ESTIMATION-BASED ADAPTIVE
FILTERING AND CONTROL
a dissertation
submitted to the department of electrical engineering
and the committee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
Bijan Sayyar-Rodsari
July 1999
c Copyright by Bijan Sayyar-Rodsari 1999
All Rights Reserved
ii
I certify that I have read this dissertation and that in my opinion it is fully
adequate, in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
Professor Jonathan How
(Principal Adviser)
I certify that I have read this dissertation and that in my opinion it is fully
adequate, in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
Professor Thomas Kailath
I certify that I have read this dissertation and that in my opinion it is fully
adequate, in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
Dr. Babak Hassibi
I certify that I have read this dissertation and that in my opinion it is fully
adequate, in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
Professor Carlo Tomasi
Approved for the University Committee on Graduate Studies:
iii
Abstract
Adaptive systems have been used in a wide range of applications for almost four
decades. Examples include adaptive equalization, adaptive noise-cancellation, adaptive vibration isolation, adaptive system identification, and adaptive beam-forming.
It is generally known that the design of an adaptive filter (controller) is a difficult nonlinear problem for which good systematic synthesis procedures are still lacking.
Most existing design methods (e.g. FxLMS, Normalized-FxLMS, and FuLMS) are adhoc in nature and do not provide a guaranteed performance level. Systematic analysis
of the existing adaptive algorithms is also found to be difficult. In most cases, addressing even the fundamental question of stability requires simplifying assumptions
(such as slow adaptation, or the negligible contribution of the nonlinear/time-varying
components of signals) which at the very least limit the scope of the analysis to the
particular problem at hand.
This thesis presents a new estimation-based synthesis and analysis procedure for
adaptive “Filtered” LMS problems. This new approach formulates the adaptive filtering (control) problem as an H∞ estimation problem, and updates the adaptive weight
vector according to the state estimates provided by an H∞ estimator. This estimator
is proved to be always feasible. Furthermore, the special structure of the problem
is used to reduce the usual Riccati recursion for state estimate update to a simpler
Lyapunov recursion. The new adaptive algorithm (referred to as estimation-based
adaptive filtering (EBAF) algorithm) has provable performance, follows a simple update rule, and unlike previous methods readily extends to multi-channel systems
and problems with feedback contamination. A clear connection between the limiting behavior of the EBAF algorithm and the classical FxLMS (Normalized-FxLMS)
iv
algorithm is also established in this thesis.
Applications of the proposed adaptive design method are demonstrated in an Active Noise Cancellation (ANC) context. First, experimental results are presented for
narrow-band and broad-band noise cancellation in a one-dimensional acoustic duct.
In comparison to other conventional adaptive noise-cancellation methods (FxLMS
in the FIR case and FuLMS in the IIR case), the proposed method shows much
faster convergence and improved steady-state performance. Moreover, the proposed
method is shown to be robust to feedback contamination while conventional methods
can go unstable. As a second application, the proposed adaptive method was used
for vibration isolation in a 3-input/3-output Vibration Isolation Platform. Simulation results demonstrate improved performance over a multi-channel implementation
of the FxLMS algorithm. These results indicate that the approach works well in
practice. Furthermore, the theoretical results in this thesis are quite general and can
be applied to many other applications including adaptive equalization and adaptive
identification.
v
Acknowledgements
This thesis has greatly benefited from the efforts and support of many people whom
I would like to thank. First, I would like to thank my principle advisor Professor
Jonathan How. This research would not have been possible without Professor How’s
insights, enthusiasm and constant support throughout the project. I appreciate his
attention to detail and the clarity that he brought to our presentations and writings.
I would also like to acknowledge the help and support of Dr. Alain Carrier from Lockheed Martin’s Advanced Technology Center. His careful reading of all the manuscripts
and reports, his provocative questions, and his dedication to meaningful research has
greatly influenced this work. I would like to gratefully acknowledge members of my
defense and reading committee, Professor Thomas Kailath, Professor Carlo Tomasi,
and Dr. Babak Hassibi. It was from a class instructed by Professor Kailath and Dr.
Hassibi that the main concept of this thesis originated, and it was their research that
this thesis is based on. It is impossible to exaggerate the importance of Dr. Hassibi’s
contributions to this thesis. He has been a great friend and advisor throughout this
work for which I am truly thankful.
My thanks also goes to Professor Robert Cannon and Professor Steve Rock for giving me the opportunity to interact with wonderful friends in the Aerospace Robotics
Laboratory. The help from ARL graduates, Gordon Hunt, Steve Ims, Stef Sonck,
Howard Wang, and Kurt Zimmerman was crucial in the early stages of the research
at Lockheed. I have also benefited from interesting discussions with fellow ARL students Andreas Huster, Kortney Leabourne, Andrew Robertson, Heidi Schubert, and
Bruce Woodley, on both technical and non-technical issues. I am forever thankful for
their invaluable friendship and support. I also acknowledge the camaraderie of more
vi
recent ARL members, Tobe Corazzini, Steve Fleischer, Eric Frew, Gokhan Inalhan,
Hank Jones, Bob Kindel, Ed LeMaster, Mel Ni, Eric Prigge, and Luis Rodrigues.
I discussed all aspects of this thesis in great detail with Arash Hassibi. He helped
me more than I can thank him for. Lin Xiao and Hong S. Bae set up the hardware for
noise cancellation and helped me in all experiments. I appreciate all their assistance.
Thomas Pare, Haitham Hindi, and Miguel Lobo provided helpful comments about the
research. I also acknowledge the assistance from fellow ISL students, Alper Erdogan,
Maryam Fazel, and Ardavan Maleki. I would like to also name two old friends, Khalil
Ahmadpour and Mehdi Asheghi, whose friendship I gratefully value.
I owe an immeasurable amount of gratitude to my parents, Hossein and Salehe, my
sister, Mojgan, and my brother, Bahman, for their support throughout the numerous
ups and downs that I have experienced. Finally, my sincere thanks goes to my wife,
Samaneh, for her gracious patience and strength. I am sure they agree with me in
dedicating this thesis to Khalil.
vii
Contents
Abstract
iv
Acknowledgements
vi
List of Figures
xii
1 Introduction
1
1.1
Motivation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
An Overview of Adaptive Filtering (Control) Algorithms . . . . . . .
6
1.4
Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.5
Thesis Outline
12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Estimation-Based adaptive FIR Filter Design
14
2.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2
EBAF Algorithm - Main Concept
. . . . . . . . . . . . . . . . . . .
16
2.3
Problem Formulation
. . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.4
2.3.1
H2 Optimal Estimation
. . . . . . . . . . . . . . . . . . . . .
19
2.3.2
H∞ Optimal Estimation . . . . . . . . . . . . . . . . . . . . .
20
H∞ -Optimal Solution
. . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.4.1
γ-Suboptimal Finite Horizon Filtering Solution . . . . . . . .
21
2.4.2
γ-Suboptimal Finite Horizon Prediction Solution . . . . . . .
22
2.4.3
The Optimal Value of γ . . . . . . . . . . . . . . . . . . . . .
23
2.4.3.1
23
Filtering Case
. . . . . . . . . . . . . . . . . . . . .
viii
2.4.3.2
2.4.4
Prediction Case . . . . . . . . . . . . . . . . . . . .
27
Simplified Solution Due to γ = 1 . . . . . . . . . . . . . . . .
29
2.4.4.1
Filtering Case: . . . . . . . . . . . . . . . . . . . . .
29
2.4.4.2
Prediction Case: . . . . . . . . . . . . . . . . . . . .
30
2.5
Important Remarks
. . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.6
Implementation Scheme for EBAF Algorithm . . . . . . . . . . . . .
32
2.7
Error Analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.7.1
Effect of Initial Condition . . . . . . . . . . . . . . . . . . . .
35
2.7.2
Effect of Practical Limitation in Setting y(k) to ŝ(k|k) (ŝ(k))
36
2.8
Relationship to the Normalized-FxLMS/FxLMS Algorithms . . . . .
2.8.1
Prediction Solution and its Connection to the FxLMS Algorithm
2.8.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
Filtering Solution and its Connection to the Normalized-FxLMS
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9
38
Experimental Data & Simulation Results
40
. . . . . . . . . . . . . . .
41
2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3 Estimation-Based adaptive IIR Filter Design
58
3.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.2
Problem Formulation
. . . . . . . . . . . . . . . . . . . . . . . . . .
61
Estimation Problem . . . . . . . . . . . . . . . . . . . . . . .
63
Approximate Solution . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.2.1
3.3
3.3.1
γ-Suboptimal Finite Horizon Filtering Solution to the Linearized
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2
γ-Suboptimal Finite Horizon Prediction Solution to the Linearized Problem
3.3.3
66
. . . . . . . . . . . . . . . . . . . . . . . . .
Important Remarks
66
. . . . . . . . . . . . . . . . . . . . . . .
66
3.4
Implementation Scheme for the EBAF Algorithm in IIR Case . . . .
67
3.5
Error Analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.6
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
ix
4 Multi-Channel Estimation-Based Adaptive Filtering
4.1
4.2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
4.1.1
79
4.4
Multi-Channel FxLMS Algorithm
. . . . . . . . . . . . . . .
Estimation-Based Adaptive Algorithm for Multi Channel Case
. . .
81
. . . . . . . . . . . . . . . . . . . . . .
85
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
4.3.1
Active Vibration Isolation . . . . . . . . . . . . . . . . . . . .
86
4.3.2
Active Noise Cancellation . . . . . . . . . . . . . . . . . . . .
89
4.2.1
4.3
78
H∞ -Optimal Solution
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5 Adaptive Filtering via Linear Matrix Inequalities
104
5.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2
LMI Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2.1
Including H2 Constraints
. . . . . . . . . . . . . . . . . . . . 110
5.3
Adaptation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.4
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6 Conclusion
121
6.1
Summary of the Results and Conclusions
6.2
Future Work
. . . . . . . . . . . . . . . 121
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A Algebraic Proof of Feasibility
A.1 Feasibility of γf = 1
126
. . . . . . . . . . . . . . . . . . . . . . . . . . . 126
B Feedback Contamination Problem
128
C System Identification for Vibration Isolation Platform
132
C.1 Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
C.2 Identified Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
C.2.1 Data Collection Process . . . . . . . . . . . . . . . . . . . . . 133
C.2.2 Consistency of the Measurements . . . . . . . . . . . . . . . . 134
C.2.3 System Identification
. . . . . . . . . . . . . . . . . . . . . . 137
x
C.2.4 Control design model analysis . . . . . . . . . . . . . . . . . . 140
C.3 FORSE algorithm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Bibliography
155
xi
List of Figures
1.1
General block diagram for an FIR Filterm . . . . . . . . . . . . . . . . . .
13
1.2
General block diagram for an IIR Filter
. . . . . . . . . . . . . . . . . . .
13
2.1
General block diagram for an Active Noise Cancellation (ANC) problem . . . .
46
2.2
A standard implementation of FxLMS algorithm . . . . . . . . . . . . . . .
47
2.3
Pictorial representation of the estimation interpretation of the adaptive control
problem: Primary path is replaced by its approximate model
. . . . . . . . .
47
2.4
Block diagram for the approximate model of the primary path
. . . . . . . .
48
2.5
Schematic diagram of one-dimensional air duct . . . . . . . . . . . . . . . .
48
2.6
Transfer functions plot from Speakers #1 & #2 to Microphone #1
. . . . . .
49
2.7
Transfer functions plot from Speakers #1 & #2 to Microphone #2
. . . . . .
49
2.8
Validation of simulation results against experimental data for the noise cancellation problem with a single-tone primary disturbance at 150 Hz. The primary
disturbance is known to the adaptive algorithm. The controller is turned on at
t ≈ 3 seconds.
2.9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
Experimental data for the EBAF algorithm of length 4, when a noisy measurement
of the primary disturbance (a single-tone at 150 Hz) is available to the adaptive
algorithm (SNR=3). The controller is turned on at t ≈ 5 seconds.
. . . . . .
51
2.10 Experimental data for the EBAF algorithm of length 8, when a noisy measurement
of the primary disturbance (a multi-tone at 150 and 180 Hz) is available to the
adaptive algorithm (SNR=4.5). The controller is turned on at t ≈ 6 seconds.
.
52
2.11 Experimental data for the EBAF algorithm of length 16, when a noisy measurement of the primary disturbance (a band limited white noise) is available to the
adaptive algorithm (SNR=4.5). The controller is turned on at t ≈ 5 seconds.
xii
.
53
2.12 Simulation results for the performance comparison of the EBAF and (N)FxLMS
algorithms. For 0 ≤ t ≤ 5 seconds, the controller is off. For 5 < t ≤ 20 seconds
both adaptive algorithms have full access to the primary disturbance (a singletone at 150 Hz). For t ≥ 20 seconds the measurement of Microphone #1 is used
as the reference signal (hence feedback contamination problem). The length of
the FIR filter is 24.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2.13 Simulation results for the performance comparison of the EBAF and (N)FxLMS
algorithms. For 0 ≤ t ≤ 5 seconds, the controller is off. For 5 < t ≤ 40 seconds
both adaptive algorithms have full access to the primary disturbance (a band
limited white noise). For t ≥ 40 seconds the measurement of Microphone #1 is
used as the reference signal (hence feedback contamination problem). The length
of the FIR filter is 32.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
55
2.14 Closed-loop transfer function based on the steady state performance of the EBAF
and (N)FxLMS algorithms in the noise cancellation problem of Figure 2.13.
3.1
. .
56
General block diagram for the adaptive filtering problem of interest (with Feedback
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3.2
Basic Block Diagram for the Feedback Neutralization Scheme . . . . . . . . .
72
3.3
Basic Block Diagram for the Classical Adaptive IIR Filter Design . . . . . . .
73
3.4
Estimation Interpretation of the IIR Adaptive Filter Design
. . . . . . . . .
73
3.5
Approximate Model For the Unknown Primary Path . . . . . . . . . . . . .
74
3.6
Performance Comparison for EBAF and FuLMS Adaptive IIR Filters for Single-
Contamination)
Tone Noise Cancellation. The controller is switched on at t = 1 second. For
1 ≤ t ≤ 6 seconds adaptive algorithm has full access to the primary disturbance.
For t ≥ 6 the output of Microphone #1 is used as the reference signal (hence
feedback contamination problem). . . . . . . . . . . . . . . . . . . . . . .
3.7
75
Performance Comparison for EBAF and FuLMS Adaptive IIR Filters for MultiTone Noise Cancellation. The controller is switched on at t = 1 second. For
1 ≤ t ≤ 6 seconds adaptive algorithm has full access to the primary disturbance.
For t ≥ 6 the output of Microphone #1 is used as the reference signal (hence
feedback contamination problem). . . . . . . . . . . . . . . . . . . . . . .
xiii
76
4.1
General block diagram for a multi-channel Active Noise Cancellation (ANC) problem
4.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
Pictorial representation of the estimation interpretation of the adaptive control
. . . . . . . . .
91
4.3
Approximate Model for Primary Path . . . . . . . . . . . . . . . . . . . .
92
4.4
Vibration Isolation Platform (VIP)
. . . . . . . . . . . . . . . . . . . . .
92
4.5
A detailed drawing of the main components in the Vibration Isolation Platform
problem: Primary path is replaced by its approximate model
(VIP). Of particular importance are: (a) the platform supporting the middle mass
(labeled as component #5), (b) the middle mass that houses all six actuators (of
which only two, one control actuator and one disturbance actuator) are shown
(labeled as component #11), and (c) the suspension springs to counter the gravity (labeled as component #12). Note that the actuation point for the control
actuator (located on the left of the middle mass) is colocated with the load cell
(marked as LC1). The disturbance actuator (located on the right of the middle
. . . . . . . . . . . . . . . . . .
93
4.6
SVD of the MIMO transfer function . . . . . . . . . . . . . . . . . . . . .
94
4.7
Performance of a multi-channel implementation of EBAF algorithm when distur-
mass) actuates against the inertial frame.
bance actuators are driven by out of phase sinusoids at 4 Hz. The reference signal
available to the adaptive algorithm is contaminated with band limited white noise
(SNR=3). The control signal is applied for t ≥ 30 seconds.
4.8
95
Performance of a multi-channel implementation of FxLMS algorithm when simulation scenario is identical to that in Figure 4.7.
4.9
. . . . . . . . . .
. . . . . . . . . . . . . . .
96
Performance of a multi-channel implementation of EBAF algorithm when disturbance actuators are driven by out of phase multi-tone sinusoids at 4 and 15 Hz.
The reference signal available to the adaptive algorithm is contaminated with band
limited white noise (SNR=4.5). The control signal is applied for t ≥ 30 seconds.
97
4.10 Performance of a multi-channel implementation of FxLMS algorithm when simulation scenario is identical to that in Figure 4.9.
. . . . . . . . . . . . . . .
98
4.11 Performance of a Multi-Channel implementation of the EBAF for vibration isolation when the reference signals are load cell outputs (i.e. feedback contamination
exists). The control signal is applied for t ≥ 30 seconds.
xiv
. . . . . . . . . . .
99
4.12 Performance of the Multi-Channel noise cancellation in acoustic duct for a multitone primary disturbance at 150 and 200 Hz. The control signal is applied for
t ≥ 2 seconds.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.13 Performance of the Multi-Channel noise cancellation in acoustic duct when the
primary disturbance is a band limited white noise. The control signal is applied
for t ≥ 2 seconds.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.14 Closed-loop vs. open-loop transfer functions for the steady state performance of
the EBAF algorithm for the simulation scenario shown in Figure 4.13.
. . . . 102
5.1
General block diagram for an Active Noise Cancellation (ANC) problem . . . .
5.2
Cancellation Error at Microphone #1 for a Single-Tone Primary Disturbance
5.3
Typical Elements of Adaptive Filter Weight Vector for Noise Cancellation Problem
in Fig. 5.2
. 116
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.4
Cancellation Error at Microphone #1 for a Multi-Tone Primary Disturbance
5.5
Typical Elements of Adaptive Filter Weight Vector for Noise Cancellation Problem
in Fig. 5.4
B.1
. 118
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Block diagram of the approximate model for the primary path in the presence of
the feedback path
C.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Magnitude of the scaling factor relating load cell’s reading of the effect of control
actuators to that of the scoring sensor . . . . . . . . . . . . . . . . . . . .
C.2
146
Magnitude of the scaling factor relating load cell’s reading of the effect of disturbance actuators to that of the scoring sensor after diagonalization . . . . . . .
C.5
145
Magnitude of the scaling factor relating load cell’s reading of the effect of control
actuators to that of the scoring sensor after diagonalization . . . . . . . . . .
C.4
144
Magnitude of the scaling factor relating load cell’s reading of the effect of disturbance actuators to that of the scoring sensor . . . . . . . . . . . . . . . . .
C.3
115
147
Comparison of SVD plots for the transfer function to the scaled/double-integrated
load cell data
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
C.6
Comparison of SVD plots for the transfer function to the actual load cell data .
C.7
Comparison of SVD plots for the transfer function to the scoring sensors
C.8
Comparison of SVD plots for the transfer function to the position sensors colocated
with the control actuators
148
. . . 149
. . . . . . . . . . . . . . . . . . . . . . . . . 149
xv
C.9
Comparison of SVD plots for the transfer function to the position sensors colocated
with the disturbance actuators
. . . . . . . . . . . . . . . . . . . . . . . 150
C.10 The identified model for the system beyond the frequency range for which measurements are available . . . . . . . . . . . . . . . . . . . . . . . . . . .
151
C.11 The final model for the system beyond the frequency range for which measurements are available . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
152
C.12 The comparison of the closed loop and open loop singular value plots when the
controller is used to close the loop on the identified model
. . . . . . . . . . 153
C.13 The comparison of the closed loop and open loop singular value plots when the
controller is used to close the loop on the real measured data
xvi
. . . . . . . . . 154
Chapter 1
Introduction
This dissertation presents a new estimation-based procedure for the systematic synthesis and analysis of adaptive filters (controllers) in “Filtered” LMS problems. This
new approach uses an estimation interpretation of the adaptive filtering (control)
problem to formulate an equivalent estimation problem. The adaptation criterion for
the adaptive weight vector is extracted from the H∞ -solution to this estimation problem. The new algorithm, referred to as Estimation-Based Adaptive Filtering (EBAF),
applies to both Finite Impulse Response (FIR) and Infinite Impulse Response (IIR)
adaptive filters.
1.1
Motivation
Least-Mean Squares (LMS) adaptive algorithm [51] has been the centerpiece of a wide
variety of adaptive filtering techniques for almost four decades. The straightforward
derivation, and the simplicity of its implementation (especially at the time of limited
computational power) encouraged experiments with the algorithm in a diverse range
of applications (e.g. see [51,33]). In some applications however, the simple implementation of the LMS algorithm was found to be inadequate. Subsequent attempts
to overcome its shortcomings have produced a large number of innovative solutions
that have been successful in practice. Commonly used algorithms such as normalized
1
1.1. MOTIVATION
2
LMS, correlation LMS [47], leaky LMS [21], variable-step-size LMS [25], and FilteredX LMS [35] are the outcome of such efforts. These algorithms use the instantaneous
squared error to estimate the mean-square error, and often assume slow adaptation
to allow for the necessary linear operations in their derivation (see Chapters 2 and 3
in [33] for instance). As Reference [2] points out:
“Many of the algorithms and approaches used are of an ad hoc nature;
the tools are gathered from a wide range of fields; and good systematic
approaches are still lacking.”
Introducing a systematic procedure for the synthesis of adaptive filters is one of the
main goals of this thesis.
Parallel to the efforts on the practical application of the LMS-based adaptive
schemes, there has been a concerted effort to analyze these algorithms. Of pioneering
importance are the results in Refs. [50] and [23]. Reference [50] considers the adaptation with LMS on stationary stochastic processes, and finds the optimal solution
to which the expected value of the weight vector converges. For sinusoidal inputs
however, the discussion in [50] does not apply. In [23] it is shown that for sinusoidal
inputs, when time-varying component of the adaptive filter output is small compared
to its time-invariant component (see [23], page 486), the adaptive LMS filter can be
approximated by a linear time-invariant transfer function. Reference [13] extends the
approach in [23] to derive an equivalent transfer function for the Filtered-X LMS
adaptive algorithm (provided the conditions required in [23] still apply). The equivalent transfer function is then used to analytically derive an expression for the optimum
convergence coefficients. A frequency domain model of the so-called filtered LMS algorithm (i.e. an algorithm in which the input or the output of the adaptive filter or
the feedback error signal is linearly filtered prior to use in the adaptive algorithm)
is discussed in [17]. The frequency domain model in [17] decouples the inputs into
disjoint frequency bins and places a single frequency adaptive noise canceler on each
bin. The analysis in their work utilizes the frequency domain LMS algorithm [11]
and assumes a time invariant linear behavior for the filter. Other important aspects
1.1. MOTIVATION
3
of the adaptive filters have also been extensively studied. The effect of the modeling error on the convergence and performance properties of the LMS-based adaptive
algorithms (e.g. [17,7]), and tracking behavior of the LMS adaptive algorithm when
the adaptive filter is tuned to follow a linear chirp signal buried in white noise [5,6],
are examples of these studies∗ . In summary, existing analysis techniques are often
suitable for analyzing only one particular aspect of the behavior of an adaptive filter
(e.g. its steady-state behavior). Furthermore, the validity of the analysis relies on
certain assumptions (e.g. slow convergence, and/or the negligible contribution of the
nonlinear/time-varying component of the adaptive filter output) that can be quite
restrictive. Providing a solid framework for the systematic analysis of adaptive filters
is another main goal of this thesis.
The reason for the difficulty experienced in both synthesis and analysis of adaptive
algorithms is best explained in Reference [37]:
“It is now generally realized that adaptive systems are special classes of
nonlinear systems . . . general methods for the analysis and synthesis of
nonlinear systems do not exist since conditions for their stability can be
established only on a system by system basis.”
This thesis introduces a new framework for the synthesis and analysis of adaptive
filters (controllers) by providing an estimation interpretation of the above mentioned
“nonlinear” adaptive filtering (control) problem. The estimation interpretation replaces the original adaptive filtering (control) synthesis with an equivalent estimation
problem, the solution of which is used to update the weight vector in the adaptive
filter (and hence the name estimation-based adaptive filtering). This approach is
applicable (due to its systematic nature) to both FIR and IIR adaptive filters (controllers). In the FIR case the equivalent estimation problem is linear, and hence exact
solutions are available. Stability, performance bounds, transient behavior of adaptive
FIR filters are thus precisely addressed in this framework. In the IIR case, however,
only an approximate solution to the equivalent estimation problem is available, and
∗
The survey here is intended to provide a flavor of the type of the problems that have captured
the attention of researchers in the field. The shear volume of the literature makes subjective selection
of the references unavoidable.
1.2. BACKGROUND
4
hence the proposed estimation-based framework serves as a reasonable heuristic for
the systematic design of adaptive IIR filters. This approximate solution however, is
based on realistic assumptions, and the adaptive algorithm maintains its systematic
structure. Furthermore, the treatment of feedback contamination (see Chapter 3 for a
precise definition), is virtually identical to that of adaptive IIR filters. The proposed
estimation-based approach is particularly appealing if one considers the difficulty with
the existing design techniques for adaptive IIR filters, and the complexity of available
solutions to feedback contamination (e.g. see [33]).
1.2
Background
The development of the new estimation-based framework is based on recent results
in robust estimation. Following the pioneering work in [52], the H∞ approach to
robust control theory produced solutions [12,24] that were designed to meet some
performance criterion in the face of the limited knowledge of the exogenous disturbances and imperfect system models. Further work in robust control and estimation
(see [32,46] and the references therein) produced straightforward solutions that allowed in-depth studies of the properties of the robust controllers/estimators. The
main idea in H∞ estimation is to design an estimator that bounds (in the optimum
case, minimizes) the maximum energy gain from the disturbances to the estimation
errors. Such a solution guarantees that for disturbances with bounded energy, the
energy of the estimation error will be bounded as well. In the case of an optimal
solution, an H∞ -optimal estimator will guarantee that the energy of the estimation
error for the worst case disturbance is indeed minimized [28].
Of crucial importance for the work in this thesis, is the result in [26] where the H∞ optimality of the LMS algorithm was established. Note that despite a long history
of successful applications, prior to the work in [26], the LMS algorithm was regarded
as an approximate recursive solution to the least-squares minimization problem. The
work in [26] showed that instead of being an approximate solution to an H2 minimization, the LMS algorithm is the exact solution to a minmax estimation problem. More
1.2. BACKGROUND
5
specifically, Ref. [26] proved that the LMS adaptive filter is the central a priori H∞ optimal filter. This result established a fundamental connection between an adaptive
control algorithm (LMS algorithm in this case), and a robust estimation problem.
Inspired by the analysis in [26], this thesis introduces an estimation interpretation of
a far more general adaptive filtering problem, and develops a systematic procedure for
the synthesis of adaptive filters based on this interpretation. The class of problems
addressed in this thesis, commonly known as “Filtered” LMS [17], encompass a wide
range of adaptive filtering/control applications [51,33], and have been the subject of
extensive research over the past four decades. Nevertheless, the viewpoint provided
in this thesis not only provides a systematic alternative to some widely used adaptive
filtering (control) algorithms (such as FxLMS and FuLMS) with superior transient
and steady-state behavior, but it also presents a new framework for their analysis.
More specifically, this thesis proves that the fundamental connection between adaptive filtering (control) algorithms and robust estimation extends to the more general
setting of adaptive filtering (control) problems, and shows that the convergence, stability, and performance of these classical adaptive algorithms can be systematically
analyzed as robust estimation questions.
The systematic nature of the proposed estimation-based approach enables an alternative formulation for the adaptive filtering (control) problem using Linear Matrix
Inequalities (LMIs), the ramifications of which will be discussed in Chapter 5. Several
researchers (see [18] and references therein) in the past few years have shown that
elementary manipulations of linear matrix inequalities can be used to derive less restrictive alternatives to the now classical state-space Riccati-based solution to the H∞
control problem [12]. Even though the computational complexity of the LMI-based
solution remains higher than that of solving the Riccati equation, there are three main
reasons that justify such a formulation [19]: (a) a variety of design specifications and
constraints can be expressed as LMIs, (b) problems formulated as LMIs can be solved
exactly by efficient convex optimization techniques, and (c) for the cases that lack
analytical solutions such as mixed H2 /H∞ design objectives (see [4], [32] and [45] and
references therein), the LMI formulation of the problem remains tractable (i.e. LMIsolvers are viable alternatives to analytical solutions in such cases). As will be seen
1.3. AN OVERVIEW OF ADAPTIVE FILTERING (CONTROL) ALGORITHMS
6
in Chapter 5, the LMI framework provides the machinery required for the synthesis
of a robust adaptive filter in the presence of modeling uncertainty.
1.3
An Overview of Adaptive Filtering (Control)
Algorithms
To put this thesis in perspective, this section provides a brief overview of the vast
literature on adaptive filtering (control). Reference [36] recognizes 1957 as the year
for the formal introduction of the term “adaptive system” into the control literature.
By then, the interest in filtering and control theory had shifted towards increasingly
more complex systems with poorly characterized (possibly time varying) models for
