Lyapunov-based robust control for linear parametrically varying systems

This dissertation considers feedback control design for linear parametrically varying (LPV) systems. Two Lyapunov-based methods are presented to perform robust performance analysis and (sub)optimal control synthesis for LPV systems, where the admissible Lyapunov function is no longer limited to a single quadratic one. The foundation is the Lyapunov stability theory and an extension of the bounded real lemma (called generalized bounded real inequality, GBRI) which captures performance in terms of the induced {dollar}lsb1{dollar} (or {dollar}Lsb2{dollar}) norm of the closed loop.; First we consider a polyhedral Lyapunov function for state feedback design of discrete-time LPV systems. By the GBRI, the {dollar}lsb1{dollar}-analysis problem is transformed into the search of a so-called contractive set with {dollar}lsb1{dollar}-performance in the state-space. Such a set, if exists, can be computed via linear programming, and its Minkowski (level) function induces a Lyapunov function. The (sub)optimal full state feedback synthesis is then solved accordingly.; Next, spline-parametrized quadratic Lyapunov functions are proposed for the {dollar}Lsb2(lsb2){dollar}-analysis problem. Compared with a single quadratic Lyapunov function, such Lyapunov functions considerably reduce the conservatism for slowly varying LPV systems. A convex covering approach is proposed to reduce the numerical complexity. The corresponding output feedback synthesis is then solved in the framework of LMI-based H{dollar}sbinfty{dollar} control, and a numerically tractable algorithm is obtained. Analysis and synthesis examples demonstrate the viability of this approach.; Finally, we apply the proposed {dollar}Lsb2{dollar}-synthesis to an automatic vehicle steering problem. An existing robust control design gives a controller with high bandwidth, which affects closed-loop robustness at high velocities. The proposed method, on the other hand, gives a gain-scheduled controller which adjusts the bandwidth based on the velocity to capture both the tracking and robustness. The stability and performance against slow velocity variations is guaranteed by a spline Lyapunov function, and is verified by frequency and time domain simulations.