Realization theory for rational systems: the existence of rational realizations

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1 Realization theory for rational systems: the existence of rational realizations Jana Němcová CWI, Amsteram, The Netherlans Jan H. van Schuppen CWI, Amsteram, The Netherlans Abstract In this paper we solve the problem of the existence of rational realizations of response maps. Sufficient an necessary conitions for a response map to be realizable by a rational system are presente. We provie also the characterization of the existence of rationally observable an canonical rational realizations for a given response map. Keywors: Rational systems, realization theory, algebraic reachability, rational observability. Mathematics Subject Classification (2000): 93B15, 93B27, 93C10 Publishe in SIAM Journal on Control an Optimization, 48(4): , DOI: / Introuction The motivation to investigate realization theory of rational systems is the use of rational systems as moels of phenomena in life sciences, in particular in systems biology. For example, rational systems occur as moels of metabolic, genetic, an signaling networks. They can be foun also in engineering, physics, an economics. Moreover, as Bartosiewicz state in [3], the theory of rational systems coul be simpler an more powerful, once it is evelope, than the theory of smooth systems. The realization problem for rational systems consiers a map from input functions to output functions an asks whether there exists a finite-imensional rational system with an initial conition such that its input/output map is ientical to the consiere map. Such a system is then calle a realization of the consiere input/output map. A generalization, which is not aresse in this paper, is to regar any relation between observe variables an ask for a realization as a rational system. Another goal of realization theory is to characterize certain properties of realizations. One wants to fin the conitions uner which the systems realizing the consiere map are observable, controllable, or minimal. The relations between realizations having these properties are also of interest since they can be applie in control an observer synthesis an in system ientification. Polynomial an rational systems are a special class of nonlinear systems amitting a more refine algebraic structure. Realization theory for iscrete-time polynomial systems was formulate by Sontag in [16]. Later, in [18], Wang an Sontag publishe their results on realization theory for polynomial an rational continuous-time systems base on the approach of formal power series in non-commuting variables an on the relation of two characterizations of observation spaces. Another approach to realization theory for polynomial continuous-time systems, motivate by the results of Jakubczyk in [11] for nonlinear realizations, is introuce by Bartosiewicz in [1, 4]. This approach is base on [16]. Furthermore, in [3], Bartosiewicz introuces the concept of rational systems an eals with the problem of immersion of smooth systems into rational systems. Since this problem is similar to the problem of rational realization, there is an analogy between our approach to the realization theory for rational systems an Bartosiewicz s results presente in [3, 4]. Compare to the realization theory for rational systems evelope by Wang an Sontag in [18], our approach is ifferent. We apply the algebraic-geometric approach rather than techniques base on formal power series. We solve the same problem of existence of rational realizations 1

2 REALIZATION THEORY FOR RATIONAL SYSTEMS 2 (compare Theorem 5.2 in [18] an Theorem 5.16 in this paper). In aition we eal with the questions of rational observability an algebraic reachability of rational realizations which are not treate in [18]. Another major ifference is that the realizations within the class of rational systems which we consier o not have to be affine in the inputs as assume by Wang an Sontag. This is motivate by the planne application of realization theory to biochemical systems where the inputs may enter in a rational way. The first step, motivate by biochemical reaction networks, in eveloping a realization theory for rational positive systems is one in [17]. We leave the problem of rational realization with the positivity constraint for further research. The organization of the paper is as follows. Terminology, notation an mathematical preliminaries are provie in Section 2. Section 3 introuces the concept of rational systems an Section 4 introuces the classes of amissible inputs which are consiere omains for response maps. The problem of rational realization is formulate, an necessary an sufficient conitions for the existence of a rational realization of a response map are presente in Section 5. Section 6 eals with canonical rational realizations. Section 7 conclues the paper. 2 Algebraic preliminaries In this section we introuce algebraic-geometric framework in which we formulate the problem of rational realization. Note that the methos of algebraic geometry are alreay known for control an system theory, see for example [6, 8, 9]. For the basic efinitions an theorems of commutative algebra an algebraic geometry see [5, 7, 12, 13, 19]. By a polynomial in finitely many ineterminates X 1,..., X n with real coefficients we mean a sum k N n c(k) n i=1 Xk(i) i where only finitely many coefficients c(k) R are non-zero. We enote the ring of all polynomials in n variables with real coefficients by R[X 1,..., X n ]. Because the fiel R is an integral omain, so is the algebra R[X 1,..., X n ]. Therefore we can efine the fiel of quotients of R[X 1,..., X n ] as the set of fractions {p/q p, q R[X 1,..., X n ], q 0}. This fiel is enote by R(X 1,..., X n ) an we refer to its elements as rational functions. Generally we use the notation Q(S) for the fiel of quotients of an integral omain S. For example, Q(R[X 1,..., X n ]) = R(X 1,..., X n ). By R/I, for a ring R an an ieal I R, we enote the quotient (factor) ring of R moulo I. Polynomial an rational functions on varieties A real affine variety X is an algebraic variety in R n, i.e. there are finitely many f 1,..., f N R[X 1,..., X n ] such that X = {(x 1,..., x n ) R n f 1 (x 1,..., x n ) = = f N (x 1,..., x n ) = 0}. We say that a variety is irreucible if we cannot write it as an union of two non-empty varieties which are its strict subvarieties. Let I R[X 1,..., X n ] be the ieal of all polynomials which are zero at every point of the variety X. By a polynomial on a variety X we mean a map p : X R for which there exists q R[X 1,..., X n ] such that p = q on X. The algebra of all polynomials on X, enote by A, is then isomorphic to R[X 1,..., X n ]/I. From Hilbert Basis Theorem an from the fact that the ieals in the quotient ring R[X 1,..., X n ]/I are in one-to-one corresponence with the ieals of R[X 1,..., X n ] containing I, every ieal in R[X 1,..., X n ]/I is finitely generate. Therefore A is a finitely generate algebra of polynomials. Since X is an irreucible variety, A is an integral omain. Therefore we can efine the fiel Q of quotients of A. The elements of Q are calle rational functions on X. On R n we consier the Zariski topology which is a topology where the close sets are efine as real affine varieties. A variety X R n is enowe with the relate topology, the Zariski topology on X. We refer to an open/close/ense set in Zariski topology as to Z-open/Z-close/Z-ense set. More etails can be foun in [10, 12]. Rational vector fiels Let X be an irreucible real affine variety, let A be the algebra of polynomials on X, an let Q enote the fiel of rational functions on X. Definition 2.1 A rational vector fiel f on X is an R-linear map f : Q Q such that f(ϕ ψ) = f(ϕ) ψ + ϕ f(ψ) for ϕ, ψ Q.

