996 PROCEEDINGSVOL. IEEE, OF 9, THE NO. 66, SEPTEMBER 1978 Relationships Between Digital Signal Processing and Control and Estimation Theory Invited Paper Akmcr-The purpose of this paper is to explore several current far deeper level than is possible in this initial effort. To aid le&aeh directions in the fields of digitrl signal procesing and modem others who may wish to follow up on some of the directions cant101 and estimation theory. We examine topics such as stability theory, tineu prediction, and pmuneter identifiition, system synthesis developedin thispaper; an extensivebibliography has been and implementation, twodimensionnl filtering, decentralized control included.In addition, theinterested reader is referred to and estimation, and image processbrg, in order to uncover some of the [ 1621 in which all of these research directions are explored brsic similarities and differences in thegods, techniques, and philosophy in substantially greater breadth and detail. of the two disciplines. Nowhere in thepaper have I madea direct attemptto INTRODUCTION define the fields of digital signal processing and control and HE WRITING of this paper was motivated by the belief estimation.Rather, I hopethat by examiningmany of the that the fields of digital signalprocessing and control issues of importance to workers in these fields, the reader will andestimation theory possess enough similarities and be able to piece together a picture of the disciplines and their differencesT in philosophy, goals, and analytical techniques to relationship to each other. As a preface to our examination, merita detailed joint examination. In orderto explore the let me mention several points concerning each field. relationship between these two fields, I found it essential to In digital signalprocessing, one of the crucial problems is concentrate on several specific research directions to provide the design of an implementable system meeting certain given a focus for my investigations. The results of this study were a design specifications such as an ideal frequency response. Here talk delivered during the 1976 IEEE Arden House Workshop the emphasis often is on the word implementable, with a fair on Digital Signal Processing, the present paper, and a far more amount of attention paid to issues such as the structure of the comprehensive manuscript [ 1621. digital filter, its complexity in terms of architecture and com- Although the paper consists of discussions of several specific putation time, the effect of finite wordlength on performance, research directions, the primary emphasis of this paper is not etc. Much of this attention is motivated by the need for ex- on results. Rather, I have been far m.ore interested in under- tremely efficient systems to perform complex signal processing standing thegoals of the research and the methods and approachtasks (e.g., the implementation of high-order recursive or non- used by workers in both fields. Understanding the goals may recursive filters) at very high data rates (for example, sampling help us to see why the techniques used in the two disciplines rates encountered in speech processing run on the order of 10 differ. Inspecting the methods and approaches may allow one kHz, and higher rates arethe rule in video and radar systems). to see areas in which concepts in one field maybe usefully In control and estimation, the emphasis has been far less on applied in theother. In summary,the primary goal of this implementationand more on developing methodsfor deter- study is to providea basis for future collaboration among mining system design specifications for estimation or control researchers in bothfields. systems. At one level these specifications are just a particular It is hoped that the above comments will help explain the class of design guidelines which can then be used to construct spirit in whichthis paper has been written. In readingthis an implementable digital system.However, there aremajor paper,the reader may find many comments that are either differences between the systems arising in the control context partially ortotally unsubstantiated. These points havebeen and the typical digital processing application. For one thing, included in keeping with the speculative nature of the study. the data rates for control systems are often far lower (e.g., in However, I have attemptedto providebackground for the aircraft control systems sampling rates on the order of 0.1 kHz speculationand have limitedthese comments toquestions are oftenencountered, although much higherrates can be which I feel representexciting opportunities for interaction found in certain casessuch as video-directed systems). More andcollaboration. Clearlythese issues must be studiedat a fundamentally, however, the signal processing to be done in a control system cannot be judged by itself, as can other signal processing systems, since it is part of a feedback loop, and the Manuscript received January 16,1978;revised May 8, 1978. Thiswork effect of the processing must be studied in the context of its was supported in part by NASA Ames under Grant NGL-22-009-124 closed loop effects. and by NSF under Grant GK-41647, and in part by a Senior Visiting Fellowship to the Department of Computingand Control, Imperial Also, many modern control and estimation techniquesinvolve College, awarded by the Science Research Council of Great Britain. the use of a state-space formulation,as opposed to input-output Theauthor is withthe Department of ElectricalEngineering and Computer Science, Massachusetts Institute of Technology, Cambridge, descriptionswhich are usually encountered in digital signal MA 02 139. processing applications. Some of the reasons for this difference 0018-9219/78/0900-0996$00.750 1978 IEEE WILLSKY:DIGITAL SIGNAL PROCESSING AND CONTROL AND ESTIMATION THEORY 997 will be made clear in the following sections, but one implica- A. State-Space Realizationsand State-SpaceDesign Techniques tion is immediately evident. The use of a state-space descrip- The basicrealization problem (for linearsystems) isas tion implies that the system under consideration is causal. In follows: we are given a (possibly time-varying) description of standardfeedback control problems this is clearly the case, the input-output behavior of a system and thus state-space formulations make a great deal of sense. As we shallsee, there are digital signal processing problems involving noncausal systems or systems in which the indepen- (1) dent variable hasnothing to dowith time and for which causalityhas no intrinsicmeaning. Thus, while wewill find where u and y mayboth be vectors.In the time-invariant several places in which statespace concepts fit in naturally case we have that the sequence of impulse response matrices in the digital signal processing context, we’ll also find others satisfies in which that is decidedly not the case. The preceding comments were made in order to provide the T(k, i)= T(k - i, 0) p Tk-i reader with some insight into the perspective I have taken in and in this case we may be given an alternative input-output writing this paper. With this as background, let us begin our description in the transform domain examination of research topics in the twofields. 00 I. DESIGN, REALIZATION,AND IMPLEMENTATION Y(z)= G(z)U(z), G(z)= Tiz-’. (3 1 In this section we investigate one subject area in which some i=O of the differences in perspective between the two disciplines The realization problem consists of finding a state space model aremost apparent. Specifically, we consider thequestion of design. However, our discussion will not deal very much with x(k + 1) = A(k)x(k)+ B(k)u(k) design methods but rather with the question of trying to pin- y(!c) = C(k)x(k)+ D(k)u(k) (4) point what researchers in the twodisciplines mean by “design” and what sorts of problems their techniques are equipped to that yields the desired input-output behavior (( 1) or (3)) when handle. x( 0) = 0. Perhaps the most obvious difference between the fields‘is in Therealization problem has been studied in detail in the the type of system representations used. In digital signal pro- controlliterature, and one aspect that has received agreat cessing, the emphasis is heavily on input-output descriptions, deal of attention is that of determining minimal realizations- while in control and estimation the emphasis is more on state- Le., models as in (4) with the dimension df x as small as pos- space models. The reasons for this difference stem from the sible. The basic idea here is that a minimal realization has no differentquestions addressed by researchers in the two dis- superfluous states that either cannot be affected by inputs or ciplines.In digital signal processing one isinterested in the do not affect the output. These concepts lead directly to the issue of implementation of a system with aspecified input- notions of controllabilityand observability. In the time: output behavior(hence the need foran input-output de- invariant case, one obtains a rather complete description. Spe- scription).Questions such as efficientimplementation and cifically, we find that the system (3), has a finite-dimensional numberof bits needed to achievethe desired levelof ac- realization if and only if G(z) is rational with each element curacy are of great importance. having nomore zeroes than poles. Furthermore,any con- Onthe other hand, in controland estimation theory the trollable,and observable time-invariant realizationis of minimal issue of implementation is not considered to nearly the same dimension,