Describing Functions for Information Channels Subject to
Packet Loss and Quantization
MASSACHUSES 1-rN fTE
OF TECHNOLOGY
by
OCT 16 2014
Eric Gilbertson
LIBRARIES
S.B., Massachusetts Institute of Technology (2008)
S.M., Massachusetts Institute of Technology (2010)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
Signature redacted
A uthor .............................. ...............................
Department of Mechanical Engineering
August 1, 2014
Signature redacted
...........
Franz S. Hover
Finmeccanica Career Development Professor of Engineering
Thesis Supervisor
Certified by .................
Signature redacted
Accepted by..........................David
. t
David E. Hardt
Chairman, Department Committee on Graduate Students
2
Describing Functions for Information Channels Subject to Packet Loss
and Quantization
by
Eric Gilbertson
Submitted to the Department of Mechanical Engineering
on August 1, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Wirelessly connected robotic systems have become widespread in the terrestrial, aerial and
underwater environments. Examples include autonomous underwater vehicle (AUV) coordination and navigation, Wide Area Measurement System (WAMS) control of power grids,
and automobile networked subsystems control. As one would expect, demand on throughput grows to fill the available channel capacity; in a clean short-range RF setting, entire
images may be transferred in each cycle, as part of a vision-based control system, whereas
in the ocean, acoustic channels with perhaps twenty bits per second allow only the most
basic sensor and command information to be shared regularly. Packet-based wireless communication systems like these that operate near their limits are necessarily quantized, and
often prone to loss. These properties directly impact the overall system performance, and
thus methodologies for understanding and designing feedback systems with quantization
and packet loss are valuable. This thesis makes several contributions to feedback control of
systems subject to quantization and stochastic packet loss in the sensor feedback channel.
First, we derive and verify describing functions (DFs) for information channels subject to quantization and packet loss. The DFs represent the loss and quantization effects
by frequency- and amplitude-dependent gains and phases, similar to transfer functions.
These DFs are unique because, unlike most other DFs that describe hardware and physical
elements, these describe stochastic information channels. DFs are presented for a general
codec algorithm, and for four commonly-used sensor-feedback codecs: Zero-Output, HoldOutput, Linear Filter, and Modified Information Filter. These are each given as closed-form
mathematical expressions of the provably optimal gains and phases for each case, with each
decoder a specific case of the general codec algorithm. Gains and phases predicted by the
models are verified by simulation for open-loop stable, open-loop unstable, minimum phase
and nonminimum phase example systems.
Second, we show how the DFs can be used as analysis tools to predict limit cycles
in dynamic feedback control systems. Computation times using the DFs are shown to be
orders of magnitude faster than those from simulation for these calculations.
Third, we propose a synthesis method to use the DFs to design a codec for the sensor
feedback channel that decreases limit cycle amplitudes induced by quantization and packet
3
loss for a large class of systems. Up to three-fold reductions in limit cycle amplitudes
are shown, with the tradeoff being slightly higher system sensitivity to disturbances and
slightly higher steady state errors to step inputs. The designed codec is of the special and
simple form of a constant times the sent signal if the signal is received and a different
constant times the previous decoded signal if the sent signal is lost. This is the equivalent
structure and computation complexity to both Zero-Output and Hold-Output decoders. A
DF for this decoder allows the constants to be solved for as functions of target limit cycle
amplitudes. The constants reduce to solutions of cubic equations, which are guaranteed to
have a real root, and thus the codec is physically realizable. The codec allows for multiple
limit cycle frequency solutions for the same amplitude solution. The analysis and synthesis
tools are verified both by numerical examples, and by a physical experiment controlling
heading of a small robotic raft where the designed decoder results in smaller limit cycles
than does a linear-filter-based decoder.
Thesis Supervisor: Franz S. Hover
Title: Finmeccanica Career Development Professor of Engineering
4
Acknowledgments
I would first like to thank my advisor Professor Franz Hover and my committee members
Professor Hardt and Professor Leonard for all their guidance on the project. Thank you to
my labmates Brooks, Chris, Josh, Mei, and Pedro for helping with the kayak experiments
in the river, with the raft experiments in the pool, and helping me with questions during lab
lunch presentations. Thanks also to my brother Matthew for helping me debug problems,
often in talks as we ran around the Charles River or hiked up a mountain. I'm also grateful
to the MIT Outing Club for getting me up in the mountains for breaks from schoolwork.
This work is supported by the Office of Naval Research, Grant N00014-09-1-0700,
the National Science Foundation, Contract CNS-1212597, Chevron Energy Technology
Company, and Finmeccanica.
5
6
This doctoral thesis has been examined by a Committee of the Department
of Mechanical Engineering as follows:
Professor Franz Hover...............................................
Chairman, Thesis Committee
Associate Professor
Professor D ave Hardt................................................
Member, Thesis Committee
Professor of Mechanical Engineering
....
.............................o
.
Professor John Leonard ...........
Member, Thesis Committee
Professor of Mechanical Engineering
8
Contents
1
Introduction
1.1
2
3
23
Motivation and Problem Statement . . . . . . . . . . . . . . . . . . . . . . 23
1.1.1
Underwater Applications . . . . . . . . . . . . . . . . . . . . . . . 24
1.1.2
Terrestrial Applications . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2
Overview and Contributions of the Thesis . . . . . . . . . . . . . . . . . . 27
1.3
Assumptions
1.4
Example Systems Studied. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Background
30
33
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2
Control with Quantization
2.3
Control with Packet Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4
Control with Quantization and Packet Loss
2.5
Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6
Control of Underwater Thrusters . . . . . . . . . . . . . . . . . . . . . . . 39
. . . . . . . . . . . . . . . . . . . . . . . . . . 33
. . . . . . . . . . . . . . . . . 37
2.6.1
Thruster Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.2
Thruster Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.7
Synopsis of Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 41
2.8
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Experiments with Packet-Loss Robust Control
3.1
42
45
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9
Hardware and Operating Environment . . . . . . . . . . .
.
45
3.3
System Model . . . . . . . . . . . . . . . . . . . ...
.
.
47
3.4
MIF Experimental Setup . . . . . . . . . . . . . . . . . .
48
3.5
Physical Connectivity . . . . . . . . . . . . . . . . . . . .
.
48
3.6
Experimental Results . . . . . . . . . . . . . . . . . . . . ..
50
3.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
51
.
.
3.2
53
4 Describing Function Derivations
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Describing Function for General Multiple Input Multiple Output Informa-
.
4.1
.
tion Channels Subject to Packet Loss . . . . . . . . . . .
4.3.1
Loss with Zero-Output Estimator (LZ) . . . . . . . . . . . . . .
63
4.3.2
Loss with Hold-Outpu t Estimator (LH) . . . . . . . . . . . . .
65
4.3.3
Loss with LQG-Type Linear Filter Estimator (LF)
. . . . . . .
67
4.3.4
Loss with Modified In formation Filter (LM) . . . . . . . . . . .
68
.
. . . .
72
.
Model Verification
Convergence
. . .
72
4.4.2
LH Verification . .
73
4.4.3
LF Verification . .
74
4.4.4
LM Verification . .
87
4.4.5
Coupled Losses . .
.
.
.
4.4.1
.
. . . . . . . . . . . . . . . . . . . . . . .
.
90
. . . .. .. . .. .. . . .. . . . .. . .
92
Quantizer Describing Functio . p . . . . . . . . . . . . . . .. . . . . .
Uniform Quantizer
.
4.5.1
....
p.................
Logarithmic Quantize
93
Combined Quantizer and Enc
95
4.5.2
95
4.6.1
Quantizer and Zero O
4.6.2
Quantizer and Hold 0
4.6.3
Quantizer and Linear Filter . . . . . . . .
97
4.6.4
Quantizer and MIF
. . . . . . . . . . .
. . 98
.
.
utput . . . . . . . . . . . . . . . . . . . .
.
.
4.6
88
..................
..
4.5
.
.
.
.
62
.
4.4
. . . . . . . . 53
Describing Functions for Spec ific State Estimation Algorithms . . . . .
.
4.3
53
10
96
Sum mary . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .
.
4.7
5 Describing Functions as Analysis Tools
. 99
Introduction . . . . . . . . . . . . . . . . . .
101
5.2
Computation Times for Analysis vs Simulation
101
5.3
Bode Plots . . . . . . . . . . . . . . . . . . .
106
5.4
Sensitivity Functions . . . . .
.
107
5.5
Limit Cycle Numerical Example
110
5.6
Summary . . . . . . . . . . .
113
6 Describing Functions as Design Tools
117
.
.
5.1
.
101
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2
SISO Design Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3
6.4
6.5
6.2.1
System Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2.2
Limit Cycle Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2.3
Solving the Nyquist Equations . . . . . . . . . . . . . . . . . . . . 120
6.2.4
Solving for Codec Parameters . . . . . . . . . . . . . . . . . . . . 130
6.2.5
Codec Implementation . . . . . . . . . . . . . . . . . . . . . . . . 131
SISO Design Numerical Example
. . . . . . . . . . . . . . . . . . . . . . 132
6.3.1
System with No Loss . . . . . .... . . . . . . . . . . . . . . . . . 133
6.3.2
System with Packet Loss and Quantization . . . . . . . . . . . . . 134
6.3.3
Codec Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3.4
Design Point Side Effects
6.3.5
Final Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
. . . . . . . . . . . . . . . . . . . . . . 142
MIMO Design Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.4.1
Limit Cycle Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.4.2
Solving The Nyquist Equation . . . . . . . . . . . . . . . . . . . . 151
6.4.3
Codec Implementation . . . . . . . . . . . . . . . . . . . . . . . . 151
MIMO System Design Numerical Example . . . . . . . . . . . . . . . . . 153
6.5.1
MIMO System with No Loss . . . . . . . . . . . . . . . . . . . . . 153
6.5.2
System with Packet Loss and Quantization
11
. . . . . . . . . . . . . 158
. 162
6.5.4
Design Performance
. . . . . . . . . .
. 168
6.5.5
Checking for Near Optimality . . . . .
. 172
Summary . . . . . . . . . . . . . . . . . . . .
. 177
.
.
.
Codec Design . . . . . . . . . . . . . .
.
6.6
6.5.3
179
7 Experiments
7.2
Experimental Setup . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 179
7.3
System Models . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 182
7.3.1
Thrusters . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 182
7.3.2
Raft Model . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 182
.
.
.
.
Introduction
Determining Model Parameter Values . . . . . . . . . . . . . . . . . . . . 185
7.5
Controller Design . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 185
7.6
Limit Cycle Analysis . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 188
7.7
Limit Cycle Experimental Tests
. .
. . . . . . . . . . . . . . . . . . . . 190
7.8
Codec/Controller Design . . . . . .
. . .... . . . . . . . . . . . . . . . .192
7.8.1
Design Requirements . . . .
. ... . . . . . . . . . . . . . . . . .192
7.8.2
Design Equations . . . . . .
. . . . . . . . . . . . . .I. . . . . . 194
7.8.3
Effect of Designed Codec on S ystem with Packet Loss . . . . . . . 195
7.8.4
Effect of Designed Codec on System with No Packet Loss
. . . . . 195
7.8.5
Limit Cycle Experiments with Designed Codec . . . . .
. . . . . 198
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 199
.
.
.
.
.
7.4
Sum mary
.
.
7.9
8
. . . . . . .
7.1
.
179
201
Conclusion
. . . . . . . . . . . . . . . . . . . . .201
. . . ...
8.1
Review of Contributions
8.2
Areas for Future Work .. . . .....
12
.......
. . . . . ..
. . . . .203
List of Figures
1-1
Schematic of Long baseline system with AUV . . . . . . . . . . . . . . . . 24
1-2
Schematic of ultra short baseline system with ROV and AUV . . . . . . . . 25
1-3
Schematic of Wide Area Measurement System (WAMS). . . . . . . . . . . 26
3-1
Autonomous surface vehicle operating in the Charles River, Boston, MA.
.
46
3-2 Experimental Micro-Modem performance data in the Charles River Basin,
which is not a power-limited environment but rather one limited by multipath 47
3-3
Example noise histograms from kayak GPS and compass sensors, with the
vehicle stationary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3-4
Kayak closed-loop heading model fit on experimental data in waves. . . . . 49
3-5
MIF-LQG control tests with acoustic communications on a day with light
wind, 27 November 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3-6
MIF-LQG control tests with varying packet loss on a day with nearly no
wind, 24 October 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3-7 MIF-LQG control tests with varying packet loss on a day with strong wind,
16 October 2012. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4-1
Block diagram for a MIMO feedback control system with an encoder, binary erasure channel, decoder, controller, and plant . . . . . . . . . . . . . 55
4-2 Block diagrams of SISO versions of the four codec algorithms . . . . . . . 63
4-3
Comparison of LZ (top left), LH (top right), LF (bottom left), and LM
(bottom right) blocks showing input and decoded signals. . . . . . . . . . . 64
4-4 Block diagrams of SISO versions of the four codec algorithms with quantizers added. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
13
4-5
Convergence of the LH describing function phase formula with g, for different values of a, and or = 0.1.
4-6
Convergence of the LM describing function phase formula with g, for different values of a, and (or = 0.1.
4-7
. . . . . . . . . . . . . . . . . . . . . . 73
. . . . . . . . . . . . . . . . . . . . . . 74
Gain (top plot) and phase co of the LH model . . . . . . . . . . . . . . . . 75
4-8 Example 1-mass, 1-input 1-output system . . . . . . . . . . . . . . . . . . 76
4-9 LF describing function gain (top plot) and phase (bottom plot) from analysis and Monte Carlo simulation, for the contact task example. . . . . . . . .. 77
4-10 Example 1-mass, 1-input 1-output inverted pendulum system . . . . . . . . 78
4-11 LF describing function gain (top plot) and phase (bottom plot) from analysis and Monte Carlo simulation, for the inverted pendulum example. .....
4-12 Example 1-input, 1-output all-pass symmetric lattice circuit system.
79
. . . . 80
4-13 LF describing function gain (top plot) and phase (bottom plot) from analysis and Monte Carlo simulation, for the nonminimum phase circuit example. 81
4-14 MIMO LF describing function gain (top plot) and phase (bottom plot) from
analysis and Monte Carlo simulation, for the 2-sensor contact task example.
82
4-15 Example 2-mass, 1-input 2-output MIMO system . . . . . . . . . . . . . . 83
4-16 MIMO LF describing function gain (top plot) and phase (bottom plot) from
analysis and Monte Carlo simulation, for the 1-input 2-output double mass
exam ple. . . . . . . . . . . .... . . . . . . . . . . . . . . . . . . . . . . . 84
4-17 Example 3-mass, 2-input 3-output, 6-state MIMO system . . . . . . . . . . 85
4-18 MIMO LF describing function gain (top plot) and phase (bottom plot) from
analysis and Monte Carlo simulation, for the 2-input 3-output three-mass
example. Legend is same for both plots. . . . . . . . . . . . . . . . . . . . 86
4-19 LM gain (top plot) and phase co (bottom plot) from analysis and Monte
Carlo simulation, for the contact task example.
. . . . . . . . . . . . . . . 87
4-20 LM gain (top plot) and phase co (bottom plot) from analysis and Monte
Carlo simulation, for the MIMO contact task example.
14
. . . . . . . . . . . 88
4-21 LM gain (top plot) and phase a (bottom plot) from analysis and Monte
Carlo simulation, for the 3-mass MIMO example. Legend is same for both
plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4-22 Gain difference between coupled loss model and independent loss model
for two-channel LZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4-23 Gain and phase differences between coupled loss model and independent
loss model for two-channel LH.
. . . . . . . . . . . . . . . . . . . . . . . 91
4-24 Describing function gain N(a/h) for the uniform quantizer [32] with bin
width h and input signal amplitude a . . . . . . . . . . . . . . . . . . . . . 92
4-25 Describing function gain for log quantizer as a function of input magnitude
a and quantizer parameter p for fine quantization (deadband p 1000 , top plot)
and coarse quantization (deadband p 2 , bottom plot). . . . . . . . . . . . . . 94
4-26 Block diagram for a MIMO feedback control system with a quantizer, encoder, binary erasure channel, decoder, controller, and plant . . . . . . . . . 95
5-1
Computation time for simulation and describing function for the SISO LZ
decoder, for
5-2
and a = 50% . . . . . . . . . . . . . . . . . . . . . 102
Computation time for simulation and describing function for the SISO LH
decoder, for
5-3
='r -
r = -l and a = 50% . . . . . . . . . . . . . . . . . . . . . 103
Computation time for simulation and describing function for the SISO LF
decoder, for ar = -l and a = 50% . . . . . . . . . . . . . . . . . . . . . 103
5-4
Computation time for simulation and describing function for the SISO LM
decoder, for cor = -I and a = 50% . . . . . . . . . . . . . . . . . . . . . 104
5-5
Computation time for simulation and describing function for the SISO LM
decoder computing normalized bode plot gains, for 100 frequencies and
a = 50%
5-6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Gain (top) and phase (bottom) of the QLF for the contact task model; a =
10% , h = 46, T= 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5-7
Gain (top) and phase (bottom) of the QLF for the contact task model; a =
60% ,h = 10, T = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
15
5-8
Gain (top) and phase (bottom) of QLH; as with LH, these properties are
independent of the plant or controller. a = 10%, h = 46, r = 0.1. . . . . . . 108
5-9
Gain (top) and phase (bottom) of QLH. a = 60%, h = 10, -r = 0.1. . . . . . 108
5-10 Block diagram of feedback control loop with additive disturbance on plant
output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5-11 Gain of the sensitivity function for the contact task model with the QLF
codec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5-12 Closed-loop step response of double-integrator example system, with LQG
design and a one-step delay.
. . . . . . . . . . . . . . . . . . . . . . . . . 111
5-13 Input amplitudes a and quantization levels h that result in a limit cycle for
the double-integrator example system . . . . . . . . . . . . . . . . . . . . 112
5-14 Time series for the QLZ system with quantization level h = 70. . . . . . . . 113
5-15 Time series for the QLH system with quantization level h = 35 . . . . . . . 114
5-16 Frequency o of limit cycles from the analysis and from simulation . . . . . 114
6-1
Block diagram of SISO control system with binary erasure channel in sensor feedback, and describing function for the quantizer-encoder-channeldecoder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6-2 Example 1-mass, 1-input 1-output system . . . . . . . . . . . . . . . . . . 132
6-3
Step responses of original system (top plot) and system with 1.1-second delay in sensor feedback (bottom plot). Nondimensional time is used, defined
as gO. T*tim e.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6-4 Nyquist plots for original system (top plot) and system with 1.1-second
delay in sensor feedback channel (bottom plot). Nondimensional time is
used, defined as gO r* time. . . . . . . . . . . . . . . . . . . . . . . . . . 136
6-5
Time series plots for QLZ (top) and QLH (bottom). Nondimensional time
is used, defined as gor* time. . . . . . . . . . . . . . . . . . . . . . . . . 137
6-6
Plot of limit cycle amplitudes for QLZ, QLH, and designed codecs. Blue
dots represent design points. . . . . . . . . . . . . . . . . . . ... . . . . . . 138
6-7
Nyquist plots for system with decoder and delay but no loss or quantization 139
16
6-8
Limit cycle magnitudes as a function of M and N . . . . . . . . . . . . . . 141
6-9
Sensitivity magnitude plots for original system and designed systems DI,
D 2 , and D 3 , with and without delay . . . . . . . . . . . . . . . . . . . . . . 144
6-10 Describing-function-derived gain and phase loci of system with delay, loss,
quantization, and designed codec . . . . . . . . . . . . . . . . . . . . . . . 146
6-11 Time series of system with designed decoder. Nondimensional time is
used, defined as gOw
* time. . . . . . . . . . . . . . . . . . . . . . . . . . 147
6-12 Step responses of original system (top plot) and system with 1.1-second
delay in sensor feedback (bottom plot) . . . . . . . . . . . . . . . . . . . . 148
6-13 Example system with two inputs and four outputs . . . . . . . . . . . . . . 152
6-14 Time response of original system to initial position displacements of 1 for
each mass, and initial velocities zero. Nondimensional time is used, defined
2
as gcO T*tim e.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6-15 Loop transfer functions magnitudes for MIMO example, at low frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6-16 MIMO Nyquist plots for original system (top plot) and system with 1.5second delay in sensor feedback channels (bottom plot) . . . . . . . . . . . 158
6-17 Time series responses of original system with 1.5-second time delay in
sensor feedback of all channels, for initial displacements of one for each
mass and zero initial velocities. Nondimensional time is used, defined as
g
2
T *tim e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .159
6-18 MIMO design time series plots using QLZ codec . . . . . . . . . . . . . . 160
6-19 MIMO design time series plots using QLH codec . . . . . . . . . . . . . . 161
6-20 Plot of limit cycle amplitudes for states X1 and X2 for QLZ, QLH, and
designed codecs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6-21 Nyquist plots of designs DI (top), D 2 (middle) and D 3 (bottom) for systems
with no nonlinearities (i.e.systems with delay and decoder but no loss or
quantization)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6-22 Nyquist plots of designs D4 (top) and DFinal (bottom) for systems with no
nonlinearities (i.e.systems with delay and decoder but no loss or quantization) 165
17
6-23 Describing-function-derived gain-phase loci of designed system with delay, quantization, and packet loss . . . . . . . . . . . . . . . . . . . . . . . 167
6-24 MIMO design time series plots using designed codec . . . . . . . . . . . . 169
6-25 Time series responses of original system with no delay, loss, or quantization with decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6-26 Time series responses of original system with delay but no loss or quantization with decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6-27 Transfer functions magnitudes for MIMO example with designed decoder,
at low frequencies.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6-28 Quantizer gain plots. Top plot shows gain at a/h=0.6 corresponds to gain at
larger value of a/h=0.9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6-29 Plot of limit cycle amplitudes for states Xi and X2 for QLZ, QLH, and
designed codecs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 174
6-30 Sensitivity plots of mean square limit cycle amplitude for small changes in
M decoder parameters. Red dot marks design point. . . . . . . . . . . . . . 175
6-31 Sensitivity plot of X1 and X2 limit cycle amplitudes for changes in M4 decoder parameter. Red dots mark design point. . . . . . . . . . . . . . . . . 176
6-32 Sensitivity plots of mean square limit cycle amplitude for small changes in
N decoder parameters. Red dot marks design point. . . . . . . . . . . . . . 177
6-33 Sensitivity plot of mean square limit cycle amplitude for large changes in
N4 decoder parameter. Red dot marks design point. . . . . . . . . . . . . . 178
7-1
Experimental test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7-2
Experimental test platform side view . . . . . . . . . . . . . . . . . . . . . 181
7-3
Experimental test platform top view, as seen from overhead camera. . . . . 181
7-4 Block diagram of experimental system.
7-5
. . . . . . . . . . . . . . . . . . . 182
Actuator describing function capturing deadband (u <0.1), unity gain(0. 1 <
u < 1), and saturation (u > 1) regimes. . . . . . . . . . . . . . . . . . . . . 183
7-6
Top view of autonomous raft used in experiments, showing thrusters and
coordinate frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
18
7-7
Raft model vs data for thruster propeller speed step inputs of 30, 50, 70 and
90 percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7-8
Nyquist plot of loop transfer function with one time step delay. Crossover
is at -0.59. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7-9
Example time series response of system to initial error of approximately-35
degrees, for one time step delay in sensor measurement. . . . . . . . . . . . 187
7-10 Nyquist plot of loop transfer function with three time step delay. . . . . . . 188
7-11 Example time series response of system to initial error of -35 degrees, for
three time step delay in sensor measurement.
. . . . . . . . . . . . . . . . 189
7-12 Analytically predicted and experimentally observed limit cycle amplitudes.
191
7-13 Regions of different dominating factors influencing limit cycles. . . . . . . 191
7-14 Experimentally observed limit cycle of magnitude approximately 1.9 for
pure quantizer, no packet loss. . . . . . . . . . . . . . . . . . . . . . . . . 192
7-15 Experimentally observed limit cycle of magnitude approximately 1 for pure
quantizer, no packet loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7-16 Experimentally observed limit cycle of magnitude approximately 0.8 for
QLZ with 15% packet loss. . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7-17 Experimentally observed limit cycles of magnitude approximately 1.8 for
QLH with 55% packet loss. . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7-18 Bode-like plots for the describing function of the designed codec.
. . . . . 196
7-19 Describing-function-derived gain-phase loci for system with designed controller with 25% packet loss and quantization ratio a/h = 0.9 . . . . . . . . 197
7-20 Nyquist plot for system with designed controller with no packet loss or
quantization.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7-21 Experimental time response for system with designed controller and delay,
with no quantization or packet loss.
. . . . . . . . . . . . . . . . . . . . . 199
7-22 Limit cycle in experiment with designed controller/codec for 25% packet
loss. Magnitude is approximately 0.9. . . . . . . . . . . . . . . . . . . . . 200
19
20
List of Tables
3.1
Comparison of RMS errors from experiments with light wind and with
strong w ind. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1
Summary of mathematical symbols used in this chapter. . . . . . . . . . . . 54
6.1
Summary of mathematical symbols used in this chapter. . . . . . . . . . . . 118
6.2
Time series responses for system with no delay, delay, and delay+loss+quantization,
for LZ, LH, and designed decoders . . . . . . . . . . . . . . . . . . . . . . 133
6.3
Nyquist plot for system with and without delay, for LZ, LH, and designed
decoders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.4
Codec coefficients and limit cycle amplitudes . . . . . . . . . . . . . . . . 138
6.5
Gain and phase margins of system with delay and different decoders, with
no loss or quantization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.6
Time series responses for system with no delay, delay, and delay+loss+quantization,
for LZ, LH, and designed decoders. Nondimensional time is used, defined
2
as gOJ T*time.
6.7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Nyquist plot for system with and without delay, for LZ, LH, and designed
decoders .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
21
22
Chapter 1
Introduction
In this chapter we introduce the real-world problem that has motivated this thesis: controlling robotic systems where sensor signals are subject to packet loss and quantization
errors. A statement of the problem is given in Section 1.1 with several applications, and a
statement of the contributions of this thesis given in Section 1.2. A review of relevant prior
work is given in Chapter 2.
1.1
Motivation and Problem Statement
Wirelessly connected robotic systems have become widespread in the terrestrial, aerial and
underwater environments. Examples include autonomous underwater vehicle (AUV) coordination and navigation([1 1], [3], [33]), network-based process control engineering ([117])
, automobile networked subsystems control ([50], [21]), and power grid control ([106],[7 1],
[105]). As one would expect, demand on throughput grows to fill the available channel capacity; in a clean short-range RF setting, entire images may be transferred in each cycle,
as part of a vision-based control system, whereas in the ocean, acoustic channels with perhaps twenty bits per second allow only the most basic sensor and command information
to be shared regularly. Packet-based wireless communication systems like these that operate near their limits are necessarily quantized, and often prone to loss. These properties
directly impact the overall system performance, and thus methodologies for understanding
and designing feedback systems with quantization and packet loss are valuable.
23
1.1.1
Underwater Applications
Many example applications exist in underwater environments, where autonomous underwater vehicles (AUVs) are beginning to perform tasks previously done by human divers or
remotely operated vehicles. AUVs have the advantage of requiring no tether, and not endangering the lives of divers, but communication underwater between the AUV and remote
sensors or other AUVs may be subject to packet loss and quantization.
One application is AUV navigation through challenging environments using remotesensor positioning measurements. Some of the most commonly-used remote-sensor systems for this application are Long Baseline (LBL) and Ultra-Short Baseline (USBL)[68].
Figure 1-1: Schematic of Long baseline system with AUV. Image credit: Kongsberg Maritime, http://www.km.kongsberg.com/ks/web/nokbg0240
In the LBL setup, shown schematically in Figure 1-1, fixed nodes on the sea floor
take acoustic range measurements to the AUV, triangulate the position, and acoustically
transmit the position back to the AUV. LBL systems are currently used in AUV positioning
for survey and inspection of deep water oil and gas facilities [102]. They have also been
used to precisely position nuclear submarines navigating through underwater canyons or
24
preparing to launch missiles [34].
Figure 1-2:
Schematic of ultra short baseline system with ROV and AUV. Image
credit: Teledyne Benthos, http: //www. benthos . com/index. php /product/
positioning-systems/benthos-dat
In the USBL setup, shown schematically in Figure 1-2, a surface vessel uses a hullmounted underwater transceiver with multiple acoustic transducers to detect the range and
angle to an underwater vehicle. The underwater vehicle emits a time-stamped acoustic
signal, and the transceiver on the ship calculates range from the signal travel time. The
multiple transducers receive different phase shifts of the same signal, and these phase differences are used to calculate the angle to the target. The ship can calculate the position
from the range and angle measurements, and transmit this measurement acoustically back
to the underwater vehicle. USBL positioning is currently used for AUV surveying and inspection of underwater oil and gas facilities [102], and has the advantage that no additional
underwater nodes need to be installed prior to operations.
Another underwater application is AUV collaborative positioning, where multiple AUVs
must communicate with each other acoustically to perform a task. One example task could
be teams of AUVs decommissioning offshore oil rigs. Offshore oil rigs are generally abandoned at the end of their operational life, and in the United States oil companies are required
to remove the rig within a year of abandonment. Current practice is to use a combination
of human divers and Remotely Operated Vehicles (ROVs) to decommission and remove
an old rig, but this is very dangerous for the divers, and ROV tether management can be
problematic. An alternative method under research is to replace the divers and ROVs with
AUVs, thus removing the danger to humans and the problems associated with tethers. This
25
may require AUVs to communicate acoustically to coordinate cutting and lifting tasks, and
the communications could thus be subject to packet loss and quantization errors.
In these underwater applications both packet loss and quantization could be significant
at the same time. For instance, if an acoustic modem used for communication has only
a few different transmission modes (as is realistic), then the operation is limited to one of
those modes and it may not be possible to configure the system such that only packet loss is
significant or only quantization is significant. Thus the case of both significant quantization
and packet loss is realistic for these applications.
1.1.2
Terrestrial Applications
Many terrestrial applications exist where networks of sensors are wirelessly connected and
subject to packet loss [104]. Such a system can be advantageous if sensors are located
far away from the plant, or if space is limited such that extensive wiring is undesirable.
However, the network introduces the possibility of packet loss and quantization errors in
signals.
