ANALYSIS OF THE PREDICTIVE DMC CONTROLLER PERFORMANCE APPLIED TO A FEED-BATCH BIOREACTOR

ANALYSIS OF THE PREDICTIVE DMC CONTROLLER PERFORMANCE APPLIED TO A FEED-BATCH BIOREACTOR

J. A. D. RODRIGUES ¹ and R. MACIEL FILHO ²

¹ Instituto de Tecnologia de Alimentos - ITAL, CEP 13.073-001, Caixa Postal 139,

Campinas-SP, Brazil. Fax: [phone]. EMail: [e-mail]

² Faculdade de Engenharia Química, Departamento de Processos Químicos, UNICAMP,

CEP 13.081-970, Caixa Postal 6066, Campinas-SP, Brazil. Fax: [phone].

EMail: [e-mail]

(Received: June 11, 1997; Accepted: October 30, 1997)

Abstract - Two control algorithms were implemented in the stabilization of the dissolved oxygen concentration of the penicillin process production phase. A deterministic and nonstructured mathematical model was used, where were considered the balances of cell, substrate, dissolved oxygen and product formation as well as kinetic of the growth, respiration, product inhibition due to excess of substrate, penicillin hydrolyze, yield factors among cell growth, substrate consumption and dissolved oxygen consumption. The bioreactor was operated in a feed-batch way using an optimal strategy for the operational policy. The agitation speed was used as manipulated variable in order to achieve the dissolved oxygen control because it was found to be the most sensitive one. Two types of control configurations were implemented. First, the PID feedback control with the parameters estimated through Modified Simplex optimization method using the IAE index, and second, the DMC predictive control that had as control parameters the model, prediction and control horizons as well as suppression factor and the trajectory parameter. A sensitivity analysis of these two control algorithms was performed using the sample time and dead time as the index to make stability evaluation. Both configurations showed stable performance, however, the predictive one was found to be more robust in relation to the sample time, as well as the dead time variations. This is a very important characteristic to be considered for the implementation of control scheme in real fermentative process.

Keywords: DMC Control, feed-batch bioreactor, dissolved oxygen, penicillin process.

INTRODUCTION

The biotechnology processes, usually, have a complex dynamic behavior being regulated through biochemical activities of the microorganisms and operating conditions. These two factors interact and only the operating variables can be direct controlled (Bailey and Ollis, 1986).

In order to make a correct process control design is needed a good knowledge of control law and an appropriate strategy selection of the manipulated process variables to obtain significant improvements. This fact is specially important in fermentation process where a feed-batch operation plus nonlinearity and nonstationary behavior difficult the stability performance of the controllers (Shimizu, 1993).

The conventional PID controller has been the most frequently used control technique in the biochemical process due to historical factors and implementation facilities. However, there are some special reasons in fermentation processes that becomes it inadequate as the presence of nonlinearity and nonstationary behavior in the biochemical systems, and time delay of the sensors involved in the measured variables. Nonconventional and advanced configurations have been developed in order to solve these weaknesses of the conventional configurations. They have good potential to be used in such situations especially due to the fact that their structure are different being based in process representations as polynomials and convolutions models (Fisher, 1991; Newell and Lee, 1989; Onken and Weiland, 1985).

Siegel and Gaden Jr (1962) showed how the dissolved oxygen concentration was affected by air inlet flow rate, agitation speed and oxygen partial pressure in this flow rate. Lee et al. (1991) implemented an adaptive control algorithm in the dissolved oxygen concentration considering the dynamic of the electrode through the time delay and manipulating two variables: air flow rate and agitation speed, being the parameters estimated on-line through the least square approach.

This work presents a comparison between the conventional PID feedback and predictive DMC configurations considering the sensitivity of the closed loops response in relation to IAE index when both the sample time and the dead time were modified. The process taken into account was the production phase of the penicillin process and the control loop considered was the dissolved oxygen concentration as controlled variable and speed agitation as manipulated variable.

THEORETICAL FUNDAMENTALS

Penicillin Process Production Phase Model

The penicillin process production phase is carried out in a feed-batch operating mode with substrate supplementary addition. The bioreactor is perfectly stirred, isothermal, with aeration and agitation systems.

The model is deterministic and nonstructured, developed upon Contois growth type kinetic, yield coefficients among growth specific rate, substrate consumption specific rate, oxygen uptake specific rate and production specific rate, endogenous and maintenance metabolism, product inhibition due excess of substrate and hydrolyze of the penicillin. Volumetric oxygen transfer coefficient was estimated through Reynolds modified number, Power number and apparent viscosity as a cell concentration function. The operation mode was feed-batch, and the parameters were obtained from the literature (Rodrigues, 1996).

