Limit cycle generation for a class of non-linear systems with jumps using a low dimensional predictive control

In this paper, it is shown that using a low dimensional non-linear predictive control scheme, provably stable limit cycles can be obtained for open-loop non-linear systems with unstable equilibrium point. A particular case, where the limit cycle may be reduced to a single point in the state space, can be obtained, which corresponds to asymptotic stabilization. The system may present hybrid nature in the sense that discontinuities (jump phenomena) on the state evolution may be handled. The proposed feedback scheme holds for classical jump-free systems as a particular case. The proposed strategy is illustrated through two examples: a jump-free system (the ball and beam) and a non-linear hybrid dynamical system including state jumps (the modified impulsive Lorenz chaotic system).