Analytic systems on an arbitrary time-scale are studied. As particular cases they include continuous-time and discrete-time systems. Several local observability properties are considered. They are characterized in a unified way using the language of real analytic geometry, ideals of germs of analytic functions, and their real radicals. It is shown that some properties related to observability are preserved under various discretizations of continuous-time systems.

1. Introduction

Local observability for nonlinear systems is defined in various ways [1–4]. As most of these concepts are introduced in the general system-theoretic setting via indistinguishability relation, they mean the same for continuous- and discrete-time systems. We show here that some of these concepts may be studied in a unified way in the framework of systems on time-scales. A time-scale is a model of time, which can be continuous, discrete, or even mixed. Calculus on time-scale is a unification of ordinary differential calculus and the calculus of finite differences. Delta differential equations may be used to model continuous- and discrete-time systems. In the discrete case, time-scale may be nonhomogeneous. It may be applied to systems that are obtained by nonuniform sampling or nonuniform Euler discretization of continuous-time systems.

We concentrate on strong, weak, and robust local observability of analytic systems. We show that the results obtained in [2, 3, 5] may be extended to systems on arbitrary time-scales. This is due to the fact that all the properties are characterized with the aid of the observation algebra of the system, which may be introduced in a universal way on all time-scales. Since the observation algebra consists of real functions defined on the state space, we can use the common procedures to derive the criteria of local observability. As in [2, 3] we use the language of local analytic geometry and real algebra to characterize weak and robust local observability. Ideals of germs of analytic functions and real radicals of these ideals are used to express the criteria.

As an application of time-scale approach to local observability we consider discretization of continuous-time systems. This means replacing the standard derivative by the delta derivative on an appropriate discrete time-scale. We show that some of the properties related to observability are preserved under this operation. We will allow arbitrary discretizations: the discrete time-scale will not have to be homogeneous. Such nonuniform discretizations behave in a better way in many computations.

In Appendices we provide necessary information on time-scale calculus, local analytic geometry, and real algebra.

2. Preliminaries

Let 𝕋 be a time-scale. We will assume that 𝕋 is forward infinite; that is, for every t₀ ∈ 𝕋 there are infinitely many elements of 𝕋 that are greater than t₀. This will allow us to compute delta derivatives of arbitrary order at t₀. Let us consider a control system with output

()

where t ∈ 𝕋, x(t) ∈ ℝⁿ, y(t) ∈ ℝ^r, and u(t) ∈ Ω—arbitrary set. For ω ∈ Ω, let f_ω be defined by f_ω(x): = f(x, ω). We assume that the maps h and f_ω for every ω ∈ Ω are analytic and that controls u are piecewise constant functions of time.

If 𝕋 = ℝ, then (1) is the standard continuous-time system

()

For 𝕋 = ℤ (1) takes the form

()

This can be rewritten in a more standard shift form as

()

As there is a simple passage from f to g and vice versa, all statements for (3) may be translated to statements for (4).

Remark 1. The equation x(t + 1) = g(x(t), u(t)) may be studied on an arbitrary set X or on an analytic manifold M, if analyticity of the system is essential. But then we cannot pass to form (3), as to do this we need a linear space structure. Thus, one can argue that (4) is more general than (3). However, we concentrate here on local analytic problems, for which ℝⁿ is general enough.

By γ(t, t₀, x₀, u) we denote the solution of the equation x^Δ = f(x, u) corresponding to control u and the initial condition x(t₀) = x₀ and evaluated at time t.

Let x₁, x₂ ∈ ℝⁿ and t₀ ∈ 𝕋. Then x₁ and x₂ are called indistinguishable at time t₀ if

()

for every control u defined on [t₀, t₁) _𝕋 for some t₁ ∈ 𝕋, and every t ∈ [t₀, t₁] _𝕋 for which both sides of the equation are defined. Otherwise x₁ and x₂ are distinguishable at time t₀.

The states x₁ and x₂ are called indistinguishable if they are indistinguishable at time t₀ for every t₀ ∈ 𝕋. Otherwise x₁ and x₂ are distinguishable. Thus, x₁ and x₂ are distinguishable if they are distinguishable at some time t₀.

Remark 2. If the time-scale is not homogeneous, indistinguishability at time t₀ may depend on t₀. Though the systems we consider have “constant coefficients,” that is, the map f does not depend on time, inhomogeneity of the time-scale results in the behavior found in time-variant systems. This may be observed even for linear systems (see [6]).

