Linear Systems - IEEE Xplore

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systems, and the brief final chapter ten points him or her towards some topics of current research. In the first chapter the author says he aims to be logically ...

IEEE TRANSACTIONS

ON INFORMATION

THEORY,

VOL.

IT-21, NO. 3, MAY1981

385

Book Reviews F’rojective Geometries over Finite Fields-J. W. P. Hirschfeld England: Oxford University Press, 1979, ‘474 pp.)

N. J. A. SLOANE,

FELLOW,

(Oxford,

IEEE

Anyone who has worked in coding theory, block designs, combinatorics, or finite groups soon realizes that the finite geometers are in control of an enormous amount of material that is fundamental to these subjects. Up to now this material has been well guarded. There were few books, and they were neither easy nor up-to-date; see for example Artin [I], Dembowski [2], Dieudonne [3], and Segre [4]. Here at last is a modern book, written with combinatorialists, group theorists, and coding theorists in mind. It is encyclopaedic in scope. This volume is the first of two, and contains the following chapters: finite fields; projective spaces; subspaces of PG( n, q); partitions of PG( n, q); canonical forms for varieties and polarities; PG(l,q); first properties of PG(2,q); ovals; arithmetic of k-arcs; arcs in ovals; cubic curves; plane (k; n)-arcs; blocking sets; PG(2, q) for small q. Each chapter is densely written and contains an immense amount of information. The volume includes a comprehensive bibliography, containing 8 19 items. I strongly recommend it. REFERENCES

[I] E. Artin, Geometric Algebra. New York: Interscience, 1957. [2] P. Dembowski, Finite Geometries. New York: Springer, 1968. [3] J. A. Dieudonnk, La Gkvnetrie des Groupes Classiques. 3rd ed. New York: Springer, 1971, [4] B. Segre, Lectures on Modern Geometry. Rome: Cremonese, 1961.

N. J. A. Sloane (S’62-M’66-SM’77-F’78) for a biography please seepage 268 of the March 1981 issue of this TRANSACTIONS.

Linear Systems-T.

Kailath 1980, 672 pp., $27.50 cloth).

(Englewood Cliffs, N.J.: Prentice-Hall,

PETER E. CAINE&

MEMBER, IEEE

The theory of linear systems is fundamental to contemporary control theory, network theory, communication theory, and indeed to system theory in general. A control engineer can claim that the only adequate theory for the analysis and synthesis of control systems is provided by linear system theory, and that the debates concerning the relative merits of such design techniques as quadratic regulator theory, modal control, the “inverse Nyquist array” method, the “geometric theory” of multivariable control system synthesis, and the characteristic loci and sequential return difference techniques, merely serve to demonstrate the richness and technical sophistication of the area. This is in contrast to optimal control theory as associated with the maximum principle. (In principle this can be applied to large classes of nonlinear systems, but in fact it is only used in a few specific areas such as trajectory optimization for space vehicles.) Furthermore, among the small number of practically applicable theories of stochastic control, one must count the linear quadratic Gaussian (LQG) theory and the closely related minimum variance control strategy (and their adaptive versions), and these are both firmly rooted in the theory of linear systems subject to random disturbances. From the viewpoint of communication engineering, the linear filtering theory of Wiener and Kolmogorov (for stationary processes) and the linear recursive filtering theory associated with Kalman and Bucy (for nonstationary processes with finite dimensional linear state models) constitute basic solutions to one formulation of the problem of reliable communication. (Of course communication engineers frequently use models other than continuous-state linear system models: for instance those arising in algebraic coding and Shannon theory.)

Hard-core linear system theory consists of the mathematical study of the properties of the equations 1=Ax+Bu,