The resulting model was solved by a fourth-order Runge-Kutta-Gill variable step algorithm. Process productivity optimization was made with relation to cell initial concentration, substrate concentration in the feed-batch flow rate and the feed-batch strategy (Rodrigues and Maciel Filho, 1996-a).

To carried out the dissolved oxygen concentration control was used the speed agitation as manipulated variable due to its sensitivity and operating implementation facility (Rodrigues and Maciel Filho, 1996-b). It was considered both the down and the up levels constraints of the agitation speed in order to make the perfectly mixture without to reduce the productivity as experimental evidences suggest (Smith et al., 1990).

Conventional PID Controller

The conventional PID control algorithm was implemented in the velocity form (Luyben, 1990; Seborg et al., 1989) and the parameters were estimated through a Modified Simplex Algorithm (Edgar and Himmelblau, 1989) considering the IAE (Integral of the Absolute Error) as the optimization index.

Predictive DMC Controller

The basic idea of the DMC algorithm make use of a time domain model adjusted by a step perturbation in that way to allow the computation of the future changes in the manipulated variable through a minimization of a performance index. Therefore, the DMC configuration calculated the control actions so that an optimization of the future trajectory is made and constraints are considered (Chang et al., 1992; Marchetti et al., 1983; Pinto, 1990).

Some variables are generically defined in the DMC algorithm synthesis:

(1)

(2)

(3)

(4)

Note that E’ is estimated based in the past control action and represents the predict out error in relation to the desired trajectory. However, its assume that perturbation will not occur between k and k+R. The system must be solved based on a least square performance optimization criterion:

(5)

In this way the control action is obtained in the period from k to k+L-1, however only D u_k will be implemented. At the sample k+1 equation (5) is used again. Therefore, a modified performance index is defined so that the constraints on the manipulated variable is considered. Then, the control law resulting is:

(6)

Therefore, the design of the DMC controller is made up estimating five parameters: model horizon (N), prediction horizon (R), control horizon (L), reference trajectory parameter (a ) and suppression factor (f).

RESULTS AND DISCUSSION

The conditions of the process simulations were the same of those used by Rodrigues and Maciel Filho (1996-a). Two different control strategies were implemented and their performance were analyzed.

In the first strategy, the parameters of the PID controller was estimated through an interactive method so that the absolute error between the set-point and the output variable (IAE) was minimized by a Modified Simplex Algorithm. The Figure 1 shows the resulting dissolved oxygen and speed agitation profiles, when such control strategy is active.

The DMC parameters was estimated through the analysis of the response of the controlled and manipulated variables as well as by the computation of the IAE index. The Figure 2 shows the resulting dissolved oxygen and speed agitation profiles.

Through these two figures, it can be observed the closed behavior of the two configurations where the unstable periods were due to changes in the feed-batch strategy (Rodrigues and Maciel Filho, 1996).

Figure 1: Dissolved oxygen and speed agitation profiles of the PID controller [K_c=0.756 10^-1 rps; t _i=0.494 10^-3 h^-1; t _d=0.223 10^-1 h; sample=0.1 h; IAE=25.2].

Figure 2: Dissolved oxygen and speed agitation profiles of the DMC controller [R=14; L=2; a =0; f=0.01; N=15; sample=0.1 h; IAE=35.0].

In the sequence was analyzed the influence of the sample time modifications (see Figures 3 and 4) and the presence of dead time in the measured variable (see Figures 5 and 6). The analysis of such figures show that DMC configuration appears to be much more robust in relation to sample time (5 times higher than the standard value) as well as to dead time (5 times higher than the standard value of the sample time). This is an important characteristic of the control configurations because, usually, in the process industry the dead time is not well established as well as it may be time variant.

Figure 3: Dissolved oxygen and speed agitation profiles of the PID controller with other sample time [K_c=0.756 10^-1 rps; t _i=0.494 10^-3 h^-1; t _d=0.223 10^-1 h; sample=0,1 h; IAE=75.4].

Figure 4: Dissolved oxygen and speed agitation profiles of the DMC controller with other sample time [N=15; R=14; L=2; a =0; f=0.01; IAE(sample=0.1 h)=35.0; IAE(sample=0.2 h)=44.6; IAE(sample=0.5 h)=61.0 ].

Figure 5: Dissolved oxygen and speed agitation profiles of the PID controller with dead time [K_c=0.756 10^-1 rps; t _i=0.494 10^-3 h^-1; t _d=0.223 10^-1 h; sample=0.1 h; IAE=68.9].

