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Find the PI and PID controller parameters using Ziegler–Nichols tuning rule and simulate the closed-loop control systems.
Solution. We build a Simulink simulation program for proportional control as illustrated in Figure 1.1. Since this system has a negative steady-state gain of , so the feedback control gain should be negative.3 Beginning the tuning process by setting and decreasing gradually to , the closed-loop control system exhibits sustained oscillation as shown in Figure 1.12. From this figure, the period of oscillation reads as 3.35. Based on Table 1.1, the proportional gain for the PI controller is and the integral time constant The proportional gain for the PID controller is , , and The PI and PID control systems are simulated where the reference signal is a unit step signal. Figure 1.13 compares the closed-loop output responses based on the PI and PID controller structures. Here the derivative control is implemented on the output only with a filter time constant . It is seen that with the derivative term, the closed-loop oscillation existing in the PI controller is reduced.
In general, the Ziegler and Nichols tuning rules using the oscillation method lead to quite aggressive responses with oscillations in closed-loop responses that are undesirable for many control applications. In Tyreus and Luyben (1992), a smaller
It is important to point out that generating sustained oscillation by increasing the controller gain is not a safe operation because a small error in the tuning process could cause the closed-loop system to become unstable. This unsafe procedure is replaced by using relay feedback control in Chapter 9, which also produces a sustained closed-loop oscillation.
Figure 1.13 Comparison of closed-loop output response using Ziegler–Nichols rules (Example 1.5). Key: line (1) PI control; line (2) PID control.
1.3.2 Tuning Rules based on the First Order Plus Delay Model
The majority of tuning rules existing in the literature are based on a first order plus delay model, which has the following transfer function:
where
The reaction curve is obtained by performing a plant step response test in open-loop operation, therefore we assume that the plant is stable, although it could be applied to integrating plant with caution. When performing this test, the plant input signal
The plant information obtained from the test includes the steady-state gain, defined as
(1.45)
Figure 1.14 Step response data. (a) Input signal. (b) Output signal. Key: line (1) the output response; line (2) steady-state output position before the response (
The time delay
