Observer-based control of discrete-time piecewise affine systems: Exploiting continuity twice

Output-based feedback control of discrete-time hybrid systems is an important problem, as in practice it is rarely the case that the full state variable is available for feedback. A typical approach for output-based feedback design for linear and smooth nonlinear systems is to use certainty equivalence control, in which an observer and a state feedback controller (using the observer state) are combined. Although for linear systems and some classes of nonlinear systems, separation principles exist to justify this approach, for hybrid systems this is not the case. In this paper, we isolate a class of hybrid systems for which a systematic design procedure for certainty equivalence controllers including a separation principle will be presented. This class consists of discrete-time piecewise-affine (PWA) systems with continuous dynamics. In the design procedure, we will exploit the continuity of the PWA dynamics twice. Firstly, it will be used to establish input-to-state stability (ISS) w.r.t. measurement errors from ISS w.r.t. additive disturbances. This is a crucial step as the latter problem is much easier to tackle than the former. Secondly, continuity will be used in the observer design procedure to obtain a significantly simplified set of LMIs with respect to existing observer design approaches for PWA systems. All the design conditions will be formulated in term of LMIs, which can be solved efficiently, as is also illustrated by a numerical example.