**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:

$$\ddot{\mathbf{y}}\ (t) + \dot{\mathbf{y}}\ (t) + \mathbf{y}(t) + \mathbf{y}(t)^3 = \mathbf{u}(t) \tag{29}$$

*An Optimization Procedure of Model's Base Construction in Multimodel Representation… DOI: http://dx.doi.org/10.5772/intechopen.96458*

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*<sup>1</sup> plot suggests submodels give the first submodels. While maximum absolute value of *U*<sup>2</sup> 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:

$$
\mu(k) = \mathbf{0}.2\sin\left(\pi k \zeta\_{\mathbf{0}0}\right) + \mathbf{0}.5\sin\left(\mathbf{3}\pi k \zeta\_{\mathbf{1}00}\right) \tag{30}
$$

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

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.

#### **5.2 Example 2**

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

$$y(t) = 0.4u(t-1)^3 \exp\left(-0.5|y(t-1)|\right) \tag{31}$$

**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).*

#### **Figure 4.**

*Real and multimodel outputs proposed reduction approach. The solid line is the plant output y, and dashed line is the reduction multimodel yR mul (second order continuous system).*


#### **Table 1.**

*Performance comparison (second order continuous system).*

*An Optimization Procedure of Model's Base Construction in Multimodel Representation… DOI: http://dx.doi.org/10.5772/intechopen.96458*

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 *α<sup>i</sup> <sup>g</sup>* <sup>¼</sup> <sup>0</sup>*:*45 and *<sup>α</sup> <sup>f</sup> <sup>g</sup>* ¼ 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 t*ð Þ � 1 and *y t*ð Þ � 1 . Therefore the feature variables considered are: *u t*ð Þ � 1 and *y t*ð Þ � 1 in the local models forming a first order model. The procedure of submodels elimination is also considered. The analysis of the singular vector *U*<sup>1</sup> and *U*<sup>2</sup> or the difference between the absolute value of *U*<sup>1</sup> and *U*<sup>2</sup> 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*<sup>1</sup> appears on submodels 5 and 16. Moreover, the maximum absolute value of *U*<sup>2</sup> 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:

$$u(t) = \sin\left(\pi t/100\right)\tag{32}$$

We have recorded in **Figure 7**, the evolution of the real process given by *y k*ð Þ and the evolution of the multimodel output reduction submodels given by *y<sup>R</sup> mul*ð Þ*t* . 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 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).*

#### **Figure 7.**

*Real and multimodel outputs proposed reduction approach. The solid line is the plant output y, and dashed line is the reduction multimodel yR mul (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

*An Optimization Procedure of Model's Base Construction in Multimodel Representation… DOI: http://dx.doi.org/10.5772/intechopen.96458*


**Table 2.**

*Performance comparison (second example).*

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:

$$\begin{split} \boldsymbol{\omega}\_{k+1}^{(1)} &= \boldsymbol{\omega}\_{k}^{(1)} + \mathbf{0.5} \frac{\boldsymbol{\varkappa}\_{k}^{(1)} \boldsymbol{\varkappa}\_{k}^{(2)}}{\boldsymbol{\varkappa}\_{k}^{(1)} \boldsymbol{\varkappa}\_{k}^{(2)}} - \mathbf{0.5} \boldsymbol{u}\_{k} \boldsymbol{\varkappa}\_{k}^{(1)}, \\ \boldsymbol{\varkappa}\_{k+1}^{(2)} &= \boldsymbol{\varkappa}\_{k}^{(2)} - \mathbf{0.5} \frac{\boldsymbol{\varkappa}\_{k}^{(1)} \boldsymbol{\varkappa}\_{k}^{(2)}}{\boldsymbol{\varkappa}\_{k}^{(1)} \boldsymbol{\varkappa}\_{k}^{(2)}} - \mathbf{0.5} \boldsymbol{u}\_{k} \boldsymbol{\varkappa}\_{k}^{(2)} + \mathbf{0.05} \boldsymbol{u}\_{k}, \\ \boldsymbol{\varkappa}\_{k} &= \boldsymbol{\varkappa}\_{k}^{(1)}, \end{split} \tag{33}$$

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-secondslong stairs augmented with random amplitude between 0 ≤*u*≤ 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 *α<sup>i</sup> <sup>g</sup>* <sup>¼</sup> <sup>0</sup>*:*45 and *<sup>α</sup> <sup>f</sup> <sup>g</sup>* ¼ 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

**Figure 8.** *Determination of the number of cluster via FSCL (biological reactor application).*

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.

