortion is caused by a quadratic nonlinearity.
This linear system and its nonlinear effect are defined by
H2 (s) =
s + 35
s + 25
U (t) = H2 (s)[I(t) + η2 I 2 (t)]
(2.9)
(2.10)
where the value used for η2 was 0.3. Figure 2.9 shows the three simulated signals to be used
during this procedure. Analysis of the first subsystem (voltage-to-current) is the easier of
the two, so it will be discussed first, followed by a description of the analysis of the second
49
Figure 2.9: Input and output signals for the two subsystems.
subsystem (current-to-velocity). Incidentally, simulations were also performed when the
nonlinearity affects the output of the linear systems, and similar results were obtained. As
a final note, the signals were all normalized with respect to their peak amplitude prior to
applying this analysis procedure. This linear scaling will not affect the results and can be
recovered after identification if needed, but it does help with some numerical issues that
arise in high degree models.
In the voltage-to-current subsystem, the input voltage is a single-frequency sine wave
(5 Hz). Because of the nonlinearity acting on this input, the output of this subsystem is
composed of linear portion and a nonlinear portion (see Figure 2.8b)
I˜ = I˜L + I˜N
(2.11)
where the tilde signifies data, the subscript L denotes the linear portion, and the subscript
N denotes the nonlinear portion. Based on the previously mentioned linear analysis, the
minimum-parameter, linear model for a single-frequency input signal is calculated from a
Volterra expansion of order 2 and degree 1. This will give a linear portion of the output,
which will be denoted
IL = h12 (ṼL )
(2.12)
where hN
M (·) is the Volterra model with order M and degree N , and the input sinusoid
50
Figure 2.10: Average validation error for subsystem 1 (voltage-to-current) simulations.
is considered a measured, single-frequency sine wave. Based on the signal decomposition
concept highlighted by equation 2.4, the remaining portion of the output can be attributed
to nonlinearity within the system
I˜N
= I˜ − IL
= I˜ − h12 (ṼL)
(2.13)
where all degrees 2 and greater have been lumped together. In order to identify this nonlinearity, several Volterra models of various degree and order must be calculated
IN = hnm11 (ṼL)
(2.14)
where the subscripts on the order and degree refer to the first subsystem. Like the approach
discussed in the previous section, the average, absolute validation error is used to determine
the best, or minimum-parameter, model of the nonlinearity. Figure 2.10 shows the errors for
this system. In the figure, it is seen that both the linear and quadratic models are unable to
converge to zero error. This is an indication that the actual nonlinearity in this subsystem
is of higher degree. Examining the error for degrees 3 and 4, they appear to overlay each
other, converging to zero at an order of 2. Since the number of model parameters is smaller
51
for a Volterra expansion of degree 3 than that of degree 4 for the same order, the minimumparameter model for the nonlinearity in the voltage-to-current subsystem is of order 2 and
degree 3 (h32 ). When considering that this particular subsystem introduced a cubic nonlinear
effect to the original sine wave input, it makes sense that the identification process found
this to be the case. This also provides a good check on the overall analysis procedure.
Bringing the analysis of the first subsystem to a close, Figure 2.11a has been included
to show the prediction and validation data sets used, along with the total response. The
top row of plots shows the linear Volterra response (IL ), which is scaled and shifted version
of the original input (ṼL). The nonlinear response shown in the middle row of plots is
˜ and the linear response. In the bottom row,
the difference between the system output (I)
the original input is shown (dotted), along with the total Volterra response overlaying the
system output. The total Volterra response is given by
I = IL + IN
= h12 (ṼL) + hnmo1o1 (ṼL)
(2.15)
where the subscript “o” in the definition of IN refers to the minimum order and degree. It
˜ but this does not necessarily
should be noted here that in perfect model matching I = I,
imply that the linear and nonlinear components are also equivalent to one another; only
their sum needs to be the same. Evidence of this is seen in this case since the linear and
nonlinear portions of the original input are nearly in phase.
In simulating the Volterra models over longer time periods, power spectra were
estimated. Spectra of the input, output, and Volterra models can be seen in Figure 2.11b.
The top plot shows that the input and linear Volterra component have matching frequency
and amplitude, as does the middle plot for the output of the system and the total Volterra
model prediction. Shown on the bottom are the linear and nonlinear components of the
total response, as identified by the Volterra models. The linear component has one peak,
which happens to be the most dominant amplitude, at the excitation frequency, while the
nonlinear component has less influence at the excitation frequency. The second harmonic
(degree 3) is most dominant in the nonlinear portion of the response. This makes sense
because the linear model identification should account for the majority of the response at
the excitation frequency, leaving the nonlinear model to capture the dominant response at
other frequencies. It should be stated here that had this system been purely linear, the
52
Figure 2.11: Volterra results for subsystem 1: (a) prediction and validation; (b)
power spectra.
nonlinear portion IN would have been zero and only the linear component would be seen
in the figure.
With the nonlinearity in the first subsystem being identified, it is now time to move
on to the second subsystem (current-to-velocity), which is slightly more complicated because
it has already been distorted and its input characteristics are different than those of V . The
basic procedure is the same, but the complication arises when considering that the input to
this subsystem has two components instead of one, and consequently, the output will have
three components: two linear versions of the two inputs and the nonlinear portion that is
unaccounted for by the linear response. This is shown in Figure 2.8c. Each of the response
components identified from the voltage-to-current subsystem are treated individually in
this second subsystem. This will allow for the minimum-parameter linear and nonlinear
components to be identified. It should also be noted that what was considered the linear
and nonlinear current components in the first subsystem will simply be considered as two
separate, “original” input signals to the second subsystem, denoted I1 (previously IL ) and
I2 (previously IN ). For this subsystem, the input signal is
I = I1 + I2
(2.16)
and the output of the subsystem is given as
Ũ = ŨL + ŨN
53
(2.17)
where the linear component ŨL is now composed of linear contributions from each of the
input’s two individual components
ŨL = UL1 + UL2 .
(2.18)
Similar to before, the nonlinear analysis of the second subsystem begins with a linear
identification using I1 as the input and the total system output Ũ as the output
UL1 = h12 (I1 ).
(2.19)
With UL1 defined, the next step requires a linear identification using IL2 as the input and
the error between Ũ and UL1 (called ÛL2 ) as the output
Ũ − UL1 = h1m∗ (I2 )
(2.20)
UL2 = h1m∗ (I2 ).
(2.21)
where the superscript “∗” on the order indicates that the order of the minimum-parameter,
linear model was determined by examining the power spectrum on the nonlinear component
in Figure 2.11b. Recalling from the linear analysis section, it was shown that the number of
peaks in the one-sided spectrum of a sinusoidal signal should be reflected in the order of the
minimum-parameter, linear Volterra model as twice that number. In the present case, there
are two peaks in I2 , which means that h1m∗ becomes h14 . Although the model parameters
are calculated from equation 2.20, this linear model will not match ÛL2 unless there is no
nonlinearity in the second subsystem. For this reason, the actual UL2 is determined from
simulating the model determined in equation 2.20 (parameters of h1m∗ ) with the input I2 ,
as shown in equation 2.21. Combining equations 2.19 and 2.21 into equation 2.18 now gives
the full linear Volterra model of the second subsystem
UL = UL1 + UL2
= h12 (I1 ) + h1m∗ (I2 )
(2.22)
where UL = ŨL . Having calculated the linear response component, the nonlinear portion
can now be determined by
UN
= Ũ − UL
= Ũ − (UL1 + UL2 )
= Ũ − (h12 (I1 ) + h1m∗ (I2 )).
54
(2.23)
Figure 2.12: Average validation error for subsystem 2 (current-to-velocity) simulations.
The next step in the analysis is to identify this nonlinear response of the current-tovoltage subsystem. As with the first subsystem, the nonlinear identification involves several
model calculations with various degree and order combinations. In the development of each
of these models, the nonlinear component UN , given in equation 2.23, is used as the output
that the Volterra expansion tries to match, given I = I1 + I2 as the input. The minimum
order m2 and degree n2 are determined from
UN = hnm22 (I)
(2.24)
by comparing the average, absolute validation error and model complexity. Figure 2.12
shows this comparison for the second subsystem. It can be seen immediately that the linear
model does not converge to zero error, implying that this subsystem cannot be approximated
as linear. Examining the other results shows that degrees 2 through 4 all converge to zero
error in orders of 4, 3, and 2, respectively. This introduces a non-uniqueness into the
model because choosing any of these combinations would give a sufficient model for the
nonlinearity of the subsystem. Now a choice will be made to find the minimum-parameter
model. Since the error is small for each, the best choice will be left to the model with the
fewest number of parameters. Carrying out the expansion for each of the order and degree
55
Figure 2.13: Volterra results for subsystem 2: (a) prediction and validation; (b)
power spectra.
combinations where the error converges to zero gives the model sizes as
|h24 | = 15
|h33 | = 20
|h42 | = 15.
From these model parameter numbers, it can be said that either h24 or h42 should be chosen
since they have equally few parameters. In this example, the model used was h24 because
of the knowledge about the system. That is, there is a quadratic nonlinearity in the second subsystem (n = 2) and there are two different input frequencies, as determined from
identification of the first subsystem (m = 4).
Completing the nonlinear identification of the second subsystem, Figure 2.13a has
been included to summarize the results of each step in the procedure, in both prediction
and validation. The “linear 1” and “linear 2” responses are examples of UL1 and UL2 ,
which are the linear responses to the two components of the current signal (I1 and I2 ), as
determined in the analysis of the voltage-to-current subsystem. The “nonlinear” response,
UN , is the error between the output velocity signal, Ũ , and the two linear components that
make up UL . In the bottom plots, the input current is displayed with the Volterra prediction
overlaying the output velocity. This shows that the model matches the output well.
As in the previous subsystem’s analysis, the power spectra corresponding to this
56
Figure 2.14: Forward running sum RMS of each subsystem.
subsystem will also be discussed. Figure 2.13b has these results. The linear Volterra
prediction is shown to match the frequencies of the input current signal in the top plot,
while the middle plot shows how well the total Volterra prediction matches with the output
velocity. Separating the Volterra solution into linear and nonlinear components, the bottom
plot shows the relative influence of each. The linear portion contains two frequencies,
corresponding to the frequencies of the input current, and the nonlinear portion contains
nonlinear influence at each of these frequencies, but with lesser influence, and at the first,
third, and fifth harmonics, in descending order. Recalling that the nonlinearities in the
system were designed to be quadratic and cubic, it should be noted that the influence of
other harmonics is common in systems with quadratic and cubic nonlinearities because of
interaction between the different frequencies.
Now that the nonlinear identification of the entire system has been completed, it is
now time to use these results to draw conclusions about which, if either, of the two subsystems contributes more nonlinearity to the overall system. The technique used to quantify
this nonlinear influence is the running sum root-mean-square (RMS). This is essentially an
integration of the power spectrum performed on the nonlinear signal component for each
subsystem (solid lines), so the subsystem that sums to the greater value will be considered more dominantly nonlinear. Figure 2.14 shows the forward sum for each of the two
57
subsystems, where it could be said that the current-to-velocity subsystem contributes more
nonlinearity to the total system response, adding to nearly 2.5 times that of the nonlinear
influence in the voltage-to-current subsystem. Had the difference between the summations
been much larger, for example, an order of magnitude or greater, this would have provided
more conclusive evidence that one system is more dominantly nonlinear than the other.
That the sums differ by only a factor of 2 actually makes it more difficult to say that one
system is “more nonlinear” than the other. This plot also draws attention to the dominant
frequencies in the nonlinear portion of the system responses. These are the jumps that occur. For instance, the actuation frequency is 5 Hz, which corresponds to the first rise in each
subsystem. Then subsystem 2 has another significant rise at 10 Hz, indicating a quadratic
component, while subsystem 1 shows evidence of a cubic nonlinearity with a large rise at
15 Hz. Quickly, had either subsystem been purely linear, its running sum RMS would have
shown a jump at only the excitation frequency.
While this analysis technique does provide a method for identifying and quantifying
nonlinearity in a given system or series of systems, the results discussed were of an arbitrarily
designed system. The system structure is the same as that assumed for an ionic polymer
actuator, allowing direct application to actual experimental results, but the results shown
above are not meant to be indicative of those in the actual polymer system. Those results
will come in the following section.
2.3.3
Methodology Comments
A comment on the extraction of physical insight utilizing this method will now be made.
It was seen in this example that the influence of quadratic and cubic nonlinearities in
a system could be identified. Being that this was a simulation exercise, it was actually
known that these were the types of nonlinearities introduced in the system, making their
physical identification simpler. On the other hand, the types of nonlinearity in the given
system may not be known at all in an experimental situation. This could lead to erroneous
conclusions regarding the type of nonlinearity in the system. For instance, the output of a
cubic nonlinearity and a dead zone nonlinearity can look remarkably similar in distortion,
while the actual influences of the nonlinearities are quite different. Therefore, caution
needs to be used when attempting to gain physical insight about the system using this
Volterra analysis procedure because discontinuous nonlinearities could skew the results.
58
Nonetheless, this method does provide information on whether a signal contains nonlinear
distortion. Without other experiments targeted to validate a particular nonlinear form,
however, certain conclusions may be difficult to draw.
Before beginning with the experimental results of an ionic polymer actuator, a slight
modification to the procedure will be employed to reduce a potential source of error propagation. Recalling that the nonlinear analysis of the second subsystem used model prediction
results from the first subsystem as inputs for the identification algorithm, if there was any
error in these results, that would have a strong tendency to carry through into the second
subsystem and skew the overall results. To avoid this problem, the measured current signal,
˜ will be used as the input to subsystem 2, rather than the model predicted outputs of the
I,
first subsystem, IL and IN . Since the minimum order of the linear model for subsystem 1
is known from knowledge of the input voltage waveform, the spectrum of the current signal
is used to determine this order for the linear model of the second subsystem. The analysis
of subsystem 2 will then be simplified to the same procedure as with the first subsystem
(equations 2.11–2.15), except that the I’s and V ’s will be replaced with U ’s and I’s, respectively. This will consequently modify the structure of Figure 2.8c to be the same as that of
Figure 2.8b, with a single input component and two output components.
2.4
Experimental Identification using the Volterra Series
Having explained the analysis procedure for an arbitrary simulation model, this section
will present experimental identification results for ionic polymer actuators. The question
to be answered in this investigation is whether one of the subsystems introduces more
nonlinearity into the overall actuation response than the other, and how this changes with
input frequency and amplitude. If one system proves more nonlinear than the other, the
result will be used as motivation for focusing characterization and modeling efforts on the
more dominantly nonlinear portion of the polymer response. Results from two different
polymer samples will be detailed, drawing conclusions about the nonlinearity in each, and
how the actuator solvent affects the response. The specifics of each sample will be discussed
first, followed by a description of the test setup. After this, the nonlinear identification
results will be presented.
59
Table 2.1: Summary of Actuator Compositions.
Actuator
Component
Ionomer
Solvent
Cation
Electrode
2.4.1
Sample 1
Sample 2
r
Nafion
-117
water
Na+
Pt - Au
r
Nafion
-117
EMI-Tf
Na+
Pt - Au
Actuator Materials
Since two different materials have been used in this study, it is important that the specific
actuator makeup be discussed, as it may affect the individual response characteristics. Each
r
-117 as the base ionomer. The first sample,
sample uses the commercially available Nafion
which will henceforth be referred to as Sample 1, uses deionized water as the solvent and
sodium as the mobile cation. The chemically deposited electrodes are composed of platinum
with a thin gold over-layer that was prepared using an impregnation-reduction process
(Bennett and Leo, 2003). The second sample (Sample 2) uses the ionic liquid 1-ethyl-3methylimidazolium trifluoromethanesulfonate (EMI-Tf) as the solvent and sodium as the
cation. It should be noted here that the use of ionic liquids as solvents has been employed
because of their high electrochemical stability, when compared to water, and the fact that
dehydration does not occur in an open-air environment, as is the case with water (Bennett
and Leo, 2005a). The electrodes of Sample 2 are the same as that for Sample 1, platinum
with a thin gold over-layer. To briefly comment on the reason for the thin gold over-layer, it
can be stated that this layer, usually about 50nm thick, is to increase surface conductivity.
In the use of other electrode compositions, such as the non-precious metal copper, the gold
layer also serves the purpose of preventing oxidation. Table 3.1 displays a summary of each
actuator’s composition, where it is seen that the only difference between the two is in the
solvent.
It should be noted here that one type of ionic polymer used in this study does suffer
from dehydration (Sample 1), but the system is examined only in the short-term, where
effects from dehydration are negligible. Therefore, the hydration state of the material can
be thought of as another input to the system that happens to be constant over the time
span of interest, essentially removing its influence. And as an extra precaution, deionized
60
Figure 2.15: Experimental setup for ionic polymer actuator testing.
water was brushed over the actuator before data was collected for each test performed.
2.4.2
Laboratory Setup
In order to conduct an experimental investigation using the described analysis procedure,
data from the two polymer samples was collected. In these experiments, voltage, current,
and tip velocity were the measured signals. For data acquisition, a dSPACE DS 2003
analog-to-digital converter and DS 2103 digital-to-analog converter (±5.0 V, 14-bit) were
used. Due to the low signal strength in the ionic liquid sample (an order of magnitude
lower current than the water sample), the A/D converters were set to ±5.0 V and had a
16-bit range. A laser vibrometer (Polytec OFV 3001 Controller and 303 Sensor Head) was
used to measure the tip velocity of the cantilever ionic polymer actuators. The sensitivity
was adjusted from test to test depending on the signal characteristics, but only the two
lowest settings (1.0 mm/s/V and 5.0 mm/s/V) were used. This was done to ensure that the
whole A/D range was used as effectively as possible. The input signal was generated by the
61
dSPACE system and passed through a Hewlett-Packard 6825A Bipolar Supply/Amplifier
to amplify the current before reaching the polymer sample. The data collection of this
voltage occurred after the amplification. The current in the amplified input signal was
measured with a current sensing circuit, which consisted of an operational amplifier and
two resistors. Finally, to hold the polymer and apply the electric field, a test fixture with
gold foil electrodes was used. Figure 3.2 has been included to show the necessary equipment,
along with the measured voltage (V), current (I), and velocity (U) signals.
Experiments were run with single-frequency sine waves spanning four orders of magnitude in frequency and a single order of magnitude in amplitude, but because of poor
signal-to-noise ratios at the low frequencies and amplitudes, the amount of useful data was
cut nearly in half. Additionally, since water-based samples cannot tolerate voltages as high
as samples with ionic liquids, a voltage ceiling limited the data further. For these reasons,
the data to be analyzed varies in frequency from 1.0 Hz to 50 Hz in half-decade increments
and in voltage from 1.0 V to 1.5 V in 0.25 V increments. The sample rate was also varied
from test to test, but was kept a constant multiple of the actuation frequency over the
frequency range, which provided the same number of points per cycle in each test. Setting
this multiple to 200 allowed full use of the sample rate range of the data acquisition system.
The actuator samples were manufactured as stated in the previous subsection. The
water-based sample (Sample 1) had a free length of 8.0 mm and a width of 2.0 mm, giving the
cantilever a 4:1 length-to-width ratio. Maintaining this ratio, while differing in dimensions
was Sample 2, which had a free length of 16.0 mm and a width of 4.0 mm. The actuator
dimensions varied from sample to sample due to material availability and timing issues, but
it is believed that maintaining the same length-to-width ratio between the two samples will
not adversely affect any conclusions drawn from the nonlinear analysis.
2.4.3
Experimental Identification Results
Comparing the identification results from two different transducer samples, this section
will be sub-divided into a section pertaining to each actuator and a section reserved for
discussion at the end. These results will follow from the analysis procedure outlined in the
beginning of this chapter. In addition to parameterizing nonlinear models at each input
amplitude and frequency, linear Volterra models were also constructed to use as another
means of comparison. In each model for each subsystem, the order of the linear model
62
for the voltage-to-current subsystem was set to 2 because the input voltage had a single
frequency, while that for the current-to-velocity subsystem was set to 6 because there were
three peaks in the spectrum of the current signal in most cases. The order was varied from
1 to 8 in each nonlinear model and degrees from 1 through 4 were tested. These values were
chosen and kept consistent for all models because convergence in the prediction error could
be achieved within these limits. Since several data sets have been collected and analyzed for
each subsystem and sample, results from a typical subsystem at one voltage and frequency
setting have been selected to portray the effectiveness of the analysis technique. After
showing the results for this characteristic data set, tables have been included to summarize
the results for each individual test condition.
Sample 1 Results
The first step in the analysis procedure is building a linear, order 2 model from the voltage
and current signals. Next, the error of this linear model output and the data is modeled
with several degree and order combinations to test for convergence. Figure 2.16a shows
these results for the test condition of 1.5 V and 5.0 Hz, which has been chosen to represent
the results of Sample 1. It can be seen in this figure that the linear and quadratic models
start off with a large error at M = 1 and appear to be slowly converging, but at inefficient
rates as far as minimizing the model size is concerned. The error of the cubic and quartic
terms begins at M = 1 with less than half the error of the lower two degrees and each
converges to near zero error in the order range shown. Seeing that the data sequences used
here were actual measurements and likely corrupted with some noise, the fact the near zero
error is achieved makes it probable that the validation error will not converge to the same
level.
Figure 2.16b has been included to show the validation error results, zooming in on
the order axis. As expected from the prediction error convergence level, the validation
error does not converge as well. It is also noticed that divergence seems to occur at higher
orders and degrees. This is most likely attributable to overly sensitive model parameters
that have matched the noise in the prediction data. Common to both the prediction and
validation error is the clear separation between degrees 2 and 3. As before, the average
error in the validation set begins in the higher two degrees at half the starting value of the
lower two degrees. This provides evidence that the nonlinearity in the voltage-to-current
63
Figure 2.16: Error of models for subsystem 1 at 1.5 V, 5.0 Hz (Sample 1): a)
prediction; b) validation.
subsystem could be a cubic mechanism since degree 3 appears to perform the best of the
four shown. Keeping with the parsimonious modeling effort, order 2 was chosen to complete
parameterization of the best model.
After selecting the best model based on the validation error, the linear and nonlinear
components were combined to yield the total Volterra model for the system. Each of these
components is displayed individually and together for one input cycle in Figure 2.17a. The
left and right columns of plots refer to prediction and validation results, respectively. The
top row of plots show the linear model fit to the output data, the middle two plots show
the nonlinear component (linear model error) and the Volterra prediction, and the bottom
plots show the combined response along with the input. In this figure, the dotted line is
the input signal (voltage), the solid line is the output signal (current), and the dashed line
is the noted Volterra model output. Comparing the linear and nonlinear components to
one another and to the measured current signal, it does appear that the nonlinearity is
cubic since there is no sign of a discontinuous nonlinearity in the data, and the nonlinear
component goes through three cycles in the time it takes the linear component and input to
go through one. The effect of measurement noise can be seen in the data, as well. Also note
that the amplitude axes vary in the results. In particular, the linear component oscillates
between ±1, while the nonlinear component oscillates at an order of magnitude lower.
The power spectrum also gives more evidence to a cubic nonlinearity, as there is a
64
Figure 2.17: Volterra model h32 results for subsystem 1 at 1.5 V, 5.0 Hz (Sample
1): a) times series; b) power spectra.
peak at 15 Hz. Figure 2.17b has these results. The top plot is the input power (overlaid
with the linear component), the middle plot is the output power (overlaid with the total
Volterra solution), and the bottom plot overlays the linear and nonlinear components. The
dotted line represents the data again, the dashed line is the linear component, the light
solid line is the nonlinear component, and the dark solid line is the total Volterra output.
By examining the input, a small peak at 15 Hz is noticed, which could be introduced from
the amplifier or the data collection process. Although its influence is small in the input
signal, the output signal shows that it has been significantly amplified, which is most likely
due to the presence of nonlinearity in the system. The bottom plot shows this more clearly,
overlaying the two signal components.
There are several small peaks in the output spectrum, but all of them except the
cubic frequency are below that at 60 Hz, which is likely electric noise. The large number of
peaks in the current spectrum is most likely due to the low signal strength of the current,
which oscillates between about ±8 mA for this case. Considering the gain of the sensing
circuit, which may also introduce some of the peaks, this corresponds to just less than a
±200 mV oscillation in the measured signal. Recalling that the lowest setting of the A/D
converters is ±5.0 V, this signal certainly does not make full use of the available range. A
more elaborate current sensing system may be required to improve these results any further,
however.
65
Table 2.2: Nonlinear Volterra model error for Sample 1 (values in percent, %).
Amplitude
(V)
1.00
1.00
1.25
1.25
1.50
1.50
Subsystem
1
2
1
2
1
2
Frequency (Hz)
5.0
10
3.7 (h23 ) 1.6 (h33 )
4.3 (h22 ) 2.7 (h42 )
2.8 (h32 ) 4.2 (h34 )
2.2 (h33 ) 2.2 (h22 )
1.7 (h32 ) 2.5 (h33 )
1.8 (h34 ) 2.0 (h43 )
1.0
12.2 (h42 )
13.0 (h33 )
6.1 (h32 )
9.6 (h32 )
6.4 (h32 )
13.0 (h12 )
50
1.8 (h33 )
2.0 (h32 )
0.6 (h33 )
1.3 (h32 )
0.7 (h18 )
1.2 (h34 )
Table 2.3: Linear Volterra model error for Sample 1 (values in percent, %).
