Classification of dynamic processes and PID controller tuning in a parameter plane

A quadruplet, defined by the ultimate frequency ωu, the ultimate gain ku, the angle φ of the tangent to the Nyquist curve at the ultimate frequency and the gain Gp(0), is sufficient for classification of a large class of stable processes, processes with oscillatory dynamics, integrating and unstable processes Gp(s). From the model defined by the above quadruplet, a two parameter model Gn(sn) is obtained by the time and amplitude normalizations. Two parameters of Gn(sn), the normalized gain ρ and the angle φ, are coordinates of the classification ρ–φ parameter plane. Model Gn(sn) is used to obtain the desired closed-loop system performance/robustness tradeoff in the desired region of the classification plane. Tuning procedures and tuning formulae are derived guaranteeing almost the same performance/robustness tradeoff as obtained by the optimal PID controller, applied to Gp(s) classified to the same region of the classification plane. Validity of the proposed method is demonstrated on a test batch consisting of stable processes, processes with oscillatory dynamics, integrating and unstable processes, including dead-time.