## Brazilian Journal of Chemical Engineering

##
*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol. 14 no. 4 São Paulo Dec. 1997

#### http://dx.doi.org/10.1590/S0104-66321997000400018

**DYNAMIC MODELLING AND ADVANCED PREDICTIVE CONTROL OF A CONTINUOUS PROCESS OF ENZYME PURIFICATION**

**E.C. Dechechi**^{2}**, M.I. Martins**^{1}**, R. Maciel Filho**^{2}** and F. Maugeri**^{1}

^{1} DEA/FEA/UNICAMP, CP 6121, CEP 13081-970

^{2} LOPCA/DPQ/FEQ/UNICAMP, CP 6066, CEP 13081-970,

Fone: (019) 239-8534, Fax: (019) 239-4717,

Campinas-SP, Brazil. email: maciel@feq.unicamp.br

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

Abstract- A dynamic mathematical model, simulation and computer control of a Continuous Affinity Recycle Extraction (CARE) process, a protein purification technique based on protein adsorption on solid-phase adsorbents is described in this work. This process, consisting of three reactors, is a multivariable process with considerable time delay in the on-line analyses of the controlled variable. An advanced predictive control configuration, specifically the Dynamic Matrix Control (DMC), was applied. The DMC algorithm was applied in process schemes where the aim was to maintain constant the enzyme concentration in the outlet of the third reactor. The performance of the DMC controller was analyzed in the feed-flow disturbances and the results are presented.Dynamic model, protein, DMC.

Keywords:

**INTRODUCTION**

Control techniques and automation of fermentor processes have been used increasingly in the last 10 years. However, there are few papers in the literature about control of enzyme recovery and purification. The purification process considered in this study consists of a continuous three-staged system with recycling of the adsorbent beads for protein purification based on principles of adsorption chromatography. The basic two-stage contactor developed originally by Pungor et al. (1987) was modified by the addition of an intermediate wash stage in order to increase purification. It is shown schematically in Figure 1. The adsorbption stage takes place in the first reactor where the liquid feed from a fermentor (*F*_{1}) comes into contact with the adsorbent. In the second reactor some of the adsorbed product may be desorbed with the addition of washing buffer (*F*_{2}). Desorption of the protein to be purified is obtained in the third reactor by maintaining appropriate conditions, and particularly, by the action of a suitable eluting solvent. The adsorbent beads are recycled to the first reactor (*F*_{R}), while the solution with the poor product and the contaminants is continuously removed from the first and second reactors (*E*_{1}*, C*_{1}* *and* E*_{2}*, C*_{2}). The third reactor is fed with a flow rate (*F*_{3}) containing an adequate solvent and leads to the output of a concentrated product (*E*_{3}).

The objective of this work is to develop the control structure, as well as to analyze the performance, of the predictive algorithm control based on the Dynamic Matrix Control (DMC) concept.

**PURIFICATION MODEL**

The mathematical model of the modified CARE (Continuous Adsorption Recycle Extract System) process was developed considering a model proposed by Rodrigues et al. (1992). An intermediate wash stage was added in relation to the original CARE and a set of material balances for the enzyme contaminants was developed. Since the process is essentially isothermal, no energy changes need to be accounted for and the pressure drop in the system is negligible. As first approximation, the three reactors are assumed to be perfectly mixed. This is a reasonable approach as shown by Pungor et al. (1987), for the conventional CARE.

The adsorption and washing processes are regarded as a reversible second-order reaction, whereas the adsorption stage is assumed to be a first-order irreversible reaction (Chase, 1984), so that they can be written as:

Adsorption:

(1)

**Figure 1:** The continuous adsorption recycle extraction system.

Desorption:

(2)

where A, B and AB are the target protein, the adsorbent and the adsorbed protein, respectively. The adsorption rate can be expressed by equation 3, where q is the protein concentration in the solid phase and C the protein concentration in the liquid phase.

(3)

At equilibrium, , so equation 3 can be rewritten to give equation 4, a Langmuir-type isotherm model, as follows:

(4)

The rate constants (k_{1}, k_{2} and k_{3}) not only represent the intrinsic adsorption and desorption kinetics, but also include contributions from both external and internal mass transfer resistance. The mechanisms can be lumped into a single mass transfer coefficient which can then be determined experimentally, as shown by Chase (1984). This approach makes the mathematical formulation more tractable at the expense of a less rigorous physical description. On the other hand, experimental results presented by Pungor et al. (1987) indicate that in such a system it is reasonable to have internal mass transfer limitations without a significant loss of accuracy. In fact, this can be justified, at least in the case of high molecular weight proteins, because these are probably mostly adsorbed at active sites located at or near the particle surface so that they block diffusion into the gel. For the protein and contaminants in the liquid phase (C, E) and the solid phase (q) at the adsorption and desorption stages, the four transient equations can be written as:

-For free enzyme (E):

(5)

(6)

-For adsorbed enzyme (q):

-For contaminants (C):

where the parameters t _{1}, y , g , d and e are defined as:

These equations form a system of ordinary equations that are solved by a 4^{th} order Runge-Kutta algorithm. Therefore, it is possible to examine the dynamic behavior of the system, so that advanced control algorithms, such as the DMC, can be developed and implemented.

The parameter reaction rate and maximum adsorption capacity for lisosyme adsorbed on Cibracon-Blue were obtained based on the results shown by Chase (1984) and Cowan et al. (1986). The operational conditions at the steady state used in this work are shown in the Table 1.

