On the Genericity of Algebraically Observable Ponlynomial Systems: the Discrete-Time Case

On the Genericity of Algebraically Observable Ponlynomial Systems: The Discrete-Time Case

Uwe Helmke Department of Mathematics University of Wurzburg¨ 97074 Wurzburg,¨ Germany [email protected] Jonathan H. Manton Department of Electrical and Electronic Engineering The University of Melbourne Victoria 3010 Australia [email protected]

May 21, 2004

Abstract A polynomial system is algebraically observable if the initial state can be expressed as a polynomial function of (a ﬁnite number of) observations. Algebraically observable systems are of interest, as polynomial high gain observers can be constructed for them. We provide insight into why algebraic observability is not a generic property for discrete-time polynomial systems and present an embedding result for polynomial maps. We also formulate a conjecture that algebraic observability is generic within a subclass of polynomial systems whose state map has a polynomial inverse.

Keywords: Polynomial systems, observability, embeddings, genericity.

1 Introduction

Consider the uncontrolled polynomial system

x(t+1) = f(x(t)), y(t) = h(x(t)) (1.1)

n n p on K , where f ∈ K [x1, . . . , xn], h ∈ K [x1, . . . , xn] are polynomials, and K denotes the ﬁeld of real or complex numbers. For such systems, various concepts of observability have

1 been proposed in the literature. The weakest concept is (set-theoretic) observability; the system (1.1) is observable if the state vector x(0) is uniquely determined from knowledge (t) ∞ of the complete observation sequence {y }t=0. In fact, it follows from the Hilbert Basis Theorem that (1.1) is observable if and only if there exists a ﬁnite integer N ≥ 1 such that for all x, y ∈ Kn,

h(f i(x)) = h(f i(y)) for i = 0, · · · , N − 1 ⇒ x = y where, by abuse of notation, f i denotes composition of f with itself i times. (t) N−1 Even if (1.1) is observable, the mapping from the observation sequence {y }t=0 to the state x(0) need not have nice properties, such as being extendible to a continuous function defined on the whole observation space Kp×N . In fact, if a polynomial map is injective, its inverse is always expressible locally as a rational function. Since rational functions are not defined everywhere unless they are in fact polynomials, another attractive concept of observability, introduced by Sontag [12], is the following. The polynomial system (1.1) is algebraically observable if there exists a finite integer N ≥ 1 n and a polynomial π ∈ K[y0, · · · , yN−1] such that for all x ∈ K

x = π h(x), h(f(x)), . . . , h(f N−1(x)) ¡ ¢ holds, that is, the initial state is expressible as a polynomial function of the N observations. Since f and h are polynomial maps this definition is equivalent to the property that the map (0) (t) N−1 from the initial state x to the finite observation sequence {y }t=0 is a proper polynomial embedding for some N. In the continuous-time case, generic observability of uncontrolled systems with f and h being smooth or real analytic vectorfields and maps, respectively, has been studied in [1, 2, 6, 14]; see also Sontag [13] for a new approach, applicable to specific system parametrizations. In fact, these authors proved genericity of a stronger version than observability, i.e. that the map from the state vectors x(0) to the finite observation sequence defined by the Lie derivatives is a proper embedding for N > 2n. It shoul be noted that generic observability results for f and h belonging to the category of smooth or real analytic vectorfields/maps do not imply analogous results over smaller categories, such as the category of polynomial vectorfields/maps. Thus the results we are interested in do not follow from earlier results for smooth or real analytic systems. The situation is even more complicated in the discrete- time case as this cannot be reduced in any reasonable sense to the continuous-time case. In fact, the discrete-time case is considerably harder to analyze than the continuous-time case and has been studied to a less extend. Even more so, and in sharp contrast to continuous- time systems, genericity of observability for systems (1.1) does not hold without imposing further, restrictive assumptions. This new difficulty is due to the fact, that f is generically not injective (while the flow map of a vector field is!). In order to verify algebraic observability for a polynomial system, one needs to check whether an associated polynomial map has a polynomial left inverse. Our first main result in this direction is positive one and asserts that a generic polynomial map g : Cn → Cm with m ≥ 2n + 1 has a polynomial inverse. Unfortunately that does not imply genericicity of algebraic observability, as we show by simple counterexamples. Thus additional assumptions are

2 needed in order to guarantee generic observability. We conjecture, that a system (1.1), drawn generically from a subclass of all such systems with the state map f having a polynomial inverse, is algebraically observable.

