On the Gap Between the Complex Structured Singular Value and Its Convex Upper Bound

Abstract

Abstract. The gap between the complex structured singular value of a complex matrix M and its convex upper bound is considered. New necessary and sufficient conditions for the existence of the gap are derived. It is shown that determining whether there exists such a gap is as difficult as evaluating a structured singular value of a reduced rank matrix (whose rank is equal to the multiplicity of the largest singular value of M ). Furthermore, if an upper bound on this reduced rank problem can be obtained, it is shown that this provides an upper bound on the original problem that is lower than the convex relaxation upper bound. An example that illustrates our procedure is given. We also give the solution of several structured-approximation problems of independent interest.

Nearest common root of polynomials, approximate greatest common divisor and the structured singular value

“…The approach is not limited to"small" perturbations in the polynomials' coefficients and general vector norms can be used in principle to represent uncertainty (in the paper quadratic and l ∞ norms are considered). The resulting optimization is non-convex and computationally demanding for large problems, although tight convex bounds (and techniques for improving them) have been reported in the literature [16], [17], [14], [20], [22], [29], [33], [34], [35]. Algorithms combining the technique presented in this work with aspects of [24] and [25] also seem possible and will be reported in a future publication.…”

“…Let also: It is shown in section 3 that Problem 2.1 is equivalent to the calculation of the structured singular value (µ) of an appropriate matrix. This is a problem arising in robust control which has been analyzed extensively over recent years [14], [17], [29], [33], [34]. A generalization of Problem 2.1 involving the computation of the numerical GCD of two polynomials is also introduced at a later section and is shown to correspond to the calculation of a generalized structured singular value subject to additional rank constraints (see section 4).…”

“…Although the problem is computationally NP hard [5], [16], tight bounds have been reported in the robust control literature using exclusively convex programming techniques. In a recent work [17] tests have been developed for certifying the absence of a duality gap between the structured singular value and its convex upper bound, along with a systematic procedure for reducing the duality gap when this is non-zero.…”

An explicit state-space solution to the one-block super-optimal distance problem

“…Early references report applications in the areas of disturbance rejection [Kwa86], robust stabilization [KN89], [Nym95] and hierarchical H ∞ design [HJ98a], [HJW97]. Applications of super-optimization in the areas of robust stabilization and structured-singular value approximations can be found in [GHJ00] and [JHMG06].…”

“…Reference [GHJ00] applies super-optimization techniques in the area of maximal robust-stabilization of LTI systems under additive perturbations: Explicit expressions for the improved robust stability radius are derived by imposing structure on the perturbation set via a uniform frequency constraint in the mostcritical direction which is identified. The method is also used in [GHJ00], [JHMG06] to derive an upper bound on the structured singular value for multivariable systems in the case of complex structured block-diagonal perturbations, which is tighter than the convex upper bound provided by the "D-iteration". In this context, the multiplicity of the largest Hankel singular value becomes a crucial consideration, which motivates the detailed analysis of the general problem presented in this paper.…”