An Optimization Procedure of Model’s Base Construction in Multimodel Representation of Complex Nonlinear Systems

Abstract

The multimodel approach is a research subject developed for modeling, analysis and control of complex systems. This approach supposes the definition of a set of simple models forming a model’s library. The number of models and the contribution of their validities is the main issues to consider in the multimodel approach. In this chapter, a new theoretical technique has been developed for this purpose based on a combination of probabilistic approaches with different objective function. First, the number of model is constructed using neural network and fuzzy logic. Indeed, the number of models is determined using frequency-sensitive competitive learning algorithm (FSCL) and the operating clusters are identified using Fuzzy K- means algorithm. Second, the Models’ base number is reduced. Focusing on the use of both two type of validity calculation for each model and a stochastic SVD technique is used to evaluate their contribution and permits the reduction of the Models’ base number. The combination of FSCL algorithms, K-means and the SVD technique for the proposed concept is considered as a deterministic approach discussed in this chapter has the potential to be applied to complex nonlinear systems with dynamic rapid. The recommended approach is implemented, reviewed and compared to academic benchmark and semi-batch reactor, the results in Models’ base reduction is very important witch gives a good performance in modeling.

1. Introduction

There are several representations of industrial processes: linear, nonlinear or other technique using fuzzy logic and/or neural networks. Nonlinear models are found in a large part and are used to properly represent the dynamics of real processes. However, their comlexcity proves a real obstacle for control or when designing an observer or in a diagnostic strategy. So, the multimodel approach is a powerful approach developed in the aim to overcome problems related to modeling and control of industrial processes which are often complex, nonlinear and/or nonstationary.

The multimodel approach supposes the representation of the nonlinear model by a set of linear models (as designed in future by submodels or model’s base) thus forming what is called a model library or model’s base. The interaction of each models of this model’s base through a certain normalized validity calculation forming the global nonlinear system in its all area of operating.

The different models of the base could be of different structures and orders and no model can represent the system in its whole operating domain. Therefore, the multimodel approach aims at lowering the system complexity by studying its behavior under specific condition. Figure 1 illustrates the concept of a process formed by a multimodel approach. This mechanism evaluates the contribution of each model in the description system’s behavior.

Figure 1.

Multimodel representation.

The decision unit estimates by means of the numerical validity of each model the selection of the most relevant models at each time. The contribution of the different model’s is made by a decision output unit that compute the multimodel output.

Several researchers have been interested in multimodel analysis and control and many applications have been proposed in different contexts. The multimodel approach has knowledgeable a sure interest since the publication of the work of [1]. The idea of the multimodel approach is to describe the complex nonlinear systems by a set of local models (linear or affine) characterizing the operation of the system in different areas of operation. In spite of its success in many fields, the multi-model approach remains confronted with several difficulties and some design problems such as the determination of the models’ base and the adequate validity computation. Several works have treated these two points in the literature. We can refer to the works presented in [2, 3, 4, 5, 6, 7, 8]. Indeed once the model library is built it will remain intact. The number of model found is, first responsible to adequately represent the nonlinear model of the system and second, it will remain static when it is used in control or diagnosis. In this context there are very few works that have tried to reduce the models’ base once it is built and this opens up a new axis of research added to the design of the multimodel approach in the representation of nonlinear systems. We can cite in this context the works of Gasso in [9, 10, 11], where the reduction of the models’ base is based on an iterative procedure which include tree operation that is: elimination of less important local submodel, merging of the neighbouringsubmodel that describe the same behavior of the nonlinear system and finally a parameters optimization of the resultant structure is made on. In [12, 13, 14, 15], The nonlinear system is modeled by a set of linear models and with the of a Gap metric in the model based control technique we can decide how many models are sufficient for control. In other context, when the process can be represented by a fuzzy model, the complexity of rule base can be minimized through two procedure: illumination of the less important rule and merging of neighboring rules that can describe the same behavior of the nonlinear system [16]. The Chiu’s classification method [17] is used in order to obtain the optimal systematic determination of model’s base [18]. The latest work is a new approach permit to optimize the number of submodels with respect to the submodel complexity is presented. In [19] the optimal number of submodels is done based on both a reinforced combinatorial particle swarm optimization and a hybrid K-means.

In this chapter book we study a very relevant modeling approach especially where do not have an adequate model of the process. We will use the data set relative to the identification of the process in its area of operating. On the other hand the input and output measurement. The multimodel approach in this case becomes a very efficient method to overcome the problem of modeling.

