Limited Data Rate in Control Systems with Networks (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari

275

Springer Berlin Heidelberg NewYork Barcelona Hong Kong London Milan Paris Tokyo

Hideaki Ishii, Bruce A. Francis

Limited Data Rate in Control Systems with Networks With 80 Figures

13

Series Advisory Board

A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis

Authors Dr. Hideaki Ishii University of Illinois Coordinated Science Laboratory 1308 West Main Street 61801 Urbana USA

Professor Bruce A. Francis University of Toronto Electrical and Computer Engineering M55 1A4 Toronto, On Canada

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek – CIP-Einheitsaufnahme Limited data rate in control systems with networks / Hideaki Ishii ; Bruce A. Francis (ed.). – Berlin ; Heidelberg ; NewYork ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in control and information sciences ; Vol. 275) isbn 3-540-43237-X

isbn 3-540-43237-X Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution act under German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Digital data supplied by author. Data-conversion by PTP-Berlin, Stefan Sossna e.K. Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper spin 10867690 62/3020uw - 5 4 3 2 1 0

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 ob jectives, ÿnd a controller that uses the channel in the feedback loop such that the overall system achieves the control ob jectives.

The theoretical question of interest is to ÿnd the minimum data

rate necessary for the channel. The motivation for this study comes from the recent growth in communication technology. The use of networks has become common practice in many control applications, connecting sensors and actuators to controllers. One of the ob jectives of the book is therefore to provide some basics of the networks used in control systems. From the theory side, we have been stimulated by the increasing attention devoted to the ÿeld 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 data rate in the channel, there is some discrete decision making in 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 ÿrst part of the book, we develop a time sequencing technique for a distributed control system where a network cable connects local controllers. 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

conÿrm the intuition that properly increasing the 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 ÿnite data rate channel in the feedback loop. Messages are sent from the sensor side to the actuator side periodically and can take only a ÿnite number of bits; there is time delay associated to the rate as well. We propose controller design methods for a continuous-time, linear plant to achieve quadratic stability in the continuous-time domain.

As a ÿrst 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 problem is to ÿnd the largest sampling period for stability.

Preface

vi

The next step is to extend the results for a system that has a quantizer in 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 ÿnal 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 diþerent types of quantizers in an example. To further reduce the data rate, we also show several variations of the quantized sampled-data system. These systems involve more hybrid decision making, unlike the simple sampling scheme in the original problem. The book is based on the Ph.D. thesis of the ÿrst author.

Acknowledgements

:

We wish to thank Ted Davison, Raymond Kwong, Steve

Morse, and Murray Wonham, who served on the thesis committee of the ÿrst author, for their helpful comments and suggestions.

We also wish to thank

Mireille Broucke and Daniel Liberzon for stimulating discussions. The ÿrst author would like to thank Yutaka Yamamoto, who introduced him to the theory of digital control in the ÿrst place.

He is also grateful to

Sean Bourdon and Guangdi Hu for sharing ideas during the course of his study in Toronto. Finally, the ÿnancial support from the Natural Sciences and Engineering Research Council of Canada and in part the National Science Foundation through the University of Illinois at Urbana-Champaign is gratefully acknowledged.

Contents

Preface

v

Notation

1

2

Introduction

1

1.1 Limited data rate . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Outline and contributions . . . . . . . . . . . . . . . . . . . . .

2 3

Control networks

7

2.1 Control over networks . . . . . . . . 2.2 How control networks work . . . . . 2.2.1 Periodic pattern . . . . . . . 2.2.2 Medium access methods . . . 2.2.3 A protocol example . . . . . 2.3 Application examples . . . . . . . . . 2.3.1 Jacking systems of train cars 2.3.2 Networks on automobiles . . 2.3.3 Process control . . . . . . . . 3

4

xi

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

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. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

The switch box problem . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . Stabilization using PTV local controllers Assignability measure analysis . . . . . . Multiple mobile robot example . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

4.1 Dwell-time switched systems and their stability . 4.2 Quadratic stabilization of sampled-data systems . 4.2.1 Problem formulation . . . . . . . . . . . . 4.2.2 Control Lyapunov function approach . . . 4.2.3 Bounds on trajectories . . . . . . . . . . . 4.2.4 Solution to the sampled-data problem . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

Distributed control over networks

3.1 3.2 3.3 3.4 3.5

. . . . . . . . .

Finite data rate control | single-input case

7 8 8 9 10 12 12 13 14 15

15 17 19 23 25

33

33 35 35 37 40 42

Contents

vii

4.3

4.4 4.5

Quantized sampled-data control . . . . . . . . . . . . . . . . . . Problem formulation . . . . . . . . . . . . . . . . . . . .

48

4.3.2

A suÆcient condition for stability . . . . . . . . . . . . .

49

4.3.3

Solution to the quantized sampled-data problem

Finite quantizers

. . . .

51

. . . . . . . . . . . . . . . . . . . . . . . . . .

60

Control over a ÿnite 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

ÿ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

4.6

Design of

4.7

Magnetic ball levitation example

. . . . . . . . . . . . . . . . .

5 Towards data rate reduction 5.1

47

4.3.1

72

83

Time-varying quantization . . . . . . . . . . . . . . . . . . . . .

83

5.1.1

83

Problem formulation . . . . . . . . . . . . . . . . . . . .

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

5.4

5.3.1

Dwell-time switched systems with logarithmic partitions

91

5.3.2

Problem formulation . . . . . . . . . . . . . . . . . . . .

93

5.3.3

Hybrid automata representation

5.3.4

Solution to the dwell-time switching problem

. . . . . . . . . . . . .

98

Finite partition dwell-time switching . . . . . . . . . . . . . . .

101

5.4.1

Dwell-time switched systems with ÿnite partitions

. . .

102

5.4.2

State feedback control . . . . . . . . . . . . . . . . . . .

103

5.4.3

State feedback under measurement noise . . . . . . . . .

104

5.4.4

Observer-based output feedback

. . . . . . . . . . . . .

106

5.4.5

Cart-pendulum system example . . . . . . . . . . . . . .

109

6 Extensions for the multiple input case 6.1

6.2

6.3

94

. . . . . .

117

Quadratic stabilization of sampled-data systems . . . . . . . . .

117

6.1.1

117

Problem formulation . . . . . . . . . . . . . . . . . . . .

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

Quantized sampled-data control . . . . . . . . . . . . . . . . . .

128

6.2.1

Problem formulation . . . . . . . . . . . . . . . . . . . .

128

6.2.2

Solution to the quantized sampled-data problem

. . . .

129

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 6.4

Design of 6.4.1

6.5

ix

S

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A suÆcient condition for

S>I

. . . . . . . . . . . . . .

6.4.2

Extension of the design method for

6.4.3

A design method for

S>I

ÿ

. . . . . . . . . .

149

. . . . . . . . . . . . . . . .

150

Two cart-pendulum system example

. . . . . . . . . . . . . . .

7 Conclusion Bibliography

147 148

152

161 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163

Symbol Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

170

Subject Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

172

Notation R R+ C Z Z+

set of real numbers nonnegative subset of set of integers nonnegative subset of

2C complement of X closure of X interior of X

Re z

X

real part of

c

X) X)

cl(

int( 2X

X B

Z

z

family of all subsets of

?

span

R

set of complex numbers

X

X

orthogonal complement of span of a subset

S

S

of a vector space

n

ball of radius

0

transpose of

A

ÿ

complex-conjugate transpose or adjoint of

rank A

rank of

Im A

image of

A

Ker A

kernel of

A

A

ÿ (A);

þ(A);

kAk kxk

fÿ fþ

g

k (A)

diag(a1 ; : : :

dxe bxc

g

k (A)

r >

0 centered at

2R

x (r )

A

set of singular values of

A

2C

mþn

max = ÿ1 ÿ ÿ2 ÿ þ þ þ ÿ ÿmin

ÿ

in nonincreasing order:

fm;ng

set of eigenvalues of A

Euclidean norm of

(A; B; C; D )

A

A

spectral norm of ; an )

x

x

=

ÿ

min

A

2R

n

diagonal matrix with diagonal entries

1

a ; : : : ; an

the smallest integer equal to or larger than

x

the greatest integer equal to or smaller than

x

linear time-invariant system with state equations

where

x

is the state,

u

x

=

Ax

+ Bu;

y

=

Cx

+ Du;

is the input, and

y

is the output.

Chapter 1 Introduction In control systems, communication arises when signals are sent from sensors to controllers and then from controllers to actuators. The systems involved in the communication are often digital. Thus, the signals take quantized values at discrete-time instants, and there is time delay inherent in transmission. A traditional assumption made on such communication in control is that there is no quantization and no time delay. the theoretical treatment of systems.

This can signiÿcantly simplify

Moreover, this assumption has been

justiÿed since, in practice, dedicated cables have been used to connect system components.

Data rate

is one of the basic characteristics of communication

systems and is the rate at which bits are transmitted over a channel in bit/sec or bps. Under the traditional view, the channel requires inÿnite data rate. Recently, as a result of the vast progress in communication technology, the use of digital, serial networks has become popular in large-scale control systems. Networks used for the connection of sensors/actuators and controllers are called

control networks or ÿeldbuses . Nowadays, many advanced protocols

are available for such networks.

While this change in hardware oþers new

potential for applications, it suggests that the ÿniteness of the data rate in communication may now be relevant. Since the channel is shared by multiple sources of data, the total amount of communication expected in the system must be known and must not exceed the physical bound on the data rate. Furthermore, in control systems, the real-time requirement is critical, and the time delay in communication has to be minimal. Consequently, it is important that the channel be shared in an eÆcient manner, or otherwise the system will not operate properly. On the other hand, the use of networks in control poses questions that diþer in nature from the traditional views. For example, (i) what is a better strategy to reduce the data rate, to increase the sampling period or to decrease the number of bits per message? (ii) In a large-scale system, network cables may oþer data exchange between system components where there was no cabling before, even at a very low transmission rate; how can such communication enhance the performance of the system?

H. Ishii and B.A. Francis (Eds.): Limited Data Rate in Control Systems with Networks, LNCIS 275, pp. 1−6, 2002.  Springer-Verlag Berlin Heidelberg 2002

2

Chapter 1.

Introduction

Data rate issues in control are also related to systems in which communication capability is constrained due to limited sources of energy or stealth reasons.

Examples of such systems are planetary rovers, multi-agent mobile

robots, unmanned autonomous vehicles, and so on. The aim of this book is to develop theories for control systems under data rate limitation.

This study has been particularly motivated by the recent

popularity in control networks and hence incorporates some of the practical aspects of this technology. The underlying theoretical interest is, however, to investigate what the minimum data rate is in control to achieve given speciÿcations.

1.1

Limited data rate

In this section, we discuss problems that arise in control systems with limited data rate and give a survey on the literature related to this topic.

We focus

on issues such as time sequencing, quantization, time delay, and distributed control. In serial networks, messages are multiplexed in time.

Through time se-

quencing of messages, all system components can share the same network cable.

On the other hand, all messages have to be quantized, or be repre-

sented by a ÿnite number of bits. two techniques:

Clearly, there is a trade-oþ between these

Given a ÿxed rate, one can use fewer bits per message and

increase the number of messages per time, and vice versa. Brockett [11] initiated an interesting study concerning time sequencing and proposed a simple model of control networks and a design method of a central controller connected to sensors/actuators by a network.

Because of

the limitation on data rate, only one of the senor/actuator pairs is allowed to communicate with the controller at each discrete instant. Quantization has been studied in the context of digital control and signal processing. Typically, quantizers round oþ the input signals uniformly with a ÿxed step size. In many previous works, the error caused by such quantization is modeled as noise, often uniform and white [71].

On the other hand, it

is known that, even in a simple feedback setup, interesting and complicated behaviors such as limit cycle and chaotic tra jectories can be observed [17, 84]. Many of the recent deterministic treatments of quantizers in feedback systems follow the inýuential paper by Delchamps [24]. Elia and Mitter [27, 28] proposed a stabilization technique for linear systems; a nonuniform quantizer is designed as a result of an optimization problem. Brockett and Liberzon [12] studied the use of uniform but time-varying quantizers.

In [94], coding and

time delay inherent in channels of networked control systems are explicitly taken into account, and a new notion of stability is introduced. There are several types of time delay characteristic of serial networks. One is called the transmission time. This is the time it takes for a transmitter to send out data:

To send

N

bits of data over a

D

bps channel, it takes

N=D

seconds. Another one is a consequence of time sequencing and is the waiting

1.2.

3

Outline and contributions

time when a node has to send out a message, but cannot due to a busy channel; this type of delay is by nature time-varying and, thus, diÆcult to determine. It is known that delay in a feedback loop can result in instability of the system, and there is a fairly large body of research concerning time-delay systems; recent works relevant to control over networks can be found in [62, 70, 98]. One application example is bilateral control of telerobotic systems (e.g., [3]), which often operate over the Internet. To reduce the amount of traÆc over the network, control functions often have to be distributed over a system.

The controllers in such systems are

local in the sense that they can measure only a limited number of outputs and can control only certain actuators. Control of distributed systems has been studied in the area of decentralized control [55, 77]. It is known that this structure may impose a severe limitation on the performance of such systems. The idea of information exchange among local controllers has been long studied (e.g., [64, 90]). However, there are not many works that deal with realistic, practical models of networks incorporating bandwidth issues [99]. Other studies dealing with data rate issues include state estimation problems [23, 60, 93] and control of linear quadratic Gaussian systems [58, 81].

1.2

Outline and contributions

In this book, we study the eþect of data rate limitation in control systems using networks. Among the issues listed in the previous subsection, we are interested in the time sequencing and quantization aspects involved in networks. The outline of the book is given as follows. In Chapter 2, we highlight some practical issues related to control networks. We ÿrst describe the characteristics of the traÆc in control systems to show how the requirements for control networks are diþerent from those for data networks. Second, the advantages of using networks in application are given. Third, common features in control networks protocols are studied brieýy, so as to provide some relations of the studies in the following chapters to practical applications. Finally, some industrial examples are given. Chapter 3 deals with a distributed control problem, where a plant is controlled by two local controllers (see Fig. 1.1).

A network cable is used to

connect the controllers so that local data can be shared. The assumption due to the limited data rate is that only one controller can transmit a message at a time. We solve the problem of ÿnding the sequencing of messages over the network to stabilize the system. It is then shown that the capability of the system can be enhanced in some sense if the traÆc is properly increased. In Chapters 4, 5, and 6, we study control methods for a system that has a ÿnite data rate channel in the feedback loop.

We are motivated by the

problem of ÿnding the minimum data rate required to achieve speciÿed control objectives.

In particular, quadratic stabilization methods for a continuous-

time, linear time-invariant plant are developed.

4

Chapter 1.

Introduction

plant

controllers

interfaces network Figure 1.1: Distributed control system

In Chapter 4, we ÿrst give a class of hybrid systems, which provides a uniform stability criterion for the systems in the following sections and chapters.

Given a stabilizing controller

K

for the single-input, continuous-time

plant represented by (A; B ) in Fig. 1.2(a), as a ÿrst step to model the digital channel, we insert a sampler

T and a zeroth order hold

S

T in the loop as in

H

Fig. 1.2(b). The ÿrst problem is to quadratically stabilize the sampled-data system, and we derive a bound on the sampling period

T

which guarantees a

prespeciÿed decay rate of the state in the continuous-time domain. This problem serves as a basis for the second problem where in the channel a quantizer

Q

is added as in Fig. 1.2(c). We propose a method to design a

quantizer and to ÿnd a bound on

T

while maintaining the quadratic stability

of the closed-loop system. Further, we give more speciÿc results for uniform quantizers and other types of quantizers. Finally, to incorporate the time delay in the communication, we consider the system in Fig. 1.3. The delay is time-varying, but has an upper bound We give an analysis on how large the maximum delay time

ý

ý.

can be for the

closed-loop system to maintain its stability. With this result we can explicitly calculate the necessary data rate for the channel. A design example of a magnetic levitation ball system is given, and we compare the data rate necessary for stabilization using several quantizers. In Chapter 5, we propose some methods to reduce the data rate necessary for stabilization by extending the results in Chapter 4. The picture of the system is more general and is depicted in Fig. 1.4, where the sensor and the actuator are connected to a network through a coder and a decoder, respectively. The ÿrst direction is to use time-varying quantizers that can change their range and thus require almost arbitrarily small number of quantization levels. The other direction for reducing the data rate is to employ nonuniform sampling. We ÿrst give a fairly simple method and then a switching control method. One of the obstacles in switching control of a continuous-time system is that fast switching or chattering may occur. Our results on quantized sampled-data control lead us to a switching control method, called dwell-time switching control; the protocol is that a switching cannot occur until a speciÿed dwell time has elapsed from the last switching.

1.2.

5

Outline and contributions

u(t)

-

-

x(t)

(A; B )

?

x(t)

u(t)

(A; B )

K

K

ÿ

T

H

(a) Continuous-time system

u(t)

ÿ

T

S

ÿ

(b) Sampled-data system

-

x(t)

(A; B )

? K

T

H

ÿ

Q

ÿ

T

S

ÿ

(c) Quantized sampled-data system

Figure 1.2:

Chapter 6 is devoted to the extension of the problems in Chapters 4 and 5 for the multiple-input case. First we formulate the sampled-data problem and the quantized sampled-data problem in a manner parallel to the single-input case. Then, the dwell-time switching control scheme is generalized as well. In Chapter 7, we discuss some future research topics on networked control systems and ÿnite data rate issues. The contributions of the book can be summarized as follows. The time sequencing techniques of messages employed in networks are treated in a distributed control setup.

We propose a problem of designing

dynamic local controllers connected to each other by a network. Stability issues are characterized, and the advantage of the use of networks is clariÿed in the context of decentralized control. Quantization methods for a centralized control system with a ÿnite data rate communication channel in the feedback loop are developed in a systematic manner. In particular, we consider the stabilization of a continuous-time plant, and the stability criterion throughout this study is quadratic stability with a guaranteed decay rate in the continuous-time domain.

Such a treatment

6

Chapter 1.

u(t)

-

Introduction

x(t)

(A; B )

? K

T

H

ÿ

ý

delay

ÿ

ý

ÿ

Q

T

S

ÿ

Figure 1.3: Quantized sampled-data system with delay

u(t)

-

x(t)

(A; B )

T

D

ÿ

Decoder

T

C

i(t)

ÿ

Coder

Figure 1.4: Switching system

of sampled-data control systems with the use of control Lyapunov functions seems to be new. We deal with a fairly general class of quantizers. For systems with quantizers, it may not be possible to achieve asymptotic stability; instead, we employ a weaker notion of stability which requires the state to go only close to the origin. Also, by limiting the region of initial states in the state space to be bounded, the number of quantization levels can be ÿnite. We propose a design method for a sampling period and a quantizer to achieve such stability and give a simple deÿnition of data rate required for control. When a message with data is sent over a network, there is transmission time delay due to the limited data rate.

We analyze the delay time that

the quantized sampled-data system can allow while maintaining stability. The continuous-time quadratic stability of the original closed-loop system is the key to this analysis. This result combines the issues related to network control, namely sampling, quantization, and time delay, and gives us a control system that uses a strictly ÿnite data rate channel in the feedback loop. Extending the ideas underneath the quantized sampled-data problem, we further develop several methods to reduce the data rate for control. Although control systems that use quantized measurements are hybrid in nature, there have been few studies in hybrid dynamical systems from the data rate point of view.

Chapter 2

Control networks In this chapter, we give a brief review of control networks. First, we study the characteristics of communication in control and the advantages of using networks there. Then, we summarize more practical issues in control networks to provide some ideas on the current technology and to describe how the networks work.

2.1

Control over networks

In control systems, the traÆc between sensors/actuators and controllers has some distinct characteristics. First, the messages are small because they contain raw data from sensors or for actuators. Second, the transmission rate is constantly high and the interval between messages is short due to the real-time aspect of control; many sensors/actuators have ÿxed, small sampling periods. Third, it is important that the time delay in communication be minimal and that the delay time be known. Control networks have to be designed to handle such traÆc. It is clear that the requirements for this type of network are diþerent from the requirements for data networks, where large data messages are transmitted occasionally for short intervals of time at high data rates. The idea of using networks in control systems is certainly not new. For example, the nuclear science community developed a parallel bus protocol called CAMAC in the early 1970s, and another standard MIL-STD-1553 has been used in military avionic applications since the mid-1970s in western countries [41].

What is new is that networks are fast enough for the real-time

requirements in control and that network interfaces are now inexpensive and small enough that, even if used in a large number, their presence is not costly. In the past decade much eþort has been made for the standardization of protocols for control networks by groups of companies and professional organizations. These protocols include Controller Area Networks (CAN) [50], Factory Instrumentation Protocol (FIP) [82], Foundation ÿeldbus [73], Lon-

H. Ishii and B.A. Francis (Eds.): Limited Data Rate in Control Systems with Networks, LNCIS 275, pp. 7−14, 2002.  Springer-Verlag Berlin Heidelberg 2002

8

Chapter 2.

Control networks

Works [68], and Process Fieldbus (Proÿbus) [8]. There are various advantages to use network cables in control systems. First of all, the overall cost for cabling, installation, and maintenance can be reduced drastically.

Moreover, detection of cable faults and system expan-

sion is much easier because of the modular nature of the network. In addition, networks can bring new functionalities that were not available previously: Networks with broadcast capability allow simultaneous availability of data to all devices connected to the network. Also, network interfaces have some processing capabilities that can be used for control purposes. As we discussed in Section 1.1, the main drawback for using a shared medium instead of dedicated wires is the limitation on data rate.

2.2

How control networks work

In this section, we summarize how control network protocols are designed in practice to meet the requirements described above.

Media access methods

make the diþerences among existing protocols and are brieýy explained. We also give some details on one of the standardized protocols.

2.2.1

Periodic pattern

In the previous section, we looked at the nature of communication in control systems. We can divide the traÆc typically involved in control into two classes. One class is of periodic signals; these signals usually have critical real-time requirements and are transmitted periodically from sensors or to actuators. The other class is of event-driven, aperiodic signals, which need not be transmitted periodically but only when necessary. These signals include emergency alarms, on/oþ type signals, and data from counters. Control networks are mostly serial and thus employ time division multiplexing. A common technique for such networks is to use a communication type with a

periodic pattern

in time [66]. The pattern consists of two phases, peri-

odic and aperiodic, corresponding to the two classes of signals, as in Fig. 2.1.

period

periodic phase

aperiodic phase

time

Figure 2.1: Periodic pattern of communication The periodic phase is deterministic in that the transmission order of messages is scheduled and ÿxed. Hence, no collision of messages occurs and the delay time in communication is known. Only during the aperiodic phase, the

2.2.

9

How control networks work

event-driven communication is allowed to take place, and thus there may be time delay.

It is important that the periodic pattern be designed to leave

enough time for the aperiodic phase so that, in any event, the delay time in aperiodic communication is suÆciently small and that the system does not run into a hazardous state. Because of the periodic and deterministic nature of communication and the critical requirement in time, it is preferred to have one or more

master

stations

in the system that poll all the nodes of sensors, actuators, and controllers in the system according to a periodic schedule. An additional advantage for such architecture is that it requires simple protocols and thus reduces the cost. The presence of a master on a network can be, however, controversial as it creates a single point of failure, that is, if the master fails the entire system comes to a halt. 2.2.2

Medium access methods

As to how to realize a collision-free transmission scheme that suits the requirements described above, there are many ways. Medium access methods are what determine how each node connected to a network gains access to the network. The choice of the method is a crucial step in deÿning network protocols because it determines the complexity and the cost of the communication system as well as the data rate. Here, we describe brieýy some of the common medium access methods and study how they can be combined for control network use. The simplest is the centralized access method, where a single master controls the network and polls all the nodes (i.e., asks nodes one by one if they have any data to transmit) according to the periodic schedule. Slave nodes can transmit a message only when they are given permission by the master. This method is deterministic and is eÆcient for periodic signals. Nevertheless, when the number of nodes is large, the polling of messages can easily lead to an unacceptably large time delay in the system. One of the popular decentralized access methods is the token passing bus technique. It is an extension of the token passing ring method, but does not require a ring topology; a token that allows message transmission is passed around among nodes in an order determined prior to the operation. This is another example of deterministic methods. Other decentralized methods are those in the class of the carrier sense multiple access with collision detection (CSMA/CD). Basically, each transmitting node ÿrst checks whether the channel is free. If not, the node waits for some time and retries.

If the channel is free, the node begins the transmission.

However, since two nodes may simultaneously start sending out messages, the nodes listen to the channel while transmitting to detect any collisions. There are several measures to handle the possible collisions, including the delayed retry (DR) method (as in the Ethernet), the collision resolution (CR) method, and the collision avoidance (CA) method [50]. Because of the event-driven nature of the media access, CSMA methods are strong on aperiodic signals

10

Chapter 2.

for control networks.

Control networks

The shortcoming is the cost due to the complexity in

protocols and interfaces. Each method above has its strong points and weak points.

In today's

advanced control network protocols, to handle eÆciently both periodic and aperiodic signals, combinations of several access methods are usually allowed. We give three examples in the following: (i) Multiple masters, each with polling capability, pass a token around in a deterministic manner. (ii) A master node is put into a CSMA type network for polling. (iii) In a centralized setup, the master polls all the nodes during the periodic phase; any nodes having aperiodic data to be transmitted ÿrst notify the master at this time and are then given permission for transmission later during the aperiodic phase. To keep the complexity low, many protocols employ variations of the centralized access method.

In such cases, of the seven layers of the Open Systems

Interconnection (OSI) reference model, only three layers are necessary, namely the physical layer, the data link layer, and the application layer.

2.2.3

A protocol example

Here we look into the speciÿcation of one of the typical ÿeldbus protocols. Some details on the rate required for the periodic real-time communication are given.

The goal of this book is to reduce the data rate necessary for

control, and one way to do so is to reduce the number of bits in quantization. To see the eþect of doing so, we calculate the transaction time required for sending each message. Factory Instrumentation Protocol (FIP) is a ÿeldbus protocol accepted internationally in the industry [22, 50, 82]. It was developed in France in the late 1980s and is standardized in Europe. FIP is currently provided by a nonproÿt organization called WorldFIP. Some parts of the protocol are shared with the Foundation ÿeldbus, which is based in North America. The maximum data rate for FIP is 2.5 M bps with twisted pair cables and 5 M bps with ÿber optics. The bus length depends on the data rate and can be up to 2000 m at the slow rate mode (31.75 k bps).

This can be, however,

extended by interconnecting up to 4 networks using repeaters. There may be about 60 stations per network. FIP employs a centralized media access method. A master, or a

trator,

bus arbi-

polls the slave stations periodically based on a predeÿned polling list.

There is a mechanism to handle event-driven messages as well, but here we take a closer look at the periodic communication, for which this protocol is mainly designed [95]. When the system is conÿgured, all signals and their periodicity are determined.

Each signal is associated with a unique identiÿer, and what the bus

2.2.

11

How control networks work

bytes

1

1

preamble

frame start delimeter

1

2

control

identiÿer

2

1

frame check

frame end

sequence

delimiter

(a) Question frame

bytes 1

preamble

1

1

frame start delimeter

control

1

1

0-128

data

data

type

length

2

1

frame check frame end

data

sequence

delimiter

(b) Response frame

Figure 2.2: Message formats of FIP

arbitrator has is a list of identiÿers. According to this list, the bus arbitrator

question frame, which contains an identiÿer. Only one station producer, and one or more stations recognize themselves consumers. The producer broadcasts a response frame with the data

broadcasts a

recognizes itself as the as the

of the signal corresponding to the identiÿer, and only the consumers on the bus accept it. After this the bus arbitrator can go on to the next identiÿer on the list. An acknowledgement message is optional; instead, if the response frame is slow (or lost), the consumers return to the status waiting for the next question frame. Note that because of the deterministic nature, none of the frames described here contains addresses of stations. The message formats for the question frame and the response frame are shown in Fig. 2.2. Both frames include the preamble, the frame start delimiter, the control, the frame check sequences, and the frame end delimiter.

The

question frame contains the identiÿer of 2 bytes, making the frame 8 bytes in total. The response frame has the data type/length bits in addition to the data; thus the total

overhead

is 8 bytes. The data size can be between 0 and

128 bytes. Now we calculate the total transaction time for sending one message with data. The protocol requires some

turnaround time

after each frame transmis-

sion (for physical layer transactions), a minimum of 4 mode. Thus, the transaction time to send transaction time =

n

64 bits (question) + (64 + 8

= 59:2 + 3:2

üs

at the 2.5 M bps

bytes of data is

ü

n)

bits (response)

2:5 M bps

ü

+2 n ü

s:

ü4

üs

(2.1)

12

Chapter 2.

For example, the transaction times are 62.4 219.2

üs

for 50 bytes, and 379.2

üs

Control networks

üs for 1 byte, 65.6 üs for 2 bytes,

for 100 bytes. We see that the eþect of

reducing the data size into half is very diþerent in the cases from 2 bytes to 1 and from 100 bytes to 50. The amount of time to send the question frame and the overhead and the turnaround time are indeed not negligible. In real-time control, the data size is typically small, and thus changing it can be less eþective in reducing data rate (compared to changing the sampling period), unless it becomes signiÿcantly smaller. Comparisons of other protocols can be found in, e.g., [51].

2.3

Application examples

As described above, there are many advantages in introducing networks into control systems, especially those requiring large amount of cabling and/or having many system components. Industrial applications include automobiles [15, 42, 50], robotic systems [16, 39], jacking systems for trains [5], automated manufacturing systems [8, 22, 65], and process control systems [69, 73]. In this section, we give some detailed examples to illustrate the changes that networks can bring into such systems.

2.3.1

Jacking systems of train cars

For maintenance, train cars are raised by jacking systems [5]. Four jacks are typically used for one car, and these are movable on the ýoor for ýexibility. Since the decoupling and coupling of the cars may be time-consuming, it is more common to lift multiple, coupled cars, up to 12 cars, at once. A level lift of the entire train is required for safety and to keep the the coupling devices and the jacks from any damage.

Hence, the control objective here is the

synchronized operation of the jacks. The conventional solution for the realization of the system has been a pointto-point, centrally controlled system as in the layout in Fig. 2.3(a), where every jack is connected to the central controller with a separate cable. However, the large amount of cabling can be costly and further causes installation and storage problems. It also requires a central controller with high computational capability for controlling a certain number of jacks. Recently, a new system has been introduced in a more distributed fashion as in Fig. 2.3(b). All jacks are connected to a single network cable, and each of them has a local controller with communication capability; depending on the information of the neighboring jacks, each jack adjusts its height. Clearly, it uses less cabling; the central controller is downsized by distributing its function to local controllers on the jacks and, as a result, merely has supervising functions. Another advantage is that expansion of the system can be done without much change in hardware. New jacks simply have to be connected to the network, and the rest can be taken care of by the software.

2.3.

Application examples

Central Control

13

þ þ

þ þ

þ þ

þ þ

þ þ

þ þ

þ þ

þ þ

þ þ

þ þ

Car 1

Car 2

þ þ

Car 12

þ þ:

jack

(a) Conventional centrally controlled type

þ þ CC

Car 1

þ þ

þ þ

þ þ

þ þ

Car 2

þ þ

þ þ

þ þ

þ þ

þ þ

þ þ

Car 12

þ þ:

jack

: local controller (b) Distributed type with a network

Figure 2.3: Jacking system

This system has employed the LonWorks networks, which uses a CSMA type access method; the jacks send out their height every time they have moved a certain amount. Thus, the system serves as an example of a real-time control system with event-driven communication.

2.3.2

Networks on automobiles

Modern cars are a good example of systems with extensive cabling in limited spaces. The cabling in a medium size car can be more than 2000 m in length and 100 kg in weight, and there can be more than 600 diþerent types of wires [50]. To ÿt these cables compactly into a car within a short period of time is a challenge for the industry. Since the early 1980s, automobile companies have been developing protocols and have implemented on production automobiles such as the 1986 Buick Riviera and the 1986 Pontiac 6000 STE [42]. In a car, there are several types of communications: One is for real-time critical control as in engines, cruise control, gear boxes, brakes, etc. Another type is for body control as in power windows control, door lock control, mirror positioning, seat positioning, lamps, and so on; the real-time requirement in these applications is much lower, but because of the large number of such systems in a car it is cost sensitive. As a result, two or more separate network cables are often used, one for real-time control, one for body control, and possibly more for diagnostics and

14

Chapter 2.

Control networks

supervisory station

controllers ÿeldbuses sensors/actuators

Figure 2.4: Hierarchical structure

other status information sharing.

However, these networks cannot operate

separately, and thus they are connected to each other through gateways. Consequently, protocols for networks on automobiles are designed to handle all of these types of communication. The Controller Area Networks (CAN) is an example of protocols designed by automobile companies. It was ÿrst implemented in Mercedes S-class cars in 1992. For the access method, it employs a CSMA/CD technique with collision resolution, which is event-driven but deterministic; see [50] for details. 2.3.3

Process control

In the area of process control or more generally in automated manufacturing, systems are often large in scale and physically distributed. In such systems, several networks may compose a hierarchical structure for the interconnection of systems at various levels; a simple system is shown in Fig. 2.4. This structure provides eÆciency and reliability in the communication [41, 65]. Of the hierarchy of networks, at the lowest level is the control networks. This connects sensors and actuators to control devices. Compared to the higher level communication, the data size of messages here is smaller and the realtime requirement is more critical. The controllers may also receive commands from supervisory systems located in a higher level. In [33], an installation of ÿeldbuses at the Monsanto Chocolate Bayou plant in Alvin, TX is reported. In their condensate recovery system, there are 14 ÿeld devices such as magnetic ýowmeters, pressure sensors, temperature sensors, control valves, and so on. Two ÿeldbus cables are used to connect them to two controllers separately and realize a distributed system. It is emphasized that devices provided from various companies work together over the networks. Fieldbuses are used in a much larger scale as well.

At the Shell Ham-

burg oil blending facility, the control system for batch and plant control was upgraded in 1997 using Proÿbus networks [83].

Thousands of temperature

sensors, solenoid valves, and level switches were connected. In some parts, a variation of Proÿbus, designed to prevent explosions caused by sparks from the communication media, is used.

Chapter 3

Distributed control over networks In this chapter, we study a distributed control system, where the controllers can communicate to each other over a limited data rate channel. We study methods for time sequencing of messages over the channel and characterize the stabilizability issues. Then we clarify the advantage of the use of networks in the context of decentralized control.

3.1

The switch box problem

In this section, we formulate our design problem for a distributed control system with two local controllers connected to a network cable through network interfaces as in Fig. 3.1. Signals not available locally for each controller can be communicated over the network, possibly at a rate slower than the sampling rate of the controllers. We give a simple model for the network, governed by the protocol described in the previous chapter, as a \switch box." Because of the sequential transmission over the network, at most one signal is allowed to be communicated through the switch box at each discrete-time instant. The behavior of the switch box is periodic and deterministic. Consider the discrete-time system in Fig. 3.2. The plant P is a linear, time-invariant (TI) system whose state equations are given by

ÿ

( + 1) = yi (k ) =

x k

( ) + Bp1 u1 (k ) + Bp2 u2 (k ); ( ) + Dpi1 u1 (k ) + Dpi2 u2 (k );

Ap x k

Cpi x k

i

= 1 ; 2;

(3.1)

where x(k ) 2 Rn is the state, and ui (k ) 2 Rmi and yi (k ) 2 Rpi are the local control input and the output of channel i = 1; 2, k 2 Z+, respectively. The local controllers Ki , i = 1; 2, are periodically time-varying (PTV) systems H. Ishii and B.A. Francis (Eds.): Limited Data Rate in Control Systems with Networks, LNCIS 275, pp. 15−31, 2002.  Springer-Verlag Berlin Heidelberg 2002

16

Chapter 3.

Distributed control over networks

plant

controllers

interfaces network Figure 3.1: A distributed control system

with the same period

ÿ

K ÿ 1, and their state equations are

N

i (k + 1) = Aki (k)ûi (k) + Bki1 (k)yi (k) + Bki2 (k)vi (k); ui (k ) = Cki (k )ûi (k ) + Dki1 (k )yi (k ) + Dki2 (k )vi (k ); where ûi (k ) 2 Rn is the state and vi (k ) 2 Rp is the signal through the network for k 2 Z+. Here Aki (k ) = Aki (NK + k ) for all k 2 Z+ and likewise û

i

ki

for other system matrices. The switch box

ÿ

y1 P

S

is a PTV, memoryless system

u1 v1 K1 v2 u2 K2

S

y2

Figure 3.2: The switch box problem with a period

N,

where

þ

1 (k) v2 (k ) v

and

S (k )

ý

þ =

S (k )

1 (k) y2 (k ) y

ý ;

k

2 Z+;

is one of the three matrices

þ

0 :=

S

ý

0

0

0

0

þ ;

1 :=

S

0 0

p2

I

0

ý

þ ;

and

2 :=

S

0

p1

I

ý

0 0

:

3.2.

17

Preliminaries

The switch box provides each local controller with the signals not accessible lo-

switching pattern s 2 f0; 1; 2gN , whose elements correspond to the indices of matrices Si ; i = 0; 1; 2. For excally. Its periodic behavior is determined by a

s = [1 2 0 0] means S has a period N = 4 and S (4k ) = S1 ; S (4k + 1) = + 2) = S (4k + 3) = S0 for k 2 Z+. Thus, at k = 0, the controller 2 K1 receives y2 (0); at k = 1, K2 receives y1 (1); at k = 2; 3, no data is sent over

ample,

S ; S (4k

the network; and this repeats periodically.

Finally, in Fig. 3.2, ü denotes the generalized plant containing It is PTV with the period We deÿne our and detectable. controllers

P

and

S.

N.

switch box problem as follows:

Given a switching pattern

s

Assume that

2 f0; 1; 2gN ,

P

is stabilizable

ÿnd PTV local

1 and K2 such that the closed-loop system is exponentially stable.

K

A technique similar to the switch box is employed in the design problems

in [11, 35]. Their setup is that a controller is connected to pairs of sensors and actuators by a network and can communicate with only one pair at each sampling time, depending on a periodic pattern. Even though their model is also simple and linear, because of the constraint on the controller structure (it must be memoryless), the stabilization problem was shown to be an NP hard nonlinear optimization problem. We have extended their problems to a decentralized control setting. The plant with the switch box is PTV and is diÆcult to control with TI local controllers. Thus, we have enlarged the class of controllers to PTV ones, and as a result our solution is much simpler.

3.2

Preliminaries

In this section, we review some preliminary results on the stabilizability of decentralized control systems. Our focus is on the control of linear PTV systems using PTV local controllers [46]. Consider a

ú -channel,

linear, PTV system ü with a period

equations are

ÿ

x(k

i

y

+ 1)

=

A(k )x(k )

(k )

=

C

i

+

Pÿ Pjÿj=1=1

j (k)uj (k); ij (k)uj (k);

K:

N

ÿ

i (k + 1) ui (k )

û

whose state

B

(k )x(k ) +

D

At each channel i, we apply a local PTV controller period

N

i

i

= 1; 2; : : :

K ;i

; ú:

= 1; 2; : : :

; ú,

(3.2)

with a

ki (k)ûi (k) + Bki (k)yi (k); ki (k)ûi (k) + Dki (k)yi (k):

=

A

=

C

We now deÿne two problems:

û The PTV problem :

Find PTV local controllers

the closed-loop system is exponentially stable;

1

K ;::: ;K

ÿ

such that

18

Chapter 3.

û The TI problem dependent of

k,

:

Distributed control over networks

Suppose all the matrices in ü and

i.e.,

=

N

NK

K1 ; : : : = 1. Find TI local controllers

; Kÿ

are in-

K1 ; : : : ; Kÿ

such that the closed-loop system is stable. We follow the formulation in [46] and give some deÿnitions. A linear, TI system

G

with a state model (A; B; C; D ) is said to be complete if rank (M ((A; B; C; D ); z ))

where

n

is the dimension of

A

and

M ((A; B; C; D ); z )

disk, then the state model of

G,

z

for

n

þþ þ ú 2C

M(

If the rank condition holds for every

ÿ

z

2C

(3.3)

;

; ) is the system matrix deÿned by A

:=

zI

ý

B

C

D

:

in the complement of the open unit

or simply

G,

is said to be weakly complete .

Note that the rank condition (3.3) has to be checked only for eigenvalues of A

since otherwise the condition is automatically satisÿed. If the rank condition

fails for some

z

system.

2C

, then

z

is an uncontrollable and unobservable mode of the

For the system ü in (3.2), its complementary subsystems üi1 :::iÿ j1 :::jþ ÿÿ are

given by the state models

0 B@ ü A;

where

f

ik ;

1; 2; : : :

g

jl

;ú

2f

Ci

þþþ

1

Bj

2 32 û 6 .. 1 7 6 4.54 þ ÿÿ

Bj

1; 2; : : :

;ú

;

g

Ciÿ

,

ü

ú

k

. . .

þþþ ..

1 ÿÿ

Di jþ

. . .

.

1 þþþ ú f

Diÿ j

Diÿ jþ

31 75CA

;

ÿÿ

g

;ú 1, i1 ; : : : ; iþ ; j1 ; : : : ; jÿ ýþ = of the matrices is not displayed for sim-

= 1; 2; : : :

. The dependence on

plicity. There are 2ÿ

;

11

Di j

2 such subsystems for ü. For example, a 2-channel ü

has only two complementary subsystems: ü12 and ü21 . First, for the design problem of TI systems, the following lemma gives a necessary and suÆcient condition [91]. Lemma 3.2.1 The TI problem has a solution if and only if ü is stabiliz-

able and detectable, and all the 2ÿ complete.

ú

2 complementary subsystems are weakly

2

For a TI system ü, z0 C is called a decentralized ÿxed mode (DFM) if it is an uncontrollable or unobservable ÿxed mode, or if the rank condition in (3.3) fails for the state model of some complementary subsystem. With this notion, the lemma above can be stated that the TI problem is solvable if and only if ü has no unstable DFM. If ü is stabilizable by local controllers, there are a number of methods for decentralized control design, e.g., [19, 30, 77, 79, 91]. We next proceed to give the solution to the PTV case. Deÿne the

operator

LN

[47] by

LN

:

82 > < f ( )g 2Z+ 7! >:64 u k

u(kN )

. . .

k

u((k

+ 1)N

ú

39 > 75= > ;

1)

: k

2Z+

N -lifting

3.3.

19

Stabilization using PTV local controllers

Hereafter, we denote the N -lifting of a signal u by u ~ := LN u and that of a ~ := L GL 1 . Note that G ~ is TI if N is a multiple of the PTV system G by G N N

ý

period of

G.