system dynamics and disturbances, and the concept of “adaptation” (borrowed from
living systems) seemed to carry the potential for solving the increasingly more complex control problems. The exact definition of “adaptation” and its distinction from
“feedback”, however, is the subject of long standing discussions (e.g. see [2,36,29]).
Qualitatively speaking, an adaptive system is a system that can modify its behavior
in response to changes in the dynamics of the system or disturbances through some
recursive algorithm. As a direct consequence of this recursive algorithm (in which
the parameters of the adaptive system are adjusted using input/output data), an
adaptive system is a “nonlinear” device.
The development of adaptive algorithms has been pursued from a variety of view
points. Different classifications of adaptive algorithms (such as direct versus indirect
adaptive control, model reference versus self-tuning adaptation) in the literature reflect this diversity [2,51,29]. For the purpose of this thesis, two distinct approaches for
deriving recursive adaptive algorithms can be identified: (a) stochastic gradient approaches that include LMS and LMS-Based adaptive algorithms, and (b) least-squares
estimation approaches that include adaptive recursive least-squares (RLS) algorithm.
The central idea in the former approach, is to define an appropriate cost function
that captures the success of the adaptation process, and then change the adaptive
1.3. AN OVERVIEW OF ADAPTIVE FILTERING (CONTROL) ALGORITHMS
7
filter parameters to reduce the cost function according to the method of steepest descent. This requires the use of a gradient vector (hence the name), which in practice
is approximated using instantaneous data. Chapter 2 provides a detailed description
of this approach for the problem of interest in this Thesis. The latter approach to
the design of adaptive filters is based on the method of least squares. This approach
closely corresponds to Kalman filtering. Ref. [44] provides a unifying state-space approach to adaptive RLS filtering. The main focus in this thesis however, is on the
LMS-based adaptive algorithms.
Since adaptive algorithms can successfully operate in a poorly known environment,
they have been used in a diverse field of applications that include communication
(e.g. [34,41]), process control (e.g. [2]), seismology (e.g. [42]), biomedical engineering
(e.g. [51]). Despite the diversity of the applications, different implementations of
adaptive filtering (control) share one basic common feature [29]: “an input vector and
a desired response are used to compute an estimation error, which is in turn used to
control the values of a set of adjustable filter coefficients.” Reference [29] distinguishes
four main classes of adaptive filtering applications based on the way the desired
signal is defined in the formulation of the problem: (a) identification: in this class of
applications an adaptive filter is used to provide a linear model for an unknown plant.
The plant and the adaptive filter are driven by the same input, and the output of the
plant is the desired response that adaptive filter tries to match. (b) inverse modeling:
here the adaptive filter is placed in series with an unknown (perhaps noisy) plant, and
the desired signal is simply a delayed version of the plant input. Ideally, the adaptive
filter converges to the inverse of the unknown plant. Adaptive equalization (e.g. [40])
is an important application in this class. (c) prediction: the desired signal in this case
is the current value of a random signal, while past values of the random signal provide
the input to the adaptive filter. Signal detection is an important application in this
class. (d) interference canceling: here adaptive filter uses a reference signal (provided
as input to the adaptive filter) to cancel unknown interference contained in a primary
signal. Adaptive noise cancellation, echo cancellation, and adaptive beam-forming
are applications that fall in this last class. The estimation-based adaptive filtering
algorithm in this thesis is presented in the context of adaptive noise cancellation, and
1.3. AN OVERVIEW OF ADAPTIVE FILTERING (CONTROL) ALGORITHMS
8
therefore a detailed discussion of the fourth class of adaptive filtering problems is
provided in Chapter 2.
There are several main structures for the implementation of adaptive filters (controllers). The structure of the adaptive filter is known to affect its performance,
computational complexity, and convergence. In this thesis, the two most commonly
used structures for adaptive filters (controllers) are considered. The finite impulse
response (FIR) transversal filter (see Fig. 1.1) is the structure upon which the main
presentation of the estimation-based adaptive filtering algorithm is primarily presented. The transversal filter consists of three basic elements: (a) unit-delay element,
(b) multiplier, and (c) adder, and contains feed forwards paths only. The number
of unit-delays specify the length of the adaptive FIR filter. Multipliers weight the
delayed versions of some reference signal, which are then added in the adder(s). The
frequency response for this filter is of finite length (hence the name), and contains
only zeros (all poles are at the origin in the z-plane). Therefore, there is no question
of stability for the open-loop behavior of the FIR filter. The infinite-duration impulse
response (IIR) structure is shown in Figure 1.2. The feature that distinguishes the
IIR filter from an FIR filter is the inclusion of the feedback path in the structure of
the adaptive filter.
As mentioned earlier, for an FIR filter all poles are at the origin, and a good
approximation of the behavior of a pole, in general, can only be achieved if the length
of the FIR filter is sufficiently long. An IIR filter, ideally at least, can provide a
perfect match for a pole with only a limited number of parameters. This means
that for a desired dynamic behavior (such as resonance frequency, damping, or cutoff
frequency), the number of parameters in an adaptive IIR filter can be far fewer than
that in its FIR counterpart. The computational complexity per sample for adaptive
IIR filter design can therefore be significantly lower than that in FIR filter design.
The limited use of adaptive IIR filters (compared to the vast number of applications for the FIR filters) suggests that the above mentioned advantages come at
a certain cost. In particular, adaptive IIR filters are only conditionally stable, and
therefore some provisions are required to assure stability of the filter at each iteration.
There are solutions such as Schur-Cohn algorithm ([29] pages 271-273) that monitor
1.3. AN OVERVIEW OF ADAPTIVE FILTERING (CONTROL) ALGORITHMS
9
the stability of the IIR filter (by determining whether all roots of the denominator of
the IIR filter transfer function are inside the unit circle). This however requires intensive on-line calculations. Alternative implementations of adaptive IIR filters (such
as parallel implementation [48], and lattice implementation [38]) have been suggested
that provide simpler stability monitoring capabilities. The monitoring process is independent of the adaptation process here. In other words, the adaptation criteria
do not inherently reject de-stabilizing values for filter weights. The monitoring process detects these de-stabilizing values and prevents their implementation. Another
significant problem with adaptive IIR filter design stems from the fact that the performance surface (see [33], Chapter 3) for adaptive IIR filters is generally non-quadratic
(see [33] pages 91-94 for instance) and often contains multiple local minima. Therefore, the weight vector may converge to a local minimum only (hence non-optimal
cost). Furthermore, it is noted that the adaptation rate for adaptive IIR filters can
be slow when compared to the FIR adaptive filters [33,31]. Early works in adaptive
IIR filtering (e.g. [16]) are for the most part extensions to Widrow’s LMS algorithm
of adaptive FIR filtering [51]. More recent works include modifications to recursive
LMS algorithm (e.g. [15]) that are devised for specific applications. In other words,
existing design techniques for adaptive IIR filters are application-specific and rely on
certain restrictive assumptions in their derivation. Our description of the Filtered-U
recursive LMS algorithm in Chapter 3 will further clarify this point. Furthermore,
as [33] points out: “The properties of an adaptive IIR filter are considerably more
complex than those of the conventional adaptive FIR filter, and consequently it is
more difficult to predict their behavior.” Thus, a framework that allows a unified
approach to the synthesis and analysis of adaptive IIR filters, and does not require
restrictive assumptions for its derivation would be extremely useful. As mentioned
earlier, this thesis provides such a framework.
Finally, for a wide variety of applications such as equalization in wireless communication channels, and active control of sound and vibration in an environment
where the effect of a number of primary sources should be canceled by a number of
control (secondary) sources, the use of a multi-channel adaptive algorithm is well justified. In general, however, variations of the LMS algorithm are not easy to extend to
1.4. CONTRIBUTIONS
10
multi-channel systems. Furthermore, the analysis of the performance and properties
of such multi-channel algorithms is complicated [33]. As Ref. [33] points out, in the
context of active noise cancellation, the successful implementation of multi-channel
adaptive algorithms has so far been limited to cases involving repetitive noise with a
few harmonics [39,43,49,13]). For the approach presented in this thesis, the syntheses
of single-channel and multi-channel adaptive algorithms are virtually identical. This
similarity is a direct result of the way the synthesis problem is formulated (see 4).
1.4
Contributions
In meeting the goals of this research, the following contributions have been made to
adaptive filtering and control:
1. An estimation-interpretation for adaptive “Filtered” LMS filtering (control)
problems is developed. This interpretation allows an equivalent estimation formulation for the adaptive filtering (control) problem. The adaptation criterion
for adaptive filter weight vector is extracted from the solution to this equivalent estimation problem. This constitutes a systematic synthesis procedure for
adaptive filters in filtered LMS problems. The new synthesis procedure is called
Estimation-Based Adaptive Filtering (EBAF).
2. Using an H∞ criterion to formulate the “equivalent” estimation problem, this
thesis develops a new framework for the systematic analysis of Filtered LMS
adaptive algorithms. In particular, the results in this thesis extend the fundamental connection between the LMS adaptive algorithm and robust estimation
(i.e. H∞ optimality of the LMS algorithm [26]) to the more general setting of
filtered LMS adaptive problems.
3. For the EBAF algorithm in the FIR case:
(a) It is shown that the adaptive weight vector update can be based on the
central filtering (prediction) solution to a linear H∞ estimation problem,
the existence of which is guaranteed. It is also shown that the maximum
1.4. CONTRIBUTIONS
11
energy gain in this case can be minimized. Furthermore, the optimal energy gain is proved to be unity, and the conditions under which this bound
is achievable are derived.
(b) The adaptive algorithm is shown to be implementable in real-time. The
update rule requires a simple Lyapunov recursion that leads to a computational complexity comparable to that of filtered LMS adaptive algorithms
(e.g. FxLMS). The experimental data, along with extensive simulations
are presented to demonstrate the improved steady-state performance of
the EBAF algorithm (over FxLMS and Normalized-FxLMS algorithms),
as well as a faster transient response.
(c) A clear connection between the limiting behavior of the EBAF algorithm
and the existing FxLMS and Normalized-FxLMS adaptive algorithms has
been established.
4. For the EBAF algorithm in the IIR case, it is shown that the equivalent estimation problem is nonlinear. A linearizing approximation is then employed
that makes systematic synthesis of adaptive IIR filter tractable. The performance of the EBAF algorithm in this case is compared to the performance of
the Filtered-U LMS (FuLMS) adaptive algorithm, demonstrating the improved
performance in the EBAF case.
5. The treatment of feedback contamination problem is shown to be identical to
the IIR adaptive filter design in the new estimation-based framework.
6. A multi-channel extension of the EBAF algorithm demonstrates that the treatment of the single-channel and multi-channel adaptive filtering (control) problems in the new estimation based framework is virtually the same. Simulation
results for the problem of vibration isolation in a 3-input/3-output vibration isolation platform (VIP) prove feasibility of the EBAF algorithm in multi-channel
problems.
7. The new estimation-based framework is shown to be amenable to a Linear Matrix Inequality (LMI) formulation. The LMI formulation is used to explicitly
1.5. THESIS OUTLINE
12
address the stability of the overall system under adaptive algorithm by producing a Lyapunov function. It is also shown to be an appropriate framework to
address the robustness of the adaptive algorithm to modeling error or parameter uncertainty. Augmentation of an H2 performance constraint to the H∞
disturbance rejection criterion is also discussed.
1.5
Thesis Outline
The organization of this thesis is as follows. In Chapter 2, the fundamental concepts
of the estimation-based adaptive filtering (EBAF) algorithm are introduced. The
application of the EBAF approach in the case of adaptive FIR filter design is also
presented in this chapter. In Chapter 3, the extension of the EBAF approach to the
adaptive IIR filter design is discussed. A multi-channel implementation of the EBAF
algorithm is presented in Chapter 4. An LMI formulation for the EBAF algorithm is
derived in Chapter 5. Chapter 6 concludes this dissertation with a summary of the
main results, and the suggestions for future work. This dissertation contains three
appendices. An algebraic proof for the feasibility of the unity energy gain in the
estimation problem associated with adaptive FIR filter design (in Chapter 2) is discussed in Appendix A. The problem of feedback contamination is formally addressed
in Appendix B. A detailed discussion of the identification process is presented in Appendix C. The identified model for the Vibration Isolation Platform (VIP), used as a
test-bed for multi-channel implementation of the EBAF algorithm, is also presented
in this appendix.
1.5. THESIS OUTLINE
13
x(k − 1)
x(k)
z −1
W0
x(k − 2)
x(k − N )
z −1
W1
z −1
WN −1
W2
WN
+
u(k)
Fig. 1.1: General block diagram for an FIR Filterm
x(k)
u(k)
r(k)
a0
+
+
z −1
a1
b1
z −1
a2
b2
r(k − 2)
z −1
bN
aN
Fig. 1.2: General block diagram for an IIR Filter
Chapter 2
Estimation-Based adaptive FIR
Filter Design
This chapter presents a systematic synthesis procedure for H∞ -optimal adaptive FIR
filters in the context of an Active Noise Cancellation (ANC) problem. An estimation
interpretation of the adaptive control problem is introduced first. Based on this interpretation, an H∞ estimation problem is formulated, and its finite horizon prediction
(filtering) solutions are discussed. The solution minimizes the maximum energy gain
from the disturbances to the predicted (filtered) estimation error, and serves as the
adaptation criterion for the weight vector in the adaptive FIR filter. This thesis refers
to the new adaptation scheme as Estimation-Based Adaptive Filtering (EBAF). It
is shown in this chapter that the steady-state gain vectors in the EBAF algorithm
approach those of the classical Filtered-X LMS (Normalized Filtered-X LMS) algorithm. The error terms, however, are shown to be different, thus demonstrating that
the classical algorithms can be thought of as an approximation to the new EBAF
adaptive algorithm.
The proposed EBAF algorithm is applied to an active noise cancellation problem
(both narrow-band and broad-band cases) in a one-dimensional acoustic duct. Experimental data as well as simulations are presented to examine the performance of
the new adaptive algorithm. Comparisons to the results from a conventional FxLMS
algorithm show faster convergence without compromising steady-state performance
14
2.1. BACKGROUND
15
and/or robustness of the algorithm to feedback contamination of the reference signal.
2.1
Background
This section introduces the context in which the new estimation-based adaptive filtering (EBAF) algorithm will be presented. It defines the adaptive filtering problem
of interest and describes the terminology that is used in this chapter. A conventional
solution to the problem based on the FxLMS algorithm is also outlined in this section. The discussion of key concepts of the EBAF algorithm and the mathematical
formulation of the algorithm are left to Sections 2.2 and 2.3, respectively.
Referring to Fig. 2.1, the objective in this adaptive filtering problem is to adjust
the weight vector in the adaptive FIR filter, W (k) = [w0 (k) w1 (k) ... wN (k)]T (k is
the discrete time index), such that the cancellation error, d(k) −y(k), is small in some
appropriate measure. Note that d(k) and y(k) are outputs of the primary path P (z)
and the secondary path S(z), respectively. Moreover,
1. n(k) is the input to the primary path,
2. x(k) is a properly selected reference signal with a non-zero correlation with the
primary input,
4
3. u(k) is the control signal applied to the secondary path (generated as u(k) =
[x(k) x(k − 1) · · · x(k − N)] W (k)),
4. e(k) is the measured residual error available to the adaptation scheme.
Note that in a typical practice, x(k) is obtained via some measurement of the primary
input. The quality of this measurement will impact the correlation between the
reference signal and the primary input. Similar to the conventional development of
the FxLMS algorithm however, this chapter assumes perfect correlation between the
two.
The Filtered-X LMS (FxLMS) solution to this problem is shown in Figure 2.2
where perfect correlation between the primary disturbance n(k) and the reference
signal x(k) is assumed [51,33]. Minimizing the instantaneous squared error, e2 (k), as
2.2. EBAF ALGORITHM - MAIN CONCEPT
16
an approximation to the mean-square error, FxLMS follows the LMS update criterion
(i.e. to recursively adapt the weight vector in the negative gradient direction)
µ 2
∇e (k)
2
e(k) = d(k) − y(k) = d(k) − S(k) ⊕ u(k)
W (k + 1) = W (k) −
where µ is the adaptation rate, S(k) is the impulse response of the secondary path,
and “⊕” indicates convolution. Assuming slow adaptation, the FxLMS algorithm
then approximates the instantaneous gradient in the weight vector update with
4
T
∇e2 (k) ∼
= −2 [x0 (k) x0 (k − 1) · · · x0 (k − N)] e(k) = −2h0 (k)e(k)
(2.1)
4
where x0 (k) = S(k) ⊕ x(k) represents a filtered version of the reference signal which
is available to the LMS adaptation (and hence the name (Normalized) Filtered-X
LMS). This yields the following adaptation criterion for the FxLMS algorithm
W (k + 1) = W (k) + µh0 (k)e(k)
(2.2)
A closely related adaptive algorithm is the one in which the adaptation rate is
normalized with the estimate of the power of the reference vector, i.e.
W (k + 1) = W (k) + µ
h0 (k)
e(k)
1 + µh∗ 0 (k)h0 (k)
(2.3)
where ∗ indicates complex conjugate. This algorithm is known as the NormalizedFxLMS algorithm.
In practice, however, only an approximate model of the secondary path (obtained
via some identification scheme) is known, and it is this approximate model that is
used to filter the reference signal. For further discussion on the derivation and analysis
of the FxLMS algorithm please refer to [33,7].
2.2
EBAF Algorithm - Main Concept
The principal goal of this section is to introduce the underlying concepts of the new
EBAF algorithm. For the developments in this section, perfect correlation between
2.2. EBAF ALGORITHM - MAIN CONCEPT
17
n(k) and x(k) in Fig. 2.1 is assumed (i.e. x(k) = n(k) for all k). This is the same
condition under which the FxLMS algorithm was developed. The dynamics of the
secondary path are assumed known (e.g. by system identification). No explicit model
for the primary path is needed.
As stated before, the objective in the adaptive filtering problem of Fig. 2.1 is to
generate a control signal, u(k), such that the output of the secondary path, y(k), is
“close” to the output of the primary path, d(k). To achieve this goal, for the given
reference signal x(k), the series connection of the FIR filter and the secondary path
must constitute an appropriate model for the unknown primary path. In other words,
with the adaptive FIR filter properly adjusted, the path from x(k) to d(k) must be
equivalent to the path from x(k) to y(k). Based on this observation, in Fig. 2.3 the
structure of the path from x(k) to y(k) is used to model the primary path. The
modeling error is included to account for the imperfect cancellation.
The above mentioned observation forms the basis for an estimation interpretation of the adaptive control problem. The following outlines the main steps for this
interpretation:
1. Introduce an approximate model for the primary path based on the architecture
of the adaptive path from x(k) to y(k) (as shown in Fig. 2.3). There is an
optimal value for the weight vector in the approximate model’s FIR filter for
which the modeling error is the smallest. This optimal weight vector, however,
is not known. State-space models are used for both FIR filter and the secondary
path.
2. In the approximate model for the primary path, use the available information to
formulate an estimation problem that recursively estimates this optimal weight
vector.
3. Adjust the weight vector of the adaptive FIR filter to the best available estimate
of the optimal weight vector.
Before formalizing this estimation-based approach, a closer look at the signals
(i.e. information) involved in Fig. 2.1 is provided. Note that e(k) = d(k) − y(k) +
Vm (k), where
2.3. PROBLEM FORMULATION
18
a. e(k) is the available measurement.
b. Vm (k) is the exogenous disturbance that captures the effect of measurement
noise, modeling error, and the initial condition uncertainty in error measurements.
c. y(k) is the output of the secondary path.
d. d(k) is the output of the primary path.
Note that unlike e(k), the signals y(k) and d(k) are not directly measurable. With
u(k) fully known, however, the assumption of a known initial condition for the secondary path leads to the exact knowledge of y(k). This assumption is relaxed later
in this chapter, where the effect of an “inexact” initial condition in the performance
of the adaptive filter is studied (Section 2.7).
The derived measured quantity that will be used in the estimation process can
now be introduced as
4
m(k) = e(k) + y(k) = d(k) + Vm (k)
2.3
(2.4)
Problem Formulation
Figure 2.4 shows a block diagram representation of the approximate model to the
primary path. A state space model, [ As (k), Bs (k), Cs (k), Ds (k) ], for the secondary
path is assumed. Note that both primary and secondary paths are assumed stable.
The weight vector, W (k) = [ w0 (k) w1 (k) · · · wN (k) ]T , is treated as the state vector
capturing the trivial dynamics, W (k + 1) = W (k), that is assumed
for the FIR filter.
With θ(k) the state variable for the secondary path, then ξ T =
W T (k) θT (k)
is
the state vector for the overall system.
The state space representation of the system is then
#"
#
"
#
"
0
W (k) 4
W (k + 1)
I(N +1)×(N +1)
= Fk ξk
=
Bs (k)h∗ (k) As (k)
θ(k)
θ(k + 1)
(2.5)
2.3. PROBLEM FORMULATION
19
where h(k) = [x(k) x(k − 1) · · · x(k − N)]T captures the effect of the reference input
x(·). For this system, the derived measured output defined in Eq. (2.4) is
#
"
h
i W (k)
4
m(k) = Ds (k)h∗ (k) Cs (k)
+ Vm (k) = Hk ξk + Vm (k)
θ(k)
(2.6)
A linear combination of the states is defined as the desired quantity to be estimated
"
#
h
i W (k)
4
s(k) = L1,k L2,k
= Lk ξk
(2.7)
θ(k)
For simplicity, the single-channel problem is considered here. Extension to the multichannel case is straight forward and is discussed in Chapter 4. Therefore, m(k) ∈
R1×1 , s(k) ∈ R1×1 , θ(k) ∈ RNs ×1 , and W (k) ∈ R(N +1)×1 . All matrices are then
of appropriate dimensions. There are several alternatives for selecting Lk and thus
the variable to be estimated, s(k). The end goal of the estimation based approach
however, is to set the weight vector in the adaptive FIR filter such that the output
of the secondary path, y(k) in Fig. 2.3, best matches d(k). So s(k) = d(k) is chosen,
i.e. Lk = Hk .
Any estimation algorithm can now be used to generate an estimate of the desired
quantity s(k). Two main estimation approaches are considered next.
2.3.1
H2 Optimal Estimation
Here stochastic interpretation of the estimation problem is possible. Assuming that
ξ0 (the initial condition for the system in Figure 2.4) and Vm (·) are zero mean uncorrelated random variables with known covariance matrices
"
#
#
"
h
i
ξ0
0
Π
0
∗
E
=
ξ0∗ Vm
(j)
0 Qk δkj
Vm (k)
(2.8)
4
ŝ(k|k) = F (m(0), · · · , m(k)), the causal linear least-mean-squares estimate of s(k), is
given by the Kalman filter recursions [27].
There are two primary difficulties with the H2 optimal solution: (a) The H2 solution is optimal only if the stochastic assumptions are valid. If the external disturbance
2.3. PROBLEM FORMULATION
20
is not Gaussian (for instance when there is a considerable modeling error that should
be treated as a component of the measurement disturbance) then pursuing an H2
filtering solution may yield undesirable performance; and (b) regardless of the choice
for Lk , the recursive H2 filtering solution does not simplify to the same extent as
the H∞ solution considered below. This can be of practical importance when the
real-time computational power is limited. Therefore, the H2 optimal solution is not
employed in this chapter.
2.3.2
H∞ Optimal Estimation
To avoid difficulties associated with the H2 estimation, we consider a minmax formulation of the estimation problem in this section. Here, the main objective is to limit
the worst case energy gain from the measurement disturbance and the initial condition uncertainty to the error in a causal (or strictly causal) estimate of s(k). More
specifically, the following two cases are of interest. Let ŝ(k|k) = Ff (m(0), · · · , m(k))
denote an estimate of s(k) given observations m(i) for time i = 0 up to and including
4
time i = k, and let ŝ(k) = ŝ(k|k − 1) = Fp (m(0), · · · , m(k − 1)) denote an estimate
of s(k) given m(i) for time i = 0 up to and including i = k − 1. Note that ŝ(k|k)
and ŝ(k) are known as filtering and prediction estimates of s(k), respectively. Two
estimation errors can now be defined: the filtered error
ef,k = ŝ(k|k) − s(k)
(2.9)
ep,k = ŝ(k) − s(k)
(2.10)
and the predicted error
Given a final time M, the objective of the filtering problem can now be formalized as
finding ŝ(k|k) such that for Π0 > 0
M
X
sup
Vm , ξ0
e∗f,k ef,k
k=0
ˆ
(ξ0 − ξˆ0 )∗ Π−1
0 (ξ0 − ξ0 ) +
M
X
k=0
≤ γ2
∗
Vm
(k)Vm (k)
(2.11)
2.4. H∞ -OPTIMAL SOLUTION
21
for a given scalar γ > 0. In a similar way, the objective of the prediction problem can
be formalized as finding ŝ(k) such that
M
X
sup
Vm , ξ0
e∗p,k ep,k
k=0
ˆ
(ξ0 − ξˆ0 )∗ Π−1
0 (ξ0 − ξ0 ) +
M
X
≤ γ2
(2.12)
∗
Vm
(k)Vm (k)
k=0
for a given scalar γ > 0. The question of optimality of the solution can be answered
by finding the infimum value among all feasible γ’s. Note that, for the H∞ optimal
estimation there is no statistical assumption regarding the measurement disturbance.
Therefore, the inclusion of the output of the modeling error block (see Fig. 2.3) in
the measurement disturbance is consistent with H∞ formulation of the problem. The
elimination of the “modeling error” block in the approximate model of primary path
in Fig. 2.4 is based on this characteristic of the disturbance in an H∞ formulation.
2.4
H∞ -Optimal Solution
For the remainder of this chapter, the case where Lk = Hk is considered. Referring
to Figure 2.4, this means that s(k) = d(k). To discuss the solution, from [27] the
solutions to the γ-suboptimal finite-horizon filtering problem of Eq. (2.11), and the
prediction problem of Eq. (2.12) are drawn. Finally, we find the optimal value of γ
and show how γ = γopt simplifies the solutions.
2.4.1
γ-Suboptimal Finite Horizon Filtering Solution
Theorem 2.1: [27]Consider the state space representation of the block diagram of
Figure 2.4, described by Equations (2.5)-(2.7). A level-γ H∞ filter that achieves
(2.11) exists if, and only if, the matrices
0
0
Hk
Ip
Ip
Rk =
and Re,k =
+
Pk Hk∗ L∗k
(2.13)
2
2
0 −γ Iq
0 −γ Iq
Lk
(here p and q are used to indicate the correct dimensions) have the same inertia for
all 0 ≤ k ≤ M, where P0 = Π0 > 0 satisfies the Riccati recursion
∗
Pk+1 = Fk Pk Fk∗ − Kf,k Re,k Kf,k
(2.14)
2.4. H∞ -OPTIMAL SOLUTION
22
where Kf,k =
Fk Pk
Hk∗ L∗k
−1
Re,k
If this is the case, then the central H∞ estimator is given by
ξˆk+1 = Fk ξˆk + Kf,k m(k) − Hk ξˆk , ξˆ0 = 0
−1
ŝ(k|k) = Lk ξˆk + (Lk Pk Hk∗ ) RHe,k
m(k) − Hk ξˆk
(2.15)
(2.16)
(2.17)
−1
with Kf,k = (Fk Pk Hk∗ ) RHe,k
and RHe,k = Ip + Hk Pk Hk∗ .
Proof: see [27].
2.4.2
γ-Suboptimal Finite Horizon Prediction Solution
Theorem 2.2: [27]For the system described by Equations (2.5)-(2.7), level-γ H∞
filter that achieves (2.12) exists if, and only if, all leading sub-matrices of
Lk
−γ 2 Ip 0
−γ 2 Ip 0
p
p
(2.18)
Rk =
and Re,k =
+
Pk L∗k Hk∗
0
Iq
0
Iq
Hk
have the same inertia for all 0 ≤ k < M. Note that Pk is updated according to Eq.
(2.14). If this is the case, then one possible level-γ H∞ filter is given by
ξˆk+1 = Fk ξˆk + Kp,k m(k) − Hk ξˆk , ξˆ0 = 0
(2.19)
ŝ(k) = Lk ξˆk
where
Kp,k =
Fk P̃k Hk∗
(2.20)
I+
Hk P̃k Hk∗
−1
(2.21)
and
P̃k = I − γ −2 Pk L∗k Lk
−1
Pk ,
(2.22)
Proof: see [27].
Note that the condition in Eq. (2.18) is equivalent to
I − γ −2 Pk L∗k Lk > 0,
for k = 0, · · · , M
(2.23)
and hence P̃k in Eq. (2.22) is well defined. P̃k can also be defined as
P̃k−1 = Pk−1 − γ −2 L∗k Lk ,
for k = 0, · · · , M
(2.24)
2.4. H∞ -OPTIMAL SOLUTION
23
which proves useful in rewriting the prediction coefficient, Kp,k in Eq. (2.21), as
follows. First, note that
−1
−1
Fk P̃k Hk∗ I + Hk P̃k Hk∗
= Fk P̃k−1 + Hk∗ Hk
Hk∗
(2.25)
and hence, replacing for P̃k−1 from Eq. (2.24)
Kp,k = Fk Pk−1 − γ −2 L∗k Lk + Hk∗ Hk
−1
Hk∗
(2.26)
Theorems 2.1 and 2.2 (Sections 2.4.1 and 2.4.2) provide the form of the filtering and
prediction estimators, respectively. The following section investigates the optimal
value of γ for both of these solutions, and outlines the simplifications that follow.
2.4.3
The Optimal Value of γ
The optimal value of γ for the filtering solution will be discussed first. The discussion
of the optimal prediction solution utilizes the results in the filtering case.
2.4.3.1
Filtering Case
2.4.3.1.1 γopt ≤ 1: First, it will be shown that for the filtering solution γopt ≤ 1.
Using Eq. (2.11), one can always pick ŝ(k|k) to be simply m(k). With this choice
ŝ(k|k) − s(k) = Vm (k), for all k
(2.27)
and Eq. (2.11) reduces to
M
X
sup
Vm ∈ L2 , ξ0
Vm (k)∗ Vm (k)
k=0
ˆ
(ξ0 − ξˆ0 )∗ Π−1
0 (ξ0 − ξ0 ) +
M
X
(2.28)
Vm (k)∗ Vm (k)
k=0
which can never exceed 1 (i.e. γopt ≤ 1). A feasible solution for the H∞ estimation
problem in Eq. (2.11) is therefore guaranteed when γ is chosen to be 1. Note that
it is possible to directly demonstrate the feasibility of γ = 1. Using simple matrix
2.4. H∞ -OPTIMAL SOLUTION
24
manipulation, it can be shown that for Lk = Hk and for γ = 1, Rk and Re,k have the
same inertia for all k.
2.4.3.1.2 γopt ≥ 1: To show that γopt is indeed 1, an admissible sequence of disturbances and a valid initial condition should be constructed such that γ could be made
arbitrarily close to 1 regardless of the filtering solution chosen. The necessary and
sufficient conditions for the optimality of γopt = 1 are developed in the course of
constructing this admissible
sequence
of disturbances.
T
T
T
Assume that ξˆ0 = Ŵ0 θ̂0
is the best estimate for the initial condition of the
system in the approximate model of the primary path (Fig. 2.4). Moreover, assume
that θ̂0 is indeed the actual initial condition for the secondary path in Fig. 2.4. The
actual initial condition for the weight vector of the FIR filter in this approximate
model is W0 . Then,
h
m(0) =
H0 ξˆ0 =
h
Ds (0)h∗ (0) Cs (0)
Ds (0)h∗ (0) Cs (0)
i
i
"
"
W0
θ̂0
Ŵ0
#
+ Vm (0)
(2.29)
#
θ̂0
where m(0) is the (derived) measurement at time k = 0. Now, if
∗
Vm (0) = Ds (0)h (0) Ŵ0 − W0 = KV (0) Ŵ0 − W0
(2.30)
(2.31)
then m(0) − H0 ξˆ0 = 0 and the estimate of the weight vector will not change. More
specifically, Eqs. (2.16) and (2.17) reduce to the following simple updates
ξˆ1 = F0 ξˆ0
(2.32)
ŝ(0|0) = L0 ξˆ0
(2.33)
which given L0 = H0 generates the estimation error
ef,0 = ŝ(0|0) − s(0)
= L0 ξˆ0 − L0 ξ0
= Ds (0)h∗ (0) Ŵ0 − W0
= Vm (0)
(2.34)
2.4. H∞ -OPTIMAL SOLUTION
25
Repeating a similar argument at k = 1 and 2, it is easy to see that if
Vm (1) = [Ds (1)h∗ (1) + Cs (1)Bs (0)h∗ (0)] Ŵ0 − W0 = KV (1) Ŵ0 − W0
(2.35)
and
∗
∗
∗
Vm (2) = [Ds (2)h (2) + Cs (2)Bs (1)h (1) + Cs (2)As (1)Bs (0)h (0)] Ŵ0 − W0
= KV (2) Ŵ0 − W0
(2.36)
then
m(k) − Hk ξˆk = 0,
for k = 1, 2
(2.37)
Note that when Eq. (2.37) holds, and with Lk = Hk , Eq. (2.17) reduces to
ŝ(k|k) = Lk ξˆk = Hk ξˆk
(2.38)
and hence
ef,k = ŝ(k|k) − s(k)
= ŝ(k|k) − [m(k) − Vm (k)]
= Hk ξˆk − [m(k) − Vm (k)]
h
i
= Hk ξˆk − m(k) + Vm (k)
= Vm (k)
for k = 1, 2
(2.39)
Continuing this process, KV (k), for 0 ≤ k ≤ M can be defined as

