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Brazilian Journal of Chemical Engineering

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

Braz. J. Chem. Eng. vol.17 n.4-7 São Paulo Dec. 2000 



M.P.Vega, E.L.Lima and J.C.Pinto
Programa de Engenharia Química / COPPE, Universidade Federal do Rio de Janeiro,
Cidade Universitária, CP: 68502, Rio de Janeiro - RJ, Brasil,


(Received: October 19, 1999 ; Accepted: April 06, 2000)



Abstract - or multivariable non linear predictive control implementations, a hybrid-neural model (lumped model) was successfully used for modeling a loop-tubular polymerization reactor (a lumped or distributed model, depending on recycle ratio). Bifurcation diagrams were computed in order to investigate the agreement between process and model, of paramount importance for model based controller implementation purposes. Performance was evaluated considering the nonlinear model predictive control of both a loop tubular reactor (lumped) SISO problem and a tubular reactor (distributed) MIMO problem.
Keywords: hybrid-neural model; loop-tubular reactor; polymerization; nonlinear model predictive control.




Although many polymerization reactions are conducted in batch reactors, the use of continuous systems is rapidly growing because they increase polymer productivity, as start-up, shut-down and cleaning procedures are less frequent. The continuous stirred tank reactor (CSTR) is the most popular configuration but it cannot be used for high conversions, due to viscosity problems that reduce the heat transfer capacity. Conversely, tubular reactors have larger heat transfer capacity as they present higher surface/volume ratios and, as a consequence, monomer conversions may reach higher values. Also, due to the simplicity of the tubular design, this configuration has small fixed and operational costs (Fleury et al., 1992). One of the main disadvantages of these reactors is related with high residence time distribution, associated with high viscosity of the reaction mixture, that difficult the control of the product quality.

Loop reactors (tubular reactors with recycle) represent a promising alternative for polymer reactions, as they offer the advantages of the tubular technology, avoiding the large residence time distribution problem and, also, presenting faster dynamic responses. Subject to the recycle magnitude these reactors can behave as traditional tubular reactors (non-recycle) or pure CSTRs (normally for recycle ratios over 35). There are only a few works related to modeling the loop-tubular polymerization reactors and almost none on control studies (Cabral, l998). Most papers in the literature are devoted to modeling the steady-state behavior of such reactors (Biesenberger et al., 1977; Lynn et al., 1971; Sala et al., 1979; Zhirkov et al., 1980; Mallikarjan, 1985; Mc Laughlin et al., 1986 and Soliman et al., 1994).

Cabral (1998) has shown that the recycle ratio effect on product quality is almost negligible, but has a significant influence on the reactor dynamics and, consequently, can be potentially used for designing useful production strategies. The possibility of varying the recycle ratio through a wide spectrum of values clearly determines an optimization problem that, depending on the desired objectives, could result in high quality operational alternatives. This aspect is not explored in this work, since the main objective is to extend some results already obtained for tubular reactors (zero recycle ratio) for the case of loop configurations. The nonlinear nature of polymerization reactions is a motivation for studying the application of nonlinear control algorithms in these systems. In this work nonlinear model based algorithms were implemented, using simple internal models. The quality of these models plays an important role on the closed-loop performance and the focus of this study is concentrated on this subject.

The paper describes the construction of simple hybrid neural models (HNMs) for the solution styrene polymerization in a loop tubular reactor. By combining simple phenomenological models with neural networks (NNs), this type of model may gather the best characteristics of both phenomenological and empirical approaches. This methodology uses a simple and flexible model (based on well known balance equations) coupled with a NN that models both nonlinear characteristics and uncertainties of the system being studied. In the polymerization reactor engineering area, many complex and often unknown phenomena have to be modeled, such as reaction rates, kinetic constants, fluid and flow behavior, physical properties, etc. As a consequence, these processes are good candidates for hybrid neural modeling, especially for on line advanced model based control design strategies.