The PTV problem is characterized by the next result [46]. Lemma 3.2.2 The PTV problem is solvable if and only if ü is stabilizable

and detectable (with a centralized controller), and for each complementary subsystem üi1 :::iÿ j1 :::jþ ÿÿ , if its input-output map is zero, then its ~ ü i1 :::iÿ j1 :::jþ ÿÿ is weakly complete.

N -lifting

If the conditions above are satisÿed, there exist stabilizing PTV local conû , where N û is deÿned as follows: For a TI NK ý N K K system G with a state model (A; B; C; D ), let T0 := D; Ti := C Ai 1 B; i ÿ 1, and deÿne

trollers of some period

ý

ú

ù (G)

:= i0 + 1 +

ù

n

;

rank (Ti0 )

where i0 := minfi; Ti 6= 0g and bac denotes the greatest integer equal to or smaller than a. Then û N K

:=

N

þ max

n ø ù

~ ü i1 :::iÿ j1 :::jþ ÿÿ

÷o

;

where the maximum is taken over all complementary subsystems of ü. For a linear system

G,

we write

G

= 0 if its output equals zero for every

input; likewise G 6= 0 means otherwise. ~ is equal to zero. N -lifting G

3.3 Even if

Clearly,

G

= 0 if and only if its

Stabilization using PTV local controllers P

in Fig. 3.2 does not satisfy the conditions in Lemma 3.2.1 or 3.2.2,

stabilization may be possible by making use of the switch box, that is, by exchange of local signals. We can see that the system ü is a PTV system, so Lemma 3.2.2 is applicable. In this section, a necessary and suÆcient condition for solvability of the switch box problem is given. First we introduce some notation for the systems in Fig. 3.2. Let be partitioned appropriately as

þ

P

=

respectively. Let their

LN Pij L

P12

P21

P22

N -lifting

~ := P ~ij := where P

P11

þ~

ý1, N

P11

~ P 21

ý

þ and

S

=

systems be

~ P 12

ý

~ P 22

and J~i :=

~ := and S

LN Ji L

ý1 N

0

J1

J2

0

þ

0 ~ J 2

for

i; j

P

and

S

ý

~ J 1 0

;

ý ;

= 1; 2. Here J~i are TI

memoryless and hence have corresponding matrix forms; in what follows, we

20

Chapter 3.

Distributed control over networks

i

i 2 f0; 1; 2gN

use the same notation J~ for these matrices. The matrices J~ ; i = 1; 2, can be constructed as follows. Decompose the switching pattern

i 2 f0; 1gN ; i = 1; 2, that satisfy s = s1 + 2s2 . ~ = diag(s ); i = 1; 2: J i i

into

s

For example, the switching pattern and

= [0 0 1].

s2

Now the

s

S

(3.4)

= [1 1 2] is decomposed to

s

of

Then we have

e

= [1 1 0]

s1

e

~ of ü is given by ü ~ := S ~ P ~ , where S ~ is a TI memoryless ü

N -lifting

system with a matrix form

3 7 2(p1 +p2 )N þ(p1 +p2 )N 17 2R : 05

2 I 6 ~ := 6 0 S e 4J~ 2

0 ~ J

0

I

e

~ has full column rank. Clearly, the matrix S For the sake of completeness, we give the state models of these systems. ~ be Let the state model of P

ø

p; p; p p

~ A

~ B

~ C

~ D

e ÷

÷

ö

:=

p;

~ A

ü~

p

~ B

B 1

û þC~p1 ý þD~ p11 p2 ; C~p2 ; D~ p21

p ~ D p22 ~ D 12

ýõ

:

~ =S ~ P ~ , so the state model of ü ~ can be expressed as By deÿnition, ü

ø

s

s

s D~ s

~ ; B ~ ; C ~ A

where

ö

:=

p

~ ; A

ü~

p

p

~ B 2

B 1

û þC~s1 ý þD~ s11 ; ; ~ ~ C D s2 s21

ýõ

2þ ý þ ý3 þ~ ý þ ~ ý 6 I ~0 7 þ ~ ý 0 Cs1 J1 ~ Cp1 = 6þ ý þ ý7 C~p1 := S e 4 ~ ~ ~ 0 5 C Cs2 Cp2 J 2 p2 0

and similarly

þ~

s ~ D s21 D 11

ij

~ The subsystems ü i; j

s ~ D s22 ~ D 12

= 1; 2.

s ~ D s22 ~ D 12

ý

(3.5)

(3.6)

I

þ~

e D~ p p21

~ := S

;

D 11

p ~ D p22 ~ D 12

ý (3.7)

:

p ; B~pj ; C~si ; D~ sij ),

~ are given by the state models (A ~ of ü

We now state our main result. It gives a necessary and suÆcient condition for the solvability of our switch box problem under mild conditions. Theorem 3.3.1 Suppose that P is stabilizable and detectable,

þ

P11 P21

Then

P

ý

6= 0;

þ

and

P12 P22

ý

6= 0:

(3.8)

is stabilizable using the network structure in Fig. 3.2 with PTV local

controllers if and only if

3.3.

21

Stabilization using PTV local controllers

(i) if

P12

= 0 and is not weakly complete, then

s

contains 1;

(ii) if

P21

= 0 and is not weakly complete, then

s

contains 2.

Proof

By Lemma 3.2.2,

P

is stabilizable with the switching and PTV local

controllers iþ (a) ü is stabilizable and detectable; ~ ~ (b) if ü 12 = 0, then ü12 is weakly complete; ~ ~ (c) if ü 21 = 0, then ü21 is weakly complete. We ÿrst show that (a) is equivalent to having

P stabilizable and detectable. ~ is staLifting preserves stabilizability and detectability of systems. Hence, P ~ ~ ~ ~ bilizable and detectable iþ P is. By deÿnition, ü = S P and S is memoryless.

e

e

~ is stabilizable iþ P is. Thus, ü ~ has column full rank, we have from (3.5) and (3.6) Since S

e

þ

zI

rank

for

z

2C

ú

s

~ C

s

~ A

ý

þ

I

= rank

0

0 ~ S

ýþ

e

zI

ú

p

~ C

p

~ A

ý

þ = rank

zI

ú

p

~ A

p

~ C

ý

~ and that of ü ~ are equivalent. Thus, . Hence, the detectability of P

(a) holds iþ

P

is stabilizable and detectable.

Next, we must show that (b) is equivalent to (i). First, observe that (i)

, ,

[If

s

does not contain 1, then

[If

P12

= 0 and

s

P12

6

= 0 or is weakly complete]

does not contain 1, then

P12

is weakly complete] :

In the following, we show that the last condition above is equivalent to (b). By hypothesis, we can see that

þ ~ ü 12 =

~ P 12

~P J 1 ~22

ý =0

,

[P12 = 0 and

J1

= 0]:

Thus,

h

(b)

, h ,

If

P12

= 0 and

J1

~ = 0, then ü 12 is weakly complete

If

P12

= 0 and

J1

~ is weakly complete = 0, then P 12

i

i

~ = 0 implies that the rank condition for weak completeness of ü 12 ~ only depends on P12 . Finally, the weak completeness property of the TI system P12 is invariant under lifting by Theorem 1 in [46]. Thus, (b) is equivalent to

because

J1

(i). Likewise, we can show that (c) is equivalent to (ii).

ÿ

If the conditions (i) and (ii) are satisÿed, then stabilizing controllers of û exist, based on Lemma 3.2.2. period at most N

K

22

Chapter 3.

Distributed control over networks

The main idea is simple. By Lemma 3.2.2, for stabilizability, we need to introduce output exchange through the switch box so that each complementary subsystem of ü is either not equal to zero, or equal to zero but not weakly complete. In the result above, under a weak hypothesis, the strategy is to ensure that no complementary subsystem is equal to zero by using the switch box. The eþect of exchange of the local output information to remove DFMs was considered in [64, 90] for stabilization of a TI plant using TI local controllers. However, a point-to-point connection between controllers for each exchange is required in the results, which may be costly and may result in complex wiring. We give some examples to illustrate our result. Example 3.3.2 For decentralized stabilization, we have seen that three choices

of controllers are available: TI controllers, PTV controllers, and PTV controllers with a switch box. Of the three, the more sophisticated the controllers are, the larger the class of plants that can be stabilized. The ÿrst example is a 2-channel plant

P

that is stabilizable using PTV

controllers but not with TI controllers and its state model is given by

021 0 0 3 21 03 þ 1 ý þ ý @40 ú1 ú35 40 15 01 10 00 00 00 A 0 0 ú2 0 1 We see that 12 = 6 0, but is not weakly complete; it has a DFM of 1. We next consider a 2-channel plant given by 021 0 0 3 21 03 þ 1 ý þ ý @40 ú1 0 5 41 05 01 10 0 00 00 A 0 0 ú2 0 1 If = 6 0, then is stabilizable only with PTV controllers with a switch box, ;

;

;

:

;

:

P

P

;

c

;

c

P

among the three control structures. In this case, weakly complete because of the DFM 1. pattern

s

6

P12 is equal to 0, but not By Theorem 3.3.1, the switching

must have 1, which gives ü12 = 0.

The case with

P12 is equal to 0 and 0 0 0 not weakly complete, we cannot apply Theorem 3.3.1 because [P12 P22 ] =

0.

c

= 0 is interesting. Although again

Nevertheless, the system is stabilizable with a switch box, and we can

check that the DFM 2 in 1.

P12 is removable by a switching pattern having As a result, ü12 has a zero input-output map and is weakly complete.

This implies that information exchange may result in a weakly complete, zero

5

complementary subsystem. As seen in the last example, even if (3.8) fails,

P

can still be stabilizable; in

general, even for the 2-channel plant case, the necessary and suÆcient condition is not known. This observation is suggestive for the case with a

ú -channel

plant. We can ÿnd a switching pattern to stabilize the system by following the idea in Theorem 3.3.1: If a complementary subsystem

Pi1 :::iÿ j1 :::jþ

ÿÿ

is

3.4.

23

Assignability measure analysis

equal to 0 and not weakly complete, then choose a pattern so that the corre~ sponding subsystem ü i1 :::iÿ j1 :::jþ ÿÿ is not equal to 0. However, the example ~ above implies that there may be a pattern that makes ü equal to i1 :::iÿ j1 :::jþ

ÿÿ

0 and weakly complete. So, the pattern choice for the general problem is not so straightforward.

3.4

Assignability measure analysis

Naturally, we expect that the use of the switch box enhances the capability of the system in some way. For a quantitative analysis of the system characteristics, we ÿrst deÿne a partial ordering in the switching patterns and then employ the assignability measure proposed in [87] for decentralized control systems. In this section, we show that our system in Fig. 3.2 can be improved in terms of assignability by allowing more signal transmission with respect to the partial ordering.

f0; 1; 2g

N

We ÿrst introduce a partial ordering on the set patterns.

Let

s1 ; s2

be obtained from s1

s1

= [1 1 2 2] and

dependence on

s

s2

2 f0; 1; 2g

of switching

s1

ÿ

s2

by setting some elements of

s1

to zero. For example, if

N

.

We deÿne

= [1 0 2 0], then

ÿ s2 .

s1

to mean that

s2

can

When necessary, we show the

of the system ü explicitly as in ü(s).

Assignable systems refer to systems not having any DFMs, that is, those that can be stabilized decentrally. The assignability measure gives the distance between a system and the set of systems having DFMs. This is an extension of the controllability measure for centralized control systems in [26]. We deÿne the assignability measure and give some preliminary results on the measure in the following. Let

Sÿ~

be the set of TI state models whose dimensions are the same as ~ that of ü as

Sÿ~ :=

ÿö

A;

ü

B1

B2

û

þ

C1

;

C2

ý þ ;

D11

D12

D21

D22

Ci

Then a metric

ýõ

2R

pi N

þ þ) on Sÿ~ is deÿned by

d( ;

d((A; B; C; D ); (A; B; C ; D ))

:=

:

þ

n

A

2R þ

; Dij

n

n

; Bj

2R

óþ ó AúA ó ó C úC

pi N

þ

2R þ

mj N

n

mj N

; i; j

;

ô

= 1; 2

:

ýó ú B óó ; D úD ó B

kþk denotes the maximum singular value. A system with a state model Sÿ~ having at least one DFM is called unassignable. Let Uÿ~ be the set of all

where in

such systems:

Uÿ~ := f(A; B; C; D) 2 Sÿ~ : 9z 2 C The

assignability measure

s.t. rank M ((A; B; C; D ); z )

for a TI state model (A; B; C; D )

Uÿ~ deÿned by d((A; B; C; D ); U ~ ) := ÿ

from the set

inf

2Uÿ

(Au ;Bu ;Cu ;Du )

d((A; B; C; D );

g:

< n

2 Sÿ~ is its distance

(Au ; Bu ; Cu ; Du )):

24

Chapter 3.

Let

Distributed control over networks

n (þ) denote the nth largest singular value of a matrix.

ÿ

The following

lemma gives a method to obtain this measure computationally [87]. Lemma 3.4.1 For a system with a state model

(A; B; C; D

ö ü )= A;

B1

B2

û þ 1ý þ

D11

D12

C2

D21

D22

;

C

;

ýõ

2 Sÿ~ ;

its assignability measure is given by d((A; B; C; D );

Uÿ~ ) = min z C minfÿn ([A ú zI 2

n (M ((A; B2 ; C1 ; D12 );

ÿ

B1 B2 ]); ÿ z )); ÿ

n ([A ú zI 0

0

0 0

C1 C2 ]

);

g

n (M ((A; B1 ; C2 ; D21 );

z )) :

Now we are ready to state our result on assignability and switching. Theorem 3.4.2 Let s1 ; s2

2 f0; 1; 2gN with s1 ÿ s2 .

Suppose

P

in Fig. 3.2 is

stabilizable using switching patterns s1 and s2 . Then the assignability measure ~ with s is greater than or equal to that for the system with s , that is, for ü 1 2

s s s (s1 ); D~ s (s1 )); Uÿ~ ) ÿ d((A~s ; B~s ; C~s (s2 ); D~ s (s2 )); Uÿ~ ):

~ ;B ~ ;C ~ d((A

~ in (3.5), we must show that for all From the state model of ü

Proof

(3.9) z

2C

n (M ((A~p ; B~p2 ; C~s1 (s1 ); D~ s12 (s1 )); z )) ÿ ÿn (M ((A~p ; B~p2 ; C~s1 (s2 ); D~ s12 (s2 )); z )); ~ ;B (ii) ÿn (M ((A p ~p1 ; C~s2 (s1 ); D~ s21 (s1 )); z )) ÿ ÿ (M ((A~p ; B~p1 ; C~s2 (s2 ); D~ s21 (s2 )); z )); 02A~ ú zI 31n 02A~ ú zI 31 p p ~ (s ) 5A ÿ ÿn @4 C ~ (s ) 5A : (iii) ÿn @4 C s1 1 s1 2 ~ (s ) ~ (s ) C C s2 1 s2 2 (i)

ÿ

Then, by Lemma 3.4.1, (3.9) follows. We ÿrst show (i). For

p p2 s1 (si

~ M ((A

~ ;B

~ ;C

~ ); D

i

= 1; 2, from (3.6) and (3.7), we have

þ~

ý

~ ú zI B s12 (si )); z ) = C~p (s ) D~ p(2s ) 2 A~s1 úi zI s12 B~i 3 p p2 ~ ~ =4 C D p1 p12 5 ~ (s )C ~ ~ (s )D ~ J J 2I 1 0i p20 31 2iA~ úp22zI p ~ 0 54 C = 40 I p1 ~ ~ 0 0 J1 (si ) C p2 A

3

p2 ~ D p12 5 : ~ D p22 ~ B

(3.10)

3.5.

25

Multiple mobile robot example

s1 ÿ s2 and (3.4), the system matrix in (3.10) for s2 can be obtained by deleting some rows in that for s1 . By Theorem 7.3.9 in [34], we have (i). In a similar manner, (ii) and (iii) can be shown.

Because

ÿ

Thus, the assignability measure for the system in Fig. 3.2 is a monotonically nondecreasing function of

s

with respect to the partial ordering on

f0; 1; 2gN .

This conÿrms our intuition that capability of a system improves with increased communication. The result itself may appear obvious from the partial ordering deÿnition. Comparison of systems can be, however, diÆcult if there is no ordering in their switching patterns. For example, systems with s1 = [1 1 2] and s2 = [2 2 1] use the same data rate over the channel, but cannot be compared analytically with respect to the assignability measure. Moreover, it is not clear how the measure may change by changing the patterns.

3.5

Multiple mobile robot example

In this section, we give a toy example of a multiple mobile robot system to illustrate our results. This example satisÿes the following conditions so that our theory is applicable: The original plant is open-loop unstable, stabilizable, and detectable, and the unstable poles are not assignable without a network structure. So, the system is stabilizable only when signal transmission over the network is allowed. We note, however, that systems with such properties rarely exist, as was shown in [21]; thus, we make assumptions that may appear artiÿcial. Consider the three cart system in Fig. 3.3.

Cart 1 receives reference

r

for the position and is the leader. Carts 2 and 3 follow it trying to keep the distance x0 between them; M is the mass of each cart; x2 + x0 is deÿned to be the distance between Carts 1 and 2 (the desired value of x2 being 0); likewise for ø. i

x3 + x0 . Cart 2 has an inverted pendulum of mass m, length l , and angle However, ø is not sensed. Each cart has a discrete-time local controller K ,

i ; xi ]

= 1; 2; 3, whose input is the measured signal [x

period

T;

0

its output goes through the 0th order hold

control input

u

i.

i

sampled at sampling H

and becomes the

Communication among the carts is to be kept low: signals x2 and x3 are measurable only to Carts 2 and 3, respectively; a communications channel is given only between Carts 1 and 3, and its behavior is determined by a switching pattern

s.

Suppose

K2 is constant (i.e., proportional-derivative control) and is designed so that Cart 2 follows Cart 1 with no consideration for the pendulum.

Thus, the pendulum cannot be stabilized locally by Cart 2. The control problem is to design s, K1 , and K3 so that the whole system, including the inverted pendulum on Cart 2, is stabilized, and x1 tracks r . The state ø is observable only to Cart 3 through signals x3 ; x3 . Hence, its stabilization is possible only by sending signals from Cart 3 to Cart 1 as

26

Chapter 3.

Distributed control over networks

x x +x

x m ÿ þ 2 l ÿ Cart x u x_ M u

+x

1

2

þ 1 ÿ Cart

x x_ M ÿ þ r ÿ 1

3

2

1

1

K

0

ÿ Cart þ 3

x x_ M

H

3

3

H

K

K

2

1

u

3

2

2

0

w

0

H

3

switch box Figure 3.3: Carts system

shown with a switch box in Fig. 3.3. Let

i be the velocity of Cart i for i = 1; 2; 3. Then v1 = x1 ; v2 = x1 ú x2 ; and v3 = x1 ú x2 ú x3 :

v

The dynamics of the linearized system are described as M v1

= 100u1 ;

M v2

= 100u2

M v3

= 100u3 ;

ú mgø;

 = (M + m)gø M lø where

g

= 9:8 m=s2 ,

M

= 3 kg,

Assignability analysis:

m

= 0:1 kg,

ú 100u2; l

= 1 m, and

K2

= [0:0001 0:05].

We computed the assignability measures for sys-

tems with switching patterns

i þþþ

z }| {

i := [1

s

Here, period

N

1 0

þþþ

0]

2 f0; 1; 2g6;

i

= 0; 1; : : : ; 6:

= 6 was arbitrarily chosen. Note that

The result is shown in Table 1 for eþectively zero for

i

= 0, and for

i

i ÿ iý

s s 1 ; i = 1; 2; : : : ; 6. = 0:8 s. We see that the measure is = 0 this jumps up to about 2 10 5 .

T

6

ü ý

As shown in Theorem 3.4.2, the measure slightly increases as we allow more 1s in the switching pattern. Based on Lemma 3.2.2, the maximum periods û of controllers were also obtained. Note, however, that the measures were N

K

computed for

K

N

= 6, and it was conÿrmed that controllers of this period

could assign all poles of the plant.

3.5.

Multiple mobile robot example

27

i

In Fig. 3.4, the assignability measure is plotted versus the index i of s ; i = 0; 1; : : : ; 6 for T = 0:2; 0:4; 0:6; 0:8 s. We again observe that the assignability of the system improves by allowing more signals on the network.

Note that

the measure also decreases by reducing the sampling period. Nevertheless, the measure is relative to state-space realizations, and thus comparison between systems with diþerent sampling periods may not be appropriate.

Table 3.1: Results for T = 0:8 switching pattern s0 = [ 0 0 0 0 0 0 ] s1 = [ 1 0 0 0 0 0 ] s2 = [ 1 1 0 0 0 0 ] s3 = [ 1 1 1 0 0 0 ] s4 = [ 1 1 1 1 0 0 ] s5 = [ 1 1 1 1 1 0 ] s6 = [ 1 1 1 1 1 1 ]

assign. measure

ü 10ý13 ý5 1:86 ü 10 ý5 1:91 ü 10 ý5 1:94 ü 10 ý5 1:97 ü 10 ý5 2:00 ü 10 ý5 2:09 ü 10

3:07

K

N

ù

36 54 30 18 18 12

−5

2.5

x 10

assign measure

2

1.5 T=0.2 T=0.4 T=0.6 T=0.8

1

0.5

0 0

2

1

4

3

5

6

index i of s

i

Figure 3.4: Assignability measure for the example

H 1 design

:

Assignability is just one measure of achievable performance,

and we observed that in terms of assignability the diþerence is very small

6

among the systems with i = 0.

1

To compare achievable performance of the

systems from a diþerent viewpoint, a simple controller was designed using the H

N = N

method based on [31] as follows.

K = 6, and T

Let K2

= K3

= [0:0001 0:05],

= 0:2 s. Design a servo controller K1 so that x1 follows

r, where reference r is a step, and ø is small with an appropriate weight.

28

Chapter 3.

Distributed control over networks

For s2 , s4 and s6 , the local controller for Cart 1 was computed. As a result of the lifting technique, we need N integrators and N delays in the design,

K

K

and thus the order of the controller is 33.

Figures 3.5, 3.6, and 3.7 show step responses of positions of the carts, the

ÿ

angle

ø , and the control input u1 for r (t) = 10; t 0. Here, to emphasize the diþerence in performance, white noise w = [w1 w2 ]0 is added to the signal

[x3 ; x3 ]0 communicated over the network. Signals

w1 ; w2 are uniformly distributed and model quantization error, assuming that 19 bits can be used [71].

The system with s2 is sensitive to noise, showing a noticeable oscillation in cart positions, while the systems with s4 and s6 both exhibit almost perfect tracking. However, there is still a slight diþerence in the behavior of

ø;

system with

does not

s6 , ø

goes to zero slowly, but that in the system with

s4

in the

and keeps oscillating. The diþerence among the systems observed in these step response plots is

Te1 r : r 7! e1 , where e1 = r ú x1 , ýr : r 7! ø, and Týw1 : w1 7! ø shown in Figs. 3.8, 3.9, and 3.10. All systems have the same sensitivity of Te1 r and Týr . This explains that all systems have similar tracking abilities. On the other hand, the sensitivity of Týw1 improves

evident in the frequency responses of systems

T

as we increase the signal transmission rate over the network. Through this part

of the simulation, we conÿrm that the achievable performance of the system improves by increasing the data rate over the network; this may not have been clear in the previous part using assignability.

Positions

15 10 5

Cart 1 Cart 2 Cart 3

0 −5 0

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

Angle θ

0.4 0.2 0

Control Input u1

−0.2 0

5

0

−5 0

Time t

Figure 3.5: Step response for

s2

= [1 1 0 0 0 0]

3.5.

Multiple mobile robot example

29

15

Positions

10 5

Cart 1 Cart 2 Cart 3

0 −5 0

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

Angle θ

0.4 0.2 0 −0.2 0

Control Input u1

5

0

−5 0

Time t Figure 3.6: Step response for

s4

= [1 1 1 1 0 0]

15

Positions

10 Cart 1 Cart 2 Cart 3

5 0 −5 0

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

Angle θ

0.4 0.2 0

Control Input u1

−0.2 0 5

0

−5 0

Time t Figure 3.7: Step response for

s6

= [1 1 1 1 1 1]

30

Chapter 3.

Distributed control over networks

10 5 0

Gain [dB]

−5 −10 −15 −20 −25

s2 s 4 s6

−30 −35 −3 10

−2

10

−1

10

0

10

1

10

Frequency [rad/sec] Figure 3.8: Frequency response for

Te1 r

0

−10

Gain [dB]

−20

−30

−40 s2 s4 s

−50

6

−60 −3 10

−2

10

−1

10

0

10

Frequency [rad/sec] Figure 3.9: Frequency response for

Týr

1

10

Multiple mobile robot example

31

100 90 80

Gain [dB]

3.5.

70 60 50 40

s2 s 4 s6

30 20 −3 10

−2

10

−1

10

0

10

Frequency [rad/sec] Figure 3.10: Frequency response for

Týw1

1

10

Chapter 4 Finite data rate control | single-input case The goal of this chapter is to design a stabilizing controller for a single-input linear time-invariant plant that has a ÿnite data rate channel in the feedback loop.

This is done in two steps: First we propose a design method for a

sampled-data controller that stabilizes the plant in a quadratic manner. Then this result is extended to a design problem of a quantizer in a sampled-data controller.

To provide stability criteria applicable to the diþerent types of

systems, we give a general class of switched systems in the ÿrst section.

4.1

Dwell-time switched systems and their stability

In this section, we introduce a class of hybrid systems, called dwell-time switched systems , and develop Lyapunov-like stability results for such sys-

tems. The objective of this section is to give stability criteria for the later sections. Consider the nonlinear switching system x(t)

where

x(t)

2 Rn

is

s0

2S

fj

fi(t) (x(t));

(4.1)

is the state of the system and

switching logic with the index set

assumed that

=

S

i(t)

2S

is globally Lipschitz continuous for each

such that

0 (0)

fs

is the state of the

and the ÿxed dwell time

= 0. The index set

S

j

T

>

0. It is

2 S and that there

may be a discrete set, as in

most switching systems, but may be a continuum as well. The switching logic generates are switching times (ii) i(t) = jk0

jk

= s0 for

k

i

from

2

on [tk ; tk+1 ), and (iii) if 0

ÿ k.

x.

It is assumed that for every

ftk gk Z+ and fjk gk Z+ ø S 2

x(tk )

such that (i)

= 0 then i(t) =

s0

i

there

ÿ tk + T , for t ÿ tk , i.e.,

tk+1

H. Ishii and B.A. Francis (Eds.): Limited Data Rate in Control Systems with Networks, LNCIS 275, pp. 33−81, 2002.  Springer-Verlag Berlin Heidelberg 2002

34

Chapter 4.

Finite data rate control | single-input case

Because of the rule (i), there is no chattering in the switching, and thus there is no need to introduce generalized solutions [85]. The trajectory the system is uniquely deÿned satisfying (4.1) for all times

=

t

tk , k

2Z

ÿ

t

+.

þ

x(

) of

0 except at switching

From the rule (ii), it follows that for the system (4.1) the origin is an

equilibrium point

in the sense that if

x(0)

= 0 then

x(t)

= 0 for

ÿ

t

natural to give stability deÿnitions for such systems as follows. Deÿnition 4.1.1 The origin of the system (4.1) is (i)

there exists

Æ >

(ii) (globally )

!

x(t)

0 as

t

0 such that

k k x(0)

asymptotically stable

!1

< Æ

implies

stable

k k x(t)

< ÷

0. It is

if, for every ÷ for all

if it is stable and if, for every

t

ÿ

x(0)

.

0,

>

0 and

2R

n

,

We also give another looser but more general deÿnition of stability. Here

B

x (r )

denotes the closed ball center

Deÿnition 4.1.2 Let

B

0 (r0 ) is

DøR D

n

attractive from

Note that if

D

=

Rn

x,

and

radius

r0

if, for every

ÿ

r.

0. For the system (4.1), the ball

x(0)

2D

,

x(t)

converges to

B

0 (r0 ).

and r0 = 0 then attractiveness of the ball is equivalent

to asymptotic stability. A simple extension of standard Lyapunov stability theory (e.g., [45], Theorem 3.1) gives the following suÆcient conditions for stability of the system (4.1). Proposition 4.1.3 Let P

R

2R

nþn

a positive-deÿnite function. Let

suppose that, for every trajectory

ö

then

Then the ball

B

@V @x

0 (r0 )

õ

fi(t)

be a positive-deÿnite matrix and

x(

(x(t))

(x) :=

þ

V

Given

DøR

n

) of the system (4.1), if

ýú D s

ö(x(t))

is attractive from

r0

0

x P x.

= r1

for

t

ÿ

x(t)

ö

and

:

Rn

2 D nB r1

ÿ

!

0,

0 (r1 )

0:

for the system (4.1), where

þmax (P ) þmin (P )

:

If the conditions in the proposition are satisÿed with a Lyapunov function V

R

(x) = nþn

from and

0

x Px

and a function

ö

of a quadratic form

is positive deÿnite, we say that the ball

D with respect to

r0

B

ö(x)

0 (r0 ) is

=

0

x J x,

where

(P; J ) for the system (4.1). In particular, if

= 0, we say the system is

quadratically stable with respect to

D

=

S

2

Rn

(P; J ).

Our main goal is to develop systems in which the communication of quires only a ÿnite data rate. That is,

J

quadratically attractive

i

re-

is ÿnite and there are ÿnite switchings

in ÿnite time. The convention of dwell time in our switching logic is to meet the property of ÿnite switchings in ÿnite time. Dwell-time switching was studied extensively in [59], but not in the context of ÿnite data rate. We see in the next section that a rather simple sampled-data system falls into this class of systems.

4.2.

35

Quadratic stabilization of sampled-data systems

4.2

Quadratic stabilization of sampled-data systems

In this section, we study a problem of stabilizing a continuous-time linear plant by a sampled-data controller. The stability to be achieved is quadratic stability deÿned in the previous section. 4.2.1

Problem formulation

Consider the continuous-time control system in Fig. 4.1(a). The pair (A; B ) represents a linear time-invariant plant given by x(t)

where A

6

2 Rn

x(t)

= 0, that

A

=

is the state and

Ax(t)

+ Bu(t);

2R

u(t)

(4.2)

is the control input. Assume that

is unstable, i.e., it has one or more eigenvalues with nonnegative

real parts, and that (A; B ) is stabilizable. The matrix

K

2 R1

n is chosen

þ

so that the closed-loop system is stable, i.e., all eigenvalues of

A

+

BK

have

negative real parts. Since

A

+ BK is stable, the system is quadratically stable with respect to

a positive-deÿnite

2

P

(A + BK ) Now let

V

of

:

V

of V

With

u

=

K x,

2 Rn

0

x P x; x

0

P

+ P (A + BK ) =

. For a trajectory

(x(t)) is determined by

Rn ü R ! R

decay rate V

(x) :=

V

2

J Rnþn , there exists n þn R that satisÿes the Lyapunov equation

some pair (P; J ). For example, given a positive-deÿnite

J

ú þ

(4.3)

J:

x(

) of the plant (4.2), the

in the following sense: The derivative

along the trajectories is deÿned by 0

(x; u) := (Ax + Bu)

we have

V

(x; K x) =

ú

Px

0

+x

P (Ax

+ Bu):

0

x J x.

Now suppose that the communication channel between the sensor and the actuator has a ÿnite data rate. The traditional way to communicate

u(t)

is to

sample and quantize. The ÿrst step to model such a setup is to add a sampler and a hold to the system as in Fig. 4.1(b). An sampling period T >

0 is deÿned by

T :õ

S

the

zeroth order hold HT

T : õd

H

and

0

u

2R

7!

v

:

7!

ideal, uniform sampler ST

d : õd (k ) := õ(kT );

õ

k

with

2 Z+

;

is given by v (t)

:=

d (k );

õ

t

2

[kT ; (k + 1)T );

k

2 Z+

;

. Observe that u(t)

=

0

u K x(kT )

for

t

2

[kT ; (k + 1)T ); k

2 Z+

:

(4.4)

36

Chapter 4.

Finite data rate control | single-input case

u(t)

-

-

x(t)

(A; B )

?

x(t)

u(t)

(A; B ) u0

ÿ

K

v (t)

K

6

ÿ

T

H

(a) Continuous-time system

ÿ

T

S

d (k )

õ

õ(t)

(b) Sampled-data system

Figure 4.1:

This system can be viewed as a dwell-time switched system (4.1) by taking f ( ) (x(t))

ut

=

Ax(t)

+

is always dwell time

T

Bu(t)

S

and

=

R.

Obviously

f0 (0)

= 0 and there

between switching times. Moreover, if

k ) = 0, it

x(t

2 [tk ; tk+1 ), from (4.4) and consequently x(tk+1 ) = 0; repeating this argument, we have u(t) = 0 for t ÿ tk . Thus, we can use the follows that

u(t)

= 0; t

stability deÿnitions in Section 4.1.

The question of interest here is, how large can

T

be while maintaining the

stability of the closed-loop system with respect to the same Lyapunov function at a decay rate similar to that of the original system? Here we allow a scalar multiplication on clear that

u0

K

and have

u0

in Fig. 4.1(b). In our treatment it becomes

= 1 is not always the best choice.

The problem of this section is stated as follows. a sampling period

T

and a scalar

u0

Given

÷

2

(0; 1), ÿnd

such that the closed-loop sampled-data

system in Fig. 4.1(b) is quadratically stable with respect to (P; ÷J ). We call this problem the sampled-data control problem . Notice that the choice of system. Larger

÷

÷

changes the decay rate of the sampled-data

guarantees faster decaying. We also note that

÷

being strictly

smaller than 1 is a key assumption; we elaborate more on this later. A similar but simpler problem is to ÿnd the sampling period for the discretized system. That is, discretize the plant to (Ad ; Bd ) via step-invariant transformation and ÿnd the sampling period for all eigenvalues of

d + Bd K

A

to have absolute values less than 1. In contrast with this simpler problem, we emphasize that in our problem the decay rate of trajectories is bounded and that the stability is in the continuous-time domain, i.e., the intersample behavior is taken into account. In our setup, since the full state is measured and there is no noise, one way to achieve better control is to send the sampled state the sampled control

K x(kT )

x(kT )

instead of

from the sensor side to the actuator side. Then,

on the actuator side, we can place an open-loop controller whose state is reset periodically. Such a scheme has an obvious advantage over ours and is consid-

4.2.

37

Quadratic stabilization of sampled-data systems

ered in [74]; however, this paper does not deal with quantization issues. We stress that the objective to study our sampled-data problem is to lay the basis for the problem with quantization in Section 4.3. 4.2.2

Control Lyapunov function approach

Our objective in the sampled-data control problem is to achieve quadratic stability for a ÿxed Lyapunov function

V

and a bound on the decay rate. In

other words, we look for a control law, or the switching logic on a continuum, so that the positive-deÿnite function

V

becomes a Lyapunov function for the

V control Lyapunov function (e.g., [48]). In the following, we limit the class of K to LQR optimal ones for technical reasons and analyze function V for such K . We note that this assumption may

sampled-data system to guarantee its stability. In this design methodology, is referred to as a

not be such a restriction since LQR optimal gains are known to have inÿnite gain margin and fairly large phase margin; in particular, 1+ ( )ý1

j K j!I úA B j ÿ ! 2 R [2, 43]. These robustness properties are suitable for our design. Let Q 2 Rn n be a positive-deÿnite matrix and R > 0. Denote by P the

1 for

þ

unique positive-deÿnite solution of the Riccati equation

A P + P A ú P BR 1 B P + Q = 0:

(4.5)

úR 1B P .

Now let the control

0

Deÿne the stabilizing feedback matrix

V (x) := x P x. u = Kx is V (x; Kx) = úx Jx, where 0

Lyapunov function be

ý

0

K

:=

ý

P,

For this

0

0

the decay rate of

V

when

J := ú(A + BK ) P ú P (A + BK ) = Q + P BR 1 B P > 0: 0

ý

By (4.5) and the deÿnition of

K, V

0

can be alternatively represented as

V (x; u) = x (A P + P A)x + 2x P Bu 1 = x (P BR B P ú Q)x + 2x P Bu = x K R(Kx ú 2u) ú x Qx: 0

0

0

ý

0

Fix

÷ 2 (0; 1).

For

0

0

0

0

0

u 2 R, let X (u) be the set of states for which u decreases

the control Lyapunov function:

n o X (u) := x 2 Rn : V (x; u) ý ú÷x Jx n = fx 2 R : x K R [(1 + ÷)Kx ú 2u] ý (1 ú ÷)x Qxg : 0

0

0

0

Q = 1=(1 ú ÷)I without loss of generality (we can x to give this Q). Now, let

In the following, we select always transform

M := (KerK )

?

Then

K:

= Im

0

e := K =kK k constitutes an orthonormal basis for M. 0

(4.6)

38

Chapter 4.

Finite data rate control | single-input case

Finally, letting := (1 + ÷)R

ÿ

X

we obtain a simpler expression for Lemma 4.2.1 For u

X

+y :

õe

there exist unique

õ

;

2X ,

+y

(u)

2R

õ

?

y

and 0

(õe + y )

, , k k 0

ú

(ÿ

;

õ

K

K

2M

?

y 0

K R[(1

Rn

0

õ(1

K R

2

õ

+y :

õ

=

õe

õ

K

?

y

, for any

K

2u]

2ÿ

2u

u

(1 + ÷)

K

B

2R 2M

?

y

+

õ

K

K

(ÿ

ú

y

:

(4.8)

2 Rn

,

0

(õe + y ) (õe + y )

(since

y

y

2+y y

?

0

K ; e)

0

õ

ÿ

:

0

u =(1 + ÷).

1)õ

ò

y

õ

and u ~0 =

;

0

2

e)

2 2 k k ý kk 2 k k ýk k

k k

;

+y

0

K

x

+ y . Then

ú ý

õ e e

ý

u

~ := (1 + ÷) For simplicity, let B

õe

M÷M

0

2u]

+ ÷)K

(1 + ÷)

1)õ

(4.8) can be expressed as

(1 + ÷)

ú ý 2ý k kú ý

(by deÿnition of

(ÿ

ñ

+ ÷)K (õe + y )

þ

, ú , ú 2ú

(u) =

x

k k ýk k

u

2ÿ

=

such that

[(1 + ÷)K õe

þ

õÿ

ú

2 1)õ

(by (4.6))

0

õe K R

X

(4.7)

;

(u).

Corresponding to the decomposition

Proof

õe

K

,

ÿ

(u) =

2R 2R 2M

k k2

2

ú

2ÿuõ

~ ), Then, for (A; B

ý k k2ô y

:

(4.9)

Figure 4.2 depicts a system equivalent to that in Fig. 4.1(b). Now the control input is

u(t)

=u ~0 e0 x(kT ) for

t

2

[kT ; (k + 1)T ); k

2 Z+

.

From (4.9), we see that the state space can be parameterized by

õ

and

kk y

for the purpose of stabilization. This is not diÆcult to picture because the continuous-time control x

in

M

u

=

Kx

depends only on the component of the state

. This observation motivates us to deÿne

þ

F (x)

Clearly,

F

kk õ y

;

where

õ

=

0

e x

and

: y

Rn

=

! R2 ú

x

by (4.10)

õe:

preserves norm. Then, from (4.9), we have the image of

F:

F

=

ý

F

X

ÿþ ý (u) =

õ ô

2

R2

:

ô

ÿ

0; (ÿ

ú

2 1)õ

ú

2ÿuõ

ý

2 ô

X

(u) under

ò :

(4.11)

4.2.

39

Quadratic stabilization of sampled-data systems

u(t)

-

x(t)

~) (A; B

?

u ~0

0

e

6 T

H

ÿ

ÿ

T

S

Figure 4.2: Equivalent sampled-data system

ÿ

ÿ=

p ýÿ (

þ ÿ 2ýuþ

1) 2

F X (u)

ÿ ÿÿ1 u

0

þ

Figure 4.3: Set

Figure 4.3 shows this set for particular the line

õ

=

ÿu=(ÿ

ú

2X

(u)

We refer to the image space of

F

X

(u)

1 and

2 Rn , 2 X

1). Clearly, for x

F

ÿ >

F (x)

as the

u >

0. It is symmetric about

,

x

F

õ-ô

(u):

plane

(4.12)

.

The following result gives a suÆcient condition for quadratic stability of the system in Fig. 4.2. We make use of of the

õ-ô

F,

so the result is expressed in terms

plane.

Lemma 4.2.2 If for every initial condition x(0) F (x(t))

2 X F

(~ u0 e

0

x(0))

for

2 Rn 2

t

the trajectory satisÿes

[0; T ];

(4.13)

then the closed-loop system in Fig. 4.2 is quadratically stable with respect to (P; ÷J ). Proof

By (4.12), the condition in (4.13) is equivalent to

for

[0; T ].

t

2

x(t)

2X

0 (~ u0 e x(0))

Since this holds for any initial condition, we have

x(t)

2

40

X

V

Chapter 4.

0 (~ u0 e x(kT )) for

2

t

Finite data rate control | single-input case

[kT ; (k + 1)T ); k

(x(t)) decreases at all times: V

(x(t); u ~0 e

0

2 Z+

. Thus, by the deÿnition of

ýú

x(0))

Xþ

( ),

0

÷x(t) J x(t):

ÿ

This implies quadratic stability with respect to (P; ÷J ).

The diÆculty encountered by the use of sampled-data controllers for quadratic stability is that during the sampling period the plant is controlled blindfold, but the control input has to be guaranteed to keep

V

decreasing during

that period. The result above states this intuition formally. It implies that, to achieve stabilization, the next step is to choose u ~0 so that, starting at the trajectory

x(t)

X

stays in

and, moreover, that this step can be done in the

4.2.3

x(0),

0 (~ u0 e x(0)) for the entire ÿrst sampling period

õ-ô

plane.

Bounds on trajectories

We give a preliminary result on the bound of the state trajectory during the sampling period, characterized in the We ÿrst give some notation. For

Observe that Suppose

k úk ! !

:= max

cT

2[0;T ]

t

cT ; dT

cT <

At

e

0 as

I

;

õ-ô

óZ ó := max ó ó t2[0;T ]

dT

0 and that since

T

:=

1 c

2

T

û (u)

ü

~ (õ; u) :=

û

ÿþ ý :=

õ ô

2

R2

These are shown in Fig. 4.4; about the line

u

e

ó ó ó

~ dý ó : B

= 0, if

(4.14)

0, then

T >

cT >

0.

1:

õ

=õ ~.

:

ö

ú új jú j j

ÿ

f

ô

aT (õ

F~

û (u)

max

õ ~)

õ ~

dT cT

(4.15)

u

û

û

ò

g

+ ý f ~ (õ; u); f ~ (õ; u); 0

(4.16)

:

is the shaded area and is a cone symmetric

The following proposition says that if control

6

0

Aü

, deÿne the functions

and the set

T

ú

t

2R

f

taking

A

1. Let

aT

F~

0, let

T >

s

Now, for õ ~

plane.

cT <

1 (which can be achieved by

small enough) then the trajectory starting on õe ~ under a constant

2R

can be bounded by a set corresponding to

sampling period. Recall that

B

x (r )

F~

û (u)

denotes the closed ball center

during the x,

radius

r.

