topology are called k-points of algebraic sets. Algebraic sets in k" can themselves be considered as topological spaces under the topology induced fro...

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ON DISCRETE-TIME

POLYNOMIAL

SYSTEMS

E. D. SONTAG* Center for Mathematical System Theory, Department of Mathematics, University of Florida, Gainesville, FL 32611, U.S.A. and Y. ROUCHALEAU* Center for Mathematical System Theory, Department of Mathematics, University of Florida, Gainesville, FL 32611, U.S.A. and Centre d'Automatique, E. N. S. Mines F-77305 Fontainebleau, France (Received I March 1976) K e y words: Nonlinear dynamical systems, system analysis, reachability, observability, canonical systems, applied algebra.

INTRODUCTION

THE FIELDof Algebraic System Theory is slowly expanding beyond the traditional confines of linear systems to include various restricted types of nonlinear systems, for instance, bilinear systems. We shall consider here a more general type of systems which have algebraic constraints on their state set and for which the state transitions are given by (arbitrary) polynomial functions of the inputs and state variables. The only technical restriction we shall make is to consider systems which begin their evolution from an equilibrium state. This paper is a summary (of a preliminary nature) of the results obtained so far on the above questions. A thorough study of the realization theory of polynomial systems is carried out in Sontag [7]. One of our main concerns in this work has been to obtain finiteness conditions, for it is our belief that finite computational procedures are an essential goal of applied algebra. In the first four parts of this paper we give the necessary mathematical preliminaries and define the systems that we study. Part 5 is concerned with teachability in bounded time, while in Part 6 we look at the problem of deciding whether two systems have the same external behavior by applying finitely many inputs. In Part 7 we show that finitely many inputs (which can be chosen quite arbitrarily) are sufficient to separate those states of a system which are distinguishable. Part 8 is an investigation concerning the conditions under which systems with the same external behavior are isomorphic.

* This research was supported in part by US Army Research Grant DAHC04-74-G-0153 and by US Air Force Grant AFOSR 72-2268 through the Center for Mathematical System Theory, University of Florida, Gainesville, FL 32611, U.S.A. This is an expanded version of the paper with the same title presented at the CNR-CISM Symposium on Algebraic System Theory, June 1975, Udine, ITALY. The editor of the proceedings, G. Marchesini, was unable to include this paper in the proceedings. 55

