Geometric state-space theory in linear multivariable control: A status report

Automaliea Vol. 15, pp. 5-13

Pergamon PressLtd. 1979. Printedin Great Britain © InternationalFederationof AutomaticControl

Geometric State-Space Theory in Linear Multivariable Control: A Status Report* W. M. WONHAMt:~ A theory based on a small number of key geometric concepts can be used for the discovery and unification of synthesis procedures for linear multivariable controls. Key Word Index--Geometric state-space theory; multivariable control systems; algebraic system theory; decoupling; regulator theory; servomechanisms.

Abstract--With the renewed emphsis in control theory on qualitative structural issues, as distinct from techniques of optimization, the last decade has brought significant growth in the application of geometric ideas to the formulation and solution of problems of controller synthesis. In this article we review the status of geometric state space theory as developed for application to systems that are linear, multivariable and time-invariant. After a brief summary of the underlying geometric concepts ((A,B)-invariant subspaces and (A,B)controllability subspaces) we outline two standard problems of feedback control that have been successfully attacked from this point of view, and briefly survey recent results in a range of other topic areas. Then we discuss certain issues of methodology, and conclude with some remarks on computational procedures.

1. I N T R O D U C T I O N

THE LAST ten years have witnessed a remarkable development in linear multivariable control, from both the theoretical and applied viewpoints. In particular, significant advances have taken place in our theoretical understanding of system structure. Under this heading we may group two major topic areas: realization theory, and system synthesis. Realization theory is concerned with the axiomatic definition of 'system', and with relating 'internal' system descriptions (i.e. state descriptions) with 'external' (i.e. input output) descrip-

*Received February 28, 1978; revised July 10, 1978. The original version of this paper was presented at the 7th IFAC Congress on A Link between Science and Applications of Automatic Control which was held in Helsinki, Finland during June 1978. The published Proceedings of this 1FAC Meeting may be ordered from: Pergamon Press Limited, Headington Hill Hall, Oxford, OX3 0BW, England. This paper was recommended for publication in revised form by associat. D. O. Anderson. +This research was partially supported by The National Research Council of Canada, Grant No. A-7399. :~Systems Control Group, Dept. of Electrical Engineering, University of Toronto, Toronto, Canada M5S 1A4. §Some of the remarks to follow are taken from the author's report {Wonham, 1977).

tions. For this a unified mathematical framework was supplied by Kalman in 1969, exploiting the connections (well known to algebraists) between linear algebra and the theory of modules over a polynomial ring. The general theme has since been pursued in the context, for example, of delay systems, bilinear systems, and for general systems from the 'categorical' point of view (Arbib and Manes, 1974). While realization theory supplies the infrastructure, it is system synthesis and design that is perhaps of most direct interest to engineers, and the remainder of this paper will be devoted to sketching the situation here, in respect to the 'geometric state-space' approach (Wonham, 1974). Before getting down to specifics it may be useful to distinguish between 'synthesis' and 'design' in the context of control systems (cf. also Rosenbrock (1974), Chap. 2). We would label 'synthesis' the process by which one establishes the qualitative structural possibilities: that is, one determines whether and how one can achieve such desirable system properties as noninteraction (among functionally distinct subsystems and their associated controllers), loop stability, regulation of specified outputs with respect to disturbances, tracking by specified outputs to reference signals in some preassigned class, and finally structural stability, namely preservation of the foregoing properties within a range of parameter variations. On the other hand, 'design' would refer to the numerical massaging (ideally, optimization) of system parameters, within the structural framework established by synthesis, to meet quantitative design specifications related to transient response, stability margin, saturation levels and SO o n .

While the distinction is obviously not clearcut,

6

W . M . WONttAM

we at least have two working definitions. The point of making them is to suggest that a control theory, qua theoretical construct, deals largely with synthesis, and thus may be expected to operate on a level of relative generality and abstraction. Its function is precisely to clarify the structural possibilities. Design may then bring in a host of special techniques, some of them largely heuristic, to ensure that a specific system works in practice. It is hoped that these simple-minded remarks may forestall futile controversy about the "gap between theory and practice'. A theory of synthesis will start from an assumed model or class of models together with an assumed class of admissible controls, and supply constructive answers to a series of precisely posed questions of the following kind: Does an admissible control exist such that the resulting (synthesized) system displays such-and-such desirable characteristics? To qualify as a theory, and not just a library of algorithms, our constructive procedures should somehow exhibit the answers in terms of a reasonably small number of basic system concepts. While these must of course be made theoretically precise, they must also embody enough intuitive content to render the process of discovering new answers, by new procedures, easy and natural for the investigator. Finally, the variety of questions amenable to attack must be wide enough to embrace a significant range of potential applications, and our constructive procedures must be capable of translation into computational algorithms: these will provide the framework for design, as noted earlier. l think it is fair to claim that at least two complementary theories exist today which go some way to meeting the criteria just described. These are the geometric state-space approach (Wonham, 1974; and references cited therein), and the polynomial matrix-frequency domain approach (Rosenbrock, 1970, 1974; Wolovich, 1974; and cited references). In neither of these theories is the notion of optimization at all central, and we omit discussion of the familiar 'lineaF quadratic" optimization problem, on which an extensive textbook literature already exists. II suffices to say that quadratic optimization is merely one method of achieving the rather limited goal of stabilization via state feedback; and that this theory, useful as it may be in design, contributes little to the primary objectives of structural synthesis. In the remainder of this paper we outline the geometric state-space theory as it is presented in Wonham (1974), together with some of its further developments in the last three years.

2. G E O M E T R I C STATE SPACt" T H E O R Y : F U N D A M E N ' I ALS*

Here we take for granted a (discrete or) continuous-time slate-variable model of the familiar type 2=Ax+Bu+Er+... y =Cx+ C'r+... z = D x + D'u+ D " r + . . .

As usual, x and u denote the state and control vectors, r ( = r ( t ) ) is a vector of disturbance inputs, y is the vector of measured output variables, and z is the vector of output variables to be regulated (i.e. to be maintained ideally at zero); the components of z are tracking errors and/or deviations from set-point values. We remark that everything that is said below applies equally well to the corresponding discrete-time system model. Also, we restrict attention to the deterministic theory, inasmuch as the stochastic theory is still in a very rudimentary state of development (Snyders and Wonham, 1975). The admissible controls are of the type u=Fx+F'r+...

incorporating state feedback and (possibly) disturbance feedforward. One must impose a constraint restricting the control to process only the measured variable y; this translates algebraically as

Ker[F, F'] ~ Ker[C, C'], where Ker means kernel (or null space). More generally (and much more usefully) one can admit dynamic compensation by introducing additional (compensator) state variables via equations -'c'c :

llc,

i!'~ :

Xc,

and lumping these with the state description given originally. By a similar device one accommodates the use of observers. What are the basic system concepts ? Everything starts from the fundamental ideas of controllabilityt and observability, or more accurately the geometric objects 'controllable subspace' and 'unobservable subspace' of the state space f . Two related families of subspaces which *Except where otherwise noted, the l~aterial in this section (including historical backgroundl is treated in detail in Wonham (1974). tin this article 'controllable" is synonymuus wilh "reachable'.

Geometric state-space theory in linear multivariable control" A status report play a basic role are the '(A,B)-invariant subspaces' and the '(A,B)-controllability subspaces'. To give a brief indication of their meaning, suppose we have simply

~=Ax+Bu,

y=x,

z=Dx.

(1)

Denote the state space by .~7 and the control vector space by 4/. Thus A : f ~ f , B : q / ~ W are linear maps. Recall that a (linear) subspace ~ is (geometrically) just a hyperplane passing through the origin 0 ~ f , of any dimension 0 < d i m ~ O) such that the resulting state trajectory x(t)=x(t; x(0), u) also satisfies x ( t ) ~ for t > 0 . Briefly, we can hold x ( . ) in ~ by suitable choice of u(-). Now it can be shown that ~" is (A,B)-invariant if and only if

A~t~= ~ + ~

(2)

i.e. for each vector v e ' t , Av can be expressed as A v = v ' + B u for suitable v ' e U and u~all; denotes the image (or range) of B. The condition (2) is actually equivalent to the following 'synthesis' property: there exists a linear state feedback map F: f ~ o k ' such that

(A + BF)'~"=~U.

(3)

Namely, if we set u = F x in (1) and consider the autonomous system ~ = ( A + B F ) x , then for this system x ( 0 ) ~ U implies x(t)~'U ( t > 0 ) ; so if x ( . ) starts in 5/~, it stays in "f': ~t- has been made invariant by suitable state feedback. Now ask: what motions of the system (1) would leave the regulated output z unchanged? For such a motion one has Dx(t)=O, i.e. x(t)EKerD (KerD c f is the subspace of states x where D x = 0 ) , and this will be true if x(t) (t>O) belongs to some (A,B)-invariant ~ ; = K e r D . A remarkable fact (easily proved) is that there always exists a largest such ~¢~, i.e. a unique ~'* of largest dimension, that contains all the other (A,B)invariant U's in Ker D. So "//* is the largest set of states x in f such that Dx =0 and, if you start in this set, you can be controlled to stay in it. There is a simple recursive algorithm that computes ~'* in a number of steps < d i m ( K e r D ) ; and then it is easy to compute an F that synthesizes "t"* in the sense of (3). What is the use of "t~*? Here is a simple, yet basic application. Suppose = Ax + Bu + Er,

y = x,

z = Ox,

7

and you want to decouple the disturbances r ( - ) from the output z( • ), using state feedback u =Fx. It turns out this is possible just when "/~* = g (the image of E), because then r ( . ) can never knock you out of "V*, and a suitable F will always lock you in ~t"*.t In other words, ~ ' * ~ is a very simple and constructive way of stating that there exists an F such that the closed loop transfer matrix from r to z vanishes identically:

D(sI - A - BF)- 1 E=_O. Obviously this latter condition is a very awkward nonlinear condition on the elements of F, being equivalent to D ( A + B F ) i - I E = O , l _ < i < d i m f ; and so the example begins to illustrate the power and intuitive naturalness of the geometric approach. An interesting practical application of this solution, to the design of a controller for a distillation column, is reported in Takamatsu, Hashimoto and Nakai (1978). With this problem we can easily go a bit further, showing that it is 'generically' (or 'almost always') solvable provided DE=O and the number of controls>the number of outputs ( d i m ~ > d i m f - d i m Ker D). We can also extend the result (cf. Bhattacharyya, 1975c) to allow disturbance feedforward (u=Fx+F'r): the condition is then ~U*+M=~', which is evidently easier to satisfy ! In formulating this problem we ignored the important issue of whether F could be chosen to guarantee, in addition, that (say) A +BF is stable. Investigation of stabilizability subject to a constraint of the type (3) leads one to introduce the :following subclass of (A,B)-invariant subspaces. Given the system (1) as before, we say that a subspace ~ = f is an (A,B)-controllability subspace (c.s.) if every state x~ .~ can be reached from the origin of Y' along a controlled state trajectory that is wholly contained in .8. Trivially 0 is a c.s. and so is , the controllable subspace + of the pair (A,B). If the system is single-input ( d i m ~ = l ) these are the only c.s. (and the c.s. concept is uninteresting); but in the multi-input case ( d i m ~ > 2 ) there are lots of c.s. of various dimensions (Warren and Eckberg, 1975; Wonham, 1974), and they serve to 'parametrize' certain design possibilities, as we now indicate. t i t is assumed that initially, say, x(O)=O. ~By definition. (AI,~' > =.~ + A,¢~+ ... + A ' - ' where n = d i m W . Thus (A[,~> is the image (range) of the controllability matrix.

8

W . M . WONHAM

First, if ,~ is an (A,B)-c.s. then ~ is (A,B)invariant, so that ( A + B F ) ~ :~ for some F. For such F it can be shown that . # = ( A +BFl~c~ ,~), namely :~ is 'generated' by its intersection with the control input vectors :~. This means there exists G:J#-,')# such that ,~ = ( A + B F I I m B G ) , so that if in (1) we set u = F x +Gv, with v a new external control input, then by manipulating v(. ) we can reach all states (and only states) in ~, starting from x = 0 . In other words, we have created a controllable 'subsystem' by means of state feedback and input gain selection; and all controllable 'subsystems' are created in this way. Next, by exploiting the equivalence between controllability and pole assignability we may characterize the class of c.s. by the following basic property: a subspace .~ ~ ~' of dimension p is a c.s. if and only if it is (A,B)-invariant and, for every (symmetric) set A of p complex numbers, there exists state feedback u = F x such that (A+BF) ~ ~ and also a ( A + B F I . ~ ) = A : namely the spectrum (set of eigenvalues) a, of A + B F restricted to ;8, coincides with A. It is this 'design' property of c.s. that makes the concept central in the applications. Many other properties of c.s. are known (Heymann, 1976; Wonham, 1974): to complete this sketch we note the crucial fact that, for any output map D:72'--,2, there exists a unique largest c.s. UJ?*cKer D. Thus you can sneak from 0 to any state x e ~ * without being seen from z, and ,~* is the largest set of states with this property. From what has been said it should be clear that 0 ~ : ~ * ~ * ~ K e r D (where the inclusions may or may not be strict). Finally, ~ * can be computed from ~:* by a simple algorithm (Wonham, 1974), again in a number of steps < dim(Ker D). To illustrate the inclusions ~ * c / " * c Ker D, it may be helpful to sketch the corresponding system decomposition in terms of signal flow. Thus one can write the state-space .Y" as a direct

x -

Xs

:+:~ ® +E, •

X 1 - R*, X 2 =: V*IR*, X 3 =XIV*

U/O

~ •

~

.~ =Xl

R+

V+ i]

V+/R*

X~ V*

F;G. 1. State space decomposition. Furthermore, as "t:*c Ker D, D has matrix D=[0

D3];

0

(5b)

also, Y2 can be chosen in such a way that

B=

Ii

00

ll

(5c)

1

B)2

The signal flow graph for the system 2 = Ax + Bu,

z = Dx,

decomposed as in (4), is shown in Fig. 1. Here the branch transmissions are easily identified from (5). What is the use of ~ * ? Historically, the first application (1970) was to the longstanding prob-, lem of noninteracting control, or decoupling. In the simplest version one is given 2 = Ax + Bu,

y-=-x,

z~ = D~x

i e k.

