Observable Space of the Nonlinear Control System on a Homogeneous Time Scale

The observability property of the nonlinear system, defined on a homogeneous time scale, is studied. The observability condition is provided through the notion of the observable space. Moreover, the observability filtration and observability indices are defined and the decomposition of the system into observable/unobservable subsystems is considered.


INTRODUCTION
The theory of dynamical systems on time scales is a new and popular research area.From a modelling point of view, dynamical systems on time scales incorporate both the continuous-and discrete-time systems as special cases, allowing us to unify the study and consider the classical results as special cases of the new theory.However, it is important to note that the discrete-time model in the time scale formalism is given in terms of the difference operator, and not in terms of the more conventional shift operator as, for example, in [1][2][3]13].The difference-based models, often referred to as delta-domain models, are not completely new for the description of discrete-time systems.They have been promoted during the last 20 years as the models closely linked to continuous-time systems, being less sensitive to round-off errors at higher sampling rates [12,20].
The properties (including observability) of linear systems, defined on time scales, were studied, for instance, in [5] and [11].In [4] the algebraic formalism in terms of differential one-forms has been developed for the study of nonlinear control systems defined on homogeneous time scales and used later to study different problems like transfer equivalence, irreducibility, reduction, and realization of nonlinear inputoutput equations [7,17,18].The formalism constructs the vector space of differential one-forms, defined over the differential field of meromorphic functions, associated with the control system.In the present paper we apply this formalism to define and construct the observable space for the nonlinear control system on a homogeneous time scale and define the observability indices of the system.Moreover, we provide the necessary and sufficient condition to check the single-experiment observability 1 of the system using the notion of the observable space.Finally, we discuss the possibility of decomposing the system into observable/unobservable subsystems.
The paper is organized as follows.Preliminary information about the time scale calculus and algebraic framework is given in Section 2. The notions of observability, observability filtration, observable space, and observability indices are provided in Section 3. In Section 4 the decomposition of the system into observable/unobservable subsystems is studied.Section 5 provides brief conclusions.

Time scale calculus
For a general introduction to the time scale calculus see [6].Here we recall only those notions and facts that we need in this paper, in particular, the concept of delta derivative for real function defined on a homogeneous time scale.
A time scale T is an arbitrary nonempty closed subset of the set R of real numbers.The standard cases comprise the continuous-time case, T = R, and the discrete-time cases, T = Z and T = τZ for τ > 0. We assume that T is a topological space with the topology induced by R. In the definition of the delta derivative, the so-called forward jump operator plays an important role.For t ∈ T the forward jump operator σ : while the backward jump operator ρ : T → T is defined by In this definition we set in addition σ (max T) = max T if there exists a finite max T. Obviously σ (t) is in T when t ∈ T. This is because of our assumption that T is a closed subset of R. The graininess functions µ : T → [0, ∞) and ν : T → [0, ∞) are defined by µ(t) := σ (t) − t and ν(t) := t − ρ(t), respectively.A time scale T is called homogeneous2 if µ = ν ≡ const.Let T κ denote a truncated set consisting of T except for a possible maximal point such that ρ(max T) < max T.
Definition 2.1.Let f : T → R and t ∈ T κ .The delta derivative of f at t, denoted by f ∆ (t), is the real number (provided it exists) with the property that given any ε > 0, there is a neighbourhood U = (t − δ ,t + δ ) ∩ T (for some δ > 0) such that Example 2.2.
• If T = R, then µ(t) ≡ 0 and the delta derivative is the ordinary time derivative.
τ is the difference operator.
For a function f : T → R one can define the 2nd delta derivative f [2] In a similar manner one defines higher-order delta derivatives f [n] , n ≥ 1.For notational convenience, denote