E . D . So~rrAo AND Y. ROUCHALEAU

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1. A L G E B R A I C

PRELIMINARIES

We review now some notions from commutative algebra and algebraic geometry. The results can be found in any of the standard references; see, for instance, Dieudonn6 [1], Mumford [4], Macdonald [3]. Although the literature usually treats only the case in which solutions are taken in algebraically dosed fields, the generalization to the present setup (namely, arbitrary infinite fields) does not present any major difficulty. We shall therefore omit the proofs here. Throughout this paper, k denotes an arbitrary but fixed infinite field. For any nonnegative integer n, k" is the set of all n-tuples of elements in k. In particular, k ° is a one-element set (0}. Similarly, k[T 1. . . . . TJ , or kiT] when there is no danger of confusion, is the ring of polynomials in the indeterminates T1. . . . . T with coefficients in k. If n = 0, kiT] is defined to be k. When f is in k i T 1. . . . . T ] and x e k" then f(x) denotes the evaluation (i.e. specialization) o f f at x. For each subset Y _ k[T 1. . . . . T,] we define V(Y): = {x e ld'lf(x) = 0 for all f e Y}. For each subset S ~_ k" we define I(S): = { f e k[T 1. . . . , T,]I f ( x ) = 0 for all x e S}. Of course, V(Y) = V((Y>), where (Y> denotes the ideal generated by the set Y. Since every ideal of kiT] is finitely generated (Hilbert's Theorem), we note that Lemma 1.1. For every Y _~ kiT] there exists a finite subset Yo - Y such that V(Y) = V(Yo). Proposition 1.2. Let V and I be defined as above. Then V, I give a Galois connection between the lattices of subsets of k i T 1. . . . . T,] and of ~. Furthermore, the induced closure operator S~--~ S: = V(I(S)) defines a topology on/d', called the Zariski topology. The dosed sets for this topology are called k-points of algebraic sets. Algebraic sets in k" can themselves be considered as topological spaces under the topology induced from k". Since the field k is infinite, a polynomial function f: k" ~ k: x ~ f(x) can tm identified in an unambiguous way with a polynomial in k [ T 1. . . . . T,]. We shall always assume such an identification and shall speak indistinctly of polynomials and polynomial functions. A polynomial function f : V ~ k, where V _ k", is by definition the restriction to V of a polynomial function f:/d' --, k. Definition 1.3. Let V ~ k" be an algebraic set. Then the k-algebra A(V) of all polynomial functions f : V -~ k with pointwis¢ operations is the coordinate ring of E The k-algebras of type A(V) are affine k-algebras. It is not difficult to verify that A(V) = k[T]/I(V). When k is algebraically dosed, affine kalgebras are characterized as finitely generated k-algebras with no nilpotent elements. The dimension of a k-algebra A is defined as the maximum possible length of chains of prime ideals. When A is an integral domain finitely generated over k, the dimension of A coincides with the transcendence degree of A over k. We say an algebraic set V _ k" is irreducible if there do not exist algebraic sets V1, V2 such that V = V1 w V2 and V # V1, V # V2. Such algebraic sets are also called (affine) varieties. The following results are well known. Proposition 1.4. An algebraic set V is irreducible if and only if I(V) is a prime ideal. I f . . . c V _ c Vi c Vi+ 1 ~ ... is a strictly monotonic chain of algebraic sets, then ... ~ I(Vi_l) ~ I(V~) I(V~+ 1) ~ -.. is also strictly monotonic. In particular, every ascending or descending chain of irreducible algebraic subsets of V has a length at most equal to the dimension of A(V). The dimension of A(k") is n.

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We remark that the last statement of Proposition (1.4) is the only place where the assumption k = field is essential. Generalizations to arbitrary integral domains are not very difficult. The (algebraic)dimension dim V of an algebraic set V is by definition the dimension of A(V). Except for the algebraically closed case, this number is only an upper bound on the geometric dimension given by the maximum length of chains of varieties included in V. Clearly, if V _~ k~ then dim V ~< n. Definition 1.5. Let V, W be algebraic sets and W _ km. A map f: V ~ W is called a morphism (or polynomial map) if there exist f l . . . . . fm in A(V) such that f(x) = (fl(x) . . . . ,f,~(x)) for all x e V. The associated comorphism A(f): A(BO ~ A(V) is the k-algebra homomorphism defined by A(f):g ---, g o f for all g ~ A(W). Theorem 1.6. Algebraic sets and morphisms define a category, which is equivalent to the dual of the category of affine k-algebras and k-algebra homomorphisms. Remark 1.7. The image of an algebraic set under a morphism is not necessarily an algebraic set. However, it can be proved (see Mumford [4], Chapter I, §8, Corollary 2]) that the image of an algebraic set is a constructible set if k is an algebraically closed field. (A set is constructible in V if it can be expressed as a finite union of sets of the form U c~ F, where U is open and F is closed in the Zariski topology of V.) Theorem 1.8. Let f : V ~ W be a morphism of algebraic sets. Then: (a) f is continuous for the Zariski topologies on V and W. (b) f is dominating, i.e. ~ ~ = W, if and only if A(f) is injective. (c) f is injective if A(f) is surjective. Remark 1.9. Injectivity o f f does not imply surjectivity of A(f), except in special but important cases, for example, when f is the restriction of an injective affine map f: k ~ ~ / ¢ ' . (Surjectivity of A(f) is in fact equivalent in general o f f being a closed embedding.) Proposition 1.10. Products exist in the category of algebraic sets. Moreover, A(V x W) is isomorphic as a k-algebra to A(V) ® A(W), the tensor product of the k-algebras A(V) and A(W). Intuitively, if V _~ kn and W _ k~ then the categorical product of V and W is the set product V x W seen as an algebraic subset of k n+m. F r o m Hilbert's theorem on ascending chains, it is known that a chain of ideals in kiT] is necessarily finite. A recent theorem gives explicit bounds for the lengths of such chains: Theorem 1.11. (Seidenberg [5]). Let K(j) be a nonnegative integer for j = 0, 1. . . . and consider strictly ascending chains of ideals I o c I 1 c . . . c I in k i T 1. . . . . Tj , where Ij has a basis of elements of degree ~

