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78
Randomized Algorithms for Analysis and Control of Uncertain Systems
 in Perspectives in Robust Control
, 2001
"... Undoubtedly, model uncertainty and robustness have been key themes in the development of modern automatic control during the last four decades. In fact, in many situations feedback control of dynamical systems allows to substantially improve typical control engineering objectives, such as accurate p ..."
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Cited by 39 (5 self)
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Undoubtedly, model uncertainty and robustness have been key themes in the development of modern automatic control during the last four decades. In fact, in many situations feedback control of dynamical systems allows to substantially improve typical control engineering objectives, such as accurate pathfollowing or effective disturbance attenuation, even if only a rather poor mathematical model of the tobecontrolled dynamics is available. On the other hand, optimizationbasedcontroller design strategies typically rely on a sufficiently accurate model of the tobecontrolled plant. Recent years have witnessedthe development of techniques for quantifying the plantmodel mismatch, such as in uncertainty estimation basedon measureddata or as resulting from model reduction to reduce complexity. Various widely used paradigms for the mathematical description
Robustness Analysis of Polynomials with Polynomial Parameter Dependency Using Bernstein Expansion
 IEEE TRANS. AUTOMAT. CONTR
, 1998
"... ..."
Robust Pole Placement in LMI Regions
 IEEE Transactions on Automatic Control
, 1999
"... This paper discusses analysis and synthesis techniques for robust pole placement in LMI regions, a class of convex regions of the complex plane that embraces most practically useful stability regions. The focus is on linear systems with static uncertainty on the state matrix. For this class of uncer ..."
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Cited by 15 (0 self)
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This paper discusses analysis and synthesis techniques for robust pole placement in LMI regions, a class of convex regions of the complex plane that embraces most practically useful stability regions. The focus is on linear systems with static uncertainty on the state matrix. For this class of uncertain systems, the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary LMI regions. The resulting tests for robust pole clustering are all numerically tractable since they involve solving linear matrix inequalities (LMIs), and cover both unstructured and parameter uncertainty. These analysis results are then applied to the synthesis of dynamic outputfeedback controllers that robustly assign the closedloop poles in a prescribed LMI region. With some conservatism, this problem is again tractable via LMI optimization. In addition, robust pole placement can be combined with other control objectives such as H 2 or H1 performance to capture realist...
Global Optimization in Generalized Geometric Programming
 Engng
, 1997
"... A deterministic global optimization algorithm is proposed for locating the global minimum of generalized geometric (signomial) problems (GGP). By utilizing an exponential variable transformation the initial nonconvex problem (GGP) is reduced to a (DC) programming problem where both the constraints ..."
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Cited by 12 (3 self)
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A deterministic global optimization algorithm is proposed for locating the global minimum of generalized geometric (signomial) problems (GGP). By utilizing an exponential variable transformation the initial nonconvex problem (GGP) is reduced to a (DC) programming problem where both the constraints and the objective are decomposed into the difference of two convex functions. A convex relaxation of problem (DC) is then obtained based on the linear lower bounding of the concave parts of the objective function and constraints inside some box region. The proposed branch and bound type algorithm attains finite fflconvergence to the global minimum through the successive refinement of a convex relaxation of the feasible region and/or of the objective function and the subsequent solution of a series of nonlinear convex optimization problems. The efficiency of the proposed approach is enhanced by eliminating variables through monotonicity analysis, by maintaining tightly bound variables thro...
WorstCase Properties of the Uniform Distribution and Randomized Algorithms for Robustness Analysis
, 1996
"... Motivated by the current limitations of the existing algorithms for robustness analysis and design, in this paper we take a different direction which follows the socalled probabilistic approach. That is, we aim to estimate the probability that a control system with uncertain parameters q restricted ..."
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Cited by 12 (0 self)
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Motivated by the current limitations of the existing algorithms for robustness analysis and design, in this paper we take a different direction which follows the socalled probabilistic approach. That is, we aim to estimate the probability that a control system with uncertain parameters q restricted to a box Q attains a given level of performance fl. Since this probability depends on the underlying distribution, we address the following question: What is a "reasonable" distribution so that the estimated probability makes sense? To answer this question, we define two worstcase criteria and prove that the uniform distribution is optimal in both cases. In the second part of the paper, we turn our attention to a subsequent problem. That is, we estimate the sizes of both the socalled "good" and "bad" sets via sampling. Roughly speaking, the good set contains the parameters q 2 Q with performance level better than or equal to fl while the bad set is the set of parameters q 2 Q with perform...
