linear optimal control syst

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1 L I N E A R O P T I M A LC O N T R O L SYSTEMS


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Linear OptimalControl Systems

II H U I B E R T K W A K E R N A A K

Twente Uniucrdy of TechnologyEnrchcde, The Nefherlur~ds

iI R A P H A E L SIVANTechnion, I m e l Institute of TechnologyHoifo, Israel

WILEY-INTERSCIENCE, a Diuision of John Wiley & Sons, Inc.New York . Chichester - Brisbane . oronto


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Copyright 0 972, by Jo!m Wiley &Sons, Inc.A l l rights reserved. Published simultaneously in Canada.Reproduclion or translation of any part of this work beyondthat permitted by Sections 107 or 108 of the 1976United StatesCopyright Act without the permission of the copyright owneris unlawful. Requests for permission or further informationshould be addressed to the Permissions Department. JohnWiley & Sons, Inc.Librnry of Corrgress Cof nlog ir~gn Publim iion Drtla:Kwakernaak, Huibert.Linear optimal control systems.Bibliography: p.1. Conlrol theory. 2. Automatic control.I. Sivan, Raphael, joint author. 11. Title

Printed i n the United Stat= o i America10 9 8 7 6


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T o ~ i l i n e , nnemorie, and MartinH . K.

In memory of my parents Yelnrda and Toua and to m y wife IlanaR . S.


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P R E F A C E

Durin g the last few years mo de m linear contro l theory has advanced rapidlyand is now being recognized as a powerful a nd eminently practical tool fo rthe solution of linear feedback co ntrol problems. T he main characteristics ofmodern linear control theory are the state space description of systems,optimization in terms of quadratic performance criteria, and incorporationof Kalman-Bucy optim al 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 m ultioutput systems a ndtime-varying s itua tions ; 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 fo r"classical," o r "conventional," con trol theory, namely, the theory that con-sists of design methods based upon suitably shaping the transmission andloop gain functions, employing pole-zero techniques. However, we do notshare such a disregard; on the contrary, we believe that the classical appro achis well-established an d prove n by prac tice, and distinguishes itself by a cnl-lection of sensible and useful goals and problem formulations.This bo ok attem pts to reconcile modern linear control theory with classicalcontrol theory. One of the major concerns of this text is to present designmethods, employing modern techniques, for obtaining control systems thatstand u p to the requirements that have been so well developed in the classicalexpositions of control theory. Therefore, among other things, an entirecha pter is devoted t o a description of the analysis of contro l systems, mostlyfollowing the classical lines of thought. In the later chapters of the book, inwhich modern synthesis methods are developed, the chapter on analysis isrecurrently referred to. Furthe rmo re, special attention is paid to subjects tha tare standard in classical control theory but are frequently overlooked inmodern treatments, such as nonzero set point control systems, trackingsystems, and control systems that have to cope with constant disturbances.Also, heavy emphasis is placed upon the stochastic nature of control problem sbecause the stochastic aspects are so essential.

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viii PrefaceW e believe t ha t modern and classical control theory can very well be taugh tsimultaneously, since they cover different aspects of the same problems.There is no inherent reason for teaching the classical theory first in under-

gradua te courses an d t o defer the modern theory, particularly the stochasticpa rt of it, t o graduate courses. In fact, we believe tha t a modern courseshould be a blend of classical, m odern, a nd stochastic control theory. Th is isthe app roach followed in this hook.The book bas been organized as follows. About half of the material,contain ingm ost of the analysis and design methods, as well as al ar ge num berof examples, is presented in unmarked sections. The finer points, such asconditions for existence, detailed results concerning convergence to steady-state solutions, and asymptotic properties, are dealt with in sections whosetitles have been mark ed with a n asterisk. TIE i~~iniarlcedsectro~ isave been sowritten that they forni a textbo ok for a tiso -se !i~ est erj irs t ourse on controltheory at the senior orfist-year grodlrate level. Th e mark ed sections consistof supplementary material of a more advanced nature. The control engineerwho is interested in applying the ma terial wiU find m ost design me thod s inthe unmarked sections but may have to refer to the remaining sections form ore detailed inform ation o n difficult points.The following background is assumed. The reader should have had ak s t course on linear systems o r linear circuits and should possess someintroductory knowledge of stochastic processes. I t is also recomme nded tha tth e reader have som e experience in digital computer programming and t ha t hehave access to a computer. We do not believe that it 1s necessary for therea der to have followed a course on classical control theory before studyingthe material of this book.A chapter-by-chapter description o f the bo ok follows.In Chapter 1, "Elements of Linear System Theory," th e description oflinear systems in terms of the ir state is the startin gpo int, w hile transfer matrixand frequency response concepts are derived from the state description.Topics important for the steady-state analysis of linear optimal systems arecarefully discussed. The y a re : controllability, stabilizability, reconstructibility,detectability, and duality. Th e last two sections of this chap ter are devoted toa description of vector stochastic processes, with special emphasis on therepresentation of stochastic processes a s the outp uts of linear differentialsystems driven by white noise. In later chapters this material is extensivelyemployed.Chapter 2, "Analysis of Con tro l Systems," gives a general description o fcontrol problems. Furthermore, it includes a step-by-step analysis of theVarious aspects of control system performance. Single-input single-output

