#1

xiv Contents

1.5.2Frequency Response, 37

1.5.3Zeroes of Transfer Matrices, 39

1.5.4Snterconnections of Linear Systems, 43

1 . 5 9 Root Loci, 51

Controllability, 53

1.6.1 * Definition of Controllability, 53

1.6.2' Controllability of Linear Time-Invariant

Systems, 55

1.6.3' The Controllable Subspace, 57

1.6.4' Stabilizability, 62

1.6.5" Controllability of Time-Varying Linear

Systems, 64

Reconstri~ctibility,65

1.7.1* Definition of Reconstructibility, 65

1.7.2* Reconstructibility of Linear Time-Invariant

Systems, 67

1.7.3' The Unreconstructible Subspace, 70

1.7.4' Detectability, 76

1.7.5" Reconstructibility of Time-Varying Linear

Systems, 78

Ditality of Linear Systeias, 79

Phase-Variable Canonical Foims, 82

Vector Stocliastic Processes, 85

1.10.1Defmitions, 85

1.10.2Power Spectral Density Matrices, 90

1.10.3The Response of Linear Systems to Stochastic Inputs, 91

1.10.4Quadratic Expressions, 94

The Response of Linear Differcatial Sjtstems to Tbite

Noise, 97

1.11.1White Noise, 97

1.11.2Linear Differential Systems Driven by White Noise, 100

1.11.3The Steady-State Variance Matrix for the Time-Invariant Case, 103

1.11.4Modeling of Stochastic Processes, 106

1.11.5Quadratic Integral Expressions, 108

Probleins, 113

Contents xv

Chapter 2 Analysis o f Linear Control Systems

119

2.1Inb.odrrction, 119

2.2The Forr~nrlationof Control Prohlerns, 121

2.2.1Introduction, 121

2.2.2The Formulation of Tracking and Regulator Problems, 121

2.2.3The Formulation o f Terminal Control Problems, 127

2.3Closed-Loop Controllers; The Basic Design Ohjectiue, 128

2.4The Stability of Control Systents, 136

2.5The Steadjz-State Analysis of tlrc Tracking Properties, 140

2.5.1The Steady-State Mean Square Tracking Error and Input, 140

2.5.2The Single-Input Single-Output Case, 144

2.5.3The Multiinput Mnltioutput Case, 155

2.6The Transient Analysis of tlre Tracking Properties,

165

2.7The Effects of Disturbances in tlre Single-fi~prrlSingle- Ontptrt Case, 167

2.8The Effects of Observation Noise in the Single-Inpnt Single-Ontpnt Case, 174

2.9Tlre Effect of Plant Paranteter Uncertainty in the Single-Inpat Single-Ostpnt Case, 178

2.10* Tlte Open-Loop Steady-State Eqsiualent Control

Schenre, 183

2.11Conclrrsions, 188

2.12P~.ohlerns,189

Chapter 3 Optimal Linear\ State Feedback Control Systems

193

3.1Introdaction, 193

3.2Stability Intprouentcnt of Linear Systcnts by State Feedhacli, 193

3.2.1 Linear State Feedback Control, 193

xvi Contents

3.2.2" Conditions for Pole Assignment and Stabilization, 198

3.3The Deterministic Linear Optirnal Regulator Problem, 201

3.3.1introduction, 201

3.3.2Solution of the Regulator Problem, 207

3.3.3Derivation of the Riccati Equation, 216

3.4Steady-State Solrrtion of tlre Deterministic Linear Optirrral Regrrlator Problenr, 220

3.4.1Introduction and Summary of Main Results, 220

3.4.2' Steady-State Properties or Optimal Regulators, 230

3.4.3" Steady-State Properties of the TimeInvariant Optimal Regulator, 237

3.4.4* Solution of the Time-Invariant Regulator Problem by Diagonalization, 243

3.5Nrrnrerical Solution of tlrc Riccnti Eqrration, 248

3.5.1Direct Integration. 248

3.5.2The Kalman-Englar Method, 249

3.5.3" Solution by Diagonalization, 250

3.5.4" Solution by the Newton-Raphson Method, 251

3.6Stochastic Lincnr Optirnal Regrrlntor nnd Tracking Problems, 253

3.6.1Regulator Problems with DisturbancesThe Stochastic Regulator Problem, 253

3.6.2Stochastic Tracking Problems, 257

3.6.3Solution of the Stochastic Linear Optimal Regulator Problem, 259

3.7Regulators and Trnckirr~Sjmterrr wit11 Norrzero Set Points and Constant Distrrrbnnccs, 270

3.7.1 Nonzero Set Points, 270

3.7.2' Constant Disturbances, 277

3.8" Asyr~ytotic Properties of Time-Inuarinrrt Optimal

Control Laws, 281
3.8.1Qsymptotic	Behavior of the Optimal Closed-

Loop Poles, 281

Contents xvii

3.8.2' Asymptotic Properties of the Single-Input Single-Output Nonzero Set Point Regulator,

297

3.8.3" The Maximally Achievable Accuracy of Regulators and Tracking Systems, 306

3.9' Sensitivitj~of'Linear State Feedback ControlSysten~s.

