Sensitivity of Piecewise Control Laws with Respect to Perturbation of the State-Space Partition

Abstract

The control design techniques for linear or hybrid systems (e.g. anti-wind up, MPC) lead often to off-line state-space partitions with convex polyhedral regions. This corresponds to a piecewise state feedback control laws associated to polyhedral partition of the state-space. In this work, we consider the perturbation in the representation of the vertices of the polyhedral regions. For such control laws, this problem is of particular interest with respect to the reduced precision representation of the state-space partition. The perturbed state-space partitions might lose one of the important property of the explicit controllers: the non-overlapping characterization. We derive a set called vertex sensitivity region to determine the admissible perturbation independently for each vertex of the polyhedral partition. A perturbation is deemed admissible if the property of the polyhedral regions is preserved. In the present work, the analysis of the sensitivity of each vertex is done under the assumption that the rest of the partition remains on the nominal configuration.