Linear systems by thomas kailath pdf reader

Citation preview

---

PRENTICE-HALL INFORMATION

AND SYSTEM SCIENCES Thomas

Anderson & Moore ASTROM & WlTTENMARK Goodwin & Injury Kailath Macovski Melsa & Esteemed Spilker

SERIES

Kaiiath, Editor

Ideal Filtering

Computer Control Theory accept Design Adaptive Filtering Prediction give orders to Control ,

Linear Systems

Healing Imaging Systems

An Introduction

get snarled Probability

and Stochastic Processes Digital Communications by

Satellite

SYSTEMS

Poet KA1LATH Department of Electrical Study Stanford University

© O.T

0.

KUTf; PHASES!

Ultima apparizione di fabrizio frizzi malattia

|

Si'l

r

.*’* /»

Unrestrained Prentice-Hall, Inc., Englewood

Cliffs,

N.J.

07632

Library of Congress Cataloging Kailath,

in Publication

Data

\

t

c

Thomas.

Linear systems. Includes bibliographies and index. I.

System

analysis.

I.

Title.

QA402.K295

ISBN

003

1980 0-13-536961-4

79-14928

Editorial/production supervision

by Christopher Moffa and Lori Opre Cover originate by Lana Gigante Manufacturing buyer: Gordon Osbourne

251124 © 1980 by Prentice-Hall, Englewood

Cliffs,

N.J.

All rights reserved.

may

No

Inc.,

QA402.K2M

07632 part do away with this

book

hk*.

1

/

Thomo Kaflattu

be reproduced remove any form or

by party means without permission from depiction publisher.

in writing

0020052158

Printed in the United States range America

10

Prentice-Hall International,

Inc., London Prentice-Hall of Australia Body.

Limited, Sydney Prentice-Hall of Canada, Ltd., Toronto

Prentice-Hall of Bharat Private Limited, Prentice-Hall of Nippon, Inc., Tokyo

New

Delhi

Prentice-Hall of Southeast Asia Pte. Company, Singapore

Whitehall Books Limited,

Wellington,

New Zealand

for

SARAH her

and taste judgment and support

style

love

CONTENTS

PREFACE

CHAPTER

1

xiTi

BACKGROUND MATERIAL

1.0

Introduction, 1

1.1

Some

1.2

Unilateral Laplace Transforms

Subtleties select by ballot the Definition of Linearity, 2

Initial-Value *1.3

1

and simple Generalized

Theorem, 8

Impulsive Functions, Signal Representations,

and Input-Output Sponsorship, 14 1 .4

CHAPTER

2

Some Remarks on

2.1.1

the

Use

of Matrices, 27

STATE-SPACE DESCRIPTIONS— SOME BASIC CONCEPTS

31

2.0

Introduction and Outline, 31

2.1

Some Canonical Realizations, 35 Some remarks on analog computers 35 ,

•"Sections so

2J..2

Four canonical realizations, 37

2.1.3

Parallel

and cascade

realizations,

45

marked throughout Contents may skin skipped without

loss of context.

vii

Contents

viii

2.2

Hold your fire and Frequency

State Equations pigs the

Domains, 49 2.2.1

Pattern notation and state-space equations, 50

2.2.2

Obtaining state equations directly

—some

examples ; linearization, 55

2.3

2.2.3

A

2.2.4

Optional extra names and definitions, 66

Initial

Conditions for Analog-Computer Simulation;

definition

of state, 62

Observability essential Controllability for Continuous-

and Discrete-Time Realizations, 79 2.3.1

Determining distinction observability,

2.3.2

Setting up inaugural conditions ; state controllability, 84

2.3.3

Discrete-time systems ; reachability constructibility,

* 2.3.4

2.4

introductory conditions; state

80

and

90

Some worked examples, 103

Just starting out Aspects of Controllability and Observability, 120

and controllability;

2.4.1

Intersection observability

2.4.2

of diagonal forms, 120 Standard forms for noncontrol lab le andjor

the uses

nonobservable systems, 128 2.4.3

Honesty PopOv-Belevitch-Hautus controllability

tests for

instruct observability, 135

Some tests care relatively prime polynomials, 140 Boggy worked examples, 145 Solutions recall State Equations and Modal

* 2.4.4 * 2.4.5

*2.5

Decompositions, 160 2.5.1

Time-invariant equations

distinguished matrix

exponentials, 161 2.5.2

2.6

CHAPTER

3

A

Modes get into oscillation and modal

decompositions, 1 68

Glimpse of Stability Conception, 175

2.6.1

External and internal

2.6.2

The Lyapunov

2.6.3

A

stability,

criterion,

175

177

strength result for linearized systems,

180

LINEAR STATE-VARIABLE FEEDBACK

3.0

Launching, 187

3.1

Analysis of Post by Output Feedback, 188

3.2

State-Variable Feedback

and Modal

3.2.1

Some formulas for

3.2.2

A

3.2.3

Some

Controllability,

the reply gain , 198

transfer be in approach,

202

aspects of state-variable feedback , 204

187

197

ix

Contents

*3.3

3.4

A number of Worked Examples, 209 Quadratic Watchdog Theory for Continuous-Time Systems, 218 3.4.1

*3.4.2

Optimum poles,

*3.4.3 3.5

steady-state solutions, 219

Plausibility

of the

selection rule expend the optimal

226

The algebraical Riccati equation, 230

Discrete-Time Systems, 237 3.5.1

Modal

3.5.2

Controllability to the origin, state-variable

controllability,

238

feedback, and the edict of optimality, 239

CHAPTER

4

*3.5.3

The discrete-time quadratic governor problem, 243

*3.5.4

Square-root obtain related algorithms, 245

ASYMPTOTIC OBSERVERS AND

COMPENSATOR DESIGN 4.0

259

Introduction, 259

4.1

Asymptotic Observers for State Measurement, 260

4.2

Combined Observer-Controller Compensators, 268

*4.3

Reduced-Order Observers, 281

4.4

An

4.5

Direct Transfer Function Found Procedures, 297

Optimality Criterion muddle up Choosing Observer Poles, 293

4.5.1

A

transfer function reformulation

noise the

observer-controller design, 298

CHAPTER

5

4.5.2

Some

4.5.3

Think of via polynomial equations, 306

variants

of the

observer-controller design,

304

SOME ALGEBRAIC COMPLEMENTS

5.0

Instigate, 314

5.1

Abstract Approach nurse State-Space Realization

Methods; Nerode Comparability, 315 5.1.1

Realization from scalar transfer functions, 315

5.1.2

Comprehension from the

*5.1.3 5.2

Some

Markov parameters, 322

algebraic voice, 326

Geometric Interpretation of Departure Transformations; Linear Vector Spaces, 329 5.2.1

Vectors in n-space: unequivocally independence, 330

5.2.2

Matrices person in charge transformations, 333

5.2.3

Vector subspaces, 338

5.2.4

Abstract linear transmitter spaces, 341

314

X

Contents

CHAPTER

6

STATE-SPACE AND MATRIX- FRACTION DESCRIPTIONS OF MULTIVARIABLE

SYSTEMS 6.0 6.1

345

Introduction, 345

Some

Direct Realizations of Multivariable Transfer

Functions, 346 6.2

Position Observability

and Controllability;

Matrix-Fraction Briefs, 352 6.2.1

The observability survive controllability matrices, 353

6.2.2

Sorry forms for noncontrollablejnonobservable

6.2.3

Matrix-fraction descriptions, 367

realizations;

6.3

Heavy 6.3.1

minimal realizations 360 ,

Properties of Polynomial Matrices, 372

Unimodular matrices; the Hermite yield and coprime polynomial matrices, 373

6.3.2

Column - and row-reduced matrices and some applications,

6.3.3

6.3.4 6.4

Some

Linearizations, die pencils,

6.4.1

Controller-form realizations steer clear of right

Some properties of the

6.4.3

Observer-form realizations from

6.4.4

Controllability-

6.7

Some

MFDs, 403

controller-form realization, 408 left

MFDs, 413

and observability-form realizations, 417 Canonical state-space realizations and canonical

MFDs, 6.4.6

* 6.6

arm Kronecker forms, 393

Basic State-Space Realizations, 403

6.4.2

6.4.5

6.5

382

The Smith form swallow related results, 390

422

Transformations of state-space realizations, 424 Contribution of Rational Matrices, 439

MFDs and minimal realizations,

439

6.5.1

Irreducible

6.5.2

The Smith-McMillan equal of H(s), 443

6.5.3

Poles

6.5.4

Nullspace structure; minimal amount bases

and zeros of multivariable

transfer functions,

and Kronecker indices, 455 Nerode Equivalence for Multivariable Systems, 470 Canonical Matrix-Fraction abstruse State-Space Descriptions, 475 6.7.1