3 REALIZATION THEORY FOR RATIONAL SYSTEMS 3 We say that the rational vector fiel f is efine at the point x X if f(o x ) O x where O x = {ϕ Q ϕ is efine at x}. The set X(f) of all points at which a rational vector fiel f is efine is given as X(f) = {x X f(o x ) O x }. Definition 2.2 The trajectory of a rational vector fiel f from a point x 0 X(f) is the map x : [0, T ) X(f) X such that for t [0, T ) an ϕ A, t (ϕ x)(t) = (fϕ)(x(t)) an x(0) = x 0. Note that it is sufficient to consier only the polynomials ϕ A in the efinition above as it is prove in [3]. Theorem 2.3 [3, Theorem 1] For any rational vector fiel f an any point x 0 X(f) there exists an unique trajectory of f from x 0 efine on the maximal interval [0, T ) (T may be infinite). Proof: The proof of this statement can be foun in [2, 3]. 3 Rational systems We consier rational systems as systems on irreucible real affine varieties with the ynamics efine by rational vector fiels an with output functions having rational components. This concept of rational systems is introuce in [3] an we recall it in this section. We consier an input space U to be an arbitrary set U R m. As an output space we consier R r. By a slight moification of [3, Definition 2] we efine rational systems as follows. Definition 3.1 A rational system Σ with an input space U an an output space R r is a quaruple Σ = (X, f, h, x 0 ) where (i) X is an irreucible real affine variety, (ii) f = {f α α U} is a family of rational vector fiels on X, (iii) h : X R r is an output map with rational components, i.e. h i Q for i = 1,..., r, (iv) x 0 X is an initial state such that all h i, i = 1,..., r an all f α, α U are efine at x 0. The states at which all components of the output function h are efine an at which at least one of the rational vector fiels f α, α U is efine is a Z-ense open subset of X, see [3]. If U is a finite set then the set of points at which all components of h an all vector fiels f α, α U are efine is also a Z-ense open subset of X. As the space of input functions we consier the set U pc of piecewise-constant functions u : [0, ] U. Let u U pc, then u = (α 1, t 1 )... (α nu, t nu ) means that for t ( i j=0 t j, i+1 j=0 t j] the input u(t) = α i+1 U for i = 0, 1,..., n u 1, t 0 = 0, an u(0) = α 1. Every input u U pc has a time omain [0, T u ] where T u = n u j=1 t j. If u = (α 1, t 1 )... (α n, t n ), v = (β 1, s 1 )... (β k, s k ) U pc, then (u)(v) U pc is an input which we get by concatenating v to u, i.e. (u)(v) = (α 1, t 1 )... (α n, t n )(β 1, s 1 )... (β k, s k ). To express that the input u was applie only on time omain [0, t] [0, T u ] we write a subinex [0, t] to u like u [0,t]. The empty input e is such input that T e = 0. Consier a rational system Σ = (X, f, h, x 0 ). The trajectory of the system Σ corresponing to a constant input u = (α, T u ) U pc is the trajectory of the rational vector fiel f α from x 0, i.e. it is the map x( ; x 0, u) : [0, T u ] X for which t (ϕ x)(t; x 0, u) = (f α ϕ)(x(t; x 0, u)) an x(0; x 0, u) = x 0 for t [0, T u ] an for ϕ A. Note that a trajectory of a vector fiel on a close interval is efine as in Definition 2.2, see [4]. The trajectory of the system Σ corresponing to an input u = (α 1, t 1 )... (α nu, t nu ) U pc with T u = n u j=1 t j is the map x( ; x 0, u) : [0, T u ] X such that x(0; x 0, u) = x 0, an x(t; x 0, u) = x αi (t i 1 j=0 t j) for t [ i 1 j=0 t j, i j=0 t j], t 0 = 0, i = 1,..., n u where x αi : [0, t i ] X is a trajectory of a vector fiel f αi from the initial state x( i 1 j=0 t j; x 0, u) = x αi 1 (t i 1 ) for i = 2,..., n u, an from the initial state x 0 for i = 1. Since a trajectory of a rational system Σ oes not nee to exist for every input u U pc, see [3], we efine the set of amissible inputs for Σ.

4 REALIZATION THEORY FOR RATIONAL SYSTEMS 4 Definition 3.2 Let Σ = (X, f, h, x 0 ) be a rational system. We efine the set of amissible inputs U pc (Σ) for the system Σ as a subset of the set of piecewise-constant inputs U pc for which there exist a trajectory of Σ. Note that for every u U pc (Σ) it hols that u [0,t] U pc (Σ) for every t [0, T u ]. We conclue the convention that a trajectory of a rational system Σ with an initial state x 0 X is for the empty input e equal to x 0, i.e. x(0; x 0, e) = x 0. The set of amissible inputs U pc (Σ) may contain only the empty input e. Remark 3.3 Every input of U pc has finitely many switching time-points. This property is neee for the proof of irreucibility of the smallest variety containing a reachable set of a rational system (see Proposition 5.5) which is use in the proof of Theorem 6.1. It is also necessary conition for proving that the set of functions analytic at the switching time-points of such inputs is an integral omain (see Theorem 4.4). This is the crucial property of a class of functions for which there exists a rational realization. 4 Amissible inputs Let U R m be an input space. We efine the sets of amissible inputs for rational systems with the values in U. These sets of inputs are the sets on which the response maps stuie with respect to the realization problem for rational systems are efine. Definition 4.1 A set Ũ pc U pc of input functions with the values in an input space U R m is calle a set of amissible inputs if: (i) u Ũ pc t [0, T u ] : u [0,t] Ũ pc, (ii) u Ũ pc α U t > 0 : (u)(α, t) Ũ pc, (iii) u = (α 1, t 1 )... (α k, t k ) Ũ pc δ > 0 t i [0, t i + δ], i = 1,..., k : u = (α 1, t 1 )... (α k, t k ) Ũ pc. Definition 4.2 Consier a set Ũ pc of amissible inputs with the values in U R m. Let u Ũ pc. We enote the erivation of a real function ϕ : Ũ pc R at the switching time-point T u of the input (u)(α, t) Ũ pc, where t > 0 is sufficiently small an α U, as (D α ϕ)(u) = t ϕ((u)(α, t)) t=0+. Let ϕ : Ũ pc R be a real function an let u = (α 1, t 1 )... (α k, t k ) Ũ pc for a set Ũ pc of amissible inputs. Consier the function ϕ u (t) = ϕ(u [0,t] ) efine for t [0, T u ]. If t [ n i=0 t i, n+1 i=0 t i], n = 0,... k 1 then ϕ u (t) = ϕ((α 1, t 1 )... (α n, t n i=0 t i)). The erivation (D α ϕ)(u) is well-efine if the function ϕ (u)(α,t) (ˆt) = ϕ((u)(α, ˆt)), ˆt [T u, T u + t] is ifferentiable at T u +. We say that the map ϕ : Ũ pc R is smooth if the erivations D α1... D αi ϕ are well-efine on Ũ pc for every i N an α j U, j = 1,..., i. To simplify the notation, the erivation D α1... D αi ϕ can be rewritten as D α ϕ where α = (α 1,..., α i ). Definition 4.3 Consier a set Ũ pc of amissible inputs with the values in U R m. We say that a function ϕ : Ũ pc R is analytic at the switching time-points of the inputs from Ũ pc if for every input u = (u 1, t 1 )... (u k, t k ) Ũ pc the function ϕ u1,...,u k (t 1,..., t k ) = ϕ((u 1, t 1 )... (u k, t k )) is analytic, i.e. we can write ϕ u1,...,u k in the form of convergent formal power series in k ineterminates. We enote the set of real functions ϕ : Ũ pc R which are analytic at the switching time-points of the inputs from Ũ pc by A(Ũ pc R). We refer to the elements of A(Ũ pc R) as to the analytic functions on Ũ pc.