and multivariable control systems are discussed in a unified framework bythe use of the concepts of mean square tracking error and m ean sq uare input.


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Preface ixChapter 3, "Optimal Linear State Feedback Co ntro l Systems," n ot onlypresents the usual exposition of the linear optimal regulator problem butalso gives a rather complete survey of the steady-state properties of theRiccati equation and the optimal regulator. It deals with the numericalsolution of Riccati equations and treats stochastic optimal regulators, op timaltracking systems, and regulators with constant disturbances and nonzeroset points. As a special feature, the asymptotic properties of steady-statecontrol laws and the maximally achievable accuracy of regulators and track-ing systems ar e discussed.Chapter 4, "Optimal Linear Reconstruction of the State," derives theKalman-Bucy filter starting with observer theory. V arious special cases, sucha s singular observer problems an d problems with colored observation noise,

are also treated. The various steady-state and asymptotic properties ofoptimal observers are reviewed.In Chapter 5, "Optimal Linear Output Feedback Control Systems," thestate feedback controllers of Chapter 3 are connected to the observers ofChapter 4.A heuristic and relatively simple proof of the se para tion principleis presented based on th e innov ation s concept, which is discussed in Ch apte r4. Guidelines are given for the des~gn f various types of output feedbackco ntr ol systems, and a review of the design o f reduced-orde r co ntrollers isincluded.

In C hapter 6, "Linear Optima l Con trolTh eory fo r Discrete-Time Systems,"the entire theory of Cha pters 1 through 5 is repeated in condensed form forlinear discrete-time con trol systems. Special attentio n is given to s tate dead-beat and output deadbeat control systems, and to questions concerning thesynchronization of the m easurements an d the co ntrol actuation.Throughout the book important concepts are introduced in definitions,and the main results summarized in the form of theorems. Almost everysection concludes with one o r m ore examples, m any of which ar e numerical.These examples serve to clarify the m aterial of the text an d, by their physicalsignificance, to emphasize the practical applicability of the results. Mostexamples are continuations of earlier examples so that a specific problem isdeveloped ov er several sections o r even chap ters. When ever nu merical valuesare used, care has been taken to designate the proper dimensions of thevarious quantities. To this end, the SI system of units has been employed,which is now being internationa lly accepted (see, e.g., Barro w, 1966; IEEEStan dard s Com mittee, 1970). A complete review of th e S I system can befound in the Reconinieiidotiotis of the International Organ izat~on or Stand-ardization (various dates).The book contains about 50 problems. They can be divided into twocategories: elementary exercises, directly illustrating the m aterial of the text;an d supplementary results, extending the m aterial of the text. A few of the


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problems require the use of a digital computer. The problems marked withan asterisk are not considered to belong to the textbook material. Suitableterm projects could consist of writing and testing the computer subroutineslisted in Section 5.8.