4.2.

41

Quadratic stabilization of sampled-data systems

ÿ

ÿ

ÿ

= fþ ~ (þ; u)

ÿ

F (x(t))

ky~k

0 Figure 4.4:

+

= fþ ~ (þ; u)

F (x(t))

þ ~

starting at

þ

x(0)

= õe ~ +y ~ for

t

2 [0; T ]

2 M? . Let x(þ) be the trajectory starting at x(0) = x0 with a constant control u 2 R. Then x(t) 2 B 0 (c (jõ ~ j + ky ~k) + d juj) for all t 2 [0; T ]. Proposition 4.2.3 Suppose x0 = õe ~ +y ~, õ ~

x

Moreover, if

cT <

T

T

2 F ~ (u)

for all

û

We show the case for õ ~

>

0 and

u >

2 [0; T ].

t

0. We have

ó ó Z t ó At ó Aü ~ ó kx(t) ú x0 k = óe x0 + e B dý u ú x0 óó 0 óZ t ó ó At ó ó ó Aü ~ ó ó ó ý e ú I þ kx0 k + ó e B dý óó u

ý c kõe ~ +y ~k + d u ý c (kõe ~ k + ky ~k) + d = c (õ ~ + ky ~k) + d u T

Tu

T

t

2 [0; T ].

0

T

T

for

y ~

1, then F (x(t))

Proof

2 R,

T

(since

kek = 1)

This proves the ÿrst claim.

r := c (õ ~ + ky ~k) + d u. 2 B( ~ k ~k) (r) for all t 2 [0; T ]. Proof From the ÿrst claim x(t) 2 B 0 (r ) = B ~ +~ (r ) for t 2 [0; T ]. Here ñ ô 2 2 B ~ +~(r) = õe + y : k(õe + y) ú (õe ~ +y ~)k ý r ñ ô 2 2 2 = õe + y : k(õ ú õ ~ )ek + ky ú y ~k ý r (since e ? (y ú y ~)) ñ ô 2 2 2 = õe + y : jõ ú õ ~ j + ky ú y ~k ý r ô ñ 2 ø õe + y : jõ ú õ~j + jkyk ú ky~kj2 ý r2 :

The second claim is shown in two steps. Let F (x(t))

Step 1

x

ûe

y

T

û; y

ûe

y

T

42

Chapter 4.

The last inclusion holds since F (x(t))

2

ÿþ ý õ

:

ô

Finite data rate control | single-input case

jkyk ú ky~kj ý ky ú y~k.

jõ ú õ~j

2

+

jô ú ky~kj ý r 2

Thus, we have

ò

2

Now we must show that the ball in Step 1 is in Fig. 4.4). If

Proof

It suÆces to show that

cT <

ü

û ~ (õ; u) deÿning we show only for

f

1, then

B( ~

Step 2

F ~ (u) in (4.16). û

ô

=

f

ý

û ~

ø F ~ (u). B( ~ ~ ) (r) is

û;ky ~k) (r )

F ~ (u) û

ý

tangent to the two lines

ý

(4.17)

û

ú ky~k = úa

T

(õ

ú õ~) ú õ~ ú

úa

T

(õ

ú õ~) ú

1

cT dT

u

ý

û

=

T (õ

þ; þ) in (4.15)

(

ú õ~) ú s

1 + a2 (õ T

ý

û ~

ú ky~k

ö

ú õ~)2 + (f ~ (õ; u) ú ky~k)2 ú r2 = (õ ú õ~)2 + úa "q

f

r:

cT

So we have

ú õ~) +

1 c

Thus, (4.17) has a single real solution under the assumption

4.2.4

=

õ

ú õ~)2 + (f ~ (õ; u) ú ky~k)2 ú r2 = 0

(õ; u)

ô

(õ; u).

=

(õ

(as illustrated in

The two cases are similar, so in the following

has a single real solution. Notice that from the deÿnition of fû ~

5

û;ky ~k) (r ):

û

û;ky k

We must show that the quadratic equation of (õ

B( ~

=

2

1 cT

õ2 r

ú r2

#2

ú 1r

:

T

cT <

5ÿ

1.

Solution to the sampled-data problem

With Lemma 4.2.2 and Proposition 4.2.3, we now see that the sampled-data stabilization problem for the system in Fig. 4.2 can be fully described in the lower dimensional

õ-ô

space. The suÆcient condition for stabilization given in

the lemma is that the trajectory starting at F

x(0)

has an image under

F

inside

X (~u0 e x(0)) during the sampling period, while the proposition gives a bound 0

in the plane for such trajectories characterized by a cone. This observation leads us to the following suÆcient condition. Proposition 4.2.4 Suppose cT < 1. If for every õ ~

F ~ (~u0 õ~) ø F X (~u0 õ~); û

2R (4.18)

then the closed-loop system in Fig. 4.2 is quadratically stable with respect to (P; ÷J ).

4.2.

43

Quadratic stabilization of sampled-data systems

The condition (4.18) is equivalent to that, for every

Proof

x(0)

2 Rn ,

Fe x(0)(~u0 e x(0)) ø F X (~u0 e x(0)): 0

0

Since

c

T

<

0

1, by Proposition 4.2.3, this implies that, for every

the trajectory of the system satisÿes

F (x(t))

2 F X (~u0 e x(0)); 0

t

Lemma 4.2.2, the system is quadratically stable.

x(0)

2 Rn ,

2 [0; T ].

By

ÿ

This result makes the sampled-data problem much simpler, and now the question is how large

can be while the inclusion in (4.18) holds for all õ ~.

T

We have seen that, for a ÿxed õ ~ , the cone symmetric about lines

= õ ~ and

õ

=

õ

Fû~ (~u0 õ~) and the set F X (~u0 õ~) are ú 1), respectively (Figs. 4.3

ÿu ~0 õ= ~ (ÿ

and 4.4). Hence the best way to ÿt them together to obtain the maximum

T

is to make them symmetric about the same line, and thus in this framework the best choice of u ~0 is u ~0

=

ÿ

ú1

(4.19)

ÿ

and not u ~0 = 1. To illustrate the diþerence, we give controllers for both cases in the following. Now we are ready to construct the stabilizing sampled-data controller. 1. Select

Q

and

R

in (4.5) so that

ÿ >

1. That this can be done will be

shown in Section 4.6. 2a. Set u ~0 = (ÿ

ú 1)=ÿ and select T T

<

T

ý

c

1

p

ÿ

d

ÿc ÿ

ú1

0 small enough that

;

öq

+1

T

>

õ

T úÿ+1ú1 2

a

(4.20) ;

(4.21)

or 2b. set u ~0 = 1 and select

T

<

T

ý

c

d

T >

1

p

ÿ

ÿ

ú1

;

öq

+1

T

ÿc

0 small enough that

T úÿ+1ú1ú 2

a

T

a

ú1 ÿ

õ :

Our main result now follows. Theorem 4.2.5 For the sampled-data controller just deÿned, the closed-loop

system in Fig. 4.2 is quadratically stable with respect to (P; ÷J ).

44

Chapter 4.

Finite data rate control | single-input case

The proof is preceded by some notation and lemmas, which will also be useful in later sections. Assume ÿ > 1 for the rest of the section. Let (

) :=

g õ; u

p( ú 1) 2 ú 2 ÿ

õ

ÿuõ

2 R so that we have from (4.11) ò ÿþ ý 2 2 2 X( ) = 2 R : ÿ 0 ÿ ( ) We write \ û~+ ÿ with " to mean that û~+ ( )ÿ ( ) for all such that + ( ) ÿ 0 and ( ) ÿ 0, that is, we work in the upper half-plane in the û~ - plane; likewise for û~ý ÿ with ( is a dummy variable here). It is clear

for

õ; u

F

f

f

ô

ô

g

õ; u

u

;ô

f

:

g õ; u

õ; u

g õ; u

õ

g õ; u

õ ô

that

õ

u

f

g

u

Fû~ ( ) ø X ( ) , u

F

u

+

û~

f

u

ÿ

g

and fû~ý

ÿ

g

with

u:

(4.22)

The suÆcient condition for closed-loop stability given in Proposition 4.2.4 can be dealt with separately for õ ~ = 0 and for nonzero õ ~ . The ÿrst lemma is for the zero case (see Fig. 4.5).

ÿ

ÿ=

F0(0)

q

1

c2T

p

ÿ = ý ÿ 1þ

F X (0)

þ

0

F0(0) ø X (0) p , F0(0) ø X (0). T ý1 Observe from (4.16) that ÿþ ý ò F0(0) = 2 R2 : ÿ max fö T g Figure 4.5: Condition :

Lemma 4.2.6 c Proof

ÿ 1þ

=

ÿ

F

F

õ

ô

ô

a

õ

and from (4.11) that F

ò ÿþ ý 2 2 2 X (0) = 2 R : ÿ 0 ÿ ( ú 1) ÿþ ý 2 ò p ñ ô = 2 R : ÿ max ö ú 1 õ ô

õ ô

ô

ô

; ô

ÿ

õ

ÿ

õ

:

4.2.

45

Quadratic stabilization of sampled-data systems

ÿ fþ~ÿ

fþ~+ g

g

þ ~

0

Figure 4.6: Condition:

Thus

F0 (0) ø X (0) holds iþ F

a

þ

ÿ ÿÿ1 u

T =

ý ÿ g and f + ÿ g with u û~

û~

f

p

1=c2 T

ú1ÿ

p

ÿ

ÿ

ú 1.

The nonzero case is shown through the following two lemmas (see Fig. 4.6). Let h(õ; u)

:=

ý

2 ú g(õ; u)2 :

û~ (õ; u)

f

ý

Lemma 4.2.7 Suppose õ ~ > 0 and õ ~ h(õ; u)

= 0, as a quadratic equation of

ÿu=(ÿ

õ,

ú 1).

Then

ýÿg

û~

f

with

u

iþ

has either a single real solution or

complex solutions.

ý Denote by õf the point that the line fû ~ crosses the õ-axis. Note2 that ý 2 2 (fû ~ ) and g are parabolas with vertices at (õf ; 0) and (ÿu=(ÿú1); ú(ÿu) =(ÿú 1)), respectively. By deÿnition, õf < õ ~ and moreover, by hypothesis, õf < ý2 2 ÿu=(ÿ ú 1). Hence, the vertex of (f ) is on the left of that of g . û~ ý ý 2 Now suppose fû ~ ÿ g with u. It then follows that (fû~ ) is above or tangent to g 2 for õ ý õf . From the position of the vertices of the parabolas, clearly ý2 ý2 2 (fû g for õ ÿ õf as well. Thus, (fû ~ ) is above ~ ) is above or tangent (at one 2 point) to g . This is equivalent to h(õ; u) = 0 not having two real solutions. Proof

ÿ

The converse can be shown similarly. Lemma 4.2.8 Suppose õ ~ > 0 and õ ~

1

p

T

<

T

ý úT1

c

d

ÿ

ÿc

ÿ

ÿ

+1

ý

ÿu=(ÿ

ú 1).

;

(ÿ

ú 1)( T ú 1) ~ ú a

ÿu

õ

T+

a

q2

T

a

Then

ú

ý ÿ g with u iþ

û~

f

(4.23)

ò

ÿ

+1

:

(4.24)

46

Chapter 4.

Proof f

Finite data rate control | single-input case

By Lemma 4.2.7, we have

ý ÿ g with u ,

h(õ; u)

= 0, as a quadratic equation of õ, has either a single real solution or complex solutions.

û~

We claim that the condition (4.25) holds iþ the bounds on

T in (4.24) are satisÿed for

d

First rewrite h(õ; u)

h(õ; u)

=

f

as

ý

ú

2

û~ (õ; u) 2

= (aT

ú

2

+ 1)õ

ÿ

2

ý ò ÿ þ T ú T Tú ú T ú þ ý T ú ú T T

g (õ; u)

2

(a

a

+ (a

So (4.25) holds iþ

ÿ þ

T (aT

a

ú ú

T cT

d

1)õ ~

ý

ú

u

ò

2

2

1)õ ~

ú

þ

ÿ

and

a

T

a

1

2

T

ú

q

2 a

T

ú

ý

ÿ

+1

ý

ú ú

(aT

1)õ ~

ÿ

ÿ

+1

1)õ ~

ý

2

u

ý

0:

(4.26)

T u, which is by deÿnition

=c

ý ú

þ

ÿu

u

T cT

ÿ

a

1

T+

q

2 a

T

ú

ý

ÿ

+1

0. That is, by some arranging,

a

1

2

+

a

and

By assumption, õ ~

2

a

ÿ

ÿu=(ÿ

+1

u

d

õ ~

c

2

a

1

ÿ

1

a

+1 +

ÿ

ÿ

d

T cT

d

õ

:

ú ÿ þ q ý Tú ú ú T Tú ý TT þ q ý ýú ú T ú T ú ÿ Tú ý ú ÿ T ú ÿ

on

d

ú ú

d

u

c

ú úT

View this as a quadratic inequality of (aT

ÿu

2

d

+ 1) (aT

ÿ

u

c

þ

ú Tú (a

ÿu

d

1)õ ~

1)õ ~

a real number. Hence, (4.26) holds iþ ÿu

T in (4.23) and

c

0.

T >

(4.25)

a

ÿ

+1 +

a

T cT

d

ýú ú ÿ

2

1). So if

1 2

ÿ

and

þ a

T ú

a

This is equivalent to (4.24) and

ú

T

a

q

+1

ÿ

T

c

u

0:

ÿ

+1

<

ú

2 a

ÿ

T

ÿ

+1 +

õ ~

0 then the lower bound

T

a

ú u

1

õ ~

>

0. Thus, the

(4.28)

0:

p

1=

ý

1

(4.27)

T =cT in (4.27) is negative. Moreover, by deÿnition, dT =cT

condition (4.27) holds iþ

ú

T

ÿ.

Thus clearly the bounds in (4.23)

and (4.24) imply the condition (4.28), which is equivalent to the condition (4.25).

4.3.

Quantized sampled-data control

47

ý ý

ú T ú ú

To show the converse, we need to prove only that the condition (4.28) implies (4.23). By d

(4.28)

)

T =cT

þ

q2

> 0 and the assumption õ ~

ú ú 2ú T 2 ) Tú , T p ÿu

0 <

ÿ

1

T

a

ú ÿ

ÿ +1

and a

T

a

ú

ÿ +1

ÿu=(ÿ

+

(a

1), we have

1)ÿu

ÿ

1

0

ÿ +1 > 1

a c

1

<

ÿ

:

ÿ +1

Proof of Theorem 4.2.5

2R

We show the case 2a, 2b being similar.

By

Proposition 4.2.4, it is suÆcient to show that the condition (4.18) holds for all õ ~

.

õ ~ = 0.

T

By the bound on c

in (4.20) and Lemma 4.2.6, (4.18) is satisÿed for

6

For õ ~ = 0, it is suÆcient to show for õ ~ > 0, the negative case being symmetric. By (4.22), the condition (4.18) is equivalent to

Since both

Fû~

+

û~

f

0

ÿ

(u ~ õ) ~ and F

X

ý ÿ g with u = u~0 õ: ~

û~

g and f

ý0 û~

ÿ

(u ~ õ) ~ are symmetric about the line õ = õ ~ by

(4.19), we have to show only f

0

g with u = u ~ õ ~ for õ ~ > 0.

Lemma 4.2.8, this is equivalent to the bounds on c

T

T

and d

However, by in (4.20) and

ÿ

(4.21), respectively.

Fû~

We give some remarks as follows. due to the size of Proposition 4.2.3.

Conservativeness in the size of T

is

(u), which was shown to give bounds on tra jectories in

In the examples in Section 4.7, we compare this sampling

period with that necessary for the stability in the discrete-time sense.

X

Next we discuss the assumption that ÷ be strictly smaller than 1. Suppose ÷ = 1. Then

(u) in (4.6) has the form

X

(u) =

f 2 Rn x

0

:

0

2x K R (K x

ú ýg

which does not depend on Q explicitly anymore. can be expressed as

F

X

(u) =

ÿþ ý õ ô

2 R2

:

ô

ÿ X

Thus, it follows that when ÷ = 1 the set F

2

0; õ

u)

0

;

Now, from (4.11), F

úk k ý u

K

õ

ò 0

(u)

:

Fû~

(u) becomes simply a strip. Clearly,

the construction in Theorem 4.2.5 does not work because cones ÿt in strips.

4.3

X

(u) don't

Quantized sampled-data control

In this section, the sampled-data control problem is studied further, taking account of the eþect of quantization. We consider the design of quantizers by

48

Chapter 4.

Finite data rate control | single-input case

extending the results in the previous section, and thus the development is done in a similar manner. A few speciÿc quantizers are examined for comparison of their characteristics. 4.3.1

Problem formulation

A quantizer is usually deÿned as a function from

R to a ÿnite or countable set.

We give a fairly general treatment of quantizers with the following deÿnition in terms of partitioning. Deÿnition 4.3.1 A

countable

2 Q0 f j gj ø R

(ii) 0 q

fQj gj of R is a set of bounded intervals with a Z+ such that (i) Qi \Qj = ;, i 6= j , and [j Qj = R,

partition

S

index set

=

, and (iii) for

2S

with

0

q

6

2S

= 0, 0

j

= 0, a

2 Qj =

quantizer

Q(õ)

=

cl( Q

j if

q

õ

2S

R ! fqj gj

:

fQj gj

). Given a partition

2 Qj 2 S S ;j

Some remarks are given as follows. (1) We use

2S

2S

and

is deÿned by

(4.29)

:

=

Z+ here for deÿniteness,

but may use other countable sets elsewhere. (2) The property (iii) of partitions is due to technical reasons. Consequently, the origin is either in a bounded cell, e.g, [a; b] with

a <

0

< b,

or in a cell

point. (3) The same notation

fg

0 , in which case it is an accumulation

is used for a quantizer and for a positive-

Q

deÿnite matrix in Section 4.2.2 due to the convention. In the following sections, we note whenever confusion may arise. The most common type is the equal length and each

uniform quantizer, where

j is the midpoint of

q

Qj

Qj

are intervals of

. Nevertheless, it is not certain

if uniform quantizers are eÆcient in quantization for the purpose of control. Here, we investigate other possible choices of quantizers for control use. u(t)

-

x(t)

~) (A; B

?

0

u ~

0

e

6 T

H

ÿ

ÿ

Q

T

S

ÿ

Figure 4.7: Quantized sampled-data system In Fig. 4.7 is shown a sampled-data system that has a quantizer

Q

to the communication channel in the system in Fig. 4.2. Recall that is deÿned by the LQR gain as

e

=

0

k k

K = K

parameter. Observe that the control input is u(t)

=u ~0 Q(e

0

x(kT ))

for

t

2

and that u ~0

[kT ; (k + 1)T ); k

2R

e

added

2 Rn

is a designable

2 Z+

:

4.3.

49

Quantized sampled-data control

The closed-loop system is a dwell-time switched system if we take uj

and

fj (x(t))

follows that

=

Ax(t)

f0 (0)

+

for

Buj

j

= 0 and that if

:= u ~0 qj

2S

.

x(kT )

(4.30)

By the deÿnition of quantizers, it = 0 then

u(t)

=

u0

= 0 for

t

Thus we can use the stability deÿnitions in Section 4.1. Extending the sampled-data problem, we are interested in how large

ÿ T

kT .

can

be while maintaining the stability of the closed-loop system with respect to the same Lyapunov function at a certain decay rate.

However, in general,

asymptotic stability is diÆcult to achieve with a quantizer; if the origin is an interior point of

Q

0 , then when the trajectory is suÆciently close to the origin the control becomes u(t) = u0 = 0, but the plant is unstable by assumption. We therefore aim at designing a quantizer for a looser stability in our problem. Since

qj ; j

2S

, are designable in the quantizer, we ÿx u ~0

as in (4.19), where

ÿ

=

ú

ÿ

1

(4.31)

ÿ

is deÿned in (4.7).

We now state the quantized sampled-data control problem . Given ÿnd a quantizer

B

Q

and a sampling period

T >

÷

0 such that, for some

R

2

(0; 1),

r0

ÿ

0,

n with respect to (P; ÷J ) for 0 (r0 ) is quadratically attractive from the closed-loop quantized sampled-data system in Fig. 4.7.

the ball

Note that the radius r0 of the attractive ball is not prespeciÿed in this problem, though as a design problem it is preferred to be so. In general, this turns out to be diÆcult. There are, however, examples of quantizers that allow us to choose r0 prior to the design. We also remark that if r0 = 0 the control objective is that the closed-loop system be asymptotically stable. 4.3.2

A suÆcient condition for stability

For the sampled-data control problem in the previous section, there is a suÆcient condition expressed in the

õ-ô

plane. In this section, the approach taken

there is generalized to the case of discrete-valued control inputs. We give a suÆcient condition for closed-loop stability, an extension of Lemma 4.2.2, that holds for any quantizer. For

e

and the quantizer

Q

:

R

the state space that is mapped to

X

j :=

= Such sets form a partition of

!f g qj

f 2R f 2R x

x

Rn

X 2S

j 2S , let

j, j

qj :

n

n

: :

0

Q(e x) 0

e x

= qj

2Q g j

, be the subset of

g

(4.32)

:

and are referred to as state partition cells . In

terms of state partition cells, the quantized sampled-data controller works as follows: If

x(kT )

2X

j then u(t) = uj for t

2

[kT ; (k + 1)T ),

k

2 Z+

.

The following lemma is used repeatedly in our development (see Fig. 4.8).

50

Chapter 4.

Finite data rate control | single-input case

rc2

E

rc1

Figure 4.8: An ellipsoid and balls

Lemma 4.3.2 Let

E

be an ellipsoid of the form

E = fx 2 Rn The largest ball contained in smallest ball containing Proof 0

x Px

ÿ

If

0

E

is

ý rc21 ,

then 2 þmin (P )rc2 = c. x x

E

0

:

is

x Px

B0 (rc1 )

ý cg ;

0

0:

pc1 =

with

B0 (rc2 ) with rc2 =

x Px

c >

r

p

c=þmax (P )

and the

c=þmin (P ).

ý þmax (P )rc21

=

c.

Similarly, if

0

x x

ý rc22 ,

ÿ

Now the idea used in Lemma 4.2.2 is extended to the state partition case in the following result. Lemma 4.3.3 Suppose the quantized sampled-data controller is designed to

have the following properties: 1. There exists r1 for t [0; T ].

2

2. For

j

ÿ 0 such that if x(0) 2 X0 then F (x(t)) 2 F X (0) [B0 (r1 )

2 S , j 6= 0, if x(0) 2 Xj

then

F (x(t))

2 F X (uj ) for t 2 [0; T ].

Then, for the closed-loop system in Fig. 4.7, the ball attractive from

Rn

with respect to (P; ÷J ), where

s

r0

Proof

for every

= r1

þmax (P ) þmin (P )

B0 (r0 ) is quadratically

(4.33)

:

It follows from the assumptions and the deÿnition of j

2S

if

x(0)

2 Xj

x(t)

2

then

(

X (0) [ B0 (r1 ); X (uj );

if

j

if

j

= 0;

6= 0;

F

in (4.10) that

4.3.

for

2 [0; T ].

t

which

x(kT )

B0 (r1 ) here is in Rn .

Note that j

2

(

2 [kT ; (k + 1)T ]; k 2 Z+.

t

Lyapunov function

B0 (r1 ).

is in

X (0) [ B0 (r1 ); X (ujk );

j

if

j

6 0; k=

This means that at each

The smallest level set containing

from Lemma 4.3.2.

E0 .

k = 0;

if

t

:

V

(x)

B0 (r1 ) is

ý þmax (P )r12 g

This is an invariant set, and hence

Finally, the smallest ball that contains

Lemma 4.3.2.

4.3.3

ÿ 0 either the control X (þ), or else x(t)

(x(t)) is decreasing, by the deÿnition of

V

E0 = fx 2 Rn stays in

Thus, denoting the cell index

k , we have

is in by

x(t)

for

51

Quantized sampled-data control

E0

x(t)

is

goes into and

B0 (r0 ),

again by

ÿ

Solution to the quantized sampled-data problem

The similarity of the quantized version of the sampled-data problem to the original problem is obvious now. This problem too is fully described in the õ-ô

plane. The diþerence in Lemmas 4.2.2 and 4.3.3 is how the initial states

in Lemma 4.2.2 and x(0) 2 Xj in Lemma 4.3.3. Fû~ (u), deÿned in (4.16), gives a bound on the trajectories starting on x(0) = õe ~ + y; y 2 M? with a constant control u 2 R. To continue the parallel discussion, deÿne the set FQj (u); j 2 S ; u 2 R, by

are classiÿed: õ ~ =

0

e x(0)

Recall that the cone

FQj (u) :=

[

Fû~ (u):

û~ 2Qj

This gives a bound on the trajectories starting in closure of

FQj (uj ) is another cone expressed as

FQj (uj )) =

ÿþ ý õ

cl(

ô

:

ô

ÿ max

n

ÿ

j

:=

oò

+ ý sup Qj (õ; uj ); finf Qj (õ; uj ); 0

f

and this set is symmetric about the line

q

Xj by Proposition 4.2.3.

õ

1

ö jsup Q j ú jinf Q j

2

a

j

j

T

=

q

ÿ j , where

+ sup

Qj + inf Qj

;

The

(4.34)

õ :

(4.35)

These are not diÆcult to check and we skip the proof; see Fig. 4.9. We can now reduce the suÆcient condition for stability to the following one. Proposition 4.3.4 Suppose cT < 1. If

(i) there exists

1 ÿ 0 such that FQ0 (0) ø F X (0) [ B0 (r1 ), and

r

Chapter 4.

52

Finite data rate control | single-input case

ÿ

ÿ finf Qj

+ fsup Qj

0

inf

Q

j

Q

j

sup

qþ

Q

þ

j

j

FQj (uj ))

Figure 4.9: Set cl(

(ii) for j

2 S , j 6= 0, FQj (uj ) ø F X (uj ),

then, for the closed-loop system in Fig. 4.7, the ball attractive from

Proof x(0)

Rn

By the deÿnition of

2 X0 ,

B0 (r0 )

is quadratically

with respect to (P; ÷J ), where r0 is as in (4.33).

FQ0 (0), the condition (i) implies that, for every 2 F X (0) [ B0 (r1 ).

the tra jectory of the system satisÿes F (x(t))

Similarly, it follows from the condition (ii) that, for every tra jectory starting

2 Xj , j = 6 0, F (x(t)) 2 F X (uj ) for t 2 [0; T ]. B0 (r0 ) is quadratically attractive. at x(0)

The symmetry of the two sets F

6

Thus, by Lemma 4.3.3,

ÿ

X (uj ) and FQj (uj ) plays an important X (uj ) is symmetric about the line

role here. We have seen that, for j = 0, F õ = ÿ=(ÿ

ú 1)uj = qj and FQj (uj ) about õ = qjÿ .

Figure 4.10 illustrates these

two sets. Thus, in our framework, to maximize the sampling period T , it is natural to take the quantizer's output values to be

(

qj =

0;

if j = 0;

qj ;

if j = 0:

ÿ

(4.36)

6

Recall that, by Deÿnition 4.3.1, q0 = 0. For this choice of

fqj gj2S , the suÆcient

condition in Proposition 4.3.4 for closed-loop stability can be further reduced to bounds on T in the following theorem. Let q j

2 S , j 6= 0.

j

:= min

fjsup Qj j; jinf Qj jg for

We are now ready to construct a stabilizing quantized sampled-data controller.

4.3.

53

Quantized sampled-data control

ÿ

FQ (uj ) j

F X (uj )

qj qjþ

0 Figure 4.10: Sets

u ~0

3.

j)

and

F

X (u

j)

for

j

6= 0

2 (0; 1). Select the matrix Q and R > 0 in (4.5) so that ÿ > 1. ú 1)=ÿ. Given fQ g 2S , set fq g 2S as in (4.36). For j 2 S ; j = 6 0, select T > 0 small enough that

1. Fix

2.

FQj (u

þ

÷

Set

= (ÿ

j

j

j

j

j

cTj <

dTj

ý

1

p

ÿ

ÿcTj ÿ

;

((

+1

ú 1)q úa jqÿ j

aTj

ú1

T >

Tj >

0, let

r0

f

= max sup

Tj

+

q

0. Set

s

Q0 ; ú inf Q0 g

T

2

a

Tj

j

There always exists such 4. If

j

úÿ+1

(4.37)

)

(4.38)

:

= inf j 2S ;j 6=0 Tj .

þmax (P )

p

þmin (P ) aT

ÿ (aT

p + 1) : ú ÿú1

(4.39)

Clearly, we may have T = 0, in which case a controller does not exist for the given quantizer. We call all quantizers that yield

T >

0

stabilizing

quantizers.

For such quantizers, we have the main theorem. Theorem 4.3.5 With the quantized sampled-data controller just deÿned, for

the closed-loop system in Fig. 4.7, the ball from

R

n

with respect to (P; ÷J ).

In Proposition 4.3.4, the

j

B0 (r0 ) is quadratically attractive

= 0 case and the nonzero

j

case have diþerent

conditions and can be dealt with separately. The proof of the theorem is based on two lemmas in the following. We ÿrst give a lemma related to

j

= 0.

54

Chapter 4.

Finite data rate control | single-input case

ÿ

F

F[ÿ

r;r

X (0)

+ (þ; 0)

=

ÿ

fr

](0) ÿ

p

=

ý

ÿ1

ÿr1

r

0

þr1

þ

F[ý ](0) and F X (0)

Figure 4.11: Sets

r;r

pÿ 0 be large enough that Q0 ø [úr; r] and T

Lemma 4.3.6 Let r

enough that

cT <

1=

ÿ

>

0 small

+ 1. Then

FQ0 (0) ø F X (0) [ B0 (r1 ); where

1=

p

(4.40)

ÿ (aT

r

aT

ú

p + 1) r: ÿ ú1

(4.41)

The condition (4.40) is equivalent to

Proof

FQ0 (0) \ (F X (0)) ø B0 (r1 ): c

Since

ÿþ ý

FQ0 (0) ø F[ý ](0) =

õ

r;r

ô

:

ô

ÿ max

ñ + ý (õ; 0); 0ô f (õ; 0); f

ý

r

r

ò ;

it is suÆcient to show that

F[ý ](0) \ (F X (0)) ø B0 (r1 ): c

r;r

Observe F

X (0) =

Figure 4.11 shows sets (F

X (0))

c

.

ÿþ ý õ ô

2 R2

F[ý ](0) r;r

:

ô

and

p

ÿ maxf F

X (0);

ÿ

(4.42)

ò

ú 1jõj; 0g

:

the dark area is

F[ý ](0) \ r;r

þ

4.3.

Quantized sampled-data control

55

p ú

p

0 r1 ] be the intersecting point of lines ô = ÿ 1 õ and ô = + + fr (õ; 0). This point exists since cT < 1= ÿ + 1 implies the slope aT of fr is

Let [õr1

larger than

ô

p ú ÿ

1. Then, by direct calculation,

óóþ ó p ú óp ú T ú

óóþ ýóó óó r1 óó = r1

(aT + 1)r

õ ô

a

1

ÿ

ýóó ó= 1 ó

1

ÿ

1

r :

ÿ

Hence, (4.42) and consequently (4.40) follow. We now have a result for nonzero

2S

Lemma 4.3.7 For j

enough that

T

c

p

<

ý ú ø X j

T

FQ

j)

j (u

a

Since

F

X

6

ÿ j = qj and that T is small

= 0, suppose that

q

ú jú j jÿ j 1)q

T +

q

ú

2

T

a

q

a

)

Qj ú1

) or (

;

+ sup Qj

ÿ j.

ÿ

FQ j Qj ÿj 2 Qj

However, the sets cl( q

j (u

Note that

q

j = inf

f

)) and

F

Thus, from (4.35),

>

X

j=

u

ÿ

Lemma 4.2.8, and the bounds on Proof of Theorem 4.3.5 c

u

=

cl(

) and

j (u

j ))

ø X F

(uj ).

j

u :

(uj ) are symmetric about the same line

ý

j

ú ÿ

1

q

Therefore, it follows from (4.43) that

Since

with

=

ý

inf Qj

f

ÿ

g

with

u

=

u

>

p

ÿ >

1.

and hence

q

From (4.30) and (4.36),

r; r ].

g

2 Qj

); the other case is

j. 0. By the bound on cT , we have aT

q

[

ÿ

ý

inf Qj

and

g

Consequently, it suÆces to show

Q0 ø ú

1 FQ

0). We show for (0;

(uj ) is a closed set, we must show that cl(

f

=

:

is a bounded interval and 0

1

By (4.34), this is equivalent to

õ

+1

ÿ

(u ).

F

is in either (0;

symmetric.

T

1

By Deÿnition 4.3.1,

Qj

Proof

hence

ÿ

j

.

;

((

+1

ÿc

T

Then

1 ÿ

d

with

2S

j

T and

(4.43)

j= u

j

ÿ

ú ÿ

1

ÿ

j

q :

ÿ ú (ÿ

1)=ÿq . We can now apply

r

j

ÿ f Q0 ú Q0g ÿ

ý T and dT imply fqj

c

Let

ÿ

j

q :

= max sup

;

g

inf

with

u

=

j.

u

0. Clearly,

T in (4.14) are nondecreasing functions of

d

ÿ

T,

56

c

T

Chapter 4.

ý cTj

and

d

T

ý dTj .

Finite data rate control | single-input case

Thus, by the bounds on

T in (4.37), we can apply

c j

Lemma 4.3.6 and obtain

FQ0 (0) ø F X (0) [ B0 (r1 ) (4.44) p p with r1 = ( ÿ (aT + 1))r=(aT ú ÿ ú 1). On the other hand, bounds on cTj in (4.37) and those on that

T in (4.38) allow us to apply Lemma 4.3.7, and it follows

d j

FQj (uj ) ø F X (uj ):

p

(4.45)

From (4.44) and (4.45), the stability of the system follows by Proposition 4.3.4; letting

=

þmax (P )=þmin (P ), we conclude that attractive for the closed-loop system. r0

r1

The size of

T

B0(r0 )

is quadratically

ÿ

Fû~ (u), as in the FQj (uj ) is conservative. An-

is conservative again because the size of

sampled-data controller, which aþects the size of

other source of conservativeness here is the size of

r0 . As can be seen in the proof, the radius of the actual attractive ball can be much smaller.

In the following, we examine two quantizers, one that uses cells of ÿxed length and the other with cells of varying length. Applying the above theorem, we see that the latter type can do better in terms of the sampling period and eÆciency in partitioning.

Uniform quantizer Here, we solve the quantized sampled-data control problem for the commonly used uniform quantizer.

Qÿ1 ÿ 3þ2

Q0 ÿ þ2

Q1

Q2

þ 2

0

Q3 5þ 2

3þ 2

þ

7þ 2

Figure 4.12: Partitioning of uniform quantizers For ù

>

0, the

uniform quantizer

the cells

Qj =

þ

(j

ú

1 2

(see Fig. 4.12), and the output values

ÿ j = qj =

q

Qþ

is deÿned by the index set

)ù; (j +

ö

q0

j

1 2

)ù

õ

= 0 and for

+

1 2aT

õ

j

6= 0

ù:

For this class of quantizers, we have a corollary of the theorem.

S

=

Z,

4.3.

57

Quantized sampled-data control

Corollary 4.3.8 Given r0 > 0, select T > 0 small enough that

T

<

T

ý

c

d

1

p

ÿ

T

ÿc ÿ

ÿ

+1

;

ú1

T (aT ú 1) 2aT + 1

a

and set

s ù = 2r0

ú aT +

pú

þmin (P ) a þmax (P )

T

q

T úÿ+1 2

ò

a

p

ÿ (a

T

ÿ

ú1

+ 1)

(4.46) ;

(4.47)

:

Then, for the closed-loop system with the uniform quantizer the ball

B0 (r0 ) is quadratically attractive from Rn Set

Proof

T

(4.37) and on

T

c j <

T

d j

ý

j = T . For j ÿ 1, cl(Qj ) = úcl(Qýj ), so the bounds on cTj in T in (4.38) are the same for j and új :

d j

1

p

ÿ

T

;

ÿ

+1

ÿc j ÿ

T

T

a j (a j

ú1

ú 1)(2j ú 1)

2aTj j + 1

We claim that (4.46) implies (4.48). those in (4.48) for of

j

Qþ in Fig. 4.7, with respect to (P; ÷J ).

ÿ 1.

j

ú aTj +

ò

Tj ú ÿ + 1

2 a

(4.48) :

The bounds in (4.46) are identical to

= 1. Moreover, the bound on

Finally, (4.39) is equivalent to (4.47).

B0 (r0 ) follows from Theorem 4.3.5.

q

T is an increasing function

d j

Therefore, the attractiveness of

ÿ

One of the noticeable features of the uniform quantizer is that the radius r0

of the ball can be prespeciÿed. This is because, in the design of uniform

quantizers, r0 and ingly.

T

are selected independently and then ù is chosen accord-

On the other hand, this quantizer may not be eÆcient in partitioning in the sense that all cells including those far away from the origin have the same size, whereas the control objective is only stability. Furthermore, as a consequence, T

may be much smaller than

j with large j (since Tj is an increasing function).

T

In this respect, we can say that the uniform quantizer requires partition cells that may be smaller than necessary. Logarithmic quantizer

In this section we give another quantizer, called the logarithmic quantizer. This type of quantizer was ÿrst proposed for the stabilization problem of discretetime systems in [27, 28]. It was shown to be the coarsest partition among the quantizers that achieve asymptotic stability.

In our continuous-time setup,

58

Chapter 4.

Finite data rate control | single-input case

Q[0 ÿ1] Q[ÿ1 ÿ1] Q0

Q[ÿ1 1] Q[0 1] Q[1 1]

;

;

ÿÆ

ÿ1 ÿÆÿ1

;

Æ ÿ1 1

0

Q[2 1]

;

;

;

Æ2

Æ

þ

Æ3

Figure 4.13: Partitioning of logarithmic quantizers

this type is very eÆcient in partitioning among the stabilizing quantizers in some sense. The

logarithmic quantizer

QÆ

is deÿned by

Z ü fú1 0 1g; we deÿne the cells by ;

1 and the index set

Æ >

S

=

;

Q0 = Q[0 0] = f0g; ü ð Q[ 1 ü1] = ö Æ 1 ; Æ 1 +1 ; j1 2 Z; ;

j

j ;

(4.49)

j

and the output values by

0 = 0; 1 ÿ q[ 1 ü1] = q[ =ö 1 ü1] 2 q

j ;

(see Fig. 4.13).

ö

j ;

Æ

ú1 aT

õ

+Æ+1

Æ

j1

;

12Z

j

(4.50)

A direct application of Theorem 4.3.5 yields the following

result. Corollary 4.3.9 Take Æ > 1 and T > 0 small enough that

cT <

dT

ý

1

p

ÿ

ÿcT ÿ

ÿ

+1

ú1

;

ú 1) úa + 1)Æ + a ú 1

2aT (aT (aT

T

+

q2 a

T

T

ò

úÿ+1

Then the closed-loop system with the logarithmic quantizer

(4.51) :

QÆ

in Fig. 4.7 is

quadratically stable with respect to (P; ÷J ). Proof

1)Æ

j1

For

=2aT

j

= [j1 ; s1 ] with

from (4.50) and

q

j

1

s

=

=

j1

Æ

ö1,

we have

jqÿ j j

= ((aT + 1)Æ +

. Thus, the bounds on

dTj

aT

ú

in (4.38) for

6= 0 are all identical to (4.51). Thus, T = T satisÿes the bounds in (4.37) and (4.38) for j = 6 0. Moreover, since Q0 = f0g, it follows that r0 = 0. This j

j

implies quadratic stability.

ÿ

Here we obtain asymptotic stability, which is a stronger result than the uniform quantizer case. One might expect that asymptotic stability is possible with other types of quantizers as well, as long as point.

Q0 = f0g is an accumulation

However, we will see shortly through an example that this is not a

suÆcient condition for asymptotic stability.

4.3.

59

Quantized sampled-data control

It is

1 that determines the coarseness of the partition of

Æ >

R

in log-

arithmic quantizers. The price we pay for a coarser partition is the size of T:

The coarser the partition (i.e., the larger

seen in the bound on that, for

Æ <

dT ,

Æ ),

the smaller

which is a decreasing function of

3, the sampling period

T

T.

Æ.

This can be

We also claim

can be larger than that for the uniform

quantizer. This can be checked directly from the bounds on

dT

.

Intuitively, the logarithmic quantizers are eÆcient in partitioning since the control objective is stability, and the cells far from the origin are much larger than those close to it. Another advantage is that the bounds on for inÿnitely many

j

cTj

and

dTj

can be reduced to the one pair in (4.51). Technically,

this means that for every

j

6

= 0 the inclusion

FQ

ø X

j (uj )

F

(uj ) in Proposi-

tion 4.3.4 can be realized tightly; the two sets can be tangent to each other by choosing the maximum

T

for (4.51). We can conclude that logarithmic

quantizers are optimally eÆcient in partitioning in this sense. The sampled-data controller designed in Theorem 4.2.5 is an extreme case of controllers with logarithmic quantizers. If we take

Æ

= 1, the bound on

in

dT

Corollary 4.3.9 becomes identical to that in (4.21). This can be explained from the tight inclusion of the cones as explained above, which is also a feature of the sampled-data controller. This does not happen with the uniform quantizers even if ù = 0. Now we take a look at an interesting variant of the logarithmic quantizer. It serves as an example of a quantizer that is not stabilizing. Suppose that we want to have a coarser partition for a quantizer. Deÿne a quantizer, similar to the logarithmic one, by

Z ü fú1 0 1g, cells ;

;

Q Q Q ü ö 0 =

[j1 ;

1]

=

[0;0]

fg 0

;

2 j (j +1)2 [Æ 1 ; Æ 1 );

=

(see Fig. 4.14), and output values

q0

j1

Æ

2Z

= 0 and

qj

1, the index set

>

=

ÿ

qj

for

6

j1

in the logarithmic quantizer. From the design procedure, for each cTj <

dTj

p

1 ÿ

ý ú

ÿcT

ÿ

1

;

ÿ

+1

2aTj (aTj Æ 2j1 +1 (aTj

We see that the bound on and moreover

!

dTj

This quantizer has

Q

0

=

1)

+ 1) + aTj

2S

there exists

ú ú 1

!1 fg

. Clearly,

0

T

0

= 0

+

q

Tj >

2

a

= inf j 2S

Tj

ú

Æ

2

j1

here and

0 such that

ÿ

+1

ò :

for

Tj

= 0, and this is

j

= [j1 ;

ö

j1

1],

as in the logarithmic quantizer, but the

sampled-data controller is not deÿned because of

Q fg

aTj

is a decreasing function of

0 as j1 not a stabilizing quantizer. T[j1 ;s1 ]

ú

j

=

= 0. Thus, the

j

diþerence here is the points at which the cells are partitioned: Æ

S

T

= 0. Thus, it follows that

is not a suÆcient condition for asymptotic stability.