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E.D. So~rrAoANDY. ROUCI'V~E~U

exists a unique f ~ k i t 1. . . . . 7"] such that d e g i f ~< m and f ( l 1. . . . . In) = A(! 1. . . . . l.) for all (l 1. . . . ', 1~) ~ ~ . 2. SYSTEMS Let k be an infinite field, Y a fixed algebraic set over k and U a fixed variety (irreducible algebraic set). Then Definition 2.1. A (polynomial, discrete time, constant) dynamical system over k, having input space U and output space Y, is E = (X, x o, p, h), where (i) X (the state space) is an algebraic set over k. (ii) x 0 (the initial state) is a fixed element of X. (iii) p (the transition map) and h (the output map) are m o r p h i s m s X x U ~ X and X ~ Y. Remarks 2.2. (i) x t = p(x t_ 1, ut- 1) is interpreted as the state at time t corresponding to the state xt- 1 and the input u t_ ~ at time t - 1; Yt = h(x) is interpreted as the output at time t. (ii) Since k= is irreducible, a system with unconstrained inputs fits in our framework. (iii) Since any algebraic set is e m b e d d e d in some affine space, there is no loss of generality in assuming that the o u t p u t set is k °. (iv) x o can be identified with a m a p from k ° into X. Definition 2.3. The dimension of a system ~ is that of the algebraic set X z. I f X z = / d , the system E is said to be unconstrained. The m a p p can be extended recursively to a m a p X x U* ~ X (where U* is the free m o n o i d on U), which we shall also denote by the letter p. Let g: U* --, X be defmed by if(to) = p(x o, to). Then: Definition 2.4. T h e input-output map of a system E is the m a p f : U* -~ Y given by f = h o y. We shall denote by gt and ft the restrictions of g and f to U t (the t-fold Cartesian p o w e r of U). It follows from the definition of p and h that at and ft are m o r p h i s m s from the variety U t into X and Y. Definition 2.5. Let X = (X, x o, p, h) and X = (J~, ~o, P,/i) be two p o l y n o m i a l systems. A morphism of systems E -* E is a m o r p h i s m T of algebraic sets such that the following d i a g r a m commutes:

X x U~X Y and 2

x

T(xo) = :~o.

u~--,2

Observe that the existence of a m o r p h i s m Z --* E implies that fz = ft3. COSYSTEMS

Definition 3.1. A polynomial cosystem E over k is a quadruple (A, ~, fl, 7) where A is a k-algebra and ~: A(Y) -~ A, fl: A ~ A ® A(U), 7: A --, k are k-algebra h o m o m o r p h i s m s . If A is an affine algebra, then E is an affine cosystem. We can associate in a straightforward m a n n e r to any cosystem an i n p u t - o u t p u t function m a p p i n g A(Y) into H (®hA(U)). Definition 3.2. A morphism of cosystems -= -~ ~ is a k-algebra h o m o m o r p h i s m r / f r o m A to

On discrete-time polynomial systems

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such that the following diagrams commute:

A

~ ,A®A(U)

A

rl ® 1A(V)

A(Y)

/

\ /i

~ ~.~i ® A(U))

