Asian Journal of Control, Vol. 17, No. 3, pp. 1–11, May 2015 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.931 A NEW TUNING METHOD FOR STABILIZATION TIME DELAY SYSTEMS USING PI  D  CONTROLLERS Sami Hafsi, Kaouther Laabidi, and Rihem Farkh ABSTRACT This paper presents a new design procedure to tune the fractional order PI  D  controller that stabilizes a frst order plant with time delay. The procedure is based on a suitable version of the Hermite–Biehler Theorem and the Pontryagin Theorem. A Theorem and a Lemma are developed to compute the global stability region of the PI  D  controller in the (k p , k i , k d ) space. Hence, this Theorem and Lemma allow us to develop an algorithm for solving the PI  D  stabilization problem of the closed loop plant. The proposed approach has been verifed by numerical simulation that confrms the effectiveness of the procedure. Key Words: Fractional PID controller, delay system, stability, tuning method. I. INTRODUCTION Fractional order dynamic systems and controls have gained increasing attention in research communities during the last three decades, mainly due to their demon- strated applications in numerous, seemingly diverse and widespread, felds of science and engineering including control theory, mechanics, physics, chemistry and signals processing [1–7,9–12,14]. The mathematical modeling and simulation of systems and processes, based on the description of their physical laws, leads to differential equations of a fractional order. In fact, it is natural that many authors have tried to solve the fractional differential equations and propose different defnitions that involve defnite fractional integrals and fractional derivatives. The earliest systematic works on fractional calculus were made by researchers such as Leibniz, Liouville and Riemann [6]. In [14] Oustaloup demon- strated the superior performance of the CRONE (Commande Robuste d’Ordre Non Entier) method over the proportional–integral–derivative (PID) controller for the control of dynamic systems. The fractional order proportional integrative derivative is a matter of almost exclusive interest not only because of the number of tunable parameters in a controller, but also its design fexibility and perfor- mance benefts from its inherent structure [3]. Podlubny proposed a generalization of the classic PI and PID Manuscript received March 17, 2013; revised September 20, 2013; accepted December 21, 2013. Sami Hafsi (corresponding author, e-mail: samihafsi@ymail.com), Kaouther Laabidi, and Rihem Farkh are with the Université de Tunis El Manar, Ecole Nationale d’Ingénieurs de Tunis, LR-11-ES20 Laboratoire d’Analyse, Concep- tion et Commande des Systèmes, 1002, Tunis, Tunisie. controllers defned as PI  and PI  D  where the order integrator  and the order differentiator  assume real non-integer values. Also he proved that this type of fractional controller can achieve better control of dynamic systems [12]. Using PI  D  allows one to choose, in addition to the parameters of the classical PID (k p , k i and k d ), the orders of integration  and derivation . It is well known that the classical PID allows a phase shift only of ±  2 , whereas the PI  D  controller may provide more intermediate information, since the phase shifts are between -  2  and  2  [3]. Numerous methods and approaches using some extensions of classical control theory have been proposed to defne new effective tuning techniques for fractional controllers. It has been shown in these methods that the fractional controllers provide a better response than the integer order controllers when used both for the control of integer-order systems and fractional-order sys- tems [1–14]. In [11] Maione and Lino introduce a new design approach for PI  controllers inspired by the clas- sical symmetrical optimum method. More recently in [3], the authors present some effective synthesis methods for fractional-order PI  , PD  and PI  D  to fulfl different design specifcations for some closed-loop systems. In [6] Hamamci use the real root boundary (RRB), the infnite root boundary (IRB) and the complex root boundary (CRB) to determine the set of global stability regions of PI  and PI  D  for different values of the fractional orders  and , but this technique, which is based on analytical expressions derived from the stability domain boundaries, is biased in the case where  +  is equal to 2. In general, real systems often involve time delay, so the dynamic behavior of many industrial plants can © 2014 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd