Reduced Complexity Controllers for LPV Systems: Towards Incremental Synthesis

International audienceExisting synthesis methods for LPV systems often result in controllers of high complexity. So far, there is no efficient and systematic remedy to this issue as there exists no convex formulation of the problem of finding a solution of reduced complexity to the general case LPV synthesis problem. In this paper, the specific case is considered when parameter-dependent signals are measured. It is proven that these measures can be exploited so that the problem of reduced-complexity controller synthesis can be written as an LMI optimization problem. A complete procedure for the controller construction is provided. The interest of the result is discussed in relation with nonlinear methods. First, an interpretation of the controller strategy is proposed with regard to the feedback linearization method. Second, it is proven that a nonlinear controller ensuring the closed loop incremental properties can be constructed

Commande robuste de systèmes non linéaires incertains.

This thesis studies the LPV approach for the robust control of nonlinear systems. Its originality is to propose for the first time a rigorous framework allowing to solve efficiently nonlinear synthesis problems.The LPV approach was proposed as an extension of the H-infinity approach in the context of LPV (Linear Parameter-Varying) systems and nonlinear systems. Although this approach seemed promising, it was not much used in practise. Indeed, beyond certain theoretical limitations, the nature itself of the obtained solutions did not seem adequate. This open question constitutes the starting point of our work.We first prove that the observed weak variation of the controllers is in fact mostly due to the information structure traditionally used for LPV synthesis, and that under reasonable assumptions, the LPV framework can overlap feedback linearization strategies. This point having been resolved, a second difficulty lies in the actual achievement of nonlinear controllers yielding performance guarantees. We propose a rigorous framework allowing to solve efficiently an incremental synthesis problem, through the resolution of an LPV problem associated to a specific information structure compatible with the one identified in the first part.This study and its corollary description of a formal framework and of a complete controller synthesis procedure, including complexity reduction methods, provide powerful arguments in favor of the LPV approach for the robust control of nonlinear systems.Cette thèse étudie l'approche LPV pour la commande robuste des systèmes non linéaires. Son originalité est de proposer pour la première fois un cadre rigoureux permettant de résoudre efficacement des problèmes de synthèse non linéaire. L'approche LPV a été proposée comme une extension de l'approche H-infini dans le contexte des systèmes LPV (« Linéaires à Paramètres Variant dans le temps »), voire non linéaires. Quoique prometteuse, cette approche pour la commande des systèmes non linéaires restait peu utilisée. En effet, au-delà même de certaines limitations théoriques, la nature des solutions obtenues semblait inadéquate. Cette question ouverte est notre point de départ. Nous montrons tout d'abord que la faible variation des correcteurs constatée est due avant tout à la nature du schéma informationnel utilisé traditionnellement lors de la synthèse LPV, et que sous des hypothèses raisonnables, le cadre LPV peut permettre de recouvrir des stratégies de type « linéarisation par bouclage ». Ce point étant acquis, une deuxième difficulté réside dans l'obtention effective de correcteurs non linéaires donnant des garanties de performance. Nous proposons un cadre rigoureux permettant de résoudre efficacement un problème de synthèse incrémentale pondérée, par la résolution d'un problème LPV associé à un schéma informationnel spécifique compatible avec celui identifié dans la première partie. Cette étude et son aboutissement à la définition d'un cadre formel et d'une procédure complète d'obtention de correcteurs, incluant des méthodes de réduction de complexité, donnent des arguments puissants en faveur de l'approche LPV pour la commande robuste de systèmes non linéaires