Many references are quoted throughout the book, but no attempt hasbeen made to reach any degree of completeness or to do justice to history.The fact that a particular publication is mentioned simply means that i t hasbeen 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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C O N T E N T S

Notation and SymbolsChapter 1 Elements of Linear System Theory 1

1.1 Introduction, 11.2 State Desc~ipfion f Linear Syslems, 11.2.1 State Description of Nonlinear and LinearDifferential Systems, 11.2.2 Linearization, 21.2.3 Examples, 31.2.4 State Transformations, 101.3 Solution of tlre State Differential Equotion of LinearSj~stenrs,11

1.3.1 The Transition Matrix and the ImpulseResponse Matrix, 111.3.2 The Transition Matrix of a Time-InvariantSystem, 131.3.3 Diagonalization, 151.3.4" The Jordan Form, 191.4 Stability, 24

1.4.1 Definitions of Stability, 241.4.2 Stability of Time-Invariant Linear Systems,271.4.3' Stable and Unstable Subspaces far Time-In-variant Linear Systems, 291.4.4" Investigation of the Stability of NonlinearSystems through Linearization, 311.5 Transform Analysis of Time-Znua~iantSystems, 33

1.5.1 Solution of the State Differential Equationthrough Laplace Transformation, 33

'See the Preface for the significance of the marked sections.xiii


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xiv Contents1.5.2 Frequency Response, 371.5.3 Zeroes of Transfer Matrices, 391.5.4 Snterconnections of Linear Systems, 431 . 5 9 Root Loci, 51Controllability, 531.6.1* Definition of Controllability, 531.6.2' Controllability of Linear Time-InvariantSystems, 551.6.3' The Controllable Subspace, 571.6.4' Stabilizability, 621.6.5" Controllability of Time-Varying Linear

Systems, 64Reconstri~ctibility,651.7.1* Definition of Reconstructibility, 651.7.2* Reconstructibility of Linear Time-InvariantSystems, 671.7.3' The Unreconstructible Subspace, 701.7.4' Detectability, 761.7.5" Reconstructibility of Time-Varying LinearSystems, 78Ditality of Linear Systeias, 79Phase-Variable Canonical Foims, 82Vector Stocliastic Processes, 851.10.1 Defmitions, 851.10.2 Power Spectral Density Matrices, 901.10.3 The Response of Linear Systems to Sto-chastic Inputs, 911.10.4 Quadratic Expressions, 94The Response of Linear Differcatial Sjtstems to T b i t eNoise, 971.11.1 White Noise, 971.11.2 Linear Differential Systems Driven byWhite Noise, 1001.11.3 The Steady-State Variance Matrix for theTime-Invariant Case, 1031.11.4 Modeling of Stochastic Processes, 1061.11.5 Quadratic Integral Expressions, 108Probleins, 113


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Contents xvChapter 2 Analysis o f Linear Control Systems 119

2.1 Inb.odrrction, 1192.2 The Forr~nrlationof Control Prohlerns, 121

2.2.1 Introduction, 1212.2.2 The Formulation of Tracking and RegulatorProblems, 1212.2.3 The Formulation o f Terminal ControlProblems, 127

2.3 Closed-Loop Controllers; The Basic DesignOhjectiue, 1282 .4 The Stability of Control Systents, 1362.5 The Steadjz-State Analysis of tlrc TrackingProperties, 140

2.5.1 The Steady-State Mean Square TrackingError and Input, 1402.5.2 The Single-Input Single-Output Case, 1442.5.3 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 ingle-Ontptrt Case, 1672.8 The Effects of Observation Noise in the Single-InpntSingle-Ontpnt Case, 1742.9 Tlre Effect of Plant Paranteter Uncertainty in theSingle-Inpat Single-Ostpnt Case, 1782.10* Tlte Open-Loop Steady-State Eqsiualent ControlSchenre, 1832.11 Conclrrsions, 1882.12 P~.ohlerns,189

Chapter 3 Optimal Linear\ State Feedback Control Systems 1933.1 Introdaction, 1933.2 Stability Intprouentcnt of Linear Systcnts by StateFeedhacli, 193

3.2.1 Linear State Feedback Control, 193


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xvi Contents3.2.2" Conditions for Pole Assignment and

Stabilization, 1983.3 The Deterministic Linear Optirnal RegulatorProblem, 201

3.3.1 introduction, 2013.3.2 Solution of the Regulator Problem, 2073.3.3 Derivation of the Riccati Equation, 216

3.4 Steady-State Solrrtion of tlre Deterministic LinearOptirrral Regrrlator Problenr, 2203.4.1 Introduction and Summary of Main Results,

2203.4.2' Steady-State Properties or OptimalRegulators, 2303.4.3" Steady-State Properties of the Time-

