Liuping Wang - PID Control System Design and Automatic Tuning using MATLAB/Simulink

Figure 1.12 Sustained closed-loop oscillation (control signal).

Table 1.1 Ziegler–Nichols tuning rule using oscillation testing data.

P
PI
PID

A proportional control will not cause sustained oscillation for first order plant and second order plant with a stable zero. Thus, the tuning rule is not applicable to these two classes of stable plants. The following example is used to illustrate an application of the tuning rule.

Assume that a continuous time plant has the Laplace transfer function:

(1.43)

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 and a larger are recommended to reduce the oscillations, which have the following values:

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 1.13Comparison of closed-loop output response using Ziegler–Nichols rules ( Example 1.5). Key: line (1) PI control; line (2) PID control.

The majority of tuning rules existing in the literature are based on a first order plus delay model, which has the following transfer function:

(1.44)

where is the steady-state gain of the system, is the time delay, and is the time constant. This is mainly because the primary applications of tuning rules are for process control where typically the process is stable with time delay. There are many methods available for obtaining a first order plus delay model. Among them is an incredibly simple procedure that is called fitting a reaction curve. This reaction curve is the so-called step response test.

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 takes a step change from an initial constant value to a normal operation value, ; the measurement of the plant output signal in response to the step input change gives us the plant step response test data or the reaction curve. The response test completes when the value of the output signal reaches a constant or the signal fluctuates around a constant value due to noise and disturbances. Figure 1.14illustrates a typical set of step response testing data, with the step change occurring at time . Figure 1.14(a) shows that the input takes a step change from the initial value to the final value and Figure 1.14(b) shows the actual output response from the steady-state output position to the steady-state output position .