2 Preliminaries

In this section we recall some basic deﬁnitions and facts from complex algebraic geometry. See e.g. the books by Mumford [11] and Hartshorne [7] for further details. Let C denote the ﬁeld of complex numbers. A complex algebraic subvariety V ⊂ Cn is the zero set n V = {z ∈ C | p1(z) = · · · = pr(z) = 0}

of a finite number of polynomials p1, · · · , pr ∈ C[z1, · · · , zn]. V is also called a Zariski– closed subset of Cn and its complement U = Cn − V a Zariski–open subset. In particular, nonempty Zariski–open subsets of Cn are open and dense with respect to the standard Euclidean topology of Cn. A constructible subset X ⊂ Cn is a Boolean combination of a finite number of Zariski–closed subsets. More explicitly, a subset of Cn is locally closed if it is the intersection of a Zariski–open and a Zariski–closed subset. Any union of finitely many locally closed subsets is a constructible set. The class of constructible subsets has nice properties. The union, the intersection and the Zariski–closure of finitely many constructible subsets is constructible. Moreover, the complement V − W of two constructible subsets is n m again constructible. If f : C → C is a polynomial map, i.e. if f = (f1, · · · , fm) with n f1, · · · , fm ∈ C[z1, · · · , zn] polynomials, then f maps constructible subsets X ⊂ C onto constructible subsets Y = f(X) ⊂ Cm. Moreover, for the Zariski–closure f(X) in Cm

dim(f(X)) ≤ dim(X)

holds for all constructible subsets X ⊂ Cn. In particular,

dim(X) = dim(X)

for each constructible subset X ⊂ Cn and corresponding Zariski–closure X. Finally, a constructible subset X ⊂ Cn is Zariski–closed if and only if X is closed in the Euclidean topology of Cn. Complex polynomial maps are special holomorphic functions and one can therefore apply differential geometric techniques to study such maps. The following definition is well–known from differential geometry.

Deﬁnition 2.1. A polynomial map f : Cn → Cm is called a proper embedding if and only if the following conditions hold:

(i) f is injective

(ii) f is an immersion, i.e. the derivative df(z) : Cn → Cm is an injective linear map, for all z ∈ Cn.

3 (iii) f : Cn → Cm is proper, i.e. the preimage f −1(K) is compact for each compact subset K ⊂ Cm. The above conditions on f imply that the image f(Cn) is a closed complex submanifold of Cm. Even more is true as the following characterization shows; see [7]. Proposition 2.2. A polynomial map f : Cn → Cm is a proper embedding if and only if f(Cn) is a Zariski–closed subset of Cm and f maps Cn algebraically isomorphic to f(Cn). Equivalently, f(Cn) is Zariski–closed and there exists a polynomial map π : Cm → Cn with n π(f(z)) = z, ∀z ∈ C . (2.1) The above result allows for a purely algebraic reformulation, which leads to the standard algebraic definition of a (proper) embedding. Proposition 2.3. A polynomial map f : Cn → Cm is a proper embedding if and only if: ∗ f (C[y1, · · · , ym]) = C[x1, · · · , xn], (2.2) where f ∗(C[y]) = {p(f(x)) ∈ C[x] | p ∈ C[y]}. (2.3) If m is sufficiently large, then there is a beautiful and more explicit recent characterization of complex polynomial proper embeddings due to Z. Jelonek that shows that they are unknotted maps. See [10] for a proof of the next result. Theorem 2.4. Let m ≥ 2n. A complex polynomial map f : Cn → Cm is a proper embedding m m if and only if there exists a polynomial automorphism F : C → C with F |Cn×{0} = f. Thus in these dimensions, proper embeddings always extend to polynomial automorphisms on Cm. The crucial condition for a complex polynomial map to qualify as an algebraic embedding is the properness. It is usually hard to verify. For the sake of completeness we state a well–known algebraic characterization. Proposition 2.5. A polynomial map f : Cn → Cm is proper if and only if it is a finite morphism, i.e. if and only if the polynomial ring C[x1, · · · , xn] is an integral ring extension over C[y1, · · · , ym]. Recall that a ring extension C[x] of f ∗(C[y]) ⊂ C[x] is called integral, if every element p ∈ C[x] satisfies a monic equation m m−1 p + am−1p + · · · + a0 = 0 (2.4) ∗ with coefficients ai ∈ f (C[y]). Equivalently, as the integral elements of C[x] with respect to f ∗(C[y]) forming a subring of C[x], we conclude: n m Corollary 2.6. A polynomial map f : C → C is proper if and only if every component xi satisfies a monic polynomial equation