Algebraic framework
In this subsection we recall some notions and facts from [4], necessary for our study.Consider a multi-input multi-output (MIMO) nonlinear control system, defined on the homogeneous time scale T, and described by the state equations where x : T → X ⊂ R n is an n-dimensional state vector, u : T → U ⊂ R m is an m-dimensional input vector, and y : T → Y ⊂ R p is a p-dimensional output vector.Moreover, f : X × U → X and h : X → Y are assumed to be real analytic functions.
Remark 2.3.Note that we are focusing neither on local nor global, but on the generic properties of the system, i.e. the properties that hold almost everywhere, except on a set of measure zero.Although the notion of generic property does not make sense, in general, for systems defined by C ∞ functions, the choice of analytic functions allows us to employ the generic approach.Morover, unlike the ring of C ∞ functions, the ring of analytic functions is integral domain, meaning that it can be embedded into its quotient field whose elements are meromorphic functions.
Assume that the map (x, u) → f (x, u) := x + µ f (x, u) generically defines a submersion, i.e. generically rank holds.Assumption (2) is not restrictive, since it is a necessary condition for system accessibility [13] and is always satisfied in the case of µ ≡ 0. Consider the infinite set of (independent) real indeterminates C := {x i , i = 1, . . ., n; u υ , υ = 1, . . ., m, k ≥ 0}.Let K denote the field of meromorphic functions in a finite number of variables from the set C .Thus for each F ∈ K there is k ≥ 0 such that F depends on x and u [0...k] .Let σ f : K → K be the forward shift operator defined by Under the submersivity assumption, σ f is injective endomorphism and so the operator σ f is well defined on K (see [4]).Furthermore, define the operator ∆ f : K → K by

The delta derivative satisfies the following properties:
(i) (iv) on a homogeneous time scale operators ∆ f and σ f commute, i.e.
An operator ∆ f satisfying the rule (iii) of Proposition 2.4 is called a σ f -derivation [9].A commutative field endowed with a σ f -derivation is called a differential field.The field K is endowed with a σ fdifferential structure determined by system (1), and there exists the differential overfield K * , called the inversive closure of K .In [4] the construction of the inversive closure K * for system (1) is given.The extension of σ f to K * is an automorphism [9].
Consider the infinite set of symbols dC = {dx i , i = 1, . . ., n; du υ , υ = 1, . . ., m, k ≥ 0} and denote by E the vector space over the field K * spanned by the elements of dC , namely where only a finite number of coefficients B υk are nonzero elements of K * .The elements of E will be called the differential one-forms.
Let us define the operator d : K * → E as follows: υ . ( Let ω = ∑ i A i dζ i be a one-form, where and One says that ω ∈ E is an exact one-form if ω = dF for some F ∈ K * .A one-form ω for which dω = 0 is said to be closed.It is well known that exact forms are closed, while closed forms are only locally exact.Integrability of the subspace of one-forms may be checked by the Frobenius theorem below, where the symbol dω i means the exterior derivative of one-form ω i and ∧ means the exterior or wedge product (for details see [8]).

