Output regulation by error dynamic feedback in hybrid systems with periodic state jumps

This work deals with output regulation in multivariable hybrid systems featuring a continuous-time linear dynamics periodically affected by instantaneous changes of the state. More precisely, given a hybrid linear plant and a hybrid linear exogenous system, with periodic state jumps, the problem consists in finding a hybrid feedback regulator, with the same characteristics, achieving global asymptotic stability of the closed-loop dynamics and asymptotic tracking of the reference generated by the exogenous system for all the initial states. Starting from a general, necessary and sufficient condition for the existence of a solution, the discussion leads to a more specific, sufficient condition which outlines the computational framework for a straightforward synthesis of the compensator. The internal model principle is shown to hold in a more general formulation than the original one, adapted to the hybrid systems considered. A numerical example is worked out with the aim of illustrating how to implement the devised technique. The geometric approach is the key methodology in attaining these results.