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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 



A.Bolsoni, E.L.Lima* and J.C.Pinto
Programa de Engenharia Química / COPPE /, Universidade Federal do Rio de Janeiro,
Cidade Universitária, CP: 68502, CEP 21945-970, Rio de Janeiro - RJ, Brazil
Phone: +(55)(21) 590-2241, Fax : +(55)(21) 290-6626


(Received:October 10, 1999 ; Accepted: April 6, 2000)



Abstract - The objective of this work is to develop a strategy of nonlinear model predictive control for industrial slurry reactors of propylene polymerizations. The controlled variables are the melt index (polymer quality) and the amount of unreacted monomer (productivity). The model used in the controller presents a linear dynamics and a nonlinear static gain given by a neuronal network MLP (multilayer perceptron). The simulated performance of the controller was evaluated for a typical propylene polymerization process. It is shown that the performance of the proposed control strategy is much better than the one obtained with the use of linear predictive controllers for setpoint tracking control problems.
Keywords: Predictive Control, Polymerization Reactors, Neural Networks, Melt Index.




Model based predictive control strategies have been used successfully in many industrial systems since the late seventies (Qin & Badgwell, l997), and different factors can be identified as responsible of this success. As an explicit model is involved in the design, the controller is able to directly cope with the most important static and dynamic characteristics of the process. Also, as the predictive control algorithm is based on the plant behavior along a certain future horizon, it can anticipate and eliminate certain process disturbances. Additionally, it is possible to include the main process physical constraints in the controller algorithm and, again, anticipate and avoid constraint violation.

Although most industrial processes present nonlinear behavior, many predictive control implementations are based on linear dynamic models. Many reasons have favored the use of linear model predictive controllers (LMPCs), including the fact that linear models are easily identified from data gathered during simple plant tests. Also, the main objective of many control problems is to keep the controlled variables as close as possible to steady state operation points (regulator problem) instead of moving from one operation point to a second one (servo problem). If the process nonlinear characteristics are not very significant, a carefully identified linear model can adequately represent its behavior near a steady state operation point. Finally, it must be mentioned that using linear models considerably simplify the implementation of the control algorithm.

Nevertheless, there are many important industrial problems where linear models cannot be used to provide an efficient representation of the process. Henson (1998) and Qin & Badgwell (1998) have reported different important processes where nonlinear model predictive controllers (NMPCs) clearly present a much better performance than their linear counterparts. Basically these algorithms find their application field in the control of strongly nonlinear processes near steady state operation points (ex. high purity distillation columns, neutralization systems) and processes with moderate nonlinear characteristics, working through a large spectrum of operating conditions (ex. multi-grade polymerization reactors, ammonia synthesis)

The performance of the predictive control algorithms is strongly related to the model capacity in representing the real process. Nonlinear phenomenological models based on first principles are generally complex and most of the time requires heavy computational work for their solution. These disadvantages explain why nonlinear empirical models identified from input/output process data are so frequently used. Artificial neural networks represent a very attractive class of empirical models and are increasingly used, not only for system identification but also for process control (Donat et. al., 1991; Nahas et. al., 1992; Van Can et. al., 1995; Tendulkar et. al., 1998).

The identification of nonlinear models requires plant tests that not always can be performed in practice. Meanwhile many processes present nonlinear static combined with almost linear dynamic behavior. As a consequence of this observation, different proposals have been published to integrate linear dynamic with nonlinear static behavior (Al-Duwaish e Karim, 1997; Qin e Badgwell, 1998; Su e McAvoy, 1993). This particular approach makes possible to use historical data to identify the nonlinear static model part (ex. a neural network), while data from simple dynamic tests can be used to identify the linear dynamic model part.

This work describes a nonlinear model predictive control strategy for propylene polymerizations in slurry reactors. The model used as reference comprises a linear dynamic model with the gain given by a static neural network, trained from steady state input/output process data. The simulated performance of the proposed control scheme is evaluated for setpoint tracking problems in a typical industrial polypropylene polymerization process and shown to be better than the performance obtained when linear predictive controllers are used.



