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

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Название PID Control System Design and Automatic Tuning using MATLAB/Simulink
Автор произведения Liuping Wang
Жанр Отраслевые издания
Серия
Издательство Отраслевые издания
Год выпуска 0
isbn 9781119469407



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IP controller structure (see Figure 1.7), and compare their closed-loop step responses.

      Solution. With the PI controller in the original structure, the closed-loop transfer function between the reference signal and the output signal is calculated using,

      (1.26)equation

       By substituting the plant transfer function (1.25) and the PI controller structure (1.16), the closed-loop transfer function is

      (1.27)equation

       With the PI controller in the IP structure, the Laplace transform of the control signal is defined by (1.24). By substituting this control signal into the Laplace transform of the output via the following equation,

      (1.28)equation

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

      (1.29)equation

Image described by caption and surrounding text.

       By comparing the closed-loop transfer function (1.27) from the original PI controller structure with the one (1.29) from the alternative structure, we notice that both transfer functions have the same denominator, however, the one from the original structure has a zero at . Because of this zero, the original closed-loop step response may have an overshoot.

       Indeed, the closed-loop step responses for both structures are simulated and compared in Figure 1.8, which shows that the original PI closed-loop control system has a large overshoot; in contrast, the IP closed-loop control system has reduced this overshoot. The penalty for reducing the overshoot is the slower reference response speed.

       The closed-loop transfer function obtained with IP controller structure can also be interpreted as a two degrees of freedom control system with a reference filter . This topic will be further discussed in Section 2.4.2.

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

      (1.30)equation

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

      (1.31)equation

      1.2.4 PID Controllers

      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 images of a PID controller is the sum of the three terms,

      (1.32)equation

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

      (1.33)equation

      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 images is expressed in the following form:

      (1.34)equation

      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)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: