LINEAR SYSTEMS
by
Panos J. Antsaklis and Anthony N. Michel
Linear Systems - Panos J. Antsaklis and Anthony N. Michel (New York, NY: McGraw-Hill, 1997). Reviewed by Shankar P. Bhattacharyya, Department of Electrical Engineering, Texas A & M University, College Station, TX 77843-3128, Email:bhatt@eesun1.tamu.eduLinear Systems Theory is the study of the behaviour of dynamic systems described by linear differential or difference equations. The subject occupies a position of central importance in several engineering disciplines such as Electrical, Mechanical, Chemical and Aerospace Engineering and plays a broad and fundamental role in the fields of Control, Communications and Signal Processing. There are several reasons for this. First, many engineering systems are intentionally built to be linear so that they may be easily calibrated, controlled and analyzed. Second, nonlinear behaviour can often be adequately accounted for by linearizing a system about prescribed operating points and considering small perturbations. Finally, a general theory of dynamic systems that is sufficiently detailed to be useful in engineering, exists only for linear systems.
Until about 1960 the subject of linear systems consisted of Laplace transforms, transfer function representations of single input single output systems and their interconnections and frequency response methods developed in the 1930's and 40's by Nyquist and Bode. In the late 1950's the state space model was developed as a natural representation, convenient for computation, simulation, realization and optimization of complex dynamic systems, including high order multiinput-multioutput systems. In particular it was shown by Kalman, in a 1960 paper presented at the first IFAC World Congress, that optimal control problems could be conveniently formulated and solved using a quadratic cost function, for linear systems described by a state space model. The role of controllability and observability in guaranteeing the existence of optimal stabilizing solutions were established in this paper. In 1967 Kalman established an important link by proving that controllability and observability were equivalent to minimality of the order of a state space realization of the transfer function matrix. These ideas were instrumental in establishing the correct procedure for analyzing the stability of interconnected multivariable feedback systems in which ``phantom" cancellations could conceal the presence of instability. The body of knowledge consisting of controllability, observability, minimal realizations, state feedback, observers and pole placement became well entrenched in the Controls curriculum by the late 1960's and came to be known as algebraic system theory.
Linear Systems by Antsaklis and Michel is a thoroughand rigorous exposition of algebraic system theory. It is written as a textbook for a one semester graduate level first course in linear systems. The seven topics covered in as many chapters are: Mathematical Descriptions of Systems, Response of Linear Systems, Controllability Observability and Special Forms, State Feedback and Observers, Realization Theory and Algorithms, Stability, and Polynomial Matrix and Matrix Fraction Descriptions of Systems. There is also an Appendix titled Numerical Considerations and each chapter ends with a large set of exercises. Most of the results in the book are derived simultaneously for continuous time as well as discrete time systems and time-varying as well as time-invariant systems. A brief description of the contents follows.
Chapter 1 is almost entirely devoted to the questions of existence, uniqueness and continuity of solutions to ordinary, not necessarily linear, differential equations described in the state space form. The treatment is rigorous, what is meant by a solution is precisely defined, and the supporting mathematics is also developed in detail. The latter consists of results such as the Ascoli-Arzela Lemma, Cauchy-Peano Existence Theorem, Gronwall Inequality, Zorn's Lemma and Weierstrass M-test, along with the usual definitions of Cauchy sequences, convergence, linear vector spaces and normed linear spaces. The technique of linearization about a nominal solution is introduced and the general results on existence, uniqueness and continuity are specialized to linear systems. The chapter concludes with a description of discrete time state space systems and an input output or external description of linear systems in which the impulse response matrix is introduced. Although most of the existence results of this chapter will not be directly used by engineering students or professors it is valuable as source of reference for researchers, particularly since it is not available in other linear systems texts. After reading this rather long chapter though (95 pages), I could not help wondering what significance differential equations with nonexistent or nonunique solutions could have in the modeling of physical systems. A few thoughts along these lines would have placed the contents of this chapter in perspective, at least for engineering students.
In Chapter 2 the solutions of linear state space systems are developed for continuous time, discrete time, time-varying and time-invariant systems. The notions of fundamental matrix or state transition matrix, eigenvalues, eigenvectors and Jordan canonical form, Laplace and z-transforms, zero state and zero input response and stability are developed. Sampled data systems and periodic systems are also covered. A large number of exercises (77) are included.
Chapter 3 introduces the notions of controllability and reachability, and observability and constructibility respectively, based on time domain considerations. The usual rank conditions on (A,B) and (C,A) are derived based on transferability of an initial state to a prescribed final state or recoverability of an initial internal state from output measurements. The relationship between the range and null spaces of the controllability and observability matrices and those of the associated Wronskians or Grammians are developed. Coordinate transformations displaying the controllable and unconcontrollabe as well as observable and unobservbable parts of a system are developed and the Kalman Decomposition Theorem displaying the four parts (controllable and observable, controllable and unobservable, uncontrollable and observable and uncontrollable and unobservable) is introduced. Next, some special forms of the (A,B) and (C,A) matrices are derived, based on controllability or observability indices. The Smith-McMillan form of a transfer matrix is introduced and some formulas connecting the controllability and observability indices and a polynomial matrix fraction decomposition of the system transfer matrix are derived. These formulas called the Structure Theorem turn out to be useful in a subsequent chapter when state space realizations are constructed. Finally Rosenbrock's system matrix is introduced and this along with the Smith McMillan form are used to define the zeros, input and output decoupling zeros and transmission zeros of multivariable systems.
The next chapter discusses pole assignment by state feedback, state observers and the design of feedback controllers obtained by implementing computed state feedback control laws as observer generated ``estimated" state feedback. A brief discussion of the linear quadratic optimal regulator and the Kalman-Bucy optimal state estimator are also given without proof.
Chapter 5 is devoted to Realization Theory and algorithms. The objective is to come up with an internal or state space description from a given input-output description. The time-varying case is treated first and here a state space construction to realize a given impulse or pulse response is given. The rest of the chapter deals with the time-invariant case for which the fundamental equivalence between controllability and observability and minimality is first proved. The order of a minimal realization is related to the rank of the Hankel matrix. A number of formulas for realizations are then given. The Structure Theorem introduced in Chapter 3 is used here and the duality between controllability and observability is exploited to derive minimal realizations with the A matrix in special forms such as block companion or block diagonal. A balanced realization, that is one for which the controllability and observability Wronskians are equal and diagonal is also derived. Such realizations are useful in order reduction as they can be used to eliminate the least controllable and observable parts of a system.
The next chapter is devoted to stability theory. The theory of Lyapunov stability of an equilibrium state of a general nonlinear system is developed and subsequently specialized to linear systems. The notions of stability, uniform stability, asymptotic stability, uniform asymptotic stability and exponential stability are carefully defined and explained. For linear systems the criteria for the various types of stability are stated in terms of boundedness and convergence of the induced norm of the state transition matrix. The Lyapunov matrix equation is developed and used to prove the fundamental result that exponential stability of a nonlinear system can be ascertained, under mild conditions, from that of the linearized part of the system. Next, several results on input-output stability are derived and stated in terms of the boundedness of the integral of the norm of the impulse response matrix. The discrete time counterpart of all of the above results are also given for time-invariant systems. This chapter also includes a proof of the stability part of the Routh Hurwitz criterion based on the interlacing or Hermite Bieler theorem and the Leonhard-Mikhailov criterion for a polynomial p(s) of degree n to have all its roots in the left half plane. The latter condition states that the plot of p(jw) should pass through n quadrants strictly moving counterclockwise as w runs from zero to infinity.
The last chapter deals with polynomial matrix descriptions (PMD's) and matrix fraction descriptions (MFD's). Many of the results described in this chapter were developed by Antsaklis and his coworkers. PMD's are general realizations that include state space realizations as well as MFD's as special cases. The topics of coprime factorizations over the ring of polynomials and over the ring of stable proper rational functions are introduced along with the associated Diophantine equation. The most important result of this chapter perhaps, is a characterization of all rational proper stabilizing feedback controllers for a given system. This result well known in the control literature as the YJBK parametrization is described in detail and several equivalent parametrizations are given. These are also related to the usual observer based state feedback controllers.
The Appendix on Numerical Considerations describes the singular value decomposition and LU decompositions of a matrix A and the role of the condition number in determining the robustness of solutions to linear equations with respect to uncertainties in parameters. This material is particularly relevant in view of the fact that several coordinate transformations involved in the theory of linear systems, such as the transformation to companion forms and those based on the controllability or observability matrix, tend to be ill conditioned.
In assessing the present book as a potential textbook for our first graduate linear systems course, I find it compares very favorably with some of the excellent texts already available such as Kailath's Linear Systems. I usually supplement this material with notes on the multivariable servomechanism problem, the Internal Model Principle developed by Wonham and Francis, the Brasch-Pearson bound on the order of pole placement controllers and some basic material on robust stability. In my opinion this rounds out the subject nicely and sets the stage for the next course which should deal with optimal and robust control.
Antsaklis and Michel have contributed an expertly written and high quality textbook to the field and are to be congratulated. Their work is especially noteworthy in view of the fact that this is a mature discipline and there are several outstanding texts already available in the field. Because of its mathematical sophistication and completeness the present book is highly recommended for use, both as a textbook as well as a reference.
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