The investigation was conducted simulating the simple models previously described by Vega et al. (1997) and Cabral (1998), which were aforesaid thought out the paper as plants. To guarantee the suitability of the HNMs, a validation procedure based on dynamic and static behavior analysis (Vega et al., l998) was used. The resulting hybrid neural models were evaluated as internal models of nonlinear predictive control strategies. Predictive control refers to a class of optimal control algorithms in which a dynamic process model is used to predict the future behavior. Nonlinear predictive control technology is indicated for process with strong nonlinear characteristics or with weak nonlinear characteristics used in a large range of operating regimes. An important aspect of process control problems is the presence of constraints on input and output variables. Predictive controllers have explicit constraint handling capability. The use of a nonlinear predictive algorithm to control tubular polymerization reactors may be justified if one realizes that this system introduces dead time to the output signals and that the dynamics is extremely nonlinear, due to the polymerization reaction.



The process studied is the production of polystyrene in toluene, initiated by benzoyl peroxyde, conducted in a loop tubular reactor, subject to extreme recycle regimes. The loop reactor operating with a recycle ratio of 35 can be considered as a CSTR. In the other hand, using a zero recycle ratio the system acts as a tubular reactor.

The loop-tubular reactor model is described by the following system of equations,




where Ci represents each balanced element (initiator concentration, monomer concentration, solvent concentration, modifier concentration and the three dead polymer moments), vz is the axial velocity, Ri are reactions rates and a is the recycle ratio.

A discrete version of the model was obtained using finite difference approximations in both, radial and axial directions, and the resulting dynamic system was integrated along the time axis, using a 4th order Runge-Kutta method. Average profiles for the state variables were calculated by numerical quadrature, using quadrature points that are the roots of an orthogonal polynomial generated by a weighting function, which allows the minimization of the integration error (Vega, 1997).

This model is based on the following assumptions, which were validated by carrying out experiments in a loop-tubular reactor unit (Vega et al., 1997 and Cabral, 1998): the flow inside reactor is laminar; physical properties are constant; velocity profiles are parabolic; axial and radial mass diffusion are negligible; reactor temperatures are equal to the jacket temperatures; the reaction mechanism follows the classical vinyl polymerization kinetics, including gel effect. Oliveira Jr. (1995) developed a complex model for this system based on the assumptions that physical properties are functions of local temperature, polymer concentration and average polymer molecular weight, allowing the computation of velocity profiles, which were shown to exhibit large spatial deformities. In spite of the complexity introduced, the results obtained showed similar behavior for both models.



The capacity of HNMs in describing nonlinear systems for model based controllers design purposes is analyzed. The nonlinear model predictive control algorithm that will be used to evaluate that capacity is now introduced. In this class of controllers a nonlinear dynamic process model is used to predict and optimize process performance. A sequence of control moves is computed to minimize an objective function, which includes predicted future values of the controlled outputs. The predictions are obtained from a nonlinear process model, a HNM in this case. The optimization problem is solved subject to constraints on input and output variables, as well as constraints imposed by the nonlinear model equations. Following the scheme indicated in Figure 1 the control law can be described as




subject to:

where y is the vector of controlled variables (monomer conversion for the SISO case; monomer conversion and weight average molecular weight for the MIMO case), yset is the vector of set point values; y' is the vector of predicted values (using the HNM); u is the vector of manipulated variables (jacket reactor temperature for the SISO case; jacket reactor temperature and feed modifier concentration for the MIMO case); umax and umin are lower and upper bounds for u; Dumax and Dumin are lower and upper bounds for Du; P is the prediction horizon; L is the control horizon; and G and L are weighing parameters matrices.

The optimization problem was solved using a Successive Quadratic Programming (SQP) algorithm. During the optimization procedure it was assumed that perturbations and modeling errors were constant throughout the prediction horizon and equal to the difference between the value of the controlled variable as predicted by the HNM and the actual measured value,




The main idea behind a hybrid approach is to combine phenomenological and empirical information in such a balance proportion for obtaining an efficient model. The balance between the two information sources may depend on the availability of that information or on some economic objective. In this work hybrid models of the loop reactor are obtained combining mass balances around the reactor volume with NNs, describing the reactions rates and also the modeling errors. These errors increase by decreasing the recycle ratio because of the growing distributed characteristic of the loop reactor, which finally behaves as a tubular reactor.

