LPV system analysis via quadratic separator for uncertain implicit systems

This paper considers a class of linear systems containing time-varying parameters whose behavior is not known exactly. We assume that the parameters vary within known intervals and there are known bounds on their rates of variation. Our objective is to give a computationally verifiable condition that guarantees stability of the system for all possible parameter variations. To this end, we first point out that the information on the rate bounds can be exploited by considering an augmented system described by an implicit model. We then develop a tool, quadratic separator, for stability analysis of uncertain implicit systems. Using this tool, a sufficient stability condition is obtained for the original linear parameter-varying (LPV) system. Moreover, we show that the stability condition, thus obtained through the quadratic separator for implicit systems, is equivalent to the existence of a Lyapunov function that depends on the parameters in a linear fractional manner. Finally, the computational aspects of the proposed stability conditions are addressed in terms of linear matrix inequalities which can be solved efficiently via interior point methods.