**DYNAMIC MATRIX CONTROL (DMC)**

The DMC algorithm was originally presented by Cutler and Ramaker (1979) and currently is considered to be a very robust advanced control algorithm (Dechechi, 1996). Basically, such an algorithm uses a linear model to calculate the required future changes in the manipulated variables that result in an optimum set-point tracking for a specified performance index. A brief development of the controller is described as follows. For a detailed description, see Luyben (1989) and Dechechi (1996).

An SISO system output can be computed from its step response model coefficients (b_{i}) by equation 17, which is obtained by rearrangement of equations 15 and 16:

**Table 1: Steady state operational conditions**

F [l/h] | E[mmol/l] | C[mmol/l] | q[mmol/l] | Volume [l] | |

feed | 0.3825 | 7.100 10^{-3 } | 5.680 10^{-2 } | 0.0 | ---- |

stage 1 | 0.3825 | 1.045 10^{-3} | 5.434 10^{-2} | 3.460 10^{-1} | 0.100 |

stage 2 | 0.2325 | 8.422 10^{-4} | 3.803 10^{-3} | 3.205 10^{-1 } | 0.100 |

stage 3 | 0.1254 | 1.692 10^{-2} | 4.659 10^{-4} | 5.607 10^{-5} | 0.100 |

recycle | 0.0250 | 1.692 10^{-2} | 4.659 10^{-4} | 5.607 10^{-5} | ---- |

- liquid phase fraction = 70 %

where NP is the Prediction Horizon, are the values of the output variable at sampling time i predicted by the convolution model in open loop form, and is the measured value. The b_{i} are the step response model coefficients (convolution model), is the variation of the manipulated variable at the k sampling time in the past and d_{k} is the dynamic correction of the load effect disturbance and model errors.

The convolution model to predict the behavior of the controlled variables in a closed-loop form can be written as equation 18:

where NC is the control horizon. In fact, NC is a controller design variable, such that, at each sampling time, NC values of are calculated and only the first is implemented at the manipulated variable.

The DMC algorithm minimizes the square of the deviation between the predicted output in the closed-loop form and the set-point values at NP future sampling time periods by solving the constrained least squares minimization problem:

where J is the performance index to be minimized, is the vector of the NC future changes to be calculated and *f* is the suppression factor or tuning parameter that penalizes the objective function for changes in the inputs and may be a interesting parameter to analyze (Dechechi, 1996).

The solution of the problem given by equation 19 for the SISO system using the principle of least square method of minimization is given by:

where A is the so called dynamic matrix (dimension [NPxNC]) of the convolution dynamic model of the process considered in the DMC formulation and I is the identity matrix. The tuning factor *f* directly affects the performance and the robustness of the system.

In this form, the DMC algorithm consists of discovering NC optimal values for the manipulated variable, but between two consecutive sampling times, once the first variation Dm_{1} is implemented, the complete procedure is repeated. The DMC is completely represented by equations 20 and 17, and the tuning parameters of the DMC controller are: NP (prediction horizon), NC (control horizon) and *f* (suppression factor). A procedure for initial estimation of these parameters for a SISO system is proposed by Maurath et al. (1988).

With initial DMC tuning parameters, the refined tuning procedure is basically concerned with varying *f* to adjust the behavior of controlled variables.

**RESULTS AND DISCUSSION**

DMC control was applied as an SISO configuration to regulatory control (load disturbances), where the enzyme concentration (E_{3}) and the feed flow rate (F_{3}), both from the third reactor, were the controlled and manipulated variables, respectively. The on-line measure of enzyme concentration (E_{3}), was obtained by computer simulations using the mathematical model described in this paper and a fourth-order Runge-Kutta numerical method, which considered 24 minutes as a time delay in the on-line enzyme analyses methods. Figures 2.a and 2.b show the Regulatory DMC control for ± 10% step load disturbance, e.g., -10% in the feed flow rate (F_{1}) (Figure 2a) and +10% in the feed enzyme concentration (E_{0}) (Figure 2b).

The effect of the dumping factor *f* on controller performance can be seen, and this draws attention to the importance of correct tuning of the parameter for good controller performance.

**Figure 2**: Behavior of controlled variable by DMC; load disturbance: -10% F_{1} (2.a) and +10% E_{0} (2.b); effect of *f* factor.

**Figure 3:** Behavior of controlled variable by DMC, load disturbance in F_{1} and E_{0} (3.a) and comparison with open-loop response (3.b).

DMC control efficiency is shown in Figure 3, where the same set of controller parameters (NP=15, NC=8, f=0.0001) were applied for step changes in both feed variables (E_{0} and F_{1}) [ Figure 3.b] .

Finally, a better illustration of DMC controller efficiency is depicted in Figure 3.b, where the same closed-loop response of positive load disturbances shown in Figure 3.a is plotted together with the open-loop response system. It can be seen that the DMC controller was very effective, even for a relatively large change in feed conditions.

**CONCLUSIONS**

Based on the operational conditions and results shown in this work, it can be concluded that a predictive control strategy such as regulatory DMC, applied in this modified enzyme purification (CARE process), presented a satisfactory performance. In fact, DMC control is interesting in this case with the inherent time delay of analysis equipment.

**ACKNOWLEDGMENTS**

The authors would like to thank CNPq, Brazil, for the financial support received for this work.

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