312

3.10Conclnsior~s,318

3.11Problems, 319

Chnpter 4 Optimal Linenr Reconstruction of the Stnte

4.1Zntrodnction, 328

4.2Observers, 329

4.2.1Full-Order Observers, 329

4.2.2* Conditions for Pole Assignment and Stabilization of Observers, 334

4.2.3" Reduced-Order Observers, 335

4.3The Optimal Obscrucr, 339

4.3.1A Stochastic Approach to the Observer Problem, 339

4.3.2The Nonsingular Optimal Observer Problem with Uncorrelated State Excitation and Observation Noises, 341

4.3.3*	The Nonsingular Optimal Observer Problem
	with Correlated State Excitation and
4.3.4"'	Observation Noises, 351
	The Time-Invariant Singular Optimal
4.3.5'	Observer Problem, 352
	The Colored Noise Observation Problem,
	356
4.3.6'	Innovations, 361

4.4"' The Duality of tlre Optimal Observer and the Optin~al Regnlator; Steady-State Properties of' the Optirnal Observer, 364

4.4.1" Introduction, 364

4.4.2* The Duality of the Optimal Regulator and

the Optimal Observer Problem, 364

xviii Contents

4.4.3* Steady-State Properties of the Optimal

Observer, 365

4.4.4* Asymptotic Properties of Time-Invariant

Steady-State Optimal Observers, 368

4.5Corrclrrsions, 373

4.6Problerrrs, 373

Chnpter 5 Optimal Linear Output Feedback Control Systems

377

5.1Introdrrction, 377

5.2The Regrrlation of Linear Systenrs ~cfitlrIncomplete

Measrrremcnts, 378

5.2.1The Structure of Output Feedback Control Systems, 378

5.2.2* Conditions for Pole Assignment and Stabilization of Output Feedback Control Systems, 388

5.3Optinral Linear Regrrlalors ~vitlr Inconrplete and Noisy Measrrrerrrmts, 389

5.3.1Problem Formulation and Solution, 389

5.3.2Evaluation of the Performance of Optimal

Output Feedback Regulators, 391

5.3.3* Proof of the Separation Principle, 400

5.4Linear Optirnal Tracking Systenrs wit11 Incorrrglete and Noisy Measrrrcnrents, 402

5.5Regrrlators and Trackirrg Systenw with Nonzero Set Points and Constant Disturbances, 409

5.5.1 Nonzero Set Points, 409

5.5.2* Constant Disturbances, 414

5.6* Serrsitiuity of Tirne-Inuariarr1 Optirnal Linear Orrtprrt Feedback Control Systenrs, 419

5.7" Linear Optinral Orrtprrt Feedback Controllers of

Reduced Dimerrsions, 427

5.7.1t Introduction, 427

5.7.2* Controllers of Reduced Dimensions, 428

5.7.3* Numerical Determination of Optimal Con-

trollers of Reduced Dimensions, 432

Contents xix

5.8Conclnsio~rs,436

5.9i'roblems, 438

Chapter 6* Linear Optimal Control Theory for Discrete-Time Systems 442

6.1Introdnction, 442

6.2Theory of Linear Discrete-Time System , 442

6.2.1Introduction, 442

6.2.2State Description of Linear Discrete-Time Systems, 443

6.2.3Interconnections of Discrete-Time and Con- tinuous-Time Systems, 443

6.2.4Solution of State Difference Equations, 452

6.2.5Stability, 454

6.2.6Transform Analysis of Linear Discrete-Time Systems, 455

6.2.7Controllability, 459

6.2.8Reconstructibility, 462

6.2.9Duality, 465

6.2.10Phase-Variable Canonical Forms, 466

6.2.11Discrete-Time Vector Stochastic Processes, 467

6.2.12Linear Discrete-Time Systems Driven by White Noise, 470

6.3Annlj!sis of Linear Discrete-Tinre Control Systems, 475

6.3.1Introduction, 475

6.3.2Discrete-Time Linear Control Systems, 475

6.3.3The Steady-State and the Transient Analysis of the Tracking Properties, 478

6.3.4Further Aspects of Linear Discrete-Time Control System Performance, 487

6.4Optinral Linear Discrete-Time State Feedbaclc Control Sj~stems,488

6.4.1Introduction, 488

6.4.2Stability Improvement by State Feedback, 488

6.4.3The Linear Discrete-Time Optimal Regulator Problem, 490

a Contents

6.4.4Steady-State Solution of the Discrete-Time Regulator Problem, 495

6.4.5The StochasticDiscrete-Time Linear Optimal Regulator, 502

6.4.6Linear Discrete-Time Regulators with Nonzero Set Points and Constant Disturbances, 504

6.4.7Asymptotic Properties of Time-Invariant Optimal Control Laws, 509

6.4.8Sensitivity, 520

6.5Optintol Linear Reconstr~tctionof the State of Linear Discrete-Time Systems, 522