OBSERVABILITY AND OBSERVABLE SPACE
Frequently the observability rank condition is used to check whether the continuous-time nonlinear system is locally weakly observable [10,14].This condition is sufficient for an arbitrary initial state and necessary for almost all initial states.Thus, we introduce the definition of observability for nonlinear systems, defined on homogeneous time scales, through the observability rank condition.
Definition 3.1.System (1) is called generically (single-experiment) observable if the rank of the observability matrix is generically equal to n, i.e. if Observe that h for k ≥ 2 and take into account that for T = τZ, τ > 0 the higher-order delta derivative can be computed explicitly as where For T = τZ, τ > 0, the following holds: Proof.Using ( 6), the arbitrary row of the left-hand side matrix in ( 7) may be rewritten as Separating the first addend of the above sum yields Now the sum ∑ i k=1 in the above equality is the linear combination of the previous rows of the matrix and therefore can be removed without changing the rank of the matrix.Since ∂ h σ i f ν /∂ x is the row of the righthand side matrix of (7) for i = 1, . . ., n − 1, the statement of the proposition holds.Remark 3.3.Since for T = R the delta derivative coincides with the classical time derivative, the condition ( 5) is equivalent to the observability rank condition in [10].By Proposition 3.2 in the discrete-time case the condition ( 5) is equivalent to the observability rank condition given in [16].
Although Definition 3.1 may be applied to check observability, it is easier to be done using a concept of observable space like in the continuous-time case [10].Moreover, the observable space, if integrable, allows us to decompose the system into observable/unobservable subsystems.In the remaining part of this section we extend the concept of observable space to the case of (MIMO) systems, defined on homogeneous time scales, and, using the notion of observable space, provide the necessary and sufficient observability condition.
Given system (1), denote by X , Y k , Y , and U the following subspaces of differential one-forms: ν , ν = 0, . . ., p, j ≥ 0 , By analogy with [10], the finite chain of subspaces where is called the observability filtration.Denote by O ∞ the limit of the observability filtration.It is easy to see that and analogously with [10] we call the subspace O ∞ of X the observable space 3 of system (1).The unobservable space of system (1), denoted by X Ō, is defined as a subspace of X , which satisfies where X /O ∞ denotes the factor-space.From ( 8), taking into account (3) and using the linear transformations, one obtains Consequently, according to (10), yielding Before studying the properties of the observable space we provide Lemma 3.4.Denote the one-forms which generate the observable space O ∞ as ω ν, j := ∂ h [ j]   ν ∂ x dx for ν = 1, . . ., p, j ≥ 0 and arrange them in the form of the following matrix: Also denote the arbitrary row of the above matrix by Ω ν .
The proof of Lemma 3.4 is given in the Appendix.
The proposition below describes the property of the subspace O ∞ .
Proof.Represent the observable space as where O ν ∞ is generated by the elements of Thus, the rows of the observability matrix with n columns can be regarded as the representation of the elements of the codistribution O ∞ .Therefore, the number of linearly independent vectors of O ∞ , i.e. dim K * O ∞ , can be found as the rank of the matrix (12).
The following theorem is a direct consequence of Definition 3.1 and Proposition 3.5 and provides the characterization of the observability of the system.Theorem 3.6.A system (1) is (single-experiment) observable if and only if O ∞ = X .Example 3.7.Consider the continuous-time model of unicycle [10] and its discrete-time approximation, based on the Euler sampling scheme, as a single model defined on the homogeneous time scale T: Using (11), the observability filtration (9) of system (13) may be computed as follows: Since the observable space O ∞ = X , the system is observable.Alternatively, one may check that direct application of Definition 3.1 yields the same result but requires more computations: where Given a system of the form (1), its observability filtration (9), like in the continuous-time case [10], defines a set of structural indices σ j for j = 1, . . ., k * by Another set of indices s i , for i = 1, . . ., p, being dual to the set σ j , j = 1, . . ., k * , is defined by and called the set of observability indices of system (1).The integer σ j represents the number of observability indices s i which are greater than or equal to j, and duality implies that σ j = card {s i | s i ≥ j}.
Observability indices determine how many delta derivatives of the respective output components one needs to use for computation of the initial state x on the basis of the inputs and outputs and their delta derivatives.The following proposition describes the key property of the observability indices.Proposition 3.8.Given a system of the form (1), one has (14), one can write Separating the last addend of the first sum in the right-hand side of (15), replacing in this sum index j by j − 1, and taking into account that The relation between indices σ j and s i can be expressed by means of a k * × p table, whose ( j, i)th element is defined by ( j = 1, . . ., k * pointing to the row and i = 1, . . ., p to the column) Thus, the indices σ j and s i are the sums of elements in the jth row and ith column, respectively, i.e.
Taking ( 16) and ( 17) into account, one obtains which completes the proof.