Let C^ω(ℝⁿ) denote the algebra of all real analytic functions on ℝⁿ and let g : ℝⁿ → ℝⁿ be analytic. Let us fix t₀ ∈ ℝⁿ. In [7] the following operator

()

was introduced as

()

where φ ∈ C^ω(ℝⁿ) and φ^′ is the gradient of φ (a row vector).

Example 3. Let φ = xⁱ be the ith coordinate function on ℝⁿ. Then φ^′ = e_i—the (row) vector of the standard basis of ℝⁿ with 1 at the i-th position. For any t₀ ∈ 𝕋 we have

()

Observe that if μ(t₀) > 0 then

()

On the other hand for μ(t₀) = 0 we obtain

, which is the standard Lie derivative of the function φ along the vector field g. In general, when operator

does not depend on t₀, we will denote it by Γ_g. This happens, for instance, if the time-scale is homogeneous.

Let be another map related to the function g and the time t₀ defined by

()

If μ(t₀) = 0, then

is the identity map. In general we have an obvious property

Proposition 4. The map is an endomorphism of the algebra C^ω(ℝⁿ).

For μ(t₀) > 0 the operators

and

are related by the following equality:

()

We also have the following generalization of the Leibniz rule.

Proposition 5.

()

This property means that is a skew derivation of the algebra C^ω(ℝⁿ) with respect to or, in other words, that is a -derivation of this algebra. For μ(t₀) = 0 it is an ordinary derivation.

3. Local Observability

Let

be a subset of C^ω(ℝⁿ) consisting of functions of the form

()

where i = 1, …, p, k ≥ 0 and

for some ω_i ∈ Ω and i = 1, …, k. If k = 0 then this function is just h_i. Let

()

Proposition 6. (i) The states x₁ and x₂ are indistinguishable at time t₀ if and only if for every , φ(x₁) = φ(x₂).

(ii) The states x₁ and x₂ are indistinguishable if and only if for every φ ∈ H, φ(x₁) = φ(x₂).

Proof. The statement (ii) was shown in [1] for continuous-time systems (𝕋 = ℝ). In this case (i) and (ii) mean the same since indistinguishability at some t₀ is equivalent to indistinguishability. For an arbitrary time-scale (ii) follows from (i). The statement (i) for an arbitrary time-scale was shown in [8]. Analyticity of the control system and the functions from H is essential in the proof.

Remark 7. Proposition 6 allows us to use the same language of analytic functions on ℝⁿ to study different observability properties of analytic systems on arbitrary time-scales as long as these properties are defined via the indistinguishability relation. It also implies that indistinguishability is an equivalence relation. This is not true for smooth systems (see [9]) or for analytic partially defined systems (see [10], where a different definition was developed to preserve this property for partially defined systems).

Let ℋ denote the subalgebra of C^ω(ℝⁿ) generated by H. It will be called the observation algebra of the system Σ. The elements of ℋ are obtained by substituting functions from H into polynomials of several variables with real coefficients. In particular, all constant functions belong to ℋ.

From Proposition 6 we get the following.

Proposition 8. The states x₁ and x₂ are indistinguishable if and only if for every φ ∈ ℋ, φ(x₁) = φ(x₂).

We say that Σ is observable if any two distinct states are distinguishable.

From the definition and Proposition 8 we obtain the following characterization.

Proposition 9. Σ is observable if and only if for any distinct x₁, x₂ ∈ ℝⁿ there is φ ∈ ℋ such that φ(x₁) ≠ φ(x₂).

The condition stated in Proposition 9 is difficult to check. This is one of the reasons that a weaker concept of local observability seems to be more interesting. There are many different concepts of local observability and one concept has often a few different names. For the first two concepts we follow the terminology used in [4].

We say that Σ is weakly locally observable at x₀ (WLO(x₀)) if there is a neighborhood U of x₀ such that for every x ∈ U, x and x₀ are distinguishable.

Remark 10. Weak local observability at x₀ is in fact a weak property. It holds, for example, for the system

()

at x₀ = 0 ∈ ℝⁿ. Since all solutions of x^Δ = 0 are constant, time does not influence indistinguishability relation. To distinguish points we have to use only the output function. Clearly, it takes different values at 0 and any other point, so we can distinguish x₀ from any of its neighbors. Observe that local observability fails at any x₀ ≠ 0, if n ≥ 2.

