Gain-Scheduled L2 Control of Discrete-Time Polytopic Time-Varying Systems

Abstract. The objective of this work is to present sufficient conditions that can guarantee stability of discrete-time Linear Parameter-Varying (LPV) systems. The proposed stability conditions are described in terms of linear matrix inequality (LMIs). LPV systems are systems whose dynamics changes according to a varying parameter, usually called the scheduling parameter. Practical examples of such systems are: aerospace structures that are constantly exposed to extreme variation of the temperature and robotic systems commonly used in pick-and-place applications. In this work, it is assumed that the system matrices of the LPV model belong to a polytope. The stability condition, derived using Lyapunov theory, is described by an LMI which depends on the time-varying parameter. Therefore, this LMI must be satisfied for each time instant. This is an infinite dimensional problem which is impossible to solve numerically. To avoid this issue, a finite set of sufficient LMI conditions is derived by imposing on the Lyapunov matrix a polytopic structure. The set of LMIs to be determined can take into consideration bounds on the rate of variation of the scheduling parameter. Thus, providing less conservative results than those obtained using stability conditions that allow the scheduling parameter to vary infinitely fast, as quadratic stability. Numerical simulations are performed to show the benefits of the proposed technique.