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1 LINEAR OPTIMAL CONTROL SYSTEMS

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Linear Optimal Control Systems I I HUIBERT KWAKERNAAK Twente Uniucrdy of Technology Enrchcde, The Nefherlur~ds i I RAPHAEL SIVAN Technion, Imel Institute of Technology Hoifo, Israel WILEY-INTERSCIENCE, a Diuision of John Wiley & Sons, Inc. New York . Chichester - Brisbane . Toronto

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Copyright 0 1972, by Jo!m Wiley &Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduclion or translation of any part of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department. John Wiley & Sons, Inc. Librnry of Corrgress Cofnlogir~g in Publimiion Drtla: Kwakernaak, Huibert. Linear optimal control systems. Bibliography: p. 1. Conlrol theory. 2. Automatic control. I. Sivan, Raphael, joint author. 11. Title Printed in the United Stat= oi America 10 9 8 7 6

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To ~iline, Annemorie, and Martin H. K. In memory of my parents Yelnrda and Toua and to my wife Ilana R. S.

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PREFACE During the last few years modem linear control theory has advanced rapidly and is now being recognized as a powerful and eminently practical tool for the solution of linear feedback control problems. The main characteristics of modern linear control theory are the state space description of systems, optimization in terms of quadratic performance criteria, and incorporation of Kalman-Bucy optimal state reconstruction theory. The significant ad- vantage of modern linear control theory over the classical theory is its ap- plicability to control problems involving multiinput multioutput systems and time-varying situations; the classical theory is essentially restricted to single- input single-output time-invariant situations. The use of the term "modem" control theory could suggest a disregard for "classical," or "conventional," control theory, namely, the theory that con- sists of design methods based upon suitably shaping the transmission and loop gain functions, employing pole-zero techniques. However, we do not share such a disregard; on the contrary, we believe that the classical approach is well-established and proven by practice, and distinguishes itself by a cnl- lection of sensible and useful goals and problem formulations. This book attempts to reconcile modern linear control theory with classical control theory. One of the major concerns of this text is to present design methods, employing modern techniques, for obtaining control systems that stand up to the requirements that have been so well developed in the classical expositions of control theory. Therefore, among other things, an entire chapter is devoted to a description of the analysis of control systems, mostly following the classical lines of thought. In the later chapters of the book, in which modern synthesis methods are developed, the chapter on analysis is recurrently referred to. Furthermore, special attention is paid to subjects that are standard in classical control theory but are frequently overlooked in modern treatments, such as nonzero set point control systems, tracking systems, and control systems that have to cope with constant disturbances. Also, heavy emphasis is placed upon the stochastic nature of control problems because the stochastic aspects are so essential. vii

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viii Preface We believe that modern and classical control theory can very well be taught simultaneously, since they cover different aspects of the same problems. There is no inherent reason for teaching the classical theory first in under- graduate courses and to defer the modern theory, particularly the stochastic part of it, to graduate courses. In fact, we believe that a modern course should be a blend of classical, modern, and stochastic control theory. This is the approach followed in this hook. The book bas been organized as follows. About half of the material, containingmost of the analysis and design methods, as well as alarge number of examples, is presented in unmarked sections. The finer points, such as conditions for existence, detailed results concerning convergence to steady- state solutions, and asymptotic properties, are dealt with in sections whose titles have been marked with an asterisk. TIE i~~iniarlcedsectro~is have been so written that they forni a textbook for a tiso-se!i~esterjirst course on control theory at the senior orfist-year grodlrate level. The marked sections consist of supplementary material of a more advanced nature. The control engineer who is interested in applying the material wiU find most design methods in the unmarked sections but may have to refer to the remaining sections for more detailed information on difficult points. The following background is assumed. The reader should have had a kst course on linear systems or linear circuits and should possess some introductory knowledge of stochastic processes. It is also recommended that the reader have some experience in digital computer programming and that he have access to a computer. We do not believe that it 1s necessary for the reader to have followed a course on classical control theory before studying the material of this book. A chapter-by-chapter description of the book follows. In Chapter 1, "Elements of Linear System Theory," the description of linear systems in terms of their state is the startingpoint, while transfer matrix and frequency response concepts are derived from the state description. Topics important for the steady-state analysis of linear optimal systems are carefully discussed. They are: controllability, stabilizability, reconstructibility, detectability, and duality. The last two sections of this chapter are devoted to a description of vector stochastic processes, with special emphasis on the representation of stochastic processes as the outputs of linear differential systems driven by white noise. In later chapters this material is extensively employed. Chapter 2, "Analysis of Control Systems," gives a general description of control problems. Furthermore, it includes a step-by-step analysis of the Various aspects of control system performance. Single-input single-output and multivariable control systems are discussed in a unified framework by the use of the concepts of mean square tracking error and mean square input.

