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

Abstract

The paper deals with the problem of output regulation for nonlinear systems in a multivariable and "nonequilibrium" context. A "chicken-egg dilemma" arising in the design of the internal model and the stabiliser units is pointed out and a general adaptive framework yielding approximate, possibly asymptotic, regulation is proposed to cope with it. It is shown that the framework allows one to deal with classes of nonlinear systems not covered by existing results and provides new insights about the use of identification tools in the design of adaptive internal models. The vision that emerges from the paper is that approximate, rather than asymptotic, regulation is the more appropriate way of approaching the problem in a multivariable and uncertain context, by thus opening new perspectives about the design of robust internal model-based regulators.

Adaptive output regulation for linear systems via discrete-time identifiers

“…The main interest for future developments concerns the synergy between identification and control and the relation between the prediction performances of the identifier and the regulation performance of the control scheme. As in [19,21], indeed, the asymptotic "optimality" of the identifier turned out here to be the key to obtain asymptotic regulation. Future researches will mainly focus on the extension of the control paradigm to more general plants modeled by nonlinear differential inclusions and the integration into the framework of more general identifiers, by eventually introducing stochastic identifiers.…”

“…The same idea was applied to a class of uncertain nonlinear exosystems in [17,18]. Finally, in [19,20,21] adaptation is cast as a system identification problem, and parameter estimation is performed by any continuous-or discrete-time algorithm satisfying some strong stability properties.…”

Approximate Nonlinear Regulation via Identification-Based Adaptive Internal Models

“…With (24) we associate the set-valued map Opt (ς,w) (j) := argmin θ∈Θ J (ς,w) (j, θ), representing, at each j, the set of optimal parameters according to (24). The choice of the remaining degrees of freedom (Ξ, ϕ, ϑ) is then made to satisfy the conditions contained in the forthcoming requirement, in which we make reference to the following cascade of the core process to the identifier:…”

“…Among the approaches to approximate regulation it is worth mentioning [15], [16], whereas practical regulators can be found in [17], [18], [13]. Adaptive designs of regulators can be found, e.g., in [19], [20], [21], where linearly parametrized internal models are constructed in the context of adaptive control, in [22] where discrete-time adaptation algorithms are used in the context of multivariable linear systems, and in [23], [24], [25] where adaptation of a nonlinear internal model is approached as a system identification problem.…”

“…Besides, unlike in [12], we ensure that the parameters of the controller are fixed a priori independently from the added internal model. On the heels of [23], [24], and contrary to canonical adaptive control designs, adaptation is not carried by means of "ad hoc" algorithms developed under structural assumption on the internal model unit and by means of Lyapunov-like arguments; rather we approach the adaptation of the internal model as a system identification problem, where the best model matching with the measured data and performance needs to be identified. We thus allow for different identification schemes to be used, by individuating a set of sufficient stability conditions that they need to satisfy to be used within the framework.…”