m (i) mi−1 xi + ami−1(f(x))xi + · · · + a0(f(x)) = 0 (2.5) n (i) for all i = 1, · · · , n, x ∈ C , aj ∈ C[y].

Thus the properness of f is equivalent to the state variables x1, · · · , xn being monic algebraic functions of y1, · · · , ym = f1(x), · · · , fm(x).

4 3 Generic Polynomial Maps

Bézout’s Theorem implies that if a polynomial map g : Cn → Cn has randomly chosen coefficients then with high probability g(x) = 0 has a finite number of solutions. Precisely, n n given d = (d1, · · · , dn) ∈ N , let P(n, d) = {g ∈ C [x1, · · · , xn] | deg gi ≤ di, i = 1, · · · , n} denote the finite dimensional vector space of all polynomial maps g = (g1, · · · , gn) whose components gi have degrees bounded by di. Then Bézout’s Theorem asserts the existence of a nonempty Zariski-open subset Ω ⊂ P(n, d) such that g ∈ Ω implies g(x) = 0 has precisely n i=1 di distinct solutions. QOne may hope that adding an extra equation will result in a g : Cn → Cn+1 which is invertible. Indeed, it is well known that a generic g will have a rational inverse, meaning there exists a rational function r : Cn+1 → Cn such that r(g(x)) = x whenever x ∈ Cn is such that r(g(x)) is defined (that is, g(x) is not a pole of r). Here, we state then sketch the proof of a stronger result.

1 Proposition 3.1. Let g1, · · · , gn ∈ C[x1, · · · , xn] be such that the polynomial map g = n n n (g1, · · · , gn) : C → C has a ﬁnite number of solutions to g(x) = c for generic c ∈ C . Let n n+k h1, · · · , hk ∈ C[x1, · · · , xn] be such that the map h = (g1, · · · , gn, h1, · · · , hk) : C → C has a polynomial inverse. Then there exists a polynomial e which is not identically zero and such that e(α1, · · · , αk) =6 0 implies the polynomial map f = (g1, · · · , gn, α1h1 + · · · + αkhk) : Cn → Cn+1 has a rational inverse.

Proof. Let R = C[x1, · · · , xn] denote the ring of polynomials and S = C[g1, · · · , gn] a subring of R. Define h = α1h1 + · · · + αkhk and note that f has a rational inverse if and only if 0 0 (S[h]) = R , where the prime denotes the field of fractions. Since the condition on the gi ensures R0 is an algebraic extension of S0, it follows that (S[h])0 = S0[h]. By assumption, 0 0 R = S[h1, · · · , hk], hence R = S [h1, · · · , hk] for the same reason. Kronecker’s “method of indeterminates” can be used to show the existence of a non-zero polynomial e such that 0 0 e(α1, · · · , αk) =6 0 implies h is a primitive element of the field extension R over S (i.e. h satisfies R0 = S0[h]) — see the proof of the “primitive element theorem” in Zariski and Samuel’s classic book on Commutative Algebra. This completes the proof.