Polynomial, discrete-time, constant systems and cosystems, with their respective morphisms, form two categories which we shall call Systems and Cosystems respectively. Affine cosystems will be a full subcategory of Cosystems. Theorem 3.3. (Duality). Systems and (Affine cosystems) °v are equivalent categories. Proof. This is a direct consequence of (1.6). 4. E Q U I L I B R I U M - S T A T E

ASSUMPTION

Let X t = gt(Ut). It is the reachable set at time t. Definition 4.1. The state x o e X z is an equilibrium state for the system Z if there exists u o ~ U such that P(Xo, Uo) = x o. We shall restrict ourselves to considering systems having an initial state which is an equilibrium state. By a change of coordinates in U and X, we can take x o = 0, u o = 0 without loss of generality; this assumption then means that p has no constant term. In other words, we assume that the system was operating in a certain steady-state mode before we disturbed it with our new sequence of inputs. Such an assumption is satisfied for most classes of systems studied so far (linear, bilinear). F r o m our point of view, the main value of the assumption is due to the following easy result: Lemma 4.2. The sequence of reachable sets {Xt} is monotonically increasing if and only if the initial state of the system is an equilibrium state. 5. R E A C H A B I L I T Y

AND QUASI-REACHABILITY

Definition 5.1. A system E is reachable iff XR: = u X t = X. E is reachable in bounded time iff t>~O X r = X for some r. Definition 5 . . The system ~ is quasi-reachable iff its state set is the closure (in the Zariski topology) of the reachable set, i.e. XQ: = u X~ = X. It is quasi-reachable in bounded time iff t~>0 , ~ = X for some r. A system ZQ with state-space XQ can be naturally defined using the restrictions of p, h to X o. Lemma 5.3. The sequence {Xt} is strict, i.e. Xt = "~,+1 implies Xt+l = Xt+2" Proof. By definition, we have X t + 1 = P(Xt × U), for all t > 0. The morphism p is continuous, hence p(X t × U) ~ p(X'~--x U). Thus X,+~ = p(X, x U) ~_ p(X, x U) = p(X, x U) = p(X,+ ~ x U). But of course P(Xt+ 1 x U) _~ p(X~+l x U) = X~+ 2. By (4.2), X,+ 2 _ Xt+ r So Xt+ 1 _ X,+ 2 ~_ Xt+ 1, and therefore -~t+l = Xt+2Proposition 5.4. Let dim E = n. Then the set of quasi-reachable states is X . Proof. Since U is irreducible by assumption, so is U t. Since X t is the closure of the image of U t under the morphism g, (of algebraic sets), Xt is also irreducible. We therefore have an increasing sequence T o _~ X , _ X 2 _~ ... of closed irreducible subsets of X. Since dimension of X is n,

