Int J Adv Manuf Technol (2002) 20:459–463 Ownership and Copyright 2002 Springer-Verlag London Limited

Experimental Nonlinear Model Based Predictive Control for a Class of Semi-Batch Chemical Reactors F. M’sahli1, R. ben Abdennour2 and M. Ksouri3 Institut Supe´rieur des Etudes Technologiques de Ksar-Hellal, 5070, Ksar Hellal, Tunisia; 2Ecole Nationale d’Inge´nieurs de Gabes, universite´ de Sfax, Gabes, Tunisia; and 3Institut National des Sciences Applique´es et de Technologie, Tunis, Tunisia

This paper investigates experimentally the application of a nonlinear model based predictive control (NLMBPC) to a class of semi-batch chemical reactors equipped with a mono-fluid heating/cooling system. We present the experimental results dealing with a strongly exothermic reaction carried out in a small pilot plant according to a procedure commonly used in industry. The application of the NLMBPC is based on a constrained optimisation problem solved repeatedly on-line. The control objective is to keep the reactor temperature within safe operating specifications by manipulating the heating power. Experimental results demonstrate that this control strategy works well in the presence of hard constraints and load disturbances. Keywords: Constraints; Semi-batch reactor

1.

Optimisation;

Predictive

Control;

Introduction

Batch processes, in particular batch reactors, are required to transfer from performing one reaction to another type of reaction as the chemical industry seeks to manufacture low-volume, high-value-added chemicals. The majority of syntheses, in the fine chemical and pharmaceutical industries, are carried out in batch reactors. Their flexibility, polyvalent character and operation resemblance to bench-scale laboratory reactors make their use attractive, but their control is difficult. This paper focuses on an application study of a model-based predictive control strategy applied to a semi-batch reactor used to produce an ester. To increase product quality and ensure reproducibility, improving the control of such a process is required [1]. The control performance is mainly dependent on the heating/cooling system associated with the reactor. Thus, it is necessary to improve the automation of these reactors. Many industrial configurations of heating/cooling systems (mono-fluid multiCorrespondence and offprint requests to: Dr F. M’sahli, Institut Supe´rieur des Etudes Technologiques de Ksar-Hellal, 5070 Ksar Hellal, Monastir, Tunisia. E-mail: [email protected]

fluid systems) are used for the temperature control of these reactors. The reactor temperature control is difficult since the choice of circulating fluid and its flowrate have to be determined in order to track the a priori defined optimal temperature trajectory [2]. So, in the case of an exothermic reaction, such as the esterification one, there are three distinct steps: heating, reaction, and cooling. Thus, to change from heating to cooling causes a very abrupt change in the dynamics of the process, so, the reactor model used is a dynamic nonlinear model. Nonlinear control systems have been the subject of important work during the last decade. Among them we can mention two main techniques. The first is based on differential geometry [3,4]. The second concerns nonlinear model based predictive control (MBPC). MBPC has been received well in the chemical process industries. A review by Bequette [5] elaborates the various approaches to handling nonlinear systems via MBPC and their pertinent robustness issues. In this paper, we consider the second family of these techniques. The MBPC strategy is based on a repeated optimisation of an open-loop performance index, subject to constraints over a finite horizon extending from the current time into the future. We note here that there are two ways of performing MBPC calculations. The first method employs separate algorithms to solve the differential equations and to carry out the optimisation. First, the manipulated variable profile is guessed, and the differential equations are solved numerically to obtain the control variable profile. The objective function is then calculated. The gradient of the objective function with respect to the manipulated variable can be found either by numerical perturbation or by solving sensitivity equations. The control profile is then updated using some optimisation algorithm, and the process is repeated until the optimal profiles are obtained. An attractive alternative, which is employed in this paper, concerns the use of a simultaneous solution and optimisation strategy. The model differential equations are discretised, and along with the algebraic model equations are included as constraints in a nonlinear programming problem. The paper is organised as follows. In Section 2, the real process description is presented. In Section 3, we develop a dynamic model of the semi-batch reactor. This model is derived from mass and energy balances. The design of the explicit

460

F. M’sahli et al.

constrained NLMBPC is proposed in Section 4. Section 5 contains experimental results.

2.

