Method 3: AMIGO PI controller tuning method

• This method can be used only for PI controller design either for an integrating process with transfer function (which appears in e.g. level control of liquid in a tank) or a first order plus time delay (FOPTD) process with transfer function  (which appears in e.g. temperature control). These two process models appear in many real world applications. Recall that more than 90% of the controllers currently in use on earth are PI controllers.
• This PI controller tuning method maximizes the integral gain subject to a constraint on the peak of the sensitivity function. We know that increasing the integral gain speeds up the closed loop system response but it also reduces the phase margin and gain margin and leads to larger overshoots in the response of the feedback system (recall that reducing the phase margin and gain margin is equivalent to increasing the peak of the sensitivity function). AMIGO method limits the peak of sensitivity function to 1.4 which means that the gain margin and phase margin of the closed loop system are at least equal to 3.5 and 45 degrees, respectively.
• Controller design algorithm: Consider the transfer function of PI controller as . If the plant model is in the form of a FOPTD transfer function then define  and calculate the parameters of PI controller from the following equation

Note that in order to find the FOPTD process model there is no need for exerting complicated system identification algorithms and such a plant model can be obtained from the step response of the process as shown in Figure 9.

Fig. 9: finding FOPTD process model from the step response of process.

Equation (1) can also be used for PI controller design for an integrating process. For this purpose first the parameters ,  and  should be found for the integrating process. In order to do this, apply a pulse as shown in Fig. 10 to the process and then calculate the process gain (), process time constant () and process delay () as detailed in Fig. 10. After that the parameters of PI controller can be calculated from Eq. (10).

Fig. 10: pulse response of an integrating process

Fig. 11: finding the process model from (vertically-scaled) pulse response