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International audience; In Marconi et al. 2007, it was shown that a solution to the output regulation problem for minimum-phase normal forms always exists. That approach, however, yields only an existence result, and no general analytic procedure is known to actually choose the regulator’s degrees of freedom, even for simple problems. In this paper we propose an adaptive regulator that, leveraging the aforementioned existence result, self-tunes online according to an optimization policy. To this aim, the regulator may employ every system identification scheme that fulfills some given strong stability properties, and the asymptotic regulation error is proved to be directly related to the prediction capabilities of the identifier.

Output regulation refers to the class of control problems in which some outputs of the controlled system must be steered to some desired references, while maintaining closed-loop stability and in spite of the presence of unmeasured disturbances and model uncertainties. While for linear systems the problem has been elegantly solved in the 70s, output regulation for nonlinear systems is still a challenging research field, and 30 years of active research left open many fundamental problems. In particular, all the regulators proposed so far are limited to very specific classes of nonlinear systems and, even in those cases, they fail in extending in their full generality the celebrated properties of the linear regulator. The aim of this thesis is to make a decisive step towards the systematic extension of the output regulation theory to embrace more general multivariable problems. To this end, we touch here three fundamental pillars of regulation theory: the structure of regulators, the robustness issue, and the adaptation of the control system. Regarding the structural aspects, we pursue here a design paradigm that is complementary to canonical nonlinear regulators and that trades a conceptually more suitable structure with a strong internal intertwining between the different parts of the regulator. For what concerns robustness, we introduce a new framework to characterize robustness of regulators relative to steady-state properties more general than the usual requirement asking a zero asymptotic error. We characterize in this unifying terms a large part of the existing approaches, and we end conjecturing that general nonlinear regulation admits no robust solution. Regarding the evolution of regulators, we propose an adaptive regulation framework in which adaptation is used online to tune the internal models embedded in the control system. Adaptation is cast as a general system identification problem, allowing for different well-known algorithms to be used.

International audience; We propose an adaptive observer scheme to estimate boundary parameters in first-order hyperbolic systems of Partial Differential Equations (PDE). The considered systems feature an arbitrary number of states travelling in one direction and one counter-convecting state. Uncertainties in the boundary reflection coefficients and boundary additive errors are estimated relying on a pre-existing observer design and a novel Lyapunov-based adaptation law.

We propose an adaptive observer scheme to estimate boundary parameters in first-order hyperbolic systems of Partial Differential Equations (PDE). The considered systems feature an arbitrary number of states traveling in one direction and one counter-convecting state. Uncertainties in the boundary reflection coefficients and boundary additive errors are estimated relying on a pre-existing observer design and a novel Lyapunov-based adaptation law.

International audience; In this paper we propose a general framework in which the robustness properties and requirements of output regulation schemes can be formally described. We introduce a topological definition of robustness relative to arbitrary steady state properties, extending the usual notion of robustness relative to the existence of a steady state in which the regulation error vanishes. We review some of the main control approaches for linear and nonlinear systems, by re-framing their robustness properties within the proposed setting. We show that the celebrated robustness property of the linear regulator, namely the ``internal model principle'' stated by Francis, Wonham and Davison in the 70's, can be generalized to nonlinear systems in a robustness property relative to the Fourier expansion of the regulation error. We then focus on nonlinear regulation, where we show that only practical regulation can be achieved robustly, while asymptotic regulation is achieved in a quite fragile way. The paper concludes with a conjecture stating that, in a general nonlinear context, asymptotic regulation cannot be achieved in a robust way with a finite dimensional regulator.

The problem of output regulation has always been tackled in frameworks in which the references to be tracked and the disturbances to be rejected are generated by an autonomous differential equation, referred to as the exosystem. This assumption, that is routinely used in the design of asymptotic regulators, plays also a fundamental role in the formulation of the regulation problem and in the definition of the basic concepts such as the steady state and the zero dynamics of nonlinear systems. In this paper we show that the concepts of steady state, zero dynamics and the output regulation problem can be equivalently defined in a framework in which the exosystem is generated by a differential inclusion.

In this paper we propose an adaptive regulator for general multivariable linear systems to deal with references and disturbances generated by an unknown exosystem. The proposed regulator merges a continuous-time internal model and a discrete-time least-squares identifier that adapts the internal model parameters. We show that, under a suitable persistence of excitation condition, asymptotic regulation is achieved whenever an upper bound on the dimension of the exosystem is known.

This letter deals with nonlinear repetitive control (RC), a technique used to reject periodic disturbances with a known and constant period. Since RC systems are defined over a state space of infinite dimension, the main theoretical problem that makes nonlinear case not trivial resides in the lack of adequate mathematical tools to study well-posedness of the closed-loop system and regularity of the solutions. Here, the stability analysis relies on recent results about the boundary control of infinite-dimensional port-Hamiltonian systems via nonlinear regulators, and the major contribution is the definition of a class of nonlinear plants for which a RC scheme is, at first, well-posed, and then exponentially stable. Moreover, an explicit proof of perfect local asymptotic tracking and disturbance rejection for exponentially stable RC systems is provided.

In this paper we review the problem of output regulation for nonlinear systems. We discuss how the robustness properties and the simple structure of the linear regulator are linear artefacts which do not extend in more general nonlinear cases. We talk about the intertwining that is necessarily present between the internal model and the stabilising parts of a nonlinear regulator, in which the role of the exosystem mixes up with those of the residual plant's dynamics. We discuss a general guideline to deal with such structural challenges, by looking at adaptation and universal approximators as a promising way to provide systematic design procedures for approximate regulators in a general nonlinear setting.

This paper deals with the problem of adaptive output regulation for a class of minimum-phase nonlinear systems. Based on the hybrid framework proposed in [1], an adaptive internal model regulator is proposed, relying on a discrete recursive weighted least squares scheme. Practical and asymptotic regulation results are presented and simulation results are used to show the effectiveness of the method in handling highly uncertain regulation problems.

This paper deals with the problem of output regulation for minimum-phase single-input single-output nonlinear systems. We propose a variation of the “regression-like” design introduced by Byrnes and Isidori (2004), showing how it is possible to move the internal model from the control input to the regulation error. In this way, we obtain a post-processing scheme that more likely resembles the general design for linear systems. Although these approaches are equivalent for single-input single-output systems, a post-processing design represents a step forward in the systematic extension of the theory to the general multivariable case.

This paper deals with the problem of adaptive output regulation of single-input single-output nonlinear systems, with respect to uncertainties in the exosystem. We endow a recently developed post-processing internal model design with a hybrid adaptive structure, which allows to use different identification schemes to adaptively tune the internal model at runtime. Practical regulation results are presented, with the regulation error that is proved to be linearly related to the prediction capabilities of the identifier.

International audience; In this paper we introduce the low-power high-gain observer, developed in [1], to solve problems of output regulation for nonlinear systems. We show how the new tool makes it possible the implementation of high dimensional controllers, that tipically arise when the ideal steady-state control that must be generated to secure zero regulation error is affected by uncertainties.