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Figure 1.8 Closed-loop step response of PI control system (Example 1.3). Key: line (1) response from the original structure; line (2) response from the IP structure.

       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)

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

      (1.31)

      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 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 1.9 shows 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 1.10 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.

Example 1.4

       Suppose that the plant is described by the transfer function:

      (1.37)

       and the PI part of the controller has the parameters:

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