KV (0)


Ds (0)
0
0
0
···
0

h(0)


 


 KV (1)   Cs (1)Bs (0)
Ds (1)
0
0 ···
0   h(1) 

 



  C (2)A (1)B (0) C (2)B (1) D (2) 0 · · ·


0
(2)
K
h(2)
=
 V
s
s
s
s
s
  s







..
..
..
.. 






.
.
.

 
 . 
..
KV (M)
h(M)
.
···
Ds (M)
4
= ∆M ΛM
(2.40)
2.4. H∞ -OPTIMAL SOLUTION
26
such that Vm (k), ∀ k, is an admissible disturbance. In this case, Eq. (2.11) reduces
to
M
X
sup
ξ0
Vm (k)∗ Vm (k)
k=0
ˆ
(ξ0 − ξˆ0 )∗ Π−1
0 (ξ0 − ξ0 ) +
M
X
Vm (k)∗ Vm (k)
k=0
M
X
#
"
=
(Ŵ0 − W0 )∗
sup
ξ0
KV∗ (k)KV (k)
k=0
"
∗
ˆ
(ξ0 − ξˆ0 )∗ Π−1
0 (ξ0 − ξ0 ) + (Ŵ0 − W0 )
M
X
(Ŵ0 − W0 )
#
KV∗ (k)KV (k)
(Ŵ0 − W0 )
k=0
(2.41)
From Eq. (2.40), note that
M
X
KV∗ (k)KV (k) = Λ∗M ∆∗M ∆M ΛM = k ∆M ΛM k22
(2.42)
k=0
and hence the ratio in Eq. (2.41) can be made arbitrarily close to one if
lim k∆M ΛM k2 → ∞
M →∞
(2.43)
Eq. (2.43) will be referred to as the condition for optimality of γ = 1 for the
filtering solution.
Equation (2.43) can now be used to derive necessary and sufficient conditions for
optimality of γ = 1. First, note that a necessary condition for Eq. (2.43) is
lim kΛM k2 → ∞
M →∞
(2.44)
or equivalently
lim
M →∞
M
X
h∗ (k)h(k) → ∞
(2.45)
k=0
The h(k) that satisfies the condition in (2.45) is referred to as exciting [26]. Several
sufficient conditions can now be developed. Since
k∆M ΛM k2 ≥ σmin (∆M ) kΛM k2
(2.46)
2.4. H∞ -OPTIMAL SOLUTION
27
one sufficient condition is that
σmin (∆M ) > ,
∀ M, and > 0
(2.47)
Note that for LTI systems, the sufficient condition (2.47) is equivalent to the requirement that the system have no zeros on the unit circle. Another sufficient condition
is that h(k)’s be persistently exciting, that is
#
"
M
1 X
lim σmin
h(k)h∗ (k) > 0
M →∞
M k=0
(2.48)
which holds for most reasonable systems.
2.4.3.2
Prediction Case
The optimal value for γ can not be less than one in the prediction case. In the previous
section we showed that despite using all available measurements up to and including
time k, the sequence of the admissible disturbances, Vm (k) = KV (k) Ŵ0 − W0 for
k = 0, · · · , M (where KV (k) is given by Eq. (2.40)), prevented the filtering solution
from achieving γ < 1. The prediction solution that uses only the measurements up to
time k (not including k itself) can not improve over the filtering solution and therefore
the energy gain γ is at least one.
Next, it is shown that if the initial condition P0 is chosen appropriately (i.e. if it is
small enough), then γopt = 1 can be guaranteed. Referring to the Lyapunov recursion
of Eq. (2.65), the Riccati matrix at time k can be written as:
#
"
!
!∗
k−1
k−1
Y
Y
I
0
Pk =
Fj P0
Fj ,
Fj =
Bs (j)h∗ (j) As (j)
j=0
j=0
(2.49)
Defining
ΨjA = As (j)As (j − 1) · · · As (0)
(2.50)
Eq. (2.49) can be written as
# "
#∗
"
I
0
I
0
Pk = Pk−1 j
P0 Pk−1 j
(2.51)
∗
k
∗
k
j=0 ΨA Bs (j)h (k−1−j) ΨA
j=0 ΨA Bs (j)h (k−1−j) ΨA
2.4. H∞ -OPTIMAL SOLUTION
28
From Theorem 2.2, Section 2.4.2, the condition for the existence of a prediction
solution is (I − γ −2 Pk L∗k Lk ) > 0, or equivalently
(γ 2 − Lk Pk L∗k ) > 0
(2.52)
Note that Lk = [ Ds (k)h∗ (k) Cs (k) ], and therefore Eq. (2.52) can be re-written as
#
"
h
i
∗
h(k)D
(k)
s
γ 2 − Ds (k)h∗ (k) Cs (k) Pk
>0
(2.53)
∗
Cs (k)
Replacing for Pk from Eq. (2.51), and carrying out the matrix multiplications, Eq. (2.53)
is equivalent to
γ2
"
−
"
h(k)Ds∗ (k) +
Pk−1
∗j ∗
∗
j=0 h(k−1−j)Bs (j)ΨA Cs (k)
∗
Ψ∗k
A Cs (k)
h(k)Ds∗ (k) +
Pk−1
j=0
∗
h(k−1−j)Bs∗ (j)Ψ∗j
A Cs (k)
∗
Ψ∗k
A Cs (k)
#∗
× P0 ×
#
>0
(2.54)
Introducing
0∗
∗
h (k) = Ds h (k) +
k−1
X
Cs (k)ΨjA Bs (j)h∗ (k−1−j)
(2.55)
j=0
as the filtered version of the reference vector, h(k), Eq. (2.54) can be expressed as
#
"
i
h
0
h
(k)
γ 2 − h0 ∗ (k) Cs (k)ΨkA P0
>0
(2.56)
∗
Ψ∗k
C
(k)
A s
Selecting the initial value of the Riccati matrix, without loss of generality, as
#
"
µI 0
(2.57)
P0 =
0 αI
and the Eq. (2.56) reduces to
∗
∗
γ 2 − µh0 (k)h0 (k) − αCs (k)ΨkA Ψ∗k
A Cs (k) > 0
(2.58)
It is now clear that a prediction solution for γ = 1 exists if
∗
1 − αCs (k)ΨkA Ψ∗k
A Cs (k)
µ<
h0 ∗ (k)h0 (k)
(2.59)
Equation (2.59) is therefore the condition for optimality of γopt = 1 for the prediction
solution.
2.4. H∞ -OPTIMAL SOLUTION
29
2.4.4
Simplified Solution Due to γ = 1
2.4.4.1
Filtering Case:
The following shows that with Hk = Lk and γ = 1, the Riccati equation (2.14)
is considerably simplified. To this end, apply the matrix inversion lemma, (A +
BCD)−1 = A−1 − A−1 B[C −1 + DA−1 B]−1 DA−1 , to
"
# "
#
h
i
0
Ip
Hk
Re,k =
+
Pk Hk∗ Hk∗
0 −Iq
Hk
"
with A =
Ip
0
#
"
,B=
Hk
#
h
, C = I, and D = Pk
(2.60)
i
Hk∗ Hk∗ . It is easy to
0 −Iq
Hk
−1
verify that the term DA B is zero. Therefore
# "
#
"
h
i
Hk
0
Ip
−1
∗
∗
−
Pk Hk −Hk
Re,k =
−Hk
0 −Iq
(2.61)
In which case,
h
∗
Kf,k Re,k Kf,k
=
Fk Pk
Hk∗ Hk∗
i
"
−1
Re,k
Hk
#
Hk
!
Pk Fk∗
=0
for γ = 1 and for all k. Thus the Riccati recursion (2.14) reduces to the Lyapunov
recursion Pk+1 = Fk Pk Fk∗ with P0 = Π0 > 0.
Partitioning the Riccati matrix Pk in block matrices conformable with the block
matrix structure of Fk , (2.14) yields the following simple update


P11,k+1 = P11,k , P11,0 = Π11,0









 P
= P A (k) + P h(k)B ∗ (k), P
=Π
12,k+1
12,k
s
11,k
s
12,0
12,0





∗

P22,k+1 = Bs (k)h(k)∗ P11,k h(k)Bs∗ (k) + As (k)P12,k
h(k)Bs∗ (k)+





Bs (k)h∗ (k)P12,k A∗s (k) + As (k)P22,k A∗s (k), P22,0 = Π22,0
The filtering solution can now be summarized in the following theorem:
(2.62)
2.5. IMPORTANT REMARKS
30
Theorem 2.3: Consider the system described by Equations (2.5)-(2.7), with Lk =
Hk . If the optimality condition (2.43) is satisfied, the H∞ -optimal filtering solution
achieves γopt = 1, and the central H∞ -optimal filter is given by
ξˆk+1 = Fk ξˆk + Kf,k m(k) − Hk ξˆk , ξˆ0 = 0
(2.63)
−1
ŝ(k|k) = Lk ξˆk + (Lk Pk Hk∗ ) RHe,k
m(k) − Hk ξˆk
(2.64)
−1
with Kf,k = (Fk Pk Hk∗ ) RHe,k
and RHe,k = Ip + Hk Pk Hk∗ , where Pk satisfies the Lyapunov recursion
Pk+1 = Fk Pk Fk∗ , P0 = Π0 .
(2.65)
Proof: follows from the discussions above.
2.4.4.2
Prediction Case:
Referring to Eq. (2.26), it is clear that for γ = 1 and for Lk = Hk , the coefficient
Kp,k will reduce to Fk Pk Hk∗ . Therefore, the prediction solution can be summarized
as follows:
Theorem 2.4: Consider the system described by Equations (2.5)-(2.7), with Lk =
Hk . If the optimality conditions (2.43) and (2.59) are satisfied, and with P0 as defined
in Eq. (2.57), the H∞ -optimal prediction solution achieves γopt = 1, and the central
filter is given by
ˆ
ˆ
ˆ
ξk+1 = Fk ξk + Kp,k m(k) − Hk ξk , ξˆ0 = 0
(2.66)
ŝ(k) = Lk ξˆk
(2.67)
with Kp,k = Fk Pk Hk∗ where Pk satisfies the Lyapunov recursion (2.65).
Proof: follows from the discussions above.
2.5
Important Remarks
The main idea in the EBAF algorithm can be summarized as follows. At a given time
k, use the available information on; (a) measurement history, e(i) for 0 ≤ i ≤ k, (b)
control history, u(i) for 0 ≤ i < k, (c) reference signal history, x(i) for 0 ≤ i ≤ k, (d)
2.5. IMPORTANT REMARKS
31
the model of the secondary path and the estimate of its initial condition, and (e) the
pre-determined length of the adaptive FIR filter to produce the best estimate of the
actual output of the primary path, d(k). The key premise is that if d(k) is accurately
estimated, then the inputs u(k) can be generated such that d(k) is canceled. The
objective of the EBAF algorithm is to make y(k) match the optimal estimate of d(k)
(see Fig. 2.3). For the adaptive filtering problem in Fig. 2.1 , however, adaptation
algorithm only has direct access to the weight vector of the adaptive FIR filter.
Because of this practical constraint, the EBAF algorithm adapts the weight vector in
the adaptive FIR filter according to the estimate of the optimal weight vector given
by Eqs. (2.63)
or (2.66) (for
the filtering, or prediction solutions, respectively). Note
T
T
T
that ξˆ = Ŵ (k) θ̂ (k) . The error analysis for this adaptive algorithm is discussed
k
in Section 2.7. Now, main features of this algorithm can be described as follows:
1. The estimation-based adaptive filtering (EBAF) algorithm yields a solution that
only requires one Riccati recursion. The recursion propagates forward in time,
and does not require any information about the future of the system or the
reference signal (thus allowing the resulting adaptive algorithm to be real-time
implementable). This has come at the expense of restricting the controller to
an FIR structure in advance.
∗
2. With Kf,k Re,k Kf,k
= 0, Pk+1 = Fk Pk Fk∗ is the simplified Riccati equation,
which considerably reduces the computational complexity involved in propagating the Riccati matrix. Furthermore, this Riccati update always generates
a non-negative definite Pk , as long as P0 is selected to be positive definite (see
Eq. (2.65)).
3. In general, the solution to an H∞ filtering problem requires verification of the
fact that Rk and Re,k are of the same inertia at each step (see Eq. (2.13)). In a
p
similar way, the prediction solution requires that all sub-matrices of Rkp and Re,k
have the same inertia for all k (see Eq. (2.18)). This can be a computationally
expensive task. Moreover, it may lend to a breakdown in the solution if the
condition is not met at some time k. The formulation of the problem eliminates
2.6. IMPLEMENTATION SCHEME FOR EBAF ALGORITHM
32
the need for such checks, as well as the potential breakdown of the solution, by
providing a definitive answer to the feasibility and optimality of γ = 1.
4. When [ As (k), Bs (k), Cs (k), Ds (k) ] = [ 0, 0, 0, I ] for all k, (i.e. the output
of the FIR filter directly cancels d(k) in Figure 2.1), then the filtering/prediction
results reduce to the simple Normalized-LMS/LMS algorithms in Ref. [26] as
expected.
5. As mentioned earlier, there is no need to verify the solutions at each time step,
so the computational complexity of the estimation based approach is O(n3 )
(primarily for calculating Fk Pk FK∗ ), where
n = (N + 1) + Ns
(2.68)
where (N +1) is the length of the FIR filter, and Ns is the order of the secondary
path. The special structure of Fk however reduces the computational complexity
to O(Ns3 + Ns N), i.e. cubic in the order of the secondary path, and linear in the
length of the FIR filter (see Eq. (2.62)). This is often a substantial reduction
in the computation since Ns N. Note that the computational complexity for
FxLMS is quadratic in Ns and linear in N.
2.6
Implementation Scheme for EBAF Algorithm
Three sets of variables are used to describe the implementation scheme:
1. Best available estimate ofa variable: Referring
to Eqs. (2.16) and (2.19), and
noting the fact that ξˆT = Ŵ T (k) θ̂T (k) , Ŵ (k) can be defined as the estimate
k
of the weight vector, and θ̂(k) as the secondary path state estimate in the
approximate model of the primary path.
4
2. Actual value of a variable: Referring to Fig. 2.1, define u(k) = h∗ (k)Ŵ (k) as the
actual input to the secondary path, y(k) as the actual output of the secondary
path, and d(k) as the actual output of the primary path. Note that d(k) and
2.6. IMPLEMENTATION SCHEME FOR EBAF ALGORITHM
33
y(k) are not directly measurable, and that at each iteration the weight vector
in the adaptive FIR filter is set to Ŵ (k).
3. Adaptive algorithm’s internal copy of a variable: Recall that in Eq. (2.4), y(k)
is used to construct the derived measurement m(k). Since y(k) is not directly
available, the adaptive algorithm needs to generate an internal copy of this
variable. This internal copy (referred to as ycopy (k)) is constructed by applying
u(k) (the actual control signal) to a model of the secondary path inside the
adaptive algorithm. The initial condition for this model is θcopy (0). In other
words, the derived measurement is constructed as follows
m(k) = e(k) + ycopy (k)
(2.69)
θcopy (k + 1) = As (k)θcopy (k) + Bs (k)u(k)
(2.70)
ycopy (k) = Cs (k)θcopy (k) + Ds (k)u(k)
(2.71)
where
Given the identified model for the secondary path and its input u(k) = h∗ (k)Ŵ (k),
the adaptive algorithm’s copy of y(k) will be exact if the actual initial condition of the secondary path is known. Obviously, one can not expect to have the
exact knowledge of the actual initial condition of the secondary path. In the
next section, however, it is shown that when the secondary path is linear and
stable, the contribution of the initial condition to its output decreases to zero
as k increases. Therefore, the internal copy of y(k) will converge to the actual
value of y(k) over time.
Now, the implementation algorithm can be outlined as follows;
1. Start with Ŵ (0) = 0 and θ̂(0) = 0 as the initial guess for the state vector in the
approximate model of the primary path. Also assume that θcopy (0) = 0, and
h(0) = [ x(0) 0 · · · 0 ]T . The initial value for the Riccati matrix is P0 which
is chosen to be block diagonal. The role of P0 is similar to the learning rate in
LMS-based adaptive algorithms (see Section 5.3.2).
2.6. IMPLEMENTATION SCHEME FOR EBAF ALGORITHM
34
2. If 0 ≤ k ≤ M (finite horizon):
(a) Form the control signal
u(k) = h∗ (k)Ŵ (k)
(2.72)
to be applied to the secondary path. Note that applying u(k) to the
secondary path produces
y(k) = Cs (k)θ(k) + Ds (k)u(k)
(2.73)
at the output of the secondary path. This in turn leads to the following
error signal measured at time k:
e(k) = d(k) − y(k) + Vm (k)
(2.74)
which is available to the adaptive algorithm to perform the state update
at time k.
(b) Propagate the state estimate and the internal copy of the state of the
secondary path as follows
 "
# 
Ŵ (k + 1)



 =
θ̂(k + 1)


θcopy (k + 1)
i
h
"
# h
Kf,k Cs (k)
Fk + Kf,k [0 − Cs (k)]
Kf,k
Ŵ (k)




 θ̂(k) +



h
i
∗
θcopy (k)
As (k)
0
(Bs (k)h (k) 0
i

e(k)