Taken into account the analysis of the evolution of the singular vectors *U*<sup>1</sup> and *U*<sup>2</sup> or the difference between the absolute value of *U*<sup>1</sup> and *U*<sup>2</sup> 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*<sup>1</sup> is signaled on the submodels 6 and 7. Against, the maximum absolute value of *U*<sup>2</sup> 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).*

*An Optimization Procedure of Model's Base Construction in Multimodel Representation… DOI: http://dx.doi.org/10.5772/intechopen.96458*

#### **Figure 10.**

*Real and multimodel outputs proposed reduction approach. The solid line is the plant output y, and dashed line is the reduction multimodel yR mul (biological reactor).*


#### **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:

$$\begin{aligned} y(k) &= 0.9722y(k-1) + 0.3578u(k-1) - 0.1295u(k-2) \\ &- 0.3103y(k-1)u(k-1) - 0.04228y(k-2)^2 + 0.1663y(k-2)u(k-2) \\ &- 0.03259y(k-1)^2y(k-2) - 0.3513y(k-1)^2u(k-2) \\ &+ 0.3084y(k-1)y(k-2)u(k-2) + 0.1087y(k-2)u(k-1)u(k-2); \end{aligned} \tag{34}$$

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 *α<sup>i</sup> <sup>g</sup>* <sup>¼</sup> <sup>0</sup>*:*42 and *<sup>α</sup> <sup>f</sup> <sup>g</sup>* ¼ 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**).

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*<sup>1</sup> and *U*<sup>2</sup> or the absolute value between *U*<sup>1</sup> and *U*<sup>2</sup> indicate that the two submodels 2 and 3 are

**Figure 11.** *Determination of cluster number (liquid level process application).*

retained and sufficient to represent the nonlinear plant (**Figure 12**). In order to validate the proposed process modeling, the following input sequence is considered:

$$u(k) = \mathbf{0}.25\sin\left(\pi k \langle \mathbf{1}00 \rangle + \mathbf{0}.5\sin\left(\pi k \langle \mathbf{3}0 \rangle\right)\right) \tag{35}$$

The results of multimodel reduction approach *y<sup>R</sup> mul* is plotted in **Figure 13**, demonstrate that the novel approach tracks the real output with a very small error.

**Figure 12.** *Determination of the optimal number of submodels (liquid level control).*

*An Optimization Procedure of Model's Base Construction in Multimodel Representation… DOI: http://dx.doi.org/10.5772/intechopen.96458*

#### **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<sup>R</sup> mul (liquid level plant).*

This is confirmed by inspecting respectively the value of NRMSE, FAV and the FIT of modeling given in **Table 4**.

#### **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:



**Table 4.** *Performance comparison (liquid level control).*

• E2 exchanger cools the heat transfer fluid.

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

**Figure 15.** *Reactor dataset for system identification.*

*An Optimization Procedure of Model's Base Construction in Multimodel Representation… DOI: http://dx.doi.org/10.5772/intechopen.96458*

**Figure 16.** *The FSCL determination of cluster number in process reactor application.*

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

**Figure 18.**

*Evolutions of the reduction multimodel output y<sup>R</sup> mul and the reactor output y k*ð Þ*.*


**Table 5.**

*Performance comparison (process reactor application).*

Based on the FSCL algorithm, we have considered five neurons in the output layer with following parameters of *α<sup>i</sup> <sup>g</sup>* <sup>¼</sup> <sup>0</sup>*:*35 and *<sup>α</sup> <sup>f</sup> <sup>g</sup>* ¼ 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.

The SVD technique is applied in order to reduce the number of submodels. The plotted singular vectors *U*<sup>1</sup> and *U*<sup>2</sup> and the absolute value between *U*<sup>1</sup> and *U*<sup>2</sup> 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<sup>R</sup> mul* 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**.
