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18
Disciplined convex programming
 Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Application Series
, 2006
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Rankone LMI Approach to Simultaneous Stabilization of Linear Systems
, 1998
"... Following a polynomial approach to control design, the costabilization by a fixed controller of a family of SISO linear systems is interpreted as an LMI feasibility problem with a rankone constraint. An LMI relaxation algorithm and a potential reduction heuristic are then proposed for addressing t ..."
Abstract

Cited by 23 (17 self)
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Following a polynomial approach to control design, the costabilization by a fixed controller of a family of SISO linear systems is interpreted as an LMI feasibility problem with a rankone constraint. An LMI relaxation algorithm and a potential reduction heuristic are then proposed for addressing this key optimization problem. This work was supported by the Barrande Project No. 97/00597/026, by the Grant Agency of the Czech Republic under contract No. 102/97/0861, by the Ministry of Education of the Czech Republic under contract No. VS97/034 and by the French Ministry of Education and Research under contract No. 10INSA96. y Corresponding author. Email henrion@laas.fr. FAX 33 5 61 33 69 69. 1 Introduction We consider the problem of simultaneously stabilizing, or costabilizing, a family of singleinput singleoutput (SISO) linear systems by one fixed controller of given order. This fundamental problem, recognized as one of the difficult open issues in linear system theory, ar...
A Newtonlike method for solving rank constrained linear matrix inequalities
 in Proc. 43rd IEEE Conference on Decision and Control
, 2004
"... Abstract — This paper presents a Newton–like algorithm for solving systems of rank constrained linear matrix inequalities. Though local quadratic convergence of the algorithm is not a priori guaranteed or observed in all cases, numerical experiments, including application to an output feedback stabi ..."
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Cited by 20 (4 self)
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Abstract — This paper presents a Newton–like algorithm for solving systems of rank constrained linear matrix inequalities. Though local quadratic convergence of the algorithm is not a priori guaranteed or observed in all cases, numerical experiments, including application to an output feedback stabilization problem, show the effectiveness of the algorithm. I.
Algebraic Approach to Robust Controller Design: A Geometric Interpretation
 Proceedings of the American Control Conference
, 1998
"... The problem of robust controller design is addressed for a singleinput singleoutput plant with a single uncertain parameter. Given one controller that stabilizes the nominal plant, the YoulaKucera parametrization of all stabilizing controllers and quadratic forms over HermiteFujiwara matrices ar ..."
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Cited by 8 (7 self)
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The problem of robust controller design is addressed for a singleinput singleoutput plant with a single uncertain parameter. Given one controller that stabilizes the nominal plant, the YoulaKucera parametrization of all stabilizing controllers and quadratic forms over HermiteFujiwara matrices are used to provide clear and simple geometric answers to the following questions: Can the plant be robustly stabilized by a nominally stabilizing controller ? How can this robust controller be designed ? Thanks to recent results on bilinear matrix inequalities, this geometric interpretation allows to state the equivalence between robust controller design and the concave minimization problem. 1 Introduction Since the pioneering work of Kharitonov, significant results have been achieved through the polynomial approach to linear systems robustness. In his monograph [1], Barmish presents a clear and comprehensive survey of existing techniques. Given a nominally stable polynomial with a single un...
LMI Approximations for the Radius of the Intersection of Ellipsoids
 Journal of Optimization Theory and Applications
, 1998
"... This paper addresses the problem of evaluating the maximum norm vector within the intersection of several ellipsoids. This difficult nonconvex optimization problem frequently arises in robust control synthesis. Linear matrix inequality relaxations of the problem are enumerated. Two randomized algor ..."
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Cited by 7 (4 self)
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This paper addresses the problem of evaluating the maximum norm vector within the intersection of several ellipsoids. This difficult nonconvex optimization problem frequently arises in robust control synthesis. Linear matrix inequality relaxations of the problem are enumerated. Two randomized algorithms and several ellipsoidal approximations are described. Guaranteed approximation bounds are derived in order to evaluate the quality of these relaxations. 1 Introduction 1.1 Problem Statement In this paper we consider the optimization problem p opt = max x x 0 x s.t. x 2 F (1) where x is a vector in R n and the set F is the intersection of m ellipsoids F = E 1 " E 2 " \Delta \Delta \Delta " Em (2) Corresponding Author. Email: henrion@laas.fr defined as E i = fx : x 0 P i x 1g (3) for P i a given symmetric positive definite matrix in R n\Thetan . Feasible set F is the intersection of m centered ellipsoids in R n , hence F is convex and centered about the origin. It i...