These systems are used, for example, in some modem automobiles using using ControlArea-Network (CAN)-based data communication [117]. In this case space is very limited,
and sensors for anti-lock braking systems, acceleration slip regulation, etc. are connected
wirelessly to controllers over a network.
Satellite
Controller
Power Grid
Figure 1-3: Schematic of Wide Area Measurement System (WAMS).
These systems have also been used in power systems applications ([106],[71]) to maintain power grid stability and prevent blackouts. In these cases, phasor measurement units
(PMUs) are installed strategically in power grids to measure AC phases and waveforms,
26
and measurements are transmitted over a network using satellites to a controller, as shown
in Figure 1-3. This is referred to as a Wide Area Measurement System (WAMS). As pointed
out in [105], power systems are often operated close to their limits of stability due to economic and environmental pressures and deregulation, thus the effect of packet loss and
quantization from WAMS could be significant if it pushed the system into a limit cycle or
to instability.
In process control, such as in wastewater treatment plants, senors measuring water levels, pH factors, temperature, and chemical oxygen demand transmit signals through a network to a system controller [117]. In all of these applications, signals in the feedback
control loop are subject to packet loss and quantization induced by the network.
Loss of a sensor measurement can also be caused at the sensor itself, and have the
same effect as if it were lost in the feedback channel. For instance, automated welding for
manufacturing is often controlled via visual servoing, though this is subject to sensor error
from optical disturbances [119]. Arc glares, welding spatters, and smoke can all cause
the sensor measurement to be effectively lost, even though it is not sent through a lossy
communication channel.
1.2
Overview and Contributions of the Thesis
The goal of this thesis is to study the performance of feedback control systems limited by
significant finite static quantization and stochastic packet loss. Unlike previous studies, the
performance metric of this work is limit cycles - i.e., self-induced oscillations in nonlinear
systems. Limit cycles are central in the performance of robotic and other systems because
they dictate the precision which can be achieved. As will be shown, systems that are close
to instability can be pushed into undesirable limit cycling by the combination of packet loss
and significant finite quantization.
In Chapter 2 we review existing methods from the literature for designing and analyzing
feedback control systems subject to quantization and packet loss. We also review describing functions - quasi-linear approximations to non-linear system components - and some
relevant applications in recent research. We then outline how describing functions can be
27
used to approximate the non-linear effects of quantization and packet loss, and in turn be
used to analyze limit cycles.
In Chapter 3 we show experimental results from tests of a packet-loss robust algorithm
- the modified information filter - from the literature. Heading control tests are conducted
on a robotic kayak in open water, with the sensor measurements transmitted to the kayak
using a shore-based acoustic modem and subject to realistic packet loss. These tests are
similar to true USBL-based control of an AUV, and serve as motivation for more detailed
experiments in a controlled lab setting. These experimental results are also documented in
[33].
In Chapter 4 we derive describing functions for information channels subject to packet
loss and quantization. These are each given as closed-form mathematical expressions of
the provably-optimal gains and phases for each case, with each decoder a specific case of
the general codec algorithm. First, a DF is derived for a general codec of a channel subject
to just packet loss. The general DF is then applied to four commonly-used codecs from
the literature: Zero-Output, Hold-Output, Linear Filter, and Modified Information Filter.
These DFs are verified by simulation for open-loop stable, open-loop unstable, minimum
phase, and nonminimum phase SISO system examples, as well as for multi-sensor channels
of several different MIMO system examples. The DFs are then combined with a standard
quantizer DF to give DFs for channels subject to both quantization and packet loss.
These DFs build on the DF methods of the literature for studying nonlinear physical
elements in control loops, and the new DFs are unique because they describe information
sent through channels, and the information itself is stochastic. These two properties of
the DF - describing non-physical and stochastic elements - distinguish the analysis from
previous work and open the possibility for DF analysis to be applied to a much broader
range of systems than has been standard practice in the past.
In Chapter 5 we show how the DFs require orders of magnitude less computation time
than do simulations for computing gains and phases for quantized and lossy information
channels. The DFs are shown to be useful analysis tools to create loop sensitivity-like
functions to study the effect of disturbances, and to create channel Bode-like gain and
phase plots. The.Bode-like plots can in turn be used to predict limit cycles of the system,
28
and a numerical example is given outlining this procedure. The example shows a case
where the new DFs predict limit cycles in a quantized lossy system that are not predicted
by a quantizer DF alone. The limit cycles are then shown by simulation to indeed exist.
In Chapter 6 we show how the DFs can be used as synthesis tools to design a channel
codec. In this design procedure we assume two operating conditions for the system: a nominal operating case and a second operating case that is close to instability. This could apply,
for instance, to an underwater vehicle navigating via LBL where a nominal operating case
is in open water, while a second case is operating in a cluttered environment where acoustic
signal performance is negatively affected, and quantization and packet loss push the system to limit cycle. For this synthesis tool, a controller is first designed for the nominal
operating case, and an acceptable limit cycle amplitude specified for the second operating
case. The design method gives a decoder that meets the limit cycle requirement with small
effect to the nominal operating performance. The design method can be iteratively applied
for smaller limit cycle amplitude specifications, until a balance is met that achieves small
enough limit cycle amplitudes with acceptable effect on nominal operating performance.
The designed codec is of the special and simple form of a constant times the sent signal
if the signal is received and a different constant times the previous decoded signal if the
sent signal is lost. This form has the equivalent structure and computational complexity
as the Zero-Output and Hold-Output decoders. A DF for this codec allows the constants
to be solved for as functions of target limit cycle amplitudes. The constants reduce to
solutions of cubic equations, which are guaranteed to have a real root, and thus the codec
is physically realizable. The codec allows for multiple limit cycle frequency solutions for
the same amplitude solution. SISO and MIMO numerical examples are given illustrating
the design procedure.
In Chapter 7 we show experimental results on a small robotic raft. We control raft
heading in an indoor pool environment and artificially add packet loss and quantization to
the heading sensor measurements. The example codecs are implemented on the raft, and
limit cycles predicted by the DFs are verified for a range of packet loss scenarios. A new
design point is then selected where the system limit cycles at a smaller amplitude than any
of the example codecs for a given packet loss probability. The design method is used to
29
achieve this design point, and verified experimentally. The results demonstrate how the
designed decoder can be used to decrease limit cycle amplitudes even when compared to a
linear-filter-based decoder.
We conclude in Chapter 8 by reviewing the contributions of the thesis and discussing
areas of potential future work.
Assumptions
1.3
For clarity, we summarize the major assumptions used in the analysis and examples of the
thesis:
* Discrete time control.
" Loss and quantization in sensor feedback channel.
" Uniform quantizer used.
" Bernoulli packet loss.
" No acknowledgments.
1.4
Example Systems Studied
In order to verify and illustrate the describing functions, we study an example set of linear
open-loop stable, open-loop unstable, minimum phase, and nonminimum phase SISO systems, and linear, minimum-phase, mass-spring-damper MIMO systems, and we list these
example systems here for clarity:
" 1-mass with one force input (Section 5.5).
" 1-mass with spring and damper, with one force input (Section 4.4.3, 4.4.4).
" Open-loop unstable inverted pendulum (Section 4.4.3).
" Nonminimum phase symmetric lattice all-pass circuit (Section 4.4.3).
30
* 2 masses connected with springs and dampers, with one force input (Section 4.4.3,
4.4.4).
* 2 masses connected with springs and dampers, with two force inputs (Section 4.4.3,
4.4.4).
* 3 masses connected with springs and dampers, with two force inputs (Section 4.4.3,
4.4.4).
1.5
Summary
This chapter described the importance of feedback control systems that are subject to packet
loss and quantization. Example systems described include underwater vehicle navigation
and coordination using acoustic communication, and power grid control. The contributions
of this thesis to the control of these types of systems were then given, and the chapters of
the thesis outlined.
31
32
Chapter 2
Background
2.1
Introduction
This chapter gives a survey of prior work in control systems subject to quantization and
packet loss, as well as work related to describing functions. We begin with an overview of
work related to control with quantization, followed by work related to control with packet
loss and then work on a combination of the two. We then review recent applications of
describing functions, and give background on control of underwater vehicle thrusters. We
close with a synopsis of the background work and explain how describing function analysis
fits into the existing research on packet loss and quantization.
2.2
Control with Quantization
The use of quantization in control systems has been studied at least since the 1950s, when
Kalman [51] observed that quantized signals cause a stabilizing controller to exhibit limit
cycles and chaotic behavior. Since then, work on quantizers has fallen into two basic
categories: work studying static (also called memoryless) quantizers, and work studying
dynamic quantizers (also called quantizers with memory). A static quantizer maintains the
same parameters (such as bin sizes and location) over time, while for a dynamic quantizer
these properties are allowed to vary over time. Curry [17], Miller [67], and Delchamps
[19] gave early results on the effect of static uniform quantizers on control systems, and
33
Wong [112] determined bounds on the number on uniform quantization intervals needed
to practically stabilize a linear system. However, a finite uniform quantizer can do no
better than practical stability. To achieve quadratic stability with a static quantizer Elia [23]
proved that, at least in the case of a SISO system, the quantizer must be logarithmic with
infinite precision. This type of quantizer is, of course, not practically realizable. Fagnani
[24] looked at methods to reach practical stability for SISO systems with more practical
static finite logarithmic quantizers.
Brockett [12] developed one of the first results using dynamic quantizers, showing that
dynamic finite quantizers can in fact achieve asymptotic stability, unlike static finite quantizers. Several works later studied bounds on the number of dynamic quantization levels
needed for closed loop quadratic (Ling [60] and Nair [74]) and exponential ([75]) stability.
Fu ([29] and [30]) developed a unique dynamic logarithmic quantizer that scales a gain on
the quantizer parameters.
Currently one of the most common means of dealing with quantization in feedback
control is to use the sector bound method developed by Fu ([28], [27]). In this method,
quantization error is treated as uncertainty and bounded using a sector bound. By doing
this, the quantization error can be considered to be additive noise, and standard control
analysis tools can be applied to study and minimize the quantization effect. Fu used the
sector bound method to determine stabilizability and H. performance criteria for SISO and
MIMO systems. Coutinho [16] used the sector bound approach to quadratically stabilize
a system using infinite logarithmic quantizers in sensor and controller channels. Bai [4]
developed a set of bilinear matrix equations for system stability, for MIMO systems with
different quantization in different channels, and Rasool [85] extended these results to include H. performance criteria. Wei [109] modified the sector bound method to account for
multiplicative noise instead of additive.
The sector bound method developed by Fu is, however, conservative in stability conditions. Less conservative approaches using quantization-dependent Lyapunov functions
(Gao [31]) have also been developed.
A final method to deal with quantization is to look at it from an information theoretic
view. Tatikonda [100] used this method, and determined the minimum bit rate required to
34
achieve stability and control objectives.
2.3
Control with Packet Loss
Much work has been done in the past 50 years studying the important problem of control
systems subject to packet loss. Most work focuses on how to design the encoder, decoder,
and/or controller to attain some stability or performance result. Some of the first work, by
Nahi [73] and Hadidi [41] studied SISO systems with packet loss in the sensor feedback
channel and acknowledgments. They derived an optimal codec scheme where a linear combination of current and past measurements are sent if a measurement is lost, with Bernoulli
or Markov loss probability. Robinson [87] extended these results to include system performance criteria. Other methods of dealing with packet loss include treating lost packets as
long delays (Zhang [118]), and looking at the system's output power spectral density as a
function of packet loss (Ling [59]).
One of the most common ways to deal with packet loss in the sensor channel is to use a
Kalman Filter decoder, where a state estimate is propagated through a system model if the
measurement is lost, and a Kalman gain is multiplied by the difference between measurement and propagated model if the measurement is received. Sinopoli [92] and Fortmann
[25] found critical packet loss probabilities for error covariance matrix convergence in this
case.
Many variations exist on the standard Kalman-Filter setup. Several authors (Smith [97],
Nilsson [77], [76], and Costa [15]) extended the Kalman Filter results to look at jump linear estimator decoders [65] that select a gain from a fixed set depending on whether the
measurement packet was received. This type of decoder was shown to be less computationally expensive than the standard Kalman filter. Wang [108] and Schenato [89] designed
general constant-gain linear filter decoders that bound the estimation error covariance by a
prescribed amount.
Another useful variant of the Kalman Filter that has been studied for these problems
is the information filter. Gupta [40],[39] developed a modified information filter codec for
sensor channel loss that involves sending a specialized information vector, which depends
35
on current and past measurements and state estimates. This was shown to be the optimal
LQG-type control for losses in the sensor channel. Jin [49] and Jiang [48] considered
the case where the sensor packet can be split up and sent through multiple independent
channels, before being recombined and decoded with a Kalman Filter-type decoder, and Li
[58] studied the Unscented Kalman Filter under packet loss.
Fewer works have considered the scenario where only the controller-actuator channel
is subject to packet loss. One of the simplest codec methods (Mahmoud [63]) is to send the
single control command and decode with a zero-output or hold-output decoder. Quevedo
[83] developed packetized predictive control (PPC), where a set of current and future control commands is sent to the actuator at each step. If the signal is lost, the actuator uses
the next control command stored in its buffer. Miklovicova [66] studied the specific PPC
case where the controller is PID, and switches between gains based on whether the control
command was successfully received (requiring acknowledgments). Song [98] and Yang
[114] extended PPC to the MIMO case with mixed packet losses, called network predictive
control.
Several works have considered the case where packet loss can occur in both the sensorcontroller and controller-actuator channels. Most studies assume a Kalman Filter decoder
for the sensor channel, and a zero-output decoder for the controller channel. Azimi-Sadjadi
[1] developed one of the first results for this system, establishing LMI stability criteria, and
Sinopoli [93] developed critical packet loss probability for convergence of the error covariance matrix. Schenato [90] showed that, if acknowledgments are used there is a critical
packet loss probability for stability and that the separation principle holds, but the separation principle does not hold in the case of no acknowledgments. Lu [62] and Yang [113] designed stable controllers that meet H. performance specifications for this system. Moayedi
studied two variations on this control scheme, where control commands are encoded in a
PPC manner [69], and when they are sent normally but with a hold-output decoder [70].
36
2.4
Control with Quantization and Packet Loss
Only in the past decade have studies considered the important real-world situation where
quantization and packet loss exist simultaneously in a feedback control system. Branicky
et al. [10] was one of the earliest to include both quantization and packet loss in a control system analysis, developing a scheduling co-design method for Networked Control
Systems (NCSs). Hadjicostis [42] treated quantization as additive noise on a signal and derived stability criteria for a simple SISO system with sensor channel packet loss. Tsumura
et al. [103] developed one of the first fundamental analyses for this problem, expressing
the minimum quantization level required for feedback stability for a given packet loss and
open-loop unstable eigenvalues. The study assumed a log quantizer with a realistic deadband at the origin, and required acknowledgments for the lossy sensor channel.
Niu et al. [78] later derived a set of linear matrix inequalities that guarantee feedback
stability for packet loss and infinite-precision logarithmic quantization in the sensor channel; this result was extended by Dai et al. [18] to include quantization and packet loss
also in the controller channel. Liu [61] added random delays to the model, and Yang [115]
considered different delays on different channels of a multi-channel system. In terms of
closed-loop performance, Rasool and Nguang [84] recently introduced a variation where
disturbance-to-penalty norm can be minimized, and Jiang [47] developed a set of LMI
constraints that guarantee system stability and H. performance criteria. Guo [38] studied
the case where the channel decoder is a general linear filter that propagates a model if the
sensor measurement is lost.
Some studies assume quantization errors are small and treat them as sector-bounded
additive noise. Huaicheng [45] used this treatment of quantization and developed LMI
stability criteria for systems with mixed delays and mixed loss probabilities in the sensor
feedback channel. Seiler [91] derived LMIs for synthesis of H. controllers for Markovian
packet loss, and Chiuso [14] and Dey [20] designed LQR gain matrices. Mahmoud [63]
developed stability conditions for the case of quantization and loss in both control and
sensor channels, with non-stationary Bernoulli packet loss probabilities.
Several works have looked at quantized and lossy controller-actuator channels. Na37
gahara [72] extended the work of Quevedo [83] to send a sparse representation of the
quantized signal, and Ishido [46] extended it to include a finite uniform quantization of
the signal.
2.5
Limit Cycles
No previous works have looked at control systems subject to packet loss and quantization
from the perspective of the effect on limit cycling. However, limit cycling is very important
for many robotic systems because it is a measure of achievable precision. Limit cycles will
not necessarily happen in all systems. For example, a system that is open-loop stable with
a very mild controller may not exhibit any limit cycles. Systems that are operated closer
to instability, though, may be prone to limit cycling in the presence of packet loss and
quantization, as will be described later in more detail.
There are three main methods from the literature used to find limit cycles: Bendixson's
Theorem, contraction theory, and describing functions. Bendixson's Theorem states that
if a region of state space has no trajectories leaving it, then trajectories must go to limiting sets, which are either equilibrium points or limit cycles [36]. The theorem has been
used to prove stability of some systems such as biological cellular concentrations [64] and
formation-keeping for mobile agents [5]. However, it does not give insights into the limit
cycle characteristics. Contraction theory is similar to Bendixson's Theorem in many ways,
and states that a system moves to a unique trajectory when initial conditions or disturbances are forgotten exponentially fast. Contraction has been used to study limit-cycling
of coupled nonlinear oscillators, e.g., [107].
A describing function (DF) is a quasi-linear approximation to a nonlinear element, and
depends explicitly on the form of its input; for sinusoidal signals in a limit cycle, the DF
depends on the input frequency or amplitude, or both [32]. DFs are very practical tools for
real-world control applications, and have been described as "an indispensable component
in the bag of tools of practicing control engineers" [95]. It is for this practical reason
that we choose to use DFs for limit cycle analysis of systems subject to packet loss and
quantization.
38
DFs have been used successfully in many applications. To name a few, in electrical engineering Sanders [88] studied limit cycles in periodically-switched circuits, and Peterchev
et al. [81] found quantization conditions that eliminate limit cycles in digitally-controlled
PWM converters. In medicine, Kinnane [56] used DFs to predict blood pressure oscillations in human patients for diagnosing cardiovascular health. In control, E. Kim et a]. [52]
developed a DF for a fuzzy logic controller, and Kim et al. [54] used DFs for identifying
frictional elements in support of feedforward control action. In ocean engineering, Yoerger
et al. [116] used DFs to explain and reduce limit cycles in underwater robots caused by
nonlinear thruster dynamics. More details about control of underwater thrusters is given in
Section 2.6.
More recently, DFs have been widely applied to stability analysis of airfoils (e.g., Tang
et al. [99]) and flame formation in combustion, e.g., Noiray et al.[79]. Gelb and Vander
Velde [32] provide a good resource, with many other DFs for a wide range of applications,
and Taylor [101] gives examples of calculating DFs for multiple-input-multiple-output systems. It is noteworthy that these prior works all address DFs for hardware and physical
elements, but not for information sent across channels, our current focus. Our work is also
unique in that the dynamic element whose describing function is to be developed is itself
stochastic.
2.6
Control of Underwater Thrusters
Because the contributions of this thesis have many applications in the control of underwater
vehicles, it is important to review previous work on ROV and AUV control. As pointed
out in Section 2.5, a key difficulty in controlling underwater vehicles is compensating for
nonlinear thruster behavior, which affects vehicle positioning. A variety of methods exist
in the literature to model and control underwater thrusters.
2.6.1
Thruster Modeling
In general in steady state the thrust from the thruster is proportional to the square of the
propeller rotational velocity. However, in the transient state this simplified model does not
39
hold as well. One approach to model the transient state thrust is to derive an equation
of motion by using an energy balance to equate axial power at the propeller disk to power
expended at the propeller shaft [116]. This leads to a nonlinear differential equation relating
propeller velocity to thrust. Another modeling method is to include the motor in the model,
with a voltage or current input [43]. A momentum balance can then be used to relate thrust
to propeller velocity. The input command can also be assumed to be propeller velocity
[110], which avoids taking into account motor voltages and currents. A more detailed
modeling approach is also possible, which takes into account variables such as propeller
blade angles, blade lift and drag forces, and fluid dynamics [43].
These modeling methods assume a model is derived in advance, but other methods
allow for the possibility of model parameters changing over time. Several studies ([2], [26])
developed techniques for adaptively identifying model parameter values in real time. These
methods, however, generally require measuring axial fluid velocity, which is difficult to do
in practice. One method approximates axial flow by a function of ambient flow velocity
and propeller shaft velocity, which are easily measurable, thus allowing parameters to be
identified in real time [53].
2.6.2
Thruster Control
The simplest, and a very common, framework to control underwater thrusters is open-loop
control with no feedback. This understandably may not meet performance requirements,
for instance in the case of strong disturbances, and perhaps the next step up in complexity
is feedback velocity control [111]. In this method the thrust is approximated to be linearly
proportional to the propeller velocity, which can be measured. Another approach is modelbased velocity control, where one of the many nonlinear relationships between thrust and
propeller velocity is assumed, and a measurement of axial flow velocity is also used. One
more approach is to use an extended Kalman filter to estimate motor torque, then estimate
thrust from this torque estimate and a hydrodynamic model [37].
Many different types of controllers can be applied to these general control frameworks.
For instance LQR [35], neural-network [55], proportional-integral [44], and proportional40
derivative [96] controllers have all been tested either in simulation or experiment on ROVs
or AUVs. More recently, fuzzy sliding mode controllers have been studied for underwater thruster control ([8],[9],[7]). In these controllers the compensator gain is variable and
determined by a fuzzy inference system. Such a design can be more robust to parametric
uncertainties and can deal with realistic motor deadband, but is prone to chatter and may
reduce system precision.
2.7
Synopsis of Literature Review
Most of the previous studies that consider both packet loss and quantization rely on infinitelength logarithmic quantizers with no lower bound on precision, and this makes their application to severely limited channels tenuous. Also, in these works all signals in a channel
are assumed lost or passed together, with no consideration of a mixed-loss scenario. The
channel decoders are generally assumed to be one of three specific algorithms: zero-output,
hold-output, or linear filter. It is unclear if stability or performance results hold if a different
decoder is used than the specific one assumed in each study.
Works that assume finite quantization levels generally assume quantization error is
small enough to be treated as additive noise. However, if quantization error is large, it is not
clear that the treatment as additive noise holds. Additive noise will not effect the Nyquist
plot of a system, but, as will be shown later, coarse quantization does effect the Nyquist
plot of a system, and thus is not well-approximated by the additive noise assumption.
Studies that assume coarse finite quantization generally use dynamic quantizers, but dynamic quantizers don't necessarily work with packet loss. A dynamic quantizer inherently
relies on information being exchanged between the quantizer and receiver any time the
quantizer changes. If the channel being used is subject to random packet loss, implementing a dynamic quantizer becomes more difficult. If the sending side changes the quantizer
parameters, but the message specifying the new parameters gets dropped en route to the
receiving side, the receiving side will not correctly decode future messages.
One way to ensure the message successfully reaches the receiving side is to use acknowledgments, i.e. to send a signal from the receiving side confirming that it has received
41
the new quantizer parameters. This strategy relies on the acknowledgment signal not being
dropped in the channel. Acknowledgments are commonly used in transmissions over the
internet using Transmission Control Protocol (TCP) [82]. This strategy works in transmissions over the internet because, in general, the acknowledgment message is much smaller
than a standard message and much less likely to be dropped by the channel. However, in
underwater acoustic communication this assumption that the acknowledgment will not be
dropped does not always hold. In general, smaller packets have a smaller probability of
being dropped, but in the underwater environment the possibility of an acknowledgment
signal being dropped cannot be ignored. Thus dynamic quantization is not always possible
in channels subject to packet loss.
With respect to underwater systems, many studies have focused on controlling underwater vehicles in the face of nonlinear thruster dynamics, and somehave looked at the effect
of limit cycles induced by these thruster nonlinearities. However, no works have considered
the effect of packet loss and quantization on limit cycling of underwater vehicles, though
these nonlinearities are potentially significant in many underwater feedback control applications such as those pointed out in Section 1.1.1. While thruster dynamics, packet loss,
and quantization could all potentially contribute to limit cycling at the same time, this thesis is concerned with the case where packet loss is high enough and/or quantization coarse
enough that the packet loss and quantization effects dominate the limit cycling.
This thesis thus advances previous work by considering systems with both significant
finite quantization and packet loss from the perspective of limit cycling, with applications
to underwater vehicle control.
2.8
Summary
In this section we reviewed prior work on control systems subject to quantization and
packet loss, as well as work related to limit cycles and underwater vehicle control. Most
previous works treat quantization errors as small, or assume infinite-precision logarithmic
quantizers. Work that considers both packet loss and quantization often assumes acknowledgments can be used. All of these assumptions may not hold in the case of severely
42
limited channels. Also, no previous work has studied control systems subject to packet loss
and quantization from the perspective of limit cycling. This review motivates the work of
this thesis, which considers feedback control subject to significant finite quantization and
packet loss from the perspective of limit cycling, with applications to underwater vehicle
control.
43
44
Chapter 3
Experiments with Packet-Loss Robust
Control
3.1
Introduction
As motivation for future work on control systems with packet loss and quantization, we first
investigated the practical performance of a packet-loss robust algorithm, the MEF strategy
proposed in Gupta et a]. [40] for a lossy link between the sensor and controller. We are
interested in this algorithm because it is the theoretical optimum LQG-type controller for
systems with packet loss in the sensor feedback channel. We implemented this algorithm
in the control of an autonomous kayak in the Charles River, Boston, MA, using artificial
packet loss in some tests, and real packet loss from acoustic modem communication in other
tests. This work in a difficult operating environment subject to wind and waves motivated
future experiments in a more controlled lab setting for systems subject to both packet loss
and quantization.
3.2
Hardware and Operating Environment
In these experiments, instead of sending the most recent measurements or the innovation
to the controller at each time step, we sent a specialized information vector, in accordance
with the MIF algorithm. We note that our implementation is a subset of Gupta's algorithm:
45
the entire message is either successfully transmitted or lost. The full MIF algorithm is given
in detail in Section 4.3.4.
Figure 3-1: Autonomous surface vehicle operating in the Charles River, Boston, MA.
We performed control experiments on a Wavesport Fuse 35 whitewater kayak, shown
in Figure 3-1, with a 220N thrust Minn Kota Riptide thruster under the bow of the boat.
The kayak is 1.8m long with a mass of 40kg including all onboard electronics and batteries.
During operation a WHOI Micro-Modem hung underneath at a depth of two meters. The
vehicle runs MOOS-IvP autonomy software [61 integrated with custom control algorithms.
Forward speed during experiments was 2.1 m/s, and the cycle time was six seconds.
We conducted experiments on the Charles River in Boston, MA, an environment characterized by sporadic acomms behavior, as shown in Figure 3-2. Packet losses can range
from 5% to over 70% depending on conditions and user settings.
The kayak was equipped with a GPS receiver to measure position and tilt-compensated
compass to measure heading angle. In designing Kalman filters, we used a sensor noise
covariance of 10m 2 for the crosstrack position based on GPS; raw compass measurements
were passed through a first-order low-pass filter having time constant 1.95s, and we modeled the noise on this signal with a variance of 10deg2 . Data supporting these settings are
given in Figure 3-3.
46
Figure 3-2: Experimental Micro-Modem performance data in the Charles River Basin,
which is not a power-limited environment but rather one limited by multipath. There is a
stone wall about 10m behind the source, a trench under the low-SNR circles in the lower
right, and a shoreline just out of the field of view on the lower right. The source is fixed
on a dock, the paths are made by a kayak towing a modem, and the blue cluster is from
a second kayak station-keeping. This SNR value indicates sound pressure level relative to
ambient noise.
3.3
System Model
We used the kayak's local PID heading controller for all our experiments, and modeled the
closed-loop behavior as a (stable) third-order, all-pole transfer function
CO
O(s)
4dd(s)
with 0 the heading and
Od
2
s 3 +c2 s +cs+
c0
(3.1)
the command. The angles are defined as deviations from a fixed
trackline heading. We arrived at the fit co = 1.7, cl = 2.5, and c2 = 7.5 in Equation 3.1,
through ten-, twenty-, and thirty-degree step responses, as shown in Figure 3-4. These
traces indicate a rise time of about four seconds, and 30% overshoot. This model holds
only for small inputs; the image also illustrates the effect of one-Hertz chop that is typical
in our light-air operations.
We used the simplest kinematic model for crosstrack error, assuming that the vehicle
was translating only in the direction it was pointing: e = V sin(O) ~ V4. The state vector
47
1000
500C
-8
-4
-6
0
E
6
4
2
0
-2
GPS Deviation [m]
8
:3200100-
0
A
d
-5
LL
0
Compass Deviation [deg]
5
Figure 3-3: Example noise histograms from kayak GPS and compass sensors, with the
vehicle stationary. The GPS is the uBlox NEO-6 sampled at five Hertz; the variance is
13.3m2 . The compass is the OceanServer OS5000 sampled at ten Hertz, and then filtered
by MOOS at a 1.95Hz cutoff; the variance is 9.6deg2
for the system model is taken as x =
[0, 4, e]T, and we
convert the system to discrete
time with a zero-order hold to obtain the state-space matrices [A, B, C, 0]. The loop time At
is set based on the delay of one acoustic link to transmit the information vector for MIF.
We assume constant time step and delay, valid for acoustic ranges less than one kilometer
where message coding and decoding times dominate.
3.4
MIF Experimental Setup
For the MIF-LQG experiments we set the LQR state penalty matrix to be diagonal with
2
Ql/ 2 = 31 and all other diagonal entries Ql/ = 10, so that error in crosstrack position is
most highly penalized. We set the control penalty as R1/2 = 316. We found through tuning
that these parameter values led to good performance of the baseline system in the absence
of packet loss.
3.5
Physical Connectivity
We conducted two different sets of experiments: one set relied completely on Wi-Fi communication with artificially-imposed packet size and losses, and the other on communica48
. ..........
..........
45
Heading Data
Model Fit
%
40 --30
. 20-
1050
0
2
4
Time [s]
6
8
10
Figure 3-4: Kayak closed-loop heading model fit on experimental data in waves. Winds
and waves were very light for these trials, with wind less than 5 knots.
