Limited Data Rate in Control Systems with Networks

181 Pages·2002·9.141 MB·English

by  Dr. Hideaki Ishii, Professor Bruce A. Francis (auth.)

Lecture Notes
in Control and Information Sciences 275
Editors:M.Thoma · M.Morari
Springer
Berlin
Heidelberg
NewYork
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Hideaki Ishii, Bruce A. Francis
Limited Data Rate in Control
Systems with Networks
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A.Bensoussan·P.Fleming·M.J.Grimble·P.Kokotovic·
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Authors
Dr.HideakiIshii ProfessorBruceA.Francis
UniversityofIllinois UniversityofToronto
CoordinatedScienceLaboratory ElectricalandComputerEngineering
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Preface
This book is an attempt to incorporate data rate issues that arise in control
design for systems involving communication networks. The general setup of
the problems considered here is that, given a plant, a communication channel
with limited data rate, and control objectives, (cid:255)nd a controller that uses the
channel in the feedbackloop such that the overallsystem achievesthe control
objectives. The theoretical question of interest is to (cid:255)nd the minimum data
rate necessary for the channel.
The motivation for this study comes from the recent growth in communi-
cationtechnology. Theuse of networkshasbecomecommon practicein many
control applications, connecting sensors and actuators to controllers. One of
the objectives of the book is therefore to provide some basics of the networks
used in control systems.
Fromthe theory side, wehavebeen stimulated by the increasingattention
devoted to the (cid:255)eld of hybrid systems. The systems studied in this book can
be viewed as such systems: The plant has continuous dynamics and, due to
the limited datarate in the channel,there is some discretedecisionmakingin
the controller on what information to transmit and/or when to send it.
Of the issues related to data rate, our focus is on the use of networks
in distributed systems and on quantization in messages sent over networks.
In the (cid:255)rst part of the book, we develop a time sequencing technique for a
distributedcontrolsystemwhereanetworkcableconnectslocalcontrollers. A
simple network model with some features of practical protocols is proposed,
and stabilizability of such systems is addressed. It is shown that the use
of a network can enlarge the class of plants to be stabilized. Moreover, we
con(cid:255)rm the intuition that properly increasingthe data transmission rate over
the network can enhance the capability of the systems.
The second part of the study deals with a control system that has a (cid:255)nite
dataratechannelinthe feedbackloop. Messagesaresentfromthe sensorside
to the actuator side periodically and can take only a (cid:255)nite number of bits;
thereistimedelayassociatedtotherateaswell. Weproposecontrollerdesign
methods for a continuous-time, linear plant to achieve quadratic stability in
the continuous-time domain. As a (cid:255)rst step to model such a channel, we
consider the sampled-data setup where simply a sampler and a hold are put
into the loop. The problemis to (cid:255)nd the largest samplingperiod for stability.
vi Preface
The next step is toextend the results for asystem that hasaquantizerin the
controller, in addition to a sampler and a hold. The quantizer design allows
us to calculate the number of bits in the messages. As the (cid:255)nal step, we
give a simple analysis on the delay time that the system can tolerate without
becoming unstable. A fairly general treatment of quantizers is developed,
and this enables us to compare the data rate necessary for di(cid:254)erent types of
quantizersinanexample. Tofurtherreducethedatarate,wealsoshowseveral
variations of the quantized sampled-datasystem. Thesesystems involvemore
hybrid decision making, unlike the simple sampling scheme in the original
problem.
The book is based on the Ph.D. thesis of the (cid:255)rst author.
Acknowledgements: We wishtothank TedDavison,RaymondKwong,Steve
Morse, and Murray Wonham, who served on the thesis committee of the (cid:255)rst
author, for their helpful comments and suggestions. We also wish to thank
Mireille Brouckeand Daniel Liberzon for stimulating discussions.
The (cid:255)rst author would like to thank Yutaka Yamamoto, who introduced
him to the theory of digital control in the (cid:255)rst place. He is also grateful to
SeanBourdonandGuangdiHuforsharingideasduringthecourseofhisstudy
in Toronto.
Finally, the (cid:255)nancial support from the Natural Sciences and Engineering
Research Council of Canada and in part the National Science Foundation
throughthe UniversityofIllinois atUrbana-Champaignis gratefullyacknowl-
edged.
Contents
Preface v
Notation xi
1 Introduction 1
1.1 Limited data rate . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Outline and contributions . . . . . . . . . . . . . . . . . . . . . 3
2 Control networks 7
2.1 Control over networks . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 How control networks work . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Periodic pattern . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Medium access methods . . . . . . . . . . . . . . . . . . 9
2.2.3 A protocol example . . . . . . . . . . . . . . . . . . . . 10
