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19
Robust Solutions To Uncertain Semidefinite Programs
 SIAM J. OPTIMIZATION
, 1998
"... In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible value ..."
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Cited by 109 (8 self)
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In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is "full," our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (Hölderstable) with respect to the unperturbed problem's data. The approach can thus be used to regularize illconditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.
Robust Solutions To Uncertain Semidefinite Programs
, 1998
"... In this paper we consider semidenite programs (SDPs) whose data depends on some unknownbutbounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible values ..."
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Cited by 83 (3 self)
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In this paper we consider semidenite programs (SDPs) whose data depends on some unknownbutbounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible values of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist, as SDPs. When the perturbation is "full", our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique, and continuous (Hölderstable) with respect to the unperturbed problems' data. The approach can thus be used to regularize illconditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation and integer programming.
Control system analysis and synthesis via linear matrix inequalities
 Control Conference, American
, 1982
"... A wide variety of problems in systems and control theory can be cast or recast as convex problems that involve linear matrix inequalities (LMIs). For a few very special cases there are \analytical solutions " to these problems, but in general they can be solved numerically very e ciently. In ma ..."
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Cited by 16 (1 self)
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A wide variety of problems in systems and control theory can be cast or recast as convex problems that involve linear matrix inequalities (LMIs). For a few very special cases there are \analytical solutions " to these problems, but in general they can be solved numerically very e ciently. In many cases the inequalities have the form of simultaneous Lyapunov or algebraic Riccati inequalities; such problems can be solved in a time that is comparable to the time required to solve the same number of Lyapunov or Algebraic Riccati equations. Therefore the computational cost of extending current control theory that is based on the solution of algebraic Riccati equations to a theory based on the solution of (multiple, simultaneous) Lyapunov or Riccati inequalities is modest. Examples include: multicriterion LQG, synthesis of linear state feedback for multiple or nonlinear plants (\multimodel control"), optimal transfer matrix realization, norm scaling, synthesis of multipliers for Popovlike analysis of systems with unknown gains, and many others. Full details can be found in the references cited. 1.
A class of lyapunov functionals for analyzing hybrid dynamical systems
 in American Control Conference, 1999. Proceedings of the 1999
, 1999
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Feedback KalmanYakubovich Lemma and Its Applications in Adaptive Control
, 1996
"... In this paper we give a survey of results related to the so called Feedback KalmanYakubovich Lemma (FKYL) giving necessary and sufficient solvability conditions for some class of bilinear matrix inequalities or conditions of feedback passivity of linear systems. Applications to adaptive and variabl ..."
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Cited by 7 (5 self)
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In this paper we give a survey of results related to the so called Feedback KalmanYakubovich Lemma (FKYL) giving necessary and sufficient solvability conditions for some class of bilinear matrix inequalities or conditions of feedback passivity of linear systems. Applications to adaptive and variable structure control systems are also discussed.
Linear Estimation in Krein Spaces  Part II: Applications
, 1996
"... We show that several interesting problems in H 1 \Gammafiltering, quadratic game theory and risk sensitive control and estimation, follow as special cases of the Krein space linear estimation theory developed in [1]. We show that all these problems can be cast into the problem of calculating the s ..."
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Cited by 4 (2 self)
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We show that several interesting problems in H 1 \Gammafiltering, quadratic game theory and risk sensitive control and estimation, follow as special cases of the Krein space linear estimation theory developed in [1]. We show that all these problems can be cast into the problem of calculating the stationary point of certain second order forms, and that by considering the appropriate state space models and error Gramians, we can use the Krein space estimation theory to calculate the stationary points and study their properties. The approach discussed here allows for interesting generalizations, such as finite memory adaptive filtering with varying sliding patterns. This work was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command under Contract AFOSR910060 and by the Army Research Office under contract DAAL0389K0109. This manuscript is submitted for publication with the understanding that the US Government is authorized to reproduce and dis...
