Constrained Nonlinear Model Predictive Control of a Polymerization Process via Evolutionary Optimization ()
Moreover, due to recent advancements of computational hardware and software tools, the usage of MPC is rapidly expanding to other control domains including electrical machines, renewable energy, aerospace and automotive control systems.
In the past two decades, the effective control of polymerization processes control has been studied by many authors [1-7]. Polymerization kinetic is usually complex due to the nonlinearity of the process. Therefore, the control of the polymerization reactor has been staying a challenging task. Due to its great flexibility, a batch reactor is suitable to produce small amounts of special polymers and copolymers. The batch reactor is always dynamic by its nature. It is essential to have a suitable dynamic model of the process. Rafizadeh [1] presented a review on the proposed models and suggested an on-line estimation of some parameters. His model consists of the oil bath, electrical heaters, cooling water coil, and reactor. Peterson et al. [2] presented a non-linear predictive strategy for semi batch polymerization of MMA. Soroush and Kravaris [3] applied a Global Linearizing Control (GLC) method to controling the reactor temperature. Performance of the GLC for tracking an optimum temperature trajectory was found to be suitable. DeSouza et al. [4]
studied an expert neural network as an internal model in control of solution polymerization of vinyl estate. In their study, they compared their neural network control with a classic PID controller. Clarke-Pringle and MacGregor [5] studied the temperature control of a semi-batch industrial reactor. They suggested a coupled non-linear strategy and extended Kalman filter method. Mutha et al. [6] suggested a non-linear model based on control strategy, which includes a new estimator as well as Kalman filter. They conducted experiments in a small reactor for solution polymerization of MMA. Rho et al. [7] assumed a first order model plus dead time to pursue the control studies and estimated the parameters of this model by on line ARMAX model. Nonlinear predictive control of the batch reactor considered [8,9] via PCA and Wiener modeling approaches, respectively. When MPC is formulated as a state feedback controller, the full state information is required which must be provided using state estimators in nonlinear H2 (e.g. EKF) or nonlinear H∞ paradigms [10-12]. Rafizadeh [13] designed a sequential linearization adaptive controller for the solution polymerization of methyl methacrylate in a Batch Reactor.
This paper presents a constrained model predictive control of an MMA reactor, based on the genetic algorithm optimization. A previously developed mechanistic model of the process was used. The model is a sequential piecewise linearization along a selected temperature trajectory. The piecewise linear model is used both for the plant output calculation through the prediction horizon and for the closed loop simulation of the controller, using a time-triggered switching mechanism. We are using an output feedback MPC, therefore, no state estimator is required, which is advantageous. The results of tracking the trajectory and eliminating noise and disturbances show a promising performance of the controller.
2. Polymerization Mechanism
Methyl methacrylate normally is produced by a free radical, chain addition polymerization. Free radical polymerization consists of three main reactions: initiation, propagation and termination. Free radicals are formed by the decomposition of initiators. Once formed, these radicals propagate by reacting with surrounding monomers to produce long polymer chains; the active site being shifted to the end of the chain when a new monomer is added. During the propagation, millions of monomers are added to 
radicals. During termination, due to reactions among free radicals, the concentration of radicals decreases. Termination is by combination or disproportionate reactions. With chain transfer reactions to monomer, initiator, solvent, or even polymer, the active free radicals are converted to dead polymer. Table 1 gives the basic free radical polymerization mechanism [14].
The free radical polymerization rate decreases due to reduction of monomer and initiator concentration. However, due to viscosity increase beyond a certain conversion there is a sudden increase in the polymerization rate. This effect is called Trommsdorff, gel, or auto-acceleration effect. For bulk polymerization of Methyl Methacrylate beyond the 20% conversion, reaction rate and molecular weight suddenly increase. In high conversion, because of viscosity increase there is a reduction in termination reaction rate.
3. Mathematical Modeling of Polymerization
The polymer production is accomplished by a reduction in volume of the mixture. The volumetric reduction factor is given by:
(1)
The instantaneous volume of mixture is given by:
(2)
The parameter 
is defined as:
(3)
During the free radical polymerization, the cage, glass, and gel effects occur. For the cage effect, the initiator efficiency factor is used. The CCS (Chiu, Carrat and Soong) model is used in this study to take into consideration the glass and the gel effects. Therefore, propagation rate constant, 
, is changing according to:
(4)
is changing as Arrhenius function, and 
is given by equation:
(5)
Similarly, termination rate constant, 
, is given by
(6)
is changing as Arrhenius function. 
and 
are adjustable parameters related to propagation and termination rate constants, respectively. All other necessary parameters and constants for this model are given in [1, 14,15].
Long Chain Approximation (LCA) and Quasi Steady State Approximation (QSSA) are used in this study. Equations are highly nonlinear and, using Taylor expansion series, these equations were converted to linearized form. The linearized state space form is given by:
(7)
The molecular properties of the produced polymer are controlled by ensuring the reaction temperature is changing according to a desired reference trajectory. This is a tracking control problem which we are solving using MPC.
Figure 1 shows the result of model validation [14]. As it is seen, the model is a good representative of the process. Equation (7) is converted to the transfer function for reaction temperature to input power:
(8)
The result of sequential linearization is 131 transfer functions along the temperature profile. See [13,14] for a more detailed description of the MMA polymerization dynamic modeling.
4. Experimental Setup
A schematic representation of the experimental batch reactor setup is shown in Figure 2 [14]. The reactor is a Buchi type jacketed, cylindrical glass vessel. A multipaddle agitator mixes the content. Two Resistance Temperature Detectors (RTDs) of PT100 type were used with accuracy of ±0.2˚C to measure the reactor temperature and the oil temperature in the oil bath. Methyl Methacrylate and Toluene were used as monomer and solvent, respectively. Benzoyl Peroxide (BPO) was used as the initiator. The heater, heats the oil circulating the oil bath, which is pumped into the reactor. Cool water is circulated in a coolant coil inside the oil bath through an electric on/off valve and acts as a safety feature to prevent the oil and consequently the reactor from overheating. The RTD outputs are converted into 0 - 10 VDC through a bridge and an instrument amplifier and are read by the data acquisition card A/D channels. The controller output is fed into a MOSFET-based power electronics switching circuit as PWM signals. The maximum heater power available is 1000 W, which is a constraint on the control signal.
5. Model Predictive Control
Due to its high performance, model predictive control method has received a great deal of attention to control chemical processes, in the last few years. Figure 3 shows block diagram of a model predictive controller.
There are three main approaches to model predictive control, MAC (Model Algorithmic Control), which is based on system’s impulse response, DMC (Dynamic Matrix Control), which uses the process step response samples, and GPC (Generalized Predictive Control), which is based on the process transfer function. In practice, it is easier to obtain step response samples rather than impulse response or a full transfer function, and therefore the DMC method is more popular. We use the DMC method in this research. The cost function is defined as:
(9)
where P, M and N1 are prediction horizon, control horizon and pure time delay, respectively. Matrices 
and 
are weighting matrices used in the weighted 2-norms.
Figure 2. Schematics of the experimental setup.