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Preface ix Chapter 3, "Optimal Linear State Feedback Control Systems," not only presents the usual exposition of the linear optimal regulator problem but also gives a rather complete survey of the steady-state properties of the Riccati equation and the optimal regulator. It deals with the numerical solution of Riccati equations and treats stochastic optimal regulators, optimal tracking systems, and regulators with constant disturbances and nonzero set points. As a special feature, the asymptotic properties of steady-state control laws and the maximally achievable accuracy of regulators and track- ing systems are discussed. Chapter 4, "Optimal Linear Reconstruction of the State," derives the Kalman-Bucy filter starting with observer theory. Various special cases, such as singular observer problems and problems with colored observation noise, are also treated. The various steady-state and asymptotic properties of optimal observers are reviewed. In Chapter 5, "Optimal Linear Output Feedback Control Systems," the state feedback controllers of Chapter 3 are connected to the observers of Chapter 4. A heuristic and relatively simple proof of the separation principle is presented based on the innovations concept, which is discussed in Chapter 4. Guidelines are given for the des~gn of various types of output feedback control systems, and a review of the design of reduced-order controllers is included. In Chapter 6, "Linear Optimal ControlTheory for Discrete-Time Systems," the entire theory of Chapters 1 through 5 is repeated in condensed form for linear discrete-time control systems. Special attention is given to state dead- beat and output deadbeat control systems, and to questions concerning the synchronization of the measurements and the control actuation. Throughout the book important concepts are introduced in definitions, and the main results summarized in the form of theorems. Almost every section concludes with one or more examples, many of which are numerical. These examples serve to clarify the material of the text and, by their physical significance, to emphasize the practical applicability of the results. Most examples are continuations of earlier examples so that a specific problem is developed over several sections or even chapters. Whenever numerical values are used, care has been taken to designate the proper dimensions of the various quantities. To this end, the SI system of units has been employed, which is now being internationally accepted (see, e.g., Barrow, 1966; IEEE Standards Committee, 1970). A complete review of the SI system can be found in the Reconinieiidotiotis of the International Organizat~on for Stand- ardization (various dates). The book contains about 50 problems. They can be divided into two categories: elementary exercises, directly illustrating the material of the text; and supplementary results, extending the material of the text. A few of the

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problems require the use of a digital computer. The problems marked with an asterisk are not considered to belong to the textbook material. Suitable term projects could consist of writing and testing the computer subroutines listed in Section 5.8. Many references are quoted throughout the book, but no attempt has been made to reach any degree of completeness or to do justice to history. The fact that a particular publication is mentioned simply means that it has been used by us as source material or that related material can be found in it. The references are indicated by the author's name, the year of publication, and a letter indicating which publication is intended (e.g., Miller, 1971b).

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ACKNOWLEDGMENTS The first author wishes to express 111s thanks to the Department of Applied Physics at the Delft University of Technology, where he worked until April, 1970, and to the Department of Applied Mathematics at the Twente Uni- versity of Technology for invaluable support during the writing of this book in terms of time granted and facilities made available. The second author extends lus thanks to the Technion, the Israel Institute of Technology, for supporting the writing oFt11e book. Time on the preparation of the manuscript was spent by the second author while he was a National Research Council Senior Research Associate at the NASA Langley Research Center, Hampton, Virginia, dur~ng the academic year 1970-1971. Without the assistance of these institutions, and their help in financing various trips to Israel, the Netherlands, and the United States, it would not have been possible to complete this book. Several typists spent their efforts on the various versions of the manuscript. Special mention should be made of the extremely diligent and competent work of Miss Marja Genemans of Delft and Mrs. Dini Rengelink of Twente. The line drawings were made by Mr. M. G. Langen of Delft, who is com- mended for his accurate and carelul work. Final thanks are due to one of the first author's former students, Mr. J. H. van Schuppen, For his comments on the text and for programming and working examples, and to Mr. R. C. W. Strijbos of Twente and Prof. J. van de Vegte, Toronto, for their comments on early versions of the manuscript. The final manuscript was read by Prof. L. Hasdorff of the Virginia Poly- technic Institute and Dr. Paul Alper of Twente; their constructive criticism and remarks are greatly appreciated. The second author is grateful to his graduate students, in particular to Victor Shenkar, for helping to correct early versions of the manuscript. H. K. R. S.

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CONTENTS Notation and Symbols Chapter 1 Elements of Linear System Theory 1 1.1 Introduction, 1 1.2 State Desc~ipfion of Linear Syslems, 1 1.2.1 State Description of Nonlinear and Linear Differential Systems, 1 1.2.2 Linearization, 2 1.2.3 Examples, 3 1.2.4 State Transformations, 10 1.3 Solution of tlre State Differential Equotion of Linear Sj~stenrs, 11 1.3.1 The Transition Matrix and the Impulse Response Matrix, 11 1.3.2 The Transition Matrix of a Time-Invariant System, 13 1.3.3 Diagonalization, 15 1.3.4" The Jordan Form, 19 1.4 Stability, 24 1.4.1 Definitions of Stability, 24 1.4.2 Stability of Time-Invariant Linear Systems, 27 1.4.3' Stable and Unstable Subspaces far Time-In- variant Linear Systems, 29 1.4.4" Investigation of the Stability of Nonlinear Systems through Linearization, 31 1.5 Transform Analysis of Time-Znua~iant Systems, 33 1.5.1 Solution of the State Differential Equation through Laplace Transformation, 33 'See the Preface for the significance of the marked sections. xiii

