Closed loop identi®cation of transfer function model for an unstable bioreactor is proposed based on an optimization method using either a step or a pulse response of PI/PID controlled bioreactor. A simple method is proposed for the initial guesses of the parameters of the ®rst order plus time delay (FOPTD) transfer function model. A PID controller is designed for the identi®ed model. Simulation study on the nonlinear model equations of an unstable bioreactor exhibiting multiple steady-states shows that the PID controller designed on the identi®ed FOPTD model gives a good closed loop response similar to the one designed based on the linearized model from the nonlinear model equations.
List of symbolsD dilution rate (1/h) k c controller gain k p open loop process gain K I substrate inhibition rate constant (g/l) K m substrate saturation constant (g/l) L time delay of the open loop system s Laplace variable S substrate concentration in the bioreactor (g/l) S f feed substrate concentration (g/l) t time (h) t s settling time of the closed loop system (h) X cell mass concentration g/l y p,cl output of the closed loop system y m,cl output of the closed loop model y p output of the open loop system Greek symbols s time constant of the open loop system s I integral time of the controller s D derivative time of the controller f damping coef®cient of the open loop system f cl damping coef®cient of the closed loop system l speci®c growth rate constant l m maximum speci®c growth rate constant c yield coef®cient D deviation variable from the steady-state condition 1 Introduction Identi®cation of transfer function model of a bioreactor is required for the design of PID controllers. Identi®cation of transfer function model can be carried out by open loop or by closed loop method. Closed loop identi®cation is preferable since the method is insensitive to disturbances and measurement noise. Further, identi®cation of transfer function model for unstable bioreactors can be carried out only by closed loop method. In this method, closed loop response for a step change in the set point is obtained by using a proportional controller. The controller gain is adjusted such that the closed loop response resembles that of an underdamped second order plus time delay model. From this response, the closed loop transfer function model is ®rst identi®ed. Then the open loop transfer function model is assumed whose parameters are extracted from the closed loop transfer function model. Kavdia and Chidambaram [1] have applied this method to identify an unstable FOPTD model of an unstable bioreactor. The method is an extension of the method proposed by Yuwana and Seborg [2] for stable system. Since the proportional controller introduces offset, the method may not be permitted in industries. For unstable systems with a ratio of time delay to time constant >0.67, the proportional controller cannot stabilize the reactor [3]. Recently Ananth and Chidambaram [4]have proposed a method to identify a FOPTD model for an unstable bioreactor by using a PID cont...
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