Amplitude
(V)
1.00
1.00
1.25
1.25
1.50
1.50
Subsystem
1
2
1
2
1
2
Frequency (Hz)
1.0 5.0 10 50
10.9 3.8 2.0 2.8
13.3 4.4 2.9 2.3
9.8 3.3 5.4 0.8
9.9 2.5 2.8 1.7
11.0 4.3 3.3 1.9
13.0 4.2 2.4 1.6
This same procedure was performed again on the current-to-velocity subsystem, but
the resulting plots have been excluded due to their similarity to those already discussed.
In comparing the running sum RMS of the nonlinear components for each subsystem at
this test condition, neither showed clear nonlinear dominance, so these plots have not been
shown. In fact, they summed to nearly identical values. What this means is that at these
particular input conditions, each subsystem contains roughly equal nonlinear components
in their respective responses.
Performing this procedure for each subsystem at all of the test conditions on Sample 1, the results summarized in Table 2.2 were obtained. In this table, the numerical
values listed are the percentages (%) of average, absolute validation error of the “minimumparameter” Volterra model, as indicated in parentheses (hN
M ). A common trend that can
be seen going from high to low frequency, but not necessarily from high to low voltage, is
that the error increases. As noted previously, the current draw in the polymer actuators
decreases with frequency and is not very high to begin with. For this reason, the larger
validation errors at low frequency are mostly attributed to noise in the measurements.
66
Figure 2.18: Actuation response at different frequencies for water-based sample.
Table 2.3 has also been included to summarize the results of the linear Volterra
model error. These are the results from the system before the nonlinear model is identified
and combined to give the total response. Again, the values are average, absolute validation
error percentages. Most often, the linear model error is reduced by adding the nonlinear
model, as can be seen in comparing the two tables at each test condition. This shows that
the signals contain some amount of nonlinear distortion. There are a few instances where
this is not the case, however. Most notably, this occurs at 1.0 Hz. This is another indication
that a high level of noise corrupts the measurements at low frequency.
Another observation not shown in the above results can be seen by comparing the
linear and nonlinear model errors with frequency. There is noticeable distortion present at
low frequency, but it has less apparent influence as the frequency increases, addressing how
the distortion changes with frequency. For example, Figure 2.18 shows that the actuation
response at 1.0 Hz appears much more distorted than at 50 Hz for Sample 1. The voltage
amplitude for this data is 1.5 V. In comparing the validation error for these test conditions,
it is seen that at 50 Hz, the voltage-to-current error goes from 1.5% to 0.7% by adding
nonlinear terms, and the current-to-velocity error goes from 1.6% to only 1.2%. Looking
now at the more visually distorted frequency results at 1.0 Hz, the error in subsystem 1
decreases from 11% down to 6.4% (the largest model improvement of the data) with the
67
Figure 2.19: Error of models for subsystem 1 at 1.5V, 5.0Hz (Sample 2): a)
prediction; b) validation.
addition of nonlinear terms. The results for subsystem 2 at 1.0 Hz are less convincing since
the error remains constant at 13%, but this is attributed to noise in both the input and
output, which can be seen in the figure.
There has not been much stated about the relative nonlinear influence of the two
subsystems yet. The main reason for this is that there were no conclusive findings that
point to one particular subsystem being dominantly nonlinear over the other for Sample 1.
Changing the solvent from water to ionic liquid may have some effect though. These results
will be discussed next.
Sample 2 Results
The results shown for the ionic liquid sample will also show the voltage-to-current subsystem
to provide a more direct comparison between the two solvents. As before, these results will
also be at 1.5 V and 5.0 Hz. Fitting Volterra models of various degree and order to the error
of the linear, order 2 model of subsystem 1 gives the results shown in Figure 2.19a. Contrary
to the results of Sample 1, Figure 2.19a does not show the initial accuracy difference between
the lowest two and highest two degree models, but near zero error convergence does occur.
This difference could be an initial sign that the distortion has changed with the solvent.
To give a better sense of how well the models perform and to ultimately select the
best order and degree, the validation data was simulated to the models computed with
68
Figure 2.20: Volterra model h34 results for subsystem 1 at 1.5V, 5.0Hz (Sample
2): a) time series; b) power spectra.
the prediction data set. These results are given in Figure 2.19b, zoomed in on low order.
The trends are clearly different here, seeing that the model error is nearly constant at each
order and degree. This means that the linear model performs as well as the nonlinear
models, indicating that the distortion seen in the water sample has either been removed
or is below the noise level in the signal. It should also be noted that the current signal of
Sample 2 was an order of magnitude lower than Sample 1. While it is not as obvious which
model parameters to choose in this case, the time series and power spectrum results are
also examined to find the best model, which was determined to be of degree 3 and order 4.
With the selection of the best minimum-parameter model, the linear and nonlinear
response components were compared over one cycle. It can be seen in Figure 2.20a that
the linear component fits the basic trend of the data, while the nonlinear component has
difficulty in matching the error between the linear model and the data. Visually examining
the error response in both the prediction and validation data, a clear trend is not discernable.
When considering the corresponding error values, it is also noticed that the linear model
has 4.0% error, while adding the nonlinear portion gives no further reduction. This is
an indication that the actual input-output relationship is predominantly linear, but these
measurements do contain a higher level of noise.
This noise can also be seen in looking at the power spectra of the data shown
in Figure 2.20b. In the current measurement (output), every peak except the excitation
69
Table 2.4: Nonlinear Volterra model error for Sample 2 (values in percent, %).
Amplitude
(V)
1.00
1.00
1.25
1.25
1.50
1.50
Subsystem
1
2
1
2
1
2
1.0
11.8 (h25 )
9.7 (h12 )
11.6 (h14 )
9.0 (h32 )
7.7 (h32 )
8.1 (h14 )
Frequency (Hz)
5.0
10
4.8 (h25 ) 5.5 (h22 )
3.6 (h32 ) 5.0 (h33 )
4.0 (h25 ) 4.9 (h23 )
4.4 (h34 ) 4.4 (h33 )
4.0 (h34 ) 3.6 (h22 )
3.2 (h34 ) 2.6 (h34 )
50
4.3 (h22 )
4.0 (h32 )
3.6 (h14 )
3.1 (h34 )
5.4 (h22 )
3.4 (h34 )
frequency is below the amplitude level of 60 Hz. As mentioned before, this is likely due to
the low signal strength in the current measurements and could be introduced either in the
data acquisition, by the amplifier, or by the current sensing circuit. Also, in comparing the
linear and nonlinear components (bottom), it can be seen that the linear response is much
more dominant.
Summarizing the results over the range of voltage amplitudes and frequencies tested
in the experiments, Table 2.4 was constructed. As before, the minimum-parameter model
for each data set is given in parentheses (hN
M ) after the percentage value of the average,
absolute validation error. Comparing these percentages to those of Sample 1 (Table 2.2)
it is noticed that the error for Sample 2 is consistently higher, owing to the decreased
signal-to-noise ratio of the current measurements.
Table 2.5 helps to examine the effect of adding this nonlinear portion of the model
to the linear model. An interesting result is noticed. Comparing the error percentages at
each frequency and amplitude condition, it is apparent that adding the nonlinear portion to
the original linear model has a fairly insignificant effect on reducing the error. Despite the
presence of more noise in the signals, this is an indication that the input-output relationships
using EMI-Tf as the solvent are predominantly linear at these operating conditions. The
linear model fits the basic trend through the noisey data, but the absence of any distortion
trend causes very little effect in the overall model prediction by adding nonlinear terms.
This gives some evidence that changing the solvent from water to EMI-Tf can reduce the
nonlinear distortion at low voltage levels, an indication that the solvent greatly affects the
actuation response properties of ionic polymer transducers.
One final comment about the results from each sample has to do with only dis-
70
Table 2.5: Linear Volterra model error for Sample 2 (values in percent, %).
Amplitude
(V)
1.00
1.00
1.25
1.25
1.50
1.50
Subsystem
1
2
1
2
1
2
Frequency (Hz)
1.0 5.0 10 50
12.7 6.3 5.6 4.3
9.7 3.7 5.1 4.2
12.0 5.1 4.9 3.6
9.4 4.4 4.7 3.3
9.2 4.4 3.7 3.0
8.3 4.0 3.2 3.7
cussing the voltage-to-current subsystem. This was done because the current-to-velocity
results were more similar to one another for each sample, and it was the voltage-to-current
results that highlighted the changing response due to using different solvents in the polymer actuators. As was the case for the water sample, the subsystem 2 results for Sample 2
show no consistent trend in the voltage and frequency ranges under investigation. Therefore, these plots were again excluded. This finding seems reasonable in regard to Sample
2 because the results showed that there was negligible nonlinear influence contained in the
measurements. Following this, there is an indication that the solvent could influence the
electrochemical characteristics of the actuators more than the mechanical characteristics.
Discussion of Results
Some common findings from the results of the two samples will be discussed to start. The
first observation has to do with the phase of the response. It was seen in each of the
samples tested that the voltage and current signals were nearly in phase with each other,
while the current and velocity signals were approximately 180 degrees out of phase with
each other. This could be an indication that the mechanism responsible for deformation
has a delay with respect to the electrical characteristics of the material. Also seen was
that the current and velocity increased with input voltage amplitude and frequency. This
is intuitive since giving more voltage at increasingly higher frequencies causes the material
to draw more current, which then deforms the actuator at a higher rate. There was also no
evidence of any clipping or saturation nonlinearities in the amplitude and frequency range
tested, implying that the material can tolerate higher voltages. To comment quickly on this
conjecture, there are no foreseeable problems with the ionic liquid sample, but the chemical
71
breakdown of water, electrolysis, will begin to occur above 1.23 V. Therefore, increasing the
voltage to higher levels, for extended periods of time, could give skewed results as more of
the input power would contribute to this effect instead of actuation.
Based on the results shown above for Sample 1 and Sample 2, it was determined that
there is no obvious answer to the original question posed of whether or not one subsystem
possesses more nonlinear mechanisms than the other. However, two new observations were
made that will require further investigation. First, both the voltage-to-current and currentto-velocity subsystems contain nonlinearity in the water-based sample, but when analyzing
results from the ionic liquid-based sample, there is almost no difference in the performance
of linear models and nonlinear models. This is an indication that the type of solvent
employed by the actuator could actually cause different nonlinear response characteristics.
This necessitates the further study of solvent interaction in the actuation response of ionic
polymer materials. The second new observation was brought out in comparing the time
response waveforms at different actuation frequencies for Sample 1 (see Figure 2.18). These
results indicated that the nonlinearities are more dominant at the lower frequencies and
appear to be negligible at higher frequencies, which provides evidence that the nonlinearity
has dynamics associated with it.
2.5
Summary
This chapter began with providing a motivating example for studying nonlinear models
of ionic polymers. Following this, the Volterra representation of a nonlinear system was
described, where it was shown how it could be employed to identify the harmonic distortion
in a system excited with a harmonic input. After the analysis method was shown to be
in working order, experimental results were compared for two actuators in different solvent
forms. By comparing these results, it was concluded that the solvent plays a significant
role in the nonlinear actuation response of ionic polymer transducers. Here it was shown
that the ionic liquid sample seems to have a linear response, while the water-based actuator
suffers from nonlinear distortion. The relative effect of this distortion (primarily cubic)
was also shown to change with excitation frequency, indicating the presence of a dynamic
nonlinearity. These findings provide even more motivation for exploring nonlinear actuation
models, in addition to highlighting the need to investigate the effect that the solvent material
72
has on the nonlinear response characteristics. These topics are discussed in the next chapter,
where the nonlinearity is more rigorously characterized and a third solvent material is added
to the study.
73
Chapter 3
Characterization of the
Nonlinearity
A more expansive characterization of the nonlinearity will be performed in this chapter. The
systems to be characterized will be mentioned first and then the characterization techniques
will be detailed. Methods pertaining to two different types of input sequences are employed
to provide verification and support for the form of the nonlinearity to be used in the model
development. For single-frequency harmonic excitations, one approach takes advantage of
the Volterra series representation of a system to identify the source of nonlinear distortion
in the measured input-output signals. This is coupled with a harmonic distortion analysis
to complete the single-frequency analysis. For the second method, multisine excitations
are used to detect the presence of nonlinear distortions over a broader frequency range. A
subsection is dedicated to each method below.
3.1
System Descriptions
Each of the nonlinear characterization techniques will be performed on two systems to be
described here. Recall from the modeling overview the two-port model form proposed by
Newbury and Leo (2002). This model has two inputs (one electrical and one mechanical)
and two outputs (one electrical and one mechanical). Figure 3.1a shows a similar system
arrangement, where i(t) is the current, f (t) is the force, v(t) is the voltage, u(t) is the
74
Figure 3.1: Polymer model arrangements: (a) full 2-input, 2-output system; (b)
modified 1-input, 2-output system under no mechanical load.
velocity, and the coupled set of equations have the form
⎧
⎫ ⎡
⎫
⎤⎧
⎨ v(t) ⎬
G11 G12 ⎨ i(t) ⎬
⎦
=⎣
⎩ u(t) ⎭
G21 G22 ⎩ f (t) ⎭
(3.1)
In the work by Newbury and Leo (2002), linear systems were identified for each Gij by
applying different boundary conditions to the polymer actuator, but the present work looks
to identify nonlinear relationships. It should be noted here that the mechanical input
and output in equation 3.1 have been reversed from the previous work of equation 1.6.
Reversing these signals took place to help emphasize one of the advantages of ionic polymer
materials, their high strain capability, in this first study of dynamic nonlinearities. Also
in the present case, the system is examined under the free deflection boundary condition,
which neglects the force input term. This allows the nonlinear analysis to be performed on
the most commonly studied configuration, where strain (or strain rate in this case) is the
mechanical output. Applying this boundary condition also has the effect of modifying the
nonlinear analysis of the overall system of two inputs and two outputs seen in Figure 3.1a
to identifying the overall system with one input and two outputs shown in Figure 3.1b.
For the present case of characterization, identification, and modeling, this singleinput, two-output system will be separated into two single-input, single-output systems
with a common input. In this arrangement, the system G11 will capture the electrical
response of the actuator (voltage) and the system G21 will capture the mechanical response
(velocity), both with no imposed mechanical load. This work begins with characterizing
these two systems, which will be discussed in the next sections.
75
3.2
Single-Frequency Analysis
The primary difference in the analysis here from that in the nonlinear analysis of the previous
chapter is that the input signal has been switched from voltage to current. This change
has two main reasons. In discussing the results from the preliminary work, it was stated
that the signal-to-noise ratio was larger in the water sample than it was in the ionic liquid
sample because the latter material draws less current. One solution to this problem was to
conduct current-controlled experiments (Robinson, 2005), which was done here to provide
a more consistent baseline for comparison. The other reason to change the electrical input
was based on a recent finding about the actuation mechanism in ionic polymer materials.
It was shown that the electromechanical responses from different actuators tend to overlay
each other when normalized with respect to charge, even for actuators with significantly
different composition, indicating that charge flow may be the actual physical mechanism
behind actuation (Farinholt and Leo, 2004a). This makes the second reason for changing
from a voltage- to a current-controlled setup to reflect this new physical understanding.
Since this particular analysis methodology was detailed in the previous chapter, only a
brief review will be given here before discussing the laboratory setup and results.
Since the Volterra analysis was extensively detailed in the previous chapter, the
development will not be reviewed here. It will just be stated that this technique is used to
characterize the nonlinear response of ionic polymers by building a model between inputoutput measurements of the actuation response, where the input current signal for each
model consists of one of a select number of frequencies. The nonlinear influence of the
system can then be judged by comparing the model error between linear and nonlinear
Volterra models at each frequency point and the degree necessary to accurately predict the
measured response. The materials under investigation and the experimental setup will be
discussed next.
3.2.1
Laboratory Setup
To conduct an experimental investigation using the described analysis procedure, data from
r
three polymer samples was collected. Each sample uses the commercially available Nafion
117 as the ionomer and cation lithium. The first sample, which will henceforth be referred to
as Sample 1, uses deionized water as the solvent and measures (20 x 2.0 x 0.2)mm. The plat-
76
Table 3.1: Summary of Actuator Compositions (viscosity values at 25o C).
Actuator
Component
Ionomer
Cation
Electrode
Solvent
Viscosity (cP)
Sample 1
Sample 2
Sample 3
r
Nafion
-117
+
Li
Pt - Au
H2 O
0.89
r
Nafion
-117
+
Li
Pt - Au
EMI-Tf
45
r
Nafion
-117
+
Li
Pt - Au
BMPyr-tris
224
inum portion of the electrodes is chemically deposited by impregnation/reduction (Bennett
and Leo, 2003) and a thin gold layer was electro-plated over it. The second sample (Sample
2) uses the ionic liquid 1-ethyl-3-methylimidazolium trifluoromethanesulfonate (EMI-Tf) as
the solvent, and Sample 3 is solvated with the ionic liquid 1-butyl-1-methyl-pyrrolidinium
tris(pentafluoroethyl)trifluorophosphate (BMPyr-tris). The viscosity of EMI-Tf is 45 cP
and that of BMPyr-tris is 224 cP. Noting that water has a viscosity of 0.89 cP, this range
spans two orders of magnitude. Each ionic liquid sample also measures (30 x 3.0 x 0.3)mm,
which is different than Sample 1, but since they maintain the 10:1 length-to-width ratio, it
is believed that they will not adversely affect any conclusions (Newbury and Leo, 2003b).
The electrodes of Samples 2 and 3 have a layer of platinum applied as in Sample 1, but the
gold layer is hot pressed on. It should be noted that the water sample will not tolerate the
hot pressing procedure, but since water samples prepared as Sample 1 are the most common
to date, it was fabricated more as a baseline case. Table 3.1 summarizes the composition
of each actuator, where it is seen again that the solvent is the only difference.
In these experiments, voltage, current, and velocity were measured. For data acquisition, a dSPACE DS 2003 analog-to-digital converter (16-bit) and DS 2103 digital-to-analog
(14-bit, ±5 V) converter were used. The A/D converters were set to ±5.0 V for Samples
1 and 2, and ±10.0 V for Sample 3 because of higher impedance levels. A Polytec OFV
3001 vibrometer controller and 303 sensor head were used to measure the tip velocity of
the cantilever ionic polymer actuators, with the sensitivity adjusted from test to test depending on the signal characteristics to ensure that the A/D range was used as effectively
as possible. The input signal was generated by the dSPACE system and passed through a
transconductance amplifier to get an input current of the desired waveform. The voltage
data collection occurred at the output of the amplifier. Finally, to hold the cantilever poly77
Figure 3.2: Experimental setup for ionic polymer actuator testing.
mer and apply the electric field, a test fixture with gold foil electrodes was used. Figure 3.2
has been included to show the necessary equipment, along with the measured voltage (V),
current (I), and velocity (U) signals.
Experiments were run with these three samples at several different test cases of
single-frequency excitations. The frequencies used in the sine wave tests scaled in frequency
based on the mechanical resonance of the first bending mode and occurred at multiples
of this resonance frequency of 1, 0.75, 0.5, 0.25, and 0.1. Amplitudes of excitation were
varied across the samples since the impedance plays a part in the voltage measurement.
The sample rate was also varied from test to test, but was kept a constant multiple of the
actuation frequency over the frequency range, which provided the same number of points
per cycle in each test. Setting this multiple to 200 allowed full use of the sample rate range
in the data acquisition system.
3.2.2
Experimental Volterra Results
Here the characterization results from the single frequency sine waves experiments will be
discussed. Because plotted results were shown in the previous chapter, and because of
78
Table 3.2: Volterra model errors for Sample 1 (values in percent, %).
Amplitude
(mA-rms)
8.0
8.0
4.0
4.0
2.0
2.0
System
I-V
I-U
I-V
I-U
I-V
I-U
1.68
6.6 (h34 ) 2.9
6.6 (h42 ) 4.9
6.2 (h35 ) 3.1
8.5 (h33 ) 6.7
8.1 (h34 ) 4.2
10.2 (h33 ) 9.1
7.2
8.3
7.6
9.1
5.6
8.9
4.19
(h35 )
(h42 )
(h35 )
(h42 )
(h36 )
(h24 )
3.9
4.3
6.3
6.8
1.8
8.2
Frequency (Hz)
8.37
7.1 (h36 ) 3.3
10.0 (h33 ) 2.8
7.4 (h35 ) 3.6
7.9 (h21 ) 4.7
9.1 (h26 ) 5.5
10.5 (h14 ) 10.1
6.3
2.6
8.1
3.6
8.4
5.9
12.56
(h36 ) 2.3
(h32 ) 1.9
(h34 ) 7.1
(h34 ) 3.2
(h26 ) 4.4
(h34 ) 5.4
7.5
1.2
4.3
2.0
7.9
1.6
16.75
(h34 ) 4.9
(h17 ) 1.2
(h36 ) 2.0
(h36 ) 1.4
(h35 ) 3.4
(h32 ) 1.6
Table 3.3: Volterra model errors for Sample 2 (values in percent, %).
Amplitude
(mA-rms)
5.0
5.0
3.5
3.5
2.0
2.0
System
I-V
I-U
I-V
I-U
I-V
I-U
5.0
4.8
2.3
2.6
5.2
4.8
1.10
(h37 )
(h37 )
(h34 )
(h34 )
(h37 )
(h28 )
1.9
2.4
0.7
1.4
1.7
1.8
2.75
2.5 (h37 ) 0.6
2.3 (h44 ) 0.99
3.8 (h33 ) 1.9
2.8 (h34 ) 1.5
6.2 (h34 ) 4.6
4.1 (h34 ) 3.1
Frequency (Hz)
5.50
8.25
2.4 (h36 ) 0.7
1.7 (h28 ) 0.7
3.5 (h28 ) 0.35 0.47 (h34 ) 0.38
2.9 (h37 ) 1.0
1.3 (h18 ) 0.6
2.8 (h28 ) 0.88 0.60 (h35 ) 0.43
4.9 (h35 ) 3.1
3.4 (h18 ) 2.2
3
3.0 (h5 ) 1.7
1.1 (h35 ) 0.84
2.8
8.7
2.7
8.2
3.3
9.8
11.0
(h34 )
(h34 )
(h37 )
(h37 )
(h35 )
(h35 )
the large number of results (15 test conditions per sample), they will be omitted here.
Instead the results for each of the test conditions will be reported in tabular form for each
sample. Tables 3.2, 3.3, and 3.4 show the Volterra modeling results for Samples 1, 2, and
3, respectively. In these tables the amplitudes are ordered highest to lowest from top to
bottom and the frequencies appear lowest to highest from left to right. Each cell of results
displays the average, absolute linear Volterra model error first and the average, absolute
nonlinear Volterra model error last, separated by the minimum-parameter nonlinear model
form in parentheses (hN
M ). To determine the effect of adding nonlinear terms to each model,
the linear error can be compared to the nonlinear error. For instance, in the first cell of
Table 3.2 (8.0 mA, 1.68 Hz), the linear model error of 6.6% is reduced to 2.9% by adding a
nonlinear Volterra model of order M = 4 and degree N = 3. Looking at all the results, it
can be seen that some of the Sample 3 results for the I-U system appear to have unusually
large errors, particularly at the lower amplitudes. It should be noted that this is caused by
low signal levels, not poor model construction.
79
2.0
6.1
1.1
3.0
1.9
5.0
Table 3.4: Volterra model errors for Sample 3 (values in percent, %).
Amplitude
(mA-rms)
1.25
1.25
1.00
1.00
0.75
0.75
System
I-V
I-U
I-V
I-U
I-V
I-U
2.0
7.1
1.9
8.2
3.5
6.7
1.2
(h34 )
(h12 )
(h28 )
(h32 )
(h35 )
(h22 )
0.9
7.1
0.6
8.1
1.7
6.7
3.0
1.5 (h35 )
7.5 (h24 )
1.7 (h35 )
8.2 (h32 )
2.5 (h35 )
13.1 (h13 )
Frequency (Hz)
6.0
9.0
0.8
1.2 (h18 ) 1.0
1.3 (h18 )
7.1
6.9 (h14 ) 6.3
8.4 (h33 )
1.1
1.6 (h18 ) 1.2
1.6 (h33 )
8.1
11.0 (h18 ) 10.3 11.7 (h33 )
1.5
2.3 (h14 ) 1.9
1.8 (h18 )
1
12.8 12.9 (h4 ) 12.4 17.8 (h18 )
1.1
6.6
1.4
10.6
1.4
16.8
12.0
1.2 (h23 ) 1.0
10.6 (h18 ) 2.8
1.3 (h33 ) 1.1
11.9 (h34 ) 7.2
1.4 (h23 ) 1.2
14.7 (h16 ) 8.9
In comparing the results for each of the actuator samples, a consistent trend can
be seen in both the electrical and mechanical responses. That is, as the viscosity in the
samples increases, the linear Volterra model error between the input current and either of
the two outputs decreases. This means that with larger linear model error, the samples
with lower viscosity benefit more from nonlinear model terms in more accurately predicting
the measured voltage with the given current signal. To help visualize this result, Figure
3.3 has been included to show these results for the three samples. The top plots show the
linear error for Sample 1 (0.89 cP), the middle plots show the error for Sample 2 (45 cP),
and the bottom plots show the results for the most viscous solvent, Sample 3, at 224 cP. In
this figure the vertical axis is the percentage of the average, absolute linear model error and
the horizontal axis has the normalized frequencies. The frequencies were normalized with
respect to the mechanical resonance for a more direct comparison, and the actual values
can be found in the tables. Three lines also appear in each of the plots, where the diamonds
indicate the lowest input current, the square points indicate the middle amplitude, and the
circles shows results for the highest amplitude.
