Parameter-dependent Lyapunov functions and the discrete-time Popov criterion for robust analysis

Key Words-Parameter-dependent Lyapunov functions; constant real parameter uncertainty; robust stability and performance. Almhmct-Parameterodependent Lyapunov functions are developed for discrete-time systems with constant real parameter uncertainty. This construction of a family of Lyapanov functions is then used to analyze robust stability with He performance bounds for state space systems. A special case of these results generalizes the classical discrete-time Popov criterion. [tr ZZ*] ta (Frobenius matrix norm) [(1/2~)f~_,, IIH(e~°)ll 2 dO] trz 1. Introduction SINCE MOST physical processes evolved naturally in continuous time, it is not surprising that the bulk of robust control theory has been developed for continuous-time systems. Nevertheless, it is the overwhelming trend to implement controllers digitally. The analysis and synthesis of robust digital controllers thus requires an understanding of how the continuous-time uncertainties transform into an uncertain discrcte-time model. A glimpse of the nature of this type of difficulty can be seen in Bernstein and Hollot (1989).