We say that Σ is strongly locally observable at x₀ (SLO(x₀)) if there is a neighborhood U of x₀ such that for every distinct x₁, x₂ ∈ U, x₁ and x₂ are distinguishable.

We say that Σ is robustly locally observable at x₀ (RLO(x₀)) if there is a neighborhood U of x₀ such that Σ is weakly locally observable at x for every x ∈ U.

Robust local observability was introduced in [3] for continuous-time systems under the name “stable local observability.” It means that the weak local observability at x₀ is stable or robust with respect to small perturbations of the initial condition x₀.

We call Σ weakly locally observable (WLO) (strongly locally observable (SLO), and robustly locally observable (RLO), resp.), when it is weakly locally observable (strongly locally observable and robustly locally observable, resp.) at every x ∈ ℝⁿ.

Let dℋ(x₀) denote the linear space of the differentials of functions from ℋ taken at x₀. The following theorem is a simple extension of the result from [1].

Theorem 11. (a) If dim dℋ(x₀) = n, then Σ is SLO(x₀).

(b) (for all x₀ ∈ ℝⁿ : dim dℋ(x₀) = n)⇒(for all x₀ ∈ ℝⁿ : Σ is SLO (x₀))⇒(∃X—a real analytic set in ℝⁿ : for all x₀ ∉ X : dim dℋ(x₀) = n).

Let us denote the condition dim dℋ(x₀) = n by HK(x₀) (Hermann-Krener condition at x₀). The second part of Theorem 11 says that if we are interested in strong local observability at large, that is, at each point, then condition HK(x₀) is satisfied almost everywhere, so the gap between sufficient condition and necessary condition for strong local observability at large is quite narrow. However, when one is interested in local observability at a particular point of the state space, the Hermann-Krener condition may be far from being necessary (see [2]).

We have a nice gradation of different local observability concepts.

Proposition 12. HK(x₀)⇒Σ is SLO(x₀)⇒Σ is RLO(x₀) ⇒Σ is WLO(x₀).

None of the implications in Proposition 12 may be, in general, reversed.

Example 13. (a) Let x ∈ ℝ, x^Δ = 0, and y = x³. Thus, H = {x³}. The system is observable, so it is also strongly locally observable at any x. But the Hermann-Krener condition fails at x = 0.

(b) Let x ∈ ℝ, x^Δ = 0, and y = x². Then H = {x²}. The system is robustly locally observable at 0, but it is not strongly locally observable at this point.

(c) Let x ∈ ℝ², x^Δ = 0, and . The system is weakly locally observable at 0, but it is not robustly locally observable at this point.

But we have an important, though obvious, global equivalence.

Proposition 14. Σ is RLO(x₀) for every x₀ ∈ ℝⁿ if and only if Σ is WLO(x₀) for every x₀ ∈ ℝⁿ. In other words, Σ is RLO if and only if Σ is WLO.

By 𝒪_x we will denote the algebra of germs at x of analytic functions on ℝⁿ (see Appendix B). Let J_x be the ideal of 𝒪_x generated by germs at x of functions from ℋ that vanish at x.

For x ∈ ℝⁿ and for an ideal J of 𝒪_x, let Z(J) be the germ at x of the zero-set of J. Since J is finitely generated (𝒪_x is Noetherian), Z(J) is well defined. Let I(Z(J)) be the ideal of 𝒪_x consisting of all germs of analytic functions that vanish on Z(J).

Lemma 15. The following conditions are equivalent:

(a)
;
(b)
arbitrarily close to x₀ there is x such that φ(x₀) = φ(x) for every φ ∈ ℋ.

Proof. (a) holds if and only if arbitrarily close to x₀ there is x such that all representatives of germs (all defined in some neighborhood of x₀) are 0 at x. This is equivalent to the fact that all functions φ ∈ ℋ take on the same values at x₀ and this x, which means precisely (b).

To characterize weak local observability we use the concept of real radical (see Appendix B).

Theorem 16.

(a)
Σ is WLO(x₀) if and only if .
(b)
HK(x₀) if and only if .

Proof. (a) Σ is not weakly locally observable at x₀ if and only if arbitrarily close to x₀ there is x such that x₀ and x are indistinguishable. By Lemma 15, the last statement is equivalent to the condition , which in turn means that . But , so from Theorem B.1 the last inequality is equivalent to the condition . This gives the equivalence of both sides in (a).

