Resolution des equations d’etat lineaires d’ordre fractionnaire

Abstract

In this work, the resolution of the fractional state space equation representing the linear fractional systems of commensurate order, for all the eigenvalues types of the state matrix A and the order of differentiation m was proposed. The explicit expressions of the homogeneous and non-homogeneous solutions of this fractional state space equation were developed. For different values of the state-space matrix A eigenvalues and the order m, the obtained solutions are linear combinations of suitable fractional fundamental functions whose Laplace transforms are irrational functions. The approximations of these irrational functions by rational functions were obtained so that the solutions of the fractional state space equation are linear combinations of classical exponential, cosine, sine, damped cosine and damped sine functions. Illustrative examples for all the eigenvalues types of the state matrix A and the order m were presented and the results obtained were very satisfactory.