One asks for state feedback control with new external inputs v~, k

t+= Fx + ~

aivi,

i=l

sum

say, where ,+'t = ~ * and ~ J ' l ( ~ ° , ~ ' 2 = ¢ "*. One can always select a feedback map F:?t'~,¢/ in such a way that ( A + B F ) ,~*~ ~*,

(A+BF)I'*c/'*;

so that, in a basis adapted to the decomposition (4), A + BF has a matrix of the form

A+BF=

[

~ll

A12

A13~

A22 A23l 0

A33.. ]

(5a)

such that each v~ controls the output vector z~ alone, without affecting the zj for j@i. It is convenient to augment this class of controls by bringing in auxiliary dynamic compensation by the means noted earlier; in this way we are led to the 'extended decoupling problem' (EDP). Then we have the major result that EDP is solvable if and only if ,~* + Ker D i = 2 ,

iek,

(6)

where ~ * is the largest c.s. contained in the subspace (~s,iKerDs. Again, the condition is intuitively quite transparent: because (6) is equivalent to D i ~ * = I m D i (iek), it says that

Geometric state-space theory in linear multivariable control: A status report the largest set of states x controllable by vi without interaction among the z's, must be large enough to allow the ith output to reach any vector of the form z = D i x ; namely, output controllability is achievable despite the noninteraction constraint. And again, the condition (6) is algorithmically verifiable. Under very reasonable conditions on the number of available controls, one finds that (6) is almost always satisfied, and furthermore one can achieve more-or-less arbitrary pole locations for the closed loop system (Wonham, 1974). Numerical applications are reported in the theses of Fabian (1974) and Yuan (1976). The second major structural problem to be treated successfully by these methods was the general multivariable servo-regulator problem (1974). One is given

2=Ax+Bu,

y=Cx,

z=Dx.

(7)

Here the dynamics incorporate not only the plant, to be controlled, but also an explicit dynamic model of the reference signals and disturbance signals (e.g. steps, ramps, sinusoids . . . . ) which the system is to be designed to track or reject respectively. This model we call the exosystern (i.e. 'outside world'). Thus if, for example, the problem requires tracking a ramp input while rejecting a step disturbance then (7) could be written in more detail

_.dlx3/

Ii 00

Lx4j

y=Cx,

010//x l + o o o,x, 0 o 0 j L s/

z=[d T-1

0

[i]

U

0]x.

Here x 1 is the state vector of the plant, (xz, x3) are the state variables of the reference generator and x4 is that of the disturbance generator: (x2,x3,x4) describe the exosystem. The regulated variable z(-) (the 'error' signal vector) is to be maintained ideally at zero; thus it is required that z(t)--,O as t--,00 from arbitrary initial states x(O); this we call output regulation.* Next, the control loop is to be stable when the exosystem is itself at the zero state; more precisely the controllable, observable 'modes' of the system must be stabilized: this is the requirement of internal stability. Finally, the state feedback matrix F corresponding to an admissible control u

*It is straightforward to incorporate direct control feedthrough in the regulation condition; i.e. in (7), z = D x + E u . For the details see W o n h a m (in press).

9

=Fx must satisfy the observation constraint that K e r F D Y , where Y c X is the unobservable subspace determined by the pair (C, A), namely ~+~= ~ Ker(CA i- 1), n = dim X. i=l

It can be shown that this constraint on F is equivalent to the use of a dynamic observer with inputs the observed output signal y ( . ) and the control signal u(-). Our problem description has been rather informal; but with some extra notation and standard algebra, the foregoing requirements can be stated crisply and lucidly in coordinate-free, geometric style (Wonham, 1974). The setup is visualized to work as follows. Each tracking (or disturbance rejection) 'task' is initiated by placing a suitable initial condition on the exosystem dynamics; the system is to be designed so that then z(t)--+O as t ~ , namely z(t)~_O for t ~ T , depending on the exponents (poles) assigned to the error response transient. If no tracking task is present the equilibrium state x = 0 is asymptotically stable, i.e. x(t)-oO (t---,oo) again with response governed by suitable exponents. In practice, time constants are to be selected so that T is small compared to the estimated time interval between successive 'tasks'. Thus the viewpoint is essentially classical. Limitations of space preclude a complete statement of the necessary and sufficient conditions for the solvability of our 'regulator problem with internal stability' (RPIS) (Wonham, 1974). These conditions can, however, be very naturally expressed in terms of the subspaces 3 " and ~'* in Ker D. Assuming..~V=0 (complete observability) for simplicity, the idea is to arrange matters so that the 'unstable modes', denoted by X+(A +BF), are placed in U * c K e r D (for output regulation), while 'splitting off' X+(A +BF) from (AIM) (for internal stability). In more detail, this can be done just when (i) X + ( A ) c ( A [ ~ > + ~ "*, namely the unstable modes of A can be rendered unobservable at the regulated output by state feedback, and (ii) the factor space ('U* c~ ( A [ ~ > ) / ~ * admits a complemefft in U * / 3 " that is invariant under the map induced. in X~ it* by A+BF, for suitable choice of F. Condition (i) ensures output regulation, while condition (ii) guarantees that regulation can be achieved subject to the requirement of internal stability. Computationally, the latter condition amounts to the solvability of a linear matrix equation of Sylvester's type, AaX--XA2+A3=O; our result is, therefore, entirely constructive. Furthermore, in expressing directly the relevant structural principle, its compact geometric form