(b)
It is enough to prove the proposition for x = 0 in ℝⁿ.

Observe that dℋ(x) = dJ_x(x) for every x. Thus, if J₀ = m₀, then dℋ(0) contains all the differentials dx_i for i = 1, …, n, which are linearly independent.

On the other hand, the condition dim dℋ(0) = n implies that in a neighborhood U of 0, there are functions φ₁, …, φ_n whose germs belong to J₀ and the differentials at 0, dφ_i(0) are linearly independent. We may write φ_i = ∑_j Φ_ijx_j for some analytic functions Φ_ij on U (sufficiently small). Then dφ_i(0) = ∑_j Φ_ij(0)dx_j. This means that the matrix Φ = (Φ_ij) is invertible at 0 and then in some neighborhood of 0. Let Ψ = Φ⁻¹. Then the germs of elements of Ψ are in 𝒪_x and dx_i = ∑_j Ψ_ijdφ_j which means that φ₁, …, φ_n generate m₀.

Let G be a subset of 𝒪_x. By D(G) we will denote the ideal of 𝒪_x generated by Jacobians ∂(φ₁, …, φ_n)/∂(x₁, …, x_n), where φ_i ∈ G. Observe that these Jacobians are well defined on germs of functions. If 𝒢 is a family of real analytic functions on an open set U in ℝⁿ, then similarly we define the Jacobian ideal D(𝒢) in 𝒪_U. Furthermore, there is a simple relation between Jacobian ideals for functions and germs of functions. If 𝒢 is a family of analytic functions on U, and x₀ ∈ U then we have

()

Now, for a point x ∈ ℝⁿ, we define a sequence of ideals in 𝒪_x related to the system Σ. Let and . It is clear that instead of ℋ_x in this definition one can take the previously defined ideal J_x; that is, . This leads to the following generalization of statement (b) of Theorem 16.

Corollary 17. HK.

The ideals will be the main tools in studying robust local observability. First we prove the basic fact.

Proposition 18. For any k ≥ 0, , and there is s ≥ 0 such that .

Proof. To prove the first part we proceed by induction. It is clear that , so assume that for some k ≥ 0. Then also and , so finally . This means that . Since the ring 𝒪_x is Noetherian the sequence of ideals must stabilize at some s.

Now we can characterize robust local observability.

Theorem 19. System Σ is RLO(x₀) if and only if for some s > 0.

The proof of Theorem 19 will rely on several lemmas. They appeared in a similar form in [3]. However, there were a few flaws in the proofs, which are now corrected.

Let 𝒢 be a family of analytic functions on some open set U ⊂ ℝⁿ. Denote by S_x(𝒢) the germ at x of the level set of 𝒢 that passes through x. Thus

()

The set-germ S_x(𝒢) is a germ of analytic set. One of the representatives of S_x(𝒢) is the analytic set in U : {y ∈ U : φ(y) = φ(x) for φ ∈ 𝒢}.

Lemma 20. Let U be an open subset of ℝⁿ and let X be an analytic set in U. Consider a family 𝒢 of analytic functions on U. If for every x ∉ X : S_x(𝒢) = {x}, then for every x ∈ X : S_x(𝒢) ⊂ X_x.

Proof. Suppose that there is x ∈ X such that S_x(𝒢)⊄X_x. This means that for every representatives and we have . Take and arbitrarily small neighborhood V of x in U. Let be a representative of S_x(𝒢) in V. Then there is such that y ∉ X. Take a sequence (y_n) of such points converging to x. We may assume that all these points belong to some (large enough) representative Y of S_x(𝒢) and that Y is an analytic set. Only finite number of points y_n may be isolated points of Y. This means that arbitrarily close to x there is a point y_n for which . Thus we get a contradiction.

Lemma 21. Let U be an open subset of ℝⁿ and let 𝒢 be a family of analytic functions on U. For every x ∈ U: if x ∉ Z(D(𝒢)), then S_x(𝒢) = {x}.

Proof. If x ∉ Z(D(𝒢)), then there are functions φ₁, …, φ_n ∈ G such that

()

Thus, the map

is injective in a neighborhood of x. This implies that S_x(𝒢) = {x}.

