Nabla derivatives associated with nonlinear control systems on homogeneous time scales

Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15-351 Bialystok, Poland z.bartosiewicz@@pb.edu.pl; m.wyrwas@pb.edu.pl Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia kotta@cc.ioc.ee; tanel.mullari@ttu.ee; maris@cc.ioc.ee Faculty of Mechanical Engineering Bialystok University of Technology, Wiejska 45C, 15-351 Bialystok, Poland e.pawluszewicz@pb.edu.pl


Introduction
In papers [2,3], an algebraic formalism for nonlinear control systems, defined on homogeneous time scales, has been developed.It has already found applications in the solution of several control problems like system reduction [17], realization of the external system description in the state space form [9]. Note that the properties of both the continuous-and discrete-time control systems can be studied, characterized and checked by the same mathematical techniques, theorems and algorithms.The formalism is based on differential one-forms and vector fields.In our set-up,the vector fields may have infinite number of terms.Though in the classical control theory the vector fields are defined in the finite-dimensional spaces, the infinite dimensional versions are not completely new.For instance, in [12], the infinite vector fields are considered.They have been also used in problems related to dynamic state feedback like in the studies of flatness [18].An inversive σ f -differential field K * of meromorphic functions in system variables is constructed and equipped with two operators, delta derivative ∆ f and forward shift σ f , defined by system equations.Then two vector spaces over K * , of one-forms and of vector fields, respectively, were introduced, and the operators ∆ f and σ f were extended to these vector spaces.
The goal of this paper is to introduce the nabla derivative operator in these vector spaces and prove a number of its properties.Note that the literature on control systems on time scales is, up to now, mostly limited by application of delta derivatives, see, for example, [4,8,9].However, some recent applications in economics, neural networks, control and other topics [1,11,14,16] have suggested that in some situations nabla derivative is more natural.In particular, backward differences are preferable in numerical and computational methods due to practical implementation (information availability) reasons.Moreover, in control and neural networks, nabla derivative is a natural tool that allows to handle delays in system [13,16].In differential geometric approach to control problems, in majority of cases (though not always), nabla derivative of a vector field is much easier to compute than the delta derivative.Though the computation of both derivatives requires to apply the backward shift operator, defined by the inverse map of the extended system equations, in the case of delta derivative, the shift is applied to a one-form whereas in the case of nabla derivative, it is applied to a function.Since in the continuous-time case both delta and nabla derivatives coincide, they yield the same concept and whenever one assumes the finite-dimensional space, both result in classical Lie derivative along the vector field, defined by system equations (see Corollary 1 below).Therefore, it is natural to assume that several distributions, important for control theory, expressed in terms of Lie derivatives of vector fields (such as accessibility distribution) can be extended into the time-scale domain using either nabla or delta derivatives.Only further studies will reveal which of those derivatives are better suited in addressing different problems.
Some extensions of the Lie derivative of the vector field into the discrete-time domain were made in [19], but in this paper, the discrete-time system (i) was described in terms of the shift operator and not via difference operator as in this paper, (ii) the vectors were assumed to belong into the finite-dimensional spaces and (iii) though the extension was in fact nabla derivative, it was not recognized as such.In [19], two possible definitions of Lie derivative are recalled, both yielding in the same result in the continuous-time case, but when extended to discrete-time case, one of them will yield nabla derivative and the other delta derivative.