Invariant Optimal Regulator, 2373.4.4* Solution of the Time-Invariant Regulator

Problem by Diagonalization, 2433.5 Nrrnrerical Solution of tlrc Riccnti Eqrration, 248

3.5.1 Direct Integration. 2483.5.2 The Kalman-Englar Method, 2493.5.3" Solution by Diagonalization, 2503.5.4" Solution by the Newton-Raphson Method,

2513.6 Stochastic Lincnr Optirnal Regrrlntor nnd TrackingProblems, 253

3.6.1 Regulator Problems with Disturbances-The Stochastic Regulator Problem, 253

3.6.2 Stochastic Tracking Problems, 2573.6.3 Solution of the Stochastic Linear Optimal

Regulator Problem, 2593.7 Regulators and Trnck irr ~ jmterrr wit11 Norrzero SetPoints and Constant Distrrrbnnccs, 270

3.7.1 Nonzero Set Points, 2703.7.2' Constant Disturbances, 277

3.8" A sy r~ yt ot ic Properties of Time-Inuarinrrt Optim alControl Laws, 2813.8.1Qsymptotic Behavior of the Optimal Closed-

Loop Poles, 281


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Contents xvii3.8.2' Asymptotic Properties of the Single-InputSingle-Output Non zero Set Po int Regulator,

2973.8.3" The Maximally Achievable Accuracy ofRegulators and Tracking Systems, 306

3.9' Sensitivitj~ f 'Linear State Feedback Co ntro lSy sten ~s .312

3.10 Conclnsior~s, 183.11 Problems, 319

Chnpter 4 Optimal Linenr Reconstruction of the Stnte4 .1 Zntrodnction, 3284.2 Observers, 329

4.2.1 Full-Order Observers, 3294.2.2* Conditions fo r Pole Assignment an dStabilization of Observers, 3344.2.3" Reduced-Order Observers, 335

4.3 The Optimal Obscrucr, 3394.3.1 A Stochastic Approach to the ObserverProblem, 3394.3.2 The Nonsingular O ptimal Observer Problemwith Uncorrelated State Excitation andObservation Noises, 3414.3.3* Th e Nonsingular Op timal Observer Problemwith Correlated State Excitation an dObserv ation Noises, 3514.3.4"' Th e Time-Invariant Singular Optim al

Observer Problem, 3524.3.5' The Colored Noise Observation Problem,356

4.3.6' Innovations, 3614.4"' The Duality of tlre Optim al Observer and the O pt in ~ alRegnlator; Steady-State Properties of' the OptirnalObserver, 364

4.4.1" Introduction, 3644.4.2* The Duality of the Optimal Regulator andthe Optimal Observer Problem, 364


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xviii Contents4.4.3* Steady-State Properties of the OptimalObserver, 3654.4.4* Asymptotic Properties of Time-Invariant

Steady-State Optimal Observers, 3684.5 Corrclrrsions, 3734.6 Problerrrs, 373

Chnpter 5 Optimal Linear Output Feedback Control Systems 3775.1 Introdrrction, 3775.2 The Regrrlation of Linear Systenrs ~cfitlr ncomplete

Measrrremcnts, 3785.2.1 The Structure of Output Feedback ControlSystems, 3785.2.2* Conditions for Pole Assignment andStabilization of Output Feedback ControlSystems, 388

5.3 Optinral Linear Regrrlalors ~vitlr Inconrplete andNoisy Measrrrerrrmts, 3895.3.1 Problem Formulation and Solution, 3895.3.2 Evaluation of the Performance of OptimalOutput Feedback Regulators, 3915.3.3* Proof of the Separation Principle, 400

5.4 Linear Optirnal Tracking Systenrs wit11 Incorrrgleteand Noisy Measrrrcnrents, 4025.5 Regrrlators and Trackirrg Systenw with Nonzero SetPoints and Constant Disturbances, 409

5.5.1 Nonzero Set Points, 4095.5.2* Constant Disturbances, 414

5.6* Serrsitiuity of Tirne-Inuariarr1 Optirnal Linear OrrtprrtFeedback Control Systenrs, 4195.7" Linear Optinral Orrtprrt Feedback Controllers ofReduced Dimerrsions, 427

