Robust stability of fractional order polynomials with complicated uncertainty structure

Abstract

The main aim of this article is to present a graphical approach to robust stability analysis for families of fractional order (quasi-)polynomials with complicated uncertainty structure. More specifically, the work emphasizes the multilinear, polynomial and general structures of uncertainty and, moreover, the retarded quasi-polynomials with parametric uncertainty are studied. Since the families with these complex uncertainty structures suffer from the lack of analytical tools, their robust stability is investigated by numerical calculation and depiction of the value sets and subsequent application of the zero exclusion condition.

Robust Stability of Fractional‐Order Linear Time‐Invariant Systems: Parametric versus Unstructured Uncertainty Models

“…There is a rich variety of tools for IO families available in the literature. For FO cases, the selection is more limited (e.g., the famous Kharitonov theorem is not valid for FO interval polynomials [46]), but several approaches have been already developed (see, e.g., [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]), and the number is still growing.…”

“…The detailed information on parametric uncertainty and robust stability analysis and also examples of the typical value sets for the IO systems can be 2 Complexity found in [24] and subsequently, for example, in [27,28]. The works [36,37,41,42,[46][47][48] have extended the idea of the value set concept also to FO uncertain polynomials (or quasi-polynomials [49,50]). The value set for the family of polynomials (2) at the frequency ω ∈ ℝ is defined as [24] p jω, Q = p jω, q : q ∈ Q , 3…”

“…In this paper, the visualization of the value set is based on suitable sampling (gridding) of the uncertain parameters and on direct calculation of related partial points of the value sets for a supposed frequency range. This sampling (brute-force) method is easily applicable for the computation of the value sets of a polynomial family even with complicated uncertainty structures [48] (or even for quasi-polynomial families [49,50]). Then, the robust stability can be analyzed via standard zero exclusion theorem with the necessary and sufficient condition.…”

“…Recently, a number of authors have already tried to solve the robustness of FO LTI systems, especially under parametric uncertainty [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. The particularized bibliographic research on advances in robust stability analysis of FO LTI systems with parametric uncertainty, starting from the pioneering contribution [51], can be found within the introduction of the paper [48]. However, there is still not many works focused on the robustness of FO LTI systems with unstructured uncertainty [52][53][54].…”

“…The principal advantage of the applied graphical robust stability analysis under parametric uncertainty is its universality, that is, the technique is applicable to various uncertainty structures which suffer from a lack of alternative methods. It works even for extremely complicated uncertainty structures as well as for time-delay systems, and above all, it works not only for IO LTI systems but also for FO LTI systems of noncommensurate orders [48]. Relatively weak restrictions include the invariant degree of polynomials in the family, pathwise-connected uncertainty bounding set, and continuous coefficient functions.…”