6.5.1introduction, 522

6.5.2The Formulation of Linear Discrete-Time Reconstruction Problems, 522

6.5.3Discrete-Time Observers, 525

6.5.4Optimal Discrete-Time Linear Observers, 528

6.5.5Innovations, 533

6.5.6Duality of the Optimal Observer and Regulator Problems; Steady-State Properties of the Optimal Observer, 533

6.6Optirnal Linear Discrete-Time Outpvt Feedback Systen~s,536

6.6.1Introduction, 536

6.6.2The Regulation of Systems with Incomplete Measurements, 536

6.6.3Optimal Linear Discrete-Time Regulators with Incomplete and Noisy Measurements, 539

6.6.4Nonzero Set Points and Constant Disturbances, 543

6.7Concl~~sions,546

6.8Problems, 547

References

553

Index

N O T A T I O N A N D SYMBOLS

Chapters are subdivided into sections, which are numbered 1.1, 1.2, 1.3, and so on. Sections may he divided into subsections, which are numbered 1.1.1, 1.1.2, and so on. Theorems, examples, figures, and similar features are numbered consecutively within each chapter, prefixed by the chapter number. The section number is usually given in parentheses if reference is made to an item in another section.

Vectors are denoted by lowercase letters (such as x and 1 0 , matrices by uppercase letters (such as A and B) and scalars by lower case Greek lellers (such as a.and p). I t has not been possible to adhere to these rules completely consistently; notable exceptions are f for time, i andj for integers, and so on. The components of vectors are denoted by lowercase Greek letters which correspond as closely as possible to the Latin letter that denotes the vector; thus the 11-dimensionalvector x has as components the scalars C,, f2, . .., f,,, the 111-dimensional vector 1/ has as components the scalars ~ 1?la~. '. ', ?7.,, and so on. Boldrace capitals indicate the Laplace o r z-transform of the corresponding lowercase time functions [X(s) for the Laplace transform of x(t), Y(z) for the z-transform of ~ ( i )etc, .].

xT

col(fl.Czr ...,fa)

( ~ l lv>l 3 ... ?1,0

kj 9 4

9 -1 (XI.

llxll dim (x)

A~

A-'

tr ( 4 det (A)

transpose of the vector x

column vector with components fl, C,, ...,6, row vector with components ?ll, ?i2,''',TI,,

partitioning of a column vector into subveclors x, and x,

norm of a vector x dimension of the vector x transpose of the matrix A

inverse of the square matrix A trace of the square matrix A determinant of the square matrix A

xxii	Notation and Symbols
diag (A1,A,,		...,A,,)	dit~gonalmatrix with diagonal entries A,,	A,, ...,A,
(el, e,,	...	,en)	partitioning of a matrix into its columns el, e,, ...,e,,
			partitioning of a malrix into its rowsf,,f,,	...,f,

:( ):

diag (Jl,J,, ...,J,J

~ { x ( O } Re (a )

In1 (a) min (a,p) min

max

=

0

A(t), A(& A

partitioning of a matrix into column blocks TI,

T?,. ..:T ,

partitioning of a matrix into row blocks U,, U,, ...,

U,"

partitioning of a matrix into blocks A, B, C,and D

block diagonal matrix with diagonal blocks J,,

J,, . ..,J",

the real symmetric or Hermitian matrix M i s positivedefinite o r nonnegative-definite, respectively

the real symmetric o r Hermitian matrix M - N is positive-definite or nonnegative-definite, respectively

time derivative of the time-varying vectol x(t)

Laplace transform of x(t)

real part of the complex number a imaginary part of the complex number a the smallest of the numbers a and ,B

the minimum with respect to a the maximum with respect to a

zero; zero vector; zero matrix

plant matrix of a finite-dimensional linear differential system

Notntion and Symbols xxiii

input matrix of a finite-dimensional linear differential system (B becomes b in the single-input case)

output matrix of afinite-dimensional linear differential system; output matrix for the observed variable (C becomes c in the single-output case)

mean square tracking or regulating error mean square input

output matrix for the controlled variable (D becomes d in the single-output case)

base of the natural logarithm

tracking o r regulating error; reconstruction error i-th characteristic vector

expectation operator

gain matrix of the direct link of a plant (Ch. 6 only) frequency

regulator gain matrix (Fbecomes f in the single-input case)

controller transfer matrix (from y to -11) plant transfer matrix (from ti to y) integer

unit matrix a;integer

return difference matrix or function

observer gain matrix (K becomes k in the singleoutput case)

plant transfer matrix (from ti to z) closed-loop transfer matrix dimension of the state x

transfer matrix o r funclion from r to zi in a control system

controllability matrix

solution of the regulator Riccati equation controller transfer matrix (from r to ti) terminal state weighting matrix reconstructibility matrix

variance matrix; solution of the observer Riccati eqhation

initial variance matrix second-order moment matrix reference variable

weighting matrix of the state weighting matrix of the input