DECOMPOSITION
For certain applications it will be useful to have system representations in which observable and unobservable state variables can be explicitly distinguished.For a continuous-time nonlinear control system the decomposition into observable/unobservable subsystems has been carried out both via differential geometric [15,21] and linear algebraic methods [10] and is proved to be always possible.For example, in [10] the decomposition was first carried out for a linearized system defined in terms of one-forms, and then, it was proved that the observable subspace of differential one-forms is always (generically) integrable.Therefore, the observable subspace of one-forms can be (at least locally) spanned by exact one-forms whose integrals define the observable state coordinates.As demonstrated in [16], for the discrete-time nonlinear control systems described in terms of the shift operator σ f the decomposition at the level of equations (state variables) is not always possible since the observable space of one-forms is not necessarily completely integrable.Moreover, the paper [19] provides a general subclass of systems with a non-integrable observable subspace.The purpose of this section is to study the possibility of decomposing the nonlinear control system defined on a homogeneous time scale into observable and unobservable subsystems.Since the delta-domain model obtained via sampling [12] behaves similarly to the continuous-time system and at the limit, when the sampling frequency increases infinitely, approaches the continuous-time system, it was our working hypothesis that the delta-domain models are, in general, decomposable into observable/unobservable parts.This would mean that the respective observable space O ∞ , as a space of differential one-forms, is completely integrable.In [10] the observable space O ∞ is proved to be integrable in the case of µ ≡ 0 (T = R).Unfortunately, unlike the case T = R for the case T = τZ, τ > 0, O ∞ is not necessarily integrable.We give a number of counterexamples.
Example 4.1.Consider the control system, defined on a homogeneous time scale: By ( 9), for this system, The next example demonstrates that the loss of integrability does not necessarily occur only at µ = 1.
Example 4.2.Consider the system The observable space of the system , again non-integrable by the Frobenius theorem.
Finally, we provide an example of the system for which the observable space O ∞ is integrable for every choice of the value of µ.
To conclude, we conjecture that the observable space O ∞ is in general integrable, except for a few possible µ values where these values correspond to the sampling frequencies at which the state transition map of the sampled system is not reversible.The following example illustrates this conjecture.18) is where we use the notation x + := x(t + µ).In order to check the reversibility of the system, one needs to verify whether the Jacobian matrix ∂ f (x, u)/∂ x is nonsingular.The Jacobian matrix of system ( 21) is One can verify that the above matrix is singular for µ = 1, implying that the state transition map (21) is not reversible at the sampling frequency equal to 1. Next, consider the state transition map of system (19), which reads as The Jacobian matrix of system (22), i.e.
is singular for µ = 3.Consequently, the state transition map (22) is not reversible at the sampling frequency equal to 3. Finally, the state transition map of system ( 20) is and its Jacobian matrix reads as where a := µ u 1 cos 2 (x 1 −x 2 ) + u 2 sin (2 (x 1 − x 2 )) .One can verify that the above matrix is nonsingular for any µ ≡ const, meaning that the state transition map ( 23) is reversible at any sampling frequency.To conclude, comparing the above result with those presented in Examples 4.1-4.3,one can observe the consistency of the sampling frequencies at which the state transition maps are not reversible and the values of µ for which the observable spaces O ∞ are not integrable.These examples support our conjecture.

CONCLUSIONS
Although the theory of continuous-and discrete-time dynamical systems as presented in the literature is different, the analysis of time scales is nowadays recognized as a right tool to unify the seemingly separate fields of discrete dynamical systems (i.e.difference equations) and continuous dynamical systems (i.e.differential equations).In the paper we studied the observability of multi-input multi-output control systems on a homogeneous time scale, which allows us to unify continuous-and discrete-time theories, presenting both of them simultaneously under the same language.The presented approach covers the continuous-and discrete-time cases in such a manner that those are special cases of the formalism.Since the delta derivative (used in our paper to describe the dynamical systems) coincides with the time derivative for the continuoustime case, the results available in the literature can be obtained from our results as a special case, namely the case in which the time scale is a set of real numbers.On the other hand, our formalism includes the description of a discrete-time system based on the difference operator description (delta-domain approach), for which the results shown in the paper are new, since previous results have been obtained for discrete-time systems considered on the basis of the shift-operator formalism.Therefore, in our paper the discrete-time systems are described in terms of the difference operator, unlike in the majority of papers where the system is described via the shift-operator.To conclude, although the computation of the delta derivative is different in the continuous-and discrete-time cases, the results obtained by means of it are the same for both time domains.
In the paper the notion of the observable space was used to provide the observability condition that can be easily checked.However, note that the definition of the observability was introduced through the observability rank condition, commonly used both in continuous-and discrete-time cases.One of the future goals is to define the observability of the nonlinear system on a homogeneous time scale, using the concept of (in)distinguishable states.Another goal is to find the conditions under which the nonlinear system defined on a homogeneous time scale is transformable into the observer form, which allows construction of an observer with linearizable error dynamics.
for some β k 's.By Lemma 5.1 and ( 27) Using (iii) of Proposition 2.4 for α k and then (i) of Proposition 2.4 for α k , we get By Lemma 5.1 Changing the summation index of the second sum for s = k + 1, separating the last addend of the second sum, and applying (27) to it, we obtain Separating the first addend of the first sum yields Denoting β 0 := α ∆ f 0 +α k−1 , we get (28).Similar arguments can be applied to the case j > i + 1.