5 REALIZATION THEORY FOR RATIONAL SYSTEMS 5 Let ϕ be an analytic function on Ũ pc. Then for every (u)(α, 0)(v) Ũ pc ϕ((u)(α, 0)(v)) = ϕ((u)(v)). (1) Theorem 4.4 The set A(Ũ pc R) of analytic functions on a set Ũ pc of amissible inputs with the values in U R m is an integral omain. Proof: To prove that A(Ũ pc R) is an integral omain we prove that for f, g A(Ũ pc R) it hols that if fg = 0 on Ũ pc then f = 0 on Ũ pc or g = 0 on Ũ pc. Consier f, g A(Ũ pc R) such that fg = 0. Then fg(u) = 0 for every u Ũ pc. Let u Ũ pc be an arbitrary input. Because f(u), g(u) R an because R is an integral omain, the equality fg(u) = f(u)g(u) = 0 implies that either f(u) = 0 or g(u) = 0. To complete the proof we have to prove that it is not possible that there exist u, v Ũ pc such that f(u) = g(v) = 0 an f(v), g(u) 0. Let us assume for a contraiction that there exist such u = (α 1, t u 1)... (α k, t u k ), v = (β 1, t v 1)... (β l, t v l ) Ũ pc. Since u, v Ũ pc, we erive from Definition 4.1 (ii) that t v i [0, t v i ], i = 1,..., l : w = (α 1, t u 1)... (α k, t u k)(β 1, t v 1 )... (β l, t v l ) Ũ pc. From Definition 4.1 (iii), there exists δ > 0 such that w = (α 1, t u 1 )... (α k, t u k )(β 1, t v 1 )... (β l, t v l ) Ũ pc for every t u i [0, t u i +δ), i = 1,..., k, an for every tv j [0, t v j +δ), j = 1,..., l. Because fg = 0 on Ũ pc, it follows that 0 = f(w )g(w ) an thus 0 = f α1,...,α k,β 1,...,β l (t u 1,..., t u k, tv 1,..., t v k )g α 1,...,α k,β 1,...,β l (t u 1,..., t u k, tv 1,..., t v k ) for t u i [0, t u i + δ), tv j [0, t v j + δ), i = 1,..., k, j = 1,..., l. Hence, f α1,...,α k,β 1,...,β l g α1,...,α k,β 1,...,β l = 0 on [0, t u i + δ) [0, t v i + δ). (2) i=1,...,k j=1,...,l Because f, g A(Ũ pc R) it hols that f α1,...,α k,β 1,...,β l an g α1,...,α k,β 1,...,β l are convergent formal power series in k + l ineterminates for all α 1,..., α k, β 1,..., β l U, an k, l N. Since i=1,...,k [0, tu i + δ) j=1,...,l [0, tv i + δ) is an open connecte set in R k+l, from (2) an from the fact that a ring of convergent formal power series over R in finitely many ineterminates is an integral omain (this follows from [20, Vol.2, Ch.7, Theorem 1]), we erive that either f α1,...,α k,β 1,...,β l = 0 on [0, t u i + δ) [0, t v j + δ), (3) or g α1,...,α k,β 1,...,β l = 0 on i=1,...,k i=1,...,k j=1,...,l [0, t u i + δ) j=1,...,l [0, t v j + δ). (4) By assuming that either (3) or (4) hols we come to a contraiction. This completes the proof. Let us assume that (3) hols. Therefore for τ i [0, t u i + δ), τ j [0, tv j + δ), i = 1,..., k, j = 1,..., l, we get that f α1,...,α k,β 1,...,β l (τ 1,..., τ k, τ 1,..., τ l ) = 0. Because f α 1,...,α k,β 1,...,β l (0,..., 0, τ 1,..., τ l ) = f β 1,...,β l (τ 1,..., τ l ), we erive that f β 1,...,β l (τ 1,..., τ l ) = 0 for τ j [0, tv j +δ), j = 1,..., l. Further, because v Ũ pc, from Definition 4.1 (iii) it follows that ɛ > 0 t j [0, t v j + ɛ), j = 1,..., l : (β 1, t 1 )... (β l, t l ) Ũ pc. Thus, f β1,...,β l can be represente as a convergent formal power series in l ineterminates with a convergence omain containing j=1,...,l [0, tv j + ɛ). Since f β 1,...,β l = 0 on j=1,...,l [0, tv j + δ), then f β1,...,β l = 0 also on j=1,...,l [0, tv j +ɛ). Therefore f β 1,...,β l (t v 1,..., t v l ) = f(v) = 0 which contraicts the assumption f(v) 0. Let us assume that (4) hols. Then g α1,...,α k,β 1,...,β l (τ 1,..., τ k, τ 1,..., τ l ) = 0 for all τ i [0, t u i + δ), τ j [0, tv j + δ), i = 1,..., k, j = 1,..., l, an thus especially for τ i = t u i, i = 1,..., k, an τ j = 0, j = 1,..., l. Therefore 0 = g α1,...,α k,β 1,...,β l (t 1,..., t k, 0,..., 0) = g α1,...,α k (t 1,..., t k ) = g(u) which contraicts the assumption g(u) 0. Corollary 4.5 Because the set A(Ũ pc R) is an integral omain, we can efine the fiel Q(Ũ pc R) of the quotients of elements of A(Ũ pc R).