An LMI Condition for Robust Stability of Polynomial Matrix Polytopes
, 2000
"... A sufficient LMI condition is proposed for checking robust stability of a polytope of polynomial matrices. It hinges upon two recent results: a new approach to polynomial matrix stability analysis and a new robust stability condition for convex polytopic uncertainty. Numerical experiments illustrate ..."
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Cited by 11 (9 self)
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A sufficient LMI condition is proposed for checking robust stability of a polytope of polynomial matrices. It hinges upon two recent results: a new approach to polynomial matrix stability analysis and a new robust stability condition for convex polytopic uncertainty. Numerical experiments illustrate that the condition narrows significantly the unavoidable gap between conservative tractable quadratic stability results and exact NPhard robust stability results. Keywords Polynomial matrix, Parametric uncertainty, Robust stability, Quadratic stability, LMI. This work has been supported by the Barrande Project No. 97/00597/026, by the Grant Agency of the Czech Republic under contract No. 102/99/1368 and by the Ministry of Education of the Czech Republic under contract No. VS97/034. y Corresponding author. Email henrion@laas.fr. FAX 33 5 61 33 69 69. Introduction Polynomial matrices appear as a key tool for studying systems control. Dynamics of many systems (e.g. lightly damped st...
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...
Application of Bernstein Expansion to the Solution of Control Problems
 University of Girona
, 1999
"... We survey some recent applications of Bernstein expansion to robust stability, viz. checking robust Hurwitz and Schur stability of polynomials with polynomial parameter dependency by testing determinantal criteria and by inspection of the value set. Then we show how Bernstein expansion can be used t ..."
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Cited by 7 (0 self)
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We survey some recent applications of Bernstein expansion to robust stability, viz. checking robust Hurwitz and Schur stability of polynomials with polynomial parameter dependency by testing determinantal criteria and by inspection of the value set. Then we show how Bernstein expansion can be used to solve systems of strict polynomial inequalities.
Robust absolute stability of timevarying nonlinear discretetime systems
, 2002
"... This paper studies the problem of robust absolute stability of a class of nonlinear discretetime systems with timevarying matrix uncertainties of polyhedral type and multiple timevarying sector nonlinearities. By using the variational method and the Lyapunov second method, criteria for robust abs ..."
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Cited by 7 (0 self)
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This paper studies the problem of robust absolute stability of a class of nonlinear discretetime systems with timevarying matrix uncertainties of polyhedral type and multiple timevarying sector nonlinearities. By using the variational method and the Lyapunov second method, criteria for robust absolute stability are obtained in different forms for the class of systems under consideration. Specifically, we determine the parametric classes of Lyapunov functions which define the necessary and sufficient conditions of robust absolute stability. We apply the piecewiselinear Lyapunov functions of the infinity vector norm type to derive an algebraic criterion for robust absolute stability in the form of solvability conditions of a set of matrix equations. Some simple sufficient conditions of robust absolute stability are given which become necessary and sufficient for several special cases. An example is presented as an application of these results to a specific class of systems with timevarying interval matrices in the linear part.
The Shape of the Solution Set for Systems of Interval Linear Equations with Dependent Coefficients
, 1998
"... A standard system of interval linear equations is defined as Ax = b, where A is an m × n coefficient matrix with (compact) intervals as entries, and b is an mdimensional vector whose components are compact intervals. It is known that for systems of interval linear equations the solution set, ..."
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Cited by 6 (4 self)
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A standard system of interval linear equations is defined as Ax = b, where A is an m × n coefficient matrix with (compact) intervals as entries, and b is an mdimensional vector whose components are compact intervals. It is known that for systems of interval linear equations the solution set, i.e., the set of all vectors x for which Ax = b for some A 2 A and b 2 b, is a polyhedron. In some cases, it makes sense to consider not all possible A 2 A and b 2 b, but only those A and b that satisfy certain linear conditions describing dependencies between the coefficients. For example, if we allow only symmetric matrices A (a ij = a ji ), then the corresponding solution set becomes (in general) piecewisequadratic.