ISA Transactions 49 (2010) 196–206 Contents lists available at ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans Fractional order phase shaper design with Bode’s integral for iso-damped control system Suman Saha a , Saptarshi Das a , Ratna Ghosh b , Bhaswati Goswami b , R. Balasubramanian c , A.K. Chandra c , Shantanu Das d , Amitava Gupta a,* a Power Engineering Department, Jadavpur University, Kolkata-700098, India b Instrumentation & Electronics Engineering Department, Jadavpur University, Kolkata-700098, India c R & D, Electronic System, Nuclear Power Corporation of India Ltd., Mumbai-400085, India d Reactor Control Division, Bhabha Atomic Research Centre, Mumbai-400085, India article info Article history: Received 23 October 2009 Received in revised form 3 December 2009 Accepted 7 December 2009 Available online 1 January 2010 Keywords: Bode’s integral Fractional order calculus Iso-damping Phase shaper abstract The phase curve of an open loop system is flat in nature if the derivative of its phase with respect to frequency is zero. With a flat-phase curve, the corresponding closed loop system exhibits an iso-damped property i.e. maintains constant overshoot with the change of gain. This implies enhanced parametric robustness e.g. to variation in system gain. In the recent past, fractional order (FO) phase shapers have been proposed by contemporary researchers to achieve enhanced parametric robustness. In this paper, a simple methodology is proposed to design an appropriate FO phase shaper to achieve phase flattening in a control loop, comprising a plant controlled by a classical Proportional Integral Derivative (PID) controller. The methodology is demonstrated with MATLAB simulation of representative plants and accompanying PID controllers. © 2009 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction PID controllers account for over 90% of controllers used in the process industry. While PID controllers produce closed loop systems with satisfactory tracking, disturbance rejection and ro- bustness of varying degrees depending upon the tuning method- ology [1,2], enhanced closed loop performance in terms of noise attenuation, for example, can only be achieved by specific loop shaping techniques e.g. [3]. As a further enhancement, a methodol- ogy to achieve increased parametric robustness through flattening of phase curve with a PID controller has been proposed by Chen et al. in [4] and extended in [5] with the use of a FO phase shaper. [4,5] propose an analytical method to obtain the constants of a PID controller to make the phase derivative zero around the tan- gent frequency (the frequency at which the Nyquist curve touches the sensitivity circle tangentially) and a simple FO element of the form s q to adjust the width of the flat-phase region. The method has been extended by Monje et al. in [6] to tune a Fractional Order PID (FOPID) controller or a PI λ D μ controller to achieve a flat-phase system with enhanced parametric robustness and iso-damped step response under varying gains. * Corresponding author. Tel.: +91 3323355813; fax: +91 3323357254. E-mail address: amitg@pe.jusl.ac.in (A. Gupta). A major issue in the use of FOPID controllers is the physical re- alization of such controllers. Vinagre et al. deal with issues related to the use of FOPID controllers for industrial applications in [7]. As reported in [7], a FO element can be realized by a lossy ca- pacitor based micro-electronic approach [8,9]; by analog circuit realization [10,11] or by integer approximation using, for exam- ple, Carlson representation [12,13] or the Crone Toolbox [14–16]. However, integer approximations are valid for specific frequency ranges. For the present work, Carlson representation is adopted which, in the simplest case, approximates a FO differ–integrator (differentiator or integrator) as a rational transfer function with a first order numerator and a first order denominator. The approach is independent of the method of realization of FO elements. The methodology presented in the paper proposes the use of a FO dif- fer–integrator for phase shaping and automatically establishes the integer approximated FO phase shaper as a rational transfer func- tion, which is effective for a frequency spread around a specified frequency viz. the gain crossover frequency. The phase shaper de- signed by this methodology produces the widest flat-phase region around the gain crossover frequency of the system comprising the plant and its controller, with the phase margin fixed above a spec- ified value. Thus, the resultant closed loop system exhibits iso- damped step response, with constant overshoot for a variation of system gain within a range. The plant may be assumed to be a First Order Plus Time Delay (FOPTD) or a Second Order Plus Time 0019-0578/$ – see front matter © 2009 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2009.12.001