# Local Observability of Systems on Time Scales

Authors: Zbigniew Bartosiewicz
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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.


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## Body

## 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

Let𝕋 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
(1)Σ:xΔ(t)=f(x(t),u(t)),y(t)=h(x(t)),
where t∈𝕋, x(t)∈ℝn, y(t)∈ℝr, and u(t)∈Ω—arbitrary set. For ω∈Ω, let fω be defined by fω(x):=f(x,ω). We assume that the maps h and fω for every ω∈Ω are analytic and that controls u are piecewise constant functions of time.If𝕋=ℝ, then (1) is the standard continuous-time system
(2)x˙(t)=f(x(t),u(t)),y(t)=h(x(t)).
For 𝕋=ℤ (1) takes the form
(3)x(t+1)-x(t)=f(x(t),u(t)),y(t)=h(x(t)).
This can be rewritten in a more standard shift form as
(4)x(t+1)=f(x(t),u(t))+x(t)=:g(x(t),u(t)),y(t)=h(x(t)).
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 equationx(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.
Byγ(t,t0,x0,u) we denote the solution of the equation xΔ=f(x,u) corresponding to control u and the initial condition x(t0)=x0 and evaluated at time t.
Letx1, x2∈ℝn and t0∈𝕋. Then x1 and x2 are called indistinguishable at timet0 if
(5)h(γ(t,t0,x1,u))=h(γ(t,t0,x2,u))
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 timet0.
The statesx1 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 timet0 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]).
LetCω(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
(6)Γgt0:Cω(ℝn)⟶Cω(ℝn)
was introduced as
(7)(Γgt0φ)(x):=∫01φ′(x+sμ(t0)g(x))ds·g(x),
where φ∈Cω(ℝn) and φ′ is the gradient of φ (a row vector).Example 3.
Letφ=xi be the ith coordinate function on ℝn. Then φ′=ei—the (row) vector of the standard basis of ℝn with 1 at the i-th position. For any t0∈𝕋 we have
(8)(Γgt0xi)(x)=eig(x)=gi(x).
Observe that ifμ(t0)>0 then
(9)(Γgt0φ)(x)=1μ(t0)(φ(x+μ(t0)g(x))-φ(x)).
On the other hand for μ(t0)=0 we obtain (Γgt0φ)(x)=φ′(x)g(x), which is the standard Lie derivative of the function φ along the vector field g. In general, when operator Γgt0 does not depend on t0, we will denote it by Γg. This happens, for instance, if the time-scale is homogeneous.
Letσgt0:Cω(ℝn)→Cω(ℝn) be another map related to the function g and the time t0 defined by
(10)(σgt0φ)(x):=φ(x+μ(t0)g(x)).
If μ(t0)=0, then σgt0 is the identity map. In general we have an obvious propertyProposition 4.
The mapσgt0 is an endomorphism of the algebra Cω(ℝn).Forμ(t0)>0 the operators Γgt0 and σgt0 are related by the following equality:
(11)σgt0φ=φ+μ(t0)Γgt0φ.
We also have the following generalization of the Leibniz rule.Proposition 5.
(12)
Γ
g
t
0
(
φ
ψ
)
=
(
Γ
g
t
0
φ
)
(
σ
g
t
0
ψ
)
+
φ
(
Γ
g
t
0
ψ
)
.This property means thatΓgt0 is a skew derivation of the algebra Cω(ℝn) with respect to σgt0 or, in other words, that Γgt0 is a σgt0-derivation of this algebra. For μ(t0)=0 it is an ordinary derivation.


## 3.   Local Observability

LetHt0 be a subset of Cω(ℝn) consisting of functions of the form
(13)Γgkt0…Γg1t0hi,
where i=1,…,p, k≥0 and gi=fωi for some ωi∈Ω and i=1,…,k. If k=0 then this function is just hi. Let
(14)H=⋃t0∈𝕋Ht0.Proposition 6.
(i) The statesx1 and x2 are indistinguishable at time t0 if and only if for every φ∈Ht0, φ(x1)=φ(x2).
(ii) The statesx1 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.
Proposition6 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).
Letℋ denote the subalgebra of Cω(ℝn) generated by H. It will be called the observation algebra of the systemΣ. 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 Proposition6 we get the following.Proposition 8.
The statesx1 and x2 are indistinguishable if and only if for every φ∈ℋ, φ(x1)=φ(x2).We say thatΣ is observable if any two distinct states are distinguishable.From the definition and Proposition8 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 Proposition9 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 thatΣ is weakly locally observable atx0 (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 atx0 is in fact a weak property. It holds, for example, for the system
(15)xΔ=0,y=x12+⋯+xn2,
at x0=0∈ℝn. Since all solutions of xΔ=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 thatΣ is strongly locally observable atx0 (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 thatΣ is robustly locally observable atx0 (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 callΣweakly 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.
Letdℋ(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) Ifdimdℋ(x0)=n, then Σ is SLO(x0).
(b)(forallx0∈ℝn:dimdℋ(x0)=n)⇒(forallx0∈ℝn:Σ is SLO (x0))⇒(∃X—a real analytic set in ℝn:forallx0∉X:dimdℋ(x0)=n).Let us denote the conditiondimdℋ(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 Proposition12 may be, in general, reversed.Example 13.
(a) Letx∈ℝ, xΔ=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) Letx∈ℝ, xΔ=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) Letx∈ℝ2, xΔ=0, and y=x12+x22. 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.By𝒪x we will denote the algebra of germs at x of analytic functions on ℝn (see Appendix B). Let Jx be the ideal of 𝒪x generated by germs at x of functions from ℋ that vanish at x.Forx∈ℝn and for an ideal J of 𝒪x, let Z(J) be the germ at x of the zero-set of J. Since J is finitely generated (𝒪x is Noetherian), Z(J) is well defined. Let I(Z(J)) be the ideal of 𝒪x consisting of all germs of analytic functions that vanish on Z(J).Lemma 15.
The following conditions are equivalent:(a)
Z(Jx0)≠{x0};(b)
arbitrarily close tox0 there is x such that φ(x0)=φ(x) for every φ∈ℋ.Proof.
(a) holds if and only if arbitrarily close tox0 there is x such that all representatives of germs φ∈Jx0 (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 AppendixB).Theorem 16.
(a)
Σ is WLO(x0) if and only if Jx0ℝ=mx0.(b)
HK(x0) if and only if Jx0=mx0.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 Z(Jx0)≠{x0}, which in turn means that I(Z(Jx0))≠I({x0}). But I({x0})=mx0, so from Theorem B.1 the last inequality is equivalent to the condition Jx0ℝ≠mx0. This gives the equivalence of both sides in (a).(b)
It is enough to prove the proposition forx=0 in ℝn.
Observe thatdℋ(x)=dJx(x) for every x. Thus, if J0=m0, then dℋ(0) contains all the differentials dxi for i=1,…,n, which are linearly independent.
On the other hand, the conditiondimdℋ(0)=n implies that in a neighborhood U of 0, there are functions φ1,…,φn whose germs belong to J0 and the differentials at 0, dφi(0) are linearly independent. We may write φi=∑j‍Φijxj for some analytic functions Φij on U (sufficiently small). Then dφi(0)=∑j‍Φij(0)dxj. This means that the matrix Φ=(Φij) is invertible at 0 and then in some neighborhood of 0. Let Ψ=Φ-1. Then the germs of elements of  Ψ are in 𝒪x and dxi=∑j‍Ψijdφj which means that φ1,…,φn generate m0.LetG be a subset of 𝒪x. By D(G) we will denote the ideal of 𝒪x generated by Jacobians ∂(φ1,…,φn)/∂(x1,…,xn), where φi∈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 𝒪U. 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
(16)D(𝒢)x0=D(𝒢x0).Now, for a pointx∈ℝn, we define a sequence of ideals in 𝒪x related to the system Σ. Let Ix(0):=(0) and Ix(k+1):=D(ℋx∪Ix(k))ℝ. It is clear that instead of ℋx in this definition one can take the previously defined ideal Jx; that is,  Ix(k+1)=D(Jx∪Ix(k))ℝ. This leads to the following generalization of statement (b) of Theorem 16.Corollary 17.
HK(x0)⇔Jx0=mx0⇔Ix0(1)=𝒪x0.The idealsIx(k) will be the main tools in studying robust local observability. First we prove the basic fact.Proposition 18.
For anyk≥0, Ix(k)⊆Ix(k+1), and there is s≥0 such that Ix(s)=Ix(s+1).Proof.
To prove the first part we proceed by induction. It is clear thatIx(0)⊆Ix(1), so assume that Ix(k)⊆Ix(k+1) for some k≥0. Then also ℋx∪Ix(k)⊆ℋx∪Ix(k+1) and D(ℋx∪Ix(k))⊆D(ℋx∪Ix(k+1)), so finally D(ℋx∪Ix(k))ℝ⊆D(ℋx∪Ix(k+1))ℝ. This means that Ix(k+1)⊆Ix(k+2). Since the ring 𝒪x is Noetherian the sequence of ideals Ix(k) must stabilize at some s.Now we can characterize robust local observability.Theorem 19.
SystemΣ is RLO(x0) if and only if Ix0(s)=𝒪x0 for some s>0.The proof of Theorem19 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.Let𝒢 be a family of analytic functions on some open set U⊂ℝn. Denote by Sx(𝒢) the germ at x of the level set of 𝒢 that passes through x. Thus
(17)Sx(𝒢)={y∈U:φ(x)=φ(y)forφ∈𝒢}x.
The set-germ Sx(𝒢) is a germ of analytic set. One of the representatives of Sx(𝒢) is the analytic set in U:{y∈U:φ(y)=φ(x)forφ∈𝒢}.Lemma 20.
LetU 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:Sx(𝒢)={x}, then for every x∈X:Sx(𝒢)⊂Xx.Proof.
Suppose that there isx∈X such that Sx(𝒢)⊄Xx. This means that for every representatives S~x(𝒢) and X~x we have S~x(𝒢)⊄X~x. Take X~x:=X and arbitrarily small neighborhood V of x in U. Let S~x(𝒢) be a representative of Sx(𝒢) in V. Then there is y∈S~x(𝒢) such that y∉X. Take a sequence (yn) of such points converging to x. We may assume that all these points belong to some (large enough) representative Y of Sx(𝒢) and that Y is an analytic set. Only finite number of points yn may be isolated points of Y. This means that arbitrarily close to x there is a point yn for which Syn(𝒢)≠{yn}. Thus we get a contradiction.Lemma 21.
LetU 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 Sx(𝒢)={x}.Proof.
Ifx∉Z(D(𝒢)), then there are functions φ1,…,φn∈G such that
(18)∂(φ1,…,φn)∂(x1,…,xn)(x)≠0.
Thus, the map x-↦(φ1(x-),…,φn(x-)) is injective in a neighborhood of x. This implies that Sx(𝒢)={x}.Lemma 22.
Assume thatXx0(k)≠Xx0(k+1) for some k≥0. Then there is a neighborhood U of x0 in ℝn and a representative X~x0(k+1) of Xx0(k+1) in U such that for every x∈U:
(19)x∉X~x0(k+1)⟹Sx(ℋ)={x}.Proof.
First observe thatZ(D(ℋ|U)) is a representative of Z(D(ℋx0)). We proceed by induction. Let k=0 and assume that Xx0(1)≠Xx0(0)=ℝx0n. Take any neighborhood U of x0. Since Xx0(1)=Z(D(ℋx0)), then Z(D(ℋ|U))=:X~x0(1) is a representative of Xx0(1) in U. If x∉X~x0(1), then by Lemma 21, Sx(ℋ)=Sx(ℋ|U)={x}.
Now assume that the statement of the lemma holds fork-1≥0. Hence, there is a neighborhood U of x0 and a representative X~x0(k)=Z(φ1,…,φs) of Xx0(k) such that if x∈U and x∉X~x0(k), then Sx(ℋ)={x}. The functions φ1,…,φs are representatives on U of generators of Ix0(k). The set X~x0(k+1):=Z(D(ℋ|U∪{φ1,…,φs})) is a representative of Xx0(k+1). Clearly X~x0(k+1)⊆X~x0(k). Take x∈U such that x∉X~x0(k+1). If x∉X~x0(k), then Sx(ℋ)={x}. Assume then that x∈X~x0(k). From Lemma 20 we get Sx(ℋ)⊆(X~x0(k))x. Then Sx(ℋ)=Sx(ℋ)∩(X~x0(k))x=Sx(ℋ|U∪{φ1,…,φs}). Since x∉Z(D(ℋ|U∪{φ1,…,φs})), then, by Lemma 21, Sx(ℋ)=Sx(ℋ|U∪{φ1,…,φs})={x}. This finishes the inductive step of the proof.Lemma 23.
