About Robustness of Internal Model-Based Control for Linear and Nonlinear Systems

“Class-Type” Identification-Based Internal Models in Multivariable Nonlinear Output Regulation

“…The internal model is thus fixed a priori as an integrator acting on the error and the cascade is then stabilised using forwarding techniques. In [23] (see also [24,25]) this approach is extended to the case in which w is periodic, with a design that, however, guarantees only approximate regulation. In that works the internal model is fixed a priori as a linear system containing some of the harmonics of w(t), and a local state feedback stabiliser is used to force a periodic steady state, with the remarkable feature that the Fourier components of the steady state error associated to the harmonics contained in the internal model are zero.…”

“…We say that the stabiliser (15) and the matrices G i of (12) satisfy the stability requirement if system (25), (26), (27) is practically ISS with respect to the input ε ⋆ with possible restrictions on the initial conditions. Namely, there exist a set O ⊆ R n ×R n ξ ×R pd ×Z, functions β s ∈ KL and ρ s ∈ K, and a positive ν s such that for all initial conditions (x(0), ξ(0), η(0), z(0)) ∈ O the trajectories of (25), (26), (27) satisfy…”

“…The overall closed-loop system is thus given by the interconnection of system (26)- (27), with input (ẽ, ε ⋆ ) and outputη, and system (25), with inputη and outputẽ. It comes thus natural to design the stabiliser to induce a small-gain condition in the aforementioned interconnection.…”

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

“…While the concept of internal model led to a definite answer to the linear regulation problem, the situation is quite different when nonlinear systems are concerned 3 , and nonlinear output regulation is still a quite open problem 4 . Without restrictive immersion assumptions 5,6,7,8,9 , indeed, the knowledge of the exosystem alone is neither sufficient nor necessary for the solvability of the problem 3,4,10 , and its role in conditioning the asymptotic behavior of the regulator mixes up with the plant's residual dynamics, thus making the celebrated robustness property of the linear regulator hard to imagine in a general nonlinear context 11 .…”

“…A different approach to the design of post-processing regulators was recently pursued by Astolfi, Praly and Marconi 22,23 , where the linear regulator is attached to a class of nonlinear systems. In particular, the authors have shown that the output regulation problem can be solved robustly 24 by a post-processing integral action whenever the steady state is made of equilibria 23 , and then Astolfi, Praly and Marconi 22 have extended the results to the case in which the steady-state signals are periodic, obtaining, however, only an approximate result stating that the Fourier coefficients in the regulation errors corresponding to the frequencies embedded in the internal model vanish at the steady state.…”