Control to facet problems for affine systems on simplices and polytopes – With applications to control of hybrid systems

In this paper, a general control-to-facet problem for affine systems on polytopes is studied: find an affine feedback law such that all trajectories of the closed-loop system leave the state polytope through an a priori specified (possibly empty) set of facets. Solutions are presented in terms of (bi)linear inequalities in the coefficients of the affine feedback. The result is applied to control synthesis for piecewise-affine hybrid systems. Using a backward recursion algorithm, a sufficient condition for reachability of hybrid systems is obtained, and a piecewise-affine controller is computed that realizes the required reachability property.