6 REALIZATION THEORY FOR RATIONAL SYSTEMS 6 5 Rational realizations Response maps In this paper we work with response maps rather than with input/output (I/O) maps since it is technically more convenient. The I/O maps are consiere to be the maps between the spaces of input an output functions (functions of time) mapping an input to an output. We call a map which escribes the outputs immeiately after applying finite parts of the inputs a response map. Definition 5.1 Let Ũ pc be a set of amissible inputs. A map p : Ũ pc R r is calle a response map if its components p i : Ũ pc R, i = 1,..., r are such that p i A(Ũ pc R). We poste extra assumptions on response maps in the efinition above because to solve the problem of realization of a response map by a rational system we use the objects like observation algebra an observation fiel of a response map, see Definition 5.9. These assumptions are necessary for well-efineness of these objects. Problem formulation A rational system which for each input gives us the same output as a response map p is calle a rational realization of p (a rational system realizing p). The realization problem for rational systems can be unerstoo as the problem of fining such a rational system for a given map. Formally we state the problem of realization of a response map by a rational system as follows: Let Ũ pc be a set of amissible inputs. Consier a response map p : Ũ pc R r. The realization problem for rational systems consists of etermining a rational system Σ = (X, f, h, x 0 ) such that p(u) = h(x(t u ; x 0, u)) for all u Ũ pc an Ũ pc U pc (Σ). Properties of rational realizations Note that the solution of a rational system Σ = (X, f, h, x 0 ) corresponing to an input u Ũ pc is a piecewise-analytic map such that the intervals on which the solution is analytic are etermine by the switching time-points of the input u. Definition 5.2 Let Σ = (X, f, h, x 0 ) be a rational realization of a response map p : Ũ pc R r where Ũ pc is a set of amissible inputs. Let A enote the algebra of polynomial functions on X. We efine the input-to-state map τ : Ũ pc X as the map τ(u) = x(t u ; x 0, u) for u Ũ pc. The map τ etermine by τ is efine as τ : A A(Ũ pc R) such that τ (ϕ) = ϕ τ for all ϕ A. To simplify further reference we state some properties of the map τ in Proposition 5.3. The proof of this proposition is omitte because it irectly follows from the efinition of τ. Proposition 5.3 Let Σ = (X, f, h, x 0 ) be a rational realization of a response map p : Ũ pc R r where Ũ pc is a set of amissible inputs. Let A be the algebra of polynomials on X, let ϕ 1,..., ϕ k A, k < be such that A = R[ϕ 1,..., ϕ k ], an let Q enote the fiel of rational functions on X. Then the map τ : A A(Ũ pc R) efine in Definition 5.2 is a homomorphism an τ (R[ϕ 1,..., ϕ k ]) = R[τ ϕ 1,..., τ ϕ k ]. Moreover, the map τ : A/Ker τ R[τ ϕ 1,..., τ ϕ k ], efine as τ ([ϕ]) = τ ϕ for every ϕ A, is an isomorphism. The map τ can be extene to an isomorphism of the fiels Q(A/Ker τ ) an R(τ ϕ 1,..., τ ϕ k ). The efinitions of algebraic reachability (Definition 5.4) an rational observability (Definition 5.6) of rational realizations are base on [3, Definition 3,4]. Definition 5.4 Let Σ = (X, f, h, x 0 ) be a rational realization of a response map p : Ũ pc R r where Ũ pc is a set of amissible inputs. The rational system Σ is sai to be algebraically reachable (from the initial state) x 0 if the reachable set is Z-ense in X. R(x 0 ) = {x(t u ; x 0, u) X u Ũ pc U pc (Σ)}

7 REALIZATION THEORY FOR RATIONAL SYSTEMS 7 Proposition 5.5 Let Σ = (X, f, h, x 0 ) be a rational system as in Definition 5.4. Then the closure Z-cl(R(x 0 )) of the reachable set R(x 0 ) in Zariski topology on X is an irreucible variety. Proof: The Zariski closure of the reachable set R(x 0 ) is the smallest variety in X containing R(x 0 ) which is given as Z-cl(R(x 0 )) = {x X ϕ(x) = 0 for all ϕ A such that ϕ = 0 on R(x 0 )}. Therefore, by consiering τ : Ũ pc X an τ : A A(Ũ pc R) efine in Definition 5.2, we erive that Z-cl(R(x 0 )) = {x X ϕ(x) = 0 for all ϕ A such that ϕ τ = 0 on Ũ pc} = {x X ϕ(x) = 0 for all ϕ A such that τ ϕ = 0} = {x X ϕ(x) = 0 for all ϕ Ker τ }. Because τ is a homomorphism an thus τ (A) is a subalgebra of A(Ũ pc R), an because A(Ũ pc R) is an integral omain, τ (A) is an integral omain. From Proposition 5.3, the map τ : A/Ker τ τ (A) is an isomorphism. This implies, since τ (A) is an integral omain, that Ker τ is a prime ieal. Accoring to [7, Chapter 5.1, Proposition 4], an affine variety V is irreucible if an only if the ieal of polynomials which vanish on V is a prime ieal. Because, the variety Z-cl(R(x 0 )) is etermine by the polynomial ieal Ker τ, an because Ker τ is a prime ieal, the variety Z-cl(R(x 0 )) is irreucible. Definition 5.6 Let Σ = (X, f = {f α α U}, h, x 0 ) be a rational system an let Q enote the fiel of rational functions on X. The observation algebra A obs (Σ) of Σ is the smallest subalgebra of the fiel Q containing all components h i, i = 1,..., r of h, an close with respect to the erivations given by rational vector fiels f α, α U. The observation fiel Q obs (Σ) of the system Σ is the fiel of quotients of A obs (Σ). The rational system Σ is calle rationally observable if Q obs (Σ) = Q. The observation algebra of a rational system is an integral omain because it is a subalgebra of an integral omain. Therefore the observation fiel of a rational system is well-efine. The observation fiel Q obs (Σ) is also close with respect to the erivations given by rational vector fiels f α, α U. Proposition 5.7 ( [3], Proposition 1) For a rational system Σ, Q obs (Σ) is a finitely generate fiel extension of R, i.e. there exist ϕ 1,..., ϕ k Q obs (Σ) such that Q obs (Σ) = R(ϕ 1,..., ϕ k ). Definition 5.8 We call a rational realization of a response map canonical if it is both rationally observable an algebraically reachable. In realization theory one is given a response map an is suppose to fin a system within a certain class of systems which correspons to this map. This map is therefore representing the unknown system. For this reason it is useful to efine objects as observation algebra an observation fiel also for response maps. Definition 5.9 Let Ũ pc be a set of amissible inputs an let p : Ũ pc R r be a response map. The observation algebra A obs (p) of p is the smallest subalgebra of the algebra A(Ũ pc R) which contains the components p i, i = 1,..., r of p, an which is close with respect to the erivations D α, α U. The observation fiel Q obs (p) of p is the fiel of quotients of A obs (p). Note that the observation fiel Q obs (p) of p is well-efine only if A obs (p) is an integral omain. This is the case for response maps because the components of a response map are the elements of A(Ũ pc R) for a set Ũ pc of amissible inputs an because A(Ũ pc R) an thus also Q(Ũ pc R) is an integral omain. For well-efineness of the observation algebra of p it is sufficient to assume that the components of p are smooth (with respect to D α erivations). Existence of rational realizations We provie sufficient an necessary conitions for a response map to be realizable by a rational system. The realizability of response maps by a polynomial system is treate in [4, Theorem 2]. The proof of that theorem an the proof of Proposition 5.14 have the same structure. The following lemma can be foun in [4] state for polynomial systems.