10

W . M . WONHAM

enjoys a simplicity not shared by the monolithic matrix rank conditions to which various other approaches to problems of similar type have led. This completes our brief introduction to the basic ideas and two major problems of the geometric state space theory as it developed to about 1974. In the next section we survey more recent results. 3. GEOMETRIC STATE SPACE THEORY: RECENT RESULTS Numerous variations on the basic themes have suggested themselves and have been pursued with success.* In regard to decoupling, we note an application to sample-data systems (Zahir and Slivinsky, 1974), and the combined study of decoupling and disturbance rejection (Chang and Rhodes, 1975; Fabian and Wonham, 1975b). Disturbances may arise internally from parameter variations, and it is of interest to study the class of variations that leave decoupling unaffected (Fabian and Wonham, 1975a). Issues of generic (i.e. 'typical') structure are relevant, not only in indicating when decoupling is likely, in principle, to be feasible (Warren and Mitter, 1975; Wonham, 1974), but also in guiding the design of adaptive controllers that maintain decoupling when system parameters vary (Yuan, 1976; Yuan and Wonham, 1975). Some study has been devoted to the servoregulator problem from the viewpoint of structural stability: namely, determining when the problem is well-posed (i.e. solvable throughout some neighborhood of a nominal data point in parameter space) and, when that is the case, establishing the structure of a controller that maintains output regulation and internal stability at parameter values throughout such a neighborhood. Well-posedness was discussed in Wonham (1974); here we note further that it amounts geometrically to requiring (in our previous notation for RPIS) that the two invariant spectra, associated with (~"* ~ ( A [ ~ ) ) / , ~ * and ~/~*/(~ * ~ ~A[~)) respectively, be disjoint, and that dim(~c~ :¢'*) be suitably small. For structural stability the controller must incorporate feedback of the regulated variable z and an 'internal model' of the exosystem (Francis and Wonham, 1975a, c, 1976; Pearson and Staats, 1974); numerical applications are reported in Sebakhy (1974) and Sebakhy and Wonham (1976). The geometric principle at the heart of these results is transver*In this section we cite almost exclusivelycontributions in which significant use is made of the geometric concepts and methods sketched in Sec. 2. Of course this is not meant to imply that only such methods have been found effectivein the problem areas considered.

sality, or the notion of ~intersection m general position'. Two subspaces y,~, .~' <:// are said to intersect transversely when dim( .~ +.9't is a maximum compatible with the given dimensions of .~, .~/' and .~'. The transversality concept plays a central role in the structural stability theory of smooth manifolds: a nice introductory account is given in Guillemin and Pollack (1974). A system a t i c treatment of the internal model principle from the viewpoint of transversality is presented in Wonham (in press): and these general ideas are expected to lead to useful insights into feedback structure in nonlinear systems too. In the linear setting the foregoing results may be translated to the frequency domain: the technique for doing so is to choose a state-space basis that exhibits ~At~ ), ~ *, .~* and then to compute the relevant polynomial system matrices (in the sense of Rosenbrock (1974)) and transfer matrices, as done in Anderson (1976), Bengtsson (1976), Francis and Wonham (1975b), Molinari (1976c) and Morse (1973a). A useful concept to emerge has been that of the transmission zeros of the open-loop system (see the references just cited): just as in the single-input, single-output case, these zeros are complex frequencies at which the system may block signal transmission. The transmission zeros are exhibited in (5at as the eigenvalues of A22, regardless of the specific choice of F provided the condition (A + B F ) I * c ~ * is satisfied. Right-half plane transmission zeros cannot be removed by feedback without illegal pole-zero cancellation (resulting in internal instability). In these terms it can be shown, for instance, that RPIS is well-posed just when the open-loop transmission zeros are disjoint from the (right-half plane) poles of the exosystem (these two sets of complex numbers coincide respectively with the two invariant spectra mentioned earlier), and when the open-loop transfer matrix is right-invertible (corresponding to the associated dimensional condition). Another longstanding synthesis problem that has now been extensively treated by the geometric approach is the model-matching problem, or that of forcing the input o u t p u t behavior of a given system to coincide with that of a desired 'model', by means of suitable dynamic compensation. Among the issues involved are those of generic solvability, stabilizability and of minimizing the compensator's dynamic order (Bengtsson, 1976; Howze, Thisayakorn and Cavin, 1976; Morse, 1973b). Closely related is the problem of solving linear equations in rational matrices and its connection with system invertibility (Emre and Silverman, 1976; Morse, 1976). In the same context it has recently been shown (Emre, Silverman and Glover, 1977) that a unify-