Dynamic Model

The first decision to be taken during the design phase of a predictive controller is related to the dynamic model that will be used. Most used models are empirical, derived from plant tests and linear identification techniques. Meanwhile, as already mention in the introduction, for NMPC the process identification from plant tests is much more difficult and some useful alternatives have been proposed.

To design the predictive controller in this work, a linear dynamic model was used, with a variable gain given by a static neural network (Pearson, l995). The original MIMO (multi-input-multi-output) system was modeled as a combination of MISO (multi-input-single-output) systems. It was assumed that the process input and output variables can be represented by a combination of a static parcel, following a nonlinear relation, and a deviation, following a linear dynamic model. This way, given a vector of input variables Uk = [u1(k), u2(k),....,unu(k)] and the output variable y(k), the deviation variables are defined as,




where usi and ys are the steady state input and output variables values. These variables follow a nonlinear relationship given by a static MLP neural network (multilayer perceptron):


The deviation variables follow an ARX (autoregressive with exogenous input) linear dynamic relation of the following form,


where A and B are parameters of the linear model, nu is the number of input variables, na is the model order related to the output y, nbi is the model order related to the input ui and nki is the dead time related to the input ui.

In practice, the linear dynamic model identification can be performed based on data obtained from tests applied in a small operating region of the plant. The static neural network can be trained from historical process data. In this case it is assumed that the available historical data completely described the nonlinear process behavior in any operating region. Given a vector of input variables Uk, the neural network gain related to the input variable ui is represented by


The dynamic model gain related to the input variable ui is given by


This gain is fixed and can be very different from the actual process gain, depending on the operating region and the process nonlinear characteristics. Modifying the ARX model parameters in such a way that the linear model gain becomes identical to the neural network gain reduces this error. So, the linear dynamic model with the gain given by the nonlinear static model takes the following form






Because the neural network gain depends on the operating region, the ARX model parameters must be adjusted at any sample time in such a way that the dynamic model gain equals the local neural network gain. When the input vector changes from Uk-1 to Uk, the neural network gain that will be incorporated to the linear model is calculated in terms of the average value Umk.

NMPC Algorithm

The adopted control strategy involves the implementation of an NLMPC algorithm that solves the following optimization problem


subject to














Matrices Q , F and G are weighting parameters, P is the prediction horizon and M is the control horizon. The controlled variable set point and predicted values at a future k time are represented by and , respectively. The difference between the controlled variable measured, y(t), and predicted, , values at the present time is included in the objective function to eliminate offset. The predicted values for the controlled variable are calculated using the nonlinear model described in the preceding section.



Process Description

The predictive control algorithm was used for the simulation of an industrial propylene polymerization process, in a slurry reactor, using a Ziegler-Natta catalyst. The process scheme is presented in Figure 1. The polymerization process is carried out in five continuous stirred tank reactors connected in series. The first reactor is fed with a diluted phase stream (solvent + catalyst + co-catalyst) and an inert stream (nitrogen). Monomer is fed into the first three reactors. The chain transfer agent (hydrogen) enters the reactors with the monomer stream. Reactor temperature is kept at the operating point through manipulation of the cooling fluid flow rate.



In this simulated study a rigorous phenomenological model formed by a set of differential-algebraic equations represented the plant. This model is a modification of the model presented by FREITAS (l998), adapted to the polypropylene production process. This model involves mass and energy balances, thermodynamic equilibrium relations and polymer moments balances.

