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Closed-loop identification of transfer function model for unstable systems

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Abstract

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.

Introduction

Many systems exhibit multiple steady states due to certain nonlinearity of the systems. Operating the system at the unstable steady state is desirable for safety or economical reasons. For purposes of designing controllers [1], many of these systems can be adequately represented by unstable first order plus time delay transfer function model [kpexp(−Ls)/(τs−1)]. Identification of such models are required for improved tuning of PI/PID controllers or for design of advanced controllers like adaptive controller or model reference controllers. Open-loop identification methods are not applicable for unstable systems.

Kavdia and Chidambaram [2] have extended the method proposed by Yuwana and Seborg [3] for stable systems to identify an unstable first order plus time delay model. Basically the method first identifies a second order plus time delay model for the closed-loop system using the response obtained for a step change in the set point of a proportional controller. From the identified closed-loop model, the open-loop transfer function is derived. Improved model parameters can be obtained by using better approximations for time delay [4].

All these methods are based on the stable response obtained using a proportional controller. For unstable plus time delay systems, a large off set is obtained by using a proportional controller [5]. Hence, use of PI or PID controller is required to eliminate the offset. However, the PI controller gives a larger overshoot. A PID controller significantly reduces the overshoot also. For L/τ⩾0.7, a PI controller cannot stabilize the unstable system [5]. For this case, a PID controller is required for stabilization. Hence, a method is required to obtain the model parameters by using the step response of the closed-loop system using a PID controller. In the present work, a method is proposed for obtaining an unstable first order plus time delay model for unstable systems using a PID controller.

Section snippets

Proposed method

For the purpose of designing controllers, an unstable first order plus time delay model is assumed to represent the systemGp[=y(s)/u(s)]=kpexp(−Ls)/(τs−1).The following PID controller is used to stabilize the system:u=kc[e+(1/τI)∫edt−τD(dy/dt)],where e=yry. The closed-loop transfer function model is given byy(s)/yr(s)=(1+τIs)exp(−Ls)[(τs−1)(τI/kckp)s+(1+τIs+τIτDs2)exp(−Ls)].In Eq. (2), it is assumed that the derivative is taken only for the process output and not for the set point value. This

Simulation results

The system is assumed to be given by Eq. (1). Let us assume L=2,kp=4 and τ=4. An offset of 170% [calculated from 100/(kckp−1)] is obtained [5] if we use proportional controller for carrying out identification method. An initial set of PID controller settings is calculated by the tuning formulae given by Clement and Chidambaram [6] as kc=0.35, τI=22.75 and τD=1.365. The response for a unit step change in the set point is obtained similar to that shown in Fig. 1. The value of time delay (L) is

Simulation application to a bioreactor

The proposed identification method is applied to a nonlinear continuous bioreactor which exhibits output multiplicity behaviour. The model equations are given by [7]dX/dt=(μ−D)X,dS/dt=(Sf−S)D−(μX/γ),whereμ=μmS/(Km+S+KIS2).Here the dilution rate (D) is the manipulated variable. The model parameters are given by Agarwal and Lim [7]:γ=0.4g/g,Sf=4g/g,μm=0.53(1/h),D=0.36(1/h),Km=0.12g/g,KI=0.4545g/g.Here X and S are concentrations of the cell and the substrate, respectively. The reactor exhibits

Conclusion

A closed-loop method is proposed for identifying the unstable first order plus time delay parameters using PI or PID controllers. Simulations results on transfer function models show that the present method gives good parameters values. Simulation results on nonlinear bioreactor problem show that the present method gives model parameters and the designed PID controller settings close to the ones obtained based on the local linearization around the operating point. For a larger delay, PID

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