Lemma 22. Assume that for some k ≥ 0. Then there is a neighborhood U of x₀ in ℝⁿ and a representative of in U such that for every x ∈ U:

()

Proof. First observe that Z(D(ℋ_|U)) is a representative of . We proceed by induction. Let k = 0 and assume that . Take any neighborhood U of x₀. Since , then is a representative of in U. If , then by Lemma 21, S_x(ℋ) = S_x(ℋ_|U) = {x}.

Now assume that the statement of the lemma holds for k − 1 ≥ 0. Hence, there is a neighborhood U of x₀ and a representative of such that if x ∈ U and , then S_x(ℋ) = {x}. The functions φ₁, …, φ_s are representatives on U of generators of . The set is a representative of . Clearly . Take x ∈ U such that . If , then S_x(ℋ) = {x}. Assume then that . From Lemma 20 we get . Then . Since x ∉ Z(D(ℋ_|U ∪ {φ₁, …, φ_s})), then, by Lemma 21, S_x(ℋ) = S_x(ℋ_|U ∪ {φ₁, …, φ_s}) = {x}. This finishes the inductive step of the proof.

Lemma 23. Let x₀ ∈ U ⊂ ℝⁿ, where U is open, and let 𝒢 be a family of analytic functions on U. If D(𝒢) = 0 (zero ideal), then arbitrarily close to x₀ there is x ∈ U such that S_x(𝒢)≠{x}.

Proof. Let

()

Then s < n and arbitrarily close to x₀ there is x ∈ U and φ₁, …, φ_n ∈ 𝒢 such that rank (∂(φ₁, …, φ_n)/∂(x₁, …, x_n)) (x) = s. This rank is preserved in some neighborhood V of x. Thus, we may assume that gradients of φ₁, …, φ_s are linearly independent at every point of V and span

for

. Then by Frobenius Theorem, V is a union of integral manifolds of codistribution d𝒢. The integral manifolds are the level sets of 𝒢 and have dimension greater than or equals to 1. This means that S_x(𝒢)≠{x}.

Lemma 24. Let U be an open subset of ℝⁿ and let φ₁, …, φ_k be analytic functions on U whose gradients are linearly independent at each point of U. Let Y = Z(φ₁, …, φ_k), x₀ ∈ Y, and let 𝒢 be a family of analytic functions on U.

If Y ⊂ Z(D(𝒢 ∪ {φ₁, …, φ_k})), then arbitrarily close to x₀ there is x ∈ Y such that S_x(𝒢)≠{x}.

Proof. Changing the coordinates we can obtain φ_i = x_i, i = 1, …, k. Let ψ₁, …, ψ_n−k ∈ 𝒢. Then for x ∈ Y we have

()

Note that Y is an analytic manifold and the last term above is actually the Jacobian of a map defined on Y. Hence, after restricting to the manifold Y, we get D(𝒢_|Y) = 0. From Lemma 23, arbitrarily close to x₀ there is x ∈ Y such that S_x(𝒢_|Y)≠{x}. But S_x(𝒢_|Y) ⊂ S_x(𝒢) so also S_x(𝒢)≠{x}.

Lemma 25. Assume that for some s ≥ 0 and . Then in every neighborhood of x₀ there is x such that S_x(ℋ)≠{x}.

Proof. Let φ₁, …, φ_k be representatives of generators of the ideal , defined on some common neighborhood U of x₀. Then is a representative of in U. In every neighborhood of x₀ one can find a regular point of the analytic set (see [11, 12]). Let be such a point and let V be a neighborhood of in ℝⁿ such that is an analytic manifold. Then Y = Z(φ_1|V, …, φ_k|V). We may assume that the gradients of φ_1|V, …, φ_k|V are linearly independent on Y. Otherwise, after possible shrinking of V, we can remove the functions whose gradients are linear combinations of the gradients of other functions. Let x ∈ Y. Then so that Y ⊂ Z(D(ℋ_|V ∪ {φ_1|V, …, φ_k|V})). The statement of the lemma follows now from Lemma 24.

Proof of Theorem 19. Sufficiency. Assume that for some s > 0. This means that . From Lemma 22 it follows that nontrivial set-germs S_x(ℋ) (i.e., different from {x}) may be found only in —some representative of . But so in some neighborhood of x₀ the level set-germs of ℋ must be trivial. This means that Σ is robustly locally observable at x₀.

Necessity. Assume that for some s ≥ 0. From Lemma 25 it follows that Σ is not robustly locally observable.

The statements of Corollary 17, Proposition 18, and Theorem 19 can be translated into the language of germs of analytic sets. Let . Because the ideals are real, we also get , so there is one-to-one correspondence between the ideals and the set-germs. We have then the following.

Corollary 26. Let x₀ ∈ ℝⁿ.

(a)
HK(x₀) if and only if .
(b)
For every k ≥ 0, , and for some s ≥ 0, .
(c)
Σ is RLO(x₀) if and only if for some s ≥ 0, .

Example 27. Consider the following system:

()

and choose x₀ = 0. Then for an arbitrary time-scale we get

and

. Thus

is the germ of a union of three planes intersecting at 0. In the next step we obtain

. A quick calculation shows that

is the germ of a union of three lines intersecting at 0. Finally,

and then

, so Σ is robustly locally observable at x₀.

4. Discretization

As we are going to consider several time-scales, we will denote the graininess function on the time-scale 𝕋 by μ_𝕋. A time-scale 𝕋 is called discrete, if μ_𝕋(t) > 0 for all t ∈ 𝕋.

Let Σ be the continuous-time system

()

and let Σ_𝕋 be its discretization

()

on a discrete time-scale 𝕋. Usually 𝕋 is equal to cℤ for some c > 0, but nonhomogeneous time-scales are allowed as well. In the discretized system the ordinary derivative is replaced by the delta derivative on the discrete time-scale.

Thus, (23) is replaced with

()

for t ∈ 𝕋.

Let ℋ denote the observation algebra of the system Σ and ℋ_𝕋 the observation algebra of the system Σ_𝕋. Observe that each generator of ℋ may be approximated by a corresponding generator of ℋ_𝕋 for μ(t) sufficiently small. This follows from the form of the operators on ℝ and 𝕋, which is used in this procedure.

A natural question is which properties related to observability are preserved under discretization.

Proposition 28. If x₁ and x₂ are distinguishable by Σ, then there is such that x₁ and x₂ are distinguishable by Σ_𝕋 at t ∈ 𝕋 whenever .

Proof. Suppose that for every there are a time-scale 𝕋 and t ∈ 𝕋 with such that x₁ and x₂ are indistinguishable by Σ_𝕋 at t. This means that that there is a sequence (𝕋_i) of time-scales and a sequence (t_i) of real numbers such that t_i ∈ 𝕋_i and when i → ∞ and for every , φ(x₁) = φ(x₂). Every function ψ ∈ ℋ may be approximated by functions from ; that is, there is a sequence of functions (ϕ_i) such that and φ_i → ψ on some compact set containing x₁ and x₂. This implies that ψ(x₁) = ψ(x₂), so x₁ and x₂ are indistinguishable by Σ.

In particular, distinguishability of x₁ and x₂ is preserved for quantum discretization, where 𝕋 = q^ℕ, if q is sufficiently close to 1.

We will show now that the Hermann-Krener rank condition is preserved under discretization. Let HK(x₀) denote the Hermann-Krener condition at x₀ for the system Σ and HK_𝕋(x₀) for the system Σ_𝕋.

Proposition 29. If HK(x₀), then there is such that for every time-scale 𝕋 if there is t ∈ 𝕋 with , then HK_𝕋(x₀).

Proof. Assume that HK(x₀) holds. Then there are ψ₁, …, ψ_n ∈ ℋ such that dψ₁(x₀), …, dψ_n(x₀) are linearly independent. Each ψ_i may be approximated by some for t ∈ 𝕋 and μ_𝕋(t) sufficiently small. Observe that the functions φ_i depend actually on the parameter μ_𝕋(t) and this dependence is continuous. For μ_𝕋(t) sufficiently small also dφ₁(x₀), …, dφ_n(x₀) will be linearly independent. This means that HK_𝕋(x₀) holds for such 𝕋.

The converse of Proposition 29 does not hold.

Example 30. Let Σ be

()

The observation algebra ℋ is generated by a single function

, which means that the Hermann-Krener condition does not hold at any point. The discretized system Σ_𝕋 is given by

()

The observation algebra ℋ_𝕋 contains now the functions

and ψ₂(x₁, x₂) = x₁. The Hermann-Krener condition is then satisfied at all points and for all discrete time-scales 𝕋.

