reduce the overshoot in output response to a step reference change, the proportional term in the PID controller may also be implemented on the plant output. In this case, the control signal is calculated using
(1.35)
Accordingly, the Laplace transform of the control signal is expressed as
(1.36)
Figure 1.10 shows a block diagram of the alternative PID controller structure (called an IPD controller).
The example below is used to illustrate the effect of the derivative term in the closed-loop control. The starting point is the PI controller designed in Example 1.3, based on which a derivative term is introduced.
Figure 1.9 PID controller structure.
Figure 1.10 IPD controller structure.
Suppose that the plant is described by the transfer function:
(1.37)
and the PI part of the controller has the parameters: , 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)
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
(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
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