We now consider how many equations are required before a generic g : Cn → Cm has a polynomial inverse. If m = n + 1 then generically g has a rational inverse. If m = n + 2 there are thus two different rational inverses, one using only the first n + 1 equations, the other using the first n equations and the last equation. Precisely, if g is chosen generically, there will exist rational functions ri(y) depending only on the components y1, · · · , yn, yn+i of n y, such that ri(g(x)) = x for i = 1, · · · , m − n and all x ∈ C satisfying hi(g(x)) =6 0, where hi is the denominator of ri. m n m Consider using the ri to construct a function r : C → C defined everywhere on C and n m such that r(g(x)) = x for all x ∈ C . Given a y ∈ C , if there exists an i such that hi(y) =6 0 then locally we can define r(y) by r(y) = ri(y). This strategy fails if y ∈ V , where V is the m variety V = {y ∈ C | h1(y) = · · · = hm−n(y) = 0}. Intuitively then, we want m to be large 1This is true if and only if the Jacobian of g is not identically zero. Note too that by Bézout’s Theorem, this condition is true for generic g.

5 enough so that V is empty. Recall that hi is a function of only y1, · · · , yn, yn+i. Therefore, the set of points (y1, · · · , yn) such that there exists a yn+i satisfying hi(y) = 0 is expected to have codimension one. Hence, if m = 2n, we expect V to contain a ﬁnite number of points, and if m = 2n + 1, we expect V to be empty. Moreover, if V is empty, it is a standard result in algebraic geometry that the ri do indeed “patch together” to form a function r which is a polynomial inverse of g. This leads to the following result, which is proved in [8] using a rigorous but less insightful argument than the one just presented.

m Theorem 3.2. Let P(n, m, d) = {g ∈ C [x1, · · · , xn] | deg gi ≤ di, i = 1, · · · , m} denote the vector space of polynomial maps g : Cn → Cm of bounded degree d ∈ Nm, with each element of d strictly positive and n ≥ 1. If m ≥ 2n + 1 then there exists a non-empty Zariski-open subset Ω ⊂ P(n, m, d) such that every g ∈ Ω has a polynomial inverse. The proof runs by showing that a generic g is 1) injective, 2) an immersion, and 3) proper. By Proposition 2.2, such a g has a polynomial inverse, hence the result follows. We believe that the bound 2n + 1 is the best possible one. Conjecture 3.3. If m ≤ 2n then a generic g : Cn → Cm of bounded degree d ∈ Nm does not have a polynomial inverse, provided the degrees di are sufficiently large. For n = 1 it is easy to construct polynomial maps g : C1 → C2 that define a proper immersion with simple normal crossings. Such maps are robustly non-injective in the sense, that any sufficiently close polynomial map is also non-injective. We give two such examples in the next section. It seems plausible that the consstruction can be extended to higher dimensions to provide a proof of the conjecture.

4 Impediments to Algebraic Observability

Given an uncontrolled polynomial system (1.1) and an integer N, deﬁne the observation map ΦN (f, h) : Cn → CpN taking x(0) to (y(0), · · · , y(N−1)). Precisely, ΦN (f, h)(x) = h(x), h(f(x)), · · · , h(f N−1(x)) . Since ΦN has a very special structure, the genericity results¡ of the previous section are¢ not applicable. In fact, this section explains why it is not true that a generically chosen f and h will result in an observation map ΦN (f, h) having a polynomial inverse. By B´ezout’s Theorem, if f : Cn → Cn is a generic non-linear polynomial, the equation f(x) = c will have more than one solution for generic c. Intuitively, this means the state map taking x to f(x) discards information; if f(x) = x2 then both 2 and −2 get mapped to 4, so no subsequent observation h(f i(x)) for i ≥ 1 can distinguish between x = 2 or x = −2. Continuing with the example f(x) = x2, if h(x) is any polynomial such that h(z)−h(−z) = 0 has a non-zero solution z, then ΦN (f, h)(z) = ΦN (f, h)(−z) for all N ≥ 1 by the above discussion, that is, the system (f, h) is not algebraically observable. Moreover, if h(x) is a generically chosen polynomial of degree 3 or more, then a simple calculation shows h(x) − h(−x) is a generically chosen polynomial of degree 3 or more but with no even monomials in it. (For example, if h(x) = x3 + 2x2 + 3x + 4 then h(x) − h(−x) = 2x3 + 6x.) Hence, a generic h will have a non-zero solution to h(x) − h(−x) = 0, proving that (f, h) is not algebraically observable for generic h of degree 3 or higher and where f(x) = x2.