The outline of this work is as follows: in Section 2, we will present the procedure used in the multimodel representation of complex nonlinear systems. Indeed, the number of models is determined using frequency-sensitive competitive learning algorithm and the operating clusters are identified using Fuzzy K- means algorithm. The structure of each local model and the validity computation are also presented to each submodel of the model’s base (number of model built). Section 3 presents the recommended approach to reduce the number of submodels. Then, focusing on the use of both two type of validity calculation for each submodel and an SVD technique is used to evaluate the adequate number that can be retained in the model’s base.

2. Multimodel system identification

In this section a two classification algorithms are used to allow the determination of the number, structure and parameters of the different models of the base. So, for the determination of model’s base, we propose both Frequency Sensitive Competitive Learning (FSCL) and Fuzzy K-means to obtain respectively the number of models and the operating regime of each one. In the second step we propose the validity computation for each submodel in the model’s base. The validity of each model is designed in such a way that it involves two types of validity, namely a simple and reinforced validity. The contribution of each submodel is done by each validity value and thus appears via an optimization technique. On the basis of the last results, one can in fact, minimize the number of submodels for those that have no effect on multimodel modeling. This can be explaining by a measure of similarity between each submodel and the global one. This technique is highlighted via an SVD technique.

2.1 Determination of the number models

Most existing clustering algorithms [20, 21], when we have not an idea about the number they cannot handle the selection of the appropriate number of cluster. However, when it is used for clustering, the FSCL algorithm automatically allocates the suitable number of units for an input data set. So, for the determination of model’s base, we propose Frequency Sensitive Competitive Learning (FSCL). Frequency Sensitive Competitive Learning (FSCL) [22] is a competitive algorithm with N neuron that is trained by a dataset of $P$ data vectors $x\left(t\right)$. In the FSCL, the competitive computing units are penalized in proportion to the frequency of their winning. Giving an input $x_i$ each time, FSCL determines the winner by:

$$u_i=\left\{\begin{array}{l}1\mathit{\text{if}}i=c\\ 0\mathit{\text{otherwise}},\end{array}\right.$$E1

Such that

$${\gamma }_c{‖x-w_c‖}^2=\underset{j}{min}{\gamma }_j{‖x-w_j‖}^2$$E2

where.

$N$: Is initial estimation of the number of clusters in the given data.

$u_{i}1\le i\le k$: The output units of dimension $k$.

$w_{i}1\le i\le k$: Weight vectors each of dimension $k$.

$x_{i}1\le i\le d$: The d-dimensional input vector from data set $P$.

$w_c$: D-dimensional weight vector corresponding to the winner.

$‖\ast ‖$: Euclidean distance;

c: index of the unit which wins the competition;

${\gamma }_j$: Conscience factor used to reduce the winning rate of the frequent winners defined as follows [23]:

$${\gamma }_j=\raisebox{1ex}{$n_j$}\!\left/ \!\raisebox{-1ex}{${\sum }_{i=1}^k{n}_i$}\right.$$E3

Where $n_j$ refers to the cumulative number of occurrences the node $j$ has won the competition.

After selecting the winner, FSCL updates the winner as follow:

$$w_j\left(t+1\right)=w_j\left(t\right)+{\alpha }_g\left(t\right)\left(x-w_j\left(t\right)\right)$$E4

Where ${\alpha }_g$ is the learning rate defined as follow:

$${\alpha }_g={\alpha }_g^i{\left(\frac{{\alpha }_g^f}{{\alpha }_g^i}\right)}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$t_{\max }$}\right.}$$E5

The convergence of the appropriate FSCL algorithm to a local minimum is studied in [24]. Using this technique with its parameters as defined so that, the maximum number of iteration designed by $t_{\max }$ and with its initial and final learning rate designed respectively ${\alpha }_g^i,{\alpha }_g^f$, convergence of the algorithm visualize in the final step that there are some of the data clusters are more densely populated than others which ideally result that there are some wining units are more often in those clusters than other. It must be considered that when the number of unit (neuron) is larger than the real number of cluster in the input data-set, the extra units are gradually driven faraway from the distribution of the data-set. If the number of cluster c is not equal to the true value, FSCL will lead to an incorrect clustering result. So, it needs to pre-assign the number of clusters c. For all the studied examples we have varied the cluster number, the learning rate for each case study and take the opportunity to determine the appropriate one in such away that ${\alpha }_g\in \left[0.1-0.5\right]$.

