On Chebyshev's Systems and Non-Uniform Sampling Related to Caputo Fractional Dynamic Time-Invariant Systems

Abstract

This paper is concerned with the investigation of the controllability and observability of Caputo fractional differential linear systems of any real orderα. Expressions for the expansions of the evolution operators in powers of the matrix of dynamics are first obtained. Sets of linearly independent continuous functions or matrix functions, which are also Chebyshev's systems, appear in such expansions in a natural way. Based on the properties of such functions, the controllability and observability of the Caputo fractional differential system of real orderαare discussed as related to their counterpart properties in the corresponding standard system defined forα=1. Extensions are given to the fulfilment of those properties under non-uniform sampling. It is proved that the choice of the appropriate sampling instants is not restrictive as a result of the properties of the associate Chebyshev's systems.