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robust semidefinite programming problems with general nonlinear parameter dependence: Approaches using the DC-representations

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AUTOMATICA 2012年第11期48卷 2937-2944页

作者： Oishi, Yasuaki Alamo, Teodoro Nanzan Univ Dept Syst Design & Engn Seto 4890863 Japan Univ Seville Dept Ingn Sistemas & Automat Seville 41092 Spain

Two conservative approaches are proposed to a semidefinite programming problem nonlinearly dependent on uncertain parameters. These approaches are applicable to general nonlinear parameter dependence not necessarily polynomial or rational. They are based on the mild assumption that the parameter dependence is expressed as the difference of two convex functions. The first approach uses constant bounds on the parameter dependence. Optimization of the bounds is reduced to convex nonsmooth minimization. The second approach uses parameter-dependent bounds for a less conservative result. Optimization of the bounds is immediate when the centroid is computable for the parameter set. Numerical examples are presented for illustration of the approaches. (c) 2012 Elsevier Ltd. All rights reserved.

关键词： robust semidefinite programming Nonlinear parameter dependence DC-representation Convexity Centroid Linear matrix inequalities Conservatism

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robust semidefinite programming Problems with General Nonlinear Parameter Dependence: Application of the DC-Representations

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IFAC Proceedings Volumes 2011年第1期44卷 7939-7944页

作者： Yasuaki Oishi Teodoro Alamo Department of Systems Design and Engineering Nanzan University Seireicho 27 Seto 489-0863 Japan Departamento de Ingeniería de Sistemas y Automática Universidad de Sevilla Camino de los Descubrimientos s/n. 41092 Sevilla Spain

Abstract Two conservative approaches are proposed to a semidefinite programming problem nonlinearly dependent on uncertain parameters. These approaches are applicable to general nonlinear parameter dependence not necessarily polynomial or rational. They are based on a mild assumption that the parameter dependence is expressed as the difference of two convex functions. The first approach uses constant bounds on the parameter dependence. Optimization of the bounds is reduced to convex nonsmooth minimization in general. The second approach uses parameter-dependent bounds for a less conservative result. Optimization of the bounds is immediate when the gravity center is computable for the parameter set. Numerical examples are presented for illustration of the approaches.

关键词： robust semidefinite programming nonlinear parameter dependence DC-representation convexity conjugate functions linear matrix inequalities conservatism

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Exploiting sparsity in the matrix-dilation approach to robust semidefinite programming

Exploiting sparsity in the matrix-dilation approach to robus...

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26th American Control Conference

作者： Oishi, Yasuaki Isaka, Yusuke Nanzan Univ Dept Informat Syst & Math Sci Seireicho Seto 4890863 Japan Univ Tokyo Dept Math Informat Tokyo 1138656 Japan

ISBN: (纸本)9781424409884

A computationally improved approach is proposed for a robust semidefinite programming problem whose constraint is polynomially dependent on uncertain parameters. By exploiting sparsity, the proposed approach gives an approximate problem smaller in size than the matrix-dilation approach formerly proposed by the group of the first author. Here, the sparsity means that the constraint of the given problem has only a small number of nonzero terms when it is expressed as a polynomial of the uncertain parameters. This sparsity is extracted with a special graph called a rectilinear Steiner arborescence, based on which a reduced-size approximate problem is constructed. The quality of the approximation can be evaluated quantitatively. This evaluation shows that the quality can be improved to any level by dividing the parameter region into small subregions.

关键词： robust semidefinite programming linear matrix inequalities matrix dilation sparsity rectilinear Steiner arborescences computational cost conservatism

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A region-dividing approach to robust semidefinite programming and its error bound

A region-dividing approach to robust semidefinite programmin...

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American Control Conference 2006

作者： Oishi, Yasuaki Univ Tokyo Dept Math Informat Bunkyo Ku Tokyo 1138656 Japan

ISBN: (纸本)1424402093

A new asymptotically exact approach is presented for robust semidefinite programming, where coefficient matrices polynomially depend on uncertain parameters. Since a robust semidefinite programming problem is difficult to solve directly, an approximate problem is constructed based on a division of the parameter region. The optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer. An advantage of this approach is that an upper bound on the approximation error is available before solving the approximate problem. This bound shows how the approximation error depends on the resolution of the division. Furthermore, it leads to construction of an efficient division that attains small approximation error with low computational complexity. Numerical examples show efficacy of the present approach. In particular, an exact optimal value is often found with a division of finite resolution.

关键词： robust semidefinite programming polynomial optimization linear matrix inequalities approximation conservatism computational complexity

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Reduction of the number of constraints in the matrix-dilation approach to robust semidefinite programming

Reduction of the number of constraints in the matrix-dilatio...

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45th IEEE Conference on Decision and Control

作者： Oishi, Yasuaki Univ Tokyo Dept Math Informat Bunkyo Ku Tokyo 1138656 Japan

ISBN: (纸本)9781424401703

In order to solve a robust semidefinite programming problem, which is a semidefinite programming problem containing uncertain parameters, an approximate problem with a small number of constraints is proposed. The proposed approximate problem is constructed by dilation of the matrices in the constraints. The minimum value of the approximate problem converges to that of the original problem as the parameter region is divided finer. Although, with the previously proposed approximate problem, the number of constraints is of exponential order in the parameter dimension, it is of linear order with the new approximate problem. Just as the previous approximate problem, the quality of approximation is expressed in terms of the resolution of the division. This result is obtained by application of the result of Ben-Tal-Nemirovski (SIAM J. Opt., vol. 12, no. 3, pp. 811-833, 2002). In the last section of the paper, a useful property is derived on polynomial optimization, which can be reduced to robust semidefinite programming.