Figure 3.3a has the results for the electrical output (voltage), which is easiest to see
the trend. With fixed-value error axes, it can be seen that with Sample 3 on the bottom, the
error stays between 1% and 4%, while the linear error in Sample 2 increases to a range of
about 2% to 7%, and Sample 1 results at the top range from 5% to 9%. Turning attention
now to the mechanical output (velocity), it can be see in Figure 3.3b that the same trend
is repeated for Samples 1 and 2, but the results for Sample 3 appear to be opposite. When
looking back in the data for this case, it was noticed that the highest viscosity material also
gave the slowest rate of actuator deflection, which resulted in low signal strength. Also,
80
Figure 3.3: Linear Volterra model error: (a) electrical output; (b) mechanical
output.
referring back to the I-U results in Table 3.4, it can be seen that adding nonlinear terms
to these linear models gives very little improvement in reducing the error, if any at all.
This provides more evidence that these results do not reflect an obscurity in the nonlinear
trends, but instead signify a sensitivity limit in the measurement equipment.
Another indication of the nonlinear effect in these results is apparent when considering the degree of the Volterra models and the lack of an improved response prediction.
For example, several of the model degrees for the I-V system in Sample 3 achieved the
best minimum-parameter results with just another linear model (N = 1), and the error is
reduced very little with the addition of more terms to the model. When examining the
results for Samples 1 and 2, the nonlinear Volterra models are nearly all of degree 3 for
the two systems, showing much more improvement when comparing the linear error to the
nonlinear error. The fact that so many of the models are third degree also provides more
evidence that the nonlinear mechanism could be cubic. This will be examined more next.
3.2.3
Harmonic Distortion Analysis
While the previous analysis concluded that the nonlinear response could be dominated by
a cubic mechanism, this will be explored further here by qualifying the nonlinear response
components. To analyze the content of the distortion at each frequency, fast Fourier transforms were calculated over three periodic time blocks with rectangular windows. This was
81
Figure 3.4: Harmonic ratio analysis: (a) quadratic terms – 2f ; (b) cubic terms
– 3f .
done so the magnitude results would show the actual amplitude of response at each frequency. The distortion analysis results of Figure 3.4 show the amplitude ratio of the two
most dominant distortion peaks (quadratic and cubic) to the actuation frequency for each of
the frequencies tested for the three different samples. The results shown are for the highest
amplitudes of Samples 2 and 3 because this is where the nonlinearity has the most influence,
but they are compared with the lowest amplitude for Sample 1 to help provide evidence that
the lower viscosity solvent still displays more nonlinearity. Figure 3.4a shows the quadratic
(2f ) distortion ratios and Figure 3.4b shows the same for the cubic distortion (3f ). In these
figures, the diamonds correspond to Sample 1 (water), the squares correspond to Sample 2
(EMI-Tf), and Sample 3 (BMPyr-tris) is represented by the circle points. The upper plots
display results from the measured voltage response and the lower plots display the velocity
results.
The most consistent trend coming from the voltage results of Figure 3.4a is that
the quadratic nonlinearity appears to be directly related to the solvent viscosity. Here the
harmonic amplitude ratios for the quadratic terms are largest for the lowest viscosity sample
(water) and smallest for the highest viscosity sample (BMPyr-tris). While the quadratic
terms do not necessarily dominate the voltage response, this is another indication that
the solvent affects the distortion. A similar trend can also be seen in the cubic terms of
the voltage in Figure 3.4b. In addition to showing more nonlinear influence with lower
82
viscosity solvents, this figure also shows that the cubic mechanism appears to be triggered
at different frequencies, related to the viscosity in the polymer actuator. This trend states
that the cubic nonlinearity begins impacting the voltage response at higher frequencies when
the solvent viscosity is lower. Qualitatively, the frequency-dependence of the nonlinearity
could be thought to act like a low-pass filter, with a break-point frequency determined by
the solvent viscosity (lower viscosity, higher frequency). In comparing Samples 1 and 2,
more support that the solvent viscosity activates the nonlinearity is provided in the recent
work by Bennett and Leo (2005b), where it was shown that the polymer morphology is
r
.
similar with water or EMI-Tf in Nafion
Turning now to the mechanical results of Figure 3.4, the quadratic ratios appear to
have surprisingly similar character at higher frequencies, but the low frequency amplification
in the water sample does not fall in line with the expected trend of more influence at
lower frequencies. Then again, this is the quadratic term which has not been shown to be
the dominant nonlinearity, so its influence may not need to follow the expected trend as
clearly. This trend does show up again for the cubic velocity ratio of Figure 3.4b, however.
Ratios for all three samples are nearly zero at high frequency, but then as the frequency
decreases, the onset of cubic nonlinear influence begins to occur first (higher frequency)
for the lowest viscosity sample (water) and last (lowest frequency) for the highest viscosity
solvent (BMPyr-tris). This was the expected result from the Volterra series analysis. These
trends will be investigated more extensively in the next section over a broader range of
frequencies and at a finer frequency resolution.
3.3
Multiple-Frequency Analysis
In this section, nonlinear characterization of ionic polymer actuators will be performed with
a broadband excitation signal, as opposed to single-frequency sine waves. An introduction
to multisine excitation design will be presented first, followed by an illustrative example
of the nonlinear detection. With the analysis in order, the experimental results will be
presented at the close of the section.
83
3.3.1
Multisine Excitations
The nonlinear distortion in a system can also be characterized over a broader frequency
range, while simultaneously measuring the frequency response, by exciting the system with
a periodic multisine signal (Schoukens et al., 2001). Multisines have the form
x(t) =
N
Xk ej2πfmax kt/N
(3.2)
k=−N
where N is the number of frequency components and fmax is the maximum frequency of the
signal. The amplitude of the signal |Xk | is often set as a constant, but any other desired form
could also be used. Additionally, the phase of each frequency component can be prescribed
by adjusting the complex value of Xk . One of the more common phase additions is to use
Schroeder phases given by (Schroeder, 1970)
φk = −k(k − 1)π/N.
(3.3)
One of the key advantages of adding various phase contributions to each frequency component is to increase the compactness of the excitation, making a more balanced signal
with less impulsive behavior, and Schroeder phases are one method of accomplishing this
task (Pintelon and Schoukens, 2001). This addition of phase can easily be seen in another
formulation for designing a multisine signal, though mathematically identical to equation
3.2, which is
x(t) =
N
Ak cos(2πfk t + φk )
(3.4)
k=1
where Ak are the signal amplitudes at each frequency fk , and φk are the added phase terms.
The frequency components are chosen from a grid of excited and unexcited lines,
where the unexcited lines are used to detect, qualify, and quantify the nonlinear distortion in
the system (Vanhoenacker et al., 2001). If the given system is purely linear, then its response
will have content only at the frequencies in the excitation signal and the response at the
unexcited nonlinear detection lines will be at the noise level. However, if the system contains
nonlinear distortions, it can be seen by examining the detection lines. The distortion can
be qualified as even or odd by choosing to excite some, but not all, of the odd-numbered
grid lines (this is an odd multisine). The remaining odd lines are then used to detect odd
distortion, while all of the even lines detect even distortion in the response. Figure 3.5 has
been included to help in the explanation of multisine signals. In this example the grid size
84
Figure 3.5: Example of multisine excitation design.
is four and the first frequency component of each grid is excited. This allows the frequency
lines of [2] and [4] (2f and 4f ) to detect even nonlinear distortions and any response at
frequency line [3] (3f ) to detect odd nonlinearities.
Quantification of the nonlinear distortion is accomplished by comparing the magnitude of the autospectrum at excited lines (linear response) and the odd and even detection
lines (nonlinear response). Aside from these capabilities, multisine excitations also provide
a measurement of the frequency response function, although the frequency resolution is
lowered due to the unexcited lines. For example, the frequency grid given in Figure 3.5
has one quarter the frequency resolution of what it would be if all lines were excited for
the linear system identification. An example of this technique is provided in the following
section.
3.3.2
Illustration of Multisine Analysis
To help illustrate how multisine excitations can characterize the levels of nonlinear distortion
in a system, an example on a representative system has been put together. This system
has an underlying linear system H(f ) with two nonlinearities in a parallel arrangement.
The first nonlinearity is an even function (x2 (t)) followed by another linear system A1 (f ),
while the second nonlinearity has an odd function (x3 (t)) preceding another linear system
A2 (f ). The input applied to the system, x(t), directly enters all three systems and the
output of each individual system is added together to form the complete system output,
y(t), as shown in Figure 3.6a.
85
Figure 3.6: Multisine example: (a) representative system structure; (b)
frequency-dependence of system components.
The underlying linear system was arbitrarily designed to have two second-order poles
(30 Hz, 2% damping; 70 Hz, 1% damping) and one second-order zero (47 Hz, 3% damping),
and the two nonlinearities have been designed to affect this system in different ways. The
frequency-dependence of the even nonlinearity acts like a low-pass filter with a critically
damped second-order pole at 8 Hz. For sufficiently low input amplitudes, this system is
most dominant at lower frequencies and its influence tapers off at higher frequencies. The
frequency-dependence for the odd nonlinearity is a second-order system with the pole at
39 Hz and 3% damping. With the low frequency gain of this system well below that of
the underlying linear system, its effect is predominantly between the two poles of H(f ) for
lower amplitude levels. Figure 3.6b shows the magnitude and phase plots for these three
systems, where the frequency bands of dominance can be seen.
In simulating this system, an odd multisine signal was designed. Like in Figure 3.5,
the grid size was set to four frequency lines, exciting the first and leaving the other three to
detect nonlinear distortions (two for even lines, one for odd lines). With a sample rate of
200 Hz, the number of points per period was taken to be 8192 with a maximum excitation
frequency of 100 Hz. This gives the frequency spacing f0 ≈ 0.024 Hz. Also, the amplitude
was chosen to be 1.3 (0.92 unit-rms) to allow some nonlinear effects to be present, but not
to completely dominate the system output.
The time series results for the simulation are given in Figure 3.7a. Here, one quarter
86
Figure 3.7: Multisine response of system: (a) time series (quarter-period); (b)
autospectra.
of a period is shown for both the multisine input signal (upper) and the system response
(lower). The appearance of this data set somewhat resembles that of a chirp signal, but the
multisine signal is periodic and designed to excite only a selected number of frequencies.
Figure 3.7b shows the various autospectra of the input and output signals. The top plot
shows that the multisine input signal has a flat spectrum over the chosen frequency range
of interest, and the bottom plot displays the spectra of the output signal. The solid line
represents the linear component, the dashed line represents the even nonlinear distortion,
and the dotted line shows the odd distortion level. It can be seen that the linear component,
which is the response occuring at the excitation frequency lines, is dominant over most of
the frequencies, but the presence of nonlinear distortion can also be seen. Recalling from
Figure 3.6b that the even nonlinearity affects the lower frequencies, it is seen here that
this low frequency influence is detected by the even detection lines. Likewise for the odd
nonlinearity, the highest levels are seen between the two poles of the underlying linear
system. It should be noted that the linear component does pick up the response of the
nonlinearities when they dominate, most notably here with the odd nonlinear detection
around 39 Hz, but this can be worked out through nonlinear system identification, where
the different system components can be separated from each other. Since this is the topic
of the next chapter, no more mention will be made here, and instead the experimental
multisine analysis will now be covered.
87
Table 3.5: Multisine excitation design parameters.
Parameter
fmax (Hz)
f0 (mHz)
T (s)
ts (s)
n
avgs
3.3.3
Sample 1
33
16.3
61
0.015
4096
4
Sample 2
25
6.1
164
0.02
8192
5
Sample 3
25
6.1
164
0.02
8192
5
Laboratory Setup
A grid size of four with the first line containing the excitation component was employed
again in experimentally characterizing the nonlinearity in the actuation response of ionic
polymer actuators. As mentioned before, the remaining three lines (two even, one odd)
will be used for nonlinear response detection. The odd multisine excites the polymer at the
frequencies (4k + 1)f0 , k = 0, 1, 2, ..., N, where f0 and fmax varied for the different samples,
but the frequency spacing was on the order of milli-Hertz. Such resolution was used here
because the nonlinearity is believed to be most dominant at low frequencies. The spectrum
amplitude was also designed to be flat over the range of frequencies and four amplitudes
of 0.5, 1.5, 2.5, 3.5 mA-rms were tested for each sample. These amplitudes were kept the
same for each sample to give a better basis of comparison, while covering nearly an order
of magnitude was believed to be sufficient to show nonlinear effects.
The individual design parameters for the excitations are displayed in Table 3.5. It
can be seen here that each of the ionic liquid samples (Sample 2 and 3) are tested for longer
periods of time (T) to get smaller frequency spacing in the results. Because of potential
dehydration effects, the water sample (Sample 1) could not be tested for this long and the
period was conservatively kept around one minute. For these tests, data from only one
period of the signal was collected for each run and water was brushed over the material
in between each test. The potential for material dehydration also affected the number of
averages, where a very conservative approach was taken again. The maximum frequency
was also higher for the water sample because it had the highest mechanical resonance.
The laboratory setup for the experiment is the same as that mentioned for the
current-controlled Volterra identification (see Figure 3.2) with the dSPACE system, laser
vibrometer, and transconductance amplifier, but the excitation signal is now one of the
88
described multisines instead of individual sine waves. Results from the samples with three
different solvents will be discussed next.
3.3.4
Experimental Multisine Results
Using the designed multisine signals, the linear response and nonlinear detection lines in
the collected data were separated and plotted in Figure 3.8. Here, Figure 3.8a contains
the autospectra for the voltage output at 3.5 mA-rms and Figure 3.8b shows the same for
the velocity output. The upper plots show results for Sample 1, the middle plots have the
results for Sample 2, and the results for Sample 3 are in the lower plots. Blocks denoting
the sample number have also been placed in the upper right corner all of the plots. In each
of these figures, the solid line indicates the linear response, the dotted line represents the
level of even nonlinear distortion, the dashed lines represents the odd nonlinear distortion,
and plus symbols of σ provide a measure of the standard deviation in the data (based on
the linear response component).
Looking first at the voltage response in Figure 3.8a, it can be seen that the linear
response is most dominant over the entire frequency range. The level of the linear measurement noise has the second highest value, particularly at higher frequencies where it drowns
out any nonlinear effects, but the odd nonlinearity is more dominant at lower frequency.
Also, the even nonlinearity is negligible as it remains below the noise level at all frequencies.
It should be noted that the standard deviation of the detection lines, while omitted here
to avoid cluttering the figure, was below that shown for the linear response. The key point
to take away from examining these results is that the nonlinear response is only significant
at low frequencies, and odd nonlinearity is detected to have much more influence in the
response than any even nonlinearity. This nonlinear effect is also shown to increase as the
frequency decreases, which can be seen by comparing the magnitude of the linear response
to that for the odd nonlinearity. Here, the relative difference is several orders of magnitude
at high frequencies, but the gap closes at an increasing rate as the frequency drops.
Also to note in these results is how the nonlinear influence is affected by viscosity of
the solvent. At the lowest frequencies, the magnitude ratio for the detected odd nonlinear
response to the linear response is nearest to unity in the least viscous polymer sample
(water – 1) and closest to zero for the sample with the most viscous solvent (BMPyr-tris
– 3). This is another indication that the solvent affects the nonlinearity in the voltage
89
Figure 3.8: Experimental multisine response characteristics (3.5 mA-rms): (a)
voltage response; (b) velocity response.
response. Regarding the onset of the nonlinear influence in the voltage response, a similar
statement can be made with these multisine results as was made in the single-frequency
results. That is, the lower the solvent viscosity is, the higher the frequency is where a
nonlinear response begins to arise. In Figure 3.8a, this can be seen by taking note of the
frequency where the odd nonlinearity begins to rise above the level of the noise in each
sample. For Sample 1 this occurs at approximately 0.9 Hz, for Sample 2 it takes place near
0.1 Hz, and for the highest viscosity solvent it is near 0.05 Hz.
Turning attention now to the velocity response, Figure 3.8b shows some of the same
trends that were seen in the voltage response. The level of the linear noise, which is
higher than the nonlinear detection lines at nearly all frequencies, indicates that the overall response has negligible nonlinear effects, and the linear autospectrum has the highest
magnitude at most frequencies. Also, the nonlinear response components in the water sample (least viscous) approach the same levels as the linear response (and noise) at a higher
frequency (∼0.05 Hz) than either of the two ionic liquid samples, though Samples 2 and 3
90
appear to show this happening at near the same frequency (∼0.02 Hz) in this case.
Some other differences here are that the effect of both nonlinear detections appear to
be relatively equivalent, and they appear to have equal magnitude with the linear response
only at the lowest frequencies, which is similar to the conclusions drawn from the singlefrequency velocity results of Figure 3.4. This indicates again that the nonlinearities are
most prevalent at low frequencies, while their influence is negligible at higher frequencies.
Considering the different viscosities, Samples 1 and 2 follow the previously mentioned trend
that the lower viscosity material will be more affected by nonlinearity beginning at a higher
frequency, but the results for Sample 3 look to be different. Recalling from the Volterra
results that the velocity measurements for Sample 3 contained more noise due to the slow
rate of motion, this could be an issue here as well, particularly since very little distortion
was seen in the sine wave measurements.
With the basic characteristics and trends of the multisine responses discussed, the
impact of the input level will now be shown. Figure 3.9 shows these results in the same
organization as before, with Figure 3.9a showing the voltage autospectra response components, Figure 3.9b showing the velocity autospectra response components, and the sample
numbers ordered 1 to 3 from top to bottom (solvent viscosity increases top to bottom).
These plots show the detected odd nonlinear contributions for each of the four current levels tested. Odd nonlinearities were chosen because, overall, they had the larger response
of the two nonlinearities as depicted in the previous results. In each plot set the dotted
line corresponds to the lowest input of 0.5 mA-rms, the dash-dot line corresponds to the
1.5 mA-rms results, the 2.5 mA-rms results are shown by the dashed line, and the highest
input of 3.5 mA-rms appears as the solid line.
Figure 3.9a shows clearly that the relative influence of the odd nonlinearity is
amplitude-dependent, particularly at low frequency. By this statement, it is meant that the
effect of nonlinearities increases with input amplitude. At higher frequencies, the nonlinear
detection lines appear more scattered, but this is in the region where nonlinear effects are
essentially negligible in the response because the magnitude is below that of the noise level
of the linear response component. The input scaling is not as clearly seen for the velocity
responses in Figure 3.9b; however, this seems reasonable because these detected nonlinear
levels were below that of the measurement noise. It is evident at low frequency for Samples
1 and 2, but the trend is less consistent at higher frequencies for all three samples, as the
91
Figure 3.9: Odd multisine components for increasing amplitudes: (a) voltage
response; (b) velocity response.
autospectra appear more noisey and nearly overlay each other. Along with the comparison
to the linear noise, this is an indication that the nonlinear components have no effect except
at the lowest frequencies. Here again the results for Sample 3 are not in agreement with
the two lower viscosity solvent materials, in terms of nonlinear influence increasing with a
decreasing solvent viscosity. But this problem has been evident throughout the discussion
of results and has been attributed to low signal strength in the slow deformation rate of the
high viscosity sample.
3.4
Summary
As was initially seen in the voltage-controlled Volterra results, these characterization results
for the current-controlled system arrangement showed that the most prevalent distortion
arises from an odd nonlinearity in the system. More specifically, it appears to be the result
of a cubic mechanism. This was shown both in single-frequency analysis with the Volterra
92
identification and harmonic analysis, and in broadband form by exciting the polymer actuators with designed multisine signals. Results from these characterization techniques provide
stronger evidence that the electrical response is dominated by a cubic nonlinearity and that
the nonlinear influence decreases as the solvent viscosity increases. Also shown was that
the viscosity of the solvent affects the frequency at which nonlinearities begin to distort the
response, where it was shown that lower viscosities give rise to nonlinear effects at higher
frequencies. This essentially says that the frequency band over which nonlinearities are more
dominant is widened for actuators manufactured with lower viscosity solvents. Regarding
the mechanical response, the high viscosity sample gave varying results, while these Volterra
results showed that the nonlinearity does increase for lower viscosity solvents. However, in
quantifying this nonlinearity with multisine excitations, it appeared that both the odd and
even nonlinearities introduce nearly equivalent levels of distortion into the velocity response.
Based on the characterization in this chapter, nonlinear model structures that require an
assumed form of the nonlinearity can now be examined.
93
Chapter 4
Nonlinear Identification Techniques
Using knowledge of the nonlinear form that was gained through the characterization, this
chapter will detail two different nonlinear identification techniques. Each method explored
can be applied to system structures with nonlinearities affecting the response through multiple arrangements, so it is thought that examining these techniques will be sufficient in
modeling the actuation response. Following a brief discussion of the model forms for which
these techniques can be applied, analysis of the two methods will be presented. After each
description, an example showing the utility of the method will be provided.
4.1
Nonlinear System Structures
To model the electromechanical response of ionic polymer actuators, three different system
arrangement schemes were tested to see which method can best predict the nonlinear distortion that has been observed in voltage and velocity measurements. The first two structures
take on different variations of the multi-input/single-output (MI/SO) technique described
by Bendat (1998), while the third structure takes a slightly different approach. Figure 4.1
shows block diagrams for the three model structures, where x(t) is the system input, y(t) is
the system output, H(f ) is the underlying linear system, N L is a nonlinear function, A(f ) is
another linear system acting like a frequency-dependent parameter of the nonlinearity, and
g(·, ·) is kept as a general nonlinear dynamic system. While each of the first two structures
has the nonlinearity in parallel with the underlying linear system, the difference between
them lies in how the nonlinearity affects the system. For Figure 4.1a the nonlinearity affects
the output by acting directly on the input, like feedthrough, but in Figure 4.1b the non94
Figure 4.1: System arrangements used for impedance modeling: (a) MI/SODirect; (b) MI/SO-Reverse; (c) DIN.
linearity affects the output and alters the system response through output feedback. The
structure in Figure 4.1c is different in that the nonlinearity modifies the input signal before
passing through the underlying linear system. In this case, the nonlinear function g(·, ·)
could be modeled in a variety of ways, such as a Hammerstein or Wiener system, but it is
assumed to be dynamic and this form has been dubbed dynamic input nonlinearity (DIN).
More specifics of these methods will be discussed in the next sections as the algorithms are
detailed. It should be noted here that in each of the methods to be discussed, it is assumed
that an underlying linear system H(f ) does exist.
4.2
Multi-Input/Single-Output Technique (MI/SO)
The multi-input/single-output analysis technique identifies linear systems, or frequencydependent parameters, in a parallel system arrangement. The analysis methodology will be
discussed first and this section will close with an illustrative example.
95
Figure 4.2: System descriptions: (a) general system; (b) decomposed system of
MI/SO nonlinear analysis technique.
4.2.1
MI/SO System Analysis
Beginning with the most general form of a single-input/single-output system as seen in
Figure 4.2a, the MI/SO technique decomposes this system into a linear and nonlinear component connected in parallel, where this linear component represents the underlying linearity
of the system. The nonlinear system is then further decomposed into a nonlinear function
in series with another linear system, shown in Figure 4.2b. It is this system arrangement to
which the analysis method is applied. The method, as will be described, is based on developing relationships between two equivalent system representations from only the measured
input and output data, and some knowledge of the nonlinearity. The initial representation
may have correlated linear and nonlinear signals, while the revised system takes advantage
of relationships between uncorrelated signals. It should be noted here that if the system
were purely linear, the nonlinear system would be zero.
Assuming that there is an underlying linear system involved, discussion of the
method will continue. The underlying linear system of the initial representation has the
output yh (t), and the output of the nonlinear system is yv (t). Together with any model
96
noise, n(t), the total system output can be given as
y(t) = yh (t) + yv (t) + n(t),
(4.1)
where yh (t) and yv (t) may be correlated and n(t) is assumed to be uncorrelated. Calculating
the autospectrum of the output gives
Syy (f ) = Syh yh (f ) + Syv yv (f ) + Snn (f ) + Syh yv (f ) + Sy∗h yv (f ),
(4.2)
where all the cross terms formed from the noise signal are zero. The trouble in analyzing a
system with this equation is that the decomposition is difficult when Syh yv (f ) = 0. Changing
the description to represent uncorrelated signals gives the revised system, which looks similar
to equation 4.1:
y(t) = yo (t) + yu (t) + n(t),
(4.3)
where the optimal linear output is yo (t) and the revised nonlinear output is yu (t). Because
all signals in equation 4.3 are uncorrelated with one another, no cross terms appear and the
output autospectrum of the system description has the form
Syy (f ) = Syo yo (f ) + Syu yu (f ) + Snn (f ),
(4.4)
which simplifies the decomposition analysis significantly.