Process Description

A schematic diagram of the experimental device used in this application is given in Fig. 1 [6]. The core is a 2-l cylindrical stainless-steel reactor (R) installed in the process control laboratory of at the School of Engineers of Gabes (Tunisia). Its temperature is regulated by means of a fluid circulating through a surrounding jacket. The circulation of the fluid is assured by the pump (P2). Depending on whether the reactor temperature has to be raised or lowered, the fluid is either heated by a heating exchanger (E1) with a set of three resistors whose electric power can be varied in the range 0–3 kW, or it is cooled in a tubular cooler (E2) whose cooling rate is changed by varying the external cooling water flow in the range 0– 1200 l h⫺1. A vertical stirrer (S), set along the reactor axis, is continuously rotated at a constant speed (usually 600 r.p.m.) in order to keep the medium as homogeneous as possible in both temperature and composition. Should a semi-batch recipe be required, a specific device (R1), located above the reactor, allows the supply, using pump (P1), of any additional reactant. The reaction we are interested in is carried out at atmospheric pressure. However, in the temperature range under consideration, solvent evaporation always takes place to a significant extent. Accordingly, a water-cooled condenser (C) must be used to return the solvent back to the reactor. Apart from the above-mentioned control variables, three other variables are measured on-line: inlet temperature Ti, outlet temperature To and reaction temperature TR. The process is interfaced to an IBM PC Pentium processor that is used for process monitoring and control.

The ester obtained is a product with a very high added value. It is used in the fine chemical industry. This reaction is highly exothermic and very difficult to operate manually. The dynamic behaviour of this reactor is modelled by a set of differential Eq. derived from mass and energy balances for the reaction mixture and the jacket. The model has been established using the following basic assumptions: 1. The reactor contents are perfectly mixed and the temperature of the reaction mixture is homogeneous. 2. The kinetic laws and the reactant concentrations of the chemical reaction are unknown. 3. The thermal losses between the reaction mixture and the outside are neglected. 4. The chemical reactions are performed in a pseudohomogeneous medium. 5. Feeding of the reactive material is performed without volume contraction. A mass and energy balance gives the following equations: Mass balance d(CL1 ) = FeCe1 ⫺ FvCv1 ⫺ 1rV ⫺ C1L (Fe ⫺ FLV) V dt

(1)

d(CL2 ) = FeCe2 ⫺ FvCv2 ⫺ 2rV ⫺ C2L (Fe ⫺ FLV) V dt

(2)

d(CL3 ) = ⫺ 3rV ⫺ CL3 (Fe ⫺ FLV) V dt

(3)

d(CLT) = FeCeT ⫺ FvCvT ⫺ CLT (Fe ⫺ FLV) V dt

(4)

dV = Fe ⫺ FLV dt

(5)

Ci ⫺ 1 = 0

(6)

冘

3. The Dynamic Model of the Semi-Batch Reactor

i

Energy balance The reaction carried in this semi-batch reactor is an esterification one. This reaction is given by the following equation:

d(VLCLPT) = Fe(eCePTe ⫺ LCPL T) ⫺ Vr⌬HR dt FvV⌬HV ⫺ +Q MV

Acid + Alcohol ) Ester + Water

(7)

where Ci is the concentration of component (i): i = 1,2, and 3, respectively, for acid, alcohol, and water. 1 = 2 = 3 = 1, for the reactants and 1 = 2 = 3= ⫺1, for the products. The complete list of symbols is given in the end of the paper. The physical parameters corresponding to this model are given in Table 1.

Table 1. Initial kinetic parameters of the reaction. C1 (mol/l) C2 (mol/l) C3 (mol/l) CT (mol/l) TR (°K) 0.3 Fig. 1. Diagram of the experimental process and its surroundings.

0.8

0

0.5

380

V (l) 2

Predictive Control for Chemical Reactors

4. Model Based Predictive Control Algorithm for the Semi-Batch Reactor The model based predictive control algorithm has been proposed as a “general purpose” adaptive control method [7–9]. The MBPC type algorithm uses the following objective function:

再冘 Ny

J(Ny,Nu,␥) = E

[y(k + j)

j=1 Nu

⫺ w(k + j)]2 + ␥

冘

⌬u2 (k + j ⫺ 1)

j=1

(8)

