Model predictive control algorithms and their application to a continuous fermenter

Model predictive control algorithms and their application to a continuous fermenter

R. G. SILVA, C. S. ANASTÁCIO and W. H. KWONG

Department of Chemical Engineering , Federal University of São Carlos, P. O. Box 676, CEP 13565-905, São Carlos - SP, Brazil, Phone: [phone], Ext 228, Fax: [phone], E-mail: [e-mail]

(Received: January 19, 1999; Accepted: April 7, 1999)

Abstract - In many continuous fermentation processes, the control objective is to maximize productivity per unit time. The optimum operational point in the steady state can be obtained by maximizing the productivity rate using feed substrate concentration as the independent variable with the equations of the static model as constraints. In the present study, three model-based control schemes have been developed and implemented for a continuous fermenter. The first method modifies the well-known dynamic matrix control (DMC) algorithm by making it adaptive. The other two use nonlinear model predictive control algorithms (NMPC, nonlinear model predictive control) for calculation of control actions. The NMPC1 algorithm, which uses orthogonal collocation in finite elements, acted similar to NMPC2, which uses equidistant collocation. These algorithms are compared with DMC. The results obtained show the good performance of nonlinear algorithms.

Keywords: Predictive control, DMC, ADMC, continuous fermenter, controllers.

INTRODUCTION

During the past several years great interest has been shown in biochemical process control for maintaining operational conditions inside a specific optimum range for each process type, microorganism and medium. In general these processes are characterized by highly nonlinear relationships that make it difficult or even impossible to use classic control strategies, which when implemented show unsatisfactory actuation.

Predictive control strategies such as MAC ("Model Algorithmic Control") (Richalet et al., 1978) and DMC (Cutler and Ramaker, 1979) are advanced control strategies based on predictions of linear convolution models. The use of these linear models is appropriate if the operational conditions of the processes are close to the nominal conditions of the steady state or if the behavior of these is approximately linear. However, as soon as the system deviates from the nominal point operation, the models inaccuracy increases. If system dynamics vary with time, a large discrepancy develops between model and process, and they do not express the real system. In these situations, it becomes necessary to estimate the process model. A way of achieving this is to use an adaptive scheme such as ADMC (adaptive dynamic matrix control) (Maiti and Saraf, 1995), which can identify the process on-line periodically under open or closed-loop conditions. For biochemical processes, control algorithms that use nonlinear models or use an adaptive scheme are more appropriate because the success of the control depends a lot on knowledge of nonlinear process dynamics. The use of SQP (successive quadratic programming) in NMPC is described by Patwardhan et al. (1992).

The present work describes the application of two NMPC algorithms and ADMC to a continuous fermentation process where productivity is the controlled variable and feed substrate concentration is the manipulated variable. These three algorithms are compared with standard DMC.

NONLINEAR MODEL PREDICTIVE CONTROL

The problem of predictive control with a nonlinear model can be expressed as:

The process is described by the following differential/algebraic equations:

where y and u are the controlled and the manipulated variable vectors, respectively, x is the state variable vector and p is the set of model parameters which may include disturbances.

The objective of NMPC is to calculate a set of L future control actions (control horizon) that minimize a performance index, F , based on the desired output trajectory in a prediction horizon of R sampling intervals. In general, the ISE (Integral Square Error) performance index is adopted. For orthogonal collocation in finite elements as well as for equidistant collocation, the resulting nonlinear programming problem from the discretization of Equation 2 can be solved using SQP, and the first manipulated movement is implemented (Kwong and Corrêa, 1996; Silva and Kwong, 1998). The process is repeated, moving inwards with each sampling instant in the moving horizon concept. The effect of modeling error and unmeasured disturbances is treated as an additive disturbance, and is estimated at the k^th sampling instant in a manner similar to DMC:

This constitutes the feedback portion of the algorithm. If a perfect process model is availiable, d is equal to the additive disturbance in the process output.

