Robust stability and stabilisation of discrete-time descriptor systems with uncertainties in the difference matrix

This study considers the problems of robust stability and stabilisation for linear discrete-time descriptor systems with norm-bounded uncertainties in all the system matrices including that in the difference matrix E. Under the assumption that the difference matrix E is rank-invariant for all admissible uncertainties, the robustness analysis problem under consideration falls into two cases: right-singular case and left-singular case. A necessary and sufficient stability condition and a sufficient stability condition are proposed for the first and second case, respectively. By using a state augmentation technique, sufficient stabilisation conditions are obtained for both cases. Furthermore, a necessary and sufficient stabilisation condition is established in the case of a non-singular matrix E. Finally, illustrative examples show that the proposed conditions are effective to design the stabilising controller for uncertain discrete-time descriptor systems with both the singular and non-singular matrixE.