Stability and stabilization of discrete-time linear time-varying systems via Lyapunov difference equations and inequalities

In this paper, the stability and stabilization issues of discrete-time linear time-varying (DLTV) systems are investigated. The recently proposed concepts of stability for continuous linear time-varying (LTV) systems are extended to DLTV systems, including uniform upper boundedness (by an exponential function) and uniform lower boundedness (by an exponential function). For the newly proposed concepts of stability, techniques for testing stability are developed. First, utilizing the scalar stable function, the Lyapunov difference inequalities (LDIs) provide both sufficient and necessary conditions for the new concepts of stability. Second, sufficient and necessary criteria based on Lyapunov difference equations are presented. Third, by introducing uniform complete controllability and uniform complete reconstructibility, the Lyapunov equations whose matrix Q is degenerated are further examined. Finally, a method for designing state feedback controller is presented.