Geometric state-space theory in linear multivariable control: A status report ing geometric concept is that of a (generalized) dynamic cover: given A : ~ S f and subspaces M, 8, ~e in ~, a subspace ~ c Y " is said to be a (generalized) dynamic cover for Lr, relative to the triple (A,~,8), if ~ e ~ + ~ and A ~ " c ~ + ~ 3 + 8. In addition it may be appropriate to impose a constraint C c J g . Thus ~ is an ( A , ~ + g ) invariant subspace whose projection ( m o d e ) contains (or 'covers') the factor space ( ~ + M ) / M . The main problem (Emre, Silverman and Glover, 1977) is to determine whether or not a cover exists and, if this is so, to characterize the covers of minimal dimension. An algorithmic solution to the problem is presented in Emre, Silverman and Glover (1977), to which the reader is referred for the details. While at first glance the concept of a cover appears rather formal, it is readily interpreted (Emre, Silverman and Glover, 1977) in the context of dynamic observers. Suppose the objects listed above are all defined with respect to the dual space ~T (say) of linear functionals on the usual state space Y" (that is, in the definition of cover replace ~ by y,v, A by AT etc.), and set = 0, ~" = ?~;'T for simplicity. Then a cover ,/~-T is just the span of linear functionals generated by a dynamic observer that estimates outputs in the span of ~ T c . T T with the aid of measured functionals (i.e. system outputs) spanning ~T. Clearly the problem of minimal covers is centrally related to the problem of minimal observers (see also Kimura, 1975a), and of minimalorder compensation for stabilization or complete pole assignment (see also Kimura, 1975b). It is remarkable that this concept turns out to be essential also in the model matching problem (Morse, 1976b) and, as explained in Emre, Silverman and Glover (1977), in problems of transfer matrix realization and of deterministic identification of state models from input output sequences. In addition links may be established to the structure of observers that operate with only partial access to system inputs, and to dual problems of output-nulling observability (Anderson, 1975; Basile and Marro, 1973; Molinari, 1976a, b). To conclude this survey we note the increasingly important topic of decentralized control, the structural principle according to which a system is jointly controlled by several agents, each working to some extent independently through its own output and control channels. Coupling effects are minimized, or may be compensated for by a 'coordinator' acting as a supervisory control in a hierarchical organization. One interesting potential application is the regulation of vehicle spacing and velocity in a long train (McLane, Peppard and Sundareswaran, 1976). As is often pointed out, simplicity of implementation, re-

11

liability of operation (Wong, 1975) and economy of data processing and communication, are among the benefits that might be gained by decentralization. One question that arises immediately is whether stabilization can be accomplished by purely local control, and a fairly complete answer, based on concepts from graph theory and the geometric state space theory, is developed in Corfmat and Morse (1976a, b). Of interest also is the decentralized approach to disturbance decoupling (Hamana and Furuta, 1975). It seems likely that this topic area will attract increasing attention.

4. ISSUES OF LANGUAGE AND COMPUTATION The 'language' in which the the geometric state space theory is usually presented is the mathematically quite standard one of abstract linear algebra, along with useful gadgets like commutative diagrams, at about the level familiar to mathematics juniors. This choice of language does not represent a predilection for the abstract per se, but rather a recognition that certain standard algebraic tools (canonical projections, factor spaces and so forth) are suited to capture systemic concepts (like 'subsystem') in a lucid and economical way. In this way manipulations at the logical level can proceed unencumbered by irrelevant coordinatizations and without a tacit commitment to any particular technique of numerical computation. And it should be recognized, for instance, that the spectrum of a matrix is no more visible by inspection than is '~~* for a triple (A,B,D): both entities must be computed by some algorithm; each can then be exhibited by a suitable choice of basis; and which of them is considered the more 'abstract' is only a matter of cultural conditioning. With the issue of language disposed of, the problems of computation can be viewed as quite separate from those of the theory itself. In fact, it is quite straightforward to translate the operations of subspace algebra into matrix manipulations; this is done in considerable detail in Wonham (1974). It is then, in turn, straightforward to translate matrix manipulations into a special-purpose interactive computer language like APL; a listing of some forty APL programs written by the author's students for use with Wonham (1974) is available on request. Such 'naive' algorithms have been found instructive and adequate for application to state descriptions of order up to around ten. What is not automatic or straightforward is the development of numerically stable algorithms for professional application to systems of high

12

W . M . WONHAM

order; but this caveat is in no way special to the 'geometric approach': it applies to any and all computations in numerical linear algebra. As state-of-the-art numerical methods become known to the control engineering community (cf. the remarks in Bierman (1976)) one may expect that sound computational procedures specific to problems of linear synthesis (and design) will follow; as recent instances we cite Laub and Moore (1978) and Moore and Laub (1978). 5. C O N C L U S I O N

Geometric state space theory is a powerful and now well-established methodology for the treatment of qualitative problems of structural analysis and synthesis in linear multivariable control. It is likely that the approach will also succeed in further exploring recent problem areas like hierarchical and decentralized control. On the quantitative side, computation of the geometric objects (subspaces) of interest is under investigation with the best current techniques of numerical linear algebra. Of major interest will be the incorporation within the framework already established of algorithms for numerical design. REFERENCES Much of the 'geometric state-space' literature prior to 1974 is cited in the monograph (Wonham, 1974); the list to follow mainly includes more recent items. B. D. O. Anderson (1975). Output-nulling invariant and controllability subspaces. Preprints, Sixth Triennial World Congress, International Federation of Automatic Control, Boston/Cambridge, Mass., 1975; Part 1B, paper no. 43.6. B. D. O. Anderson (1976). A note on transmission zeros of a transfer matrix. IEEE Trans. Autom. Control, AC-21(4), 589-591. M. A. Arbib and E. G. Manes (1974). Foundations of system theory: decomposable systems. Automatica 10, 285.-302. G. Basile and B. Marro (1973). A new characterization of some structural propeties of linear systems: unknowninput observability, invertibility and functional controllability. Int. J. Control 17(5), 931 943. G. Bengtsson (1973). A theory for control of linear multivariable systems. Rpt. 7341, Div. Aut. Control, Lund Inst. of Technology. G. Bengtsson (1976). Feedback realization of linear multivariable systems. Rpt. 7608(C), Dept. of Aut. Control, Lund Inst. Technology. S. P. Bhattacharyya (1975a). On calculating maximal (A, B) invariant subspaces. IEEE Trans. Autom. Control AC20(2), 264 265. S. P. Bhattacharyya (1975b). Regulation of linear systems. IEEE Trans. Autom. Control AC-20(2), 265-266. S. P. Bhattacharyya (1975c). Compensator design based on the invariance principle. IEEE Trans. Autom. Control AC-20(5),708 711. G. J. Bierman (1976). Additional comments on "Multistage least-square parameter estimators)). IEEE Trans. Aurora. Control AC-21(6), 883 885. M.F. Chang and 1. B. Rhodes (1975). Disturbance localization m linear systems with simultaneous decoupling, pole assignment, or stabilization. IEEE Trans. Autom. Control AC-20(4), 518-.523. J. P. Corfmat and A. S. Morse (1976a). Control of linear systems through specified input channels. S I A M J. Control Opt. 14(1), 163 175.

