Static state feedback linearization of nonlinear control systems on homogeneous time scales

The paper addresses the problem of static state feedback linearization for nonlinear control systems defined on homogeneous time scales. Necessary and sufficient conditions for generic local linearizability of the considered systems by static state feedback and state transformation are presented in terms of a sequence of subspaces of differential one-forms related to the system.

of today. Time scale calculus is a general framework that allows unification of the study of continuous-and discrete-time systems [8]. The term "time scales" refer to the way dynamic systems behave over time. Most engineering applications assume time to be either continuous or (uniformly) discrete, both of which are merged in time scale formalism into a general framework, and follow from the latter as special cases. The main concept of time scale calculus, the delta derivative, is the generalization of both time derivative and the difference operator. Though time scale calculus accommodates more possibilities, regarding the control theory, the most important special cases are continuous-and (uniformly) discrete-time systems that are the examples of systems defined on homogeneous time scale.
The description of the dynamics that relies on the difference operator is often called delta-domain description, see for instance [13,17,18,22,30,37,39]. The delta-domain approach has been promoted as an effective tool for dynamic system modeling and control. Compared to models based on the shift operator, the delta-domain models are less sensitive to round-off errors and do not yield ill-conditioned models when signals are sampled at a high sampling rate, see [23,31,33]. In [1], the authors demonstrate that the numerical properties of structure detection are improved by a delta-domain model and such delta-domain models provide models closely linked to the continuous-time systems. The delta-domain approach allows to address the question of preservation of the system properties under Euler discretization scheme since it treats the discretized systems as a special case of a system defined on a time scale.
Though the literature on time scale calculus is rich (see the references in [3,6,9,16,19,20,32,34,38]), there is not so many papers addressing the control problems for nonlinear systems defined on time scale [7,12,28]. In the earlier papers [4,5] the algebraic formalism has been developed that allows to address various analysis and control problems for nonlinear systems, defined on homogeneous time scales. This approach has been successfully applied to address the irreducibility, reduction [28] and realization of the input-output equations in the state space form [12].
The theories of continuous-and discrete-time dynamical systems as presented in the literature are different, but the analysis on time scales is nowadays recognized as the right tool to unify the seemingly separate fields of discrete dynamical systems (i.e., difference equations) and continuous dynamical systems (i.e., differential equations), and also to present both continuous-and discrete-time theories in the same language. The goal of this paper is to study the iconic problem of static state feedback linearization for nonlinear systems defined on homogeneous time scales where the state transformation method based on one-forms is used. The presented approach covers the continuous-and discrete-time cases in such a manner that those are the special cases of the formalism. Since delta derivative (used in our paper to describe the dynamical systems) coincides with the time derivative for the continuous-time case, the earlier results for continuous-time systems, see for instance [15,25], follow from our results as a special case, namely the case in which the time scale is the set of real numbers. Note, however, that our results do not recover those given in [2,21,26,29] for the discrete-time systems, defined in terms of the shift operator since the latter is not a delta-derivative. Instead, our formalism includes the description of a discretetime system based on the difference operator description (delta-domain approach), for which the results shown in the paper are new. Using the difference operator in the discrete case allows to unify the theories of static state feedback linearization for continuous-and discrete-time nonlinear control systems.
Static state feedback linearization is a much studied research topic since 1980s, both for continuous-and discrete-time cases, see [24,26,27] and the references therein. Many different techniques that speak different languages (including differential geometry and algebra) have been used in linearization studies that make their comparison a difficult task. The paper [35] establishes the explicit relationship between the two sets of necessary and sufficient solvability conditions, given in terms of integrability of certain vector spaces of differential one-forms and involutivity of distributions of vector fields. In particular, it has been demonstrated that the distributions, when they are involutive, are the maximal annihilators of the corresponding codistributions (vector spaces of one-forms). Moreover, in [35], two methods have been compared from the point of view of computational complexity and it was shown that regarding the computation of the state transformation method based on one-forms is simpler. By this reason, we decided to rely on this method in unification. Finally, note that the results of [11] are claimed in [11] to be dual to those of [2] though not shown in detail.
The paper is organized as follows. Section 2 recalls the notions from time scale calculus and algebraic formalism used in the paper, while in Sect. 3 the properties of the sequences of subspaces of differential one-forms are presented, in terms of which our main results are formulated. The basic results given in Sect. 3 together with the facts presented in Sect. 4 contain the key contribution of the paper. Section 4 is devoted to give the main results, i.e., the conditions for the generic local static state feedback linearization of control systems defined on homogeneous time scales. Finally, some illustrative examples that describe our results are presented.

