Francis Equations vs Invariant Subspace Algorithm for Hybrid Output Regulation

In this paper, we deal with the output regulation problem for linear hybrid systems in the presence of unpredictable jumps, hence with arbitrary time domains. In particular we provide necessary and sufficient conditions for the solvability of the hybrid regulation problem, which non-trivially extend the classical conditions of regulation for LTI non-hybrid systems. Interestingly, differently from the latter, the former conditions are intrinsically nonlinear, since the dynamics of the interconnected system restricted to the error-zeroing invariant subspace may not be limited, as in the non-hybrid case, to those of the exosystem E. Then, we explore the relation between the discussed necessary and sufficient conditions and the constructive approach provided by the subspace invariant algorithm.