General Rules for Optimal Tuning the PI  D  Controllers With Application to First-Order Plus Time Delay Processes Farshad Merrikh-Bayat* Department of Electrical and Computer Engineering, University of Zanjan, Zanjan, Iran For two main reasons optimal tuning the PI  D  controllers is a challenging task: First, the search space is very large in dealing with such controllers, and second, there is not any generally applicable method for stability testing of the linear feedback systems containing both time delay and fractional-order controllers. Hence, easy-to-use and effective rules for optimal tuning such controllers are highly demanded. In this paper, explicit formulas for optimal tuning the parameters of the PI  D  controller, when it is applied in series with a first-order plus time-delay process in a standard output-feedback system, are proposed. Keywords: fractional-order PID, optimal tuning rule, command following, disturbance rejection INTRODUCTION P ID controllers have been successfully used in a wide vari- ety of industrial applications during the past six decades (Zhuang and Atherton, 1993; ˚ Astr ¨ om and H ¨ agglund, 2005). It is a well-known fact that one main reason for this great suc- cess is the availability of easy-to-use and effective rules, such as the Ziegler–Nichols-type formulas, for the tuning of these con- trollers. According to the high achievement of PID controllers, many researchers have tried to develop new generations of this type of controller to reach a better performance and robustness. One of these attempts led to the so-called fractional-order PID (FOPID) controller, which was proposed by Podlubny (1999b). Transfer function of the FOPID controller, which is also known as the PI  D  controller, is given by: C(s) = K p  1 + 1 T i s  + T d s   (1) where K p ,T i ,T d ∈R and ,  ∈R + are the tuning parameters, and the controller design problem is to calculate the value of these unknown parameters such that some predetermined con- trol objectives are met. Note that the fractional powers of the Laplace variable, s, in Equation (1) are commonly interpreted in the time domain using either the Gr ¨ unwald–Letnikov or the Riemann–Liouville or the Caputo fractional operator (Podlubny, 1999a). As it can be observed, the FOPID controller given in Equa- tion (1) has five parameters to select, that is two more than the conventional PID controllers, and it is the main reason for the superiority of FOPIDs to PIDs. The PI  D  controllers have received a considerable attention during the past decade both from the aca- demic and industrial point of view (see, for example, Oustaloup et al., 1995; Petras, 1999; Xue et al., 2006; Ma and Hori, 2007; Bhambhani and Chen, 2008; Chen et al., 2008; Hamamci, 2008; Monje et al., 2008) and the references therein for more informa- tion on this subject). Various methods have been developed by researchers for the tuning of PI  D  controllers during the past few years. A frequently used strategy for tuning these controllers is based on the min- imisation of a suitably chosen cost function (in the time or the frequency domain). For example, Podlubny (1999b) proposed the application of a PI  D  controller in series with a fractional-order plant, and then calculated the error to the unit step command as a function of unknown parameters. Then, using the gradient method for optimisation, the unknown parameters of the PI  D  controller are calculated such that some predefined time-domain ∗ Author to whom correspondence may be addressed. E-mail address: f.bayat@znu.ac.ir Can. J. Chem. Eng. 9999:1–11, 2012 © 2012 Canadian Society for Chemical Engineering DOI 10.1002/cjce.21643 Published online in Wiley Online Library (wileyonlinelibrary.com). | VOLUME 9999, 2012 | | THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING | 1 |