8 REALIZATION THEORY FOR RATIONAL SYSTEMS 8 Lemma 5.10 Let Σ = (X, f, h, x 0 ) be a rational realization of a response map p : Ũ pc R r where Ũ pc is a set of amissible inputs an let τ : Ũ pc X be as in Definition 5.2. Then for any ϕ from the algebra A of polynomials on X an for any α U where U is the set of values of the inputs of Ũ pc it hols that D α (ϕ τ) = (f α ϕ) τ. Proof: Let u Ũ pc an let α U. Because Ũ pc U pc (Σ), the trajectories of the rational system Σ corresponing to the input (u)(α, s) with sufficiently small s > 0 an to all restrictions of (u)(α, s) to shorter time-omains are well-efine. From Definition 4.2 we get that for arbitrary ϕ A D α (ϕ τ)(u) = s (ϕ τ)((u)(α, s)) s=0+ = s ϕ(τ((u)(α, s))) s=0+. By the efinition of an input-to-state map τ, τ((u)(α, s)) = x(t u + s; x 0, (u)(α, s)). accoring to Definition 2.2, Further, D α (ϕ τ)(u) = (f α ϕ)(x(t u + s; x 0, (u)(α, s))) s=0+. Finally, by the continuity of rational function f α ϕ along the trajectory of Σ etermine by the input (u)(α, s) an by the properties of a trajectory, we get that D α (ϕ τ)(u) = (f α ϕ)(τ(u)). Proposition 5.11 Let p : Ũ pc R r be a response map realizable by a rational system Σ = (X, f, h, x 0 ). Let τ : Ũ pc X be as in Definition 5.2. Then the map τ ext : A obs (Σ) A obs (p) efine as τ extϕ = ϕ τ for every ϕ A obs (Σ) is a well-efine surjective homomorphism, i.e. τ ext(a obs (Σ)) = A obs (p). Proof: Note that τext is efine in the same way as τ but on a ifferent omain, see Definition 5.2. Obviously, τext is a homomorphism. We prove that τext is well-efine an that it is surjective. By Definition 5.6, 5.9, the observation algebras of a system Σ an of a map p are generate by h i, f α1... f αj h i an p i, D α1... D αj p i, respectively, such that j N, α 1,..., α j U, an i = 1,..., r. Since τext is a homomorphism, we prove that τext(a obs (Σ)) = A obs (p) by proving that the generators of A obs (Σ) an A obs (p) are mappe to each other by τext. To prove that τext is well-efine it is sufficient to prove that τext is well-efine for the generators of the algebra A obs (Σ). Since Σ is a rational realization of p, we know that p = h τ an that p is well-efine. Because h A obs (Σ), p = τexth which implies that τext is well-efine at h. Let h num, h en A be such that h = hnum h en. For a rational vector fiel f α f, it hols that (f α h) τ = (f α h num h en ) τ = (f αh num τ)(h en τ) (f α h en τ)(h num τ) (h en τ) 2, an further, by Lemma 5.10, that (f α h) τ = D α(h num τ)(h en τ) D α (h en τ)(h num τ) (h en τ) 2. Therefore, τext(f α h) = (f α h) τ = D α ( hnum h en τ) = D α (h τ) = D α (p). As p is a response map, the erivations D α of p are well-efine an consequently τext is well-efine at f α h A obs (Σ) Q. In the next proposition we state necessary conitions for a response map to be realizable by a rational system. Proposition 5.12 Let p : Ũ pc R r be a response map realizable by a rational system Σ = (X, f, h, x 0 ). Let τ ext : A obs (Σ) A obs (p) be as in Proposition Then (i) Q obs (p) = τ (Q(A obs (Σ)/Ker τ ext)), where τ : A obs (Σ)/Ker τ ext A obs (p) is an isomorphism erive from the map τ ext : A obs (Σ) A obs (p), (ii) Q obs (p) is finitely generate.

9 REALIZATION THEORY FOR RATIONAL SYSTEMS 9 Proof: (i) Accoring to Proposition 5.11, the map τext : A obs (Σ) A obs (p) is a surjective homomorphism which is not necessarily injective. Then the map τ : A obs (Σ)/Ker τ ext A obs (p) efine as τ ([ϕ]) = τext(ϕ) for every ϕ A obs (Σ), is an isomorphism. Since the algebras A obs (Σ)/Ker τext an A obs (p) are integral omains, we can construct the fiels of fractions of A obs (Σ)/Ker τext an A obs (p). We exten the isomorphism τ of the algebras A obs (Σ)/Ker τext an A obs (p) to the isomorphism τ of the fiels Q(A obs (Σ)/Ker τext) an Q obs (p). Then Q obs (p) = τ (Q(A obs (Σ)/Ker τext)). (ii) The fiel Q(A obs (Σ)/Ker τext) is a fiel isomorphic to a subfiel F of Q obs (Σ). From [5, Chapter V, Section 14.7, Corollary 3] it follows that if G is a finitely generate fiel containing R, then every subfiel F of G containing R is finitely generate. Then, since F Q obs (Σ) Q an Q is a finite fiel extension of R, it follows that F is also finitely generate. Because a fiel isomorphic to a finitely generate fiel is finitely generate, there exist ϕ 1,..., ϕ k Q(A obs (Σ)/Ker τext) such that Q(A obs (Σ)/Ker τext) = R(ϕ 1,..., ϕ k ). (5) From (i), Q obs (p) = τ (Q(A obs (Σ)/Ker τext)) where τ is an isomorphism. Then, by (5), Q obs (p) = τ (R(ϕ 1,..., ϕ k )) = R( τ ϕ 1,..., τ ϕ k ). Thus Q obs (p) is finitely generate. In the following proposition we prove that the generators of the observation fiel Q obs (p) of a response map p can be chosen from A(Ũ pc R). This allows us to reformulate the necessary conition for a response map p to be realizable by a rational system which is state as the conition (ii) in the proposition above in the following way: Q obs (p) is finitely generate by the elements from A(Ũ pc R). Proposition 5.13 Let p : Ũ pc R r be a response map. The observation fiel Q obs (p) is finitely generate if an only if it is finitely generate by the elements from A(Ũ pc R), i.e. there exist finitely many ϕ 1,..., ϕ k A(Ũ pc R) such that Q obs (p) = R(ϕ 1,..., ϕ k ). Proof: ( ) Let Q obs (p) be finitely generate by the elements from A(Ũ pc R). Then it is obviously finitely generate. ( ) Let Q obs (p) be finitely generate. There exist ϕ 1,..., ϕ k Q obs (p) such that Q obs (p) = R(ϕ 1,..., ϕ k ). As ϕ i Q obs (p), i = 1,..., k we know that ϕ i = ϕi,num ϕ i,en where ϕ i,num, ϕ i,en A obs (p) for i = 1,..., k. We efine the fiel F = R(ϕ 1,num, ϕ 1,en,..., ϕ k,num, ϕ k,en ). Because ϕ i,num, ϕ i,en A obs (p) for i = 1,..., k, an because A obs (p) is a subalgebra of A(Ũ pc R), it follows that ϕ i,num, ϕ i,en A(Ũ pc R) for i = 1,..., k, an (6) R[ϕ 1,num, ϕ 1,en,..., ϕ k,num, ϕ k,en ] A obs (p). (7) By (6) the fiel F is generate by the elements from A(Ũ pc R). From the efinition of F it is obvious that F Q obs (p), an from (7) we get by taking the quotients that F Q obs (p). Therefore Q obs (p) = F an thus the fiel Q obs (p) is finitely generate by the elements from A(Ũ pc R). In the following proposition we specify sufficient conitions for a response map to be realizable by a rational system. Proposition 5.14 Let p : Ũ pc R r be a response map. If there exists a fiel F Q(Ũ pc R) such that (i) F is finitely generate by the elements from A(Ũ pc R), (ii) F is close with respect to D α erivations, i.e. i N, α j U, j = 1,..., i : D α1... D αi F F,

