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

Another form of PI controller, perhaps more convenient for model-based controller design as in Chapter 3, is described by:

(1.30)

This form of PI controller is identical to the original PI controller structure when the parameters of and are selected as

(1.31)

A PID controller consists of three terms: the proportional (P) term, the integral (I) term, and the derivative (D) term. In an ideal form, the output of a PID controller is the sum of the three terms,

(1.32)

where is the feedback error signal between the reference signal and the output , and is the derivative control gain. The Laplace transfer function of the PID controller is

(1.33)

If the design is sound, the sign of is positive. If the sign of is negative, the derivative control term is to be neglected and instead a PI controller should be chosen.

Analogously to the proportional plus derivative controller described in Section 1.2.2, for most of the applications the derivative control is implemented on the output only with a derivative filter. For this reason, the control signal is expressed in the following form:

(1.34)

Figure 1.9shows the block diagram of the PID controller structure.

To 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 1.10shows 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 1.11Step responses of PID control system ( Example 1.4). (a) Responses for PID structure. (b) Responses for the IPD structure. Key: line (1) , line (2) ( , ).