Realization Theory of Discrete-Time Nonlinear Systems - CiteSeerX

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VI D. E. KnutJr, “Matroid partitioning,” Report, Stan-Cf&73-342, 1201 S. Seshu and M. B. Reed, Linear Grqhs and Electrical Network Stanford Univ.. 1973. Reading, JUA: Addison-Weslev. 1961. 191 E. S. Kuh and R. A. RoJuer, “The state variable approach to PJI .I. Tow~“0rde.r of complexi@ of Jinear active networlrs,” Pra: network analysis,” Proc. IEEE, vol. 53, pp. 672-686, July 1%5. IEE, vol. 115. DD. 1259-1262. Sent. 1968. 1101E. L. Lawler, “Matroid intersection algorithms,” Math Progr., vol. [221 R. I. W ilson,’ ‘Ati introduc~on ‘to matroid theory,” Amer. Math. 9, pp. 31-56, 1975. Monthly, vol. 80 no. 5, pp. 500-525, May 1973. FJI S. K. Mark and M. N. S. Swamy, ‘“Die generalized tree for state

variables in linear active networks,” Circ. T/L and Appt., vol. 4, pp. 87-92, 1976. W I W. Mayeda, Graph Z+eoty. New York: W iley, 1972. LJ31 G. J. Minty, ‘On the axiomatic foundations of the theories of directed linear graphs, electrical networks and network-programming,” J. Math. Mech., vol. 15, no. 3, 1966. [J41 T. Gzawa, ‘“To~JoguxJ conditions for the solvability of active En; networks, Circuat T?zeory and Appl., vol. 4, pp. 125-136,

1151 B. P&men, “JZJe&ricaJ networks and matroids,” Ph.D. dissertation, TecJmicaJ University of Denmark, Lyngby, 1976. “Circuits in the union of matroids. An algorithmic apW I -, proa&,” 1978 (to be submitted). 1171 E. J. Purslow, “Solvability and analysis of linear active networks by use of the state equations,” IEEE Trans. Circuit TJ~oty, vol. CT-17, Nov. 1970. W I A. Recski, ‘On partitional matroids with applifations,” CoZl.Math Sot. J. Bolyai, vol. 10, Infinite and finite sets, Keszthely (Hungary), 1973, North-Holland-American Else&r, Amsterdam, New York 1974, pp. 1169-1179. “Matroids and independent state variables,” presented at 1191 -, f’;$

Second European Con$ Circuit .Theoty and Design, Geneva,

+

Bjern Petersen was born in Denmark in 1948. He received the M.Sc. degree in electrical en-

gineering from the Technical University of Denmark, Lyngby, Denmar Jc, in 1976, and is currently working towards the Ph.D. degree at the Institute of Circuit Theory and Telecommunication, and the Department of Mathematics, Technical University of Denmark. Until February 1977, he was an Associate Professor at the Department of Mathematics, University of Denmark Lyngby. He has carried out research in the fields of circuit theory, graph theory, matroid theory, and computational complexity. His fields of interest are linear active network tlmory and matroid algorithms.

Realization Theory of Discrete-Time Nonlinear Systems: Part I - The Bounded Case EDUARDO D. SONTAG

Abstmct-A statespace realization theory is presented for a wide class of diwete time input/output behaviors. Although ill many ways restricted, thJst!JassdaeeJncJudeasparucuJarcssestbo¶etreatedhtthelJterature (bear, multilinear, internally biiear, homogeneous), as Well as certain nonanalytic nonlhearities. The thry is conceptually simple, and matrixtJleorelk! algDrlthms are stmlgJltforward Finite-reaBzablllty of these behaviors by state&k systems is shown to be equivalent both to the existence of high-order input/output equations and to realizability by more general types of systems.

INTRODUCTION

HIS WORK deals with some aspects of realization theory of deterministic nonlinear discrete-time systems. The realization theory of linear systems is by now a

T

Manuscript received September 11, 1977; revised July 6, 1978. This research was supported in part by U. S. Army Research Grant DAH CO4-74-G-0153 througli the Center for Mathematical System Theory, Gainesville, FL 32611. The author is with the Department of Mathematics, Rutgers University, New Brunswick, NJ 08903.

successful part of system theory, which has resulted in a deep understanding of behavior and has permitted the application of state-spacemethods of analysis and synthesis. It may be reasonable to expect, then, that a corresponding theory will eventually derive analogous benefits for nonlinear systems. For the most part, this paper presents a “lineaxized” realization theory via systems which are linear in state variables -but arbitrarily nonlinear in inputs, state-uffine systems. W h ile such systems are highly restrictive vis a vis general nonlinear m o d e ls, they do include those for which a detailed realization theory has been developed, in particular, linear, internally bilinear, and m u ltilinear systems. The importance of S-A representations in the analysis of certain nuclear reactors, heat-transfer processes, and population m o d e ls, among others, has been m a d e explicit by various authors (see, for instance, [34]); other applications being currently explored are in the areas of image processing and in stochastic filtering. Moreover, in some

