On controllability and observability of a class of fractional-order switched systems with impulse

This paper is devoted to investigating two structural properties for a class of fractional-order switched and impulsive systems. The structural properties, i.e., observability and controllability, are explored mainly based on an algebraic approach. More specifically, firstly, according to the Laplace transform and mathematical induction, the general solution of such hybrid fractional-order systems is obtained over every impulsive interval. Next, applying the solution derived and relevant matrix theory, several necessary and sufficient controllability and observability criteria that take the form of a row of Gramian matrices are analytically established in terms of a deterministic impulse-switching time sequence. Resorting to the property of matrix Mittag-Leffler function, the developed controllability and observability Gramian criteria are further converted to some easy-test Kalman-type rank conditions. Finally, a numerical example illustrating the theoretical controllability and observability conditions is given.