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 (worst-case) 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 (Hölder-stable) with respect to the unperturbed problem's data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.

In this paper we consider semidenite programs (SDPs) whose data depends on some unknown-but-bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worst-case) 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 (Hölder-stable) with respect to the unperturbed problems' data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation and integer programming.

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 ("multi-model control"), optimal transfer matrix realization, norm scaling, synthesis of multipliers for Popov-like...

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 path-dependence 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 Lyapunov 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 software. Examples are presented to show the effectiveness of this class of Lyapunov functionals in analyzing hybrid dynamical systems.

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 AFOSR91-0060 and by the Army Research Office under contract DAAL03-89-K-0109. This manuscript is submitted for publication with the understanding that the US Government is authorized to reproduce and dis...

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.

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 structural-digraph approach, passing through time-domain, polynomial-matrix-domain frequential and geometric approaches. 1.

descriptor systems Linear Descriptor Systems Linear time-invariant systems in generalized state-space 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., semi-discretized Navier-Stokes equations), • circuit simulation, VLSI chip design, in particular modeling of interconnet via RLC networks, • simulation of MEMS and NEMS (micro-/nano-electro-mechanical systems). PARALLELE

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