Method for designing PIλ Dμ stabilisers for minimum-phase fractional-order systems

This paper deals with the problem of designing the PI ^λ D ^μ-type controllers for minimum-phase fractional systems of rational order. In such systems, the powers of the Laplace variable, s, are limited to rational numbers. Unlike many existing methods that use numerical optimisation algorithms, the proposed method is based on an analytic approach and avoids complicated numerical calculations. The method presented in this paper is based on the asymptotic behaviour of fractional algebraic equations and applies a delicate property of the root loci of the systems under consideration. In many cases, the resulted controller is conveniently in the form of P, I ^λ, PD ^μ or PI ^λ D ^μ. Four design examples are explained and the results are compared with existing fractional-order PIDs. These results confirm the usefulness of the proposed method.