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the choice of basically led to a pole-zero cancellation in the control system. Such a pole-zero cancellation will limit, as discussed in the next chapter, the control system performance in the disturbance rejection, particularly when is large4. In Skogestad (2003), the IMC-PID tuning rules are modified to reduce the integral time constant as

      (1.48)

while and are unchanged from (1.47).

      The IMC-PID controller tuning rules are also extended to integrating systems in Skogestad (2003). Although the system has an integrator as part of its dynamics, integral control is still required for disturbance rejection (see Chapter 2).

      Assuming that the system has the integrator with delay model:

      (1.49)

      then a PI controller is recommended with the following parameters:

      (1.50)

      If the transfer function for the integrating system has the form:

      (1.51)

      then a PID controller is recommended to have the following parameters:

      (1.52)

      If the system has a double integrator with the transfer function

      (1.53)

      then a PID controller is recommended with the following parameters:

      (1.54)

      The IMC-PID controller tuning rules will be studied in Examples 2.1 and 2.2.

      

      1.4.2 Padula and Visioli Tuning Rules

      Several sets of tuning rules were introduced in Padula and Visioli (2011) and Padula and Visioli (2012). These tuning rules are based on the first order plus delay model:

Table 1.4 Padula and Visioli tuning rules (PI controller).

Table 1.5 Padula and Visioli tuning rules (PID controller).

      They were derived using optimization methods for minimizing an error function together with the sensitivity peak in the frequency domain (see Chapter 2).

Here, we only include two sets of the tuning rules introduced for disturbance rejection in their paper. Tables 1.4 and 1.5 present the tuning rules for PI and PID controllers, respectively. Each table contains two sets of rules. For the specification of ( corresponds to the sensitivity peak (see Chapter 2)), the set of tuning

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