2
MIF-LQG &Acomms
RMS 1.2
-60
0
10
20
30
40
Time Steps
50
60
Figure 3-5: MIF-LQG control tests with acoustic communications on a day with light
wind, 27 November 2012. Green vertical bars indicate packet successes. Each time step
represents six seconds.
tion through acoustic modems. At every time step the kayak transmitted GPS and compass
measurements to shore over Wi-Fi, with essentially no delay. The shore randomly decided
whether the packet would be lost, based on a pre-set probability. The shore then simulated
a delay, and transmitted the control packet over Wi-Fi to the kayak.
In the acoustic modem set of experiments, sensor signals were transmitted from kayak
to shore using an acoustic modem, and control command packets sent from shore to kayak
over Wi-Fi.
49
Constant
RMS 2.13
Set Heading-'-
2
1-iorT---0
-----
- --
-- ------------------
No Loss
0.41
E
_--__l 30% _Loss _- _--
0
--
--
70% Loss
10
_0A41_
0.38
20
40
30
Time Steps
50
60
Figure 3-6: MIF-LQG control tests with varying packet loss on a day with nearly no wind,
24 October 2012. Several apparent flyers were removed from this plot (gaps).
3.6
Experimental Results
Crosstrack errors over time are shown in Figure 3-5 for trials with acoustic-modems on
27 November 2012. Errors are shown in Figure 3-6 for Wi-Fi-only MIF-LQG trials on 24
October 2012 for calm wind conditions, and in Figure 3-7 on 16 October 2012 for strong
wind conditions. Table 3.1 summarizes the results.
Results from 27 November act mainly as a demonstration that we can control the kayak
with sensor signals via the actual acoustic modem. In this trial packet loss rate was about
5% and wind conditions were zero to five knots. 24 October had close to zero wind, resulting in very low crosstrack errors; RMS levels were essentially the same for trials with no
loss, 30% packet loss, and 70% packet loss. These errors were also significantly lower than
that of a run with no crosstrack feedback (labeled Constant Set Heading), as expected.
16 October had significant chop with sustained winds of fifteen knots, gusting to thirty.
We see a bias due to the disturbances, and a reduction in control fidelity with increased
packet loss. Most dramatically, in the 70%-oss case the vehicle cannot maintain the track-
line at all. Underlying this failure is the fact that the boat is operating well outside of the
linear model; heading commands from the LQR exceed sixty degrees late in the run. At
this signal level, progress toward the trackline is sub statianty
50
e-eimated, so that cor-
30 No Packet Loss
__-_-_-_-_-_-_ MS .6
01
-30%
Loss
10.8
50% Loss
141
70% Loss
2.
I-
liii
I
10
20
30
40
Time Steps
50
60
Figure 3-7: MIF-LQG control tests with varying packet loss on a day with strong wind, 16
October 2012.
Packet Loss Probability
RMS Error (light wind)
RMS Error (strong wind)
0%
30%
0.43
0.41
7.6
10.8
50%
70%
-_14.1
0.38
28.3
Table 3.1: Comparison of RMS errors from experiments with light wind and with strong
wind.
rective actions are too small. Meanwhile, the error signal to the controller is erroneously
well-behaved in the absence of packets containing crosstrack error information.
These results highlight the effect of packet loss on feedback control systems, and motivate more detailed studies in a controlled lab environment on systems with both packet
loss and quantization.
3.7
Summary
This chapter detailed the results from an experimental test of an example packet-loss robust algorithm, the modified information filter. Heading control tests were conducted on
a robotic kayak in open water, with the sensor measurements transmitted to the kayak using a shore-based acoustic modem and subject to realistic packet loss. These tests were
51
similar to true USBL-based control of an AUV, and serve as motivation for more detailed
experiments in a controlled lab setting.
52
Chapter 4
Describing Function Derivations
4.1
Introduction
In this chapter we derive and verify by simulation describing functions for information
channels subject to packet loss and quantization. We start by deriving a DF for a general
codec with multiple sensor inputs, and then apply the DF to four specific codec cases: Zero
Output, Hold Output, Linear Filter, and Modified Information Filter. The first three are the
most commonly-used codecs in the literature, and the last one is the theoretically optimum
codec for an LQG-type system subject to packet loss. Monte Carlo simulations verify gains
and phases computed by the DFs for all four codecs, for a variety of example systems. DFs
for several types of quantizers are shown, and we prove that these DFs can be combined
with the packet-loss DFs to describe cases with both packet loss and quantization. For
clarity, Table 4.1 lists the mathematical symbols used in this chapter and their definitions.
4.2
Describing Function for General Multiple Input Multiple Output Information Channels Subject to Packet
Loss
We first derive a describing function for a multi-sensor binary-erasure information channel
for a Multiple Input Multiple Output (MIMO) system, as shown in Figure 4-1. We examine
53
Symbol
Sk
P(k)
Q(k)
C
n
9
0
k
A
coT
I
Kf
y
h
G
Y
g
A
'Tk
a
<D
Oi
a,
Dk
J
K1 ...K5
z
Zk
Hk
Fk
Mk
Definition
at time step k
output
Decoded
Decoder function
Decoder function
Controller
Number of states of system
Scalar binary packet loss variable
Matrix of binary packet loss variables
Time step variable
Frequency
Time step size
Decoder DF gain
Decoder DF phase
Identity matrix
Linear filter gain matrix
Measurement at time step k
Quantization bin size
System model matrix
System model matrix
Time horizon for DF
MIF encoded message at time step k
MIF control signal variable
Packet loss probability
Expected value of 0
Input phase lag of channel i
Input amplitude of channel i
DF intermediate variable
DF derivation intermediate variable
Intermediate variables
DF solution intermediate variable
Zero output decoded output at time k
Hold output decoded output at time k
Linear Filter output decoded output at time k
MIF output decoded output at time k
Table 4.1: Summary of mathematical symbols used in this chapter.
54
the case where this channel is in the feedback path of a control system, though the same
describing function will, in theory, apply to a lossy controller channel as well. In this
system multiple sensor signals are encoded and sent through a mixed-loss binary-erasure
channel, where different signals may be lost with different loss probabilities. The decoder
updates a state estimate based on received signals and past state estimates, and sends this
new state estimate to the controller. A quantizer block will be added to the system in a later
section.
Controller
Plant
C~z)
Decoder
P1(z)
Encoder
Binary Erasure Channel
Describing Function
Figure 4-1: Block diagram for a MIMO feedback control system with an encoder, binary
erasure channel, decoder, controller, and plant. The describing function represents the
encoder-channel-decoder blocks.
We consider two general cases of the encoder-channel-decoder sequence. In the first,
most general case, Sk, the decoded signal at time step k, can depend on all previous measurements, all previous decoded signals, and packet success histories. In the second, less
general case, the state estimate takes a form given by
Sk = P(k) + Q(k)Skl
(4.1)
where P(k) and Q(k) define how the signals are encoded, lost in the channel, and decoded.
In this case, P(k) and Q(k) can depend on the current measurement, a system model, and
binary variables describing whether sensor signals were successfully transmitted through
the channel or were lost. The only direct dependence on previous state estimates or measurements is through the Q(k)Sk-l term. To derive the DFs for these two cases, we assume
the system has n sensors and n states. If there are fewer sensors than states, the decoder
55
can treat the missing sensor signals as being lost in the binary-erasure channel. Thus, the
assumption of n sensors and n states will still hold.
To understand the meaning of these representations it is helpful to look at how Sk, P(k)
and Q(k) are defined for several standard state estimation algorithms for systems subject to
packet loss. We will consider the Zero Output (LZ), Hold Output (LH) and Linear Filter
(LF) as example estimators with the structure defined in Equation (4.1), and the Modified
Information Filter (LM) as an example estimator with the more general structure. We
consider these algorithms because LZ, LH, and LF are the most commonly-used algorithms
in the literature for packet-based control with loss, and LM is the theoretical optimal LQGtype control for packet-based systems with lossy sensor feedback. We will briefly describe
these algorithms here, and analyze them in more detail in later sections.
In LZ a received signal is passed directly and a lost signal is decoded as zero. In
LH a received signal is also passed directly, but a lost signal is decoded as the previous
estimate of that signal. The LF algorithm propagates the previous state estimate through a
system model if a signal is lost, and computes a weighted sum of the measurement and the
propagated model if the signal is received. The LM algorithm encodes the current and past
sensor measurements into a specialized information vector. The decoder uses this vector if
the signal is received, and propagates the previous state estimate through a system model if
the signal is lost.
The LZ decoder is typically used in regulation applications where states are regulated
to zero and where computation power is limited. It is also common in applications with
packet loss in the controller-actuator channel. The LH decoder could be used in the same
applications as LZ, though because it is slightly more sophisticated it can also be used in
non-regulation applications such as following reference signals. LF and LM are appropriate
for applications where more computation power is available, with packet loss in the sensor
feedback channel. The main difference in application is the LM requires the sensor side to
be able to encode the measurement, while LF just requires a decoder.
Table (4.2) defines the first three state estimation algorithms in terms of P(k) and Q(k)
from Equation (4.1).
56
Decoder
P(k)
Q(k)
LZ
Ekyk
0
LH
Ek&
(I - 9 k)
LF
EkKfyk
(-KfekC+I)G
The LM estimator, with state estimate Mk at timestep k is defined as
Mk = Ok [Y-(gk+ Wk)] + (I-k)GMk_1.
The vectors
A
(4.2)
and Wk depend on past measurements and state estimates and are defined
in a Section 4.3.4. The variable yk is the nx1 vector of sensor measurements at time step
k and G is defined as G = A + BK for system model given by matrices A, B, and C, and
controller gain matrix K. For LF and LM we assume constant error covariance matrices,
which results in a constant linear filter gain matrix Kf and constant inverse error covariance
matrix Y. I is the nxn identity matrix. The binary erasure channel is modeled with the
diagonal matrix
Ok
defined as
Ok=
[91,k
0
0
-.
0
0
0
J
(4.3)
On,k
where the Bernoulli variable ei,k indicates whether the packet from sensor i was successfully received at time step k, defined as:
i,k =
1,
0,
- a(
if transmission success, probability 1
if transmission failure, probability a;
In this packet loss model sensor signal losses can be either coupled or independent. For
independent losses, each Bernoulli variable 0 i,k is unique. For coupled losses, Bernoulli
variables are shared between the coupled channels. For instance, if channels 1 and 2 are
always lost or passed together, then 0 1,k = 6 2,k, such that
57
0
0
0
0
1,k
0
0
0
0
0
63,k
0
0
0
0
0*'.
0
0
0
elk
o
Ok=
6
(4.5)
0
0
6
n,k
For use in future equations we will define the expected value of the binary erasure
channel matrix as
E[Ok] =
(D, where 4) is the constant matrix
I-ai
1-a]
0
0
[= 0
(4.6)
0
0
0
1-an
Computing 4D depends on the type of packet loss in the model. For this work we assume
Bernoulli loss, but if another loss type were assumed, such as Markov, the effect would be
accounted for in computing 4).
Note that in the case of channels with coupled loss, the expected matrix 4D is the same
as if the channels had independent losses with the same probability of loss. For example,
4) is the same if 01,k and 8 2,k are independent both with loss probability a,, or if both
are coupled with loss probability a1 . Because the describing functions depend on 4) and
not on
0
k,
this means independent or coupled losses both result in the same describing
function. This outcome is verified by simulation in section 4.4.5, where it is shown that
describing function gains and phases for LZ and LH are the same for the case of coupled
or independent loss for a 2-sensor example case.
In general, because of random packet losses, the mapping from sensor measurement to
state estimate is highly nonlinear. However, we can approximate the mapping by a quasilinear describing function. This DF, as will be shown in later sections, can then be used as
an analysis tool to study system limit cycling, and as a synthesis tool to design the system
decoder.
To derive the DF for state estimate Sk, we assume sensor input signals of the form
[al sin(w(kr + 0 1)...a, sin(o(kr +
n))]',
for n input signals of amplitude ai (i = 1..n), all
58
of the same input frequency o, sampling timestep 'r, time index k, and potentially different
phase shifts
4i.
We use the prime notation [-]' to denote the transpose operation.
The DF that best approximates the decoded output signals is the one with the smallest
squared-error between expected output and DF output. This is written as the optimization
objective
min J , where J
A,,Ts
E
(Dk(As,
s)
- Sk)'(Dk (AI T) - Sk)
(4.7)
-k=1
Here g is the number of time steps the minimization is taken over, and the DF is defined as
the nxl vector
Dk(As, Ts) = [As,I sin((o(kr - Ts,1))...As,, sin(co(kr - Ts,,))]'
(4.8)
with DF amplitudes given by As = [As, ... As,n] and phase shifts Ts = [Ts, ... Ts,]. Clearly g
has to cover many cycles for good accuracy in D&, but we will show that practical results
are still easy to obtain.
The objective expands as
gn
As,i sin2 (o(kT - Ts,i))
E
-
J=
k=1 i=1
+
2 1E[SkD(AsTs)1
k=1
M2
E [SSk] .
(4.9)
k=1
M3
Term m3 is not a function of either Ts or As, so it has no bearing on the DF solution and we
drop it. Term ml can be simplified via trigonometric identities to
59
i=1
cos(2koarr)
A
g
sin(2(oTs,i)
\
i=1
/
k=1
/
-
- cos(2o)Ts,i)
Ai
=
sin(2kt)J.
k=1
(4.10)
/
2mi
(
n
If we multiply and divide each inner summation by 2 sin(cor) and use trigonometric multiplication identities, then
2ml 2E
n
2i (g- cos(2wTsi)(sin(w(2g +1)r) - sin(or))
g
2sin (or)
As
n 2
EA i
(4.11)
icr
sin(2oTsi) (cos(or) - cos(w(2g + 1),r))
.
Expanding Dk(As, Ts) results in the new expression for m2:
m2
=
(E[Sk])'sin(okr) [As,, cos(o.Ts,1) ...As,n cos(wTs,,)]'
- L (E[Sk])'cos(k'r) [As,, sin(coTs,1) ... As,n sin(CTs,n)]'k=1
In the case where the estimate can be represented by Equation (4.1) we observe that, if
we initialize So = 0, the term Sk can be written compactly as
(PQ
k-1 /k
P(k) +
Q()) P(l)
(4.12)
and thus the expected value of Sk is simply
E[Sk]=
E[P(k)]+
(
E[Q(j)]
E[P(l)]).
(4.13)
The expansion of ml depends on cos(2oTi) and sin(2Ti), and thus to simplify the
objective function J for future computations we have similarly rewritten m2 in terms of
cos(coTi) and sin(coTi) using trigonometric angle summation identities. We also write the
objective function in terms of sine and cosine of the coTi and 2coTi terms, which allows later
60
simplifications. The objective works out to be
=
A2 [Ki +K 2 cos(2)T,i )+K 3 sin(2oT,i)]
-
J
2
As,i [K4,i cos((oTs,i) + K5 ,i sin(COT,i)]
(4.14)
where K1 , ...,K 5 ,i are given by
K1 = g/2
(4.15)
K2 =
sin(o.(2g + 1)r) - sin(cor)
4sin(or)
(4.16)
cos(Coz) - cos(w(2g + 1),)
4 sin(oT)
(4.17)
K3
-
K4sJ =
K5sJ =
E[Sk] sin(kaor) (i)
-
(4.18)
E [Ski] cos(kowr) (i).
(4.19)
In the notation for K4s,i and K5s,i, [-] (i) indicates the ith element of the vector [-]. The
s in the subscript denotes the S estimator used. In later DF derivations we will use the
subscript z for the LZ estimator, h for LH, f for LF, and m for LM. To identify solutions for
the optimization problem, we set derivatives dJ/dA,,i and dJ/dTs,i to zero. This results,
respectively, in the two surprisingly clean sets of constraints
)
s~ = K4s,icos(o)T,i) +K5s,isin(coTs7,)
K1 + K2 cos(2o3T,i) + K3 sin(2coT,1
Asi
=
K5sicos(oTs,i) - K4s,i sin(wTs,)
K3 cos(2(oTs,i) -K 2 sin(2wTs,i)
(4.21)
We can equate the expressions and then use trigonometric double angle identities to
rewrite and cancel terms for Ts,i:
61
1
(4.22)
Ts= - (arctan(zs,i) + 0i)
=
(K4s,iK3 - K2K5sj - K1K5sj)
(K3 K5 si + K2 K4s,i - KIK4 s,i)
(K5s,i - K4s,izs,i) 1+
-K 3z,.- 2K2zs,i+K3
,
where zji
'
(4.23)
Because this is the unique solution to an unconstrained problem, we conclude that global
optimality is achieved.
4.3
Describing Functions for Specific State Estimation Algorithms
To illustrate the application of the describing function method, we next use Equations (4.22)
and (4.23) to derive the DFs for the LZ, LH, LF, and LM state estimators. For simplicity we
will assume all signals have zero phase shift, i.e.
4j =
0. To visualize these DFs we show
examples of single input single output (SISO) cases of each algorithm. Block diagrams of
the algorithms are shown in Figure 4-2, and describing functions in Figure 4-3.
These examples are analyzed in more detail in Section 4.4. Note from Figure 4-2 that
the output of LZ and LH are the measurement estimate, while that of LH and LM is the control signal. This is done to reduce the describing functions to SISO cases. MIMO versions
would have the outputs of LF and LM as the state estimates, as will be seen later. Also note
that only LM has an encoder, while all algorithms contain decoders. For reference, Figure
4-4 shows where the quantizers will fit into the block diagram for the SISO versions QLF,
QLM, QLH, and QLZ. In the cases of QLF, QLH, and QLZ the measurement is quantized,
and for QLM the encoded measurement in quantized.
62
Control
Command
Control
Command
Plant
Cntrle
..
L
nMeasurementn
o
oOuertut
j
Binary Erasure
Channel
t---------------------
a
menMeasurement
SL H
D escribing
:Function
ou ut
D escribing
F unction
Binary Erasure
Channel
J
r ------
Control
Controller
Plant
-
Cntrole
i
Control
Command
Controller
Plant
Measurement
I
EstimateI
Plant
Measurement
ae
Estimate
- --
-
-
---
Measurement
I iter
r
Binary Erasure
Channel
Deco der
escribing
F unction
I
Binary Erasure
Channel
Mo
LMV
Describing
Function
4- - - -
w-w
----w- w
1
Figure 4-2: Block diagrams of SISO versions of the four codec algorithms. Signals are
labeled in red. Note that the output of LZ and LH are the estimate of the measurement,
while that of LH and LM is the control signal. Also, only LM contains an encoder.
4.3.1
Loss with Zero-Output Estimator (LZ)
The zero-output estimator (LZ) passes the input signal if the packet transmission was successful, and generates zero if the packet was unsuccessful. For this algorithm, P(k) =
Ok[al...a,] sin(okr) and Q(k) = 0, as defined in Table 4.2. Thus, the state estimator is
written as
Zk =
a,] sin(cokT)
Ok [al ...
(4.24)
with the LZ state estimate at time step k given by Zk and
E [Zk] = E [Ok ] [a . . a] sin(wkZ).
Substituting this into Equations (4.18) and (4.19) yields
63
(4.25)
1
1
0.8-
-
0.6-
-
Sent Signal
Decoded Signal
DF Approximation
0.6-
0.4~0
0.2
0-
0
-0.2-
-0.2-
-
-0.4-
k
-0.4-
-0.6-
-0.6
-0.8-
-0.8
-1
0
50
100
Time Steps
0
Signal
Decoded Signal
---_DF Approximation
100
Time Steps
-
0.4
0.4
0.2
0.2
2
0t
---Sent Signal
---
-
0.6
150
~--Decoded Signal
-
0.8
-
0.6
DF Approximation
0
0
2-0.2
-0.2
-0.4-
-0.4
-0.6-
-0.6
-0.8-
-0.8
-1
50
-ent
--
0.8
2
-1
200t
150
1
E
Received Signal
DF Approximation
0.4-
0.2
a
Sent Signal
-
0.8-
0
50
100
Time Steps
200
150
50
0
100
Time Steps
150
200
Figure 4-3: Comparison of LZ (top left), LH (top right), LF (bottom left), and LM (bottom
right) blocks showing input and decoded signals. For the LZ DF, the system has 50%
packet loss. The optimal describing function parameters for LZ are Az = 0.5 and T = 0.
For LH, the system has 90% packet loss, and the fit has gain Ah = 0.9 and delay Th = 8 time
steps. For LM and LF, the systems have 70% packet loss. The optimal describing function
parameters for LF are gain Af = 0.154 and T = 25.85 time steps. For LM, the fit has gain
Am = 0.148 and Tm = 27.82 time steps.
[<D[ai...an](KI+K2) ] (i)
K4zi
=
K5z'i
= [CD[al...
an]K31](i).
(4.26)
(4.27)
These values lead to cancellations in Equation (4.2), which results in Zz,j = 0. Substituting Zz,j, K4z,i and Kszj into Equations (4.23) and (4.22) yields the describing function
parameters
Az,i = 4D[ai...an] and
64
T
(4.28)
Control
Control
Command
Command
Cntrle
Controller
Plant
Estimated
Estimated
Measurement
MMeasurement
Plant
Measurement
Measurementt uQuantized
Measurement
Quantized
Measurement
Zero
L C utput I
Binary Erasure
Channel
Quantizer
Hold
rQLZi
uni~]IDescribing
Function
CQuantizer
output
C LH
D escribing
F unction
I
Binary Erasure
Channel
I--------------------------
Control
Control
Command
Controller
ComPman
Plant
Pln
IController
Measurement
Measurement
State
. --
Estimate
%_
.
- - - -Quantized
Measurement
w.- -
-Li
uaQLF
n'ea
7
Quantizer
nterr
z
Binary Erasure
Channel
1Describing
State
-
Quantized
Encoded
Measurement
Estimate
Decoder
I Function
Binary Erasure
Channel
MIF
Encoder I
I
I
Encoded
Measurement
Quantizer I
I Describing
i Function
Figure 4-4: Block diagrams of SISO versions of the four codec algorithms with quantizers
added. Signals are labeled in red. Note that the output of QLZ and QLH are the estimate
of the measurement, while that of QLH and QLM is the control signal. Also, only QLM
contains an encoder.
The DF gain is thus
' =
ai
1-
ai.
(4.29)
These gain and phase shift parameters intuitively make sense, because the estimator Zk
for the LZ algorithm is already in the form of a gain multiplying a phase-shifted sinusoid.
4.3.2
Loss with Hold-Output Estimator (LH)
We next consider the hold-output estimation algorithm, LH, again assuming an input signal
of the form [aI...an]sin(cok-r).
As before, we model the binary erasure channel with a Bernoulli variable matrix Ek
indicating whether each signal was successfully received at time step k, and the hold-output
decoder with variable Hk. Hk is given recursively by
65
Hk = O[al ... a,] sin(okT) + (I - Ek)Hk_.
(4.30)
(3
In the context of Equation (4.1), P(k) = Ok[al...a,] sin(okr) and Q(k) = (I - Ok).
Taking expectations, we see
E[P(k)]
= 4[ai...aj]sin(okr)
(4.31)
E[Q(k)]
=
(4.32)
(I -<D).
If we substitute these expected values into Equation (4.13), we find the expected value
of Hk is
k
E[Hk]
=
(I_ 4)k
([al... a,] sin(wlr).
(4.33)
This expected value of Hk can now be used to solve for K4h,j and K5h,j. Using Equations
(4.18) and (4.19) we find
g
K4hJ
k
]=1
(I - D)klD[a ...a,} sin(olr) sin(o(k),r) (i)
=
(4.34)
k=
g
K5hi
k
1: E - (Dk-I(D[al...
an]sin(co1,)cos((D(k),r) (i).
(4.35)
-k=1 -=1
Now the DF parameters are given by
where Zh,i
=
1
= -
(4.36)
arctan(zh,1
)
Th,,
(K4hiK3 - K2K5hi - K1K5hi)
(K3K5h,i + K2K4h,i - K1K4h,i)
(K5h,i - K4h,izh,i)
Ah
+h
-K3zh,i - 2K2zh,i
+ K3
66
(4.37)
4.3.3
Loss with LQG-Type Linear Filter Estimator (LF)
We will next derive a DF for a LQG-type linear filter estimator. We refer to this case as LF.
For this we follow the usual LQG-type construction. The prior estimate propagates with
Xkjkl1 = Aik-1k_1 +BUk-1,
where A and B are constant system matrices, and Ukl1 is the in-
put. If the signal is received, we update the posterior as skik = xkIk_1 + Kf (yk
-
CIk_1) =
(I - KfC)^Ik-_1 + Kfyk where Kf is the filter gain matrix, Y is the measurement, and C
is a constant output gain matrix. If the signal is not received, we perform no update:
Xkjk = Xklk_1.
Finally, the control command is
Uk =
KU Xkjk, where Ku is a constant gain
matrix.
Although Kf can be optimally updated at each time step, depending on whether or not
the packet is received, many previous works have pointed out that real-time applications
generally use constant gains, e.g., [94] and [76]. Some reasons to use constant gains include less computation time required and simplicity of implementation in networks where
complexity issues are important. Thus we have fixed Kf in our analysis here. Clearly a constant Kf could be designed if some properties of the packet loss were known, as outlined in
[89], but this too is outside our present scope.
Define Fk
-k Xkj
and G=A + BKu. Then we can easily derive
Fk = Ok [(I - KfC)GFk_1 + Kfyk] + (I - Ok)GFk_1.
(4.38)
This equation can be rewritten as
Fk = OkKfYk + (-KfC+I)GFk-1=
OkKfyk and Q(k) = (-KfekC
I)G. Thus, assuming a sinusoidal input as before and taking expectations we find
E[P(k)]
= cIKf[al...a,] sin(kcor)
E[Q(k)]
= (-Kf4IC+I)G.
67
+
In accordance with Equation (4.1), this gives P(k)
(4.39)
We next use these expected values to find E[Fk] from Equation (4.13). This yields
kkI
((-Kf4C+I)G)-
E [F] =
(4.40)
Kf [al...an] sin(cofl).
l= I
This expected value of Fk can now be used to solve for K4f,; and K 5f,j. Using Equations
(4.18) and (4.19) we find
K
K4f,i
g
k
Kf [a .. .an] sin(olr)sin(.)(k)T ) (i)
((-KgIC + I)G)k ~
=
-k=l1=1
K5f,
=
((-KfDC+1)G)k-lDKf[a...an]sin(clr)cos(o(k)) (i).
-
-k=l1 =1
Now the DF parameters are given by
Tf,i =
-
(4.41)
arctan(zf,;)
(K4fiK3 - K2K5fj - K1K5fj)
where zfi = (K3 K
'
K35f,i +K2K4fi - K, K4f, i)
(K5f,i - K4fizfj)
1+Zfj
-K3zf, -2K2zf,i+K3
4.3.4
(4.42)
Loss with Modified Information Filter (LM)
The MIF-LQG strategy was proposed in Gupta et a]. [40] for a lossy link between the
sensor and controller. In this strategy, instead of sending the most recent measurements
or the innovation to the controller at each time step, a specialized information vector is
sent. We note that the implementation considered here is a subset of Gupta's algorithm: the
entire sensor measurement is either successfully transmitted or lost.
Let the sensor noise covariance be R, and the process noise covariance be
Qp.
The
measurement at time k is yk, and the control Uk. The control gain matrix is K, designed in
this case as an LQR. At each time step we implement the following:
68
Sensor Side: Initializing suitably, make the following calculations:
C
Ykjk
=
Ykjk-1+CTRS
ykc
=
Yklk-1AYlk-
k
=
CT Rsl Yk
gk
=
Ak + Yklk-1, and send it
Yk+1Ik
=
QP1- Q
_
A(ATQ-A
+Ykjk)-1ATQP.
Controller Side:
1. Update Y and compute k as in the first two steps on the sensor side. From a zero
initial value, update ' by
Tk= Yk IlBUkl + 'k~k-1.
2. If the message is successfully received, update the state estimate as
k 1
Xk
(k+--k).
Else, update it as
Xk =AUk-1 +BUk_1.
3. Compute the new control Uk = -Ku
k.
4. Evolve Y as in the last step on the sensor side.
If we assume the error covariance matrix is constant, then this implies Ykjk, Ykjk-1, and y
are constant, with YkIk = Y1, Ykk-1 = Y2 , and y= Y2AY 1. This also yields the simplification
Tk = Y2BUk_1 -+ yTk_1. Then, defining Mk as the state estimate at time k, the control
algorithm can be written as
Mk = Ok
[Y1' (gk + Wk)] + (I 69
Ok)GMk-1
(4.43)
with the Bernoulli variable matrix Ok defined as before. In order to find the expected value
of Mk for use in Equation (4.13) we must first derive closed-form representations of the
vectors g and Wk. For sinusoidal inputs, the specialized information vector g is defined as
g = CT R- [ai ...an] sin(wmkr) + Ygk-1
(4.44)
and can be rewritten as
sin(olt) CTR-1
yk
gk= [a...any
(4.45)
7
=0
To simplify this expression, we observe that the sinusoidal term can be rewritten as the
imaginary part of a complex exponential function. This gives
k sin(oft)
_
7
[k (eimi'
1
i.
Imi[
=0o
(4.46)
.
1=0
1
Now, we use finite geometric series relations to expand the summation as
k sin(rIT)
[I - (1
e
)k+1
Im['(4.47)
1=0
(4.47)
We next rewrite the exponential terms in trigonometric form, yielding
k sin(ot) =
_
IM [I
y-
yk 1 (cos((k+l)w1r) + isin((k+ )co))
I - y-(cos(wor)+ isin(or))
(4.48)
We now multiply numerator and denominator by the complex conjugate of the denominator, and take the imaginary part of the result. Substituting this result back into Equation
(4.45) gives
I
k=
k
cos((k+ )c
Yk+1I
)
sin((r)
cosk+
_ Cos(ST)
Ik+)
sin((ii
s
70
C( R) ' R~[ai...an,]. (4.49)
The variable 'k
is defined as
Tk = Y2BUk- 1
+
(4.50)
7'k-
and can be rewritten as
k-i
T
Y= yk-1
E
1=0
y-'Y 2BKuM.
(4.51)
Using this expression for Tk and Equation (4.43), we next write out the first ten terms
each one is a function of MO and g for k = 0...9, with no dependencies on M1 ,M 2 ,.
-
MO, ... ,M9 . We reapply Equations (4.43) and (4.51) repeatedly to each of these terms until
With this set of terms it is possible to identify patterns in the coefficients of Mo and g.
