Multiple Robust Controller Design based on Parameter Dependent Lyapunov Functions

This paper tackles the problem of simultaneously designing a partition of an uncertain set and its corresponding set of multiple controllers that optimize the worst-case performance of a linear time invariant system under parametric uncertainty. The parametric uncertainty region is assumed to be convex polytopic, which is also partitioned into a set of convex polytopic local regions. It is desired that all plants that belong to a local region are to be controlled by a single controller, which is designed, based on a parameter dependent Lyapunov function, to give an optimal worst-case performance for that region. The total performance is evaluated as the maximum of worst-case performances for all the local regions. The performance is minimized with respect to a fixed number of convex polytopic local regions, as well as the same number of controllers. Even though the formulated problem is nonconvex, and thus it is difficult to ensure global optimality, descent algorithms are provided to update the local regions and the multiple controllers so that they guarantee monotonic non-increasing total performance.