Remark 31. Hermann-Krener rank condition is equivalent to the property that the ideal is maximal; that is, it is generated by the coordinate functions. One can show that this property is preserved when the ideal is replaced with the ideals corresponding to systems Σ_𝕋 if 𝕋 contains t with μ_𝕋(t) sufficiently small. In characterizations of weak and robust local observability there appear real radicals of ideals. It is not clear whether desired properties of the radicals like maximality (for WLO(x₀)) or nonproperness (for RLO(x₀) are preserved under discretizations. Thus preservation of weak and robust local observability under Euler discretization is still an open problem.

We finish this discussion with a positive example.

Example 32. Let Σ be

()

The observation algebra of Σ is generated by

The discretization gives Σ_𝕋

()

where

()

The observation algebra of Σ_𝕋 is also generated by

Thus, Σ and Σ_𝕋 are both weakly locally observable at x = 0. They are not weakly locally observable at any other point.

Example 27 describes a positive behavior of robust local observability under discretization. In fact, the calculations are the same for all time-scales.

5. Conclusions

We have shown that the methods of real analytic geometry and real algebra developed for continuous time systems may be used for systems on arbitrary time-scales, in particular on the scale of integers and on the quantum scales. Different concepts of local observability for systems on arbitrary time-scales have been considered. We have established relations between these concepts and provided characterizations of weak and robust local observability with the aid of certain ideals of the ring of germs of analytic functions and real radicals of those ideals. Equivalent geometric characterizations have been given. Observation algebras from which the ideals are obtained and the ideals themselves depend on the time-scale on which the systems is defined, but once the ideals are computed, the procedures and the criteria of local observability are the same for all time-scales. This allows for unified treatment of observability of systems on arbitrary time-scales.

The language of time-scales allows for a natural description of discretization of continuous-time systems: the ordinary derivative is replaced by delta derivative on a discrete time-scale 𝕋. The paper contains preliminary results on preservation of properties related to observability under discretization. In particular Hermann-Krener rank condition is preserved. Preservation of other properties, in particular weak and robust local observability, is stated as an open problem. To solve the problem one will have to study limit properties of real radicals for rings of germs of analytic functions. This will be a subject of a future research.

Acknowledgment

This work was supported by the Bialystok University of Technology Grant S/WI/2/2011.

Appendices

A. Calculus on Time-scales

A time-scale 𝕋 is an arbitrary nonempty closed subset of the set ℝ of real numbers. In particular ℝ, hℤ for h > 0 and q^ℕ : = {q^k, k ∈ ℕ} for q > 1 are time-scales. We assume that 𝕋 is a topological space with the relative topology induced from ℝ. If t₀, t₁ ∈ 𝕋, then [t₀, t₁] _𝕋 denotes the intersection of the ordinary closed interval with 𝕋. Similar notation is used for open, half-open or infinite intervals.

For t ∈ 𝕋 we define the forward jump operator σ : 𝕋 → 𝕋 by σ(t): = inf {s ∈ 𝕋 : s > t} if t ≠ sup 𝕋 and σ(sup 𝕋) = sup 𝕋 when sup 𝕋 is finite; the backward jump operator ρ : 𝕋 → 𝕋 by ρ(t): = sup {s ∈ 𝕋 : s < t} if t ≠ inf 𝕋 and ρ(inf 𝕋) = inf 𝕋 when inf 𝕋 is finite; the forward graininess function μ : 𝕋 → [0, ∞) by μ(t): = σ(t) − t; the backward graininess function ν : 𝕋 → [0, ∞) by ν(t): = t − ρ(t).

If σ(t) > t, then t is called right-scattered, while if ρ(t) < t, it is called left-scattered. If t < sup 𝕋 and σ(t) = t, then t is called right-dense. If t > inf 𝕋 and ρ(t) = t, then t is left-dense.

The time-scale 𝕋 is homogeneous, if μ and ν are constant functions. When μ ≡ 0 and ν ≡ 0, then 𝕋 = ℝ or 𝕋 is a closed interval (in particular a half-line). When μ is constant and greater than 0, then 𝕋 = μℤ + c, for some c ∈ ℝ.

Let 𝕋^κ : = {t ∈ 𝕋 : t is nonmaximal or left-dense}. Thus 𝕋^κ is obtained from 𝕋 by removing its maximal point if this point exists and is left-scattered.

Let f : 𝕋 → ℝ and t ∈ 𝕋^κ. The delta derivative of f at t, denoted by f^Δ(t), is the real number with the property that given any ε there is a neighborhood U = (t − δ, t + δ) _𝕋 such that

()

for all s ∈ U. If f^Δ(t) exists, then we say that f is delta differentiable at t. Moreover, we say that f is delta differentiable on 𝕋^k provided f^Δ(t) exists for all t ∈ 𝕋^k.