The loop reactor plant, represented by Eq. (1) is reduced to the following HNM representation (lumped system):


The quality of the resulting models strongly depends on the quality of the involved NNs. It is well known that the construction of an efficient NN is a function of many factors. The amount and appropriateness of the available training data is an important factor. In addition, the optimal NN structure is not easy to pre-specify, the optimization of the NN weights can lead to local minimum, different learning algorithms can result in contrasting generalization characteristics and alternative convergence criteria for training can also result in different solutions.

All these factors transform the identification task in almost an art. The last step in the iterative identification procedure is validation against an unknown data. In this work the HNMs were validated in terms of traditional methods (Pollard et al., l992), in terms of their complex static and dynamic behaviors, as proposed by Vega et al. (l998), and in terms of the resulting controllers performance. In the work of Vega et al. (l998), it was observed that the use of traditional validation tests was not enough to guarantee successful HNMs for control purposes. It was observed that the complex dynamic behavior displayed by the hybrid model might be completely different from the one displayed by the plant, resulting in poor control efficiency. Good controller performance was obtained when model and plant showed similar bifurcation diagrams.

The study of a loop reactor, a SISO problem, where temperature and conversion were the input and output variables, respectively, was performed. This SISO problem has been already studied for the traditional tubular reactor (Vega et al., l997). For this last distributed system, this work investigates a MIMO problem, where the input variables are temperature and feed modifier concentration, and the output variables are conversion and weight average molecular weight.

Loop Reactor

For this lumped system the HNM possess a NN with one hidden layer architecture. Hyperbolic tangent activation functions were employed in the hidden layer and a linear activation function in the output layer. This NN gives one step ahead prediction of the monomer consumption rate, based on actual conversion (x) and temperature (T) data:


The unknown NN outputs (Eq.7) were calculated from plant data using a discrete version of Eq. (5)


Plant data were generated from couple step-random jacket temperature perturbations (Figure 2). This kind of disturbance, that provides both steady state and dynamic information, allows the implementation of increasing thermal levels in order to acquire conversion data comprising the range between 0-80%. The common approach of scaling all data used for the NN input-output information is employed, in order to avoid large weights, non uniform learning and local minimum.



The influence of the data number on the quality of the HNM generalization capacity was analyzed in detail, as different NN parameter initializations were performed. In addition, it was verified that with 200 conversion-temperature data points the HNM successfully represented the loop reactor. The best NN architecture was found to be 2-2-1 (2 input neurons, 2 hidden neurons and 1 output neuron).

The dynamic behavior of the reactor, as computed from dynamic simulation and confirmed through detailed stability analysis of the process model, is stable and the dynamic is trivial’s S-shaped behavior. Bifurcation diagrams were obtained using AUTO (Doedel, l986) and the stability analysis of the different attractors were performed as described in Vega et al. (1998).

It can be seen from Figure 3 that the HNM approach was successfully used to model the loop reactor (lumped system) as both present the same S-shaped behavior.



Tubular Reactor

Due to the distributed nature of the system, it was necessary to use actual and five past time history data as input signals for the NN. This architecture guarantees the one to one (injective) relation of the training data (Vega et al., l998). Three different NNs were identified. The first one, predicting the one-step ahead non linear monomer consumption rate (output) has as input variables the present and five past conversion and jacket temperature data. The second one, predicting the non linear first order momentum of dead polymer formation rate, has as input data the present and five past information of the first order momentum of dead polymer and the jacket temperature. Finally the third one, predicting the non linear second order momentum of dead polymer formation rate , employed present and five past data of the second order momentum of dead polymer, the jacket temperature and the feed modifier concentration. As in the case of the SISO model (Eq.5), the unknown output data were obtained from discrete versions of the corresponding lumped equations.

