A nonlinear programming technique to compute a tight lower bound for the real structured singular value

Abstract

The real structured singular value (RSSV, or real μ) is a useful measure to analyze the robustness of linear systems subject to structured real parametric uncertainty, and surely a valuable design tool for the control systems engineers. We formulate the RSSV problem as a nonlinear programming problem and use a new computation technique, F-modified subgradient (F-MSG) algorithm, for its lower bound computation. The F-MSG algorithm can handle a large class of nonconvex optimization problems and requires no differentiability. The RSSV computation is a well known NP hard problem. There are several approaches that propose lower and upper bounds for the RSSV. However, with the existing approaches, the gap between the lower and upper bounds is large for many problems so that the benefit arising from usage of RSSV is reduced significantly. Although the F-MSG algorithm aims to solve the nonconvex programming problems exactly, its performance depends on the quality of the standard solvers used for solving subproblems arising at each iteration of the algorithm. In the case it does not find the optimal solution of the problem, due to its high performance, it practically produces a very tight lower bound. Considering that the RSSV problem can be discontinuous, it is found to provide a good fit to the problem. We also provide examples for demonstrating the validity of our approach.