Set-point weighted PID controllers for unstable systems

The methods proposed by Astrom and Hagglund for selection of set-point weighting parameters for stable systems are extended to unstable first-order plus time-delay (FOPTD) systems. Since these methods require the real part of the dominant pole, an analytical solution is derived for calculation of dominant poles of the closed-loop system given the PID controllers settings. For PID controller settings given by DePaor and O'Malley and Clement and Chidambaram, the set-point weighting parameters are calculated and simple equations are given for the parameters in terms of ε (ratio of time delay to dominant time constant). Simulation results on an unstable FOPTD transfer function model and higher-order transfer function model show that the overshoot is significantly reduced by employing the set-point weighting parameters in the PID controller. The performances of the closed-loop servo responses for all the extended methods are compared.