Time scale calculus and algebraic formalism
For a general introduction to the calculus on time scales, see [8]. Here, we recall only those notions and facts that will be used later.
A time scale T is a nonempty closed subset of R. We assume that the topology of T is induced by R. The forward jump operator σ : T → T is defined as σ (t) := inf{s ∈ T | s > t}, σ (max T) = max T if there exists a finite max T, the backward jump operator From now, we assume that T is a homogeneous time scale.

Remark 1
In [5], we called time scale homogeneous if only μ is a constant function. The reason is that when one only needs the forward shift and delta derivative, it is not necessary to have constant ν. However, if we also need the backward shift and nabla derivative, we have to assume additionally ν to be constant since μ being constant does not yield the latter. For example, in the set of natural numbers μ ≡ 1, but ν(1) = 0 whereas ν(k) = 1 for k ≥ 1; so ν is not constant.
Let us now recall the definition of the delta derivative f Δ of a real valued function f .

Definition 1
The delta derivative of a function f : T → R at t ∈ T is the real number f Δ (t) (provided it exists) such that for each ε > 0 there exists a neighborhood U (ε)

Remark 2
The delta derivative is a natural extension of time derivative in the continuous-time case and forward difference operator in the discrete-time case. Therefore, for where is the usual forward difference operator.
For a function f : T → R, we can define the second delta derivative f [2] := ( f Δ ) Δ provided that f Δ is delta differentiable with derivative f [2] : T → R. Similarly, we define higher order delta derivatives f For f : If f and f Δ are delta differentiable functions, then for homogeneous time scale one has f σ Δ = f Δσ . Now, we recall some definitions and facts given in [4,5] that will be used in the paper.

Differential field and space of one-forms
Consider now the control system, defined on a homogenous time scale T, where (x(t), u(t)) ∈ X × U, X × U is an open subset of R n × R m , m n, and f : X × U → R n is analytic. We assume that the input applied to system (1) is infinitely many times delta differentiable, i.e., u [k] exists for all k 0. Let us define f (x, u) := x + μf (x, u) and assume that there exists a map ϕ : , we can uniquely compute (x, u) as an analytic function of (x, z). For μ = 0, this condition is always satisfied with ϕ(x, u) = u.
Note that for T = R, the Eq. (1) becomes an ordinary differential equation and when T = hZ, for h > 0, it becomes the difference equation. Both T = R and T = hZ are homogeneous time scales.
For notational convenience, (x 1 , . . . , x n ) will simply be written as x, and (u [ and let K be the (commutative) field of meromorphic functions in a finite number of the variables from the set C. Let σ f : K → K be an operator defined by where .k] + μu [1..k+1] , for k 0 and by (1) Under the assumption about the existence of ϕ such that Φ is an analytic diffeomorphism, σ f is an injective endomorphism.
The field K can be equipped with an operator Δ f : K → K defined by Remark 3 Note that in both continuous-time and discrete-time case, the operator Δ f corresponds to the total delta derivative of a function with respect to the dynamic of the system, i.e., for control u(·) applied to system (1), for solution x(·) of (1) corresponding to control u and F ∈ K we get On the right hand side of (4), the operator Δ f is applied to function F which depends on indeterminates from the set C and substituting x(·) and u [l] (·) instead of x and u [l] , respectively, we get relation (4) where the time appears.
The operator Δ f satisfies, for all F, G ∈ K, the conditions An operator satisfying the generalized Leibniz rule is called a "σ f -derivation" and a commutative field endowed with a σ f -derivation is called a σ f -differential field [14]. Therefore, under the assumption about the existence of ϕ such that Φ is an analytic diffeomorphism, K endowed with the delta derivative Δ f is a σ f -differential field. The σ f -differential field K is called inversive, if every element of K has a preimage in K with respect to σ f , i.e., σ −1 f (F) is defined for all F ∈ K, see [14]. For μ = 0, σ f = σ −1 f = id and K is inversive. Though K is not inversive in general, it is always possible to embed K into an inversive σ f -differential overfield K * , called the inversive closure of K [14]. Since σ f is an injective endomorphism, it can be extended to K * such that σ f : K * → K * is an automorphism. For μ = 0, the overfield K * consists of meromorphic functions in a finite number of the independent variables [4,5]. Let z := (z 1 , . . . , z m ). Then, , where ψ is a certain vector valued function, determined by f in (1) and the extension z = ϕ(x, u). Although the choice of variables z is not unique, all possible choices yield isomorphic field extensions. We extend the operator Δ f to new variables by The extension of operator Δ f to K * can be made in analogy to (3). Such operator Δ f is now a σ f -derivation of K * . A practical procedure for construction of K * (for μ = 0) is given in [5], where the explicit construction of an inversive σ f -differential overfield K * is unique up to an isomorphism, see [5]. Note that K * = K for μ = 0. We assume that for system (1) the assumption of independence of controls holds. 2 We will consider regular state feedbacks, which are defined as follows:

