Robust stability and stabilization for descriptor systems with uncertainties in all matrices

This paper is concerned with the robust stability and stabilization via state feedback for a class of continuous linear time-invariant descriptor systems with norm-bounded perturbations in derivative matrix E and other systems matrices A and B. Due to the fact that matrix E being rank invariant is an efficient necessary condition to the solvability of stabilization via state feedback, perturbations in matrix E can be described in a new form that is different from the general form studied in previous works. Problems under consideration fall into 2 cases, ie, one is expressed by a constant matrix left multiplying an invertible uncertain matrix, and the other is produced by a dual form of the first case. Then, we transfer the problem of robust stability into the question of a feasible solution for linear matrix inequality and necessary and sufficient conditions for stability are derived for the 2 cases. In terms of the stabilization problem, by using a new state augmentation technique, we transfer the descriptor system with uncertainties in the derivative matrix into an augmented descriptor system that the derivative matrix has no uncertainties, and we find that there are no properties changed during this augmentation process. Next, basing on the theory of stability, we receive sufficient conditions to the original stabilization problem for the 2 cases. Finally, 2 numerical examples are given to illustrate the effectiveness and merits of the main results.