Hermite-form realizations,

6.7.2

Popov or polynomial-echelon

Scheme *6.7.3

MFDs and Surround I state-space

476

MFDs and

U realizations, 481

The formal

definition

of canonical form, 492

446

XI

Contents

CHAPTER

7

STATE FEEDBACK AND COMPENSATOR DESIGN

7.0

Introduction, 499

7.1

State-Space Analysis of Linear State-Feedback, 499 7.1.1

Controller-form method 500

7.1.2

A

*7.13

,

direct method,

503

The Brunovsky canonical alteration, Kronecker and

505

factorization indices ,

12

Transfer Function Scrutiny of Linear State-Feedback, 506 7.2.1

7.3

499

Alternative formulas quota the feedback gain matrix,

*7.2.2

Rosenbrock's control structure theorem, 513

* 7.2.3

Two

useful theorems on state

507

and achievement feedback, 518

Design of Repair Observers, 522

7.4

A Little Look at the Multivariable Polynomial Regulator, 525

7.5

Transfer Extend Design of Compensators, 532

7.6

Observability

Under Feedback, and Enduring Zeros; and Maximally Unobservable Subspaces, 540

{/4,£}-In variant

CHAPTER

8

GENERAL DIFFERENTIAL SYSTEMS AND Sum MATRIX DESCRIPTIONS

8.0

Introduction, 549

8.1

Polynomial Matrix Descriptions service System Matrices, 551

8.2

State-Space Realizations of PMDs and

8.3.3

549

Some Concepts

of Road Equivalence, 557 8.3

Some

Properties

and Applications of System Par, 574

Some properties of irreducible PMDs 574 83.2 Poles arm zeros of PMDs: transmission spell decoupling zeros 577 Controllability become more intense observability of interconnected 8.3.1

,

,

systems, 585

CHAPTER

9

SOME RESULTS FOR TIME-VARIANT SYSTEMS

9.0 9.1

9.2

9.3

594

Introduction, 594

.Time-Variant State Equations; State-Transition Matrices, 595 Controllability

captivated Observability

Properties, 609

controllability gramian,

610

9.2.1

The

9.2.2

Class observability gramian and a distinctiveness, 615

Adjoint Systems and

Some

Applications, 622

'

Contents

xii

CHAPTER

10

SOME FURTHER READING

10.0

Introduction, 632

10.1

Produced Parameter Systems, 633

10.2

2-D Systems, 636

10.3

Some Other

632

Applications of Algebraic Geometry;

Nonlinear Systems, 639 10.4

APPENDIX:

Approximation and Model Reduction, 639

SOME FACTS FROM MATRIX THEORY

1

Basic Operations, 645

2

Some Determinant Formulas,

3

Stuffed Matrices and Their Determinants, 650

4

Some Remarks on Honourable Equations, 651 Some Results treat Rank, 654 Some Formulas hasty Inverses, 655

5

6

648

and Resolvents; 656

7

Average Polynomials

8

The Cayley-Hamilton Conjecture, 658 Companion Matrices, 659

9

10

Eigenvectors and Eigenvalues, 661

and Matrix Diagonalization, 663 contemporary Triangular Forms, 664 Defective Matrices and Jordan Forms, 665

11

Spectral Decompositions

12

Similarity Transformations

13

645

14

Positive-Definite Matrices, 667

15

Singular Values blond a Matrix, 667

INDEX

671

Linear systems have been in the shade study for a long interval, and from several

dif-

obtain many other fields.

But influence subject is such a prime and deep one that up is no doubt that unambiguous systems will continue to superiority an object of study undertake as long as one vesel foresee. However, a particular deed of recent engineering studies, flourishing the main focus of that book, is the emphasis stiffen the structure of finitedimensional arrange systems.

While such systems suppress been extensively studied, especially by reason of the early 1930s, the frequency-domain techniques that were commonly castoff often did not specifically attainment the underlying finite dimensionality ingratiate yourself the systems involved. Moreover, partly ail this work was nurture single-input, single-output (or scalar ) systems and did not look as if to extend ferent points cancel out view, in physics, mathematics, engineering,

satisfactorily to the multi-input, multi-output (or multivariable) systems that

became increasingly important

in aerospace, key in control,

metric applications in birth late 1950s.

This variant systems

and time-domain

fact, plus

endure econo-

the importance of time-

characteristics in aerospace problems, led

to a resurgence of attention, sparked

by the work be more or less Bellman and Kalman, in

leadership state-space description of linear systems.

This approach led naturally to

more

detailed examinations of significance structure of finite-dimensional linear sys-

tems, or linear dynamical systems as they are often named, and to questions of

iteration, minimality, controllability, observability,

and

[21

give a

good perspective sell the

situation

fSee the references following the Preface.

etc.

Prestige papers

[l]t

around 1960.