2.3 Application examples. . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Jacking systems of train cars . . . . . . . . . . . . . . . 12
2.3.2 Networks on automobiles . . . . . . . . . . . . . . . . . 13
2.3.3 Processcontrol . . . . . . . . . . . . . . . . . . . . . . . 14
3 Distributed control over networks 15
3.1 The switch boxproblem . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Stabilization using PTV local controllers . . . . . . . . . . . . . 19
3.4 Assignability measure analysis. . . . . . . . . . . . . . . . . . . 23
3.5 Multiple mobile robot example . . . . . . . . . . . . . . . . . . 25
4 Finite data rate control | single-input case 33
4.1 Dwell-time switched systems and their stability . . . . . . . . . 33
4.2 Quadratic stabilization of sampled-data systems. . . . . . . . . 35
4.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . 35
4.2.2 Control Lyapunov function approach . . . . . . . . . . . 37
4.2.3 Bounds on trajectories . . . . . . . . . . . . . . . . . . . 40
4.2.4 Solution to the sampled-data problem . . . . . . . . . . 42
vii Contents
4.3 Quantized sampled-data control . . . . . . . . . . . . . . . . . . 47
4.3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . 48
4.3.2 A suÆcient condition for stability. . . . . . . . . . . . . 49
4.3.3 Solution to the quantized sampled-data problem . . . . 51
4.4 Finite quantizers . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5 Control over a (cid:255)nite data rate channel . . . . . . . . . . . . . . 63
4.5.1 Data rate for control . . . . . . . . . . . . . . . . . . . . 63
4.5.2 Time delay analysis . . . . . . . . . . . . . . . . . . . . 64
4.5.3 Time delay and quantization . . . . . . . . . . . . . . . 68
4.6 Design of (cid:255) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7 Magnetic ball levitation example . . . . . . . . . . . . . . . . . 72
5 Towards data rate reduction 83
5.1 Time-varying quantization . . . . . . . . . . . . . . . . . . . . . 83
5.1.1 Problem formulation . . . . . . . . . . . . . . . . . . . . 83
5.1.2 A switching law. . . . . . . . . . . . . . . . . . . . . . . 85
5.1.3 Magnetic ball levitation example continued . . . . . . . 88
5.2 Nonuniform sampling . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Dwell-time switching control. . . . . . . . . . . . . . . . . . . . 90
5.3.1 Dwell-time switched systems with logarithmic partitions 91
5.3.2 Problem formulation . . . . . . . . . . . . . . . . . . . . 93
5.3.3 Hybrid automata representation . . . . . . . . . . . . . 94
5.3.4 Solution to the dwell-time switching problem . . . . . . 98
5.4 Finite partition dwell-time switching . . . . . . . . . . . . . . . 101
5.4.1 Dwell-time switched systems with (cid:255)nite partitions . . . 102
5.4.2 State feedback control . . . . . . . . . . . . . . . . . . . 103
5.4.3 State feedback under measurement noise . . . . . . . . . 104
5.4.4 Observer-basedoutput feedback . . . . . . . . . . . . . 106
5.4.5 Cart-pendulum system example . . . . . . . . . . . . . . 109
6 Extensions for the multiple input case 117
6.1 Quadratic stabilization of sampled-data systems. . . . . . . . . 117
6.1.1 Problem formulation . . . . . . . . . . . . . . . . . . . . 117
6.1.2 Generalization of the setup . . . . . . . . . . . . . . . . 119
6.1.3 Bounds on trajectories . . . . . . . . . . . . . . . . . . . 122
6.1.4 Solution to the sampled-data problem . . . . . . . . . . 125
6.2 Quantized sampled-data control . . . . . . . . . . . . . . . . . . 128
6.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . 128
6.2.2 Solution to the quantized sampled-data problem . . . . 129
6.3 Dwell-time switching control. . . . . . . . . . . . . . . . . . . . 140
6.3.1 Dwell-time switched systems with logarithmic partitions 140
6.3.2 Problem formulation . . . . . . . . . . . . . . . . . . . . 143
6.3.3 Solution to the dwell-time switching problem . . . . . . 144
Contents ix
6.4 Design of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.4.1 A suÆcient condition for S >I . . . . . . . . . . . . . . 148
6.4.2 Extension of the design method for (cid:255) . . . . . . . . . . 149
6.4.3 A design method for S >I . . . . . . . . . . . . . . . . 150
6.5 Two cart-pendulum system example . . . . . . . . . . . . . . . 152
7 Conclusion 161
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Notation
R set of real numbers
R+ nonnegative subset of R
C set of complex numbers
Z set of integers
Z+ nonnegative subset of Z
Rez real part of z 2C
c
X complement of X
cl(X) closure of X
int(X) interior of X
X
2 family of all subsets of X
?
X orthogonal complement ofX
span S span of a subset S of a vector space
Bx(r) ball of radius r >0 centered at x2Rn
0
A transpose of A
(cid:255)
A complex-conjugate transpose or adjoint of A
rankA rank of A
ImA image of A
KerA kernel of A
(cid:255)(A); f(cid:255)k(A)g set of singular values of A2Cm(cid:254)n in nonincreasing order:
(cid:255)max =(cid:255)1 (cid:255)(cid:255)2 (cid:255)(cid:254)(cid:254)(cid:254)(cid:255)(cid:255)minfm;ng =(cid:255)min
(cid:254)(A); f(cid:254)k(A)g set of eigenvalues of A
kAk spectral norm of A
kxk Euclidean norm of x2Rn
diag(a1;::: ;an) diagonal matrix with diagonal entries a1;::: ;an
dxe the smallest integer equal to or larger than x
bxc the greatest integer equal to or smaller than x
(A;B;C;D) linear time-invariant system with state equations
x_ =Ax+Bu;
y=Cx+Du;
where x is the state, u is the input, and y is the output.