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the preceding sequence cannot have more than n + 1 distinct sets, by (1.4). It follows therefore from (5.3) that this sequence cannot increase after step n. Remark 5.5. We have just proved that a quasi-reachable system is quasi-reachable in bounded time. The same property does not hold, however, for reachability, as can be seen from the following example. Let U = X = Y=R,p(x,u)=x+ u2 - 2 u , h ( x ) = x , x o=0.ThenX t={x~Rlx>1 -t}. The system is clearly reachable, but not in bounded time. We can nevertheless obtain the following characterization of reachable states: Proposition 5.6. Let k be an uncountable, algebraically closed field. Then the reachable states form a constructible set if and only if they can be reached in bounded time. Proof. Since t f is a variety and gt a morphism of algebraic sets, it follows immediately from (1.7) that X t is a constructible set. If X R is reached in bounded time, it is equal to an X,, hence constructible. The proof of the converse statement, based on Noether's Normalization Theorem, is too long to be given here. 6. FINITE CHARACTERIZATION OF THE INPUT-OUTPUT MAP In this part we study questions related to the specification of polynomials systems. Specifically, we ask: how many experiments are needed to characterize the input-output map of a system? Recall that if Y~ is known to be linear, U = k, and if dim ~z ~< n, then to determine the map fx it is enough to know the values of f,~(co) for the 2n inputs (0. . . . . 0, 1, 0 , . . . , 0) of length 2n. To determine uniquely a polynomial of degree r, in general its values at r + 1 points must be known. Hence, to obtain similar results for nonlinear systems, some a priori estimates are needed for the degree of the polynomials involved in the definition of Z. Using such estimates, one of the contributions of this paper is to effectively compute a bound on the number of input-output experiments sufficient to decide whether or not two systems have identical external behavior. Theorem 6.1. Let Z, Z be systems of dimension ~< n. Write f : =f~, f : = ft" Then the following statements are equivalent: (i) f = f, (ii) f2n = f2n" Proof. By definition Y c k v for some p. Hence the map f:e~--~f(co) - f(co) is well defined. The map f can be realized as f~, where Z is defined as X': = X x ~ , Xo: = (Xo, ~o), p((x, ~), u): = (p(x, u), p(x, u)), and h(x, ~): = h ( x ) - h($). The theorem is then an immediate consequence of the following Lemma 6.2. Let f = f~, where X is a system of dimension r. Suppose that fr - Y ~ Y, a constant function. Then ft = Y for all t > 0. Proof. Since ft = h°gt for all t, it follows that ft = Y if and only if hlX t =- y. By hypothesis, hlX ~ - y. Since h is continuous, it follows that hlX r = y. By (3.4), X = XQ _ X,, for all t. Therefore h IX , - y for all t. Observation 6.3. In general, knowledge of the map ft for some t < 2n is not enough to identify a system of dimension n. Indeed, consider affine systems with U: = Y: = k, X: = k~ and x o = 0:.

~x(t + 1) = Fx(t) + gu(t) E: [y(t) = hx(t) + c

O n discrete-time polynomial systems

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Then ft is completely determined by c, a I . . . . . a t where a~ = hF s- 10. It is well k n o w n from linear system theory that the data c,a~ . . . . . a2, (but no p r o p e r subset of it) is sufficient to determine f. Definition 6.4. deoree (E) <<.s iff there exist representations of p and h by polynomials having (ordinary) degree ~< s separately in each variable ui, i = 1. . . . . m (where U _ k ~) and having degree ~< s jointly in the variables xj. If degree (Z) ~< s then ft: Ut ~ Y admits a representation by polynomials of degree ~

OBSERVABILITY

Definition 7.1. T w o states x 1, x 2 are distinguishable if and only if there exists an input sequence o9 ~ U* such that h~'(Xl) # h°'(x2), where h°'(x): = hop(x, ~0). A system is said to be abstractly observable when any pair of states are distinguishable. Proposition 7.2. F o r any system E, there exists a finite set of input sequences 091, o92. . . . . o~q such that h'~'(xl) = h°"(x2) for all i~ {1 . . . . . q}, if and only if x 1 and x 2 are not distinguishable. Proof. An input sequence o9 enables us to define a m o r p h i s m l~: X x X - , k P : ( x l , x ~ ) ~ h~'(Xl) - h°'(xz). If we c o m p o s e this m a p with the p coordinate projections k p ~ k: y ~Yi, i = 1. . . . . p, we get p m o r p h i s m s l°"i• X x X ~ k. T h e states x I and x 2 are distinguishable if and only if one of these m o r p h i s m s is nonzero at (xl, x2). Thus the states which are not distinguishable form exactly the algebraic subset of X x X defined as V({l~.lo9 ~ U*, 1 <<.i <~ p}). It follows from the Hilbert Basis T h e o r e m (1.1) that this can be defined as the set of c o m m o n zeros of finitely m a n y of the l~"s. It is therefore sufficient to apply the corresponding finitely m a n y input sequences to determine whether any two states are distinguishable or not. Corollary 7.3. Distinguishable states can be separated in b o u n d e d time. We have thus p r o v e d that to test whether any two states are distinguishable or not it is sufficient to apply input sequences of b o u n d e d length to the system. In fact, we have even shown that it is enough to apply finitely m a n y input sequences. But our result is just an existence statement and does not tell us h o w to choose these input sequences. T h e following t h e o r e m answers this question. Theorem 7.4. Given a p o l y n o m i a l system E there exist two integers t and r (which can be c o m p u t e d effectively from Z) such that the following p r o p e r t y is true: If L is a subset of U of cardinality r, then h°'(xl) = h~'(x2) for all o9 e/~ implies h°'(xl) = h°J(x2) for all o9 in U*. Proof. Let V: = {(xl, x2)e X x Xth°/(xa) = h~'(x2), for all og~ U j, j <<.n, 1 <~ i <<.p}. The degrees of the polynomials h~(x) in the variables x are b o u n d e d for all 09 e U" because of their recursive definition. It follows that Vt = Vt+ x . . . . for some c o m p u t a b l e index t ( T h e o r e m (1.1 1)). Thus x I cannot be distinguished from x 2 if and only if (xa, x 2 ) e Vt. But (xl, x 2 ) e Vt m e a n s that