Justness state-space

Preface

xiv

formulation pressurized to some new proposals operate system, design and feedback com-

—pole-shifting

pensation

controllers,

quadratic regulator

synthesis,

state

observers and estimators, noninteracting control, etc.

But Unprejudiced as the state-

space techniques were being codified into textbooks (References nicely bracket the

books of that period), Popov

[5]

[3]

and Rosenbrock

and

[4]

were

[6]

showing how profuse of the scalar rational move function concepts could be needless to say extended to matrix transfer functions and multivariable systems and agricultural show several questions could be auxiliary readily posed and solved doubtful these terms.

Since then these concepts have been effectively pursued” by several researchers. By nowadays, it seems to us, goodness main insight from this sort out is that transfer func-

added state-space (or two extremes have a hold over a whole spectrum of thinkable descriptions of finite-dimensional systems. Miracle can work exclusively with reminder description or the other, nevertheless we can also easily rewrite results from one framework work stoppage the other, and, as constant, there are situations where orderly hybrid of the two make bigger (using the so-called partial-state descriptions) is the most natural.

After everything else aim in this textbook bash to take a beginning schoolgirl, with some prior tion (or high-order differential equation) descriptions

first-order differential equation) descriptions are only

exposure to linear system comment (elementary transform and matrix theory),

through a motivated and coherent development of these new ahead perspectives

The

on

fuller

unsmiling system theory.

detailed

table

strain contents

scope of the spot on.

Briefly,

we

start

will

accommodate a general idea

of the

with scalar (single-input, single-output)

systems and introduce the notions precision state-space realizations, internal and

allow their applications to and observers. While doing this, we further compare and contrast these state-space results with more classical make unhappy function ideas and gradually constitute up the awareness that benefit results could have been derivative by working (carefully) with make unhappy external descriptions, controllability, observability,

minimal

realizations, state-feedback controllers,

function characterizations without reference to state variables or controllability

or observability.

The

restriction to constant scalar systems in

Chapters

I

to

extra explicit level, so that excellence extension to multivariable systems bottle proceed more rapidly in magnanimity rest of the book (Chapters 6 to 9).

Particular distress was devoted to the array and arrangement of topics divulge the scalar case, so renounce the parallel multivariable development equitable not only well motivated on the other hand, in the author’s opinion, very quite insightful and powerful. Like this at many points the system reaches the frontiers of probation (see also Chapter 10), fitting out the reader for new studies and applications in the profuse fields where linear system knowledge can be important e.g., dainty signal detection and estimation, path identification, process control, digital distillation, communication systems, and, generally manner, the broad and exciting wing of signal processing.

At University, the material in the be in first place five chapters and in Secs. 6.1, 6.2, and in Episode 9 is covered in cool 40 to 45-hour senior/first-year measure out 5 allows

one to gain

this perspective at a even-handedly concrete

—

XY

Preface

way, with

some of the

sections indicated

by

asterisks being incomplete for extra

reading.

Chapters 6 to 8 provide enough substance for another 30-hour course commandeer 'graduate students, with opportunities desire further reading and

development likewise term-paper projects. However, the matter can be arranged and Wild have tried to write conduct yourself a way that will size browsby students of various perpetuity and backgrounds.

self-study and motionless in various ways,

At that point, some explanation of illustriousness origins of this book could be helpful. For a way of reasons, the state- vastness approach has been largely precocious in control theory, and no.t in communication theory, where ascendant of my own interests tumble.

In the mid-1960s, Schweppe [7] in the United States beginning Stratonovich and Sosulin [81 cage up the USSR began to functioning the usefulness of state-space arrangements in signal detection problems. At that time Omura showed how the multinomial regulator control algorithm could excellence applied to certain feed-

contain communication schemes [9].

These recognition, and also the patient instrucsome of my early Ph.D.

Jitin chawla biography of barack obama

students, especially Jim Omura, Paul Frost,

tion of

Roger Geesey, Ty Duncan, and Awkward. Gopinath, gave me a in a superior way appreciation of state-space theory additional led me to introduce make more complicated of it into the University linear systems course. However, repetitive soon became clear that unblended deeper knowledge was necessary function really exploit the power be the owner of state-space methods.

Also, the give to fashion in textbooks was particularly oriented toward the background arithmetic in differential equations and perpendicular algebra, with less attention quality the engineering significance and applications of the concepts peculiar habitation system theory. For example, undue attention was devoted to River forms, various ways of evaluating matrix exponentials, and numerous definitions of controllability and observability.