J. P. Corfmat and A. S. Morse (1976b), Decentralized controt of linear multivariable systems. Automatica 12i3), 479 495. E. Emre and L. M. Sitverman (1976). Minimal dynamic inverse for linear systems with arbitary initial states. IEEE Trans. Autom. Control AC-21(5), 766 770. E. Emre, L. M. Silverman and K. Glover (1977). Generalized dynamic covers for linear systems with applications to deterministic identification and realization problems. IEEE Trans. Aurora. Control AC-22(1), 26 35. E. Fabian (1974). Decoupling, disturbance rejection and sensitivity. Ph.D. Thesis, Dept. of Electl. Engrg., Univ. of Toronto. E. Fabian and W. M. Wonham (1975a). Decoupling and data sensitivity. IEEE Trans. Autom. Control AC-20(3t, 338344. E. Fabian and W. M. Wonham (1975b). Decoupling and disturbance rejection. IEEE Trans. Autom. Control AC20(3/, 339-40l. B. A. Francis and W. M. Wonham (1975a). The internal model principle for linear multivariable regulators. Appl. Math. Opt. 2(2), 17(~194. B. A. Francis and W. M. Wonham (1975b). The role of transmission zeros in linear multivariable regulators. Int. J. Control 22(5), 657-681. B. A. Francis and W. M. Wonham (1975c). The internal model principle of linear control theory. Preprints, Sixth Triennial World Congress, International Federation of Automatic Control, Boston/Cambridge, Mass., 1975: Part 1B, paper no. 43.5. B. A. Francis and W. M. Wonham 11976). The internal model principle of control theory. Automatica 12(5), 457 465. V. Guillemin and A. Pollack (1974). Differential "Iopology. Prentice-HMl, Englewood Cliffs, N.J. F. Hamano and K. Furuta (1975). Localization of disturbances and output decomposition in decentralized linear multivariable systems. Int. J. Control 22(4), 551--562. M. Heymann (1976). Controllability subspaces and feedback simulation. S I A M J. Control Opt. 14(4), 769--789. J. W. Howze, C. Thisayakorn and R. K. Cavin (1976). Model following using partial state feedback. IEEE Trans. Autom. Control AC-21(6), 844- 846. H. Kimura (1975a). Design of observers for linear functions of the state: a geometric approach. Rpt. 75/30, Dept. Comp. & Control, Imperial College, London. H. Kimura (1975b). Pole assignment by gain output feedback. IEEE Trans. Autom. Control AC-20(4), 509 516. A. J. Laub and B. C. Moore (1978). Calculation of transmission zeros using QZ techniques. Automatica 14{6), 557 566. A. Locatelli (1974). A note on trajectory insensitivity. Ric. Autom. 511), 70 77. P. J. McLane, L. E. Peppard and K. K. Sundareswaran (1976). Decentralized leedback controls tor the brakeless operation of multilocomotive powered trains. IEEE Trans. Autom. Control AC-21(3), 358 363. B. P. Molinari (1976a). Extended controllability and observability for linear systems. IEEE Trans. Aurora. Control AC-21(I), 136-138. B. P. Molinari (1976b). A strong controllability and observability m multivariable control. IEEE Trans, Aurora. Control AC-2I(5), 761 764. B. P. Molinari (1976c). Zeros of the systems matrix. IEEE Trans. Autom. Control AC-21(5), 795-797. B. C. Moore and A. J. Laub (1978). Computation of supremal (A,B)-invariant and (A, B)-controllability subspaces. IEEE Trans. Aurora. Control AC-25(5), to appear. A. S. Morse (1973a). Structural invariants of linear multivariable systems. S l A M d. Control 1 !(3), 44(~465. A. S. Morse (1973b). Structure and design of linear model following systems. IEEE Trans. Autom. Control AC-18(4), 346 354. A. S. Morse (1976). Minimal solutions of transfer matrix equations. IEEE Trans. Autom. Control AC-21(I), 131 133. J. B. Pearson and P. W. Staats, Jr (1974). Robust controllers for linear regulators. IEEE Trans. Aurora. Control AC19(3j, 231 234.

Geometric state-space theory in linear multivariable control: A status report H. H. Rosenbrock (1970). State-Space and Multivariable Theory. Wiley, New York. H. H. Rosenbrock (1974). Computer-Aided Control System Design. Academic, New York. O. A. Sebakhy (1974). Synthesis of linear multivariable regulators. Ph.D. Thesis, Dept. of Electl. Engrg., Univ. of Toronto. O. A. Sebakhy and W. M. Wonham (1976). A design procedure for multivariable regulators. Automatica 12(5), 467~-78. J. Snyders and W. M. Wonham t1975). Regulation of linear stochastic systems. SIAM J. Control Opt. 13(4), 853-864. T. Takamatsu, I. Hashimoto and Y. Nakai (1978). A geometric approach to multivariable control system design of a distillation column. Preprints, Seventh Triennial World Congress, International Federation of Automatic Control (IFAC), Pergamon, Oxford (U.K.). pp. 309-317. M. E. Warren and A. E. Eckberg (1975). On the dimensions of controllability subspaces: A characterization via polynomial matrices and Kronecker invariants. SIAM J. Control Opt. 13(2), 434-445. M. E. Warren and S. K. Mitter (1975). Generic solvability of Morgan's problem. IEEE Trans. Autom. Control AC-20(2), 268 269.

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"W. A. Wolovich (1974). Lineal" Multivariable Systems. Springer. New York. P. K. Wong (1975). On the interaction structure of linear multi-input feedback control systems. Rpt. ESL-R-625, Electronic Systems Lab., Dept. of Electl. Engrg. and Comp. Sci., Mass. Inst. Tech. W. M. Wonham (1974). Linear Multirariahle Control: A Geometric Apprmwh. Springer, New York. W. M. Wonham (1977). Recent progress in multivariable control. In: NASA CP-003, Washington, D.C., pp. 157 164. W. M. Wonham (in press). Linear Multi~ariable Control: A Geometric Approach. 2nd edn. Springer-Verlag, New York. To appear Spring, 1979. J. S-C. Yuan (1976). Adaptive decoupling control of linear multivariable systems. Ph.D. Thesis, Dept. of Electl. Engrg., Univ. of Toronto. J. S-C. Yuan and W. M. Wonham (1975). An approach to on-line adaptive decoupling. Proc. 1975 IEEE Con(. on Decision and Cmttrol. pp. 853 857. K. M. Zahir and C. Slivinsky (1974). State variable feedback in computer-controlled multivariable systems. IEEE Trans. Aurora. Control AC-19(4), 404~1-07.