Abstract
The backward shift and nabla derivative operators, defined by the control system on homogeneous time scale, in vector spaces of one-forms and vector fields are introduced and some of their properties are proven. In particular the formulas for components of the backward shift and nabla derivative of an arbitrary vector field are presented.
Highlights
In papers [2, 3], an algebraic formalism for nonlinear control systems, defined on homogeneous time scales, has been developed
The formalism is based on differential one-forms and vector fields
Two vector spaces over K∗, of one-forms and of vector fields, respectively, were introduced, and the operators ∆f and σf were extended to these vector spaces
Summary
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]. 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). 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. 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
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