Implemented Control Strategy

For the process described above, the controlled variables are the melting flow index (MI) and the amount of unreacted monomer, known as off-gas (OG). The manipulated variables are hydrogen volumetric fraction in the monomer stream (F) and the catalyst concentration of the stream entering the diluted phase vessel (C). The first reactor temperature (T1) and the pressures of the first, second and third reactors (P1, P2 and P3) exert a significant effect upon the MI and where considered as measured disturbances. Table 1 presents the sample times and the time delays of the measurements associated with the input and output variables. The adopted sampling time was 30 min.



The same model used to represent the plant, but with a different propagation kinetic constant simulating a plant model mismatch, was used to estimate the MI values at the sampling times where there were not measurement values available. Two simpler models were considered for the predictive control algorithm. One MISO model predicts MI values from the knowledge of F, C, T1, P1, P2 and P3 values. Another SISO model predicts OG from the knowledge of the catalyst concentration in the diluted feed vessel input stream. These two models are dynamic with static gains given by neural networks, as described in Section 2.1. It should be noted that the control algorithm presented in Section 2.2 includes one term representing the difference between the actual and the predicted controlled variable. In the case of MI, that is measured less frequently, this term was considered to be the difference between the last measured available value and the model-predicted value at the time of that measurement.



Model Identification

The parameters of a linear dynamic model (ARX) were initially identified from process input and output data gathered in a small operating region. Afterwards, a static neural network was trained from steady state data covering the entire process-operating region. As described in Section 2, the internal model used by the controller is a combination of the dynamic characteristics of a linear model with the static characteristics of a nonlinear one. The static gain of the linear model is adjusted at each sampling time in such a way that it equals the corresponding nonlinear model gain.

Figures 2 and 3 compare the proposed and the constant gain dynamic model performance, in terms of MI and OG variables, for two different operating regions. Region 1 corresponds to the region where the data used for identification were gathered and region 2 corresponds to a completely different operating region. Results indicate that for the region used for the process identification both models show similar behavior. However, for the other operating region, the proposed nonlinear model is significantly better. The squared error sum for both models and both operating regions can be found in Table 2.





Controller Performance

The proposed predictive control algorithm was evaluated for set point changes in MI, while keeping OG in an appropriate operating zone. To this purpose, the term that represents the difference between OG set point and predicted value in Equation (10) is activated only when its value exceed the zone limits. Simulations were performed under typical operating conditions for the studied polymerization process, considering the presence of measured disturbances and noise on the input and output variables. The results obtained with the predictive control algorithm based on a linear dynamic model with local nonlinear gain (NMPC) were compared with the results obtained with a predictive controller based on the linear dynamic model with constant gain (LMPC). These results were obtained in two different operating regions, one where the linear identification was performed (Figures 4 and 5) and the other, a totally different one (Figures 6 and 7).






As can be seen in Figure 4, for the region where the linear model was identified both algorithms present similar behavior. When the algorithms operate in a different region, Figure 6a indicates a neat superiority of the nonlinear controller. Two important aspects should be mentioned in relation to these results: the performance of the proposed algorithm does not involve excessive control actions, that remain all the time far from the constraints (0.05%<F<0.30%; 0.53kg/m3<C<0.60kg/m3); in both cases, the OG remains within it zone of constraints.



It is a common practice in the polymerization industry to use the same reactor for the production of different polymer grades. This practice imposes heavy performance demands on the control system, as it must be robust enough to cope with the characteristics of different operating regimes. The polypropylene industry is an example of multi-grade production practice. In this work, some control aspects of the propylene polymerization process in slurry reactors were studied. A nonlinear model predictive controller was proposed, where the internal model is a gain varying linear dynamic model. It is based on the combination of an ARX model and a static neural network in a block oriented structure. The main advantage of this model is the simplicity of the identification step, which significantly reduces the time required for identification and implementation. The proposed nonlinear control strategy compared favorably with a similar linear one when they were implemented to control the reactor in a wide range of operating points (grades).



The authors would like to thank CAPES (Fundação Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), CNPq (Conselho Nacional de Pesquisa) and Polibrasil Resinas S.A. for their financial and technical support.



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