A note on the relationship between single- and multi-experiment observability for discrete-time nonlinear control systems

The connection between the concepts of the single-experiment and the multi-experiment unobservability of a nonlinear discrete-time control system is studied. The main result claims that if the system is single-experiment unobservable and the observable space is integrable, then the system is also multi-experiment unobservable. For the proof of the main result a novel mathematical technique, the so-called algebra of functions, is used.


INTRODUCTION
Observability is a fundamental property of the control system.Different applications rely on different observability concepts.In some situations, one is given just a single sequence of inputs and the corresponding sequence of outputs to find the arbitrary initial state.If this is possible, the system is called single-experiment observable.In some other cases, one is allowed to use several input sequences together with the corresponding output sequences to determine the initial state.The associated observability concept is called multi-experiment observability.Obviously, if the system is single-experiment observable, it is also multi-experiment observable.For continuous-time systems also the converse is proved to hold, at least for analytic systems [5].In the discrete-time case, for the converse to hold, one has to assume additionally that the (analytic) system is reversible [5].However, for discrete-time nonlinear control systems, simple bilinear nonreversible examples exist, demonstrating that the system may be single-experiment unobservable but multi-experiment observable [2] (see also a more complicated example in [5]).The goal of this short paper is to study further the connection between two observability notions, and relate it with the integrability of the observable space.Note that the possible non-integrability of the observable space is purely a discrete-time phenomenon [2,3] like the possibility that the analytic system may be multi-experiment observable but not single-experiment observable.The main result of this paper claims that if the system is single-experiment unobservable and the observable space is integrable, then the system is also multiexperiment unobservable.No reversibility assumption is made in proving this result.

PRELIMINARIES
Consider a single-input single-output nonlinear discretetime control system, described by the state equations where and h : R n → Y are the real-analytic functions.Notice that in this paper we use symbols + and [i] instead of the arguments t + 1 and t + i, respectively, to simplify the exposition, so x + := x(t + 1), x := x(t), and Assume that the map (x, u) → f (x, u) generically defines a submersion, i.e. generically rank holds.
Over the field K one can define a difference vector space Under the assumption (2), K is a difference field [2].

Observability
Both in the single-or multi-experiment context, the observability property that is easiest to characterize for nonlinear systems is local weak observability.In the rest of the paper, local weak observability will simply be called observability.Further discussion for discrete-time systems can be found in [1].
The chain of subspaces where O k := X ∩ Y k + U is called the observability filtration.If we denote by O ∞ the limit of the observability filtration, it is easy to see that and we can introduce the following definition.
Definition 1.The subspace O ∞ is called the observable space of system (1).
If the observable space is integrable, the system can be decomposed into the observable and unobservable subsystems.Unfortunately, unlike in the continuoustime case for discrete-time systems, O ∞ is not necessarily integrable [2].If O ∞ is integrable, and therefore, has locally an exact basis {dz 1 , . . ., dz r }, one can complete the set {z 1 , . . ., z r } to a basis {z 1 , . . ., z r , z r+1 , . . ., z n } of X , where z 1 , . . ., z r are observable coordinates and z r+1 , . . ., z n are unobservable coordinates.Then, in these coordinates, the system can be decomposed into observable and unobservable subsystems For each control sequence ω ∈ R k , define f ω : X → X inductively by f e (x) = x for the empty sequence and Definition 2. System (1) is said to be multi-experiment observable if any two distinct states x 1 , x 2 can be distinguished by some input sequence µ.
To prove the main result of the paper, we apply a special mathematical technique, the so-called algebra of functions.

