Computers and Chemical Engineering 30 (2006) 1416–1423 Comparison of some well-known PID tuning formulas Wen Tan a,∗ , Jizhen Liu a , Tongwen Chen b , Horacio J. Marquez b a Department of Automation, North China Electric Power University, Zhuxinzhuang, Dewai, Beijing 102206, PR China b Department of Electrical & Computer Engineering, University of Alberta, Edmonton, AB, Canada T6G 2V4 Received 10 June 2005; received in revised form 15 January 2006; accepted 3 April 2006 Available online 22 May 2006 Abstract Criteria based on disturbance rejection and system robustness are proposed to assess the performance of PID controllers. A simple robustness measure is defined and the integral gains of the PID controllers are shown to be a good measure for disturbance rejection. An analysis of some well-known PID tuning formulas reveals that the robustness measure should lie between 3 and 5 to have a good compromise between performance and robustness. © 2006 Published by Elsevier Ltd. Keywords: PID tuning; Robustness; Structured singular value; Performance 1. Introduction PID controllers are widely used in industry due to their simplicity and ease of re-tuning on-line (Astrom & Hagglund, 1995). In the past four decades, there are numerous papers dealing with the tuning of PID controllers. A natural question arises: how can the PID settings obtained by different meth- ods be compared? A simple answer is to use step responses of the closed-loop systems and compare the overshoot, rise time and settling time. An alternative is to use the integral error as a performance index. However, these time domain performance measures do not address directly another important factor of a closed-loop system—robustness. It is a well-known fact that models used for controller tuning or design are often inaccurate, so a PID setting based on optimization assuming an accurate model will generally not be guaranteed to be robust. For a fair comparison of different PID settings, both time domain perfor- mance and frequency domain robustness should be considered (Shinskey, 1990). For a single-input-single-output (SISO) process, gain and phase margins are good measures of system robustness. Ho, Gan, Tay, and Ang (1996) and Ho, Hang, and Zhou (1995) compared gain-phase margins of some well-known PID tuning formulas. ∗ Corresponding author. Tel.: +86 10 51963934; fax: +86 10 80798468. E-mail addresses: wtan@ieee.org (W. Tan), tchen@ece.ualberta.ca (T. Chen), marquez@ece.ualberta.ca (H.J. Marquez). They show that the load-based tuning methods give gain margins of about 1.5 and phase margins range from 30 ◦ to 60 ◦ , while the setpoint-based tuning methods give gain margins of about 2 and phase margins of about 65 ◦ . In this paper, we will propose a simple method to ana- lyze system robustness and performance, and then compare some well-known PID tuning formulas and observe some inter- esting results. It should be noted that the purpose of this paper is to find a simple robustness measure so that we can compare different PID settings. Though we adopt the well- known methods in checking system robustness and time domain performance, our main contribution can be summarized as follows: (1) We propose to use a simple method to measure the robust- ness of a system. The measure is more suitable for the purpose of comparing PID controllers than other measures, since it is applicable to multivariable systems and it bounds the sensitivity and the complementary sensitivity functions simultaneously. (2) By using the measure, we examine the well-known PID tuning formulas and draw some interesting conclusions. All the symbols used in this paper are common in the area of robust control. The definition of the H ∞ norm and the structured singular value can be referred to Zhou and Doyle (1998), and they can be easily computed with the aid of MATLAB  toolbox 0098-1354/$ – see front matter © 2006 Published by Elsevier Ltd. doi:10.1016/j.compchemeng.2006.04.001