60

Chapter 4.

Finite data rate control | single-input case

Q[ÿ1 ÿ1] Q[0 ÿ1] Q0 ;

Q[0 1]Q[1 1]

;

ÿÆ4 ÿÆ ÿ1

;

0

Q[2 1]

;

;

Æ4

1Æ

þ

Æ9

Figure 4.14: Partitioning of the modiÿed logarithmic quantizers

4.4

Finite quantizers

The stability criterion considered in the quantized sampled-data control problem is a global one. This, in turn, resulted in inÿnitely many partition cells in the quantizers since the cells are deÿned to be bounded intervals. In this section, we turn our attention to a more local stability criterion and use a

ÿnite

special class of quantizers, called

fQ g of R is said to be ÿnite if for every r > fQ k g =1 with N 2 N such that [úr; r] ø [ =1 Q k .

Deÿnition 4.4.1 A partition

0 there is a ÿnite subset

Finite quantizers

quantizers.

j

j 2S

N k

j

N k

j

are quantizers deÿned by ÿnite partitions.

For example, uniform quantizers are ÿnite, but logarithmic ones are not. The problem in this section is as follows: Given subset

D of R

n

, and

r0 >

loop system in Fig. 4.7, the ball respect to (P; ÷J ). trajectory

÷

2

(0; 1), a bounded

0, ÿnd a ÿnite quantizer such that, for the closed-

B

D

with 0 (r0 ) is quadratically attractive from Also, ÿnd a ÿnite subset N of such that, for every

S

þ) starting in D, x(t) 2 [ N X D, let E0 be a level set of V

x(

j 2S

for

j

For the given

E0 := fx 2 R

n

:

V

(x)

S

t

ÿ 0.

containing

ý c0 g ;

c0 >

D:

0;

DøE

that is,

c0 is chosen large enough that 0. The design of ÿnite quantizers is similar to that of general quantizers.

1. Given a ÿnite partition

fQ g j

j 2S ,

follow the design procedure of quan-

tizers in Subsection 4.3.3 and obtain 2. Select

S øS N

such that

E0 ø

fq g j

[X N

j 2S

and

fT g j

=0 .

j 2S ;j 6

(4.52)

j;

j 2S

where 3. Set

T

fX g is the state partition in (4.32). j

= minj 2SN ;j 6=0 Tj .

E 0 := fx 2 R : x P x ý r02 þmin (P )g (this is the largest level set inside B0 (r0 ) by Lemma 4.3.2).

4. Take

r0

as in (4.39) and

r

n

0

4.4.

61

Finite quantizers

In contrast to the general quantizer case,

is always positive here, and

T

hence ÿnite quantizers can stabilize the system under only a mild condition.

2

Theorem 4.4.2 Suppose c0 > r0 þmin (P ). Then, for the closed-loop system

in Fig. 4.7 with the sampled-data controller using the ÿnite quantizer

SN

Q

and

just deÿned,

B0 (r0 ) is quadratically attractive from D with respect to (P; ÷J ); (ii) more speciÿcally, E0 and Er0 are invariant sets, and every trajectory starting in E0 goes into Er0 in ÿnite time. Proof Let r = maxfsup Q0 ; ú inf Q0 g ÿ 0. Clearly, Q0 ø [úr; r ]. By bounds on cTj in (4.37) and on dTj in (4.38), we can apply Lemmas 4.3.6 and (i) the ball

4.3.7 and obtain

F 0 (0) ø F X (0) [ B0 (r1 ); F j (uj ) ø F X (uj ); j 2 SN ; j 6= 0; p p with r1 = r ( ÿ (aT + 1)) p =(aT ú ÿ ú 1). Q

(4.53)

Q

Notice that r0 = r1 þmax (P )=þmin (P ). Thus, the assumption implies c0 > By Lemma 4.3.2, it follows that 0 (r1 ) 0 . Moreover, from (4.52), 0 (r1 ) . 0 2SN

B

2

r1 þmax (P ).

B

ø E ø [j

Xj

øE

x(þ) starting at x(0) 2 2 X0 then F (x(t)) 2 F X (0) \ B0 (r0 ) for t 2 [0; T ], or else there exists j 2 SN such that x(0) 2 Xj and F (x(t)) 2 F X (uj ) for t 2 [0; T ]. Now, following an argument similar to Lemma 4.3.3, we have that B0 (r0 ) is quadratically attractive from D for the closed-loop system. Since E0 is a level set covered by [j N Xj , it is an invariant set. Also, it can be easily veriÿed that all trajectories starting in E0 go into any level sets containing B0 (r1 ) in ÿnite time and stay inside; by Lemma 4.3.2, n 2 the psmallest such set is fx 2 R : x P x ý r1 þmax (P )g. Because of r0 = þmax (P )=þmin (P ), this set is the same as Er0 . r1 ÿ

E0 ,

Thus, we have from (4.53) that, for every trajectory if

x(0)

2S

0

Note that, if the assumption has to take either

2

c0 > r0 þmin (P )

D larger or r0 smaller.

does not hold initially, one

On the other hand, the size of

N

may

be larger than necessary because the ÿnite partition covers a ball containing

E0 , while it needs to cover only E0 .

We give corollaries for uniform quantizers and for a ÿnite version of loga-

rithmic quantizers. Here, we take

D

B

= 0 (R0 ) with R0 > 0 so that comparison is easier later in examples. In this case, the level set 0 in (4.52) is given with 2 c0 = R0 þmax (P ) from Lemma 4.3.2. We start with the result for uniform quantizers.

E

Corollary 4.4.3 Given R0 > 0 and r0 > 0 with R0 > r0 , select T small

enough that inequalities in (4.46) hold and set ù

&

N

=

s

R0

þmax (P )

ù

þmin (P )

ú

>

1 2

'

0 as in (4.47). Let

;

(4.54)

62

Chapter 4.

and

S

f ö

1; : : :

ö g S N

. Then, for the closed-loop system in Fig. 4.7 with

the uniform quantizer and

0 (r0 ) is quadratically attractive from

N

= 0;

Finite data rate control | single-input case

;

N , the ball 0 (R0 ) with respect to (P; ÷J ).

B

Proof

Since

R0 > r0 , c0

only that the union of

[X ÿ

=

02 max (P ) > r02 þmin (P ).

X 2S j ,j

N,

and

j

2SN

=

õe

E0 ø B0 p 0 (

+y :

2 ú ú

õ

r

õ

N

covers the level set

(

N

1

)ù; (N +

2

1 2

, and the rest follows

)ù

õ ; y

2M

?

ò ;

min(P )) by Lemma 4.3.2, it suÆces to show

c =þ

j jý The least

þ

Thus we have to show

E0

R þ

from Corollary 4.3.8. Since

j

B

ö

) j jý

0 þmin (P ) c

õ

+

N

õ

1 2

ù:

ÿ

for this is (4.54).

Next we consider the ÿnite logarithmic quantizer. For this, we introduce

õ0 to give more freedom in the partition. 0 > 0, we deÿne the ÿnite logarithmic quantizer as follows. be = Z+ 1; 0; 1 , let the partition cells , j , be

a new parameter õ

S

ü fú

g

Q0 Q[0 0] ú Q[ ü1] ö 0 0 =

j1 ;

=

=(

;

[õ

Æ

j1

Q 2S

Given

Æ >

j

0 0 ); +1 ); Æ

õ ;õ

;õ

1 and

Let the index set

(4.55)

j1

(see Fig. 4.15) and let the output values be

ö ú ö

0 = q[0 0] = 0; õ0 Æ q[ 1 ü1] = 2 a q

;

j ;

1

+Æ+1

õ

(4.56) Æ

j1

(4.57)

:

T

For this class of quantizers, we have the next result.

Q[1 ÿ1] Q[0 ÿ1] ;

;

Q0

Q[0 1] Q[1 1] Q[2 1]

0

þ0 þ0 Æ

ÿþ0 Æ2 ÿþ0 Æ ÿþ0

;

;

Q[3 1]

;

þ0 Æ 2

;

þ0 Æ 3

þ

þ0 Æ 4

Figure 4.15: Partitioning of the ÿnite logarithmic quantizers

0 0

Corollary 4.4.4 Given Æ > 1 and R ; r

with

0

R

> r

enough that the inequalities in (4.51) hold and set

s

0

õ

=

r

p pú ú

min (P ) a 0 þ (P ) max þ

ÿ

T

ÿ (aT

1

+ 1)

0

>

0, select

T

small

4.5.

63

Control over a ÿnite data rate channel

and

ï s

& N

=

log

Æ

R0

þmax (P )

õ0

þmin (P )

!' (4.58)

:

Then, for the closed-loop system in Fig. 4.7 with the ÿnite logarithmic quantizer and

SN

=

attractive from

f0; 1; : : : ; N ú 1g ü fú1; 0; 1g, the ball B0 (r0 ) is quadratically B0 (R0 ) with respect to (P; ÷J ).

This can be shown similarly to Corollary 4.4.3, and hence we skip the proof.

4.5

Control over a ÿnite data rate channel

Signals communicated over a channel with ÿnite data rate must be sampled and quantized, but are also delayed.

When such a channel is used in the

feedback loops of control systems, these eþects on the signals have to be taken into account in the controller design and in determining the necessary data rate of the channel. In this section, we ÿrst characterize the data rate used by the quantized sampled-data systems dealt in the previous section.

Then analyses on the

robustness against time delay in the sampled-data systems are given; based on these analyses, one can determine the bandwidth necessary for the channel. 4.5.1

Data rate for control

With a stabilizing ÿnite quantizer and its index set

SN

rate used for the communication between the quantizer can be ÿnite: Messages are sent every

T

in Fig. 4.7, the data Q

and the hold

H

T

seconds and it is guaranteed that

there are only ÿnitely many diþerent messages to be communicated. Here we give a simple deÿnition of data rate that is necessary for this communication. For a ÿnite set

SN ,

denote the number of elements in the set by

Then, given a stabilizing ÿnite quantizer

Q

with an index set

the data rate required for the transmission of the control input data rate for control with

Q

:=

log2

jSN j

T

bps:

SN , u

jSN j.

we deÿne

by (4.59)

We note that the data rate in this deÿnition is only what is used by the data transmission for the control purpose out of the entire data rate available in the communication system. Clearly it excludes all overheads that may be necessary in practice. The data rate necessary for the communication system is another issue and is discussed further in the next subsection. To clarify the diþerence in the two data rate notions, we call the one deÿned above the data rate for control or in the sense of (4.59). For ÿnite uniform and logarithmic quantizers, the number of cells required

is

jSN j = 2N + 1.

However,

N

is a function of

T,

and it is rather diÆcult to

64

Chapter 4.

see what is the best choice of

T

Finite data rate control | single-input case

for a quantizer to minimize the data rate in

the sense deÿned above. In Section 4.7, we design the uniform and logarithmic quantizers for an example and ÿnd those that require the minimum rates.

4.5.2

Time delay analysis

Communication over a physical channel is always accompanied with a certain amount of delay. To begin with, there is transmission delay, which is determined by the size of the message: To send takes

N=D

N

bits over a

D

bps channel, it

seconds. This is a limitation due to the ÿniteness of the data rate

of the communication system. Moreover, networks are shared by many nodes and, depending on the media access method, time delay due to busy channel can exist; this type is not deterministic and varies over the time. In this subsection, we analyze the sampled-data system studied in Section 4.2 and ÿnd the time delay that the system can tolerate in the feedback loop while it maintains stability. The results will be extended to the case with quantization in the next subsection. Suppose the sampled-data controller in Fig. 4.2 is designed based on Theorem 4.2.5.

There we obtained a quadratically stable system with respect

to (P; ÷J ). This stability is in the continuous-time domain; though the measurement is not updated between the sampling times, the Lyapunov function V

(x) =

0

x Px

is guaranteed to decrease at a certain rate. This fact suggests

that, even if an update arrive late but no later than the next sampling time, the Lyapunov function will decrease from that point and, thus, that the system can maintain a certain level of control. Our goal for this subsection is to ÿnd how large the delay can be.

-

u(t)

x(t)

~) (A; B

?

u ~0

v (t)

0

e

6 He;T

ÿ

delay

vd (t)

ýý

ÿ

ST

õd (t)

ÿ

õ(t)

Figure 4.16: Sampled-data system with time delay Now consider the system in Fig. 4.16, where the channel has a varying time delay smaller than or equal to Let the sampled signal

õd

õd (t)

ý

2 [0; T ).

We call

ý

the

maximum delay time .

be expressed as

(

=

õ(kT )

if

0

otherwise;

t

=

kT , k

2 Z+;

4.5.

and let the delay that occurred while

2 Z+

. So the delayed signal

k

65

Control over a ÿnite data rate channel

(

vd (t)

extended

We deÿne the :

He;T

vd

=

7!

v

:

v (t)

õd (kT )

õd (kT )

if

0

otherwise:

t

=

zeroth order hold

(

0

=

vd (kT

is transmitted be

is

vd

+ ýk )

+ ýk ,

kT

t

if

t

2 2

2

[0; ý ],

2 Z+

;

by

He;T

if

k

ýk

[kT ; kT + ýk ),

k

2 Z+

;

[kT + ýk ; (k + 1)T ),

k

2 Z+

:

This hold is synchronized with the sampler and, at each sampling time, sets the output to zero until the next message arrives (see Fig. 4.17). Notice that if

ýk

= 0 it works the same with the original hold

HT .

þþ ý (t)

v(t)

d

T

0

2T

3T

4T

Figure 4.17: Signals The

time delay analysis problem

T

0

t

and

õd

2T

3T

4T

t

v

is to ÿnd the maximum delay time

ý

2

[0; T ) such that the closed-loop system in Fig. 4.16 is asymptotically stable. In particular, given trajectory k

2 Z+

þ

x(

.

ó

2

(0; 1), we would like to ÿnd

) of the closed-loop system,

V

such that, for every

ý

(x((k + 1)T ))

We begin with a lemma similar to Lemma 4.2.2. Let c :=

and let

ó

2

(0; 1).

Lemma 4.5.1 Suppose for every ý0

(i)

óP

ú

ý ( ý 0)e

e

c T

ü

0

A ü0

Aü0

Pe

(ii) for every initial condition u(t)

ÿ

0;

x(0)

(

2

2R

n

, if the control

0

u ~0 e x(0)

if

t

if

t

2 2

2 X F

(x(kT )) for

÷þmin (J )=þmax (P ),

0

(~ u0 e

x(0))

for

u

is

[0; ý0 ); [ý0 ; T ];

then the trajectory satisÿes F (x(t))

óV

[0; ý ]

0

=

ý

t

2

[ý0 ; T ]:

(4.60)

(4.61)

66

Chapter 4.

Finite data rate control | single-input case

Then the closed-loop system in Fig. 4.16 is asymptotically stable. In particular, V

(x(kT ))

Proof

ý

k

ó V

(x(0)) for

k

2 Z+

.

By (4.12) and the condition in (4.61), for every

[0; ý ] under the control in (4.60), the deÿnition of V

Xþ

x(t)

( ), 0

(x(t); u ~0 e

x(0))

2X

ýú

0

(~ u0 e

x(0))

0

÷x(t) J x(t)

for

for

t

2

x(0)

2

t

2 Rn

and

ý0

2

[ý0 ; T ]. Hence, by

[ý0 ; T ]:

So, by the comparison lemma (e.g., Lemma 2.5 in [45]), we have V

(x(t))

ý

We now show that for V

t

e

ýc(týü0) V (x(ý0 ))

2

(x(T ))

for

t

[ý0 ; T ] under (4.60)

ý

óV

(x(0)) for

x(0)

2

[ý0 ; T ]:

2 Rn

(4.62)

(4.63)

:

From (4.62), a suÆcient condition for this is

ýc(T ýü0) V (x(ý0 )) ý óV (x(0))

e

for

2 Rn

x(0)

(4.64)

:

Now (4.64)

,

ýc(T ýü0) V (eAü0 x(0)) ý óV (x(0)) n (since u(t) = 0 for t 2 [0; ý0 ]) for x(0) 2 R , eýc(T ýü0) x(0)eA ü0 P eAü0 x(0) ý óx(0) P x(0) for x(0) 2 Rn , e c(T ü0) eA ü0 P eAü0 ý óP: e

0

ý

ý

0

0

Thus, (i) yields (4.63). Therefore, in the closed-loop system, every

x(0)

2

Rn .

This implies

V

(x(t))

!

V

ý !1

(x(kT ))

0 as

t

k

ó V

(x(0)),

k

.

2Z

+,

for

ÿ

We have a few remarks on the condition (i) in the lemma. It is clear that if ó >

eýcT then there exists

ý >

0 such that the condition holds for

On the other hand, the constant

c

is proportional to

÷.

quadratic stability, the larger ÷, the faster the decrease in control is applied. This may imply that with a larger

ý0

2

[0; ý ].

By the deÿnition of ÷

V

when the correct

the period of time

to apply the correct control can be shorter to recover from the increase in V

during the delay time.

We will discuss more on this in the example in

Section 4.7. In the following theorem, we show that for any

ý

satisfying (i) the system is

asymptotically stable with an additional technical condition. In the construction of the original controller, we make use of the bounds on the trajectories in Proposition 4.2.3. Due to the conservativeness in these bounds, it is not diÆcult to show that the condition (ii) in the above lemma is satisÿed for every ý0

2

[0; T ].

4.5.

67

Control over a ÿnite data rate channel

Theorem 4.5.2 If ý > 0 is small enough that

ý

ý

and if for every ý0

þ

1

kAk

ln

öq

ÿcT dT (ÿ

ú 1)

õý

úÿ+1ú1

2

a

T

(4.65)

2 [0; ý ] óP

ú eý ( ý 0) e c T

0

A ü0

ü

Aü0

Pe

ÿ 0;

(4.66)

then the closed-loop system in Fig. 4.16 is asymptotically stable. We must show that, with the original controller and

Proof

(4.65), the condition (ii) in Lemma 4.5.1 holds for all stability of the system follows. Fix that

ý0

x(0)

2

[0; T ]. Given

starting from

x(0)

2

ý

satisfying

[0; T ] so that

2 R , there exist õ~ 2 R and y~ 2 M such 2 [ý0 ; T ], we have a bound on the trajectory n

x(0)

= õe ~ +y ~. Then, for

ý0

t

?

as

ó ó Z t ó At ó As ~ 0 ó kx(t) ú x(0)k = óe x(0) + e B ds u~0 e x(0) ú x(0)óó ü0 óZ t ó ó At ó ó ó As ~ ó ó ó ý e ú I þ kx(0)k + ó e B dsóó þ ju~0 e0 x(0)j ü0 ó ó ó ó Aü0 Z týü0 As 0 ~ dsó þ ju e B ~0 e x(0)j ý cT kx(0)k + óóe ó 0 óZ týü0 ó ó ó Aü0 As ~ 0 ó ý cT kx(0)k + ke k þ ó e B dsó ó þ ju~0 e x(0)j 0

ý c kx(0)k + e 0 d ju~0 e x(0)j ý c (jõ~ j + ky~k) + e 0 d ju~0 e x(0)j: kAkü

T

kAkü

T

F ~ (u) in (4.16),

In the deÿnition of by

û

F~

0

T

0

T

replace

dT

by ekAkü0 dT and denote this

û;ü0 (u). By Proposition 4.2.3, it follows from the bound above that the

trajectory satisÿes

F (x(t))

Note here that

cT <

2F

0

u0 e x(0)) e0 x(0);ü0 (~

for

t

2 [ý0 ; T ]:

(4.67)

1 holds from the original design.

On the other hand, by (4.65) kAkü0

e

dT

ý

ÿcT ÿ

öq

ú1

õ

2

a

T

úÿ+1ú1

:

Thus, following the construction of the original controller with of

F ~ (u), we have that for every x(0) 2 R

n

û

F

0

u0 e x(0)) e0 x(0);ü0 (~

ø F X (~u0 e x(0)) 0

F~

û;ü0 (u)

instead

(4.68)

68

Chapter 4.

Finite data rate control | single-input case

by Proposition 4.2.4. Now, from (4.67) and (4.68), we obtain that, for every initial condition all

ý0

2 [0; T ].

x(0), F (x(t))

2 F X (~u0 e x(0)) for t 2 [ý0 ; T ]. 0

This holds for

ÿ

We remark that this result relies on the quadratic stability of the original system in the continuous-time domain; even if the messages are delayed, the Lyapunov function can be decreasing at the sampling points. The use of the extended hold

He;T

is the key to the analysis, though it may be rather naive

not to have any control during the delay time.

4.5.3

Time delay and quantization

The quantized sampled-data systems designed in Sections 4.3 and 4.4 have the same quadratic stability property as the sampled-data systems in Sections 4.2. Thus, an analysis similar to the one in the previous subsection on the maximum time delay can be carried out for this class of systems. We derive a result for the case with ÿnite quantizers. This result gives us a stabilization method for linear time-invariant plants using a strictly ÿnite data rate channel. Let us brieýy review our ÿnite quantizer design in Section 4.4. Given a plant (A; B ), a controller region Q

DøR

n

K,

matrices

P

and

J, ÷

2

(0; 1), and a bounded

, the method yields a sampling period

with a ÿnite index set

S

N , and a radius

resulting closed-loop system, the ball

D with respect to (P; ÷J ).

-

u(t)

r0

B0 (r0 )

T,

a ÿnite quantizer

of the attractive ball. For the

is quadratically attractive from

x(t)

~) (A; B

?

u ~0

0

e

6 He;T

ÿ

delay

ýý

ÿ

Q

ÿ

ST

ÿ

Figure 4.18: Quantized sampled-data system with time delay Now we consider the system in Fig. 4.18, where

He;T

is the extended hold

deÿned in the previous subsection, and the varying delay element with the maximum delay time

ý >

analysis problem is to ÿnd

0 is added to the original system of Fig. 4.7. The

B

such that for the closed-loop system the ball 0 (r0 ) maintains to be attractive (but not necessarily quadratically attractive) from ý

D.

The approach taken here is similar to the one in the previous subsection; the following lemma is analogous to Lemma 4.5.1. Let and let

ó

2 (0; 1).

c

:=

÷þmin (J )=þmax (P )

4.5.

69

Control over a ÿnite data rate channel

Lemma 4.5.3 Suppose the sampled-data system with a ÿnite quantizer has

the following properties:

ú eýc(T ýü0 ) eA ü0 P eAü0 ÿ 0 for every ý0 2 [0; ý ]; there exists r1 ÿ 0 such that if x(0) 2 X0 and if u(t) = 0 for t 2 [0; T ] then F (x(t)) 2 F X (0) [ B0 (r1 ) for t 2 [0; T ]; for j 2 SN , j = 6 0 and for ý0 2 [0; ý ], if x(0) 2 Xj and if the control is

(i)

0

óP

(ii)

(iii)

(

u(t)

then

F (x(t))

=

0

j

u

if

t

if

t

2 [0; ý0 ); 2 [ý0 ; T ];

(4.69)

2 F X (uj ) for t 2 [ý0 ; T ].

B0 (r0 ) is attractive from p D, where r0 = r1 þmax (P )=þmin(P ). In particular, if x(kT ) 2 Xj , j 2 SN , j = 6 0, then V (x((k + 1)T )) ý óV (x(kT )).

Then, for the closed-loop system in Fig. 4.18, the ball

Following a discussion similar to the proof of Lemma 4.5.1,

Sketch of Proof

we obtain from the conditions (i) and (iii) that for every under some

u(t)

j

in (4.69),

2 SN .

V

(x(T ))

ý

óV

x(0)

(x(0)). Here note that

2 Xj , j 6= 0, 2 Xj with

x(T )

x(0) 2 X0 þ) õ 0, V (x(t)) decreases or else x(t) is in B0 (r1 ) for t 2 [0; T ]. The ball B0 (r0 ) is the smallest ball containing a level set with B0 (r1 ) inside by Lemma 4.3.2. Clearly, B0 (r0 ) is an invariant set and moreover trajectories starting in D go in to it in ÿnite time. ÿ

The condition (ii), on the other hand, implies that, for every

under

u(

This lemma is applicable to any ÿnite quantizer. Note that the condition (ii) in the lemma holds automatically with the original design in Subsection 4.4 because when u(t)

= 0 for

t

r1

x(0)

that is obtained in the

2 X0

2 [0; T ] whether or not there is time delay.

the control will be

Thus, the size of the

attractive ball is not an issue to be worried about. In the following, we state our main theorems of the chapter for the uniform and logarithmic quantizers. The procedure of the analysis is to design

2 (0; 1).

a quantizer based on Corollary 4.4.3 or 4.4.4 and then to set

ó

theorems give the conditions on the maximum delay time

We begin with

ý.

The

the uniform quantizer case. Theorem 4.5.4 Given a stabilizing ÿnite uniform quantizer for which the ball

B0 (r0 ) is quadratically attractive from B0 (R0 ) with respect to (P; ÷J ), suppose that

ý >

ý

ý

þ

ö

0 is small enough that 1

kAk

ln

ÿc

T

T (ÿ ú 1)

d

and that, for every

ý0

2 [0; ý ],

q

õý

T (aT ú 1) ú a + a2 ú ÿ + 1 T T 2aT + 1 ýc(T ýü0 ) eA ü0 P eAü0 ÿ 0. Then, óP ú e

a

closed-loop system in Fig. 4.18, the ball

0

for the

B0 (r0 ) is attractive from B0 (R0 ).

70

Chapter 4.

Finite data rate control | single-input case

The proof for this result is straightforward following that for Theorem 4.5.2. We next state the result for the logarithmic quantizer case as a theorem as well. Theorem 4.5.5 Given a stabilizing ÿnite logarithmic quantizer with Æ > 1

for which the ball

B

0 (r0 ) is quadratically attractive from to (P; ÷J ), suppose that ý > 0 is small enough that

ý

ý

1

kAk

ln

þ

ö

ÿcT dT (ÿ

ú 1)

B0 (R0 ) with respect

õý

q ú 1) ú aT + a2T ú ÿ + 1 (aT + 1)Æ + aT ú 1 2aT (aT

2 [0; ý ], óP ú eýc(T ýü0 ) eA ü0 P eAü0 ÿ 0. Then, for the closed-loop system in Fig. 4.18, the ball B0 (r0 ) is attractive from B0 (R0 ). and that, for every

0

ý0

These theorems allow us to control a linear time-invariant plant over a channel with a strictly ÿnite data rate: Every

T

seconds, a message containing

jSN j) bits of data is sent, and the data rate of the channel has to be larger than log2 (jSN j)=ý bps. For the calculation of the actual necessary data rate

log2 (

for the channel, in addition, we have to take account of computation time,

coding/decoding time, the propagation delay, the bits in the overhead of each message, and so on.

4.6

Design of

We have seen that

ÿ

ÿ

plays an important role in the design of the sampled-data

controllers. In particular, large

ÿ

can result in a small sampling period, as

can be observed in the bounds on cT and dT in (4.20) and (4.21), respectively. However, in general, it is diÆcult to obtain a prespeciÿed value of ÿ by changing

Q

and

R.

On the other hand, we have made the critical assumption

In this section, we show that design

ÿ

ÿ

arbitrarily close to 1.

So far, we have also made the assumption that (0; 1). (1

Here we allow

ú ÷)Q.

Q

1.

ÿ >

is always greater than 1 and give a method to Q

= 1=(1

ú ÷)I

with

to be any positive-deÿnite matrix, and set

Qú

In view of (4.6), we introduce a change in coordinates using

÷

2

:=

1=2

Qú

and deÿne :=

Qú

û B

:=

Qú

û := K The correct deÿnition of

ÿ

ÿ (Q; R)

1=2

ý1=2 ;

AQú

~ B;

(4.70)

ý1=2: KQ ú

is therefore (see (4.7))

ÿ (Q; R)

showing its dependence on

1=2

û A

= (1 + ÷)R

and

óó ó

Q

:= (1 + ÷)R R

explicitly. Observe

ó

ý1=2óó2 =

K Qú

kKû k2 ;

1+÷ 1

ú÷

R

óó ó

ó

ý1=2 óó2 :

KQ

(4.71)

4.6.

71

Design of ÿ

For this general Q case, the results in the previous sections hold by replacing û and so on. matrix A with A

2 (0; 1), then

Theorem 4.6.1 If (A; B ) is stabilizable, A is unstable, and ÷

inf

Q >

0;

R >

1+÷

g=

0

1

ú÷

(4.72)

:

By (4.71), the minimization problem (4.72) is equivalent to

Proof

1+÷ 1 Fix

fÿ(Q; R) :

Q >

ú÷

0 and R

ÿ

inf

R >

óó ó

ò

ó

ó2

9Q > 0; R > 0) R óóK Qý1=2óó ý ò

: (

0. Let

ó

ò

ÿ RkK Qý1=2k2.

ý1=2óó2 = R(Rý1 B P Q 0

KQ

=

1=2

ý

þmax (Q

ò

:

By deÿnition,

1=2

ý

P BR

1=2

ý

)(Q

1

ý

P BR

0

1=2

ý

B PQ

ý ò:

1

ý

)

) (4.73)

This is equivalent to P BR

1

ý

0

B P

ý òQ = ò (úA P ú P A + P BR 0

1

ý

0

B P)

(4.74)

by the Riccati equation in (4.5). This in turn is equivalent to 0

+ PA

A P

Here

P

is positive deÿnite, and

inequality (4.75) holds only if

k

ú

ö

1

A

ò

ú

1

õ

P BR

ò

1

ý

ý 0:

0

B P

(4.75)

is an unstable matrix by assumption. Thus,

ÿ 1.

Next we show that for every ò > 1 there exist matrices Q and R such that ý1=2 2 ~ . Find the unique, ò . Take any ò > 1 and positive-deÿnite Q

R KQ

k ý

positive-deÿnite solution

P

of

0

+ PA

A P

ú P BB P + Q~ = 0: 0

(4.76)

Now set Q

~+ =Q

1

ò

ú1

0

P BB P;

R

ò

=

ú1 ò

:

Then 0=

0

A P

+ PA

ú P BB P + Q~ 0

ö

=

A P

+ PA

ú PB

=

A P

0

+ PA

ú P BR

0

1

ú

1

ý

1

õ

ò 0

B P

R

1

ý

+ Q:

0

B P

~ +Q

(4.77)

72

Chapter 4.

Thus,

P , Q,

using this

and R satisfy the Riccati equation (4.5), so R

P.

(4.73). Thus, the inÿmum

Finite data rate control | single-input case

kK Qý1=2k2 is deÿned

Now (4.77) implies (4.75), and this implies ÿrst (4.74) and then 1=2 2 R KQ ò . Since ò > 1 is arbitrary, we conclude that

ò

ý k ý

k

ÿ

equals 1.

ý

ú

In the proof, we saw that, given ò > 1, we can design ÿ ò (1 + ÷)=(1 ÷). ~ is a free parameter, and the control input is one dimensional; thus, P Here, Q can be any LQR solution by (4.76). This gives some freedom in the design, as shown in the numerical example in Section 4.7. The parameter

in

X (u)

÷

2 (0; 1) deÿnes the decay rate of the Lyapunov function

in (4.6): the larger

÷

the faster the decay to the origin. The above

theorem, however, says that faster decay rate can result in larger

ÿ,

which in

turn makes the sampling period smaller. This trade-oþ is discussed further in the numerical example as well.

4.7

Magnetic ball levitation example

We view our results in the previous sections as being theoretical in nature. Nevertheless, it is always interesting to see what practical issues arise when theory is applied to an example. Here, we apply the sampled-data controller to a magnetic ball levitation system depicted in Fig. 4.19. A steel ball of mass

M

is levitated in air by an electromagnetic force generated by the electromagnet. The control objective is to keep the position by controlling the voltage

v.

y

of the ball at an equilibrium

The current in the coil is denoted by i, and the

resistance and inductance of the magnet are R and L, respectively. The force produced by the coil is K i(t)2 =y (t)2 , where K > 0, and g is acceleration due to gravity. We take1 Nm2 /A2 .

M

= 0:068 kg,

R

= 10 ø,

= 0:41 H, and

L

K

= 3:27

ü 10ý5

The dynamics of the system is given by

where x0

g

My 

=

Mg

v

=

L

ú

di dt

2

2 y

;

+ Ri;

= 9:8 m/s2 . We take the state as

of the system for the nominal voltage

Ki

x

v0

:= [y

0

y i]

. Then the equilibrium

= 10:0 V is

2 q 3 2 3 u0 K 7:00 ü 10 3 R Mg 6 7 5: x0 = 4 0 5=4 0 u0 1:00 R Now, letting ùx := x ú x0 and ùv := v ú v0 , we have the linearized system ý

ùx = 1 These

Aùx

are values from a real system [63].

+ B ùv;

4.7.

73

Magnetic ball levitation example

i(t)

+

R v (t)

L

ú

y (t)

M

Figure 4.19: Magnetic ball levitation system

where

2 0 q 6 Rg Mg A = 42 v0

1 0

K

0 The eigenvalues of First,

K

and

ÿ

A

are

=

v0

2 3 0 B = 405: 1 L

úR L

f52:9; ú52:9; ú24:2g.

were designed using the method in Section 4.6. We took = 8:18;

ò

and obtained

þ(A)

0

3 7 ú2 Rg 5 ; 0

2 Q = 4

~ Q

1:15 ü 107 2:17 ü 105

2:17 ü 105 4:11 ü 103

ú5:52 ü 104

R

= 0:878;

K

= 1:04 ü 104

ü

These matrices yield

= 0:01 diag(1; 1; 1)

ú1:04 ü 103 196

û

ú49:7

3

ú5:52 ü 104 ú1:04 ü 103 5 ; 265

(4.78)

:

ý1=2k2 = 8:18 as expected.

k

R KQ

These values were ar-

rived at by some trial-and-error with a view to a reasonable transient response. The eigenvalues of

A

+ BK are

þ(A

For

÷

+ BK ) =

= 0:100, we obtained

ÿ

fú24:2; ú52:9; ú67:6g:

= 10:0. We ÿx

K

and

÷

throughout most of

this section. In the following, we design controllers for the sampled-data and quantized sampled-data setups. Sampled-data control:

For the sampled-data controller in Section 4.2, we

calculated the maximum sampling period

Tmax

satisfying bounds on

cT

and

74

Chapter 4.

ô

Finite data rate control | single-input case

8 7 ô

=

6

ý ~0 õ ~) û (õ; u

f

ô

=

+ ~0 õ ~) û (õ; u

f

5 4 3 2

=

ô

0

g (õ; u ~ õ ~)

1 0 −2

4

3

2

1

0

−1

õ ~

Figure 4.20: Functions

(

max =

T

ý + û~ , fû~ , and

f

ú 1) 1: 3 76 ü 10ý3 3 57 ü 10ý3

~0 = (ÿ T for the two cases u

d

õ

g

for õ ~ =1

=ÿ;

: :

for u ~ 0 = (ÿ for u ~0 = 1:

There is a slight advantage in using u ~0 = (ÿ

ú 1)

ú 1)

=ÿ;

=ÿ ,

as expected.

For comparison, we also calculated the sampling period for the stability of

the system in the discrete-time domain. We discretized the plant (A; B ) via step-invariant transformation and obtained (Ad ; Bd ). The maximum sampling periods for all eigenvalues of

(

max =

T

Note here that

0

u

d + Bd u0 K to be less than 1 were

A

ü 10ý2 1 78 ü 10ý2 2:03 :

for u ~ 0 = (ÿ for u ~0 = 1:

ú 1)

=ÿ;

= (1 + ÷)~ u0 as in Subsection 4.2.2. These are roughly 5

times larger than the sampling period required for quadratic stability.

We

stress that the discrete-time stability does not have a guaranteed decay rate. From this point in the example, we ÿx u ~0 = (ÿ

controller with

T

= 3:76

ü 10ý3.

Functions

ú 1)

ý + û~ , fû~ ,

f

=ÿ .

and

g

We designed a for õ ~ = 1 are

graphed in Fig. 4.20. Note that the functions are tangent to each other. The time response plots of ùx, ùi, and ùv are given in Fig. 4.21 for the initial condition 0

ùx(0) = [ùy (0) ùy (0) ùi(0)] = [0:700

ü 10

3

ý

0

0 0]

:

(4.79)

This is typical [63]. The solid lines are for the sampled-data system and the dashed lines are for the full state feedback case (u =

K x).

We see that in

4.7.

75

Magnetic ball levitation example

−3

∆y

1

x 10

0.5

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

∆i

0.2

0.1

0 0

∆v

10

5

0 0

time t Figure 4.21: Time responses for the sampled-data system

both cases

y

goes to the equilibrium in about 0:25 seconds. In the transient

response, the position for the sampled-data system decreases slightly faster than that for the continuous-time system.

Eþect of ÿ in the design of þ:

The parameter

÷

2

(0; 1) determines

the decay rate of the Lyapunov function for the sampled-data controllers and enters into the design in (4.6). The result in Section 4.6 says that the larger ÷,

the larger

ÿ;

in other words, a faster decay rate results in larger

further implies a smaller sampling period, by the bound on

T

c

ÿ,

which

as in (4.20).

We conÿrmed this observation using the current example as follows: The gain

K

was ÿxed as above, and we varied

maximum sampling period

Tmax .

÷

2 (0; 1) and calculated ÿ

and the

The results are shown in Fig. 4.22.

We have to mention, however, that it is not clear how the choice of aþect the decay rate.

In the simulations above, we used

÷

÷

might

= 0:1, which is

fairly small. Nevertheless, in the time response plots, the trajectory for the full state feedback case and that for the sampled-data control case in Fig. 4.21 look almost the same, and hence there is little diþerence in the decay rate.

The maximum delay time analysis:

In Subsection 4.5.2, an analysis

on the robustness of the sampled-data system against varying time delay was given. The maximum delay time ý was calculated for the case ÷ = 0:1 and ó = 0:999 for T (0; 3:76 10 3 ); this is shown in Fig. 4.23. Here the solid

ü ý

2

line is the bound on

ý

calculated from the condition in (4.66) in Theorem 4.5.2 10 4

and the dashed line is the one from (4.65). The largest ý is ý = 4:70 obtained at T = 3:76 10 3 , which is eþectively the maximum T .

ü ý

ü ý

76

Chapter 4.

Finite data rate control | single-input case

200

σ

150 100 50 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−3

5

x 10

4

T

max

3 2 1 0 0

ε

Figure 4.22:

ÿ

and the maximum

As discussed in Subsection 4.5.2, size of the

÷

ý.

Here the largest

= 0:1 case, these

that at

÷

ý

ý

÷

T

versus

÷

is a parameter that has an eþect on the

is plotted in Fig. 4.24 for several

÷

2 (0; 1); as in

were obtained at around the maximum

= 0:4 the maximum delay time is about 100

T.

We see

üs.

In Subsection 2.2.3, we showed the delay time, or the transaction time, for the communication of one data message using a practical control networks protocol. According to the formula in (2.1), with the delay time of 100

üs,

about 12 bytes of data can be transmitted periodically over a 2.5 M bps channel using the FIP protocol. Hence, we can conclude that the bounds obtained above are fairly realistic for this example. Quantized sampled-data control:

We design the quantized sampled-data

controllers in Section 4.3. Here, in particular, we compare the characteristics of the uniform and logarithmic quantizers. The maximum sampling periods for these quantizers are

(

Tmax

=

ü 10ý3 3:76 ü 10ý3 2:43

for the uniform quantizer; for the logarithmic quantizer:

We have seen that, in the design of the uniform quantizer T

Qþ ,

parameters

ý Tmax and r0 > 0 can be chosen independently; for a ÿxed T , ù in (4.47)

is a linear function of

r0 .

The controller with a logarithmic quantizer too

has two parameters,

and

T,

Æ

but, in contrast, there is a trade-oþ between

them as seen in (4.51). For each Fig. 4.25. As Tmax

= 3:76

Æ

Æ >

1, there is

approaches 1 from above, 10 3 as given above.

ü ý

T

Tmax , and this is shown in grows larger. Note that for Æ = 1,

4.7.

77

Magnetic ball levitation example

−5

5

x 10

maximum delay τ

4

3

2

1

0 0

0.5

1

1.5

2

2.5

3

3.5

4 −3

x 10

T

Figure 4.23: Time delay analysis:

versus

ý

Sampled-data control with ÿnite quantizers:

T

for

÷

= 0:1

As discussed in Sec-

tion 4.4, when ÿnite quantizers are used, the data size of the message sent from the sensor to the actuator is ÿnite. Moreover, the data rate in the sense of (4.59) for this communication is ÿnite. Here, we are interested in ÿnding the minimum data rate required for the uniform and logarithmic quantizers in our framework. Following the construction in Corollaries 4.4.3 and 4.4.4, we ÿx and

r0

= 3. First, for the uniform quantizer, we computed ù,

the data rate in the sense in (4.59) for

T < Tmax

are given in Fig. 4.26. We see that ù and T,

N

= 2:43

ü 10ý3.

Next, we computed Æ ,

N,

T

=

R0 = 10 and then

The plots

are almost linear functions of

and as a result the data rate decreases exponentially as

minimum data rate is about 7780 bps at

N,

T

increases. The

Tmax .

and the data rate for control using the logarithmic

quantizer in a similar fashion. The results are shown in Fig. 4.27. (The plot on the top is the same as Fig. 4.25 with opposite coordinates.) Here, the growth of

N

is exponential, but the rate is slow. Hence, for sampling periods not too

close to

Tmax , the data rate is a decreasing function of T , and there is a clear advantage over the uniform quantizer. The minimum data rate for this system is about 1810 bps at T = 3:07 10 3 .

ü

ý

We designed the two quantizers with the parameters to achieve the minimum data rate. For the uniform quantizer, we obtained ù = 0:364 and N = 2:38 105 and for the logarithmic quantizer õ0 = 1:86, Æ = 1:79, and

ü

N

= 24. We computed the time responses for these two systems with the

same initial condition ùx(0) as in (4.79). These are shown in Fig. 4.28 for the system with the uniform quantizer and in Fig. 4.29 for the one with the loga-

78

Chapter 4.

Finite data rate control | single-input case

−4

1

x 10

0.8

τ

0.6

0.4

0.2

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.8

0.7

0.9

1

ε

Figure 4.24: Time delay analysis: the largest

ý

versus

÷

rithmic quantizer. We have chosen

r0 = 3 in the design so that the switching in the control input ùv is visible in the steady state. As a result, in both plots, the position ùy only goes down to about 0:2 10 3 and seems to reach

ü

ý

a steady state there. To achieve a steady state closer to 0, we can redesign with a smaller r0 . The number of cells used in these simulations are 7 and 6 for the uniform case and the logarithmic case, respectively. The diþerence may seem small, given that the logarithmic quantizer uses a more eÆcient partition scheme. This is however because the initial state is not so far away from cell redesigning the controllers with a smaller

r0

X

0 . Again, will likely make the diþerence in

the number of cells to be used larger. The conservativeness in the design can be found as follows. The norm of initial condition x(0) in (4.79) is about 2:25 in the state space corresponding to û B û ). This means that the state is already in the prespeciÿed ball of radius (A; r0 = 3. In the steady state, this norm goes down to about 0.193 and 0.130 for the responses in Figs. 4.28 and 4.29, respectively. Thus, trajectories go

into a ball much smaller than the one prespeciÿed by r0 . This phenomenon is expected since r0 is a rather large estimate.

Finite data rate control:

Finally, we carried out the time delay analysis

on the sampled-data system with ÿnite quantizers based on the results in Subsection 4.5.3. We used

ó = 0:999. For the uniform quantizer that achieved the minimum data rate, the maximum delay time ý was 2:92 10 5, while for the logarithmic quantizer it was ý = 3:68 10 5 . So both are in the order of

10

üs.

ü ý

ü ý

These values were calculated using plots similar to Fig. 4.23.

4.7.

79

Magnetic ball levitation example

−3

4

x 10

3.5

Sampling period T

3

2.5

2

1.5

1

0.5

0 0

10

5

30

25

20

15

δ

Figure 4.25: Trade-oþ between

Æ

and

T

It was observed above in the analysis for the sampled-data systems without quantizers that

÷

serves as a parameter that has inýuence on

ý,

as seen in

Fig. 4.24. This is also true for the case with the quantizers. For example, for ÷

= 0:4, the sets of parameters that achieved the minimum data rate were for

the uniform quantizer T

= 1:88

ü 10ý3,

2:55 ü 10ý1 , N

ü

data rate for control = 1:01 104 bps, ù = 5 5 = 2:76 10 , and ý = 6:02 10

ü

ü

ý

and for the logarithmic quantizer T N

ü 10ý3, data rate for control = 2 33 ü 103 bps, = 24, and = 7 72 ü 10ý5 .

= 2:41

So clearly

:

ý

ý

Æ

= 1:76,

:

can be increased by changing ÷.

Now we look at the logarithmic quantizer case here. of cells necessary is

jSN j

Since the number

= 2N + 1 = 49, each message to be sent over the

channel contains less than 1 byte of data. In Subsection 2.2.3, we presented the expected delay time for the communication system realized by the FIP protocol with a 2.5 M bps channel. This is about 63

üs

for 1 byte of data,

according to the formula in (2.1). Therefore, it is clear that the maximum delay time calculated for the logarithmic quantizer above is suÆciently large using this communication system with a strictly ÿnite data rate.

Chapter 4.

80

Finite data rate control | single-input case

0.8

∆

0.6 0.4 0.2 2.5

0 5 x 10

0.5

0 4 x 10

0.5

0

0.5

1

1.5

2

2.5 x 10

3

N

2 1.5

Data rate [bps]

1 10

1

1.5

2

2.5 x 10

3

5

0

1

1.5

2

2.5 x 10

Sampling period T [s]

3

Figure 4.26: Data rate for control for the uniform quantizer

30

δ

20 10

0

0

0.5

1

1.5

2

2.5

3

3.5

4 x 10

150

3

N

100

50

Data rate [bps]

0

0

0.5

1

1.5

2

2.5

3

3.5

4 x 10

5000

3

4000 3000

2000 1000

0

0.5

1

1.5

2

2.5

Sampling period T [s]

3

3.5

4 x 10

3

Figure 4.27: Data rate for control using the logarithmic quantizer

Magnetic ball levitation example

81

−3

∆y

1

x 10

0.5

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

∆i

0.2

0.1

0 0

8

∆v

6 4 2 0 −2 0

time t

Figure 4.28: Responses for the system with a uniform quantizer

−3

∆y

1

x 10

0.5

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

∆i

0.2

0.1

0 0

8 6

∆v

4.7.

4 2 0 −2 0

time t Figure 4.29: Responses for the system with a logarithmic quantizer

Chapter 5

Towards data rate reduction In this chapter, we show a few variations of the quantized sampled-data system in Fig. 4.7 that can potentially decrease the data rate for control. The deÿnition of data rate in (4.59) suggests two ways: (i) by decreasing the number of bits, or the number of partition cells, and (ii) by increasing the time between samplings.

In the following sections, we propose methods for time-varying

quantization and nonuniform sampling in our system. Combinations of these methods are possible, but can be rather complicated and are not attempted.

5.1

Time-varying quantization

One way to reduce the data rate is to decrease the number of bits, that is, to use fewer cells in the quantizer. This is possible by introducing time-varying quantizers that use cells to cover only the necessary region in the state space at the time. Here, we employ the technique proposed by Brockett and Liberzon [12, 52]. We show that a ÿnite quantizer with a sampling period

T

can

be modiÿed to use only a few cells along with additional data of 1 bit for stabilization. Further, this method has the feature that the states go to the origin, not only into a small ball. The price to be paid is, however, the decay rate; it is not quadratic anymore. Although this approach is applicable to any ÿnite quantizers, we use the ÿnite logarithmic quantizer in the following construction.

5.1.1

Problem formulation

Consider the sampled-data system in Fig. 5.1 with the ÿnite logarithmic quantizer deÿned in Subsection 4.4. Recall that the parameters that determine the

H. Ishii and B.A. Francis (Eds.): Limited Data Rate in Control Systems with Networks, LNCIS 275, pp. 83−116, 2002.  Springer-Verlag Berlin Heidelberg 2002

84

Chapter 5.

-

u(t)

Towards data rate reduction

x(t)

~) (A; B

?

u ~0

0

e

6 HT

ÿ

QÆ;û0 (k)

ÿ

ST

ÿ

Figure 5.1: Quantized sampled-data system

Q

Æ;þ0 (k1 )

Q

Æ;þ0 (k2 )

Figure 5.2: The zoom-in feature (k1

< k2 )

quantizer are Æ >

Given

õ0

and

N,

instants

t

=

T >

0;

õ0 >

0; and

N

2 N:

a ÿnite logarithmic partition is deÿned as in (4.55).

see that, for a ÿxed

Q

the union of

1;

j, j

kT , k

N,

the size of

õ0

2 S , covers. Here, we allow õ0 to change at sampling 2 Z+, and thus denote õ0 (k). We call õ0 (k) the state of N

the quantizer, and, moreover, we denote the quantizer by dependence on

We

determines the size of the region that

õ0 (k )

explicit.

QÆ;û0 (k) ,

making its

The key idea behind the time-varying quantizer lies in the zoom-in feature. For any ÿxed

õ0 ,

the closed-loop system with a ÿnite quantizer is stable with

some attractive ball. smaller

Once the trajectory is inside the ball, by choosing a

appropriately, it can go into a smaller ball and so on, approaching

õ0

the origin. In Fig. 5.2, we show how the logarithmic partition of the quantizer zooms in to have ÿner cells around the origin. The time-varying quantizer problem is stated as follows: Given a bounded set

DøR

n

, ÿnd a switching law for the state

nite quantizer starting in

QÆ;û0 (k)

þ)

õ0 (

of the time-varying ÿ-

such that, for the closed-loop system, every trajectory

D eventually goes to the origin.

Time-varying quantizers were ÿrst proposed in [12] for stabilization problems of both continuous-time and discrete-time systems. The continuous-time problem considered there is somewhat diþerent from ours. First, the quantized signal is a continuous-time one, and thus chattering is allowed to occur. This

5.1.

85

Time-varying quantization

setup necessitates solutions in the sense of Filippov [85]. From our viewpoint, this is problematic since a bound on the necessary data rate cannot be derived. Second, asymptotic stability can be achieved by having a zoom-out mode in the quantizer; if the initial state is not within the range of the quantizer, it enlarges the range until the state is captured and then goes into the zoom-in mode. In practice, a local stability result is suÆcient, and hence we leave this zoom-out feature out of the system. 5.1.2

A switching law

In this section, we show how the parameter õ0 of the ÿnite logarithmic quantizer (see (4.55)) changes the characteristics of the closed-loop system and then give a stabilizing switching law for

õ0 . We construct the controller based on the ÿnite logarithmic quantizer in

Corollary 4.4.4. Parameters

T

tizer is designed for a given

õ0 .

1. Fix

÷

2. Take

2 (0; 1). Æ >

T

c

Select matrix

1 and

<

p

and

T >

1 ÿ

ÿ

+1

ÿcT T ý ÿú1

d

N

Q

and

2N

N,

Note that 4. Given

let

ü >

ÿ >

ò

q T T ú 1) ú a + a2 ú ÿ + 1 T T (aT + 1)Æ + aT ú 1 2a (a

:=

pT ú

a

p

ÿ (a

ÿ

T

1.

ú1

s

+ 1)

þmin (P ) þmax (P )

(5.1) :

:

large enough that N >

For this

0 in (4.5) so that

;

ü0

N

R >

0 small enough that

3. Let

Choose

are ÿrst determined and then the quan-

SN

=

log

1

Æ ü0 :

f0; 1; : : : ; N ú 1g ü fú1; 0; 1g and N ü := Æ ü0 :

(5.2)

(5.3)

1, from (5.2).

fQj gj2SN be the logarithmic partition in (4.55) and f j gj2SN in (4.57) Deÿne the level set E0 (þ) of V by E0 (õ0 ) := fx 2 Rn : V (x) ý c0 (õ0 )g; õ0 > 0;

õ0 > 0, let output values q

where c0 (õ0 ) := (õ0 Æ

N )2 þmin (P ).

86

Chapter 5.

T

Note that the choices of

and

N

Towards data rate reduction

do not depend on

õ0 .

Lemma 5.1.1 For the closed-loop system in Fig. 5.1 with the ÿnite logarith-

E0 (õ0 ) and E0 (õ0 =ü) are invariant sets, and every E0 (õ0 ) goes into E0 (õ0 =ü) in ÿnite time.

mic quantizer just deÿned, trajectory starting in Let

Proof

s r0 (õ0 )

as in (4.39).

=

õ0

Notice here that

þmax (P )

p

pT + 1) T ú ÿú1 ÿ (a

þmin (P ) a r0 (õ0 )

=

õ0 =ü0 .

To apply Theorem 4.4.2,

2 SN , j 6= 0, the bounds on cTj in (4.37) j and dTj in (4.38) are satisÿed; (ii) E0 (õ0 ) ø [j 2SN Xj (õ0 ); and (iii) c0 (õ0 ) > we must show (i) for

=

T

T, j

2

r0 (õ0 ) þmin (P ).

First (i). This is immediate since for a logarithmic partition, the bounds on

Tj

c

and

d

Tj

are all identical and are equal to those in (5.1).

Next (ii). By Lemma 4.3.2 and the deÿnition of c0 (õ0 )

E0 (õ0 ) ø B0

ïs

c0 (õ0 )

!

þmin (P )

=

B0 (õ0 ÆN ):

By the deÿnition of state space cells in (4.32), we also have

[

j2SN Thus, clearly,

Xj (õ0 ) =

ñ

õe

+y :

õ

2 (úõ0 ÆN ; õ0 ÆN ); y 2 M?

E

Xj

ô

:

2 SN

(õ0 ), j . 0 (õ0 ) is contained in the union of cells Finally we show (iii). Since ü > 1, c0 (õ0 ) > c0 (õ0 =ü). On the other hand,

ö

c0 (õ0 =ü)

=

õ0

N Æ

ö õ2

=

ü

õ0

õ2

þmin (P )

þmin (P )

ü0

2

= r0 (õ0 )

þmin (P )

(by deÿnition of c0 (õ0 ))

(by deÿnition of (since

r0 (õ0 )

=

ü

in (5.3))

õ0 =ü0 ).

D = E0 (õ0 ). Thus, for E0 (õ0 ) and E0 (õ0 =ü) are invariant sets, and moreover trajectories starting in E0 (õ0 ) go into E0 (õ0 =ü) in ÿnite time. ÿ Now, the conditions in Theorem 4.4.2 are satisÿed with the closed-loop system,

We now describe the quantizer. Given

switching law

for the state

D, take õ00 > 0 large enough that D ø E0 (õ00 ):

þ) of the time-varying

õ0 (

5.1.

Clearly, such

õ00

always exists. Take

(

õ0 (k

for

87

Time-varying quantization

ÿ

k

+ 1) =

õ0 (0)

x(kT )

and

2E

if

õ0 (k );

otherwise;

0 (õ0 (k )=ü);

(5.4)

1.

D

Theorem 5.1.2 Given

time-varying quantizer converges to the origin.

ÿ

Proof k

õ00

õ0 (k )=ü;

The stability result for the system using

for

=

D

stays in

Denote this time by

E

now follows.

, for the sampled-data system in Fig. 5.1 with the

QÆ;û0 ( )

Let k0 = 0. By (5.4), k0 . However, by

starting in

QÆ;û0 ( )

deÿned above, every trajectory starting in

õ0 (k )

DøE

0 (õ(k0 ))

=

õ0 (k0 )

as long as

2E

x(kT ) =

D

0 (õ0 (k0 )=ü)

0 (õ0 (k0 )) and Lemma 5.1.1, a trajectory

and goes into

E

0 (õ0 (k0 )=ü)

in ÿnite time.

ÿ

= t1 and the next sampling instant by t = k1 T t1 . Then, by (5.4), õ0 (k ) = õ0 (k1 ) = õ0 (k0 )=ü as long as x(kT ) = 0 (õ0 (k1 )=ü) for

ÿ

k

k1 .

t

Applying Lemma 5.1.1 again, the trajectory in

this set and enters

E

0 (õ0 (k1 )=ü) in ÿnite time.

E

2E

0 (õ0 (k1 ))

Repeating this argument, we obtain an increasing sequence that x(t)

Since

ü >

2E

0 (õ0 (kl ))

for

t

2

[kl T ; kl+1 T ),

l

f g 2Z+ kl

such

2Z

+:

=

õ0 (k0 ) l

ü

!

0 as

l

!1

:

ÿ

Therefore, the trajectory goes to the origin. The advantage of using this quantizer is that

D

l

1, õ0 (kl )

of

stays in

N

is independent of the size

and thus can be chosen to be small. Moreover, asymptotic stability can

be achieved with a ÿnite data rate. The drawback is that the stability is not quadratic any more and thus that the decay rate for the trajectory can be appreciably slower than that using the original (time-invariant) quantizer. In particular, if at some set

E

0 (õ0 (k )=ü),

t

2

[kT ; (k + 1)T ) the trajectory is in the smaller level

the Lyapunov function may not be decreasing for some time.

Moreover, if the trajectory is already in a much smaller level set for some

l

ÿ

the period of

1 at lT

t

=

kT ,

E

l

0 (õ0 (k )=ü

)

the Lyapunov function may continue to increase for

in the worst case. For this reason, using

ü

close to 1 may not

be eÆcient for the decay rate. How does the time-varying ÿnite quantizer change the necessary data rate for the closed-loop system? The data to be sent from the sensor side to the actuator side at each sampling instant are

û

the index of the cell in which

x(kT )

is,

88

Chapter 5.

û

one of the two states

{ zoom in : if x(kT ) 2 E0 (õ0 (k )=ü), { hold : otherwise.

The parameter case

Towards data rate reduction

N

N

must satisfy (5.2), but can be as small as

N

jS j = 2

N

= 1, in which

+ 1 = 3 and this requires 2 bits per message. The additional

data (zoom in/hold) requires only one bit. Therefore, data rate using

5.1.3

QÆ;û0 (k)

=

log2

jS j + 1 = log (2 2

N

N

T

+ 1) + 1

bps:

T

Magnetic ball levitation example continued

The use of a time-varying quantizer is expected to reduce the data rate. It is not clear from the theory how eþective this quantizer can be compared to the time-invariant one. For comparison, we computed the necessary data rate for the magnetic levitation system in Section 4.7. As in the design of other controllers there, we used the same

K

in (4.78).

The time-varying quantizer is based on the logarithmic quantizer, so the maximum sampling period is Tmax

The minimum

N

in (5.2),

ü

= 3:76

ü 10ý

3

:

in (5.3), and the data rate for

T

plotted in Fig. 5.3.

2 (0

; Tmax )

are

Minimum N

15 10

5 0

0

0.5

1

1.5

2

2.5

3

3.5

4 x 10

15

3

µ

10

5

Data rate [bps]

0

0

0.5

1

1.5

2

2.5

3

3.5

4 x 10

5000

3

4000

3000 2000

1000

0

0.5

1

1.5

2

2.5

3

3.5

4 x 10

Sampling period T [s]

3

Figure 5.3: Data rate for control using the time-varying quantizer We see that

N

= 1 is allowed for

T

ý 1 08 ü 10ý , and there is a local ' 2400 bps at = 1 08 ü 10ý .

minimum data rate (log2 (2N + 1) + 1)=T

:

3

T

:

3

5.2.

89

Nonuniform sampling

The global minimum is about 1460 bps and is achieved at with

N

T

= 2:86

ü 10ý3

= 4. Since the minimum data rate using the logarithmic quantizer

was about 1810 bps, we see that it is indeed advantageous to use the timevarying quantizer. We note that this reduction is possible at the expense of the decay rate of the trajectories; in particular,

ü

closer to 1 can result in a

slower decay rate. We see in the plot that local minima in the data rate are always accompanied by

ü

' 1.

Thus, there is a certain trade-oþ between the

data rate and the decay rate.

5.2

Nonuniform sampling

As described in Chapter 2, many control networks protocols employ periodic sampling for transmission of control signals. viewpoint, it may be worthwhile to look into

However, from the data rate

nonuniform

sampling (e.g., [75])

so that messages are transmitted over the network only when necessary. This can be advantageous on a shared network. If the maximum data rate required for the system is known, the data rate while the system is in operation varies over time and can be less than the bound. A ÿnite quantizer output values determined by

Q

f j gj2SN . q

T

= min

is deÿned by an index set

SN , cells fQj gj2SN , and

In the design procedure, the sampling time

j2SN Tj ,

where

j

T

(4.37) and (4.38). By Lemma 4.3.7, this implies that trajectories in

Xj at

t

= 0 remain in

X ( j ) for at least 2 [0 j ]. u

T

is

is chosen to satisfy the bounds in t

x(t)

starting

;T

STj as follows. k and that x(tk ) is in Xjk . Then the next sampling time is given by t = tk + Tjk . We can deÿne a nonuniform zeroth-order hold HTj , synchronized with STj . Replacing the uniform sampler

Thus, we are motivated to deÿne a nonuniform sampler

Suppose that a sampling takes place at

t

=

t

and hold with these, we obtain a system with the same stability characteristics that the original system has. As a result, the data rate changes over time and is at most equal to what the original system needs: Data rate using The use of

j

T

instead of

T

Tj

S

and

H

Tj

ý log2 jSN j T

:

in sampling can be eþective especially if

j

T

> T

for many j ; this is the case for the uniform quantizers as we see in an example shortly. For other quantizers, such as the logarithmic quantizers, where T = for all

j,

T

j

we propose another nonuniform sampling scheme called dwell-time

switching control in the next section.

Magnetic ball levitation example continued For the magnetic ball levitation system using the uniform quantizer, we calculated the sampling time

j

T

for the nonuniform sampler and hold.

The design of the uniform quantizer is based on Corollary 4.3.8. In its proof, the bounds on

Tj

c

and

Tj

d

are given in (4.48). The calculation is based

90

Chapter 5.

ÿ

on these bounds for j that T1 = 2:43

ü

ý3

Towards data rate reduction

1; the result is shown in Fig. 5.4.

It is observed

is the minimum and thus T = T1 and that Tj quickly

10

increases to about 3:75

ü

ý3.

10

Notice that the uniform quantizer has partition cells of the same size. However, with this type of quantizer, when the state is far from the origin, the sampling can be slower.

This characteristic can reduce the data rate during

the operation of the system. This is in contrast with the logarithmic quantizer case, where the sampler must be uniform but the data rate is saved through the nonuniform partitioning. −3

4

x 10

T

j

3.5

3

2.5

2 0

20

40

60

80

100

j Figure 5.4: Sampling times Tj , j

5.3

ÿ

1, for the uniform quantizer

Dwell-time switching control

Systems whose control is based on quantized measurements have a hybrid nature: They have continuous dynamics accompanied by discrete signals taking values in a countable set. This aspect is not well represented in the quantized sampled-data control problem, for the setup is more classical with a ÿxed sampling period; messages are transmitted from the sensor side to the actuator side periodically, whether or not the message contains a quantized value diþerent from the previous one.

This may be ineÆcient from the data rate viewpoint

because some messages are likely not to have any new information. In contrast, many switching control methodologies in hybrid systems employ schemes where switchings occur only when the tra jectory of the plant enters a new state partition cell. Therefore, there is no ÿxed sampling period. Instead, the controller, or the switching logic, attentively observes the track of the tra jectory.

5.3.

91

Dwell-time switching control

In this section, we are interested in extending the results of the quantized sampled-data control problem from a hybrid systems viewpoint and study a stabilization method for a continuous-time plant via switching control. One of the major obstacles in switching control of a continuous-time system is that chattering, or arbitrarily fast switching, may occur. This is not only physically harmful to the actuators and/or the plant, but also impossible to realize with a ÿnite data rate channel. In addition, it presents mathematical diÆculties concerning existence of solutions of diþerential equations. We employ the switching control strategy based on state partitioning. The state space is partitioned into cells, and each cell has a corresponding constant control input. Of the several methods to prevent chattering [59, 86], we here employ a dwell-time switching controller, the protocol being that switching cannot occur until a speciÿed dwell time has elapsed. Some papers treating continuous-time switching controllers, e.g., [12, 52, 78], handle the potential of chattering by permitting solutions in the sense of Filippov [85]. By contrast, our switching with dwell time prevents chattering, and control signals are piecewise constant. Although the development can be done based on any of the stabilizing quantizers, here we use the logarithmic quantizer, which has some optimal properties, as seen in the previous chapter. We note that the quantizer itself does not appear explicitly; it is rather the state-space partition induced by the quantizer that we use. For this reason, we call the state partition cells simply cells in this section.

5.3.1

Dwell-time switched systems with logarithmic partitions

In Section 4.1, we deÿned a class of hybrid systems called dwell-time switched systems. In this section, we give a subclass of such systems where switching logics are equipped with a logarithmic partition.

R . Let e 2 R be a M denote the 1-dimensional subspace generated by

We deÿne a logarithmic partition of the state space vector of unit length, let e,

and let

Æ >

n

n

1. The partition is

X00 := M ; X := fx 2 R ?

ü

n

j

The picture is Fig. 5.5 for

n

:

0

e x

2 ö[Æ

= 2. Thus,

j

;Æ

X00

j

+1 )g;

is an (n

j

2 Z:

ú 1)-dimensional sub-

space, so it has empty interior, while the other cells, being bounded by parallel hyperplanes, have nonempty interiors. For simplicity, we rename the cells to

X

j; j

2 S , in the following manner:

Let

X[0 0] := X00 ; X[ 1 1] := X 1 ;

S := Z ü fú1; 0; 1g and

;

ü

j ;ü

In particular, we denote

j

X0 := X[0 0]. ;

[j1 ;

ö1] 2 S :

92

Chapter 5.

Towards data rate reduction

x2

Xÿ1+ X0+ X1ÿ

M

X1+

X0ÿ Xÿ1ÿ x1

Figure 5.5: Logarithmic partition of

R2

Now consider the nonlinear switching system

where

x(t)

x(t)

2R

n

=

fi(t) (x(t));

2S

is the state of the system, and i(t)

switching logic with a ÿxed dwell time Lipschitz continuous for each The switching logic (how

times as follows: Set

t0

j

2S

x(t)

T >

(5.5) is the state of the

0. It is assumed that

and that

f0 (0)

fj

is globally

= 0.

generates i(t)) is deÿned in terms of switching

= 0 and let j0 denote the index of the cell

x(t0 )

is in.

The equation x(t)

has a unique solution, say

=

fj0 (x(t))

ÿ , on the interval [t ; 0

x

1

). If

x(t0 )

= 0, then no

switching will ever occur; this is the trivial case. Otherwise, the ÿrst switching time is t1

( =

þ

t0

+ T;

fÿ

min

t

t0

+T :

x

ÿ (t)

if j0 = 0 and

2 X g =

int(

j0 )

;

6

x(t0 )

if j0 = 0.

Letting cl( ) denote closure, we set i(t)

= j0 for

2

[t0 ; t1 );

8 > <0 : the index of the cell X , where > : ( ) 2 cl(X 1 ) and 1 6= t

;

j1

j

x t1

j

j1

j0 ;

if

x(t1 )

2X

otherwise:

0;

6

= 0;

5.3.

93

Dwell-time switching control

This rather complicated deÿnition is to avoid the situation where trajectory is on the hyperplane

X

at

x

= t0 + T and will leave it the next moment; the cell that the state is entering cannot be deÿned. 0

t

Proceeding, the equation x(t)

has a unique solution, say

=

fj1 (x(t))

ÿ , on the interval [t ; 1

x

1

). Again, if

x(t1 )

= 0, then

no further switching will occur. Otherwise, the second switching time t2 is

(

t1

=

t2

+ T;

fÿ

min

t

t1

+T

Set i(t)

= j1 for

2

if j1 = 0 and

2 X g

: xÿ (t)

=

int( j1 )

;

6

[t1 ; t2 );

;

j2

x t2

þ

Continuing, we obtain i( ) on [0; jectory

þ

x(

j2

1

if

x(t2 )

j1 ;

= 0 then

2X

0;

f k gk2Z 2 1 + ÿ

) and switching times

x(0)

= 0;

otherwise:

) is uniquely deÿned satisfying (5.5) for all

switching times. It is clear that if

6

if j1 = 0:

8 > <0 : the index of the cell X , where > : ( ) 2 cl(Xj2 ) and j2 6= t

x(t1 )

x(t)

t

t

. The tra-

[t0 ;

) except at the

= 0 for all

t

0.

We call this class of systems dwell-time switched systems with a logarithmic partition. This class of systems falls into the dwell-time switched systems, and

hence the stability deÿnitions in Section 4.1 can be applied. 5.3.2

Problem formulation

The problem in this section is to design a dwell-time switched system with a logarithmic partition that stabilizes a given linear time-invariant plant. Consider the feedback system in Fig. 5.6. The pair (A; B ) represents the plant

where

x(t)

2 Rn

x(t)

=

is the state and

Ax(t) u(t)

+ Bu(t);

2R

(5.6)

is the control input; single input

is assumed as in the previous chapter. To avoid triviality, it is assumed that A

6

= 0 and

Besides of

A

is an unstable matrix, but that (A; B ) is stabilizable.

The controller is termed a dwell-time controller with dwell time

Rn

T,

the controller is deÿned in terms of a logarithmic partition

and a corresponding set

U f j gj2S =

u

u(t)

=

it

u ( );

so the plant equation becomes x(t)

=

Ax(t)

T

>

of control values. The decoder

in Fig. 5.6 is governed by the law

+ Bui(t) :

0.

fXj gj2S T

D

94

Chapter 5.

u(t)

-

Towards data rate reduction

x(t)

(A; B )

T

D

ÿ

T

C

i(t)

Decoder

ÿ

Coder

Figure 5.6: Dwell-time switching control scheme

This has the form (5.5), and therefore the switching logic of the preceding section deÿnes the coder

T mapping

C

x

to i.

We can now state the dwell-time switching control problem with the logarithmic partition: Find a dwell time

T >

Rn , and corresponding control values U

0, a logarithmic partition =

fuj gj2S

fXj gj2S

of

such that the closed-loop

dwell-time switched system in Fig. 5.6 is asymptotically stable. One historical solution to this problem is time-optimal control [13]. One could specify a control bound, say

juj ý 1, and then compute the time-optimal

bang-bang control to steer from an initial state to the origin. The control would switch a ÿnite number of times, with the state reaching the origin in ÿnite time (after which, the control is set to

u

= 0). One objection to this is a practical

one: The switching surfaces can be complicated, requiring great computational eþort to determine when the control should switch. By contrast, the switching surfaces in our setup are hyperplanes. Our solution is based on the results for the quantized sampled-data control systems. The system in Fig. 5.6 can be converted to a system similar to a sampled-data one with a sampler, a hold, and a logarithmic quantizer. Then, a suÆcient condition for stabilization, expressed in the two-dimensional

õ-ô

space, follows. Since this condition is the same as that in Lemma 4.3.3, the remaining construction follows directly from the main result for the logarithmic quantizer. 5.3.3

Hybrid automata representation

The dwell-time switched system with a logarithmic partition, deÿned in Section 5.3.1, is a fairly complicated example of a hybrid system, having real-time, or continuous, dynamics and discrete events. The objective of this section is to place this class of systems into the more general context of hybrid dynamical systems, so as to relate the terminology in the ÿeld to our systems. The hybrid automata model is one of the reasonably general classes of hybrid systems [86]. It is deÿned as a combination of state-space models described by diþerential equations for the continuous dynamics and ÿnite automata for the discrete dynamics. In the following, we express the dwell-time switched system with a logarithmic partition as a hybrid automaton.

5.3.

95

Dwell-time switching control

The discrete states for the dwell-time switched system correspond to the state partition cells, is inÿnite.

X[0 0] ;

and

X[

1] , j

j;ü

2Z

, and thus the discrete state space

So, strictly speaking, this system does not ÿt into the class of

hybrid automata. We emphasize that the purpose of this section is merely to represent our switching system in a diþerent format. We also note that the origin, contained in the cell

X[0 0] ;

, is a special state

since once the state trajectory comes there, no switching will ever occur. Thus,

X[0 0] X[1 0] f g X[2 0] X[0 0] r f g

we deÿne two subcells in

as

;

;

:=

;

:=

0

;

0

;

:

As a hybrid automaton, the dwell-time switched system can be described

LX AEIF

as a six-tuple (

;

;

in the following. Its

ûL

;

;

;

), where the deÿnitions of the symbols are listed

state transition graph

L f 2L

In the graph, each

ûX

locations

is the set of discrete states or =

=

R

n

ü R+

is shown in Fig. 5.7.

[1; 0]; [2; 0]; [j;

ö

1] :

j

2 Zg

:

is a vertex.

l

is the space of the continuous state

is the state of the system in (5.5) and

symbols

is the set of

given by

xc

= [x0 ; xT ]0 , where

x

is the timer state, which times

xT

the dwell time and is thus governed by

ûA 2L ûE

and is deÿned by

= 1.

xT

A f g =

al

l2L ,

where each symbol

al

represents the event that the discrete state transitions to a new state l

and serves as a label of the edges in the graph.

events

is the set of

or transitions, which are the edges. Each edge is

G J 2 L( 2 A X üf g G X ü 1 6 ÿöþ ý þ ýõ J 2R

deÿned by (l; al0 ; l; l

ll0 ;

0

ll0

;

=

;l

0

), where

al0

;

cl(

l0 )

T

cl(

l0

)

[T ;

x

=

xT

;

When the continuous state cell and dwell time from state

l

to

l

0

is

T

x

0

xc

û

The mapping

:

is in

if

l

= [2; 0];

if

l

= [1; 0]; [2; 0];

n

x

guard

enabled. During jump set

I L! :

);

; xT

G

ll0 ,

ÿ

6 ò

T

l

=

0

l ;

:

that is, when

x

is in a new

has passed since the last transition, the transition

J

state x ^c speciÿed by the reset to 0.

;

the transition,

xc

as (xc ; x ^c )

; the timer state is

2J

jumps to a new

2X speciÿes the subset of the state space in

which the continuous state

xc

must stay while the discrete state is l , i.e.,

96

Chapter 5.

a[2;0]

G

xT

[2;0][2;0]

Towards data rate reduction

:= 0

[2; 0] x _

=

f[2;0] (x)

=0

x _T

=1 location invariants x(t) Rn

2 2 [0; T ]

xT (t)

G

xT

[2;0][1;0]

xT

:= 0

[2;0]l

xT

a[2;0] al

[2;0]l 0

:= 0

x _

:= 0

=

x _T

f[1;0] (x)

a[2;0]

G

al 0

[1; 0]

G

l [2;0]

xT

G

a[1;0]

G

:= 0

l

=0

xT

=1

0

[2;0]

:= 0

location invariants x(t) = 0 xT (t)

l x _

=

x _T

:= 0

xT

G

fl (x)

+

xT

:= 0

a[1;0]

G

l

location invariants

2 I (l)

l

a[1;0]

l [1;0]

=1

xc (t)

2R

xT

G

ll

0

:= 0

al

al 0

0

[1;0]

x _

=

x _T

0

fl 0 (x)

=1

location invariants

G

0

xc (t)

l l

xT

2 I (l ) 0

:= 0

Figure 5.7: The hybrid automaton of the dwell-time switched system (l; l 0 f[j; ö1]

:

j 2

Z

g)

2

5.3.

97

Dwell-time switching control

xc (t)

2I

(l (t)) at all

t

ÿ

8 >:R ü [0 (R ü [0

0. It is deÿned as

+;

n

l

The set

I

ûF

;T

(l ) is called the

to occur when

xc

S ]) (X

; T ];

n

[j;ü1]

üR

+ );

location invariant

if

l

= [1; 0];

if

l

= [2; 0];

if

l

= [j;

ö

1],

j

2Z

of l . A transition is

is the set of functions given by

þ

xc

(t)

F f g =

l2L .

Fl

At each location

öþ

=

Fl

x(t)

ýõ þ =

xT (t)

fl (x(t))

state

xc (t)

t

2

,

1

:

At each

, the local trajectory is given by (l; xc ; ý ), where the continuous

evolves for

2

t

[0; ý ] and satisÿes xc (t)

for all

2L

ý

The trajectory of this hybrid automaton is deÿned as follows.

2L

l

= [x0 ; xT ]0 satisÿes the diþerential equation

ý

x(t) xT

l

enforced

violates the condition given by the location invariant.

the continuous state

location

:

xc (t)

2I

=

(l );

Fl (xc (t))

[0; Æ ). Then, the trajectory of the hybrid automaton is given as a

sequence of local trajectories accompanied with events: (l0 ; xc0 ; ý0 ) Here the events

f g alk

t1

!

al

1

!2

al

(l1 ; xc1 ; ý1 )

(l2 ; xc2 ; ý2 )

Z+ occur at switching times

k2

= ý0 ;

t2

and, at each switching time

= ý0 + ý1 ; tj ; j

t3

!3 þþþ

al

:

given by

= ý0 + ý1 + ý2 ; : : :

;

= 1; 2; : : : , the continuous state satisÿes two

conditions speciÿed by the guard and the jump as xc(j ý1) (tj ) xcj (tj )

2 Gþ ÿ1 lj

=

lj ;

xj ý1 (tj )

0

ý :

The discrete state space is not ÿnite, as in the deÿnition of hybrid automata. Nevertheless, the dwell-time switched systems can be well represented in this framework. As an automaton, the dwell-time switched system has some characteristics as follows. The automaton is deterministic and events

alk

are internally induced in the

following sense. At each location, the local trajectory is unique. Each time the location invariant is violated, there is only one guard that allows (and hence forces) one transition to occur.

Therefore, for any given initial continuous

state, the trajectory is unique and is deÿned on [0;

1

). Moreover, because of

98

Chapter 5.

Towards data rate reduction

the presence of dwell time, local trajectories are always deÿned with Only one switching, or an event, can take place in time

T,

ý

ÿ T.

and multiple events

at one time instant can not occur. As seen in the graph in Fig. 5.7, all the discrete states are connected to each other by edges in both directions except between [1; 0] and the rest. The states [1; 0] and [2; 0] are special. There is no possible transition from [1; 0], while only [2; 0] has a selþoop, as a result of the rather complicated switching law. Also, all transitions starting in [2; 0] are forced at

5.3.4

t

=

T.

Solution to the dwell-time switching problem

The approach taken here to solve the dwell-time switching control problem is to reduce the problem to a quantized sampled-data control one. We ÿrst obtain a system equivalent to the one in Fig. 5.6. We discuss its similarity to a sampled-data system; the main diþerence in the two setups is the sampling time. We then give a suÆcient condition for the stability of this equivalent system, which is identical to the one we used for solving the quantized sampleddata control problem. Thus, the solution for the dwell-time switching control problem follows immediately. Let us recap how the system in Fig. 5.6 evolves: If of the cell in which

x(0)

lies, then the control is

u(t)

j0

=

for

t

= 0 up to

ÿ T . At the ÿrst switching time t = t1 , the trajectory is in or enters Xj1 with j1 6= j0 , and the control switches to u(t) = uj1 for t 2 [t1 ; t2 ).

= t1 the cell t

This is repeated for all switching times cell

denotes the index

uj0

x(t)

is in, we have

u(t)

=

u ( ),

it

k . Denoting by j (t) the index of the t 2 [tk ; tk+1 ).

t

where i(t) = j (tk ) for

With this switching scheme in mind, we can convert the dwell-time switching system in Fig. 5.6 to that in Fig. 5.8(a), where

fQj gj

fqj gj

2S

Æ is the logarithmic

Q

R and output values ø R. Its output takes the form qj(t) , where j (t) 2 S is the index of the

quantizer in Section 4.3.3, deÿned by cells

2S

in

T 0 is the system that is governed by the dwell-time switching j t to i(t). In the decoder DT , DT 0 is the nonuniform zeroth order hold synchronized with CT 0 and is deÿned by DT 0 (i(t)) = qi(t) ; u ~0 is a cell

x(t)

is in;

C

logic, mapping

q ( )

scalar. Thus, the control input is u(t)

=u ~0 qi(t) = u ~0 QÆ (e

0

k ));

x(t

t

2 [tk ; tk+1 ):

Æ , the cells Qj can be obtained as Qj = ý1 from (4.32) and the output values as qj = u ~0 uj , j 2 S , from (4.30). Note that, for the quantizer

Q

0

e

Xj

This system is very similar to the sampled-data control system in Fig. 4.7.

Q = QÆ commutes with the uniform sampler T , and thus the system in Fig. 4.7 is equivalent to the one in Fig. 5.8(b), where QÆ and ST are interchanged.

Note that there the quantizer S

As we compare the two systems in Figs. 5.8(a) and 5.8(b), the uniform

T is replaced by CT 0 , which has a nonuniform sampling time, and T is replaced by DT 0 . We note that the switching in CT 0 is intermittent and slower than the sampling in ST . The discrete-time signal taking discrete values between QÆ and HT in Fig. 4.7 is represented now by i(t). sampler

S

likewise

H

5.3.

99

Dwell-time switching control

-

u(t)

ÿ

u ~0

ÿ

T0

D

it

q ( )

T

x(t)

~) (A; B

T0

C i(t)

ÿ

Æ

Q

jt

q ( )

ÿ

0

e

õ(t)

ÿ

T

D

C

(a) Dwell-time switching system

-

u(t)

x(t)

~) (A; B

?

u ~0

0

e

6

ÿ

T

H

ÿ

T

S

Æ

Q

ÿ

(b) Sampled-data system

Figure 5.8: Comparison of systems

We follow the design methodology for the quantized sampled-data system as summarized in the following. Fix a positive-deÿnite matrix

2 Rn

(4.5) and obtain the solution K

=

úR

ý1

0

B P,

and set

e

=

÷

P. 0

2 (0; 1).

n and

þ

Q

We design

e

as usual: Given

0, solve the Riccati equation

2 R1

þn be the LQR optimal one, ~ . Replace B = (1 + ÷) K B with B .

Then let

k k

K = K

R >

K

k k

The problem is reduced to designing the dwell-time switching controller so that the closed-loop system is quadratically stable with respect to (P; ÷J ).

þ) of the system, the decay rate of V (x) := x P x ý ú÷x (t)J x(t), t ÿ 0, where J = Q + P BR 1 B P . Recall

That is, for any trajectory is

V

(x(t); u(t))

0

x(

0

ý

0

that in the continuous-time system with the full state feedback (i.e., in Fig. 4.1(a), the decay rate is given by

V

(x(t); K x(t)) =

u

=

úx (t)J x(t).

K x)

0

The following lemma gives a suÆcient condition for quadratic stability of the system in Fig. 5.8(a) with a dwell-time switching controller characterized by

T >

0,

fXj gj

2S

, and

U

=

fuj gj

2S

. We make use of operator

so the result is expressed in terms of the

õ-ô

F

in (4.10),

plane. This key lemma is parallel

to Lemma 4.3.3 for the quantized sampled-data control problem. Lemma 5.3.1 Suppose the dwell-time controller is designed to have the fol-

lowing property: For every t

2

j

2 S , if x(0) is in Xj , then F (x(t)) 2 F X (uj ) for

[0; T ]. Then the dwell-time switched system in Fig. 5.6 is quadratically

100

Chapter 5.

Towards data rate reduction

stable with respect to (P; ÷J ).

Proof

Let j0 denote the index of the cell in which x(0) lies. By assumption, x(0), the trajectory x( ) of the switched system satisÿes

þ

starting at

F (x(t))

2 F X (uj0 )

for

t

2 [0; T ]:

By (4.12), this is equivalent to x(t)

This implies that

X

øX

2 X (uj0 )

Xj0 ø X (uj0 )

for

2 [0; T ]:

t

and therefore, since

that cl( j0 ) (uj0 ). By the deÿnition of switching times, if t1 x(t)

2 cl(Xj0 ) ø X (uj0 )

Hence, whether the switching occurs at

t

> T,

for

=

T

t

or

X (uj0 )

is a closed set,

then

2 [T ; t1 ]: t > T , x(t)

is in

[0; t1 ].

X (uj0 ) on

Repeating this argument, we obtain

2 X (ujk )

x(t)

where

jk

for

t

2 [tk ; tk+1 ]; k ÿ 1;

is the index of the cell which

x(t)

implies that the control Lyapunov function the deÿnition of

fX (uj )gj V

2S

V

is in or enters at

t

=

(x(t)) decreases for all

tk . t

This

ÿ 0 by

in (4.6); in particular, we have

(x(t); u(t)) =

V

(x(t); ui(t) )

ý ú÷x (t)J x(t): 0

ÿ

Hence, by Proposition 4.1.3, the system is quadratically stable.

With the logarithmic quantizer in the sampled-data system, asymptotic stability can be achieved by Corollary 4.4.4. The construction of the quantized sampled-data controller is based on the suÆcient condition in Lemma 4.3.3. For the logarithmic quantizer, this condition holds with r0 = 0. This is equivalent to the suÆcient condition above in Lemma 5.3.1, and hence the dwell-time control problem can be solved by constructing the logarithmic quantizer and sampling period

T

QÆ

as in Corollary 4.4.4.