冎

subject to ⌬ulow ⱕ ⌬u(k) ⱕ ⌬uhigh, ulow ⱕ u(k) ⱕ uhigh

(9)

where w(t + j) is a sequence of future set points, Ny is the maximum prediction horizon, Nu ⬍ Ny is the control horizon and ␥ is a control-weighting factor. ⌬u(k + j), j 苸[1,Nu], is a sequence of future control increments computed by the optimising problem at time k; ⌬u(k + j) = 0 for j ⬎ Nu . ulow,uhigh,⌬ulow, and ⌬uhigh are, respectively, the lower limit, upper limit, lower derivative limit, and higher derivative limit of the control input [6,10]. At time step k, the model differential Eqs (1)–(5) and (7) are discretised, and along with the algebraic equation (Eq (6)) are included in Eq. (9) as constraints for nonlinear programming, Eq. (8). The NLMBPC optimiser computes the present and future manipulated variable changes so that the predicted output yˆ (k + j/k) follows the selected reference trajectory w(k + j/k) in a satisfactory manner. Although the optimisation problem solved at time k results in an optimal sequence only u(k/k), is ⌬uTk = [⌬u(k/k),. . .,⌬u(k + Nu ⫺ 1/k)], implemented for the real process over the time (k,k + 1): u(k/k) = u(k ⫺ 1) + ⌬u(k/k)

(10)

At time step (k + 1), the new measurement y(k + 1) is used together with u(k/k) to compute the new estimate yˆ (k + 1/k), the horizons Ny and Nu are shifted forward by one step and a new optimisation problem is solved at time step (k + 1) with new initial conditions yˆ (k + 1/k) to update the predictions and compute ⌬uk. To solve the constrained NLMBPC optimisation problem, we have used an ellipsoidal cutting plane algorithm [6]. At each sampling time, the control problem is solved iteratively computing the control input vector ⌬uk using the ellipsoidal cutting plane algorithm. In order to keep the dimensionality of the nonlinear problem low, we have used an orthogonal collocation to discretise the differential equations.

5.

For experimental requirements, some control interfaces have been added to link the process to a mini computer Pentium processor. Figure 2 shows the hardware used for the identification and the control of the process. The exchange of data between the process and the computer was made through 12bit A/D and D/A converters. To study the performance of the NLMBPC algorithm, different temperature profiles have been used in order to examine tracking of these set point profiles for the mixture temperature. A typical profile has been considered: 1. Preheating of the reaction mixture according to a predefined profile until the temperature reaction TR = 110°C is reached: duration 30 min. 2. Control of the mixture temperature at the constant value TR = 110°C during reactive feeding and maintenance of the mixture temperature TR constant after the end of reactant feeding, to complete the reaction: duration 1 h. 3. Cooling to ambient temperature, according to a predefined temperature profile, to avoid by-product formation: duration 30 min. It is clear that this type of operating mode causes abrupt and large changes in the process dynamics. This temperature profile is the most widely used in the fine chemical industry for the operation of highly exothermic reactions in a semi-batch jacketed reactor. For the real-time implementation requirements of the NLMBPC algorithm, we have used an extended Kalman filter (EKF) to estimate the concentration of products C1,C2,C3, and CT [11]. These estimates are used for the discretisation procedure and to predict the output trajectory over the prediction horizon into the future. The tuning parameters for NLMBPC are Ny = 6, Nu = 1, the input weight ␥ = 0.1. The chosen constraints are: 0 ⱕ u(k) ⱕ 3000 W (for all times) and 兩⌬u(k)兩 ⱕ 60 W also for all times. The predictive model controllers are tuned so that the manipulated variable u(k) and the control change ⌬u(k) satisfy the imposed constraints. Figures 3 and 4 show, respectively, the closed-loop response of the process and the input signal for NLMBPC (without disturbance). The solid line is the output and the dotted line is the reference signal. It is clear that the nonlinear model based system shows a closer correspondence to the reference signal. When a load disturbance signal (supply

Experimental Results

The objective of the system is to control the reactor temperature TR by operating the heating power as required. The main disturbance affecting the control performance comes from the temperature of the supply water for the cooling coil. Some experimental results are also given that show the qualities of control strategy proposed in Section 4.

461

Fig. 2. Hardware used for process control.

462

F. M’sahli et al.

that no temperature overshoot occurs at the end of the heating phase. This is a key performance of the technique, as it is well known that it is very difficult to avoid such an overshoot in a batch reactor. Experiments have shown the efficiency, the flexibility, and the good performance of the implemented NLMBPC algorithm in controlling the temperature of a reactor equipped with a mono-fluid system, and it also has a good temperature tracking performance.

