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From a closed-loop setpoint experiment using a PI controller with gain Kc0, one can easily, while the experiment is running, read off the following 3 parameters: tp, yp (overshoot), yu (undershoot). In addition, one must know the setpoint (ys) and the initial steady-state (y0). One may use yinf (final steady-state) instead of yu, but with yu one can stop the experiment when the first undershoot is reached. The main limitation with the closed-loop experiment is that the system must be at steady-state before the experiment starts.

Other data could also be used, but are more difficult to identify. For example, to find the rise time (tr), one first needs to find the steady-state value, yinf, so one needs to save the response and go back and find tr afterwards.

COMMENTS FROM S. SKOGESTAD, 20 Dec. 2010 / 17 May 2016: In this paper, the one-step procedure is recommended. However, it turns out that after correction, the two-step procedure, where one first identifies a process model and then finds the controller settings, gives almost identical results See equations (5.7)-(5.9) in:Sigurd Skogestad and Chriss Grimholt, ''The SIMC Method for Smooth PID Controller Tuning'', Chapter 5 in: R. Vilanova, A. Visioli (eds.), PID Control in the Third Millennium, Advances in Industrial Control, DOI 10.1007/978-1-4471-2425-2_5, © Springer-Verlag London Limited 2012

The advantage with the two-step procedure is that one can check if the values for k, theta and tau1 obtained from the closed-loop experiment are reasonable. For example, if one knows the delay theta from a separate expreiment, then one may adjust tau1 such that the sum theta+tau1 (as given by (5.7()+(5.9) remains constant), -------------------------------------------------------------------------------------------------- This contains details about the method which as mentioned was publised in Chapter 5 in 2012 (see above). First a Correction to the JPC 2010-paper: Equations (36) and (37) are not consistent in terms of the time delay estimate, and should be replaced by the following: First, from (19) compute tau/theta = 2A |b/(|-b)| Based on this value, estimate the time delay If tau/theta<8 use: theta = 0.43 tp (36a') If tau/theta>8 use: theta = 0.305 tp (36b') which for better accuracy and smooth transition can be replaced by: theta = tp*[0.309 + 0.209*exp(-0.61*tau/theta)] (36c') The estimate of the time constant is then obtained from (19) tau = theta*2A |b/(1-b)| (37') end correction ------------------------------------------------------------------------------------------------- THEN TO DETAILS ABOUT THE RECOMMENDED TWO-STEP PROCEDURE (The text below was written in 2010, and as mentioned you can alternatively read the book chapter from 2012) Shamsuzzoha and Skogestad (2010) presented a new closed-loop method for PI controller tuning based om the SIMC method. They claimed that a two-step approach, where one first obtains a process model, was inferior, but if one makes some minor change in the time delay estimate, then it turns out that this claim is incorrect. Rather than a new tuning method, it can then be regarded as a simple closed-loop method for identifying a first-order with delay (FOD) process model: STEP 1: Obtain FOD model. From a closed-loop setpoint experiment using a P-controller with gain Kc0, obtain the following data (see Figure 1 in Shamsuzzoha and Skogestad (2010)): • Overshoot = (yp - yinf)/(yinf - y0) • Time to reach overshoot (first peak), tp • Relative steady state output change, b =(yinf-y0)/(ys-y0). The overshoot should be between 0.1 and 0.6, with 0.3 as a good value. If one does not want to wait for the system to reach steady state, one can stop th experiment at the first undershoot and use the estimate (yinf-y0) = 0.45((yp-y0) + (yu-y0)) where yu is the output at the first undershoot. Based on these data compute A = [1.152(overshoot)^2 - 1.607(overshoot) + 1.0] r = 2A*|b/(1-b)| The estimated values of the process gain, time delay and time constant are then (Grimholt, 2010) k = (1/Kc0) * |b/(1-b)| theta = tp*[0.309 + 0.209*exp(-0.61*r)] tau = theta*r The value for theta is somewhat changed compared to Shamsuzzoha and Skogestad (2010). It is based fitting the data from Fig. 8 in Shamsuzzoha and Skogestad (2010) for an overshoot of 0.3. The value for tau is equal to the value for the integral time (taui) given in Shamsuzzoha and Skogestad (2010). STEP 2: Controller tuning. PI or PID settings can be obtained by any tuning method, although it should be noted that above estimates have been based on the SIMC PI tunings rules (Skogestad, 2003), Kc = (1/k)* tau/ 4(tauc+theta) taui = min [tau, 4(tauc+theta)] with the closed-loop time constant (tuning parameter) selected as tauc=theta. With SIMC tunings, the two-step approach gives identical controller gain (Kc) and very similar integral time (taui) as the one-step approach in Shamsuzzoha and Skogestad (2010). Details can be found in section 8 of Grimholt (2010). One advantage of the two-step approach is that one get more insight into the process model and can also compare with results from open-loop experiments. However, one should be careful about combining parameters from different experiments, because the main objective is to use the model for control purposes. In particular, from the SIMC PI-tuning rules, we see that we with tauc=theta can get similar controller settings even if the individual model parameters k, tau and theta are quite different. This is also confirmed by Grimholt (2010) who considered example processes E1-E13 from Skogestad (2003). References C. Grimholt, "Verification and improvements of the SIMC nethod for PI control", Project Report, Department of Chemical Engineering, NTNU (2010) (available at the home page of S. Skogestadl). M. Shamsuzzoha and S. Skogestad, "The setpoint overshoot method: A simple and fast closed-loop approach for PID tuning". Journal of Process Control, 20 (2010), 1220-1234. S. Skogestad, "Simple analytic rules formodel reduction and PID controller tuning", J. Process Control 13 (2003), 291-309.