State-feedback Abstractions for Optimal Control of Piecewise-affine Systems

In this manuscript, we investigate symbolic abstractions that capture the behavior of piecewise-affine systems under input constraints and bounded external noise. This is accomplished by considering local affine feedback controllers that are jointly designed with the symbolic model, which ensures that an alternating simulation relation between the system and the abstraction holds. The resulting symbolic system is called a state-feedback abstraction and we show that it can be deterministic even when the original piecewise-affine system is unstable and non-deterministic. One benefit of this approach is the fact that the input space need not be discretized and the symbolic-input space is reduced to a finite set of controllers. When ellipsoidal cells and affine controllers are considered, we present necessary and sufficient conditions written as a semi-definite program for the existence of a transition and a robust upper bound on the transition cost. Two examples illustrate particular aspects of the theory and its applicability.