Stability region analysis using simulations and sumofsquares programming
 In Proceedings of the American Control Conference
, 2007
"... Abstract — The problem of computing bounds on the regionofattraction for systems with polynomial vector fields is considered. Invariant sets of the regionofattraction are characterized as sublevel sets of Lyapunov functions. Finite dimensional polynomial parameterizations for the Lyapunov funct ..."
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Cited by 4 (0 self)
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Abstract — The problem of computing bounds on the regionofattraction for systems with polynomial vector fields is considered. Invariant sets of the regionofattraction are characterized as sublevel sets of Lyapunov functions. Finite dimensional polynomial parameterizations for the Lyapunov functions are used. A methodology utilizing information from simulations to generate Lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed. The suitability of the Lyapunov function candidates are assessed solving linear sumofsquares optimization problems. Qualified candidates are used to compute provably invariant subsets of the regionofattraction and to initialize various bilinear search strategies for further optimization. We illustrate the method on several small examples drawn from the literature. I.
LMI Relaxations for Robust Stability of Linear Systems with Saturating Controls
"... A method is proposed for studying robust stability of uncertain linear systems subject to actuator saturation. It hinges upon successive LMI relaxations and therefore can readily be implemented using widely available numerical tools. The technique is illustrated on two examples and is shown to signi ..."
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Cited by 3 (1 self)
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A method is proposed for studying robust stability of uncertain linear systems subject to actuator saturation. It hinges upon successive LMI relaxations and therefore can readily be implemented using widely available numerical tools. The technique is illustrated on two examples and is shown to significantly improve previously published results. Introduction It is now widely recognized that both uncertainty and actuator saturation are ubiquitous features that must explicitly be taken into account when designing control laws for linear systems. On the one hand, theoretical achievements in the study of uncertain systems led to the socalled robust controllers, see [14] for a good introduction. On the other hand, several new results recently emerged in the realm of systems with constrained or saturated inputs, see [12] for an overview. Quite surprisingly, a very few works have been devoted so far to combining both features, in spite of their relevance in practical control problems. Motiv...
A Predictive Controller with Artificial Lyapunov Function for Linear Systems with Input/State Constraints
, 1998
"... This paper copes with the problem of satisfying input and/or state hard constraints in setpoint tracking problems. Stability is guaranteed by synthesizing a Lyapunov quadratic function for the system, and by imposing that the terminal state lies within a level set of the function. Procedures to max ..."
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Cited by 3 (0 self)
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This paper copes with the problem of satisfying input and/or state hard constraints in setpoint tracking problems. Stability is guaranteed by synthesizing a Lyapunov quadratic function for the system, and by imposing that the terminal state lies within a level set of the function. Procedures to maximize the volume of such an ellipsoidal set are provided, and interiorpoint methods to solve online optimization are considered. Key words: Predictive control, Constraints, Lyapunov function, Setpoint control, Optimization problems, Interiorpoint methods, Quadratically constrained quadratic programming. 1 Introduction The necessity of satisfying input/state constraints is a feature that frequently arises in control applications. Constraints are dictated for instance by physical limitations of the actuators or by the necessity to keep some plant variables within safe limits. In recent years, several control techniques have been developed which are able to handle hard constraints, see e....
Loworder Robust Controller Synthesis for Interval Plants
 International Journal of Control
, 1998
"... This paper deals with robust controller synthesis for SISO linear plants subject to interval parametric uncertainty, a longstanding open problem of control theory. Based on HermiteFujiwara matrices and the Generalized Kharitonov's Theorem, a necessary and sufficient condition is derived for th ..."
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Cited by 3 (2 self)
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This paper deals with robust controller synthesis for SISO linear plants subject to interval parametric uncertainty, a longstanding open problem of control theory. Based on HermiteFujiwara matrices and the Generalized Kharitonov's Theorem, a necessary and sufficient condition is derived for the existence of a robustly stabilizing controller of given order. This condition is formulated as a nonconvex rankone LMI feasibility problem in the controller parameters. Two heuristics are then proposed to handle this key optimization problem, namely a potential reduction algorithm and a Frank and Wolfe gradient algorithm. Both algorithms hinge upon standard semidefinite programming techniques. Several numerical examples bear out the usefulness of our approach for designing robust controllers of small order at low computational cost. 1 Introduction We focus on the problem of robust stabilization of an uncertain singleinput singleoutput plant whose parameters belong to given real intervals. ...