Remark 4
The regular static state feedback defined above is not globally invertible, but the condition rank ∂φ ∂v = m guarantees local invertibility with respect to v for each fixed point x ∈ X . Moreover, the map ϑ : From now on Consider the infinite set of differentials of indeterminates dC * = {dζ i , ζ i ∈ C * }.
. . , m, k 0, can be treated as differentials of coordinate functions which select x i or u [k] j from the set of all variables, respectively. Define E := span K * dC * . Any element of E is a vector of the form where only a finite number of coefficients B jk and C s are nonzero elements of K * . Since F is a meromorphic function in a finite number of variables from the set C * , then an operator d : K * → E can be defined in the standard manner: The elements of E are called differential one-forms. One says that ω ∈ E is an exact one-form if ω = dF for some F ∈ K * . We will refer to dF as to the total differential (or simply the differential) of F.
The operator σ f : E → E is invertible and the inverse operator σ −1 f : E → E is given by [5]), For the homogeneous time scale T, we have for Note that the following relation between operators Δ f and σ f holds where id denotes the identity operator. We will use sometimes the more compact notations

Vector fields and p-forms
Let E be the dual vector space of E, i.e., the space of linear mappings from E to K * . The elements of E are of the form where a i , b jk , c s ∈ K * and are called vector fields.
∈ E and the vector field X ∈ E of the form (9), we get The delta-derivative X Δ f and the forward-shift X σ f of X ∈ E may be defined uniquely by the equations and Let E p be the set of differential p-forms, see [4]. Every p-form α ∈ E p has a unique representative of the form The exterior product (alternatively called the wedge product) of a p-form where F i 1 ···i p , G j 1 ··· j q ∈ K * and ζ i l , ζ j s ∈ C * , l = 1, . . . , p, s = 1, . . . , q, see [4].
Let E be the space of all forms, i.e., Exterior differential d is an operator d : E → E that satisfies the following properties: then dF coincides with the ordinary differential, see (7).
The properties (i)-(iii) define uniquely the operator d.
The operators Δ f : K * → K * and σ f : K * → K * , related to system (1), induce the operators Δ f : E p → E p and σ f : and where ζ i 1 , . . . , ζ i p ∈ C * and A i 1 ···i p ∈ K * . Proposition 2 [4] Let ω ∈ E p , p 1. Then, for a homogeneous time scale T: Now let us consider an operator i X : E → E associated with a vector field X ∈ E that satisfies the following properties: The operator i X is called interior product with respect to X . Note that i X (dζ i ) = X, dζ i so that for a differential 2-

