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Figure 1.16 Closed-loop unit step response with PI controller (Example 1.6). Key: line (1) Ziegler–Nichols tuning rule; line (2) Cohen–Coon tuning rule; line (3) Wang–Cluett tuning rule.

Example 1.7

       A continuous time plant has the transfer function:

      (1.57)

       The unit step response of this transfer function model is shown in Figure 1.17. In this figure, a line is drawn to reflect the maximum slope of the reaction curve and the points of interest are marked by two arrows.

       Using MATLAB command ginput(2), with a click on the bottom point, we will find the coordinates ; and a click on the top point, we will find . Find the PI and PID controllers using the reaction curve based tuning rules.

      Solution. The steady state gain The time delay is and the parameter The PI controller parameters are calculated using the reaction curve based methods (see Tables 1.2, 1.3, and 1.6) and are summarized in Table 1.8. The closed-loop control systems with the PI controllers are simulated using the plant model (1.57). The unit closed-loop step responses are shown in Figure 1.18. From this figure, we can see that both PI controllers from Ziegler–Nichols and Cohen–Coon tuning rules failed to produce a stable closed-loop system. However, the PI controller using Wang–Cluett tuning rule gives a stable closed-loop system.

       The PI control systems will be subsequently used as examples for closed-loop stability analysis. The Nyquist plots of the PI controllers used in this example are analyzed in Example 2.3 of Chapter 2.

      The following example illustrates the applications of the Padula and Visioli tuning rules.

Figure 1.17 Unit step response (Example 1.7).

       Table 1.8 PI controller parameters with reaction curve.

Ziegler–Nichols		6.4		108
Cohen–Coon		6.5667		75.9231
Wang–Cluett		3.8867		127.2154

Figure 1.18 Closed-loop unit step response with PI controller (Example 1.7). Key: line (1) Ziegler–Nichols tuning rule; line (2) Cohen–Coon tuning rule; line (3) Wang–Cluett tuning rule.

Example 1.8

       Consider the same third order system with time delay used in Example 1.7. Find the PI and PID controller parameters using Padula and Visioli tuning rules and simulate their closed-loop step response.

      Solution. The parameters used in the tuning rules are , , and . To evaluate the PI controller performance, Table 1.4 is used to calculate the controller parameters. For , we have and . For , we have and .

       With sampling interval (s), the closed-loop step responses are compared in Figure 1.19, where the IP controller structure is used to reduce overshoot to the step reference signal. It is clearly seen that the closed-loop system with is stable; however, with it is not.

       Now, we evaluate the closed-loop performance for a PID controller with filter, where the filter time constant is chosen to be . Based on Table 1.5, the PID controller parameters are calculated as for , , , , and for , , , .

Figure 1.19 Comparison of closed-loop responses using Padula and Visioli PI controller (Example 1.8). (a) Control signal. (b) Output. Key: line (1) tuning rule with ; line (2) tuning rule with .

Figure 1.20 Comparison of closed-loop responses using Padula and Visioli PID controller (Example 1.8). (a) Control signal. (b) Output. Key: line (1) tuning rule with Скачать книгу