5.7.1t Introduction, 4275.7.2* Controllers of Reduced Dimensions, 4285.7.3* Numerical Determination of Optimal Con-

trollers of Reduced Dimensions, 432


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Contents xix5.8 Conclnsio~rs,4365.9 i'roblems, 438

Chapter 6* Linear Optimal Control Theory for Discrete-Time Systems 4426.1 Introdnction, 4426.2 Theory of Linear Discrete-Time System, 4426.2.1 Introduction, 4426.2.2 State Description of Linear Discrete-TimeSystems, 4436.2.3 Interconnections of Discrete-Time and Con-tinuous-Time Systems, 4436.2.4 Solution of State Difference Equations, 4526.2.5 Stability, 4546.2.6 Transform Analysis of Linear Discrete-TimeSystems, 4556.2.7 Controllability, 4596.2.8 Reconstructibility, 4626.2.9 Duality, 4656.2.10 Phase-Variable Canonical Forms, 4666.2.11 Discrete-Time Vector Stochastic Processes,

4676.2.12 Linear Discrete-Time Systems Driven byWhite Noise, 4706.3 Annlj!sis of Linear Discrete-Tinre Control Systems,4756.3.1 Introduction, 4756.3.2 Discrete-Time Linear Control Systems, 4756.3.3 The Steady-State and the Transient Analysis

of the Tracking Properties, 4786.3.4 Further Aspects of Linear Discrete-TimeControl System Performance, 4876.4 Optinral Linear Discrete-Time State FeedbaclcControl Sj~stems,4886.4.1 Introduction, 4886.4.2 Stability Improvement by State Feedback,4886.4.3 The Linear Discrete-Time Optimal

Regulator Problem, 490


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a Contents6.4.4 Steady-State Solution of the Discrete-TimeReg ulator Problem, 4956.4.5 The StochasticDiscrete-Time Linear OptimalRegulator, 5026.4.6 Linear Discrete-Time Regu lators with Non-zero Set Points and Constant Disturbances,5046.4.7 Asym ptotic Prop erties of Tim e-InvariantOptimal Control Laws, 5096.4.8 Sensitivity, 520

6.5 Optintol Linear Reconstr~tction f the State of LinearDiscrete-Time Systems, 5226.5.1 intr od uct ion , 5226.5.2 The Formulation of Linear Discrete-TimeReconstruction Problems, 5226.5.3 Discre te-Tim e Observers , 5256.5.4 Optimal Discrete-Time Linear Observers,5286.5.5 Innovation s, 5336.5.6 Duality of the Optimal Observer andRegulator Problems; Steady-State Prop-erties of the Optimal Observer, 533

6.6 Optirnal Linear Discrete-Time Outpvt FeedbackSys t en~s , 366.6.1 Int rod uc tion , 5366.6.2 Th e Regulation of Systems with IncompleteMeasurements, 5366.6.3 Optim al Linear Discrete-Time Regulatorswith Incomplete and Noisy Measurements,5396.6.4 Nonzero Set Points and Constant Distur-bances, 5436.7 Concl~~s ions ,466.8 Problems, 547

References 553Index


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

Ch apters ar e subdivided into sections, which a re numbered 1.1, 1.2, 1.3, an dso on. Sections may h e divided into subsections, which a re num bered 1.1.1,1.1.2, and so on. Theorems, examples, figures, and similar features arenum bered consecutively within each chapte r, prefixed by the ch ap ter num ber.T he section numb er is usually given in p arentheses if reference is ma de to a nitem in another section.Vectors are denoted by lowercase letters (such as x and 1 0 , matrices byuppercase letters (such as A and B) and scalars by lower case Greek lellers(such as a.an d p). I t has n ot been possible t o ad here to these rules completelyconsistently; nota ble exceptions are f for time, i an d j fo r integers, and so on .The components of vectors are denoted by lowercase Greek letters whichcorrespond as closely as possible to the Latin letter that denotes the vector;thus the 11-dimensional vector x has as co mp one nts the scalars C,, f2 , . . . , f,,,the 111-dimensional vector 1/ has a s c om po ne nts t he s ca la rs ~ 1 ~ .la. ' ' ' , 7.and so on. Boldrace capitals indicate the Laplace or z-transform of thecorresponding lowercase time functions [X(s) fo r the Laplace transform ofx(t), Y(z) for the z-transform of ~ ( i ) ,tc.].

