1. Simple Approximations of Semialgebraic Sets and their Applications to Control
- Author
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Didier Henrion, Fabrizio Dabbene, Constantino Lagoa, CNR Institute of Electronics, Computer and Telecommunication Engineering [Torino] (CNR | IEIIT), CNR Istituto di elettronica e di ingegneria dell'informazione e delle telecomunicazioni (CNR | IEIIT), National Research Council of Italy | Consiglio Nazionale delle Ricerche (CNR)-National Research Council of Italy | Consiglio Nazionale delle Ricerche (CNR), Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Czech Technical University in Prague (CTU), Department of Computer Science and Engineering [University Park], Pennsylvania State University (Penn State), Penn State System-Penn State System, Institute of Electronics, Computer and Telecommunication Engineering (IEIIT-CNR), Politecnico di Torino = Polytechnic of Turin (Polito)-Consiglio Nazionale delle Ricerche [Torino] (CNR), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), and Université Fédérale Toulouse Midi-Pyrénées
- Subjects
Semialgebraic set ,0209 industrial biotechnology ,Polynomial ,Positive polynomial ,Stability (learning theory) ,Linear matrix inequalities ,02 engineering and technology ,Systems and Control (eess.SY) ,01 natural sciences ,020901 industrial engineering & automation ,FOS: Mathematics ,FOS: Electrical engineering, electronic engineering, information engineering ,[INFO.INFO-SY]Computer Science [cs]/Systems and Control [cs.SY] ,Applied mathematics ,Almost everywhere ,0101 mathematics ,Electrical and Electronic Engineering ,Sampling ,Mathematics - Optimization and Control ,Finite set ,Approximation ,Mathematics ,Hierarchy (mathematics) ,010102 general mathematics ,Solution set ,Control and Systems Engineering ,Optimization and Control (math.OC) ,Semialgebraic set Approximation Sampling ,Computer Science - Systems and Control ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] - Abstract
International audience; Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the solution set of linear matrix inequalities or the Schur/Hurwitz stability domains. These sets often have very complicated shapes (non-convex, and even non-connected), which renders very difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrect-angles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly non-convex approximations , based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encounter in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set. Two main applications of the proposed approach are considered. The first one aims at reconstruction/approximation of sets from a finite number of samples. In the second one, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples.
- Published
- 2017
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