The pattern identification in this step is the intellectual crux of deriving the MIF describing
function, and is non-trivial. We find the closed form expression for Mk is given by
k
(4.52)
Mk = MOVk + I (Vk-lOkYC'gl)
1=0
where
k-2
Vk =
m=1
[ (OkYC'Y 2BK
+ (I- Ok)G) mNk,mj + (EkY
1
Y 2BKu + (I - Ok)G)k
(4.53)
with VO = 1, VI = (Okyj-1-Y2BKu + (I - Ok))O), and
Nk,m =
E
(OkYY2BKu)'yi
i+j=k-m,j;>i
j
i
(
l
.
(4.54)
*
r-i
We can simplify the expression for Mk by initializing MO = 0. We next take expectations, and see that the expected value of Mk is given by
71
k k-l-2
+ (I -<D)G)m
E[Mk] = EY~Y2B
1=0 m=0
1
(4.55)
x
-i) +(cYC Y2BKu+ (I- c)G)k 4Y
(Y)-(Y2BK)-
1
1 gl.
1
This expected value of Mk can now be used to solve for K4m,i and K5m,i. Using Equations
(4.18) and (4.19) we find
K4m,i
E[Mk] sin(o(k)v) (i)
=
-k=l
K5m,i
=-
E[Mk] cos (co(k),r) (i).
-k=1
Now the DF parameters are given by
1
(4.56)
Tm,i = - arctan(zm,j)
K4miK3 - K2K5m,i - K1K5m,i
K3K5m,i + K2K4m,i - KiK4m,i
(K5mi - K4m,izm,i)
Am, i
4.4
4.4.1
.- K3zm,i - 2K2zm,i
1+ zm
+K3
.2
(4.57)
Model Verification
Convergence
The DFs will be most accurate if at least one period of the input sine is sampled, and if at
least four samples are taken each period. This leads to a target range for nondimensional
frequency
< WIC < -.
-
2
g
72
(4.58)
Figure 4-5 shows the derived values of Th for a SISO case for g between 10 and 1000,
at frequency coz = 0.1 and for a range of packet loss probabilities. Similar convergence
is seen in plots of Ah, for different input frequencies, and for the LF model.
Because
of this, we assume g=1000for all LH and LF DF computations in the rest of the paper
Computation cost scales strongly with g, and there is no reason to choose a value higher
than necessary.
0.7
0.60.50.4-
---0.6
-0.8
E 0.3-
=0.4
0.2
0.1
-0.1-
-0.2
1
1.5
2
Log1 0 g)
2.5
3
Figure 4-5: Convergence of the LH describing function phase formula with g, for different
values of a, and on = 0.1.
Figure 4-6 shows the derived values of Tm for the SISO contact force example problem
described later, for g between 6 and 80, at frequency oro = 0.1 and for a range of packet
loss probabilities. Similar convergence is seen for Am. Because of this, we assume g = 80
for all LM computations for the rest of the paper. Computation cost for LM scales even
more strongly with g than in the LH and LF cases, because the LM DF contains several
additional nested summations.
4.4.2
LH Verification
We will now verify by simulation the LH, LF, and LM describing functions over the valid
frequency range, using the values of g previously determined for convergence. We need
not verify LZ by simulation, because of its simplicity. The model and simulation results
for LH are given in Figure 4-7 for a representative range of packet losses (40%, 60%, and
80%), and we see good agreement for gains and phases.
73
-0.8
-0.9
-0.6
-0.8
-1.2 -
-1.4
9-1.6
0.9
1
1.1
1.3
1.2
Log (g)
1.4
1.5
1.6
Figure 4-6: Convergence of the LM describing function phase formula with g, for different
values of a, and ar = 0.1.
Note that LH is a universal descriptor, and does not depend on a system model. Also,
the LH model can apply independently to each channel of a multi-sensor system, and thus
model verification for one channel, as shown here, is sufficient verification for the multichannel case as well.
4.4.3
LF Verification
To verify the LF and LM describing functions, we study several different SISO and MIMO
examples that are open-loop stable, open-loop unstable, minimum phase, or nonminimum
phase, with different numbers of inputs and outputs. Unlike LH and LZ, LF and LM depend
on a plant model and are not universal descriptors. This is why it is necessary to verify them
for specific example cases.
For the first example, we study a system with single input to and single output from the
describing functions. Consider a two-state mass-spring-damper system with an actuator
force F acting on the mass, and only position feedback, as shown in Figure 4-8. This
is a simple model of a robotic contact task on a compliant surface. The description is
mX + bi + kx = F, where b is the surface damping coefficient and k is its spring constant.
We use the values m = 4, b = 2, and k = 1.
For LF, we fix the state and control penalties and solve for the LQR controller gain
K = [-0.38 - 0.86]; for fixed process and sensor noise, the constant linear filter gain is
Kf = [0.49 0.1 1]r. We discretized the system model using r = 0.1 and a zero-order hold.
74
....
.......
........
......
...
1
0.8F
Ca
0.6h
Sim a=0.4
-Model
a=0.4
--- 0.6
-0.6
--- 0.8
-0.8
---
C
0.4
0.2F
-2
-1.5
-1
Log10 of Input m
-0.5
0
(1
-10
-20
(D
W~
- -- Sim (x=0.4
-Model
x=0.4
--- 0.6
-0.6
- - -0.8
-0.8
-30-40-
-50-
'
*I
-2
-1.5
-1
Log10 of Input
o
-0.5
0
Figure 4-7: Gain (top plot) and phase o of the LH model. These curves are universal: they
do not depend on the plant model or the controller.
75
X
Figure 4-8: Example 1-mass, 1-input 1-output system. Mass is connected by a spring and
damper to a wall, and acted on by a force.
The closed-loop system has damping ratio 0.56 and a natural frequency of 0.60 rad/sec.
The bottom left plot in Figure 4-3 refers to this model, with 70% packet loss, 0) = 1, and
a = 1. We give the DF gain and phase in Figure 4-9, taking the position measurement
signal to the control command. The image covers a range of frequencies and packet loss
rates; simulations confirm the analytical model. Note that the Monte Carlo simulations are
much more expensive for the LF than for the LH, and thus we plot less simulation data in
Figure 4-9 than in Figure 4-7.
For a second SISO example we study an inverted pendulum system which is open-loop
unstable. The system is shown in Figure 4-10. In this system an actuator torque T acts on
the pendulum arm. The massless arm is of length L and the end has mass m acted on by
gravity. The position of the arm with respect to vertical is given by 6. For this example we
assume 0 is small, such that we can use the approximation sin(6) = 6. The equation of
motion for this system is T
=
mL 2 6 - mgLO. We use the values m = 2 and L = 1.
For LF, we fix the state and control penalties and solve for the LQR controller gain
K = [- 1.36 - 0.43]; for fixed process and sensor noise, the constant linear filter gain is
Kj = [0.64 1. 9 9 ]T. We discretized the system model using r = 0.1 and a zero-order hold.
We give the DF gain and phase in Figure 4-11, taking the position measurement signal
to the control command. The image covers a range of frequencies and packet loss rates;
simulations confirm the analytical model.
For a third SISO example we study an all-pass symmetric lattice circuit system which
76
...
..........
.........
.
....
0.18
0.16
-
0.14
-
o
012
o
Sim
0
Model a=0.4
0.6
0.6
a=0.4
0 0.8
1--0.8
S-0.1
0.08
S
0.06-
0.04-
0.02o0
-2
-1.5
-1
-0.5
Log,, of Input at
0
-140.
-160
-
-180
-200
Sim
---
-220
a=0.4
Model
0 0.6
a=0.4
0.6
0 0.8
--
-240 - -0.8-
-20
-2
-1.5
-1
Log
of Input cn
-0.5
0
Figure 4-9: LF describing function gain (top plot) and phase (bottom plot) from analysis
and Monte Carlo simulation, for the contact task example.
77
g
T
Figure 4-10: Example 1-mass, 1-input 1-output inverted pendulum system.
is nonminimum phase, meaning it contains zeros in the right half plane. The system is
shown in Figure 4-12, and is used in electrical engineering applications to change a system
phase while maintaining unity gain [22]. In this system the voltage vi is input to control
the output voltage v0 , with the circuit consisting of resistors, capacitors, and inductors. The
equation for this system is given by
CLVO + CRO + 1 = CLVi - CRi + 1
(4.59)
and has two left-half-plane poles and two right-half-plane zeros. We assume for this exampleR = L= C = 1.
For LF, we fix the state and control penalties and solve for the LQR controller gain
K = [-0.65
- 0.37]; for fixed process and sensor noise, the constant linear filter gain is
Kf = [0.0024 -0.
0022
]T. We discretized the system model using v = 0.1 and a zero-order
hold. We give the DF gain and phase in Figure 4-13, taking the output voltage measurement
signal to the input voltage command. The image covers a range of frequencies and packet
loss rates; simulations confirm the analytical model.
We next verify the LF model for a multi-sensor system. We consider the same plant
as in the SISO mass-spring-damper system, but assume sensor measurements (xi and x2)
78
..
....
....
. .......
.....
l."', - '. ..
. ...
..........
.
..
..........
...
.....
..
....
....
..
......
0 Sirn a=0.4
2.5
0.6
0.6
0.8
0.8
0
S --
2
a=0.4
Model
--
0
--
-
-
0.5
-2
-1.5
-1
-0.5
Log, 0 of input aT
0
-170
-180
-190
-200
-210
S
E -220
-230 -
0
Sim a=0.4
0
0.6
--
Model
a=0.4
-0.6
o
-240
0.8
-0.8
-250
-2
-1.5
-1
-0.5
Log1 0 of Input wT
0
Figure 4-11: LF describing function gain (top plot) and phase (bottom plot) from analysis
and Monte Carlo simulation, for the inverted pendulum example.
79
R
Vo
Vi
L
L
-AAA
R
Figure 4-12: Example 1-input, 1-output all-pass symmetric lattice circuit system.
are taken of each state (i.e. position and velocity of the mass). Both sensor measurements
are sent through the channel and the 2-element state estimate is output. The signals have
significantly different loss probabilities, with 30% loss probability for x, and 70% for x2.
We fix the state and control penalties and solve for the LQR controller gain
Ku = [-7.73
- 9.95].
(4.60)
For fixed process and sensor noise, the constant linear filter gain is
Kf=
0.055 0.019
[0.014 0.034
J
(4.61)
We discretize the system model using 'r = 0.1 and a zero-order hold, and again see good
agreement between simulation and DF predicted gains and phases.
We additionally verify the LF model on a multi-sensor MIMO system with 1 input and
2 outputs. We consider a system with two independent masses on a frictionless surface,
with a spring connecting them to each other, a damper connecting one mass to a wall, and
a force acting on the other mass. This setup is shown schematically in Figure 4-15, with
the input force on m2, and output measurements x1 and x2.
This system is described mathematically by the equations
m1p1 +bx1 +kx1 - kx2 = 0
(4.62)
= F
(4.63)
-kx 1 + m2x2 +
80
kX2
(1.11/
0.06
0.05
o
o
Sim a=0.4
Moddel a=0.4
0.6
-
L-M
3t0.04
1
0.6
0 0.8
-0.8
0.03
0.02
0.01
-2
-1.5
-1
-0.5
Log10 of Input m
0. 5
o
Sim x=0.4
-Model
a=0.4
-50 [
0
- -100
0
0.6
-0.6
0 0.8
I
--
0.8
a.-150
-200[
-2
-2
-1.5
-1
-0.5
Log, 0 of Input on
0
0.5
Figure 4-13: LF describing function gain (top plot) and phase (bottom plot) from analysis
and Monte Carlo simulation, for the nonminimum phase circuit example.
81
0
0
0
0 Sim x1,A=0.3
Model
0 Sim x2,a=0.7
- Model
0.6-
-
0.5W0.4
0 0.3
-
0.2
0.1
0
-1.5
-2
-0.5
-1
Log, 0 of Input ot
0
150-
o
Sim x1 ,a=0.3
Model
o Sim x2,a=0.7
-Model
--
100
50
0
-50-
-10 -2
-1.5
-1
-0.5
Log10 of Input on
0
Figure 4-14: MIMO LF describing function gain (top plot) and phase (bottom plot) from
analysis and Monte Carlo simulation, for the 2-sensor contact task example.
82
...
.......
..
........
.
.....
....
..
.. ....
..
....
. .....
.
........................
...........
..........
Figure 4-15: Example 2-mass, 1-input 2-output MIMO system. Masses are connected by a
spring to each other, a damper to a wall, and acted on by a force.
and can be represented in state space with state vector [s]
For this example we choose parameter values k
=
S2 s 3 S4] = [xi x 1
1, b = 2, ml = 0.5, and m2
=
x2
2].
4.
We fix the state and control penalties and solve for the LQR controller gain K=
-[0.72
0.17 0.65 2.50]. For fixed process and sensor noise, the constant linear filter
gain is
Kf =
0.014
0.0030
0.019
0.0014
0.0026
0.0029 0.0088
0.0037
0.019
0.0090
0.037
0.010
0.00094 0.0033
0.0090
0.093
.(4.64)
To verify the DF gains and phases we consider an example scenario where sensors for
states S2 and S4 do not exist (i.e. measurements lost with probability 100%), and measurements from sensors si and S3 have loss probabilities 30% and 10% respectively. For this
scenario we assume arbitrary input amplitudes for si and s3 and arbitrary phase differences
to allow for effects of process and sensor noise. We assume states S2 and S4 are shifted by
r/2 radians in phase from states sI and S3, respectively, because of their position-velocity
relationship. For the same reason we also assume the magnitudes of S2 and S4 are equal
to the magnitudes of si and S3 multiplied by the input frequency. Figure 4-16 shows good
agreement between DF and Monte Carlo gains and phases for this scenario.
To further verify LF, we consider a final additional example case, where three masses
are connected by springs to each other, by a damper to a wall, and acted on by two forces,
as shown in Figure 4-17. This system has two inputs, six outputs, and six sensor channels.
83
...
.. ................
2.5-
o Sims, a=0.3
-Model
Sims a=1
Model
o Sim s3 a=0.1
0
o
2
-
',
1.5
Model
0 Sim
a=
s4
- Model
0.5
0
-0.5
-1
of Input at
-1.5
-2
Log
Sim
g
150L
-
100-
0
s
a=0.3
Model
1
Sim s a 1
a y aa
50-
--
Model
--
Model
-
S4 O'IM
s4a=1
Model
0~ Simss a=0-. 1
0
-ml
-50 -100-
150
-2
-1.5
-1
Log, 0 of
Input
-0.5
w
0
Figure 4-16: MIMO LF describing function gain (top plot) and phase (bottom plot) from
analysis and Monte Carlo simulation, for the 1 -input 2-output double mass example.
84
..... . .
..
....
....
..
X2
,
-4-+
b
X3
k2
mF
F2
Figure 4-17: Example 3-mass, 2-input 3-output, 6-state MIMO system. Masses are connected by springs to each other, a damper to a wall, and acted on by a force on m 1 and a
force on m2.
This system is described mathematically by the equations
m1jY+ bk1 + k1 x1 -k
-klx 1 +
+
m2x2
k1 x 2
-
k2 x 3
1 x2
=
F1
(4.65)
+ k 3x 2
=
F2
(4.66)
-
0
(4.67)
-k 2x 2 + k2x 3 + m 3X3
and can be represented in state space with state vector
[SI
s2 S3
S4 S5
s6]'
= [XI )I
X2
e2 X3 )3]'.
(4.68)
For this example we choose parameter values k1 = 1, b = 2, m, = 0.5, m2 = 4, k2
=
1,
and m3 = 2.
We fix the state and control penalties and solve for the LQR controller gain
Ku=-
0.63 0.32
0.18
0.61
1.14 3.09
0.11
0.86 0.07
0.35
0.25
0.99
For fixed process and sensor noise, the constant linear filter gain is
85
(4.69)
o
3
-
2. 5-
-
Model
0 Sim s, =0.1
Model
o Sim s a=1
o
2li0
1.
-
51
o
0. 5
o
Sim s, a=0.3
Model
Sim s, a=1
Model
0 Sims
Model
0 Sim s
a=0.5
a=1
7-Model
.am
0
-2
-1.5
-1
-0.5
Log,,) of Input corT
0
100
50
0
-50
Q_ -100
-150
-200
-2
-1.5
-1
Log,, of Input o
-0.5
0
Figure 4-18: MIMO LF describing function gain (top plot) and phase (bottom plot) from
analysis and Monte Carlo simulation, for the 2-input 3-output three-mass example. Legend
is same for both plots.
Kf =
0.009
0.002 0.013
0.0004
0.011
0.005
0.002
0.002 0.006
0.002
0.002
0.002
0.012
0.007
0.026
0.007
0.015
0.008
0.00002 0.002 0.005
0.008
0.002
-0.0009
0.012
0.002 0.015
0.002
0.028
0.007
0.005
0.002 0.009
-0.0006
0.005
0.011
(4.70)
To verify the DF gains and phases we consider an example scenario where sensors
for states S2, S4, and s 6 do not exist (i.e. measurements lost with probability 100%), and
measurements from sensors si, S3, and s5 have loss probabilities 30%, 10%, and 50%
respectively. We plot DF gains and phases in Figure 4-18 and again see good agreement
between model and Monte Carlo simulation.
86
- __........
...
...........
....
....
....
..
....
............
......
........
.. .......
.....
......
.
.
....
..
.....
.........
...........
1
0.9
0.8-
0 Sim no=0.4
0.7 -
-- Model a=0.4
-Modela=0.4
0 0.6
0.6-
-0.6
;0.5 -
-0.8
0
0 0.8
-
0.4
-
0.3
0.2-
0.1
Log1o of Input ot
5
-140
-
-130
-150-1600)
-170
0 Sim a=0.4
S-180
-200
Model a=0.4
-
(D-190
0 0.6
-
-0.6
0
-210
0.8
-0.8
-220
-230-240-2
-2
-1.5
ipt
-1
d -0.5
Log1 0 of Input codt
L
0
Figure 4-19: LM gain (top plot) and phase o (bottom plot) from analysis and Monte Carlo
simulation, for the contact task example.
4.4.4
LM Verification
To verify LM we consider some of the same examples used for LF. We first consider the
one-mass SISO example case used for LF, with the same plant model and controller gains.
The model and simulation results for LM are given in Figure 4-7 for the SISO version of
the example, and again we see good agreement. The bottom right plot in Figure 4-3 refers
to this model, with 70% packet loss, co = 1, and a = 1. As with LF, the Monte Carlo
simulations are much more expensive for the LM than for the LH, and thus we plot less
simulation data in Figure 4-19 than in Figure 4-7.
We also plot the two-sensor case with the same plant and controller in Figure 4-20. In
this case the two inputs to the DF are the states, position and velocity, and the two outputs
are the estimates of the states. As in the MIMO LF simulation, we assume significantly
different loss probabilities in each channel (30% for one and 70% for the other), and see
87
0.9
0.8
0.7
0.6-
0
0.5-
-
0.4
-
Sim xgoc=0.3
05odel
-
M
0 Sim x2 a=0.7
Model
Logl 0 of Input
o
.
0.3
0
-5-10 --
-15
-
-20
-2
(25
-
-30 -
Si ,a=0.3
Model
0 Sim x 2 (=0.7
Model
-35- .-40-
-2
-1 5
Log~c
of Input
mr
Figure 4-20: LM gain (top plot) and phase w (bottom plot) from analysis and Monte Carlo
simulation, for the MIMO contact task example.
good agreement between simulated and analytically-determined gains and phase losses.
Finally, we verify LM for the 3-mass, 2-input, 3-output system described previously.
The same packet loss scenario is assumed, and good agreement between model and DF
prediction is again seen in Figure 4-21.
4.4.5
Coupled Losses
To verify the mathematical assertion that coupled losses lead to the same describing functions as independent losses, we compare the gains and phases found by simulation for the
LH and LZ models for coupled and independent channels. For these simulations we assume
two measurements are passed through a lossy channel and decoded either by LZ or LH. In
the coupled case the measurements are lost or passed together, and in the independent case
they are lost independently with equal probability. For LZ we plot the root mean square
88
.....
- .........
..
.........
.....
. ....
......
..
..
- -------
..........
.. ......
7
-
6
0 Sim sI
Model
a=0.3
0 Sims 2 ,
=1
Model
a Sims3, a=0.1
Model
o Sims4, c=1
Model
0 Sim s5, a=0.5
Model
0 Sims6, a=1
5
0
-
4
3
2
0
*
-2
-1.5
Model
-
0
1
-
-
-
-
-
a e
-
-1
-0.5
Log10 of Input on
On
30
1 501
0
0050--
0.
0
50-
-
-11 00
-2
-1.5
-1
Log, 0 of Input
-0.5
0
0o3
Figure 4-21: LM gain (top plot) and phase co (bottom plot) from analysis and Monte Carlo
simulation, for the 3-mass MIMO example. Legend is same for both plots.
89
difference Glz between the gains for the coupled and independent channels, defined as
GIZ =
(Gj; - Gic) 2 + (G 2 ; - G2c)2
(4.71)
where G1 ; is the gain for channel I with independent loss, G1, is the gain for channel 1 with
coupled loss, G2i is the gain for channel 2 with independent loss, and G2c is the gain for
channel 2 with coupled loss. We expect Glz to be zero over the range of valid frequencies,
and indeed this is seen in Figure 4-22. For this simulation we have assumed a time step of
0.1 seconds and test packets loss probabilities of a = 0.4,0.6 and 0.8.
a=0.8
o
0)
0
C
0)
.2
0
0
0
GO
0
0
0
0
o0
0
0
0
0
0
0
0
(x=0.6
0
C
0 .1-------------------------------------o6=0.4
-2
-1.5
-1
Log 0 of Input
on
-0.5
0
Figure 4-22: Gain difference between coupled loss model and independent loss model for
two-channel LZ.
We perform similar simulations for the case of the LH decoder, and also see zero mean
square differences in gain and phase as expected, as shown in Figure 4-23.
4.5
Quantizer Describing Functions
We next review the describing functions of several common types of quantizers. These
DFs will eventually be combined with the lossy channel DFs to describe channels with
both packet loss and quantization.
90
..
..........
.............
II
a=0.8
0
%-
C
U)
11)
0
0
p
0
p
0
G0
a=0.6
0
0.1 0.
0 0
-2
-1.5
0 0
0
0
0
0
0
-1
Log, 0 of Input on
-0.5
0
a=0.8
0
0
0
0
0)
-o
a,
0
C
0)
0)
a--0.6
0
0)
(U
0~
00
2--------------------------------------ac=0.4
0
0
-2
0
-1.5
p
0
-1
Log, 0 of Input
p
C
on
-0.5
0
0
0
Figure 4-23: Gain and phase differences between coupled loss model and independent loss
model for two-channel LH.
91
4.5.1
Uniform Quantizer
Quantizers have well-known describing functions [32], and we will treat here the uniform
round-down quantizer with bin size h and input amplitude a. The results in this paper can
easily be extended to a logarithmic quantizer [115], for which we show the DF in Section
4.5.2.
The uniform quantizer for a SISO channel has describing function N(a/h), which carries no frequency dependence or phase:
4h
N ()
(2i - I)h
nq
ra .
2
2a
1
(4.72)
2q + 1
2
ha
h -
2nq 2
where nq is an integer defining which quantization bin a/h is in. Figure 4-24 shows the
uniform quantizer DF gain as a function of a/h. As the ratio a/h increases, the quantization
becomes finer and the gain approaches unity.
.
.
1 5
5 --
0
2
4
6
8
10
a/h
Figure 4-24: Describing function gain N(a/h) for the uniform quantizer [32] with bin width
h and input signal amplitude a.
The MIMO version of the quantizer DF for input [al,..., anI' and sensor quantization
bin sizes hl,..., hn is simply a diagonal matrix given by
92
N(ai/1hj) 0
... an]/hh)
0
0
0
0
N.
aN([a
0
(4.73)
N(anlhn)
For simplicity in future derivations we will assume a constant quantization bin size h
for all sensor channels.
4.5.2
Logarithmic Quantizer
The logarithmic quantizer qiog with quantization density p is defined as
qjOg (v) = p ,
<V <
(4.74)
with the SISO describing function [32]
4
nq
N(og (a) = -- E (p2-pa- )
2a
(l+P)Pi_)
(4.75)
2
.
(+p)pnq
(I +p)pnq-1
a
<
2
-<
2
-a
2
We plot the DF of the finite log quantizer in Figure 4-25 as a function of input amplitude a, quantizer parameter p, and deadband region. The top plot is for a scenario with
fine quantization (minimum quantization bin size p 1 000), and the bottom plot for coarse
quantization (minimum bin size p 2 ). We see that, for fine quantization, the DF gain starts
out near one for low a and is different than one at high a, which matches intuition that
the log quantizer is most accurate near the origin. For coarse quantization the DF gain is
different than one over a larger range of a, which is consistent with it distorting the input
signal more.
The MIMO version of this DF is constructed in a similar manner to that of the MIMO
uniform round-down quantizer DF. Because we can formulate the log quantizer in a similar
way as the uniform quantizer, the analytical describing function results could be applied to
either. For the remainder of the paper we assume for simplicity that uniform quantizers are
used, though results could be applied to log quantizers as well.
93
1
0.8
Z
0.6
0.4-
-
p = 0.25
-0.5
-
0.2[
0
2
4
6
a
0.75
8
10
1
0.8
Z
0.6
C
0.4
-p = 0.25
-0.5
0.75
0.2
0
2
4
a
6
8
10
Figure 4-25: Describing function gain for log quantizer as a function of input magnitude
a and quantizer parameter p for fine quantization (deadband p 100 0 , top plot) and coarse
quantization (deadband p 2 , bottom plot).
94
4.6
Combined Quantizer and Encoder-Channel-Decoder
Describing Functions
We now consider the scenario where sensor signals are quantized before being sent through
the lossy feedback channel, as illustrated in the block diagram in Figure 4-26 for the general MIMO case. We aim to derive a DF for the entire quantizer-encoder-channel-decoder
sequence, based on the DF for the quantizer and the DF for the encoder-channel-decoder.
Controller
Plant
C(z)
Ir
m
-pw
w
~
"W
:
-
Decoder
MI
P(z)
WuWWm
" g o " W Wm~
Quantizer
Binary Erasure Channel
ip
i
1
sj
Encoder
a
- W
Wi -WWpg I Mopo W N
W
Describing Function
Figure 4-26: Block diagram for a MIMO feedback control system with a quantizer, encoder, binary erasure channel, decoder, controller, and plant. The describing function represents the quantizer-encoder-channel-decoder blocks.
The quantizer DF is independent of input frequency and the encoder-channel-decoder
DF is independent of input amplitude. These facts, however, do not imply directly that the
aggregate DF is simply the cascade of the encoder-channel-decoder DF and the quantizer
DF. We will show, however, that it is possible to simply cascade the DFs in the cases of LZ,
LH, LF, and LM. We refer to the DFs for the quantizer cascaded with the encoder-channeldecoder DFs as QLZ, QLH, QLF, and QLM, respectively.
4.6.1
Quantizer and Zero Output
To prove the quantizer and LZ DFs can be cascaded, quantize the input sine signals with
bin size h. The decoded signal becomes
Zk
=
kh [al...a sin(kr)
hn
where [.] represents the round-down (or floor) function.
95
J(4.76)
Using the same derivation as described in Section 4.2, the gain and phase for the QLZ
are given by
1
Tz,=
-
where Zqz,i
(4.77)
arctan(zqz,i)
K4qz,iK3 - K2K5qz,i - K1K5qz,i
K3 K5qz,i + K2K4qz,i - KiK4qz,i
=
1 +qzi
(K5qz,i - K4qz,izqz,i)
'
-K3z 2i -
2
and
K2Zqz,i + K3
Here K1 , K2 , and K3 are as above, and there are two new scalars:
K4z,i
=
[4D gh [[al...anIin(Wkr)
sin(cokr)] (i)
(4.79)
_[al
cos(okr)](i).
(4.80)
k=t
-[c
K5qz,i =
h
.. a,] in(k)
k=1-
If we now apply the quantizer DF gain, we see that K4qz,i = N(ai/h)K4z,i and K5qz,i
=
N(ai/h)Kz,i. This leads to cancellations in Equation (4.78) that give zz,i = zz,j. Substitut-
ing K4qz,i,K5qz,i, and zqz,i into Equations (4.77) and (4.78) yields
Aqz,i = N(ai//h)A
and
ai
ITqz,i
,.
(4.81)
Thus the quantizer and LZ DFs can be cascaded and the quantizer DF does not affect the
phase.
4.6.2
Quantizer and Hold Output
By similar reasoning as used in to derive the QLZ DF, to show the quantizer DF and LH
DF can be cascaded we must show that K4qh,i = N(ai/h)K4h,i and K5qh,i = N(ai/h)Kh,i- If
we quantize the input sine signals with bin size h, the decoded signal from LH becomes
96
I
(4.82)
+ (I- Ok)Hk _1
.
[ai...han] sin(wkr)
Hk = Okh
Using Equations (4.18) and (4.19) we find
= [(DE(,
-
K4qh,i
4
[
)k-1h
k=1 1=1
K5qh,i
LE-(I -
=
[al...han] sin(oft)
I
sin (okr) ](i)
(D)k-lh [a1...an] sin(wlr) cos((wkTr)] (i).
(4.83)
(4.84)
IU
k=1 1=1
If we now apply the quantizer DF gain, we see that K4qh,i = N(ai/h)K4 h,i and K5qh,i
=
N(ai/h)K5hj. Thus the quantizer and LH DFs can be cascaded and the quantizer DF does
not affect the phase. The QLH DF parameters are thus
Aqh,i
ai
4.6.3
= N(ai/h)Ah-2
ai
and
ITqh,i =i
(4.85)
Quantizer and Linear Filter
As with QLZ and QLH, to show the quantizer DF and LF DF can be cascaded we must
show that K4qf,i = N(ai/h)K4f,i and K5qf,i = N(ai/h)K5f,j. If we quantize the input sine
signals with bin size h, the decoded signal from LF becomes
Fk
= Ok
[(I-KfC)GFk-I + Kfh [al...an] sin(ok r)] +(I- Ek)GFk-1.