By comparing equations 4.1 and 4.3, another useful expression can be obtained,
shown in Fourier transforms as
Y (f ) = Yh (f ) + Yv (f ) + N (f )
(4.5)
= Yo (f ) + Yu (f ) + N (f ).
(4.6)
Then equating these equations leads to
Yh (f ) + Yv (f ) = Yo (f ) + Yu (f )
(4.7)
since the noise terms cancel each other. This equation now allows for better comparison
between the two different system representations. Figure 4.3a shows the initial system with
correlated linear and nonlinear terms. In this system, some equations of interest are
Yh (f ) = H(f )X(f )
(4.8)
Yv (f ) = A(f )X(f )
(4.9)
v(t) = g[x(t)],
(4.10)
97
Figure 4.3: Equivalent MI/SO representations: (a) initial (correlated) system;
(b) revised (uncorrelated) system.
where g(·) is an arbitrary (zero-memory) nonlinear system. In accordance with the MI/SO
technique, the system shown in Figure 4.3a can be thought of as a 2-input/1-output linear
system with potentially correlated inputs x(t) and v(t). Revising this system for analysis
gives another equivalent system representation given in Figure 4.3b. The uncorrelated
inputs to this 2-input/1-output system are x(t) and u(t). This revised system is what will
now be the focus of analysis. Once the signal decomposition and system identification are
complete, the actual (original) system can then be extracted.
As a quick aside, this notion of correlated and uncorrelated inputs will be elaborated.
Consider a parallel system structure as in Figure 4.3, where the nonlinearity is v(t) = x3 (t).
In the case of a harmonic input x(t) = cos(ωt), the calculated nonlinear system output
will contain content at the frequencies ω and 3ω because x3 (t) = (3cos(ωt) + cos(3ωt))/4.
Here it can be seen that v(t) and x(t) are correlated through the content at frequency ω.
By applying this identification algorithm, calculation of u(t) will remove this content at
frequency ω, leaving this signal uncorrelated with the input x(t).
The first step in the methodology is to determine the optimal linear system Ho (f ).
This can be found having only measurements of the system input x(t) and output y(t) from
98
the well-known equations
Ho(f ) =
=
X ∗ (f )Y (f )
X ∗ (f )X(f )
Sxy (f )
.
Sxx (f )
(4.11)
(4.12)
This is commonly referred to as the H1 method. Once this system is calculated, the revised
(optimal) linear output yo (t) follows directly from
Yo (f ) = Ho (f )X(f ).
(4.13)
It should be noted here that yo (t) = yh (t) = y(t) only if there is no nonlinearity or noise
in the system. A measure of the relative amount of linearity in the overall system can be
calculated either directly from input/output measurements or using the autospectrum of
Yo (f ) as
2
(f ) =
γxy
=
Syo yo (f )
Syy (f )
|Sxy (f )|2
.
Sxx (f )Syy (f )
(4.14)
(4.15)
The first expression (equation 4.14) comes from the notion of conditioned spectral density
functions and partial coherence functions discussed by Bendat (1993), while equation 4.15
is the common coherence function.
With the underlying linearity in the revised system calculated, the next step in the
analysis procedure is to determine properties of the nonlinear component, which begins
considering the initial MI/SO system with correlated signals. Based on any knowledge or
assumption of the nonlinearity, for which the characterization results from the Volterra
analysis or multisine excitations can be used, the output of the nonlinear system, v(t), can
be calculated as in equation 4.10. Any linear relationship hidden in this nonlinear system
can then be found, transforming the overall system into a 2-input/1-output linear system.
Similar to calculation of the optimal linear system Ho (f ), the following equations apply
L(f ) =
2
(f ) =
γxv
Sxv (f )
Sxx (f )
|Sxv (f )|2
.
Sxx (f )Svv (f )
(4.16)
(4.17)
where L(f ) is any linear system contained within the nonlinear function g[x(t)]. Then by
removing this linear component from the nonlinear system output, any correlation between
99
with the system input x(t) is eliminated. The result is a signal U (f ) that is uncorrelated
with X(f ) and becomes the second input to the revised system of Figure 4.3b by the
relations
2
(f )]
Suu (f ) = Svv (f )[1 − γxv
(4.18)
U (f ) = V (f ) − L(f )X(f ).
(4.19)
By referring to this as the “nonlinear system” from here on will not imply any more
nonlinearity, only that the input to this other linear system was formed from a nonlinear
operation. This will also help distinguish between the two linear systems. That being said,
the cross-spectrum of the nonlinear system can be calculated knowing the uncorrelated
input as
Suy (f ) = Svy (f ) −
Svx (f )
Sxy (f )
Sxx (f )
(4.20)
Note the reversed subscripts on the cross-spectrum between X(f ) and V (f ). The reason
for this can be seen when considering equations 4.16 and 4.19 with the definition of the
cross-spectrum Suy (f ) = U ∗ (f )Y (f ).
Calculation of both the cross- and autospectra of the nonlinear system leads to the
calculation of its corresponding linear system and contribution to the total system output
Y (f ) by
Suy (f )
Suu (f )
Yu (f ) = A(f )U (f ).
A(f ) =
(4.21)
(4.22)
Using these quantities now allows calculation of the relative influence of the revised nonlinear
system on the overall system through the (partial) coherence function
2
(f ) =
qxy
=
Syu yu (f )
Syy (f )
|Suy (f )|2
,
Suu (f )Syy (f )
(4.23)
(4.24)
where this “nonlinear coherence” is the same as the linear coherence between the signals
2 (f ) = γ 2 (f )).
u(t) and y(t) (qxy
uy
The only remaining signal to be determined from Figure 4.3b is the model noise,
n(t). Recalling the definition of the output autospectrum of the revised system given in
equation 4.4, the noise autospectrum is found by a simple rearrangement as
Snn (f ) = Syy (f ) − Syo yo (f ) − Syu yu (f ),
100
(4.25)
making it the portion of the output that is not modeled by either the linear or nonlinear
system. Note that this noise is not necessarily the same as some disturbing noise in the
output measurement. The effect of the measurement noise is minimized by averaging,
leaving the noise of Snn (f ) to be in the model. This could be thought of as a measure of
the uncertainty in the model developed.
As has been done with the other contributing signals, the relative effect of the noise
on the system can be determined with the coherence
Snn (f )
Syy (f )
2
2
(f ) − qxy
(f )],
= [1 − γxy
2
γny
(f ) =
(4.26)
(4.27)
thus, completing the analysis of the revised (uncorrelated) system.
Turning now to the initial (correlated) system, the underlying linear system H(f )
can be calculated by removing any component from yv (t) that is correlated to x(t) from the
optimal linear system Ho (f ) as
H(f ) = Ho (f ) − L(f )A(f )
= Ho (f ) − C(f ),
(4.28)
(4.29)
where the system C(f ) represents the correlated component. Other signal of interest in the
actual system can then be calculated according to
Yh (f ) = Yo (f ) − C(f )X(f ) = H(f )X(f )
(4.30)
Yv (f ) = Yu (f ) + C(f )X(f ) = A(f )V (f ).
(4.31)
Some general remarks regarding the MI/SO technique will now be made. The first
is about the assumed system structure. Recalling the series arrangement of the nonlinear component in Figure 4.2b, the nonlinear system comes before the linear system. Had
the location of these system been reversed, the method would no longer apply in terms
of identifying the linear system A(f ). To determine the linear system in this case, more
knowledge or assumptions of the overall system would need to be made. Secondly, this analysis methodology can also be applied when more than one nonlinearity is in the system by
indexing the above relations, beginning with vi (t) = gi [x(t)] and following script. However,
the process does become more involved when calculating the mutually uncorrelated input
signals since each successive Vi+1 (f ) must also have any correlation with Vi(f ) removed.
101
This same technique can be applied in two different ways. The first is the Direct
method in the system of Figure 4.1a. Here, x(t) is the measured input to the system and
y(t) is the measured output. Alternatively, the Reverse method can be applied to identify a
connection of systems in feedback, as in Figure 4.1b. In this case, the measured input and
output are reversed prior to performing the analysis, using the measured output as x(t) for
the identification and the measured input as y(t). As a quick example, consider a system
described by the nonlinear differential equation
y¨∗ (t) + 2ζωn y˙∗ (t) + ωn2 (y ∗ (t) + α∗ y ∗3 (t)) = x∗ (t),
(4.32)
where x∗ (t) is an input force, y ∗ (t) is the displacement output, ζ is the damping, ωn is the
natural frequency, and α is a nonlinear stiffness term that can be a function of frequency.
Using the Direct MI/SO method, no knowledge of α∗ can be obtained. However, by reversing
the input and output measurements for the analysis, the linear system and α∗ can be
identified (fitting the form of Figure 4.3b). To carry this out, let x = y ∗ , y = x∗ , and
α = ωn2 α∗ to change equation 4.32 to the form
y(t) = ẍ(t) + 2ζωn ẋ(t) + ωn2 x(t) + αx3 (t).
(4.33)
Now from knowledge of the nonlinear form, x3 (t) can be calculated from the measured
x(t) (= y ∗ (t)) and the Fourier transform of this signal and the others can then be taken.
This gives
Y (f ) = (ωn2 − ω 2 + j2ζωn ω)X(f ) + A3 (f )X3 (f ) + N (f )
= A0 (f )X(f ) + A3 (f )X3 (f ) + N (f )
(4.34)
(4.35)
where ω is the frequency, A3 (f ) is the Fourier transform of α, X3 (f ) is the Fourier transform
of x3 (t) (note that X3 (f ) = [X(f )]3), and any other uncorrelated content is found in N (f ).
In relation to the original system in equation 4.32 and the MI/SO analysis methodology,
H(f ) = [A0 (f )]−1 =
ωn2
−
ω2
1
+ j2ζωn ω
(4.36)
is the underlying linear system and A3 (f ) is the linear system of the nonlinear component in
the feedback path. As with the Direct method, the Reverse MI/SO method can be extended
to multiple nonlinearities.
The two methods discussed above are each used to identify the components of a
nonlinear system of the forms given in Figures 4.1a–b. Once the identification is complete,
102
Figure 4.4: Representative system for MI/SO identification example.
various methods can then be applied to fit models to the individual systems. This will allow
for simulations to be performed on the full nonlinear model and for comparisons to be made
between measurements and simulated output.
4.2.2
Example MI/SO Identification
To help illustrate how this multi-input/single-output technique works to identify a nonlinear
system, an example will now be given. The system structure of this representative system
is shown in Figure 4.4 to have parallel linear and nonlinear systems, with the nonlinear
system consisting of a static nonlinear function and another linear system. Because the
analysis procedure is the same for both Direct- and Reverse-MI/SO techniques, only one
example will be shown here.
The underlying linear system H(f ) has been arbitrarily defined to have two secondorder poles (30 Hz, 2% damping; 90 Hz, 1% damping) and one second-order zero (55 Hz,
3% damping), while the linear system of the nonlinear component A(f ) has been defined
as a critically damped, second-order system, like a low-pass filter with the breakpoint at 10
Hz. Frequency response plots of the two linear systems are given in Figure 4.5, with H(f )
as the solid line and A(f ) as the dashed line. This figure shows that the nonlinearity is
dominant at lower frequencies and essentially negligible at higher frequencies, qualitatively
similar to experimental results with the ionic polymer system.
To collect simulation data on this system for use in the identification procedure,
a mean-zero Gaussian signal at 0.25 unit-rms was applied with a sample rate of 1.0 kHz.
In order to more accurately imitate experimental conditions, measurement noise was also
added to the output. This was also a mean-zero Gaussian signal, but the unit-rms level was
reduced to 0.0005, which comes out to nearly 10% of the output signal. Data was collected
for 10 averages of 4096 data points each and a Hanning window was used in estimating all
of the spectral quantities.
103
Figure 4.5: Frequency response plots of system components for MI/SO example.
Two different sets of identification will now be analyzed. The difference between the
two are in the assumption of the nonlinear form. One set of results that will be presented
is with the correct assumption (cubic: v(t) = x3 (t)) and the other set is with an incorrect
assumption (quadratic: v(t) = x2 (t)). This will help show the utility of the method, while
also emphasizing certain characteristic results that give an indication of whether the proper
assumption was made.
First the identified underlying linear system of Figure 4.6 will be examined. Results
from the correct nonlinear form are given in Figure 4.6a on the left side and those from the
incorrect nonlinearity are on the right in Figure 4.6b. In each of the sets of plots, the top
plot shows the magnitude, the middle plot shows the phase in degrees, and the bottom plot
shows the coherence. The solid line represents the actual underlying linear system H(f ),
the dashed line represents the identified optimal linear system Ho (f ), and the dotted line is
the simulation model. The optimal linear system is the same in both sets of plots because
it is related to the linear correlation between only the input and output, without regard to
the nonlinearity.
Recalling the system definitions of the simulation model in Figure 4.5, it can be seen
that the magnitude and phase of Ho (f ) take after which ever linear system is more dominant
2
(f ) is highest over the frequency
over different frequency ranges, and the linear coherence γxy
104
Figure 4.6: MI/SO example – Identified underlying linear system: (a) correct
assumption; (b) incorrect assumption.
range that H(f ) is more dominant. The difference between these two sets of results is
2 (f ) + q 2 (f )), which is a measure
highlighted by looking at the cumulative coherence (γxy
xy
2
(f ) → x(t), y(t)), plus the linear
of linear correlation between the input and the output (γxy
2
(f ) → u(t), y(t)). It can
correlation between the assumed nonlinearity and the output (qxy
be seen that the cumulative coherence for the correct nonlinear assumption is near unity over
the entire frequency range, while the incorrect assumption leads to very little improvement
over the underlying linear coherence. Evidence of this is also seen in the magnitude and
phase plots, where the correct assumption leads to H(f ) overlaying the simulation model,
and the incorrect assumption has H(f ) and Ho (f ) nearly overlaying each other. From these
results it can be concluded that one metric for assessing whether or not the correct nonlinear
form was assumed is to check that the cumulative coherence improves the linear coherence
over the frequency range of interest.
Turning now to identification of the nonlinearity frequency-dependence, Figure 4.7a
shows the results for the correct nonlinear assumption and Figure 4.7b shows them for
the incorrect assumption. As in Figure 4.6, these plots show the magnitude, phase, and
coherence, where the coherence here is only the linear correlation between the assumed
nonlinearity and the output. Again, this coherence is a good place to look in determining
if the proper nonlinearity was assumed. Figure 4.7a has higher values at the frequencies
where the linear coherence suffers, but Figure 4.7b is nearly zero over the entire range. Also
105
Figure 4.7: MI/SO example – Identified frequency dependence of nonlinearity:
(a) correct assumption; (b) incorrect assumption.
similar to the H(f ) results, the correctly assumed nonlinearity allows the identified A(f )
to overlay the simulation model, while the incorrect assumption does not. When examining
the magnitude and phase of Figure 4.7a it can be noticed that there are two small spikes at
the poles of the underlying linear system that could cause discrepancies when attempting
to fit a model, but the coherence values are near zero here. This implies that the coherence
function could be used as a frequency weighting when fitting a model to this system.
One last set of results that will be shown are the various output autospectral functions. As mentioned in the discussion of the MI/SO technique, these quantities can be used
to determine the coherence functions. Figure 4.8a has these results for the correct nonlinear
assumption and Figure 4.8b has the results for the incorrect assumption. The top two plots
are the quantities for the optimal system, this is with uncorrelated inputs x(t) and u(t), and
the bottom plots are the original, or actual, system with possibly correlated inputs x(t) and
v(t). In each of the plots, the autospectrum of the output signal is the large dotted line, the
linear system is the solid line, the nonlinear system is the dashed line, and the noise is the
small dotted line. The output autospectrum for all four plots is the same because it comes
from the measured output and the noise spectrum for each set of nonlinear assumption
results is the same, but the individual system output spectra are different.
In the optimal system Syo yo (f ) includes linear correlation from both inputs, while
Syh yh (f ) includes only the known input x(t). These two quantities would actually be iden106
Figure 4.8: MI/SO example – Autospectral functions: (a) correct assumption;
(b) incorrect assumption.
tical if there were no nonlinearity in the system. The other spectra that differ from the
optimal to original systems are the nonlinear system output autospectra for the same reason as just mentioned, Syv yv (f ) may include some linear correlation to x(t), while Syu yu (f )
has none. In comparing the two sets of results, the key difference that can be seen is in
the relative significance between Snn (f ) and the nonlinear system outputs Syu yu (f ) and
Syv yv (f ). When the correct assumption of the nonlinearity is made, the noise level is far
lower than the other outputs as in Figure 4.8a, but when the nonlinearity is incorrectly
assumed, Figure 4.8b shows that the noise level is actually higher than that of the nonlinear system. This is another indication that the wrong nonlinear form was assumed. Recall
that this noise level does not necessarily reflect measurement noise, which is dealt with in
averaging the Fourier transforms, but rather is an indication of model error or uncertainty.
One final note for the correctly assumed nonlinearity is that in the original system plot, the
individual output components match the output autospectrum over the frequency range
where they dominate.
4.3
Dynamic Input Nonlinearity Technique (DIN)
The next identification algorithm to be detailed has a dynamic nonlinearity on the input to
an underlying linear system. This system arrangement is different than the MI/SO system
107
Figure 4.9: System arrangements for low amplitude data.
because the individual systems act in series. While the analysis technique is presented,
the nonlinear system is kept general since no assumption on its form needs to be made
in identifying the underlying linear system. Following this analysis procedure, an example
system identification is provided.
4.3.1
DIN System Analysis
This technique takes a different route than the Direct and Reverse MI/SO methods to
develop a system model and will be referred to as the Dynamic Input Nonlinearity (DIN)
method for simplicity. This name comes from the system layout shown in Figure 4.1c,
where the nonlinearity is not in parallel with the underlying linear system as in the MI/SO
methods, but instead alters the input prior to passing through the underlying linear system.
And to allow for frequency-dependence, as was shown in the characterization results, this
method has been termed “dynamic.” Similar to the MI/SO technique, this method uses only
measurements of the input and output, and some knowledge of the nonlinearity. However,
unlike the MI/SO technique, this method requires two sets of input-output data: one at a
low input level where the nonlinear response is considered negligible and one at a higher
input level when the output is thought to be affected somewhat by the system nonlinearity.
One disadvantage of the DIN technique when compared to the MI/SO methods is that some
modeling of the system takes place during the identification instead of only at the end. This
places more emphasis on fitting a good linear model and also presents one area where error
could be introduced.
108
The first step in the DIN modeling process is to identify the underlying linear system
shown in Figure 4.9a. This is done using the low amplitude input-output data (x̃b (t) and
ỹ b (t)) to calculate the frequency response function. In this identification scheme, a tilde
will be used to signify data, a superscript b will denote low amplitude, and a superscript B
will denote high amplitude. The low amplitude frequency response function is given by
H̃(f ) =
b
(f )
S̃xy
b (f )
S̃xx
,
(4.37)
where Sxy (f ) and Sxx (f ) are cross- and auto-spectra of the data, respectively. This equation
is the same as equation 4.12. Once this system is identified, a model fitting routine is applied
to get a system description H(f ) that will allow simulations to be made. One restriction
on this model is that it have relative degree r = 0. This is required because during the next
steps, the linear system must be inverted and if H(f ) does not have r = 0, then H −1 (f )
will not have proper form.
With H(f ) defined, the output can be simulated using the input data x̃b (t) as
Yhb (f ) = H(f )X̃ b(f ).
(4.38)
In the ideal case when the measurement noise n(t) is negligible and the amplitude is low
enough that a linear approximation is sufficient, it will hold that Yhb (f ) ≈ Ỹ b (f ). This can
2 (f ) as
be checked in two ways. The first is simply using the linear coherence function γxy
defined in equation 4.15, and the second is comparing the simulated output to the data as
N (f ) = Ỹ b (f ) − Yhb (f ) ≈ 0.
(4.39)
From here on, it will be assumed that the measurement noise is low.
Assuming that equation 4.39 holds, the underlying linear system can now be simulated to the higher amplitude input x̃B (t) to get yhB (t). Since this input has been selected
to include nonlinear effects, it will be true that YhB (f ) = Ỹ B (f ). Instead, the measured
output is made up of a component directly from the applied input, YhB (f ), and a component
due to the input nonlinearity, YnB (f ), given by
Y B (f ) = YhB (f ) + YnB (f ).
(4.40)
Figure 4.9b shows this system with two input components. At this point, the nonlinear
output component can be determined by comparing the higher amplitude output data to
109
Figure 4.10: Nonlinear system structures: (a) Hammerstein model; (b) Wiener
model.
the linear simulation output
YnB (f ) = Ỹ B (f ) − YhB (f ).
(4.41)
Now the only unknown in the system is the component of the input that caused the linear
model error. This is where the linear system is inverted and simulated with the nonlinear
output component to determine this signal
XnB (f ) = H −1 (f )YnB (f ).
(4.42)
One important aspect to consider by inverting the linear model is the introduction of error
from unmodeled, high frequency dynamics. Regarding the polymer actuator, this inversion
will likely be very amenable to the electrical response model because at high frequency, the
impedance levels off and the response is similar to a resistor. The mechanical response, on
the other hand, may have a larger amount of error introduced, due to modes outside the
measured frequency range. This is something that will have to be considered in the polymer
model development.
Provided that the linear model inversion does not cause too many problems, results
from the characterization study can now be applied to assume a nonlinear form for g[x(t), f ]
and different structures can be analyzed. Keeping in line with the MI/SO technique assumptions, a place to start involves a static nonlinearity and a linear system. Two of the
most common types are the Hammerstein system with a static nonlinearity on the input to
a linear system, and a Wiener system with a static nonlinearity on the output of a linear
system. Figure 4.10 has been included to show these system arrangements. With knowledge
of the nonlinearity and the known quantities X(f ) and Xn (f ), the signal acting between
the static nonlinearity and the linear system can be calculated and used to identify (using
equation 4.37) and model the system.
110
Figure 4.11: Representative system for DIN modeling example.
Figure 4.12: Frequency response plots of system components for DIN example.
4.3.2
Example DIN Identification
As with the MI/SO identification technique, an illustrative example will now be shown
for the DIN methodology. The representative system used for this example is given in
Figure 4.11. In this figure it can be seen that the dynamic nonlinearity on the input is
a Hammerstein system, where the static nonlinearity is again a cubic function. Like the
MI/SO identification example, this DIN example will also consider two different sets of
simulation data: one with the correct nonlinear assumption x3 (t) and one with an incorrect
assumption of x2 (t).
The representative system definitions are slightly different here because of the restriction that the underlying linear system have relative degree zero. To this end, the simulation
model linear system H(f ) has only one second-order pole (11 Hz, 2% damping) and one
second-order zero (21 Hz, 7% damping). The frequency-dependence of the nonlinear has
also been modified this time to have two first-order poles (2.7 Hz and 3.0 Hz), but still
111
Figure 4.13: DIN example: (a) identification of linear system for two input
amplitudes; (b) model fit to low amplitude, identified system.
serves as a low-pass filter. Frequency response plots of these two illustrative systems are
given in Figure 4.12. In this figure, the underlying linear system is the solid line and that
of the nonlinear component is dashed.
Data collection for this simulation system was similar to the previous example. Here
again, a mean-zero Gaussian signal was the input to the system, and simulation data was
collected for 10 averages of 4096 points. The sample rate for this example was 200 Hz.
An input level of 0.9 unit-rms was used for the level where contribution from the input
nonlinearity was effectively negligible and this was increased by a factor of ten to 9 unitrms to collect distorted data. Also as before, random measurement noise was added to the
output signal at a level of about 10% of the low amplitude output.
The linear identification of the system is shown in Figure 4.13a for each of the two
input levels. The linear identification here is just the frequency response function. Results
for the low input are dashed and the solid line is for the high input. The effect of the
nonlinearity is clear when examining all three plots of magnitude, phase, and coherence.
If the system were truly linear, any increase in input amplitude would scale linearly in
the output and this factor would cancel itself out when calculating the frequency response
function. This would mean that the two lines would overlay each other, but since this is not
the case, the source of the nonlinearity will now be explored. Recalling the design of the
system, it can also be seen that the high amplitude results differ from the low amplitude
112
Figure 4.14: DIN example – inverse system simulations: (a) low amplitude; (b)
high amplitude.
results over the frequency range where the nonlinear system is more dominant.
The first major step of the DIN analysis is to fit a model (r = 0) to the system
identified as the underlying linear system (low amplitude simulation). Based on the identification results, the system appears to be second-order in both the numerator and the
r
denominator, so this will be the first guess in the model fitting routine. The Matlab
function invfreqs.m, which performs a least-squares fit, is used in this stage. Inputs to the
function are the complex-valued frequency response function (identification results), frequency vector, order of numerator and denominator, weighting function (coherence), and
the number of iterations. The iterations were set to 5,000, but the model was within the
default tolerance range of 1% far before reaching this limit. Figure 4.13b shows the results
for this model as the solid line, along with the identified frequency response (dashed) and
the simulation model (dotted). As it can be seen, the three lines lie nearly on top of one
another, indicating a good model fit. Where the identification and model do not completely
overlay the simulation model is at frequencies above the anti-resonance. This is the region
where the magnitude is lowest and where the added measurement noise adversely affects
the identification.
