A new local control strategy for control of discrete-time piecewise affine systems

In this paper we propose an enhanced local control strategy for the control of discrete-time piecewise affine systems on full-dimensional polytopes. The control strategy is divided into a local control and a supervisory control problem. The local control problem is to reach and cross one selected facet of a polytope ensuring that the next sample of the time-discrete trajectory is picked up in the adjacent polytope. The procedure is based on conditions, given as inequalities, for the discrete-time gradient of the system, evaluated in the vertices of the polytopes. Solving an optimization problem with respect to the inequality conditions, a performance index is minimized. The performance index represents a minimal retention period of the trajectory in the polytope or the minimal quadratic sum of the input signal. The control law is then obtained by a simple matrix inversion. The supervisory control problem is to find a suitable combination of polytopes and local control strategies that transfers the trajectory to the polytope that contains the operating point.