2.2 Fuzzy k-means clustering

In order to establish the operating cluster, the use of Fuzzy k-means is considered. This last algorithm was defined by [25] and improved by [26]. The determination of different cluster centers and dada set $x_i$ assigned to each clusters for each centers is done by the use of the minimization objective function mentioned by:

$${\mathit{J}}_{\mathit{m}}=\sum _{\mathit{j}=1}^{\mathit{K}}\sum _{\mathit{i}=1}^{\mathit{N}}{\mu }_{\mathit{ij}}^{\mathit{m}}{‖{\mathit{x}}_{\mathit{i}}-{\mathit{c}}_{\mathit{j}}‖}^2,1\le \mathit{m}\le \infty ;$$E6

Where:

${\mu }_{\mathit{ij}}$: represents degree of membership of $x_i$ to cluster $j$ and stands for the local model’s activation degree for that observation ${\sum }_{j=1}^K{\mu }_{\mathit{ij}}=1$;

$N$: is the number of data points,

$K$: Number of cluster or local models;

$x_i$: i^th data point;

$m$: (real number greater than 1) is the “fuzzy exponent” that influences the membership values and represents the overlapping shape between clusters.

$c_j$: Center of cluster $j$.

The algorithm consists of the following steps:

1. Initialize the membership matrix $U=\left[{\mu }_{\mathit{ij}}\right]$ with random value between 0 and 1.

2. Calculate fuzzy centers using:

   $$c_j=\frac{{\sum }_{i=1}^N{\mu }_{\mathit{ij}}^m{x}_i}{{\sum }_{i=1}^N{\mu }_{\mathit{ij}}^m}.$$E7

3. Compute a new matrix U using

   $${\mu }_{\mathit{ij}}={\left[\sum _{r=1}^K{\left(\frac{‖x_i-c_j‖}{‖x_i-c_r‖}\right)}^{2/\left(m-1\right)}\right]}^{-1}$$E8

4. The iteration will stop if:

   $$‖U\left(k-1\right)-U\left(k\right)‖<\xi .$$E9

Where $\xi$ is a termination criterion between 0 and 1.

2.3 Local linear models

By identifying the cluster number and the datset for each cluster, the next step focuses primarily on obtaining cluster sub-models. This last step requires two phases: the first for the structure of submodel and the second deal with its parameters identification. For the different obtained vectors representative of cluster, a parametric estimation uses the Recursive Least-Square method is retained. The ARX model being chosen for each cluster whose equation is given by:

$$y\left(k\right)=-\sum _{i=1}^{n_a}{a}_{i}y\left(k-i\right)+\sum _{j=1}^{n_b}{b}_{j}u\left(k-j\right)$$E10

Where $a_i$ and $b_j$ are parameters of the ith submodel. $n_a$ and $n_b$ are the lags respectively in input and output. The order of each model is determined by the instrumental determinants’ ratio-test [27]. For every order value $d$, the instrumental determinants ‘ratio $\mathit{RDI}\left(d\right)$ is computed and the retained order d isthe value for which the ratio $\mathit{RDI}\left(d\right)$ quickly increases for the first time. This method consists in building an information matrix $Q_d$ giving by:

$$Q_d=\frac{1}{n_d}\sum _{k=1}^{n_d}\left[\begin{array}{l}u\left(k\right)\\ u\left(k+1\right)\\ u\left(k-d+1\right)\\ u\left(k+d\right)\end{array}\right]{\left[\begin{array}{l}y\left(k+1\right)\\ u\left(k+1\right)\\ y\left(k+d\right)\\ u\left(k+d\right)\end{array}\right]}^T$$E11

Where $n_d$ is the observations’ number and the instrumental determinants’ ratio RDI (d) is given by the following relation:

$$\mathit{RDI}\left(d\right)=\left|\frac{det\left(Q_d\right)}{det\left(Q_{d+1}\right)}\right|$$E12

2.4 Computation of validity

In the proposed approach the validity computation of each submodel is proposed by this expression:

$$v_i^{\mathit{mul}}={\alpha }_i{v}_i^{\mathit{\text{simp}}}+{\beta }_i{v}_i^{\mathit{\text{renf}}}$$E13

Where $v_i^{\mathit{\text{simp}}}$ and $v_i^{\mathit{\text{renf}}}$ are respectively simple and reinforced validity. The simple validity is defined by:

$$v_i^{\mathit{\text{simp}}}\left(k\right)=\frac{1-r_i^{\mathit{norm}}\left(k\right)}{N-1},i=1,N_m$$E14

And the reinforced validity is expressed by

$$v_i^{\mathit{\text{renf}}}\left(k\right)=v_i\left(k\right)\prod _{j=1j\ne i}^N\left(1-v_j\left(k\right)\right)$$E15

The $r_i^{\mathit{norm}}$ is the normalized residue given by:

$$r_i^{\mathit{norm}}\left(k\right)=\frac{r_i\left(k\right)}{\sum _{i=1}^N{r}_i\left(k\right)},i=1,N_m$$E16

The residue is expressed as the distance between the process output $y$ and the considered local output $y_i$ of the submodel $M_i$ which has the following formula:

$$r_i=\left|y-y_i\right|,i\in \left[1,N_m\right]$$E17

Where $N_m$ is the number of submodel in the models’ base.

