Figure 1.7 IP controller structure.
Applying a Laplace transform to this equation leads to the Laplace transform of the controller output in relation to the reference and the output as
(1.24)
Figure 1.7 shows a block diagram of this PI closed- loop control configuration. This type of implementation is called an IP controller in the literature, which is an alternative PI controller configuration. In Section 2.4, this PI controller structure is examined in the context of a reference filter within the framework of a two degrees of freedom control system. To demonstrate how this simple modification in the PI controller configuration can reduce the overshoot effect, we examine the following example.
Assume that the plant is described by the transfer function:
(1.25)
and the PI controller has the parameters: , 2. Find the closed-loop transfer function between the reference signal and the output signal for the original PI controller structure (see Figure 1.5) and the 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)
By substituting the plant transfer function (1.25) and the PI controller structure (1.16), the closed-loop transfer function is
(1.27)
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)
re-grouping and simplification lead to the closed-loop transfer function:
(1.29)
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
(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
(1.32)
where
(1.33)
If the design is sound, the sign of
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
(1.34)
Figure 1.9 shows the block diagram of the PID controller structure.
To