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h~(xl) = h~(x2) as morphisms from Uj into k. Since the degrees of the polynomials representing these morphisms are bounded, the result follows from (1.12). In other words, the computational aspects of our problem are reduced to those of Seidenberg's Theorem (1.11). 8. SYSTEM ISOMORPHISM This section is concerned with the "uniqueness theorem for canonical realizations". This theorem, which is known to hold for several classes of systems (e.g. automata, linear systems) states that two "canonical" realizations of a given input/output map are "isomorphiC'. The appropriate notion of system isomorphism in the present context is that given by (2.5); a counterexample will show, however, that a naive definition of "canonical" as "reachable + (abstractly) observable" is not sufficient to give us the desired result. Counterexample 8.1. We wish to construct two systems E, E with f~ = f t both reachable and abstractly observable but such that E is not isomorphic to ~. The construction is based on the fact that (in the category of algebraic sets) a morphism which is bijective (one-to-one and onto) is not necessarily an isomorphism. Let q: V ~ W be a morphism between varieties V, W such that q is a bijective map but V is not isomorphic to W. For instance, let V: = k,

W: = {(x,y)~k2lx 3 - y2 __ 0}, q : z ~ (z2, z3). (It is a standard algebraic-geometric fact that V :p W.) Given any q, V, W as above, let U: = V, Y: = W and define ~ : = (V, Xo, prz, q),

~: = (W, q(Xo), q°Pr2, lw); here x 0 e V is arbitrary while pr 2 : V x V ~ V and P~2: W x V ~ V denote the projection on the second factor. Since q induces a system morphism E ~ E, it follows that f~ = ft" Using that q is bijective, it is easy to verif,y that both systems are reachable and abstractly observable. Finally, E is not isomorphic to Z, because their state-sets V, W are not isomorphic. The difficulty in proving the uniqueness theorem is most easily understood in terms of cosystems. In view of Theorem (1.8), the notion of reachability (rather quasi-reachability) is dualized into a notion of "observability" for cosystems. The notion of abstract observability for systems, however, does not have a nice dualization (see Remark 1.9)). Categorical problems like systemisomorphism are easier to study in algebraic categories like Cosystems rather than in geometricaltopological categories like Systems. Therefore, it is natural to require a stronger notion of system observability, corresponding to "cosystem reachability". This stronger notion, which we shall call "algebraic" observability, can also be motivated by system-theoretic considerations. An intuitive definition of observability for a system E is that there should exist a procedure which permits the determination of an arbitrary initial state of Z on the basis of appropriate input-output experiments (see a detailed discussion in Kalman I-2, Chapter 10]). The word "procedure" is, of course, overly vague. In the classical context of linear systems it is well-known that any possible procedure is equivalent to a linear procedure. Already