${\alpha }_i,{\beta }_i$: are respectively the adequate values which can be calculated as:

$$\begin{array}{l}\underset{{\alpha }_i,{\beta }_i}{\mathit{Min}}\left[\left(\left(\sum _{i=1}^{N{}_m}\left({\alpha }_i{v}_i^{\mathit{\text{simpl}}}+{\beta }_i{v}_i^{\mathit{\text{renf}}}\right)y_i\right)-y\right)\right]\\ \sum _{i=1}^n{\alpha }_i=1,\sum _{i=1}^n{\beta }_i=1.\end{array}$$E18

A Least Square Estimation is used in order to prove the value of ${\alpha }_i$ and ${\beta }_i$. In fact, the multimodel output $y_{\mathit{mul}}$ is calculated by a fusion of the models’ outputs $y_i$ weighted by their respective multi-validity indexes $v_i^{\mathit{mul}}$ which is illustrated by the following expression:

$$y_{\mathit{mul}}\left(k\right)={\sum }_{i=1}^{N_m}{v}_i^{\mathit{mul}}\left(k\right)y_i\left(k\right)={\theta }^T\phi \left(k\right)$$E19

Where we introduce the following regressor vector:

$${\phi }^T\left(k\right)=\left[y_1\left(k-1\right),,y_i\left(k-1\right)\right],i=1,N$$E20

And the parameters vector:

$${\theta }^T\left(k\right)=\left[{\alpha }_1,{\alpha }_i,{\beta }_1,{\beta }_i\right],i=1,N$$E21

The recursive least squares estimate of $\theta$ is:

$$\left\{\begin{array}{l}\theta \left(k\right)=\theta \left(k-1\right)+P\left(k\right)\phi \left(k\right)e\left(k\right)\\ P\left(k\right)=P\left(k-1\right)-\frac{P\left(k-1\right)\phi \left(k\right){\phi }^T\left(k\right)P\left(k-1\right)}{1+{\phi }^T\left(k\right)P\left(k-1\right)\phi \left(k\right)}\\ e\left(k\right)=y\left(k\right)-{\theta }^T\left(k-1\right)\phi \left(k\right)\end{array}\right.$$E22

Where $P$ denote the covariance matrix and $e$ is an error of modeling.

3. Reduction of the model’s base

Based on the new validity calculation of each submodel, the need of each one in the model’s base is identified via the numerical values $\alpha$ and $\beta$. Thus, the impact of each submodel above the global model is also determined by the numerical values $\alpha$ and $\beta$. Each submodel can contribute in one way or another via its new validity and one can thus determine the adequate number of submodels where their contribution in modeling is important compared to the other submodels. A comparison is made on to have a good performance by comparing each submodel to the global model. The submodel whose contribution is important will be retained. To solve this problem, the following matrix $X$ defined by Eq. (23) have to be determined first and an analysis using singular value decomposition analysis should be carried out.

$$X=\left[\begin{array}{l}\left({\alpha }_1{v}_1^{\mathit{\text{simp}}}+{\beta }_1{v}_1^{\mathit{\text{renf}}}\right)y_1\circ y\left({\alpha }_2{v}_2^{\mathit{\text{simp}}}+{\beta }_2{v}_2^{\mathit{\text{renf}}}\right)y_2\circ y\\ \left({\alpha }_i{v}_i^{\mathit{\text{simp}}}+{\beta }_i{v}_i^{\mathit{\text{renf}}}\right)y_i\circ y\end{array}\right]$$E23

SVD analysis is a numerical algorithm that decomposes an $N_m\times m$ matrix into three unique component matrices:

$$X=U\mathrm{\Sigma }{V}^T$$E24

Where $U$ is left singular vectors with $N_m\times N_m$ matrix, $V$ is right singular vectors with $N_m$$\times$$m$ matrix and $\mathrm{\Sigma }$is singular value with $m\times m$ matrix. Among these component matrices, $U$ vectors provide the information of retained submodel in an orthogonal form. $U_1$ and $U_2$ which is the first two rows in the $U$ vectors indicate the two most combinations of required submodel. Another approach that can facilitate this analysis is modified version of principal component analysis (PCA). In modified PCA analysis, the differences between $U_1$ and $U_2$ are also calculated by:

$$Z_i=\left|U_{1i}\right|-\left|U_{2i}\right|$$E25

Where $Z_i$ denotes the absolute value of difference between the best two $U$ vectors. The result of this analysis is easier to interpret as only one function is plotter versus submodel.