ADAPTIVE DYNAMIC MATRIX CONTROL

Due to the large number of step response coefficients in the process model, recursive identification at each sampling instant, as used in the adaptive control theory, is infeasible in DMC. It is not always possible to know when to update the dynamic matrix. For example, if a large persistent disturbance enters the process, it may temporarily shift the operational point beyond the range of the existing DMC. This may result in a degraded controller performance, and the process may become oscillatory or unstable. An adaptive DMC may be able to eliminate the disturbance quickly when its presence is detected. By rearranging the convolution model, we obtain:

where {h_i}are the pseudo impulse response coefficients.

Maiti and Saraf (1995) proposed an on-line procedure for detecting disturbances and identifying pseudo impulse response coefficients, which consists basically of the following steps:

Step 1 - Check to see if ADMC is active according to the following conditions:

(a) steady state (check whether Du_k-1< e, where i = 1,2, k, N, e.g., 10 to 15).

(b) the magnitude of change in process response to check if the disturbance is large (check whether Dy > Dy _max).

(c) slope reversal of process response after entry of the disturbance. The step-like disturbance causes the process response to shoot in a positive or negative direction depending on the nature of the disturbance. The direction of the change must be reversed once the controller starts to eliminate the disturbance. It is at this point that the adaptive scheme starts to update the process model. To identify this slope reversal, the product of slopes at two consecutive instants is calculated after the entry of the disturbance. This is repeated at other successive sampling points until the product becomes negative.

Step 2 - If ADMC is inactive, use DMC; if not, go to step 3.

Step 3 - Estimate the pseudo impulse response coefficients, {h_i} on the identification horizon IH using Equation 6, and the remaining coefficients can be predicted using a fitted model.

CONTINUOUS FERMENTER

The model of a continuous fermenter assumes a constant volume and well-mixed condition (Henson and Seborg, 1992). Dilution rate D and feed substrate concentration S_f are available as manipulated inputs. Biomass concentration X, substrate concentration S and product concentration P are the process state variables. A variety of fermentations can be described by the following unstructured model:

where m is the specific growth rate, Y_X/S is the cell-mass yield with respect to the limiting substrate and a and b are yield parameters for the product. The specific growth rate model is allowed to exhibit both substrate and product inhibition as shown below:

This model contains four model parameters: the maximum specific growth rate m_m,the product saturation constant P_m ,the substrate saturation constant K_m and the substrate inhibition constant K_i. The nominal model parameters and operating conditions are given by Henson and Seborg (1992). If the fermenter deviates significantly from the operating conditions where the data was collected, the previously determined model parameters may no longer be valid. Cell-mass yield Y_X/S and the maximum specific growth rate m_m tend to be especially sensitive to changes in the operating conditions. From a process control perspective, these two model parameters can be viewed as an unmeasured disturbance because they may exhibit significant time-varying behavior.

Productivity Q, in relation to the product, can be defined as the amount of product cells produced per unit time in following way:

Figures ¹ and ² show the variation in productivity Q as a function of S_f or three values of m_m and Y_X/S. Note that small changes in m_m and Y_X/S can have dramatic effects on optimum productivity. If the fermenter is operated at high feed substrate concentrations, no cells are produced and productivity drops to zero. This phenomenon is known as washout. Unfortunately, the optimum productivity is near the washout region. A well-designed controller will maintain the fermenter near the optimum while avoiding washout.

Figure 1: The effect of dilution rate (a) and feed substrate concentration (b) on productivity for three values of the maximum growth rate.

Figure 2: The effect of dilution rate (a) and feed substrate concentration (b) on productivity for three values of cell-mass yield.

