Approximate regulation for nonlinear systems in presence of periodic disturbances

Abstract

In this work we present a new approach to the problem of output regulation for nonlinear systems in presence of periodic disturbances, possibly with an infinite number of harmonics. We show that, by adding a linear internal model, approximate regulation is achieved if the disturbance is small enough. Nominally all the harmonic included in the internal model are absent in the periodic steady state regulation error. Furthermore we show that the regulation error can be made arbitrarily small (in the L2 sense) by enlarging the dimension of the internal model. The novel approach relies on forwarding technique. An example is provided to show the efficacy of the result.

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

“…Consistently, our main result aims to relate the performances on the regulation side with the performances of the corresponding identified model, expressed in terms of prediction error. Asymptotic regulation, in turn, will follow only when a right model exists that is in the "range" of the identifier used (and in this sense we observe how this design philosophy matches with the results of [23]).…”

“…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.…”

“…In this paper we propose a framework based on a postprocessing structure. We deal with the chicken-egg dilemma by leveraging on the intuition, coming from the linear case and the approaches of [22,23], that the internal model can be fixed in advance on the basis of the expected class of functions η ⋆ resulting after considering all the possible plant's and exosystem's uncertainties and all the possible choices of the stabiliser inside a prescribed set. The class (say C ⋆ η ) of functions η ⋆ results from an overall assessment about the knowledge of the plant and the exosystem, the expected uncertainty in their models and the expected set of stabilisers that will be adopted, all treated equally.…”

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

“…The limits of the P P P 0 -robustness of the linear regulator motivate seeking for a regulation objective P P P for which a regulator that is P P P-robust relatively to more general topological spaces (F, τ F ) more likely could be constructed. As proposed in [23], in this section we let F = C 1 (X × U), τ F = τ C 1 and we consider a regulator of the kind (9) with (φ, θ) possibly nonlinear and with n im = (2d + 1)p e , for some arbitrary d ∈ N. We then choose a basis for R n × R nη in which (Φ, G) read as…”

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

“…Nevertheless, no conceptual progress has been made in terms of extensions to larger classes of systems compared to their pre-processing counterparts. 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.…”