Value-Set-Based Approach to Robust Stability Analysis for Ellipsoidal Families of Fractional-Order Polynomials with Complicated Uncertainty Structure

Abstract

This paper presents the application of a value-set-based graphical approach to robust stability analysis for the ellipsoidal families of fractional-order polynomials with a complex structure of parametric uncertainty. More specifically, the article focuses on the families of fractional-order linear time-invariant polynomials with affine linear, multilinear, polynomic, and general uncertainty structure, combined with the uncertainty bounding set in the shape of an ellipsoid. The robust stability of these families is investigated using the zero exclusion condition, supported by the numerical computation and visualization of the value sets. Four illustrative examples are elaborated, including the comparison with the families of fractional-order polynomials having the standard box-shaped uncertainty bounding set, in order to demonstrate the applicability of this method.

Robust Stability of Fractional-Order LTI Systems with Independent Structure of Ellipsoidal Parametric Uncertainty

“…2-4). However, this holds true only for simple uncertainty structures (such as independent one, which is studied in this contribution, and affine linear one), not for more complicated ones [5], [15], [16], [17]. The value sets for the range of frequencies from 0 to 2 with step 0.1 are shown in Fig.…”

“…They would also enter the fourth quadrant for even higher values of ω. According to the zero exclusion condition [5], [17], the family of FO polynomials (1), ( 2) is robustly stable since the zero (complex plane origin) is excluded from the value sets and all necessary preconditions (invariant degree of the polynomial family, pathwise connected Q, continuous coefficient functions, and at least one stable member of the family) are fulfilled.…”

“…Unfortunately, these techniques are not directly applicable to fractional-order systems [12]. In this contribution, the robust stability is tested via the numerical calculation and subsequent plot of the value sets, and the application of the zero exclusion principle [5], [15], [16], [17].…”

Robustness Analysis for Multi-Agent Consensus Systems with Application to DC Motor Synchronization

“…Recently, the theory and applications of robust control in the presence of uncertainty and time delays have been addressed in the literature using a variety of control approaches [18][19][20][21], for example. The problem of the robustness of a multi-agent consensus control in the presence of uncertainty and time delays has been studied using various approaches [22][23][24][25].…”

Investigation of Robust Stability for Fractional-Order LTI Systems with Multilinear Structure of Ellipsoidal Parametric Uncertainty

“…The commonly used engineering approach is to consider these bounds as real intervals; in other words, the bound has a shape of a box. However, there are some other alternative shapes available, such as a diamond, a sphere, or, more generally, an ellipsoid [2][3][4][5][6][7][8][9][10].…”