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12
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 of paramet ..."
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Cited by 77 (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 of paramet ..."
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Cited by 55 (2 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
, 1993
"... 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 efficiently. In many c ..."
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Cited by 13 (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 efficiently. 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...
A Class of Lyapunov Functionals for Analyzing Hybrid Dynamical Systems
 IEEE Trans. Aut. Control
, 1999
"... In this paper, we introduce a new class of Lyapunov functionals for analyzing hybrid dynamical systems. This class can be thought of as a generalization of the Lyapunov functional introduced by Yakubovich for systems with hysteresis nonlinearities which incorporates path integrals that account for t ..."
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Cited by 5 (2 self)
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In this paper, we introduce a new class of Lyapunov functionals for analyzing hybrid dynamical systems. This class can be thought of as a generalization of the Lyapunov functional introduced by Yakubovich for systems with hysteresis nonlinearities which incorporates path integrals that account for the energy loss or gain every time a hysteresis loop is traversed. Hence, these Lyapunov functionals capture the pathdependence of the "stored energy" in hybrid dynamical systems and are therefore less conservative over previously published approaches in analyzing such systems. More importantly, we show that searching over the proposed class of Lya punov functionals to prove some specification (e.g., stability) can be cast as a semidefinite program (SDP), which can then be efficiently solved (globally) using widely available sofware. Examples are pre sented to show the effectiveness of this class of Lyapunov functionals in analyzing hybrid dynamical systems.
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...
Gessen, “LMI approach to stabilization of a linear plant by a pulse modulated signal
 Int. J. Hybrid Systems
, 2003
"... ABSTRACT: 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 ..."
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Cited by 1 (1 self)
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ABSTRACT: 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. AMS (MOS) subject classification: 93D15, 93D10, 15A39 1.
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 1 (0 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.
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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Cited by 1 (0 self)
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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,
Modern Control Theory A historical perspective
"... Abstract. The purpose of this paper is to present a brief sketch of the evolution of modern control theory. Systems theory witnessed different stages and approaches, which will be very shortly presented. The main idea is that, at present, Control Theory is an interdisciplinary area of research where ..."
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Abstract. The purpose of this paper is to present a brief sketch of the evolution of modern control theory. Systems theory witnessed different stages and approaches, which will be very shortly presented. The main idea is that, at present, Control Theory is an interdisciplinary area of research where many mathematical concepts and methods work together to produce an impressive body of important applied mathematics. A general conclusion is that the main advances in Control of Systems would come both from mathematical progress and from technological development. We start with frequencydomain approach and end our historical perspective with structuraldigraph approach, passing through timedomain, polynomialmatrixdomain frequential and geometric approaches. 1.
SIMULATION Outline
, 2005
"... descriptor systems Linear Descriptor Systems Linear timeinvariant systems in generalized statespace form: arise, e.g., in E ˙x(t) = Ax(t) + Bu(t), • control and simulation of coupled systems, y(t) = Cx(t) + Du(t), • control of multibody (mechanical) systems, • manipulation of fluid flow (e.g., s ..."
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descriptor systems Linear Descriptor Systems Linear timeinvariant systems in generalized statespace form: arise, e.g., in E ˙x(t) = Ax(t) + Bu(t), • control and simulation of coupled systems, y(t) = Cx(t) + Du(t), • control of multibody (mechanical) systems, • manipulation of fluid flow (e.g., semidiscretized NavierStokes equations), • circuit simulation, VLSI chip design, in particular modeling of interconnet via RLC networks, • simulation of MEMS and NEMS (micro/nanoelectromechanical systems). PARALLELE