4. Model validity tests

In order to compare the modeling performances in both cases with all the number of submodels in the library and in the reduction submodel case, there are three different performance criteria considered: The normalized mean-square error (NMSE), the best fit (FIT) and the variance accounted for (VAF) criterion defined respectively as follows:

The normalized root mean squared error (NRMSE) is defined as:

$$\mathit{\text{NRMSE}}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$max\left(y\right)$}\right.\sqrt{\frac{1}{M}\sum _k\left(y-y_s\right)}$$E26

The best fit is defined as:

$$\mathit{FIT}=\left(1-\frac{‖y\left(k\right)-y_S‖}{‖y\left(k\right)-y_{\mathit{\text{mean}}}‖}\right).100\%$$E27

The variance accounted for is defined as:

$$\mathit{VAF}=max\left\{\left.1-\frac{var\left(y\left(k\right)-y_S\right)}{var\left(y\left(k\right)\right)},0\right\}.100\%\right.$$E28

Where $y$ denotes the measured output and $y_S$ is the estimated output of the multimodel.

5. Simulation examples

Numerical simulations are presented to compare the proposed approach to the conventional multimodel design with full submodels number and investigate the impact to model highly nonlinear systems.

5.1 Example 1: second order continuous system

Taken from [28], the first simulation example considered is the system whose evolution is described by the following equations:

$$\stackrel{\mathrm{..}}{y}\left(t\right)+\stackrel{.}{y}\left(t\right)+y\left(t\right)+y{\left(t\right)}^3=u\left(t\right)$$E29

The system has been sampled at a period of 0.2 s. A 1500 sampled data output $y$ are used for the identification of the system. The adequate number of clusters is determined using the FSCL algorithm. Figure 2 gives the results. With twelve neurons used in the output layer six centers move away from the observation data. We can conclude that the number of cluster is equal to six. An identification procedure is than applied in order to determinate the efficient transfer function of each local model for the library. So, the required data set for each data is used in a parametric and structure identification where the RDI index gives that the order of each model is equal to two. An SVD technique is applied in order to reduce the model’s base. Figure 3 shows the elements of $U$ vector plotted versus submodels. The maximum absolute value of $U_1$ plot suggests submodels give the first submodels. While maximum absolute value of $U_2$ plot gives the secondary submodels. In the case study submodels 2 and 4 can be retained to represent the whole real process. The difference $Z$ between the best two $U$ vectors is also given prove the same retained submodels. Based in these results the following input sequence is considered to validate the proposed process modeling:

Figure 2.

Determination of the number of cluster via FSCL (second order continuous system).

Figure 3.

Determination of the optimal number of submodels (second order continuous system).

$$u\left(k\right)=0.2sin\left(\raisebox{1ex}{$\mathit{\pi k}$}\!\left/ \!\raisebox{-1ex}{$80$}\right.\right)+0.5sin\left(\raisebox{1ex}{$3\mathit{\pi k}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.\right)$$E30

The results given in Figure 4 demonstrate that the novel approach tracks the real output with a very small error.

Figure 4.

Real and multimodel outputs proposed reduction approach. The solid line is the plant output $y$, and dashed line is the reduction multimodel $y_{\mathit{mul}}^R$ (second order continuous system).

Numerical performance comparison values are given in Table 1. Illustrate that the novel approach has good properties in modeling compared to the whole process with 6 submodels. With two submodels the modeling approach is accuracy with small NRMSE.

Number of submodel	VAF (%)	FIT (%)	NRMSE
6 submodels	100.0000	99.9985	7.5795e-004
2 submodels	100.0000	99.9951	0.0014

Table 1.

Performance comparison (second order continuous system).