10 REALIZATION THEORY FOR RATIONAL SYSTEMS 10 (iii) Q obs (p) F, then p has a rational realization. Proof: Consier a response map p : Ũ pc R r. We assume that there exists a fiel F Q(Ũ pc R) satisfying (i)-(iii). Let ϕ 1,..., ϕ k : Ũ pc R A(Ũ pc R) be such that F = R(ϕ 1,..., ϕ k ). Since F is close with respect to D α erivations, for any α = (α 1,..., α j ) such that α 1,..., α j U, j N, an for every ϕ i, i = 1,..., k, there exists vi α R(X 1,..., X k ) such that D α ϕ i = vi α (ϕ 1,..., ϕ k ). (8) Since Q obs (p) F, the components of the map p = (p 1,..., p r ) are elements of F an thus for every p j, j = 1,..., r there exists w j R(X 1,..., X k ) such that We prove that a rational system Σ = (X, f, h, x 0 ) where X = R k, k f α = i=1 v α i p j = w j (ϕ 1,..., ϕ k ). (9) x i, α U, h j (x 1,..., x k ) = w j (x 1,..., x k ), j = 1... r, x 0 = (ϕ 1 (e),..., ϕ k (e)), e is the empty input, is a rational system realizing p. Let us efine Ψ(t) = (ϕ 1,..., ϕ k )(u [0,t] ) for u Ũ pc, t [0, T u ]. It is well-efine because ϕ i A(Ũ pc R), i = 1,..., k are efine for every u Ũ pc an because u [0,t] Ũ pc if u Ũ pc an t [0, T u ]. Especially, Ψ(0) = (ϕ 1,..., ϕ k )(u [0,0] ) is well-efine because u [0,0] = e Ũ pc an the functions ϕ 1,..., ϕ k A(Ũ pc R) are efine properly at e. Consier a constant input u = (α, T u ) Ũ pc. Note that from Definition 4.1(i), (ii) there exists ɛ > 0 such that for every t, τ 0 such that t + τ [0, T u + ɛ] it hols that u = (α, t + τ) Ũ pc. If t + τ T u then u = u [0,t+τ], an if T u < t + τ then u = u [0,T u]. In both cases we refer to the corresponing inputs as to the inputs u [0,t+τ], t + τ [0, T u + ɛ]. Then, Ψ(0) = (ϕ 1,..., ϕ k )(u [0,0] ) = (ϕ 1 (e),..., ϕ k (e)) = x 0, an t Ψ(t) = τ Ψ(t + τ) τ=0+ = τ (ϕ 1(u [0,t+τ] ),..., ϕ k (u [0,t+τ] )) τ=0+. Because (u [0,t] )(α, τ) = u [0,t+τ] an because D α ϕ(u [0,t] ) = τ ϕ((u [0,t])(α, τ)) τ=0+, the erivation t Ψ(t) equals (D αϕ 1 (u [0,t] ),..., D α ϕ k (u [0,t] )). Furthermore, ue to (8), t Ψ(t) = (vα 1 (ϕ 1,..., ϕ k )(u [0,t] ),..., v α k (ϕ 1,..., ϕ k )(u [0,t] )). Finally, as Ψ(t) = (ϕ 1,..., ϕ k )(u [0,t] ), we get that t Ψ(t) = (vα 1 (Ψ(t)),..., v α k (Ψ(t))), an Ψ(0) = x 0. Hence Ψ(t) = x(t; x 0, u [0,t] ) for a constant input u Ũ pc an for t [0, T u ]. This also proves that for u = (α 1, t 1 )... (α j, t j ) Ũ pc an for t [0, t 1 ], we have Ψ(t) = x(t; x 0, u [0,t] ). If we consier u = (α 1, t 1 )... (α j, t j ) Ũ pc an if we take t [t 1, t 1 + t 2 ] instea of t [0, t 1 ], we get that Ψ(t 1 ) = x(t 1 ; x 0, u [0,t1]) an tψ(t) = (vα2 1 (Ψ(t)),..., vα2 k (Ψ(t))) for t [t 1, t 1 + t 2 ] with the same reasoning as before. For t [t 1, t 1 + t 2 ] we have that Ψ(t) = x(t; Ψ(t 1 ), u [t1,t 1+t]) = x(t; x(t 1 ; x 0, u [0,t1]), u [t1,t 1+t]). In the analogous way we stuy the cases for t [t 1 + t 2, t 1 + t 2 + t 3 ],..., t [t t i 1, t t i ],.... Finally, Ψ(t) = x(t; x 0, u [0,t] ) for an arbitrary u Ũ pc an t [0, T u ]. Thus the trajectories of Σ are escribe by Ψ. To prove that the rational system Σ is a realization of the response map p, we have to prove that p(u) = h(x(t u ; x 0, u)) for every u Ũ pc. Consier an arbitrary u Ũ pc. Due to (9), p(u) = (p 1,..., p r )(u) = (w 1 (ϕ 1,..., ϕ k ),..., w r (ϕ 1,..., ϕ k ))(u).

11 REALIZATION THEORY FOR RATIONAL SYSTEMS 11 Further, by the efinitions of h j, j = 1,..., r an Ψ it follows that p(u) = (h 1 (ϕ 1,..., ϕ k ),..., h r (ϕ 1,..., ϕ k ))(u) = (h 1 (Ψ(T u )),..., h r (Ψ(T u ))). Finally, since Ψ(T u ) = x(t u ; x 0, u) for u Ũ pc, we erive that p(u) = h(x(t u ; x 0, u)) for u Ũ pc. Remark 5.15 Proposition 5.14 can be state as an equivalence. The proof of the other implication is the same as the sufficiency proof of Theorem The main theorem of this section solving the problem of the existence of rational realizations for response maps is base on the three propositions above. Theorem 5.16 (Existence of rational realizations) A response map p : Ũ pc R r has a rational realization if an only if Q obs (p) is finitely generate. Proof: ( ) See the part (ii) of Proposition 5.12 for this statement an the proof. ( ) From Proposition 5.14, the existence of a fiel F Q(Ũ pc R) finitely generate by the elements from A(Ũ pc R), containing Q obs (p), an close with respect to D α erivations implies the rational realizability of p. Since we assume that Q obs (p) is finitely generate, it follows from Proposition 5.13 an Definition 5.9 that Q obs (p) has those properties. By following the steps of the proof of Proposition 5.14 for F = Q obs (p), we construct a rational realization of p. Example This example is motivate by an example state in [4]. The proceure to construct a rational system realizing a given response map is following the steps mae in the proof of Proposition Let Ũ pc be a set of amissible inputs with the values in R. We etermine a rational system Σ realizing a map p : Ũ pc R efine as p(u) = exp( T u 0 u(s) (1+s) 2 s). By u(s) we enote the value of an input u Ũ pc at a time s [0, T u ]. Firstly we compute D α erivations of p. Consier α 1, α 2 R an u Ũ pc. For t 1, t 2 > 0 sufficiently small (u)(α 1, t 1 )(α 2, t 2 ) Ũ pc. Then (D α1 p)(u) = τ p((u)(α 1, τ)) τ=0+ [ ( Tu = τ exp u(s) (1 + s) 2 s + = 0 + p(u)α 1 [exp 1 = α 1 p(u) (1 + T u ) 2, 0 ( Tu+τ Tu+τ T u α 1 T u α 1 (1 + s) 2 s (1 + s) 2 s )] [ τ=0+ )] τ τ=0+ Tu+τ T u 1 (1 + s) 2 s ] τ=0+ (D α2 D α1 p)(u) = 1 = D α2 (α 1 p(u) (1 + T u ) 2 ) = α 1 1 (1 + T u ) 2 D 1 α 2 p(u) + α 1 p(u)d α2 (1 + T u ) [ ] 2 α 1 α 2 = (1 + T u ) 4 p(u) + α 1 1p(u) τ (1 + T u + τ) 2 = α 1α 2 p(u) (1 + T u ) 4 + 2α 1p(u) (1 + T u ) 3. We can compute the erivations (D αi... D α1 p)(u) for any i N, α j R, j 1,..., i. If we efine ϕ 1 (u) = p(u) an ϕ 2 (u) = 1 + T u then for any i N an α j R, j 1,..., i it hols that (D αi... D α1 p)(u) R(ϕ 1 (u), ϕ 2 (u)). Therefore, by Definition 5.9, Q obs (p) R(ϕ 1, ϕ 2 ) an consequently Q obs (p) is finitely generate. Hence, accoring to Theorem 5.16, there exists a rational system realizing p. We construct a rational system Σ = (X, f, h, x 0 ) realizing p by following the proof of Proposition Let us consier the fiel F = R(ϕ 1, ϕ 2 ). It is finitely generate, contains Q obs (p), τ=0+