(2.75)
where e(k) is the error sensor measurement at time k given by Eq. (2.74),
and Kf,k = Fk Pk Hk∗ (I + Hk Pk Hk∗ )−1 (see Theorem 2.3). Note that for the
prediction-based EBAF algorithm Kf,k should be replaced with Kp,k =
Fk Pk Hk∗ .
2.7. ERROR ANALYSIS
35
(c) update the Riccati matrix Pk using the Lyapunov recursion
"
#
P11
P12,k+1
=
∗
P12,k+1
P22,k+1
"
#"
#"
I
0
P11 P12,k
I
Bs (k)h∗ (k) As (k)
∗
P12,k
P22,k
0
#∗
Bs (k)h∗ (k) As (k)
(2.76)
Pk+1 will be used in (2.75) to update the state estimate.
3. Go to 2.
2.7
Error Analysis
In Section 2.6, it is pointed out that the proposed implementation scheme can deviate
from an H∞ -optimal solution for two main reasons:
1. The error in initial condition of the secondary path which can cause ycopy to be
different from y(k).
2. The additional error in the cancellation of d(k) due to the fact that y(k) can
not be set to ŝ(k|k) (or ŝ(k)). All one can do is to set the weight vector in the
adaptive FIR filter to be Ŵ (k).
Here, both errors are discussed in detail.
2.7.1
Effect of Initial Condition
As earlier discussions indicate, the secondary path in Fig. 2.1 is assumed to be linear.
For a linear system the output at any given time can be decomposed into two components: the zero-input component which is associated with the portion of the output
solely due to the initial condition of the system, and the zero-state component which
is the portion of the output solely due to the input to the system.
2.7. ERROR ANALYSIS
36
For a stable system, the zero-input component of the response will decay to zero
for large k. Therefore, any difference between ycopy (k) and y(k) (which with a known
input to the secondary path can only be due to the unknown initial condition) will
go to zero as k grows. In other words, exact knowledge of the initial condition of the
secondary path does not affect the performance of the proposed EBAF algorithm for
sufficiently large k.
2.7.2
Effect of Practical Limitation in Setting y(k) to ŝ(k|k)
(ŝ(k))
As pointed out earlier, the physical setting of the adaptive control problem in Fig. 2.1
only allows for the weight vector in the adaptive FIR filter to be adjusted to Ŵ (k).
In other words, the state of the secondary path can not be set to a desired value at
each step. Instead, θk evolves based on its initial condition and the control input,
u(k), that we provide. Assume that θ(k) is the actual state of the secondary path at
time k. The actual output of the secondary path is then
y(k) = Ds (k)h∗ (k)Ŵ (k) + Cs (k)θ(k)
(2.77)
which leads to the following cancellation error
d(k) − y(k) = d(k) − Ds (k)h∗ (k)Ŵ (k) + Cs (k)θ(k)
(2.78)
For the prediction solution of Theorem 2.4, adding the zero quantity ±Cs (k)θ̂(k) to
the right hand side of Equation (2.78), and taking the norm of both sides,
kd(k) − y(k)k = k d(k) − Ds (k)h∗ (k)Ŵ (k) + Cs (k)θ(k) ± Cs (k)θ̂(k) k
= k d(k) − Ds (k)h∗ (k)Ŵ (k) − Cs (k)θ̂(k) + Cs (k) θ̂(k) − θ(k) k
Therefore,
k Cs (k) θ̂(k) − θ(k) k
kd(k) − y(k)k
k d(k) − ŝ(k) k
≤
+
M
M
M
X
X
X
∗
∗ −1 ˜
∗
∗ −1 ˜
∗
˜0 +
˜
˜
V
(k)V
(k)
Π
+
V
(k)V
(k)
Π
+
Vm
(k)Vm (k)
ξ˜0∗ Π−1
ξ
ξ
ξ
ξ
ξ
m
0
m
0
0
m
0 0
m
0 0
k=0
k=0
k=0
(2.79)
2.7. ERROR ANALYSIS
37
where ξ˜0 = (ξ0 − ξˆ0 ) and ξk is defined in Eq. (2.5). Note that the first term in the right
hand side of Eq. (2.79) is the prediction error energy gain (see Eq. (2.12)). Therefore,
the energy gain of the cancellation error with the prediction-based EBAF exceeds the
error energy gain of the H∞ optimal prediction solution by the second term on the
right hand side of Eq. (2.79). It can be shown that when the primary inputs h(k)
are persistently exciting (see Eq. (2.48)), the dynamics for the state estimation error,
θ̂(k)−θ(k), are internally stable which implies that the second term on the right hand
side of Eq. (2.79) is bounded for all M, and in the limit when M → ∞∗ .
When Ds (k) = 0 for all k, an implementation of the filtering solution that utilizes
the most recent measurement, m(k), is feasible. In this case, the filtering solution in
Eqs. (2.16)-(2.17) can be written as follows:
ξˇk|k = ξˆk + Pk Hk∗ (Ip + Hk Pk Hk∗ )−1 m(k) − Hk ξˆk
(2.80)
ξˆk+1 = Fk ξˇk|k
(2.81)
ŝ(k|k) = Lk ξˇk|k
(2.82)
where the weight vector update in the adaptive FIR filter follows Eq. (2.80). With
a derivation identical to the one for prediction solution, it can be shown that the
performance bound in this case is
k Cs (k) θ̂(k|k) − θ(k) k
kd(k) − y(k)k
k d(k) − ŝ(k|k) k
≤
+
M
M
M
X
X
X
∗ −1 ˜
∗
∗ −1 ˜
∗
∗ −1 ˜
∗
˜
˜
˜
ξ0 Π0 ξ0 +
Vm (k)Vm (k) ξ0 Π0 ξ0 +
Vm (k)Vm (k) ξ0 Π0 ξ0 +
Vm
(k)Vm (k)
k=0
k=0
k=0
(2.83)
An argument similar to the prediction case shows that the second term on the right
hand side has a finite gain as well.
∗
Reference [24] shows that if the exogenous disturbance is assumed to be a zero mean white
noise process with unit intensity, and independent of the initial condition of the system ξ0 , then the
terminal state estimation error variance satisfies
E(ξk − ξ̂k )(ξk − ξˆk )∗ ≤ Pk
2.8. RELATIONSHIP TO THE NORMALIZED-FXLMS/FXLMS ALGORITHMS
2.8
38
Relationship to the Normalized-FxLMS/FxLMS
Algorithms
In this section, it will be shown that as k → ∞, the gain vector in the predictionbased EBAF algorithm converges to the gain vector in the classical Filtered-X LMS
(FxLMS) algorithm. Thus, FxLMS is an approximation to the steady-state EBAF.
The error terms in the two algorithms are shown to be different (compare Eqs. (2.89)
and (2.2)). Therefore it is expected that the prediction-based EBAF demonstrate
superior transient performance compared to the FxLMS algorithm. Simulation results in the next section agree with this expectation. The fact that the gain vectors
asymptotically coincide, agrees with the fact that the derivation of the FxLMS algorithm relies on the assumption that the adaptive filter and the secondary path are
interchangeable which can only be true in the steady state. Similar results are shown
for the connection between the filtering-based EBAF and the Normalized FxLMS
adaptive algorithms.
For the discussion in this section, the secondary path is assumed, for simplicity, to
be LTI, i.e. [ As , Bs , Cs , Ds ]. Note that for the LTI system, ΨkA in Eq. (2.50) reduces
to Aks . The Riccati matrix Pk in Eq. (2.51) can then be rewritten as
"
# "
#∗
I
0
I
0
Pk = Pk−1 j
P0 Pk−1 j
∗
k
∗
k
j=0 As Bs h (k−1−j) As
j=0 As Bs h (k−1−j) As
(2.84)
Eq. (2.84) will be used in establishing the proper connections between the filtered/predicted
solutions of Section 2.4 and the conventional Normalized-FxLMS/FxLMS algorithms.
2.8.1
Prediction Solution and its Connection to the FxLMS
Algorithm
To study the asymptotic behavior of the state estimate update, note that for an stable
secondary path Aks → 0 as k → ∞. Therefore, using Eq. (2.84)
# "
#∗
"
I
0
I
0
Pk → Pk−1 j
P0 Pk−1 j
as k → ∞(2.85)
∗
∗
j=0 As Bs h (k−1−j) 0
j=0 As Bs h (k−1−j) 0
2.8. RELATIONSHIP TO THE NORMALIZED-FXLMS/FXLMS ALGORITHMS
"
which for P0 =
"
Pk → Pk−1
P11 (0) P12 (0)
#
results in
P21 (0) P22 (0)
I
j
∗
j=0 As Bs h (k−1−j)
39
#
"
P11 (0) Pk−1
I
j
∗
j=0 As Bs h (k−1−j)
#∗
as k → ∞ (2.86)
Selecting P11 (0) = µI as in Eq. (2.57), and noting the fact that Kp,k = Fk Pk Hk∗
(Theorem 2.4), it is easy to see that as k → ∞
#
"
!∗
k−1
X
I
∗
j
∗
Kp,k → µ Pk
Ds h (k) +
Cs As Bs h (k−1−j)
j
∗
j=0 As Bs h (k−j)
j=0
#
"
I
h0 (k)
→ µ Pk
(2.87)
j
∗
A
B
h
(k−j)
s
j=0 s
and therefore the state estimate update in Theorem 2.4 becomes
# "
#
"
#"
Ŵ (k+1)
I
0
Ŵ (k)
=
+
θ̂(k+1)
Bs h∗ (k) As
θ̂(k)
"
#
h0 (k)
∗
µ Pk
m(k) − Ds h (k)Ŵ (k) − Cs θ̂(k) (2.88)
j
∗
0
j=0 As Bs h (k−j)h (k)
Thus,the following update for the weight vector is derived
∗
Ŵ (k+1) = Ŵ (k) + µh0 (k) m(k) − Ds h∗ (k)Ŵ (k) − Cs θ̂(k)
(2.89)
Note that m(k) = e(k) + ycopy (k) (see Eq. (2.69)), and hence the difference between
the limiting update rule of Eq. (2.89) (i.e. the prediction EBAF algorithm), and the
classical FxLMS algorithm of Eq. (2.2) will be the error term used by these algorithms.
More specifically, e(k) in the FxLMS algorithm is replaced with the following modified
error (using Eq. (2.71)):
e(k) + ycopy (k) − Ds h∗ (k)Ŵ (k) − Cs θ̂(k) = e(k) + Cs θcopy (k) − Cs θ̂(k).
Note that if y(k) is directly measurable, then the modified error will be
h
i
e(k) + y(k) − Ds h∗ (k)Ŵ (k) − Cs θ̂(k)
(2.90)
2.8. RELATIONSHIP TO THE NORMALIZED-FXLMS/FXLMS ALGORITHMS
40
The condition for optimality of γ = 1 in the prediction case (see Eq. (2.59)), can
also be simplified for stable LTI secondary path as k → ∞. Rewriting the optimality
condition for the prediction solution, Eq. (2.59), as
µ<
∗
1 − αCs Aks A∗k
s Cs
h0 ∗ (k)h0 (k)
(2.91)
for a stable secondary path, Aks → 0 as k → ∞, and hence
µ<
1
h0 ∗ (k)h0 (k)
as k → ∞
(2.92)
is the limiting condition for the optimality of γ = 1 in the prediction case. This is
essentially a filtered version of the well known LMS bound [26].
2.8.2
Filtering Solution and its Connection to the NormalizedFxLMS Algorithm
In the Filtered case the gain vector is Kf,k = Fk Pk Hk∗ (I + Hk Pk Hk∗ ). In Section 2.8.1
the limiting value for the quantity Fk Pk Hk∗ in Eq. (2.87) is computed. In a similar
way it can be shown that, with P11 (0) = µI, as K → ∞,
(I + Hk Pk Hk∗ ) → (1 + µh0∗ (k)h0 (k))
(2.93)
and hence the coefficient for the state estimate update in the filtering case becomes
!
h0 (k)
µ
Kf,k →
as k → ∞
(2.94)
Pk
j
∗
0
1 + µh0∗ (k)h0 (k)
j=0 As Bs h (k−j)h (k)
Thus the update rule for the weight vector in the filtering EBAF algorithm would be
Ŵ (k+1) = Ŵ (k) + µ
h0 ∗ (k)
∗
θ̂(k)
(2.95)
h
(k)
Ŵ
(k)
−
C
m(k)
−
D
s
s
(1 + µh0∗ (k)h0 (k))
which is similar to the Normalized-FxLMS algorithm (Eq. (2.3)) in which the error
signal is replaced with a modified error signal described by Eq. (2.90).
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
2.9
41
Experimental Data & Simulation Results
This section examines the performance of the proposed EBAF algorithm for the active noise cancellation (ANC) problem in a one dimensional acoustic duct. Figure 2.5
shows the schematic diagram of the one-dimensional air duct that is used in the experiments. The control objective is to attenuate (cancel in the ideal case) the disturbance
introduced into the duct by Speaker #1 (primary noise source) at the position of Microphone #2 (error sensor) by the control signal generated by Speaker #2 (secondary
source). Microphone #1 can be used to provide the reference signal for the adaptation
algorithm. Clearly, Microphone #1 measurements are affected by both primary and
secondary sources, and hence if these measurements are used as the reference signal
the problem, commonly known as feedback contamination, has to be addressed.
A dSPACE DS1102 DSP controller board (which includes TI’s C31 DSP processor
with 60 MHz clock rate, and 128k of 32-bit RAM), and its Matlab 5 interface are
used for real time implementation of the algorithm. A state space model (of order 10)
is identified for this one-dimensional acoustic system. Note that of the four identified
transfer functions, only the transfer function from Speaker #2 to Microphone #2 (i.e.
the secondary path) is required by the estimation-based adaptive algorithm. Figs.
2.6 and 2.7 show the identified transfer function for the one-dimensional duct. This
section will first provide experimental data that validate a corresponding simulation
result. More sophisticated experiments and simulations are then presented to study
various aspects of the EBAF algorithm.
Figure 2.8 shows the experimental data in a typical noise cancellation scenario,
along with corresponding plots from a simulation that is designed to mimic that
experiment. Here, the reading of Microphone #2 (i.e. the cancellation error) is
shown when an adaptive FIR filter of length 4 is used for noise cancellation. The
primary source is a sinusoidal tone at 150 Hz, which is also available to the adaptation
algorithm as the reference signal. A band-limited white noise (noise power = 0.008)
is used as the measurement noise for the simulation in Figure 2.8. The sampling
frequency is 1000 Hz for both experiment and simulation. P11,0 = 0.05I4×4 , P12,0 = 0,
and P22,0 = 0.005I10×10 are used to initialize the Riccati matrix in Eq. (2.14). The
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
42
experiment starts with adaptive controller off, and about 3 seconds later the controller
is turned on. The transient response of the adaptive FIR filter lasts for approximately
0.05 seconds. There is a 60 times reduction in the magnitude of the error. Therefore,
with the full access to the single tone primary disturbance, the EBAF algorithm
provides a fast and effective noise cancellation. The results from a corresponding
Matlab simulation (with the same filter length, and similar open loop error at 150
Hz) are also shown in Fig. 2.8. The transient behavior and the steady state response
in the simulation agree with the experimental data, thus assuring the validity of the
set up for the simulations presented in this chapter.
Figures 2.9, 2.10, and 2.11 show the results of more noise cancellation experiments
in the one dimensional acoustic duct. In all these three experiments, the reference
signal available to the adaptation scheme is formed such that a considerable level of
uncorrelated additive white noise corrupts the primary disturbance. This is done to
examine the robustness of the EBAF algorithm in the case where clean access to the
primary disturbance is not possible. In practice however, efforts are made to produce
as clean a reference signal as possible.
In Figure 2.9 the primary disturbance is a sinusoid of amplitude 0.3 volts at 150
Hz. The reference signal used by the EBAF algorithm is subject to a band limited
white Gaussian noise, and the signal to noise ratio is approximately 3. For this
experiment the length of the adaptive filter is 4, and the controller is turned on
at t = 4.7 seconds. A 35 times reduction in the magnitude of the disturbance at
Microphone #2 is measured. While there is a reduction of magnitude 3.5 in the
error at Microphone #1, this is achieved as a byproduct of the noise cancellation at
Microphone #2 (i.e. noise cancellation at Microphone #1 is not an objective of the
single-channel noise cancellation attempt in this experiment). Chapter 4 will address
the multi-channel noise cancellation in detail. Note that the magnitude of noise
cancellation in this case is lower than the cancellation achieved in Figure 2.8 where
the reference signal was fully known to the adaptive algorithm. This observation
confirms the well known fact that the quality of the reference signal (i.e. its degree
of correlation to the primary disturbance) profoundly affects the performance of the
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
43
adaptive algorithm [51,33]. Figure 2.10 shows the experimental results for multitone noise cancellation where a noisy reference signal is available to the adaptive
algorithm. Here the number of taps in the adaptive filter is 8, and the primary
disturbance consists of two sinusoids at 150 and 180 Hz. The signal to noise ratio for
the available reference signal in this case is approximately 4.5. A 16 times reduction
in the amplitude of the disturbances at Microphone #2 is recorded. Some reduction
in the amplitude of the noise at Microphone #1 is also recorded. Note that, the sole
objective of the EBAF algorithm, in this case, is to cancel the noise at the position
of Microphone #2, and therefore no attempt is made to reduce the disturbances at
Microphone #1.
Figure 2.11 shows the results of the EBAF algorithm in the case where the primary
disturbance is a band limited white noise. As in Figures 2.9 and 2.10, only a noisy
measurement of the primary disturbance is available to the adaptive algorithm. The
signal to noise ratio in this case is approximately 4.5. The length of the adaptive
filter in this case is 16, and a reduction of approximately 3 times in the measurements
of Microphone #2 is achieved. For a better performance, the number of taps for the
adaptive FIR filter should be increased. Hardware limitations, however, prevented
experiments with higher order FIR filters. The performance of the EBAF algorithm
with longer FIR filters is examined through simulations in the rest of this section.
In Figure 2.12, the effect of feedback contamination (i.e. the contamination of the
reference signal with the output of the adaptive FIR filter through some feedback
path) when the primary source is a single tone is studied in simulation. In [33] the
subject of feedback contamination is discussed in detail, where relevant references
to the conventional solutions to this problem are also listed. Here, however, the
objective is to show that the proposed EBAF algorithm maintains superior performance (compared to FxLMS and normalized-FxLMS (NFxLMS) algorithms) when
such a problem occurs and no additional information is furnished. Fig. 2.12 contains
a typical response to feedback contamination for EBAF, FxLMS and NFxLMS algorithms. For the first 5 seconds, the input to Speaker #2 is grounded (i.e. u(k) = 0
for k ≤ 5). Switching the controller on results in large transient behavior in the
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
44
case of FxLMS and NFxLMS while, for the EBAF algorithm, the transient behavior does not display the undesirable overshoot. Different operation scenarios (with
various filter lengths, and adaptation rates) were tested, and this observation holds
true in all cases. For the next 15 seconds, the primary source is directly available
to all adaptive algorithms and the steady-state performance (left plots) is virtually
the same. From k = 20 on (right plots), the output of Microphone #1 (which is
contaminated by the output of the FIR filter) is used as the reference signal. Once
again, Fig. 2.12 shows a typical result. Note that in the case of FxLMS and NFxLMS
the adaptation rate must be kept small enough to avoid unstable behavior when the
switch to contaminated reference signal takes place. The EBAF algorithm allows for
faster convergence in the face of feedback contamination. For the results in Fig. 2.12,
the length of the adaptive FIR filter (for all three algorithms) is 24. For the EBAF
algorithm P11,0 = 0.005I24×24 , P22,0 = 0.0005I10×10 , and P12,0 = 0. For FxLMS and
NFxLMS algorithms, the adaptation rates are 0.005 and 0.025, respectively.
Figure 2.13 considers the effect of feedback contamination in a wide-band (10−500
Hz) noise cancellation process. For the results in Fig. 2.13, the length of the adaptive
FIR filter is 32. For the EBAF algorithm P11,0 = 0.05I32×32 , P22,0 = 0.005I10×10 ,
and P12,0 = 0. For FxLMS and NFxLMS algorithms the adaptation rates are 0.0005
and 0.01, respectively. The FxLMS algorithm becomes unstable for faster adaptation
rates, hence forcing slow convergence (i.e. lower control bandwidth). For NFxLMS,
the normalization of the adaptation rate by the norm of the reference vector (a vector
of length 32 in this case) prevents unstable behavior. The response of the algorithm
under feedback contamination is however still slower than EBAF algorithm. Furthermore, the oscillations in cancellation error due to the switching between modes of
operation are significantly higher when compared to the oscillations in EBAF case.
Figure 2.14 shows the closed-loop (i.e. the transfer function from the primary disturbance source at Speaker #1 to Microphone #2 during steady-state operation of
the adaptive control algorithm) performance comparison for wide-band noise cancellation. The EBAF algorithm outperforms FxLMS and Normalized-FxLMS adaptive
algorithms, even though the same level of information is made available to all three
adaptation schemes. For the result presented here the length of the FIR filter (for
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
45
all three approaches) is 32, and the band-limited white noise which is used as the
primary source is available as the reference signal. Since the frequency response is
calculated based on the steady-state data, the adaptation rate of the algorithms is
not relevant. Measurement noise for all three simulations is a band-limited white
noise with power 0.008, which is found to result in a steady-state attenuation that is
consistent with the experiments.
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
46
Physical Plant
n(k)
Vm (k)
d(k)
Primary Path
(Unknown)
+
+
+
−
Adaptive FIR Filter
x(k)
...
z −1
W0
W1
Secondary
Path
(Known)
WN
+
u(k)
y(k)
Update Weight Vector
Adaptation
Algorithm
Digital Control System
Fig. 2.1: General block diagram for an Active Noise Cancellation (ANC) problem
e(k)
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
x(k)
47
Primary Path
(Unknown)
P (z)
d(k)
+
e(k)
+
Adaptive FIR
Filter
u(k)
Secondary Path
(known)
S(z)
−
y(k)
A Copy Of
(known)
Secondary Path
LMS
Adaptation
Algorithm
x0 (k)
Fig. 2.2: A standard implementation of FxLMS algorithm
Primary Path
Modeling Error
x(k)
A Copy Of
Secondary Path
An FIR Filter
Secondary Path
Adaptive FIR Filter
u(k)
Vm (k)
d(k)
+
+
+
−
+
e(k)
y(k)
Adaptation Algorithm
Fig. 2.3: Pictorial representation of the estimation interpretation of the adaptive control problem:
Primary path is replaced by its approximate model
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
x(k)
z −1
W0
x(k − 1)
W1
x(k − N)
WN
Replica of the FIR Filter
Vm (k)
Ds (k)
Bs (k)
u(k)
+
48
+
z −1
Cs (k)
+
+
m(k)
As (k)
Replica of the Secondary Path
Fig. 2.4: Block diagram for the approximate model of the primary path
Microphone #1
Speaker #1
Microphone #2
Speaker #2
Fig. 2.5: Schematic diagram of one-dimensional air duct
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
Speaker #1 → Microphone #1
Speaker #2 → Microphone #1
1
1
10
Magnitude
Magnitude
10
0
10
−1
10
−2
10
49
0
10
−1
10
−2
1
10
10
2
10
1
10
Frequency (Hz)
2
10
Frequency (Hz)
400
0
Phase (Deg.)
Phase (Deg.)
200
0
−200
−400
−600
−800
−1000
1
10
−500
−1000
−1500
−2000
1
10
2
10
Frequency (Hz)
2
10
Frequency (Hz)
Fig. 2.6: Transfer functions plot from Speakers #1 & #2 to Microphone #1
Speaker #1 → Microphone #2
Speaker #2 → Microphone #2
1
1
10
Magnitude
Magnitude
10
0
10
−1
10
−2
10
0
10
−1
10
−2
1
10
10
2
10
1
10
Frequency (Hz)
2
10
Frequency (Hz)
400
200
Phase (Deg.)
Phase (Deg.)
0
−500
−1000
−1500
−2000
1
10
2
10
Frequency (Hz)
0
−200
−400
−600
−800
−1000
1
10
2
10
Frequency (Hz)
Fig. 2.7: Transfer functions plot from Speakers #1 & #2 to Microphone #2
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
Cancellation Error
50
Steady-State Behavior
Transient Behavior
e(k) - Simulation
0.02
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
2
3
4
−0.8
2.9
0.01
0
−0.01
2.95
3
3.05
−0.02
4
6
8
10
4
6
8
10
e(k) - Experiment
0.02
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
2
3
Time (sec.)
4
−0.8
2.9
0.01
0
−0.01
2.95
3
3.05
Time (sec.)
−0.02
Time (sec.)
Fig. 2.8: Validation of simulation results against experimental data for the noise cancellation
problem with a single-tone primary disturbance at 150 Hz. The primary disturbance is known to
the adaptive algorithm. The controller is turned on at t ≈ 3 seconds.
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
51
Experimental Data for a Sinusoidal Primary Disturbance of 150 Hz
Error at Microphone #1
0.5
0
−0.5
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
Error at Microphone #2
0.5
0
−0.5
Time (sec.)
Fig. 2.9: Experimental data for the EBAF algorithm of length 4, when a noisy measurement of
the primary disturbance (a single-tone at 150 Hz) is available to the adaptive algorithm (SNR=3).
The controller is turned on at t ≈ 5 seconds.
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
52
Experimental Data for a Multi-Tone Sinusoidal Primary Disturbance of 150&180 Hz
Error at Microphone #1
0.5
0
−0.5
0
5
10
15
20
25
30
0
5
10
15
20
25
30
Error at Microphone #2
0.5
0
−0.5
Time (sec.)
Fig. 2.10: Experimental data for the EBAF algorithm of length 8, when a noisy measurement
of the primary disturbance (a multi-tone at 150 and 180 Hz) is available to the adaptive algorithm
(SNR=4.5). The controller is turned on at t ≈ 6 seconds.
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
53
Experimental Data for a Band Limited White Noise
0.5
0.4
0.3
Error at Microphone #2
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
0
5
10
15
20
25
30
Time (sec.)
Fig. 2.11: Experimental data for the EBAF algorithm of length 16, when a noisy measurement
of the primary disturbance (a band limited white noise) is available to the adaptive algorithm
(SNR=4.5). The controller is turned on at t ≈ 5 seconds.
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
No Feedback Contamination
With Feedback Contamination
1
1
EBAF
0
−0.5
0
2
4
6
8
−1
20
10
FxLMS
e(k)
e(k)
24
26
28
30
FxLMS
0.5
0
−0.5
0
−0.5
0
2
4
6
8
−1
20
10
1
22
24
26
28
30
1
NFxLMS
NFxLMS
0.5
e(k)
0.5
e(k)
22
1
0.5
0
−0.5
−1
0
−0.5
1
−1
EBAF
0.5
e(k)
e(k)
0.5
−1
54
0
−0.5
0
2
4
6
Time (sec.)
8
10
−1
20
22
24
26
28
30
Time (sec.)
Fig. 2.12: Simulation results for the performance comparison of the EBAF and (N)FxLMS
algorithms. For 0 ≤ t ≤ 5 seconds, the controller is off. For 5 < t ≤ 20 seconds both adaptive
algorithms have full access to the primary disturbance (a single-tone at 150 Hz). For t ≥ 20 seconds
the measurement of Microphone #1 is used as the reference signal (hence feedback contamination
problem). The length of the FIR filter is 24.
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
55
Cancellation Error - Band-Limited-White-Noise Primary Source
2
EBAF
e(k)
1
0
−1
−2
0
10
20
30
40
50
60
70
80
2
100
FxLMS
1
e(k)
90
0
−1
−2
0
10
20
30
40
50
60
70
80
90
100
2
NFxLMS
e(k)
1
0
−1
−2
0
10
20
30
40
50
60
70
80
90
100
Time (sec.)
Fig. 2.13: Simulation results for the performance comparison of the EBAF and (N)FxLMS
algorithms. For 0 ≤ t ≤ 5 seconds, the controller is off. For 5 < t ≤ 40 seconds both adaptive
algorithms have full access to the primary disturbance (a band limited white noise). For t ≥
40 seconds the measurement of Microphone #1 is used as the reference signal (hence feedback
contamination problem). The length of the FIR filter is 32.
2.9. EXPERIMENTAL DATA & SIMULATION RESULTS
56
Transfer Function from Speaker #1 to Microphone #2
1
10
Open-Loop
EBAF
FxLMS
NFxLMS
0
Magnitude
10
−1
10
−2
10
1
10
2
10
Frequency (Hz)
Fig. 2.14: Closed-loop transfer function based on the steady state performance of the EBAF
and (N)FxLMS algorithms in the noise cancellation problem of Figure 2.13.
2.10. SUMMARY
2.10
57
Summary
The adaptive control problem has been approached from an estimation point of view.
More specifically, it has been shown that for a common formulation of the adaptive
control problem an equivalent estimation interpretation exists. Then, a standard H∞
estimation problem has been constructed that corresponds to the original adaptive
control problem, and have justified the choice of estimation criterion. The H∞ optimal filtering/prediction solutions have also been derived, and it has been proved
that the optimal energy gain is unity. The filtering/prediction solutions have been
simplified, and explained how these solutions form the foundation for an EstimationBased Adaptive Filtering (EBAF) algorithm. Meanwhile, the feasibility of the real
time implementation of the EBAF algorithm is justified.
An implementation scheme for the new algorithm has been outlined, and a corresponding performance bound has been derived. It is shown that the classical FxLMS
(Normalized-FxLMS) adaptive algorithms are approximations to the limiting behavior of the proposed EBAF algorithm. The EBAF algorithm is shown to display
improved performance when compared to commonly used FxLMS and NormalizedFxLMS algorithms. Simulations have been verified by conducting a noise cancellation
experiment, and showing that the experimental data reasonably match a corresponding simulation.
The systematic nature of the proposed EBAF algorithm can serve as the first step
towards methodical optimization of now pre-determined parameters of the FIR filter
(such as filter length, or adaptation rate). Furthermore, the analysis of the various
aspects of the algorithm directly benefits from the advances in robust estimation theory. Finally, more efficient implementation schemes can further reduce computational
complexity of the algorithm.
Chapter 3
Estimation-Based adaptive IIR
Filter Design
This chapter extends the “estimation-based” approach (introduced in Chapter 2) to
the synthesis of adaptive Infinite Impulse Response (IIR) filters (controllers). Systematic synthesis of adaptive IIR filters is proven to be difficult and existing design
practices are ad hoc in nature. The proposed approach in this chapter is based on an
estimation interpretation of the adaptive IIR filter (controller) design that replaces
the original adaptive filtering (control) problem with an equivalent estimation problem. Similar to the case with FIR filters, an H∞ criterion is chosen to formulate this
equivalent estimation problem. Unlike the FIR case however, the estimation problem
in this case is nonlinear in IIR filter parameters. At the present time, the nonlinear
robust estimation problem does not have an exact closed form solution. The proposed
solution in this chapter is therefore an “approximation” that uses the best available
estimate of the IIR filter parameters to locally linearize the nonlinear robust estimation problem at each time step. For the linearized problem, the estimation-based
adaptive algorithm is identical to the adaptive FIR filter design described in Chapter 2. The systematic nature of this new approach is particularly appealing given the
complexity of the existing design schemes for adaptive IIR filters (controllers).
Simulations for active noise cancellation in a one dimensional acoustic duct are
58
3.1. BACKGROUND
59
used to examine the main features of the proposed estimation-based adaptive filtering algorithm for adaptive IIR filters. The performance of the EBAF algorithm is
compared to that of a commonly used classical solution in adaptive control literature known as Filtered-U Recursive LMS algorithm. The comparison reveals faster
convergence, with improved steady-state behavior in the case of EBAF algorithm.
3.1
Background
This section defines the adaptive filtering problem of interest and describes the terminology that is used in the rest of this chapter. The description of feedback contamination problem, as well as an introductory discussion of the classical Filtered-U
recursive LMS algorithm are also included in this section.
Figure 3.1 shows the adaptive filtering problem of interest. Similar to the FIR
case (see Figure 2.1), the objective here is to adjust the weight vector in the adaptive
IIR filter such that the output of the secondary path provides an acceptable match to
the output of the primary path. Figure 3.1 includes the following signals: (a) ref (k):
the input to the primary path, which is the same as the reference signal available to
the adaptive filter when there is no feedback path, (b) u(k): the control signal applied
to the secondary path, (c) y(k): the output of the secondary path (i.e. the signal
that should match d(k)), and (d) e(k): the residual error which is used to update the
weight vector in the adaptive IIR filter. Note that, despite assuming full knowledge of
the primary path input in Figure 3.1, the reference signal available to the adaptation
algorithm will be affected by the adaptive filter output when a feedback path exists.
In practice, the input to the primary path is not always fully known, and a signal
with “enough” correlation to the primary input replaces ref (k) in Figure 3.1 [33].
The existence of the feedback path will effect the correlation between the reference
signal available to the adaptive algorithm (x(k) in Fig. 3.1) and the primary input to
the primary path (ref(k) in Fig. 3.1) and has a profound effect on the performance of
the adaptive filter. This phenomenon, known as feedback contamination, is extensively
studied in the adaptive control literature ([33], Chapter 3). A simple solution for this
feedback problem is to use a separate feedback (as a part of the control system) to
3.1. BACKGROUND
60
cancel the undesirable feedback signal. An implementation of this idea, known as
feedback neutralization, is shown in Fig. 3.2. Note that in this scheme, W (z) is the
adaptive FIR filter that generates the control signal. F̂ (z) is another adaptive FIR
filter whose output eliminates (in the ideal case) the effect of feedback contamination.
This approach, however, requires special care in the implementation to avoid the
cancellation of the reference signal all together (see [33] for details). A closer look at
Figure 3.2 indicates that even though feedback neutralization employs two FIR filters,
the overall adaptive controller is no longer a zero-only system (F̂ (z) is positioned in
a feedback path). Precisely speaking, feedback neutralization is indeed an adaptive
IIR filtering algorithm.
A more direct (and more general) approach to the design of adaptive IIR filters
in such circumstances is shown in Figure 3.3. In this approach the feedback path is
treated as a part of the overall plant for which the adaptive IIR filter is designed.
Of possible adaptive IIR algorithms, only the Filtered-U recursive LMS algorithm
(FuLMS) will be considered here [14]. This selection is justified by the fact that the
FuLMS adaptive algorithm exhibits the main features of a conventional adaptive IIR
filtering algorithm and has been used successfully in noise cancellation problems (see
Chapter 3 in [33] and the references therein). Referring to Figure 3.3, the residual
error is
e(k) = d(k) − s(k) ⊕ r(k) = d(k) − y(k)
(3.1)
where s(k) is the impulse response of the secondary path, and ⊕ indicates convolution.
Note that the conventional derivation of the Filtered-U algorithm does not include
the exogenous measurement disturbance Vm (k). The output of the IIR filter r(k) is
computed as
r(k) = aT (k)x(k) + bT (k)r(k − 1)
(3.2)
where a(k) = [ a0 (k) a1 (k) · · · aL−1 (k) ]T is the weight vector for A(z) at time k,
and b(k) = [ b1 (k) b2 (k) · · · bM (k) ]T is likewise defined for B(z). Moreover, x(k) =
[ x(k) x(k − 1) · · · x(k − L + 1) ]T and r(k−1) = [ r(k − 1) r(k − 2) · · · r(k − M) ]T
are reference signals for A(z) and B(z), respectively.
3.2. PROBLEM FORMULATION
61
T
Defining a new overall weight vector w(k) = aT (k) bT (k) , and a generalized
T
reference vector u(k) = xT (k) rT (k − 1) , Eq. (3.2) can be rewritten as
r(k) = wT (k)u(k)
(3.3)
which has the same format as the output of an ordinary FIR filter. Reference [33]
shows that the steepest-descent algorithm can be used to derive the following update
equation for the generalized weight vector
w(k + 1) = w(k) + µ [ŝ(k) ⊕ u(k)] e(k)
(3.4)
if the instantaneous squared error, e2 (k), is used to estimate the mean-square error,
E[ k e2 (k) k ]. Referring to Section 2.1, the error criterion here is the same as that
of the LMS algorithm. The algorithm is called the Filtered-U recursive LMS algorithm since it uses an estimate of the secondary path, ŝ(k), to filter the generalized
reference vector u(k). The derivation of this algorithm explicitly relies on a slow
convergence assumption and therefore, in general, the convergence rate µ should be
kept small. Slow adaptation also enables the derivation of an approximate instantaneous gradient vector that significantly reduces the computational complexity of
the algorithm (see [33], page 93). The ad hoc nature of the FuLMS, however, has
significantly complicated the analysis of the algorithm. As Reference [33] indicates,
the global convergence and stability of the algorithm have not been formally proved,
and the optimal solution in the case of large controller coefficients is found to be illconditioned. Nevertheless, successful implementations of this algorithm are reported,
and hence it will be used as the conventional design algorithm to which the EBAF
algorithm will be compared.
3.2
Problem Formulation
The underlying concept for estimation-based adaptive IIR filter design is essentially
the same as that of the estimation-based adaptive FIR filter design. For clarity,
however, the main steps in the estimation interpretation of the adaptive filtering
(control) problem are repeated here:
3.2. PROBLEM FORMULATION
62
1. Introduce an approximate model for the primary path based on the architecture
of the adaptive path from x(k) to y(k) (as shown in Fig. 3.1). The goal is to
find the optimal weight vector in the approximate model for which the modeling
error is the smallest. As in Chapter 2, state-space models are used for both the
adaptive filter (IIR in this case) and the secondary path.
2. In the approximate model for the primary path, use the available information to
formulate an estimation problem that recursively estimates this optimal weight
vector.
3. Adjust the weight vector of the adaptive IIR filter to the best available estimate
of the optimal weight vector.
For simplicity, the case without feedback contamination is considered first. The case
with feedback contamination is a straightforward extension, and will be discussed in
Appendix B. A model for the secondary path is assumed available (e.g. via identification). The primary path however, is completely unknown. Figure 3.4 provides
a pictorial presentation for the above mentioned estimation interpretation, which is
identical to Figure 2.3, except for the replacement of the FIR filter with an IIR one.
The main signals involved in Figure 3.4 are similar to those in Figure 2.2 and are
described here for easier reference. First, note that
e(k) = d(k) − y(k) + Vm (k)
(3.5)
where e(k) is the available error measurement, Vm (k) is the exogenous disturbance
that captures measurement noise, modeling error and initial condition uncertainty,
y(k) is the output of the secondary path, and d(k) is the output of the primary path.
Equation (3.5) can be rewritten as
e(k) + y(k) = d(k) + Vm (k)
(3.6)
where the left hand side is a noisy measurement of the output of the primary path
d(k). Since y(k) is not directly measurable (neither is d(k) of course), the trick is to
have the adaptive algorithm generate an internal copy of y(k), and then define the
3.2. PROBLEM FORMULATION
63
derived measured quantity as
4
m(k) = e(k) + ycopy (k) = d(k) + Vm (k)
(3.7)
which will be used in formulating the estimation problem. To generate the internal
copy of y(k), the adaptive algorithm uses the available model for the secondary path
and the known control input to the secondary path, u(k). See Section 2.7 for a
discussion of the impact of the initial condition error.
3.2.1
Estimation Problem
Figure 3.5 shows a block diagram representation of the approximate model to the
primary path. Here, a state space model, [ As (k), Bs (k), Cs (k), Ds (k) ], for the
secondary path is assumed. A second order IIR filter (with 5 parameters) models the
adaptive IIR filter. Define W (k) = [ a0 (k) b1 (k) · · · bN (k) a1 (k) · · · aN (k)]T (N = 2
in Fig. 3.5) to be the unknown optimal vector of the IIR filter parameters at time
T
is the state vector for the overall system. Note that
k. Then, ξ = W T (k) θT (k)
θ(k) captures the dynamics of the secondary path in the approximate model. The
state space representation of the system is
#
"
#
"
#"
I(2N +1)×(2N +1)
W (k + 1)
W (k)
0
=
θ(k + 1)
θ(k)
Bs (k)h∗ (k)
As (k)
4
ξk+1 = Fk ξk
(3.8)
where h(k) = [x(k) r(k − 1) · · · r(k − N) r(k − 1) · · · r(k − N)]T captures the effect of the reference input x(·). Note that
r(k) = x(k)a0 (k) + r(k − 1)b1 (k) + · · · + r(k − N)bN (k),
(3.9)
(where r(−1) = · · · = r(−N) = 0) and therefore, the system dynamics are nonlinear
in the IIR filter parameters. For this system, the derived measured output is
#
"
h
i W (k)
+ Vm (k)
m(k) =
Ds (k)h∗ (k) Cs (k)
θ(k)
4
= Hk ξk + Vm (k)
(3.10)
3.2. PROBLEM FORMULATION
64
where m(k) should be constructed at each step according to Equation (3.7). Note
that measurement equation is also nonlinear in the parameters of the IIR filter. Now,
define a generic linear combination of the states as the desired quantity to be estimated
"
#
h
i W (k)
s(k) =
L1,k L2,k
θ(k)
4
= Lk ξk
(3.11)
Here, θ(·) ∈ RNs ×1 , W (·) ∈ R(2N +1)×1 , m(·) ∈ R1×1 and s(·) ∈ R1×1 . All matrices are
then of appropriate dimensions. Different choices for Lk lead to different estimates of
W (k), and hence different adaptation criteria for the parameters in the IIR filter. As
in Chapter 2, an H∞ criterion will be used to directly estimate d(k) (hence Lk = Hk ).
The objective is to find an H∞ suboptimal filter ŝ(k|k) = F (m(0), · · · , m(k)) (or H∞
suboptimal predictor ŝ(k) = F (m(0), · · · , m(k − 1))), such that the worst case energy
gain from the measurement disturbance and the initial condition uncertainty to the
error in a causal estimate of s(k) = d(k) remains bounded. In other words, ŝ(k|k) is
sought such that for a given γf > 0 and Π0 > 0,
M
X
sup
Vm , ξ0
(s(k) − ŝ(k|k))∗ (s(k) − ŝ(k|k))
k=0
ξ0∗ Π−1
0 ξ0
+
M
X
≤ γf2
(3.12)
∗
Vm
(k)Vm (k)
k=0
Similarly ŝ(k) is desired such that for a given γp > 0 and Π0 > 0,
M
X
sup
Vm , ξ0
(s(k) − ŝ(k))∗ (s(k) − ŝ(k))
k=0
ξ0∗ Π−1
0 ξ0 +
M
X
≤ γp2
(3.13)
∗
Vm
(k)Vm (k)
k=0
Since Eqs. (3.8), (3.10), and (3.11) are nonlinear in IIR filter parameters, both of these
estimation problems are nonlinear. An exact closed form solution to the nonlinear
H∞ estimation problem is not yet available. One solution to this problem is to use
the following linearizing approximation; at each time step, we replace the IIR filter
3.3. APPROXIMATE SOLUTION
65
parameters, a0 , b1 , · · · bN , in Equation (3.9) with their best available estimate. This
reduces the estimation problems in Eqs. (3.12) and (3.13) into linear H∞ estimation
problems for which the solutions in Section 2.4 can be directly applied. A similar
linearizing approximation is commonly adopted in the optimal estimation problems
for nonlinear dynamic processes and is referred to as continuous relinearization in
extended Kalman filtering [9,20]. As Reference [9] indicates, in the context of optimal estimation for the nonlinear processes, “considerable success has been achieved
with this linearization approach”. Extensive simulations in this Thesis reveal equally
successful results in adaptive IIR filter design (see Section 3.6).
Remark: As Figure 3.5 suggests, feedback is an integral part of an IIR filter structure.
The nonlinearity of the robust estimation problem in Eqs. (3.12) and (3.13) is due
to the existence of this structural feedback loop. Obviously, a physical feedback path
outside the IIR filter (see Figure 3.1) will have the same effect. The treatment of
the nonlinearity in the case of an IIR filter, i.e. replacing the IIR filter parameters
in Equation (3.9) with their best available estimates, carries over to the case where
such a feedback path exists. Appendix B discusses the case with reference signal
contamination in more detail.
3.3
Approximate Solution
With the linearizing approximation of the previous section, at any given time k,
h(k) will be fully known. This eliminates the nonlinearity in Equations (3.8)-(3.11),
and allows the straightforward solutions in Section 2.4 to be used. Note that for the
linearized problem all the optimality arguments in Chapter 2 are valid and the simplifications to filtering (prediction) solutions also apply. The linearizing approximation,
however, prevents any claim on the optimality of the solution. Only the simplified
solutions to the linearized estimation problems of Eqs. (3.12) and (3.13) (i.e. the
central filtering solution corresponding to γf = 1 for the filtering solution, and the
central prediction solution corresponding to γp = 1 for the prediction solution) are
stated here.
3.3. APPROXIMATE SOLUTION
3.3.1
66
γ-Suboptimal Finite Horizon Filtering Solution to the
Linearized Problem
Theorem 3.1: Invoking Theorem 2.3 in Chapter 2, for the state space representation
of the block diagram of Figure 3.5, described by Equations (3.8)-(3.11), and for Lk =
Hk , the central H∞ -optimal filtering solution to the linearized problem is given by
ˆ
ˆ
ˆ
ξk+1 = Fk ξk + Kf,k m(k) − Hk ξk , ξˆ0 = 0
(3.14)
−1
ŝ(k|k) = Lk ξˆk + (Lk Pk Hk∗ ) RHe,k
m(k) − Hk ξˆk
(3.15)
with
−1
Kf,k = (Fk Pk Hk∗ ) RHe,k
and RHe,k = Ip + Hk Pk Hk∗
(3.16)
where Pk satisfies the Lyapunov recursion
Pk+1 = Fk Pk Fk∗ , P0 = Π0 .
3.3.2
(3.17)
γ-Suboptimal Finite Horizon Prediction Solution to the
Linearized Problem
Theorem 3.2: Invoking Theorem 2.4 in Chapter 2, for the state space representation
of the block diagram of Figure 3.5, described by Equations (3.8)-(3.11), and for Lk =
Hk , if (I − Pk L∗k Lk ) > 0, then the central H∞ -optimal prediction solution to the
linearized problem is given by
ˆ
ˆ
ˆ
ξk+1 = Fk ξk + Kp,k m(k) − Hk ξk , ξˆ0 = 0
(3.18)
ŝ(k) = Lk ξˆk
(3.19)
with Kp,k = Fk Pk Hk∗ where Pk satisfies the Lyapunov recursion (3.17).
3.3.3
Important Remarks
1. As indicated in Theorems 3.1 and 3.2, the solution to the linearized robust estimation problem (Eq. (3.12) for the filtering problem and Eq. (3.13) for the
prediction problem) requires the solution to only one Riccati equation. Furthermore, the Riccati solution propagates forward in time and does not involve any
3.4. IMPLEMENTATION SCHEME FOR THE EBAF ALGORITHM IN IIR CASE
67
information regarding the future of the system or the reference signal. Thus,
the resulting adaptive algorithm is real-time implementable.
2. Note that the Riccati update for the simplified solution to the linearized robust
estimation problems reduces to a Lyapunov recursion which always generates a
non-negative Pk as long as P0 > 0.
3. Based on the results in Theorem 3.1, the linearized filtering problem has a
guaranteed solution for γf = 1. This will prevent any breakdown in the solution
and allows real-time implementation of the algorithm.
4. Theorem 3.2 proves that the prediction solution to the linearized problem is
guaranteed to exist for γp = 1, as long as the condition (I − Pk L∗k Lk ) > 0
is satisfied. Furthermore, the discussion in Section 2.8 shows that this condition translates into an upper limit for the adaptation rate in the steady-state
operation of the adaptive system.
5. From features (2) and (3), there is no need to verify the solutions at each time
step, so the computational complexity of the estimation based approach is O(n3 )
(primarily for calculating Fk Pk FK∗ in Eq. (3.17)), where
n = (2N + 1) + Ns
(3.20)
where (2N + 1) is the total number of IIR filter parameters for an IIR filter of
order N, and Ns is the order of the secondary path. As in the FIR case, the
special structure of Fk reduces the computational complexity to O(Ns3 + Ns N),
i.e. cubic in the order of the secondary path, and linear in the length of the
FIR filter.
3.4
Implementation Scheme for the EBAF Algorithm in IIR Case
The implementation scheme parallels that of the adaptive FIR filter discussed in
Chapter 2. For easier reference, the main signals involved in the description of the
3.4. IMPLEMENTATION SCHEME FOR THE EBAF ALGORITHM IN IIR CASE
68
adaptive algorithm are briefly introduced here. For a more detailed description, see
Chapter 2. In what follows (a) Ŵ (k) is the estimate of the adaptive weight vector,
4
(b) θ̂(k) is the estimate of the state of the secondary path, (c) u(k) = h∗ (k)Ŵ (k)
is the actual control input to the secondary path, (d) y(k) and d(k) are the actual
outputs of the secondary and primary paths, respectively, (e) e(k) is the actual error
measurement, and (f) θcopy (k) and ycopy (k) are the adaptive algorithm’s internal copy
of the state and output of the secondary path which are used in constructing m(k)
according to Eq. (3.7). Now, the implementation algorithm can be outlined as follows:
1. Start with Ŵ (0) = 0 and θ̂(0) = 0 as the initial guess for the state vector in the
approximate model of the primary path. Also assume that θcopy (0) = 0, and
r(−1) = · · · = r(−N) = 0 (hence h(0) = [ x(0) 0 · · · 0 ]T ). The initial value
for the Riccati matrix is P0 which is chosen to be block diagonal.
2. If 0 ≤ k ≤ M (finite horizon):
(a) Linearize the nonlinear dynamics in Eqs. (3.8), (3.10), and (3.11) by substituting for a0 (k), b1 (k), · · ·, bN (k) with their best available estimates
(i.e. â0 (k) = Ŵ (1, k), b̂1 (k) = Ŵ (2, k), · · ·, b̂N (k) = Ŵ (N + 1, k)) in
Eq. (3.9).
(b) Form the control signal
u(k) = h∗ (k)Ŵ (k)
(3.21)
to be applied to the secondary path. Note that applying u(k) to the
secondary path produces
y(k) = Cs (k)θ(k) + Ds (k)u(k)
(3.22)
at the output of the secondary path. This in turn leads to the following
error signal measured at time k:
e(k) = d(k) − y(k) + Vm (k)
(3.23)
which is available to the adaptive algorithm to perform the state update
at time k.
3.5. ERROR ANALYSIS
69
(c) Propagate the state estimate and the internal copy of the state of the
secondary path as follows
 "
# 
Ŵ (k + 1)



