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

      Accordingly, the Laplace transform of the control signal is expressed as

      (1.36)equation

Block diagram depicting PID controller structure. Block diagram depicting IPD controller structure.

       Suppose that the plant is described by the transfer function:

      (1.37)equation

       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)equation

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

      (1.39)equation

       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

Image described by caption and surrounding text.

       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 images, images and images, where the controller transfer function images is expressed as

      (1.41)equation

      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)equation

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