Continuing with the DIN methodology, the model is simulated next to the higher
input data to get the linear model output yhB (t). Then the nonlinear component ynB (t) can
be backed out from comparing the data of the higher input, ỹ B (t), to this linear model
113
Figure 4.15: DIN example – Identified frequency-dependence of nonlinearity:
(a) correct assumption; (b) incorrect assumption.
simulation. From here the linear model is inverted and simulated with the nonlinear output
to get the nonlinear component of the input signal xB
n (t), which is shown in Figure 4.14
for the first couple of seconds. The case with the low amplitude, where it was assumed
that no nonlinear effects were present is given in Figure 4.14a and Figure 4.14b has the
results for the higher input case. The upper plots compare the input data (solid line) to the
inverse-simulated input from the model (dotted line), and the lower plots show the error
between the two, which is the nonlinear component of the input. For the lower input level,
Figure 4.14a shows that the error is small with the maximum error being approximately an
order of magnitude lower than the maximum response of the inverse system. This actually
makes sense here because the noise that was added to the output data was nearly 10% of
the output. Figure 4.14b shows that nonlinearity has a significant effect. The nonlinear
component (linear error) has almost the same vertical scale as the data itself, indicating
that the signal is quite contaminated with distortion from the nonlinearity.
Now an assumption of the nonlinear form must be made. Using a Hammerstein
system as the nonlinearity, a signal v(t) is computed from x(t) and the assumed nonlinearity.
The frequency-dependence can then be determined from a linear identification between v(t)
and xB
n (t), and the nonlinear identification of the system will be complete. In Figure 4.15a
these results are given for a case where the nonlinear assumption was correct (v(t) = x3 (t))
and Figure 4.15b has the results for an incorrect assumption (v(t) = x2 (t)). The top
114
two plots show the identification of A(f ) (solid) in magnitude and phase, along with the
2 (f ). The
simulation model (dotted), and the bottom plots show the linear coherence, γvx
agreement between the identification and the simulation model in Figure 4.15a is a good
indication that the correct assumption of the nonlinearity was made. The coherence function
is also very high over the lower frequencies before the error due to measurement noise
corrupts the data. As mentioned in the previous example, this coherence function serves as
a good weighting function for the next step when fitting a model occurs. Turning now to
Figure 4.15b, it is clear from all three sets of plots that the identification and simulation
model are not even close to matching. The magnitude and phase plots do not overlie and
the coherence is well below any acceptable level for the entire frequency range. One last
note is that when this technique is applied to the polymer actuator, a simulation model
(truth model) will not exist for comparison, so the coherence will be used as a measure of
how well the nonlinearity has been identified.
4.4
Summary
Two nonlinear identification techniques that can be applied to a variety of system structures
were discussed in this chapter. The MI/SO technique identifies a nonlinear system with
a parallel arrangement, while the DIN technique has the nonlinearity in series with the
underlying linear system. Example analyses were also shown for each methodology, drawing
attention to different factors to consider when determining which method or assumptions
will produce a better model. These techniques will be applied and compared to actuation
data for an ionic polymer transducer in the next chapter.
115
Chapter 5
Dynamic System Modeling
Now that the chosen nonlinear identification techniques are in working order, this chapter is
dedicated to applying these methods to an ionic polymer actuator. The goal is to obtain a
dynamic nonlinear model for both the electrical and mechanical response, with simulations
being able to predict both the scaling in the frequency response and the harmonic distortion
seen in the sinusoidal time response. A section is devoted to each, where comparisons will
be made between the different model structures and any model improvements are discussed.
5.1
Data Collection and Modeling Preliminaries
Throughout the characterization work, results from polymer samples in three different solvent forms were shown, but here the models will be developed for only one material. The
ionic liquid EMI-Tf was chosen as the candidate for model development for two reasons.
First, this sample gives very repeatable results, and second, noticeable distortion has been
observed in its response. The water sample would have been ideal for modeling the nonlinearity because it shows the most distortion, but the potential to have skewed results from
solvent dehydration eliminated it from consideration. The sample in the high viscosity solvent was also not chosen because it did not exhibit the nonlinear behavior that was seen
with the lower viscosity solvents.
The data used to construct these models was collected using the same laboratory
setup as was used in the characterization study (see Figure 3.2), so it will not be mentioned
again here, though specifics of the data will be detailed now. For the random data that was
used with the identification techniques, sets of 16 averages with 16,384 points each were
116
collected at 100 Hz. Such large data blocks were used because the nonlinear effects are
dominant at the lower frequencies and in order to have an ample number of data points for
accurate model fitting, long records were required to get sufficiently small frequency spacing
(6.1 mHz). Four different current levels were also used in these experiments: 0.5, 1.5, 2.5,
and 3.5 mA-rms. As will be seen in the modeling results, this range is sufficient to show
evidence of nonlinearity in the frequency response of both voltage and velocity. Based on
the identification techniques employed in the development, the models will be built using
frequency-domain data.
Sinusoidal data was also collected, primarily for the purpose of validating the models
constructed from Gaussian excitations. Eight frequency settings were used here: 0.11, 0.55,
1.1, 2.75, 5.5, 8.25, 11.0, and 13.75 Hz, where it should be noted that the first mechanical
resonance was at 11 Hz. As in the characterization results, the sample rate varied for each
frequency to give a constant number of points per cycle for all of the data sets. This constant
multiple was again set to 200. Four different input levels were also used for the sine wave
data. These were 0.5, 2.0, 3.5, and 5.0 mA-rms, which cover the range of the random data,
with one higher amplitude to show more distortion. The discussion to follow will discuss
how this data was used in model construction and validation.
When fitting models to the identified linear systems and any frequency-dependence of
the nonlinear components, the model form used will be the general linear system description
of
m
(s − zi )
,
G(s) = k ni=1
i=1 (s − pi )
(5.1)
where k is the gain, zi are the zeros, and pi are the poles. Since not all of the components
are modeled using the same techniques, descriptions for each of the individual systems will
be discussed during the appropriate time below. For each system, results from the MI/SO
technique are presented first, followed by those for the DIN technique.
5.2
Modeling the Electrical Response
In this section results from the two identification methods will be applied to construct
nonlinear models for the electrical impedance of an ionic polymer actuator with current
as the input and voltage as the output. Before beginning the model development, the
particular distortion of this sample will be mentioned. Figure 5.1a shows scaling in the
117
Figure 5.1: Evidence of voltage nonlinearity in EMI-Tf sample: (a) frequency
response; (b) time response.
frequency response from two different inputs. The dotted line represents data from a 0.5
mA-rms random input signal and the solid line shows the results when the input is increased
to 3.5 mA-rms. It can be seen that the two lines match each other at high frequencies,
indicating a linear response, but the lines begin to separate as the frequency decreases,
which is a sign of nonlinearity. The coherence function in the bottom plot is also fairly
high over the frequency range, but there is some evidence that the larger input causes a
decrease at low frequency, which is another indication of nonlinearity. Figure 5.1b shows
the time response to sinusoidal inputs at 7 mA (5.0 mA-rms) of two different frequencies.
The left plots are at 0.11 Hz, where distortion is present in the voltage response and the
right plots are in the linear region at 11 Hz, where the output is an amplitude-attenuated
and phase-shifted version of the input waveform.
5.2.1
Impedance – MI/SO Identification
As discussed during the identification chapter, there are two methods for identifying a
nonlinear system using the MI/SO technique: direct and reverse. It was also previously
mentioned that the cumulative coherence can be used as a metric to determine how well the
nonlinear assumption improves the model, so this will be used with the experimental data
to help choose which MI/SO identification method gives the best results. Using knowledge
gained from the characterization results, a cubic nonlinearity is used as the first assumption
118
Figure 5.2: MI/SO voltage results with cubic nonlinearity: (a) direct; (b) reverse.
of the nonlinear form. For the direct method this has v(t) = x3 (t) affecting the voltage
output by feedthrough and in the reverse method v(t) = y 3 (t) affects the output through
feedback.
Performing the analysis of each method on the highest input data gives the results
in Figure 5.2. The direct method results are in Figure 5.2a and the results for the reverse
method are given in Figure 5.2b. Shown in the top two plots of each figure is the frequency
response for the optimal linear system Ho(f ) (dotted), the underlying linear system H(f )
(solid), and the frequency-dependence of the nonlinearity A(f ) (dashed). The original linear
coherence is given in the lower plot as the dotted line, and the cumulative coherence appears
as the solid line.
The first thing to notice in these two plots is that the cumulative coherence and linear
coherence of the direct method fall almost completely on top of each other, indicating that
using a cubic nonlinearity in the direct system arrangement gives virtually no improvement
over the linear system Ho (f ). However, when examining the reverse method coherence plot,
it can be seen that the cumulative coherence is an improvement over the linear coherence at
low frequencies, which is where the nonlinearity has the most influence. This is an indication
that the reverse MI/SO method will offer a better model structure for the polymer actuator
than the direct method. It should be noted here that several other nonlinear forms, including
x2 (t), x(t)|x(t)|, and x2 (t) + x3 (t) as both individual and combined nonlinearities, were also
119
Table 5.1: MI/SO Voltage Model Parameters (Hz).
Gain
1.01e13
H(f ) - initial
Zeros
Poles
-717.8 (-0.111±j1.470)e8
-8.538
-6.259
-2.190
-0.2588
-0.1428
-0.0267
H(f ) - final
Gain
Zeros
Poles
0.0550 -11.01
-8.738
-2.359 -0.2085
-0.1054 -0.0219
Gain
-0.1385
A(f )
Zeros
-18.94
-0.0529
Poles
-1.732
-0.1539
assessed, but only nominal improvement was seen, if any at all. So keeping with a simpler
form, the modeling will continue using the reverse MI/SO identification with v(t) = y 3 (t)
as the nonlinear form.
Using the identification results of Figure 5.2b, transfer function models were fit to
r
function invfreqs.m was used in this step with the
H(f ) and A(f ) next. The Matlab
identification results as the frequency response data and the coherence as the weighting
functions. Various numerator and denominator orders were fit to see which model matched
the identification results best. For the underlying linear system the best result came with an
initial fit having four zeros and five poles to the 0.5 mA-rms results. However, looking more
closely at these results it was discovered that one of the zeros and two of the poles were at
very high frequencies, well outside the range of interest here. This being the case, the H(f )
system model was reduced to three zeros and three poles, all first-order, by simply removing
these less effective frequencies from the model, adjusting the gain, and slightly moving the
remaining poles and zeros to account for the loss of the higher frequency parameters. This
r
function fminsearch.m from the optimization
process was simplified using the Matlab
toolbox that searches for a local minimum using the initial fit as the starting point. Table
5.1 has been included to show the parameters for each model, where the zeros and poles
listed are in Hertz. Examining the frequency response results, Figure 5.3a shows that both
the initial higher-order fit and the reduced model overlay the identification results. In this
figure, the dots represent the frequency points from the MI/SO identification, the dashed
line is the initial fit, and the solid line is the final third-order fit.
Modeling the frequency-dependence of the nonlinearity took place on the 3.5 mArms data because at higher amplitudes the nonlinearity has more effect (the identified
A(f ) systems are actually very similar at low frequency for both input levels). The model
120
Figure 5.3: MI/SO voltage modeling results: (a) underlying linear system H(f );
(b) frequency-dependence of nonlinearity A(f ).
matched well on the first attempt to the identification results with two first-order zeros and
two first-order poles given in Table 5.1. This model can also be seen in Figure 5.3b. Here
the dots again represent identification frequency points and the solid line represents the
second-order model fit. It can also be seen in the figure that the phase points appear to
jump around, particularly in the high frequency range, some of which is the result of phase
wrapping. This is not believed to cause any problem with the resulting model because it is
already known that the nonlinearity is more dominant at lower frequencies and the model
was fit using the partial coherence function as a weighting, which placed more emphasis on
the low frequency range. It can also be seen that the magnitude drops off at high frequency,
showing that it would have less impact on the output.
Now that the individual components of the overall nonlinear system have been identified and modeled, the next step is to simulate the system and compare the results with
those measured in the laboratory. In this comparison, the model was simulated to the measured input data. Figure 5.4a shows the magnitude, phase, and coherence for the measured
and simulated frequency response functions at 0.5 and 3.5 mA-rms. In this figure, the dotted line is the low amplitude data, the dash-dot line is the high amplitude data, the dashed
line is the low amplitude simulation, and the solid line is the high amplitude simulation.
An immediate observation is that the simulation model does not accurately predict the frequency response scaling seen in the magnitude and phase of the measured results. Although
121
Figure 5.4: MI/SO voltage modeling simulation results: (a) frequency response;
(b) time response.
this does not match well, one interesting result is that the coherence functions do match
very well with increasing the input level, but this could be due to increased signal-to-noise
ratio.
While the frequency response results do not accurately predict the measured response, the time responses could still conceivably match the distortion. Figure 5.4b has
these results at 3.5 mA-rms for 0.11 Hz (nonlinear region) and 11 Hz (linear region). The
measured response appears as the dashed line and the simulated response is the solid line.
It is again seen that in the frequency range where nonlinearity is prevalent, the model does
not accurately predict the measured response. Interesting to note here is that the frequency
response did not accurately predict the magnitude for 3.5 mA-rms, but the time response
is actually very close in amplitude to the measured data. It is the phase of the nonlinearity
that appears to be inaccurate here because the shape of the simulated waveform is completely different from the measured voltage, yielding 30% error. The high frequency results
do show good correlation between measured and simulated data, which is an indication
that if only higher frequencies are going to be considered, then a linear model would be
sufficient.
The results in Figure 5.4b only show the 3.5 mA-rms case, but it is worthwhile
to show how the model performs at different amplitude levels. To this end, the average,
122
Figure 5.5: Average percentage error for sinusoidal response of MI/SO simulation model (voltage).
absolute error percentage calculated as
ε=
E[|ydata − ymodel |]
× 100%,
ȳp,data
(5.2)
with ȳp,data the average peak value of the data, is plotted in Figure 5.5 for all eight frequencies and four amplitudes that were tested with sine wave inputs. The circles denote the
lowest amplitude results (0.5 mA-rms), then come the squares (2.0 mA-rms), the diamonds
(3.5 mA-rms), and the triangles (5.0 mA-rms). These error results again confirm that the
model performs well in the linear frequency range for all amplitudes tested, but does not
hold up as well when the nonlinearity is more effective. This is particularly seen in the
transition frequency range between linear and nonlinear dominance (∼0.5 Hz), where the
characteristic shape of the error changes.
Being that the reverse MI/SO model performed poorly in comparison to the measured response, this could mean that this particular parallel model structure is not best
suited to the electrical response of ionic polymer actuators. The next section will discuss
the modeling results where the nonlinearity affects the input to the underlying linear system.
123
Figure 5.6: DIN voltage modeling results: underlying linear system H(f ).
5.2.2
Impedance – DIN Identification
A model structure with a dynamic nonlinearity on the input, rather than in parallel with
the underlying linear system will now be explored. The first step in this identification and
modeling procedure is to fit a linear model to the frequency response function identified
r
function
from the low amplitude data. As in the previous modeling work, the Matlab
invfreqs.m was used here inputting the identified frequency response and coherence, and
varying the order of the numerator and denominator. However, in this technique it is
important that the model have relative degree zero because it must be inverted in a later
step.
Parameterizing a model to the identified frequency response again required that a
higher order be selected to get a good initial fit. Like before, a fourth-order numerator and
fifth-order denominator gave the best matching results. Looking at the poles and zeros of
this model, it was discovered that one of the poles was at a very high frequency, while the
other four poles and zeros were all in the dynamic range of interest. For this reason, and
because it is needed that r = 0, the high pole was simply removed and the gain was scaled
accordingly to arrive at the reduced model with four first-order zeros and four first-order
poles. Figure 5.6 shows that the final, reduced model (solid) completely overlays the initial
model (dashed), which both predict the identified frequency response (dots) very well at all
124
Table 5.2: DIN Voltage Model Parameters for H(f ).
Gain
2.01e15
H(f ) - initial
Zeros (Hz) Poles (Hz)
-5.602
-5.73e15
-2.726
-4.033
-0.8520
-1.378
-0.0622
-0.1120
-0.021
Gain
0.0558
H(f ) - final
Zeros (Hz) Poles (Hz)
-5.602
-4.033
-2.726
-1.378
-0.8520
-0.1120
-0.0622
-0.021
frequencies. Since the four poles and zeros of the final fit are identical to four of the poles
and zeros in the initial fit, it seems that eliminating the high frequency pole was a good
choice in reducing the model because it had no noticeable effect on the dynamics in the
frequency range of interest. The individual results for these models are provided in Table
5.2, where it can be seen that the removed pole was 15 orders of magnitude higher than the
next highest pole.
With the model of the underlying linear system in good working form, it can now be
simulated to a higher amplitude input sequence (3.5 mA-rms) where nonlinear effects are
present in the response. Once this is done, the linear model output is subtracted from the
measured output to obtain the nonlinear component of the output voltage. Simulating this
quantity to the inverse model H −1 (f ) then gives the input component of the nonlinearity.
Figure 5.7 has the first few seconds of these results for both the low amplitude (a) and the
high amplitude (b). In this set of results, the upper plots show the DIN inverse simulation
(solid) with the measured input (dotted) and the lower plots give the nonlinear component
of the input, xb,B
n (t). It can be seen in Figure 5.7a that the low amplitude inverse simulation
begins with a large error before quickly reaching near zero. Averaging the absolute value
of the nonlinear input component over the entire time period and comparing that to the
absolute value of the measured input current gives an average error of 5.0%. Performing the
same error calculation on the higher input results gives an average error of 10.8%, which
is double that of the low amplitude results. This is a sign that the higher input data does
contain nonlinear distortion.
The next step in the DIN process is to model the nonlinearity between the high
amplitude, measured input xB (t) and the nonlinear component of the input, xB
n (t), that
was determined from the inverse system simulation. The two nonlinear structures explored
125
Figure 5.7: DIN voltage modeling inverse simulations: (a) low amplitude; (b)
high amplitude.
here are a Hammerstein system and a Wiener system. Recalling the form of these systems
from Figure 4.10, the frequency-dependence of the nonlinearity in the Hammerstein and
Wiener system can be respectively identified from
AH (f ) =
AW (f ) =
SvxB
(f )
n
Svv (f )
SxB v (f )
SxB xB (f )
(5.3)
(5.4)
where, under the assumption of a cubic nonlinearity, v(t) = [xB (t)]3 for the Hammerstein
1/3 for the Wiener system. Results for these linear systems idensystem and v(t) = [xB
n (t)]
tified on the polymer actuator are given in Figure 5.8, shown with a dot at each frequency
point. It can be seen that the magnitude and phase of both systems follow similar trends
with the magnitude highest at lower frequencies and dropping off at high frequency, while
the phase is at nearly 180 degrees. Where the results differ is in the coherence functions.
The Hammerstein system of Figure 5.8a has values primarily in the range from 0.6 to 0.7,
while the Wiener system in Figure 5.8b is close to 1.0, especially in the low frequency range.
Since these coherence functions are a measure of linear correlation between their respective
inputs and outputs, this could be an indication that the Wiener model is more appropriate
for the polymer actuator. But to be certain, the modeling will continue on both systems
and the better model will be chosen from the simulation results.
Modeling of these linear systems was performed next using invfreqs.m for the least126
Figure 5.8: DIN voltage identification and modeling of nonlinear system: (a)
Hammerstein assumption; (b) Wiener assumption.
squares fitting. The coherence functions were again used as weighting, but it is likely that
the Hammerstein system will not benefit much from any low frequency weighting like the
Wiener system will. Fitting each system had good results using one first-order pole and one
first-order zero. For the Hammerstein linear system these were z = -38.96 Hz and p = -2.391
Hz, and for the Wiener linear system they were z = -44.99 Hz and p = -2.478 Hz. Being that
the magnitude and phase relationships are very similar for these two systems, it makes sense
that they have similar model parameters. Figure 5.8 shows the model predictions as the
solid lines overlaying the identified systems, indicating a good fit for both the Hammerstein
and Wiener models.
Now that the two different nonlinear system structures have been modeled, they can
be placed in the full model and simulations can be performed. The first simulations to
be compared are with mean-zero Gaussian current signals. Simulated frequency response
functions calculated for both the 0.5 mA-rms and 3.5 mA-rms signals are plotted with the
measured frequency responses in Figure 5.9. Results for the system with the Hammerstein
nonlinearity on the input are in Figure 5.9a and those with the Wiener nonlinearity on the
input are given in Figure 5.9b. In each of these sets of figures, the dotted and dash-dot
lines represent measured results for low and high input amplitudes, respectively, and the
model simulations are dashed for the low input and solid for the high input. For the low
amplitude, which has been assumed to be the linear case, both system simulations match
127
Figure 5.9: DIN voltage model simulation results of frequency response: (a)
Hammerstein assumption; (b) Wiener assumption.
well with the data, but increasing the input has different effects. The system with the
Hammerstein nonlinearity under-predicts the impedance shift seen in the data, while the
system with the Wiener nonlinearity over-predicts this shift. As far as general trends are
concerned, it appears that the Hammerstein system more closely matches the magnitude,
phase, and coherence of the data. However, this conclusion is in disagreement with the
identification results shown in Figure 5.8, where the coherence for the frequency-dependence
of the nonlinearity was much higher for the Wiener system than it was for the Hammerstein
system. It should also be noted here that these simulation results for both DIN systems
compare better with the measured frequency response data than the MI/SO model discussed
previously.
With conflicting conclusions thus far about whether the Hammerstein or Wiener
nonlinear assumption is better, it will be left to the sinusoidal time responses to determine
the best nonlinear model structure for the impedance of the ionic polymer actuator. To
examine the model performance in the linear and nonlinear frequency ranges, Figure 5.10
has been included. This figure displays four sets of results with an input of 3.5 mA-rms.
The two upper plots show results for the system with the Hammerstein nonlinearity and the
lower two plots show the results for the Wiener system. On the left side are low frequency
(0.11 Hz) results in the nonlinear region, whereas the right side shows the linear response
at high frequency (11 Hz). Also, data is presented as the dashed lines and the solid lines
128
Figure 5.10: DIN voltage model simulation results of time response.
are the simulation results.
Looking first at the higher frequency, the results are as expected. That is, the
model performs well in the linear region because the low-pass frequency-dependence of the
nonlinearity makes its influence negligible. The story is different in the nonlinear region,
however. Both simulation models have trouble accurately predicting the measured voltage
here. The Hammerstein system simulation is in phase with the measured voltage, but
the Wiener system is nearly 180 degrees out of phase. Not only is the phase of the overall
system incorrect for the Wiener system, but the phase of the nonlinearity is also inaccurately
predicted, as can be seen from the mismatch in the distortion shape of the waveform. While
the Hammerstein system over-predicts the voltage magnitude at low frequency (this was
expected from the frequency response simulation), the overall phase of the voltage output
and of the nonlinearity appear to be accurate.
Based on these results, the conclusion has been reached that the system with a
Hammerstein nonlinearity on the input best represents the voltage response of the ionic
polymer system. Although identifying the frequency-dependence of the input nonlinearity
favored the Wiener nonlinearity, this conclusion came about from comparing the full system
simulations to data in both the time-domain and frequency-domain. While the DIN model
with a Hammerstein nonlinearity does perform best, there is still room for improvement, so
129
Table 5.3: DIN Voltage Model Parameters for A(f ).
A(f ) - initial
Gain
Zeros (Hz) Poles (Hz)
-4.212e-4
-39.53
-2.391
Gain
-4.212e-4
A(f ) - final
Zeros (Hz) Poles (Hz)
-39.53
-2.391
-2.550
-1.387
-0.00196
-0.00928
modifications made to this system model will be discussed next. It is also worth mentioning
that choosing a model here with r = 0, allowing for inversion, is actually quite safe from
poor performance. In some systems there could be problems with simulations matching
measurements due to unmodeled, high frequency dynamics, but the impedance of an ionic
polymer actuator levels off and is flat at high frequencies, where its electrical characteristics
behave like a resistor.
5.2.3
Impedance Model Improvements
By performing simulation studies on the various model structures, it was determined that
the impedance of an ionic polymer actuator is best modeled by an underlying linear system
with a nonlinearity on the input in the form of a Hammerstein system. Since the simulation
results showed that some work could still be done to improve the performance of this model,
some approaches to this effect will now be explored.