Letx0∈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 Sx(𝒢)≠{x}.Proof.
Let(20)s=maxx∈Uφi∈𝒢rank∂(φ1,…,φn)∂(x1,…,xn)(x).
Then s<n and arbitrarily close to x0 there is x∈U and φ1,…,φn∈𝒢 such that rank(∂(φ1,…,φn)/∂(x1,…,xn))(x)=s. This rank is preserved in some neighborhood V of x. Thus, we may assume that gradients of φ1,…,φs are linearly independent at every point of V and span d𝒢(x-) for x-∈V. 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 Sx(𝒢)≠{x}.Lemma 24.
LetU be an open subset of ℝn and let φ1,…,φk be analytic functions on U whose gradients are linearly independent at each point of U. Let Y=Z(φ1,…,φk), x0∈Y, and let 𝒢 be a family of analytic functions on U.
IfY⊂Z(D(𝒢∪{φ1,…,φk})), then arbitrarily close to x0 there is x∈Y such that Sx(𝒢)≠{x}.Proof.
Changing the coordinates we can obtainφi=xi, i=1,…,k. Let ψ1,…,ψn-k∈𝒢. Then for x∈Y we have
(21)0=det[∇φ1(x)⋮∇φk(x)∇ψ1(x)⋮∇ψn-k(x)]=det[I0(∂ψi∂xj(x))i=1,…,n-kj=1,…,n]=det(∂ψi(x)∂xj)i=1,…,n-kj=k+1,…,n.
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 Sx(𝒢|Y)≠{x}. But Sx(𝒢|Y)⊂Sx(𝒢) so also Sx(𝒢)≠{x}.Lemma 25.
Assume thatXx0(s)=Xx0(s+1) for some s≥0 and Xx0(s)≠∅. Then in every neighborhood of x0 there is x such that Sx(ℋ)≠{x}.Proof.
Letφ1,…,φk be representatives of generators of the ideal Ix0(s), defined on some common neighborhood U of x0. Then X~x0(s):=Z(φ1,…,φk) is a representative of Xx0(s) in U. In every neighborhood of x0 one can find a regular point of the analytic set X~x0(s) (see [11, 12]). Let x- be such a point and let V be a neighborhood of x- in ℝn such that Y:=V∩X~x0(s) is an analytic manifold. Then Y=Z(φ1|V,…,φk|V). We may assume that the gradients of φ1|V,…,φk|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 x∈X~x0(s+1):=Z(D(ℋ|U∪{φ1,…,φk})) so that Y⊂Z(D(ℋ|V∪{φ1|V,…,φk|V})). The statement of the lemma follows now from Lemma 24.Proof of Theorem19.
Sufficiency.Assume that Ix0(s)=𝒪x0 for some s>0. This means that Xx0(s)=∅. From Lemma 22 it follows that nontrivial set-germs Sx(ℋ) (i.e., different from {x}) may be found only in X~x0(s)—some representative of Xx0(s). But Xx0(s)=∅ 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 X~x0(s)=X~x0(s+1) for some s≥0. From Lemma 25 it follows that Σ is not robustly locally observable.The statements of Corollary17, Proposition 18, and Theorem 19 can be translated into the language of germs of analytic sets. Let Xx(k)=Z(Ix(k)). Because the ideals Ix(k) are real, we also get J(Xx(k))=Ix(k), so there is one-to-one correspondence between the ideals and the set-germs. We have then the following.Corollary 26.
Letx0∈ℝn. (a)
HK(x0) if and only if Xx0(1)=∅.(b)
For everyk≥0, Xx0(k)⊇Xx0(k+1), and for some s≥0, Xx0(s)=Xx0(s+1).(c)
Σ is RLO(x0) if and only if for some s≥0, Xx0(s)=∅.Example 27.