Time scale calculus
For a general introduction to the calculus on time scales, see [6,7].Here we give only those notions and facts that we need in our paper and most of them was taken from [6,7].http://www.mii.lt/NAThe main task is to introduce the concept of derivatives for real functions defined on a time scale.
A time scale T is an arbitrary nonempty closed subset of the set R of real numbers.The standard cases comprise T = R, T = Z and T = hZ for h > 0, but also T = q Z := {q k | k ∈ Z} ∪ {0} is a time scale.However, the set of rational numbers and the open interval (a, b), a < b, are not the examples of time scales.For t ∈ T, the forward and backward jump operators σ, ρ : T → T are defined by σ(t) = inf{s ∈ T | s > t} and ρ(t) = sup{s ∈ T | s < t}, respectively.In addition, we set σ(max T) = max T if there exists a finite max T, and ρ(min T) = min T if there exists a finite min T. Since T is a closed subset of R, both σ(t) and ρ(t) are in T when t ∈ T. Finally, the graininess functions µ, ν : T → [0, ∞) are defined by µ(t) = σ(t) − t and ν(t) = t − ρ(t) for all t ∈ T, respectively.A time scale is called homogeneous if µ and ν are constant functions.
Let T κ denote truncated set consisting of T except for a possible left-scattered maximal point.The reason to omit a maximal left-scattered point is to guarantee uniqueness of f ∆ , defined below.Definition 1.The delta derivative of a function f : T → R at t ∈ T κ is the real number f ∆ (t) (provided it exists) such that for each ε > 0, there exists a neighborhood Moreover, we say that f is delta differentiable on T κ provided f ∆ (t) exists for all t ∈ T κ .
Let T κ denote truncated set consisting of T except for a possible right-scattered minimal point.The reason to omit a maximal right-scattered point is to guarantee uniqueness of f ∇ , defined below.Definition 2. The nabla derivative of a function f : T → R at t ∈ T κ is the real number f ∇ (t) (provided it exists) such that for each ε > 0, there exists a neighborhood U (ε) of t, U (ε) ⊂ T such that for all τ ∈ U (ε), Moreover, we say that f is nabla differentiable on T κ provided f ∇ (t) exists for all t ∈ T κ .
The delta and nabla derivatives of higher order are defined inductively.The kth-order delta derivative of function f is denoted by f [k] , and the kth-order nabla derivative of function f is denoted by f {k} .
For f : T → R, we define f σ := f • σ : T → R and f ρ := f • ρ : T → R and call them respectively the forward and backward time shift of f .Denote If f and f ∆ are delta differentiable functions, then for a homogeneous time scale, one has f σ∆ = f ∆σ .Similarly, if f and f ∇ are nabla differentiable functions, then for a homogeneous time scale, one has f ρ∇ = f ∇ρ .Theorem 1. (See [6].)Let f : T → R, g : T → R be two delta (nabla) differentiable functions defined on T, and let t ∈ T. Then: for t ∈ T κ such that ρ(σ(t)) = t.If, in addition, f ∇ is continuous on T κ , then f is delta differentiable at t and (2) holds for any t ∈ T κ .