Because the model was constructed from data used to calculated the frequency response and because simulation results showed an over-prediction of the low frequency amplitude, the first place chosen to improve the model was the magnitude scaling of the frequency
response. The simulated frequency response at low amplitude predicted the measured response well, but it was the response at the high excitation level that suffered. Being that the
frequency-dependence of the nonlinearity was also modeled from the high input data, it is
this system that will be modified first. The new model for A(f ) was parameterized using the
initial model from the DIN identification as a starting point and adding two new poles and
zeros. The iterations were performed using fminsearch.m, where the objective function was
the frequency response error between the measured and simulated systems for the 3.5 mArms input. Table 5.3 shows the model parameters, and Figure 5.11a shows the frequency
response of the identification data (dots), the initial model (dashed), and the final model
130
Figure 5.11: DIN voltage model improvements: (a) frequency-dependence of
nonlinearity; (b) linear system in direct input path.
(solid). Noting that the gain of the model is negative, it makes sense that the optimization
increased the low frequency magnitude because this would reduce the over-prediction of the
initial model in this range.
This modified model of A(f ) allowed the frequency response to match very well for
the higher input, but when examining the effect this new system had on the sinusoidal
response, it was discovered that more work still needed to be done. The error was still low
at the higher frequencies (< 10%), but the error was near a factor of 2 (∼100%) at the
lowest frequency. Since the inaccuracy remaining was an over-prediction at low frequency,
a linear system B(f ) was inserted in the path between the applied input and the addition of
the nonlinear input xn (t) to form the nonlinear input to the underlying linear system. This
new system acts directly on the input. Recalling the DIN methodology, the nonlinear input
that passes through the underlying linear system is the sum of the applied input itself, x(t),
and a nonlinear component, xn (t), so including B(f ) in the model affects x(t) directly to
decrease the low frequency magnitude. This system was designed to behave like a high-pass
filter with a first-order zero at -0.1019 Hz, a first-order pole at -0.1273 Hz, and a gain of
1 (see Figure 5.11b). These values were also determined from iteration, where the initial
guess was determined from performing a direct MI/SO analysis from the applied input to
the nonlinear input. Throughout the iteration, both the frequency response results and
the sinusoidal error plots were examined because a trade-off existed between matching the
131
Figure 5.12: DIN voltage model improvements: (a) frequency response; (b)
sinusoidal response.
sinusoidal data or matching the frequency response data. The final model parameters were
chosen in the middle, so results in both the time- and frequency-domain are not as good as
they could have been individually.
Figure 5.12 displays the improved voltage model results of both the frequency response and the sinusoidal responses that validate the form. The frequency responses in
Figure 5.12a show the low amplitude data as dotted, the high amplitude data as dashdotted, the low amplitude simulation as dashed, and the high amplitude simulation as the
solid line. It can be seen in this figure that the relative scaling between low and high inputs
for the data and model appear to be close, but the actual magnitude and phase do not overlay one another. It was mentioned before that this mismatch was the cost of achieving more
accurate sinusoidal predictions from the model. These results are given in Figure 5.12b,
with the individual plots showing the dotted measured response with the solid simulated
response. The top two plots are at 0.5 mA-rms, before distortion is present, and the bottom
plots are at 3.5 mA-rms, after the nonlinearity affects the waveform. This helps show that
the model is able to capture the amplitude-dependence of the nonlinearity. The left side is
in the nonlinear region at 0.11 Hz and the right side is in the linear region at 11 Hz. In
this linear region, the model predicts the measured response well, but at lower frequencies,
while capturing the distortion shape, the magnitude is still over-predicted. To show that
the distortion is well represented by the model, an inset in the lower left plot was included
132
Figure 5.13: DIN voltage model improvements – sinusoidal response error: (a)
overall percent error; (b) amplitude and phase metrics.
in Figure 5.12b, showing that when normalized to their respective peak amplitudes, the
model output and measured voltage overlay each other.
To see how this improved voltage model performs over all the frequency and amplitude combinations tested, Figure 5.13 has been included. The results in Figure 5.13a show
the average, absolute error defined in equation 5.2 for each data set. The circles represent
the lowest amplitude results, where they are seen to have the lowest error because the system response is predominantly linear at this current level (0.5 mA-rms), as well as in the
high frequency range where the error for all amplitudes is below 10%. Squares show the
error points at 2.0 mA-rms, diamonds represent the 3.5 mA-rms simulations, and triangles
denote the error at the maximum input level tested of 5.0 mA-rms. At first glance, the error
does still seem high at 0.11 Hz. This error could have been brought down much lower, but
tuning the model to match this sinusoidal data any more closely was much more detrimental
to the simulated frequency response prediction. However, it should be mentioned that the
0.11 Hz error for the 2.0 and 3.5 mA-rms data has been lowered significantly from where it
was before including B(f ) in the model (∼100%). Also interesting to note is how the error
for 5.0 mA-rms at 0.11 Hz is well below that of 2.0 and 3.5 mA-rms. This error tends to
follow the same trend as the other input levels until the lowest frequency is reached and it
drops off. Reasons for this are unclear, but it could be an indication that the model will
hold up better to increased levels of input excitations.
133
Looking at some other error metrics may also offer insight into the final model results,
so Figure 5.13b has been included. The data sets are represented by the same symbols as
in Figure 5.13a, but the upper plot here shows the signed amplitude error and the lower
plot shows the correlation coefficient between each data set and simulation result. The
amplitude error was calculated according to
εA =
ȳp,data − ȳp,model
× 100%,
ȳp,data
(5.5)
where ȳp,data is the average peak value of the data and ȳp,model is the average peak value
of the model. Containing the sign of the error, this plot shows that the error due to
magnitude differences is the result of over-prediction, as was seen in Figure 5.12b. The
correlation coefficient, ρ, helps give some sense of the phase error involved in the model
prediction. Ranging from −1 ≤ ρ ≤ 1, perfectly correlated, and in phase, signals will have
a value of +1, while perfectly correlated signals 180 degrees out of phase will have a value
of -1. The results in Figure 5.13b give an indication that the model predicts the measured
phase well. Due to the distortion and multiple frequencies in the response, this concept
loses some meaning, but as a point of reference, two perfect sine wave with 10 degrees of
phase difference give ρ = 0.98 and with 30 degrees of difference, ρ = 0.87. Being that the
displayed results are all greater than 0.98, this does provide support that the model can
predict the phase.
Now that the final form of the voltage model has been discussed and validated, along
with its modifications for improved output prediction, this section will come to a close. The
next section will cover the tip velocity modeling of the cantilever polymer actuator.
5.3
Modeling the Mechanical Response
With the voltage model complete, the deformation will now be modeled. This section will
again discuss results from the two different identification techniques detailed in the previous
chapter. The mechanical system is described with current as the input and tip velocity as
the output. Like in the beginning of the previous section, this modeling will begin with an
illustration of the nonlinearity in the response. Figure 5.14 provides the evidence for the
need to construct a nonlinear velocity model. Scaling in the frequency response is shown in
Figure 5.14a, where the dotted line is the lower input of 1.5 mA-rms and the solid line is
the higher input of 3.5 mA-rms. It can be seen that the frequency responses overlay each
134
Figure 5.14: Evidence of velocity nonlinearity in EMI-Tf sample: (a) frequency
response; (b) time response.
other over most of the frequency range, but begin to separate below 0.1 Hz. Opposite to the
voltage response that decreased with increasing current, the low frequency velocity increases
with the current. Looking at how this nonlinearity affects the time response, Figure 5.14b
has been included. Here, both results are at the highest amplitude of 7 mA (5.0 mA-rms).
The left side shows the large harmonic distortion seen in the low frequency range at 0.11
Hz, and the right side plots show the linear region at 11 Hz.
It should be noted that results from the lowest input of 0.5 mA-rms were not included
here because the velocity response was very slow at this input level and the signal strength
suffered somewhat. Therefore, the data to be used as the low input in this section is that
from 1.5 mA-rms. This will be used in the modeling to follow.
5.3.1
Deformation – MI/SO Identification
The multi-input, single-output identification technique will be applied first to the velocity
response. Based on the characterization results, there was no single nonlinearity that appeared more dominant, so the identification here will look at a system arrangement with
both quadratic and cubic nonlinearities. After trying individual and combined scenarios,
the best results were obtained with two individual nonlinearities. This corresponds to having two different parallel nonlinear systems, with the primary nonlinearity being a square
law and the secondary nonlinearity being a cubic law. This ordering was determined from
135
Figure 5.15: MI/SO velocity results with quadratic and cubic nonlinearity: (a)
direct; (b) reverse.
the cumulative coherence, which was higher when only a quadratic nonlinearity was in the
model than it was when only a cubic nonlinearity was in the model. While this was the
case, the identification results in Figure 5.15 show that the cumulative coherence did not
improve by much for either the direct or reverse system structures. This is an indication
that either the nonlinear assumption is incorrect or the parallel system structure may not
accurately capture the dynamics.
The direct system results are given in Figure 5.15a and the reverse system is shown in
Figure 5.15b. In both sets of results, the dotted line is the optimal linear system Ho (f ), the
solid line is the underlying linear system H(f ), the dashed line is the frequency-dependence
of the primary (quadratic) nonlinearity A1 (f ), and the dash-dot line is the frequencydependence of the secondary (cubic) nonlinearity A2 (f ). Knowing that the cumulative
coherence shows little improvement, it can also be seen that the magnitude of the identified
nonlinear systems is at least two orders of magnitude below that of the underlying linear
system and the phase points show no discernable trend for model parameterization in either
the direct or reverse MI/SO results.
One more check will now be performed on this data before making any decisions on
how to proceed with the MI/SO modeling effort. This involves examining the autospectral
functions of the different output signals of the model structures. Recalling from the MI/SO
example in the previous chapter, the noise spectrum can be compared to that of the nonlin136
Figure 5.16: MI/SO autospectral velocity results with quadratic and cubic nonlinearity: (a) direct; (b) reverse.
earities to look for frequency ranges where the nonlinearities have an impact on the system
response (larger magnitude than the noise). Figure 5.16a shows these results for the direct
system arrangement and Figure 5.16b has the results for the reverse structure. In each
set of figures, the dark solid-dotted line represents the measured output, the solid line is
the linear system, the dashed line is the primary nonlinear system, the dash-dot line is the
secondary nonlinear system, and the light solid-dotted line represents the noise, or model
uncertainty. The difference between the upper and lower sets of plots is that the upper
shows the optimal system with uncorrelated inputs and the lower shows the actual system
with correlated inputs. It can be seen in all four plots that the magnitude of the noise
spectrum is larger than that for either of the two nonlinear systems, indicating a problem
between the model assumptions and the actual system.
Based on the autospectral results in Figure 5.16, those of the identified systems in
Figure 5.15, and the voltage model comparisons that showed the DIN model structure outperforming the MI/SO structure, it has been concluded that the parallel system structure
of the multi-input, single-output identification technique is not sufficient to model the nonlinear response of ionic polymer actuators. Because of this, model development with this
method will now be stopped and the structure with a dynamic nonlinearity on the input of
a linear system will be explored.
137
Figure 5.17: DIN velocity modeling results: underlying linear system H(f ).
5.3.2
Deformation – DIN Identification
The first step in this methodology is to identify and model the underlying linear system
using data from an excitation level that has a predominantly linear response. In this case,
this is the 1.5 mA-rms random signal. Moving along with the linear identification and
relative degree zero modeling, Figure 5.17 shows the results. In order to accurately model
the higher frequency modes and lower frequency shape, an initial fit of ten poles and ten
zeros was required using invfreqs.m. This initial tenth-order model appears in the figure
as the dashed line that overlays the identified system shown with the dots very well except
below approximately 0.1 Hz, where it begins to over-predict the magnitude. Taking a closer
look at the model parameters in Table 5.4, it can be seen that some of the poles and one of
the zeros are well outside of the dynamic range of interest. This, along with the inability
for the model to match the low frequency response, which is especially of interest in this
nonlinear modeling, means that the initial model had to be modified.
Reducing and modifying the initial H(f ) model was accomplished by removing the
highest pole and zero from the model, fixing the two most dominant second-order poles
and zeros, and iterating on the remaining first-order model parameters. As in the electrical
modeling section, the multi-dimensional, unconstrained, nonlinear optimization (NelderMead) function fminsearch.m was used. The resulting model shown in Figure 5.17 as the
138
Table 5.4: DIN Velocity Model Parameters for H(f ).
Gain
1.88e9
H(f ) - initial
Zeros (Hz)
Poles (Hz)
-4.297±j49.69 -3.415±25.43
-30.72±j25.91 -0.4679±11.10
-2.623±j21.93
-1.934e8
-8.047
-3.288e4
-1.558
-795.5
0.0271
-9.737
121.7
-2.201
-0.4122
Gain
-0.1920
H(f ) - final
Zeros (Hz)
Poles (Hz)
-4.297±j49.69 -3.415±25.43
-2.623±j21.93 -0.4679±11.10
-31.34
-90.00
-6.490
-7.122
-1.590
-2.287
-0.0167
-0.3948
0.0152
-0.0395
solid line accurately predicts the measured response over the entire frequency range. There
is a small phase mismatch after the second mode, but it is believed that this will not cause
any problems in the range of frequencies for which sinusoidal data was collected. As a
check, the final model was simulated to the random input sequences at 1.5 mA-rms and 3.5
mA-rms and the average, absolute error was computed to be 16% for the low amplitude and
17% for the high amplitude. This shows that the linear model performs well, but because
there is not much difference between the error from the low and high input levels, this could
be an indication that the random data collected does not span a large enough input range
to accurately portray the mechanical nonlinearity. Based on this finding, when validating
the model with sinusoidal inputs, the highest input level of 5.0 mA-rms will be used.
A more serious problem that can be seen with this model is apparent from looking
at the model parameters listed in Table 5.4. This is the non-minimum phase zero at low
frequency that is required in the model. Considering that the next step in this methodology
is to invert this underlying linear system model, having a non-minimum phase zero will result
in an unstable inverse system, which means that the inverse simulation of the output cannot
be completed as this method stands.
A solution to this problem can be found using the sinusoidal data with this nonminimum phase linear system, however. While the random output signal cannot be inverse
simulated with the model because of stability issues, the sine wave data can be, in a sense,
but with a loss of frequency resolution. By decomposing the measured tip velocity at each
frequency and amplitude into its most dominant components (linear, quadratic, and cubic)
and evaluating the linear system at each of these frequencies to determine the magnitude
139
Figure 5.18: Example of DIN inverse system augmentation using Fourier transform components.
and phase, the distorted input to the underlying linear system can be estimated. This will
be shown for an example here to help with the explanation.
Assuming that an output signal y(t) contains distortion from a quadratic and cubic
nonlinearity, calculating its Fourier transform will give the amplitude and phase of each
of these components as a complex number, from which the original signal could be reconstructed (via the inverse Fourier transform). And since the sought input signal is a linear
transformation, based on the non-minimum phase system, it too can be reconstructed using
the Fourier transform characteristics with the magnitude and phase relations of the linear
system. The following equations help in showing how this can be done:
|Y (ω)|
cos (ωt − Y (ω) − H(ω))
|H(ω)|
|Y (2ω)|
cos (2ωt − Y (2ω) − H(2ω))
x2 (t) =
|H(2ω)|
|Y (3ω)|
cos (3ωt − Y (3ω) − H(3ω))
x3 (t) =
|H(3ω)|
x(t) = x1 (t) + x2 (t) + x3 (t).
x1 (t) =
(5.6)
(5.7)
(5.8)
(5.9)
Here, |Y (·)| and Y (·) are the amplitude and angle of the measured output velocity estimated from the Fourier transform, |H(·)| and H(·) are the magnitude and phase of the
linear system evaluated at each multiple of the excitation frequency ω, and x(t) is the distorted input that passes through the underlying linear system to produce the measured
output y(t). Each xi (t) corresponds to the linear, quadratic, and cubic nonlinearities, respectively. It should be noted that the division of |H(·)| can take place because it is simply
a non-zero number resulting from evaluating the model at a particular frequency point.
Figure 5.18 has been included to visually show how this signal decomposition performs on the most distorted case (7 mA, 0.11 Hz). The plot on the left is what was calculated
140
to be the nonlinear input to the underlying linear system that causes the distortion in the
measured output. This measured output is shown on the right side as the dotted line, along
with the simulated output resulting from the calculated input. It can be seen that these
results match well, but not perfectly. The reason they do not overlay each other exactly
is that only the three most dominant frequencies of the Fourier transform were considered here. This error could be lowered by including more terms, but for the sake of the
present model development, only frequencies resulting from quadratic (2ω) and cubic (3ω)
nonlinearities will be included with the fundamental (ω).
Being that the non-minimum phase zero altered the DIN methodology with the
inversion, the remainder of the identification will also be augmented since only individual
sine wave data can be used in the development. At this point in the identification procedure,
assumptions of the nonlinearity are made. Based on the velocity characterization results,
both quadratic and cubic nonlinearities will be included. Proceeding with the method,
principles from the MI/SO technique will now be employed to aid in the identification
of the frequency-dependence of the nonlinearities. This is the idea of identifying linear
systems using uncorrelated input signals. The individual complex components of x(t) at ω,
2ω, and 3ω are known from equations 5.6–5.8 and can now be used with knowledge of the
nonlinearities to estimate their frequency-dependence.
The results to follow will look first at the Hammerstein system since the analysis
has been somewhat complicated by the fact that the DIN methodology cannot be applied
directly and because there are two nonlinearities under consideration. And if this nonlinear
form does not yield an acceptable model, then the Wiener system will be considered. The
Hammerstein system was chosen as the first nonlinear form to try here because it was this
form that was most successful with modeling the voltage response.
Applying the nonlinearities determined from the characterization results, gives v1 (t) =
x2 (t) and v2 (t) = x3 (t). Now taking the Fourier transform of these nonlinear signals will
yield the complex relations for the amplitude and angle at the harmonic frequencies of 2f
and 3f from v1 and v2 , respectively. These relations can then be used with the Fourier components from the “inverse” simulations of the output through the underlying linear system
to identify their associated frequency-dependencies. This identification process is just like
a sine-dwell test, but the with far fewer frequency points. Subtracting the measured input
from the inverse-simulated nonlinear input gives the nonlinear portion xn (t) from which
141
Figure 5.19: DIN velocity identification and modeling of nonlinear systems: (a)
quadratic frequency-dependence; (b) cubic frequency-dependence.
the harmonic components are then extracted. This signal can be separated into a portion
from the quadratic nonlinearity, xn1 (t), and a portion from the cubic nonlinearity, xn2 (t),
by separating the harmonic content. Only harmonic content at 2f will be used to identify
the linear system between v1 (t) and xn1 (t), denoted A1 (f ), and only harmonic content at
3f will be used in identifying the frequency-dependence A2 (f ) between v2 (t) and xn2 (t).
This is where the concept of uncorrelated inputs is employed because v2 (t), for example,
will also have frequency content at f , but it is not considered in identifying A2 (f ).
Figure 5.19 shows the results of these identified frequency points as circles, along
with the modeling results as solid lines. It can be seen that both the quadratic frequencydependence in Figure 5.19a and the frequency-dependence of the cubic nonlinearity in Figure
5.19b have very similar trends that fit well with one first-order pole and zero. The low-pass
filter characteristic of each system was also the expected result based on the magnitude
scaling seen in Figure 5.14a that occurs only at low frequency. These models were parameterized with invfreqs.m and did not require additional reduction or modification after the
first attempt to match the identification results shown. The specific parameters for these
models can be found in Table 5.5.
Now that the model has been fully parameterized, simulations can be conducted to
test its performance. Figure 5.20 displays these results. The frequency response results
are shown in Figure 5.20a, where the dotted line represents the low amplitude data, the
142
Table 5.5: DIN Velocity Model Parameters for Ai (f ) (Hz).
Gain
Zero
Pole
A1 (f )
-2.80e-3
-3.304
-0.1184
A2 (f )
-4.39e-4
-10.17
-0.3069
Figure 5.20: DIN velocity modeling simulation results: (a) frequency response;
(b) sinusoidal response.
dash-dot line represents the high amplitude data, the dashed line shows the low amplitude
simulation, and the high amplitude simulation is shown by the solid line. It is seen here that
all of the results match well in the linear frequency range above 1.0 Hz, but down below
0.1 Hz, where the nonlinearities are known to have the greatest influence, the magnitude
scaling of the model under-predicts the measured response. The trend of increasing velocity
with increasing current is reproduced, however. It should be noted here that based on an
initial inspection of the simulated response, the signs of the gains shown in Table 5.5 were
reversed.
Turning attention now to the sinusoidal time responses of Figure 5.20b, it is shown
that the magnitude is predicted well for both frequencies at 5.0 mA-rms. The dashed line
depicts the measured data and the solid line shows the simulation response in this figure,
with input current in the upper plots and the cantilever tip velocity in the lower plots. The
results on the left side display the nonlinear response region at 0.11 Hz. Here, the model
143
Figure 5.21: DIN velocity model error.
response does portray some of the distortion features seen in the data, but there appears
to be a mismatching phase issue in the nonlinearity because the harmonics do not impact
the response at the same point along in each cycle as is seen in the measurement. The fact
that the shape is approximated indicates that the chosen model structure is appropriate for
the velocity model, however. On the right side, the plots show the response in the linear
region at 8.25 Hz, where both the magnitude and phase of the response are predicted well
by the model.
It should be noted here that the frequency chosen for the linear results has changed
from 11 Hz, which was used in all the previously shown results. This was done because
the error at 11 Hz was very large for these results. Checking back with simulations of only
the underlying linear model, the same large error was again seen, pointing out that the
error is not an artifact of the nonlinear modeling. Since 11 Hz is very near the first bending
resonance, it is believed that the large error is associated with an error in the data collection.
That is, the measurement point may have moved slightly between the experiments with
random excitations and sinusoidal excitations. Being that the slope of the magnitude and
phase is very steep near the resonance, any small change in the measurement point location
would have a large impact on the measured tip response. Therefore, it is assumed that this
happened here and that the model, as described, is in proper working order.
144
This large error near resonance is better shown in Figure 5.21, which shows the
average, absolute error for all frequencies and amplitudes tested. The smallest amplitude of
0.5 mA-rms is denoted by the circles, the 1.5 mA-rms input by the squares, the 3.5 mA-rms
input by the diamonds, and the largest input of 5.0 mA-rms is shown by the triangles. As
mentioned, there is a large error near the resonance at 11 Hz, but other than that, the
model performs as expected. The error is low in the higher frequency range because the
response is predominantly linear, but then the error increases as the frequency lowers into
the nonlinear region. Although the model does account for nonlinearities, the error is still
greater in the low frequency range because of the noted phase mismatch.
Based on these results, it was concluded that the DIN system arrangement with two
Hammerstein nonlinearities (quadratic and cubic) can be used to accurately predict the tip
velocity response of the ionic polymer actuator, so there is no need to consider the Wiener
system arrangement. But there was some room for improvement in the output prediction,
so the full velocity model will be modified next.
5.3.3
Deformation Model Improvements
For the velocity model, sinusoidal results showed that there appears to be an incorrect phase
prediction in one or both of the nonlinearities. Also since this model was constructed primarily from the sinusoidal harmonic data, the frequency-dependence of these nonlinearities
will be the starting point for improving the deformation model. This modification began
by manually shifting the poles and zeros of each Ai (f ) to see how the shape of the response changed. It was determined from these adjustments that the phase of the quadratic
nonlinearity had the greatest effect on improving the prediction of the distortion.
With this knowledge, optimization routines were run on the model parameters of
A1 (f ) to minimize the error between the measured data and simulation response at 5.0
mA-rms and 0.11 Hz. However, these results showed that the minimum error occurred when
an undistorted wave fit through the data. This meant that adjusting the quadratic term
alone would not match the distortion, so the optimization was performed again including
the model parameters for A2 (f ), but similar results were obtained this time as well. This
is an indication that possibly more harmonic terms may be required for this model to more
accurately predict the measured distortion. This being the case, several manual iterations
were performed from the optimized results to put the appropriate distortion back in the
145
Figure 5.22: DIN velocity model improvements: (a) frequency-dependence of
quadratic nonlinearity; (b) frequency-dependence of cubic nonlinearity.
Table 5.6: DIN Improved Velocity Model Parameters (Hz).
Gain
0.68
B(f )
Zeros
-25.00
Poles
-17.00
Gain
0.3020
A1 (f )
Zeros
–
Poles
-1.000
-0.0136
Gain
0.9019
A2 (f )
Zeros
–
Poles
-5.001
-0.3232
simulated response. Once this was done, the higher frequency zero was replaced with an
extra pole in each Ai (f ) before the resonance, while negligibly affecting the lower frequency
response. This was done to ensure that any contribution from the nonlinear components
would not impact the linear response region near and above the first resonance.
Figure 5.22a shows the initial (dashed) and final (solid) models for the frequencydependence of the quadratic nonlinearity, along with the identified system (circles), and
Figure 5.22b shows the same for the cubic nonlinearity. The effect of the added pole can be
seen from the decrease in magnitude and phase at high frequency. It should be noted that
the phase of the final models is 180 degrees off from the initial models because, as noted
previously, the signs of the gains were reversed to allow the model better output prediction.
The actual parameters of these final models are provided in Table 5.6, where in comparison
to Table 5.5, it can be seen that the gains of each Ai (f ) have the opposite sign.
Another area that could be improved is the error in the higher frequency range.
146
Figure 5.23: DIN velocity model improvements: (a) frequency response; (b)
sinusoidal response.
