Closed-loop identification of transfer function model for unstable systems

A simple method of identifying first order plus time delay transfer function model is proposed for unstable systems. The method is based on a single experiment on a closed-loop system with a step change in the set point of a PI or PID controller. The step response and derived analytical formulae are used to calculate the steady-state gain, time delay and time constant of the unstable system. Simulation results are given for transfer function models and on nonlinear model equation of an unstable bioreactor. The identified model parameters and/or the designed PID controllers settings are compared with those of the exact or the linearized model used.