An LMI approach to D-stability and D-stabilization of linear discrete singular systems with state delay

The problems of $\mathcal{D}$ ‐stability and $\mathcal{D}$ ‐stabilization for linear discrete singular systems with state delay are addressed in this paper. By using the state augmentation technique, a necessary and sufficient $\mathcal{D}$ ‐stability condition and a sufficient $\mathcal{D}$ ‐stabilization condition for linear discrete singular time‐delay systems are firstly proposed. Then, based on these conditions, quadratic $\mathcal{D}$ ‐stability and $\mathcal{D}$ ‐stabilization conditions for linear discrete singular time‐delay systems with norm‐bounded uncertainties and interval uncertainties are obtained, respectively. Both the $\mathcal{D}$ ‐stability and $\mathcal{D}$ ‐stabilization conditions are proposed in terms of linear matrix inequalities. Finally, illustrative examples show that the proposed conditions are effective to check the $\mathcal{D}$ ‐stability and to design the $\mathcal{D}$ ‐stabilizing controller for linear discrete singular time‐delay systems.