Published in IET Control Theory and Applications Received on 22nd February 2008 Revised on 22nd November 2008 doi: 10.1049/iet-cta.2008.0062 ISSN 1751-8644 Method for designing PI l D m stabilisers for minimum-phase fractional-order systems F. Merrikh-Bayat 1 M. Karimi-Ghartemani 2 1 Department of Electrical Engineering, Zanjan University, Zanjan, Iran 2 Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran E-mail: f.bayat@znu.ac.ir Abstract: This paper deals with the problem of designing the PI l D m -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 l , PD m or PI l D m . Four design examples are explained and the results are compared with existing fractional-order PIDs. These results confirm the usefulness of the proposed method. 1 Introduction PIDs are one of the most popular controllers used in industry [1]. Recently, according to the advances in fractional-order modelling, there has been an increasing attention to the problem of controlling these systems. Among others, an extension of the classical PID controllers, commonly known as fractional-order PID (FOPID) or PI l D m [2–4], and the so-called CRONE [5, 6] are of more interest. The transfer function of a FOPID is given by C (s) ¼ k p þ k i s l þ k d s m where k p , k i , k d [ R, and l, m [ R þ , and the controller design problem is to find the unknown parameters k p , k i , k d , l and m such that some predetermined control objectives are met. Such a controller has five parameters to tune and provides more flexibility and strength to achieve the design specifications (two parameters more as compared with the conventional PID controller). The frequently used strategy for tuning PI l D m controllers is based on the minimisation of a suitably chosen cost function, commonly in time domain. For example, Podlubny [4] proposed the application of a PI l D m controller in series with a fractional- order plant, and then calculated the tracking error to the unit step as a function of unknown parameters. Then, using the gradient method for optimisation, the unknown parameters of the PI l D m controller are obtained in order to reach some predefined time-domain specifications. The main drawback of such an approach is the need for calculating the time- domain response, which may be very heavy even for a simple fractional-order system. That is why Podlubny [4] considered the fractional-order transfer function of the plant with no zeros. After all, that arrives at a controller that is guaranteed to have a satisfactory behaviour only for the step input. Monje et al. [7] proposed a method of tuning FOPIDs which is based on a numerical optimisation algorithm (the same as the one used in Matlab’s function FMINCON) for constrained minimisation in frequency domain. The solutions offered by this method depend on initial estimates of the unknown parameters and consequently may arrive at suboptimal solutions which even lead to instability. An extension of the well-known Ziegler – Nichols rules for FOPIDs applied to integer-order plants is studied in [8]. This is also based on a numerical minimisation algorithm and its application is limited to first-order plants possibly with time delay. Although these rules provide an effective approach of designing FOPIDs, but they usually lead to results poorer than those expected to achieve [8]. A similar approach is presented in [9] for fractional-order plants. IET Control Theory Appl., 2010, Vol. 4, Iss. 1, pp. 61–70 61 doi: 10.1049/iet-cta.2008.0062 & The Institution of Engineering and Technology 2010 www.ietdl.org Authorized licensed use limited to: Queens University. Downloaded on February 16,2010 at 09:56:55 EST from IEEE Xplore. Restrictions apply.