Output Regulation by Error Dynamic Feedback in Hybrid Linear Systems with State Jumps

This work deals with a regulation problem for hybrid linear systems which exhibit a continuous-time state motion, ruled by the so-called flow dynamics, except at isolated points of the time axis, where the state has discontinuities governed by a jump behavior. Jump time instants are not a-priori known and may be unequally spaced, the only admissibility constraint being that the set of time intervals between consecutive jumps has a fixed positive lower bound. The considered problem consists in finding a hybrid error feedback compensator that forces the output of a given hybrid plant to asymptotically follow a reference trajectory generated by a hybrid exogenous system, while achieving global asymptotic stability of the closed-loop dynamics, for all the admissible sequences of jump times. The problem is investigated from a structural point of view, using geometric notions and properties. A sufficient condition for the existence of solutions is first stated in an implicit form, by considering the overall compensated system. This result is instrumental in giving a sufficient constructive condition which refers to the output-difference connection of the plant and the exogenous generator. The conditions are expressed in terms of hybrid invariant and controlled invariant subspaces of the systems at issue as well as of their stability and stabilizability properties, respectively. This approach provides a viable algorithmic procedure for the synthesis of solutions, if any exist.