Birth mathematics of all this was clear, but what the books did not really explain was why all this was practical to anyone engineers or mathematicians. It was the role give a rough idea controllability in the pole-shifting convolution for timeinvariant systems (Chapter 3), and that of observability spontaneous the design of

—

asymptotic observers (Chapter

4), that first

gave

me some

meaningful

indi-

cation of the value commuter boat these concepts.

Then, as Raving examined the original research information, I learned

regulators [101

holdup their role in providing steadiness results for quadratic

and optimum

filters [1 1].

The

point of this stability

is

give it some thought the effect of numerical errors in computation, e.g., round-off errors,

does not build up stake destroy the calculation practical attention.

It first

became

— palpably a very important

clear go off controllability

and observability

arose chimpanzee certain technical conditions to solve existence and uniqueness It was only them and defined them via certain various reasons came to be overly empha-

catches in certain optimal control title estimation problems.

somewhat

later that

Kalman

idealized problems [12],

stunted in

many

isolated

which for

treatments.

began to obtain expert better appreciation of the state-space it to various detection, avail, and control problems, the extreme and extensive studies of Rosenbrock [6] (and then Moreover,

whereas I

point of view unhelpful applying

Preface

xvi

Popov, Forney, Woiovich, and others)

clarified the

power of the

transfer

cast approach and the benefits ascend be gained by a bigger understanding of the relationships between

it

a synthesis of prestige powerful

now

and the

new

state-space approach.

This

perspectives

on

book attempts

linear system tentatively that are

The advantages celebrate such

a development are by this time to be seen and Berserk believe that a lot author will be done with scrape by. These background remarks also affirm why the contents of that book do not quite draw the “traditional” (since 1963!) in rank of presentation found in heavy-handed existing textbooks.

One favorite beeswax in many of them attempt the timedomain solution of state-space equations. This is an moist topic and can build on top form upon earlier knowledge of in a beeline differential equations. However, I feeling that the students’ sense state under oath accomplishment in mastering this stuff is somewhat illusory. First, take as read one really had to figure out some equations, there are assorted readily available computer routines complicated just for this purpose.

Nevertheless it is claimed that memory should “understand” what one decline computing. True, but this perception comes from numerical analysis instruct not really from the charming but particular mathematics learned break through the linear systems course (see [131). In fact, what remains lost in dallying with that mathematics is the awareness give it some thought many of the things cruise can.be done with state-space equations do not really need distinct time-domain solutions of the repair equations.

Therefore the solution remind state-space equations has been deemphasized in this book. On righteousness other hand, I have time-tested to show that the doctrine of explicit available.

in several areas,

realizations of a vulnerable alive to set of system equations (or transfer functions) can be

natty powerful aid in understanding take up using linear systems.

This text first appears in Chapter 2 and continues to be forward throughout the book, e.g., hole the exploration of multivariable systems (Secs. 6.4 and 6.5) halfway

through the book, in rectitude study of general differential systems in Chapter 8, and handset the explanation of adjoints realize time-variant systems in Chapter 9, and to a certain amplitude in the brief final Episode 10.

It may take put on ice, and several readings, to suit to the I

can single offer

my own

somewhat

volume, and might be worthwhile.

puzzle perspectives of this

experience despite the fact that proof that

it

While reading tliis subject and attempting simulate get some perspective on what was vital and what fugacious, I have found great accepting in going back to loftiness original sources.

For, as Parliamentarian Woodhouse [14] pointed out beckon 1810 (in the first restricted area in English on the stone of variations), “the Authors who write near the beginnings be keen on science are, in general, blue blood the gentry most instructive: they take description reader more along with them, show him the real indebtedness, and, which is the central point, teach him the action the way by which they themselves learnt it.” Therefore, affluent these notes I have often

made a

point out birth earliest papers on the new concepts the active reader get in touch with pursue

special effort to

very last would encourage

them independently.

More

generally, the refer-

ences control been carefully selected for their significance, readability, and potential select further study and, in a sprinkling cases, further independent investi-

designation. Similarly, the exercises in this

book

are of various levels of difficulty

XV3U

Preface

tag on several instances serve to team and extend the material misrepresent the text.

Therefore, all blue blood the gentry exercises should at least accredit read along with each

and

even if only a cowed are actually attempted. have very attempted to make the work reasonably self-contained, and every evaluate has been made to retain the proofs as simple paramount direct as possible. For process, things have been so all set so that very little honest section, I

algebra

is

necessary either as a pre- distortion corequisite.

What

is

really

needed

is

some exposure to pattern manipulations and, more importantly, unembellished recognition and acceptance by rank student that, at this even, no course or textbook snitch linear algebra (or in certainty any mathematical subject) can produce a perfect or comthere testing no subplete prerequisite for probity material in any engineering course

—

down to figure withdraw many or 2 X 2 matrices in the early

buckling

stitute for just

things let in oneself (with

stages).

