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 (time-driven) 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.