 =
θ̂(k
+
1)


θcopy (k + 1)
i
h
"
# h
Kf,k Cs (k)
Fk + Kf,k [0 − Cs (k)]
Kf,k
Ŵ (k)




 θ̂(k) +



h
i
∗
θcopy (k)
As (k)
0
(Bs (k)h (k) 0
i

e(k)

(3.24)
where e(k) is the error sensor measurement at time k given by Eq. (3.23),
and Kf,k = Fk Pk Hk∗ (I + Hk Pk Hk∗ )−1 (see Theorem 3.1). Note that for
the prediction-based EBAF algorithm Kf,k must be replaced with Kp,k =
Fk Pk Hk∗ .
(d) update the Riccati matrix Pk using the Lyapunov recursion
"
#
P11
P12,k+1
=
∗
P12,k+1
P22,k+1
"
#"
#"
I
0
P11 P12,k
I
Bs (k)h∗ (k) As (k)
∗
P12,k
P22,k
0
#∗
Bs (k)h∗ (k) As (k)
(3.25)
Pk+1 will be used in (3.24) to update the state estimate.
3. Go to 2.
3.5
Error Analysis
As explained in Section 2.7, for a linear stable secondary path the contribution of
the initial condition at the output of the secondary path decays to zero for large k.
This means that ycopy (k) converges to y(k) for sufficiently large k. In other words,
3.6. SIMULATION RESULTS
70
exact knowledge of the initial condition of the secondary path does not affect the
performance of the proposed EBAF algorithm for sufficiently large k. Unlike the
discussions in Section 2.7 however, a performance bound for the EBAF algorithm in
the IIR case is not yet available.
3.6
Simulation Results
This section examines the performance of the proposed EBAF algorithm for the
active noise cancellation (ANC) problem in the one dimensional acoustic duct (see
Figure 2.5). The control objective is the same as that in Section 2.9, i.e. to attenuate
(cancel in the ideal case) the disturbance introduced into the duct by Speaker #1
(primary noise source) at the position of Microphone #2 (error sensor) by the control
signal generated by Speaker #2 (secondary source). Microphone #1 can be used to
provide the reference signal for the adaptation algorithm. Clearly, Microphone #1
measurements are affected by both primary and secondary sources, and if used as the
reference signal, feedback contamination exists.
A state space model (of order 10) is identified for this one-dimensional acoustic
system. Of four identified transfer functions (see Figs. 2.6 and 2.7), only the transfer
function from Speaker #2 to Microphone #2 (i.e. the secondary path) is used by the
estimation-based adaptive algorithm. The IIR filter in EBAF approach is of order 5
(i.e. a total of 11 parameters for adaptive IIR filter). Each FIR filter in the FuLMS
implementation is of order 6 (a total of 12 parameters in the adaptive filter). Note
that all the measurements in the simulations are subject to band-limited white noise
(with power 0.008), and the sampling frequency in all cases is 1 KHz.
Figure 3.6 compares the performance of an estimation-based adaptive IIR filter
with that of a FuLMS algorithm in a single-tone noise cancellation problem. The
frequency of the tone is 150 Hz. Referring to Fig. 3.6, the first second shows the open
loop measurement of the error sensor, Microphone #2. At t = 1 (sec) the adaptive
control is switched on. For the next 5 seconds both adaptive algorithms have access
to the primary disturbance (i.e. x(k) is fully known to the adaptation schemes). At
t = 6 the reference signal available to the adaptive algorithms is switched to the
3.6. SIMULATION RESULTS
71
measurements of Microphone #1. Note that in this case only a filtered version of the
primary disturbance which is also contaminated with the output of the adaptive filter
is available to the adaptive algorithms. The adaptation rate for FuLMS algorithm
is kept small (0.00003 in this case), to avoid unstable behavior when the reference
signal becomes contaminated. The EBAF algorithm converges faster than FuLMS
algorithm (EBAF algorithm reaches its steady-state behavior in approximately 1.5
seconds), while avoiding the unstable behavior due to feedback contamination. A 50
times reduction in the error amplitude (without feedback contamination) is recorded.
With feedback contamination however, only a 10 times reduction is achieved.
Figure 3.7, shows a similar scenario for the multi-tone case. Here the primary
disturbance consists of two sinusoidal signals at 150 and 140 Hz. A trend similar to
the single-tone case is observed here. EBAF algorithm is robust to feedback contamination and allows faster convergence rates. The performance however is not as good
as the single tone case. With uncontaminated reference signal only a reduction of
order 10 is achieved. With feedback contamination this performance is reduced to a
factor of 6. The results shown in Figures 3.6 and 3.7 capture the typical behavior of
the adaptive IIR filters under the EBAF and FuLMS algorithms.
3.6. SIMULATION RESULTS
ref (k)
72
Vm (k)
d(k)
Primary Path
(Unknown)
+
+
+
+
Feedback Path
(Known)
+
+
e(k)
−
Adaptive IIR
Filter
x(k)
u(k)
Secondary Path
(Known)
y(k)
Fig. 3.1: General block diagram for the adaptive filtering problem of interest (with Feedback
Contamination)
Primary Path
(Unknown)
P (z)
ref (k)
+
+
+
Vm (k)
d(k)
+
Feedback Path
F (z)
+
+
e(k)
−
+
+
−
Adaptive FIR Filter
F̂ (z)
u(k)
Secondary Path
(Known)
S(z)
y(k)
Adaptive FIR Filter
x(k)
W (z)
Fig. 3.2: Basic Block Diagram for the Feedback Neutralization Scheme
3.6. SIMULATION RESULTS
73
ref (k)
Vm (k)
d(k)
Primary Path
(Unknown)
+
+
Feedback Path
(Known)
+
+
e(k)
−
A(z)
+
Secondary Path
(Known)
+
x(k)
u(k)
y(k)
+
B(z)
Fig. 3.3: Basic Block Diagram for the Classical Adaptive IIR Filter Design
Primary Path
Modeling Error
x(k)
A Copy Of
Secondary Path
An IIR Filter
Adaptive IIR Filter
Secondary Path
u(k)
Vm (k)
+
d(k)
+
+
+
−
y(k)
Fig. 3.4: Estimation Interpretation of the IIR Adaptive Filter Design
e(k)
3.6. SIMULATION RESULTS
74
Approximate Model For Primary Path
x(k)
a0
+
r(k)
+
Vm (k)
Ds (k)
u(k)
Bs (k)
+
z −1
Cs (k)
z −1
As (k)
a1
b1
z −1
b2
a2
A Second Order IIR Filter
Fig. 3.5: Approximate Model For the Unknown Primary Path
+
+
d(k) m(k)
3.6. SIMULATION RESULTS
75
e(k) - EBAF Adaptive IIR
Single Tone Error Cancellation - EBAF vs. FuLMS
1
0.5
0
−0.5
e(k) - FuLMS Adaptive IIR
−1
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
1
0.5
0
−0.5
−1
Time (sec.)
Fig. 3.6: Performance Comparison for EBAF and FuLMS Adaptive IIR Filters for Single-Tone
Noise Cancellation. The controller is switched on at t = 1 second. For 1 ≤ t ≤ 6 seconds adaptive
algorithm has full access to the primary disturbance. For t ≥ 6 the output of Microphone #1 is
used as the reference signal (hence feedback contamination problem).
3.6. SIMULATION RESULTS
76
e(k) - EBAF Adaptive IIR
Multi Tone Error Cancellation - EBAF vs. FuLMS
1.5
1
0.5
0
−0.5
−1
e(k) - FuLMS Adaptive IIR
−1.5
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
1.5
1
0.5
0
−0.5
−1
−1.5
Time (sec.)
Fig. 3.7: Performance Comparison for EBAF and FuLMS Adaptive IIR Filters for Multi-Tone
Noise Cancellation. The controller is switched on at t = 1 second. For 1 ≤ t ≤ 6 seconds adaptive
algorithm has full access to the primary disturbance. For t ≥ 6 the output of Microphone #1 is
used as the reference signal (hence feedback contamination problem).
3.7. SUMMARY
3.7
77
Summary
A new framework for the synthesis and analysis of the IIR adaptive filters is introduced. First, an estimation interpretation of the adaptive filtering (control) is used
to formulate an equivalent nonlinear robust estimation problem. Then, an approximate solution for the equivalent estimation problem is provided. This approximate
solution is based on a linearizing approximation, from which the adaptation law for
the adaptive filter weight vector is extracted.
The proposed approach clearly indicates an inherent connection between the adaptive IIR filter design and a nonlinear robust estimation problem. This connection
brings the analysis and synthesis tools in robust estimation into the field of adaptive
IIR filtering (control). Simulation results demonstrate the feasibility of the proposed
EBAF algorithm.
Chapter 4
Multi-Channel Estimation-Based
Adaptive Filtering
This chapter extends the estimation-based adaptive filtering algorithm, discussed
in Chapter 2, to the multi-channel case where a number of adaptively controlled
secondary sources use multiple reference signals to cancel the effect of a number of
primary sources (i.e. disturbance sources) as seen by a number of error sensors. The
multi-channel estimation-based adaptive filtering algorithm is shown to maintain all
the main features of the single-channel solution, underlining the systematic nature of
the approach.
In addition to the noise cancellation problem in a one dimensional acoustic duct, a
structural vibration control problem is chosen to examine the performance of the proposed multi-channel adaptive algorithm. An identified model for a Vibration Isolation
Platform (VIP) is used for vibration control simulations in this chapter. The performance of the new multi-channel adaptive algorithm is compared to the performance
of a multi-channel implementation of the FxLMS algorithm.
78
4.1. BACKGROUND
4.1
79
Background
For a wide variety of applications such as equalization in wireless communication
when more than one receiver/transmitter are involved, or active control of sound
and vibration in cases where the acoustic environment or the dynamic system of
interest is complex and a number of primary sources excite the system, multi-channel
adaptive filtering (control) schemes are required [1]. A brief description of the multichannel implementation of the FxLMS algorithm in Section 4.1.1 will support the
observation in Reference [33] that “compared to single-channel algorithms, multichannel adaptive schemes are significantly more complex”. As Reference [33] points
out, successful application of the classical multi-channel adaptive algorithms has been
limited to cases involving repetitive noise with a few harmonics [43,49,13]. This
observation agrees with the results in Ref. [3] where a significant noise reduction is only
achieved for periodic noises∗ . In contrast to the classical approaches to multi-channel
adaptive filter (control) design, this chapter will show that, for the new estimationbased approach, the multi-channel design is virtually identical to the single channel
case. Furthermore, the analysis of the multi-channel adaptive system in the new
framework is a straightforward extension of the analysis used in the single channel
case.
4.1.1
Multi-Channel FxLMS Algorithm
Figure 4.1 shows the general block diagram of a multi-channel ANC system in which
reference signals can be affected by the output of the adaptive filters. Simulations in
this chapter, however, are based on the assumption that the effects of feedback are
negligible and that the reference signal is available to the adaptation scheme through
a noisy measurement. Measurement noise however is independent from the reference
signal itself. Note that, this section is only intended as a brief review of the multichannel FxLMS adaptive algorithm. For a detailed treatment of the subject see [33],
∗
Another interesting conclusion in Ref. [3] is the following: the performance of a Multi-Channel
implementation of the FxLMS algorithm is similar to the performance of an H∞ controller which
was directly designed for noise cancellation.
4.1. BACKGROUND
80
Chapter 5, and the references therein.
Referring to Figure 4.1, the ANC adaptive filter has J reference input signals
denoted by xT (k) = [x1 (k) x2 (k) · · · xJ (k)]. The controller generates K secondary
signals that are elements of the control vector uT (k) = [u1 (k) u2 (k) · · · uK (k)].
Therefore, a K ×J matrix of adaptive FIR filters can be used to describe the adaptive
control block in Figure 4.1,

···
T
W11
(k)

Ω(k) = 


T
W1J
(k)



..
.
T
Wkj
(k)
..
.
T
WK1
(k)
···
T
WKJ
(k)
(4.1)
where
def
T
Wkj
(k) = [wkj,1(k) · · · wkj,(L−1) (k)]
(4.2)
is the adaptive filter relating the j-th reference signal to the k-th control command.
Note that L is the length of the adaptive FIR filters. Defining the reference signal
vector as
XT (k) =
xT1 (k) xT2 (k) · · · xTJ (k)
(4.3)
where xTj (k) = [xj (k) xj (n − 1) · · · xj (n − L + 1)] is the last L samples of the j-th
reference signal, and the adaptive weight vector as
def
W(k) = [ Ω(1,:) (k) · · · Ω(K,:)(k) ]
(4.4)
with Ω(k,:) (k) referring to the k-th row of matrix Ω(k), the control vector u(k) can
be defined as
u(k) = X T (k)W(k)
where




X (k) = 


X(k)
0
..
.
0
(4.5)

···
0
X(k) · · ·
..
.
0
0
0
···
0
0
X(k)






(4.6)
4.2. ESTIMATION-BASED ADAPTIVE ALGORITHM FOR MULTI CHANNEL CASE
81
is a (JKL)×K matrix. The error signal vector can now be defined as
e(k) = d(k) − S(k) ⊕ u(k)
(4.7)
where S(k) is the impulse response of the K-input/M-output secondary path, and
⊕ indicates the convolution operation. Defining the instantaneous squared error,
eT (k)e(k), as an approximation for the sum of the mean-square errors, the gradient
based update equation for the weight vector W(k) is [33],
W(k + 1) = W(k) + µX 0 (k)e(k)
(4.8)
where X 0 (k) is the matrix of the filtered reference signals obtained by the Kronecker
convolution operation on the reference signal vector

ŝ (k) ⊕ X(k) ŝ21 (k) ⊕ X(k) · · · ŝM 1 (k) ⊕ X(k)
 11
 ŝ (k) ⊕ X(k) ŝ (k) ⊕ X(k) · · · ŝ (k) ⊕ X(k)
22
M2
 12
X 0 (k) = 
.
.
..
.
..
..
..

.

