Set-valued methods for linear parameter varying systems

We consider stability analysis and state feedback synthesis for Linear Parameter Varying (LPV) systems. LPV systems are defined as linear systems whose dynamics depend on exogenous time-varying parameters. Stability analysis of LPV systems involves testing the stability of a family of possible parameter trajectories. Similarly, stabilization and optimal control of LPV systems involves constructing feedback for a family of possible parameter trajectories. A further constraint is that the feedback depends on parameter trajectories in a causal manner. Previous work has considered LPV systems with unconstrained (i.e., arbitrarily fast) parameter variations. We consider the utilization of known constraints on the parameter variations, such as rate constraints. For discrete-time LPV systems whose parameters take on a finite set of values, we provide constructive computational algorithms which nonconservatively assess stabilizability. These algorithms lead to parameter-dependent Lyapunov functions for stability and parameter-dependent nonlinear state-feedback for stabilization. We further provide algorithms to construct, whenever possible, nonlinear feedback to achieve a specified level of disturbance rejection as measured by signal magnitude. We also consider the dynamic game of minimizing a transient response as measured by a norm summation and provide algorithms to construct the optimal cost and near optimal nonlinear feedback.