Robust control of nonlinear systems

Cette thèse étudie l'approche LPV pour la commande robuste des systèmes non linéaires. Son originalité est de proposer pour la première fois un cadre rigoureux permettant de résoudre efficacement des problèmes de synthèse non linéaire. L'approche LPV a été proposée comme une extension de l'approche H-infini dans le contexte des systèmes LPV (« Linéaires à Paramètres Variant dans le temps »), voire non linéaires. Quoique prometteuse, cette approche pour la commande des systèmes non linéaires restait peu utilisée. En effet, au-delà même de certaines limitations théoriques, la nature des solutions obtenues semblait inadéquate. Cette question ouverte est notre point de départ. Nous montrons tout d'abord que la faible variation des correcteurs constatée est due avant tout à la nature du schéma informationnel utilisé traditionnellement lors de la synthèse LPV, et que sous des hypothèses raisonnables, le cadre LPV peut permettre de recouvrir des stratégies de type « linéarisation par bouclage ». Ce point étant acquis, une deuxième difficulté réside dans l'obtention effective de correcteurs non linéaires donnant des garanties de performance. Nous proposons un cadre rigoureux permettant de résoudre efficacement un problème de synthèse incrémentale pondérée, par la résolution d'un problème LPV associé à un schéma informationnel spécifique compatible avec celui identifié dans la première partie. Cette étude et son aboutissement à la définition d'un cadre formel et d'une procédure complète d'obtention de correcteurs, incluant des méthodes de réduction de complexité, donnent des arguments puissants en faveur de l'approche LPV pour la commande robuste de systèmes non linéaires.This thesis studies the LPV approach for the robust control of nonlinear systems. Its originality is to propose for the first time a rigorous framework allowing to solve efficiently nonlinear synthesis problems.The LPV approach was proposed as an extension of the H-infinity approach in the context of LPV (Linear Parameter-Varying) systems and nonlinear systems. Although this approach seemed promising, it was not much used in practise. Indeed, beyond certain theoretical limitations, the nature itself of the obtained solutions did not seem adequate. This open question constitutes the starting point of our work.We first prove that the observed weak variation of the controllers is in fact mostly due to the information structure traditionally used for LPV synthesis, and that under reasonable assumptions, the LPV framework can overlap feedback linearization strategies. This point having been resolved, a second difficulty lies in the actual achievement of nonlinear controllers yielding performance guarantees. We propose a rigorous framework allowing to solve efficiently an incremental synthesis problem, through the resolution of an LPV problem associated to a specific information structure compatible with the one identified in the first part.This study and its corollary description of a formal framework and of a complete controller synthesis procedure, including complexity reduction methods, provide powerful arguments in favor of the LPV approach for the robust control of nonlinear systems

Toward nonlinear tracking and rejection using LPV control

International audience(Quasi) LPV control and more generally L2 gain control methods, referred to as nonlinear H ∞ control ones, are usually applied in order to ensure reference tracking and disturbance rejection. In this paper, we exhibit a counterexample that reveals that these specifications can not be ensured by these methods. We then propose a new LPV based approach in order to a priori ensure these specifications by combining the LPV method with the incremental L 2 gain analysis of nonlinear performance. Its benefit is illustrated on the counterexample

Convex Conditions for Model Reduction of Linear Parameter Varying Systems

International audienceComplexity being one of the main limitations of LPV methods, the need for efficient model reduction techniques is highly motivated. Yet, so far, there exists no convex formulation of the general problem of finding a reduced model of any given complexity. In this paper, we focus on the case when the reduced model is supposed to have a special structure and we then derive convex conditions. Thus, for a system modeled by an LFT on a repeated scalar parameter structure, we prove that the problem can be formulated as an LMI optimization problem in the case when the reduced model is supposed to depend only on some parameters of the original system in the same manner as the plant whereas the dependence on the other parameters has been removed. The method is applicable to quadratically stable systems. A complete construction procedure is provided and a measure of the associated model reduction error is given. The method is illustrated in the context of missile control

Toward nonlinear tracking and rejection using LPV control

(Quasi) LPV control methods and more generally L2 gain control methods, referred to as nonlinear H infinity control methods, are usually applied in order to ensure reference tracking and disturbance rejection. In this paper, we exhibit a counterexample that reveals that these specifications can not be ensured by these methods. We then propose a new LPV based approach in order to a priori ensure these specifications by combining the LPV method with the incremental L2 gain analysis of nonlinear performance. Its benefit is illustrated on the counterexample

An Enhanced Information Structure for Linear Parameter-Varying Design: Application to Reichert's Missile Benchmark

International audienceThis paper is concerned with the application of linear parameter-varying (LPV) methods. Its purpose is to investigate the interest of a new information structure for the LPV controllers. The proposed improvement consists in extending the traditional information structure by introducing, beside the signals usually measured, special signals supposed available for control. This enhances the design in two directions: first, the performance of the obtained controller is improved by a more accurate adjustment to the LPV system parameter value; second, this structure enables the implementation of a controller of reduced complexity in relation to the LPV system parameter. The advantages of the proposed structure are illustrated on the single-axis missile control problem proposed by Reichert which has been intensively studied in the existing literature