y=Cx+Du,

with u an input function taking values in WP(i.e., u(.): R + IWP),x a state function taking values in W”, y an output function taking values in Rm, and A, B,C, and D being matrices of appropriate dimension which may be parameterized by time. The analysis of these equations-as distinct from the analysis of general ordinary differential equations (ODE’s), automata, and the engineering analysis of linear systems-began around 1960. This was stimulated by Kalman’s use of linear-state space models in control system design and by his solution to the recursive linear least squares estimation problem. The early phases of the theory consisted of Kalman’s isolation of the concepts of controllability and observability, the structural results (principally the state-space isomorphism theorem) of Kalman, Gilbert and Youla, and other results revealing the importance of these properties for linear input-state-output realizations of linear inputoutput systems. The year 1963 saw the publication of the textbook Linear System Theory by Zadeh and Desoer [ 11,this is often regarded as a classic by virtue of its influence upon the field and the elegance and clarity of its exposition. Since the middle sixties the field has yielded several basic results (for instance the result that for time invariant systems controllability is equivalent to the possibility of arbitrary pole placement by state feedback), witnessed the development of theories underlying the various linear multivariable regulator design methods listed earlier-especially as in the textbooks of Rosenbrock and Wonhan-and seen the emergence of entirely new theoretical topics such as algebraic and geometric (linear) system theory. Despite this plethora of results, developments, and applications, it was possible during the last decade to teach the elements of the theory and give an overview of the subject by using, say, Desoer’s short book [2] together with Chen’s relatively elementary text [3], both published in 1970, or by using Brockett’s text [4], which was also published in 1970. In fact given the apparently semi-infinite amount of material on the topic, it was extremely useful to have available such condensed presentations of the fundamentals of the subject. However, an update on the subject was due and Professor Kailath’s Linear Systems is a wholly admirable attempt to give a textbook exposition of the subject as it stands at the end of the seventies. As is revealed by the list of chapter headings below, the book is intended to be a classroom text for theoretically oriented senior and graduate engineering students, a digestion and organization of recent advances (again in textbook form), and a guide to current research topics and adjacent areas of interest. To be specific the chapter headings are: Background Material; State Space Descriptions- Some Basic Concepts; Linear State Variable Feedback; Asymptotic Observers and Compensator Design; Some Algebraic Complements; State Space and Matrix-Fraction Descriptions of Multivariable Systems; State Feedback and Compensator Design; Genera1 Differential Systems and Polynomial Matrix Descriptions; Some Results for Time Variant Systems; Some Further Reading; and an Appendix entitled Some Facts from Matrix Theory. Two features of the book that are advertised by the author in the preface and that are certainly among its distinguishing features, are its particular pedagogical style and the way the interplay between state space and transfer function ideas is continually emphasized. Pedagogically the book is organized in a way that is intended to correspond to a path of learning and discovery on the part of the student. One major manifestation of this approach is that univariate (single-input single-output) systems are treated first and multivariate (multi-input multi-output) systems do not make their official appearance until the sixth chapter. In the first five chapters topics such as state space representations (or realizations), observability and controllability, state feedback, and observer and compensator design, are introduced and developed for univariate (single-input single-output) linear systems. As in the remainder of the book, this is carried out with a generous amount of motivating discussion, many instructive exercises, and a wealth of references to the literature. While

386 there are detailed alphabetical indices by subject and by name, the nonalphabetical arrangement of references at the end of each chapter makes it difficult to follow up on the citations made to over 330 authors. In addition, there are careful discussions of priority, and as is the author’s wont, interesting allusions to history and antiquity (Diophantus appears, but the Babylonians do not as they did in the author’s A View of Three Decades of Linear Filtering Theory [5]).

Especially nice features of the first five chapters are as follows: chapter two presents the idea of the state as a set of initial conditions for analog-computer simulations of linear systems, discusses why it took so long for the ideas of observability and controllability to emerge given that analog simulation goes back to Kelvin in the last century, and interrelates various types of canonical forms for linear systems in an informative way; chapter three discusses the nature of various control techniques when applied to discrete time parameter systems; and chapter four treats feedback and compensator design procedures for scalar linear systems by the direct use of transfer functions. All of the topics mentioned above are picked up again in chapters six through nine, the idea being that the reader is better equipped to face the technicalities of the multivariable development once the ideas have been encountered in the univariate setting. Such a style of development in a text is the converse of the mathematical one in which ideas are developed to the maximum generality once the prerequisites have been assembled. Whether one likes it or not is in the end a matter of taste; however, I think that it will be very popular with students and that such a development of the material is well suited to classroom presentation. In fact linear system theory, with its potential for a division into univariate and multivariate theory, lends itself to an approach emphasizing the increasing stages of sophistication of the reader. As stated earlier, the second distinguishing feature of the book is the manner in which the interplay between transfer function (i.e., inputoutput) descriptions and state space descriptions is continually developed and emphasized. As far as the presentation of new technical material in textbook form is concerned, I am sure that chapter six (on state space and matrix fraction descriptions) will be regarded as one of the main strengths of the book. Throughout the volume the author presents with great clarity a large amount of the computational algebra of linear system theory, displaying many interesting and useful identities along the way. This skill is deployed to maximum effect in chapter six. The complicated algebraic operations and relations concerning matrix fraction descriptions of linear systems (the matrix analogs of rational transfer functions) are laid out with an impressive and systematic clarity. (One is tempted to refer to this chapter as “what you always wanted to know about multivariable systems but were afraid to ask.“) Chapter seven replays chapters three and four in the multivariable setting by carrying out state feedback and compensator design using both the state space and matrix fraction techniques developed in chapter six. We remark that the fact that this is a text concerning linear system theory rather than control system synthesis is demonstrated by the relatively small fraction of the volume devoted to the quadratic regulator problem or to frequency domain methods. Despite this the author manages to include some interesting material devoted to the asymptotic behavior of pole-zero patterns in quadratic regulator theory. Chapter eight is a comparatively short chapter devoted to the analysis of so-called general differential systems and polynomial matrix systems, For these classes of systems-which are in certain technical senses more general than matrix fraction descriptions-various notions of system equivalence and associated rules of transformation are developed with an admirable clarity. It is hard to tell whether in the future this chapter will be regarded as a useful systematization of a complex topic or just a baroque extension of the main core of the subject. Chapter nine informs the reader of aspects of the theory of time varying systems, and the brief final chapter ten points him or her towards some topics of current research. In the first chapter the author says he aims to be logically consistent rather than mathematically rigorous and this approach may result in some frustrations for the more mathematical reader. For instance, the book avoids a formal definition of a linear input-output system, or more importantly, of a linear input-state-output system. Of course the latter may be given as the joint linear operation of the system on the system