(4.86)
Using Equations (4.18) and (4.19) we find
K4qf,i
=
g
K5qf,i
I
[a1...an]sin(wlr)I
((Kf C + I,)G)k-l Kfh
[f
k
k I
Y Y, ((Kf (DC + I,) G) - (DKf h
h
I
sin(wo(k)] (i)
[al ...
an] sin(coft)
h
cos(co(k)'r) (i).
I
-k=l 1=1
97
If we now apply the quantizer DF gain, we see that K4qf,i = N(ai/h)K4f,i and K5qf,i
=
N(ai/h)Kf,j. Thus the quantizer and LF DFs can be cascaded and the quantizer DF does
not affect the phase. The QLF DF parameters are thus
Aqf, = N(a /h)Af,
4.6.4
and Tqf,i
ai
ai
=
T5, 1 .
(4.87)
Quantizer and MIF
As with QLZ, QLH and QLF, to show the quantizer DF and LM DF can be cascaded we
must show that K4qm,i = N(ai/h)K4m,i and K5qm,i = N(ai/h)K5m,i. If we quantize the input
sine signals with bin size h, the effect of the quantization is seen in the variable gk. For gk
we have
yh
gq,k = Yk I
[[ai.an]sin(Dl)
(4.88)
CTR-1.
After applying the quantizer DF gain we find
(4.89)
gq,k = N([a1...an]/h)gk.
For E[Mk] we can see from Equation (4.55) that there is a term gj multiplying through
the summations. Thus, because the quantizer becomes a multiplicative gain for gj, it also
become a multiplicative gain for E[Mk]. Thus, using Equations (4.18) and (4.19) we find
g
K4mJi
N([a1...an]/h)E[Mk]sin(ct(k),r) (i)
=
-k=1
g
K5m,i
=
-
[
N(a...an]/h)E[Mk]cos(c(k)r)
(i).
-k=l
We see that K4qm,i = N(ai/h)K4m,i and K5qm,i = N(ai/h)Km,i. Thus the quantizer and
LM DFs can be cascaded and the quantizer DF does not affect the phase. The QLM DF
parameters are thus
98
ai
= N(aih) m,i
aiI
and Tqm,i = Tm,i.
(4.90)
By a very similar derivation, we can also show that cascading the quantizer describing
function with LM also holds if the message is quantized after it is encoded.
4.7
Summary
In this chapter we derived a general describing function for a multi-sensor information
channel of a MIMO feedback control system subject to packet loss. The DF approximates
the encoder-channel-decoder system as a gain and phase lag dependent on input frequency,
much like a transfer function. The DF was applied to four control/estimation algorithms
commonly used in systems subject to packet loss: Zero Output, Hold Output, LQG-type
Linear Filter, and Modified Information Filter. The DF models for these four systems
were validated by simulation. DFs for several types of common quantizers were presented,
and we proved that these quantizer DFs can be cascaded with the DFs for the packet loss
control/estimation algorithms.
99
100
Chapter 5
Describing Functions as Analysis Tools
5.1
Introduction
The DFs are useful analysis tools because they predict the frequency-domain behavior of
the quantized and lossy channel. In this chapter we illustrate two uses of the DFs - as tools
to predict system sensitivity to disturbances, and as tools to predict system limit cycles.
In theory these performance results could be predicted by simulation, but we show that
computing channel gains and phases - necessary to predict limit cycles and find sensitivity functions - is orders of magnitude faster using DF models than by using simulation.
Example Bode-like plots and sensitivity-like function magnitudes computed using the DFs
are shown, and a numerical example is given where theoretically-predicted limit cycle amplitudes and frequencies are verified by simulation.
5.2
Computation Times for Analysis vs Simulation
We consider SISO versions of the four example controller/codecs LZ, LH, LF, and LM for
a set of computation time tests comparing simulation and DF models. Tests are run on a
Lenovo T500 ThinkPad laptop computer with Intel Centrino 2 processor. For each case
we first compute the gain and phase using the DF model with increasing values of the parameter g until we see convergence. We plot the calculated gain and phase vs computation
time required for the final value of g. We then compute the gain and phase via Monte Carlo
101
0.90.80.7-
0.6
--
C0.0
0.4 -0.3
--
0 LZ Model
0.2-
-
Simulation
0.1-01'
0
0.5
1
1.5
2
Computation Time (sec)
2.5
3
Figure 5-1: Computation time for simulation and describing function for the SISO LZ
decoder, for or = -1 and a = 50%. The LZ model converges in less than 10-4 seconds,
while simulation takes close to 3 seconds.
simulation. The gain and phase are calculated by simulating a sinusoid sent through a lossy
channel with the given codec, and using the simplex method to fit a new sinusoid to the
decoded signal. We calculate gain and phase for increasing number of sinusoid periods the
decoded sinusoid is fit to.
I and a = 0.5 as a representative case. Plots
For all systems we plot results for (odt =
for LZ and LH are universal plots and do not depend on a system model. Plots for LF and
LM are made for the contact task system described previously. We give results for gain
only; phase results are similar. Figure 5-1 shows results for LZ, where model computation
time is less than 10-4 seconds, and simulation mean converges after approximately 3 seconds. Figure 5-2 shows the results of LH, where simulation takes at least 1000 times as
long to converge as the DF model (1.6 seconds vs 5x10-4 seconds). In Figure 5-3, for the
contact task example the LF again converges 40 times faster than simulation, and Figure
5-4 shows that the LM converges approximately twice as fast as simulation.
The models for LZ, LH and LF clearly provide a significant speed advantage over simulation for calculating a gain at a single input frequency. The LM model is also faster than
simulation, but the advantage is not as pronounced. However, the advantage of using the
LM model becomes much more apparent when we consider calculating gains at multiple
102
1.11.081.061.041.02C
as
1
0.980.96LH Model
0.94-
0.94
-Simulation
0.92-
0.9
0
0.2
0.4
1.2
0.8
1
0.6
Computation Time (sec)
1.4
1.6
Figure 5-2: Computation time for simulation and describing function for the SISO LH
decoder, for or = -1 and a = 50%. The LH model converges in 5x10- 4 seconds, while
simulation takes close to 1.6 seconds.
0I
0.1-
0.05
o
LF Model
-Simulation
0
2
6
8
4
Computation Time (sec)
10
12
Figure 5-3: Computation time for simulation and describing function for the SISO LF
decoder, for ar = -1 and a = 50%. The LF model converges in 0.32 seconds, while
simulation takes close to 13 seconds.
103
0.56
0.54
0.52
0.5
:0.48-
0.46
0.44
0 LM Model
0.42
-
Simulation
0.4
0.30
S
5
10
15
20
Computation Time (sec)
25
30
Figure 5-4: Computation time for simulation and describing function for the SISO LM
decoder, for orr = -1 and a = 50%. The LM model converges in 15 seconds, while the
simulation takes close to 30 seconds.
1.02
0.980.96N
0.94-
E 0.920
z
0.9-
o
0.88-
-
LM Model
Simulation
0
-
0.86500
1000
1500 2000 2500 3000
Computation Time (sec)
3500
4000
Figure 5-5: Computation time for simulation and describing function for the SISO LM
decoder computing normalized bode plot gains, for 100 frequencies and a = 50%. The
LM model converges in 16 seconds, while the simulation takes close to 3000 seconds.
104
frequencies, such as when creating a Bode plot. This scenario is the most realistic use of
the describing function models, because the gains and phases at multiple frequencies would
be needed to create system Nyquist plots to predict limit cycles.
To calculate gains at multiple frequencies, computation times for all simulations scale
linearly with the number of frequencies evaluated. Computation times for LZ, LH, and LF
also scale linearly with the number of frequencies, and thus LZ, LH and LF models again
have a significant speed advantage over simulation. Computation time for LM is a fixed
time proportional'to g 2 , plus a time linearly proportional to the number of frequencies. The
fixed time proportional to g 2 is three orders of magnitude greater than the time proportional
to the number of frequencies. For example, in Figure 5-4, the fixed time is 15.48 seconds,
and the time proportional to the number of frequencies is 0.02 seconds. Thus, while the
total computation time to evaluate the gain at one frequency is 15.50 seconds, the time
required for 100 frequencies is only 16.98 seconds. If done by simulation this would take
approximately 3000 seconds (100x30), as shown in Figure 5-5. The LF model similarly has
a fixed computation time proportional to g 2 , but this time is an order of magnitude smaller
than the computation time proportional to the number of frequencies sampled, in the SISO
case. In the example in Figure 5-3 the fixed time is 0.01 seconds, compared to the total
computation time of 0.32 seconds.
For MIMO cases with a single input frequency and n states, simulation computation
time scales proportional to the time required for the simplex algorithm to converge for 2n
dimensions. The LZ and LH model computation times scale linearly with n, because the
channels are independent. For LF and LM, the fixed computation times are proportional
to the time required to compute an nxn matrix raised to the power g, for n system states.
Thus, this fixed time is proportional to ng.
The computation time analysis results show that the describing functions are in general
at least several orders of magnitude faster than simulations to compute gains and phases for
systems subject to packet loss, and thus the DFs are practical analysis tools.
105
Gain A/a
160
140
120
100
-1
80
-0.5
0
0.5
600.
40
-1
-0.5
0
Log of Input
w
0
0.5
1
Figure 5-6: Gain (top) and phase (bottom) of the QLF for the contact task model; a =
1O%, h = 46, -r 0.1.
5.3
Bode Plots
The main utility of the DFs as analysis tools is predicting limit cycles, and to do this one
intermediate step is creating system Bode plots. Because the DFs for the quantized lossy
channels are not exactly transfer functions, and in fact depend on both input frequency and
input amplitude, we refer to the gain and phase plots in this section as Bode-like plots.
They still present frequency responses of the channels though, and are still useful to create
Nyquist plots and predict limit cycles.
As examples, Figure 5-6 shows a Bode-like plot of gain versus input magnitude and
input frequency (top plot), and phase versus input frequency (bottom plot) for the specific
case of a = 0.1, h = 46 with QLF and the SISO contact task model considered in Chapter
4. Figure 5-7 shows the same plots for a higher packet-loss case, with a = 0.6, h = 10. The
overall shape is generated by the combined LQG-like structure, the lossy channel, and the
quantizer; the first two of these have inherent frequency dependence.
Figures 5-8 and 5-9 show example Bode-like plots for QLH, a universal DF that does
not depend on a system model, for the same two a,h pairs. The plots show that QLH acts
106
Gain A/a
700.6
60
0.5
$50
0.4
40
0.3
0.2
20
0.1
10
-1
0
-0.5
Log, 0 of Input c
0.5
1
40
20
0 --20-
0- -40
-1
-0.5
Log
0
0
of Input to
0.5
1
Figure 5-7: Gain (top) and phase (bottom) of the QLF for the contact task model; a
60%, h = 10),c = 0. 1.
like a pure quantizer at low input frequencies, while at high frequencies the packet loss
creates phase loss.
The information from these Bode-like plots can be combined with true Bode plots of
other elements in the feedback control loop to create loop Nyquist plots. Limit cycles,
in turn, can then be predicted from the Nyquist plots. A numerical example is given in
Section 5.5 illustrating how the DFs and Nyquist plots can be used to predict limit cycles
in a simple SISO system.
5.4
Sensitivity Functions
The describing functions can also be used to analyze the effect of process noise on a feedback control system subject to packet loss and quantization. If we assume an additive
disturbance acting on the feedback channel, as shown in the block diagram in Figure 5-10,
then the effect of this disturbance on the output is characterized by the system sensitivity
function, defined as
107
Gain A/a
1.2
0.8
-)
10
10'
0.6
8
0.4
.
CL
0.2
-1
0.5
0
Log, 0 of Input w
-0.5
1
-2-4-
-1
0
-0.5
0.5
1
Log,0 of Input (o
Figure 5-8: Gain (top) and phase (bottom) of QLH; as with H, these properties are independent of the plant or controller. a = 10%,h = 46,'r = 0.1.
Gain A/a
1.2
1
E 4C
0K30
.S
0.4
0.2
-0.5
-1
0
Log, of Input
o
0.5
1
n
-10
-20
-30
-1
-0.5
0
Log10 of Input to
0.5
1
Figure 5-9: Gain (top) and phase (bottom) of QLH. a = 60%, h = 10,
108
--
--------------- -----
= 0.1.
Disturbance
Input
Plant
Controller
P,
C~
Figure 5-10: Block diagram of feedback control loop with additive disturbance on plant
output.
S= I(5.1)
1
1 +CPiGa5
for controller C, plant P, and describing function Gdf. The function in this case is not
exactly the same as a sensitivity function, since the describing function Gdf may depend on
input amplitude and is not a true transfer function, but the sensitivity-like function can still
be used as a similar analysis tool to the true sensitivity function in the frequency domain.
As an examnple application, we show the sensitivity function magnitude plots for the
one-mass contact task example with QLF and the two packet loss/quantization scenarios
considered previously. In the first case, shown on the left of Figure 5-11, the feedback
channel has loss a = 60%, h = 10, and r = 0.1. In the second case, shown on the right of
Figure 5-11, the feedback channel has loss a = 10%, h = 46, and r = 0.1.
In both cases the sensitivity function has lowest magnitude for higher input frequency,
which is consistent with the plant acting as a low-pass filter. In each case the system is
most sensitive to disturbances at frequencies around 10- 5rad/sec, with the 10% loss case
having slightly higher peak sensitivity magnitude of 1.35, compared to 1.25 of the 60%
loss case. The higher packet loss case is sensitive to a larger range of input magnitudes to
the quantizer, which would be expected for a less-perfect sensor-feedback system.
109
Gain A/a
Gain A/a
70
70
60
1.3
60
1.4
1.35
1.25
50
(0
1.25
0)4
1.2
201.15
20
1.15
:C3 30
3.1
_
1.1
1.5
10
20
20
10
0.95
-1
-0.5
0
Log10 of Input o
0.5
10
-1
1
-0.5
0
Log,, of Input c
0.5
1
Figure 5-11: Gain of the sensitivity function for the contact task model with the QLF codec.
For left plot a = 60%, h = 10,'r = 0.1, and right plot a = 10%, h = 46, = .1.
5.5
Limit Cycle Numerical Example
The DFs derived previously are useful as analysis tools to identify limit cycles in dynamic
systems. In practice, the addition of a quantized lossy channel will not necessarily cause a
system to limit cycle, and thus there is no guarantee that analysis via describing functions
will predict a limit cycle for an arbitrary system. The system Nyquist plot must be sufficiently close to the critical point in order for the addition of a quantized lossy channel to
cause it to limit cycle. In this case the DF analysis method will correctly predict the limit
cycle.
As a numerical example, we consider a simple double integrator with the position signal
quantized and sent through a lossy channel before being decoded and sent to a controller.
This scenario is not unlike a robot attempting to maintain a constant range to a leader,
where the remote range measurement is quantized and lossy.
We consider four different setups: QLF, QLH, QLZ, and a pure quantizer (Q) block.
To fairly compare these, we first design LQR and constant Kalman gains in continuous
time for the LQG that is used in the QLF system. We then implement the resulting transfer
function of the LQG as the compensator for the QLZ and QLH systems. Thus, in the
absence of packet loss and quantization, the closed-loop systems are all equivalent. After
mapping over to discrete time with -r = 0.1 and ZOH, we erode the stability margin of the
110
...............
....
111
111.111,
11
-- - ..
....
...
.......
...
....
. . ........
..
....
..- - - . ..........
.....
2-
1.5
0
0
1-
0.5-
0'
0
1
2
3
4
5
Time (sec)
6
7
8
Figure 5-12: Closed-loop step response of double-integrator example system, with LQG
design and a one-step delay.
system by injecting a one-step time delay; delay causes phase loss.
For this example, we reduce the independent variables by fixing an exponential relationship between quantization level h and packet loss rate a, given by a = e-ph. This
formula ties very fine quantization to higher packet loss rates, and vice versa. The tradeoff
is documented in the RF literature, e.g. [86], and also is consistent with what we have observed in shallow-water acoustic communication tests. For this example, we set p = 1/20,
for which value some typical (h, a) pairs are (1, 95%) and (90, 1.1%). In general, p is a
parameter to be tuned to match the environmental conditions and the system.
The equivalent LQG transfer function from sensor output to control signal is
T (s) =
2
2720s +7071
22s701(5.2)
s +33.75s +519.6
The closed-loop step response of T(s) cascaded with the double integrator plant and the
delay is given in Figure 5-12.
For a given (h, a) pair, limit cycling occurs when the Nyquist plot of the loop system
passes through the critical point; this will happen for certain frequencies and amplitudes.
Stability of a limit cycle is assessed by behavior near the intersection points of the plant
and the negative inverse of the describing function [95].
111
1009080
70
5040 -
- .'Q
QLZ Model
-QLH
Model
Model
0 QLZ Sim
QLH Sim
30
20 -
ro
,
'-QLF
10 ,,0
10
0 QLF Sim _
20
30
40
50
60
70
Quantization Bin Size h
80
90
100
Figure 5-13: Input amplitudes a and quantization levels h that result in a limit cycle for
the double-integrator example system. Plots are made for Q, QLZ, QLH, and QLF blocks
based on the describing function formulas and on and simulation. Thick lines represent
stable limit cycles and thin lines unstable ones. No stable limit cycle solutions exist for the
QLZ case when h < 45 (a > 11%), for QLF when h < 50 (a > 8%), or for QLH at low h.
Figure 5-13 shows model and corroborating simulation results for limit cycle magnitude; the range of h here corresponds to a between 5% and 60%. For high h, packet loss is
low, and all of the stable limit cycles tend toward that of the pure quantizer, as one would
expect. With lower h and consequently higher a, QLZ has a gain reduction from the lossy
channel, but also a gain effect from the quantizer. The effect of these is to diminish the limit
cycle, at least down to h = 44. For QLH and QLF the situation is much more complex, as
phase effects of the lossy channel and the decoder also come into play with decreasing h.
Even so, it is surprising that QLH creates an entire family of large, stable branches (of
which only the first two are shown in the figure) while the QLF model yields one smaller
limit cycle similar to that for QLZ, down to h = 50.
The simulations are helpful to explain the attributes of physical response along the
various branches. We consider a limit cycle to exist if at least ten consecutive oscillations
at a nearly constant frequency and amplitude are observed. A time series with clear limit
cycling is shown in Figure 5-14 for the QLZ setup and h = 70, yielding slight bias and
limit cycle amplitude of approximately 70. This signal is typical for QLZ and QLF above
their cusps in h, and for QLH at the higher h. In contrast, a more confusing time series is
112
..
..
....
..
..
..
..
....
..
...
.....
............
......
...
..
........
......
..
..
..
............
..
100
80 11
'l
"
1
1I
1
i
20
0
E
<
-20
-40
-60
'
-80
-100
0.2
0.4
0.6
1
1.2
0.8
Time Steps
1.4
1.6
1.8
2
X 104
Figure 5-14: Time series for the QLZ system with quantization level h = 70.
illustrated in Figure 5-15, for the QLH setup and h = 35. This trace has highly variable
amplitudes and frequencies - so no limit cycles can be easily identified - and is typical
for the QLZ and QLF systems below their cusps, and for QLH at the lower h. Eventually
all of the systems become unstable for very low h, because a approaches unity, and the
open-loop dynamic behavior fails to provide stability.
Figure 5-16 summarizes the nondimensional frequency for the limit cycles. The single
frequency pertaining to the multiple solution branches of QLH appears to reduce as h is
decreased. At higher values of h the limit cycle frequency for QLZ and QLH is close to
that for
Q,
consistent with the quantizer dominating the system. Most surprisingly, the
frequency is much higher for the QLF case, despite the fact that in terms of magnitude,
QLZ and QLF behavior is similar at high h. These observations serve to highlight the
complexities of controlling even simple systems through limited feedback channels.
5.6
Summary
We have shown that DFs are useful analysis tools for systems subject to packet loss and
quantization. The DF gains and phase lags for the four example control/estimation algorithms were shown to be orders of magnitude faster to compute using the models than by
using simulation. Bode-like plots were derived for the example control/estimation DFs with
113
1000-
500-
0
E
-500-
-10000.2
0.4
0.6
0.8
1
1.2
Time Steps
1.4
1.6
1.8
2
x 104
Figure 5-15: Time series for the QLH system with quantization level h = 35. Red lines
are drawn for reference at amplitude ratios +/- 38.5, and +/- 70 as predicted with the QLH
describing function.
-- Q
QLZ Model
-
7-
-QLH
-
6
Model
QLF Model
0 QLZ Sim
0 QLH Sim
o QLF Sim
C)
5-
0
0
0
40
50
60
70
Quantization Bin Size h
80
90
U-
4
3-
20
30
100
Figure 5-16: Frequency o of limit cycles from the analysis and from simulation. The QLZ
model curve is nearly on top of the Q line.
114
quantization, and these bode plots are useful to predict limit cycling. Sensitivity-like plots
were also presented, showing how the DFs can be used to analyze system response to disturbances. Finally, a numerical example was given where the different control/estimation
strategies with quantization were used to control a plant and compared. The DFs predicted
limit cycles in each case, which were verified by simulation.
115
116
Chapter 6
Describing Functions as Design Tools
6.1
Introduction
The channel describing function can be used not only as an analysis tool, but also as a
design tool to ensure a system limit cycles at a predetermined amplitude. In some applications, such as an underwater vehicle navigating through a cluttered environment, it may
be important that the vehicle stays within a certain distance of its target trajectory. A quantized and lossy sensor feedback channel may, however, cause the vehicle to limit cycle at
an amplitude larger than this safe distance. Using traditional LTI control design methods,
a controller would be designed and, only afterwards, simulated or analyzed with describing functions to see if undesirable limit cycles existed. The describing functions derived
earlier are useful analysis tools that could identify these limit cycles much more quickly
than simulation. However, this iterative method may still not result in limit cycles of small
enough magnitude for the application. In this case, it would be important for the control
engineer to specify a maximum acceptable limit cycle amplitude, and design a codec to
meet this specification. This is the motivation behind the controller-codec design method
described. For clarity, Table 6.1 lists the mathematical symbols used in this chapter and
their definitions.
117
Symbol
As
coTs
GQ
at
Oat
h
Definition
Decoded output at time step k
Decoder function
Decoder function
Controller
Plant
Designed decoder gain if packet received
Designed decoder gain if packet lost
Time step variable
Frequency
Time step size
Decoder DF gain
Decoder DF phase
Quantizer DF gain
Design target limit cycle amplitude
Design limit cycle frequency
Quantization bin size
K1 ... K14
Co...C5
Intermediate variables
Intermediate variables
g
Time horizon for DF
Sk
P(k)
Q(k)
C
P
M
N
k
Co
Table 6.1: Summary of mathematical symbols used in this chapter.
6.2
SISO Design Equations
6.2.1
System Assumptions
To illustrate how a decoder can be designed using describing functions, we first consider a
SISO system with the sensor feedback channel subject to quantization and Bernoulli packet
loss, as shown in Figure 6-1. Design for a MIMO system will be described later.
Input
1Controller
==4
Quantizer
Decoder
I
P ant
Binary Erasure Channel
Encoder
I
I
Describing Function
Figure 6-1: Block diagram of SISO control system with binary erasure channel in sensor
feedback, and describing function for the quantizer-encoder-channel-decoder.
118
In this design method we assume two operating conditions exist for the system: a nominal case with no packet loss or quantization, and a second case subject to packet loss and
quantization. The goal is to design a decoder that achieves desirable performance in the
nominal case, and meets a maximum limit cycle amplitude requirement in the loss and
quantization case. The design procedure can be summarized as follows:
1. Design controller to achieve desirable performance in no loss, no quantization case.
2. Pick baseline decoder for loss and quantization case (i.e. QLZ, QLH, etc.) and
determine limit cycle amplitudes using DFs.
3. If LC amplitude too large, pick smaller target limit cycle amplitude.
4. Find corresponding decoder using design equations.
5. If no solution, pick larger LC amplitude midway between current target amplitude
and previous amplitude and return to step 4.
6. Check effect of decoder on Nyquist plot of no-loss, no-quantization case.
7. If unstable, or if stable but other performance metric unacceptable, pick larger target
LC amplitude midway between current target and previous and return to step 4.
8. If stable, end, or pick smaller LC amplitude and return to step 4.
At the end of this design procedure, a decoder will be found that strikes a balance
between good performance in the no-loss, no-quantization case, and acceptable limit cycle
amplitudes in the case with loss and quantization.
Steps 1 and 2 are dependent on the specific application, and left to the control engineer.
Once a controller and target limit cycle are chosen, the decoder that results in the target limit
cycle amplitude must be designed. To design the decoder, assume the encoder-channeldecoder can be written in the form Sk = P(k) + Q(k)Sk-l for state estimate Sk, where P(k)
and Q(k) are of the form defined. This is an important assumption that allows the general
describing function equations to be used, and assumes no encoder. Then, for the given
controller C(z) and plant P1 (z), the describing function for the feedback channel can be used
119
to design P(k) and Q(k) to result in the target limit cycle amplitude. Many potential P(k)
and Q(k) values may exist for a given limit cycle specification. We examine a particular
design where it is assumed for simplicity that E[P(k)] = Msin(kor) and E[Q(k)] = N, for
some constants M and N to be designed. The describing function for Sk is assumed to have
amplitude As and time shift Ts, which we write in complex form as
A -
6.2.2
Tswti.
(6.1)
Limit Cycle Criteria
For this system, a limit cycle will occur at frequency wt and target amplitude at if the
Nyquist plot of the system passes through the critical point. This happens if the following
two conditions are met
GQ(at/h)IC(jotr)IjP(jOw) I-
at
= 1
ZC(jw1 ) + ZPI(jw 1) + Tco =
C
(6.2)
(6.3)
where GQ(at /h) is the quantizer gain for limit cycle magnitude at and quantization bin size
h.
6.2.3
Solving the Nyquist Equations
,
If As and Ts are expanded using equations 4.23 and 4.22, and solved for terms K4 and K5
then the two conditions can be rewritten as
at(K 3q+K2 +K1)
GQ(at/h) IC(il) I JP(jwt) 1|l +q 2
K5 =
at (K3- qK2 + qK)
GQ(a 1 h) IC(jt|) IIP(jcot)|
where
120
1+ q 2
(6.5)
(6.5
q = tan(lr - ZC(Ujw) - ZP(jco)).
(6.6)
From the definitions of K4 and K5 , and the assumptions for E[Q(k)] and E[P(k)], it can
be shown that
g
K4
k
(
=
(6.7)
Nk-iMsin(jotr) sin(kotv)
k=1 j=1
g
(6.8)
E -kMsin(jcotr) cos(kwt -r).
K5 =
k
-
N
k=1lj=1
By using standard trigonometric summation identities, rewriting some trigonometric
terms in exponential form, and using geometric series relations, equations 6.7 and 6.8 can
be combined with equations 6.4 and 6.5 to solve for N and M, which will define the system
decoder.
From Equation 6.7 we first define
9
k
K13=E
(
rr)) sin(kcot r)
Nk-M sin(It
(6.9)
k=1I 1=1
which can be simplified as
k sin(Iwt r)'\snk'i
g (N
iNN
K 13 = M
k=1 \
sin(kotr)
(6.10)
\=1
and
K13 = ME (fs(k,N)Nk sin(kotZ))
(6.11)
k=1
where we define
k sin(lotr)
fs (k, N).
N
From Equation 6.8 we also define
121
(6.12)
g
(6.13)
= -M L (f,(k,N)Nkcos(kct T)).
k=1
K14
The term K13 can be expanded by noting that
(
-
~cos(o:'r))
N22 +
C(WT)
2 + (sin (ctT)) 2
cos(Wr))
Nk+ I
N
(sin (wtV))
2
sin(( tr)
g N
N
I k=1
Nk sin(katT)] (6.14)
cos((k+ 1)ot r)
sin(kot T)
N
-1s(OtT)
1 (sin((k+
1)cot'r)sin(kcotr))]
-
N
-
Nk+
1
(1
1r)
cos((k+1)wcr) sin(wir) _ sin((k+
)
L(@(
=
'
E fs(k,N)Nk sin(kcot)
(6.15)
For further simplification of K 13 we note that
L
(Nk sin(kowtr))
k=1
=
fS (g,
(6.16)
N
and
I (cos((k+ 1)ct T)sin(korT))
k=1
g
(cos(kcotr) cos(cor) - sin(kcotr) sin(kcotr)) sin(kcotT)
k=1
Cos (cot r)
9
g
2 cod)sin(2kwcr) - sin(wor) I sin2 (kor, r)
2
k=I
k=1 I-I
)
j-2-2COS (2ko.)
122
(6.17)
Cos(WOrt)
2 sin(wtr)
[
-si(COT)
1
12
J
sin(ct(2g+ 1),r) sin(wtr) ]
2sin(cotr)
-
2
cos (cotr) - cos(co (2g + 1)r)
(6.18)
We also note that
k
r)
([sin(kcotr) coscot r + sin(cot) cos(kcotrT)] sin(kcot
=
cos(cotr) E sin2 (kotr) + sin(orC) I sin 2 (2kot'r)
k=1
sin(c (2g+1)r)-sin(ogr)[
2 sin (wt r)
) _
sin(wtr ) cos(cotr) - cos(ot (2g + 1)T)
+ 2
2 sin(ort)
1
1
2
)
COS(cotr)
=
(6.19)
k=1
Combining all of these terms, K13 can be rewritten as
1
_
M
(I1
cos(otr)
2+ (sin (wr))
2
[A sin( ct r) -B
N
1-cos(otr)) 11
N
NJ
)
K13
(6.20)
where
1
)
A =fsg
-
1
[cos(oIT) cos(Ct) - cos(,(2g + 1))
N
2
2 sin(ort)
- sin
- 1
t)
2
sin(wt(2g + 1)r) - sin(cotr)
2sin(0tr)
/1]
)
s2
J
and
123
(6.21)
B = cos((O.t)
[2
-
2
(sin(wt(2g+ 1)r) - sin(cotr)
2\sin (ot)
)I
s+~mr
2
cos (cot ) - cos(cot (2g + 1),r) .(6.22)
2sin(otr)
We further rewrite K13 for future computations as
1
K 13
M
2
COS(Wt)
+
sin(wt) 2
sin(w1 1)
N
(f
NjKs(wt)
1(1)
1 - COs(o)rt) I K9(ot)
N
N
Following similar derivations,
M
can be written as
-J
_
(
COS(Cot)
2
1
sin(wor)
+ ( si(O r
N
2
1
i
( -
cos((k+1)cor)) cos(koT)
k=1 (N
s
cos(otr) 1
N Lsin((k 1)ct)cos(tr)J
N
)
K 14
K14
(6.23)
(6.24)
and as
K14 _
M
-1
(
_ cos(wcr)
sin(wo t)
2
(sfltr))
2
N
1
(f
'N)
1-cos(aotr)
N
In this case we define
-1 Ki I (O)t))
N
fc for convenience
I K12(Ot)
N
(6.25)
the same as f, except with cosine instead of
sine. The term fc can be simplified by
124
e
L
=Re
(ix
(6.26)
)
e2+x
ie +
)
fc(g,N) = Re
g)
(1
cos((k+1 ))
Nk+ I
+ sin((kN)cootv)
_ Cos(cot r)
+
N )
(1
NOWr)
-
sif(OtT)
N
Nk+ I
2 +
sin(tr)
Wjr
2
-
1.