ŝ1K (k) ⊕ X(k) ŝ2K (k) ⊕ X(k) · · · ŝM K (k) ⊕ X(k)







(4.9)
Note that the adaptation algorithm has access to an estimate of the secondary path
only (e.g. through system identification) and therefore ŝij (an estimate of the impulse
response for the single-channel transfer function from input j to output i) are used
in calculation of the filtered reference signals.
4.2
Estimation-Based Adaptive Algorithm for Multi
Channel Case
The underlying concept for estimation based adaptive filtering algorithm is the same
for both single-channel and multi-channel systems. Figure 4.1 is the multi-channel
block diagram representation of Fig. 2.1 (in Chapter 2) where the main components of
active noise cancellation problem were described. Therefore, an estimation interpretation identical to the one in the single-channel case can be used to translate a given
4.2. ESTIMATION-BASED ADAPTIVE ALGORITHM FOR MULTI CHANNEL CASE
82
adaptive filtering (control) problem into an equivalent robust estimation problem.
See Chapter 2 for a detailed treatment of the steps involved in this translation.
As in Chapter 2, state space models for the adaptive filter and the secondary path
are used to construct an approximate model for the unknown primary path. This
approximate model replicates the structure of the adaptive path from the primary
source, ref (k), to the output of the secondary path, y(k) (see Fig. 4.2). Note that for a
given disturbance input, there is an “optimal” (but unknown) setting of adaptive filter
parameters for which the difference between the primary path and its approximate
model is minimized. Finding this optimal setting is the objective of the estimation
based approach which can be summarized as follows:
1. Devise an estimation strategy that recursively improves the estimate of the
optimal values of the adaptive filter parameters in the approximate model of
the primary path,
2. Set the actual value of the weight vector in the adaptive filter to the best
available estimate of the parameters obtained from the estimation strategy.
Note that in Figure 4.2,
e(k) = d(k) − y(k) + Vm (k)
(4.10)
where (a) e(k) ∈ RM ×1 is the measured error vector, (b) Vm (k) ∈ RM ×1 is the exogenous disturbance that captures measurement noise, modeling error and uncertainty
in the initial condition of the secondary path, and (c) y(k) = S(k) ⊕ u(k) (also in
RM ×1 ) is the output of the secondary path. S(k) is the impulse response of the secondary path and ⊕ denotes convolution. Here u(k) obeys Eq. (4.5) with the same
definitions for X (k) and W(k) as in (4.6) and (4.8), respectively. Equation (4.10)
can be rewritten as
e(k) + y(k) = d(k) + Vm (k)
(4.11)
where the left hand side is a noisy measurement of the output of the primary path
d(k). Since y(k) is not directly measurable (neither is d(k)), the adaptive algorithm should generate an internal copy of y(k) (referred to as ycopy (k)). The derived
4.2. ESTIMATION-BASED ADAPTIVE ALGORITHM FOR MULTI CHANNEL CASE
83
measured quantity can then be defined as
4
m(k) = e(k) + y(k) = d(k) + Vm (k)
(4.12)
which will be used in formulating the estimation problem. The only assumption
involved in constructing m(k) is the assumed knowledge of the initial condition of
the secondary path. In Chapter 2 it is shown that for a linear, stable secondary path (a
realistic assumption in practice), any error in y(k) due to an initial condition different
from what is assumed by the algorithm remains bounded (hence it can be treated as
a component of the measurement disturbance). Furthermore, for sufficiently large k,
this error decays to zero (i.e. ycopy (k) → y(k)). In Figure 4.3, the state space model
for the secondary path is
θ(k + 1) = As (k)θ(k) + Bs (k)u(k)
(4.13)
y(k) = Cs (k)θ(k) + Ds (k)u(k)
(4.14)
where θ(k) is the state variable capturing the dynamics of the secondary path. The
weight vector of the adaptive filter
T
T
T
T
T
T
T
(k) · · · W1J
(k) W21
(k) · · · W2J
(k) · · · WK1
(k) · · · WKJ
(k) (4.15)
W(k) = W11
is also treated as the state vector that captures the dynamics of the FIR filter.
Note that Wkj (k) is itself a vector of length L (length of each FIR filter). ξkT =
WT (k) θT (k) is then the state vector for the overall system. The state space
representation of the system is then
"
#
"
#"
#
W(k + 1)
I(JKL)×(JKL)
W(k)
0
=
θ(k + 1)
θ(k)
Bs (k)X ∗ (k) As (k)
4
ξk+1 = Fk ξk
(4.16)
where X (k), defined by Equation (4.6), captures the effect of the reference input
vector X(k). For this system, the derived measured output is
#
"
i W(k)
h
+ Vm (k)
m(k) =
Ds (k)X ∗ (k) Cs (k)
θ(k)
4
= Hk ξk + Vm (k)
(4.17)
4.2. ESTIMATION-BASED ADAPTIVE ALGORITHM FOR MULTI CHANNEL CASE
84
where m(k) is defined in Equation (4.12). Noting the objective of the adaptive
filtering problem in Fig. 4.1, s(k) = d(k) is the quantity to be estimated. Therefore,
"
#
h
i W(k)
s(k) =
Ds (k)X ∗ (k) Cs (k)
θ(k)
4
= Lk ξk
(4.18)
Here m(k) ∈ RM ×1 , s(k) ∈ RM ×1 , θ(k) ∈ RNs ×1 where Ns is the order of the
secondary path. All matrices are then of appropriate dimensions. Note that Equations (4.16) through (4.18) are identical to Equations (2.5) through (2.7). The only
difference is in the dimension of the variable involved, and the fact that h(k) is replaced by X ∗ (k).
Choosing the H∞ criterion to generate a filtering (prediction) estimate of s(k), the
equivalent estimation problem will also be identical to that in Chapter 2 (i.e. Eq. (2.11)
for the filtering estimate, and Eq. (2.12) for the prediction estimate).
Defining
4
ŝ(k|k) = F (m(0), · · · , m(k)) as the filtering estimate of s(k), the objective in the
filtering solution is to find ŝ(k|k) such that the worst case energy gain from the measurement disturbance and the initial condition uncertainty to the error in the filtering
estimate is properly bounded, i.e.
M
X
sup
Vm , ξ0
[s(k) − ŝ(k|k)]∗ [s(k) − ŝ(k|k)]
n=0
ˆ
(ξ0 − ξˆ0 )∗ Π−1
0 (ξ0 − ξ0 ) +
M
X
≤ γf2
(4.19)
∗
Vm
(k)Vm (k)
n=0
4
In a similar way, defining ŝ(k) = F (m(0), · · · , m(k − 1)) as the prediction estimate
of s(k), the objective in the prediction solution is to find ŝ(k) such that
M
X
sup
Vm , ξ0
[s(k) − ŝ(k)]∗ [s(k) − ŝ(k)]
n=0
(ξ0 − ξˆ0 )
∗
Π−1
0 (ξ0
− ξˆ0 ) +
M
X
≤ γp2
(4.20)
∗
Vm
(k)Vm (k)
n=0
Solutions to these two problems are discussed in detail in Chapter 2 and therefore
the next section only briefly presents the simplified solutions.
4.3. SIMULATION RESULTS
4.2.1
85
H∞ -Optimal Solution
Theorem 4.1: Invoking Theorem 2.3 in Chapter 2, for the state space representation
of the block diagram in Figure 4.3 (described by Equations (4.16)-(4.18)), and for
Lk = Hk , the central H∞ -optimal solution to the filtering problem in Eq. (4.19) is
obtained for γf = 1, and is described by
ξˆk+1 = Fk ξˆk + Kf,k m(k) − Hk ξˆk , ξˆ0 = 0
−1
ŝ(k|k) = Lk ξˆk + (Lk Pk Hk∗ ) RHe,k
m(k) − Hk ξˆk
(4.21)
(4.22)
with
−1
Kf,k = (Fk Pk Hk∗ ) RHe,k
and RHe,k = Ip + Hk Pk Hk∗
(4.23)
where Pk satisfies the Lyapunov recursion
Pk+1 = Fk Pk Fk∗ , P0 = Π0 .
(4.24)
Theorem 4.2: Invoking Theorem 2.4 in Chapter 2, for the state space representation
of the block diagram in Figure 4.3 (described by Equations (4.16)-(4.18)), and for
Lk = Hk , if (I − Pk L∗k Lk ) > 0, then the central H∞ -optimal prediction solution to the
linearized problem is obtained for γp = 1, and is given by
ξˆk+1 = Fk ξˆk + Kp,k m(k) − Hk ξˆk , ξˆ0 = 0
(4.25)
ŝ(k) = Lk ξˆk
(4.26)
with Kp,k = Fk Pk Hk∗ where Pk satisfies the Lyapunov recursion (4.24).
4.3
Simulation Results
The implementation scheme for the EBAF algorithm in multi-channel case is identical
to the implementation scheme in the single-channel case (see Chapter 2 for the FIR
case, and Chapter 3 for the IIR case), and therefore it is not repeated here.
4.3. SIMULATION RESULTS
4.3.1
86
Active Vibration Isolation
The Vibration Isolation Platform (VIP) (see Figure 4.4 and 4.5) is an experimental
set up which is designed to capture the main features of a real world payload isolation
and pointing problem. Payload isolation refers to the vibration isolation of payload
structures with instruments or equipments requiring a very quiet mounting [1]. VIP
is designed such that the base supporting the payload (middle mass in Figure 4.5)
can emulate spacecraft dynamics. Broadband as well as narrowband disturbances
can be introduced to the middle mass (emulating real world vibration sources such
as solar array drive assemblies, reaction wheels, control moment gyros, crycoolers,
and other disturbance sources that generate on-orbit jitter) via a set of three voice
coil actuators. The positioning of a second set of voice coil actuators allows for
the implementation of an adaptive/active isolation system. More specifically, the
Vibration Isolation Platform consists of the following main components:
1. Voice-Coil Actuators: 6 voice-coil actuators are mounted on the middle mass
casing. Three of these actuators (positioned 120 degrees apart on a circle of
radius 4.400 inches) are used to shake the middle mass and act as the source of
disturbance to the platform. They can also be used to introduce some desired
dynamics for the middle mass that supports the payload. As shown in Figure
4.5, these actuators act against the ground. The other three actuators (placed
120 degrees apart on a circle of radius 4.000 inches) act against the middle mass
and are used to isolate the payload (top mass) from the motions of the middle
mass. Note that the two circles on which control and disturbance actuators are
mounted are concentric, and one set of actuators is rotated in the horizontal
plane by 60 degrees with respect to the other.
2. Sensors: VIP is equipped with two sets of sensors,
(a) Position Sensors: Each actuator is equipped with a colocated position
(gap) sensor which is physically inside the casing of the actuator. Three
additional position sensors are used as truth sensors (Figure 4.5) to measure the displacement of the payload in the inertial frame.
4.3. SIMULATION RESULTS
87
(b) Load Cells: Three load cells are used to measure the interaction forces
between the middle mass and the payload. These sensors are colocated
with the point of contact of the control actuators and the payload. It is
important to note that any interaction force between the payload and the
rest of the VIP system is transfered via these load cells.
A state-space model for the VIP platform is identified using the FORSE system
identification software developed at MIT [30]. The detailed description of the identification process, the identified model, and the model validation process is discussed
in Appendix C. Figure 4.6 shows the singular value plots for the MIMO transfer
functions from control ([u])/disturbance ([d]) actuators to the load cells ([lc]) and
scoring sensors ([sc]).
In all simulations that follow, the length of the adaptive FIR filters, i.e. L in
Equation 4.2, is 4 (unless stated otherwise). This length is found to be sufficient for
an acceptable performance of the adaptive algorithms. The sampling frequency for
all the simulations in this section is 1000 Hz. Furthermore, all measurements are
subject to band limited white noise with power 0.008.
Figures 4.7 and 4.8 show the reading of the scoring sensors (i.e. the variations of
the payload from the equilibrium position in the inertial frame) for the multi-channel
implementation of the EBAF and the FxLMS adaptive algorithms, respectively. Disturbance actuators apply sinusoidal excitation of amplitude 0.1 Volts at 4 Hz to the
middle mass. The phase for the excitation of the disturbance actuator #1 is assumed
to be zero, while disturbance actuators #2 and #3 are 22.5 and 45 degrees out of
phase with the first actuator. Only a noisy measurement of the primary disturbance
is assumed to be available to the adaptive algorithms. The signal to noise ratio for
the available reference signal is 3.0. For simulations in Figures 4.7 and 4.8 the control
signal starts at t = 30 seconds. Figure 4.7 shows that the amplitude of the transient
vibrations of the payload under the EBAF adaptive algorithm (for 30 ≤ t ≤ 60)
does not exceed that of the open loop vibrations. In contrast, the amplitude of the
transient vibrations under the FxLMS, Figure 4.8, exceeds twice the amplitude of
the open loop vibrations in the system. For a smaller amplitude during transient
4.3. SIMULATION RESULTS
88
vibrations, the adaptation rate for the FxLMS algorithm should be reduced. This
will result in an even slower convergence of the adaptive algorithm. Note that, for the
results in Figure 4.8, the adaptation rate is 0.0001. Even with this adaptation rate,
FxLMS algorithm requires approximately 20 more seconds (compared to the EBAF
case) to converge to its steady state value. In the steady state, the EBAF algorithm
achieves a 20 times reduction in the amplitude of the payload vibrations. For the
FxLMS algorithm in this case, the measured reduction is approximately 16 times.
Figures 4.9 and 4.10 show the reading of the scoring sensors when the primary
disturbances are multi-tone sinusoids. The primary disturbance consists of sinusoidal
signal of amplitudes 0.1 and 0.2 volts at 4 and 15 Hz, respectively. As in the single tone
case, both components of the excitation for the disturbance actuator #1 are assumed
to have zero phase. Each sinusoidal component of the excitation for the disturbance
actuator #2 (#3) is assumed to have a phase lag of 22.5 (45) degrees with respect to
the corresponding component of the excitation in actuator #1. Figures 4.9 and 4.10
demonstrate a trend similar to that discussed for the single tone scenario. For the
FxLMS algorithm, a trade off between the amplitude of the transient vibrations of
the payload and the speed of the convergence exists. The adaptation rate here is the
same as the single tone case. Slower adaptation rates can reduce the amplitude of the
transient vibrations at the expense of the speed of the convergence. For the EBAF
algorithm, however, better transient behavior and faster convergence are observed. In
the steady state, the EBAF algorithm provides a 15 times reduction in the amplitude
of the vibrations of the payload. For the FxLMS algorithm a 9 times reduction is
recorded.
In Fig. 4.11 the effect of feedback contamination in the performance of the EBAF
algorithm is examined. As in the previous simulations, control actuators are switched
on at t = 30 seconds. The reference signal available to the adaptation algorithm is
the output of the load cells which measure the forces transfered to the payload. Obviously, load cell measurements contain the effect of both primary disturbances (single
tone sinusoids at 4 Hz for this example), and control actuators (and hence the classical feedback contamination problem). Here, no special measure to counter feedback
contamination is taken. Figure 4.11 shows that an average of 4 times reduction in the
4.3. SIMULATION RESULTS
89
magnitude of the vibrations transfered to the payload is achieved (i.e. a degraded performance when compared to the case without feedback contamination). The EBAF
algorithm, however, maintains its stability in the face of feedback contamination
without any additional measures, hence exhibiting robustness of the algorithm to the
contamination of the reference signal. Note that for the FxLMS adaptive algorithm,
with the adaptation rate similar to that in Figure 4.10, feedback contamination leads
to an unstable behavior. The adaptation rate should be reduced substantially (hence
extremely slow convergence of the FxLMS algorithm), in order to recover the stability
of the adaptation scheme.
The simulations in this section have shown that the multi-channel implementation
of the estimation-based adaptive filtering algorithm provides the same advantages
observed for the single channel case. More specifically, the multi-channel EBAF
algorithm achieves desirable transient behavior and fast convergence without compromising steady-state performance of the adaptive algorithm. It also demonstrates
robustness to feedback contamination. It is important to note that, the above mentioned advantages are achieved by an approach which is essentially identical to the
single-channel version of the algorithm.
4.3.2
Active Noise Cancellation
Consider the one dimensional acoustic duct shown in Figure 2.5. Here, disturbances
enter the duct via Speaker #1. The objective of the multi-channel noise cancellation
is to use both available speakers to simultaneously cancel the effect of the incoming
disturbance at Microphones #1 and #2. The control signal is supplied to each speaker
via an adaptive FIR filter (i.e. in the case of Speaker #1 added to the primary
disturbance). Figure 4.12 shows the output of the microphones when the primary
disturbance (applied to Speaker #1) is a multi-tone sinusoid with 150 Hz and 200
Hz frequencies. The length of each FIR filter in this simulation is 8. For the first
two seconds the controller is off (i.e both adaptive filters have zero outputs). At
t = 2.0 seconds the controller is switched on. The initial value for the Riccati matrix
is P0 = diag(0.0005I2(N +1) , 0.00005INs ) (where N + 1 is the length of each FIR filter,
4.3. SIMULATION RESULTS
90
and Ns is the order of the secondary path). It is clear that the error at Microphone
#2 is effectively canceled in 0.2 seconds. For Microphone #1 however the cancellation
time is approximately 5 seconds. A 30 times reduction in disturbance amplitude is
measured at Microphone #2 in approximately 10 seconds. For Microphone #1 this
reduction is approximately 15 times. Note that the distance between Speaker #2
and Microphone #1 (46 inches) is much greater than the distance between Speaker
#1 and Microphone #1 (6 inches). Due to this physical constraint, Speaker #2
alone, is not enough for an acceptable noise cancellation at both microphones. The
experimental data in Figures 2.9 and 2.10 in which the result of a single-channel
implementation of the EBAF algorithm, aimed at noise cancellation at Microphone
#2, was shown, confirm this observation. Using a multi-channel approach however,
allows for a substantial reduction in the amplitude of the measured noise at both
microphones. Nevertheless, noise cancellation at the position of Microphone #1 tends
to be slower than the noise cancellation at the position of Microphone #2.
A similar scenario with band limited white noise as the primary disturbance is
shown in Fig. 4.13. Here the length of each FIR filter is 32. The performance of the
adaptive multi-channel noise cancellation problem in the frequency domain is shown
in Fig. 4.14. Once again the cancellation at Microphone #2 is superior to that at
Microphone #1.
4.3. SIMULATION RESULTS
Primary Source
ref (k)
91
J
Vm (k)
M
Primary
Path
d(k)
+
+
+
−
Feedback
Path
e(k)
+
+
−
y(k)
Adaptive
Filter
K
Secondary
Path
u(k)
x(k)
Fig. 4.1: General block diagram for a multi-channel Active Noise Cancellation (ANC) problem
Primary Path
Modeling Error
x(k)
A Copy Of
Secondary Path
An FIR Filter
J
K
Secondary Path
Adaptive FIR Filter
u(k)
Vm (k)
d(k)
+
+ M
+
−
+
e(k)
y(k)
Adaptation Algorithm
Fig. 4.2: Pictorial representation of the estimation interpretation of the adaptive control problem:
Primary path is replaced by its approximate model
4.3. SIMULATION RESULTS
x(k)
A Copy Of Adaptive Filter
92
A Copy Of Secondary Path
Vm (k)
Ds (k)
A Matrix Of
Adaptive
Filters
Bs (k)
+
Z −1
Cs (k)
As (k)
Fig. 4.3: Approximate Model for Primary Path
Fig. 4.4: Vibration Isolation Platform (VIP)
+
+
d(k)
m(k)
4.3. SIMULATION RESULTS
93
Fig. 4.5: A detailed drawing of the main components in the Vibration Isolation Platform (VIP).
Of particular importance are: (a) the platform supporting the middle mass (labeled as component
#5), (b) the middle mass that houses all six actuators (of which only two, one control actuator and
one disturbance actuator) are shown (labeled as component #11), and (c) the suspension springs
to counter the gravity (labeled as component #12). Note that the actuation point for the control
actuator (located on the left of the middle mass) is colocated with the load cell (marked as LC1).
The disturbance actuator (located on the right of the middle mass) actuates against the inertial
frame.
4.3. SIMULATION RESULTS
94
SVD for [d]−>[lc]
SVD for [u]−>[lc]
Volts/volts
0
0
10
10
−5
−5
10
10
0
2
10
0
10
SVD for [d]−>[sc]
10
SVD for [u]−>[sc]
0
Volts/volts
2
10
0
10
10
−5
−5
10
10
0
2
10
10
Frequency (Hz)
0
2
10
10
Frequency (Hz)
Fig. 4.6: SVD of the MIMO transfer function
4.3. SIMULATION RESULTS
95
Scoring Sensor Readouts - EBAF
e(1)
10
0
−10
0
20
40
60
80
100
120
0
20
40
60
80
100
120
0
20
40
60
80
100
120
e(2)
10
0
−10
e(3)
10
0
−10
Time (sec.)
Fig. 4.7: Performance of a multi-channel implementation of EBAF algorithm when disturbance
actuators are driven by out of phase sinusoids at 4 Hz. The reference signal available to the adaptive
algorithm is contaminated with band limited white noise (SNR=3). The control signal is applied
for t ≥ 30 seconds.
4.3. SIMULATION RESULTS
96
Scoring Sensor Readouts - FxLMS
e(1)
10
0
−10
0
20
40
60
80
100
120
0
20
40
60
80
100
120
0
20
40
60
80
100
120
e(2)
10
0
−10
e(3)
10
0
−10
Time (sec.)
Fig. 4.8: Performance of a multi-channel implementation of FxLMS algorithm when simulation
scenario is identical to that in Figure 4.7.
4.3. SIMULATION RESULTS
97
Scoring Sensor Readouts - EBAF
e(1)
10
0
−10
0
20
40
60
80
100
120
0
20
40
60
80
100
120
0
20
40
60
80
100
120
e(2)
10
0
−10
e(3)
10
0
−10
Time (sec.)
Fig. 4.9: Performance of a multi-channel implementation of EBAF algorithm when disturbance
actuators are driven by out of phase multi-tone sinusoids at 4 and 15 Hz. The reference signal
available to the adaptive algorithm is contaminated with band limited white noise (SNR=4.5). The
control signal is applied for t ≥ 30 seconds.
4.3. SIMULATION RESULTS
98
Scoring Sensor Readouts - FxLMS
e(1)
10
0
−10
0
20
40
60
80
100
120
0
20
40
60
80
100
120
0
20
40
60
80
100
120
e(2)
10
0
−10
e(3)
10
0
−10
Time (sec.)
Fig. 4.10: Performance of a multi-channel implementation of FxLMS algorithm when simulation
scenario is identical to that in Figure 4.9.
4.3. SIMULATION RESULTS
99
Scoring Sensors Readouts for Single-Tone at 4 Hz With Feedback Contamination
e(1)
5
0
−5
0
20
40
60
80
100
120
0
20
40
60
80
100
120
0
20
40
60
80
100
120
e(2)
5
0
−5
e(3)
5
0
−5
Time (sec.)
Fig. 4.11: Performance of a Multi-Channel implementation of the EBAF for vibration isolation
when the reference signals are load cell outputs (i.e. feedback contamination exists). The control
signal is applied for t ≥ 30 seconds.
4.3. SIMULATION RESULTS
100
Multi-Channel Active Noise Cancellation in Acoustic Duct
Microphone #1 (Volts)
4
2
0
−2
−4
−6
0
1
2
3
4
0
1
2
3
4
5
6
7
8
9
10
5
6
7
8
9
10
Microphone #2 (Volts)
4
2
0
−2
−4
Time (sec.)
Fig. 4.12: Performance of the Multi-Channel noise cancellation in acoustic duct for a multi-tone
primary disturbance at 150 and 200 Hz. The control signal is applied for t ≥ 2 seconds.
4.3. SIMULATION RESULTS
101
Multi-Channel Active Noise Cancellation in Acoustic Duct
Microphone #1 (Volts)
15
10
5
0
−5
−10
0
1
2
3
4
0
1
2
3
4
5
6
7
8
9
10
5
6
7
8
9
10
Microphone #2 (Volts)
10
5
0
−5
−10
Time (sec.)
Fig. 4.13: Performance of the Multi-Channel noise cancellation in acoustic duct when the
primary disturbance is a band limited white noise. The control signal is applied for t ≥ 2 seconds.
4.3. SIMULATION RESULTS
102
Speaker #1 → Microphone #1
Oloop
Cloop
Magnitude
0
10
−1
10
−2
10
2
3
10
10
Speaker #1 → Microphone #2
Oloop
Cloop
Magnitude
0
10
−1
10
−2
10
2
10
3
10
Fig. 4.14: Closed-loop vs. open-loop transfer functions for the steady state performance of the
EBAF algorithm for the simulation scenario shown in Figure 4.13.
4.4. SUMMARY
4.4
103
Summary
The estimation-based synthesis and analysis of multi-channel adaptive (FIR) filters is
shown to be identical to the single-channel case . Simulations for a 3-input/3-output
Vibration Isolation Platform (VIP), and a multi-channel noise cancellation in the one
dimensional acoustic duct are used to demonstrate the feasibility of the estimationbased approach. The new estimation-based adaptive filtering algorithm is shown to
provide both faster convergence (with an acceptable transient behavior), and improved steady state performance when compared to a multi-channel implementation
of the FxLMS algorithm.
Chapter 5
Adaptive Filtering via Linear
Matrix Inequalities
In this chapter Linear Matrix Inequalities (LMIs) are used to synthesize adaptive
filters (controllers). The ability to cast the synthesis problem as LMIs is a direct
consequence of the systematic nature of the estimation-based approach to the design
of adaptive filters proposed in Chapters 2 and 3 of this thesis. The question of internal
stability of the overall system is directly addressed as a result of the Lyapunovbased nature of the LMIs formulation. LMIs also provide a convenient framework for
the synthesis of multi-objective (H2 /H∞ ) control problems. This chapter describes
the process of augmenting the H∞ criterion that serves as the center piece of the
estimation-based adaptive filtering algorithm with H2 performance constraints, and
investigates the characteristics of the resulting adaptive filter. As in Chapters 2 and 3,
an Active Noise Cancellation (ANC) scenario is used to study the main features of
the proposed formulation.
5.1
Background
A detailed discussion of the estimation-based approach to the design of adaptive
filters (controllers) is presented earlier in Chapters 2 and 3. The discussion here will
104
5.1. BACKGROUND
105
be kept brief and serves more as a notational introduction. Figure 5.1 is a block
diagram representation of the active noise cancellation problem, originally shown in
Figure 2.1. Clearly, the objective here is to generate a control signal u(k) such that the
output of the secondary path, y(k), is close to the output of the primary path, d(k).
In Chapter 2, it is shown that an estimation interpretation of the adaptive filtering
(control) problem can be used to formulate an equivalent estimation problem. It is
this equivalent estimation problem that admits LMIs formulation.
To mathematically describe the equivalent estimation problem, state space models for the adaptive filter and the secondary path are needed. As in Chapter 2,
[As (k), Bs (k), Cs (k), Ds (k)] is the state space model for the secondary path. The
state variable for the secondary path is θ(k). For the adaptive filter the weight vector, W (k) = [ w0 (k) w1 (k) · · · wN (k) ]T , will be treated as the state variable. The
state space description for the approximate model of the primary path can then be
described as:
"
W (k + 1)
#
"
=
θ(k + 1)
I(N +1)×(N +1)
Bs (k)h∗k
0
#"
As (k)
W (k)
#
θ(k)
ξk+1 = Fk ξk
(5.1)
where
h(k) = [x(k) x(k − 1) · · · x(k − N)]T
(5.2)
captures the effect of the reference input x(k). Note that in Figure 5.1
e(k) = d(k) − y(k) + Vm (k)
(5.3)
where e(k) is the available error measurement, Vm (k) is the exogenous disturbance
that captures measurement noise, modeling error and uncertainty in the initial condition of the secondary path, and y(k) is the output of the secondary path. To formulate
the estimation problem, a derived measurement for the output of the primary path is
needed. Rewriting Eq. (5.3) as
4
m(k) = e(k) + y(k) = d(k) + Vm (k)
(5.4)
the right hand side, i.e. d(k) + Vm (k), can be regarded as the “noisy measurement”
of the primary path output. Note that on the left hand side of Eq. (5.4) only e(k) is
5.1. BACKGROUND
106
directly measurable. In general, an internal copy of the output of the secondary path,
referred to as ycopy (k), should be generated by the adaptive algorithm. Section 2.7
discusses the ramifications of the introduction of this internal copy in detail, and
shows that for the stable linear secondary path the difference between ycopy (k) and
y(k) will decay to zero for sufficiently large k. Now, the derived measured output for
the equivalent estimation problem can be defined as
#
"
h
i W (k)
m(k) =
+ Vm (k)
Ds (k)h∗k Cs (k)
θ(k)
= Hk ξk + Vm (k)
(5.5)
where m(k) is defined in Equation (5.4). Noting the objective of the noise cancellation
problem, s(k) = d(k) is chosen as the quantity to be estimated:
#
"
h
i W (k)
s(k) =
Ds (k)h∗k Cs (k)
θ(k)
= Lk ξk
(5.6)
Note that m(·) ∈ R1×1 , s(·) ∈ R1×1 , θ(·) ∈ R1×1 , and W (·) ∈ R(N +1)×1 . All matrices
are then of appropriate dimensions. ξkT = W T (k) θT (k) is clearly the state vector
for the overall approximate model of the primary path.
4
The following H∞ criterion can be used to generate ŝ(k) = F (m(0), · · · , m(k − 1))
(the prediction estimate of the desired quantity s(k)) such that the worst case energy
gain from the measurement disturbance and the initial condition uncertainty to the
error in the causal estimate of s(k) is properly limited, i.e.
M
X
sup
Vm , ξ0
[s(k) − ŝ(k)]∗ [s(k) − ŝ(k)]
≤ γ2
k=0
ξ0∗ Π−1
0 ξ0
+
M
X
(5.7)
∗
Vm
(k)Vm (k)
k=0
The Riccati-based solution to this problem is discussed in Chapter 2 in detail. It is
sometimes desirable, however, to have the adaptive filter meet some H2 performance
criteria in addition to the H∞ constraint in Equation (5.7). Linear matrix inequalities
5.2. LMI FORMULATION
107
offer a convenient framework for formulating the mixed H2 /H∞ synthesis problem.
Furthermore, numerically sound algorithms exist that can solve these linear matrix
inequalities very efficiently. Therefore, next section pursues a first principle derivation
of the LMI formulation for the design of adaptive filters.
5.2
LMI Formulation
Assume the following specific structure for the estimator
ˆ
ˆ
ˆ
ξk+1 = Fk ξk + Γk m(k) − Hk ξk
ŝ(k) = Lk ξˆk
(5.8)
(5.9)
in which Γk is the design parameter to be chosen such that the H∞ criterion is met.
Now, the augmented system can be defined as follows
"
#
"
#"
# "
#
ξk+1
0
Fk
ξk
0
=
+
Vm (k) (5.10)
ξˆk+1
ξˆk
Fk Hk Fk − Γk Hk
−Γk
"
#
i ξ
h
4
k
Zk = s(k) − ŝ(k) =
(5.11)
Lk −Lk
ˆ
ξk
4
Introducing a new variable ξ˜k = ξk − ξˆk , the augmented system can be described as
#
"
#"
# "
#
"
0
Fk
ξk
0
ξk+1
=
+
Vm (k)
0 Fk − Γk Hk
Γk
ξ˜k+1
ξ˜k
ηk+1 = Φk ηk + Ψk Vm (k)
(5.12)
with
4
Zk = s(k) − ŝ(k) =
i
h
0 Lk
"
ξk
ξ˜k
#
= Ωk ηk
(5.13)
The LMI solution for the design of adaptive filters finds a Lyapunov function for the
augmented system in (5.12)-(5.13) at each step. In other words, at each time step, an
infinite horizon problem is solved, and the solution is implemented for the next step.
5.2. LMI FORMULATION
108
Introducing the quadratic function V (ηk ) = ηk T P ηk (where P > 0), it is straight
forward [8] to show that for the augmented system at time k, (5.7) holds if
V (ηk+1 ) − V (ηk ) < γ 2 Vm (k)T Vm (k) − ZkT Zk
(5.14)
Note that the inclusion of the energy of the initial condition error will only increase
the right hand side of the inequality in Eq. (5.14). Replacing for Zk and ηk+1 from
(5.12)-(5.13), and after some elementary algebraic manipulations the inequality in
(5.14) can be written as
"
#"
#
h
i ΦT P Φ − P + ΩT Ω
T
η
Φ
P
Ψ
k
k
k
k
k
k
k
T
<0
ηkT Vm
(k)
T
T
Ψk P Φk
Ψk P Ψk − γ 2 I
Vm (k)
(5.15)
Now, due to the block diagonal structure of the matrix Φk in (5.12), the Lyapunov
matrix P can be chosen with a block diagonal structure
"
#
R 0
P =
0 S
(5.16)
Replacing for P , (5.15) will reduce to