Example A.1. If 𝕋 = ℝ, then f^Δ(t) = f^′(t). If 𝕋 = hℤ, then f^Δ(t) = (f(t + h) − f(t))/h. If 𝕋 = q^ℕ, then f^Δ(t) = (f(qt) − f(t))/((q − 1)t).

A function f : 𝕋 → ℝ is called rd-continuous provided it is continuous at right-dense points in 𝕋 and its left-sided limits exist (finite) at left-dense points in 𝕋. If f is continuous, then it is rd-continuous.

A function f : 𝕋 → ℝ is called regressive, if 1 + μ(t)f(t) ≠ 0 for all t ∈ 𝕋.

A function F : 𝕋 → ℝ is called an antiderivative of f : 𝕋 → ℝ provided F^Δ(t) = f(t) holds for all t ∈ 𝕋^κ. Let a, b ∈ 𝕋. Then the delta integral of f on the interval [a, b) _𝕋 is defined by

()

Riemann and Lebesgue delta integrals on time-scales have been also defined (see, e.g., [13]). It can be shown that every rd-continuous function has an antiderivative and its Riemann and Lebesgue integrals agree with the delta integral defined above.

Example A.2. If 𝕋 = ℝ, then , where the integral on the right is the usual Riemann integral. If 𝕋 = hℤ, h > 0, then for a < b.

B. Basic Real Geometry

We assume that the reader is familiar with the concepts of germs of functions and sets, and with fundamentals of the sheaf theory and theory of analytic sets. Necessary definitions can be found, for example, in [14, 15]. If B is a “global” object (a set, a function, or a family of functions), B_x will always denote its germ at the point x (but the precise meaning of the germ will depend on the meaning of the object). If C is a germ, will denote one of its representatives. By 𝒪_x we denote the algebra of germs of real analytic functions at x, where x ∈ ℝⁿ (n fixed throughout the paper), and by m_x the (only) maximal ideal of 𝒪_x, consisting of all germs in 𝒪_x that vanish at x. By 𝒪 we denote the sheaf of germs of real analytic functions on ℝⁿ.

If U is an open subset in ℝⁿ, then 𝒪_U will mean the algebra of real analytic functions on U. If A is a subalgebra of 𝒪_U and x₀ ∈ U, then means the set of germs at x₀ of functions from A. Of course is again an algebra over ℝ. If I is an ideal of 𝒪_U, then means the ideal of generated by the germs at x₀ of function from I.

Consider a set-germ A in ℝⁿ (at some point x). Then J(A) denotes the ideal of 𝒪_x consisting of germs (at x) of real analytic functions that vanish on A. If I is an ideal of 𝒪_x, then Z(I) will denote the zero set-germ of I (at x). Let us recall that Z(I) is defined as the intersection of the set-germs Z(φ_i), i = 1, …, k, where φ₁, …, φ_k are generators of the ideal I. Since only finite intersections of set-germs are defined, we must use here the property that 𝒪_x is Noetherian.

We have a natural duality between ideals and set-germs. If I₁ ⊂ I₂, then Z(I₂) ⊂ Z(I₁).

Let P be any commutative ring with a unit and let I be an ideal of P. Then the real radical of I, denoted by

, is the set of all a ∈ P for which there is m ∈ ℕ, k ∈ ℕ ∪ {0} and b₁, …, b_k ∈ P such that

()

The real radical is an ideal in P and it contains I. If I is a proper ideal of P, then also

is proper. An ideal I is called real if

Theorem B.1 (see [16].)Let x ∈ ℝⁿ. If I is an ideal of 𝒪_x, then

()

Theorem B.1 implies that there is a 1 : 1 correspondence between germs of analytic sets at x and real ideals of 𝒪_x.

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Local Observability of Systems on Time Scales

Abstract

1. Introduction

2. Preliminaries

3. Local Observability

4. Discretization

5. Conclusions

Acknowledgment

Appendices

A. Calculus on Time-scales

B. Basic Real Geometry

References

References

Information

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Local Observability of Systems on Time Scales

Abstract

1. Introduction

2. Preliminaries

3. Local Observability

4. Discretization

5. Conclusions

Acknowledgment

Appendices

A. Calculus on Time-scales

B. Basic Real Geometry

References

References

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