Pre-processing nonlinear output regulation with non-vanishing measurements

Sufficient Conditions for Output Reference Tracking for Nonlinear Systems: a Contractive Approach

“…Furthermore, the extension to multi-input multi-output (and possibly more input than regulated output) is not trivial. See for instance, [11], [34].…”

Output regulation by postprocessing internal models for a class of multivariable nonlinear systems

“…We observe that in the linear case the chicken-egg dilemma is broken by the fact that, no matter how the stabilization gains are chosen, if a compact attractor  exists then all the closed-loop trajectories have the same modes of the exosystem (the closed-loop system (8), indeed, is a linear stable system driven by the exosystem). Therefore, a function satisfying (6) and (19) can be fixed a priori according to the knowledge of the exosystem dynamics. A similar situation also takes place in a nonlinear setting iḟ = 0.…”

“…almost everywhere. Condition (19) expresses indeed the internal model property, that is, the property of the regulator to generate all the ideal steady-state control actions η ⋆ 1 (t) needed to keep the regulation error e to zero. We also observe that the chicken-egg dilemma is strongly present in (19), since η ⋆ 1 and its derivatives, and thus the correct value of φ to be implemented, depend on the closed-loop trajectories and thus, in particular, on stabilization gains K ξ , K ζ , and K η , the latter dependent from the Lipschitz constant L φ of φ.…”

“…Apart from the chicken-egg dilemma, fulfilling condition (19) in general requires a very precise knowledge of the function η ⋆ 1 and its derivatives, which strongly depend on the plant and exosystem dynamics. Therefore, uncertainties in the model of the plant and exosystem strongly reflect into uncertainties in the right internal model function to implement, potentially ruining the asymptotic performance of the regulator [24].…”

“…Nevertheless, pre-processing regulators have some structural limitations that prevent their extension to larger classes of systems of those mentioned before. In particular, it is not clear, at a conceptual level, how a pre-processing regulator could handle in a systematic way additional measured outputs that are necessary to obtain closed-loop stability (or even minimum-phase), but that need not to vanish at the ideal error-zeroing steady state, unless filtering them out at the steady state employing redundant internal models, as recently proposed by Wang and Marconi 19 . On the other hand, there is not even a clear road map on how to handle systems having more inputs than errors.…”