SIMULATION RESULTS

For evaluation of the performance of predictive controllers, a control problem of type SISO (single-input/single-output) with variations in parameters m_m and Y_X/S was used. The objective was to maintain productivity, Q , of the fermenter at a value of 3.7303 g/l/h, corresponding to a value near maximum productivity, 3.9321 g/l/h, for the nominal model parameters. The manipulated variable was feed substrate concentration, S_f. The tuning parameters for controllers NMPC1 and NMPC2 were R = 5 (prediction horizon), L = 1 (control horizon) and T = 1 (sampling time), and for DMC and ADMC were N = 20 (Number of samples), R = 5, L = 1, IH_min = 10, IH_max = 20, T = 1.

Figures ³ to 6 show the closed-loop response of Q and S_f for changes in the two mentioned parameters. As the controller objective was to maintain productivity at the closest possible desired productivity, the superiority of the nonlinear controllers was observed in Figure 6. Controllers NMPC1 and NMPC2 presented different values for the manipulated variable when compared to ADMC and DMC. This was due to the fact that the former drove fermenter operation to another feasible operational point (see Figure 2b) with the same productivity.

Figure 3: Controller comparison for -0.02 h^-1 disturbance in m_m.

Figure 4: Controller comparison for -0.04 h^-1 disturbance in mm.

Figure 5: Controller comparison for -0.04 g/g disturbance in Y_X/S.

Figure 6: Controller comparison for -0.08 g/g disturbance in Y_X/S.

Figure 7: NMPC2 closed-loop response.

In order to investigate the performance of controller NMPC2, a servo control problem of MIMO (multiple input-multiple output) type was considered. The control objective was to move the system from the given initial condition to the optimum operational point (Table ¹). As can be observed, NMPC2 maintained a high level of closed-loop performance in the servo control problem.

CONCLUSIONS

Although ADMC and DMC presented quite closed responses, showing the validity of the adaptation outlined, the study was limited to small changes in m _m and Y_x/s. For accentuated changes in these parameters, ADMC performance could probably be better. The NMPC2 algorithm, which uses equidistant collocation, results in a problem of smaller dimension and computational time for the computation of control actions as compared with the NMPC1 algorithm, an important factor for on-line applications.

As can be observed, all the controllers were shown to be viable for control of the productivity of the continuous fermenter despite appreciable variations in process dynamics and strong interactions between the manipulated variables.

ACKNOWLEDGEMENTS

The authors wish to thank CNPq for its financial support of this study.

REFERENCES

Bequette, B. W., "Nonlinear Control of Chemical Processes: A review," Ind. Eng. Chem. Res., 30, 1391-1413 (1991).

Cutler, C. R., Ramaker, B. L., "Dynamic Matrix Control - A Computer Control Algorithm," AICHE National Meeting, Houston, TX (1979).

Henson, M. A., Seborg, D. E., "Nonlinear Control Strategies For Continuous Fermenters," Chem. Eng. Sci., 47, 821-835 (1992).

Kwong, W. H., Corrêa, R. G., "Controle Preditivo Com Modelo Não Linear de um Fermentador Contínuo." In: Anais do XI SINAFERM, São Carlos, São Paulo, vol. 1, pp.339-344 (1996).

Maiti, S. N., Saraf, D. N., "Adaptive Dynamic Matrix Control of a Distillation Column With Closed-Loop Online Identification," J. Proc. Cont., 5, 315-327 (1995).

Patwardhan, A. A., Wright, G. T., Edgar, T. F., "Nonlinear Model-Predictive Control of Distributed-Parameter Systems," Chem. Eng. Sci., 47, 721-735 (1992).

Richalet, J., Rault, A., Testud, J. L., Papon, J., "Model Predictive Heuristic Control: Applications to Industrial Processes," Automatica, 14, 413-428 (1978).

Silva, R. G., Kwong, W. H., "Controle Preditivo Não Linear de Processos Químicos." In: Anais em CD-ROM do 12º COBEQ, Porto Alegre, Rio Grande do Sul (1998).

Table 1: Initial conditions and optimum operational point.

Publication Dates

• Publication in this collection
  15 Sept 1999
• Date of issue
  June 1999

History

• Accepted
  07 Apr 1999
• Received
  19 Jan 1999