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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.