Fig. 3. Closed-loop response for constrained NLMBPC.

Fig. 4. Heating power for constrained NLMBPC.

6. Conclusion In this paper we have presented the application of a nonlinear model based predictive control (NLMBPC) algorithm for the temperature control of a semi-batch reactor. The design of the NLMBPC strategy was based on a model obtained from input– output data as well as from first principles. This approach has been applied experimentally to a pilot reactor equipped with a mono-fluid heating–cooling system. The experimental results presented demonstrate a good performance of the developed control strategy. Good tracking performance of the implemented NLMBPC algorithm has been achieved. The experimental results show the efficiency and the flexibility of the developed methodology for controlling real processes.

References

Fig. 5. Closed-loop response for constrained NLMBPC (disturbance case).

Fig. 6. Heating power for constrained NLMBPC (disturbance case).

water temperature of the cooling coil) was added to the real process, the nonlinear controller shows a slight undershoot before returning quickly to the new set point. The results of this experiment are shown in Fig. 5 for the output and in Fig. 6 for the input signal. For all experiments, it can be seen

1. R. Berber, “Control of batch reactor: a review”, in (ed) Berber R. Methods of Model Based Predictive Control, pp. 459–494, Kluwer, 1995. 2. C. Kravaris, R. A. Wright and J. F. Carrier, “Non linear controllers for trajectory tracking in batch processes”, Computers and Chemical Engineering, 13, pp. 73–82, 1989. 3. A. Isidori, Non Linear Control Systems, 3rd edn, SpringerVerlag, 1996. 4. C. Kravaris and J. C. Kantor, “Geometric methods for non-linear process control – I Background; II Controllers synthesis”, Industrial Chemical Research, 29 (12), pp. 2295–2324, 1990. 5. B. W. Bequette, “Non linear control of chemical processes: a review”, Industrial Chemical Research, 30, pp. 1391–1413, 1991. 6. F. M’sahli, F. Bouani, R. Ben Abdennour and M. Ksouri, “Adaptive non linear model predictive control using autoregressive plus Volterra models”, 3rd European Robotics, Intelligent Systems and Control Conference (EURISCON ’98), Athens, Greece, 22–25 June 1998. 7. S. L. De Oliveira, V. Nevistic and M. Morari, “Control of nonlinear systems subject to input constraints”, Proceedings of the IFAC non-linear control design Symposium, Lake Tahoe, CA, vol. 1, pp. 15–20, 1995. 8. F. J. Doyle III, B. A. Ogunnaike and R. K. Pearson, “Nonlinear model-based control using second-order Volterra models”, Automatica, 31 (5), pp. 697–714, 1995. 9. T. C. Tsang and D. W. Clarke, “Generalized predictive control with input constraints”, Proceedings IEE, Control Theory and Applications, 135 (6), pp. 451–460, 1988. 10. E. F. Camacho, “Constrained generalized predictive control”, IEEE Transactions on Automatic Control, 38(2), pp. 327–332, 1993. 11. L. Ljung, “Asymptotic behavior of the extended Kalman filter as a parameter estimator for linear systems”, IEEE Transactions on Automatic Control, 24(1), pp. 36–51, 1979.

Predictive Control for Chemical Reactors

Nomenclature

Cep CLp Cei CLi Cvi CLt Cvt Fe FLV

feed calorific capacity (J g⫺1 K⫺1) calorific capacity of liquid phase (J g⫺1 K⫺1) concentration of feed component (i) (mole l⫺1) concentration of component (i) in liquid phase (mole l⫺1) concentration of component (i) in vapour phase (mole l⫺1) total concentration in liquid phase (mole l⫺1) total concentration in vapour phase (mole l⫺1) feed volumetric flow (l min⫺1) liquid equivalent vapour flow (l min⫺1)

Fv Mv Q r T Te V ⌬HR ⌬HV L V e

vapour volumetric flow (l min⫺1) molar mass of vapour phase (g mole⫺1) heat power (J min⫺1) reaction speed (mole l⫺1 min⫺1) mixture temperature (K) feed temperature (K) instantaneous volume mixture (l) enthalpy of reaction (J mole⫺1) enthalpy of vaporization (J mole⫺1) volumetric mass of liquid phase (g l⫺1) volumetric mass of vapour phase (g l⫺1) feed volumetric mass (g l⫺1)