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xiv Contents 1.5.2 Frequency Response, 37 1.5.3 Zeroes of Transfer Matrices, 39 1.5.4 Snterconnections of Linear Systems, 43 1.59 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 1.10.1 Defmitions, 85 1.10.2 Power Spectral Density Matrices, 90 1.10.3 The Response of Linear Systems to Sto- chastic Inputs, 91 1.10.4 Quadratic Expressions, 94 The Response of Linear Differcatial Sjtstems to Tbite Noise, 97 1.11.1 White Noise, 97 1.11.2 Linear Differential Systems Driven by White Noise, 100 1.11.3 The Steady-State Variance Matrix for the Time-Invariant Case, 103 1.11.4 Modeling of Stochastic Processes, 106 1.11.5 Quadratic Integral Expressions, 108 Probleins, 113

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Contents xv Chapter 2 Analysis of Linear Control Systems 119 2.1 Inb.odrrction, 119 2.2 The Forr~nrlation of Control Prohlerns, 121 2.2.1 Introduction, 121 2.2.2 The Formulation of Tracking and Regulator Problems, 121 2.2.3 The Formulation of Terminal Control Problems, 127 2.3 Closed-Loop Controllers; The Basic Design Ohjectiue, 128 2.4 The Stability of Control Systents, 136 2.5 The Steadjz-State Analysis of tlrc Tracking Properties, 140 2.5.1 The Steady-State Mean Square Tracking Error and Input, 140 2.5.2 The Single-Input Single-Output Case, 144 2.5.3 The Multiinput Mnltioutput Case, 155 2.6 The Transient Analysis of tlre Tracking Properties, 165 2.7 The Effects of Disturbances in tlre Single-fi~prrl Single- Ontptrt Case, 167 2.8 The Effects of Observation Noise in the Single-Inpnt Single-Ontpnt Case, 174 2.9 Tlre Effect of Plant Paranteter Uncertainty in the Single-Inpat Single-Ostpnt Case, 178 2.10* Tlte Open-Loop Steady-State Eqsiualent Control Schenre, 183 2.11 Conclrrsions, 188 2.12 P~.ohlerns, 189 Chapter 3 Optimal Linear\ State Feedback Control Systems 193 3.1 Introdaction, 193 3.2 Stability Intprouentcnt of Linear Systcnts by State Feedhacli, 193 3.2.1 Linear State Feedback Control, 193

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xvi Contents 3.2.2" Conditions for Pole Assignment and Stabilization, 198 3.3 The Deterministic Linear Optirnal Regulator Problem, 201 3.3.1 introduction, 201 3.3.2 Solution of the Regulator Problem, 207 3.3.3 Derivation of the Riccati Equation, 216 3.4 Steady-State Solrrtion of tlre Deterministic Linear Optirrral Regrrlator Problenr, 220 3.4.1 Introduction and Summary of Main Results, 220 3.4.2' Steady-State Properties or Optimal Regulators, 230 3.4.3" Steady-State Properties of the Time- Invariant Optimal Regulator, 237 3.4.4* Solution of the Time-Invariant Regulator Problem by Diagonalization, 243 3.5 Nrrnrerical Solution of tlrc Riccnti Eqrration, 248 3.5.1 Direct Integration. 248 3.5.2 The Kalman-Englar Method, 249 3.5.3" Solution by Diagonalization, 250 3.5.4" Solution by the Newton-Raphson Method, 251 3.6 Stochastic Lincnr Optirnal Regrrlntor nnd Tracking Problems, 253 3.6.1 Regulator Problems with Disturbances- The Stochastic Regulator Problem, 253 3.6.2 Stochastic Tracking Problems, 257 3.6.3 Solution of the Stochastic Linear Optimal Regulator Problem, 259 3.7 Regulators 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

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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 3.10 Conclnsior~s, 318 3.11 Problems, 319 Chnpter 4 Optimal Linenr Reconstruction of the Stnte 4.1 Zntrodnction, 328 4.2 Observers, 329 4.2.1 Full-Order Observers, 329 4.2.2* Conditions for Pole Assignment and Stabilization of Observers, 334 4.2.3" Reduced-Order Observers, 335 4.3 The Optimal Obscrucr, 339 4.3.1 A Stochastic Approach to the Observer Problem, 339 4.3.2 The 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 Observation Noises, 351 4.3.4"' The Time-Invariant Singular Optimal Observer Problem, 352 4.3.5' 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

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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 4.5 Corrclrrsions, 373 4.6 Problerrrs, 373 Chnpter 5 Optimal Linear Output Feedback Control Systems 377 5.1 Introdrrction, 377 5.2 The Regrrlation of Linear Systenrs ~cfitlr Incomplete Measrrremcnts, 378 5.2.1 The Structure of Output Feedback Control Systems, 378 5.2.2* Conditions for Pole Assignment and Stabilization of Output Feedback Control Systems, 388 5.3 Optinral Linear Regrrlalors ~vitlr Inconrplete and Noisy Measrrrerrrmts, 389 5.3.1 Problem Formulation and Solution, 389 5.3.2 Evaluation of the Performance of Optimal Output Feedback Regulators, 391 5.3.3* Proof of the Separation Principle, 400 5.4 Linear Optirnal Tracking Systenrs wit11 Incorrrglete and Noisy Measrrrcnrents, 402 5.5 Regrrlators 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

