Due to the strong nonlinearity and transition dynamics between different operating points of the high purity distillation column process, it is difficult to use a single model for modeling such a process. Therefore, the multiple model based approach is introduced for modeling the high purity distillation column plant under the framework of the expectation maximization (EM) algorithm. In this paper, autoregressive exogenous (ARX) models are adopted to construct the local models of this chemical process at different operating points, and the EM algorithm is used for identification of local models as well as the probability that each local model takes effect. The global model is obtained by aggregating the local models using an exponential weighting function. Finally, the simulation performed on the high purity distillation column demonstrates the effectiveness of the proposed method.
The increasing complexity of the process control systems poses a great challenge for process modeling and parameter identification and control. In practice, most of the industrial processes show a strong nonlinearity and a dynamic nature. Therefore, it is of interest for both academic researchers and industrial practitioners to study how to model these systems and to achieve satisfactory or even optimal control performance. Linear modeling technologies have been quite sophisticated after decades of development; however, due to the inborn nonlinearity of some production processes, the performance of single linear model based controllers or optimizers may be compromised or even unsatisfactory [
The multiple model approach is a good compromise which produces complex models or controllers by piecing a number of simpler subsystems together. Some multiple modeling strategies and applications have been investigated because they are able to represent complex nonlinear processes using the linearization technique [
The distillation column is considered as one of the most important unit operations in chemical engineering [
The remainder of this paper is organized as follows: the high purity distillation column used to test the multiple model method is described in Section
Distillation is simply defined as a process in which a liquid or vapor mixture of two or more substances is separated into its component fractions of desired purity, by the application and removal of heat [
A typical twoproduct distillation column on a pilot plant scale is shown in Figure
two components (binary separation);
one feed and two products;
constant relative volatility;
constant molar flows (same vapor flow on all stages);
no vapor holdup;
total condenser.
Notation of the distillation column in reflux and boilup construction.
Variable  Description  Unit  

1 

Feed rate  kmol/min 
2 

Distillate top product composition  mole fraction 
3 

Distillate bottom product composition  mole fraction 
4  LT  Reflux flow  kmol/min 
5  VB  Boilup flow  kmol/min 
The twoproduct distillation column in reflux and boilup construction.
The high purity distillation column as mentioned above is a classic chemical production process; the plant may transit among several operating points. A bank of ARX models is employed to approximate the nonlinear dynamic process around each operating point. Consider the following ARX model [
Around the small region of each of the operation points, the ARX local model can be used to approximate the process dynamics. Here parameter
Based on the fact that the observed output
Here
The missing data set is the random variable
The EM algorithm is an efficient iterative procedure for maximum likelihood estimation in the presence of missing data. The main strength of the EM algorithm lies in its ability to handle missing variables in the identification data set [
Initialization: give an initial value of the model parameter vector.
Iteration: step 2 and step 3 are repeated until the changes in parameters after each iteration are within a specified tolerance level.
The above steps ensure that the log likelihood function of the observed data increases at every step. So the EM algorithm is guaranteed to converge to some maximum of the likelihood function [
The flowchart of the EM algorithm for estimating the parameters.
As mentioned above, if we compute the
Using the Bayesian theory, the joint probability density function such as the first term of (
Similarly, the model identity
The derivation of the third term in (
Then we can substitute (
The expectation of the complete data in the procedure of the EM algorithm is
From (
The last probability density function to be calculated denotes the conditional probability that the
Now we give the identification procedure under the frame of the EM algorithm as follows.
As a result, the new
For the local model validity width
Here a nonlinear numerical optimization is required to obtain the optimal
Nonlinear system identification is much more complex than linear system identification. In this section, a high purity distillation column is used to test the performance of the multiple model approach based on the EM algorithm.
It has been shown that the purer the products get, the more nonlinear the system becomes [
Here, the process input variables available for control purpose are the reflux
In the simulation, white noise with a variance of
The observed input data of the distillation column.
The observed output data of the distillation column.
The scheduling variable of the distillation column.
The comparison of the distillation column model.
The weight of each local model at different operating points.
To further test the performance of the identified multiple model, the crossvalidation is also carried on the case. The data used for crossvalidation as shown in Figures
The observed input data of the distillation column in crossvalidation simulation.
The observed output data of the distillation column in crossvalidation simulation.
The scheduling variable of the distillation column in crossvalidation simulation.
The comparison of the distillation column model in crossvalidation simulation.
A multiple model modeling algorithm based on EM algorithm is proposed and applied for a high purity distillation column reaction process. Taking a full consideration of the dynamic, nonlinearity, and complexity, the parameters of each local model are identified under different operating points. The global model can be obtained by using a weighting function, which adopts an exponential form about scheduling variable under the framework of EM algorithm. Simulation results show that the fusion model of multiple models based on the EM algorithm is very effective in the chemical process of a high purity distillation column.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the financial support by the National Natural Science Foundation of China (nos.: 21206053, 21276111, and 61273131) and the paper is partly supported by the 111 Project (B12018), the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), the Fundamental Research Funds for the Central Universities (JUDCF10064), and Jiangsu Innovation Program for Graduates (CXZZ11_0464).