The Stability And Semistability Of Singular Discrete-time Systems

Semistability is a new concept, which is a generalization of stability and lies between Lyapunov stability and asymptotic stability. Semistability is the property such that the solutions of a system converge to stable equilibrium points determined the initial conditions. Inspired by some existing ideas, methods and techniques, this paper investigates the semistability of normal discrete-time systems and singu-lar discrete-time systems. Through the research and analysis of the stability and semistability of discrete-time systems, using the definition of semistability of spec-trum, a new sufficient condition for semistability of discrete-time systems is proposed based on the Lyapunov equation and a rank condition. Moreover, the definition of semistability and a sufficient condition of singular discrete-time systems are raised by a similar approach. Finally, the delay positive systems are transformed into sin-gular systems without time-delay, and a necessary and sufficient condition is given for remaining positivity of states of a delay singular system. By using linear matrix inequality method, a criterion for stability of a delay positive singular system is obtained.The main results of this paper are given as follows.First, the definition of semistability of a normal discrete-time system is given by using the associated method of Lyapunov, and a new sufficient condition for semistability of a normal discrete-time system is proposed, then an example is given in order to verify our method;Second, the definition of semistability of a singular discrete-time system is given. A sufficient condition for semistability of a singular discrete-time system is obtained. Finally, an example is given to verify the obtained conclusion;Third, the delay positive systems are transformed into singular systems without time-delay, and a necessary and sufficient condition for remaining positivity of states of a delay singular system is proved. By using linear matrix inequality method, a criterion for stability of a delay positive singular system is given.