An analytical method on the stabilization of fractional-order plants with one fractional-order term and interval uncertainties using fractional-order PI λ D μ controllers

Abstract

In this study, we propose an analytical method to determine the stabilization of fractional-order plants with one fractional-order term and interval uncertainties using fractional-order ${PI}^{λ}$ $D^{μ}$ controllers. An auxiliary function related to the characteristic function of the closed-loop system is defined, and the real vertices and edges of the value set with respect to the auxiliary function are given by the Minkowski sum method. It can effectively reduce the computation burden caused by redundant vertices. The concept of switching frequencies is proposed, and the calculation method of switching frequencies is provided. In order to test the position relationship between the value set and the origin within a finite frequency interval, the upper and lower limits of the frequency interval are calculated and the mathematical expressions of the vertices in each frequency interval determined by the switching frequencies and limits are given. Based on the analysis of the position relationship, we propose the necessary and sufficient condition of the stabilization criterion. Finally, the stabilization problems of the numerical examples are analysed to verify the effectiveness of the proposed stabilization criterion.

Keywords

Fractional-order plants stabilization interval uncertainties value set

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