Consistency and Lyapunov Stability of Linear Discrete Descriptor Time Delay Systems: A Geometric Approach

Abstract

Discrete descriptor time delayed systems are dynamic systems described by the combination of algebraic and differential equations with retarded arguments. In the linear discrete descriptor time delay systems (LDDS), the smoothness and continuity concepts are not fully applicable in the stability investigations. A possible solution is to utilize consistent initial conditions x0 capable of generating the causal solution sequence $(\mathrm{x}(k):k\geq 0)$. In this study, a geometric description of such initial conditions is investigated. New delay dependent asymptotic stability conditions were derived. For the simplification of the practical implementation, these conditions were expressed in terms of system matrices $(E,\ A_{0},\ A_{1})$ only. Therefore, complicated algebraic transformations are not required when the stability of such systems is analyzed.