Computing stabilising regions: FOPID controller methodology for interval fractional-order plants with time delay

Recently, certain methods have inaccurately presented the computation of the robust stability region of Fractional-Order Proportional Integral Derivative (FOPID) controllers for interval fractional-order plants with an interval time delay. Motivated by the necessity for accuracy and reliability as well as the growing importance and widespread application of fractional-order controllers in control systems, which offer increased flexibility in controller design, this paper focuses on introducing a dependable procedure for determining the stabilising region of FOPID controllers applicable to interval fractional-order plants with an interval time delay. An interval fractional-order plant is characterised by a fractional-order transfer function wherein the coefficients are uncertain and confined within specified intervals in both the numerator and denominator. Moreover, an interval time delay pertains to a delay term whose value varies within a specified interval. To comprehensively analyze the robust stability of the resulting system, we introduce two theorems and two lemmas in this study. Following this, we propose a method to address the challenge of achieving robust stabilisation for interval plants with an interval time delay using FOPID controllers. Moreover, an additional function is introduced to improve the performance of the systems. To illustrate the proposed procedure, we provide numerical examples and an experimental verification.