463

Experimental Nonlinear Model Based Predictive Control for a Class of Semi-Batch Chemical Reactors F. M’sahli1, R. ben Abdennour2 and M. Ksouri3 Institut Supe´rieur des Etudes Technologiques de Ksar-Hellal, 5070, Ksar Hellal, Tunisia; 2Ecole Nationale d’Inge´nieurs de Gabes, universite´ de Sfax, Gabes, Tunisia; and 3Institut National des Sciences Applique´es et de Technologie, Tunis, Tunisia

This paper investigates experimentally the application of a nonlinear model based predictive control (NLMBPC) to a class of semi-batch chemical reactors equipped with a mono-fluid heating/cooling system. We present the experimental results dealing with a strongly exothermic reaction carried out in a small pilot plant according to a procedure commonly used in industry. The application of the NLMBPC is based on a constrained optimisation problem solved repeatedly on-line. The control objective is to keep the reactor temperature within safe operating specifications by manipulating the heating power. Experimental results demonstrate that this control strategy works well in the presence of hard constraints and load disturbances. Keywords: Constraints; Semi-batch reactor

1.

Optimisation;

Predictive

Control;

Introduction

Batch processes, in particular batch reactors, are required to transfer from performing one reaction to another type of reaction as the chemical industry seeks to manufacture low-volume, high-value-added chemicals. The majority of syntheses, in the fine chemical and pharmaceutical industries, are carried out in batch reactors. Their flexibility, polyvalent character and operation resemblance to bench-scale laboratory reactors make their use attractive, but their control is difficult. This paper focuses on an application study of a model-based predictive control strategy applied to a semi-batch reactor used to produce an ester. To increase product quality and ensure reproducibility, improving the control of such a process is required [1]. The control performance is mainly dependent on the heating/cooling system associated with the reactor. Thus, it is necessary to improve the automation of these reactors. Many industrial configurations of heating/cooling systems (mono-fluid multiCorrespondence and offprint requests to: Dr F. M’sahli, Institut Supe´rieur des Etudes Technologiques de Ksar-Hellal, 5070 Ksar Hellal, Monastir, Tunisia. E-mail: [email protected]

fluid systems) are used for the temperature control of these reactors. The reactor temperature control is difficult since the choice of circulating fluid and its flowrate have to be determined in order to track the a priori defined optimal temperature trajectory [2]. So, in the case of an exothermic reaction, such as the esterification one, there are three distinct steps: heating, reaction, and cooling. Thus, to change from heating to cooling causes a very abrupt change in the dynamics of the process, so, the reactor model used is a dynamic nonlinear model. Nonlinear control systems have been the subject of important work during the last decade. Among them we can mention two main techniques. The first is based on differential geometry [3,4]. The second concerns nonlinear model based predictive control (MBPC). MBPC has been received well in the chemical process industries. A review by Bequette [5] elaborates the various approaches to handling nonlinear systems via MBPC and their pertinent robustness issues. In this paper, we consider the second family of these techniques. The MBPC strategy is based on a repeated optimisation of an open-loop performance index, subject to constraints over a finite horizon extending from the current time into the future. We note here that there are two ways of performing MBPC calculations. The first method employs separate algorithms to solve the differential equations and to carry out the optimisation. First, the manipulated variable profile is guessed, and the differential equations are solved numerically to obtain the control variable profile. The objective function is then calculated. The gradient of the objective function with respect to the manipulated variable can be found either by numerical perturbation or by solving sensitivity equations. The control profile is then updated using some optimisation algorithm, and the process is repeated until the optimal profiles are obtained. An attractive alternative, which is employed in this paper, concerns the use of a simultaneous solution and optimisation strategy. The model differential equations are discretised, and along with the algebraic model equations are included as constraints in a nonlinear programming problem. The paper is organised as follows. In Section 2, the real process description is presented. In Section 3, we develop a dynamic model of the semi-batch reactor. This model is derived from mass and energy balances. The design of the explicit

460

F. M’sahli et al.

constrained NLMBPC is proposed in Section 4. Section 5 contains experimental results.

2.

