1
LINEAR OPTIMAL
CONTROL SYSTEMS
Linear Optimal
Control Systems
I
I
HUIBERT KWAKERNAAK
Twente Uniucrdy of Technology
Enrchcde, The Nefherlur~ds
i
I RAPHAEL SIVAN
Technion, I m e l Institute of Technology
Hoifo, Israel
.
-
.
WILEY-INTERSCIENCE, a Diuision of John Wiley & Sons, Inc.
New York Chichester Brisbane Toronto
Copyright
01972, by Jo!m
Wiley &Sons, Inc.
A l l 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~gin Publimiion 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 America
10 9 8 7 6
To ~ i l i n eAnnemorie,
,
and Martin
H. K.
In memory of my parents Yelnrda and Toua and to my wife Ilana
R. S.
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 advantage of modern linear control theory over the classical theory is its applicability to control problems involving multiinput multioutput systems and
time-varying situations; the classical theory is essentially restricted to singleinput 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 consists 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 cnllection 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
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 undergraduate courses and to defer the modern theory, particularly the stochastic
part of it, t o 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 steadystate solutions, and asymptotic properties, are dealt with in sections whose
have been so
titles have been marked with an asterisk. TIE i~~iniarlcedsectro~is
written that they forni a textbook for a tiso-se!i~esterjirstcourse 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
k s t course on linear systems or linear circuits and should possess some
introductory knowledge of stochastic processes. I t 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.
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. I t 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 tracking 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~gnof 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 deadbeat 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~onfor Standardization (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
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).
ACKNOWLEDGMENTS
The first author wishes to express 111sthanks 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 University 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, d u r ~ n gthe 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 commended 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 Polytechnic 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.
CONTENTS
Notation and Symbols
Chapter 1 Elements of Linear System Theory
1.1
Introduction, 1
1.2
State Desc~ipfionof 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-Invariant Linear Systems, 29
1.4.4" Investigation of the Stability of Nonlinear
Systems through Linearization, 31
1.5
Transform Analysis of Time-Znua~iantSystems, 33
1.5.1 Solution of the State Differential Equation
through Laplace Transformation, 33
1
'See the Preface for the significance of the marked sections.
xiii
xiv
Contents
1.5.2
1.5.3
1.5.4
1.59
Frequency Response, 37
Zeroes of Transfer Matrices, 39
Snterconnections of Linear Systems, 43
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
1.10.1
1.10.2
1.10.3
Stocliastic Processes, 85
Defmitions, 85
Power Spectral Density Matrices, 90
The Response of Linear Systems to Stochastic 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
Contents
Chapter 2 Analysis o f Linear Control Systems
2.1
Inb.odrrction, 119
2.2
The Forr~nrlationof Control Prohlerns, 121
2.2.1 Introduction, 121
2.2.2 The Formulation of Tracking and Regulator
Problems, 121
2.2.3 The Formulation o f 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~prrlSingleOntptrt 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
xv
119
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
3.1
Introdaction, 193
3.2
Stability Intprouentcnt of Linear Systcnts by State
Feedhacli, 193
3.2.1 Linear State Feedback Control, 193
193
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 TimeInvariant 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 DisturbancesThe 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 ClosedLoop Poles, 281
Contents
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
xvii
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
5.1
Introdrrction, 377
5.2
The Regrrlation of Linear Systenrs ~cfitlrIncomplete
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 Controllers of Reduced Dimensions, 432
377
Contents
5.8
Conclnsio~rs,436
5.9
i'roblems, 438
xix
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 Continuous-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
a
Contents
6.4.4
6.4.5
6.4.6
6.4.7
6.4.8
References
Index
Steady-State Solution of the Discrete-Time
Regulator Problem, 495
The StochasticDiscrete-Time Linear Optimal
Regulator, 502
Linear Discrete-Time Regulators with Nonzero Set Points and Constant Disturbances,
504
Asymptotic Properties of Time-Invariant
Optimal Control Laws, 509
Sensitivity, 520
6.5
Optintol Linear Reconstr~tctionof 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 Properties 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 Disturbances, 543
6.7
Concl~~sions,
546
6.8
Problems, 547
553
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). 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)
( ~ l lv>l 3 . . . ?1,0
...
kj
9
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
.
xxii
Notation and Symbols
diag (A1, A,, . . . , A,,)
(el, e,,
, en)
...
dit~gonalmatrix with diagonal entries A,, A,,
partitioning of a matrix into its columns el, e,,
partitioning of a malrix into its rowsf,,f,,
. . . ,A,
. . .,e,,
. . . ,f,
partitioning of a matrix into column blocks TI,
T?,. . . : T ,
partitioning of a matrix into row blocks U,, U,, .
U,"
:( ):
diag (Jl,J,,
.. ,
partitioning of a matrix into blocks A, B, C,and D
. . . ,J,J
block diagonal matrix with diagonal blocks J,,
J,,
. . . ,J",
the real symmetric or Hermitian matrix M i s positivedefinite or nonnegative-definite, respectively
the real symmetric or Hermitian matrix M - N is
positive-definite or nonnegative-definite, respectively
time derivative of the time-varying vectol x(t)
~{x(O}
Re (a)
In1 (a)
min (a,p)
min
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
max
=
the maximum with respect to a
0
zero; zero vector; zero matrix
plant matrix of a finite-dimensional linear differential
system
A(t), A(& A
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 (Fbecomes f in the single-input
case)
controller transfer matrix (from y to -11)
plant transfer matrix (from ti to y)
integer
unit matrix
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
a;
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
Notntion and Symbols
transition matrix
numerator polynomial
angular frequency
A
Hz
kg
kmol
m
N
rad
S
v
n
ampere
hertz
kilogram
kilomole
meter
newton
radian
second
volt
ohm
xxv