Algebra of functions
The main elements of the algebra of functions are binary relations, operations, and operators.Here those concepts, necessary to understand this paper, are recalled; for more details see [4,6,7].
Let X ⊆ R n be a vector space.Denote by ℑ X the set of vector functions with the domain X.Consider two arbitrary functions α : X → S and β : X → W from ℑ X , where S ⊆ R s and W ⊆ R r are some vector spaces.One can define the binary relation of partial preorder for functions α and β as follows.
The rule of operation × is simple Definition 6.We say that the functions α, β ∈ ℑ X form a pair and denote this as (α, β ) ∈ ∆ if there exists a function f * such that β ( f (x, u)) = f * (α(x), u) for every (x, u) ∈ X ×U.
Definition 8.The function α ∈ ℑ X is called a maximal f -invariant function if for every f -invariant function α * the following holds: α * ≤ α.
Note that Definitions 3-8 hold sometimes only locally.In such a case all claims hold for some open and dense subset of X rather than for X.Consider for instance the following example1 .
Example 1.Consider the functions α = arctan x 1 x 2 and β = arctan x 2 x 1 .To verify whether the functions are equivalent or not, first we show that Since there exists a function γ such that γ x 1 x 2 = α, according to Definition 3, x 1 x 2 ≤ α.Moreover, there exist γ = tan(•) such that γ (α) = x 1 x 2 , which demonstrates that α ≤ x 1 x 2 .As a consequence, according to Definition 4, the relation (3) holds.In a similar manner one can show that β ∼ = x 2 x 1 .But since arctan is a multivalued function, in both cases we can speak only about local equivalence.Even if we consider only the main branch of arctangent, the equivalence still holds only locally.Taking γ = x −1 , one can easily show that x 1 x 1 .However, the latter equivalence is violated for x 1 = 0 or for x 2 = 0.As a result, α ∼ = β holds only locally.
One may define the operator M as follows.
Definition 9. Given a function β ∈ ℑ X , the function M(β ) is defined by the following two conditions: where α is an arbitrary function.
The function M(β ) exists for every function β and is unique [8], therefore, M(β ) may be understood as an operator acting on functions from ℑ X .The operator M can be computed in the following way.Let β be a scalar function such that its composition with the function f can be represented as where a 1 (x), a 2 (x), . .., a s (x) are arbitrary functions and b 1 (u), b 2 (u), . .., b s (u) are linearly independent.Then If (5) does not hold, the procedure to compute the operator M(β ) is given in the theorem below.

Theorem 1 ([8]
).Let the composition β ( f ) be represented as where the function χ ∈ V ⊂ R s+1 is a function, α 1 (x), α 2 (x), . . ., α s (x) are the functions satisfying the following condition: there exist inputs u = c 1 , u = c 2 , . . ., u = c r , such that every function α i (x) may be expressed via the family of composite functions the following functional inequality holds: One of the properties of the operator M is

MAIN RESULT Theorem 2 ([6,7]). The system is multi-experiment unobservable if and only if there exists an f -invariant noninjective function α satisfying the condition α ≤ h.
Observability criterion.Let the vector function α be the maximal f -invariant function satisfying α ≤ h.In order to find α, one has to define the sequence of vector functions α i as follows: It is easy to note that the functions α 0 := h, α 1 , α 2 , . . .form the non-increasing sequence: The sequence converges, thus for the first k, which satisfied the equivalence relation α k ∼ = α k+1 , we define α := α k .According to Theorem 2, system (1) is observable if the function α is injective, otherwise the system is unobservable.
The following examples show how the observability criterion can be practically used.
Example 2. Consider the control system Compute: To compute M(h), one has to find the composition h( f (x, u)) = ux 1 + x 4 .Thus, now in (5) the functions and due to (7) can be computed as ), one has to find the composition x 2 • f (x, u) = x 3 , which yields that for M(x 2 ) the functions a 1 (x) = x 3 , b 1 (u) = 1 and M(x 2 ) = x 3 .As a result, one obtains: Note that according to Definition 5, the product of two functions is their maximal bottom and, as a consequence, x 3 × x 3 = x 3 .(We always keep only the functionally independent components of the result, trying to simplify the result as much as possible.)As a result, α 3 = x 3 × x 1 × x 4 × x 2 and α 2 ∼ = α 3 , yielding that the function Example 3. Consider the system (10) For this system the sequence of functions α i , i = 1, 2, . . ., is the following: Obviously α 1 ∼ = α 2 , consequently α = α 1 = (x 1 − x 3 ) × x 2 is the maximal f -invariant function, satisfying the condition α ≤ h.Since α is not injective, system (10) is multi-experiment unobservable.
We are now ready to present the main result of the paper.
Theorem 3. If system (1) is single-experiment unobservable and the observable space O ∞ is integrable, then (1) is multi-experiment unobservable.
Example 4 (Continuation of Example 3).The observable space of this system O ∞ = span K {dx 1 − dx 3 , dx 2 } is integrable and the system is single-experiment unobservable.As was shown in Example 3, the system is also multi-experiment unobservable, which confirms the statement of Theorem 3.
The example below demonstrates that the converse statement of Theorem 3 is not valid.Obviously, if the system is multi-experiment unobservable, it is also single-experiment unobservable.However, one cannot say anything about integrability of O ∞ .(13) In order to verify that the system is multi-experiment unobservable, compute the sequence of functions α i , defined by (8): Obviously α 1 ∼ = α 2 , consequently α = α 1 = x 3 × x 1 × x 2 is the maximal f -invariant function satisfying α ≤ h.Since α is not injective, system (13) is multi-experiment unobservable.The observable space of system (13) is O ∞ = span K {udx 1 − dx 2 , dx 3 }.Although system (13) is also single-experiment unobservable, its observable space O ∞ is not integrable.