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Contents xix 5.8 Conclnsio~rs, 436 5.9 i'roblems, 438 Chapter 6* Linear Optimal Control Theory for Discrete-Time Systems 442 6.1 Introdnction, 442 6.2 Theory of Linear Discrete-Time System, 442 6.2.1 Introduction, 442 6.2.2 State Description of Linear Discrete-Time Systems, 443 6.2.3 Interconnections of Discrete-Time and Con- tinuous-Time Systems, 443 6.2.4 Solution of State Difference Equations, 452 6.2.5 Stability, 454 6.2.6 Transform Analysis of Linear Discrete-Time Systems, 455 6.2.7 Controllability, 459 6.2.8 Reconstructibility, 462 6.2.9 Duality, 465 6.2.10 Phase-Variable Canonical Forms, 466 6.2.11 Discrete-Time Vector Stochastic Processes, 467 6.2.12 Linear Discrete-Time Systems Driven by White Noise, 470 6.3 Annlj!sis of Linear Discrete-Tinre Control Systems, 475 6.3.1 Introduction, 475 6.3.2 Discrete-Time Linear Control Systems, 475 6.3.3 The Steady-State and the Transient Analysis of the Tracking Properties, 478 6.3.4 Further Aspects of Linear Discrete-Time Control System Performance, 487 6.4 Optinral Linear Discrete-Time State Feedbaclc Control Sj~stems, 488 6.4.1 Introduction, 488 6.4.2 Stability Improvement by State Feedback, 488 6.4.3 The Linear Discrete-Time Optimal Regulator Problem, 490

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a Contents 6.4.4 Steady-State Solution of the Discrete-Time Regulator Problem, 495 6.4.5 The StochasticDiscrete-Time Linear Optimal Regulator, 502 6.4.6 Linear Discrete-Time Regulators with Non- zero Set Points and Constant Disturbances, 504 6.4.7 Asymptotic Properties of Time-Invariant Optimal Control Laws, 509 6.4.8 Sensitivity, 520 6.5 Optintol Linear Reconstr~tction of the State of Linear Discrete-Time Systems, 522 6.5.1 introduction, 522 6.5.2 The Formulation of Linear Discrete-Time Reconstruction Problems, 522 6.5.3 Discrete-Time Observers, 525 6.5.4 Optimal Discrete-Time Linear Observers, 528 6.5.5 Innovations, 533 6.5.6 Duality of the Optimal Observer and Regulator Problems; Steady-State Prop- erties of the Optimal Observer, 533 6.6 Optirnal Linear Discrete-Time Outpvt Feedback Systen~s, 536 6.6.1 Introduction, 536 6.6.2 The Regulation of Systems with Incomplete Measurements, 536 6.6.3 Optimal Linear Discrete-Time Regulators with Incomplete and Noisy Measurements, 539 6.6.4 Nonzero Set Points and Constant Distur- bances, 543 6.7 Concl~~sions, 546 6.8 Problems, 547 References 553 Index

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NOTATION AND 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 10, matrices by uppercase letters (such as A and B) and scalars by lower case Greek lellers (such as a. and p). It has not been possible to adhere to these rules completely consistently; notable exceptions are f for time, i and j 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-dimensional vector 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 or 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) (~ll> vl3 . . . 9 ?1,0 kj 9 -1 (XI. 4 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

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xxii Notation and Symbols diag (A1, A,, . . . , A,,) (el, e,, . . . , en) (: :) diag (Jl, J,, . . . , J,J ~{x(O} Re (a) In1 (a) min (a, p) min max = dit~gonal matrix with diagonal entries A,, A,, . . . , A, partitioning of a matrix into its columns el, e,, . . . , e,, partitioning of a malrix into its rowsf,,f,, . . . , f, 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 Mis positive- definite or nonnegative-definite, respectively the real symmetric or Hermitian matrix M - N is positive-definite or nonnegative-definite, respec- tively 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 0 zero; zero vector; zero matrix A(t), A(& A plant matrix of a finite-dimensional linear differential system

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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 or 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 (F becomes 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 single- output case) plant transfer matrix (from ti to z) closed-loop transfer matrix dimension of the state x transfer matrix or 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

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nriv Notntion and Symbols covariance function of the stochastic process v variable of the Laplace transform sensitivity malrix or function time transmission input variable stochastic process observation noise, measurement noise constant disturbance equivalent disturbance at the controlled variable disturbance variable intensity of a white noise process white noise process weighting matrix of the tracking or regulating error weighting matrix of the input stale variable reconstructed state variable initial state output variable; obsenied variable z-transform variable controlled variable compound matrix of hystem and adjoint differential equations delta function sampling interval scalar controlled variable scalar output variable; scalar observed variable time difference; time constant; normalized angular frequency i-th characteristic value scalar input variable scalar stochastic process i-th zero scalar state variable i-th pole weighting coefficient of the integrated or mean square input spectral density matrix of the stochastic process u characteristic polynomial closed-loop characteristic polynomial

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A Hz kg kmol m N rad S v n transition matrix numerator polynomial angular frequency ampere hertz kilogram kilomole meter newton radian second volt ohm Notntion and Symbols xxv

New York Chichester Brisbane Toronto brPage 3br Copyright 0 1972 by Jom Wiley Sons Inc All rights reserved Published simultaneously in Canada Reproduclion or translation of any part of this work beyond that permitted by Sections 107 or 108 of the ID: 23079

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1 LINEAR OPTIMAL CONTROL SYSTEMS

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Linear Optimal Control Systems I I HUIBERT KWAKERNAAK Twente Uniucrdy of Technology Enrchcde, The Nefherlur~ds i I RAPHAEL SIVAN Technion, Imel Institute of Technology Hoifo, Israel WILEY-INTERSCIENCE, a Diuision of John Wiley & Sons, Inc. New York . Chichester - Brisbane . Toronto

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Copyright 0 1972, by Jo!m Wiley &Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduclion or translation of any part of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department. John Wiley & Sons, Inc. Librnry of Corrgress Cofnlogir~g in Publimiion Drtla: Kwakernaak, Huibert. Linear optimal control systems. Bibliography: p. 1. Conlrol theory. 2. Automatic control. I. Sivan, Raphael, joint author. 11. Title Printed in the United Stat= oi America 10 9 8 7 6

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To ~iline, Annemorie, and Martin H. K. In memory of my parents Yelnrda and Toua and to my wife Ilana R. S.

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PREFACE During the last few years modem linear control theory has advanced rapidly and is now being recognized as a powerful and eminently practical tool for the solution of linear feedback control problems. The main characteristics of modern linear control theory are the state space description of systems, optimization in terms of quadratic performance criteria, and incorporation of Kalman-Bucy optimal state reconstruction theory. The significant ad- vantage of modern linear control theory over the classical theory is its ap- plicability to control problems involving multiinput multioutput systems and time-varying situations; the classical theory is essentially restricted to single- input single-output time-invariant situations. The use of the term "modem" control theory could suggest a disregard for "classical," or "conventional," control theory, namely, the theory that con- sists of design methods based upon suitably shaping the transmission and loop gain functions, employing pole-zero techniques. However, we do not share such a disregard; on the contrary, we believe that the classical approach is well-established and proven by practice, and distinguishes itself by a cnl- lection of sensible and useful goals and problem formulations. This book attempts to reconcile modern linear control theory with classical control theory. One of the major concerns of this text is to present design methods, employing modern techniques, for obtaining control systems that stand up to the requirements that have been so well developed in the classical expositions of control theory. Therefore, among other things, an entire chapter is devoted to a description of the analysis of control systems, mostly following the classical lines of thought. In the later chapters of the book, in which modern synthesis methods are developed, the chapter on analysis is recurrently referred to. Furthermore, special attention is paid to subjects that are standard in classical control theory but are frequently overlooked in modern treatments, such as nonzero set point control systems, tracking systems, and control systems that have to cope with constant disturbances. Also, heavy emphasis is placed upon the stochastic nature of control problems because the stochastic aspects are so essential. vii

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viii Preface We believe that modern and classical control theory can very well be taught simultaneously, since they cover different aspects of the same problems. There is no inherent reason for teaching the classical theory first in under- graduate courses and to defer the modern theory, particularly the stochastic part of it, to graduate courses. In fact, we believe that a modern course should be a blend of classical, modern, and stochastic control theory. This is the approach followed in this hook. The book bas been organized as follows. About half of the material, containingmost of the analysis and design methods, as well as alarge number of examples, is presented in unmarked sections. The finer points, such as conditions for existence, detailed results concerning convergence to steady- state solutions, and asymptotic properties, are dealt with in sections whose titles have been marked with an asterisk. TIE i~~iniarlcedsectro~is have been so written that they forni a textbook for a tiso-se!i~esterjirst course on control theory at the senior orfist-year grodlrate level. The marked sections consist of supplementary material of a more advanced nature. The control engineer who is interested in applying the material wiU find most design methods in the unmarked sections but may have to refer to the remaining sections for more detailed information on difficult points. The following background is assumed. The reader should have had a kst course on linear systems or linear circuits and should possess some introductory knowledge of stochastic processes. It is also recommended that the reader have some experience in digital computer programming and that he have access to a computer. We do not believe that it 1s necessary for the reader to have followed a course on classical control theory before studying the material of this book. A chapter-by-chapter description of the book follows. In Chapter 1, "Elements of Linear System Theory," the description of linear systems in terms of their state is the startingpoint, while transfer matrix and frequency response concepts are derived from the state description. Topics important for the steady-state analysis of linear optimal systems are carefully discussed. They are: controllability, stabilizability, reconstructibility, detectability, and duality. The last two sections of this chapter are devoted to a description of vector stochastic processes, with special emphasis on the representation of stochastic processes as the outputs of linear differential systems driven by white noise. In later chapters this material is extensively employed. Chapter 2, "Analysis of Control Systems," gives a general description of control problems. Furthermore, it includes a step-by-step analysis of the Various aspects of control system performance. Single-input single-output and multivariable control systems are discussed in a unified framework by the use of the concepts of mean square tracking error and mean square input.

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Preface ix Chapter 3, "Optimal Linear State Feedback Control Systems," not only presents the usual exposition of the linear optimal regulator problem but also gives a rather complete survey of the steady-state properties of the Riccati equation and the optimal regulator. It deals with the numerical solution of Riccati equations and treats stochastic optimal regulators, optimal tracking systems, and regulators with constant disturbances and nonzero set points. As a special feature, the asymptotic properties of steady-state control laws and the maximally achievable accuracy of regulators and track- ing systems are discussed. Chapter 4, "Optimal Linear Reconstruction of the State," derives the Kalman-Bucy filter starting with observer theory. Various special cases, such as singular observer problems and problems with colored observation noise, are also treated. The various steady-state and asymptotic properties of optimal observers are reviewed. In Chapter 5, "Optimal Linear Output Feedback Control Systems," the state feedback controllers of Chapter 3 are connected to the observers of Chapter 4. A heuristic and relatively simple proof of the separation principle is presented based on the innovations concept, which is discussed in Chapter 4. Guidelines are given for the des~gn of various types of output feedback control systems, and a review of the design of reduced-order controllers is included. In Chapter 6, "Linear Optimal ControlTheory for Discrete-Time Systems," the entire theory of Chapters 1 through 5 is repeated in condensed form for linear discrete-time control systems. Special attention is given to state dead- beat and output deadbeat control systems, and to questions concerning the synchronization of the measurements and the control actuation. Throughout the book important concepts are introduced in definitions, and the main results summarized in the form of theorems. Almost every section concludes with one or more examples, many of which are numerical. These examples serve to clarify the material of the text and, by their physical significance, to emphasize the practical applicability of the results. Most examples are continuations of earlier examples so that a specific problem is developed over several sections or even chapters. Whenever numerical values are used, care has been taken to designate the proper dimensions of the various quantities. To this end, the SI system of units has been employed, which is now being internationally accepted (see, e.g., Barrow, 1966; IEEE Standards Committee, 1970). A complete review of the SI system can be found in the Reconinieiidotiotis of the International Organizat~on for Stand- ardization (various dates). The book contains about 50 problems. They can be divided into two categories: elementary exercises, directly illustrating the material of the text; and supplementary results, extending the material of the text. A few of the

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problems require the use of a digital computer. The problems marked with an asterisk are not considered to belong to the textbook material. Suitable term projects could consist of writing and testing the computer subroutines listed in Section 5.8. Many references are quoted throughout the book, but no attempt has been made to reach any degree of completeness or to do justice to history. The fact that a particular publication is mentioned simply means that it has been used by us as source material or that related material can be found in it. The references are indicated by the author's name, the year of publication, and a letter indicating which publication is intended (e.g., Miller, 1971b).

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ACKNOWLEDGMENTS The first author wishes to express 111s thanks to the Department of Applied Physics at the Delft University of Technology, where he worked until April, 1970, and to the Department of Applied Mathematics at the Twente Uni- versity of Technology for invaluable support during the writing of this book in terms of time granted and facilities made available. The second author extends lus thanks to the Technion, the Israel Institute of Technology, for supporting the writing oFt11e book. Time on the preparation of the manuscript was spent by the second author while he was a National Research Council Senior Research Associate at the NASA Langley Research Center, Hampton, Virginia, dur~ng the academic year 1970-1971. Without the assistance of these institutions, and their help in financing various trips to Israel, the Netherlands, and the United States, it would not have been possible to complete this book. Several typists spent their efforts on the various versions of the manuscript. Special mention should be made of the extremely diligent and competent work of Miss Marja Genemans of Delft and Mrs. Dini Rengelink of Twente. The line drawings were made by Mr. M. G. Langen of Delft, who is com- mended for his accurate and carelul work. Final thanks are due to one of the first author's former students, Mr. J. H. van Schuppen, For his comments on the text and for programming and working examples, and to Mr. R. C. W. Strijbos of Twente and Prof. J. van de Vegte, Toronto, for their comments on early versions of the manuscript. The final manuscript was read by Prof. L. Hasdorff of the Virginia Poly- technic Institute and Dr. Paul Alper of Twente; their constructive criticism and remarks are greatly appreciated. The second author is grateful to his graduate students, in particular to Victor Shenkar, for helping to correct early versions of the manuscript. H. K. R. S.

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CONTENTS Notation and Symbols Chapter 1 Elements of Linear System Theory 1 1.1 Introduction, 1 1.2 State Desc~ipfion of Linear Syslems, 1 1.2.1 State Description of Nonlinear and Linear Differential Systems, 1 1.2.2 Linearization, 2 1.2.3 Examples, 3 1.2.4 State Transformations, 10 1.3 Solution of tlre State Differential Equotion of Linear Sj~stenrs, 11 1.3.1 The Transition Matrix and the Impulse Response Matrix, 11 1.3.2 The Transition Matrix of a Time-Invariant System, 13 1.3.3 Diagonalization, 15 1.3.4" The Jordan Form, 19 1.4 Stability, 24 1.4.1 Definitions of Stability, 24 1.4.2 Stability of Time-Invariant Linear Systems, 27 1.4.3' Stable and Unstable Subspaces far Time-In- variant Linear Systems, 29 1.4.4" Investigation of the Stability of Nonlinear Systems through Linearization, 31 1.5 Transform Analysis of Time-Znua~iant Systems, 33 1.5.1 Solution of the State Differential Equation through Laplace Transformation, 33 'See the Preface for the significance of the marked sections. xiii