(6.27)
2
From Equation 6.23, M is solved to be
(I
(fs (g,
M = K13 sin
- "
k)
sn(not-r)'
-
2
-
(6.28)
-K8(C0t))
K(w)
Now that M is solved for, we must next solve for N. This is possible because we started
with two equations and two variables (M and N). To solve for N, we next take the ratio of
[f
/1 \ K8 1
cos(otr) K9)
N
N
I
K11
)
K14 sin(ot
N
K13 \
)
Equation (6.23) and (6.25), giving
(sin(otr)
N
_
\f
N
N
cos (otr)
N
K12
N)
(6.29)
and
K 14
K 13
sin(ot r)
(fs (g, -) - -K(0))
(
COS(Wt)
- I-COS(O)
2
+ (
)2
(f
-
sin(wr)
K 9 ()
(6.30)
(g,A) - -Kii (Cot))
(1which can be rewritten as
125
COS(LOr)
2
-
+
1 -cos(wr)
(sin(w,r))
IK12(Ot)
2
sin((or)
N
K13 \
C-
COS(Wot))
K 14
)
9
N
ED
cos((otr)) K1)
(6.31)
-
N
N
) NI
K1]
1-- N
where
C
sin((g+l)cotr)(I -Nos(cor))
(1-N9+1 cos((g+ )o)r))Nsin(or )-N9+I
(1- Ncos(wor))2 + (Nsin(cotr)) 2
(6.32)
and
D
(1
2 sin((g+ 1)CO tr)sin(otr)
1
Ng+' cos((g+ 1)co,1 ))(1 -Nos(cotr))+N9+
2 + (Nsin(cotr)) 2
(1 - Nos(Wtr))
(6.33)
1-2Ncos(rtT)+N 2
K 14
E sin(cotr)
-
Multiplying by the denominators and expanding gives
(N- cos(otr ))K9 [(1 - Ncos(otr ))2 + (Nsin(otr))2]
K13
=[-
1-2Ncos(cotr)+N 2
2
2
(N- cos(tr))K2 [(1 -Ncos(orT)) + (Nsin(wcr))
N[(.1 -NAeos~ort&)) +(Nivsinkt))
'
F sin (ctr)
(6.34)
N 2 [(1 - Ncos(Cotr)) 2 + (Nsin(wotr)) 2]
where
126
wt) (I - NCOS (cotr))
))N2 sin (cor) - N9+2 sin ((g + 1)m
E =1N9+ 1 COS ((g + 1) or
2
2
)
N ((1 -Nos((otr)) + (Nsin(otr))2
2-2Ncos(otr)+N
2
'
K 8 ((1 -Nos(Ct-r)) 2 + (Nsin(wotr)) 2
)
N 2 ((1 -Nos(,otr)) 2 + (Nsin(cotr)) 2
and
N(1-Ncos(ot r)-N9+l cos((g+ 1) o r)+N_+
F =
2
cos((g+ 1)wrt) COS(w, T))
cos((g + 1)cot r))(1 -Nos((otr))
I -(1N9+1
N 2(1-Ncos(ot ))2 + (Nsin(corr))2
N+N9+3 sin((g + 1)cotr) sin(ortr)
N2(1 - Nos(or r)) 2 + (Nsin(trr)) 2
2
(N+K1) ((1 - Ns(w'r)) 2 + (Nsin(otr))2
N2 (I - NCOS (ot r))2 + (N sin (cot)) 2
'
1-2Ncos(wto)+N
Rearranging terms and multiplying again by denominators gives a polynomial expression for N as
127
Ng+3K5 sin(cr)
-cos((g+ 1) colr)sin(cotr)+sin((g+ 1) cotr)cos(cot-r)
+Ng+2
(- sin((g + 1) cor))
K5 (sin(cotr)(sin(otr) - K8 )+cos(cotr)K9 +2K 9 cos(cotr) + N3 K5(-K9)
)
+N
sin(WoVr)
+NK (sin((tr)2K8 Cos(o tT) - K9 - 2K9 cOs2(wot)
+
Ng+3 (-sin(otr)(cos((g+
1)o 1r)cos(cor)sin((g+ 1)cor)sin(wt'r))
+Ng+
+N
sin(t-r))(- oscotr) -Kio+2cos(
+N 3 (sin(otr) +K 12 ) +N
(-Ks sin(cor) +K 9 cos(cot T)
1tr))
- sin (Cotr)(- Cos((g +1) ort))
-K 1 2 cos(ojr) - 2Ki2 cos(flr))
-2KIO sin(cot) cos(cwt') +K
12 +2K1 2cos2(otr))
+K 12 (Kio sin(otr)) - cos(cor)).
This polynomial can be further rewritten in standard form. Thus, we have the following
relations for N and M:
( cos(cor) + 1 )K6
1
M =
9(k-C
( K7
WsiWt)
(6.35)
-N(
-
N
CsNg+3 +C4Ng+2 + C3N 3 + C2N 2 + C 1N + Co = 0
where K6.. -K10 are defined as
128
(6.36)
at (K3q+ K 2 + KI)
GQ(at/h)|C(jot) IIP(jot )|
K6 =
1 + q2
(6.37)
K7 = K7a +K7b
(6.38)
K8 = - cos(cotr)K3 - sin(wtr)(KI+K2)
(6.39)
K9 = -sin(cotr)K
(6.40)
at ((1 -q 2 +q2 )K3 - qK2 +qK)
K10 =
K11
3 +cos( 1 ,r)(KI+K 2 )
GQ(at/h)IC(jt)IIP(jot) IV 1 +q
= cos(wotr)(Ki- K2 )+ sin(wr)K3
(6.41)
2
(6.42)
K 12 = - cos(ot T)K3 + sin(cor r)(Ki - K2 )
(6.43)
with
K7a
K7b
(1 -N+
=
1 cos((g+1)wotr))Nsin(or
1 - 2Ncos(wt'r) +N
2
)
(6.44)
Ng+' sin ((g + 1) otr )(I - N 2Cos (o T)).
1 -2Ncos(cor)+N
(6.45)
The coefficients Co.. .C5 , which are independent of N and M, are defined as
=
(-K
)cos(cotr)
8 K- -K1)sin(otr)+ (K12+K
9
K6
K6
C1 = 2(K8
K6
+K,
)sin(cotr)cos(cotr)
C2
=
+2(-K- K- K12)cos 2(Cor) - K9 - -K 12
K6
K6
K10
K1
sin(orT)(sin(cotr)-K)+3cos(o-r)(K 9 +K
K6
K6
+sin(cor)(cos(a 1 r) -K1)
C3
=
-K9
C4
=
-K-
C5
K6
12
)
Co
-K 12 -sin(otr)
sin(cotr) sin((g +1 )ot r)
K)
-
sin(cotr)cos((g+ 1)otr)
K10 sin(cotT) sin(gcotr) + sin(wtr) cos(cotr).
129
6.2.4
Solving for Codec Parameters
Equation 6.36 is in terms of only N, not M and thus can be solved directly. For convergence
of the describing function g is assumed to be large, and thus there are two possible solutions
for N from this equation.
Case 1: INI > 1. In this case, because g is large, terms C3 N 3, C2 N 2 , CIN, and Co are
small by comparison and can be neglected. Equation 6.36 thus simplifies to the linear
equation
C5Ng+ 3 + C4Ng+ 2 =0
(6.46)
with the simple solution N = --.C5 However, if INI > 1, this implies K7 >> 1, which leads
to M << 1. A controller with a value of M very close to zero would not work, because
received signals would essentially be converted to zero. Thus, N is designed such that
INI < 1.
Case 2: INI < 1. The solution INI = I could be a solution in particular cases, but not
necessarily in general and thus we constrain our focus to the case of INI < 1. In this case,
because g is large, terms C5Ng+ 3 and C4Ng+
2
are small and can be neglected. Equation
6.36 thus simplifies to the cubic equation
C3N 3 +C 2N 2 +CIN +Co = 0,
(6.47)
which has a closed-form solution for N, with at least one real solution. The three solutions
for N are given by
N=-
(6.48)
C2+UJI+
3C 3
Uk2+
where
130
3
A 2 -4Ag
Ai+
2
-
1
U1
=
U2
=
2
U3
=
2
Ao =
C2-3C 3 C
A1 = 2c3 - 9C3C2 C1 + 27C Co.
Because N and the controller are being designed, we are free to choose any frequency
Ot
that results in the smallest real solution for N being less than one in magnitude. If no
such frequency exists, then the system will not limit cycle at the desired amplitude and a
different limit cycle amplitude must be chosen. If a frequency does exist that results in
|NI < 1, then the corresponding solution for M is given by
)
(I - 2cos (OD, ) + 1 )K6
(fsin(o)tr)
K7 - (K 9 (1-cs(Wlr)
6.2.5
Codec Implementation
To implement this encoder/estimator, P(k) and Q(k) are defined as follows:
6
P(k)
=
Q(k)
=
k
M
-yk
(6.50)
N
(6.51)
1- aat
where yk is the sensor measurement. This way, when the system is limit cycling, E[P(k)]=
Msin(kotr) and E[Q(k)] = N as desired, for a channel with packet loss probability a and
packet transmission Bernoulli variable 6 k as defined previously. For this choice of P(k) and
Q(k), the system will limit cycle at frequency cot and amplitude at.
131
6.3
SISO Design Numerical Example
F
Figure 6-2: Example 1-mass, 1-input 1-output system. Mass is connected by a spring and
damper to a wall, and acted on by a force.
To illustrate the SISO design method, we consider a simple numerical example of a
mass-spring-damper system where position is controlled by a force input, and our goal is
to regulate position to zero. The system is the same as that considered in Section 4.4 with
parameters m = 4, k = 1, b
=
2, and is shown again in Figure 6-2 for convenience. In this
example the nominal operating condition has a time step of 0.01 seconds with no delay,
but a second operating condition is subject to a delay of 1. 1 seconds in the sensor feedback
channel, and is subject to quantization with bin size h =1 and packet loss probability of
1.4%. This example could represent, for example, an AUV navigating via LBL nodes
through open water (first scenario) and through a cluttered environment (second scenario).
For ease of navigating through the design example, all time series plots are collected
in Table 6.2 and Nyquist plots in Table 6.3. The plots are for the three cases of channel
considered (original system with no delay, system with delay, and system with delay, loss,
Time series plots are given in terms of nondimensional time, defined here as gor
*
and quantization), and the three decoders considered (LZ, LH, and designed decoder).
time, where cod is the dominant frequency of the plant. For SISO systems cod is the natural
frequency of the plant, for instance
k/rn for a mass-spring-damper case. For MIMO
systems it can be any of the natural frequencies of the plant. This is a useful choice for
nondimensionalization because it includes the key dimensional parameters of the system
- the time step and plant dynamics - and the user-chosen convergence parameter g, and
132
LZ
1.2
.2
-
I2
Design
LH
1.2-
1
1
.80.8
Original
08
26
0.A
0.4
0.4
0.2
0.2
0.2
1.6
i.6
1.6
1.4
1A4
1 4
1D2
1.2
1.2
a
nn60m
100
SioTm
0Nn
IOn
nd
0Tm
en
120
110
2.12a
n100
40
a"4a
20
I4
Tie10
D
N
1
Delay
A%-
1
0.
E5
0.6
0.41
Z6.
0.6
500
00
12d2
O 22
i-
1od-
T
2 222
dm2 s"
2
Soo
0.6
0.4
T-
F.2222222
22220
20022122 212 224M2
n2 T
nd-
25
Delay+
Loss+Quant
-1
-..
-
-'.
Z5
5
2222
2
1
2M9030DO
2,2
4
22222222222-2222222
0
0
0
am
7 2.0 100
_n20
300
0400
000
T00
500
Soo
Table 6.2: Time series responses for system with no delay, delay, and delay+loss+quantization, for LZ, LH, and designed decoders. Top two rows are step responses, with bottom row an initial condition response for regulation control. Nondimensional time is used, defined as go * time.
will allow more direct comparison of systems with different time steps, plant dynamics,
and convergence parameter values. For this example, as will be described later, we use
g = 1000, cod=0.5, and T= 0.01.
6.3.1
System with No Loss
We assume for this example that position is sensed, and the system is controlled with a
proportional-integral (PI) controller with proportional gain Kp = 1 and integral gain Ki =
0.39. Figure 6-3 (top plot) shows the step response of this system. The gains are chosen to
133
LH
0
Delay
*__
*
Original
Design
-
LZ
Table 6.3: Nyquist plot for system with and without delay, for LZ, LH, and designed decoders. Nyquist plots are in general not defined for systems with loss and quantization, and
thus are not shown for those cases.
give the system a 40-degree phase margin, with resulting step response characteristics of
15% overshoot, 25-sec settling time, and no steady state error. If the system is subject to a
1.1-second delay, the step response is more oscillatory but still stable, as shown in Figure
6-3 (bottom plot).
Figure 6-4 shows the Nyquist plot of the system with no delay (top plot), and with delay
(bottom plot). The original system is stable, consistent with the time responses. However,
the delayed system has a crossover point of -0.95, and is close enough to instability that the
addition of packet loss and quantization will cause it to limit cycle.
6.3.2
System with Packet Loss and Quantization
We next examine the effect of packet loss and quantization on the second operating condition (with the 1.1-second delay). The simplest way to deal with lost packets is to use
either a zero-output or hold-output decoder. However, in this example case these decoders
result in large limit cycles. Using the QLZ and QLH describing functions, the predicted
134
1.2
1
0.8
< 0.6
0.4
0.2
0
20
40
60
80
100
120
140
Nondimensional Time
1.8
a
1.6 1
.
<0.
0.411'
0.2
01
0
500
1500
1000
Nondimensional Time
2000
Figure 6-3: Step responses of original system (top plot) and system with 1.1-second delay
in sensor feedback (bottom plot). Nondimensional time is used, defined as gCOr *time.
135
2.5
2
1.5
Ce
0.5s
0 - - - - - - - - - - -X
E
-1
-1.5-2
-2.5
-2
-1
0
Real
1
2
2.5
2
1.5
Ce
0.5-
----
0 ---------
7:M
- --------
---------
-- 0.
-1
-1.5-2-2.5
-2
-1
0
Real
1
2
Figure 6-4: Nyquist plots for original system (top plot) and system with 1.1-second delay
in sensor feedback channel (bottom plot). Nondimensional time is used, defined as gco T*
time.
136
......
..........
.....
-.............
. ....
....
.
....
.
.........
..
2h_
1.5
0.5
0
E
x
-0.5
-1
-1.5 F
-
II -I
11 1 VII 11I-11111 -I -I100
200
300
400
500
Nondimensional Time
600
700
100
200
300
400
500
Nondimensional Time
600
700
2.5
2
1.5
1
I
0.5
0
x -0.5
-1
-1.5
-2.56
Figure 6-5: Time series plots for QLZ (top) and QLH (bottom). Nondimensional time is
used, defined as gO 2 T * time.
maximum limit cycle amplitudes are 1.79 for QLZ and 1.91 for QLH.
Figure 6-5 confirms these predictions with simulation time series plots of QLZ (top
plot) and QLH (bottom plot). Dashed black lines indicate predicted maximum limit cycle
amplitudes and vertical red lines indicate packet losses. The plots show generally good
agreement with predicted and observed limit cycles.
6.3.3
Codec Design
We desire the limit cycle amplitude to be as small as possible, and the SISO design procedure will allow a decoder to be designed that significantly reduces the limit cycle amplitude.
Following the design procedure, we start with the QLZ limit cycle amplitude of 1.79 and
test smaller design points, as shown schematically in Figure 6-6 and Table 6.4. In Table 6.4,
A(D) represents predicted limit cycle amplitude (for LZ and LH), and designed limit cycle
amplitude (for DI, D 2 , D 3). Note that all design points (D 1 , D 2 , D 3 ) include the effect of the
137
Decoder
LH
LZ
D1
D2
D3
M
1
1
0.969
0.883
N
1
0
0.107
0.107
A(D)
1.91
1.79
1.0
0.85
0.840
0.107
0.7
)
Table 6.4: Codec coefficients and limit cycle amplitudes. A(D) represents predicted limit
cycle amplitude (for LZ and LH), and designed limit cycle amplitude (for D 1 , D2 , D 3
quantizer. Notationally we refer to the decoder in the form Sk
=
MOkYk + N(1
-
Ok)Sk- 1,
for measurement yk and decoded signal Sk at time step k. This way we can refer to the
constant coefficients M and N for the design.
D3
0
D2
QLZQLH
Di
@4--------------.-
------------
1
0.5
1.5
2
Limit Cycle Amplitude
Figure 6-6: Plot of limit cycle amplitudes for QLZ, QLH, and designed codecs. Blue dots
represent design points.
We first chose the design point DI
1 as a significant decrease in amplitude. The
quantizer gain for a/h = 1 is high enough that no larger values of a/h result in the same
-
gain, and thus no larger undesigned-for limit cycles will result. This results in X = 0.958
0.00009i and codec M = 0.969 and N = 0.107. This design mathematically results in a
limit cycle of the desired amplitude, but it is important to consider the effect of the decoder
on the system with no loss or quantization. Figure 6-7 (top plot) shows the Nyquist plot
of this system, and it is seen to be stable, with a gain margin of 1.09, phase margin of 5.4
degrees, and no encirclements of the critical point.
Continuing the design procedure, we tested a smaller design point D 2 = 0.85. This
point was chosen as a smaller point that will still result in no larger limit cycle amplitudes,
as determined by the quantizer describing function plot. The Nyquist plot of the system
with the resulting decoder but no loss or quantization is also seen to be stable (Figure 67 (middle plot)). We next picked a smaller design point, D 3 = 0.7. The corresponding
Nyquist plot of this system is stable (Figure 6-7 (bottom plot)).
138
..........
....
0.5F
0
E
-0.5F
-1
-1
-0.5
0
0.5
1
1.5
Real
1 .5r-
0.5h
0
E
-0.5
1.-
-1
-0.5
0
Real
0.5
1
1.5
-1
-0.5
0
Real
0.5
1
1.5
1.5 r-
0.5a
0
E
-0.5-
1
Figure 6-7: Nyquist plots for system with decoder and delay but no loss or quantization.
Plots are for designs DI (top) to D 3 (bottom).
139
We settled with this as the design point with the smallest limit cycle solution because,
based on the quantizer describing function plot, smaller design points would result in undesigned-for larger limit cycle amplitudes with the same quantizer gain. The resulting solution for D 3 is A = 0.830 - 0.00008i and codec M = 0.840 and N = 0.107 with a predicted
frequency ( = 0.59rad/s.
The designed points are summarized in Figure 6-8 (top plot) in terms of the entire
design space, with limit cycle amplitudes plotted as a function of M and N parameters for
the case of delay, loss, and quantization. Decoders QLZ and QLH are also shown. The plot
shows three regions: LC solutions, No solution, and Unstable. In the unstable region the
value of M is high enough that the delayed system goes unstable in the absence of loss and
quantization and thus these values of M and N are unacceptable. In the No Solution region
no limit cycle solution exists, and thus the analysis says nothing about performance. The
region of interest is the LC Solutions region, where the analysis holds.
The plot shows a stronger dependence on M than N, and this is probably the case
because the low packet loss probability makes the value of M more important than N.
However, N still has an effect. For example, the LZ and LH decoders have the same M but
different N values, and LZ has a limit cycle at an amplitude of 1.79 while LH limit cycles at
1.91. The bottom plot of Figure 6-8 is given at a different scale to illustrate that the contour
lines are not horizontal, and that there is a dependence on N.
The region of smallest limit cycle amplitude of the plots is in a region along the line
connecting points (N,M) = (0,0.84) and (N, M) = (1,0.82). The design point D 3 is in this
region, corroborating it as a decoder that leads to approximately the smallest limit cycle
magnitude in the design space. While all design points along this line lead to the same
limit cycle amplitude, they result in different negative side effects. As will be shown later,
solutions with smaller values of parameter M, corresponding to the bottom right part of
Figure 6-8, result in higher steady state error to step inputs and higher sensitivity function
magnitudes at low frequencies. Thus, the solution with the smallest limit cycle amplitude,
smallest steady state error, and smallest sensitivity function magnitude will be in the upper
left end of this solution line, where design D 3 is located.
While this plot alone could potentially be used to design the system decoder, it is much
140
Limit Cycle Amplitude
1.15
3
1.1
1.05
2.5
2 0.95
2
0.9
0.85
-
1.5
0.8
No Solution
1
0.750.70
0.2
0.4
1
0.8
0.6
N
1.2
Limit Cycle Amplitude
1
3
2.5
2
1.5
0
1
0
0.5
1
N
Figure 6-8: Limit cycle magnitudes as a function of M and N. Design points, QLZ, and
QLH are also shown. Bottom plot with different scale illustrates contour lines are not
horizontal.
141
Decoder
QLH
QLZ
DI
D2
D3
LC Amplitude
1.91
1.79
1.0
0.85
Gain Margin
1.05
1.05
1.09
1.20
Phase Margin [deg]
3.3
3.3
5.4
11.7
Steady State Error
0%
0%
3%
13%
0.70
1.26
15.5
18%
Table 6.5: Gain and phase margins of system with delay and different decoders, with no
loss or quantization.
more computationally expensive to generate, and would still need to be used in addition to
Nyquist plots to determine the effect of the decoder on the nominal system. Moreover, the
plot does not carry over to MIMO systems, whereas the general design procedure does, as
will be explained in a later section.
6.3.4
Design Point Side Effects
While each design point represents a smaller resulting limit cycle amplitude, the decreases
in amplitude come at the price of potentially undesirable side effects to the no-loss scenario.
The main side effects in this example are effects on the gain and phase margin, system
sensitivity, and steady state error to step responses.
The gain and phase margins resulting from the designed decoders are shown in Table
6.5 for the system with delay but no loss or quantization. While these gain and phase
margins don't directly apply to the system with loss and quantization, they represent a
sort of metric of how the delayed system is affected by the decoders. It is interesting to
note that the design points actually increase the gain and phase margin of the no-loss noquantization system, making it more stable. All phase margins are very small, with the
largest corresponding to design D 3 . Design D3 also has the largest gain margin.
Another side effect of decreasing the limit cycle amplitudes is that sensitivity is increased at low frequencies. Figure 6-9 shows the output sensitivity functions of the system
with unity feedback and the design points D1 , D 2 , and D 3 , with and without delay. We use
the sensitivity function defined as
142
I
S= I +CPGd
(6.52)
for controller C, plant P, and decoder gain Gd.
For low frequencies, as would be expected from disturbances, the designed codecs
slightly increase the system sensitivity as compared to unity feedback. This is because, in
the case of no loss and no quantization, the codec acts as a gain that is less than one, and
the sensitivity function is inversely proportional to this gain. The clear spike in the sensitivity plot of the delayed system at frequency 0.5 rad/sec is coincident with the resonance
frequency co of the system given by
-
or =
Vm
(6.53)
This frequency is 0.5 because k = 1 and m = 4 for this system. The codec apparently
reduces the sensitivity of the delayed system at this frequency, however it has little effect
at higher frequencies. At higher frequencies the sensitivity function magnitude approaches
unity for delayed and non-delayed systems, consistent with the plant acting as a low pass
filter.
The final side effect we consider is the effect on steady state error to a step response in
the no-loss scenario. While the ultimate application for this example is a regulation task
where position will be controlled about a zero reference point, and -steady state error will
not be seen, it is still important to have a control system without too large steady state
error to a step input. Because the original system has integral control, no steady state
error results. But the design points add a gain to the feedback channel and this non-unity
feedback induces a non-zero steady-state error to a step response. The steady state error
for this system is defined as
Ess = I -
__CP
1 +CPGd
(6.54)
where C, P, and Gd are the transfer function magnitudes at low frequencies for the con-
troller, plant, and decoder respectively. For low frequencies the term CPl
large because of the integrator, leading the steady state error to simplify to
143
becomes very
Original
--
1.5 -
D1
1 -
0.5
C:
0)
C
0,
_j
2
D3
- - - Original+Delay
- .Di +Delay
0 -... D2+Delay
--..
D+Delay
-0.5
-1
000
-1.5
-2
0
-1.5
-1
-0.5
Log, 0(Frequency)
0
0.5
-Original
-0.2
D2
3~
-D
-
-0.4
--- Original+Delay
-0.6 S-. . D +Delay
2
.
D . .D+Delay
Ca -0.8
. .
.D+Delay
0,
-1
-1.2
-1.4
-
-1.6
-0
-'0
-2
-1.5
-1
Logl(Frequency)
-0.5
,
Figure 6-9: Sensitivity magnitude plots for original system and designed systems DI, D 2
and D 3 , with and without delay. Bottom plot is zoomed-in version of low-frequency portion
of top plot.
144
Ess = 1 -.
1
Gd
(6.55)
With unity feedback gain Gd = 1 meaning Ess = 0, so there is no steady state error.
However, the non-unity gain for Gd from the designed codecs leads to nonzero steady state
errors. These errors are summarized in Table 6.5. Design DI results in only a 3% steady
state error, while D 3 results in 18% error.
One potential method to deal with these steady state errors to step inputs could be to
scale the step input commands. For instance, using design D 3 , for which the steady state
error is 18%, the desired reference input could simply be scaled by the factor 1/ 1.18 = 0.85
to result in a step response with zero steady state error. This scaling would not affect the
limit cycle performance because it would be done outside the feedback loop.
6.3.5
Final Design
Choosing the final decoder design requires striking a balance between low limit cycle amplitude for the lossy case and low side effects for the no-loss case. Table 6.5 allows comparisons to be made among the design parameters to help choose the final design. Ultimately
the final choice would depend on the specific application, but in this case we assume it is
most important to have the smallest limit cycle amplitude possible without a large steady
state error to step responses. None of the designs have a major effect on system sensitivity,
thus we don't consider sensitivity to be an influential design parameter. Design D 3 results
in the largest gain and phase margins and thus the most stable system. However, these are
still fairly small margins and again not as influential design parameters as steady state error
in this case. Thus we choose design D 2 as a compromise design, with a-two-fold reduction
in limit cycle amplitude at the price of 13% steady state error to a step response.
We verify that the designed codec D 2 indeed results in a limit cycle as designed by first
plotting the describing-function-derived gain-phase loci of the resulting system in Figure
6-10. The top plot represents the system with the equivalent complex representation of
the decoder gain and phase given by A = 0.872 - 0.00008i. As designed, the plot passes
through the critical point, at a frequency of co = 0.59. Note that, because this plot was
145
0.5-
-1-1.5
R
l-0.5
0.5
0 ----
--
-
--
-0.5
-1.
-0.5
Real
Figure 6-10: Describing-function-derived gain and phase loci of system with delay, loss,
quantization, and designed codec. Top plot is for positive X, and bottom plot for negative
X, where X is the complex representation of decoder gain and phase at the limit cycle
frequency.
derived using describing functions, it is only valid at the critical point itself.
Because the values of M and N for the decoder are functions of the tangent of the angle
of X, we must also look at the gain-phase loci using -X as well to see if this plot intersects
the critical point. This is because the tangent of an angle is the same as the tangent of the
supplement of that angle, and multiplying a complex number by -I results in an angle that
is the supplement of the original angle. The resulting plot is shown in the bottom plot of
Figure 6- 10. This plot does not intersect the critical point, and thus we conclude there are
no additional frequency solutions for the limit cycle.
We plot the time series response of the system with quantization, loss, delay, and the
designed decoder in Figure 6-11 and see close agreement between designed maximum limit
cycle amplitude of 0.85 (shown with black horizontal dashed lines) and actual amplitude.
The designed limit cycle amplitude represents a significant two-fold reduction in limit cycle
146
2.5
21.5-
-H0.57
-
CL -1
--0.5-
-50
100
200
300
400
500
Nondimensional Time
600
700
Figure 6-11: Time series of system with designed decoder. Nondimensional time is used,
defined as g(Od * time.
amplitude compared to the zero-output or hold-output decoders.
Finally, we plot the step responses of the nominal system with no loss and no quantization with and without delay to verify the steady state error predictions. Figure 6-12 shows
step responses for the system with no delay (top plot) and with delay (bottom plot), and
indeed the steady state error is approximately 13%.
In general we can conclude that the designed decoder significantly reduces limit cycle
amplitudes in the second operating scenario, with minimal effect on the performance in the
nominal scenario. The main negative effect of the decoder is that it adds a small amount
of steady state error to the system step response. But if limit cycle amplitude is a more
important performance metric, and the application is a regulation problem around a zero
set point, then this designed decoder may be appropriate.
6.4
MIMO Design Equations
A similar design method can also be implemented for a MIMO case. As with the SISO
case, two operating conditions are assumed, a nominal case with no quantization and no
packet loss, and a second case subject to quantization and packet loss. The MIMO design
method is, at a high level, the same as the procedure outlined in Section 6.2.1: a decoder
147
1.2
0.8
E
< 0.6
0.40.2
00
20
40
60
80
100
Nondimensional Time
120
140
100
200
500
300
400
Nondimensional Time
600
700
1.8
1.61.4
1.2*0
<
E
0.8
x
0.6-0.4
0.2
0
Figure 6-12: Step responses of original system (top plot) and system with 1.1-second delay
in sensor feedback (bottom plot). Systems have decoder but no packet loss or quantization.
Nondimensional time is used, defined as go)hT * time.