Consider the following system:(22)x1Δ=x1u,x2Δ=x2,x3Δ=0,y=x12+x22+x32,
and choose x0=0. Then for an arbitrary time-scale we get ℋ=ℝ[x12,x22,x32] and Ix0(1)=Jx0=(x1x2x3). Thus Xx0(1) is the germ of a union of three planes intersecting at 0. In the next step we obtain Ix0(2)=(x12x22,x22x32,x12x32)ℝ=(x1x2,x2x3,x1x3). A quick calculation shows that Xx0(2) is the germ of a union of three lines intersecting at 0. Finally, Ix0(3)=mx0 and then Ix0(4)=𝒪x0, 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-scale𝕋 by μ𝕋. A time-scale 𝕋 is called discrete, if μ𝕋(t)>0 for all t∈𝕋.LetΣ be the continuous-time system
(23)x˙=f(x,u),y=h(x)
and let Σ𝕋 be its discretization
(24)xΔ=f(x,u),y=h(x)
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
(25)x(t+μ𝕋(t))-x(t)μ𝕋(t)=f(x(t),u(t)),y(t)=h(x(t))
for t∈𝕋.Letℋ denote the observation algebra of the system Σ and ℋ𝕋 the observation algebra of the system Σ𝕋. Observe that each generator of ℋ may be approximated by a corresponding generator of ℋ𝕋 for μ(t) sufficiently small. This follows from the form of the operators Γgt0 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.
Ifx1 and x2 are distinguishable by Σ, then there is μ^>0 such that x1 and x2 are distinguishable by Σ𝕋 at t∈𝕋 whenever μ𝕋(t)<μ^.Proof.
Suppose that for everyμ^>0 there are a time-scale 𝕋 and t∈𝕋 with 0<μ𝕋(t)<μ^ such that x1 and x2 are indistinguishable by Σ𝕋 at t. This means that that there is a sequence (𝕋i) of time-scales and a sequence (ti) of real numbers such that ti∈𝕋i and μ𝕋i(ti)→0 when i→∞ and for every φ∈ℋ𝕋iti, φ(x1)=φ(x2). Every function ψ∈ℋ may be approximated by functions from ℋ𝕋iti; that is, there is a sequence of functions (ϕi) such that φi∈ℋ𝕋iti and φi→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 ofx1 and x2 is preserved for quantum discretization, where 𝕋=qℕ, if q is sufficiently close to 1.We will show now that the Hermann-Krener rank condition is preserved under discretization. LetHK(x0) denote the Hermann-Krener condition at x0 for the system Σ and HK𝕋(x0) for the system Σ𝕋.Proposition 29.
IfHK(x0), then there is μ^>0 such that for every time-scale 𝕋 if there is t∈𝕋 with 0<μ𝕋(t)<μ^, then HK𝕋(x0).Proof.
Assume thatHK(x0) holds. Then there are ψ1,…,ψn∈ℋ such that dψ1(x0),…,dψn(x0) are linearly independent. Each ψi may be approximated by some φi∈ℋ𝕋t for t∈𝕋 and μ𝕋(t) sufficiently small. Observe that the functions φi depend actually on the parameter μ𝕋(t) and this dependence is continuous. For μ𝕋(t) sufficiently small also dφ1(x0),…,dφn(x0) will be linearly independent. This means that HK𝕋(x0) holds for such 𝕋.The converse of Proposition29 does not hold.Example 30.
LetΣ be
(26)x˙1=-u,x˙2=3x12u,y=x13+x2.
The observation algebra ℋ is generated by a single function ψ(x1,x2)=x13+x2, which means that the Hermann-Krener condition does not hold at any point. The discretized system Σ𝕋 is given by
(27)x1(t+μ𝕋(t))=x1(t)-μ𝕋(t)u(t),x2(t+μ𝕋(t))=x2(t)+3μ𝕋(t)x12(t)u(t),y(t)=x1(t)3+x2(t).
The observation algebra ℋ𝕋 contains now the functions ψ1(x1,x2)=x13+x2 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 idealJx0 is maximal; that is, it is generated by the coordinate functions. One can show that this property is preserved when the ideal Jx0 is replaced with the ideals corresponding to systems Σ𝕋 if 𝕋 contains t with μ𝕋(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.
LetΣ be
(28)x˙1=x2u,x˙2=-x1u,y=x12+x22.
The observation algebra of Σ is generated by φ(x1,x2)=x12+x22.The discretization givesΣ𝕋(29)x1Δ=x2u,x2Δ=-x1u,y=x12+x22,
where
(30)xΔ(t)=x(t+μ𝕋(t))-x(t)μ𝕋(t).
The observation algebra of Σ𝕋 is also generated by φ(x1,x2)=x12+x22.Thus,Σ and Σ𝕋 are both weakly locally observable at x=0. They are not weakly locally observable at any other point.Example27 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-scale𝕋. 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.



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