Observability conditions for the semiglobal output regulation of non-minimum phase nonlinear systems

In this paper, we present further results on the semiglobal output regulation of non-minimum phase nonlinear systems. In our recent work, we have given a sufficient condition for the existence of an internal model based dynamic regulator for a certain class of nonlinear non-minimum phase systems. Specifically, stabilizability by output feedback of a certain auxiliary system was required, together with a uniform observability condition on the cascade of the auxiliary system and a suitably defined internal model. The contribution of the present work is to characterize the observability condition for the cascade. Results are given in terms of uniform complete observability and the existence of a uniform canonical flag. It turns out that a key condition (which holds in both the UCO and the canonical flag approach) is a nonlinear equivalent of the non-resonance condition between the structure at infinity of the given internal model, and the modes of the auxiliary system. Since the zeros of the internal model include the modes of the exosystem, and the modes of the auxiliary plant include the zeros of the original plant, we recover a familiar condition for the solvability of the linear servomechanism problem.