Analytical method on stabilisation of fractional-order plants with interval uncertainties using fractional-order PIλ Dμ controllers

This paper proposes an analytical criterion to determine the stabilisation of a fractional-order plant with interval uncertainties in the coefficients using the fractional-order PI $ ^\lambda $ λD $ ^\mu $ μ controller. First, the nominal function and the disturbance function are defined according to the characteristic function of the closed loop system. The vertex functions of the value set with respect to the disturbance function are given by using Minkowski sum. The test frequency interval is divided into several intervals, and the vertex functions are unchanged in these intervals determined by the switching frequencies. Therefore, the calculation method of these frequencies is given. Second, the calculation method on the finite test frequency interval is proposed to simplify the number of frequency intervals. Third, the analytical condition that the origin is located on the edge of the value set is investigated. Based on this condition, the sufficient condition and necessary condition for the stabilisation of fractional-order plant using the PI $ ^\lambda $ λD $ ^\mu $ μ controller are offered. Moreover, the analytical methods on a fractional-order plant with interval uncertainties using the fractional-order PI $ ^\lambda $ λ controller and the fractional-order PD $ ^\mu $ μ controller are also discussed. Finally, three examples are given to validate the effectiveness of the proposed methods.