In order to generate an uniform distribution over the region of interest (0-80% conversion and 5000-25000 weight average molecular weight), coupled step-random perturbations of jacket temperature and feed modifier concentration were used to generate the training data. Results for conversion, first order momentum of dead polymer and second order momentum of dead polymer are shown in Figure 4.



These perturbations were produced in order to generate the appropriate number of data points for efficient NN training: 300, 200 and 200 patterns of conversion, first order and second order momentum of dead polymer, respectively.

Figure 5 shows the bifurcation diagram of a HNM predicting successfully the monomer conversion. However, Figure 6 shows bifurcation diagrams of a HNM using a NN trained with different initial weight values, but using the same input-output data as the HNM shown in Figure 5. The differences in relation to Figure 5 are clear examples that an exhaustive search is required, during the training stage, to guarantee that the global minimum has been reached.




For the other two HNMs, Figures 7 and 8 show bifurcation diagrams, for the first order and second order momentum of dead polymer, with dynamic behavior similar to the plant's model.




These validation results shows that care must be taken to assure that the model satisfactorily represent the plant over a wide set of operating points. A model based controller using the HNMs of Figures 5, 7 and 8 will certainty present nice robust characteristics.



The final validation tests for the developed HNMs were as internal models in nonlinear model predictive control algorithms, as presented in Section 3.

SISO Control Problem

The control tests performed for the loop reactor (lumped system) consist of a sequence of set point changes, combined with perturbation rejection, for the plant initially operating at 330 K and with a monomer conversion of 0.082. The sequence is: (t = 0, xset = 0.20); (t = 40, xset = 0.40); (t = 80, xset = 0.80); (t = 120, ); (t = 160, xset = 0.80, ).

Figure 9 (controlled variable and manipulated variable) shows the successful predictive control implementation for those control tests. It must be observed that the closed-loop behavior was satisfactory over a wide spectrum of conversion set points and disturbance rejections.



MIMO Control Problem

As monomer conversion is not influenced by feed modifier concentration, the natural variable pairing for this process was: (temperature-monomer conversion) and (feed modifier concentration-weight average molecular weight).

The control test for the system initially operating at a temperature of 340 K, were: (t = 0, x = 0.40, Mw = 20000); (t = 60, xset = 0.60, Mw = 15000); (t = 100, ); (t = 160, xset = 0.70, ).

Figure10 (controlled variables and manipulated variables) shows the performance of the controller. The –70% disturbance applied in the initiator feed concentration produces two effects: a decrease in the conversion and an increase in the weight average molecular weight. The +70% step in the initiator feed concentration, coupled with a set point change from 60 to 70%, makes the conversion to become higher and the weight average molecular weight to decrease.



These results are, again, a clear indication that using nonlinear models, the predictive controllers performs satisfactorily over a wide operating region. This performance represents a final validation of the developed HNMs. Any special effort to arrive at the best controller adjustments was done, as it was not the objective of this work.



In the present paper, simple non linear models for control purposes were analyzed. An advantage of the HNMs approach (operational simplicity) was explored. However, it was shown that the identification task is almost an art. It is required an iterative work of training and validation, taking into account the range and number of data points, the optimization algorithms, weights and biases initialization, NN architectures, bifurcation diagrams, and controller implementation performance. The last two validation approaches were analyzed in this study. The process studied is the production of polystyrene in toluene, initiated by benzoyl peroxyde in a loop-tubular reactor (a lumped or distributed model, depending on recycle ratio). The HNM for the loop reactor (lumped system) was of simple construction. This same feature is expected for the MIMO lumped reactor system. Besides, it was shown that the HNM construction (lumped model) for the MIMO tubular reactor (distributed model) was more complex. This result confirmed the one obtained for the SISO tubular reactor (Vega et al., 1998). Bifurcation diagrams were computed in order to investigate the agreement between process and HNM, of paramount importance for implementing model based controllers.



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