An Analytical Approach to the Stability Conditions of a New Class of Fractional-Order Control Systems by the Lambert-Tsallis Function

Abstract

This paper introduces a novel framework for the stability analysis of fractional-order control systems with closed-loop characteristic equations in the form of any trinomial given by $a_{n} s^{n} + a_{m} s^{m} + a_{0} = 0$ , $\forall \, n, m \in \mathbb {Q}^{*}$ with $a_{n}\cdot a_{m} \cdot a_{0} \neq 0$ , using an innovative approach that differs from those currently employed in the literature, such as Matignon’s stability theorem. Unlike Matignon’s method, which promotes a change of variable to provide only numerical solutions, the proposed approach utilizes the Lambert-Tsallis equation to provide analytical solutions to the characteristic equation. Furthermore, it is possible to define the system’s stability criterion analytically, enabling the designer to set controller parameters that satisfy the system’s stability conditions. To the best of our knowledge, no other analytical method for the generalized solution of fractional-order trinomials exists in the current literature. As an example, the stability conditions for a benchmark case from the literature are presented analytically and confirmed a posteriori by a numerical method. Finally, a set of control systems composed of both integer- and fractional-order controllers and plants, suitable for analysis by the proposed method, is presented.

Keywords

• Analytical solution
• control’s stability
• fractional-order control system
• fractional-order polynomial
• Lambert-Tsallis function
• trinomial equation