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

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1.10, where both the proportional control and derivative control are implemented on the output only.

      3 When , , and , the implementation of the PID controller puts proportional control, integral control, and derivative control with filter on the feedback error .

      4 When , , and , the PID controller becomes the case where no derivative filter is used in the implementation. This will severely amplify the measurement noise.

      It is worthwhile emphasizing that the parameters and only affect the closed-loop response to the reference signal and they play no role in the closed-loop stability. We have examined the cases where and are either 0 or 1. However, we can extend the results to the situations where the parameters are between 0 and 1 and expect a compromised result. Upon understanding their roles, we can choose the appropriate coefficients according to the actual applications.

      1.2.6 Food for Thought

      1 The PID controllers are expressed in terms of the parameters , and . What are the possible signs of , and ?

      2 When you increase the magnitude of , do you expect the action of proportional control to decrease or increase? When you increase , do you expect the action of integral control to decrease or increase? when you increase , do you expect the action of derivative control to decrease or increase?

      3 What are the roles of integrator in a PID controller?

      4 Can you implement the integrating control on output only? If not, explain the reason.

      5 In many applications, we will put the proportional control on the feedback error, which is the original PI controller. Can you reduce the overshoot by using a ramp reference signal in the early part of the response?

1.3 Classical Tuning Rules for PID Controllers

      This section will discuss the classical tuning rules that have existed for the past several decades and have withstood the test of time. Although all tuning rules are rule-based, there is still certain knowledge assumed for the system to be controlled.

      1.3.1 Ziegler–Nichols Oscillation Based Tuning Rules

      Ziegler–Nichols oscillation based tuning rules are to use closed-loop controlled testing to obtain the critical information needed for determining the PID controller parameters.

In the closed-loop control testing, the controller is set to proportional mode without integrator and derivative action. The sign of must be the same as the steady-state gain of the plant for the reason of introducing negative feedback in the control system. With the proportional closed-loop control, the feedback gain is set to be a very small value in magnitude to begin the experiment. The value of is gradually increased until the control signal exhibits sustained oscillation (see Figure 1.12). There are two parameters obtained from this test: the value of that has caused the oscillation and the period of the oscillation. We denote this particular as and the period as . With these two parameters, the PID controller parameters are presented for the Ziegler–Nichols tuning rules in Table 1.1.

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.

Example 1.5

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

      (1.43)

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