High Accuracy Algorithms for the Solutions of Semidefinite Linear Programs
, 2001
"... hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means, i ..."
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Cited by 2 (0 self)
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hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. ii The University of Waterloo requires the signatures of all persons using or photocopying this thesis. Please sign below, and give address and date. iii Abstract We present a new family of search directions and of corresponding algorithms to solve conic linear programs. The implementation is specialized to semidefinite programs but the algorithms described handle both nonnegative orthant and Lorentz cone problems and Cartesian products of these sets. The primary objective is not to develop yet another interiorpoint algorithm with polynomial time complexity. The aim is practical and addresses an often neglected aspect of the current research in the area, accuracy. Secondary goals, tempered by the first, are numerical efficiency and proper handling of sparsity. The main search direction, called GaussNewton, is obtained as a leastsquares solution to the optimality condition of the logbarrier problem. This motivation ensures that the direction is welldefined everywhere and that the underlying Jacobian is wellconditioned under standard assumptions. Moreover, it is invariant under affine transformation of the space and under orthogonal transformation of the constraining cone. The GaussNewton direction, both in the special cases of linear programming and on the central path of semidefinite programs, coincides with the search directions used in practical implementations. Finally, the MonteiroZhang family of search directions can be derived as scaled projections of the GaussNewton direction. iv
LMI approach to stabilization of a linear plant by a pulsemodulated signal
 INT. J. HYBRID SYSTEMS
, 2003
"... The paper concerns stabilization of an unstable linear plant by a pulse modulator in feedback. The problem is reduced to finding a solution of some linear matrix inequalities (LMI). The conditions obtained guarantee that all the control system’s solutions starting in some neighborhood of a zero equ ..."
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Cited by 2 (1 self)
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The paper concerns stabilization of an unstable linear plant by a pulse modulator in feedback. The problem is reduced to finding a solution of some linear matrix inequalities (LMI). The conditions obtained guarantee that all the control system’s solutions starting in some neighborhood of a zero equilibrium vanish as time increases. The neighborhood description is found from the LMI.
Parameterdependent Sprocedure and Yakubovich lemma
 5th RussianSwedish Control Conference, 29–30
, 2006
"... The paper considers a linear matrix inequality (LMI) that depends on a parameter varying in a compact topological space. It turns out that if a strict LMI continuously depends on a parameter and is feasible for any value of that parameter, then it has a solution which continuously depends on the par ..."
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The paper considers a linear matrix inequality (LMI) that depends on a parameter varying in a compact topological space. It turns out that if a strict LMI continuously depends on a parameter and is feasible for any value of that parameter, then it has a solution which continuously depends on the parameter. The result holds true for LMIs that arise in Sprocedure and Yakubovich lemma. It is shown that the LMI which is polynomially dependent on a vector of parameters can be reduced to a parameterindependent LMI of a higher dimension. The result is based on the recent generalization of Yakubovich lemma proposed by Iwasaki and Hara and another generalization formulated in this paper. The problem of positivity verification for a nonSOS polynomial of two variables is considered as an example. To illustrate control applications, a method of parameterdependent Lyapunov function construction is proposed for nonlinear systems with parametric uncertainty. Comment: The paper was presented at the 5th RussianSwedish Control Conference, Lund,
PASSIFICATION OF NONSQUARE LINEAR SYSTEMS
"... Necessary and sufficient conditions for feedback passivity (passifiablity) of nonsquare linear systems published in Russian and Western literarure are surveyed. New output Gpassifiability conditions for nonsquare linear systems are given. The proofs are based on YakubovichKalmanPopov (KalmanYak ..."
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Cited by 1 (1 self)
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Necessary and sufficient conditions for feedback passivity (passifiablity) of nonsquare linear systems published in Russian and Western literarure are surveyed. New output Gpassifiability conditions for nonsquare linear systems are given. The proofs are based on YakubovichKalmanPopov (KalmanYakubovich) lemma. 1