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To precisely track the reactor temperature in the entire working condition, the constrained Hammerstein-Wiener model describing nonlinear chemical processes such as in the continuous stirred tank reactor (CSTR) is proposed. A predictive control algorithm based on the Kalman filter for constrained Hammerstein-Wiener systems is designed. An output feedback control law regarding the linear subsystem is derived by state observation. The size of reaction heat produced and its influence on the output are evaluated by the Kalman filter. The observation and evaluation results are calculated by the multistep predictive approach. Actual control variables are computed while considering the constraints of the optimal control problem in a finite horizon through the receding horizon. The simulation example of the CSTR tester shows the effectiveness and feasibility of the proposed algorithm.

Temperature control of the batch reactor during chemical processes [

The Hammerstein-Wiener model can describe the processes in the continuous stirred tank reactor (CSTR), pH neutralization reaction process, and other nonlinear chemical processes. Many research results were achieved in the nonlinear model predictive control (NMPC) [

Considering the Hammerstein-Wiener system of state, output, and intermediate mass constraints, this paper proposes the predictive control algorithm of the batch reaction based on the Kalman filter for the constrained Hammerstein-Wiener system by combining the literature [

The batch reactor process is generally operated under a closed reactor condition so that the reactants are added to the reactor once to react in the given condition. The intensifying reaction will cause the heat to accumulate and the temperature to rise and further intensify. Obviously, this process, as well as the interlayer temperature, is unsteady. As the control process is complex, the batch reactor control system is generally required to design the cascade structure, in which the main circuit of the outer ring takes the interlayer temperature as the control input and the reaction temperature as the control output, using

CSTR.

The Hammerstein-Wiener model consists of two static, nonlinear modules and one dynamic linear module. Its model structure is shown in Figure

Hammerstein-Wiener model structure.

Consider the following discrete-time multivariable Hammerstein-Wiener absolute system:

It is assumed that the system controllability, state observability, and the origin as the equilibrium point of the system, the parameters of the system states, intermediate variables, and the input constraints are defined as follows:

The Hammerstein-Wiener model can be divided into the Hammerstein model [

The considered output signal

The actual control variable

Considering the system in formula (

Thus, the NMPC issue of the constrained Hammerstein system is defined as follows:

The sequential quadratic programming is applied to calculate online the optimization problem in formula (

According to the principle of rolling optimization, the NMPC law based on the observation state is defined as follows:

In formula (

Control law (

The observation error of the defined state is

The NMPC closed-loop structure of the whole system is shown in Figure

NMPC closed-loop system structure of the constrained hammerstein system.

This section mainly introduces the interference model [

The input interference model is the key to achieving unbiased control and to improving the tracking performance. To improve the tracking performance and to realize the unbiased control, the system is characterized by introducing the interference model. The model is built in the input interference along with the influence of heat produced by the materials in the chemical reactor on the output. Davison and Smith [

After the establishment of the input interference model, the input interference

According to the measured value of the current output, the system state

Then, the predictive value of the augmented state at the next moment is

For the sake of discussion, the following are set:

The predictive control algorithm flowchart of the constrained Hammerstein-Wiener model with the feedforward compensation is shown in Figure

The predictive control algorithm of the constrained Hammerstein-Wiener model with the feedforward compensation.

The experimental platform is the CSTR tester that can simulate in real time the entire process of the batch reaction. The main parameters selected include the volume of the reactants: 200 L, the spin speed of the stirrer: 90

The process temperature curve rising at 0.2°C/s and remaining constant at 120°C.

The predictive control algorithm control result diagram referred to in the paper

The traditional PID control algorithm control result diagram

Compared with the PID control, the PID control method cannot make the reactor temperature follow the process curve well because of the inherent nonlinearity and time lag characteristics of the object. However, the method in this paper achieves a good control effect.

In practical production, different products are produced through different processes, requiring corresponding control tasks to be changed, that is, selecting different process curves while keeping the algorithm unchanged. In the same simulation platform, the process curve is set to rise at 0.3°C/s and be constant at 120°C, with other parameters unchanged. The control result is shown in Figure

The process temperature curve rising at 0.3°C/s and remaining constant at 120°C.

The predictive control algorithm control result diagram referred to in the paper

The traditional PID control algorithm control result diagram

For batch production, the different natures of the reactants result in different exothermic characteristics in the entire reaction process, requiring the adopted control algorithm to have good robustness. In other words, the original control result remains even when the material property changes. Now, the reactant

Comparison of the result after changing the special heat of the material.

The predictive control algorithm control result diagram referred to in the paper after changing the special heat of the material

The traditional PID control algorithm control result diagram after changing the special heat of the material

The traditional PID control algorithm is difficult to use in controlling the reaction temperature in a satisfactory range because of the specific heat change of the reactant. However, the method proposed in this paper shows good robustness, indicating practical significance to the complex and changeable strong, nonlinear, chemical process system.

With the temperature tracking issue in the batch reaction process as the study object, as well as with the condition that the value of the heat production cannot be measured and its influence law on the output is unknown in the whole process of the reactant reaction, this paper proposes the predictive control algorithm of the batch reaction based on the Kalman filter for the constrained Hammerstein-Wiener system. Combined with state observer, the paper designs the output feedback optimal control law of the linear subsystem and characterizes the unmodeled dynamic system with the output interference model, which compensates for the impact of the heat production from the reaction on the system output and improves the accuracy of temperature tracking. The simulation experiment is conducted to verify the control results of this algorithm on different process curves and determine its robustness. The proposed algorithm, when compared with the traditional PID algorithm, fully demonstrates effectiveness and feasibility.

This work is partially supported by the National Natural Science Foundation of China (no. 61074020). The authors also wish to thank the associate editor and the anonymous reviewers for their valuable and constructive criticisms and suggestions.