. We consider least-squares problems where the coefficient matrices A; b are unknown-butbounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution, and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A; b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation. Key Words. Least-squares, uncertainty, robustness, second-order cone...

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.

This paper is concerned with the robust control problem of plants subject to real parametric uncertainties. The proposed technique builds upon the use of parameter-dependent quadratic Lyapunov functions. Such Lyapunov functions are used to derive sufficient conditions for the existence of controllers ensuring robust performance of the closed-loop system. These conditions lead to a complete synthesis technique, based on a relaxation algorithm reminiscent of µ-synthesis schemes. It alternates analysis phases and synthesis phases both characterized by tractable conditions in the form of Linear Matrix Inequalities (LMIs). The major advantage of the proposed technique is to produce robust controllers whose order is the same as the original plant. It allows to bypass the frequency sampling and curve fitting steps often critical in µ synthesis algorithms. A simple illustrative application demonstrates that the approach in this paper compares favorably to traditional µ-synthesis.

We consider linear systems with unspecified parameters that lie between given upper and lower bounds. Except for a few special cases, the computation of many quantities of interest for such systems can be performed only through an exhaustive search in parameter space. We present a general branch and bound algorithm that implements this search in a systematic manner and apply it to computing the minimum stability degree. 1 Introduction 1.1 Notation R (C) denotes the set of real (complex) numbers. For c 2 C, Re c is the real part of c. The set of n \Theta n matrices with real (complex) entries is denoted R n\Thetan (C n\Thetan ). P T stands for the transpose of P , and P , the complex conjugate transpose. I denotes the identity matrix, with size determined from context. For a matrix P 2 R n\Thetan (or C n\Thetan ), i (P ); 1 i n denotes the ith eigenvalue of P (with no particular ordering). oe max (P ) denotes the maximum singular value (or spectral norm) of P , define...

A wide variety of problems in control system theory fall within the class of parameterized Linear Matrix Inequalities (LMIs), that is, LMIs whose coefficients are functions of a parameter conned to a compact set. Such problems, though convex, involve an innite set of LMI constraints, hence are inherently difficult to solve numerically. This paper investigates relaxations of parameterized LMI problems into standard LMI problems using techniques relying on directional convexity concepts. An in-depth discussion of the impacts of the proposed techniques in quadratic programming, Lyapunov-based stability and performance analysis, µ analysis and Linear Parameter Varying control is provided. Illustrative examples are given to demonstrate the usefulness and practicality of the approach.

This paper presents a new approach to finite-horizon guaranteed state prediction for discrete-time systems affected by bounded noise and unknown-but-bounded parameter uncertainty. Our framework handles possibly nonlinear dependence of the state-space matrices on the uncertain parameters. The main result is that a minimal confidence ellipsoid for the state, consistent with the measured output and the uncertainty description, may be recursively computed in polynomial time, using interior-point methods for convex optimization. With n states, l uncertain parameters appearing linearly in the state-space matrices, with rank-one matrix coefficients, the worst-case complexity grows as O(l(n + l) 3:5 ). With unstructured uncertainty in all system matrices, the worst-case complexity reduces to O(n 3:5 ).

In this paper, we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations. The proposed technique is based on the combination of a quadratic embedding of the uncertainty, and the S-procedure. This formulation leads to convex optimization problems that can be essentially solved in O(n 3)—n being the size of unknown vector — by means of suitable interior point barrier methods, as well as to closed form results in some particular cases. We further show that the uncertain linear equations paradigm can be directly applied to various state-bounding problems for dynamical systems subject to set-valued noise and model uncertainty.

Pierre Apkarian k Abstract We discuss a partially augmented Lagrangian method for optimization programs with matrix inequality constraints. A global convergence result is obtained. Applications to hard problems in feedback control are presented to validate the method numerically.

A number of techniques have been developed in recent years for the analysis and design of controllers which are robust with respect to structured complex uncertainty. In particular the complex synthesis procedure has been successfully applied to a number of engineering problems. However the presence of real parametric uncertainty in the problem description substantially complicates matters, so that standard complex synthesis techniques are no longer adequate. In this paper we develop a procedure to tackle the mixed (real and complex) synthesis problem. This procedure involves a "D,G-K iteration" between computing the mixed upper bound and solving an H1 optimal control problem, and has guaranteed convergence to a local minimum of the (nonconvex) problem. The procedure has been implemented in software, and several controller designs are compared with the corresponding complex µ synthesis designs.