Process Description

A schematic diagram of the experimental device used in this application is given in Fig. 1 [6]. The core is a 2-l cylindrical stainless-steel reactor (R) installed in the process control laboratory of at the School of Engineers of Gabes (Tunisia). Its temperature is regulated by means of a fluid circulating through a surrounding jacket. The circulation of the fluid is assured by the pump (P2). Depending on whether the reactor temperature has to be raised or lowered, the fluid is either heated by a heating exchanger (E1) with a set of three resistors whose electric power can be varied in the range 0–3 kW, or it is cooled in a tubular cooler (E2) whose cooling rate is changed by varying the external cooling water flow in the range 0– 1200 l h⫺1. A vertical stirrer (S), set along the reactor axis, is continuously rotated at a constant speed (usually 600 r.p.m.) in order to keep the medium as homogeneous as possible in both temperature and composition. Should a semi-batch recipe be required, a specific device (R1), located above the reactor, allows the supply, using pump (P1), of any additional reactant. The reaction we are interested in is carried out at atmospheric pressure. However, in the temperature range under consideration, solvent evaporation always takes place to a significant extent. Accordingly, a water-cooled condenser (C) must be used to return the solvent back to the reactor. Apart from the above-mentioned control variables, three other variables are measured on-line: inlet temperature Ti, outlet temperature To and reaction temperature TR. The process is interfaced to an IBM PC Pentium processor that is used for process monitoring and control.

The ester obtained is a product with a very high added value. It is used in the fine chemical industry. This reaction is highly exothermic and very difficult to operate manually. The dynamic behaviour of this reactor is modelled by a set of differential Eq. derived from mass and energy balances for the reaction mixture and the jacket. The model has been established using the following basic assumptions: 1. The reactor contents are perfectly mixed and the temperature of the reaction mixture is homogeneous. 2. The kinetic laws and the reactant concentrations of the chemical reaction are unknown. 3. The thermal losses between the reaction mixture and the outside are neglected. 4. The chemical reactions are performed in a pseudohomogeneous medium. 5. Feeding of the reactive material is performed without volume contraction. A mass and energy balance gives the following equations: Mass balance d(CL1 ) = FeCe1 ⫺ FvCv1 ⫺ 1rV ⫺ C1L (Fe ⫺ FLV) V dt

(1)

d(CL2 ) = FeCe2 ⫺ FvCv2 ⫺ 2rV ⫺ C2L (Fe ⫺ FLV) V dt

(2)

d(CL3 ) = ⫺ 3rV ⫺ CL3 (Fe ⫺ FLV) V dt

(3)

d(CLT) = FeCeT ⫺ FvCvT ⫺ CLT (Fe ⫺ FLV) V dt

(4)

dV = Fe ⫺ FLV dt

(5)

Ci ⫺ 1 = 0

(6)

冘

3. The Dynamic Model of the Semi-Batch Reactor

i

Energy balance The reaction carried in this semi-batch reactor is an esterification one. This reaction is given by the following equation:

d(VLCLPT) = Fe(eCePTe ⫺ LCPL T) ⫺ Vr⌬HR dt FvV⌬HV ⫺ +Q MV

Acid + Alcohol ) Ester + Water

(7)

where Ci is the concentration of component (i): i = 1,2, and 3, respectively, for acid, alcohol, and water. 1 = 2 = 3 = 1, for the reactants and 1 = 2 = 3= ⫺1, for the products. The complete list of symbols is given in the end of the paper. The physical parameters corresponding to this model are given in Table 1.

Table 1. Initial kinetic parameters of the reaction. C1 (mol/l) C2 (mol/l) C3 (mol/l) CT (mol/l) TR (°K) 0.3 Fig. 1. Diagram of the experimental process and its surroundings.

0.8

0

0.5

380

V (l) 2

Predictive Control for Chemical Reactors

4. Model Based Predictive Control Algorithm for the Semi-Batch Reactor The model based predictive control algorithm has been proposed as a “general purpose” adaptive control method [7–9]. The MBPC type algorithm uses the following objective function:

再冘 Ny

J(Ny,Nu,␥) = E

[y(k + j)

j=1 Nu

⫺ w(k + j)]2 + ␥

冘

⌬u2 (k + j ⫺ 1)

j=1

(8)

