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.Keywords Robust control · Real structured singular value · Nonlinear programming · Modified subgradient algorithm R. Kasimbeyli published under the name Rafail N. Gasimov until 2007.

A detailed comparative analysis of μ lower bound algorithms

“…Any kind of uncertainties can be considered, but this approach is especially relevant for purely real problems, since the number of decision variables significantly increases when complex uncertainties are considered. A formulation similar to (3) is considered in [22] in the case of (possibly repeated) real uncertainties:…”

“…Moreover, [34] shows that the power algorithm is usually more efficient to solve (2) in terms of accuracy and computational time. Finally, the approach of [22] is very similar to the one of [19,20,21] which is already included in the comparison. Note also that some techniques could be used in addition to any of the lower bound algorithms mentioned in Section III-A: the µ-sensitivities introduced in [35] allow to reduce the number of uncertainties and thus the computational time at the price of a slight loss of accuracy, while branch-and-bound schemes [36] allow to tighten the gap between µ upper and lower bounds at the price of an increase in the computational time.…”

Computation of Bifurcation Margins Based on Robust Control Concepts

“…The key idea behind Program 3.2 is to enforce singularity of the LFT (3.5b) by using directly the determinant condition represented by constraint (3.7b). In [40] this is listed among the known methods for the computation of µ LB , and examples of related algorithms can be found in [22,47]. The approaches presented in those works, however, are limited to the case of linear systems, i.e., they represent alternatives to well-established µ lower bounds algorithms such as the power iteration [36] and the gain-based method [44].…”

An extension of the structured singular value to nonlinear systems with application to robust flutter analysis

“…[34] this is listed among the known methods for the computation of LB , and examples of related algorithms can be found in Refs. [17,37]. The approaches presented in these references, however, are limited to the case of linear systems, i.e., they represent alternatives to well-established lower bounds algorithms such as the power iteration [29] and the gain-based method [35].…”