Sequence of subspaces of one-forms
In this section, we introduce the sequence of subspaces of E being the main tool for characterizing solvability/solution of numerous problems in control theory, for example accessibility (irreducibility), realization and feedback linearization.
Introduce, in analogy with [15], the sequence of subspaces (H k ) of E defined by Then, H 1 := span K * {dx}. Note that dim K * H 0 = n + m and dim K * H 1 = n. Let The relative degree of a one-form ω ∈ H 0 is defined as: If such integer does not exist, set r = ∞. The relative degree of a meromorphic function A(x, u) is defined to be the relative degree of the one-form d A(x, u).
The following proposition is a simple consequence of the construction of H k .
Proposition 3 (i) For k 0, H k is the space of one-forms that have relative degree greater than or equal to k. (ii) There exists an integer 0 < k * n such that, for 0 k k * , H k+1 H k and H k * +1 = H , k * + 1.
Proof Point (i) is a simple consequence of definition (15). Concerning (ii), the existence of the integer k * comes from the fact that each H k is a finite-dimensional K * -vector space so that, at each step, either its dimension decreases or H k+1 = H k . Moreover, k * n = dim K * H 1 .
Define H ∞ := H k * +1 . Then from the construction of H k , we get Now, we will show that the coordinate transformation yields an inversive closure isomorphic to the original one. Consequently, we have some correspondence between the constructed subspaces of one-forms. Letx = ξ(x), ξ = (ξ 1 , . . . , ξ n ) and ξ : X → X ⊂ R n be an analytic diffeomorphism. Then,x Note that x and z − , 1, belong to some open subsets of R n and R m , respectively. Let K * be the field of functions in variables from the set C : Moreover, K * is equipped with operators Δf and σf defined in a similar manner as operators Δ f and σ f (see definitions (3) and (2), respectively). Observe that if U = R n , then K * = K * . It is easy to see that the relation similar to (8) holds also for the operators Δf and σf , i.e., Let E := span K * {dx, du [k] , dz − , k 0, 1}. Then, ξ induces the isomorphisms ξ * : K * → K * and ξ * : E → E by If G ∈ K * and G depends only on u [0..k] , z −1 , . . . , z − and does not depend on x, then ξ * ( G) = G.
In general, we have and Moreover, and for 2 Therefore, we get Note that u [k] , k 0 and z − , 1 are the same for K * and K * . Consequently,

Proposition 6
Let ω ∈ E. Then, The proof is given in the Appendix.

Proposition 7
For subspaces H k , H k defined by (15) and (22), respectively, we have, for k 0 Proof The proof is by the induction principle. Since the diffeomorphism ξ induces isomorphism ξ * : E → E and for arbitrary ω (23)  and it is equivalent to ξ * ( ω) ∈ H n+1 . Hence using the induction principle, we get that (23) holds for all k 0.