We are ready to construct the stabilizing dwell-time controller. 1. Fix u ~0

÷

2 (0; 1). Select matrix Q and R > 0 in (4.5) that give ÿ > 1. ú 1)=ÿ.

Set

= (ÿ

2. Design a logarithmic quantizer as in Corollary 4.4.4: Take obtain

T >

0, cells

fQj gj

2S

, and output values

fqj gj

2S

.

Æ >

1 and

5.4.

101

Finite partition dwell-time switching

3. Obtain the state partition

fX g

j j 2S

through (4.32):

X00 = M = Ker ñ X = 2 R : 2 ö[ K;

ü

j

n

x

0

e x

j

Æ ;Æ

j

+1 )ô ;

j

2 Z+

:

Their corresponding control inputs are given by (4.30)

00 = u~0 q0 = 0; u =u ~0 q[ 1] ;

u

ü

j

j;ü

j

2 Z+

:

We state the stability result for the dwell-time controller as a theorem. For the dwell-time switching controller just deÿned, the closed-loop system in Fig. 5.8(a) is quadratically stable with respect to (P; ÷J ). Theorem 5.3.2

The proof of this theorem follows from Corollary 4.3.9 of Theorem 4.3.5; the proof of the theorem makes use of the suÆcient condition in Lemma 4.3.3, which is equivalent to the suÆcient condition in Lemma 5.3.1. The motivation to extend the quantized sampled-data controller to the dwell-time switching controller was to reduce the data rate. In fact, this scheme guarantees to decrease the number of message transmissions over the channel since the time between switchings is larger than or equal to T . We note, however, that the switching times can be any real numbers and thus i(t) can not be transmitted perfectly with a ÿnite data rate. In this respect, we consider our results to be theoretical, and this method may not be directly applicable using real control networks. To get a ÿnite data rate, one could time-sample i(t) appropriately or use an event-driven type protocol with some measures to take care of the possible time delay.

5.4

Finite partition dwell-time switching

The dwell-time controller of Section 5.3.4 employs a partition of the state space with inÿnitely many cells: The stringent requirement of x(t) 0 demands ÿner and ÿner resolution near the origin in the state space. In this section, we want to ÿnd a dwell-time controller deÿned by ÿnitely many cells such that all trajectories starting in a given bounded subset of the state space eventually go into the ball 0 (r0 ) of a prespeciÿed (small) radius r0 . This notion is called \practical stability" in [28]. In Subsection 5.4.1, we deÿne a state partition with ÿnite cells, similar to the logarithmic one. The dwell-time controller for this new partition employs a switching logic simpler than the one for the original partition. For the sake of completeness, we describe this switching law ÿrst. In Subsection 5.4.2, we present a stabilization result for this ÿnite partition case. The dwell-time switching control up to these subsections deals only with the state feedback case with no noise in the measurement. In Subsection 5.4.3, we extend our results to measurement noise rejection and then in Subsection 5.4.4 to output feedback control.

!

B

D

102

Chapter 5.

5.4.1

Towards data rate reduction

Dwell-time switched systems with ÿnite partitions

Repeating Subsection 5.3.1, we deÿne a

ÿnite logarithmic partition

bounded region

n

õj

:=

õ0 Æ

j

,

in the state space

2Z

j

R

n

subspace in

D

+.

spanned by

ü

j

X

ý

j

,

:=

n

Let

kk

with

0 and

õ0 >

of a given 1, and let

Æ >

M

= 1 and denote by

e

The partition cells are now deÿned by

n

x

:=

n

x

0

:

e x

:

0

(

õ0 ; õ0 )

;

[õj ; õj +1 )

e x

;

j

1 covers

;N

0

=

[0;0]

j

=

[j1 ;ü1]

and set the index set

N

. We rename the cells to

+ j

, and

as

ÿ

=

=

j

ÿ

00 ,

the

(5.7)

+:

large enough so that the union of the cells

N

= 0; 1; : : :

j

e.

2R

e

.

Xÿ f 2 R 2ú g X f 2R 2ö g 2Z 2N X X ú D X X X X X X X ú S f ú gü fú g 00

Here, we take

Then take

R

00 ;

ü

=

j1

0; 1; : : :

;

j1

= 0; 1; : : : 1

;N

1;

;N

1; 0; 1 . Both notations are

used throughout the section.

Every cell in a ÿnite logarithmic partition has interior, and thus the dwelltime switched systems equipped with such a partition can be deÿned in a simpler manner as follows. Consider the switching system x(t)

where

x(t)

2R

switching logic

n

=

dwell time j

2S

N

The switching logic determines = 0 and let

t0

x(t)

0 (x(t))

=

fj

(5.8)

2S

is the state of the system and i(t)

with a ÿxed

Lipschitz continuous for each Set

fi(t) (x(t));

j0

and also that i(t)

from

x(t)

f0 (0)

ÿ

x

= min

2

fÿ t

t0

+T :

ÿ

x

(t)

fj

is in. The system

1

). The ÿrst switching

2 X0 g =

int(

j

)

:

Then set i(t) = j0 for t [t0 ; t1 ) and j1 to be the index of the cell x(t1 ) cl( j1 ) with j1 = j0 .

2 X

6

Repeating this procedure, we obtain

f g tk

Z+,

k2

and the trajectory

þ

x(

þ

i(

is globally

= 0.

x(t0 )

, on [t0 ;

time t1 is given by t1

is the state of the

in the following manner:

be the index of the cell that

has a unique solution, say

N

0. We assume that

T >

) on [0;

1

X1 j

, where

), the switching times

) satisfying (5.8) for all

t

ÿ

0 except at the

switching times. These are all uniquely determined, and the switching logic is causal. Note that if

x(0)

= 0, then

equilibrium.

x(t)

= 0 for

t

ÿ

0, so the origin is an

This class of systems belongs to the dwell-time switched systems in Section 4.1. However, because of the ÿnite partition scheme, it is impossible to deal with asymptotic stability. In the following construction, we are concerned with attractive balls for such systems.

5.4.

103

Finite partition dwell-time switching

u(t)

n(t)

-

- ?d

x(t)

(A; B )

x ^(t)

DT

ÿ

CT

i(t)

Decoder

ÿ

Coder

Figure 5.9: Dwell-time control system

5.4.2

State feedback control

In this section, a stabilization problem for a linear time-invariant plant using a dwell-time controller with a ÿnite logarithmic partition is studied. Consider the feedback system in Fig. 5.9. The pair (A; B ) represents the plant x(t)

where

x(t)

2R

n

is the state,

=

u(t)

Ax(t)

2R

+ Bu(t);

is the control input, and

n(t)

2R

n

, is

the measurement noise. We assume the plant to be stabilizable. The dwell-time controller, composed of the coder is deÿned by three components: a dwell time

T,

fX g 2SN , and a corresponding set U := fu g 2SN j

CT

j

maps

j

x

to

i

j

CT

and the decoder

DT ,

a ÿnite logarithmic partition of control values. The coder

and is governed by the switching logic deÿned in the previous

subsection. The decoder is memoryless and is given by Here we consider the case

þ) õ 0, and,

n(

given

÷

DT (i(t))

=

ui(t) .

2 (0; 1), R0 ; r0

>

0, the

problem is to ÿnd a dwell-time controller such that, for the closed-loop system in Fig. 5.9, the ball

B0 (r0 ) is attractive from B0 (R0 ) with respect to (P; ÷J ).

This is a modiÿed problem of the design of dwell-time controller with a (countable) logarithmic partition in the previous subsection.

The stability

result relies on the techniques developed for the sampled-data control systems with ÿnite quantizers. Given

R0 ; r0

with

R0 > r0 >

0, we start the design procedure by following

Steps 1 and 2 in Section 5.3.4;

Æ >

parameters to be designed are

N

1 and

T

are determined in these steps. The

2 N and õ0 > 0. Then, deÿne the partition ÿ , X ü , j 2 f0; 1; : : : ; N ú 1g, as in (5.7) and their corresponding control cells X00 j

inputs as

ÿ = 0; ü = öu~ u

u00 j

ü

0 q[j; 1] ;

j

2 f0; 1; : : : ; N ú 1g:

The theorem below gives the parameters for the dwell-time controller that ensures stability.

104

Chapter 5.

Towards data rate reduction

Deÿne the dwell-time controller using

Theorem 5.4.1

õ0

N

=

p

pú

a

&

ÿ (a

s

+ 1)

ï s

Æ

=

ú1

ÿ

log

þmin (P ) þmax (P )

R0

þmax (P )

õ0

þmin (P )

r0 ;

!'

Then, for the closed-loop system in Fig. 5.9 with partition using

SN ,

the ball

B0 (r0 )

:

þ) õ

n(

0 and the ÿnite

B0 (R0 )

is quadratically attractive from

with respect to (P; ÷J ). The proof of this theorem follows directly from Corollary 4.4.4, in a manner similar to the countable partition case in Theorem 5.3.2. 5.4.3

State feedback under measurement noise

So far in our theory, perfect measurement with no noise has been assumed on the sensor side. However, in practice this is diÆcult to expect since the measurement is often obtained by a digital device and hence gives only quantized values, at a ÿner level than that of the coder. As a result, the measured state, denoted by x ^(t), may indicate a cell in which the actual state

x(t)

is not

contained. Although, if the magnitude of the noise is small, it is unlikely that this error can destabilize the closed-loop system, quadratic stability may not be guaranteed in the strict sense. Moreover, the noise can make it diÆcult for the system to maintain the small ball prespeciÿed in the design to be attractive. This motivates us to extend the design of dwell-time controllers for the case where measurement noise is present. Consider again the system depicted in Fig. 5.9, where x ^

:=

x+n

n(t)

is the noise and

is the measured state. We assume that the noise is bounded, that is,

k

for some

ký

ÿ

2

n0 > 0, n(t) n0 for t 0. The problem here is, given ÷ (0; 1), with R0 > r0 > 0, and n0 > 0, to design a dwell-time controller with a ÿnite partition such that 0 (r0 ) is quadratically attractive from 0 (R0 ) with

R0 ; r0

B

B

respect to (P; ÷J ). We propose a design method based on the ÿnite partition dwell-time controller. The design parameters here are design we obtain

Æ >

1 and

2 Rn ,

e

2

õ0 > 0 and N N . From the original among others. Now we deÿne a new

partition as follows. Given

n0 >

0, take

õ0 >

0 such that õ0 >

and deÿne j

õ0 := = 1; 2; : : : by

õ0

ú n0

and

j := Æõjý1 ;

õ

õ0

j

õ

Æ Æ

:=

:=

õ

+1

ú1 õ0

(5.9)

n0

+

n0 .

j ú n0 ;

Then deÿne

õ

j := õj ú n0 :

j

õ ;

j

õ ;

j

õ

for

5.4.

105

Finite partition dwell-time switching

fõj gj Z+

The new partition is now deÿned by the new set

2

control inputs are given by

00 := 0;

öu~0 õj ;

ü

ÿ

j :=

u

u

= 1; 2; : : :

j

and (5.7). The

:

We note that the cells deÿned above is smaller than those of the original partition in the previous subsection. In addition, we deÿne another set of cells, larger than those deÿned above, by

ñ

X 00 := fx 2 Rn : X j := x 2 Rn : ÿ

ü

ô

2 (úõ0 ; õ0 )g ; e x 2 ö[õj ; õj +1 ); 0

e x 0

;

j

= 1; 2; : : :

:

The dwell-time switching controller deÿned using this partition works similarly to the original one; the switching hyperplanes are only located diþer-

X 00 ÿ

ently. It is clear that the cells

and

Xj ; ü

= 1; 2; : : : , overlap and hence

j

do not comprise a partition. However, these two sets of cells guarantee that if

2 Xj then x(t) 2 X j for t ÿ 0 and j = 1; 2; : : : , and likewise with X00 and X 00 . This is the key characteristic used by the controller for our result in ü

ü

x ^(t)

ÿ

ÿ

this section. In the following, we refer to these two sets of cells as the ÿnite

(logarithmic) partition with overlapping cells . The next lemma gives a closed form for sequence is increasing.

ö

Lemma 5.4.2 (i) For j = 0; 1; : : : ,

õj

(ii)

0 > 0 and õj

õ

Proof

=

Æ

1

> õj ý

j

+1

0 ú n0 Æ ú 1 Æ

õ

for

j

õj

=

õ

fõj gj Z+

and shows that this

2

+ n0

Æ Æ

+1

ú1

:

(5.10)

= 1; 2; : : : .

(i) By deÿnition,

ú n0 = Æõj 1 ú n0 = Æ (õj 1 ú n0 ) ú n0 = Æõj 1 ú n0 (1 + Æ ): õj

ý

ý

(5.11)

ý

So the closed form is õj

=

0 ú n0 (1 + Æ)

j

Æ õ

X1 jý

=0

l

= (ii) By deÿnition of

0

õ

j

Æ

l

ú1

j

0 ú n0 (1 + Æ) Æ ú 1 : Æ

Æ õ

and the assumption on +1

0

õ

in (5.9),

0 = õ0 ú n0 > Æ ú 1 n0 ú n0 ;

õ

Æ

106 but since j

Chapter 5.

Æ >

1 we have

õ0 >

Towards data rate reduction

0. It is also clear from (5.11) that

õj > õj

= 1; 2; : : : .

ý1 for

ÿ

We must show now that the new partition is suÆcient to make the ball

B0 (r0 ) attractive from B0 (R0 )

under the presence of suÆciently small noise.

The main concern is to ensure that starting in

r0

is large enough so that all trajectories

B0 (R0 ) eventually go into and stay in B0 (r0 ).

Theorem 5.4.3 Assume that n0 is small enough that

n0 <

pT

ú

pT

ú

a

p

ÿ (a

ÿ

T

ú1

+ 1)

Let

p

s

ú1

s

þmin (P ) Æ

ú1

þmax (P )

2Æ

þmin (P )

(5.12)

r0 :

ú n0 ; q ù (P ) 2 3 max ú n0 ÆÆý+11 R0 ù (P ) min 7 N = 6logÆ ; 66 7 Æ+1 õ0 ú n0 7 Æý1

õ0

=

a

ÿ (a

T

ÿ

+ 1)

þmax (P )

(5.13)

r0

(5.14)

and deÿne the dwell-time controller using the ÿnite partition with overlapping

kn(þ)k ý n0 , the ball B0 (r0 ) is quadratically attractive from B0 (R0 ) with respect to (P; ÷J ). cells. Then, for the closed-loop system in Fig. 5.9 with

The proof of this theorem is similar to that of Theorem 5.4.1, which is the case without noise. In fact, if

n0

= 0, the results coincide.

Inequality (5.12) gives a lower bound on the radius trajectories go in for the given bound

n0

r0

of the ball that all

on the noise. In particular,

n0 < r0 =2.

As mentioned above, the new partition cells are smaller than the original ones; this is observed in (5.10). Thus, to cover the same bounded region

D,

the partition with overlapping cells requires more cells. We can conclude that our result on noise rejection is achieved at the expense of data rate.

5.4.4

Observer-based output feedback

In this section we are concerned with the output feedback case of the dwelltime controller design. It has been an underlying assumption in our dwell-time controller that computation is possible on both ends of the channel, that is, in the coder and the decoder, and that the data rate is limited in the channel. Thus, in the case of output feedback, we may as well put a standard observer on the sensor side so that the estimated state can be used for quantization. Then, the error between the actual state

x

and the estimated state x ^ can

be treated as measurement noise, and we can use the results in the previous subsection.

5.4.

107

Finite partition dwell-time switching

-

u(t)

y (t)

(A; B; C ) u(t)

T

D

ÿ

ÿ

T

C

i(t)

Decoder

x ^(t)

?

ÿ

Observer

Coder

Figure 5.10: Output feedback dwell-time control system

In Fig. 5.10, the output feedback dwell-time control system is given. Here the triple (A; B; C ) represents the plant whose state space equations are

where

x(t)

2 Rn

x(t)

=

Ax(t)

y (t)

=

C x(t);

is the state,

u(t)

2R

+ Bu(t);

is the control input, and

y (t)

2 Rp

is

the output. We assume that (A; B ) is stabilizable and (C; A) is detectable. The standard observer design is possible because of the detectability: There exists

L

2 Rnþp

such that

A

ú LC

is a stable matrix. Then the observer can

be realized as x ^(t)

where x ^(t) we have

2 Rn

z (t)

=

Ax ^(t)

+ Bu(t) + L (y (t)

is the estimated state. Let

= (A

ú LC )z (t).

z

:=

ú C x^(t)) ; x

ú x^ be the error, and then

The problem in this section is to ÿnd a control strategy based on the dwell-

÷ 2 (0; 1), and R0 ; r0 > 0, for the system in B0(r0 ) is quadratically attractive from B0 (R0 ) with respect to some

time controller such that, given Fig. 5.10,

(P; ÷J ). Hence, we want to ÿnd the parameters for the ÿnite partition with

2

õ0 > 0, N N , and the size of the overlap n0 > 0. The diÆculty in the use of an observer is that the two states may grow

overlapping cells:

large during the transient response due to the error between them. Thus we have to ÿnd a partition that covers the region in the state space through which the estimated state may travel. Our (naive) control strategy is (i) to wait and not to apply any control until the error becomes small enough, and then (ii) to start the dwell-time control so that, from this point, the state will go into a small ball centered at the origin in a quadratic manner. We describe this strategy formally and show that it is indeed stabilizing. Take

n0 >

0 such that

n0 <

and then set

õ0

Æ

ú 1 apT ú

2Æ

p

ÿ (a

ÿ

ú1

T + 1)

s

þmin (P ) þmax (P )

as in (5.13). Set x ^(0) = 0. Since

z

r0 ;

is the state of a stable

108

Chapter 5.

system, there exist

az >

0 and

Towards data rate reduction

0 such that

þz >

kz (t)k ý a eý z kz (0)k for t ÿ 0: ù t

z

Now take

T0 >

0 such that R1

N

:=

ý

ùz T0

az e

1

kAk + þ

2 := 6 66log

R0

ý n0 , and then let

az R0

q

z

R1

kLC k(1 ú eý

ùz T0

ùmax (P ) ùmin (P )

Æ

3 77

);

ú n0 ý+11 7: Æ

Æ

ú n0 +1 ý1 Æ

õ0

(5.15)

Æ

(5.16)

(5.17)

Theorem 5.4.4 For the dwell-time switching controller using the ÿnite par-

tition with the overlapping cells that we just deÿned, if the control input given by

( u(t)

then

=

is

2 [0; T0 );

0;

if

ui(t) ;

otherwise,

t

u

B0(r0 ) is quadratically attractive from B0 (R0 ) with respect to (P; ÷J ).

Proof Ax ^(t)

Since

u(t)

= 0 for

2

t

[0; T0 ) and x ^(0) = 0, the observer x ^(t) =

+ LC z (t) has a solution on [0; T0 ) as

Z

x ^(t)

t

=

ý

A(t

e

ü)

LC z (ý ) dý :

0

Now let R2

We claim that R0 ,

R1

= max

kx^(t)k:

2[0;T0 ]

t

in (5.16) gives an upper bound on

R2 .

From (5.15),

and x ^(0) = 0, we have

kz (ý )k ý a eý ý a eý z z

and thus R2

ý ý =

Z max

2[0;T0 ]

t

1

kAk + þ

=

R1 :

R0 ;

e

kAk(týü ) kLC ka eýùz ü R

az R0 z

kx(0) ú x^(0)k

kAk(týü ) kLC kkz (ý )k dý

t

0

ùz ü

e 0

max

2[0;T0 ]

t

Z

t

ùz ü

0 dý

z

kLC k(1 ú eý

ùz T0

)

kx(0)k ý

5.4.

109

Finite partition dwell-time switching

Hence we must ÿnd a ÿnite partition with overlapping cells that covers

E0 of V containing B0 (R1 ) is ô ñ E0 = x 2 Rn : V (x) ý R12 þmax (P ) :

B0 (R1 ).

The smallest level set

Therefore, similarly to Theorem 5.4.3, the number of cells to cover by

N

E0 is given

in (5.17).

x(t) = x ^(t) + z (t) for t ÿ T0 with kz (t)k ý n0 , x(t) B0 (r0 ) eventually by construction, following Theorem 5.4.3. ÿ

We can see that since goes into

As described above, we have taken the viewpoint that some computation resources are available on the sensor side. This leads us to the use of an observer; to take account of the error between the actual state and the estimated one, our control strategy is to wait until the error becomes small and then to start control employing the partition with overlapping cells. This partition has been shown to handle small noise in the measurement. A stabilization problem for the quantized output feedback case is also considered in [28]. Since the design of an observer gain is a dual problem to that of the feedback gain, an observer incorporating a quantizer is proposed: One of its inputs is the quantized error between the actual output and the estimated one; the estimated state is then sent to the quantizer as in our method. A question that arises here is why we would need an extra quantizer in the observer. On the other hand, the output feedback stabilization in [12] using a time-varying quantizer relies on the existence of

K

such that

A

ú BK C

is

stable. However, the static output feedback problem has not been solved.

5.4.5

Cart-pendulum system example

In this section, we apply the dwell-time switching controller to the familiar cart-pendulum system in Fig. 5.11. Here, the cart is of mass x,

and the force

of mass

m,

u

length l , and angle

2 3 2 x 0 66x77 660 4 ø 5 = 40  ø

We set input

M

u

and position

0

= 2 kg,

m

1

ø.

The linearized state space model is

32 3 2 x 6 6 x7 07 7 6 7 +6 5 4 5 4 ø 1

0

0

0

úmg=M

0

0

0

(M + m)g=M l

= 0:4 kg, and

l

0

ø

3 7 7 5 u:

0 1=M 0

ú1=M l

= 0:5 m. The states

x

and

ø

and the

have units of m, radians, and N.

First,

was designed using the result in Section 4.6. We used

ÿ

÷

= 0:1;

ò

= 12:3;

and obtained

ÿ

in (4.51) on

T and dT over all

plot of

M

is applied by an actuator. On top of the cart is a pendulum

Æ >

= diag(50; 10; 0:1; 0:1)

= 15:0. For this setup, the maximum

c

1 versus maximum

parameters.

~ Q

Æ >

T.

T

meeting the bounds

1 was about 0:026. Figure 5.12 shows a

We can see the trade-oþ between these two

110

Chapter 5.

m

ü u

Towards data rate reduction

l

M x

Figure 5.11: A cart with an inverted pendulum

T

Ideally, chose

T

and

Æ

should both be as large as possible. As a compromise, we

= 0:021 and

Æ

= 1:97. For

0

õ

follows:

= 0:1, the resulting parameters are as

ú0:0644 ú 0:101 ú 0:971 ú 0:208] ; fõ0; õ1 ; õ2 : : : g = f0:1; 0:197; 0:386; : : : g; fu0 ; u1 ; u2 ; : : : g = fö0:146; ö0:287; ö0:564; : : : g; X00 = fx 2 R4 : e x 2 (úõ0 ; õ0 )g; Xj = fx 2 R4 : e x 2 ö[õj ; õj+1 )g; j 2 Z+: e

ü

ü

0

=[

ü

ÿ

0

ü

0

û B û ).) (Note that the state space is of (A; In the following, we show simulation results for three diþerent dwell-time switching systems: (A) the original system using the ÿnite partition as deÿned above, (B) the system using the partition with overlapping cells for noise rejection control in Section 5.4.3, and (C) the system with an observer from Section 5.4.4. A. Dwell-time control with ÿnite partition:

The time response plots

are given in Fig. 5.13 for the initial condition [x(0) We used

0

x(0) ø (0) ø (0)]

0

= [0 0 0:1 0]

:

0 = 0:1 so that the smallest control input u+ 0

õ

is visible in the plot.

Figure 5.13(a) shows the responses in solid lines. The number of cells needed for this initial condition and

0

õ

is

N

= 11, but only 9 of them were used in

these plots. The ideal response for full state feedback (u =

K x,

no switching,

no state partition) is in dashed lines. Although the transient responses in these plots look similar, the behaviors in the steady state diþer; while the state of the optimal system goes to the origin, that of the system with the dwell-time controller, as expected, only

5.4.

Finite partition dwell-time switching

111

0.03

Max dwell time T

0.025

0.02

0.015

0.01

0.005

0 0

10

5

30

25

20

15

δ Figure 5.12: Trade-oþ between Æ and maximum dwell time T

stays around the origin. We see a periodic change in the state and also in the control input due to the dwell-time switching. We can say from these plots that the proposed controller is eÆcient in partitioning the space of the signals transmitted from the sensor to the actuator without causing much distortion in performance.

Here, as mentioned, the

response required only 9 cells, and it is fairly close to the optimal one, which virtually requires signals of real numbers. To get the state closer to the origin, we could make õ0 smaller, though this might result in a larger N . We next calculated responses for a larger dwell time without changing the rest of the controller. The case for T = 0:126 is shown in Fig. 5.13(b) in solid lines.

The response is stable though not necessarily quadratically stable; we

don't know how conservative our result is for quadratic stability with respect to (P; ÷J ).

However, compared to the plots in Fig. 5.13(a), we may say that

the transient response is more oþ from the optimal one.

For larger T , the

system became unstable.

B. Noise rejection control:

We compare the diþerence in the performance

between the system using the original partition and the one using the partition with overlapping cells. To make the measurement noise as malicious as possible

ý1=2 e, where n

in putting x and x ^ in diþerent cells, we used n(t) = n1 (t)Qú

ún0 ; n0 ] and n0

uniformly distributed over [

= (Æ

1 (t)

is

ú 1)=(Æ + 1)õ0 ü 0:7 = 0:0228.

This noise is in the direction of e after a change in coordinate using some matrix Qú . This noise is bounded as

ý1=2 ek ô 0:045.

kn(t)k ý n0 kQú

Then, the

112

Chapter 5.

Towards data rate reduction

0.2

x

0.1 0 −0.1 0 0.1

2

4

6

8

10

12

2

4

6

8

10

12

2

4

6

8

10

12

θ

0.05 0

−0.05 0 10

u

5

0 −5 0

time t (a) For

T

= 0:021

0.2

x

0.1 0 −0.1 0 0.1

2

4

6

8

10

12

2

4

6

8

10

12

2

4

6

8

10

12

θ

0.05 0 −0.05 0 10

u

5 0 −5 0

time t (b) For

T

= 0:126

Figure 5.13: Time responses for dwell-time control

5.4.

Finite partition dwell-time switching

113

dwell-time controller using the partition with overlapping cells is deÿned using

fõ0 ; õ1 ; õ2 ; : : : g = f0:1; 0:129; 0:186; : : : g; fuü0 ; uü1 ; uü2 ; : : : g = fö0:113; ö0:155; ö0:238; : : : g: In the simulations, the dwell time of T = 0:105 was used (thus, no guarantee for quadratic stability) to make the diþerence between the two systems more apparent. Time responses for the two systems are given in Fig. 5.14, for the system with the original partition in (a) and for the system using the one with overlapping cells in (b).

In each ÿgure, the responses of position x, angle ø,

and control u are shown. The initial condition is

[x(0) x(0) ø(0) ø(0)] = [0 0 0:1 0]:

The numbers of cells used were 9 for Fig. 5.14(a) and 14 for Fig. 5.14(b). We can see that, after an initial transient, the system with the original partition exhibits a larger change in the cart position compared to the other system; the magnitude in the change of ø seems to be slightly larger as well. This is because, closer to the origin, the cell sizes are smaller, and thus with the original partition the noise can easily aþect the measurement to give a wrong cell index, whereas the proposed system can take account of such noise.

C. Observer-based control:

For the system in Fig. 5.10, we assumed that

the sensor measurement is available only for x and ø, i.e.,

þ

C =

ý

1

0

0

0

0

0

1

0

:

2 R4þ2 for the observer so that the eigenvalues of A ú LC were placed at ú0:2; ú0:15; ú0:4, and ú0:3, and obtained 2 3 0:575 0:00499 60:0736 ú1:96 7 L = 6 4 0:611 0:475 7 5: We designed L

0:149

23:6

Thus the time constant for the observer error is about 5 seconds. very large, but is chosen so that errors are visible in the plots.

This is

Moreover,

of course, we don't need to estimate x and ø since these are available in the measurement; however, for the sake of presentation, we used all the estimated states for control. We calculated time responses for three diþerent systems for the initial condition

[x(0) x(0) ø(0) ø(0)] = [0:01 0 0:1 0]:

(x(0) is diþerent from the previous simulations so that there is some error between x and x ^ initially.)

In Fig. 5.15, the continuous-time case with no

114

Chapter 5.

Towards data rate reduction

0.2

x

0.1 0 −0.1 0

5

10

15

20

5

10

15

20

5

10

15

20

0.1

θ

0.05 0 −0.05 0

15

u

10 5 0 −5 0

time t (a) The original partition

0.2

x

0.1 0 −0.1 0

5

10

15

20

5

10

15

20

10

15

20

0.1

θ

0.05 0 −0.05 0

15

u

10 5 0 −5 −10 0

5

time t (b) The partition with overlapping cells Figure 5.14: Time responses under measurement noise

5.4.

115

Finite partition dwell-time switching

1

x

0.5 0 −0.5 0

5

10

15

20

5

10

15

20

10

15

20

θ

0.2

0

−0.2 0

u

10

5 0 −5 0

5

time t Figure 5.15: Time responses for observer-based continuous-time control

quantization is shown as a reference. For

x and ø, the actual states are in solid

lines and the estimated ones in dashed lines. The error between them becomes small fairly quickly.

Figure 5.16(a) shows the responses for the dwell-time

control system using the original partition, and Fig. 5.16(b) shows those for the system using the one with overlapping cells. For these systems, we simply used

u(t) = ui(t)

small.

for all

t ÿ 0, i.e., T0 = 0, since the number of cells used was

Compared to the noise rejection plots, the diþerence between the responses of the two dwell-time control systems is much smaller. This is because, by the time the actual state and the estimated state come close to the origin, the error between them is not large enough to put them into diþerent cells. However, both plots are comparable to that of the continuous-time one, and this fact is supportive for our approach using an observer. The numbers of cells used are 9 for the original partition case and 12 for the case with overlapping cells. Because the responses are similar, in the latter case we had to pay the price for using the more expensive partition.

116

Chapter 5.

Towards data rate reduction

1

x

0.5 0 −0.5 0

5

10

15

20

5

10

15

20

10

15

20

θ

0.2

0

−0.2 0 10

u

5 0 −5 0

5

time t (a) Dwell-time control using the original partition

1

x

0.5 0 −0.5 0

5

10

15

20

5

10

15

20

10

15

20

θ

0.2

0

−0.2 0

10

u

5 0 −5 0

5

time t (b) Dwell-time control using the partition with overlapping cells

Figure 5.16: Time responses for observer-based dwell-time control

Chapter 6

Extensions for the multiple input case In this chapter, we extend the limited data rate control methods for the singleinput plant in Chapters 4 and 5 to the multi-input plant case. The development is in a similar order: ÿrst the sampled-data control, then the quantized sampled-data, and ÿnally the dwell-time switching control. Other problems dealt in the previous chapters can be extended as well, but are not studied here.

6.1

Quadratic stabilization of sampled-data systems

In this section, we design a sampled-data controller to stabilize a continuoustime plant with an

m-dimensional

input. The stability criterion is quadratic

stability in the continuous-time domain with a guaranteed decay rate.

6.1.1

Problem formulation

The continuous-time system in Fig. 6.1(a) is the

m-dimensional

input version

of the one in Fig. 4.1(a). The linear time-invariant plant (A; B ) is given by the state equation x(t)

where A

x(t)

2 Rn

is the state, and

Ax(t)

u(t)

+ Bu(t);

2 Rm

is the control input. Assume that

6= 0 and A is unstable, i.e., it has one or more eigenvalues with nonnegative

real parts, that rank B = K

=

2 Rmþn

m,

and that (A; B ) is stabilizable.

The matrix

is a stabilizing state feedback gain, so all eigenvalues of

A

+

BK

have negative real parts.

H. Ishii and B.A. Francis (Eds.): Limited Data Rate in Control Systems with Networks, LNCIS 275, pp. 117−160, 2002.  Springer-Verlag Berlin Heidelberg 2002

118

Chapter 6.

-

Extensions for the multiple input case

x(t)

u(t)

(A; B )

K

ÿ

(a) Continuous-time system

u(t)

-

x(t)

(A; B )

?

0

U

v (t)

K

6

ÿ

T

H

T

S

d (k )

õ(t)

ÿ

õ

(b) Sampled-data system

Figure 6.1:

By the stability of

A + BK ,

the system is quadratically stable with respect

2

J Rnþn , n þ n there exists a positive-deÿnite solution P R of the Lyapunov equation 0 (A + BK )0 P + P (A + BK ) = J . Now let V (x) := x P x, x Rn . For a

to some pair (P; J ). For example, given a positive-deÿnite matrix

trajectory

þ

x(

:

V

V

u

=

ú

) of the system, the decay rate of

The derivative

With

2

K x,

Rn ü Rm

!R

of

V

(x; K x) =

2

(x(t)) is determined by

J:

along the trajectories is given by

V

(x; u) := (Ax + Bu)

we have

V

ú

0

Px

0

+x

P (Ax

+ Bu):

(6.1)

0

x J x.

Now, as in the single-input case, we turn our interest to the sampled-data system in Fig. 6.1(b). This represents a simple setup where the communication channel between the sensor and the actuator has a ÿnite data rate. Here,

T is the uniform sampler with a sampling period

S

T :õ

S

The

zeroth order hold

d : õd (k ) := õ(kT );

õ

T is given by

H

T : õd

H

7!

7!

v

:

v (t)

:=

d (k );

õ

t

2

k

2 Z+

T:

:

[kT ; (k + 1)T );

6.1.

Quadratic stabilization of sampled-data systems

and U0

2 Rmþm .

119

Observe that

u(t) = U0 K x(kT )

for t

2 [kT ; (k + 1)T ); k 2 Z+:

This sampled-data system is one type of dwell-time switched system in (4.1):

Let fu(t) (x(t)) = Ax(t) + Bu(t) and

S

=

Rm .

there is always dwell time T between switchings. then u(t) = 0 for t

ÿ tk ,

and thus x(t) = 0 for t

Clearly, f0 (0) = 0 and

Furthermore, if x(tk ) = 0

ÿ tk .

Hence, the stability

deÿnitions in Section 4.1 can be applied. We are interested in ÿnding the largest sampling period while the closedloop system maintains stability with respect to the same Lyapunov function 0

V (x) = x P x at a certain decay rate. The follows: Given ÷

sampled-data control problem

2 (0; 1), ÿnd a sampling period T

is as

and a matrix U0 such that

the closed-loop sampled-data system in Fig. 6.1(b) is quadratically stable with respect to (P; ÷J ). Since the problem for the single-input case is solved in Section 4.2, one might argue that Heymann's lemma [32] is applicable: be controllable and b

2 Im B, we can ÿnd a matrix K0

Assuming (A; B) to such that the single-

input plant (A + BK0 ; b) is controllable; then stabilization results should be immediate from those for the single-input case.

This is not possible under

our assumption that feedback loops can be closed only over a ÿnite data rate channel.

6.1.2

Generalization of the setup

In the single-input sampled-data control problem, we have taken the approach to use V

as a control Lyapunov function and look for a controller so that for

any tra jectory V decreases at all times. In this section, we generalize the setup given in Subsection 4.2.2 for the m-input case. Here as well, we limit the class of K to LQR optimal ones.

This is an

important technical assumption, but may not be such a restriction on K . If the matrix R is diagonal, the optimal gains are known to have robustness properties such as an inÿnite gain margin and a fairly large phase margin [2,43]. These are helpful in our design for stabilization with quantization and delay. Given positive-deÿnite matrices Q

2 Rn

n

þ

and R

2 Rm

m,

þ

let P

2 Rn

n

þ

be the unique positive-deÿnite solution of the Riccati equation 0

A P + PA

The LQR optimal gain is K := 0

ú P BR

úR

ý1

ý1

0

B P + Q = 0:

(6.2)

0

B P and let the control Lyapunov func-

tion be V (x) := x P x. The continuous-time system in Fig. 6.1(a) is quadratically stable with respect to (P; J ), where J := Q + P BR Fix ÷

2 (0; 1).

For u

2

Rm ,

u decreases V :

X (u) :=

n x

we denote by

2 Rn

:

ý1

0

B P.

X (u) the set of states for which

V (x; u)

ý ú÷x J x 0

o :

(6.3)

120

Chapter 6.

Extensions for the multiple input case

This can be expressed as

X

f 2 Rn

(u) =

x

0

:

0

x K R[(1

ú ý ú

+ ÷)K x

2u]

by (6.1) and (6.2). To simplify the problem, we select loss of generality. Let the subspace

M

Rn

of

M

Note that since rank B =

M

dim

basis

f1 2 Rn

M

. Let

was

=

E

e

=

þþþ k k

:= [e1

E

0

m]

e

.

K = K

?

=

e ;::: ;e

of

g ú

÷)x Qx

= 1=(1

Q

(6.4)

; ÷)I

m.

= Im K

m.

0

(6.5)

:

Hence, there is an orthonormal

m g ø Rn

þ

In the single-input case, the notation

The parameter ÿ in Subsection 4.2.2 is generalized to the matrix S deÿned by S

:= (1 + ÷)E

X

We now obtain the following form for

2 Rm

Lemma 6.1.1 For u

X

(u) =

ñ

Eõ

+y :

õ õ

, , , ,

+y

2X

0

0

0

0

0

K R[(1

õ E K R[(1 0

õ S 0

õ

ñ

?

; y

(S

(S

and

0

I )õ

2õ

y

2M

?

õ

ú

+ ÷)K (E õ + y )

+ ÷)K E õ

ú ú I )õ

ú ôý 1 ý

ý

[(1 + ÷)K E ] 0

2õ

S [(1

(6.6)

(u).

2 Rm 2 M ú ú

(u)

(E õ + y ) 0

õ

2 Rm

m

þ

K RK E :

;

S [(1

Corresponding to the decomposition

there exist unique Eõ

0

2 Rm

,

0

Proof

without

be

:= (KerK )

m,

0

(1

õ õ 0

2u

õ õ

1u

ý

+ ÷)K E ]

=

such that

2u]

0

2u]

Rn

ú ý +y

0

+y

0

x

=

Eõ

y

?

, for any

2:

y

2M

?

U

and

0

2 Rn

,

0

E E S)

=

I)

ÿ

with

0 := (K E ) 1 U0 K E =(1 + ÷):

~ U

x

(by (6.4))

(by the deÿnition of

~ := (1 + ÷)BK E and replace For simplicity, we use B

:

+ y . Then

0

(since

ýk k y

M÷M

ý k k2ô

(E õ + y ) (E õ + y )

y y

1u

ý

+ ÷)K E ]

ý

(6.7)

A system equivalent to the original one in Fig. 6.1(b) is shown in Fig. 6.2. In this setup, the control input becomes u(t)

~ E 0 x(kT ) for =U 0

t

2

[kT ; (k + 1)T ),

k

2 Z+

.

(6.8)

6.1.

121

Quadratic stabilization of sampled-data systems u(t)

-

x(t)

~) (A; B

?

0

~ U

E

6 T

H

ÿ

ÿ

T

S

0

õ(t)

Figure 6.2: Equivalent sampled-data system ~ ), In this system, for (A; B

X

(u) =

This set

X

ñ

Eõ

+y :

õ

X

(u) is

2 Rm 2 M ; y

?

0

; õ

ú ú

(S

I )õ

components in the state space. Hence, we deÿne

þ

F (x)

Notice that

X

Su

ý k k2 ô y

:

(u) plays a crucial role in our controller construction. We see in its

expression above that, for stabilization, parameters

F

0

2õ

õ

ÿþ ý

x

where

;

y

=

õ

0

E x

preserves norm. The image of

F

õ

(u) =

Clearly, for

ý

kk

=

2 Rn

:

ô

õ

x

2

2X

Rm ;

(u)

We refer to the image space of

F

ô

ÿ

,

0;

0

õ

and

: y

kk ! Rm+1 ú

and

Rn =

(S

ú ú I )õ

are the essential

y

by

E õ.

x

(u) under

F

F

0

2õ

is Su

ý

2 ô

ò :

(u):

space ;

õ-ô

õ

X

2 X

F (x)

as the

F

(6.9)

it is of (m + 1)-dimension.

A suÆcient condition for the stability of the system is given by the following lemma. Lemma 6.1.2 If, for any initial condition x(0) F (x(t))

2 X F

~ E (U 0

0

x(0))

2 Rn 2

for

t

, the trajectory

x

[0; T ];

satisÿes (6.10)

then the closed-loop system in Fig. 6.2 is quadratically stable with respect to (P; ÷J ).

2

By (6.9), the condition in (6.10) is equivalent to

Proof

for

X 0 Xþ t

[0; T ].

~ (U

E x(kT ))

( ),

V

0

x(t)

2X

~ E 0 x(0)) (U 0

Since this holds for any initial condition, we have for

t

2

[kT ; (k + 1)T ); k

(x(t)) decreases at all times: V

~ E 0 x(0)) (x(t); U 0

2 Z+

ýú

.

x(t)

2

Thus, by the deÿnition of

0

÷x(t) J x(t):

122

Chapter 6.

Extensions for the multiple input case

ÿ

This implies quadratic stability with respect to (P; ÷J ). A crucial assumption made in the 1-input case is clear later that

S > I

ÿ >

X (u) becomes what is known as a hyperboloid 2 S > I and u 2 R . Given ô ÿ 0, we call the set ÿ þ ý ò õ õ : 2 F X (u) ô

Then,

1. It will become

is an important condition we have to make as well.

of one sheet [76]. An

F

example plot is given in Fig. 6.3 for

the projection of

F

X (u)

onto the

õ-space

for

ô.

sets in the following as well. Note that for each ý1 ellipsoid with a center at õ = (S I) S u.

ú

We use this term for other ô

ÿ

0 the projection is an

2

β

1.5

1

0.5

1 0.5 0

0 −1

α

2

−0.5 −0.5

0

0.5

α

1

−1

1

Figure 6.3:

F

X (u):

a hyperboloid

For future use, we give its alternative form

ÿþ ý X (u) = õô : õ 2 Rm ; ô ÿ 0; î ð î 1 õ ú (S ú I ) Su (S ú I ) õ ú (S ú I )

F

ý

0

1

ý

ð

Su

ý u S (S ú I ) 0

1

ý

Su

+ô

2

o :

(6.11) 6.1.3

Bounds on trajectories

In this section, we present a result on bounds for the state trajectories during the sampling period.

6.1.

123

Quadratic stabilization of sampled-data systems

For

T >

0, let cT

Note that

:= max

2[0;T ]

t

!

cT ; dT

Suppose

cT <

0 as

ó At óe T

1. Let

!

ú

ó Ió ;

F~

2R

m

s

ÿþ

û (u) :=

2R ý 2 R +1

and

m

u

õ ô

dT

0, and since

aT

For õ ~

ó óZ t ó ó Aü ~ ó := max ó e B dý ó ó: t2[0;T ] 0

1

2

:=

c

T

A

ú

6

= 0, if

:

ô

ÿ k ú ký 0;

õ

0, then

cT >

0.

1:

(6.13)

, let

m

T >

(6.12)

õ ~

1

ö

kk õ ~

aT

+ô+

dT cT

õò

kk u

:

(6.14) This is a circular cone with an axis

õ

=õ ~ , as shown in Fig. 6.4.

2

β

1.5

1

0.5

1 0.5 0

0 −1

−0.5 −0.5

0

0.5

α1

The

m-input

−1

1

Figure 6.4: A ball inside the cone

α2

F~

û (u)

counterpart of Proposition 4.2.3 is given in the following.

2

2M 2 2

þ

? . Let x( ) be the 0 = E õ~ + y~, õ~ R ; y~ trajectory starting at x(0) = x0 with a constant control u R . Then

Proposition 6.1.3 Suppose x

x(t)

Moreover, if

cT <

2B

x0

kk kk

(cT ( õ ~ +

1, then F (x(t))

y ~

2 F~

) + dT

û (u)

for

kk

m

u

t

2

)

for

[0; T ]:

t

m

[0; T ]:

(6.15)

(6.16)

124

Chapter 6.

Proof

It follows that

kx(t) ú x0

óó k = óóe ó ý óe

At

At

Extensions for the multiple input case

Z x0

t

+

e

Aü

0

úI

óó þ k

x0

~ dý u B

óóZ k + óó

ú x0

t

óó óó

~ dý B

Aü

e 0

ý c kE õ~ + y~k + d kuk ý c (kE õ~k + ky~k) + d kuk = c (kõ ~ k + ky ~k) + d kuk T

T

T

t

part.

2 [0; T ].

k

u

T

T

for

óó óó þ k

T

0

The last equality holds since

=

E E

I.

Thus we obtain the ÿrst

The second part is shown in two steps.

Step 1 Proof

F (x(t))

For

B

2 B( ~

û;ky ~k) (cT

ñ ñ =

Eû ~ +~ y (r )

=

= Now, by

T

ñ

Eõ

+y :

Eõ

+y :

Eõ

+y :

ô

k(E õ + y) ú (E õ~ + y~)k2 ý r2 ô kE (õ ú õ~)k2 + ky ú y~k2 ý r2 (since E (õ ú õ ~ ) ? (y ú y ~)) ô 2 2 2 kõ ú õ~k + ky ú y~k ý r :

jkyk ú ky~kj ý ky ú y~k, B

Thus, if

k k ky~k) + d kuk) for t 2 [0; T ].

( õ ~ +

0, we have

r >

x(t)

Eû ~ +~ y (r )

2B

F (x(t))

ø

ñ

Eõ

Eû ~ +~ y (r ),

2

ÿþ ý õ ô

:

+y :

kõ ú õ~k2 + jkyk ú ky~kj2 ý r2

then

kõ ú õ~k

2

+

jô ú ky~kj ý r 2

2

ò =

ô

:

B( ~

5

û;ky ~k) (r ):

To prove (6.16), we must show that the ball in Step 1 is contained in the cone

F ~ (u). û

Step 2

If

1,

cT <

0 [ )=@ B

F ~ (u û

~) (û; ~ ø

k k

(cT ( õ ~

~) + d +ô T

1 ÿþ ý \ k)A 2R

ku

õ

m+1

ô

~ú0 ø

ò

:

ô

ÿ0

:

(6.17)

Proof cT

k k

First we show the inclusion

~) + d ( õ ~ +ô T

kuk.

B( ~ ~) (r) = û;ø

ó

Compare the cone

ÿþ ý õ ô

:

ÿ 0, and let F ~ (u) in (6.14) and the balls

~ in (6.17). Fix ô û

kõ ú õ~k

2

+

jô ú j ý r ~2 ô

2

ò

:

r

:=

6.1.

125

Quadratic stabilization of sampled-data systems

It is clear that, for each

ô

ÿ 0, the projection of these sets onto the õ-space

are balls. Thus, to show that the ball above is in the cone, we can deal only with their radii and show

ö

õ2

d kõ~k + ô + T kuk

1

T

c

Here under the assumption 1 2

T

a

ö

T

c

d kõ~k + ô + T kuk

= = =

ø

1 2

T T

a

c

þø

1

1

ú c2T 1

1

ú cT

ÿ 0:

r

h

ú c2T

Thus, inclusion

2

h

÷2

ú ô~)

+ cT (ô

Tr

c

2

1

1

+ cT (ô

r

2

2

ô

ÿ 0:

ú ô~)2 ú r2

+ (ô

T

c

for all

1

<

õ2

=

ÿ r2 ú jô ú ô~j2

T

2

a

ú

~) ô

+ 2cT r (ô

+ (ô

÷2

ú ô~)2 ú r2

+ (1

ú cT )(ô ú 2

ú ô~) + (ô ú ô~)2

~)2 ô

ú (1 ú cT )r 2

ý 2

i

i2

T r + (ô ú ô~)

c

(6.18)

ó

~ holds in (6.17). Furthermore, for every ô

ÿ

0 there is

such that equality holds in (6.18). Hence, the equality in (6.17) follows. A plot of the ball in Step 1 and the cone for

m

ô

5ÿ

Fû~ (u) in (6.14) is given in Fig. 6.4

= 2. They are tangent to each other as shown in Step 2.

6.1.4

Solution to the sampled-data problem

The sampled-data control problem is now reduced to a problem in the

õ-ô

space of (m + 1)-dimension. In Lemma 6.1.2, the suÆcient condition for stabilization is that the trajectory starting at

x(0)

has an image under

F

inside a

hyperboloid during the sampling period. On the other hand, Proposition 6.1.3 gives a bound for the trajectories characterized by a cone in the

õ-ô

space.

This observation leads us to the following suÆcient condition.

T

Proposition 6.1.4 Suppose c

<

1. If for every õ ~

2 Rm

Fû~ (U~0 õ~) ø F X (U~0 õ~);

(6.19)

then the closed-loop system in Fig. 6.2 is quadratically stable with respect to (P; ÷J ). Proof x(0)

The condition (6.19) is equivalent to the condition that, for every

2 Rn ,

FE x(0) (U~0 E x(0)) ø F X (U~0 E x(0)): 0

0

0

126 Since

Chapter 6.

T

c

<

Extensions for the multiple input case

1, Proposition 6.1.3 implies that for every

x(0)

2 Rn

the trajectory

of the system satisÿes

2 F X (U~0 E x(0)); 0

F (x(t))

2 [0; T ]:

t

By Lemma 6.1.2, it follows that the system is quadratically stable with respect

ÿ

to (P; ÷J ).

The sampled-data control problem is now reduced to a geometrical one of ÿnding

T such that a cone is inside a hyperboloid in an (m + 1)-dimensional m , the cone ~ õ space. We have seen that, for a ÿxed õ ~ û~ (U 0 ~ ) and the ý1 ~ ~ hyperboloid F (U0 õ ~ ) have axes õ = õ ~ and õ = (S I) S U0 õ ~ , respectively. Hence, to obtain the largest T , it is clearly advantageous to put them on the

2R

X

F

ú

same axis. To do so, set ~ U 0

=

S

1

ý

(S

ú I ):

(6.20)

We are now ready to construct the stabilizing controller. 1. Fix

÷

2 (0; 1).

Select

and

Q

R

so that

S > I.

In Section 6.4, we discuss

this condition and give a method to obtain such eigenvalues of ÿ2

S

k

by

ÿ þ þ þ ÿ ÿm > 1.

2. Select

T >

= 1; 2; : : :

ÿ ;k

; m,

Q

and

R.

Denote the

in nonincreasing order:

ÿ1

ÿ

0 small enough that

T

<

T

ý

c

d

p

1

c

ÿ

;

8s < ( T m mú1: +1

ÿ1

ÿ

m ú 1)(aT

ÿ

úÿ ú1 2

ÿ1

9 = 1 + 1) ú 1;

(6.21)

:

(6.22)

The main theorem of the section follows. Theorem 6.1.5 The closed-loop system in Fig. 6.2 with T deÿned above is

quadratically stable with respect to (P; ÷J ). Proof

Since

must show

ÿ1 >

1,

c

T

p

1=

<

2 in (6.21). Thus, by Proposition 6.1.4, we

Fû~ (U~0 õ~) ø F X (U~0 õ~ ) for all õ ~

2 Rm .

Fix õ ~ and let r (ô ) c0 (ô )

:=

1

ö

T

a

0

:= õ ~ (S

kõ~k + ô + T cT d

ú I )õ~ + ô 2 :

k

~ õ U 0~

(6.23)

õ

k

;

6.1.

Quadratic stabilization of sampled-data systems

for

ÿ 0.

ô

Then from (6.14)

F

~ õ û~ (U 0 ~) =

whose projection onto the

F

X

~ õ (U 0 ~) =

ÿþ ý õ ô

ÿþ ý

ò

õ

:

ô

õ-space

:

is a ball for each

ÿ 0;

ô

ÿ 0; kõ ú õ~k ý r(ô )

ô

Ec0 (ø) := fõ 2 Rm

: (õ

lent to

Also from (6.11)

ú õ~) (S ú I )(õ ú õ~) ý c0 (ô )

(õ

ô

ò

;

ÿ 0, namely,

ú õ~) (S ú I )(õ ú õ~) ý c0 (ô )g : 0

ý

2

r (ô )

ÿ 0.

ÿ 0.

Bû~ (r(ô )) ø Ec0 (ø) for all ô ÿ 0.

Then for (6.23) we must show

ô

ô

;

0

whose projection is an ellipsoid for each

for all

127

c0 (ô ) þmax (S

This is equiva-

(6.24)

ú I)

This inequality can be shown as follows.

Observe c0 (ô )

and

r (ô )

=

ý =

ý =

1

T

a

1

T

a

1

ö

ú I )õ~ + ô 2 ÿ þmin (S ú I )kõ~ k2 + ô 2 2 2 = (ÿm ú 1)kõ ~k + ô 0

=õ ~ (S

õ

þ

kõ~k + ô + T kU~0õ~k cT d

óó ö T

kõ~k + ô + T cT d

þ

kõ~k + ô +

d

I

1

úS ú

ý1

1

óó k ~ký õ ý õ

~0 = (by U

S

ý1

(S

ú I) = I ú S

õ

a

T

a

ô

c

ÿ

ÿ

ô

a

2

ÿ1

õ

ÿ1

ÿ

a

ÿ1

2

ú1

)

kõ~k

m 2 0Ts 1 3 ( m ú 1)( T ú + 1) 1 4k ~k + + @ ú 1A k ~k5 ú1 T 0 s 1 1 ( m ú 1)( T ú + 1) @ + k ~kA T

a

ý1

ÿ1

õ

:

(from (6.22))

128

Chapter 6.

Extensions for the multiple input case

Now, using these inequalities, we have

c0 (ô)

ÿ

ú ú (ÿ1

mú

(ÿ

ö =

1

2

0

1)r(ô)

kk

2

1) õ ~

+ô

2

s

ú úT @ ÿ1

1

ô +

2

a

m

(ÿ

ú

ÿ1

2 q ÷m ý a ý÷ õ ÷ ý T ÷ ý ú úT 64 ú aT ú÷ý ÿ1

1

2

a

1

2

1

1)( 2

(

1 1 1 1 a2

ô

1

It follows from (6.21) that a

2

T

ú

1 +1)

T

T ú ú

1)(a

2

1

3

k k7 5

1 k kA

2

ÿ1 + 1)

õ ~

2

õ ~

:

(6.25)

ÿ1 + 1 > 0 and therefore that the right-hand

ÿ

side in (6.25) is nonnegative. Thus, (6.24) follows.

ú

T

in (6.22) is the same with that

1)=ÿ.

Hence, this theorem generalizes

Notice that for m = 1 the bound on d ~ = u in (4.21) for the case U ~0 = (ÿ 0

Theorem 4.2.5. However, the proof is much simpler. This is mainly because we don't deal with the cones and hyperboloids themselves here. The pro jections of these ob jects on the õ-space are merely balls and ellipsoids for each ô and are

Fû

therefore easy to work with. Nevertheless, T given in the result may be again conservative since the cone

~ (u), which serves as a bound on tra jectories in

the construction, may be larger than necessary.

6.2 6.2.1

Quantized sampled-data control Problem formulation

In this section, we study the eþect of quantization in sampled-data systems by generalizing the single-input case studied in Section 4.3. We ÿrst give the

Aj ø R

deÿnition of quantizers for the m-dimensional case.

þþþ ü Am ø Rm

Given m bounded intervals is a bounded

S

=

deÿned by Q(õ) = q

j

if õ

2 Qj 2 S ;j

Aü 1

Q \ Qj ; 6 6 2 Qj m quantizer R ! f j gj2S

Rm , (ii) 0 2 Q0 , and (iii) for j = 0, 0 m with q = 0, a and fqj gj 2S ø R 0

with a countable

Qj

fQ g

m partition j j 2S of R is a set 1 index set = Z+ such that (i) i

Deÿnition 6.2.1 A

[j2S Qj fQj gj2S

, j = 1; 2; : : : ; m, we say the set

rectangle .

.

= cl(

One simple and commonly used quantizer is the

j

are rectangles of equal size and each q

of bounded rectangles =

, i = j , and

). Given a partition

Q :

q

uniform quantizer,

is the midpoint of

Qj

2

is

where

. We remark

that it is not certain if uniform quantizers are eÆcient in quantization for the purpose of control.

1 We use S = Z+ here for deÿniteness, but may use other countable sets for S elsewhere. 2 The notation Q is used for both a positive-deÿnite matrix in (6.2) and a quantizer. In

the following sections, we note whenever confusion may arise.

6.2.

129

Quantized sampled-data control

-

u(t)

~) (A; B

x(t)

?

0

~ U

E

6 T

H

ÿ

Q

ÿ

0

ÿ

T

S

Figure 6.5: Quantized sampled-data system

In Fig. 6.5 is shown a sampled-data system that has a quantizer

Q

added

to the system in Fig. 6.2. Observe that u(t)

~ Q(E 0 x(kT )) for =U 0

t

2 [kT ; (k + 1)T ); k 2 Z+:

The closed-loop system is a dwell-time switched system if we take ~ q j := U 0 j

(6.26)

u

and f

f

j (x(t)) = Ax(t) + Buj for

0 (0) = 0 and it follows that if

j

2 S.

x(kT )

By the deÿnition of quantizers, clearly

= 0 then

u(t)

=

0 = 0 for t ÿ kT .

u

Thus,

we can use the stability deÿnitions in Section 4.1. Extending the sampled-data problem, we want to know how large

T

can be while the stability of the closed-

loop system does not change with respect to (P; ÷J ). For quantizers that have the origin as an interior point of the cell

Q0 ,

asymptotic stability is not possible. We therefore aim at designing a quantizer for a looser stability in our problem. Since qj ; j ~ = S ý1 (S quantizer, we ÿx U I ) as in (6.20).

ú

0

2 S,

are designable in the

We now state the quantized sampled-data control problem . Find a quantizer Q

and a sampling period

from

Rn

T >

0 such that, for some

r

0 ÿ 0, B0 (r0 ) is attractive

with respect to (P; ÷J ) for the closed-loop quantized sampled-data

system in Fig. 6.5.

Again, the radius r0 of the attractive ball is not prespecifed in this problem,

though as a design problem it is preferred to be so. In general, this turns out to be diÆcult. There are, however, examples of quantizers that allow us to choose

6.2.2

0

r

prior to the design.

Solution to the quantized sampled-data problem

In the sampled-data control problem, a suÆcient condition is expressed in the õ-ô

space. We generalize the approach to the quantized control case, where

the control input takes only discrete values. In this section we give a suÆcient condition for closed-loop stability, an extension of Lemma 6.1.2.

130

Chapter 6.

For

and

E

:

Q

Rm

Extensions for the multiple input case

!f g qj :

X

f 2R f 2R

j :=

=

n

x

n

x

Rn

Such sets form a partition of

X 2S

j 2S , let

qj

space that is mapped to

j, j

0

:

Q(E x) 0

:

E x

, be the subset of the state

= qj

2Q g

g

(6.27)

j :

and are referred to as

state partition cells .

In

terms of state partition cells, the quantized sampled-data controller works as follows: If

x(kT )

2X

j then u(t) = uj for t

2

[kT ; (k + 1)T ),

k

2 Z+

.

Now the idea used in Lemma 6.1.2 is extended to the state partition case. Lemma 6.2.2 Suppose the quantized sampled-data controller is designed to

have the following properties: (i) There exists for

t

(ii) For

2 2S 6 ,

j

j

1

r

[0; T ].

ÿ

0 such that if

= 0, if

x(0)

2X

x(0)

2 X0

Rn

with respect to (P; ÷J ), where

s

r

x(0)

j , then

t

2

x(kT )

2 B0

(0)

(r1 );

[0; T ].

(r0 ) is quadratically attractive

(6.28)

V

2 XX [ B0

2 Z+

(0)

(r1 );

from Lemma 4.3.2. Lemma 4.3.2.

= 0;

if

j

= 0;

2S

, if

6

Rn .

Thus, denoting the cell

6

if jk = 0;

. This means that at each

t

ÿ

0 either the control

(x(t)) is decreasing, by the deÿnition of

=

.

j

j

if jk = 0;

(ujk );

E0 f 2 R

E0

if

(r ) here is in

(r1 ). The smallest level set containing

stays in

2

max (P ) : min (P )

(uj );

(

[kT ; (k + 1)T ]; k

Lyapunov function is in

t

(r1 )

is in by jk , we have

x(t)

t

(uj ) for

(0)

þ

2 XX [ B0 B0 1

[0; T ]. Note that the ball

index which

for

F

þ

0 = r1

(

x(t)

for

F

2 X [B0

It follows from the assumption and (6.9) that for every

2X

Proof

B0

F (x(t))

2 X

j then F (x(t))

Then, for the closed-loop system in Fig. 6.5, from

then

x

n

:

V

(x)

ý

B0

(r1 ) is

max (P )r12

þ

( ), or else

x(t)

g

This is an invariant set, and hence

Finally, the smallest ball that contains

Xþ

x(t)

E0 B0 is

goes into and

(r0 ), again by

ÿ

6.2.

131

Quantized sampled-data control

The similarity of the quantized version of the sampled-data problem is obvious now to the original problem. This problem too is fully described in the

õ-ô

space. The main diþerence between Lemmas 6.1.2 and 6.2.2 is how

the initial states are classiÿed: õ ~ =

0

E x(0)

in Lemma 6.1.2 and

x(0)

Lemma 6.2.2.

2 Xj

in

Fû~ (u) gives a bound on the trajectories 2 M under a constant control u 2 Rm . To continue the parallel discussion, let us deÿne the set F j (u); j 2 S ; u 2 Rm , In Proposition 6.1.3, the cone

starting on

x(0)

=

Eõ ~

+

?

y; y

Q

by

F

[ Fû

(u) :=

Qj

~ (u):

û~ 2Qj

This gives a bound on the trajectories starting in

Xj

by Proposition 6.1.3.

We can now reduce the suÆcient condition for stability to the following one. Proposition 6.2.3 Suppose cT < 1. If

(i) there exists (ii) for

j

r1

ÿ 0 such that F

2 S , j 6= 0, F

Qj

Q0

(0)

ø F X (0) [ B0 (r1 ), and

ø F X (uj ),

(uj )

then, for the closed-loop system in Fig. 6.5, from

Rn

Proof

with respect to (P; ÷J ), where By the deÿnition of

F

Q0

r0

B

0 (r0 ) is quadratically attractive is as in (6.28).

(0), the condition (i) implies that, for every

x(0)

2 X0 ,

x(0)

2 Xj , j 6= 0, F (x(t)) 2 F X (uj ).

the trajectory of the system satisÿes

2 X

[B

F (x(t)) F (0) 0 (r1 ). Similarly, it follows from the condition (ii) that for every trajectory starting at

Thus, by Lemma 6.2.2, the ball

is quadratically attractive.

In the single-input case,

F

Qj

B0 (r0 )

ÿ

(u), which is a union of cones, is another

cone. Thus, the stabilization problem can be easily reduced to ÿtting a cone inside a hyperboloid. This strategy cannot be generalized to the straightforwardly. Although

F

Qj

(uj ); j

2 S,

m-input

case

is a union of circular cones and

is a cone too, it is not circular. The key idea in our construction is to use a circular cone that contains

F

Qj

(uj ).

This simpliÿcation, of course, may

introduce additional conservativeness in the result. For a ÿxed

ô

ÿ

F j (uj ) 2 Rm and

0, a rough sketch of the projection of

Q

on the

ÿ 0, Fû~ (uj ) is a ball centered at õ = õ~, and the radius of this ball is a linear function of kõ ~ k and ô , as in (6.14). Thus, in the plot, the projection of F j (uj ) is a union of balls whose centers are in Qj ; the ball centered at the corner of Qj furthest from the origin has the largest radius. õ-space

for

m

= 2 is given in Fig. 6.6. Recall that for õ ~

the projection of the cone Q

We denote this radius by

j (ô ).

r

ô

132

Chapter 6.

Extensions for the multiple input case

þ2 qj

rj (ÿ )

Qj qjþ

FQj (uj ) Cj (uj ) þ1

0

FQj (uj ) and Cj (uj ) on the õ-space

Figure 6.6: The projections of

To avoid dealing with the complicated set circular cone that contains it. Let

we use a larger but

2 cl(Qj )g; q := arg minfkõk : õ 2 cl(Qj )g: j q

Since 0 in

Qj :

2 Q0 , q0 = 0.

j := arg maxfkõk :

FQj (uj ),

Let

ÿ = 0, and for j 6= 0 let qjÿ 2 Rm

q0

q

see Fig. 6.6. The radius

r

j (ô ) =

Cj (uj ) :=

ÿþ ý õ ô

:

be the midpoint

ÿ j := 2 (q j + q j ); 1

j (ô ) corresponds to q j , and hence from (6.14)

r

Finally, let

õ

ô

1

ö

T

a

õ

kqj k + ô + T kuj k cT d

:

ÿ 0; kõ ú qjÿ k ý kqj ú qjÿ k + rj (ô )

This is a circular cone with an axis

õ

=

ÿ j . For the ÿxed

q

ô

ò :

(6.29)

ÿ 0, the projection

of this set is a ball, represented by the large circle in Fig. 6.6. It can be easily veriÿed that

Cj (uj ) ó FQj (uj ):

(6.30)

Now the two conditions in Proposition 6.2.3 will follow from the stronger conditions

6.2.

(i)0 there exists (ii)0 for

(S

133

Quantized sampled-data control

j

r1

ÿ 0 such that C0 (0) ø F X (0) [ B0(r1 ), and

6= 0, Cj (uj ) ø F X (uj ). C

X

The cone j (uj ) and the hyperboloid F (uj ) have axes õ = qjÿ and õ = ý1 I) S uj = qj , respectively. Thus, in this strategy, the natural choice of

ú

j is

q

ÿ j = qj :

The use of

q

(6.31)

Qj

(uj ) greatly simpliÿes the problem. How-

Cj (uj ) instead of F

ever, of course, this introduces conservativeness, which was not present in the

single-input case. Thus, the solution in the following is not a strict generalization of that in Section 4.3. We are now ready to construct a stabilizing quantized sampled-data controller.

2

1. Fix ÷ (0; 1). Select matrices ~ = S ý1 (S U I ). 0

fQj gj

2. Given 3. For

ú

2S

, set

fqj gj

2S

Q

and

R

in (6.2) so that

S > I.

Set

as in (6.31).

2 S ; j 6= 0, select, if it exists, Tj > 0 small enough that

j

T

c j <

ý

T

d j

1

p

(6.32)

;

8s < ( m T mú1: ÿ1

ÿ

+1

9

2 kq k + aTj kqj ú qj k = m ú 1)(aTj ú ÿ1 + 1) ú j ;: ÿ1 ú 1 kqj k

ÿ

c j

ÿ

(6.33) If all 4. If

j exist, set

T

T >

T

= inf j 6=0 Tj .

0, let

s

r0

=

kq0 k

þmax (P )

pÿ

1 (aT + 1)

þmin (P ) a

T

ú

p

ÿ1

ú1

(6.34)

:

~ ) and the feedback gain We remark here that, for the plant (A; B quantizer

Q

not exist or

fQj g; fqj g; fTj g).

is characterized by the triple ( T

Clearly,

K, T

the

may

= 0, in which case a controller of our type does not exist for the

given quantizer. We call all quantizers that yield

T >

0

stabilizing

quantizers.

For such quantizers, we have the main theorem. Theorem 6.2.4 For the closed-loop system in Fig. 6.5 with the quantized

sampled-data controller just deÿned, the ball from

Rn

with respect to (P; ÷J ).

B0 (r0 ) is quadratically attractive

134

Chapter 6.

Proof

Since

ý

T

and

d

Extensions for the multiple input case

T and dT in (6.12) are nondecreasing function of

T

T, c

c

T

d j.

In Proposition 6.2.3, the

= 0 case and the nonzero

j

ý

j

T

c j

case

have diþerent conditions, and hence the proof is divided into two steps. The ÿrst step is for

Step 1

= 0.

j

Set

Then

F

have

Q0

F

Since

ÿ

q0

+ 1)

q0

1

ÿ1

a

:

(0) 0 (r1 ). = 0 by deÿnition, it follows that

= 0 from (6.26). Thus from (6.30)

u0

F

pT ú k k Tú ÿ1 (a

ø X [B

(0)

Q0

Proof

p

=

r1

ÿþ ý

øC

(0)

õ

0 (0) =

:

ô

ô

ÿ k ký 0;

= 0 in (6.31). Then we

ö

1

õ

q0

+

ô

T

a

1+

1

õ

ò

k k q0

T

a

On the other hand, from (6.11)

F

X

ÿþ ý õ

(0) =

:

ô

õ

2 Rm ÿ

0

0;

; ô

õ

(S

ú

I )õ

ý

ô

2

:

(6.35)

ò :

This is a hyperboloid and hence, by Lemma 4.3.2, the largest cone inside F

X

(0) is given by

C

ÿþ ý õ

ÿ

0 :=

:

ô

õ

2

Rm ;

ô

ÿ

0;

ò p ÿ úkk ø X 1

ÿ1

ô

õ

F

(0):

(6.36)

Observe that

F

Q0

(0)

ø X [B F

(0)

,F

0 (r1 )

Q0

(0)

\ X (F

(0))

c

øB

0 (r1 ):

Thus, from (6.35) and (6.30), it is suÆcient to show

C \ C cøB C C C C ÿþ ý ò 2 Rm r ÿ

0 (0)

( 0)

0 (r1 ):

(Figure 6.7 shows the side view of cones

0 (0) and

C

ÿ

( 0

)c .)

The intersection of the boundaries of õ

ô

for some

ô

r1

>

1

:

ÿ 0 ; the dark area is

0 (0) and

C \ 0 (0)

ÿ 0 is in the hyperplane

õ

p ú

p ú

0 and is a sphere. (In Fig. 6.7, this set is the two intersecting

points.) This set always exists because Let [õr1

(6.37)

T

c

<

1=

ÿ1

1 implies

a

T

>

ÿ1

0 r1 ] be one of the points in this set. Then, by direct calculation,

ô

óóþ ýóó óó r1 óó = k r1 õ ô

q0

k T úp

óóþ óó p 1ú1

ýóó ó= 1ú1 ó

(aT + 1)

a

ÿ

1

ÿ

r1 :

1.

6.2.

135

Quantized sampled-data control

ÿ

ÿ

=

aT kþk ÿ (aT

+ 1)kq 0 k

C0 (0) C0þ ÿr1

ÿ

=

p

ý1 ÿ 1 kþk

þr1

0 Figure 6.7: Cones

þ

C0 (0) and C0

ÿ

5

Hence, (6.37) follows.

2 S is as follows. Step 2 For j 2 S , j 6= 0, F j (uj ) ø F X (uj ). The result for nonzero

Proof

j

Q

From (6.30), it suÆces to show

Cj (uj ) ø F X (uj ):

(6.38)

Let cj (ô )

0

:= qj (S

ú I )qj + ô 2 :

We then have from (6.29) and (6.11)

Cj (uj ) = F

X (uj ) =

The projections of rj (ô ))

and

ÿþ ý õ

ÿþ ý ô

õ ô

: :

Cj (uj )

ô

ÿ 0; kõ ú qj k ý kqj ú qj k + rj (ô )

ô

ÿ 0;

and

Ecj (ø) := fõ 2 Rm

F

(õ

ú qj ) (S ú I )(õ ú qj ) ý cj (ô )

ÿ 0.

õ-space

are

ò :

Bqj (kqj ú qj k +

ú qj ) (S ú I )(õ ú qj ) ý cj (ô )g ; 0

respectively. So for (6.38) we must show ô

;

0

X (uj ) onto the

: (õ

ò

Bqj (kqj ú qj k + rj (ô )) ø Ecj (ø) for all

By Lemma 4.3.2, this is equivalent to (ÿ1

ú 1)(kqj ú qj k + rj (ô ))2 ý cj (ô ) for all ô ÿ 0:

This condition can be shown directly as follows.

(6.39)

136

Chapter 6.

Extensions for the multiple input case

Observe

ú

0 j (ô ) = qj (S

c

ÿ

2 j +ô

I )q

ú )k j k + m ú 1)k j k +

þmin (S

I

= (ÿ and

j (ô ) =

r

ý =

ý

ö

1

T

k jk +

þ

q

a

k jk +

1

T

q

þ

a

k jk +

1

q

T

T cT

d

+

ô

ô

+

ô

+

T cT

d

q

2

q

T cT

ô

ô

2

2

õ

k jk u

ý óó ó 1ó I úS kqj k ö õ ý ý

d

2

ú

1

(by

~ q = (I j =U 0 j

u

ú

S

ý1

)qj )

k jk

1

q

m

2 s 3 (ÿm ú 1)(a2 ú ÿ1 + 1) T 4ô + kqj k ú aT kqj ú qj k5

a

1

T

a

ÿ1

ÿ

ú1

(from (6.33)): From these inequalities, it follows that

ú ( ú 1)(k j ú j k + j ( )) 2 s ÿ ( m ú 1)k ~k + ú ú 1 4 + ( m ú 1)( úT ú1 j (ô )

c

ÿ1

q

ÿ

õ

2

q

ô

2

r

ÿ1

T

2

÷1 ý1 a2T

q

1

T

2

T

a

ú

ÿ1

+1

>

a

ÿ1

(÷m ý1)(a2 T ý÷1 +1) ÷1 ý1

a

From (6.32),

ÿ

ô

2

a

2 ö õ ÿ1 ú 1 64ô ú = 1ú

2

ô

ú ÷aT

1 ý1 2

2

32

k jk7 q

32 ÿ1 + 1) kqj k5

5

:

0 and thus the right-hand side is nonnegative.

5ÿ

Therefore, (6.39) follows. There are two sources of conservativeness on the size of

T

in this design.

One is, as in the sampled-data control problem, the size of the cone The other is from the use of the cone

Cj (

j ) instead of

u

F

Fû(

~ u).

Qj (uj ) though, as

mentioned above, this convention greatly simpliÿes the proof. We examine two speciÿc quantizers in the following.

Uniform quantizers Given ù

>

0, the

m-dimensional

version of the

uniform quantizer

Qj

[ 1

þö

jm ] =

j1

Qþ

is

S = Zm, the cells õ ö õ õ þö õ ö õ õ 1 1 1 1 ú 2 ù + 2 ù ü þþþ ü m ú 2 ù m + 2 ù

deÿned by the index set

;

j1

j

;

j

6.2.

137

Quantized sampled-data control

ý2

ý1

Figure 6.8: Uniform partitioning of

(see Fig. 6.8), and the output values

j1 jm ]

q[

2 3 j1 6 . 7 = 4 . 5ù .

j

for [j1

R2

m

þ þ þ jm ] 2 S .

For this class of quantizers, we have a corollary of Theorem 6.2.4. Corollary 6.2.5 For m = 1; 2; 3, given r0 > 0, select T > 0, if it exists, small

enough that c

T

T

d

<

p

1 +1

;

8s r p 9 < = 2 (ÿm ú 1)(aT ú ÿ1 + 1) m a m ÿ c ý mT : ú 2+ ú T ;; ÿm ú 1 ÿ1 ú 1 4 2 ÿ1

and set

s ù = 2r0

þmin (P ) a þmax (P )

T

ú

pÿ

p

ÿ1

ú1

1 (aT + 1)

:

Then, for the closed-loop system with the uniform quantizer T

and ù, the ball

(P; ÷J ).

B0 (r0 ) is quadratically attractive from Rn

(6.40)

(6.41)

(6.42) Qþ deÿned by with respect to

138

Chapter 6.

Proof

Set

for every

j

=

Tj

T.

2 S , j 6= 0.

óó ó

Extensions for the multiple input case

We must show that the bounds (6.32) and (6.33) hold Clearly, (6.40) implies (6.32) for all such

j.

Notice

ð 3ó ó2 î õ2 !1=2 m ö óó óóó6 j 1 j +. 21 ù 7óóó ïX 1 ù jm ] ó = ó4 óó î .. 1 ð 5óóó = k=1 j k j + 2 óó j2m3j +óó 2 ù óó óó ù 61.7óó ùp jm ] ó = ó óó 2 4 .. 5óóó = 2 j

q

[j1

;

j

j

óó ó

q

[j1 jm ] ú q[j1

m

:

1

So the bound in (6.33) for [j1

Zm+ .

þ þ þ jm ] 2 S is identical to that for [jj1 j þ þ þ jjm j] 2 m j 2 Z+ , j = 6 0. For

Thus, it suÆces to show that (6.33) holds for every

such

j,

the second term in the bound in (6.33) is

0óó ó k jk + T k j ú jk 1 Bó = @ k jk k k óóó p m q

a j

q

q

q

Its maximum over when, e.g., with

j

j

= [1 0

= [1 0

þþþ

holds for every Since

j

j

j

j

0

213óó ó 1 6 . 7ó + 4 .. 5ó + ó 2 1 ó

aTj

p

m

2

1 C A

p 2 Z+ , j 6= 0 is 2 + m=4 + aTj m=2 and is obtained þ þ þ 0]. The bound in (6.41) is identical to that in (6.33)

0]. This can be checked directly for

2 Zm + , j 6= 0.

m

= 1; 2; 3. Thus, (6.33)

kq0 k = ù=2, (6.42) is identical to (6.34). Therefore, B0 (r0 ) follows from Theorem 6.2.4.

the quadratic

ÿ

attractiveness of

We see in the bounds in the corollary that fact, it is easily veriÿed that, if

T >

is positive, and this implies that m

:

decreases as

T

m

increases. In

0 exists, then the right-hand side of (6.41)

m

ý

3. This is why our result is only for

= 1; 2; 3 and is one of the limitations of our approach.

Logarithmic quantizers Here we give another example of stabilizing quantizers, called the logarithmic quantizers. It is shown that they are eÆcient in partitioning. We adopt the following notation for intervals in 0

< a < b,

we interpret the interval

s [a; b)

8 > <(ú ú ) := f0g > :[ )

a] ;

b;

s [a; b

;

a; b

Now the Æ >

m-dimensional

s

=

if

s

= 0;

if

s

= 1:

QÆ

S = Zm üfú1; 0; 1gm.

set the cells as

jm s1 sm ] =

For

s

2 fú1; 0; 1g and

ú1;

if

logarithmic quantizer

1, take the index set as

Q[j1

;

R.

as

is deÿned as follows. Given For [j1

þ þ þ jm s1 þ þ þ sm ] 2 S ,

1 [Æj1 ; Æj1 +1 ) ü þ þ þ ü sm [Æjm ; Æjm +1 )

s

6.2.

139

Quantized sampled-data control

ý2

ý1

Figure 6.9: Logarithmic partitioning of

R2

(see Fig. 6.9) and the output values as

2 j1 3 s1 Æ 1+Æ 6 4 ... 7 5: sm ] = 2

j1 jm s1

q[

mÆ

s

jm

The stability result for such quantizers is as follows. Corollary 6.2.6 Select T > 0, if it exists, small enough that

T

c

d

T

<

1

p

+1

;

8s 9 < (ÿm ú 1)(a2 ú ÿ1 + 1) (aT + 2)Æ ú aT = ÿm cT T ý ú ;: ÿm ú 1 : ÿ1 ú 1 Æ +1 ÿ1

Then, the closed-loop system with the logarithmic quantizer is quadratically stable with respect to (P; ÷J ). Proof

For

j

(6.43)

(6.44)

Æ deÿned by

Q

T

2 S , j 6= 0, we have j =

q

Æ

+1 2

q

j and

q

j = Æq j :

dTj in (6.33) for j 6= 0 are all identical to (6.44). Thus, j = T satisÿes the bounds in (6.32) and (6.33) for j 6= 0. Moreover, since q 0 =

Thus, the bounds on T

0, it follows that r0 = 0. This implies quadratic stability by Theorem 6.2.4.

ÿ

140

Chapter 6.

Extensions for the multiple input case

The logarithmic quantizers have several distinct advantages over the uniform quantizers. First, the stability achieved here is asymptotic. This can never be accomplished by the uniform ones. Second,

Æ

serves as a parameter

to be chosen by the designer. It determines the coarseness of the state parti-

T . As in the single-input case, there is a trade-oþ Æ, the smaller T . Such ýexibility does not exist in the design of uniform quantizers. Furthermore, for a relatively small Æ , the maximum T tion, as well as the size of here: the larger

for the logarithmic quantizers is larger than that for the uniform one. The partitioning in logarithmic quantizers is eÆcient for the control objective of stability. Unlike the uniform partition, the cell size grows larger for those further from the origin. Furthermore, the bounds on inÿnitely many

j

means that for all

cTj

and

dTj

for

can be reduced to the one pair in (6.44). Technically, this

j 6= 0 the inclusion C (uj ) ø F X (uj ) can be realized tightly;

the two sets can be tangent to each other.

We also note that the sampled-data control problem corresponds to the extreme case

Æ

= 1. Then the bounds on

dT

in Theorem 6.2.4 and Corol-

lary 6.2.6 become identical. This is because the tight inclusion just explained is the feature of the sampled-data controller as well. Finally, unlike the case for the uniform quantizers, there is no limitation on the dimension

m

of the

control input.

6.3

Dwell-time switching control

One application of the quantized sampled-data control problem is the dwelltime switching control problem, which is studied in Section 5.3 for the singleinput case.

The main feature of the switching systems is that the sam-

pling/switching is nonuniform, but the time between switchings is guaranteed to be larger than a ÿxed time

T.

From the data rate viewpoint, this can be ad-

vantageous because switchings occur only when necessary and not periodically; thus the data rate can be reduced during the operation of the system. In this section, we consider the extension of the problem to the

m-input

case. The development is similar to the single-input case in that the results are mostly based on those of sampled-data systems with logarithmic quantizers. The diÆculty lies in the switching scheme, which is more complex due to the presence of many accumulation points in the logarithmic partition. In the next subsection, we deÿne the dwell-time switched system with the logarithmic partition.

Then we formulate the dwell-time switching control

problem. In the last subsection, we give a solution to this problem, based on the results from the quantized sampled-data control.

6.3.1

Dwell-time switched systems with logarithmic partitions

The dwell-time switched systems equipped with a logarithmic partition, deÿned in Section 5.3.1, are generalized to the multi-dimensional case. First, we

6.3.

141

Dwell-time switching control

introduce a logarithmic partition of the state space based on a subspace dimension

M of

m.

M be an m-dimensional closed subspace in Rn with an orthonormal basis fe1 ; : : : ; em g. Let E := [e1 þ þ þ em ]. We deÿne the set of cell indices by S := Zm ü fú1; 0; 1gm. Given Æ > 1, the partition is Let

X[j1 jm s1 sm ] :=

ñ

x

2 Rn

The picture is Fig. 6.10 for Here, in particular, Cells with at least one n

n

2 sk [Æjk ; Æjk +1 ); [j1 þ þ þ jm s1 þ þ þ sm ] 2 S : õ

=

0

E x; õk

= 3 and

X0 := X[0

sk

index set of such cells by of dimension

:

úm+1

m

k

= 1; 2; : : :

;m

ô

;

= 2.

M

? , and this cell contains the origin. 0] = being equal to 0 have empty interiors; we denote the

S0 ø S . Other cells are bounded by linear varieties m ÿ 2 and by hyperplanes if m = 1; these have

if

nonempty interior.

x3 þ2

M

þþ þþþ

x2

þ1

X

[j1 j2 11]

x1

Figure 6.10: Logarithmic state partition of

R3

Now that the partition is generalized, dwell-time switched systems can be deÿned and treated in a similar manner as in the 1-dimensional

M case.

Consider the nonlinear switching system x(t)

where

x(t)

2 Rn

=

fi(t) (x(t));

is the state of the system and i(t)

= [i1 (t)

þ þ þ im(t) s1 (t) þ þ þ sm (t)] 2 S

(6.45)

142

Chapter 6.

Extensions for the multiple input case

is the state of the switching logic with a ÿxed dwell time that

is globally Lipschitz continuous for each

fj

The switching logic (how

x(t)

2S

T >

0. It is assumed

f0 (0) = 0. generates i(t)) is deÿned in terms of switching j

and that

times as follows: Set t0 = 0 and let j0 denote the index of the cell x(t0 ) is in. The equation x(t)

has a unique solution, say

=

fj0 (x(t))

ÿ,

1

on the interval [t0 ; ). If x(t0 ) = 0, then no switching will ever occur; this is the trivial case. Otherwise, the ÿrst switching time is

t1

(

t0

=

x

+ T;

fÿ

min

t

t0

ÿ x (t)

+T :

if j0

2 X g =

int( j0 )

;

if j0

2S 2S =

0 and

6

x(t0 )

= 0;

0.

Set i(t)

= j0 for

2

[t

);

0 1 8 > : ( 1 ) 2 cl(Xj1 ) and t

;t

if

;

j

6

= j0 ;

j1

x t

x(t1 )

2X1 j

with j1

2S

0;

otherwise:

This rather complicated deÿnition is to avoid the situation where trajectory

X

x(t)

2S

is on j with j 0 at t = t0 + T and will leave it the next moment; the cell that the state is entering may not be deÿned. Proceeding, the equation x(t)

has a unique solution, say

=

1 (x(t))

fj

ÿ , on the interval [t1 ;

1

). Again, if x(t1 ) = 0, then no further switching will occur. Otherwise, the second switching time t2 is

(

t2

t1

=

x

+ T;

fÿ

min

t

t1

ÿ (t)

+T :

x

if j1

2 X1 g =

int( j )

;

if j1

2S 2S =

0 and

x(t1 )

6

= 0;

0.

