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Abstract

The set of controllers stabilizing a linear system is generally non-convex in the parameter space. In the case of two-parameter controller design (e.g. PI control or static output feedback with one input and two outputs), we observe however that quite often for benchmark problem instances, the set of stabilizing controllers seems to be convex. In this note we use elementary techniques from real algebraic geometry (resultants and Bezoutian matrices) to explain this phenomenon. As a byproduct, we derive a convex linear matrix inequality (LMI) formulation of two-parameter fixed-order controller design problem, when possible.

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Abstract

We consider the class of nonlinear optimal control problems (OCPs) with polynomial data, i.e., the differential equation, state and control constraints, and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI- (linear matrix inequality)-relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assumptions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments.

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In the complex plane, the frequency response of a univariate polynomial is the set of values taken by the polynomial when evaluated along the imaginary axis. This is an algebraic curve partitioning the plane into several connected components. In this note it is shown that the component including the origin is exactly representable by a linear matrix inequality if and only if the polynomial is stable, in the sense that all its roots have negative real parts.

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Abstract

Uncertainty in the integrators of 2-1 sigma-delta modulators causes imperfect cancellation of first stage quantization noise, and reduces signal-to-noise ratio in analogue-to-digital converters. Design of robust matching filters based on convex optimization over uncertain linearized statespace representations gives complicated models and high-order designs. This letter describes a polynomial design method leading to simpler multilinear models and fixed-order filters. The modulators are cast as a polynomial polytope, and filters satisfying an bound arise from solving linear matrix inequalities (LMIs). Results at low frequency showthe proposed filter outperforming the nominal one, with a performance close to the estimated optimum.

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Résumé

A l'aide de quelques exemples illustratifs, des pistes sont évoquées pour combiner les méthodes polynomiales (algèbre, géométrie algébrique) et l'optimisation convexe (inégalités matricielles linéaires, LMI) dans le but de développer des outils numériques de résolution de problèmes basiques en automatique, et en particulier pour la commande robuste des systèmes linéaires. Dans le chapitre 2, nous évoquons les liens étroits entre ensembles semi-algébriques convexes et LMI,ainsi que la notion sous-jacente de convexité cachée remettant en question la traditionnelle dichomotime entre convexité et non-convexité. Dans le chapitre 3, nous décrivons les méthodes classiques permettant d'approcher les problèmes de commande linéaire robuste à l'aide des polynômes, en insistant sur l'interaction entre algèbre et géométrie. Le chapitre 4 mentionne les différents outils logiciels développés dans ce cadre. Finalement le chapitre 5 contient quelques suggestions d'axes de recherche cohérents avec ces développements.

Abstract

In this technical note, the problem of designing fixed-order robust $H_{infty}$ controllers is considered for linear systems affected by polytopic uncertainty. A polynomial method is employed to design a fixed-order controller that guarantees that all the closed-loop poles reside within a given region of the complex plane. In order to utilize the freedom of the controller design, an $H_{infty}$ performance specification is also enforced by using the equivalence between robust stability and $H_{infty}$ norm constraint. The design problem is formulated as a linear matrix inequality (LMI) constraint whose decision variables are controller parameters. An illustrative example demonstrates the feasibility of the proposed design methods.

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This document is an introduction to the Matlab package SDLS (Semi-Definite Least-Squares) for solving least-squares problems over convex symmetric cones. The package is shortly presented through the addressed problem, a sketch of the implemented algorithm, the syntax and calling sequences, a simple numerical example and some more advanced features. The implemented method consists in solving the dual problem with a quasi-Newton algorithm. We note that SDLS is not the most competitive implementation of this algorithm: efficient, robust, commercial implementations are available (contact the authors). Our main goal with this Matlab SDLS package is to provide a simple, user-friendly software for solving and experimenting with semidefinite least-squares problems. Up to our knowledge, no such freeware exists at this date.

Résumé

Des résultats existant sur la synthèse de correcteurs d'ordre fixé sont étendus au cas de la synthèseLPV SISO avec garantie de performance H sur l'ensemble de variation de paramètre de séquencement. Les caractéristiques principales de l'approche sont: (a) l'utilisation de polynômes pour la modélisation des systèmes; (b) l'utilisation de conditions LMI flexibles autorisant une dépendance polynomiale en la variable de séquencement des fonctions de transfert du système en boucle ouverte et du correcteur; (c) l'existence d'un découplage au niveau des conditions LMI entre les variables de Lyapunov et les variables du correcteur, ce qui permet à la fois l'utilisation de fonctions de Lyapunov dépendant de paramètres et la synthèse d'un correcteur d'ordre fixé. La procédure de synthèse est intégrée à l'outil de simulation ATOL développé par la Snecma dans le but de mettre en place des techniques de synthèse LPV systématiques et de complexité réduite, pour la commande de moteurs d'avion.