We present a new semidefinite programming approach to FIR lter design with arbitrary upper and lower bounds on the frequency response magnitude. It is shown that the constraints can be expressed as linear matrix inequalities (LMIs), and hence they can be easily handled by recent interior-point methods. Using this LMI formulation, we can cast several interesting filter design problems as convex or quasi-convex optimization problems, e.g., minimizing the length of the FIR filter and computing the Chebychev approximation of a desired power spectrum or a desired frequency response magnitude on a logarithmic scale.

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In this paper we study the properties of the analytic central path of a semide nite programming problem under perturbation of a set of input parameters. Speci cally, we analyze the behavior of solutions on the central path with respect to changes on the right hand side of the constraints, including the limiting behavior when the central optimal solution is approached. Our results are of interest for the sake ofnumerical analysis, sensitivity analysis and parametric programming. Under the primal-dual Slater condition and the strict complementarity condition we show that the derivatives of central solutions with respect to the right hand side parameters converge as the path tends to the central optimal solution. Moreover, the derivatives are bounded, i.e. a Lipschitz constant exists. This Lipschitz constant can be thought of as a condition number for the semide nite programming problem. It is a generalization of the familiar condition number for linear equation systems and linear programming problems. However, the generalized condition number depends on the right hand side parameters as well, whereas it is well-known that in the linear programming case the condition number depends only on the constraint matrix. We demonstrate that the existence of strictly complementary solutions is important for the Lipschitz constant to exist. Moreover, we give an example in which the set of right hand side parameters for which the strict complementarity condition holds is neither open nor closed. This is remarkable since a similar set for which the primal-dual Slater condition holds is always open. Key words: analytic central path, semide nite programming, sensitivity, condition number.

Since 1984 there has been a concentrated e ort to develop e cient interior-point methods for linear programming (LP). In the last few years researchers have begun to appreciate a very important property of these interior-point methods (beyond their e ciency for LP): they extend gracefully to nonlinear convex optimization problems. New interior-point algorithms for problem classes such as semide nite programming (SDP) or second-order cone programming (SOCP) are now approaching the extreme e ciency of modern linear programming codes. In this paper we discuss three examples of areas of control where our ability to e ciently solve nonlinear convex optimization problems opens up new applications. In the rst example we show how SOCP can be used to solve robust open-loop optimal control problems. In the second example, we show how SOCP can be used to simultaneously design the set-point and feedback gains for a controller, and compare this method with the more standard approach. Our nal application concerns analysis and synthesis via linear matrix inequalities and SDP. Submitted to a special issue of Journal of Process Control, edited by Y. Arkun & S. Shah, for papers presented at the 1997 IFAC Conference onAdvanced Process Control, June 1997, Ban. This and related papers available via anonymous FTP at

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This paper presents a new, iterative algorithm for designing full order LTI controllers for systems with real parameter uncertainty. Robust stability isdetermined for these systems using the Popov analysis criterion and multiplier, and robust performance is investigated using a bound on the output energy. Control design to minimize the robust performance metric naturally leads to Bilinear Matrix Inequalities, which can be decoupled to a large extent. However, coupling remains in the problem since we simultaneously optimize the parameters of both the Popov stabilitymultiplier and the compensator. We present a heuristic, iterative algorithm to solve this design problem, and demonstrate that it works effectively on two numerical examples. In the process, we illustrate that the key advantages of this control design approach are the high reliability of the numerical techniques and the relative simplicity of implementing the algorithm. 1

We consider convex optimization problems with linear matrix inequality (LMI) constraints, i.e., constraints of the form F (x) =F0+x1F1+ +xmFm 0; (1.1) where the matrices Fi = F T

Abstract. A general framework for the least squares approximation of symmetric-de nite pencils subject to generalized eigenvalues constraints is developed in this paper. This approach can be adapted to di erent applications, including the inverse eigenvalue problem. The idea is based on the observation that a natural parameterization for the set of symmetric-de nite pencils with the same generalized eigenvalues is readily available. In terms of these parameters, descent ows on the isospectral surface aimed at reducing the distance to matrices of the desired structure can be derived. These ows can be designed to carry certain other interesting properties and may beintegrated numerically.

In this paper, we discuss fast randomized algorithms for determining an admissible solution for robust linear matrix inequalities (LMIs) of the form F (x# \Delta) 0# where x is the optimization variable and \Delta is the uncertainty, which belongs to a given set \Delta. The proposed algorithm is based on uncertainty randomization: it finds a solution in a finite number of iterations with probability one, if a strong feasibility condition holds. Otherwise, it computes a candidate solution which minimizes the expected value of a suitably selected feasibility indicator function. The theory is illustrated by examples of application to uncertain linear inequalities and quadratic stability of interval matrices.