关键词： robust semidefinite programming polynomial optimization linear matrix inequalities approximation conservatism computational complexity

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A Matrix-Dilation Approach to robust semidefinite programming and Its Error Bound

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SICE Journal of Control, Measurement, and System Integration 2010年第5期3卷 372-380页

作者： Yasuaki Oishi Department of Systems Design and Engineering Nanzan University

An asymptotically exact approach is presented for robust semidefinite programming, where coefficient matrices polynomially depend on uncertain parameters. Since a robust semidefinite programming problem is difficult to solve directly, an approximate problem is constructed based on a division of the parameter region. The optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes higher. An advantage of this approach is that an upper bound on the approximation error is available before solving the approximate problem. This bound shows how the approximation error depends on the resolution of the division. Furthermore, it leads to construction of an efficient division that attains small approximation error with low computational cost. Numerical examples show efficacy of the present approach. In particular, the exact optimal value is often found with a division of a finite resolution.

关键词： robust semidefinite programming polynomial optimization linear matrix inequalities approximation computational cost

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Relaxations for robust linear matrix inequality problems with verifications for exactness

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2005年第2期27卷 365-395页

作者： Scherer, CW Delft Univ Technol Delft Ctr Syst & Control NL-2628 CD Delft Netherlands

robust semidefinite programming problems with rational dependence on uncertainties are known to have a wide range of applications, in particular in robust control. It is well established how to systematically construct relaxations on the basis of the full block S-procedure. In general, such relaxations are expected to be conservative, but for concrete problem instances they are often observed to be tight. The main purpose of this paper is to suggest novel computationally veri. able conditions for when general classes of linear matrix inequality relaxations do not involve any conservatism. If the convex set of uncertainties is finitely generated, we suggest a novel sequence of relaxations which can be proved to be asymptotically exact. Finally, our results are applied to the particularly relevant robustness analysis problem for linear time-invariant dynamical systems affected by uncertainties that are full ellipsoidal or repeated and contained in intersections of disks or circles. This leads to extensions of known results on relaxation exactness for small block structures, including an elementary proof for tightness of standard structured singular value computations for three full complex uncertainty blocks.

关键词： robust semidefinite programming higher-order relaxations exactness tests robust performance analysis structured singular value linear matrix inequalities

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Parameterized LMIs in control theory

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2000年第4期38卷 1241-1264页

作者： Apkarian, P Tuan, HD ONERA CERT Control Syst Dept F-31055 Toulouse France Toyota Technol Inst Dept Control & Informat Tempa Ku Nagoya Aichi 4688511 Japan

A wide variety of problems in control system theory fall within the class of parameterized linear matrix inequalities (LMIs), that is, LMIs whose coefficients are functions of a parameter confined to a compact set. Such problems, though convex, involve an infinite set of LMI constraints and hence are inherently di cult to solve numerically. This paper investigates relaxations of parameterized LMI problems into standard LMI problems using techniques relying on directional convexity concepts. An in-depth discussion of the impact of the proposed techniques in quadratic programming, Lyapunov-based stability and performance analysis, mu analysis, and linear parameter-varying control is provided. Illustrative examples are given to demonstrate the usefulness and practicality of the approach.

关键词： linear matrix inequalities robust semidefinite programming directional convexity robustness analysis parametric uncertainty linear parameter-varying control

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Stochastic algorithms for exact and approximate feasibility of robust LMIs

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2001年第11期46卷 1755-1759页

作者： Calafiore, G Polyak, BT Politecn Torino Dipartimento Automat & Informat I-10129 Turin Italy Russian Acad Sci Inst Control Sci Moscow 117806 Russia

In this note, we discuss fast randomized algorithms for determining an admissible solution for robust linear matrix inequalities (LMIs) of the form F(x, Delta) less than or equal to 0, where 0 is the optimization variable and Delta is the uncertainty, which belongs to a given set Delta. The proposed algorithms are based on uncertainty randomization: the first algorithm finds a robust solution in a finite number of iterations with probability one, if a strong feasibility condition holds. In case no robust solution exists, the second algorithm computes an approximate solution which minimizes the expected value of a suitably selected feasibility indicator function. The theory is illustrated by examples of application to uncertain linear inequalities and quadratic stability of interval matrices.

关键词： linear matrix inequalities (LMIs) quadratic stability robust semidefinite programming stochastic algorithms uncertainty randomization

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Tractable Approximations of LMI robust Feasibility Sets

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IEEE CONTROL SYSTEMS LETTERS 2024年 8卷 472-477页

作者： Alamo, Teodoro Mammarella, Martina Dabbene, Fabrizio Sznaier, Mario Univ Seville Escuela Super Ingn Dept Ingn Sistemas & Automat Seville 41092 Spain CNR IEIIT Politecn Torino I-10129 Turin Italy Northeastern Univ Dept Elect & Comp Engn Boston MA 02115 USA

In this letter, we introduce novel tractable approximations for robust Linear Matrix Inequality (LMI) problems. We present various Quadratic Matrix Inequalities (QMIs) that enable us to characterize the effect of ellipsoidal uncertainty in the robust problem. These formulations are expressed in terms of a set of auxiliary decision variables, which facilitate the derivation of a generalized S-procedure result. This generalization significantly reduces the conservatism of the obtained results, compared with conventional approaches.

关键词： Linear matrix inequalities quadratic matrix inequality robust semidefinite programming

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