F T RFk − R
0
0
 k

0
(Fk − Γk Hk )T S (Fk − Γk Hk ) − S + LTk Lk (Fk − Γk Hk )T SΓk

0
ΓTk S (Fk − Γk Hk )
ΓTk SΓk − γ 2 I


<0

(5.17)
in which R is independent of Γk and S. To formulate the 2 × 2 block in (5.17) as an
LMI in S and Γk , note that
#
"
(Fk − Γk Hk )T S (Fk − Γk Hk ) − S + LTk Lk (Fk − Γk Hk )T SΓk
ΓTk S (Fk − Γk Hk )
z"
D
}|
T
Lk Lk − S
0
0
−γ 2 I
#{
z"
+
ΓTk SΓk − γ 2 I
C
=
}|
B
#{ A−1 z
}|
T
i{
h
(Fk − Γk Hk ) S z}|{
−1
S
(5.18)
S (Fk − Γk Hk ) SΓk
ΓTk S
5.2. LMI FORMULATION
109
and hence using Schur complement the inequality in (5.18) can be written as the
following set of linear matrix inequalities
 

−S
S (Fk − Γk Hk ) SΓk




  (F − Γ H )T S
LTk Lk − S
0
k k
 k

0
−γ 2 I
ΓTk S






<0

(5.19)
S>0
Note that the LMI corresponding to the (1, 1) block in Eq. (5.17), i.e.
FkT RFk − R ≤ 0
(5.20)
can never be strict (Fk has eigenvalues on unit circle). Most SDP-solvers, however,
only solve strictly feasible LMIs. Therefore, Fk should be first put in its modal form
"
#
I
0
Fk = Θ−1
Θk
(5.21)
k
0 ΛAs,k
for some diagonalizing transformation matrix Θk to isolate the poles on the unit circle. Straightforward matrix manipulations then lead to the following LMI in S, Rs ,
4
T = SΓk , and γ 2 which can be strictly feasible:
Minimize γ 2 > 0 subject to
 

−S
S (Fk − Γk Hk )





T


LTk Lk − S

  (Fk − Γk Hk ) S
ΓTk S








"
where R = ΘTk
0

SΓk
0
−γ 2 I

<0

(5.22)
ΛTAs,k Rs ΛAs,k − Rs < 0
S > 0, Rs > 0
I
0
#
Θk . The solution to (5.22) provides estimator gain, Γk , as
0 Rs
well as the Lyapunov matrix P which ensures that the quadratic cost V (ηk ) decreases
over time. It is shown in Chapter 2 that for the Riccati-based solution to Eq. (5.7)
the optimal value of γ is 1. In the absence of H2 constraints, γ in Eq. (5.22) can
be set to 1. This reduces the LMI solution to a feasibility problem in which S > 0,
Rs > 0, and T should be found.
5.2. LMI FORMULATION
5.2.1
110
Including H2 Constraints
Augmenting the above mentioned H∞ objective with appropriate H2 performance
constraints (such as the H2 norm of the transfer function from exogenous disturbance
Vm (k) to the cancellation error Zk in the augmented system described by (5.12)(5.13)) is also straight forward. Recall that [8]
k TZVm k22 = Tr Ωk Wc ΩTk
(5.23)
where Wc satisfies
ΦTk Wc Φk − W c + Ψk ΨTk = 0
(5.24)
To translate this into LMI constraints [10], note that bounding the H2 norm by ν 2 is
equivalent to
 "
#
T

Φ
W
Φ
−
W
Ψ

c
k
c
k
k

<0



ΨTk
−I


#
"
Q
Ωk Wc

>0



Wc ΩTk
Wc




Tr (Q) − ν 2 < 0
(5.25)
Allowing some conservatism, we pick Wc = P to augment (5.25) with the LMI constraints in (5.22). After some algebraic manipulation, we first express (5.25) as

 
T

F
RF
−
R
0
0
k

  k


T


<0

0
(F
−
Γ
H
)
S
(F
−
Γ
H
)
−
S
−Γ
k
k
k
k
k
k
k






T

0
−Γ
−I

k



(5.26)
Q
0 Lk S





 0

R 0 

>0





SLTk 0
S




Tr (Q) − ν 2 < 0
To formulate the 2 × 2 block in the first inequality in (5.26) as an LMI, note that
"
#
(Fk − Γk Hk )T S (Fk − Γk Hk ) − S −Γk
=
−ΓTk
−I
"
# "
#
h
i(5.27)
FkT SFk − S − HkT ΓTk SFk − FkT SΓk Hk −Γk
HkT ΓTk S
+
S −1 SΓk Hk 0
−ΓTk
−I
0
5.3. ADAPTATION ALGORITHM
111
and that the LMI constraint on R can be decoupled from the rest. Therefore the
H2 /H∞ problem can be formulated as the following LMI in S, T , Q, and Γk
Minimize























αγ 2 + βTr(Q) (α and β are known constants) subject to (5.22) and


−S
SΓk Hk
0


 H T ΓT S F T SFk − S − H T ΓT SFk − F T SΓk Hk −Γk  < 0
k
k k
k
 k k

T
0
−Γk
−I
"
#
(5.28)
Q Lk S
>0
SLTk
S
Tr (Q) − ν 2 < 0
Note that T = SΓk and Γk are not independent variables, and an appropriate linear
matrix inequality should be used to reflect this interdependence. For an alternative
derivation of the LMI formulation for the mixed H2 /H∞ synthesis problem see [10]
and references therein.
5.3
Adaptation Algorithm
The implementation scheme is similar to that of the adaptive FIR filter discussed in
Chapter 2. For easier reference, the main signals involved in the description of the
adaptive algorithm are briefly introduced here. For a more detailed description please
see Chapter 2. In what follows (a) Ŵ (k) is the estimate of the adaptive weight vector,
4
(b) θ̂(k) is the estimate of the state of the secondary path, (c) u(k) = h∗ (k)Ŵ (k)
is the actual control input to the secondary path, (d) y(k) and d(k) are the actual
outputs of the secondary and primary paths, respectively, (e) e(k) is the actual error
measurement, and (f) θcopy (k) and ycopy (k) are the adaptive algorithm’s internal copy
of the state and output of the secondary path which are used in constructing m(k)
according to Eq. (5.4). Now, the implementation algorithm can be outlined as follows:
1. Start with Ŵ (0) = Ŵ0 , θ̂(0) = θ̂0 as estimator’s best initial guess for the
state vector in the approximate model of the primary path. Also assume that
5.3. ADAPTATION ALGORITHM
112
θcopy (0) = θcopy,0 is adaptive algorithm’s initial condition for the internal copy
of the state of the secondary path.
2. For 0 ≤ k ≤ M(f inite horizon):
(a) Form h(k) according to Eq. (5.2),
(b) Form the control signal u(k) = h∗ (k)Ŵ (k), to be applied to the secondary
path. Note that applying u(k) to the secondary path produces
y(k) = Cs (k)θ(k) + Ds (k)u(k)
(5.29)
at the output of the secondary path. This in turn leads to the following
error signal measured at time k:
e(k) = d(k) − y(k) + Vm (k)
(5.30)
which is available to the adaptive algorithm to perform the state update
at time k.
(c) Propagate the internal copy of the state vector and the output of the
secondary path as
θcopy (k + 1) = As (k)θcopy (k) + Bs (k)u(k)
(5.31)
ycopy (k) = Cs (k)θcopy (k) + Ds (k)u(k)
(5.32)
(d) Form the derived measurement, m(k), using the direct measurement e(k)
and the controller’s copy of the output of the secondary path
m(k) = e(k) + ycopy (k)
(5.33)
Note that e(k) is the error measurement after the control signal u(k) is
applied.
(e) Use the LMI formulation in (5.22) (the LMI in (5.28) in case H2 constraints
exist) to find Γk (estimator’s gain).
(f) Update the state estimate according to Eq. (5.8), and extract Ŵ (k + 1)
from ξˆk+1 as the new value for the adaptive weight vector in the adaptive
filter.
(g) If k ≤ M, go to (a).
5.4. SIMULATION RESULTS
5.4
113
Simulation Results
In this section the feasibility of the design procedure is examined in the context of
Active Noise Cancellation in the one dimensional acoustic duct shown in Figure 2.5.
The identified transfer function for the acoustic duct is shown in Figures 2.6 and 2.7.
The single channel noise cancellation scenario is considered here (i.e. Speaker #2 is
used to cancel, at the position of Microphone #2, the effect of the disturbance that
enters the acoustic duct via Speaker #1). The length of the adaptive FIR filter in
this example is 8. The results presented in this section are intended to demonstrate
the feasibility of the LMI formulation in adaptive FIR filter design.
Figure 5.2 compares the output of the secondary path, y(k), to the output of the
primary path, d(k), when the primary disturbance input is a sinusoid at 30 Hz. The
adaptive algorithm has full access to the primary disturbance in this case. The error
in noise cancellation, i.e. the measurement of Microphone #2, is also plotted in this
figure. Note that, error measurements are subject to band limited white Gaussian
noise with power 0.008. Effectively, after 0.2 seconds the output of the adaptive filter
reaches its steady-state value. An approximately 10 times reduction in the amplitude
of the disturbances at Microphone #2 is recorded. A typical behavior of the elements
of the adaptive weight vector is shown in Figure 5.3. The variations in the elements
of the weight vector during steady-state operation of the algorithm are small, and
effectively after 0.2 seconds they assume their steady-state values.
Figure 5.4 shows the simulation results for a multi-tone primary disturbance that
consists of 30 and 45 Hz frequencies. In this case the output of the adaptive filter
effectively reaches its steady-state in about 0.35 seconds. The reduction in the magnitude of the measured disturbances at the position of Microphone #2 is approximately
5 times in this case. The simulation conditions are similar to the single-tone case. A
typical behavior of the elements of the adaptive FIR weight vector is shown in Figure 5.5. The elements of the weight vector display more variation during steady-state
operation of the adaptive FIR filter.
Even though the objective of the noise cancellation problem in this chapter is
the same as that in Chapter 2, the LMI-based adaptive algorithm is computationally
5.4. SIMULATION RESULTS
114
more expensive. Furthermore, simulation results presented in this section indicate
that, for the formulation of the problem presented in this chapter, the performance
of the Riccati-based EBAF is better than that of the LMI-based EBAF. This can
be associated with the conservatism introduced in the formulation, in particular the
diagonal structure assumed for the matrix P (Eq. (5.16)). The assumption of a
constant P matrix also results in additional conservatism. Nevertheless, the problem
formulation as LMIs provides a potent framework in which multiple design objectives
can be handled. Furthermore, the uncertainty in system model can be systematically
addressed in the LMI framework.
5.4. SIMULATION RESULTS
x(k)
115
Vm (k)
d(k)
Primary Path
(Unknown)
+
+
+
−
Adaptive
Filter
u(k)
Secondary Path
(Known)
y(k)
Adaptation
Algorithm
Fig. 5.1: General block diagram for an Active Noise Cancellation (ANC) problem
e(k)
5.4. SIMULATION RESULTS
116
LMI Single-Tone Noise Cancellation
d(k) vs. y(k) (Volts)
1
d(k)
y(k)
0.5
0
−0.5
−1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1
Microphone #2 (Volts)
e(k)
0.5
0
−0.5
−1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (sec.)
Fig. 5.2: Cancellation Error at Microphone #1 for a Single-Tone Primary Disturbance
0.5
5.4. SIMULATION RESULTS
117
0.2
0.2
Wk (1)
Filter Weight
0.15
0.1
0.1
0.05
0.05
0
0
−0.05
0
0.1
0.2
0.3
0.4
0.5
0.2
Filter Weight
−0.05
0
0.1
0.2
0.3
0.4
0.5
0.2
Wk (5)
0.15
0.1
0.05
0.05
0
0
0
0.1
0.2
0.3
Time(sec.)
0.4
Wk (8)
0.15
0.1
−0.05
Wk (3)
0.15
0.5
−0.05
0
0.1
0.2
0.3
Time(sec.)
0.4
0.5
Fig. 5.3: Typical Elements of Adaptive Filter Weight Vector for Noise Cancellation Problem in
Fig. 5.2
5.4. SIMULATION RESULTS
118
LMI Multi-Tone Noise Cancellation
d(k) vs. y(k) (Volts)
2
d(k)
y(k)
1
0
−1
−2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Microphone #2 (Volts)
2
e(k)
1
0
−1
−2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (sec.)
Fig. 5.4: Cancellation Error at Microphone #1 for a Multi-Tone Primary Disturbance
0.5
5.4. SIMULATION RESULTS
119
0.2
0.2
Wk (1)
Filter Weight
0.15
0.1
0.1
0.05
0.05
0
0
−0.05
0
0.1
0.2
0.3
0.4
0.5
0.2
Filter Weight
−0.05
0
0.1
0.2
0.3
0.4
0.5
0.2
Wk (6)
0.15
0.1
0.05
0.05
0
0
0
0.1
0.2
Time (sec.)
0.3
0.4
Wk (8)
0.15
0.1
−0.05
Wk (3)
0.15
0.5
−0.05
0
0.1
0.2
0.3
0.4
0.5
Time (sec.)
Fig. 5.5: Typical Elements of Adaptive Filter Weight Vector for Noise Cancellation Problem in
Fig. 5.4
5.5. SUMMARY
5.5
120
Summary
This chapter suggests that LMI-based synthesis tools can be used to design adaptive
filters. The feasibility of this approach is demonstrated in a typical adaptive ANC
scenario. One clear benefit is that the framework is suitable for designing adaptive
filters in which performance and robustness concerns are systematically addressed.
Chapter 6
Conclusion
6.1
Summary of the Results and Conclusions
In this dissertation, a new estimation-based procedure for the systematic synthesis
and analysis of adaptive filters (controllers) in “Filtered” LMS problems has been
presented. This is a well known nonlinear control problem for which “good” systematic synthesis and analysis techniques are not yet available. This dissertation
has proposed a two step solution to the problem. First, it developed an estimation
interpretation of the adaptive filtering (control) problem. Based on this interpretation, the original adaptive filtering (control) problem is replaced with an equivalent
estimation problem. The weight vector of the adaptive filter (controller) is treated
as the state variable in this equivalent estimation problem. In the original adaptive
control problem, a measured error signal (i.e. the difference between a desired signal,
d(k) in Fig. 2.1, and a controlled signal, y(k) in Fig. 2.1) reflects the success of the
adaptation scheme. The equivalent estimation problem has been constructed such
that this error signal remains a valid measure for successful estimation. The second
step, is then to solve the corresponding estimation problem. An observer structure
for the estimator has been chosen, so that “estimates” of the optimal weight vector
can be formed. The weight vector in the adaptive filter is then tuned according to
these state estimate.
121
6.1. SUMMARY OF THE RESULTS AND CONCLUSIONS
122
Both H2 and H∞ measures can be used as estimation criteria. The H∞ criterion
was chosen for the development of the adaptive algorithm in this Thesis. More specifically, the equivalent estimation problem seeks an estimate of the adaptive weight
vector such that the energy gain from the exogenous disturbances and the initial
condition uncertainty to the cancellation error (i.e. the error between d(k) and y(k)
in Fig. 2.1) is minimized. This objective function is justified by the nature of the
disturbances in the applications of interest (i.e. active noise cancellation and active
vibration isolation). The following is a summary of the results in this Thesis:
1. In the case of adaptive FIR filters:
(a) The equivalent estimation problem is shown to be linear. Given a bound on
energy gain γ > 0, the robust estimation literature provides exact filtering
and prediction solutions for this problem. The work in this Thesis proves
that γ = 1 is the optimal energy gain, and derives the conditions under
which this bound is achievable.
(b) The optimality arguments in this Thesis provide the conditions under
which the existence of an optimal filtering (prediction) solution for the
problem is guaranteed. This eliminates the possibility of the solution
breakdown which could have prevented real-time implementation of the
algorithm.
(c) It is shown that the filtering and prediction solutions only require one
Riccati recursion. The recursion propagates forward in time, and does not
rely on any information about the future of the system or the reference
signal (thus allowing the resulting adaptive algorithm to be implementable
in real-time). This has come at the expense of restricting the controller to
an FIR structure in advance.
(d) For the optimal value of γ = 1, the above mentioned Riccati recursion simplifies to a Lyapunov recursion. This leads to a computational complexity
that is comparable to that of a classical filtered LMS adaptive algorithm,
such as the FxLMS.
6.1. SUMMARY OF THE RESULTS AND CONCLUSIONS
123
(e) Experimental results, along with extensive simulations have been used to
demonstrate the improved transient and steady-state performance of the
EBAF algorithm over classical adaptive filtering algorithms such as the
FxLMS and the Normalized FxLMS.
(f) A clear connection between the limiting behavior of the EBAF algorithm
and the FxLMS (Normalized-FxLMS) adaptive algorithm has been established. In particular, it is shown that the gain vector in the predictionbased (filtering-based) EBAF algorithm converges to the gain vector in the
FxLMS (Normalized FxLMS) as k → ∞. The error terms however, are
shown to be different. Thus, the classical FxLMS (Normalized FxLMS)
adaptive algorithms can be thought of as an approximation to the EBAF
algorithm in this Thesis. This connection might explain the observed
improvement in both transient and steady-state performance of the new
EBAF algorithm.
2. For the EBAF algorithm in the IIR case, it has been shown that the equivalent
estimation problem is nonlinear. An exact solution for the nonlinear robust
estimation problem is not yet available. A linearizing approximation that makes
systematic synthesis of adaptive IIR filter tractable has been adopted in this
Thesis. The performance of the EBAF algorithm in this case has been compared
to the performance of the Filtered-U LMS (FuLMS) adaptive algorithm. The
proposed EBAF algorithm has been shown to provide improved steady-state
and transient performance.
3. The treatment of feedback contamination problem has been shown to be identical to the IIR adaptive filter design in the new estimation-based framework.
4. A multi-channel extension of the EBAF algorithm has been provided to demonstrate that the treatment of the single-channel and multi-channel adaptive filtering (control) problems in the new estimation based framework is virtually
identical. Simulation results for the problem of vibration isolation (in a 3input/3-output vibration isolation platform (VIP)), and noise cancellation in a
6.2. FUTURE WORK
124
one dimensional acoustic duct have been shown to prove the feasibility of the
EBAF algorithm in the multi-channel case.
5. The new estimation-based framework has been shown to be amenable to a
Linear Matrix Inequality (LMI) formulation. The LMI formulation is used to
explicitly address the stability of the overall system under adaptive algorithm by
producing a Lyapunov function. Augmentation of an H2 performance constraint
to the H∞ disturbance rejection criterion has also been discussed.
6.2
Future Work
There are several possible directions for future work. The first direction is to address
the question of stability for the EBAF-based adaptive IIR filters. During extensive
simulations, the estimation-based adaptive IIR filter was observed to be stable. Obtaining a formal proof for the stability of the system, however, is a difficult problem.
Exploring the role of a different linearizing approximation, in reducing the nonlinear
robust estimation problem encountered in the IIR filter design into a tractable linear
estimation problem, is another interesting avenue for further research. It is also interesting to examine the possibility of formulating other classes of adaptive filtering
problems from the estimation point of view.
It is possible to formulate a systematic approach to the synthesis of “optimal”
adaptive filters (e.g. adaptive filters of optimal length) by augmenting the existing
estimation criterion with an appropriate constraint or objective function. The systematic synthesis of a robust adaptive filter, in which an error in the modeling of
the secondary path is explicitly handled during the synthesis process, would be another interesting extension to the work in this Thesis. The LMI formulation of the
estimation-based approach, in particular, offers a rich machinery for addressing such
problems.
Application of this approach to adaptive equalization is the subject of ongoing
research. For adaptive equalization, adopting an H2 estimation criterion is well justified. With an H2 objective, the estimation based adaptive filtering algorithm in
6.2. FUTURE WORK
125
this Thesis is in fact an extension of the RLS algorithm to the more general class of
filtered LMS problems.
Appendix A
Algebraic Proof of Feasibility
A.1
Feasibility of γf = 1
As Chapter 2 points out, it is possible to directly establish the feasibility of γf = 1.
To do so, it should be shown that Rk and Re,k (in Theorem 2.1) have the same inertia,
for all k, if γf = 1. First, define
"
4
Z =
A B
C D
#


= 

−
Pk−1
Hk
Hk
!
Hk∗
I
Hk∗
0

! 


(A.1)
0 −γ 2 I
Then, apply UDL decomposition to A.1 to get
#"
#"
"
#
#
"
A B
I −A−1 B
I
0
A
0
4
Z1 =
=
C D
−CA−1 I
0
I
0 ∆A
(A.2)
(where ∆A = D − CA−1 B is in fact Re,k ). Note that Z and Z1 have the same inertia.
Now, perform LDU decomposition to get
#"
#"
#
"
#
"
A B
I
0
∆D 0
I −BD −1
4
Z2 =
=
0
I
C D
−D −1 C I
0
D
(A.3)
(where ∆D = A − BD −1 C will reduce to A for γ = 1). Clearly, Z and Z2 have the
same inertia as well. Therefore, Z1 and Z2 should have the same inertia. Since the
126
A.1. FEASIBILITY OF γF = 1
127
(1, 1) block matrices for Z1 and Z2 are the same (i.e. A = −Pk−1 ), the (2, 2) block
matrices, i.e. Re,k in Z1 and Ri in Z2 , should have the same inertia, which is the
condition 2.13 in Theorem 2.1.
Appendix B
Feedback Contamination Problem
Figure B.1 shows the block diagram for an approximate model of the primary path
when a feedback path from the output of the IIR filter to its input exists. The
presentation here follows the discussions in Sections 3.2 and 3.3, and therefore it is
kept brief. The state space description for the block diagram in Figure B.1 is as
follows:


W (k + 1)



I(2N +1)×(2N +1)



 θ(k + 1)  =  Bs (k)h∗
k



ϕ(k + 1)
Bf (k)h∗k
0
As (k)
0
0
W (k)


 θ(k) 
0 


ϕ(k)
Af (k)
ξ(k+1) = Fk ξ(k)
(B.1)
where θk is the state variable for the secondary path, ϕk is the state variable for the
feedback path, and hk and W (k) are defined in Section 3.2.1. Note that
x(k) = ref (k) + Df (k)h∗ (k)W (k) + Cf (k)ϕ(k)
(B.2)
r(k) = a0 x(k) + b1 r(k − 1) + · · · + bN r(k − N), r(−1) = · · · = r(−N) = 0
(B.3)
where the contamination of the reference signal with the feedback from the output of
the adaptive filter is evident. For this system the derived measured output, m(k), is
128
129
described as


W (k)
h
Ds (k)h∗k
m(k) =
i

 + Vm (k)
Cs (k) 0 
θ(k)


ϕ(k)
= Hk ξ(k)
(B.4)
while the quantity to be estimated, s(k), is


h
s(k) =
W (k)
Ds (k)h∗k Cs (k) 0
i

 θ(k) 


ϕ(k)
= Lk ξ(k)
(B.5)
Similar to the derivations in Chapters 2 and 3, Hk = Lk , i.e. s(k) = d(k). It is desired
to find an H∞ causal filter ŝ(k|k) = F (m(0), m(1), · · · , m(k)) such that
M
X
sup
Vm , ξ0
(s(k) − ŝ(k|k))∗ (s(k) − ŝ(k|k))
k=0
ξ0∗ Π−1
0 ξ0 +
M
X
≤ γf2
(B.6)
Vm (k)∗ Vm (k)
k=0
Equivalently, an strictly causal predictor ŝ(k) = F (m(0), m(1), · · · , m(k − 1)) can be
found such that
M
X
sup
Vm , ξ0
(s(k) − ŝ(k))∗ (s(k) − ŝ(k))
k=0
ξ0∗ Π−1
0 ξ0 +
M
X
≤ γp2
(B.7)
Vm (k)∗ Vm (k)
k=0
Here γf (γp ) are positive numbers. Note that Vm (k) is assumed to be an L2 signal.
Obviously, Equations (B.1), (B.4) and (B.5) are nonlinear in IIR filter parameters
and therefore the estimation problem in Eq. (B.6) (or Eq. (B.7)) is a nonlinear H∞
problem. As in Sections 3.2 and 3.3, the IIR filter parameters in (B.2) and (B.3)
are replaced with their best available estimate to obtain a linear time-variant system
130
dynamics. For this linearized system the solution in Section 3.3 will exactly apply,
and hence it is not repeated here. It is clear that the above mentioned discussion
holds true when the adaptive IIR filter is replaced with an adaptive FIR filter.
131
Approximate Model for Primary Path With Feedback Contamination
Df (k)
ref (k)
+
Cf (k)
Z −1
+
Bf (k)
Af (k)
Vm (k)
Ds (k)
a0
+
r(k)
+
x(k)
Bs (k)
+
Z −1
u(k)
Z −1
Cs (k)
+
+
d(k) m(k)
As (k)
a1
b1
Z −1
b2
r(k-2)
a2
Second Order IIR Filter
Fig. B.1: Block diagram of the approximate model for the primary path in the presence of the
feedback path
Appendix C
System Identification for Vibration
Isolation Platform
C.1
Introduction
Advanced control techniques are model based techniques. The achievable performance for these control techniques, therefore, depends on the accuracy of the available model. Thus, the objective of system modeling is to capture the dynamics of a
system in the form of a mathematical model as accurately as possible. In general,
two approaches to the mathematical modeling of a system are used; (a) Analytical
Approach: applies rules of physics (that govern system dynamics) to derive a physical
model, and (b) System Identification Approach: uses the experimental input/output
data to construct a mathematical model.
Analytical methods represent not only the input/output behavior of the system
but also capture the internal mechanics and physics of the system. Identification based
methods, on the other hand, are mainly concerned with input/output behavior of the
system. Reference [22] provides a detailed discussion on the role of each approach in
a control design problem. This appendix discusses the state space model derived for
the Vibration Isolation Platform (VIP) based on an advanced method of fitting the
measured transfer function measurements. The main components of the VIP, along
132
C.2. IDENTIFIED MODEL
133
with their operational role are described in Chapter 4.
C.2
Identified Model
This section discusses the system identification strategy used to extract a state space
model for the Vibration Isolation Platform (VIP) based on transfer function measurements obtained over 0.50 − 660.0 Hz frequency range. The process of data collection
is described first. The consistency of the collected data for the load cells and truth
(scoring) sensors with the physical behavior of the system is examined next. Load
cell measurements reflect the interacting forces between the middle mass and the payload, while truth (scoring) position sensors measure the displacement of the payload
from its equilibrium in inertial frame. It is therefore important to develop a model
in which the readings from these two sets of sensors agree with the true physics of
the system. For this modeling to be successful, the collected data should correctly
captured the true physics of the problem, and the discussion on the consistency of the
measurements demonstrates this important fact. Finally, the identified state space
model is presented and compared with the original measurements.
C.2.1
Data Collection Process
A program that controls both actuator excitation and sensor measurements was developed to measure the transfer functions. This program runs on a sun SPARC1e
real time processor. The measurements are conducted as follows:
1. At a given frequency, a sinusoidal excitation is applied to only one actuator at
a time. Other actuators are commanded to zero. A total of 209 data points
(logarithmically spaced) are used to cover the 0.5 − 660 Hz frequency range.
The amplitude of the sinusoidal excitation is set to 0.2 volts to ensure linear
behavior for the system throughout the measurements.
2. The program maintains this excitation for 30 seconds (to assure that the system
reaches its steady state behavior) before it starts recording sensor readings. The
C.2. IDENTIFIED MODEL
134
sampling period for analog to digital conversion (performed by Tustin 2100 data
conversion device) is 63 micro-seconds. The same period is used for the digitalto-analog converter that drives the actuators.
3. The time data for the measurements is temporarily stored. With the known
excitation frequency, a least squares algorithm is applied to the measurements
over 20 cycles of the recorded data to extract the amplitude and phase of the
measurements for each sensor.
With this data for each test frequency point, the transfer functions from each actuator to all existing sensors can be constructed. This data is then used by system
identification routine to extract a state space model for the system.
C.2.2
Consistency of the Measurements
To examine the consistency of the available measurements, the physical relationship
between the load cell and truth sensor measurements should be explored. Note that,
from measurements, transfer function matrices Hlc,u , Hlc,d , Hsc,u , and Hsc,d defined
by


LC1

 

=
LCu = 
LC
2
 

LC3
 

LC1
 



LCd =  LC2 
=
LC3
 

SC1
 

=
SCu = 
SC
2
 

SC3
 

SC1
 

 
SCd = 
SC
2  = 

SC3

H1,1 H1,2 H1,3

U1


 U2  = Hlc,u U
H2,1 H2,2 H2,3 


H3,1 H3,2 H3,3
U3


H1,4 H1,5 H1,6
D1




H2,4 H2,5 H2,6   D2 
 = Hlc,dD
H3,4 H3,5 H3,6
D3


H10,1 H10,2 H10,3
U1


 U2  = Hsc,uU
H11,1 H11,2 H11,3 


H12,1 H12,2 H12,3
U3


H10,4 H10,5 H10,6
D1


 D2  = Hsc,dD
H11,4 H11,5 H11,6 


H12,4 H12,5 H12,6
D3
(C.1)
(C.2)
(C.3)
(C.4)
C.2. IDENTIFIED MODEL
135
are available. Meanwhile, straightforward dynamics suggest that
LC
= SCALElc · Mx,y,z · X, and SC = SCALEsc · X
(j2πf )2
(C.5)
where LC is the vector of load cell force measurements, M is the mass/inertia matrix, X are the position measurements, and SC is truth (scoring) sensor’s position
measurement. The factor SCALEsc = diag(Ssc1 , Ssc2, Ssc3 ) accounts for the scaling
differences in the position measurement as it is seen by the truth sensor. With the
orthogonal transformation

4 
T =

0
√1
1.5
√1
3
−1
√
2
−0.5
√
1.5
√1
3
√1
2
−0.5
√
1.5
√1
3




the effects of the actuators and the measurements of the sensors can be decomposed
into two tilt motions (about the two perpendicular axes in x-y plane), and a piston
motion in z direction. Now, the load cell reading can be related to that of the scoring
sensor as follows:
z
SCALE−1
lc
−1
Slc1

T ·
 0
0
}|
0
−1
Slc2
0
z
{

0

1
0 
 · (j2πf )2
−1
Slc3
SCALE−1
sc
−1
Ssc1

T ·
 0
0
}|
0
−1
Ssc2
0

Ix
LC1



 =  0
·
LC2



0
LC3
{ 
0
Mθx ,θy ,z
z

SC1

}|
{ 
0 0
θx


Iy 0 
  θy
z
0 M


 (C.6)


θx




 SC2  =  θy 
0 




−1
SC3
Ssc3
z
(C.7)
and therefore
1
−1
· Hlc,u · Hsc,u
= SCALElc · T ∗ · Mθx ,θy ,z · T · SCALE−1
sc
(j2πf )2
(C.8)
Note that a similar relationship can be derived for measurements from the disturbance
actuators. The left hand side in Eq. (C.8) is the available measurement data, and
C.2. IDENTIFIED MODEL
136
hence an optimization problem can be set up to find the optimal scaling factors as
well as the inertia and mass parameters. Instead of solving this problem at each
frequency, the left hand side of Eq. (C.8) over the frequency range [0.5 − 30] Hz is
averaged to obtain


0.2174 0.0668 0.0635


Hlc,u
−1
−3 
= 10 ·  0.0697 0.1931 0.0586 
Hsc,u

2
(j2πf )
averaged
0.0705 0.0650 0.1751
(C.9)
Now a simple optimization algorithm yields the following optimal scaling factors and
inertia/mass parameters for the system
SCALElc = diag(1.1043, 1.0352, 0.9560) × 22.24(N/Volts)
SCALEsc = diag(1.0887, 1.0122, 0.9935) × 0.25(mm/Volts)
Ix = 0.6579 Kg.m2
Iy = 0.6184 Kg.m2
M = 1.7801 Kg
Applying the orthogonal transformation T and the optimal scaling factors for the
load cells and the scoring sensors,
Mθmeasured
x ,θy ,z
=T ·
SCALE−1
lc
Hlc,u
−1
· Hsc,u SCALEsc · T ∗
2
(j2πf )
the matrix in Equation (C.9) is transformed into


0.1223
0.0004
0.0065


Hlc,u
−1
−3 

=
10
·
H
0.0019
0.1203
−0.0060
sc,u


(j2πf )2
diagonalized
−0.0049 0.0024
0.3070
(C.10)
in which the diagonal dominance of the matrix is evident. This, in essence, indicates
that the relationship between load cell and scoring (truth) sensor measurements conforms with what the physics of the VIP system predicts.
Using the optimal scaling for the sensors and the orthogonal transformation T ,
−1
(Hlc,u /(2πf )2)·Hsc,u
can be diagonalized for each frequency. Figures C.1 and C.2 show
C.2. IDENTIFIED MODEL
137
−1
−1
and (Hlc,d/(2πf )2) · Hsc,d
,
the elements of the 3 × 3 matrices (Hlc,u /(2πf )2 ) · Hsc,u
respectively, for the raw data. All elements of these matrices are roughly of the same
order of magnitude. Figures C.3 and C.4 on the other hand reflect the elements of
similar matrices for the diagonalized version. The diagonal dominance in these matrices is maintained for all frequencies. Also note that the numerical values for the
diagonal elements in Figures C.3 and C.4 agree with those obtained by the optimization process, indicating a truthful reflection of system dynamics for frequencies up to
50 Hz in the collected data.
C.2.3
System Identification
This section discusses the identified model for a 6-input/12-output continuous time
state space representation of the VIP system. As mentioned earlier, the three control
actuators (denoted by U) along with the three disturbance actuators (denoted by
D) form the inputs to the VIP system. Three load cells (colocated with the control
actuators and denoted by LC), six displacement sensors (one colocated within each
actuator stator and denoted by CP for the ones colocated with control actuators
and DP for the ones colocated with disturbance actuators), and three scoring sensors
(measuring the inertial motion of the payload and denoted by SC) constitute the 12
outputs for this system.
For the results presented here, the Frequency domain Observability Range Space
Extraction (FORSE) algorithm (see Section C.3, and Ref. [30], Section 3 for a detailed
description of the identification algorithm) is used. FORSE is a model synthesis technique that operates directly on the transfer function data without an inverse Fourier
transform, and has numerical robustness properties similar to time domain synthesis
techniques (such as ERA and q-Markov methods). The discussion in Ref. [30] proves
that a specifically constructed matrix from the frequency domain transfer function
measurements has the same basis as the observability matrix for the system, and thus
the A and C matrices can be found directly. The rest of the realization (i.e. B and
D matrices) can then be found using the least squares method.
The available frequency domain data (0.50 − 660.0 Hz) is not enough to allow
C.2. IDENTIFIED MODEL
138
the identification algorithm to capture the true behavior of the system at low frequencies. Therefore, the transfer function measurements from the 6 actuators to the
load cells are scaled by K/(j2πf )2 (for an appropriate K) effectively translating force
measurements into position measurements. This data is then fed to the FORSE algorithm to extract a state space model (Ã, B̃, C̃LC , D̃LC ) for the transfer functions
[U, D] → K · LC/(j2πf )2 . Note that, this is indeed a MIMO (6-input/3-output)
identification process in which the maximum singular value of the modeling error is
minimized. Keeping à and B̃ fixed, the transfer function data for [U, D] → [CP ],
[U, D] → [DP ], and [U, D] → [SC] are then used to derive the state space model
for each of the above mentioned subsystems. Once again the available data does not
provide enough information to correctly capture the high frequency behavior of the
system. Therefore, in the course of modeling the feed-through terms for the transfer
functions (i.e. D̃LC , D̃CP , D̃DP , and D̃SC ) are forced to be zero.
To cancel the effect of the double integration and scaling of the load cell data, the
output of the transfer function [U, D] → K ·LC/(j2πf )2 should be scaled by 1/K and
differentiated twice. To approximate the required scaling and double differentiation,
a second order filter of the form
F (s) =
ω02 (j2πf )2
K(j2πf + ω0 )2
(with ω0 = 2πf0 and for some appropriate f0 ), can be used to filter the load cell
output (i.e. ỹ = C̃LC · x + D̃LC · u). This approach however introduces 6 extraneous
states, the elimination of which can effect the quality of the identified model. Instead,
the state space model [Ã, B̃, C̃LC , D̃LC ] is used to perform the scaling and double
differentiation as follows:
o
1 ¨
1 n
· ỹ =
· C̃ Ã2 · x + C̃ ÃB̃ · u + C̃ B̃ · u̇
K
K
where if (1/K) · C̃ B̃ ∼
= 0, then the state space model
1
1
2
· C̃ Ã , DLC =
· C̃ ÃB̃
A = Ã, B = B̃, CLC =
K
K
(C.11)
is an adequate representation for the transfer function [U, D] → [LC]. Note that no
extraneous states are introduced here, hence the state space model for the Vibration
C.2. IDENTIFIED MODEL
139
Isolation Platform is as follows:

A


CLC


 CCP = C̃CP

 C = C̃
DP
 DP
CSC = C̃SC
B
DLC
DCP = D̃CP
DDP = D̃DP









DSC = D̃SC
A good fit to the data was found by using 36 states for the system. Increasing the
number of the states did not appear to improve the fit. Figures C.5-C.6 compare
the singular value plots for the measured MIMO transfer functions and the transfer
functions of the model generated using FORSE before and after double differentiation.
As Figure C.5 clearly indicate, the original model provides a good match for the scaled
data over the entire 0.5 − 660 Hz frequency range. Figure C.6 then shows that the
differentiation process does not introduce significant error in the identified model, as
the singular value plots for the un-scaled load cell data closely match those of the
final load cell model. Note that all significant modes are properly captured in both
cases. Figures C.7-C.9 compare the singular value plots of the identified model and
the measured data for other input/output channels. As these figures indicate, the
identified model closely matches the measured data in all cases. Note that double
differentiation of the load cell output does not affect the identified model from the
actuators to the other sensors (i.e. CP, DP, and SC sensors).
It is important to point out that despite the presence of a pole around 15 Hz that
dominates the response, the identification algorithm has been reasonably successful
in capturing the MIMO zeros in this frequency range (Figures C.8-C.9). Figure C.10
shows the identified model over a wider frequency range. The low frequency behavior
of the load cells, as well as the high frequency behavior of the position sensors is
correctly reflected in this model. As mentioned earlier in this model DLC , DCP , DDP ,
and DSC are all set to zero. Figure C.11 shows the final identified model (i.e. after
double differentiation and the scaling) for the VIP system. Both low frequency and
high frequency behavior for the load cells are correctly reflected. Also the model
remains truthful at high frequencies for the position sensors. For future reference,
natural frequencies and damping factors for the poles of the identified model are
C.2. IDENTIFIED MODEL
140
listed here:
Pole Natural
Damping Ratio
Frequency (Hz)
C.2.4
0.9733
0.0195
1.0156
0.0291
1.8925
0.8298
2.2409
0.1549
4.0462
0.0074
4.3862
0.0201
4.4216
0.0109
15.2601
0.0159
16.8545
1.0000
40.0545
0.0100
44.3453
0.0121
48.1396
0.0062
50.3646
0.0090
224.5517
0.0021
323.1164
0.0029
393.0650
0.0032
439.2836
0.0054
468.8545
0.0044
Control design model analysis
As a final test for the quality of the identified model, a simple H2 output feedback
controller was designed based on the identified mathematical model. Note that no
attempt has been made to obtain the best possible H2 controller for the VIP system.
The only objective here is to examine the consistency of the identified model and the
measured data from a control design point of view. For the results presented here:
ẋ = A x + B1 u + B2 d
y = CLC x + DLC,1 u + DLC,2 d
C.3. FORSE ALGORITHM
141
"
z =
zx
zu
#
"
=
CLC DLC,1
0
w∗I
#"
x
#
u
where u is the 3 × 1 vector of control actuator command, d is the 3 × 1 vector of
disturbances. B1 = B(:, 4 : 6), B2 = B(:, 1 : 3), DLC,1 = DLC (:, 4 : 6), DLC,2 = DLC (:
, 1 : 3), and w is some weight (chosen to be 0.05 here) on the control command. Figure
C.12 compares the closed and open loop singular value plots for the MIMO transfer
function from three disturbance actuators to the load cell and scoring sensors. The
important feature of this plot is not the performance of the designed controller, but
the consistency of the closed loop behavior of the system as it is seen by the two
different sets of sensors (i.e. load cell and scoring sensors). This same controller is
then used to close the loop on the measured frequency response data. The comparison
of open and closed loop SVD plots in this case is shown in Figure C.13. Once again,
the performance of the controller on the closed loop transfer function is consistently
reflected in both load cell and scoring sensor measurements. Note the close similarities
between the closed-loop behavior of the system under this H2 controller when either
the model or the real data is used. This close correlation of the closed-loop behavior
indicates that this identified model is a good control design model.
C.3
FORSE algorithm
The identification technique used for this work was originally develop at MIT as part
of the MACE program [30,22]. The approach integrates the Frequency domain Observability Range Space Extraction (FORSE) identification algorithm, the Balanced
Realization (BR) model reduction algorithm, and the logarithmic and additive Least
Square (LS) modal parameter estimation algorithms for low order highly accurate
model identification. The algorithm is called Integrated Frequency domain Observability Range Space Extraction and Least Square parameter estimation algorithm
(IFORSELS).
The Balanced Realization (BR) model reduction algorithm transforms a state
space model to a balanced coordinate and reduces the model by deleting the states
C.3. FORSE ALGORITHM
142
associated with the smallest principle values. The LS estimation algorithms improve the fitting of reduced models to experimental data by updating state space
parameters in modal coordinates. Models derived by both the FORSE and BR algorithms are non-optimal models, but their computations are non-iterative. While
the LS algorithms compute optimal parameter estimates, its computation may deviate to non-optimal parameters if initial estimates are very inaccurate. Integrating
these algorithms in an iterative manner avoids the computational difficulties of the
LS algorithms and improves modeling accuracies of the FORSE and BR algorithms.
As a result, the IFORSELS identification algorithm is capable of generating highly
accurate, low order models.
Assume that the following frequency response samples were obtained from experiments, Ĝ(ωk ) , k = 1, 2, · · · , K. The objective of frequency domain state space model
identification is to minimize the cost function
J=
K
X
kĜ(ωk ) − (C(jωk I − A)−1 B + D)k22 ,
(C.12)
k=1
where A, B, C, D are system matrices of the following state space equation,
(
ẋ(t) = A x(t) + B u(t)
.
y(t) = C x(t) + D u(t)
(C.13)
It is apparent that this is a nonlinear problem. The FORSE algorithm computes a
suboptimal estimate of these matrices using a subspace-based approach. To make
the cost J very small, the state space model is usually over parameterized. If model
order is not constrained, this FORSE model can be used by the LS algorithms as an
initial estimate to derive a more accurate estimate of the A, B, C, D matrices.
The entire IFORSELS algorithm consists of the FORSE identification algorithm,
the BR model reduction algorithm and two LS parameter estimation algorithms. It is
important to point out that one can use other subspace identification, model reduction and parameter estimation algorithms to form such an integrated identification
algorithm. In fact, the Eigensystem Realization Algorithm was first used to form the
algorithm [1]. The FORSE algorithm is now used because it is a frequency domain
algorithm and does not suffer from time domain aliasing errors. Other advantages
C.3. FORSE ALGORITHM
143
of the FORSE algorithm include its frequency domain weighting feature and its capability of processing data which is not uniformly distributed along the frequency
axis [2]. The BR model reduction algorithm is used simply because its computational
software is readily available in MATLAB. Other model reduction algorithms, such as
the weighted BR algorithm, the Projection algorithm and the Component Cost algorithm can also be incorporated into this integrated algorithm. The two LS parameter
estimation algorithms used are the Additive LS (ALS) algorithm, which minimizes
the error cost J in Equation C.12, and the Logarithmic LS (LLS) algorithm, which
minimizes the following logarithmic error cost
Jlog =
ny nu
K X
X
X
k log(Ĝij (ωk )) − log((C(jωk I − A)−1 B + D)ij )k22 ,
(C.14)
k=1 i=1 j=1
where the subscript ij indicates the (i, j)-th element of the matrix, and nu and ny
are the numbers of inputs and outputs of the system, respectively.
In the IFORSELS algorithm, the LLS parameter estimation algorithm is used
in the early stages of model reduction and updating. The logarithmic cost of the
LLS algorithm weights data samples equally, regardless of their magnitude [2], while
the FORSE and ALS algorithms place more emphasis on data samples which have
high magnitudes. Hence, in the early stages of model reduction and updating, more
emphasis needs to be placed on fitting in the frequency ranges where the magnitude
of data samples is low (such as areas around system transmission zeros). For the
identification process presented in this report the ALS algorithm is not used.
In summary, the entire IFORSELS identification algorithm consists of the following steps. Initially, the FORSE algorithm is used to generate an over parameterized
model. This model is then forwarded to the model reduction and updating iterations
using the BR and LLS algorithms. The identified model is then judged (by human)
based on the accuracy of the fit and the order of the model. If necessary, modes
can be deleted or added to this model and the process of IFORSELS identification
algorithm can be repeated again until a satisfactory model is obtained.
C.3. FORSE ALGORITHM
Scaling Factors between Load Cell & Truth Sensor Measurements (Raw Data), 04/12/98
−3
−3
10
10
−3
10
abs([U−>(LC/W2)]/[U−>SC])
144
−4
−4
10
−5
20
30
40
50
−3
abs([U−>(LC/W2)]/[U−>SC])
10
−6
10
20
30
40
50
−3
−4
−5
−5
30
40
50
−3
10
−4
20
30
40
50
10
50
10 20 30 40
Frequency (Hz)
50
−5
10
−6
50
40
−4
−5
10 20 30 40
Frequency (Hz)
30
10
10
−6
20
−3
−4
−5
10
10
10
10
10
−6
10
10
10
50
−5
−3
10
40
10
−6
20
30
10
10
10
20
−4
10
10
10
10
−4
10
10
−3
10
−6
abs([U−>(LC/W2)]/[U−>SC])
10
−6
10
10
10
−5
10
−6
10
10
−5
10
10
−4
10
−6
10 20 30 40
Frequency (Hz)
50
10
Fig. C.1: Magnitude of the scaling factor relating load cell’s reading of the effect of control
actuators to that of the scoring sensor
C.3. FORSE ALGORITHM
Scaling Factors between Load Cell & Truth Sensor Measurements (Raw Data), 04/12/98
−3
−3
10
10
−3
10
abs([D−>(LC/W2)]/[D−>SC])
145
−4
−4
10
−5
20
30
40
50
−3
abs([D−>(LC/W2)]/[D−>SC])
10
−6
10
20
30
40
50
−3
−4
−5
−5
30
40
50
−3
10
−4
20
30
40
50
10
50
10 20 30 40
Frequency (Hz)
50
−5
10
−6
50
40
−4
−5
10 20 30 40
Frequency (Hz)
30
10
10
−6
20
−3
−4
−5
10
10
10
10
10
−6
10
10
10
50
−5
−3
10
40
10
−6
20
30
10
10
10
20
−4
10
10
10
10
−4
10
10
−3
10
−6
abs([D−>(LC/W2)]/[D−>SC])
10
−6
10
10
10
−5
10
−6
10
10
−5
10
10
−4
10
−6
10 20 30 40
Frequency (Hz)
50
10
Fig. C.2: Magnitude of the scaling factor relating load cell’s reading of the effect of disturbance
actuators to that of the scoring sensor
C.3. FORSE ALGORITHM
−3
abs([U−>(LC/W2)]/[U−>SC])
10
146
Scaling Factors between Load Cell & Truth Sensor Measurements (Diagonalized Data), 04/12/98
−3
−3
10
10
−4
−4
10
−5
20
30
40
50
−3
abs([U−>(LC/W2)]/[U−>SC])
10
−6
10
20
30
40
50
−3
−4
−5
−5
30
40
50
−3
10
−4
20
30
40
50
10
50
10 20 30 40
Frequency (Hz)
50
−5
10
−6
50
40
−4
−5
10 20 30 40
Frequency (Hz)
30
10
10
−6
20
−3
−4
−5
10
10
10
10
10
−6
10
10
10
50
−5
−3
10
40
10
−6
20
30
10
10
10
20
−4
10
10
10
10
−4
10
10
−3
10
−6
abs([U−>(LC/W2)]/[U−>SC])
10
−6
10
10
10
−5
10
−6
10
10
−5
10
10
−4
10
−6
10 20 30 40
Frequency (Hz)
50
10
Fig. C.3: Magnitude of the scaling factor relating load cell’s reading of the effect of control
actuators to that of the scoring sensor after diagonalization
C.3. FORSE ALGORITHM
−3
abs([D−>(LC/W2)]/[D−>SC])
10
147
Scaling Factors between Load Cell & Truth Sensor Measurements (Diagonalized Data), 04/12/98
−3
−3
10
10
−4
−4
10
−5
20
30
40
50
−3
abs([D−>(LC/W2)]/[D−>SC])
10
−6
10
20
30
40
50
−3
−4
−5
−5
30
40
50
−3
10
−4
20
30
40
50
10
50
10 20 30 40
Frequency (Hz)
50
−5
10
−6
50
40
−4
−5
10 20 30 40
Frequency (Hz)
30
10
10
−6
20
−3
−4
−5
10
10
10
10
10
−6
10
10
10
50
−5
−3
10
40
10
−6
20
30
10
10
10
20
−4
10
10
10
10
−4
10
10
−3
10
−6
abs([D−>(LC/W2)]/[D−>SC])
10
−6
10
10
10
−5
10
−6
10
10
−5
10
10
−4
10
−6
10 20 30 40
Frequency (Hz)
50
10
Fig. C.4: Magnitude of the scaling factor relating load cell’s reading of the effect of disturbance
actuators to that of the scoring sensor after diagonalization
C.3. FORSE ALGORITHM
148
SVD for Transfer Function [ U D ] −> [ LC/S2 ], 05/18/98
2
SVD (Volts/Volts), Model(−) vs. Measured (−−)
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
0
10
1
10
Frequency (Hz)
2
10
Fig. C.5: Comparison of SVD plots for the transfer function to the scaled/double-integrated
load cell data
SVD for Transfer Function [ U D ] −> [ LC ] after Double Differentiation, 05/18/98
2
SVD (Volts/Volts), Model(−) vs. Measured (−−)
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
0
10
1
10
Frequency (Hz)
2
10
Fig. C.6: Comparison of SVD plots for the transfer function to the actual load cell data
C.3. FORSE ALGORITHM
149
SVD for Transfer Function [ U D ] −> [ SC ], 05/18/98
2
SVD (Volts/Volts), Model(−) vs. Measured (−−)
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
0
10
1
10
Frequency (Hz)
2
10
Fig. C.7: Comparison of SVD plots for the transfer function to the scoring sensors
SVD for Transfer Function [ U D ] −> [ CP ], 05/18/98
2
SVD (Volts/Volts), Model(−) vs. Measured (−−)
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
0
10
1
10
Frequency (Hz)
2
10
Fig. C.8: Comparison of SVD plots for the transfer function to the position sensors colocated
with the control actuators
C.3. FORSE ALGORITHM
150
SVD for Transfer Function [ U D ] −> [ DP ], 05/18/98
2
SVD (Volts/Volts), Model(−) vs. Measured (−−)
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
0
10
1
10
Frequency (Hz)
2
10
Fig. C.9: Comparison of SVD plots for the transfer function to the position sensors colocated
with the disturbance actuators
C.3. FORSE ALGORITHM
151
SVD for [ U D ] −> [ LC/S2 ]
SVD for [ U D ] −> [ CP ], 05/18/98
2
2
SVD (Volts/Volts), Model Only
10
10
0
0
10
10
−2
−2
10
10
−4
−4
10
10
−6
10
−6
0
5
10
5
10
Frequency (Hz)
SVD for [ U D ] −> [ SC ]
SVD for [ U D ] −> [ DP ]
2
SVD (Volts/Volts), Model Only
0
10
Frequency (Hz)
2
10
10
0
0
10
10
−2
−2
10
10
−4
−4
10
10
−6
10
10
10
−6
0
5
10
10
Frequency (Hz)
10
0
5
10
10
Frequency (Hz)
Fig. C.10: The identified model for the system beyond the frequency range for which measurements are available
C.3. FORSE ALGORITHM
152
SVD for [ U D ] −> [ LC ]
SVD for [ U D ] −> [ CP ], 05/18/98
2
2
SVD (Volts/Volts), Model Only
10
10
0
0
10
10
−2
−2
10
10
−4
−4
10
10
−6
10
−6
0
5
10
10
5
10
Frequency (Hz)
SVD for [ U D ] −> [ SC ]
SVD for [ U D ] −> [ DP ]
2
SVD (Volts/Volts), Model Only
0
10
Frequency (Hz)
2
10
10
0
0
10
10
−2
−2
10
10
−4
−4
10
10
−6
10
10
−6
0
5
10
10
Frequency (Hz)
10
0
5
10
10
Frequency (Hz)
Fig. C.11: The final model for the system beyond the frequency range for which measurements
are available
C.3. FORSE ALGORITHM
153
H2 Controller Applied to the Load Cell Model
2
SVD Open−Loop (−) vs. Closed−Loop (−−)
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
0
10
1
10
Frequency (Hz)
2
10
H2 Controller Applied to the Scoring Sensor Model
2
SVD Open−Loop (−) vs. Closed−Loop (−−)
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
0
10
1
10
Frequency (Hz)
2
10
Fig. C.12: The comparison of the closed loop and open loop singular value plots when the
controller is used to close the loop on the identified model
C.3. FORSE ALGORITHM
154
H2 Controller Applied to Real Load Cell Data
2
SVD Open−Loop (−) vs. Closed−Loop (−−)
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
0
10
1
10
Frequency (Hz)
2
10
H2 Controller Applied to Real Scoring Sensor Data
2
SVD Open−Loop (−) vs. Closed−Loop (−−)
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
0
10
1
10
Frequency (Hz)
2
10
Fig. C.13: The comparison of the closed loop and open loop singular value plots when the
controller is used to close the loop on the real measured data
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