Stability and Robustness of Singular Systems of Fractional Nabla Difference Equations

Abstract

In this article, we study the stability and robustness of a class of singular linear systems of fractional nabla difference equations whose coefficients are constant matrices. Firstly, by assuming that the singular fractional system has a unique solution for given initial conditions, we study the asymptotic stability of the equilibria of the homogeneous system. We also prove conditions on the input vector under which the solution of the non-homogeneous system converges. Next, since it is known that existence and uniqueness of solutions depend on the invariants of the pencil of the system, by taking into consideration the fact that small perturbations can change the invariants, we perturb the singular fractional system and obtain bounds on the perturbation effect of the invariants of the pencil. In addition, by using this result, we study the robustness of solutions of the system. Finally, we give numerical examples based on a real singular fractional nabla dynamical system to illustrate our theory.