Robust global controller design for discrete-time descriptor systems with multiple time-varying delays

Descriptor systems are ubiquitous in many complex practical systems including spacecraft, electrical networks, economic systems, robotic, vibrational, and structural systems. Time delays are ubiquitous in those complex systems. If not addressed properly at the control design stage, they will result in singularities such as oscillation, instability, degraded performance. This article proposes a novel linear matrix inequality (LMI)-based global sliding mode control (GSMC) approach for uncertain discrete-time descriptor systems with multiple time-varying delays. Strict LMI conditions are established based on free-weighting matrices and the Lyapunov-Krasovskii functional to guarantee admissible closed-loop dynamics, that is, regular, causal, and stable. The sufficient condition for the asymptotic stability of the sliding mode dynamics is obtained based on the Lyapunov stability theorem and LMI constraint. The GSMC approach is then formulated based on a global sliding surface to ensure robust performance of the closed-loop system against parametric uncertainties and time delays, while eliminating the reaching phase. The global sliding surface ensures robust performance without the need for a reaching phase, thus establishing sliding around the surface right from the beginning and reducing the control effort. The effectiveness and validity of the proposed approach is assessed using a numerical example.