Linear systems - IEEE Xplore

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was possible during the last decade to teach the elements of the theory and give an overview ... output) systems do not make their official appearance until the sixth chapter. ... references to the literature. While there are ... Of course the latter may be ... Digital Contrd Systems-B.C. Kuo (New York: Holt, Rinehart and. Winston ...
PROCEEDINGS OF THE IEEE, VOL.

70,

NO.

1, JANUARY 1982

underlying supply side economic factors such as the structure of the defense industry and factors of production; demand factors, and the resultant market performance; defense research and development, and associated acquistion concerns affecting industrial mobilization capabilities; increasing concerns at the level of the many subcontractors and parts suppliers associated with defense contracting; and sectoral differences andmultinationalconsiderationsassociatedwithprocurement and defense industrial base considerations. Examples of claimeddecreasingeconomicefficiency andreduced strategic responsiveness at all Department of Defense (DoD) levels are cited in the f i s t eight chapters of the book. A summary assessment of problems andfuturetrends is presentedinChapter9.Asummary assessment of the approximate costs of current inefficiencies is highlighted. Among major needs cited are production competition, multiyear funding, budget stability, automation, labor stability at plant level, standardization of systems and subsystems across services, higher productionrates,nonduplicatedfacilitiesandlabor,andshortermore $3 bilrealistic development schedules. Gansler estimates that at least lion annually inefficiencies result from neglectof these needs. The last three chaptersof the text aredevoted t o definition of criteria for improvements to the present situation, approaches taken by other nations, and specific implementable recommendations that satisfy constraints imposed by social, economic, and political structure currently extant in the democratic private enterprise systemof the United States. Gansler’s claim is that there exist fwe meaningful criteria for effective use of the defense industry and that many contemporary problems result from independent optimizationof a single criterion or an immature approach at simultaneous optimization. He advocates maximization of productionefficiency over a given planninghorizonwithconstraints upondeterrentandbattlecapability, surgecapability,technological advancement for futuremilitary advantage, adverse effects upon society, is andthe politicalprocess.Thisconstrainedoptimizationstrategy not selected without consideration of other approaches such as nationalization of the defense industry, treating itas a public utility, designating sources for each product, encouraging open competition, separating research and development (R & D) from production, combining civilian and defense business, having an interdependent multinational defense industry, structuring the industry for mediumto long duration conflict, and oligopoly with little or no explicit government involvement with industry structure. The fial chapter of the book presents a number of specific recommendations for the future of the defense industrial base. The primary recommendations include: coordination of government policies-sector by sector, integration of civilian and military operations, 0 recognitionof the“dualeconomy” distinguishinglarge and small contractors, 0 develop policies to address international interdependence, 0 improve planning for production surge, 0 consider cost as a majordesign and acquisition criterion, 0 institutionalize industrial-base considerations. 0 0

Withoutquestion this is an importantbook.Itformulatesand analyzestrends and issueswhichmaysignificantlyweakennational security over thenearterm. Specificpolicyproposalsaremade and interpreted in terms of a number of competing concerns affecting our society and ultimately national andworld peace perspectives. The book should be of considerable interest to economists, systems enpineers,and management scientists due to its penetrating systemic policy analysis of a highly critical national resource. It holds much of interest for a general readership as well. Reprinted from B E E Transactions on System, Man,and Cybernctics, June 1981.

Linear Systems-T. Kailath (Engleewood Cliffs, NJ: Prentice-Hall, 1980, 672 pp., $27.50, cloth). Reviewed by Peter E. Canes, McGill University, Montreal, P.Q., Cam&.

The theory of linear systems is fundamental t o 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,andthatthedebates concerning the relativemerits of such design techniques as quadraticregulatortheory,modalcontrol,the “inverse Nyquist m a y “ method, the “geometric theory” of multivariable control system synthesis, and the characteristicloci and sequential

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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 onlyusedinafewspecificareassuch as trajectory optimization for space vehicles.) Furthermore, among the small number of practically applicabletheoriesofstochasticcontrol,onemust countthe linear quadraticGaussian(LQG)theoryandthecloselyrelatedminimum variance control strategy (and their adaptiveversions), and theseare both firmly rooted in the theory of linear systems subject to random disturbances. From the viewpoint of communication engineering, the linear fdtering theory of Wiener and Kolmogorov (for stationary processes)and the linear recursive filtering theory associated with Kalman and Bucy (for nonstationaryprocesseswithfinitedimensionallinearstatemodels) constitute basic solutions to oneformulation of the problem of reliable use communication.(Ofcoursecommunicationengineersfrequently models other than continuous-state linear system models: for instance those arising in algebraic codingand Shannon theory.) Hard-core linear system theory consists of the mathematical study of the properties of the equations i = A x + B u ,y = C x + D U ;

with u an input functiontaking values in RP(i.e.,u ( - ) :R +RP),x a state function taking values in R n , y an output function taking values inRm, 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 generalordinarydifferentialequations (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 therecursive linear least squares estimation problem. The early phases of the theory consisted of Kalman’s isolation of the concepts of controllability and observability,thestructuralresults(principallythestate-spaceisomorphism theorem) of Kalman, Gilbert and Youla, and other results revealing the importance of these properties for linear input-stateautputrealizations of linear input-output 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 thatfortime invariantsystems controbbility is equivalent to the possibility of arbitrary pole placement by state feedback),witnessed the development of theoriesunderlying the various linear multivariable regulator design methods listed earlier-especially as in the textbooks of Rosenbrock and Wonham-and seen the emergence as algebraicandgeometric of entirelynewtheoreticaltopicssuch (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 [ 21 together with Chen’s relatively elementary text [ 31, both published in 1970, or by using Brockett’s text [ 4 ] , which was also published in 1970. In fact given the apparently semi-infiiite amourrt of material on thetopic, it was extremelyuseful t o haveavailablesuchcondensed presentations of the fundamentals of the subject. However, an update on thesubject was dueand 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 theoreticallyorientedseniorandgraduateengineeringstudents,a digestionandorganization of recentadvances(againin textbook form),andaguide to currentresearchtopicsandadjacentareas 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; GeneralDifferentialSystems and PolynomialMatrixDescriptions;Some Results for TimeVariantSystems;SomeFurtherReading;andan 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 particularpedagogicalstyle 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 t o correspond to a path of learning and discovery on the partof the student. One major

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manifestation of this approach is that univariate (single-input singleoutput) systems are treated fmt and multivariate(multi-inputmultioutput) systems do not make their official appearance until the sixth chapter. In the first fwe chapters topics such as state space representations (or realizations), observability and controllability, state feedback, and observer and compensator design, are introducedand 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, manyinstructive exercises, and awealthof references to the literature. While 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 f m t five chapters are as follows: chapter two presents the idea of the state as a set of initial conditions foranalogcomputer 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 parametersystems; and chapter four treatsfeedback and compensator design proceduresfor 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 t o 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 thatsuch 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 t o 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 developedand 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 systemtheory, 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 transferfunctions) are laid out withan 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 boththestate space and matrixfractiontechniques 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 volumedevoted to the quadratic regulator problem or to frequency domain methods. Despite this the author manages to includesomeinterestingmaterialdevoted to the asymptotic behavior of pole-zero patterns in quadratic regulator theory. Chapter eight is a comparatively short chapter devoted to the analysis of s d e d general differential systems imd polynomial matrix systems. For these classes of systems-which are in certain technical senses more general than matrix f i c t i o n 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 timevarying systems, and the brief final chapter ten pointhim or her towards some topics of current research. In the first chapter the author says he a i m s to be logically consistent rather than mathematically rigorous and this approach may result in

PROCEEDINGS OF THE IEEE, VOL.

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JANUARY

1982

some frustrations for the more mathematical reader. For instance, the book avoids a formal definitionofa linear input-output system, or more importantly, of a linear input-stateautput system. Of course the latter may be given as the joint linear operation of the system on the system state and an appropriate segment of the input function,see e.g., [ 11, [ 21. Furthermore the formal definition of the notion of state in terms of its semigroup property (see e.g., [ 11 , [ 2)) is omitted. Instead the author gives an informal discussion of the notion of state which I think rather misleadingly brings in thenotions of minimality and minimal statistics; chapter five then goes on to the moreadvanced topic of constructing system states using the notion of Nerode equivalence. In my opinion the formal defintions 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 L- Laplace transform and of impulsive input functions for differential equations, the author uses the notation t 3 0+,t 3 0-, etc. Now whereas x(O+) has a well defined meaning for a function x(.)(= the limit of x ( t ) as t lends to zero from the right, when this limit exists), x ( t ) f o r t > 0+, for example, does not. This notation leads the author (p. 11) to write “x ( t ) = e - z f + e-2 r , t o -,” as the solution to Y ( t ) + k ( t ) = 6 ( t ) ,t > 0-, ~(0%) = 1,” when the solution to this differential equation for all time t with the given boundary conditions is evidently x ( t ) = e-2t + 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(0). Another irritation of an analytic nature is the discussion of bounded input bounded output (bibo) stability for systems described by y ( t ) = h(s)u(t - s) ds. Since h(.) lie in unspecified function spaces the author “proves” that the system is bib0 stable if and only if h ( - ) is integrable, i.e., Ih(s) 1 ds < =. Now this suggests that L’ [0, =) (integrable Lebesgue measurable functions) and L“[O, -)(bounded Lebesgue measurable functions) are duals. However, this is notthe case, for although the dual of L’ is L”, the dual of L“ is a much larger space than L ’ . On the other hand the charcterization of bib0 stability becomes true if all functions under discussion are restricted to lie in L”[O, =) at the start. My penultimatecomments concern English andtypography. 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 u p 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 thistreatise. As farastyographicalerrorsareconcerned, I can hardly find any. The author and all who assisted him are to be congratulatedon very carefulproofreading. Theauthorinforms 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 expositon of much of the last decade’s research. I am sure it will be a huge success. REFERENCES [ 11 L. A. Zadeh and C. A. Desoer, Linear System Theory. New York: McGraw Hill, 1963. [ 2 1 C. A. Desoer, Notes fora Second Course on Linear Systems. New [3] [4]

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York: Van Nostrand Reinhold, 1970. C. T. Chen, Introduction to Linear SystemTheory. New York: Holt, Rinehart, and Winston, 1970. R. W. Brockett, Finite DimemionaI Linear Systems New York: John Wiley, 1970. T. Kailath, “A view of three decades of linear filtering,” ZEEE 7kan.s. Znfonn. Theory, vol. IT-20, no. 2 , pp. 145-181, Mar. 1974.

Reprinted from IEEE Trclnsactions on Information Theory,May 1981.

Digital Contrd Systems-B.C. Kuo (New York:Holt,Rinehartand Winston, 1980, 730 pp., $27.85). Reviewed by Chin-ShungHsu, WashWA. ington, State U n i v m * t y pullman, , Significant progress made during recent years in the numerical aspects of controland systemtheory as wen as miaocomputer technology prompts a great desire to develop a Wtal control course for undergraduate seniors. This will not only introduce students to anew dimension of technical knowledge which undoubtedly is of paramount