While the believed cause for this error was already mentioned, it could be possible to
improve this response region by inserting a linear system in the direct input path of the
nonlinear input system. This is the same as the added B(f ) system in the voltage model.
Like in the previous case, this new system has one first-order pole (-17.00 Hz) and one
first-order zero (-25.00 Hz), but opposite to B(f ) in the voltage model, this time the system
has a low-pass effect to reduce the higher frequency magnitude where the error is larger.
These model parameters were obtained through iteration, while attempting to lower the
high frequency error without adversely affecting the lower frequency response. Table 5.6
also displays the parameters of this filter. It should be noted that since predicting the
frequency response is also a goal of the modeling effort, no modifications will be made to
the identified underlying linear system H(f ).
Having discussed the model improvements to the velocity model, Figure 5.23 provides validation of the model structure, showing how the performance has increased. In
Figure 5.23a, the dotted and dashed lines represent the low amplitude data and simulation,
respectively, while the dash-dot and solid lines represent the same for the high amplitude.
The model predicts the response well in the linear range up to approximately 20 Hz, where
the addition of B(f ) can be seen to lower the magnitude and phase slightly. In the low
frequency region, magnitude scaling is present in the simulated response, but it still does
not exactly overlay the measured response. Because the simulated trend is consistent with
147
Figure 5.24: DIN velocity model improvements – sinusoidal response error: (a)
overall percent error; (b) amplitude and phase metrics.
the measurements and because the sinusoidal response can also predict the distortion well,
this minor discrepancy is not a major concern of the final model.
These sinusoidal time responses are given in Figure 5.23b. Here the dashed line
shows the data and the solid line depicts the simulation results. The upper plots show
results from a low input (2.0 mA-rms) and the lower plots show the tip velocity response of
the highest current signal at 5.0 mA-rms. As before, the left side shows the response where
nonlinearities prevail (0.11 Hz) and the right side plots are in the linear response regime
(8.25 Hz). The simulated distortion looks similar to the initial result of Figure 5.20b,
but the noticeable ripple on the leading side of the waveform more closely predicts the
phase of the measured nonlinearity. Looking at the linear response, the phase is also more
accurately predicted here, but the magnitude is slightly under-predicted. Also comparing
the results of low input to high input at the low frequency, it can be seen that the model can
track the progression of the nonlinearity with increasing amplitude because the upper plot
mostly resembles an undistorted sine wave, while the lower plot contains much distortion.
It should be mentioned here that truncating the Fourier series after the cubic frequency for
model construction did not capture the ripple on the trailing side of the waveform, so it is
reasonable that the model cannot predict this characteristic (recall Figure 5.18).
The overall improvement to the sinusoidal inputs can be seen in Figure 5.24a. Here,
as in Figure 5.21, the circles show the average, absolute error at the lowest amplitude (0.5
148
mA-rms), followed by the squares, the diamonds, and the triangles at the highest input
(5.0 mA-rms). It can be seen immediately that adding the system B(f ) has proved useful,
as it has effectively lowered the high frequency error by more than 10%. Improvements
from the other modifications can also be seen since the error for all amplitudes is much
closer together in the linear region than it was initially and the error at 0.11 Hz for 5.0
mA-rms has been brought down below 25%. As in the voltage modeling results, both the
individual amplitude error and correlation coefficient are given in Figure 5.24b with the
same symbols of Figure 5.24a. The amplitude error results show that the velocity model
does well predicting the measured response since most values are less than 10%, and all
values are below 21%. There is an indication that the phase error of the model is much
larger than in the voltage model. The prediction is accurate in the linear frequency region,
but as was seen in Figure 5.23b, the phase of the nonlinear terms introduces some error.
Here, the correlation coefficient varies between 0.85 and 0.95 at the lowest frequency, where
it was greater than 0.98 in the voltage model.
This concludes the discussion of the velocity model development and its validation.
Now that both the electrical and mechanical models have been constructed and validated,
the next section will make some general concluding comments about the overall modeling
process and results.
5.4
Modeling Comments
Throughout the modeling procedure for both the electrical and mechanical responses, modifications were made to the final forms to improve the simulation performance. As this
modeling work comes to a close, Figure 5.25 has been included to show the final structure
of each system. The voltage model is shown in Figure 5.25a and the final velocity model
form is provided in Figure 5.25b. Both final model structures are augmentations of the
DIN methodology, but these figures show that each of the final models actually has aspects
of the MI/SO methodology as well. This can be seen if the block structures are separated
just before the underlying linear system H(f ), where the portion directly connected to the
input x(t) replicates the parallel MI/SO structure. And while neither of the models used
this parallel system structure directly, each model development discussed how concepts of
the two methods were incorporated together to obtain the final forms.
149
Figure 5.25: Final nonlinear model forms: (a) electrical system; (b) mechanical
system.
In regard the nonlinear characterization results of the EMI-Tf sample, some parallels
can be drawn to the model development. Recall the harmonic ratio analysis of Figure 3.4.
This plot showed the amplitude ratio of the harmonic components (2f and 3f ) to the
fundamental (f ) for voltage and velocity. In the voltage results for the quadratic term,
there was no discernable trend with frequency for where any nonlinearity was triggered,
but in the cubic term there was a sharp increase as the frequency lowered from 0.55 Hz to
0.11 Hz. Considering the voltage model, these findings are evident since no quadratic term
is necessary to predict the distortion in the measured waveform and the magnitude of the
frequency-dependence of the cubic nonlinearity is highest at 0.11 Hz, and rolls off as the
frequency increases. The same can be said for the frequency-dependence of the nonlinearities
in the velocity model. Characterization results showed fairly equivalent contributions from
quadratic and cubic nonlinearities (both were used in the model) and the magnitude of
their frequency-dependence again rolls off at frequencies above 0.11 Hz.
Also notable is that the nonlinear elements for the electrical and mechanical response
were both modeled as Hammerstein systems. While no physical model for these nonlinearities exists, it was through simulation comparisons with other nonlinear forms, including
Wiener models, that indicated that the Hammerstein system best represented the response
of the systems. Selection of this form may, therefore, offer insight for physical models
150
of the nonlinear response of ionic polymer materials by placing the frequency-dependent
parameter after the nonlinear function that operates on the applied current signal.
One final comment will now be made about the characterization method and the
most effective model structure. Using low order Volterra series was shown to accurately
predict the harmonic distortion in the voltage and velocity response of the ionic polymer
actuators. Being that the Volterra representation is a polynomial expansion of the input, it
seems quite reasonable that the model structure with a dynamic nonlinearity on the input
predicted the measured response better than structures where nonlinearities contribute
through a parallel connection, bypassing the underlying linear system (direct MI/SO), or
by acting on the output directly (reverse MI/SO).
5.5
Summary
This chapter has discussed the modeling of the electrical and mechanical responses of a
cantilever ionic polymer actuator in the free deflection boundary condition. Two different
identification techniques were employed to develop a model for both the voltage and tip
velocity of the actuator, where results of each were compared to arrive at the most appropriate candidate form. The ideal model structure for both system responses was determined
to be an underlying linear system with a dynamic nonlinearity on the input (DIN) in the
form of a Hammerstein system. A cubic nonlinearity was required for the voltage model
to accurately predict the measured response, while cubic and quadratic nonlinearities were
needed to accurately predict the nonlinearity seen in the velocity. The identified frequencydependence of each nonlinearity was also determined to be most dominant at low frequencies
before rolling off at higher frequencies, where the response is predominantly linear. Model
improvements and iterations were then conducted on the identified systems to optimize
their performance, and the final forms were validated against frequency response data and
time series sinusoidal data.
151
Chapter 6
Position Control of a Square-Plate
Actuator
To demonstrate the potential that ionic polymer transducers have in shape control applications for flexible structures, such as deformable mirrors, this chapter will consider a
square-plate polymer actuator. Building on the analyses employed in the previous chapters,
the nonlinear response between input current and the deflection of the center-point will be
characterized and a model will be developed in the first section. After the final form of the
model has been validated against measured data, a controls investigation will be performed
to assess the closed-loop performance of the system, and a section is dedicated to comparing
closed-loop simulations to experimental results.
6.1
System Model Development
In this section, the development of the polymer plate actuator will be described. Leading
up to the modeling, the first section will detail the experimental setup and data collection,
and the second section will discuss the nonlinear characterization.
6.1.1
Data Collection
The data collected for model construction of the polymer plate was similar to that discussed
before. A dSPACE DS 2103 digital-to-analog converter (14-bit, ±10 V) was used to supply
the input signal, which then passed through a transconductance amplifier, powered by an
Agilent E3648A DC power supply. The resulting signal was then split and applied to tabs
152
Figure 6.1: Laboratory setup for plate actuator experiments.
manufactured on each of the four corners of the plate actuator. This was done to ensure
that the polymer was actuated as uniformly as possible, while still allowing freedom on the
boundaries to produce a strong displacement signal. The fixture was also designed with
four movable clamps with gold foil electrodes. The displacement of the center point of the
(33.5 x 34.5 x 0.3)mm plate was measured with a Polytec OFV 303 sensor head and 3001
vibrometer controller, where the sensitivity was adjusted for each test to maximize usage
of the A/D range. All measured signals (current, voltage, and deflection) were collected
with a dSPACE DS 2003 analog-to-digital converter (16-bit, ±10 V). Figure 6.1 has been
provided to show this testing equipment.
r
as the ionomer, the ionic liquid
The polymer sample was fabricated with Nafion
EMI-Tf as the solvent material, lithium as the cation, and a composite electrode. A first
r
and the outer layer is made from
layer of the electrode is created with RuO2 and Nafion
hot pressing a gold leaf onto each side of the sample (Akle et al., 2004). As mentioned with
the experimental apparatus, tabs extended from each of the four corners of the sample to
provide spaces to clamp and actuate the polymer. These tabs can be seen on a picture of
153
Figure 6.2: Close-up of polymer plate actuator.
the sample in Figure 6.2.
For the identification and modeling, a mean-zero Gaussian input was again used,
giving current levels of 130, 195, and 265 mA-rms. Because the bending modes of this
sample are at much higher frequencies than the cantilever sample, but nonlinearities are
still believed to be in the low frequency range, random data was collected over two different
frequency bands. A sample rate of the 67 Hz was used for the low frequency range with
data blocks of 16,384 points, giving 4.0 mHz resolution, while a sample rate of 2.0 kHz was
used for the high frequency range with 16,384 points, which gave a frequency resolution of
0.12 Hz. Eight averages were collected at the low frequency range and 16 averages were
collected at the higher frequency band. A lower number of data blocks were collected at
the lower frequency band because of the length of time involved, but the coherence was
high over most of the range, so it is not thought that any modeling results will suffer. In
the model development, the low amplitude frequency response functions from these two
frequency bands will be stitched together where they overlap.
To validate the model that will be built, sine wave data was collected in the low
frequency range, where it is believed that nonlinearities are present. High frequency data
could not be collected because of the large magnitude roll off that gave low signal strength
for the measurements. The frequencies tested were 0.01, 0.05, 0.1, 0.5, 1.0, 5.0, and 10.0
Hz, at respective sample rates of 2.0, 10.0, 20.0, 100, 200, 1000, and 2000 Hz. Amplitudes
of 19.2, 38.3, 115.0, 192.0, 268.0, and 345.0 mA were also tested at each frequency setting,
although data at the lowest frequency (0.01 Hz) for the highest two amplitudes was not
collected because the voltage began to approach the limit of the electrochemical stability
window of the solvent (∼ ±2.0 V) at the 192-mA amplitude setting for this frequency.
154
Figure 6.3: Harmonic ratio analysis for plate deflection.
Additionally, some of the higher frequency data (10 Hz), especially at low amplitude (19,
38 mA), could not be used because of the weak signal level. While random data will be
used in the modeling and identification, this sinusoidal data will be used in the nonlinear
characterization, discussed next.
6.1.2
Nonlinear Characterization
Recalling the results from characterizing the nonlinear response of the cantilever actuator, it
was determined that the Volterra series analysis, harmonic ratio comparison, and multisine
excitation methods all revealed similar trends for how the different nonlinearities affect
the measured response and how they should be incorporated into the model development.
Based on this similarity in characterization techniques, only the harmonic ratio analysis will
be applied to characterize the deflection response of the plate actuator. This method was
chosen because sinusoidal response data had already been collected for model validation, and
it requires the least amount of time between the two single-frequency techniques (Volterra
series, harmonic ratios).
As expected from the previous work with the cantilever benders, the two most dominant frequencies in the deflection response of the plate actuator, aside from the fundamental,
were those corresponding to a quadratic (2f ) and a cubic (3f ) nonlinearity. This was deter-
155
mined from examining the Fourier transforms estimated from the measured signals at each
testing condition. The harmonic ratios were calculated by dividing the amplitude of the two
nonlinear frequencies by the amplitude of the fundamental. Figure 6.3 contains these results
for the quadratic nonlinearity in the upper plot and the cubic nonlinearity in the lower plot.
In each plot, the circles represent the harmonic ratio at the lowest input level (19 mA)
and the noted input amplitude increases with the symbols of squares, diamonds, downward
triangles, and up to 268 mA with the right-pointing triangles. It should be recalled here
that data from 0.01 Hz at 268 mA was not included in these plots because the measured
voltage was exceeding the electrochemical stability window of the solvent material, which
could have led to erroneous conclusions about the nonlinearities in the normal operating
range.
Figure 6.3 also repeats the trend seen with the cantilevers that the nonlinear influence
is negligible at higher frequencies (linear region) and increases with decreasing frequency
(nonlinear region). This can be seen in the figure since all of the data sets nearly overlay
each other with values close to zero at frequencies above approximately 1.0 Hz, but begin to
increase at lower frequencies. Figure 6.3 also indicates that the nonlinear influence of both
the quadratic and cubic nonlinearities has a growing effect on the deflection response as the
input level increases. There is evidence that this effect differs between the nonlinearities
shown, however. The influence of the quadratic nonlinearity appears to impact the response
at the same frequency (between 0.2 and 1.0 Hz) for all amplitudes, but with an effect
that increases with the input level, while the cubic mechanism appears to be triggered to
begin at higher frequencies for higher input amplitudes. One final observation from this
characterization analysis is that both of the nonlinearities shown have relatively equivalent
response levels, meaning that inclusion of both quadratic and cubic nonlinear forms will
likely be required to model this system, but this will be explored further in the following
discussion.
6.1.3
Deflection Modeling
To begin the model development, Figure 6.4 has been included to show the distortion present
in the deflection response of the center point of the polymer plate actuator. Looking first at
the frequency response in Figure 6.4a, there is very little indication that any nonlinearity is
present because the magnitude scaling that was seen with the cantilever sample is not seen
156
Figure 6.4: Evidence of deflection nonlinearity in plate actuator: (a) frequency
response; (b) time response.
here. The solid line (265 mA-rms) appears to overlay the measured response of the dotted
line (130 mA-rms), which differs by a factor of 2 in input level. One small indication from
these plots that nonlinearity may be present is the low frequency drop in coherence seen
for the higher input. Figure 6.4b shows some sinusoidal time responses, where evidence of
distortion can be seen. The upper two plots show the deflection from a smaller input level
of 19.2 mA (13.6 mA-rms), and the lower two plots show the response at 192 mA (136
mA-rms). Also, the left side shows a deflection response in the nonlinear frequency range
(0.01 Hz), while the plots on the right side show a linear response (1.0 Hz). Although the
frequency response showed little evidence of nonlinearity in the measured deflection, Figure
6.4b shows that distortion is present in the time response, that it appears to change with
input level, and that it is frequency-dependent. This data will be used again in validating
the model.
The candidate system structure identified from modeling the cantilever actuator was
an underlying linear system with a dynamic nonlinearity on the input (Hammerstein system), so based on its success in predicting the measured response, it will again be employed
here. The first step in this procedure is to identify and model the underlying linear system
H(f ). This is done by estimating the frequency response function with the low amplitude
random data (130 mA-rms). As mentioned previously, these frequency points are a composite of low frequency data (∆f = 4.1 mHz) and high frequency data (∆f = 0.12 Hz). The
157
Figure 6.5: Plate modeling result for underlying linear system H(f ).
stitching of these two data sets was made at 1.0 Hz because the coherence was high for each
and the magnitude and phase go through no significant change near this frequency. Figure
6.5 shows this composite system response with dots at each frequency point. It can be seen
in this figure that no data points appear from 20–70 Hz. These points were removed for
modeling purposes because they correspond to a resonance of the fixture and the coherence
was very low in this region.
A model to this system was parameterized using invfreqs.m as an initial fit with the
coherence function as the frequency weighting, but poor results were obtained for various
orders of the numerator and denominator. A constant weighting was also attempted with
no improvement. To increase the accuracy of the model, individual segments were parameterized by separating the frequency response into four regions: below dominant resonances
(0–90 Hz); dominant resonances (90–260 Hz); frequencies just above dominant resonances
(260–500 Hz); and frequencies well above dominant resonances (500–1000 Hz). Obtaining
accurate predictions for these segments, the individual models were then combined, keeping
only the second-order poles and zeros of these smaller systems, and leaving the final location
of the first-order poles and zeros to be determined through iteration with fminsearch.m. The
final model included 13 zeros and 14 poles (r = 1), where it should be noted that because
one zero was non-minimum phase, model inversion was not possible, which loosened the
158
Table 6.1: Plate Model Parameters for H(f ).
Gain
-0.0457
Zeros (Hz)
-74.53±j851.6
-37.24±j307.2
-9.020±j220.3
-7.540±j144.1
-730.0
-596.6
-407.0
-2.600
750.0
Poles (Hz)
-874.6±j933.3
-38.47±j856.7
-36.10±j334.7
-9.822±j226.5
-9.521±j198.2
-4.743±j121.4
-0.0200
-4.268e-4
constraint that r = 0 for this identification technique. These final model parameters are
provided in Table 6.1 and Figure 6.5 shows that the model response (solid line) predicts the
measured response well over the entire frequency range.
Being that the final model has a non-minimum phase zero and cannot be inverted,
the model development will proceed using the augmented DIN methodology discussed in
section 5.3.2 when modeling the velocity response of the cantilever actuator. This procedure
begins with decomposing the measured, high amplitude output signal (containing distortion)
into its most dominant frequency components (f , 2f , and 3f ) at each of the sine wave
measurement points. These individual frequency components are then inverse-simulated
through the underlying linear system H(f ) to obtain a description of the nonlinear input
(recall equations 5.6–5.9). Once these nonlinear input signals have been computed for each
test frequency, their content at f , 2f , and 3f can be respectively compared to the content
at those frequencies for i(t), i2 (t), and i3 (t) to identify a coarse estimate of the frequencydependence for each component of the computed nonlinear input, x(t).
The identified frequency points and modeling results for the quadratic and cubic
terms are given in Figure 6.6. In this figure, the circles appear as the identified frequencydependence of the nonlinearity, the dashed line is the initial model parameterized using
invfreqs.m, and the solid line is the final system after modifications were made to better
predict the measured response. These modifications were made by manually adjusting the
number and the locations of the poles and zeros, based on the simulation error and distortion
shape of the wave at low frequency. For the A1 (f ) system specifically, Figure 6.6a shows
that the initial model did not roll off at higher frequency, which led to reducing the number
159
Figure 6.6: Plate modeling results – frequency-dependence of nonlinear components: (a) quadratic term; (b) cubic term.
Table 6.2: Plate Model Parameters for input system (Hz).
Gain
0.0065
A1 (f )
Zeros
-0.0105
Poles
-0.4100
-0.1150
-0.0790
A2 (f )
Gain
Zeros
1.905e-6
–
Poles
-0.0145
Gain
1.0022
B(f )
Zeros
-0.0220
-0.0129
Poles
-0.0300
-0.0194
of zeros in the improved model. As shown in the characterization results, the nonlinearities
have negligible effect in the higher (linear) frequency region. The low-pass filter form of
A2 (f ) was already set up not to interfere with higher frequency dynamics of the underlying
linear system, but to modify the waveform of the model output at low frequency, Figure
6.6b shows that the frequency of the single pole was reduced farther for the cubic term. The
individual parameters for the final systems shown here can be found in Table 6.2, where it
can be explicity seen that all the poles and zeros are below 0.5 Hz.
Similar to the cantilever modeling, one last augmentation made to the model for the
polymer plate actuator was the addition of a system in the direct input path to the underlying linear system. This system, B(f ), was included in the final form of the deflection model
to help adjust the magnitude and phase of the predicted output at the lowest frequency
measured, while having little effect on the higher frequency response. Figure 6.7 shows this
160
Figure 6.7: Plate modeling results – linear system in direct input path.
system (solid line), along with the identified frequency-dependence between i(t) and x1 (t)
as the circles, and the initial model parameterized by invfreqs.m (dashed line). Having zero
phase and unit magnitude above 1.0 Hz, it can be seen that i(t) will not be affected in the
linear frequency range of the plate system. Table 6.2 also lists the final parameter values
for B(f ).
Now that the final model has been developed, its response can be compared to
measured data to provide validation for its form. Figure 6.8 displays these results for both
the frequency-domain and the time-domain. Looking first at the frequency response results
of Figure 6.8a, the response of the model overlays the measured response for nearly the
entire frequency range. The visible high frequency difference, mostly in the phase, was also
seen in the modeling of the underlying linear system, and because the coherence is also lower
here, it is not suspected that this will effect the model adversely. At low frequency there is
also evidence of model inaccuracies, but this discrepancy is introduced from the nonlinear
components in the model because this is the frequency range where the nonlinearities are
most dominant. Examining these results, it is seen that the simulated coherence drops off
below 1.0 Hz, also where the phase error begins, and the magnitude begins to over-predict
the response below approximately 0.1 Hz. Since these frequencies coincide with the noted
frequencies with nonlinear influence from the characterization results, it is assumed that
161
Figure 6.8: Plate modeling results: (a) frequency response; (b) time response.
this model has predicted the measured frequency response well.
Select sinusoidal time responses are given in Figure 6.8b. Here, the dotted lines
correspond to measured data and the solid lines denote the simulated response. The upper
plots in this figure show the deflection at 19 mA and the lower plots show the deflection
at 115 mA, while the two left plots contain responses in the nonlinear frequency range at
0.01 Hz and the two plots on the right side show responses in the linear region at 1.0 Hz.
Although the model suffers somewhat in predicting the distorted shape at lower frequency,
the progression of the nonlinearity with increasing amplitude is tracked and the model is
able to predict the magnitude and phase in the linear region. The model under-predicts
the low amplitude deflection and the magnitude is over-predicted at the higher amplitude,
while the overall phase appears to be approximated well. There is also a discrepancy in
the distortion shape of the deflection. Although the simulated response does not match the
rounding of the peaks, it does match the positive-sloped edge of the waveform. It should be
mentioned here that the model could have been tuned to predict the distorted shape better,
but there was a trade-off between that aspect and more closely matching the magnitude
and overall phase. And for the purposes of linear control, the topic of the next section, the
decision was made that matching the overall magnitude and phase with the model would
yield a more effective control system.
One last set of results to be discussed before beginning the control investigation is
162
Figure 6.9: Plate modeling results – sinusoidal response error: (a) overall percent error; (b) amplitude and phase metrics.
the error. Figure 6.9a shows the overall percent error, calculated from equation 5.2, and
Figure 6.9b separates the error into an amplitude metric and a phase metric. In these
figures, the circles correspond to the lowest input level (19 mA), the squares denote the
results for 38 mA, the diamonds show the results for 115 mA, and triangles represent the
error for the highest input (192 mA). The error percentage of the output is below 20% for
nearly all of the test conditions shown in Figure 6.9a, except for the highest two amplitudes
at the lowest frequency. This is where the nonlinear response has the most effect. At 192
mA and 0.01 Hz, the error peaks at 150%, but as noted during the experimental setup
discussion, the voltage for this particular test condition was approaching the limits of the
electrochemical stability window. For this reason, the large error here provides an upper
limit on the excitation range for the plate model.
Figure 6.9b displays different error metrics. As seen in Figure 6.8b, the upper plot
here shows that the model under-predicts the deflection at 0.01 Hz for the lower amplitudes
(positive value) and over-predicts the response for the higher two input levels (negative
value). At the intermediate frequencies, the amplitude is predicted within approximately
±20%. The correlation coefficient ρ appears in the lower plot of Figure 6.9b and it gives an
indication of the phase error in the model, where a value of +1 means that the measured
and simulated responses are perfectly in phase. This plot shows that the overall phase of
the model predictions are very near that of the measured deflection, except at the lowest
163
frequency for 115 and 192 mA, but this error is most likely due to the phase mismatch of
the nonlinear components that do not allow for accurate prediction of the distortion shape.
However, as mentioned previously, this was the cost of more accurately predicting the overall
phase over the measured input range, which initially differed by almost 90 degrees. It should
also be noted that the seemingly larger error at higher frequencies and low inputs is mostly
attributable to measurement noise, as the magnitude of the deflection response is very low.
Having completed the development and validation for the plate actuator model, the
next section will look at improving the open-loop performance shown here through the use
of feedback control.