5.2 Example 2

The non linear plant considered given by [11] as a second simulation is described by the following nonlinear equation:

$$y\left(t\right)=0.4u{\left(t-1\right)}^{3}exp\left(-0.5\left|y\left(t-1\right)\right|\right)$$E31

The input $u$ of the system is formed by the concatenation of piecewise constant signals with variable amplitude and duration. A set of 2500 data points is used to build the model. The FSCL algorithm is used to search the adequate number of clusters. With the following parameters of ${\alpha }_g^i=0.45$ and ${\alpha }_g^f=0.041$ we consider a 500 training iteration of the 2500 data. By the use of 36 neurons, fourteen clusters centers move away from the data which results that the number of cluster is equal to 22 (Figure 5). This result is also confirmed by [11]. According to Eq. (31), the non-linearity of the system are due to the variables $u\left(t-1\right)$ and $y\left(t-1\right)$. Therefore the feature variables considered are: $u\left(t-1\right)$ and $y\left(t-1\right)$ in the local models forming a first order model. The procedure of submodels elimination is also considered. The analysis of the singular vector $U_1$ and $U_2$ or the difference between the absolute value of $U_1$ and $U_2$ given in Figure 6 show that the number of submodels can be reduced only to three submodels. In fact, the maximum absolute value of $U_1$ appears on submodels 5 and 16. Moreover, the maximum absolute value of $U_2$ appears on submodels 4 and 5. The submodels 4, 5 and 16 are those retained in future to validate the proposed process modeling with the following input sequence:

Figure 5.

Determination of the number of cluster (FSCL c = 36) and clustering results (c = 22) (second example).

Figure 6.

Determination of the optimal number of submodels (second example).

$$u\left(t\right)=sin\left(\mathit{\pi t}/100\right)$$E32

We have recorded in Figure 7, the evolution of the real process given by $y\left(k\right)$ and the evolution of the multimodel output reduction submodels given by $y_{\mathit{mul}}^R\left(t\right)$.The envisaged approach always promise best results of modeling. Table 2 compares the performances of the structures with respectively 22 and 3 local submodels. Weremark that the deletion of 19 local submodels does not affect significantly the performances of the results in multiple model process representation. The new approach isable to identify and accurate the nonlinear process. The multimodel approach and the new approach both achieve the performance of VAF =99.9999 and VAF = 99.9854. So, clearly there is no significant difference between the performances of modeling.

Figure 7.

Real and multimodel outputs proposed reduction approach. The solid line is the plant output $y$, and dashed line is the reduction multimodel $y_{\mathit{mul}}^R$ (second example).

Number of submodel	VAF (%)	FIT (%)	NRMSE
22 submodels	99.9999	99.8798	0.0078
3 submodels	99.9854	98.7907	0.0086

Table 2.

Performance comparison (second example).

5.3 Process validation

5.3.1 Biological reactor

In order to highlight the interest and contributions of our modeling approach, based on the same system that is the bio-reactor, we compared our results with those given by [3, 29]. Being a good academic example of nonlinear system, the biological reactor has been treated in some works for the purpose of illustration in different approaches of modeling and controlling non-linear systems [30, 31]. The nonlinear model is a bioreactor where its expression is given by the Contois [32] model in its discrete form as developed in the following expression:

$$\begin{array}{l}{x}_{k+1}^{\left(1\right)}=x_k^{\left(1\right)}+0.5\frac{x_k^{\left(1\right)}{x}_k^{\left(2\right)}}{x_k^{\left(1\right)}{x}_k^{\left(2\right)}}-0.5u_k{x}_k^{\left(1\right)},\\ x_{k+1}^{\left(2\right)}=x_k^{\left(2\right)}-0.5\frac{x_k^{\left(1\right)}{x}_k^{\left(2\right)}}{x_k^{\left(1\right)}{x}_k^{\left(2\right)}}-0.5u_k{x}_k^{\left(2\right)}+0.05u_k,\\ y_k=x_k^{\left(1\right)},\end{array}$$E33

In this equation y denote the output and the control input denoted u. Based on equation given above, the system is excited by a signal of the form of 4-seconds-long stairs augmented with random amplitude between $0\le u\le 0.7$. So that a collection of 3018 experimental data set are used. The first part of data with a number of 2416 is used in the identification procedure to give the adequate number of submodels with his structure and order. The second type of data is of 602 training point used to validate the modeling strategy. The FSCL algorithm is used to search the adequate number of clusters. Considering a 500 training iteration of the 2416 data by the use of 15 neurons with following parameters of ${\alpha }_g^i=0.45$ and ${\alpha }_g^f=0.041$. Figure 8 presents the cluster centers repartition. Six clusters centers move away from the data which results that the number of cluster is equal to nine. Thus with only 9 linear models against 10 model obtained by [4] and against 196 models obtained by the modeling application proposed in [29], we have succeeded firstly in designing a new multimodel structure.

