IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 35, NO. 1, FEBRUARY 2005 23 Relay Feedback Tuning of Robust PID Controllers With Iso-Damping Property YangQuan Chen, Senior Member, IEEE, and Kevin L. Moore, Senior Member, IEEE Abstract—A new tuning method for proportional-inte- gral-derivative (PID) controller design is proposed for a class of unknown, stable, and minimum phase plants. We are able to design a PID controller to ensure that the phase Bode plot is ﬂat, i.e., the phase derivative w.r.t. the frequency is zero, at a given frequency called the “tangent frequency” so that the closed-loop system is robust to gain variations and the step responses exhibit an iso-damping property. At the “tangent frequency,” the Nyquist curve tangentially touches the sensitivity circle. Several relay feedback tests are used to identify the plant gain and phase at the tangent frequency in an iterative way. The identiﬁed plant gain and phase at the desired tangent frequency are used to estimate the derivatives of amplitude and phase of the plant with respect to frequency at the same frequency point by Bode’s integral relation- ship. Then, these derivatives are used to design a PID controller for slope adjustment of the Nyquist plot to achieve the robustness of the system to gain variations. No plant model is assumed during the PID controller design. Only several relay tests are needed. Simulation examples illustrate the effectiveness and the simplicity of the proposed method for robust PID controller design with an iso-damping property. Index Terms—Bode’s integral, ﬂat phase condition, iso-damping property, proportional-integral-derivative (PID) controller, PID tuning, relay feedback test. I. INTRODUCTION A CCORDING to a survey [1] of the state of process con- trol systems in 1989 conducted by the Japan Electric Mea- suring Instrument Manufacturer’s Association, more than 90 of the control loops were of the proportional-integral-deriva- tive (PID) type. It was also indicated [2] that a typical paper mill in Canada has more than 2,000 control loops and that 97% use PI control. Therefore, the industrialist had concentrated on PI/PID controllers and had already developed one-button type relay auto-tuning techniques for fast, reliable PI/PID control yet with satisfactory performance [3]–[7]. Although many dif- ferent methods have been proposed for tuning PID controllers, the Ziegler–Nichols method [8] is still extensively used for de- termining the parameters of PID controllers. The design is based Manuscript received February 27, 2004; revised on May 30, 2004. This paper was recommended by Associate Editor S. Phoha. Y. Chen is with the Center for Self-Organizing and Intelligent Systems (CSOIS), Department of Electrical and Computer Engineering, College of Engineering, Utah State University, Logan, UT 84322-4160 USA (e-mail: yqchen@ece.usu.edu). K. L. Moore was with the Center for Self-Organizing and Intelligent Sys- tems (CSOIS), Department of Electrical and Computer Engineering, College of Engineering, Utah State University, Logan, UT 84322-4160 USA. He is now with the Research and Technology Development Center, The Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20723-6099 USA (e-mail: kevin.moore@jhuapl.edu). Digital Object Identiﬁer 10.1109/TSMCB.2004.837950 on the measurement of the critical gain and critical frequency of the plant and using simple formulae to compute the controller parameters. In 1984, Åström and Hägglund [9] proposed an au- tomatic tuning method based on a simple relay feedback test which uses the describing function analysis to give the critical gain and the critical frequency of the system. This information can be used to compute a PID controller with desired gain and phase margins. In relay feedback tests, it is a common practice to use a relay with hysteresis [9] for noise immunity. Another commonly used technique is to introduce an artiﬁcial time delay within the relay closed-loop system, e.g., [10], to change the os- cillation frequency in relay feedback tests. After identifying a point on the Nyquist curve of the plant, the so-called modiﬁed Ziegler–Nichols method [4], [11] can be used to move this point to another position in the complex plane. Two equations for phase and amplitude assignment can be obtained to retrieve the parameters of a PI controller. For a PID controller, however, an additional equation should be intro- duced. In the modiﬁed Ziegler–Nichols method, , the ratio be- tween the integral time and the derivative time , is chosen to be constant, i.e., , in order to obtain a unique solu- tion. The control performance is heavily inﬂuenced by the choice of as observed in [10]. Recently, the role of has drawn much attention, e.g., [12]–[14]. For the Ziegler–Nichols PID tuning method, is generally assigned as a magic number four [4]. Wallén, Åström, and Hägglund proposed that the tradeoff be- tween the practical implementation and the system performance is the major reason for choosing the ratio between and as four [12]. The main contribution of this paper is the use of a new tuning rule which gives a new relationship between and instead of the equation proposed in the modiﬁed Ziegler–Nichols method [4], [11]. We propose to add an extra condition that the phase Bode plot at a speciﬁed frequency at the point where sensitivity circle touches Nyquist curve is locally ﬂat which implies that the system will be more robust to gain variations. This additional condition can be expressed as , which can be equivalently expressed as (1) where is the frequency at the point of tangency and is the transfer function of the open loop system including the controller and the plant . The above equivalence in (1) is mathematically explained in detail 1083-4419/$20.00 © 2005 IEEE