Set i(t)

= j1 for

2

[t

);

1 2 8 > : ( 2 ) 2 cl(Xj2 ) and t

;t

if

;

j

x t

þ

Continuing, we obtain i( ) on [0; jectory

þ

x(

1

j2

6

= j1 ;

x(t2 )

j

with j2

2S

0;

otherwise:

) and switching times

) is uniquely deÿned satisfying (6.45) for all

the switching times.

2X2

t

f g 2Z 2 1+ k

. The tra-

[t0 ;

) except at

tk

6.3.

143

Dwell-time switching control

For the system (6.45), the origin is an equilibrium point, and the state the switching logic is a piecewise constant signal with at least time

T

i

of

between

switchings. Hence, we can apply the stability deÿnitions in Section 4.1. The notable diþerence in this

m-dimensional

extension from the original

X

one is that the set of cells with no interior is countable, whereas it is only

0 in the 1-dimensional case. This is what makes the switching law somewhat more complicated. 6.3.2

Problem formulation

The problem in this section is to design a dwell-time switched system with a logarithmic partition that stabilizes a given linear time-invariant plant. Consider the system depicted in Fig. 6.11. The plant (A; B ) is given by x(t)

where that

x(t)

A

2 Rn

=

is the state and

Ax(t)

u(t)

+ Bu(t);

2 Rm

(6.46)

is the control input. It is assumed

6= 0 and A is an unstable matrix, but that (A; B ) is stabilizable.

also assume that rank B =

We

m.

u(t)

-

x(t)

(A; B )

ÿ

T

D

T

C i(t)

Decoder

ÿ

Coder

Figure 6.11: Dwell-time switching control system The concept of dwell-time control and the design problem for the

m-input

plant is identical to those for the single-input case in Section 5.3. The controller is called a dwell-time controller with dwell time the logarithmic partition values. The decoder

fXj gj2S

of

T >

Rn

and the set

=

it

T is governed by

0. Other parameters are

U

=

fuj gj2S

of control

D

u(t)

u ( );

so the plant equation becomes x(t)

=

Ax(t)

+ Bui(t) :

This has the form (6.45), and therefore the switching logic of the preceding section deÿnes the coder

T mapping

C

x

to i.

The dwell-time switching control problem with the logarithmic partition is as follows: Find a dwell time

T

>

0, a logarithmic partition

Rn , and corresponding control values U

=

fuj gj2S

fXj gj2S

of

such that the closed-loop

dwell-time switched system in Fig. 6.11 is asymptotically stable.

144

Chapter 6.

6.3.3

Extensions for the multiple input case

Solution to the dwell-time switching problem

Similarly to the single-input plant case, the solution presented in this section relies on the results for the quantized sampled-data control systems.

The

system in Fig. 6.11 is ÿrst converted to a system similar to a sampled-data one with a sampler, a hold, and a logarithmic quantizer. We give a suÆcient condition for stabilization, which is identical to that in Lemma 6.2.2. Thus, the remaining construction follows directly from the main result for the logarithmic quantizer. The switching scheme in the system in Fig. 6.11 can be summarized as follows. If j0 denotes the index of the cell in which x(0) lies, then the control is u(t) uj0 for t = 0 up to t = t1 T . At the ÿrst switching time t = t1 , the

õ

trajectory is in or enters the cell u(t)

õ uj1 for t 2 [t1 ; t2 ).

by

j (t)

for

t

ÿ Xj1 with j1 6= j0 , and the control switches to

This is repeated for all switching times

the index of the cell

2 [tk ; tk+1 ).

x(t)

is in, we have

u(t)

=

u ( ),

it

k . Denoting

t

where i(t) = j (tk )

With this switching scheme in mind, we can convert the dwell-time switch-

Æ is the logarithmic quantizer,

ing system in Fig. 6.11 to that in Fig. 6.12(a);

fQj gj

Q

Rm and output values fqj gj ø Rm . Its output takes the form qj (t) , where j (t) 2 S is the index of the cell x(t) is in; CT 0 is the deÿned by cells

2S

in

2S

system that is governed by the dwell-time switching logic, recovering j (t) from

j t and then mapping it to i(t). In the decoder DT , DT 0 is a nonuniform ~ zeroth order hold synchronized with CT 0 and is given by DT 0 (i(t)) = qi(t) ; U 0 q ( )

is a nonsingular matrix. Thus, the control input is u(t)

~ Q (E 0 x(t )) = U ~ q =U 0 Æ k 0 i(t) ;

be obtained as

Qj = E Xj

Æ , cells

2 [tk ; tk+1 ):

Qj

and output values qj , j ~ ý1 u from (6.26). from (6.27) and qj = U j 0

Note that, for the quantizer 0

t

Q

2 S , can

This system is very similar to the sampled-data control system in Fig. 6.5. Q = QÆ commutes with the uniform sampler T , and thus the system in Fig. 6.5 is equivalent to the one in Fig. 6.12(b), where QÆ and ST are interchanged.

Note that there the quantizer S

As we compare the two systems in Figs. 6.12(a) and 6.12(b), the uniform

T is replaced by CT 0 , which has a nonuniform sampling time, and T is replaced by DT 0 . We note that the switching in CT 0 is intermittent and slower than the sampling in ST . The discrete-time signal taking discrete values between QÆ and HT in Fig. 6.5 is represented now by i(t). sampler

S

likewise

H

To follow the design methodology for the quantized sampled-data system, we design ÷

E

as in the previous sections. We summarize this procedure: Given

2 (0; 1) and positive-deÿnite matrices Q 2 Rn

Riccati equation (6.2) and obtain the solution LQR optimal gain,

K

=

can be obtained. Replace

úR B

ý1

0

2 Rm m , solve the Then let K 2 Rm n be the

n and

þ

P.

M = (KerK ) kK kB .

B P , and set ~ = (1 + ÷) with B

þ

R

þ

?

, from which

E

We want to design the dwell-time switching controller so that the closedloop system is quadratically stable with respect to (P; ÷J ). That is, for any

6.3.

145

Dwell-time switching control

-

u(t)

~ U 0

ÿ

it

q ( )

T

ÿ

T0

D

x(t)

~) (A; B

ÿ

T0

C i(t)

ÿ

Æ

Q

jt

q ( )

õ(t)

E

0

ÿ

T

D

C

(a) Dwell-time switching system

-

u(t)

x(t)

~) (A; B

?

~ U 0

E

6

ÿ

T

H

ÿ

T

S

Æ

Q

0

ÿ

(b) Sampled-data system

Figure 6.12: Comparison of systems

trajectory

þ) of the system, the decay rate of V (x) := x P x is V (x(t); u(t)) = 0

x(

ú÷x (t)J x(t), where J = Q + P BR B P . Recall that for u = K x the decay rate is given by V (x(t); K x(t)) = úx (t)J x(t). 0

ý1

0

0

The following lemma gives a suÆcient condition for quadratic stability of

the system in Fig. 6.12(a) with a dwell-time switching controller. Lemma 6.3.1 Suppose the dwell-time controller is designed to have the fol-

lowing property: For every t

2 [0; T ].

j

2 S , if x(0) is in Xj , then F (x(t)) 2 F X (uj ) for

Then the dwell-time switched system in Fig. 6.12(a) is quadratically

stable with respect to (P; ÷J ). Let j0 denote the index of the cell in which x(0) lies. By assumption, x(0), the trajectory x( ) of the switched system satisÿes

Proof

starting at

þ F (x(t)) 2 F X (uj0 )

for

t

2 [0; T ]:

By (6.9), this is equivalent to x(t)

2 X (uj0 )

Xj0 ø X (uj0 ) Xj0 ) ø X (uj0 ).

This implies that that cl(

for

t

2 [0; T ]:

and therefore, since

X (uj0 )

is a closed set,

146

Chapter 6.

Extensions for the multiple input case

By the deÿnition of switching times, if t1 x(t)

2 cl(Xj0 ) ø X (uj0 )

Hence, whether the switching occurs at

t

> T,

for

=

T

t

or

then

2 [T ; t1 ]: t > T , x(t)

is in

[0; t1 ].

X (uj0 ) on

Repeating this argument, we obtain

2 X (ujk )

x(t)

where

jk

for

t

2 [tk ; tk+1 ]; k ÿ 1;

is the index of the cell which

x(t)

implies that the control Lyapunov function the deÿnition of

fX (uj )gj V

2S

is in or enters at

V

t

=

(x(t)) decreases for all

tk . t

This

ÿ 0 by

in (6.3); in particular, we have

(x(t); u(t)) =

V

(x(t); ui(t) )

ý ú÷x (t)J x(t): 0

ÿ

Hence, by Proposition 4.1.3, the system is quadratically stable. We are ready to construct the stabilizing dwell-time controller.

2

1. Fix ÷ (0; 1). Select matrices ~ = S ý1 (S U I ). 0

ú

Q

and

R

in (6.2) that give

2. Design a logarithmic quantizer as in Corollary 6.2.6: Take obtain

T >

0, cells

fQj gj

2S , and output values

fqj gj

S > I.

Æ >

Set

1 and

2S .

3. Obtain the state partition cells and their corresponding control inputs through

Xj = fx 2 Rn

:

0

E x

2 Qj g;

j

2 S;

as in (6.27) and u0

~ q = 0; =U 0 0

uj

~0 qj ; =U

j

2 S ; j 6= 0;

as in (6.26), respectively. The solution for the dwell-time switching control problem for the

m-input

case is now stated as a theorem. Theorem 6.3.2 For the dwell-time switching controller just deÿned, the

closed-loop system in Fig. 6.12(a) is quadratically stable with respect to (P; ÷J ). The proof of this theorem follows from Corollary 6.2.6 of Theorem 6.2.4. The proof of the theorem makes use of the suÆcient condition in Lemma 6.2.2, which is equivalent to the suÆcient condition in Lemma 6.3.1.

6.4.

147

Design of S

6.4

S

Design of

So far in our design we have assumed that the matrix the matrices

and

Q

corresponds to

ÿ >

R,

is greater than identity, i.e., Q

a design method to obtain

ÿ

2 (0; 1).

which depends on This condition

1 in the 1-input case in Section 4.6. There we showed in

Theorem 4.6.1 that for any ÷

S,

S > I.

and

ý

R

ò (1

we have + ÷)=(1

ÿ >

ú ÷)

1; moreover, we proposed for any given

ò >

1 and

The goal of this section is to answer the question whether we can

obtain a similar result for the multi-input case. It becomes clear, however, that this is not so simple. First an example is given where

does not hold. We next show a

S > I

direct extension of the 1-input case result and see its limitation that allows such an example. Then we look for a class of systems where any pair of S > I

Q

and

R.

Finally we derive a method to ÿnd

for any plant (A; B ).

Here singular values of a matrix 1; 2; : : :

; q,

where

q

2 C mþn

X

Q

S > I

and

are denoted by

R

ÿi (X ), i

fm; ng and are ordered in nonincreasing order: ÿ1 (X ) ÿ ÿ2 (X ) ÿ þ þ þ ÿ ÿq (X ):

Q

= 1=(1

=

= min

In this chapter, we have made the assumption that the matrix form

holds for that give

ú ÷)I .

For a more general treatment, we allow

ú

positive-deÿnite matrix and set

Qú

û A

:=

Qú

û B

:=

Qú

Q

Q

has the

to be any

:= (1 ÷)Q. In view of (6.4), we introduce 1=2 a change in coordinates using Qú and deÿne

û := K The correct deÿnition of

S

1=2

1=2

ý

AQú

;

1=2 ~ B;

K Qú

(6.47)

1=2

ý

:

is (see (6.6))

S (Q; R)

showing its dependence on

Q

0 û0 û E; := (1 + ÷)E K RK

and

R

explicitly. For this general

results in the previous sections hold by replacing the matrix

A

Q case, the û and so with A

on. Example 6.4.1 We give an example of a 2-input system and show that S > I

does not always hold. Let

2 ú1 A = 4 1 0

and set

÷

= 0:1 and

First we use

R

0

3 25 ;

0

2

1

0

2 0 B = 4 0 10

= I2 .

2 61:54 Q1 = 478:40

78:40 109:4

3 167:65

126:8

167:6

273:6

126:8

3 15 ; 0 0

148

Chapter 6.

and obtain

S,

whose eigenvalues are þ(S (Q1 ; R))

Thus

S > I

Extensions for the multiple input case

f

=

g

1:27; 0:657

:

does not hold.

On the other hand, for

= diag(1; 2; 1000) we have

Q2

þ(S (Q2 ; R))

=

f

g

3:67; 1:23

:

It is now clear that, unlike in the 1-input case, the choice of matter to obtain

6.4.1

Q

and

R

does

S > I.

5

S>I

A suÆcient condition for

In this section we give a suÆcient condition for

S > I.

It is for later use, but

we gain some insight into this condition. û RK û ); Lemma 6.4.2 þk (S ) = (1 + ÷)þk (K 0

Proof

Let

ing to

þk , k

:=

þk

û RK û) þk (K 0

= 1; 2; : : :

û 0 RK û = write K

; m.

and let

Set

xk

k

= 1; 2; : : :

2 Rn þþþ

be the eigenvector correspond-

= [x1

X

; m:

2 Rn

xm ]

m . Then we can

þ

û 0 . From (6.5), , and hence Im X = Im K 0 m û we also have Im K = Im E . Thus, there is yk such that xk = E yk , k

= 1; 2; : : :

X diag(þ1 ; : : : ; þm )X

; m.

The vector

because by the deÿnition of

2R

is an eigenvector of

yk

0

and

S

0

E E

=

S

corresponding to

þk

I

0 û0 û Ey = (1 + ÷)E K RK k

S yk

= (1 + ÷)þk E

0

xk

= (1 + ÷)þk yk for

k

= 1; 2; : : :

Theorem 6.4.3 If þm (P BR

if

Q

=

I

then for

÷

2

1

ý

0

û RK û) þm (K 0

1B0P

ú

0

B P

By Lemma 6.4.2,

û RK û) > ÷)þm (K

ý

(0; 1)

þm (P BR

Proof

ÿ

; m.

ú ÿ

Q) >

Q)

ú

0 then

S (Q; R) > I .

,

2÷ 1+÷

S (Q; R) > I :

is equivalent to

S > I

In particular,

þmin (S )

=

(6.48)

þm (S )

= (1 +

1. Here

= = =

1=2

ý

þm (Qú

1

1

ú ú

÷

þm (Q

n

0

1=2

ý

B P Qú

P BR

1=2

ý

þm (Q

1

ý

1=2

ý

÷

1

1

P BR

1

ý

û) ) (by deÿnition of K

0

(P BR

1

ý

1=2

ý

B PQ

0

B P

ú

) (by deÿnition of

1=2

ý

Q)Q

)+1

o

:

Qú )

6.4.

149

Design of S

Hence

m

û 0 RK û) (1 + ÷)þ (K

Thus, if

Q

=

I,

,

1

>

m(

þ

1=2

ý

Q

(P BR

1

ý

ú

0

B P

the condition (6.48) follows. For general

tion for the right-hand side of (6.49) is þm (Q

ú

1=2

ý

(P BR

1

ý

ú

0

B P

1=2

ý

Q)Q

)

1=2

ý

Q)Q

)

>

ú

2÷

:

1+÷ (6.49)

Q,

a suÆcient condi-

0:

(6.50)

ÿ

is symmetric and Qý1=2 is nonsingular, by Theorem 4.5.8 in [34], the two matrices Qý1=2 (P BRý1 B 0 P Q)Qý1=2 and P BRý1 B 0 P Since

P BR

1B 0 P

ý

ú

Q

ú

have the same number of nonnegative eigenvalues. Thus, (6.50) is equivalent to þm (P BRý1 B 0 P Q) 0.

ú ÿ

Q

ÿ

Note here that þk (P BR

since rank P BRý1 B 0 P =

1

ý

m

ú

0

B P

and

Q) <

Q >

0 if

m

+1

ý ý k

n

0 (see Theorem 4.3.1 in [34]). Hence, we

can see the condition in Theorem 6.4.3 is a sensitive one. We now calculate þm (P BRý1 B 0 P Q) for Example 6.4.1. For Q1 ,

ú

we denote the solution of the Riccati equation by þ(P1 BR

1

ý

0

B P1

ú

Q1 )

=

fú

3:26;

ú

R

=

I

and

and obtain

P1

g

1:79; 16:6

:

The second largest one is negative. Similarly, for

=

R

I

þ(P2 BR

and

1

ý

Q2

0

B P2

we have

ú

Q2 )

=

fú

g

1:00; 0:123; 0:392

:

The second largest eigenvalue is positive, and thus for this case 6.4.2

S > I

follows.

ÿ

Extension of the design method for

In this section we ÿrst give a direct extension of Theorem 4.6.1 in Section 4.6, which guaranteed ÿ S < ò (1

+ ÷)=(1

ú

>

÷)I

1. As in the 1-input case, it gives a useful design tool for for any

ò >

1; however, it shows only that

always holds, whereas our interest is to have

þmin (S ) >

1.

2

Theorem 6.4.4 If (A; B ) is stabilizable, A is unstable, and ÷

inf Sketch of Proof

f

þmax (S (Q; R))

:

Q >

0;

R >

g

0 =

1+÷ 1

ú

÷

þmax (S ) >

(0; 1), then

:

û, By Lemma 6.4.2 and the deÿnition of K

þmax (S (Q; R))

û 0 RK û) = (1 + ÷)þmax (K

1=2

ý

= (1 + ÷)þmax (Qú =

1+÷ 1

ú

÷

1=2

ý

þmax (Q

P BR

P BR

1

ý

1

ý

0

0

1=2

ý

B P Qú

1=2

ý

B PQ

)

):

1

150 Fix

Chapter 6.

Q >

0 and

R >

0 and let

Extensions for the multiple input case

ÿ þmax (Q 1=2 P BR 1 B P Q 1=2). ý

ò

follows as in Theorem 4.6.1.

ý

0

ý

The rest

ÿ

An interesting problem here is to ÿnd a class of systems that always yield S > I.

The following proposition gives a suÆcient condition on the plant

(A; B ) so that

S > I

holds for any pair of

Proposition 6.4.5 If rank A S (Q; R) > I .

ý

n

úm

and

Q

R.

then, for every

Q >

0 and

R >

0,

The following results will be used in the proof. One matrix analogue of

jRe z j ý jz j for z 2 C

k(A + A

ÿ

is

k ý kAk for A 2 C n

n , which is rather

þ

)=2

obvious. The lemma below gives another one. [29] For

Lemma 6.4.6

ö

þk

n

A

[1] For

Lemma 6.4.7

X

2 Cn

+ Aÿ

õ

2 X; Y

n,

þ

ý ÿk (A);

2 Cn

ÿi

+ 1.

þm (P BR

= 1; 2; : : :

; n:

+j 1 (X Y ) ý ÿi (X )ÿj (Y );

n,

þ

Proof of Proposition 6.4.5

k

i

ý

+

j

ý

Notice by the Riccati equation (6.2)

1 B P ú Q) = þm (A P + P A)

ý

0

0

úþn+1 m (úA P ú P A) ÿ ú2 ÿn+1 m(úP A) (by Lemma 6.4.6) = ú2 ÿn+1 m (P A) ÿ ú2 þ1(P ) ÿn+1 m (A) (by Lemma 6.4.7): =

0

ý

ý ý

ý

Since rank A

ý n ú m,

we have

rem 6.4.2 it follows that 6.4.3

þm (P BR

S > I.

S > I

0

0. Thus by Theo-

ÿ

S>I

A design method for

We have seen that

1 B P ú Q) ÿ

ý

cannot be achieved automatically for some systems.

So we may have to turn our interest to ÿnding a method that gives some and

R

that yield

S > I.

Q

In this section we show one such method that has a

certain degree of freedom in the choice of the two parameters

Q

and

R.

The design procedure is as follows: 1. Set

Q

=

I.

2. Let the singular value decomposition of

B

þ ý

B

where

X

2

Rnþn and

Y

2

=

X

üB 0

be 0

Y ;

Rmþm are unitary and üB is a diagonal

matrix with nonzero diagonal entries since rank B =

m.

6.4.

151

Design of S

3. Let

:= (A +

As

take

r

0

A

)=2. If

ÿ 0 then

þn (As )

take any

r >

0; otherwise,

that satisÿes 0

Note that

÷

÷

< r <

þn (As )

2

ú ÷2 )

2 (1

(6.51)

:

2 (0; 1), so such r always exists.

4. Set =

R

The free parameter is 1, the bound on

r

rY

2 0 üB Y

in Step 3. Even if

>

0:

(6.52)

þn (As ) <

0, by taking

close to

÷

can be arbitrarily large.

r

Theorem 6.4.8 If (A; B ) is stabilizable and rank B = m then, for Q = I and R

in (6.52),

P

[49, 96].

S (Q; R) > I .

The following lemma gives a lower bound on the smallest eigenvalue of

Lemma 6.4.9 Given a stabilizable pair (A; B ) and a positive-deÿnite R

Rm

m , let

þ

Deÿne

As

2 Rn

n be the solution of

þ

P

ÿ

þn (As )

PA

ú2÷=(1 + ÷) by Theorem 6.4.3. 1

ý

0

B P

ú P BR

1B0P +

ý

I

2

= 0.

+ (þn (As )2 + þ1 (BRý1 B 0 ))1=2 : ý1 0 þ1 (BR B )

Since

Proof of Theorem 6.4.8

þm (P BR

+

= (A + A0 )=2. Then þn (P )

Q) >

0

A P

=

Q

I

we must show

þm (P BR

1B0P

ý

ú

Observe

ú Q) = þm (P BR 1 B P ) ú 1 ÿ þn (P )2 þm (BR 1 B ) ú 1 (by Theorem 4.5.9 in [34]): ý

0

ý

0

(6.53) Here note BR

Thus

þk (BR

1

ý

1 B 0 ) = 1=r for

ý

k

B

0

=

ý

þ1

0

r Im

X

0

= 1; 2; : : :

0

; m.

0

X :

Applying Lemma 6.4.9 to (6.53)

yields

2

þn (P ) þm (BR

1

ý

B

0

)

ú1=

1

ÿ

1

=

r

2

þn (P )

(

ú1

rþn (As )

r

n 1=2 r

þn (As )

ö

+r

+

2

î

þn (As )

rþn (As )

2

+

1 r

õ1=2 )2

ú1

ð1=2o2 ú 1

+1

:

(6.54)

152

Chapter 6.

Extensions for the multiple input case

þn (As ) ÿ 0 then (6.54) is ÿ 0 for any r > 0. Thus we show that (6.54) ú2÷=(1 + ÷) for þn (As ) < 0 case. This is equivalent to

Here if is

>

r

1=2

þn (As )

î

+

rþn (As )

This inequality holds for any

2

+1

ö

ð1=2

>

1ú÷

õ1=2

1+÷

ÿ

0 satisfying (6.51).

r >

Therefore we now know that for any plant (A; B ) we can obtain

S > I

and

thus that the dwell-time switching controller design is possible. We apply this design method to Example 6.4.1 (÷ = 0:1). We set Since

þn (As )

=

ú1:70 < 0,

r

= 3:10 ü 10

3

ý

=

rY

=

I.

2 ý3 = 3:50 ü 10 2 (A) (1 ú ÷) ÷

<

4þn

and R3

Q3

we take

þ

2 0 üB Y =

3:15 ü 10ý1

ý

0

3:15 ü 10ý3

0

:

Then we obtain þ(S (Q3 ; R3 ))

=

f1:70;

1:37g;

which conÿrms the result.

6.5

Two cart-pendulum system example

ü

1

u

m l

k

1

M

1

ü

1

2

m

2

l

2

u

M

2

2

1

d x

x

2

1

Figure 6.13: Two cart system In this section, we use the example of two carts with inverted pendula in Fig. 6.13 and design sampled-data controllers proposed in the previous sections. There are Carts 1 and 2, which are connected by a spring and a damper with constants

k

and

d,

respectively. Cart

i

has mass

Mi ,

position

xi ,

and an

6.5.

actuator that applies force of mass

mi ,

length

li ,

ui

to one end of it. On top of Cart

and angle

øi .

x

where the matrices

2 0 k 6 ÿ M 6 1 6 0 6 6 6 Mk1 l1 A = 6 0 6 6 6 Mk2 6 4 0

and

A

B

1

1 1

0

Ax

Md2 0 ÿ Mdl

3 0 0 7 7 0 7 7 0 7 7 7; 0 7 7 1 M2 77 0 5 2 2

ÿ M2 l2

+ Bu;

and the state are given as

0

0

ÿ k

=

is a pendulum

i

Its linearized state-space model is

0 0 1 0 0 0 0 0

ÿ mM11g 0 (M1 +m1 )g M1 l1

ÿ Md1 0 Mdl

2 M2 l2 0 1 6 6 M 1 6 0 6 6 6ÿ M11 l1 B = 6 0 6 6 0 6 6 4 0

i

153

Two cart-pendulum system example

0 0 0 0

2x 3 1 6 x _ 17 6 7 6 ÿ1 7 6 6 ÿ_1 7 7 7: x = 6 6 x2 7 6 7 6 x _ 27 6 4 7 5

0

0

Mk1 0 ÿ Mkl

Md1 0 ÿ Mdl

2 2

2 2

1 1

1 1

0

1

ÿ Mk2 0 Mkl

ÿ Md2 0 Mdl

3

0 0 0 0 0

0 07 7 07 7 07 7 7; 07 7 07 7 15 0

ÿ mM22g 0 M2 m2 g M2 l2

(

+

)

ÿ2

_

1

ÿ2

Here we use identical carts and set Mi = 2 kg, = 1; 2, k = 10 kg/s2 , and d = 5 kg/s. First note that rank A = 7

Section 6.4.2 may not give

> n

S > I,

úm

mi

= 0:4 kg,

li

= 0:5 m for

= 6, thus the design method of

S

however, ~ Q

= 13:6;

ò

= diag(100; 20; 0:1; 0:1; 50; 10; 0:1; 0:1)

and obtained

2 175:9 57:8 232 50:5 6 57 :8 65 :0 189 41:1 6 6 232 189 861 187 6 6 50 :5 41 :1 187 40 :7 Q = 6 6 ÿ66:5 ÿ47:3 ÿ163 ÿ35:6 6 6 ÿ46:7 ÿ32:3 ÿ102 ÿ22:4 6 4

ÿ156 ÿ101 ÿ230 ÿ50 9 ÿ34 8 ÿ22 0 ÿ50 9 ÿ11 3 :

ÿ

:

0:926 R = 0 K

=

ÿ

:

0 0:926

:

:

:

18:1

10:6

+ BK ) =

A

ÿ66 5 ÿ47 3 ÿ163 ÿ35 6 : :

:

119 51:7 208 45:3

ÿ46 7 ÿ32 3 ÿ102 ÿ22 4 : :

:

51:7 49:5 167 36:3

ÿ160 ÿ100 ÿ230 ÿ50 9 :

208 167 774 168

ÿ34 83 ÿ22 077 ÿ50 977 ÿ11 377 : : : :

45:3 36:3 168 36:5

;

ÿ28 0 ÿ23 4 ÿ111 ÿ24 1

The eigenvalues of þ(A

:

þ

:

15:7

in

according to Proposition 6.4.5. We used,

3:55

17:7 ÿ26:4

10:4 ÿ21:7

15:4 ÿ105

3:47 ÿ22:8

; 7 7 7 7 5

þ :

+ BK are

fú6:3892; ú5:8242; ú5:1045; ú4:8185; ú 1:3789 ö 1:0324i; ú2:2705 ö 1:7545ig:

154 For

Chapter 6.

÷

= 0:05 we obtained S

=

Extensions for the multiple input case

þ15 0 0

ý

0

:

> I;

15:0

so the method did work in this case. We ÿx

and

K

throughout the example

÷

and designed controllers for the sampled-data and quantized sampled-data setups. Sampled-data control:

For the sampled-data controller in Section 6.1,

the maximum sampling period for the bounds on Tmax

= 2:21

ü 10

2

ý

T and dT to be met was

c

:

For comparison, we also calculated the maximum sampling period for the discrete-time stability of the closed-loop system. To do so, we ÿrst discretize the plant (A; B ) via step-invariant transformation to obtain (Ad ; Bd ) and then

see if all eigenvalues of Ad + Bd U0 K have absolute values less than 1. Note ~ (K E )ý1 from (6.7). Here we looked for the sampling that U0 = (1 + ÷)K E U 0 periods for three cases:

8 > <0 155 max = 0 144 > :0 152 :

~ = for U 0 ~ = for U 0

:

for

:

T

U0

=

S

1 (S

ý

ú

I );

I; I:

These sampling periods are about 7 times larger than that for quadratic stability with a bound on the decay rate.

ü

We designed the controller with T = 2:21 10ý2 . In Fig. 6.14, the hy~ ~ perboloid F (U0 õ ~ ) and the cone ~ ) are shown for õ ~ = [1 1]0 . The û~ (U0 õ ~ ~ õ hyperboloid is drawn as a mesh object, and we see that û ~) F (U ~ (U0 õ 0 ~) holds as in Proposition 6.1.4.

X

F

The initial response plots of

x

for the initial state

ü

F

i , øi , and ui ,

i

ø X

= 1; 2, are given in Fig. 6.15

û = ü0 0 0 2 0 0 0 ú0 1 0û

x1 (0) x1 (0) ø1 (0) ø1 (0) x2 (0) x2 (0) ø2 (0) ø2 (0)

:

:

:

(6.55) The response for the full state feedback case (i.e.,

u

=

K x)

is shown in dashed

lines as well. It can be observed that the trajectories of the two systems are similar, but the diþerence is that the control inputs of the sampled-data system are piecewise constant. Quantized sampled-data control:

Here we design the sampled-data

controllers with the uniform and logarithmic quantizers in Section 6.2. The maximum sampling periods for the two setups were

(

Tmax

=

ü 10 2 21 ü 10 7:91 :

3

for the uniform quantizer;

2

for the logarithmic quantizer:

ý ý

6.5.

155

Two cart-pendulum system example

14 12 10

ô

8 6 4 2 0 6 4 2 õ2

0 −2 −4

−3

−4

−2

−1

0

2

1

3

4

5

6

õ1

Figure 6.14: The cones

We ÿrst examined the uniform quantizer for the two cart-pendulum system. Given r0 , the parameter ù for the uniform quantizer is a function of T as seen in (6.42). We ÿxed the radius of the attractive ball to be r0 = 4. A plot of ù versus for

T

T

=

is given in Fig. 6.16; this is almost linear. We designed a quantizer = 7:91 10 3 and obtained ù = 0:194. The initial responses

ü

Tmax

ý

of the system for the initial condition in (6.55) are in Fig. 6.17 along with the responses of the full state feedback system in dashed lines. As expected, the state goes close to the origin and then shows an oscillating behavior. We remark that r0 = 4 was chosen so that this behavior is visible in the plots. Next, the logarithmic quantizer was designed based on our theory. The plot of

Æ

versus

T

is shown in Fig. 6.18. It looks similar to the one we obtained in

Section 4.7, and this one too exhibits the trade-oþ between the coarseness of the quantization and the sampling period. Note that Æ = 1 is achieved when T = Tmax = 2:21 10 2 . We chose T = 2:00 10 2 and obtained a fairly large Æ = 1:29. The time

ü

ý

ü

ý

responses for the same initial condition are given in Fig. 6.19 together with the responses of the full state feedback case in dashed lines. We stress that the system achieves quadratic stability, which is diþerent from what the system with the uniform quantizer could achieve. Hence, the state goes to the origin, in fact, in a similar manner to the full state feedback system. We also remark that the responses for the dwell-time control system look similar to the ones shown here, though the number of switchings may be fewer at times.

156

Chapter 6.

Extensions for the multiple input case

0.3

x

1

0.2 0.1 0 0

0.5

1

1.5

2

2.5

3

3.5

4

0.5

1

1.5

2

2.5

3

3.5

4

0.5

1

1.5

2

2.5

3

3.5

4

0.2

θ1

0.1 0 −0.1 0 30

u1

20 10 0 0

time t (a) Cart 1

0.1

x

2

0 −0.1 −0.2 0

0.5

1

1.5

2

2.5

3

3.5

4

0.5

1

1.5

2

2.5

3

3.5

4

0.5

1

1.5

2

2.5

3

3.5

4

θ

2

0.1

0

−0.1 0

−5

u

2

0

−10 −15 0

time t (b) Cart 2

Figure 6.15: Time responses for the sampled-data system

Two cart-pendulum system example

157

0.3

0.25

∆

6.5.

0.2

0.15 0

0.002

0.004

0.006

Dwell time T Figure 6.16: ù versus

T

0.008

0.01

158

Chapter 6.

Extensions for the multiple input case

0.4

x1

0.2 0 −0.2 0

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

0.2

θ1

0.1 0 −0.1 0

u1

20 10 0 0

time t (a) Cart 1

0.1

x2

0 −0.1

−0.2 0

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

θ2

0.1

0

−0.1 0 5

u2

0

−5

−10 −15 0

time t (b) Cart 2 Figure 6.17: Time responses for the system with the uniform quantizer

Two cart-pendulum system example

159

15

10

δ

6.5.

5

0 0

0.005

0.01

0.02

0.015

Dwell time T Figure 6.18: Trade-oþ between

T

and

Æ

0.025

160

Chapter 6.

Extensions for the multiple input case

0.3

x

1

0.2 0.1 0 0

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

0.2

θ

1

0.1 0 −0.1 0

u

1

20 10 0 0

time t (a) Cart 1

0.1

x

2

0 −0.1 −0.2 0

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

θ

2

0.1

0

−0.1 0 5

u

2

0 −5 −10 −15 0

time t (b) Cart 2 Figure 6.19: Time responses for the system with the logarithmic quantizer

Chapter 7

Conclusion Motivated by the new technology of control networks, we studied the eþective use of limited data rate in the communication of control systems.

Of the

issues related to data rate, our focus was on time sequencing of messages over the networks and on quantization. The underlying theoretical interest in our study was to ÿnd the minimum data rate necessary to achieve given control objectives. From this viewpoint, we observed two types of trade-oþs. One was in the distributed control problem between capability of systems and data rate: By appropriately increasing the transmission rate between local controllers, the capability of a system in terms of assignability measure (see Section 3.4) can increase, and vice versa. This conÿrms our intuition that better control can cost more data rate. The other trade-oþ was found in the quantized sampled-data control problem between the number of quantization cells and the sampling period for achieving a speciÿc stability criterion.

However, in this case, we observed

through examples that increasing the sampling period was not always the best strategy to reduce data rate. This result may oþer a somewhat diþerent view from that in traditional digital control, where quantization is not considered explicitly; the only measure of data rate there is the sampling period. Considering the fast pace in the growth of network technology, systems in which the data rate for control has to be minimized may be rare. Nevertheless, we conclude through our study that a variety of interesting theoretical problems can be found in control by taking data rate issues in the communication into account. We list some directions for future research in the following. We showed in Chapter 3 that the capability of distributed systems with networks can be enhanced in terms of assignability measure by using the common channel appropriately. This result itself does not imply that data exchange can improve the performance of such systems. Other measures for decentralized systems may be useful to ÿnd out more on performance.

H. Ishii and B.A. Francis (Eds.): Limited Data Rate in Control Systems with Networks, LNCIS 275, pp. 161−162, 2002.  Springer-Verlag Berlin Heidelberg 2002

162

Chapter 7.

Conclusion

The studies in Chapters 4, 5, and 6 are all based on control Lyapunov functions of the quadratic form

V

(x) =

is a solution of the Riccati equation.

0

x P x;

in particular, we use

P

that

One research direction of interest is

to use diþerent types of Lyapunov functions.

This can lead us to employ

other stability criteria as well. The problems in these chapters also assume the plants to be linear time-invariant. Another research topic is to consider nonlinear plants (e.g. [52]). In the examples for the ÿnite quantizers, we optimized the data rate for a ÿxed

P

and for a ÿxed type of quantizer (uniform or logarithmic). It is clear

that as an optimization problem this is somewhat local. Hence, one direction of research is to develop an optimization method that ÿnds among such

P

the one that requires the minimum data rate. Another problem is to generate a partition for the quantizer that minimizes the data rate for a given

P.

A

result related to computing a partition of the state space can be found in [6] for the model predictive control of hybrid systems via mixed-integer nonlinear optimization techniques. In the solutions of the quantized sampled-data problems, there are many sources of conservativeness, especially in the multi-input case study in Chapter 6. One of the limitations of our approach is that the uniform quantizer can be stabilizing only if

m

= 1; 2; 3. Since our motivation is to reduce the data

rate as much as possible, it is of importance to improve this conservativeness. In this book, distributed control and quantization issues are dealt with separately. To explicitly incorporate the ÿniteness of data rate in distributed systems, we need to combine the two problems. However, it is known that nonlinear decentralized control systems are extremely diÆcult to work with. One setup that may be viewed as distributed is the case where there are two sensors and one actuator connected by a network; this is an information fusion type problem.

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Symbol Index c

F ~ (u), 40, 123 F j (u), 51, 131

, 2X , vii

?

, 0,

ÿ

û

, vii

Q

kþk, vii dþe, bþc, vii

fj (x),

33

F (x),

38, 121

(A; B ), 35, 117

He;T

(A; B; C; D ), vii

HT

õ0 ,

62, 102

õ0 (k ), õj ,

int, vii

102 104

, 40, 123

aT

B

Im, vii

84

õj ; õj ; õj ,

x (r ),

~, B

vii

38, 120

C , vii c,

65

c0 ,

C

60

j (uj ), 132

cl, vii CT

, 94, 143

, 40, 123

cT

D, 60, 102 ù, 56, 136 Æ,

58, 91, 138, 141

diag, vii DT ,

E, e,

37

36, 119

ü

û ~ (õ; u), 40

f

37, 119

K,

37, 119

K1 ; K2 ,

15

Ker, vii

fþ

þ(A); LN ,

g, vii

k (A)

18

M ((A; B; C; D ); z ),

M, 37, 120 ü; ü0 ,

85

N,

16

n0 ,

104

NK ,

û , N K

18

16 19

n(t),

103

P

(matrix), 37, 119

P

(plant), 15

Q

(matrix), 37, 119

120

E0 , 60 ÷,

J,

93, 143

, 40, 123

dT

, 65

, 35, 118

Q(õ)

Q

(quantizer), 48, 128

j , 48, 128

Qþ , QÆ ,

56, 136 58, 98, 138

171

Symbol Index QÆ;û0 (k) ,

84

T,

33, 35, 92, 118, 142

qj ,

48, 128

ý,

qj ,

132

Tj ,

ÿ qj ,

q

j

51, 132

, 52, 132

R,

37, 119

R; R + , vii R0 , r0 ,

61, 103 49, 103, 129

rank , vii Re, vii ó,

64 53, 133

U0 ,

119

u0 ,

35 ü , j ü uj ,

u00 ; u ÿ

u00 ;

101 103, 105

~ , 120 U 0 u ~0 ,

38

uj ,

49, 93, 129, 143

U , 93, 143

65

rj (ô ),

131

S

(matrix), 120

S

(switch box), 16

s,

17

s [a; b),

138

S , 33, 48, 91, 128, 141 S , 60, 102 N

38

ÿ (A);

(x), 35, 118

X00 ; X , 91 X[0 0] ; X[ 1 1] , 91 X00 ; X , 102 X 00 ; X , 105 X , 49, 130 X[ 1 m 1 m ] , 141 X (u), 37, 119 ü j

;

j ;ü ü j ü

ÿ

ÿ

j

j

fÿ

k (A)g, vii

ÿk ,

126 ~ 20 ü, span, vii ST

(x; u), 35, 118

V

j

ü, 17 ÿ,

V

, 35, 118

j

s

Z; Z+, vii

s

Subject Index õ-ô

plane/space, 39, 121

aperiodic signal, 8

dwell-time switching control problem, 94, 143

assignability measure, 23

- output feedback, 107

attractiveness, 34

- with a ÿnite partition, 103 - noise rejection, 104

- quadratic, 34 automated manufacturing, 14 automobile, 13

equilibrium point, 34 event-driven signal, 8

CAMAC, 7

Factory Instrumentation Protocol

carrier sense multiple access method (CSMA), 9

ÿeldbus, 1

cart-pendulum system, 109

Foundation ÿeldbus, 7

(FIP), 7, 10

centralized access method, 9 chattering, 91 coder, 94, 143 complementary subsystem, 18 completeness, 18 - weak, 18 control Lyapunov function, 37 control network, 1 - protocol, 7{12 Controller Area Networks (CAN), 7, 14 data network, 7 data rate, 1 - for control, 63 decay rate, 35 decentralized access method, 9 decentralized ÿxed mode (DFM), 18

hold, 35, 118 - extended, 65 hybrid automaton, 94 hybrid system, 90 hyperboloid, 122 index set, 33, 48, 128 jacking systems, 12 lifting operator, 18 logarithmic partition, 91, 141 - ÿnite, 102 - with overlapping cells, 105 LonWorks, 8, 13 Lyapunov equation, 35, 118 magnetic ball levitation system, 72, 88, 89

decoder, 93, 143

master, 9

distributed control system, 3, 15

maximum delay time, 64

dwell time, 33, 92, 142

measurement noise, 103

dwell-time switched system, 33 - with a logarithmic partition, 93, 140

medium access method, 9 MIL-STD-1553, 7 multiple mobile robot system, 25

Subject Index

173

nonuniform sampling, 89

sampled-data control problem, 36,

notation, vii

119 sampler, 35, 118

observer, 107

sampling period, 35, 118

overhead, 11 partition, 48, 128 - ÿnite, 60 periodic pattern, 8 periodic signal, 8 process control, 14 Process Fieldbus (Proÿbus), 8, 14 projection onto

õ-space, 122

stability, 34 - asymptotic, 34 - quadratic, 34 state partition cell, 49, 130 switch box, 16 switch box problem, 17 switching logic, 33, 92, 142 switching pattern, 17

PTV problem, 17

switching time, 33, 92, 142

quantization, 2, 47, 128

system matrix, 18

quantized sampled-data control problem, 49, 129 quantizer, 48, 128 - ÿnite, 60 - logarithmic, 58, 98, 138, 144 - ÿnite, 62 - stabilizing, 53, 133 - state of, 84 - time-varying, 83 - uniform, 56, 136 rectangle, 128 Riccati equation, 37, 119

TI problem, 18 time delay, 2, 64 time delay analysis problem, 65, 68 time sequencing, 2, 15 time-optimal control, 94 time-varying quantizer problem, 84 token passing bus method, 9 transaction time, 11 transmission time, 2 two cart-pendulum system, 152 zoom in, 84