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xiv Contents 1.5.2 Frequency Response, 37 1.5.3 Zeroes of Transfer Matrices, 39 1.5.4 Snterconnections of Linear Systems, 43 1.59 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 1.10.1 Defmitions, 85 1.10.2 Power Spectral Density Matrices, 90 1.10.3 The Response of Linear Systems to Sto- chastic Inputs, 91 1.10.4 Quadratic Expressions, 94 The Response of Linear Differcatial Sjtstems to Tbite Noise, 97 1.11.1 White Noise, 97 1.11.2 Linear Differential Systems Driven by White Noise, 100 1.11.3 The Steady-State Variance Matrix for the Time-Invariant Case, 103 1.11.4 Modeling of Stochastic Processes, 106 1.11.5 Quadratic Integral Expressions, 108 Probleins, 113

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Contents xv Chapter 2 Analysis of Linear Control Systems 119 2.1 Inb.odrrction, 119 2.2 The Forr~nrlation of Control Prohlerns, 121 2.2.1 Introduction, 121 2.2.2 The Formulation of Tracking and Regulator Problems, 121 2.2.3 The Formulation of Terminal Control Problems, 127 2.3 Closed-Loop Controllers; The Basic Design Ohjectiue, 128 2.4 The Stability of Control Systents, 136 2.5 The Steadjz-State Analysis of tlrc Tracking Properties, 140 2.5.1 The Steady-State Mean Square Tracking Error and Input, 140 2.5.2 The Single-Input Single-Output Case, 144 2.5.3 The Multiinput Mnltioutput Case, 155 2.6 The Transient Analysis of tlre Tracking Properties, 165 2.7 The Effects of Disturbances in tlre Single-fi~prrl Single- Ontptrt Case, 167 2.8 The Effects of Observation Noise in the Single-Inpnt Single-Ontpnt Case, 174 2.9 Tlre Effect of Plant Paranteter Uncertainty in the Single-Inpat Single-Ostpnt Case, 178 2.10* Tlte Open-Loop Steady-State Eqsiualent Control Schenre, 183 2.11 Conclrrsions, 188 2.12 P~.ohlerns, 189 Chapter 3 Optimal Linear\ State Feedback Control Systems 193 3.1 Introdaction, 193 3.2 Stability Intprouentcnt of Linear Systcnts by State Feedhacli, 193 3.2.1 Linear State Feedback Control, 193

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xvi Contents 3.2.2" Conditions for Pole Assignment and Stabilization, 198 3.3 The Deterministic Linear Optirnal Regulator Problem, 201 3.3.1 introduction, 201 3.3.2 Solution of the Regulator Problem, 207 3.3.3 Derivation of the Riccati Equation, 216 3.4 Steady-State Solrrtion of tlre Deterministic Linear Optirrral Regrrlator Problenr, 220 3.4.1 Introduction and Summary of Main Results, 220 3.4.2' Steady-State Properties or Optimal Regulators, 230 3.4.3" Steady-State Properties of the Time- Invariant Optimal Regulator, 237 3.4.4* Solution of the Time-Invariant Regulator Problem by Diagonalization, 243 3.5 Nrrnrerical Solution of tlrc Riccnti Eqrration, 248 3.5.1 Direct Integration. 248 3.5.2 The Kalman-Englar Method, 249 3.5.3" Solution by Diagonalization, 250 3.5.4" Solution by the Newton-Raphson Method, 251 3.6 Stochastic Lincnr Optirnal Regrrlntor nnd Tracking Problems, 253 3.6.1 Regulator Problems with Disturbances- The Stochastic Regulator Problem, 253 3.6.2 Stochastic Tracking Problems, 257 3.6.3 Solution of the Stochastic Linear Optimal Regulator Problem, 259 3.7 Regulators 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

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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 3.10 Conclnsior~s, 318 3.11 Problems, 319 Chnpter 4 Optimal Linenr Reconstruction of the Stnte 4.1 Zntrodnction, 328 4.2 Observers, 329 4.2.1 Full-Order Observers, 329 4.2.2* Conditions for Pole Assignment and Stabilization of Observers, 334 4.2.3" Reduced-Order Observers, 335 4.3 The Optimal Obscrucr, 339 4.3.1 A Stochastic Approach to the Observer Problem, 339 4.3.2 The 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 Observation Noises, 351 4.3.4"' The Time-Invariant Singular Optimal Observer Problem, 352 4.3.5' 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

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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 4.5 Corrclrrsions, 373 4.6 Problerrrs, 373 Chnpter 5 Optimal Linear Output Feedback Control Systems 377 5.1 Introdrrction, 377 5.2 The Regrrlation of Linear Systenrs ~cfitlr Incomplete Measrrremcnts, 378 5.2.1 The Structure of Output Feedback Control Systems, 378 5.2.2* Conditions for Pole Assignment and Stabilization of Output Feedback Control Systems, 388 5.3 Optinral Linear Regrrlalors ~vitlr Inconrplete and Noisy Measrrrerrrmts, 389 5.3.1 Problem Formulation and Solution, 389 5.3.2 Evaluation of the Performance of Optimal Output Feedback Regulators, 391 5.3.3* Proof of the Separation Principle, 400 5.4 Linear Optirnal Tracking Systenrs wit11 Incorrrglete and Noisy Measrrrcnrents, 402 5.5 Regrrlators 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

