%0 Journal Article
%T On the Solution of a Nonlinear Semidefinite Program Arising in Discrete-Time Feedback Control Design
%A El-Sayed M. E. Mostafa
%J Journal of Applied Mathematics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/683797
%X A sequential quadratic programming method with line search is analyzed and studied for finding the local solution of a nonlinear semidefinite programming problem resulting from the discrete-time output feedback problem. The method requires an initial feasible point with respect to two positive definite constraints. By parameterizing the optimization problem we ease that requirement. The method is tested numerically on several test problems chosen from the benchmark collection (Leibfritz, 2004). 1. Introduction In this paper, the following nonlinear semidefinite programming (NSDP) problem is considered: where , , are sufficiently smooth matrix functions, where means that is positive definite. This problem is assumed to be nonlinear and generally nonconvex. The origin of the considered NSDP problem is the static output feedback (SOF) design problem in which the unknown is partitioned as , where represents the state variable and represents the control variable. In the last decade NSDP has attracted the attention of many authors in the optimization community. For instance, Jarre [1] introduced an interior-point method for nonconvex semi-definite programs. Leibfritz and Mostafa [2] proposed an interior-point trust region method for a special class of NSDP problems resulting from the continuous-time SOF problem. Ko£¿vara et al. [3] considered an augmented Lagrangian method for a similar NSDP problem. Sun et al. [4] investigated the rate of convergence of the augmented Lagrangian approach for NSDP. Correa and Ramirez [5] proposed sequential semi-definite programming method for nonlinear semi-definite programming. Yamashita and Yabe [6] study local and superlinear convergence of a primal-dual interior point method. Freund et al. [7] proposed a sequential semidefinite programming approach for a nonlinear program with nonlinear semidefinite constraints. The design problem of optimal output feedback controllers is one of the most studied problems over the last four decades in the community of systems and control. Clearly, this is due to the existence of numerous practical applications in system and control engineering, finance, and statistics. The benchmark collection [8], for instance, contains over 160 applications from system and control engineering only. M£¿kil£¿ and Toivonen in the survey [9] summarized several special purpose algorithms such as the Levine-Athans method, the dual Levine-Athans method, and the descent Anderson-Moore method as well as Newton¡¯s method for finding the local solution of a matrix optimization problem that corresponds to the
%U http://www.hindawi.com/journals/jam/2014/683797/