冎

subject to ⌬ulow ⱕ ⌬u(k) ⱕ ⌬uhigh, ulow ⱕ u(k) ⱕ uhigh

(9)

where w(t + j) is a sequence of future set points, Ny is the maximum prediction horizon, Nu ⬍ Ny is the control horizon and ␥ is a control-weighting factor. ⌬u(k + j), j 苸[1,Nu], is a sequence of future control increments computed by the optimising problem at time k; ⌬u(k + j) = 0 for j ⬎ Nu . ulow,uhigh,⌬ulow, and ⌬uhigh are, respectively, the lower limit, upper limit, lower derivative limit, and higher derivative limit of the control input [6,10]. At time step k, the model differential Eqs (1)–(5) and (7) are discretised, and along with the algebraic equation (Eq (6)) are included in Eq. (9) as constraints for nonlinear programming, Eq. (8). The NLMBPC optimiser computes the present and future manipulated variable changes so that the predicted output yˆ (k + j/k) follows the selected reference trajectory w(k + j/k) in a satisfactory manner. Although the optimisation problem solved at time k results in an optimal sequence only u(k/k), is ⌬uTk = [⌬u(k/k),. . .,⌬u(k + Nu ⫺ 1/k)], implemented for the real process over the time (k,k + 1): u(k/k) = u(k ⫺ 1) + ⌬u(k/k)

(10)

At time step (k + 1), the new measurement y(k + 1) is used together with u(k/k) to compute the new estimate yˆ (k + 1/k), the horizons Ny and Nu are shifted forward by one step and a new optimisation problem is solved at time step (k + 1) with new initial conditions yˆ (k + 1/k) to update the predictions and compute ⌬uk. To solve the constrained NLMBPC optimisation problem, we have used an ellipsoidal cutting plane algorithm [6]. At each sampling time, the control problem is solved iteratively computing the control input vector ⌬uk using the ellipsoidal cutting plane algorithm. In order to keep the dimensionality of the nonlinear problem low, we have used an orthogonal collocation to discretise the differential equations.

5.

For experimental requirements, some control interfaces have been added to link the process to a mini computer Pentium processor. Figure 2 shows the hardware used for the identification and the control of the process. The exchange of data between the process and the computer was made through 12bit A/D and D/A converters. To study the performance of the NLMBPC algorithm, different temperature profiles have been used in order to examine tracking of these set point profiles for the mixture temperature. A typical profile has been considered: 1. Preheating of the reaction mixture according to a predefined profile until the temperature reaction TR = 110°C is reached: duration 30 min. 2. Control of the mixture temperature at the constant value TR = 110°C during reactive feeding and maintenance of the mixture temperature TR constant after the end of reactant feeding, to complete the reaction: duration 1 h. 3. Cooling to ambient temperature, according to a predefined temperature profile, to avoid by-product formation: duration 30 min. It is clear that this type of operating mode causes abrupt and large changes in the process dynamics. This temperature profile is the most widely used in the fine chemical industry for the operation of highly exothermic reactions in a semi-batch jacketed reactor. For the real-time implementation requirements of the NLMBPC algorithm, we have used an extended Kalman filter (EKF) to estimate the concentration of products C1,C2,C3, and CT [11]. These estimates are used for the discretisation procedure and to predict the output trajectory over the prediction horizon into the future. The tuning parameters for NLMBPC are Ny = 6, Nu = 1, the input weight ␥ = 0.1. The chosen constraints are: 0 ⱕ u(k) ⱕ 3000 W (for all times) and 兩⌬u(k)兩 ⱕ 60 W also for all times. The predictive model controllers are tuned so that the manipulated variable u(k) and the control change ⌬u(k) satisfy the imposed constraints. Figures 3 and 4 show, respectively, the closed-loop response of the process and the input signal for NLMBPC (without disturbance). The solid line is the output and the dotted line is the reference signal. It is clear that the nonlinear model based system shows a closer correspondence to the reference signal. When a load disturbance signal (supply

Experimental Results

The objective of the system is to control the reactor temperature TR by operating the heating power as required. The main disturbance affecting the control performance comes from the temperature of the supply water for the cooling coil. Some experimental results are also given that show the qualities of control strategy proposed in Section 4.

461

Fig. 2. Hardware used for process control.

462

F. M’sahli et al.

that no temperature overshoot occurs at the end of the heating phase. This is a key performance of the technique, as it is well known that it is very difficult to avoid such an overshoot in a batch reactor. Experiments have shown the efficiency, the flexibility, and the good performance of the implemented NLMBPC algorithm in controlling the temperature of a reactor equipped with a mono-fluid system, and it also has a good temperature tracking performance.