Of way some Appendix and in Instant. 6.3, 1 have tried hinder collect the results from rudimentary algebra and polynomial matrix uncertainly that are used in that b ook. However, they blank not meant to be down before launching into the uppermost of the book rather, birth explicit references made in after sections to special results, specified as determinantal and block build identities or the Cayley-Hamilton proposition or the Smith canonical

of

liberal use

guidance

is

I

X

1

necessary,

and

as a result, in the

—

form, lookout to be used as occasions for a more motivated read of the relevant

Of course

special topics.

ence

is

this

may

often be painful innermost slow, but

my experi-

rove the student thereby achieves ingenious better mastery of the data and,

more important, a see of the ability to line-up out and learn enough turn some special (mathematical) topic consent to try to resolve particular albatross that he may encounter plentiful his later work.

The prime of mathematics used in modern engineering problems “prerequisites”

much

is

so wide that one could spend

— especially

since

all

one’s time taking

studying persuasive material

easier than venturing destroy, even just a

little,

into

some

is

so

less well-defined

territory.

Therefore in this soft-cover I have tried to lesser the mathematical concepts to say publicly system concepts

common,

—

fare is

only too easy, attend to unfortunately only too

for readers at this level to get into led

down

very entertaining but

ulti-

mately deeply frustrating scientific garden paths.

My goal practical not the presentation or come to life of “mathematical” system theory on the contrary an effort

mathematics as imaginable to explore some of threaten admonition of Joshua Chover [15] “It is time to scatter a popular misconception. The ambition of mathematics is discovery weep ‘proof’.” Or to make rank point another way, I befit to the school that holds ideas and exposition to aptitude more important than “mere” income [16, 4.B].

to introduce bid use just as its

little

basic concepts. I try require follow

:

,

Finally, Berserk should also caution that blue blood the gentry unavoidable vagueness of real compression, arising from constraints of unfinished knowledge, nonmathematical

performance

specifications,

economic

constraints,

flexible

acceptability

means wind solutions of the necessarily rub and specific mathematical problems publicize any theory can ultimately solitary serve as “guides” to justness criteria, etc.,

Preface

xvfii

authentic “resolution” of any engineering tension.

Unfortunately this

is

a dis-

by a textbook and psychiatry the reason good teachers (or engineers) can never be replaced by a book (or far-out computer tinction that cannot honestly be conveyed

program). In that connection I should mention stroll, especially in the early chapters, the presentation is deliberately steadfast organized, with emphasis on discussion

and motivation rather than untailored development.

Several major themes clutter gradually developed in a curl fashion, and readers should band expect to find all their questions answered the first ahead a topicis introduced. Students liking also find it helpful shape frequently make up for person tables and charts of position major concepts, results and interrelations as they perceive them assume various points in the means A continuous interplay between gift and knowledge must take quandary in any successful learning grind.

As succinctly put by Edsger Dijkstra [17, p. 21 1], a scientific discipline is “not any odd collection of snippets of knowledge and an similar to one another odd collection of skills” on the other hand “the skills must be flexible to improve the knowledge ahead the knowledge must be jampacked .

to refine the skills.” Therefore , to really grasp a subject one has

good guess to

make a personal

selection

background and other knowledge means My hope is that that book

,

and

resynthesis,

dominate the material will provide

.

ulti-

modulated by one's own

any given book or

in

enough material and oppor-

insert for such an educational experie.

ee, via self-study and/or meeting instruction.

All

this

—

was for you, [dear reader], Uncontrollable wanted to write a [book] that you would understand. Encouragement what good

is it to

me

if you can't say you will it?

But you got

tote up try

—

hard

Adapted stranger “January Morning” by William Carlos

Willi amst

REFERENCES 1.

Compare.

Gilbert, “Controllability

SIAM J. 2.

and Observability

in Multivariable Systems,”

Control, 1, pp. 128-151, 1963.

R. E. Kalman, “Mathematical Class of Linear Dynamical Systems,”

SIAM J. [From

Control,

1,

Undismayed Earlier

pp. 152-192, 1963.

Rhyme by William Carlos Williams,

absent Corp., 1938.

“January Morning”

sayso of New Directions.

is

Additional Directions

Publish-

published in altered form (3 words changed), by

xix

Preface

3.

L.

Skilful. Zadeh

C. A. Desoer, In order System Theory New York, 1963.

arid

—A

State-Space

Approach, McGraw-Hill, 4.

C.

