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

Choosing and 1, find the closed-loop transfer function for the PID control system shown in Figure 1.9 and simulate its closed-loop performance. Also, find the closed-loop transfer function of the IPD control system shown in Figure 1.10 and simulate its closed-loop step response.

      Solution. The control signal from Figure 1.9 has the Laplace transform given by Equation 1.34. Substituting into the following equation,

      (1.38)

       re-grouping and re-arranging lead to the closed-loop transfer function:

      (1.39)

       There are two zeros in the closed-loop transfer function: caused by the integral control, and caused by the derivative control. Figure 1.11(a) shows the closed-loop step responses for and respectively. With the increase in , the oscillation in the closed-loop response is reduced. However, there is a large overshoot in the output response.

       Using the IPD structure as shown in Figure 1.10, we calculate the closed-loop transfer function by using the Laplace transform of the controller output given in Equation 1.36. Substituting this control signal into the plant output (see Equation 1.38), the closed-loop transfer function is

(1.40)

Figure 1.11 Step responses of PID control system (Example 1.4). (a) Responses for PID structure. (b) Responses for the IPD structure. Key: line (1) , line (2) (, ).

       With this implementation, the denominator of the closed-loop transfer function is the same; however, there is only one zero at caused by the derivative control. Figure 1.11(b) shows the closed-loop step responses with an IPD controller. In comparison with the responses from the previous case, it is seen that the overshoot in the closed-loop responses has been eliminated, however their response speed becomes slower.

      1.2.5 The Commercial PID Controller Structure

      In the PID controller design, the following structure is commonly used for determining the parameters , and , where the controller transfer function is expressed as

      (1.41)

      However, as demonstrated in this section, there are several variations in PID controller structure available for the realization of the control system, and different realization leads to different control system performance with the same set of PID controller parameters.

      In order to be more flexible to the users, the commercial PID controllers (see Alfaro and Vilanova (2016)) from manufacturers such as ABB, Siemens, and National Instruments take the following general form with the Laplace transform of the control signal:

      (1.42)

      where the coefficients and are for the weighting on the reference signal, and as before, the coefficient determines the appropriate derivative filtering action. There are several special combinations of the parameters , , and that are commonly encountered.

      1 When , , and , the PID controller becomes identical to the case shown in Figure 1.9, where the derivative control with filter is implemented on the output only.

      2 When and , the PID controller becomes the IPD controller shown in Figure 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