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when considering linear systems over rings, however, the definition of "procedure" becomes more delicate, and is in fact directly related to the existence of "observers" which are systems over the same rings, see Sontag [6]. It is natural in the present context to consider an algebraic notion of observability. In other words, we shall only allow algebraic operations on the input-output data. This is formalized as follows: Definition 8.2. A system E is algebraically observable if there exist an integer s and ~ = (co1..... %) e(U*) s, such that for every costate q ~ A(X) there exists q ~ A(YO with qoH ~ = q, where He: X ~ ys is given by x~--~ (h°'l(x). . . . . h°"(x)). Clearly this definition amounts to requiring that A(H~): A(Y ~) ~ A(X) be a surjective comorphism. The image of this comorphism is called the observation algebra of the system. Now the uniqueness theorem can be proved. Theorem 8.3. Let E, ~. have the same input-output map. Assume that Z is quasi-reachable and that E is algebraically observable. Then there exists a unique morphism T: E ~ E. If ~: is quasireachable [resp. reachable] then T is dominating [-onto]. If E is observable then T is injective. If E is algebraically observable and E is quasi-reachable then T is an isomorphism. Proof. Let the dimension of E be n. It follows from quasi-reachability of E and from (5.4) that g : U" ~ X is dominating. So, by (1.8), ~ = A(g,): A(X) -. A(U') is injective. From the definition of algebraic observability we obtain s, ~ such that ~ = A(~/g): A(YO ~ A(,~) is surjective. Consider the diagram: . A(X)

A(YO

i~

A(U")

Clearly there exists a (unique) k-algebra homomorphism ~/making the diagram commutative. It follows from the equivalence of categories (1.6) that there exists a (unique) T: X --, X such that A(T) = r/. A routine calculation shows that T gives the desired morphism. Uniqueness of T is clear, and the rest of the proof is straightforward. As an application, we consider the class of state-affine systems: U = k', Y = kp, X = k", h is linear and p(x, u) is of the form pl(u) + P2(U)X(these systems are sometimes called variable-structure systems). In this case the maps h" are affine maps, so that the notions of abstract observability and algebraic observability coincide (see 1.9). Then: Corollary 8.4. Let E, Z be observable variable structure systems. Assume that f~ = ft- Then Xo is isomorphic to EQ. CONCLUSIONS

We have presented in this paper several preliminary results relating to reachability, observability, identification, and isomorphism of polynomial systems. One of the main problems not discussed here, the problem of minimizing a given system, requires an extension of the present framework, because the category Systems is too "small" to permit needed operations (quotients, etc.). This amounts to saying that Affine cosystems is too "small" and should be replaced by Cosystems. By dualizing the latter, we obtain a category Scheme Systems containing the original category Systems. Recent investigation shows that a satisfactory theory of minimal realizations exists for scheme systems. Although it is not at all clear that the "physiCal" realization of such generalized systems would be of interest, only by embedding the theory of nonlinear systems

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into such a framework can we gain the necessary perspective to analyse minimization problems. Current research efforts are directed toward this goal and toward the study of the realization problem; see Sontag [7]. Acknowledgement--Wewish to thank Prof. R. E. Kalman for numerous editorial improvements in this paper.

1. 2. 3. 4. 5. 6. 7.

REFERENCES D1EUDO~a~J., Cours de G~om~trie AIg~brique (2 vols), Presses Universitaires de France (1974). KALMANR. E., Lectures on controllability and observability, CIME Summer Course, 1968, Cremonese, Roma (1969). MACDONALDI. G., Algebraic Geometry: Introduction to Schemes, Benjamin (1968). MUMFORD,D. Introduction to Algebraic geometry, Harvard Lecture Notes, (1968). SEXDENaERGA., On the length of a Hilbert ascending chain, Proc. Amer. Math. Soc. 29, 443 (1971). SONTAGE. D., Linear systems over commutative rings: a survey, Ricerche di Automatica (1976). To be published. SOWrAG E. D., On the realization theory of polynomial input-output maps, Ph.D. Dissertation, University of Florida (1976).