12 REALIZATION THEORY FOR RATIONAL SYSTEMS 12 an is close with respect to D α erivations. The number of generators of F equals 2 which implies that the state-space X can be taken as R 2. To etermine a family of rational vector fiels f = {f α α R} we compute v α 1 (ϕ 1, ϕ 2 ) = D α ϕ 1 = D α p = αϕ 1 1 ϕ 2 2 an v α 2 (ϕ 1, ϕ 2 ) = D α ϕ 2 = 1. The last equality D α ϕ 2 = 1 hols because D α ϕ 2 (u) = [ τ ϕ 2((u)(α, τ)) ] τ=0+ = [ τ (1 + T u+ τ)] τ=0+ = [1] τ=0+ = 1 for any u Ũ pc. The output map h is etermine by a map w such that w(ϕ 1, ϕ 2 ) = p = ϕ 1, an the initial point x 0 is given as x 0 = (ϕ 1 (e), ϕ 2 (e)) where e is the empty input. Finally, the rational realization Σ = (X, f, h, x 0 ) of p is given as X = R 2, f α (x 1, x 2 ) = v1 α (x 1, x 2 ) + v2 α (x 1, x 2 ) = α x 1 x 1 x 2 x 2 2 h(x 1, x 2 ) = w(x 1, x 2 ) = x 1, x 0 = (ϕ 1 (e), ϕ 2 (e)) = (1, 1). 6 Canonical rational realizations x 1 + x 2, α R, The systems which we call canonical are calle minimal by Bartosiewicz in [4]. Therefore [4, Theorem 3] escribing the conitions uner which a map has a minimal polynomial realization correspons to our Theorem 6.1 consiering rational systems. The proofs are analogous. Theorem 6.1 Let p : Ũ pc R r be a response map. The following statements are equivalent: (i) p has an rationally observable rational realization, (ii) p has a canonical rational realization, (iii) Q obs (p) is finitely generate. Proof: (i) (ii) Let p : Ũ pc R r be a response map. Consier a rational system Σ = (X, f, h, x 0 ) realizing p which is rationally observable. Then Σ is canonical if it is algebraically reachable from the initial state x 0, i.e. if Z-cl(R(x 0 )) = X. Let us enote X = Z-cl(R(x 0 )). From Proposition 5.5, the variety X is an irreucible real affine variety. If X = X, then the proof is complete. Let us assume that X X. Because X is an irreucible real affine variety, we can consier X to be a state-space of a rational system Σ. Let I be an ieal of polynomials of A which vanish on X. Then the quotient ring A/I can be ientifie with A, the algebra of polynomials on X. We enote the corresponing bijection by Ψ : A/I A. This is an one-to-one an onto mapping which preserves sums an proucts. If we consier ϕ A an ϕ A such that Ψ([ϕ]) = ϕ, it hols that ϕ X = ϕ. More etails can be foun in [7, Chapter 5.2]. There exist polynomials ϕ 1,..., ϕ k from A such that A = R[ϕ 1,..., ϕ k ]. Moreover, A is an integral omain an we can efine the fiel Q of rational functions on X by Q = R(ϕ 1,..., ϕ k ). We efine the rational system Σ = (X, f, h, x 0) by consiering the state-space X an by eriving the family f of rational vector fiels, an output function h an an initial state x 0 from the rational system Σ. As the initial state x 0 of Σ we consier x 0 = x 0 Z-cl(R(x 0 )) = X. The output function h of Σ is efine as ( ) h h = (h 1,..., h 1,num r) =,..., h r,num h 1,en h r,en ( Ψ([h1,num ]) = Ψ([h 1,en ]),..., Ψ([h ) r,num]) Ψ([h r,en ]) = (h 1 X,..., h r X ) = h X, ( h1,num X = h 1,en X,..., h ) r,num X h r,en X where h i,num, h i,en A an h i,num, h i,en A for i = 1,..., r are such that h i = h i,num h an i,en h i = hi,num h i,en. The output function h is efine on a Z-ense subset of X because X is irreucible