xTcol(fl .Czr . . ., a)( ~ l l >l 3 . . ?1,0kj -1 (XI.4llxlldim (x)A~A-'t r ( 4det (A)

transpose of the vector xcolumn vector with components fl, C,, . .. 6,row vector with components ?ll, ?i2, ' ' , I,,partitioning of a column vector into subveclors x,and x,norm of a vector xdimension of the vector xtranspose of the matrix Ainverse of the sq uare ma trix At race of the square matr ix Adeterminan t of the squ are matrix A


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xxii Notation and Symbolsdiag (A1, A,, . . . , , , )(el , e,, . . ., n )

( :diag ( J l ,J,, . . ., ,J

~ { x ( O }R e ( a )In1 (a)min (a, )minmax=

di t~gonalmatrix with diagonal entries A,, A,, . . . , ,partitioning of a matrix into its columns el , e,, . . ., ,,partitioning of a malrix into its rowsf,,f,, . . .,f,

partitioning of a matrix into column blocks T I ,T?,. .:T ,

partitioning of a matrix into row blocks U,, U,, .. ,U,"partitioning of a m atrix into blocks A , B, C, nd Dblock diagonal matrix with diagonal blocks J,,

J,, . . ., ",the real symmetric or H ermitian matrix M i s posit ive-definite o r nonnega tive-definite, respectivelythe real symmetric or Hermitian matrix M - N ispositive-definite or nonnegative-definite, respec-tivelytime derivative of the time-varying vectol x( t )Laplace transform of x(t)real pa rt of the complex number aimaginary pa rt of the complex num ber athe smallest of the numbers a and ,Bthe minimum with respect to athe maximum with respect to a

0 zero; zero vector; zero matrixA( t ) , A(& A plant matrix of a finite-dimensional linear differentialsystem


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Notntion and Symbols xxiiiinput matrix of a finite-dimensional linear differentialsystem (B becomes b in the single-input case)ou tpu t matrix of afinite-dimen sional linear differential

system; output matrix for the observed variable(C becomes c in the single-output case)mean square tracking o r regulating errormean square inpu tout pu t matrix for the controlled variable (D becomesd in the single-output case)base of the natural logarithmtracking o r regulating e rror; reconstruction erro ri-th characteristic vector

expectation operatorgain matrix of the direct link of a plant (Ch. 6 only)frequencyregulator gain matrix (F ecomes f in the single-inputcase)controller transfer matrix (from y to -11)plan t transfer matrix (from t i to y)integerunit matrixa;ntegerreturn difference matrix or functionobserver gain matrix (K ecomes k in the single-output case)pla nt transfer matrix (from t i to z)closed-loop transfer matrixdimension of the state xtransfer matrix o r funclion from r t o zi i n a controlsystemcontrollability matrixsolution of the regulator R iccati equationcontroller transfer matrix (from r to ti)terminal state weighting matrixreconstructibility matrixvariance matrix; solution of the observer Riccatieqhationinitial variance matrixsecond-order moment matrixreference variableweighting m atrix of the sta teweighting matrix of the input


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nr iv Notntion and Symbols

covariance function of the stochastic process vvariable of the Laplace transformsensitivity malrix or functiontimetransmissioninput variablestochastic processobservation noise, measurement noiseconstant disturbanceequivalent disturbance at the controlled variabledisturbance variableintensity of a white noise processwhite noise processweighting ma trix of the tracking or regulating er ro rweighting matrix of the inputstale variablereconstructed state variableinitial stateoutput variable; obsenied variablez-transform variab lecontrolled variablecom pound ma trix of hystem and adjoint differentialequationsdelta functionsam pling intervalscalar controlled variablescalar output variable; scalar observed variabletime difference; time constant; normalized angularfrequencyi-th characteristic valuescalar input variablescalar stochastic processi-th zeroscalar state variablei-th poleweighting coefficient of the integ rated or mea n s qua reinputspectral density ma trix of th e stoch astic process ucharacteristic polynomialclosed-loop characteristic polynomial


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AHzkgkmolmNradSvn

transition matrixnum erator polynomialangular frequency

amperehertzkilogramkilomolemeternewtonradiansecondvoltohm

Notntion and Symbols xxv

linear optimal control syst

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