Fig. 3. Closed-loop response for constrained NLMBPC.

Fig. 4. Heating power for constrained NLMBPC.

6. Conclusion In this paper we have presented the application of a nonlinear model based predictive control (NLMBPC) algorithm for the temperature control of a semi-batch reactor. The design of the NLMBPC strategy was based on a model obtained from input– output data as well as from first principles. This approach has been applied experimentally to a pilot reactor equipped with a mono-fluid heating–cooling system. The experimental results presented demonstrate a good performance of the developed control strategy. Good tracking performance of the implemented NLMBPC algorithm has been achieved. The experimental results show the efficiency and the flexibility of the developed methodology for controlling real processes.

References

Fig. 5. Closed-loop response for constrained NLMBPC (disturbance case).

Fig. 6. Heating power for constrained NLMBPC (disturbance case).

water temperature of the cooling coil) was added to the real process, the nonlinear controller shows a slight undershoot before returning quickly to the new set point. The results of this experiment are shown in Fig. 5 for the output and in Fig. 6 for the input signal. For all experiments, it can be seen

1. R. Berber, “Control of batch reactor: a review”, in (ed) Berber R. Methods of Model Based Predictive Control, pp. 459–494, Kluwer, 1995. 2. C. Kravaris, R. A. Wright and J. F. Carrier, “Non linear controllers for trajectory tracking in batch processes”, Computers and Chemical Engineering, 13, pp. 73–82, 1989. 3. A. Isidori, Non Linear Control Systems, 3rd edn, SpringerVerlag, 1996. 4. C. Kravaris and J. C. Kantor, “Geometric methods for non-linear process control – I Background; II Controllers synthesis”, Industrial Chemical Research, 29 (12), pp. 2295–2324, 1990. 5. B. W. Bequette, “Non linear control of chemical processes: a review”, Industrial Chemical Research, 30, pp. 1391–1413, 1991. 6. F. M’sahli, F. Bouani, R. Ben Abdennour and M. Ksouri, “Adaptive non linear model predictive control using autoregressive plus Volterra models”, 3rd European Robotics, Intelligent Systems and Control Conference (EURISCON ’98), Athens, Greece, 22–25 June 1998. 7. S. L. De Oliveira, V. Nevistic and M. Morari, “Control of nonlinear systems subject to input constraints”, Proceedings of the IFAC non-linear control design Symposium, Lake Tahoe, CA, vol. 1, pp. 15–20, 1995. 8. F. J. Doyle III, B. A. Ogunnaike and R. K. Pearson, “Nonlinear model-based control using second-order Volterra models”, Automatica, 31 (5), pp. 697–714, 1995. 9. T. C. Tsang and D. W. Clarke, “Generalized predictive control with input constraints”, Proceedings IEE, Control Theory and Applications, 135 (6), pp. 451–460, 1988. 10. E. F. Camacho, “Constrained generalized predictive control”, IEEE Transactions on Automatic Control, 38(2), pp. 327–332, 1993. 11. L. Ljung, “Asymptotic behavior of the extended Kalman filter as a parameter estimator for linear systems”, IEEE Transactions on Automatic Control, 24(1), pp. 36–51, 1979.

Predictive Control for Chemical Reactors

Nomenclature

Cep CLp Cei CLi Cvi CLt Cvt Fe FLV

feed calorific capacity (J g⫺1 K⫺1) calorific capacity of liquid phase (J g⫺1 K⫺1) concentration of feed component (i) (mole l⫺1) concentration of component (i) in liquid phase (mole l⫺1) concentration of component (i) in vapour phase (mole l⫺1) total concentration in liquid phase (mole l⫺1) total concentration in vapour phase (mole l⫺1) feed volumetric flow (l min⫺1) liquid equivalent vapour flow (l min⫺1)

Fv Mv Q r T Te V ⌬HR ⌬HV L V e

vapour volumetric flow (l min⫺1) molar mass of vapour phase (g mole⫺1) heat power (J min⫺1) reaction speed (mole l⫺1 min⫺1) mixture temperature (K) feed temperature (K) instantaneous volume mixture (l) enthalpy of reaction (J mole⫺1) enthalpy of vaporization (J mole⫺1) volumetric mass of liquid phase (g l⫺1) volumetric mass of vapour phase (g l⫺1) feed volumetric mass (g l⫺1)

463