148
is derived that achieves an acceptable and stable response in the no-loss, no-quantization
case, and meets acceptable limit cycle amplitudes in the loss and quantization case. The
design procedure is given as the following:
1. Design controller to achieve desirable performance in no loss, no quantization case.
2. Pick baseline decoder for loss and quantization case (i.e. QLZ, QLH, etc.) and
determine limit cycle amplitudes using DFs.
3. If LC amplitudes too large, pick smaller target limit cycle amplitude for one channel,
holding others constant at baseline levels.
4. Find corresponding decoder using design equations.
5. If no solution, pick larger LC amplitude midway between current target and previous
and return to step 4.
6. Check effect of decoder on Nyquist plot of no-loss, no-quantization case.
7. If unstable, pick larger target LC amplitude midway between current target and previous and return to step 4.
8. If stable, return to step 3 for next channel, or pick smaller LC amplitude for current
channel and return to step 4, or end if last channel.
The decoder derivation is slightly different in the MIMO case than the SISO case. For
a system with n states, assume a decoder Sk = P(k) + Q(k)Sk_1, similar to that of the SISO
case, with E[P(k)] = Msin(kor)In and E[Q(k)] = NIn, where I,, is the nxn identity matrix.
It is assumed that each state limit cycles at the same frequency co, but at different target
amplitudes al, ... , an,t. Independent packet loss probabilities
Xj
are assumed for channel i,
with expected success probability matrix
[
=
-ai
...
0
1-a
0
0
149
2
...
..
0
I-
(6.56)
a~n
6.4.1
Limit Cycle Criteria
For this system a limit cycle will occur if the following conditions, analogous to SISO
Equations (6.2) and (6.3), are met:
j-1+det(1n+P(jot)C(jot)FD) =
1
(6.57)
Z(-1+det(In+P(jo
1 )C(jw1 )FD))) =
7r
(6.58)
where F is the nxn diagonal quantizer describing function matrix defined as
GQ(a,t/h)
0
0
0
..
0
0
0
GQ(an,tlh)jI
F=
.(6.59)
Here it is assumed for simplicity that each signal is quantized with the same quantization bin sizes. Both Equations (6.57) and (6.58) will be satisfied if the following condition
is met:
det(In +Pi(j3t )C(jcot)FD) = 0.
(6.60)
In general there is no closed-form simplification for the determinant of the sum of two
arbitrary matrices [57]. However, if we design the decoder to be a diagonal matrix, then
each channel will be independent and scalar decoders can be independently designed for
each channel. This simplifies the problem so that equation 6.60 can easily be solved by
optimization methods. The describing function matrix D is thus written as
D(wt)
=
AS's,10)'
0
0
0
...
0
L 0
0
As,,neTS'O
.
(6.61)
When evaluated at a specific frequency, the matrix D is composed of complex numbers
on the diagonal. For use in future equations, we thus rewrite D as
150
D(o) =
A1
0
0
0
...
0
0
0
Xn
.
(6.62)
The only unknown variables in equation (6.60) are now A1 , ... , A. and w,.
6.4.2
Solving The Nyquist Equation
Equation (6.60) can be set up as an optimization problem to solve for the value of ot
and A,
... ,
In that result in the codec P(k) matrix as close to the identity matrix times the
measurement as possible. This will ensure the codec has the least possible effect on the
no-loss system. The Simplex method is used to search through different values of co and
A. Once a solution is found, each complex A value is converted to a gain and phase, and
equations (6.48) and (6.49) are used to solve for the corresponding values of M and N for a
decoder. Each Xi will have a unique decoder pair defined by Mi and Ni. For feasibility we
require
6.4.3
INI
< 1 as in the SISO case.
Codec Implementation
Because we use an independent decoder for each channel, the decoder matrices P(k) and
Q(k) are defined as
P(k)
=
EkV-AMyk
Q(k)
=
(In - Ok)(In -'1)
(6.63)
1N
(6.64)
where
0
A =
...
0
... ... o
0
... 0
151
1
(6.65)
H>X 1
i-
F1
X2
F2
Figure 6-13: Example system with two inputs and four outputs. Two masses are connected
by a spring to each other, and dashpot to a wall, and acted on by two input forces.
and
M =
MI
0
..
0
0
..
..
0
0
... 0
NI
N =
0
(6.66)
Mn
... 0
0
...... 0
0
... 0
.(6.67)
Nn
Thus the full decoder is written as
Sk=
MI
0
..
0
0
...
...
0
0
...0
]
kYk+
NI
0
... 0
0
...
...
Mn j0
...0
0
(In--®k)Sk-I.
(6.68)
Nn
The designed system will thus limit cycle at the desired amplitudes in the packet-loss
case, and the effect of the codec on the no-loss case will be small if P(k) is close to the
identity matrix times the measurement, as designed.
152
6.5
MIMO System Design Numerical Example
As a numerical example, we consider the two-mass MIMO system described in Section
4.4.3 to illustrate the MIMO design method. We consider the case where there are two
force inputs, one for each mass, as illustrated in Figure 6-13, and we are trying to regulate
positions
x1
and
x2
to zero error with respect to their reference values. In this example
the nominal operating condition has no delay and no packet loss with a time step of 0.1
seconds, but a second operating condition that must be accounted for has a sensor feedback
delay of 1.5 seconds in all sensor channels and is subject to quantization and independent
packet losses of 1% in each channel. This example could represent, for example, two AUVs
maintaining a formation while following a leader, with ranges communicated acoustically
to a centralized controller. The first operating scenario could be in open water, and the
second scenario is operation in a cluttered environment that negatively affects acoustic
signals. We first design an LQR controller for the nominal operating condition, and then
design a codec for the lossy scenario. We choose an LQR controller because this is a
regulation task, which is well suited for LQR control.
For ease of navigating through the design example, all time series plots are collected
in Table 6.6 and Nyquist plots in Table 6.7. The plots are for the three cases of channel
considered (original system with no delay, system with delay, and system with delay, loss,
and quantization), and the three decoders considered (LZ, LH, and designed decoder). As
with the SISO example, nondimensional time is used, defined as gco. r * time. For this
example we use values g = 1000, cod = 1, and r = 0.1.
6.5.1
MIMO System with No Loss
For this example we assume sensors exist for all four states, and that the system has two
inputs, a force input on each mass. We design the LQR controller with the
Q matrix equal
to a four by four identity matrix and the R matrix a two-by-two identity matrix. This way
state error and control action are penalized equally. This yields the gain matrix
153
LZ
LH
w.
Design
w.6
0.
0
I"
Iw- -0~ o w00 "W
~~We
Original
-0.4
06
0 4
0 i
02
0
oao
SW
-
5w
toga
-
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2ow
goo
T_
300D
3W
00
0.4
042
Delay
0
ism
1-
2
0
-
05
1.5
T-
. 11
N,,,d-
.,W
T-
I8
1
-1
-02
*4:
-02
1.4i
1.5
Q 5
0,..
Delay+
Loss+Quant
111
II
l
a
i
NDr4kw-rW TI,,la
IIS-2,6
lii
-a.
Z 15
2
I
V iRlill lilt 11411111 It I I i I I
f
......
...
... .... . ............
go
A
I
I
l
1111;1 I
2
2
-2
.1l t .
.... ..I
-..
.....
li
10
0
0
I 111111 II
2523.5
-
1 I1
05
1
222222
Time series responses
t11 ll
11
I
Table 6.6:
2
for system with no delay,
delay,
5
2
5
r.0
and de-
lay+loss+quantization, for LZ, LH, and designed decoders. Nondimensional time is used,
defined as g
Vf *time.
154
. . I 1.
I
LH
LZ
Design
Original
Delay
-1
-5500.
a0
52-.! 0.5
R0a
Table 6.7: Nyquist plot for system with and without delay, for LZ, LH, and designed decoders. Nyquist plots are in general not defined for systems with loss and quantization, and
thus are not shown for those cases.
Kiqr-
0.61
0.32 0.26 0.80
0.52
0.10 0.52 2.14
(6.69)
For this controller we see the following time responses for mass positions Xi and X2
subject to initial position errors of positive one, and zero initial velocities, given in Figure
6-14. The transient errors in each case settle to zero after approximately 150 time steps.
Because this is a proportional controller, however, there will be steady state errors to
step responses. These errors aren't extremely important in this case since this is a regulation
problem where positions are being controlled to zero, but it is still insightful to check that
these steady state errors to step inputs are not too large. For a MIMO system there are
many different combinations of step inputs possible that could yield different steady state
errors, but a good general method to gauge the size of steady state errors is to look at
the magnitudes of the loop transfer functions for different channels. In this case there are
four outputs being controlled and four reference inputs, so this means 16 channels between
155
0.8x 0.6:cx 0.41
0.2
0
-0.2-0.41
o500
E
1000
1500
1000
1500
Nondimensional Time
0.8
0.6!
0.40.20-0.2
-0.4--
0
500
Nondimensional Time
Figure 6-14: Time response of original system to initial position displacements of 1 for each
mass, and initial velocities zero. Nondimensional time is used, defined as gob
T* time.
156
...........
...
....
From V
From X1
0
5
5
0
0
0
H-
-4
0
Cz
-2
0
-5
-4
0
-2
--4
0
0
-2
-2
-4
-4
-5
-4
-2
0
-4
From V2
5
-2
0
-4
-fl
0
-2
0
-4
-2
0
-2
0
-2
0
-2
0
0
0
5
F-
0
From X2
5
-2
1
-2
-4
0
-4
5
5
5
5
0
0
0
0
-j
0
I-
-4
-2
0
-51
-4
0.5
C!71
0
0
0
-2-4
-4
-2
0
-2
0
-1.5
-4
-51
-4
0
-1
-1
-2
-51
-0.5
0
0
-0.5
-4
-2
-0.5
-2
0
-1
-4
Logl(Frequency)
Figure 6-15: Loop transfer functions magnitudes for MIMO example, at low frequencies.
reference inputs and outputs.
The steady state error for any channel is roughly inversely proportional to the magnitude
of the loop transfer function for that channel at low frequency, so a large transfer function
magnitude would result in a small steady state error. We plot the 16 transfer function
magnitudes in Figure 6-15 for a range of low frequencies from 10-4 rad/sec to I rad/sec.
In this plot V and V2 are the states for the velocity of X, and velocity of X2 , respectively.
The outputs of greatest importance are Xi and X2 , since these are the two positions
we are regulating. Figure 6-15 shows that the transfer functions from any input to X, or
X2 have large and increasing magnitudes for lower frequencies, which would yield small
steady state errors as desired. The transfer function magnitudes at low frequencies are not
as large from any of the inputs to VI and V2 , so these would have some steady state error
to step inputs. We conclude, because of the large transfer function magnitudes to outputs
X 1 and X2 , that this is a reasonable controller for the regulation case with acceptable steady
state error to step inputs.
157
0.5
-0
-----. ---.-
-
E
-0.5
-2
-1.5
-1
Real
-0.5
0
0.5
0.5-
0-------------0
----------
-------
E
-0.5
-2
-1.5
-1
-0.5
0
0.5
Real
Figure 6-16: MIMO Nyquist plots for original system (top plot) and system with 1.5second delay in sensor feedback channels (bottom plot).
Next we consider the second operating condition, when the system is subject to a 1.5second sensor feedback delay in all channels. Figure 6-16 shows the Nyquist plot of the
system with no delay (top plot), and with delay (bottom plot). The original system is stable,
consistent with the time responses. However, the delayed system has a crossover point of
-0.88, and is close enough to instability that the addition of packet loss and quantization
will cause it to limit cycle.
Time responses shown in Figure 6-17 for the same initial conditions as in Figure 6-14
confirm that the system with 1.5-second delays in all channels is indeed stable, but takes
much longer to settle than the original no-delay system.
6.5.2
System with Packet Loss and Quantization
We next examine the effect of packet loss and quantization on the second operating condition (1.5-second delay). The simplest way to deal with lost packets is to use a zero-output or
158
1
0.8
0.6
-
0.4
X-
0,2qL
-0.4
0.5
1.5
2
Nondimensional Time
2.5
1
1.5
2
Nondimensional Time
2.5
3
X 104
1
0.8
0.6
0.4
E
0.2
0
-0.4
0
3
.
0
x
Figure 6-17: Time series responses of original system with 1.5-second time delay in sensor
feedback of all channels, for initial displacements of one for each mass and zero initial
velocities. Nondimensional time is used, defined as g co T * time.
159
0.5
-10
' -0.5
Nondimensional Time
X 10
4
0
Nondimensional Time
X 10
Figure 6-18: MIMO design time series plots using QLZ codec. Top plot is for X, response and bottom plot for X2 response. Red lines indicate a packet loss in one of the
four channels, and dashed horizontal black lines indicate predicted limit cycle amplitudes.
Nondimensional time is used, defined as gwo T * time.
hold-output decoder. However, these decoders result in large limit cycles. Using the QLZ
and QLH describing functions, the predicted maximum limit cycle amplitudes for these decoders are al = 0.88, a2 = 3.81 for QLZ, and al = 0.72, a2 = 5.49 for QLH, where a, and
a2
represent the limit cycle amplitudes for states X1 and X 2 , respectively. Figures 6-18 and
6-19 confirm these predictions from the simulated time series responses of the system for
QLZ and QLH, respectively. For these figures, packet losses are marked for a loss in any
channel. It is interesting to point out that even though the individual channel packet loss
probabilities are low, the whole system will be affected by a loss in any individual channel
and thus the low loss probabilities are still significant.
160
>- -0.
2
-1 50n'a
.
' m2 2.5 x1
Nondimensional Time
x 10
6
-2
-6
0
I
III liii 1TIH
011
111
111
III 'j~ j II
10
-ax
0.5
1
1.5
2
Nondimensional Time
2.5
3
x 0
Figure 6-19: MIMO design time series plots using QLH codec. Top plot is for X1 response
and bottom plot for X2 response. Red lines indicate a packet loss in one of the four channels.
Nondimensional time is used, defined as gffl T * time.
161
6.5.3
Codec Design
Ideally the limit cycle amplitudes in this scenario would be as small as possible, and the
MIMO design procedure will allow a decoder to be designed that significantly reduces
the limit cycle amplitudes. Following the design procedure, we start with the QLZ point
of (al,a2) = (0.88,3.8) and first attempt to reduce a2 . We first chose the design point
Di = (0.88,1.04) as a significant decrease in amplitude a2. The value 1.04, specifically, is
chosen such that no amplitudes greater than 1.04 have the same quantizer gain, and thus
there would be no larger limit cycle amplitudes inadvertently designed for.
For D1 we find A, values
2
A
(6.70)
-2.55 +0.16i
=
= -6.18+0.08i
(6.71)
= 0.44-0.0004i
(6.72)
(6.73)
1.29-0.004i
=
(6.74)
with the corresponding decoder
P(k) = Ok
Q(k) = (I- E)
1.82
0
0
0
0
6.35
0
0
0
0
0.42
0
0
0
0
1.23
y
0.38
0
0
0
0
0.12
0
0
0
0
0.01
0
0
0
0
0.03
(6.75)
(6.76)
Figure 6-20 shows this design point in relation to the QLZ, QLH, and other design
points.
162
D
0.9 - DD
2 x
0.8
3
D1
-
DFinal
0.7-
XD
QLH
1 4
X
5.5
0.3 -
0 Viable Design
X Unviable
-
0.2
'
0
0
Figure
signed
design
design
-
- '
'3.9
-
0.4 -
1 .3
L)
1
3
4
2
X2 Limit Cycle Amplitude
Design
'-
'
0.5-
-
E 0.6 -
5
6
6-20: Plot of limit cycle amplitudes for states X1 and X2 for QLZ, QLH, and decodecs. Blue dots represent viable design points, while red exes represent unviable
points. Euclidean norms of limit cycle amplitudes are marked for QLZ, QLH, and
points, showing significant reduction for design point.
The design point D 1 will mathematically result in the desired limit cycle amplitudes
for the lossy system, but practically it is important that the designed decoder has minimal
effect on the system in the case of no loss or quantization. One way of examining this effect
is to plot the Nyquist plot of the system with the decoder but no loss and no quantization,
and check for stability. Figure 6-21 (top plot) shows this Nyquist plot. The system is
stable, with a crossover point of -0.95 and no encirclements of the critical point, and could
potentially be pushed closer to the critical point.
In the next iteration of the design process, we chose a new point with smaller a2 value,
D2 = (0.88,0.7). This time the Nyquist plot corresponding to this design encircles the critical point, as shown in Figure 6-21 (middle plot), and the system is unstable. We continued
the design process choosing a new point midway between the previous two points until we
reached a design point with a Nyquist plot that does not encircle the critical point, given by
D3 = (0.88,1.02). The corresponding Nyquist plot is shown in Figure 6-21 (bottom plot)
and this time is stable, with a crossover point of -0.97.
Because the crossover point is now very close to the critical point, we do not try to
decrease a2 any more, and next focus on decreasing a,. We chose design point D4
163
=
-1.5
-2
-1.5
-1
-0.5
Real
0
0.5
1
E
1.5
1
0-
--------- x
-
--
--
-
0.5-
-
-0.5
-1-1.5-2 -1,5
-1
-0.5
Real
0
0.5
1
E
1.51-
0.50 - ----- ----- -- -- ---- ------ -----0.5-1-
-2
-1.5
-1
-0.5
Real
0
0.5
1
Figure 6-21: Nyquist plots of designs D, (top), D 2 (middle) and D 3 (bottom) for systems
with no nonlinearities (i.e.systems with delay and decoder but no loss or quantization).
Design D 2 is unstable because Nyquist plot encircles critical point, but Di and D 3 do not
encircle the critical point and are stable.
164
..
............
...
...
0.5
-0.5
-2
-1.5
-1
-0.5
0
0.5
1
0.5-1.5
------ ----
- --Real
-
0 - - --------0.5-
-1.5-2
-1.5
-1
-0.5
0
0.5
1
Figure 6-22: Nyquist plots of designs D 4 (top) and DFinal (bottom) for systems with no
nonlinearities (i.e.systems with delay and decoder but no loss or quantization). Design D 4
is unstable because Nyquist plot encircles critical point, but DFinal does not encircle the
critical point and is stable.
(0.7, 1.02), and see the corresponding Nyquist plot (shown in Figure 6-22 (top plot)) encircles the critical point and is unstable. We next chose an intermediate design point between
D 3 and D 4 , given by DFinal = (0.79,1.02).
The corresponding Nyquist plot (shown in
Figure 6-22 (bottom plot)) is stable, with a crossover point of -0.98.
We settled on this point as the final design, which gives a significant reduction in limit
cycle amplitudes as compared to QLZ or QLH. As shown in Figure 6-20, if we use the
Euclidean norm of the limit cycle amplitudes as a metric, the design point has norm 1.3,
compared to 3.9 for QLZ and 5.5 for QLH. By using the Euclidean norm we assume both
limit cycle amplitudes are equally important to decrease, which is the case in this example.
The values of X for this design are given by
165
A 1 = -1.60+0.18i
(6.77)
= -5.66+0.07i
(6.78)
= 0.11+000li
(6.79)
X3
A4 =
(6.80)
1.35 -0.003i
(6.81)
and the decoder matrices
P(k) = Ok
Q(k)
=
(-
Ok)
0.97
0
0
0
0
6.35
0
0
0
0
0.12
0
0
0
0
1.33
(6.82)
y
0.54
0,
0
0
0
0.11
0
0
0
0
-0.11
0
0
0
0
0.02
(6.83)
The decoder algorithm is thus
Sk =Ok
0.97
0
0
0
0
6.35
0
0
0
0
0.12
0
0
0
0
1.33
yk + (I-k)
0.54
0
0
0
0
0.11
0
0
0
0
-0.11
0
0
0
0
0.02
Sk-1.
(6.84)
It is noteworthy that the decoder matrices have non-obvious values, with numbers ranging from -0.11 to 6.35. These values do, indeed, cause the describing-function-derived
gain-phase loci of the quantized lossy system to pass through the critical point, as shown
in Figure 6-23 (top plot). Because this plot is derived using describing functions, it is only
166
0.80.6
0.40.20
-
-
- - -
--
-
--------
-0.2-0.4-0.6-0.8-1.2
-1
-0.8
-0.6
Real
-0.4
-0.2
0
-1.2
-1
-0.8
-0.6
Real
-0.4
-0.2
0
0.80.60.40.2-
--
0.2-0.4-0.6-0.8-
Figure 6-23: Describing-function-derived gain-phase loci of designed system with delay, quantization, and packet loss. Top plot is for all positive )s, and bottom plot for
(Xi, -A 2 , X3, -X 4 ). Top plot passes through critical point, marked by red x, as designed,
and bottom plot passes close to critical point.
valid at the critical point.
It is interesting to note the resemblance of part of these curves to the arches seen in
the quantizer describing function. This is because, while the limit cycle amplitudes are
specified for positions X, and X2 , we assume the velocity limit cycle amplitudes are the
product of the system frequency and position amplitude. Thus, as different frequencies are
swept through to generate the plot, the quantizer gain for the velocity states will change,
resulting in the arches seen in the plot.
We must also check that no additional frequencies are solutions to the gain-phase locus
plot passing through the critical point. As in the SISO case, because the designed codec
depends on the tangent of the desired codec phase, and the tangent of the phase is the same
as the tangent of the supplement of the phase, there may be multiple phase solutions for a
given codec. Shifting the phase by 180 degrees is equivalent to multiplying the equivalent
167
complex number representation of the gain and phase by -1. Thus, because the system has
four independent codec channels, there are 24 = 16 possible combinations of positive and
negative values of the designed A values.
The top plot of Figure 6-23 shows the case where all values are positive, with a predicted
limit cycle frequency of 1.01 rad/sec. The bottom plot of Figure 6-23 shows the case where
A 2 and A4 are multiplied by -1. In this case the resulting gain-phase loci approximately
crosses the critical point at a frequency of 2.1 rad/sec. Of the 16 possible plots, five cross
the critical point at a frequency of 1.01 rad/sec, two cross at 2.1 rad/sec, and the remaining
nine do not intersect the critical point. The two plots in Figure 6-23 are representative of
these cases.
We conclude that the limit cycles at the designed amplitudes have two solutions for
possible frequencies: 1.01 rad/sec or 2.1 rad/sec.
6.5.4
Design Performance
We plot time series responses of the system with quantization, loss, delay, and the designed
codec in Figure 6-24 and see reasonably close agreement between designed limit cycle
amplitudes (shown as dashed black lines) and actual amplitudes. In these plots the red
dashes indicate a lost packet in any of the four channels. The plot for X1 shows limit
cycling at a frequency of approximately 2 rad/sec, while X2 oscillates at a frequency of
approximately 1 rad/sec. These frequencies are consistent with the two frequency solutions
for the designed codec. While both frequencies may exist at the same time in the solutions,
.
the dominant frequencies are 1 rad/sec for X2 and 2 rad/sec for X 1
It is also important to check that the decoder does not have an undesirable effect on the
original no-loss scenario. Time series responses are given in Figures 6-25 and 6-26 for the
original system with the designed decoder, with the same initial conditions as before, and
the main effect is seen to be a longer settling time (300 time steps as opposed to 150 time
steps). The system is still stable, though, and this side effect of the decoder is deemed an
acceptable price to pay for significantly reducing the limit cycle amplitudes in the second
operating scenario.
168
-0.5
x11
-D
'a 0
x
.
Nondimersonal Time
2..
Figure 6-24: MIMO design time series plots using designed codec. Top plot is for Xi
response and bottom plot for X2 response. Red lines indicate a packet loss in one of the four
channels, and black dashed lines indicate designed limit cycle amplitudes. Nondimensional
time is used, defined as gT*
time.
169
0.80.6E
0.40.20
-0.2
0
- .4r
0
500
1000
0 500
1000
1500
2000
2500
Nondimensional Time
3000
3500
4000
0.810.6CL
O.4
E
0.2
0:!
-0.2-0545
1500 2000 2500 3000
Nondimensional Time
3500
4000
Figure 6-25: Time series responses of original system with no delay, loss, or quantization
with decoder. System is still stable, with longer settling time than in case without decoder.
Nondimensional time is used, defined as gwob * time.
170
..........
.....
.. I...........
...........
12
0.8
0.6
0.4
0.2
0
-0.2
-0.4
0
0.5
1
1.5
2
Nondimensional Time
2.5
1.5
2
1
Nondimensional Time
2.5
3
x 10,
1-
0.8-
0.6
0.4
E
0,2
-0.2-0.4
0
0.5
3
x 10
Figure 6-26: Time series responses of original system with delay but no loss or quantization
with decoder. System is still stable. Nondimensional time is used, defined as gO)T *time.
171
From X
0
5
5
5
0
0
0
-4
5
CD
Cz
0
-2
0
0
-4
5
-2
0
-2
0
5
-4
-2
0
-4
From V2
-2
0
0
I-
-5
-4
5
-2
-54
-4
5
0
0
0)
0
0
-10
-4
5
-45
0
>P
0
-2
0
0
-0.5
-4
-4
0.5
-2
0
0
-2
0
-0.5
-4
-2
-4
-4
-2
-4 20
5
0
0
2
0
0
0
I-
-4
0.5
-4
-
-o
From X2
From V
-2
0
-4
0.5
-2
0
~-4
-3-4
-2
0
-1
-2
0
-0.
-4
-2
0
-2
0
Log (Frequency)
Figure 6-27: Transfer functions magnitudes for MIMO example with designed decoder, at
low frequencies.
We finally check the effect of the designed decoder on no-loss system steady state
errors to step inputs, as done for the original controller. The transfer function magnitudes
are given in Figure 6-27, and we see similar magnitudes as for that of the original system.
Magnitudes are large and increasing for decreasing frequencies for transfer functions to
outputs X1 and X2 as before, and thus we conclude the steady state errors to step inputs are
acceptable as before.
6.5.5
Checking for Near Optimality
We finally check that the designed limit cycles are nearly locally optimal for this example.
This can be done by examining the quantizer describing function plot near the design point.
On this plot (Figure 6-28 for reference), any amplitudes between 0.53 < a/h < 0.7 have a
DF gain that also corresponds to a higher amplitude a/h. For instance, the quantizer gain
at a/h = 0.6 is the same as the gain at a/h = 0.9 (shown in the top plot of Figure 6-28).
172
1.1
7 ------
---
--- ---
- -- - ---
-- - ---- - -- - --
z
C
0
o
0.6
1.2 7 --------
0.9
a/h
---------
1.0 7 -- -- - -- -- -- - --- -
- - - - ---------- - - ---
- ---
z
0
0.7
1.04
a/b
Figure 6-28: Quantizer gain plots. Top plot shows gain at a/h=0.6 corresponds to gain at
larger value of a/h=0.9. Bottom plot shows region 0.7 < a/h < 1.0 has gain that does not
correspond to larger amplitude solutions.
However, amplitudes in the range 0.7 < a/h < 1.04 have a quantizer gain that does not
correspond to a higher amplitude (shown in the bottom plot of Figure 6-28) . Thus, an
amplitude in the former range would not be a valid solution because it would not represent
the largest limit cycle amplitude.
We conclude that, in the local region around the final design point DFin the limit cycle
amplitudes are lower bounded by 0.7. This space is shown in Figure 6-29. This plot shows
that, indeed, the designed point is close to the local lower bounds for limit cycle amplitudes.
Furthermore, we can examine the local sensitivity of the mean square limit cycle amplitude as a function of independent variations in the decoder parameters to see how close
the design is to local optimality. Figure 6-30 shows the sensitivity plots for variations in
MI...M 4
and Figure 6-32 for variations in N1 ... N4 . For each of these plots only one decoder
parameter is varied by plus or minus 50 percent of the designed value while the other seven
173
0.95
D
0.9
*
D
D2
QLZ
0.85
E
<
0
Q
0.8-
ODF
g
.
Final
---.-.--.-.-.-.-.-.-.-.-
.- - --- -- --
.E
X<
QLH
LC Solutions
0.75
0.7_1
0.65
-
-
'
-
4
0 Viable Design
Unviable Design
-X
0.6-
No LC Solutions
0.550
1
4
2
3
X2 Limit Cycle Amplitude
5
6
Figure 6-29: Plot of limit cycle amplitudes for states Xi and X2 for QLZ, QLH, and designed codecs. Blue dots represent viable design points, while red exes represent unviable
design points. Regions with and without limit cycle solutions are shown.
are held constant. The red dot in the plots indicates the design point corresponding to DFinThe plots show fairly low sensitivity to changes in Mi, M2 , and M3 , but higher sensitivity
to changes in M4 . The value of M4 is very close to locally optimal, with the only possible
improvement seen if M4 were slightly decreased. This would result in at best a 6% improvement in mean square limit cycle amplitude (from 1.30 to 1.22), so the current design
is indeed fairly close to the local optimum.
The design point corresponding to the local minimum seen in the M4 plot corresponds
to the limit cycle amplitudes (a,, a2) = (0.93,0.78) at M4 = 1.15, as shown in Figure 6-31.
This point actually has a larger limit cycle amplitude al than does the QLZ starting point
for the design method at (ai, a2) = (0.88,3.8), and this is why the design method missed
this point. The method is to alternately attempt to decrease al and a2, starting from the
initial point and moving in the direction of the local lower bound of (0.7,0.7).
It may be possible to find this locally optimal point using a more sophisticated optimization search method with a cost function that somehow takes into account both the
limit cycle mean square amplitude and how close the design point is to being viable - i.e.
how close the MIMO Nyquist plot of the no-loss system with the designed decoder is to
encircling the critical point. Such a cost function would necessarily depend on the deter174
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M
0.14
0.16
0.8
0.18
1
1.4
1.2
1.6
1.8
2
M4
3
Figure 6-30: Sensitivity plots of mean square limit cycle amplitude for small changes in M
decoder parameters. Red dot marks design point.
175
1.5
a
C
E
E
E
E
0.8
1
1.2
1.4
1.6
1.8
2
M4
Figure 6-3 1: Sensitivity plot of X1 and X2 limit cycle amplitudes for changes in M4 decoder
parameter. Red dots mark design point.
minant of the sum of the identity matrix and the loop transfer matrix of the system, as
determined by the MIMO Nyquist stability criterion, and on the quantizer and lossy channel describing functions. It would also depend on some measure of closeness to instability
for the MIMO Nyquist plot, similar to a gain or phase margin. Gain and phase margin are
not well-defined for MIMO systems, though, and the closeness measure may be difficult
to establish. Moreover, this function would in general be nonconvex because of the dependence on the nonconvex quantizer describing function, which would affect the decoder and
hence affect the determinant. Thus there would be no guarantee that such a method would
necessarily find the global optimum.