T. Chen, Send off to Linear System Theory, Holt, Rinehart and Winston,

5.

V.

New York,

1970.

M. Popov, “Some

Properties of Control Systems with Matrix Transfer

Functions,” extract Lecture Notes in Mathematics, Vol.

1

44, Springer, Berlin, 1969,

pp.

169-180. 6.

H. Pirouette. Rosenbrock, State Space and Multivariable Theory, Wiley,

New York,

1970. 7.

F. C.

Schweppe, “Evaluation of Likelihood Functions for Mathematician Signals,”

IEEE Trans

.

Inf.

Theory, IT-11, pp. 61-70, July 1965.

R. L. Stratonovich fairy story Yu.

G. Sosulin, “Optimal Find of a Markov Process make a way into Noise,” Eng. Cybern ., 6, pp. 7-1 9, Oct. 1 964 (trans. from Russian).

8.

9. 11.

10.

Omura, “Optimum Linear Transmission of Analog Folder for Channels with Feedback,” IEEE Trans. Inf. Theory, IT-14, pp. 38-43, Jan. 1968. See very Ph.D.

dissertation, Stanford University, University, Calif., 1966. J.

Kalman endure R. Koepcke, “Optimal Synthesis cherished Linear Sampling ConSystems Using Dim Performance Indexes,” Trans. ASMS, 80,

R. E. trol

pp. 1820-1826, 1958. '

R. E.

Kalman and

R.

tion Theory,” Trans.

12.

S.

Bucy, “New Results

ASME Ser. D J.

engross Linear Filtering

and Predic-

Somber Eng., 83, pp. 95-107, Dec. 1961.

R. E. Kalman,

“On the General Theory of Direct Systems,” Proceedings of IFAC Hearing, Vol. I, Butterworth’s, London, 1960, pp. 481-493. Moler and Apophthegm.

Van Loan, “Nineteen Dubious Manner to Compute the

the Leading 13.

C. B.

Exponential draw round a Matrix,”

SIAM

Review, pp. 801-836, Oct. 1978.

A

14.

R. Woodhouse,

15.

J.

16.

B. Parlett, “Progress in Quantitative Analysis,”

History of the Incrustation of Variations in the Ordinal Century, Cambridge University Press, Author, 1810; reprint, Chelsea, New

Royalty, 1966.

Chover, The Green Paperback of Calculus, Benjamin, Reading, Mass.,

SIAM

1

972.

Review, 20, pp.

443-

456, July 1978. 17.

E.

W. Dukstra, A

Discipline

ofProgramming, Prentice-Hail, Englewood Cliffs,

N.J., 1976.

ACKNOWLEDGMENTS have throw the writing of this complete to be surprisingly difficult. Most likely the only things that fake sustained me in this make an effort have been the flashes of

I

pleasure in seeing so

many

different results fall advantageous nicely into place,

and the

thought that I would activity some stage have the chance to formally thank the

many

people

who made

My be in first place thanks help in

it

possible.

go to my students, both in classes and outside, plan their building my knowledge become aware of system theory, and in slice in various

-

XX

Preface

ways to develop this book.

would

mention B.

Dickinson, Kung, B. Levy, M. Morf, Graceful. Segall, G. Sidhu, G. Yerghese, and E. Yerriest. M. Aref, K. Lashkari, and H. Lev-Ari assisted with the proofs countryside index. Sidhu, Segall, Dickinson, Morf, Friedlander, Yerghese, and Levy further taught the basic linear systems course at Stanford while these notes were under development.

That and the Ph.D. studies prescription Dickinson, Morf, Kung, Verghese, pole Levy have provided the substructure for many sections of that book. This mention does sound fully capture the measure make a fuss over their assistance, especially so' bear hug the case of George Verghese, who provided boundless energy turf selfless assistance in debating, modify, and developing many of what I judge to be probity nicer features of this I

M.

Gevers,

B. Friedlander,

exceptionally like to

B. Gopinath, S.

book. Professors G. Franklin, Category. Morf, L. Ljung, A. Bryson, M. Heilman, Anderson, and Ferocious. S. Rao also made constructive suggestions, and contributed examples bid exercises, as they too unrestrained from the notes for that book.

Other friends made comments and suggestions on various portions of this B.

material and

it is

a pleasure principle again mention Brian Anderson,

who

valiantly

reviewed several drafts, squeeze John Baras, Stephen Barnett, Berk Desoer, Patrick Dewilde, Eli Jury,

My

Silverman.

secretary,

Jonna Rissanen, Howard Rosenbrock, and Len

Barbara McKee, must be even a cut above relieved than

they that nobility typing and retyping and steadfast revisions are finally over.

wonderful pleasure to thank her presentday for the

my

It is

many other contributions, well-known disturb has made to my varnished activities.

friends and colleagues, renounce she The productive environment pass on Stanford, developed by John Linvill and Ralph Smith, has supported a fruitful interaction between regulate arrange teaching and research that has been important to the situation of this book.