IEEE TRANSACTIONS

ON INFORMATIONTHEORY,VOL.

IT-21,NO.

3,MAY 1981

state and an appropriate segment of the input function, see e.g., [ 11, [2]. Furthermore the formal definition of the notion of state in terms of its semigroup property (see e.g., [I], [2]) is omitted. Instead the author gives an informal discussion of the notion of state. which I think rather misleadingly brings in the notions of minimal&y and minimal statistics; chapter five then goes on to the more advanced topic of constructing system states using the notion of Nerode equivalence. In my opinion the formal definitions of linearity, system state, and time invariance (also omitted), are both fundamental and illuminating. In his discussion in chapters one and two of the C, and !% Laplace transform and of impulsive input functions for differential equations, the author uses the notation t > 0 + , t > 0 - , etc. Now whereas x(0 + ) has a well defined meaning for a function x( .) (as the limit of x(t) as t lends to zero from the right, when this limit exists), x(t) for t > 0 + , for example, does not. This notation leads the author (p. 11) to write “x(t) = e-” + ep2’, t>O- ,” as the solution to “n(t) +2x(t)=8(t), t>O-, x(0-) = I,” when the solution to this differential equation for all time t with the given boundary conditions is evidently x(t) = e -*I + e -*‘l( t), where l(t) is one for t > 0, zero for t < 0, and one or zero at t = 0 depending upon our arbitrary assignment of x(O). Another irritation of an analytic nature is the discussion of bounded input bounded output (bibo) stability for systems described by y(t) = jrh(s) u(t -s) ds. Since h( .) lie in unspecified function spaces the author “proves” that the system is bibo stable if and only if h( .) is integrable, i.e., /$‘I h(s)\ a!s < co. Now this suggests that L’[O, co) (integrable Lebesgue measurable functions) and Lm [0, co) (bounded Lebesgue measurable functions) are duals. However, this is not the case, for although the dual of L’ is Lm, the dual of Lm is a much larger space than L’. On the other hand the characterization of bibo stability becomes true if all functions under discussion are restricted to he in Lm [0, co) at the S.tEUt

My penultimate comments concern English and typography. I am sure that readers will find that Professor Kailath’s clear prose and easy style make this book an enjoyable one to read. One minor complaint concerns the liberal use of the qualification “some” throughout the book, as for instance in the chapter titles (five out of ten including the appendix, see the list given earlier) and section headings (four out of eight in chapter six). The suggestion is that the author does not claim to cover everything; however, I cannot imagine the reader who could complain that “everything” had not been said on the topics under discussion, especially given the heroic amount of work that has obviously gone into this treatise. As far as typographical errors are concerned, I can hardly find any. The author and all who assisted him are to be congratulated on very careful proofreading. The author informs us that there is a solutions manual that also contains an errata list. In conclusion this is a highly informative compendious pedagogically imaginative textbook. It presents in an integrated form both the basic ideas of linear system theory and a systematic exposition of much of the last decade’s research. I am sure it will be a huge success. REFERENCES [ll L. A. Zadeh and C. A. Desoer, Linear System Theory. McGraw Hill, 1963. PI C. A. Desoer, Notes for a Second Course on Linear Systems. Van Nostrand Reinhold, 1970. 131 C. T. Chen, Introduction to Linear System Theory. New Rinehart, and Winston, 1970. 141 R. W. Brockett, Finite Dimensioml Linear Systems. New Wiley, 1970. 151 T. Kailath, “A View of Three Decades of Linear Filtering,” Inform. Theov, vol. IT-20, no. 2, pp. 145-181,

New York: New York:

York: Holt, York: John IEEE Trans.

Mar. 1974.

Peter Gaines (M’74) obtained the B.A. degree from Oxford University, England, in 1967, and the Ph.D. and D.I.C. degrees from the University of London, England, in 1970. He has worked at University of Manchester Institute of Science and Technology, England, Stanford University, Stanford, CA, University of California at Berkeley, University of Toronto, Canada, and Harvard University, Cambridge, MA. He is currently Associate Professor of Electrical Engineering at McGill University, Montreal, PQ, Canada. He is a member of the IEEE, SIAM, and AMS and has served as an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL.

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