Figure 8.

Determination of the number of cluster via FSCL (biological reactor application).

Taken into account the analysis of the evolution of the singular vectors $U_1$ and $U_2$ or the difference between the absolute value of $U_1$ and $U_2$ given by Figure 9 we can conclude that submodels 2, 6 and 7 are sufficient to represent the process reactor. In fact, the maximum absolute value of $U_1$ is signaled on the submodels 6 and 7. Against, the maximum absolute value of $U_2$ is signaled on the submodels 2 and 6. This difference in size of the model’s base helps to highlight our approach which guarantees a satisfied representation with a smaller number of models compared to a same studied system. Knowing that a large number of model base risk of constituting a handicap in terms of command calculation and/or analysis. This result shows that the proposed concept for reduction in multiple-linear modeling identical in accuracy to the modeling paradigm using the classical multimodel approach. The modeling results of the proposed approach compared to the full model’s base is shown in Figure 10 and the performance comparison is given in Table 3. The two curves can hardly be distinguished from each other and there is no significant difference between the performances. The results described in this section prove the efficiency and the precision of the proposed modeling strategy and show that the method works well with various processes. Hence, there is a potential for improved quality and flexibility of final product if the cost of the model development can be reduced. In fact, for example, in the high on-line computational need to solve an optimal control actions in nonlinear model predictive control, which results in a non-convex optimization, can be compared with the new proposed concept of modeling. We can reduce the on-line computation in the NMPC scheme by transforming the NLMPC problem into a LMPC and a quadratic programming can be used to handle constraints. Limiting this paper only to modeling, this last observation will be studied in future works based on the results given in [33].

Figure 9.

Determination of the optimal number of submodels (biological reactor).

Figure 10.

Real and multimodel outputs proposed reduction approach. The solid line is the plant output $y$, and dashed line is the reduction multimodel $y_{\mathit{mul}}^R$ (biological reactor).

Number of submodel	VAF (%)	FIT (%)	NRMSE
9 submodels	99.9999	99.8798	0.00478
3 submodels	99.9854	98.7907	0.0086

Table 3.

Performance comparison (biological reactor).

5.3.2 Liquidlevel process

The model is obtained through identification of a laboratory-scale liquid-level system [34] is used to illustrate the advantages of the proposed modeling method. The model of the plant is described by the following NARX model:

$$\begin{array}{l}y\left(k\right)=0.9722y\left(k-1\right)+0.3578u\left(k-1\right)-0.1295u\left(k-2\right)\\ -0.3103y\left(k-1\right)u\left(k-1\right)-0.04228y{\left(k-2\right)}^2+0.1663y\left(k-2\right)u\left(k-2\right)\\ -0.03259y{\left(k-1\right)}^{2}y\left(k-2\right)-0.3513y{\left(k-1\right)}^{2}u\left(k-2\right)\\ +0.3084y\left(k-1\right)y\left(k-2\right)u\left(k-2\right)+0.1087y\left(k-2\right)u\left(k-1\right)u\left(k-2\right);\end{array}$$E34

The application of the FSCL algorithm is highlighted on the set of data collected from the process in order to determine the number of models of the library. Using 8 neurons with following parameters of ${\alpha }_g^i=0.42$ and ${\alpha }_g^f=0.01$, after 500 iterations of the 1600 data set, the algorithm leads to a concentration of 4 clusters centers and 4 centers have moved away from the dataset, which leads to conclude that the considered process can be modeled by 4 submodels (Figure 11).

Figure 11.

Determination of cluster number (liquid level process application).

Based on clustering results, the RDI value for each cluster lead to a value 2 and a least square estimation is used to develop the different submodels of the Model’s base. In order to reduce the number of submodels, the SVD technique is applied on the formed matrix given by Eq. (23), so the plotted singular vectors $U_1$ and $U_2$ or the absolute value between $U_1$ and $U_2$ indicate that the two submodels 2 and 3 are retained and sufficient to represent the nonlinear plant (Figure 12). In order to validate the proposed process modeling, the following input sequence is considered:

Figure 12.

Determination of the optimal number of submodels (liquid level control).

$$u\left(k\right)=0.25sin\left(\raisebox{1ex}{$\mathit{\pi k}$}\!\left/ \!\raisebox{-1ex}{$100$}\right.\right)+0.5sin\left(\raisebox{1ex}{$\mathit{\pi k}$}\!\left/ \!\raisebox{-1ex}{$30$}\right.\right)$$E35

The results of multimodel reduction approach $y_{\mathit{mul}}^R$ is plotted in Figure 13, demonstrate that the novel approach tracks the real output with a very small error. This is confirmed by inspecting respectively the value of NRMSE, FAV and the FIT of modeling given in Table 4.