Remark 6
The dimension of the subspace H k , k 0, is invariant under diffeomorphism ξ , i.e., dim U) is an analytic diffeomorphism. Then, the field K * of meromorphic functions in a finite number of variables from the set C : φ(x, v)), and E can be constructed similarly as K * and E, respectively. Then, ϑ induces the isomorphisms ϑ * : K * → K * and ϑ * : E → E by the formulas similar to (17) and (18). Moreover, the sequence ( H k ) can be defined by analogy with (22). Subspaces H k and H k , for k 0, are isomorphic similarly as in Proposition 7. In particular, Since ϑ is a diffeomorphism, one gets Note that the relative degrees of the one-forms are invariant under regular static state feedback. This fact is well known for continuous-time systems, see Proposition 8.8 in [36], and carries easily over for systems, defined on homogeneous time scales. Then, the following property characterizes the subspaces H k : Then, there exists a list of integers r 1 , . . . , r m and one-forms ω 1 , . . . , ω m ∈ H 1 whose relative degrees are, respectively, r 1 , . . . , r m such that are linearly independent over the field K * ; in particular m i=1 r i = n.
where a i s, j , b i j ∈ K * and [b i j ] has the inverse in the ring of m ×m matrices with entries in K * .
Proof For 1 i m and 1 j r i , we take ω i, j = ω Hence Thus, α ∈ H k+1 if and only if the coefficients a σ f i satisfy the following system of linear equations: System (26)  The proof of Proposition 8 provides a procedure to compute bases of the subspaces H k , k 2. Let ω i, j , 1 i m, 1 j r i be a basis of H 1 satisfying (25). Observe that there exists an integer number M such that one-forms ω i, j depend on (x, u, u [1] , . . . , u [ι] , z −1 , . . . , z −κ ) ∈ X × U × R ι+κ ⊂ R M . Let S be an open and dense subset of X × U × R ι+κ such that the forms ω Δ j f i , 1 i k * , 0 j r i − 1 evaluated at (x, u, u [1] , . . . , u [ι] , z −1 , . . . , z −κ ) ∈ S are linearly independent over R.
Let us recall that a codistribution on V ⊂ R M is a map H : The codistribution H is locally integrable if for each point p ∈ V there exist some neighbourhood V of p and exact one-forms defined on V such that for every q ∈ V the one-forms evaluated at q form the basis of H q . If p → dim H p is constant and H is generated by one-forms ω 1 , . . . , ω r (i.e., at each p ∈ V , H p is spanned by ω 1 ( p), . . . , ω r ( p)), then local integrability of H is equivalent to the Frobenius condition, i.e., dω k ∧ ω 1 ∧ · · · ∧ ω r = 0, for 1 ≤ k ≤ r , see for instance [10].
Observe that the spaces H k are not codistributions in the sense defined above. To associate with H k a codistribution on S, for p ∈ S, we set H p to be the space of ω( p), where ω ∈ H k is well defined at p. This codistribution is generated by the one-forms given in Proposition 8 and S is the subset of X × U × R ι+κ on which these one-forms are well defined and linearly independent at every point. We say that H k is locally integrable if the codistribution associated with H k is locally integrable.
Observe that under Assumption 1, all H k , k 2, define constant-dimensional codistributions on S.
Remark 7 Later, we shall restrict variables x and u to a neighbourhood of some (x,ū) ∈ X × U. This will result in H k and H ∞ restricted to such a neighborhood. Let us notice that H ∞ = {0} if and only if H ∞ restricted to all such neighbourhoods is equal 0.

Feedback linearization
Definition 3 System Σ of the form (1) is said to be linearizable by static state feedback if there exist a state analytic diffeomorphism ξ : and a static state feedback (6) such that ϑ : is an analytic diffeomorphism and, in new coordinates, we havê where the pair (A, B) locally takes form (25). Note that "locally" means in some neighborhood of arbitrary point (x,ū) ∈ T . By Frobenius's Theorem, there is no loss of generality if we assume that the basis {ω i, j , 1 i m, 1 j r i } contains exact one-forms. Thus, every ω i, j can be integrated, i.e., there exist ξ i, j such that (1) can be written as: . .
where ∂f i ∂x s, j = a i s, j and ∂f i ∂u j = b i j . Under the feedback u(t) = φ(x(t), v(t)), wherê  [15] which is a differential-form version of a vector-field characterization presented in [24,27]. For discrete-time systems, the results of Theorem 2 are new because they are described in terms of difference operator while the shift operator is used for discrete time in the literature, see for instance [2]. Now let us present examples that illustrate our results. The first example has been suggested by one of the reviewers.
By formula (24), H 0 = span K * {dx 1 , dx 2 , du}. From Proposition 3 (i), it easily follows that H 1 = span K * {dx 1 , dx 2 }, since relative degrees of dx 1 and dx 2 are obviously greater or equal to 1, and relative degree of du equals to 0. To compute the next elements of the sequence H k , the algorithm in the proof of Proposition 8 has to be applied repeatedly.
Remark 10 Note that for the systems defined on the homogeneous time scale with μ > 0, by (8) and (15), we get ω ∈ H k+1 if and only if ω ∈ H k and σ f (ω) ∈ H k . Hence and, consequently, H k+1 in (15) can be alternatively defined as:

Conclusions
In the paper, the necessary and sufficient conditions for the generic local static state feedback linearizability of nonlinear control systems defined on homogeneous time scales are given. Our main contribution has been to show the properties of subspaces of differential one-forms that contain considerable structural information about the system. Then, one of the main results, i.e., necessary and sufficient conditions for generic local linearizability by static state feedback, are formulated in terms of these subspaces. Our future goal is devoted to extend the results to non-homogeneous but regular time scales.