**1. Introduction**

The structured singular value (SSV) was first introduced by J. C. Doyel [1] and Safonov [2] as a mathematical tool, which is widely used to investigate the robustness, performance and stability of linear feedback systems in control. In control system analysis, the problem associated with the determination of stability and robustness in the presence of uncertainties is among the most fundamental issue in control and it has attracted a reasonable amount of researchers in the last three decades. Much of the research work has been done in robustness analysis for two different class of problems from system theory which involves the uncertainties. For this purpose, two different kinds of approaches has been developed.

One of the approach is based upon the frequency-based robust stability conditions in the form of the small gain condition. The small gain condition is most useful for analyzing those problems from system theory which are associated with the norm bounded unstructured or complex structured uncertainties. An example of such an approach is based upon the structured singular values introduced in [1,3].

An another approach is largely inspired by Kharitonov work [4]. The main aim of the work by Kharitonov is to determine the stability robustness with a finite number of

**Citation:** Rehman, M.-U.; Alzabut, J.; Ateeq, T.; Kongson, J.; Sudsutad, W. The Dual Characterization of Structured and Skewed Structured Singular Values. *Mathematics* **2022**, *10*, 2050. https://doi.org/10.3390/math 10122050

Academic Editors: Adolfo Ballester-Bolinches and Carlo Bianca

Received: 15 April 2022 Accepted: 6 June 2022 Published: 13 June 2022

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conditions. This approach also aims to study the problems in control when real parametric uncertainties consisting of real-valued uncertain parameters are involved, for a more details see, e.g., [5–7].

In principle, both types of methodologies can be modified to deal with real and complex parametric uncertainties. Indeed, the early developments in structured singular values deals with the uncertainties which are only pure complex. However, the extensions have been made so that the real-valued uncertainties can also be considered [8,9]. The Kharitonov type approaches deal with the *μ*-value problems while considering the realvalued uncertainties [10–12].

The exact computation of SSV is impenetrable which makes it NP-hard [13]. The NP-hard nature of SSV broach to foster methods for approximation of its lower and upper bounds. However, the SSV lower bounds are computed using the generalization of power method [14,15]. Moreover, the balanced AMI technique developed by [16] is the utilization of bounds introduced by [17] and is particularized in a preeminent style in [16]. The given matrix, under consideration, is first balanced while imploging a variation of Osborne's [18] generalized to crank the repeated real/complex scalars and the number of full blocks. Further, Perron approach is a determination for balancing the given matrix. The Perron eigenvector methodology is established on the idea apt by Safonov [2].

The D-Scaling upper bound presented in [1] is the most extensively used paper for the approximation of the upper bound of structured singular values. The D-Scaling for complex structures acquiring full complex blocks is close to the original SSV. For more details, we suggest the reader to consult [19] and the reference therein. Meanwhile, for non-trivial complex structures, the D-Scaling upper bound turns out to be more flexible [1,19].

The SSV theory for mixed real/complex cases is an extension of SSV that acquiesce the structure to consist of real and complex parts. The computation of upper bounds for the mixed SSV presented by [17] is also known as (*D*, *q*)-Scaling upper bound of skewed structured singular values *ν*, and is quite apart from actual mixed SSV [20].

The investigation for the non-fragile asynchronous *H*∞ control while considering the stochastic memory systems with Bernoulli distribution has been recently studied by [21]. An efficient algorithm for the computation of budget allocation procedure for the selection of top candidate solution for objective performance measure has been extensively studied by [22].

In this article, we give an analytical treatment for the dual characterization of structured singular values and skewed structured singular values. We present some new results for the computation of an upper bounds of these quantities.

**The rest of the paper is organized as:** In Section 2, we provide the preliminaries of our article. In particular, we give the Definitions of the block diagonal structure, structured singular values and skewed structured singular values for a set of block diagonal matrices and subset of positive definite matrices. In Section 3, we present new results on the computation of the dual characterization of structured singular values and skewed structured singular values. The computation of upper bounds of skewed structured singular values are also presented in Section 3 of our article. Finally, Section 4 is about the conclusion of our presented work.
