Robust D-Stability Analysis of Fractional-Order Controllers

This paper focuses on analyzing the robust $\mathcal{D}$-stability of fractional-order systems having uncertain coefficients using fractional-order controllers. Robust $\mathcal{D}$-stability means that each polynomial in a family of an uncertain fractional-order system has all its roots in a prescribed region of the complex plane. By employing the concept of the value set, two distinct methodologies are introduced for scrutinizing the robust $\mathcal{D}$-stability of the system. Although the outcomes of both approaches are equivalent, their computational appeal may differ. The first approach entails a graphical technique for the analysis of robust $\mathcal{D}$-stability, while the second approach furnishes a robust $\mathcal{D}$-stability testing function based on the shape properties of the value set, thereby establishing necessary and sufficient conditions for verifying the robust $\mathcal{D}$-stability of fractional-order systems using fractional-order controllers. Finally, a numerical example is provided to validate the results presented in this paper.