6 The choice f(x) = x2 is not an “unrepresentative” choice. Indeed, consider the system f(x) = x3 + x2 − x − 1 and h(x) = x3 − x. Define the map F (x) = (f(x), h(x)). The image F (C) has a simple self-intersection point at (0, 0). Thus, small perturbations of f and h will also result in an F (C) having a simple self-intersection point. Thus, there exists an open set of systems (f, h) for which ΦN (f, h) is not injective. Although we have not attempted a proof, we claim that such examples can be found in higher dimensions, hence it cannot be expected that a generic system (f, h) is algebraically observable. A second and more subtle impediment to (f, h) being algebraically observable is if there exists an x such that f i(x) = x for some i ∈ {1, · · · , N − 1}. For such initial conditions, the state sequence x(t) repeats itself. This can cause two problems. The first is in verifying that ΦN (f, h) is an immersion. Indeed, by the chain rule, the differential of y(t) = h(f t(x(0))) = (t) (0) t (t) h(x ) with respect to x is the linear operator Dh|x(t) ◦ Df |x(0) . Thus, if x repeats itself too quickly, the sum of the range spaces of the Dh|x(t) may not span the whole space, preventing ΦN (f, h) from being an immersion. The second problem arises if there exists an infinite number of periodic points. The extreme case is if f is the identity map. In this case, ΦN (f, h)(x) = (h(x), · · · , h(x)), hence ΦN (f, h) provides no new information for N > 1. That is, (f, h) is algebraically observable if and only if h has a polynomial inverse, a very uninteresting case. In general, assume that for a given i < N, the variety V = {x | f i(x) − x = 0} contains an infinite number of points. Then, assuming the initial state lies in V , since the observations will repeat themselves after at most the ith one, rather than being able to use all N observations to determine the initial state, only the first i observations can be used. Precisely, if x ∈ V then {z | ΦN (f, h)(z) = ΦN (f, h)(x)} ⊃ {z ∈ V | Φi(f, h)(z) = Φi(f, h)(x)}, so Φi(f, h) restricted to V must be injective if ΦN (f, h) is to have any chance of being injective. Note though that this second problem is a non-generic one; if f is chosen generically then f(x) − x = 0 will have only a finite number of solutions by Bézout’s Theorem. While we have not attempted a proof, we expect that for a fixed i and generic f, f i(x) − x = 0 has but a finite number of solutions too.

Conjecture 4.1. Algebraic observability is a generic property for the subclass of polynomial systems (f, h) of bounded degrees, defined by the constraint that f is a polynomial automorphism of Cn that has only finitely many periodic points of period less than or equal 2n + 1. We note that the class of polynomial automorphisms is thin within the set of all polynomial vectorfields (of the same degrees). In fact, the celebrated Jacobian conjecture states that polynomial automorphisms are characterized by having a constant nonzero Jacobian. The above conjecture is of interest even in the special case where f(x) = Ax is assumed to be linear. For example, it is known that for any invertible matrix A there exists a polynomial observation function h, such that (f, h) is observable [9]. A corresponding result for algebraic observability, possibly under additional generic assumptions on A, seems to be unknown.

7 References

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