Figure 6: Dissolved oxygen and speed agitation profiles of the DMC controller with dead time [N=15; R=14; L=2; a =0; f=0.01; sample=0.1 h; IAE(delay=0)=35.0; IAE(delay=0.1 h)= 35.0; IAE(delay=0.5 h)=37.5].

CONCLUSIONS

The control configurations implemented in this work presented good performance. Some similar behavior of both control strategies can be seen as a consequence of the detailed procedure to obtain the controller parameters, especially when similar disturbances were used. In such situation the PID algorithm is more adequate to be used because it is easier to be implemented. However, the DMC strategy is much more convenient where there are changes in the sample time as well as when dead time is considered. When such strategy is considered the system is much more robust in spite of poor knowledge of dead time and changes in sample time.

NOMENCLATURE

A Dynamic DMC matrix, dimensionaless

a_i DMC parameter of the step perturbation, dimensionaless

Co₂ Relative dissolved oxygen concentration, % of CO₂ saturation

E’ Error vector of the predict trajectory, dimensionaless

e Error of the measured variable in relation to set-point, % of CO₂ saturation

f Suppression factor, dimensionaless

h_i DMC parameter of the impulse perturbation, dimensionaless

IAE Integral of the absolute error, % of CO₂ saturation

k Actual time, h

Kc Proportional PID gain, rps

L Control horizon, dimensionaless

N Convolution or model horizon, dimensionaless

P Vector of the predict trajectory, dimensionaless

R Prediction horizon, dimensionaless

S Vector of the predict trajectory, dimensionaless

Ta Sample time, h

W₁, W₂ DMC constraints matrix of the manipulated variable, dimensionaless

y_SP Set-point, % of CO₂ saturation

Greek letters:

a Parameter of the reference trajectory, dimensionaless

D u Vector of the manipulated variable, rps

t d Derivative PID gain, h

t i Integral PID gain, h^-1

REFERENCES

Bailey, J.E. and Ollis, D.F., Biochemical Engineering Fundamentals. McGraw Hill, New York (1986).

Chang, C.M. et al., Improved DMC Design for Nonlinear Process Control, AIChE Journal, 38, 607 (1992).

Edgar, T.F. and Himmelblau, D.M., Optimization of Chemical Process. McGraw Hill, Singapore (1989).

Fisher, D.G., Process Control: An Overview and Personal Perspective. Canadian Journal of Chemical Engineering, 69, 5 (1991).

Lee, S.C. et al., Adaptive Control of Dissolved Oxygen Concentration in a Bioreactor. Biotechnology Bioengineering, 37, 597 (1991).

Luyben, W.L., Process Modeling, Simulation and Control for Chemical Engineers. McGraw Hill, Singapore (1990).

Marchetti, J.L. et al., Predictive Control Based on Discrete Convolutions Models. Industrial Engineering Chemical Process Design and Development, 22, 488 (1983).

Newell, R.B. and Lee, P.L., Applied Process Control. Prentice Hall, Brisbane (1989).

Onken, U. and Weiland, P., Control and Optimization. In: REHM, H.J. & REED, G. ed. Biotechnology. VHC, Weinheim, v. 2, cap. 30 (1985).

Pinto, J.M., Controle por Matriz Dinâmica em Coluna de Destilação. M.Sc. Dissertation, COPPE/UFRJ (1990).

Rodrigues, J.A.D., Dinâmica e Estratégias de Otimização e Controle do Processo de Penicilina. Dr. Thesis, LOPCA/DPQ/FEQ/UNICAMP (1996).

Rodrigues, J.A.D. and Maciel Filho, R., Optimal Feed Rates Strategies with Operating Constraints for the Penicillin Production Process. Chemical Engineering Science, 51, 2859 (1996-a).

Rodrigues, J.A.D. and Maciel Filho, R., Production Phase Sensitivity Analysis of The Penicillin Process by Factorial Design. Brazilian Journal of Chemical Engineering, 13, 107 (1996-b).

Seborg, J.M. et al., Process Dynamics and Control. Wiley & Sons, New York (1989).

Siegel, S.D. and Gaden Jr., E.L., Automatic Control of Dissolved Oxygen Levels in Fermentations. Biotechnology Bioengineering, 4, 345 (1962).

Smith, J.J. et al., The Effect of Agitation on the Morphology and Penicillin Production of Penicillium chrysogenum. Biotechnology Bioengineering, 35, 1011 (1990).

Publication Dates

• Publication in this collection
  06 Oct 1998
• Date of issue
  Dec 1997

History

• Accepted
  30 Oct 1997
• Received
  11 June 1997