6.2
Output Feedback Control
As in most control applications, linear methods should always be tested first to see how the
system responds, and nonlinear control should only be considered in cases where the system
fails to reach some desired specifications. To this end, simulations of a linear control system
will be performed first before making a decision on how to proceed with any nonlinear
control. The control will be developed using feedback of the output position of the centerpoint of the polymer plate actuator.
6.2.1
Linear Control Simulations
Since the application theme of this chapter has been deformable mirrors, it will be important
that the designed control system is able to track a reference position (focal adjustment).
It will also be important to have some control over how fast the system can make the
desired adjustments, and stability of the closed-loop system must be ensured. For these
reasons, the chosen linear control method is proportional-integral (PI) control. One of the
most common linear control techniques is PID, which also includes a derivative term, but
because this term tends to increase the noise, it was not included here. The form of the
control is
D(s) = kp +
kp s + ki
ki
=
s
s
(6.1)
where kp is the proportional gain and ki is the integral gain. In designing this controller, the
polymer system will be assumed linear and the description of the system will be assumed
to have the form of the modeled underlying linear system H(s).
164
During the design process, the control gains were varied and properties of the loop
gain D(s)H(s) and the closed-loop system
D(s)H(s)
Y (s)
=
R(s)
1 + D(s)H(s)
(6.2)
were examined for stability and performance. There was also one other criterion that had to
be considered in this design, which is the saturation limit (electrochemical stability window)
of the ionic polymer. For the EMI-Tf solvent in the actuator, this limit is near ±2.0 V. To
incorporate this into the control design, a “worst-case” value of the impedance was taken
from the measured current and voltage sine wave data (0.011 V/mA) and compared to the
closed-loop transfer function between the reference input, r(t), and the control input, u(t),
over the bandwidth of the closed-loop system. This system is given by
D(s)
U (s)
=
,
R(s)
1 + D(s)H(s)
(6.3)
and any control current producing a voltage that exceeded the 2.0-V limit was then iterated
upon by lowering the gains.
The maximum performance with these design iterations, while maintaining conservative stability margins, was achieved with kp = 45 and ki = 47. The gain margin (GM) was
11.2 at 125 Hz and the phase margin (PM) was 160 degrees at 2.5 Hz. This gain cross-over
frequency can also be used as an estimate of the system bandwidth, which is relatively low
here. When considering that an integrator was added to the negative slope of the open-loop
system, it does make sense that the closed-loop system has a low bandwidth. This reduced
speed of response is also one of the consequences of using integral control.
This design did perform well in the nonlinear range for sinusoidal inputs, but because
the desired application is for shape control, examining the step response could also prove
useful in the design. With the stated gains, simulations showed that nearly 8.0 Amps were
required to track a step input, which would certainly exceed the allowable limit of the
actuator (and the hardware capabilities). Therefore, a redesign was needed. Considering
the control effort of the step response this time, the gains were selected to be kp = 1.0 and
ki = 3.1, giving a much more conservative gain margin of 46 at 125 Hz and a phase margin
of 75 degrees at a farther reduced 0.42 Hz. Table 6.3 summarizes these design properties.
The open-loop (dotted), loop (dashed), and closed-loop (solid) transfer functions of this
linear approximated system are illustrated in Figure 6.10a. The D(s)H(s) response in this
165
Figure 6.10: Simulated transfer functions of square-plate actuator system with
PI-control design: (a) system response; (b) control effort.
Table 6.3: PI-Compensator design summary.
kp
ki
GM
PM
1.0
3.1
46 (at 125 Hz)
75o (at 0.42 Hz)
plot highlights the increased negative slope at low frequencies that reduces the closed-loop
bandwidth. Figure 6.10b has also been included to show the control effort response used in
determining whether the system saturated or not.
The performance of this linear controller will now be assessed on the nonlinear model
developed in the previous section. First, the closed-loop system was simulated to a sinusoidal
reference command with a frequency and amplitude chosen in the nonlinear operating range
(0.01 Hz, 200 mA). Figure 6.11 shows the reference signal as the small dashed line, the openloop simulation as the large dashed line, and the closed-loop performance as the solid line.
The upper plot contains the deflection response, while the lower plot shows the control
current. The distorted shape of the open-loop response that is slightly bowed out, or
rounded, on the negative-sloped side of the wave is nearly eliminated by the linear PIcompensator. The control effort plot draws more attention to the distorted waveform that
is required to track the reference sinusoid, which is closely approximated by the closed-loop
166
Figure 6.11: Simulation results for open- and closed-loop sine response using
PI-control in the nonlinear operating range.
response. This control scheme can also track reference sine waves at higher frequencies
with small error, but these results have been omitted here because Figure 6.11 is the more
interesting and encouraging result. Although this PI-control does not contain an internal
model of a sinusoid, it is interesting to note that it can perform well tracking the reference.
However, when considering that the frequencies in the nonlinear range are very low, this
result seems more reasonable.
Looking now at the step response of the system under linear control, Figure 6.12
has been provided. As in the sinusoidal results, the reference signal appears as the small
dashed line, the open-loop response is the large dashed line, and the solid line represents the
closed-loop response. Deflection is again shown in the upper plot with the corresponding
control current appearing in the lower plot. It can be seen that the open-loop response
is very slow, requiring over 1500 seconds to complete its rise and settling to steady-state.
On the other hand, the controlled response has a rise time on the order of 0.5 seconds and
settles to within 2% of the reference in just over 5.5 seconds. The inset on the right side
of the figure zooms in on the initial part of the closed-loop response. Increasing the speed
of response by two orders of magnitude does come at a cost, however. The inset of the
closed-loop control current reaches almost 300 mA at its peak before settling near zero.
Although this current level did exceed the electrochemical stability window of the ionic
167
Figure 6.12: Simulation results for open- and closed-loop step response using
PI-control.
liquid solvent in the low frequency sine wave data, it should be mentioned here that the
large current is only required for less than one second. In relation to the 0.01 Hz sine wave,
where this current level was maintained for a much longer period, it is not believed that
this will cause any problems here. This large current requirement also corresponds to 50%
overshoot in the response, while experience with these actuators, particularly in clamped
conditions, points out that this is unlikely to occur experimentally.
The results in Figure 6.11 indicate that this linear control technique has the ability to
accurately track a reference sinusoidal signal in the presence of nonlinear distortion. From
this point, experiments will now be conducted to confirm if the actual polymer system can
be controlled with a linear method.
6.2.2
Experimental PI-Control
Because the linear simulations showed that the controller was able to compensate for the
harmonic distortion at low frequency and effectively track a reference step input, this control
system will now be implemented experimentally. The gain values quoted in Table 6.3 will
be used as a starting point in these experiments and adjustments will be made accordingly
to improve the response where necessary. Figure 6.13 shows the control block diagram of
the experiment, where it should be noted that the same laboratory equipment was used
168
Figure 6.13: Block diagram of experimental control investigation.
as previously mentioned in the data collection section. A reference command position is
provided to the closed-loop system, which is used with negative output feedback of the
actual position to calculate the error signal. This error then enters the compensator, where
the control signal is computed. Next, the control current passes through a gain block
to produce a voltage signal proportional to the control current. Then this voltage enters
the transconductance amplifier to source the needed current before the laser vibrometer
measures the deflection of the polymer plate actuator.
Before considering various reference input signals, the most important result of this
experimental control study is to confirm that the linear PI-control can track a sinusoid
in the presence of nonlinear distortion. Figure 6.14 displays these results for a 120-µm,
0.01-Hz sine wave. The open-loop deflection measurement (large-dashed) has clear signs of
nonlinearity as the positive peaks are pointed and the negative peaks are more rounded.
Appearing as the small-dashed line, the reference signal is tracked accurately in magnitude
and phase as the solid line of the closed-loop response measurement overlays the command
signal in the upper plot. The control current of the closed-loop system in the bottom plot
also provides an indication of the ability for the system to eliminate nonlinear response
characteristics, as this signal itself is a distorted sine wave. These results have the designed
proportional gain of kp = 1.0, but the integral gain has been tuned to ki = 0.17 for an
improved response.
Another interesting result brought out by Figure 6.14 is that the wave shape appears to differ from the measurements shown in the development and validation of the
169
Figure 6.14: Experimental results of PI-controller tracking a sine wave in the
nonlinear operating range.
model. The likely cause of this discrepancy is that the polymer sample changed in some
way. This could be a product of different clamping conditions, producing changes in tension through the material, or partial breakdown of some solvent molecules. Throughout the
expansive experimentation with this actuator, there were times when the voltage exceeded
the electrochemical stability window, which could also change the system properties. An
attempt was made to avoid this, but it did occur in a few select instances. On the positive
side, however, the ability of the control system to compensate for this modified nonlinear distortion is another indication that a linear approximate model may be sufficient for
designing a linear controller.
Moving now to the study of how this closed-loop system can track and hold position
changes, Figure 6.15 can be examined. The reference appears as the small dashed line,
the open-loop is the large dashed line, and the closed-loop response is shown as the solid
line. Each of the plots on the top show the deflection response and the associated control
currents are in the bottom plots. Figure 6.15a shows step response results qualitatively
similar to the closed-loop simulation. That is, a slow open-loop rise to steady-state (150 s)
and a controlled rise time two orders of magnitude faster at 1.5 seconds, where the position
is held to within 0.5 µm. Though different from the simulation model, it was expected
from experience that there would be very little overshoot, as is seen here (∼2%). The
170
Figure 6.15: Experimental comparison of open- and closed-loop tracking: (a)
step input; (b) smoothed pulse input.
peak response of the control current is also near the same 250-mA level of the simulated
response, though the reference step is much smaller here. This discrepancy could be partly
attributable to the previously mentioned changes to the sample, such as solvent breakdown.
The gains in this result were tuned to kp = 7.0 and ki = 0.65.
Figure 6.15b shows the response of the plate actuator to a smoothed pulse. This
reference signal goes through an s-curve for 10 seconds up to a constant, holds this position
for 50 seconds, and then goes through another s-curve down to zero and holds that position.
The open-loop response shows that it takes the entire holding period for the system to reach
the desired level. However, the controlled response can track the reference command through
the transitions, as well as in the constant holding positions. The gains here were kp = 7.0
and ki = 0.75. It should be noted that better tracking performance at higher reference
positions could be achieved with the smoothed pulse signals because they did not require
the same initial spike in the current and voltage that limited the performance range of the
step responses.
The next results to be detailed are a series of step responses at different reference
position levels. The step values range from 20 µm up to 100 µm in 20-µm increments, and
appear as the dotted lines. Figure 6.16a shows the measured plate deflection (solid line) at
each level for the gains of kp = 3.0 and ki = 0.55. These values were tuned for maximum
performance at the largest position level and then used for smaller steps to show how a
171
Figure 6.16: Experimental PI-control results for a step input: (a) constant
gains; (b) variable gains.
Table 6.4: Controlled step response results.
Step
(µm)
20
40
60
80
100
kp
3.0
3.0
3.0
3.0
3.0
Constant Gains
ki
Mp (%) tr (s)
0.55
8.5
2.0
0.55
8.8
1.8
0.55
9.1
1.8
0.55
8.8
1.8
0.55
8.8
1.6
ts (s)
12.0
13.0
13.0
13.5
14.0
kp
8.0
5.0
3.0
3.0
3.0
ki
1.10
1.10
0.55
0.55
0.55
Variable Gains
Mp (%) tr (s)
2.5
0.9
7.5
1.2
9.1
1.8
8.8
1.8
8.8
1.6
ts (s)
4.0
8.5
13.0
13.5
14.0
constant gain control scheme would perform over a range of inputs. It should be noted
that better tracking could not be achieved at 100 µm because of actuator saturation in the
voltage. Also, the maximum control effort did not exceed a peak of 330 mA.
Figure 6.16b shows similar results, but rather than having a constant set of gains
tuned at the maximum position, these results allow the gain to be tailored at each individual
command position. For actual implementation of the best possible performance over a
range of references, the command position could serve as the scheduling variable of a gain
scheduling controller used to optimize the performance for each setting. Table 6.4 has been
included to summarize the gains and performance specifications of these closed-loop step
responses. It should be mentioned that rise time has been calculated from 10% to 90% of
the reference and settling time has been calculated as the time for the response to remain
172
Figure 6.17: Experimental PI-control results for a smoothed pulse: (a) constant
gains; (b) variable gains.
within ±2% of the reference. From these tabulated results, it can be seen that the gains
are the same for the highest three reference signals. In tuning the gains for the variable
gain controller, it was determined that insignificant improvement could be obtained in the
response at 60, 80, and 100 µm, so the gains were kept the same. The major difference
between varying the gains based on the command position or keeping them constant can
be seen in the results for the lowest two step responses. Here, the overshoot, rise time,
and settling time are all reduced by increasing the gains, indicating that a variable gain
controller is required to maximize the performance for a range of command positions.
Repeating this concept of comparing the response for constant gains to variable
gains over several command positions, Figure 6.17 contains the results for the smoothed
pulse signal. The reference signal is the dotted line at 10, 40, 80, and 120 µm, and the
closed-loop measured deflections are depicted by the solid lines. Figure 6.17a shows the
results when the controller gains are tuned to the maximum position and kept constant
for smaller reference command signals, while Figure 6.17b contains the results for variable
gains tuned for each different position. These gain values are summarized in Table 6.5. The
difference in these two plots is small, but upon close examination it can be seen that at 80
µm with a constant gain, the response overshoots the plateau, while no overshoot is present
in the variable gain response. Also at the two lowest positions, varying the gains allow the
closed-loop system to track the increasing s-curve with nearly half the error, and for all the
173
Table 6.5: Controlled smoothed pulse response results.
Step
(µm)
10
40
80
120
Constant Gains
kp
ki
7.0
0.75
7.0
0.75
7.0
0.75
7.0
0.75
Variable Gains
kp
ki
8.0
2.25
8.0
2.25
13.0
10.0
7.0
0.75
levels the command signal can be tracked to ±0.5 µm. Because this tracking error range is
relatively constant for command positions that span an order of magnitude, it is likely that
it is introduced either from the electronics or from ambient vibration. It could be possible
to further reduce this 0.5-µm error by performing the control experiments on an isolation
table, such as an optical bench, or by passing the input to the polymer through a low-pass
filter. Also worth mentioning is that the control current remained below 100 mA for all
positions tested. From these results and those of the step responses, it was seen that varying
the gains for different positions most benefited the smallest changes in position, though the
overall improvement was not outstanding.
These results have shown that a linear control scheme can accurately track desired
reference commands to a polymer system that exhibits nonlinear response characteristics
in the low frequency regime. Because of the noted tracking capability, there is no need to
investigate more complicated nonlinear control techniques. This result also shows promise
to the applicability of ionic polymer actuators because simpler linear control is very effective
for at least some cases, including the square-plate actuator with clamped corners.
6.3
Summary
Further validation of the selected model structure was provided in this chapter by analyzing
a more complex system geometry with different boundary conditions. A square-plate actuator with clamped corners was considered here, where a model consisting of an underlying
linear system with a dynamic input nonlinearity was shown to capture the nonlinear response. Characterization showed that both quadratic and cubic nonlinearities were present
in the response, and the influence of each term increased with input amplitude, but the cubic
term also began to impact the response at higher frequencies for higher input levels. Once
174
the model was complete, linear control simulations were performed using a proportionalintegral controller. Experiments confirmed that the chosen linear technique could track a
reference sinusoid in the nonlinear operating range, as well as track various other reference
commands that were meant to mimic focus adjustments for potential deformable mirror
applications.
175
Chapter 7
Summary and Conclusions
Having completed the presentation of the research, this final chapter will provide a synopsis
of the most notable accomplishments. The conclusions of the research will be summarized
first, followed by an account of how this work has contributed to the field of electroactive
polymer systems. To end this chapter, some recommendations for future work will be stated.
7.1
Research Conclusions
This dissertation has discussed various aspects relating to the characterization, modeling,
and control of ionic polymer actuators. The first chapter provided background information
for these transducer materials and detailed several of the proposed models for actuation.
The purpose of reviewing the modeling literature was to show that several model forms have
been proposed, from researchers with diverse areas of expertise, and drawing from different
physical concepts, but these models have limitations. In particular, many of the models
are stated to be linear approximations of response characteristics that have been shown
to contain nonlinearity. The inability of these models to accurately predict the nonlinear
dynamics provided the motivation to study this topic.
The nonlinear investigation began with characterizing the electrical and mechanical
actuation response of ionic polymer transducers. Through the use of the Volterra representation, harmonic ratio analyses, and multisine excitations, it was determined that harmonic
distortion was present in the time response and that this distortion translated into shifting
of the magnitude and phase in the frequency response. The primary source of nonlinearity in
the current-controlled voltage response was a frequency-dependent cubic mechanism, while
176
the velocity response of the tip of a cantilever sample was shown to have both quadratic
and cubic components, again depending on the frequency. To further investigate these nonlinearities, experiments were conducted on samples with three different solvent materials.
From these results, it was concluded that lower viscosity solvents triggered the nonlinear
response at higher frequencies. Also stated was the qualitative analog between the nonlinear frequency-dependence and a low-pass filter, with the break-point frequency determined
by the solvent viscosity (lower viscosity, higher frequency). Results also showed that the
influence of the nonlinear response contribution grew with input amplitude.
With this knowledge of the nonlinear response properties, block-oriented, nonlinear
identification techniques were applied to develop a model that can capture the nonlinear dynamics. The two methods explored were based on the multi-input, single-output (MI/SO)
technique and the dynamic input nonlinearity (DIN) approach. Assuming the nonlinear
forms determined from the characterization study, models were constructed and compared
with measured data in the time-domain and the frequency-domain. For both the voltage
response and the velocity response, the ideal model structure was shown to have a dynamic
nonlinearity on the input to an underlying linear system. Without a physical model for the
nonlinearity, the form of these elements were Hammerstein systems, having a static nonlinearity with a frequency-dependent parameter for each harmonic distortion component.
This choice was made based on simulation comparisons with other nonlinear forms.
Having identified the candidate model structure, its form was validated on a more
complex, application-oriented polymer system, with the application being a membrane optical device. Characterization analyses and model development were conducted on a squareplate polymer actuator that was clamped on all four corners. Actuating this sample from
each of the clamped corners and measuring the deflection of the center-point, quadratic and
cubic nonlinearities were seen in the response. Similar to the cantilever results, the effect
of these nonlinearities increased with input amplitude, while the frequency-dependent behavior was most dominant at low frequencies and the response was predominantly linear at
higher frequencies. A control study was also performed on this sample, where simulation
and experimental results showed that a proportional-integral controller was able to compensate for the nonlinear harmonic response at low frequency and track various reference
signals set up to replicate adjustments in the focus of the device. The tracking was also
shown for fixed and variable gain schemes.
177
7.2
Contributions
The contributions of this research were briefly presented in the first chapter of this document, but will be restated here, discussing in more detail how they have advanced the
state-of-the-art in electroactive polymer research.
• Characterization of Solvent-induced Nonlinear Response. Improving the response
characteristics and expanding the operation range of ionic polymer transducers by
tailoring the solvent materials is an active area of research. However, a study of how
the nonlinear actuation behavior is affected by these new solvents had not yet been
conducted. The characterization results presented here showed that the nonlinear response properties of the voltage and tip velocity are strictly low frequency phenomena
that depend on the input current level. It was also shown that solvent viscosity plays
a key role in determining the frequency range over which the nonlinearities influence
the response. From these results it was determined that lower viscosity solvents trigger the nonlinear mechanisms at higher frequencies. The knowledge gained from this
investigation offers insight into the physical mechanisms involved with ionic polymer
actuation and can benefit future modeling efforts of these underlying processes, while
also suggesting a means to tailor the response for certain applications.
• Development of Nonlinear Dynamic Actuation Model. Motivated from the leading
models’ inability to predict the nonlinear actuation response of ionic polymer materials, research was performed to identify a candidate model architecture that can
account for the nonlinear dynamics. Through comparisons and validation, the chosen
system model consists of an underlying linear system with a dynamic nonlinearity on
the input. While the nonlinear terms required to accurately predict the output differ
between the electrical and mechanical responses, the basic structure is the same. This
structure is a parallel connection of linear and nonlinear terms (cubic in the voltage;
quadratic and cubic in the velocity), where the nonlinear terms appear as Hammerstein systems with a frequency-dependent parameter following a static nonlinearity.
This model form was further validated against the more complex polymer system of
a square-plate actuator. Obtaining a system description that captures the nonlinear
behavior is essential for ionic polymer transducers to be incorporated into system level
design, and developing this input-output model is the first step in the process.
178
• Proof of Viability for Control of Structures. Continuing with this notion, the applicability of ionic polymers as active transducer materials must be proven by addressing
existing design obstacles. One area, in particular, that ionic polymers show promise
is in the control of flexible structures, such as deformable mirrors. With this in
mind, a control study was performed with a square-plate polymer actuator, and results showed that a linear proportional-integral compensator could sufficiently track
a low frequency reference sine wave in the nonlinear operating range. As such, the
performance of this proportional-integral controller was further assessed by tracking
reference inputs that were aimed at simulating focal adjustments of a lens. Showing
that both a fixed and variable gain framework lead to tracking capabilities with less
than 0.5 microns of error helps illustrate the potential that ionic polymer systems
have in this particular engineering field. The result that linear control performs well
in the presence of nonlinear distortion is also valuable for implementation because,
depending on the configuration or application, there may not be any need to develop
more advanced control schemes to reach the desired specifications.
7.3
Future Work Recommendations
By stating the conclusions of the research and discussing the contributions to the field, the
presentation of this research topic was brought to a close. Based on these results and other
knowledge gained throughout the process, the final section of this document will present
some ideas for future research with active polymer systems. These recommendations are
the following:
• Characterizing the nonlinear response of ionic polymer materials over a broader range
of fabrication components would be useful for material scientists and model developers. While the research presented here focused specifically on the solvent, exploring
other potential causes for nonlinear mechanisms, such as the cation form and the
electrode composition would yield further insights into understanding the response
characteristics of these transducers.
• The model developed here was empirically derived from measurements of the input
and output of the system, but the actual physical processes will need to be understood
and modeled before ionic polymers reach their fullest potential, particularly for design.
179
Using results from the characterization and the identified model structure, it could be
possible to look for physical interpretation in the electrical, chemical, and mechanical
mechanisms that give rise to the observed and modeled behavior.
• It was noted in an earlier chapter that the force term of the two-port model would
not be considered in the present development. However, it is critical to include the
force as an input or output of a system model for design purposes. Knowing the
force response characteristics and maximum capabilities will provide engineers with
necessary information to select appropriate transducers for their specific applications.
• Building on realizing a full transducer model, incorporating the sensing response into
the development is also an important area to consider. Several aspects of ionic polymers show promise to sensing applications, including their light weight, high compliance, and ability to be cut to any size or shape. Having a better understanding of the
sensing response could open many more doors for ionic polymer materials.
• While a control investigation was performed with a plate actuator set up to mimic
a deformable mirror application, it would be advantageous to consider attaching the
actuator to an actual structure to assess its performance. Its ability to either reject
low frequency disturbances or control the position or focus of a membrane system
would be ideal applications to help push these materials into system implementation.
• The control application in this work also applied an equivalent input to each corner
of the plate actuator, measuring the deflection only at the center-point. It would be
interesting to study the attainable shapes of the plate by examining the response at
multiple points, however. After the profiles were known for a uniform input signal, the
next step would then be to consider applying different control signals to each corner
of the plate, which would likely result in increased deformation and shape change
capabilities.
• Ultimately, with a complete description of the sensing and actuation properties of
ionic polymer transducers, the goal is to design an intelligent system that uses ionic
polymer sensors and actuators. Once this is achieved, ionic polymers will have arrived
as viable transducer materials for a broad range of applications.
180
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Vita
Curt S. Kothera was born on January 4, 1979 to Sharyn and Clark Kothera, in Ravenna,
Ohio. In 1987, his family moved to Charlotte, North Carolina, where he graduated from
Myers Park High School in 1997. He attended Virginia Tech afterward and graduated
Summa Cum Laude with a Bachelor of Science degree in mechanical engineering in May
2001. Also within this time, Curt studied abroad for the Fall 1999 semester at Luleå
Tekniska Universitet in Luleå, Sweden. After completing his undergraduate degree, he
worked for the summer at Northrop Grumman’s Electronic Sensors and Systems Sector
in Baltimore, Maryland, before returning to Virginia Tech for graduate studies in August
2001. Here, Curt studied under Dr. Donald J. Leo in the Center for Intelligent Material
Systems and Structures (CIMSS), where he was granted a Master of Science degree in
December 2002 and a Doctor of Philosophy degree in December 2005, both in mechanical
engineering. During his graduate career, Curt was also selected to participate in the Space
Scholars program at the Air Force Research Laboratory’s Space Vehicles Directorate in
Albuquerque, New Mexico, where he spent the summers of 2002 and 2003.
Address: Center for Intelligent Material Systems and Structures
310 Durham Hall
Blacksburg, VA 24061
This dissertation was typeset with LATEX 2ε1 by the author.
1 A
LT
EX 2ε is an extension of LATEX. LATEX is a collection of macros for TEX. TEX is a trademark of
the American Mathematical Society. The macros used in formatting this dissertation were written by Greg
Walker, Department of Mechanical Engineering, Virginia Tech.
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