Research Article: Robust stability of fractional order polynomials with complicated uncertainty structure

Date Published: June 29, 2017

Publisher: Public Library of Science

Author(s): Radek Matušů, Bilal Şenol, Libor Pekař, Xiaosong Hu.

http://doi.org/10.1371/journal.pone.0180274

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.

Partial Text

Fractional order control represents promising and attractive research topic, which has been widely studied recently. In fact, the field of fractional order calculus itself is not new [1]–[4], but the true boom of scientific works has exploded in various application areas over the last few years [5], [6]. The applications of fractional order calculus can be found, among others, in physics [7], [8], bioengineering [9]–[12], viscoelastic materials [13], [14] and also [12], chaotic systems [15]–[18], electronic circuits and fractance devices [19], [20], ultracapacitors [21], robotics [22]–[25], signal processing [26], [27], and many other areas. Certainly, the field of automatic control is no exception to this trend, quite the opposite [28]–[31]. On the other hand, the robustness of control systems can already be seen as one of the classical (and fundamental) problems in control engineering theory [32]–[36] and practice [37]. Naturally, the combination of robust and fractional order control is nowadays really appealing research discipline both for linear [38]–[61] and nonlinear [62]–[64] systems.

General continuous integro-differential operator (differintegral) is defined as [5], [28], [30]:
Datα={dαdtαRe α>01Re α=0∫at(dτ)−αRe α<0(1) where α is the order of the differintegration (typically α ∈ ℝ) and a and t are the limits of the operation. The differintegral can be defined in various ways. The three most common are Riemann-Liouville, Grünwald-Letnikov and Caputo definitions. A level of complicatedness of the relations among coefficients of the polynomial Eq (4) (in other words the complexity of the coefficient functions ρi and their interconnections) is a crucial factor for the decision on a suitable tool for robust stability analysis both for integer and fractional order systems with parametric uncertainty. According to this, one can distinguish among several kinds of uncertainty structures. Standard classification for integer order systems is [33], [65], [66]: As mentioned above, the complicated structures of uncertainty suffer from the lack of suitable techniques for robust stability analysis. However, a graphical method based on the combination of the value set concept and the zero exclusion condition [33] represents a universal tool, which can be applied to a wide range of uncertainty structures, including the most complex ones. Besides this, it can be used also for various regions of stability (robust D-stability). More details on parametric uncertainty, related robust stability analysis and several examples of the typical value sets for the integer order systems can be found in [33] and subsequently e.g. in [65], [66]. The works [49]–[53] extended the concept of the value set to fractional order uncertain polynomials (or quasi-polynomials [58]). In order to show the practical applicability of the graphical approach to robust stability analysis discussed hereinbefore, four illustrative examples with families of fractional order (quasi-)polynomials are presented in this Section. The first three examples deal with multilinear, polynomial and general uncertainty structure, successively, and the last one focuses on a family of retarded quasi-polynomials. This article was focused on a graphical approach to robust stability investigation for families of fractional order polynomials or even quasi-polynomials with complicated uncertainty structure. The four illustrative examples demonstrated the application of the values set concept and the zero exclusion condition for the families of fractional order polynomials with multilinear uncertainty structure, polynomial uncertainty structure, general uncertainty structure, and for the family of the fractional order retarded quasi-polynomials. The obtained results showed the effectivity of the method for robust stability analysis of fractional order polynomials with various complex uncertainty structures. The potential directions for future research can be seen in robust stability analysis of e.g. fractional order anisochronic systems with internal delays and uncertain parameters [67], fractional order systems with spherical uncertainty [70] or fractional order systems with complicated uncertainty structures combined with the uncertain fractional orders.   Source: http://doi.org/10.1371/journal.pone.0180274

 

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