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Résumé

Cet article développe la commande d'un réacteur de dénitrification décrit par un ensemble d'équations aux dérivées partielles (EDP) de type hyperbolique. L'ensemble d'EDP est transformé en un ensemble d'équations aux dérivées ordinaires (EDO) en utilisant la méthode des caractéristiques. Le nouvel ensemble d'EDO est ensuite discrétisé par la méthode des différences finies et linéarisé autour d'un point d'équilibre. On obtient ainsi la représentation en espace d'état linéaire du modèle original. On utilise alors la propriété de séparation pour faire la synthèse d'un contrôleur par retour d'état et d'un observateur d'ordre complet en résolvant deux problèmes d'optimisation H2.

Mots-Clés / Keywords

Abstract

This paper addresses the control design problem for a certain class of continuous-time nonlinear systems subject to actuator saturations. The system under consideration consists of a system with two nested nonlinearities of different type: saturation nonlinearity and cone-bounded nonlinearity. The control law investigated for stabilization purposes depends on both the state and the cone-bounded nonlinearity. Constructive conditions based on LMIs are then provided to ensure the regional or global stability of the system. Different points, like other approaches issued from the literature, are quickly discussed. An illustrative example allows to show the interest of the approach proposed.

Mots-Clés / Keywords

Abstract

The Activated Sludge Model no. 1 (ASM1) [6], which is the most widely accepted model of wastewater treatment plants (WWTPs), is used to describe the biotransformation processes of nitrification-denitrification. Due to its complexity and nonlinearity, this model is however not wellsuited to the design of a control structure. The objective of this work is threefold: (1) to reduce the ASM1, (2) to linearize the reduced model, and (3) to design a robust control algorithm, in order to improve the plant performance and reject perturbations. This procedure is illustrated with data from the COST benchmark [7].

Abstract

The design of state feedback gain-scheduled controllers for linear parameter-varying systems with saturating actuators is addressed in the paper. The parameters can vary arbitrarily fast inside a polytope with known vertices. Sufficient conditions for the existence of gain-scheduled controllers assuring asymptotic stability for initial conditions inside a region of the state space are provided in terms of parameter-dependent linear matrix inequalities. A complete characterization of the solutions of these inequalities is given in terms of homogeneous polynomially parameter-dependent matrices of arbitrary degree that can be obtained from finite linear matrix conditions, written in terms of the vertices of the polytope. A procedure based on an extension of Polya's Theorem produces a sequence of sufficient conditions which tend to the necessity as the level of relaxation increases. The scheduled controller is a homogeneous polynomially parameter-dependent state feedback gain of arbitrary degree that quadratically stabilizes the closed-loop system and provides an estimate of the domain of attraction of the origin, as illustrated by numerical examples.