Figure 13.

Real and multimodel outputs reduction approach. The solid line is the plant output $y$ and the dashed line is the reduction multimodel $y_{\mathit{mul}}^R$ (liquid level plant).

Number of submodel	VAF (%)	FIT (%)	NRMSE
4 submodels	100.0000	99.9747	0.0049
2 submodels	99.9987	99.6333	0.0130

Table 4.

Performance comparison (liquid level control).

5.4 Experimental validation

The pilot unit that we are going to study is a process installed in the laboratory of process control at the Engineering school of Gabes (Tunisia) (Figure 14) it consists mainly on:

• A 2 l jacketed reactor equipped with a drain valve to empty its contents.

• A stirrer connected to an adjustable speed motor 0–3000 tr/min.

• A tube condenser. Cooling is provided by tap water;

• Two pumps P1 and P2: P1 ensures the supply of alcohol while P2 ensures the circulation of the heat transfer fluid;

• Two reservoirs R1 and R2: the reservoir R1 is used to store the alcohol which will be used as a reagent while the reservoir R2 allows the collection of the condensa at the outlet of the condenser;

• A heater resistor E1 provides variable power from 0 to 3500w;

• E2 exchanger cools the heat transfer fluid.

Figure 14.

Process reactor.

The reaction carried in this semi batch reactor is an esterification reaction, which is given by the following scheme:

$$\text{Acide}+\text{Alcohol}\leftarrow \to \text{Ester}+\text{Water}$$E36

For identification experiments, it has proved in previous works [35, 36]. Then the reaction was heated as quickly as possible with the maximum power (3 KW) up to a temperature of 110 °C, that is to say close to desired temperature. The collected data used for the identification of the process are plotted in Figure 15. Where the process sampling time is 180 s.

Figure 15.

Reactor dataset for system identification.

Based on the FSCL algorithm, we have considered five neurons in the output layer with following parameters of ${\alpha }_g^i=0.35$ and ${\alpha }_g^f=0.01$. Because that two centers have moved apart from the data as illustrated in Figure 16 we can terminate the adequate number of cluster which is equal to three. For ach cluster, the data set is used to determine the appropriate local model, after having carried out a structure and parametric identification as well as the determination of its order using the RDI procedure.

Figure 16.

The FSCL determination of cluster number in process reactor application.

The SVD technique is applied in order to reduce the number of submodels. The plotted singular vectors $U_1$ and $U_2$ and the absolute value between $U_1$ and $U_2$ in Figure 17 indicate that the two submodels1 and 3 are retained and sufficient to represent the nonlinear plant. The fusion of each model by the new technique of validity computation leads to the results given by Figure 18. The results of multimodel reduction approach $y_{\mathit{mul}}^R$ demonstrate that the novel approach tracks the real output with a very small error. This is confirmed by inspecting respectively the value of NRMSE, FAV and FIT given in Table 5.

Figure 17.

Determination of the optimal number of submodels (process reactor).

Figure 18.

Evolutions of the reduction multimodel output $y_{\mathit{mul}}^R$ and the reactor output $y\left(k\right)$.

Number of submodel	VAF (%)	FIT (%)	NRMSE
3 submodels	98.3043	86.9121	0.0318
2 submodels	98.3043	86.9121	0.0318

Table 5.

Performance comparison (process reactor application).

6. Conclusion

A new practice method to reduce the number of model in the model’s base is designed for multimodel strategy. The proposed technique is an automatic procedure for decreasing the number of submodel in the model’s base. From an initial number of submodel determined using frequency-sensitive competitive learning algorithm (FSCL) and the K- means algorithm, the reduction model procedure repose on the use of an adequate validity computation for each submodel and an analysis of an SVD technique is made on to select the adequate number of submodel. The proposed design has been applied to numerical examples shows its effectiveness compared to conventional approachesThe novel approach is also tested for a real process presents comparable results than those of the literature. A real time application is made on in order to model a process reactor using the same technique shows very remarkable performance. In the context of our approach, we do not seek to study the structure distribution of clusters. In future work, we will study the influence of the clustering algorithm such as the kohonen network or the kmeans algorithms and other competitive learning. The influence of the simple or reinforced validity is also a subject for future work in multimodel reduction procedure. The multiagent model predictive control can be very useful for future works in order to reduce the time consumption with this structure of modeling.