13 REALIZATION THEORY FOR RATIONAL SYSTEMS 13 an because h i,en (x 0 ) 0 for i = 1,..., r an thus h i,en / I for i = 1,..., r. We efine the relation between the rational vector fiels f = {f α : Q Q α U} an the rational vector fiels f = {f α : Q Q α U} as f α Ψ([q num ]) Ψ([q en ]) = Ψ([(f αq) num ]) Ψ([(f α q) en ]) for q = qnum q en Q where q num, q en A an q en / I. Note that we assume that (f α q) num, (f α q) en A are such that f α q = (fαq)num (f αq) en for a consiere q Q. Since for ϕ A such that Ψ([ϕ]) = ϕ it is true that ϕ X = ϕ, it follows that for q = qnum Q we have that f α Ψ([q num]) Ψ([q en ]) = (fαq)num X (f αq) en = (f X α q) X, an finally if q = Ψ([qnum]) Ψ([q en ]), q en / I, f αq = (f α q) X. The well-efineness of the rational vector fiels f α, q α U follows from the fact that (f num α q en ) X is inepenent on the choice of representants q num, q en A of classes [q num ], [q en ] A/I. To verify this we consier arbitrary ϕ 1, ϕ 2 I. Thus ϕ 1 an ϕ 2 are ientically zero on X an therefore (q num + ϕ 1 ) X = q num an (q en + ϕ 2 ) X = q en. Moreover as ϕ 1 = 0 on X, ϕ 1 is ientically zero on the trajectories of the system Σ an therefore ϕ 1 τ = 0 on Ũ pc. Then also D α (ϕ 1 τ) = 0 on Ũ pc an by Lemma 5.10, (f α ϕ 1 ) τ = 0 on Ũ pc. Therefore f α ϕ 1 = 0 on X ( an hence )(f α ϕ 1 ) ( X = 0. Analogically, (f α ϕ 2 ) X = 0. Finally, ) as f α q is an R-linear map, f num+ϕ 1 α q en +ϕ 2 X = (fαq num+f αϕ 1)(q en +ϕ 2) (f αq en +f αϕ 2)(q num+ϕ 1) (q en +ϕ 2) 2 X = ( ) ( ) (fαq num)q en (f αq en )q num q X = f num α q en X. q 2 en Consier a trajectory x of Σ from x 0 etermine by the input u = (α, T u ) Ũ pc, an a trajectory x of Σ from x 0 etermine by the input u = (α, T u ) Ũ pc. We assume without loss of generality that T u = T u. Then t (ϕ x )(t; x 0, u ) = (f αϕ )(x (t; x 0, u )), ϕ (x (0)) = x 0 for t [0, T u ], ϕ A, an t (ϕ x)(t; x 0, u) = (f α ϕ)(x(t; x 0, u)), ϕ(x(0)) = x 0 for t [0, T u ], ϕ A. Since X = Z-cl(R(x 0 )), both trajectories x an x stay in X. Let ϕ A an let ϕ A be such that ϕ = Ψ([ϕ]). Because f αψ([ϕ]) = (f α ϕ) X, x 0 = x 0, an because ϕ = ϕ on X, it follows that (f α ϕ) X (x (t; x 0, u )) = (f α ϕ)(x (t; x 0, u )) = t (ϕ x )(t; x 0, u ), an ϕ(x (0)) = x 0. Therefore, by Theorem 2.3, the trajectories of the systems Σ an Σ are the same. Then also the reachable sets of both systems coincie an thus X = Z-cl(R(x 0 )) = Z-cl(R (x 0)). So, Σ is algebraically reachable. From the equality of trajectories of Σ an Σ an from the efinition of the output function h of Σ, it follows that the system Σ is a rational realization of p. More precisely, let u Ũ pc, then p(u) = h(x(t u ; x 0, u)), because Σ realizes p, = (h X )(x(t u ; x 0, u)), because Z-cl(R(x 0 )) = X, = (h X )(x (T u ; x 0, u)), from the equalities of the trajectories of Σ an Σ, = h (x (T u ; x 0, u)), by the efinition of h. Thus, the system Σ is an algebraically reachable rational realization of p. We prove that Σ is also rationally observable. Let us compute the observation fiel Q obs (Σ ) of the system Σ. Firstly, the observation algebra A obs (Σ ) is the smallest algebra containing the elements h i, f αh i for i = 1,..., r an α = (α 1,..., α k ) such that k N, α j U, j = 1,... k. As h i = h i X an f αh i = f α h i X for i = 1,..., k an α = (α 1,..., α k ), α j U, k N, we get that ϕ = ϕnum ϕ en A obs (Σ), where ϕ num, ϕ en A, if an only if ϕ = Ψ([ϕnum]) Ψ([ϕ en ]) A obs (Σ ). Furthermore, since A obs (Σ) an A obs (Σ ) are integral omains, ϕ = ϕnum ϕ en Q obs (Σ), ϕ en / I if an Q obs (Σ ). Because the system Σ is rationally observable an because Q(Ψ(A/I)) = Q(A ), we erive that the system Σ is also rationally observable. (ii) (iii) We assume that p has a canonical rational realization Σ = (X, f, h, x 0 ). From Theorem 5.16, Q obs (p) is finitely generate. (iii) (i) We assume that the observation fiel Q obs (p) of p is finitely generate. From Proposition 5.13, Q obs (p) is finitely generate by the elements of A(Ũ pc R). Let Q obs (p) = R(ϕ 1,..., ϕ k ) where ϕ i A(Ũ pc R), i = 1,..., k. By the efinition of observation fiel, Q obs (p) is close with respect to D α erivations α = (α 1,..., α i ), i N, α j U, j = 1,..., i. To prove that there exists a only if ϕ = Ψ([ϕnum]) Ψ([ϕ en ]) q en

14 REALIZATION THEORY FOR RATIONAL SYSTEMS 14 rationally observable rational realization of p, we construct a rational realization Σ = (X, f, h, x 0 ) of p such that Q obs (Σ) = Q where Q enotes the fiel of rational functions on X. The fiel F = Q obs (p) = R(ϕ 1,..., ϕ k ) fulfills the conitions (i)-(iii) of Proposition By following the proof of Proposition 5.14, with F = Q obs (p), we construct a rational realization Σ = (X, f, h, x 0 ) of p as This realization is such that X = R k, f α = k vi α (X 1,..., X k ), α U, X i i=1 h j (X 1,..., X k ) = w j (X 1,..., X k ), j = 1... r, x 0 = (ϕ 1 (e),..., ϕ k (e)). p j = w j (ϕ 1,..., ϕ k ), j = 1... r, an D α ϕ i = v α i (ϕ 1,..., ϕ k ), i = 1... k. Because X = R k, we get for the fiel Q of rational functions on X that Q = R(X 1,..., X k ). To consier h j an f α h j, j = 1... r is the same as to consier p j an D α p j but in ifferent coorinates. Therefore, Q obs (Σ), as a fiel of quotients of the smallest subalgebra of Q containing all h j, f α h j where j = 1,..., r, α = (α 1,..., α k ), k N, α i U, i = 1,..., k, equals R(X 1,..., X k ) in analogy to the relation Q obs (p) = R(ϕ 1,..., ϕ k ). That means that Q obs (Σ) = R(X 1,..., X k ) = Q which proves the rational observability of Σ. Corollary 6.2 Let p be a response map. Accoring to Theorem 5.16 an Theorem 6.1 the following statements are equivalent: (i) p is realizable by a rational system, (ii) p has a rational realization which is rationally observable, (iii) p has a rational realization which is canonical, thus both rationally observable an algebraically reachable. 7 Concluing remarks We have provie sufficient an necessary conitions for a response map to be realizable by a rational system. Since the proof of Proposition 5.14 concerning sufficient conitions for the existence of a rational realization for a given response map is constructive, it provies an algorithm for constructing a rational system realizing a response map. We prove that for a given response map the problems of the existence of a rational/rationally observable rational/canonical rational realization are equivalent. The proof of Theorem 6.1 stating this equivalence is also constructive. It provies an algorithm for the construction of a canonical rational realization from a rationally observable rational realization. Moreover it shows that if we consier as a fiel F from Proposition 5.14 the observation fiel of a response map to be realize, then the algorithm for constructing a rational realization from Proposition 5.14 gives as a result a rationally observable rational realization. Aitional results which we got by stuying minimal rational realization can be foun in [15]. For some remarks on algebraic reachability of rational systems see [14]. We work with irreucible real affine varieties for two reasons. Firstly, irreucibility simplifies the technical etails of the proofs. Seconly, working with the real varieties allows us to have a better geometric unerstaning of the state-spaces of rational systems an it is also sufficient for applications for example in systems biology. It is possible to generalize the results of this paper for reucible varieties. Algebraic framework we use can be useful from computational point of view. The proceures to check the properties of rational systems such as rational observability an algebraic reachability, an minimality of rational realizations are formulate in [15]. They can be implemente by using alreay existing computer algebra packages. The proceures for constructing rational realizations of esire properties are still to be evelope. Further research is require on the realization theory for rational positive systems. There are several issues to be overcome in that research.

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