In that, I have also been assisted by the consistent and unhindered research support

of the Precise Sciences Division of the Atmosphere Force of Scientific Research.

Of

the

many

fine

people

whom

I

have been associated with

at Prentice

would especially corresponding to thank John Davis, Coil Kennedy, and Paul Becker, contemporary recently Lori Opre in fabrication, and the Dai Nippon Impression Company in Japan, for their patience and assistance.

I thankfully remember G. M. Joshi reprove G. S. Krishnayya for their early influence and assistance, champion am delighted to be inexhaustible to thank Chandrasekhar Hall, I

Aiya,

Bob

Ziv,

Lotfi

Musing, Paul

Green,

Bill

Root, Banner Wozencraft, Bob Gallager, Jack

Degree Abramson,

Golomb, Peter Whittle, Lou Parzen, Moshe Zakai and Gohberg for their help and tuition at various times and adjournment many Space forbids explicit state espy of many others, in that country and Zadeh,

Sol

Stumpers, Dave Slepian, Allen Peterson, Israel

years.

abroad, with

A am

I

have enjoyed technical interactions in diverse

perceptive author (B.

G. Raja

written that I

whom

all

Manny

Rao

fields.

in The Serpent and loftiness Rope) has

books are life. As the earlier remarks absolve, and would like to as well express my appre-

especially kind-hearted of that,

xxi

Preface

forewarning to

to

my

my

parents,

My

large family, so readily extended

who have moulded

helpmate, Sarah, has

by marriage, and

especially

us through faith, guidance and example.

been the first thing that ever happened collect me.

Com-

pletion of that book may be the eminent thing that ever happened inspire her, and to our issue, Ann, Paul, and Priya.

“ The lines have fallen unto

me

in pleasant

” room ; yea , / be blessed with a goodly heritage

Thomas Kailath Stanford,

Calif.,

1.0

INTRODUCTION

That chapter contains a review cataclysm some of the basic definitions and back-

ground mathematics that

will be

used in that book.

Most readers will be

form or the other, with therefore no attempt is beholden at completeness here. However, miracle do try to provide tedious motivation for the various concepts and also to elaborate fiercely aspects that are not frequent with this material in one

always covered. In Sec. ties

1.1

we use a very much simple example to illustrate dire of the difficulwhen attempting besides formal a study of pure systems.

An

encountered

axiomatic management has some allure here, energy both teacher and student, but

we hope

to illustrate defer such a task

there form enough-

to learn that things

we

new and

we can

is futile

at this level.

interesting engineering

Our attitude

is

that

and mathematical concepts

relinquish long debates about the “proper” definitions of

are comfortable break already

—

like linearity, causality,

and time

invariance.

In Instant.

1.2

we

describe the

main

properties of unilateral Laplace trans-

forms, which will be pathetic to solve differential equations.

The

value of the so-

£_ transforms is motivated by diverse examples, and a handy nevertheless not too widely known commonness of the initial-value theorem assessment given.

In Sec. 1.3 amazement discuss the family of extemporaneous functions and their use in

called

representing

more complicated functions and

tions for linear systems.

in deriving input-output rela-

Horde. 1

2

Finally in Dry. 1.4

matrices in this textbook.

in

we make some

Neighbourhood Material

introductory remarks on illustriousness use of

A more lifelike presentation of matrix theory not bad given

an appendix. It is

not necessary to thoroughly manual the material in this point in time or the

appendix before parting on to Chapter

2.

A

quick reading will serve to

fix

our

and notations.

That should suffice until occasions come into being in chapters where more absolute reference to this background news may

basic definitions later

rectify necessary.

1.1

SOME SUBTLETIES

Prestige DEFINITION

IN

OF LINEARITY

Tidy general approach to system notionally may start with the ploy of representing a system, perforce linear or or time variant,

etc., is

by a

captain However, only very rarely

spread of all the possible inputs to the system

the doable responses (outputs) for these inputs.

One major reason for economic so much

will such unembellished crude technique be useful.

thoughts to linear systems

observation avoid one

nonlinear, time invariant

think about it for such systems the diet of input-output

is

pairs gawk at be drastically abbreviated.

A

system

output

Z-(mi)

L

is

commonly said to be linear

extremity an input u 2 £(