The proposed design method, though, has the advantage of simple implementation
while still taking into account both the limit cycle mean square amplitude and the viability
of design points.
Figure 6-32 shows fairly low sensitivity to changes in the N decoder parameters for the
range tested (plus or minus 50 percent of the designed value). However, this is partly due
to the design values being close to zero and thus only minor changes were tested. The limit
cycle mean square magnitude is still somewhat sensitive to more significant changes in N
values. For example, if we look at changes in N4 , the most sensitive parameter, in the range
-3 to +3, as shown in Figure 6-33, we see the mean square limit cycle magnitude decreasing
176
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-0.08
-0.06
0.1
3
oeer hsdcesei
0.015
tl mlleog
0.02
0.025
htw
0.03
N4
Figure 6-32: Sensitivity plots of mean square limit cycle amplitude for small changes in N
decoder parameters. Red dot marks design point.
by about 1% to 1.285 for N4 = -3. However, this decrease is still small enough that we
conclude the designed values for N are fairly close to the local optimum.
6.6
Summary
We have proposed a method to use the DFs as synthesis tools to design a system decoder. In
the case of packet loss and quantization in the feedback channel, the design method ensures
the system limit cycles at a user-determined amplitude. Meeting a specified limit cycle
amplitude is important because limit cycle behavior may be unpredictable via classical
design methods and only seen after a design is checked via simulation or via DFs used as
analysis tools. However, if a given limit cycle is deemed acceptable, this method allows that
limit cycle to be designed for. Furthermore, an iterative method was given to decrease the
limit cycle amplitude. Numerical examples were given illustrating the design method on
SISO and MIMO systems, showing up to three times reduction in limit cycle amplitudes. It
177
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C,
1 .305-
E
1.3_j
1.295[
C,
C
tv
1.29[
1.285[
1.28'
-3
-2
-1
0
N
1
2
3
4
Figure 6-33: Sensitivity plot of mean square limit cycle amplitude for large changes in N 4
decoder parameter. Red dot marks design point.
is noteworthy that the designed decoder is very simple, having the equivalent structure and
computational complexity as the Zero-Output and Hold-Output decoders, but is still able
to achieve strong results. The numerical examples show the designed decoders have nonobvious parameter values which lead to significant reductions in limit cycle amplitudes.
178
..
...
......
... ..........
Chapter 7
Experiments
7.1
Introduction
In this chapter we test the theoretical channel describing function results on the control of
a small autonomous floating raft subject to quantization and packet loss, in a controlled
lab setting. We consider the SISO case where we control the raft heading, with heading
measurements subject to packet loss and quantization. For given quantization and packet
loss parameters we use channel describing function analysis to predict limit cycle amplitudes for QLZ, QLH, QLF, and QLM systems, and then design a new controller/codec that
results in a smaller limit cycle than those from the other codecs/controllers. The purpose of
this experiment is to show that a physical control system that operates close to instability in
the absence of quantization and packet loss can be inadvertently pushed into a limit cycle
by the addition of quantization and packet loss. Moreover, the experiment shows that it is
possible to predict the limit cycle amplitudes using DFs, and to design a codec to reduce
the effect of the limit cycle.
7.2
Experimental Setup
In these experiments an autonomous raft floats in a 1-m deep water tank, with an overhead
camera taking raft heading measurements. Figure 7-1 shows the experimental setup. The
camera sends the measurements to a shore computer, which artificially adds quantization
179
Overhead
Camera
Quantization and/or
packet loss artificially
added to measurement
% Camera senses
%raft heading
Controller
Control command
Water tank
Figure 7- 1: Experimental test setup. We control heading of a small autonomous raft in a
lab pool environment using an overhead camera sensor and shoreside controller.
and/or packet loss to the measurement, computes a control command, and sends the control
command to the raft over XBee. The raft, shown in Figures 7-2 and 7-3, is controlled by
eight separate motors located equidistant from the center and point fore, aft, left, and right.
A large additional motor on the sterm of the raft can provide additional thrust for contact
force experiments, but is not used in the heading control experiments. The raft is 0.5m
long, 0.3m wide, and weighs 15 kg with all onboard electronics and batteries. An April tag
[80] is mounted on the top, allowing the camera to sense raft heading.
Figure 7-4 shows a block diagram of the feedback control system. The camera senses
the heading and sends it to the shore computer. The measurement is then encoded, quantized, and passed through an artificial binary erasure channel. The output is then decoded,
the state estimated, the estimate passed to the controller, and the control command sent to
the raft.
180
Figure 7-2: Experimental test platform side view. Raft is equipped with April tag on top so
overhead camera can sense heading.
Figure 7-3: Experimental test platform top view, as seen from overhead camera.
181
Controller
Contrller
Delay
Saturation/
Deadband=
Pat
Pnt.
Decoer/Quantizer
El~
lstimator
Sno
Ssr
.
Encoder
Qatz
Lossy Channel
Figure 7-4: Block diagram of experimental system. Thrusters are represented by the delay
block and saturation/deadband block. Communication channel is given by the encoder,
quantizer, lossy channel, and decoder blocks.
7.3
System Models
7.3.1
Thrusters
The raft thrusters are modeled with a delay block and saturation/deadband block. The delay
is characteristic of underwater thrusters - the thruster may spin but there is a delay before
thrust is felt by the raft. The saturation/deadband is characteristic of most motors - below
a certain small positive command level the motor doesn't rotate, and above a certain level
the motor saturates, reaching its maximum speed. Figure 7-5 shows the describing function
gain for the thruster in this experiment. In this plot the control input is normalized so the
maximum motor propeller speed is attained at an input of one. For inputs greater than
one, the motor propeller speed is still equal to one, and thus the describing function gain is
equal to 1/u for input u. We found the deadband for the motor is at 0.1, meaning control
commands u < 0.1 result in no motor movement, i.e. gain of zero.
7.3.2
Raft Model
We derive a model of the raft plant from control command to heading, based on the raft
thruster configuration shown in Figure 7-6. In this setup the raft has eight identical thrusters
mounted radially, such that four are engaged for positive rotation and the other four engaged
for negative rotation. For this model we assume the raft moves as a mass with damping
provided by the water, acted on by a net thrust, as described by the equation
182
1
Unity Gain
0.8-
Saturation
0.60
0.4-
0.2Deadband
00
0.5
2.5
2
1 5
1
Normalized Input Control Command
3
Figure 7-5: Actuator describing function capturing deadband (u < 0.1), unity gain(0. 1 <
u 1), and saturation (u > 1) regimes.
JA = -b6 +4rFt
(7.1)
for raft inertia J, water damping coefficient b, radius r from raft center to thruster, and individual thruster force Ft. The control command sent to the raft is a voltage that commands
propeller speed. In steady state, thrust force is proportional to the square of propeller speed
[116], given by the equation
Ft = CtK2I9
(7.2)
for propeller speed n and model constant Ct dependent on thruster and propeller geometry.
We linearize this model, as similarly done in [116], by assuming thruster force is approximately linearly dependent on propeller speed, for an appropriately-chosen proportionality
constant Ct, different than Ct. We also assume a lag between input propeller speed and
steady-state thrust, as is characteristic of underwater thrusters. For simplicity we model
this lag as a pure delay. The final model is thus represented by the continuous-time transfer
function from control command u to raft heading 9
-
Ke-s'
183
(7.3)
~0~
I
I
I
I
I
I
p.
Figure 7-6: Top view of autonomous raft used in experiments, showing thrusters and coordinate frame.
184
....
...
....
I...
................
.....
..
..
............
...
15i
- - - Heading Data
10-
-Model
5
0.-
0
0.2
0.4
0.6
0.8
1
Time [s]
Figure 7-7: Raft model vs data for thruster propeller speed step inputs of 30, 50, 70 and 90
percent.
for some constants K, c and Td.
7.4
Determining Model Parameter Values
To find the constants of this model we performed heading step responses for control commands within the unity-gain regime of the actuator (i.e. not in the deadband or saturation
zones). In each test we started the raft at rest and sent a constant control command of 0.3,
0.4, 0.7, or 0.9. Figure 7-7 shows the unfiltered step responses of heading vs time and a
model fit. We fit the model to the first second of the step response, as a relevant time frame
for future control experiments conducted at a timestep of 0. 12 seconds. For the model we
see acceptable agreement for a 0.36-second delay, K = 99.7 and c = 2. 1. We note that,
while we assume a constant dt of 0. 12 for analysis, in reality the dt varied between 0. 115
and 0. 125, and we have assumed the average value of 0. 12.
7.5
Controller Design
We designed an LQG controller in discrete time to control raft heading, for use in
and
QLF
QLM, and derived the equivalent compensator for use in QLZ and QLH. This way, in
185
the absence of packet loss and quantization all systems will behave identically, but with
packet loss and quantization they will behave differently. We first formulated the system in
state space, with two heading states (heading and heading rate), and three control command
states (command and two delayed command states). For the LQG controller/estimator we
assume sensor and process noise covariances of one, and choose LQR gains R = 1, and
Q=
10
0
0 0 0
0
10 0 0 0
0
0
1
0 0
0
0
0
1 0
L0
0
0 0 1J
(7.4)
to penalize state error much more than control command. We used the constant time step of
dt = 0.12, as determined by the camera sensor update rate. This yields the constant steady
state Kalman gain matrix Kf
=
[1.28 3.48 0 0 0]' and control matrix
Ku= -[0.086
0.057 0.73 0.80 0.89].
(7.5)
In this set of experiments we assume the system is designed to operate in nominal
conditions of a one time-step delay in sensor feedback, as may be typical in underwater
communication, with possible delays of up to three time steps in the sensor channel. If the
system has a one time-step delay in sensor feedback, then Figure 7-8 shows the Nyquist
plot of the loop transfer function of this system, without accounting for quantization, packet
loss, or saturation. This system is stable, with a crossover of -0.59. An example time
response of this system for an initial error or -35 degrees is shown in Figure 7-9.
If the delay in the sensor feedback channel is three time steps, then the system is pushed
closer to instability, though still stable. Figure 7-10 shows the Nyquist plot of this system,
with a crossover of -0.93 at a frequency of wl = 1.06 rad/s. The system is stable because it
has no encirclements of the critical point. Figure 7-11 shows an experimental time response
of the system with a three time step delay, illustrating how it is still stable in this situation.
However, this system is now close enough to instability that the addition of quantization
186
0.8
.1
0.6
0.4
0.2
Mn
01
C)
CD
E
-0.4
+
.. ........
. .......
-0.6
-0.8
-1
-1
-0.5
0.5
0
Real Axis
1
Figure 7-8: Nyquist plot of loop transfer function with one time step delay. Crossover is at
-0.59.
5
0
-5
'.5;
S-10
)
-15
-20
-25
-30
0
10
5
15
Time [s]
Figure 7-9: Example time series response of system to initial error of approximately-35
degrees, for one time step delay in sensor measurement. System stabilizes after 15 seconds.
187
1 ~~
~
~
~
~
---- ----_ --
0.8
-
0.6
0.40.2-
0-+
-0.2-0.4-0.6-0.8 -~
-1
. ..........
-1-0.5
---
0
Real Axis
..
0.5
1
Figure 7-10: Nyquist plot of loop transfer function with three time step delay. Crossover is
at -0.93.
and packet loss in the sensor feedback channel will push the system into a limit cycle.
7.6
Limit Cycle Analysis
To analyze limit cycles in this system, we first examined the case where sensor signals are
quantized, but there is no packet loss. In this case all controllers are equal, and the quantizer
and saturation DFs will potentially cause the system to limit cycle. Both the quantizer and
saturation DFs only depend on the input amplitude, and are independent of input frequency.
Thus, in order for the system to limit cycle the product of the gains from these DFs must
equal the inverse of the Nyquist crossover point. In this case, for quantizer gain Q(a1 /h),
saturation gain G(ui), and limit cycle magnitudes al and ul we have
Q(ai/h)G(ul) = 1.07.
(7.6)
Because the plant transfer function is between the control command and sensor output,
the control and heading limit cycle amplitudes are related to each other by the magnitude
of the plant transfer function evaluated at the limit cycle frequency. This can be expressed
as
188
A'
20
10
0)
CD
-10
-20
-300
5
15
20
Time [s]
10
25
30
Figure 7-11: Example time series response of system to initial error of -35 degrees, for
three time step delay in sensor measurement. System stabilizes after 30 seconds.
K
K
al = uiG(u1 )
(01
(7.7)
o2+c2
To determine the limit cycle amplitudes we must consider two cases, whether the control signal is causing the thruster to saturate or not. We will neglect the case when the
control signal is in the deadband range, because that would not result in a limit cycle,
because the raft would not move.
Case 1: ul < 1. In this case G(ul)
=
1. Thus, from Equation 7.6 we have Q(a;/h)
=
1.07. From the quantizer describing function plot we see this occurs at values al/h = 0.57,
1.06, 1.67, and 1.87.
Case 2: ul > 1. In this case ujG(uj) = 1. Thus, using parameter values for K, c, and w0,
Equation 7.7 simplifies to give a, = 40.1. In this case, the limit cycle ratio is given simply
by 40.1/h.
In the case of packet loss and quantization we use the describing functions QLZ, QLH,
QLF, and QLM to analytically predict the limit cycles for a range of packet losses. For
these predictions we assume a value h = 6 that experimentally results in minimal or no
motor saturation for moderate to low packet loss, such that observed limit cycles can be
attributed to the effects of the communication channel.
189
Figure 7-12 shows predicted limit cycle amplitude ratios for
Q,
QLZ, QLH, QLF, and
QLM controller/decoders for a range of packet losses, as computed using the DFs and the
system model. Solid lines represent stable limit cycles and dashed lines unstable limit
cycles.
7.7
Limit Cycle Experimental Tests
We tested the limit cycle predictions by independently implementing each controller/decoder
scheme for a given packet loss rate and determining resulting limit cycles based on the time
responses.
For each test we set the raft stationary in the middle of the pool at zero heading, then
turned on the control system for heading control. We then recorded raft heading measurements with the overhead camera as the raft oscillated. We tested packet loss probabilities
up to 55%, and observed that above this level motor saturation began to significantly affect
the response. Figure 7-13 shows the approximate regions of dominating factors influencing
raft behavior. For low packet loss rates the quantization dominates, while for medium rates
packet loss becomes more important. At higher packet loss rates, it is more likely that a
longer string of packets will be successively lost, resulting in larger errors, and subsequent
motor saturation as the controller compensates for the larger errors.
The circles in Figure 7-12 represent observed limit cycle amplitudes for the given controller/codec and packet loss rate. Example time responses for some of these limit cycles
are given in Figure 7-14 for a pure quantizer, Figure 7-16 for QLZ with 15% packet loss,
and Figure 7-17 for QLH with 55% loss. In these plots red vertical lines indicate a lost
packet. We see all systems have the same limit cycle amplitudes of 1.0 and 1.9 in the
absence of packet loss, because in this case the systems were designed to be equivalent.
With increasing packet loss, predicted limit cycles generally decrease in magnitude. To the
right of the cusps of the graph the limit cycle method does not tell us the behavior of the
system, though experimentally we find this means the systems generally behave erratically,
eventually becoming unstable.
Interestingly we found experimentally that QLM, while stable with pure packet loss,
190
-Q
2.5--
Model
QLZ
-QLH
-QLF
-2
0
-QLM
-------------
----
----
0 Q Exp
L
QLM
-
QLH
E01.5 -0
# Design Pt
E
-
-- - - - - - - - - -- 9- - - -
0
0.2
0.4
0.6
Packet Loss Probability
a
0.8
1
Figure 7-12: Analytically predicted and experimentally observed limit cycle amplitudes.
Circles represent experimental results. Target point for controller design also shown.
Dominant Limit Cycle Factor
C
0
4-#
C
ru
Packet
Loss
Motor
Saturation
0(
0.5
Packet Loss Probability a
i
Figure 7-13: Regions of different dominating factors influencing limit cycles. Quantization
dominates at low packet loss rates, while motor saturation dominates at high packet loss
rates.
191
2
1.5
C:
T
0.5
0
I0N-0.5-
-3'
-2
-2.5
0
10
20
30
40
50
Time [s]
60
70
80
90
Figure 7-14: Experimentally observed limit cycle of magnitude approximately 1.9 for pure
quantizer, no packet loss.
goes unstable in the presence of quantization. In QLM an information vector is quantized
and not just the measurement, and the effect of quantization is apparently much more pronounced on the decoded signal because so much more information is put in the packet. The
developers of the MIF algorithm [40] proved this was the optimal LQG-type codec for pure
packet loss, but made no claims about the effects of quantization.
Ignoring QLM, we see that QLZ has cusps for the lowest packet loss rate, consistent
with it being the least sophisticated and worst-performing algorithm. The QLH and QLF
cusps occur for higher packet loss rates, as expected for the more sophisticated controllers.
QLF is apparently superior in the sense that no limit cycles around magnitude 1.9 are
predicted, as are for QLH. This makes sense because QLF uses knowledge of a system
model while QLH does not.
7.8
Codec/Controller Design
7.8.1
Design Requirements
Now let us assume in this example that the system will be operated in the presence of 25%
packet loss, and that it is desirable to have a lower limit cycle amplitude than is predicted
192
1.5r
1
C3
C
0.5
I
0
N
E
0
z -0.5
-1
-1.50
5
15
10
25
20
Time [s]
Figure 7-15: Experimentally observed limit cycle of magnitude approximately 1 for pure
quantizer, no packet loss.
1.5[
1
Ca
0.5
a)
0
0
z
-0.5F
-1
0
2
4
6
8
10
Time [s]
12
14
16
18
Figure 7-16: Experimentally observed limit cycle of magnitude approximately 0.8 for QLZ
with 15% packet loss. Red vertical lines indicate lost packets.
193
2.5
21.5
Z
0.5
N
-1
-1.5
0
5
10
15
Time [s]
20
25
Figure 7-17: Experimentally observed limit cycles of magnitude approximately 1.8 for
QLH with 55% packet loss. Red vertical lines indicate lost packets.
by the controllers/codecs considered. The smallest limit cycle amplitude predicted for this
packet loss rate is 1.0, but assume we want to design a system with limit cycle amplitude
0.9. This design point is labeled with the x in Figure 7-12. The controller/codec design
method described in Chapter 4 allows us to achieve this goal.
7.8.2
Design Equations
In this case we use the existing compensator used from the QLZ and QLH experiments,
Czh, as a starting point for the controller/codec design. We use equations (6.49) and (6.48)
and set up an optimization problem to choose qr that results in a value for P as close as
possible to one times the measurement. This will ensure the codec has a small effect on
the system in the event of no packet loss. We set constraints that at/h = 0.9 and bound the
limit cycle frequency by reasonable experimental values 0.1 < q < 2.
Using the Nelder-Mead Simplex optimization method we find a feasible solution for
N = 0.23 and M = 1.187 at q, = 1.06. The magnitude of N is less than one, as is required
for feasibility of the design method. To actually implement this codec, we see using equations (6.50) and (6.51) with the given packet loss rate and desired limit cycle amplitude that
the codec is defined by
194
Sk = eko. 5 8 yk + (1 - ek)0. 92 Sk
(78
(7.8)
for state estimate Sk, measurement Zk, and packet loss variable ek at time step k. Practically,
this means that if the measurement is received, we decode by multiplying the measurement
by 0.58, and if the packet is lost we decode by multiplying the previous state estimate by
0.92. This can be thought of as a hybrid between QLZ and QLH. In QLZ we multiply
the measurement by one and the previous state estimate by zero, while in QLH we multiply the measurement by one and the previous state estimate by one. In this case we've
allowed those coefficients to be different numbers than zero or one. Because of the design constraints used in the optimization problem setup, we can use this QLZ and QLH
compensator Czh, along with the new codec.
7.8.3
Effect of Designed Codec on System with Packet Loss
To understand this new codec, we show its Bode plot in Figure 7-18, created by using the
describing function of the codec. These plots have similar general shapes to the LH Bode
plots, but much higher gain at lower frequency. Because this codec was designed for a
specific limit cycle frequency, we are really only concerned with its gain and phase at that
frequency, marked by the vertical dashed line intersection point in the plot. We see that the
codec has a gain of 3.9 and phase of -50 degrees at the target frequency.
We plot the corresponding gain-phase loci of the system in Figure 7-19 for the case of
25% packet loss and quantization ratio a/h = 0.9 for the system with the new codec. As designed for, the plot passes through the critical point (-1,0) for the positive X corresponding
to the designed codec (top plot). We also check the gain-phase loci for the case of negative
X to check for multiple frequency solutions (bottom plot). This plot does not pass through
the critical point, and thus we conclude only one frequency solution exists.
7.8.4
Effect of Designed Codec on System with No Packet Loss
The new codec also affects the system performance in the no-loss no-quantization case.
In this case it is simply a gain of 0.58 on the feedback channel, because this is what the
195
C9
3
2
1
-2
-1.5
-1
Log,, of Input wT
-0.5
0
U
-10
-20
0 -30
-2
-.-
-2
-1.5
1
-.
-1
-0.5
-40
-50
-60
'7n
Logl 0 of Input o
0
Figure 7-18: Bode plots for the describing function of the designed codec.
196
1
0.80.60.4
0.2
--- --------------- ------------
-
0
-0.4
-
-0.2
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
Real Axis
-0.2
0
-0.8
-0.6
-0.4
-0.2
0
1
0.8
0.6
0.4
X-
0.2
0
-0.2
-0.4-0.6-0.8-1
-1
Real Axis
Figure 7-19: Describing-function-derived gain-phase loci for system with designed controller with 25% packet loss and quantization ratio a/h = 0.9, for positive X (top plot) and
negative X (bottom plot). Red x marks critical point.
197
0.80.6
0.4-
0
E
+
0.2-
-0.2
-0.4
-0.6-
__
-1
-0.5
J_
_-
0
Real Axis
0.5
1
Figure 7-20: Nyquist plot for system with designed controller with no packet loss or quantization.
measurement is multiplied by in the general lossy case. We plot the Nyquist plot of this
no-loss case in Figure 7-20, and see the system is stable (no encirclements of the critical
point) with a crossover at -0.54.
An example experimental time series response of this system with no loss and no quantization is shown in Figure 7-21. We started this system with an initial error of around
35 degrees and it stabilizes after about 37 seconds. This system takes slightly longer to
stabilize than the original controller for the same input error, but is still stable and, as we
will show, meets the limit cycle amplitude designed for in the presence of quantization and
packet loss.
7.8.5
Limit Cycle Experiments with Designed Codec
We next implemented this controller/codec on the raft in the presence of quantization and
25% packet loss and saw a limit cycle with a magnitude of approximately 0.9, as desired,
shown in Figure 7-22.
The design represents a significant improvement when compared to QLZ, QLH, and
QLM. At the packet loss rate designed for, QLZ behaves erratically and has no LC solution,
QLH has a largest limit cycle amplitude twice as large as the designed decoder, and QLM
198
20
10-
0-1D
-20
-30-
-0
5
10
1,5
20
Time [s]
2'5
30
35
Figure 7-21: Experimental time response for system with designed controller and delay,
with no quantization or packet loss.
also behaves erratically with no LC solution.
This design represents a 10% reduction in limit cycle amplitude from the best alternative, QLF. This may appear to be only a modest improvement, but the designed codec
does not require implementing a Kalman filter, only implementing one of two different
gains and a compensator. This could be important in applications where simpler computations are attractive, such as if a small robot has limited onboard computing capability, or in
applications where faster computation is attractive.
In this example the best alternative codec (QLF) for the design condition a = 0.25
had only one stable limit cycle branch. However, if the best codec had two stable limit
cycle branches, such as the QLH in this example, then the design point could potentially
be chosen to avoid the larger branch and lead to a much more significant reduction in limit
cycle amplitude.
7.9
Summary
We performed experiments with an autonomous robotic raft in an indoor lab pool setting
to test analytical limit cycle predictions and to test the controller/codec design method.
We performed SISO experiments controlling raft heading subject to artificially-injected
199
1.5
CD
-0.5
N
0
z
-0.5-
0
5
10
Time [s]
15
20
Figure 7-22: Limit cycle in experiment with designed controller/codec for 25% packet loss.
Magnitude is approximately 0.9. Red lines indicates lost packets.
quantization and packet loss with the different control/estimation algorithms and verified analytically-predicted limit cycles. We then demonstrated the controller/codec design
method on this system to make it limit cycle at a smaller amplitude for a given packet loss
rate.
200
.......
... ...
Chapter 8
Conclusion
In this final chapter we summarize the contributions of the thesis, and discuss possible areas
of future work.
8.1
Review of Contributions
In this thesis a new class of describing functions was derived for stochastic information
channels, and applied to the analysis and design of feedback control systems. The analysis
and design methods advance the state of the art in control system engineering for systems
subject to quantization and stochastic packet loss.
The describing functions derived in this thesis build on the DF methods of the literature
for studying nonlinear physical elements in control loops, and the new DFs are unique because they describe information sent through channels, and the information itself is stochas-
tic. These two properties of the DF - describing non-physical and stochastic elements
distinguish the analysis from previous work and open the possibility for DF analysis to be
applied to a much broader range of systems than has been standard practice in the past.
The new DFs were verified by simulation for open-loop stable, open-loop unstable,
minimum phase, and nonminimum phase SISO system examples, as well as for multisensor channels of several different MIMO system examples. The DFs were shown to
be useful analysis tools in several respects. In the event that a system is subject to external
disturbances, the DFs are useful to compute sensitivity-like functions to measure the system
201
sensitivity to disturbances. If a system is close to instability, the DFs can be used to predict
limit cycles that may result from a quantized and lossy sensor feedback channel. The
analysis for both of these cases relies on the provably-optimal effective gains and phases
predicted by the DF for the quantized and lossy information channel, and we showed that
computing these gains using the DFs is orders of magnitude faster than computing them
via simulation. A numerical example was given showing agreement between limit cycles
predicted by DF analysis and by simulation, for a feedback control system with packet loss
and quantization in the sensor feedback channel.
The DFs were also shown to be useful synthesis tools. Unlike standard DFs for physical elements, the DFs for information channels can easily be tuned, so to speak, to have
different frequency responses by changing the decoder parameters of the channel. This is
the basis behind the DF synthesis method. In this method we assume two different operating regimes: a nominal condition, and a condition that is close to instability. This could
represent, for instance, an AUV navigating via LBL in open water (the nominal condition),
and in a cluttered environment (the second condition), where acoustic communication performance may be negatively affected. The synthesis method allows the control engineer to
specify a maximum acceptable limit cycle amplitude for the second condition, and design
a decoder that meets this requirement with small effect on the nominal condition performance. The decoder has the same simple equivalent structure and computational complexity as the Zero-Output and Hold-Output decoders. Furthermore, the method allows this
limit cycle amplitude to be decreased for a large class of systems. Numerical examples of
the design method were shown for SISO and MIMO systems leading to up to three times
reduction in limit cycle amplitudes.
Finally, the DFs, analysis tools, and synthesis tools were applied experimentally on a
small robotic raft in a lab environment. The raft heading was controlled while quantization
and packet loss were artificially inserted in the sensor feedback channel. The DFs were
used to predict limit cycle amplitudes using several different codecs, and the limit cycles
verified experimentally. A new desired design point was then selected at a smaller limit
cycle amplitude than those of the tested codecs, and a new decoder designed and tested to
meet this requirement. The results demonstrated how the designed decoder can be used to
202
decrease limit cycle amplitudes even when compared to a linear-filter-based decoder.
8.2
Areas for Future Work
The describing function analysis and synthesis tools developed in this thesis are directly
applicable today to some practical systems. Wide Area Measurement Systems on power
grids offer perhaps the best short-term application. These systems have the traits that they
are commonly operated close to instability and are subject to quantization and packet loss
in the sensor feedback channel. This leads to the real possibility that these systems could
undergo unforeseen and undesired limit cycling. Currently many WAMS systems are used
primarily for monitoring and observation, despite the potential for feedback control to be
employed [13]. The design and synthesis tools developed in this thesis could potentially
help these WAMS systems to be transitioned to feedback control setups, and realize their
full control potential. While final designs for systems may ultimately be based on detailed
simulations, the describing function tools derived in this research would provide good starting points for decoder system designs.
Further into the future, underwater multi-robot systems stand to benefit from this research. Though currently there are no multi-robot underwater systems decommissioning
oil rigs or tracking oil plumes, that may change in the coming years. A better understanding
of the effects of quantization and packet loss on these systems, as given in this research,
may enable operators to be more confident in allowing teams of vehicles to operate without
direct human control in challenging communication environments.
Expanding on the theoretical aspects of this work, other encoding/decoding control
algorithms exist that would be amenable to the describing function analysis presented here.
In the sensor feedback channel, for instance, Gupta [39] has developed other variants of the
MIF algorithm where sensor measurement are encoded, sent through a lossy channel, and
decoded by a modified Kalman Filter. The method used to derive the describing function
for the MIF would in theory work for any other codec where the decoded output Sk is
defined by a linear recurrence sequence, as is true in the MIF case. The most difficult step
in deriving the describing function for the MIF was finding the patterns in the sequence,
203
and this pattern-finding step would also be required for other codecs defined by linear
recurrence sequences. The result for E [Sk] would likely be of a similar form to that of E[Sk]
for the MIF, and thus the derivations in this thesis could guide derivations of other similar
codecs.
The describing function analysis could also be applied to controller-actuator codec
algorithms. The Hold-Output and Zero-Output algorithms are commonly-used decoding
techniques for systems with packet loss in the controller-actuator channel, and the results
of this thesis could thus be applied to these systems. Other more sophisticated algorithms
exist as well, such as PPC [83] and S-PPC [72]. These involve both encoders and decoders,
and in theory describing functions could be derived for these algorithms.
Looking beyond feedback control systems, describing functions for stochastic systems
potentially have much broader applications. Any system with stochastic variables could
potentially be modeled using stochastic describing functions. To name a few examples,
ocean wave behavior, stock market options pricing, Brownian particle motion, and turbulent transport are all processes that are functions of stochastic variables, and stochastic
describing functions may help to better understand such systems.
204
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