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Contents xix 5.8 Conclnsio~rs, 436 5.9 i'roblems, 438 Chapter 6* Linear Optimal Control Theory for Discrete-Time Systems 442 6.1 Introdnction, 442 6.2 Theory of Linear Discrete-Time System, 442 6.2.1 Introduction, 442 6.2.2 State Description of Linear Discrete-Time Systems, 443 6.2.3 Interconnections of Discrete-Time and Con- tinuous-Time Systems, 443 6.2.4 Solution of State Difference Equations, 452 6.2.5 Stability, 454 6.2.6 Transform Analysis of Linear Discrete-Time Systems, 455 6.2.7 Controllability, 459 6.2.8 Reconstructibility, 462 6.2.9 Duality, 465 6.2.10 Phase-Variable Canonical Forms, 466 6.2.11 Discrete-Time Vector Stochastic Processes, 467 6.2.12 Linear Discrete-Time Systems Driven by White Noise, 470 6.3 Annlj!sis of Linear Discrete-Tinre Control Systems, 475 6.3.1 Introduction, 475 6.3.2 Discrete-Time Linear Control Systems, 475 6.3.3 The Steady-State and the Transient Analysis of the Tracking Properties, 478 6.3.4 Further Aspects of Linear Discrete-Time Control System Performance, 487 6.4 Optinral Linear Discrete-Time State Feedbaclc Control Sj~stems, 488 6.4.1 Introduction, 488 6.4.2 Stability Improvement by State Feedback, 488 6.4.3 The Linear Discrete-Time Optimal Regulator Problem, 490

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a Contents 6.4.4 Steady-State Solution of the Discrete-Time Regulator Problem, 495 6.4.5 The StochasticDiscrete-Time Linear Optimal Regulator, 502 6.4.6 Linear Discrete-Time Regulators with Non- zero Set Points and Constant Disturbances, 504 6.4.7 Asymptotic Properties of Time-Invariant Optimal Control Laws, 509 6.4.8 Sensitivity, 520 6.5 Optintol Linear Reconstr~tction of the State of Linear Discrete-Time Systems, 522 6.5.1 introduction, 522 6.5.2 The Formulation of Linear Discrete-Time Reconstruction Problems, 522 6.5.3 Discrete-Time Observers, 525 6.5.4 Optimal Discrete-Time Linear Observers, 528 6.5.5 Innovations, 533 6.5.6 Duality of the Optimal Observer and Regulator Problems; Steady-State Prop- erties of the Optimal Observer, 533 6.6 Optirnal Linear Discrete-Time Outpvt Feedback Systen~s, 536 6.6.1 Introduction, 536 6.6.2 The Regulation of Systems with Incomplete Measurements, 536 6.6.3 Optimal Linear Discrete-Time Regulators with Incomplete and Noisy Measurements, 539 6.6.4 Nonzero Set Points and Constant Distur- bances, 543 6.7 Concl~~sions, 546 6.8 Problems, 547 References 553 Index

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NOTATION AND 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 10, matrices by uppercase letters (such as A and B) and scalars by lower case Greek lellers (such as a. and p). It has not been possible to adhere to these rules completely consistently; notable exceptions are f for time, i and j 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-dimensional vector 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 or 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) (~ll> vl3 . . . 9 ?1,0 kj 9 -1 (XI. 4 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

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xxii Notation and Symbols diag (A1, A,, . . . , A,,) (el, e,, . . . , en) (: :) diag (Jl, J,, . . . , J,J ~{x(O} Re (a) In1 (a) min (a, p) min max = dit~gonal matrix with diagonal entries A,, A,, . . . , A, partitioning of a matrix into its columns el, e,, . . . , e,, partitioning of a malrix into its rowsf,,f,, . . . , f, 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 Mis positive- definite or nonnegative-definite, respectively the real symmetric or Hermitian matrix M - N is positive-definite or nonnegative-definite, respec- tively 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 0 zero; zero vector; zero matrix A(t), A(& A plant matrix of a finite-dimensional linear differential system

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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 or 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 (F becomes 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 single- output case) plant transfer matrix (from ti to z) closed-loop transfer matrix dimension of the state x transfer matrix or 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

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nriv Notntion and Symbols covariance function of the stochastic process v variable of the Laplace transform sensitivity malrix or function time transmission input variable stochastic process observation noise, measurement noise constant disturbance equivalent disturbance at the controlled variable disturbance variable intensity of a white noise process white noise process weighting matrix of the tracking or regulating error weighting matrix of the input stale variable reconstructed state variable initial state output variable; obsenied variable z-transform variable controlled variable compound matrix of hystem and adjoint differential equations delta function sampling interval scalar controlled variable scalar output variable; scalar observed variable time difference; time constant; normalized angular frequency i-th characteristic value scalar input variable scalar stochastic process i-th zero scalar state variable i-th pole weighting coefficient of the integrated or mean square input spectral density matrix of the stochastic process u characteristic polynomial closed-loop characteristic polynomial

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A Hz kg kmol m N rad S v n transition matrix numerator polynomial angular frequency ampere hertz kilogram kilomole meter newton radian second volt ohm Notntion and Symbols xxv

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