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cases the canonical realization of a given input/output .behavior a d m its a S-A structure; this is the case with bounded polynomial responses, those whose output values at any given instant are arbitrary sums and products of previous input values, subject only to the restriction that there is a bound-hence the name-to the exponents to which each single input can be raised in calculating outputs. Bounded responses were originally defined in a polyn o m ial context, but the present work treats directly a more general case, which has the advantage of including many types of nonanalytic nonlinearitites (piecewise polyn o m ial, in particular). The first part of this paper deals with an abstract realization theory, while the second presents a concrete matrix realization algorithm which both generalizes and unifies those known in the literature for the various classesof systems.These two parts result in particular in a self-contained realization methodology for S-A systems, and serve also as an introduction to a more general (and strictly nonlinear) realization theory. Part three provides further finiteness results, including a generalization of the linear system fact that finite realizability is equivalent to the existence of a (high-order) input/output difference equation, and studying the relationship between stateaffine realizability and more general realizability of a bounded response.The paper closes with some remarks in part four.

W h e n U is a subset of a vector space k”, the notation uu) for the vector u in U will m e a n the ith coordinate of u-the notation z.+being reserved for the ith element of a sequenceof vectors. If A, B are sets, [A, B] will denote the set of all maps f: A +B. W h e n B is a vector space, [A, B] will denote the corresponding vector space, with the pointwise operations: (f+ rg)(a): =f(a)+ rg(a). A. Response Maps

An. “input/output m a p ” sends input sequences into output sequences.W h e n such a m a p is causal, it can be equivalently specified by its associated “response map,” which describes how past inputs affect present outputs; this latter object is defined directly. Definition 1.1: A response is a m a p f : U + + Y. A strict& causal response has, for each t, ft(u,; * * ,u,) independent of z+; for a memovless response, ft(u,,- * * ,u,) depends on& on u,,; an equilibrium response is one for which there is some ii in U with f (iiw) = f (w) for all w in U+. A response is polynomial (resp. anaZytic) iff U is a subset of k “‘, Y = kP, and f, is a polynomial function in mt variables (resp. k= R or C, U is an analytic m a n ifold, Y = kp, and each f, is analytic), for all t > 1. The interpretation of the above is that output values yt at tim e t are functions of inputs ui, * * * ,u, at times 1, * * * , I, “strictly” causal m e a n ing that present inputs cannot affect present outputs. “Equilibrium” is equivalent to “shift invariance” of the corresponding input/outp.ut map, where the “equilibrium input value” ii plays the role of a “zero input.” I. ABSTRACTREALIZATION THEORY W h e n U is Euclidean space R” (or an open set thereof), This section develops a realization theory which will Y is just R, and each f, is a real analytic map, it is give the theoretical basis for the algorithms presented customary to representf by a “Volterra series” later. The following notational conventions will hold throughout this work. (1.2) For any set S and integer t > 0, S’ will be the set of all isO sequencesw = (wi, * - - , w,) of length t, wi in S; note So= set containing only the empty sequence e. The set S* where Lti is a homogeneous degree i polynomial in the (resp. S+) is the union of all S’, t 2 0 (resp. t > 1); ]w] coordinates of (ui,. * - ,u,). O ther representations in this denotes the length of w. For notational convenience, a case are the “m u ltidimensional z-transform” [I] and the w= (w,, * * - J,,,) will be also denoted as wiw2- - - wt. (This “regular transfer function” [8]. Each of these alternative should not be confused with the product of the wi, when representations has its computational advantages dependsuch a product would be also defined.) The expression a’ ing on the problem to be solved. Since they all give the will, correspondingly, m e a n a - * - a (t times). The con- same information about f, and since the passagebetween catenation of two sequencesv = vi - * - v, and w = w, - * * w, them is well understood, this paper uses the more abstract isvw:=v,---v,w,* - wr of length s + t. For any function f definition in (l.l), which has the advantage of allowing defined on S +, the restriction off to sequencesin S’ isf,, for nonanalytic nonhnearities in f. while the maps vwf(wv) (w in U’) are denoted by f,, B. Systems and the maps vwf (vw) are denoted by f”. Definition 1.3: A system 2 =(X, P, Q,.?) is defined by a An arbitrary field k will be fixed throughout the discussion; “vector space” will m e a n k-vector space, “linear” vector space X, maps P : X X U+X and Q : X X U+ Y will m e a n k-linear, etc. (Some.results will be stated only and an X in X (called, respectively, the state space, next for k = reals or complexes, but most are valid without any state, transition or control map, output or measurement restrictions on k.) Recall that uffine manifoZd= transate of map, and the initial state). The dimension of Z is the a subspace, and affine m a p = linear + translation. Unless (possibly infinite) dimension of X; zero-dimensional sysotherwise stated, U will denote an arbitrary set (of input tems are called memo@ess. A state-output system has Q values) and Y an arbitrary vector space (of output values). independent of inputs U. W h e n there is an U in U with

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Existance of a morphism Z,+Z, forces equality of responses; the following is a partial converse to this fact. Theorem 1.8: Let Z, and Z, be S-A systems having the same response map. Assume that Z, is span reachable and Z, is observable. Then there is a unique morphism T: 2,+&. Furthermore, T is onto if Z, is also span reachable, and one-to-one if Z, is observable. Proof By span reachability, any state x in X, can be written as Zrigi(wi), Zr,= 1, for some finite set of input 4 + 1= P(x,, 4 sequences wj and scalars ri. Write T(x): =ZrigZ(wi). To see that this gives a well-defined map T: X,+X,, consider any other expression x = Zsigi(wi), Zs, = 1 (for the same set of wi’s, adding zeroes if necessary to the ri,si). Since with the initial condition xi= X. State-output systems, in Ql(*,w) is affine, ~riQ~(g,(Wi>,W)=~siQ,(g,(wi,w)), i.e., whichy, is a function of x, alone, are the ones found most Zr,f(wiw)=Zs,f(wiw), for any w in U+. This implies that frequently in the control literature, and are called “Moore Q,(Zrig2(wi),w)= Q,(Zs,g,(w,), w) for all w in U+ and machines” in automata theory. In the state-affine case one thus, by observability of Z,, that Zrig,(wi)=Zsig,(wi). has equations Thus T is well defined, and it is clearly affine. Uniqueness is clear, since T( g,(w,)) = g2(wi) is forced by the definition of morphism. The last two statements follow by analogous arguments. Remark 1.9: If a given S-A system Z is not span reachwith P’(u), G(u) linear maps and G(u), I(u) vectors for able, one may of course restrict P and Q to the affine each u in U. The notations F, G, H, I will be used freely span x of the reachable states (considered as a vector instead of P, Q. The symbolic notation x’ = P(x, u), y = space in itself, once that an arbitrary x in x is choosen as Q(x,u) (where the prime indicates a time-shift operator) origin), resulting in a span-reachable realization of the same response. If C is not observable, a dual construction will be used freely. The extended transition map P* : X X U*+X is defined gives an observable realization: it is sufficient to form the recursively by P*(x, e): = x, P*(x, wu): = P(P*(x, w), u); P’ quotient of X by the subspace of all states indistinguishis the restriction to sequences of length t. With a slight able from zero (i.e., all x with Q(x, *)= Q(0, e)); P and Q abuse of notation, P* will be denoted also by-P. For any naturally induce maps in the quotient. Thus existence of a w=vu in U+, where v is in U* and u in U, one writes single S-A realization of anf already implies existence of a Q +(x, w) (or just Q(x,w)): = Q(P(x,v),u), and Qf for the span-canonical one. Such a realization is constructed berestriction to X x U’. The reachability map of Z is g : U*+ low, for anyf. X where g(w): = P(X, w); the reachable states are those in Definition 1.10: The image realization of the response f its image. Z is said to realize its responsefz : U++ Y: w+ has X: =[ U+, Y], X: =f, P(b,u): = b,,, and Q(b,u): = b(u). The above realization is an S-A, in fact state-linear, Q 1 there are (finitely many) vectors a, in Y with f(w) = W(w)a,

(1.16)

for all w in U’. Conventions 1.17: W ithout loss of generality it will be always assumed that 6, is the unit constant function: as(u)= 1 for all u in U. This will greatly simplify notations. Furthermore, the family of functions J will be assumed linearly independent (i.e., if there are scalars ri with Xr$$(u) = 0 identically on U then all ri = 0); if this were not the case, a maximal linearly independent set can be extracted from a given J. For a type J= {S,, . . . ,a,}, [J] denotes the set of integers (0,. . - J}. An obvious and trivial example of bounded response is just any m a p 6, : U+ Y, inducing a memoryless response f(uz* * * u,)=6()(uJ- * * a,( u, _ i) 6 i (u,). More interesting exa m p les follow. Example 1.18: As explained in the introduction, the terminology “bounded” has its origins in the m a in (and motivating) example: polynomial bounded responses.This case corresponds to U=a subset of k”, m > 1, and the Si

being a set of possible m o n o m ials in the m input variables (for instance, J= all m o n o m ials of degree .

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Proposition 1.30: W ith X=0, an S-A system is span-re-

achable iff

Proof

span { G ,,cu in

[J]‘}

=

span { Va,cu in

[J]‘}

=X.

Consider the affine m a n ifolds X,:=affinespanof

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unclear whether these dualities have any system-theoretic m e a n ing in the S-A case. Numerical Examples 1.36: Consider a system Z t with U = R, X = R3, X = 0, and transitions x;=x2 if u