Differential fields
Let us recall the facts for a σ f -differential field given in [2] and introduce the concept of a ρ f -differential field.
From now on we assume that T is homogeneous.Note that for homogeneous time scales, ν = µ = const.Consider now the control system, defined on T, where x is a state, u is a control (input) of the system and function f : U → R n is analytic.Let us define f (x, u) := x + µf (x, u) and assume that there exists a map ϕ : U → R m such that Φ = ( f , ϕ) T is an analytic diffeomorphism5 from the set U onto U .This means that from (x, z) = ( f (x, u), ϕ(x, u)) = Φ(x, u) 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.In the case µ > 0, system (3) can be rewritten in the following equivalent form: For notational convenience, (x 1 , . . ., x n ) will simply be written as x, and for k 0, (u m ) will be written as u [k] .For i k, let u [i...k] := (u [i] , . . ., u [k] ).We assume that the control (input) applied to system (3) is infinitely many times delta differentiable, i.e. u [0...k] exists for all k 0. Consider the infinite set of real (independent) indeterminates 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 F ∈ K depends on x and u [0...k] .For particular cases when We assume that (x, u) ∈ U and the other variables are restricted in such a way that σ f is well defined.Under the assumption about the existence of ϕ such that Φ = ( f , ϕ) is an analytic diffeomorphism, σ f is injective endomorphism.
The field K can be equipped with a delta derivative operator ∆ f : K → K defined by where F ∈ K depends on x and u [0...k] .The more compact notations F σ f and F ∆ f will be sometimes used instead of σ f (F ) and ∆ f (F ).Note that when x is considered as the function defined on the homogeneous time scale T, then the notations x σ and x ∆ will be used to denote the functions x σ : Whereas the notations x σ f and x ∆ f are used for the image of coordinate function F (x) = x with respect to operators σ f and ∆ f .Similarly, in the case u [k] , k 0.
The delta derivative ∆ 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 [10].Therefore, under the assumption about the existence of ϕ such that Φ = ( f , ϕ) is an analytic diffeomorphism, K endowed with the delta derivative = 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 [10].Since σ f is an injective endomorphism, it can be extended to K * , so that σ f : K * → K * is an automorphism.
Let ρ f : K * → K * be an operator defined by Hence one gets ρ f (F ) = F • ρ f for F ∈ K * .It was shown in [3] that, for µ = 0, the inversive closure of K may be constructed as the field of meromorphic functions in a finite number of the independent variables , where the new variables are related by σ f as follows: z , where Ψ is a certain vector valued function, determined by f in (3), 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 ∆ f to new variables by The extension of operator ∆ f to K * can be made in analogy to (7).Such operator ∆ f is now a σ f -derivation of K * .A practical procedure for construction of K * (for µ = 0) is given in [3].Additionally, the field K * can be equipped with a nabla derivative operator ∇ f : where F ∈ K * .Note that for µ > 0, we get and for µ = 0, ρ f = id, so on homogeneous time scales, we get the following relation between the operators ∆ f and ∇ f : Moreover, applying the operator σ f to (10), one gets The nabla derivative ∇ f satisfies, for all F, G ∈ K, the conditions: http://www.mii.lt/NATherefore, the operator ∇ f is a "ρ f -derivation", and the commutative field K * endowed with a ρ f -derivation is called a ρ f -differential field [10].
Similarly as in the case of the operators σ f and ∆ f , the more compact notations F ρ f and F ∇ f will be sometimes used instead of ρ f (F ) and ∇ f (F ).
From now on Consider the infinite set of differentials of indeterminates dC * = {dζ i , ζ i ∈ C * } and 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 * .The elements of E are called differential one-forms.Let d : K * → E be defined in the standard manner: 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 .
If ω = i A i dζ i is a one-form, where A i ∈ K * and ζ i ∈ C * , one can define the operators ∆ f : E → E and σ f : E → E by and By ( 13) and ( 14) we get then we have where where The operator σ f : E → E is invertible, and the inverse operator for By ( 17) and ( 18) we get Since (19) holds, then Note that, in order to find the components of ω ρ f and ω ∇ f , we have to know the formula for inverse of the map Φ.
), so, in the general case, one cannot give the explicit formulas for components as it was given for components of ω ∆ f and ω σ f in ( 15) and ( 16).http://www.mii.lt/NAProposition 2. Let T be the homogeneous time scale and F ∈ K * .Then Proof.Formulas (20a) and (20b) have been shown in [3].Then by ( 10) and (20b) we get so (20d) holds.Next, by ( 10) and (20a) we have so (20c) also holds.
For one-forms, similarly as for functions, the more compact notations ω ∆ f and ω σ f , ω ∇ f and ω ρ f will be used instead of ∆ f (ω) and σ f (ω), ∇ f (ω) and ρ f (ω), respectively.The relations between ∆ f and ∇ f operators given for functions from K * in ( 11) and ( 10) hold also for one-forms.Proposition 3. Let T be the homogeneous time scale and ω ∈ E. Then Proof.Note that if ω = i A i dζ i , A i ∈ K * and ζ i ∈ C * , then by the previous properties we have Additionally, similarly as in (22), one can prove that (21b), (21c) and (21d) hold.
Nonlinear Anal.Model.Control, 21(4):547-563 4 The space of vector fields 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 called the vector fields, and they are of the form where j ∈ E and the vector field X ∈ E of the form (23), we get Note that though the (formal) sum ( 23) is infinite, ω, X is always a sum with only finitely many nonzero terms in (24).The delta-derivative X ∆ f and forward-shift X σ f of X ∈ E may be defined uniquely by the equations and ω, X respectively, where ω is an arbitrary one-form.Note that ρ f (ω), X ∈ K * , so ρ f (ω), X σ f and ρ f (ω), X ∆ f are well defined.
Remark 1.Note that for µ = 0, we have σ f = ρ f = id and In [2], we show that evaluating (25) and ( 26) with the elements of canonical basis (i.e. with the elements from the set dC * ), we obtain two sets of explicit formulas that define the coefficients of X ∆ f and X σ f , respectively.Note that, in order to find the components of the vector field X ∆ f and X σ f , we have to know the formula for Φ −1 .Proposition 4. (See [2].)Let X ∈ E .Then for arbitrary ω ∈ E, http://www.mii.lt/NANote that for µ = 0, formula (27) gives X σ f = X, so σ f = id.Now let us define the backward-shift and nabla-derivative of the vector field X.The backward-shift X ρ f and nabla-derivative X ∇ f of X ∈ E may be defined uniquely by the equations respectively, where ω is an arbitrary one-form.
Example.Consider the system described by x ∆ 2 = u.For µ > 0, the system can be rewritten in the form (4), i.e.
We shall need x 2 and x σ 2 are different than 0, so we define the subset U of R 3 that satisfies the following inequalities: Then the inversive closure of K can be chosen as the field of meromorphic functions in a finite number of variables x 1 , x 2 , u [k] , z − , k 0, 1, where z −1 = σ −1 f (z) and z − = σ −1 f (z − +1 ).We construct the field extension in two different ways (choosing z as u or x 1 ) and compute X ρ f , X ∇ f , X ∇ 2 f using the formulas given in Proposition 7. Let X = ∂/∂u be an element of E .Then X z − = X x1 = X x2 = X u [k] = 0, 1, k 1, and X u = 1.

Conclusions
The paper may be understood as continuation of papers [2,3,4], focused on the development of algebraic tools that allow to study the properties of nonlinear control systems, defined on time scales.In this paper, we have extended the backward shift and nabla derivative operators, defined by the control system on homogeneous time scale, for oneforms and vector fields, and proved a number of their properties.The operator of nabla derivative (applied to the vector fields) is useful tool.In future studies, the nabla derivative operator can be applied, for instance, in computation of the accessibility distribution and checking accessibility property of the system, defined on time scales, extending the results for nonlinear continuous-time systems from [15].In the continuous-time case, the nabla derivative is equal to the delta derivative, whereas in the discrete-time case the two derivatives are different.The unification aspect of time scale approach entails considerable simplification in the software development on NLControl website [5].