A Local Control Strategy for the Control of Discrete-Time Piecewise Affine Systems on Full-Dimensional Polytopes

In this paper, we propose a new local control strategy for the control of discrete-time piecewise affine systems on full-dimensional polytopes. The local control problem is to reach and cross one selected facet of a polytope, ensuring that the next sample of the discrete-time trajectory is picked up in the corresponding 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. If all conditions are fulfilled, a simple transformation and a matrix inversion lead to the desired control law. The advantage of this approach is that such a control law is valid on the entire polytope, for all trajectories leaving through the specified facet. The proposed control strategy is used to minimize the so called Bonanza effect in automotive powertrains with backlash