On the combination of kernel principal component analysis and neural networks for process indirect control

Figures & data

Figure 1. The architecture of indirect neural control.

Figure 2. The principle of neural network model.

Figure 3. The new architecture of indirect neural control.

Table 1. The usual kernel functions.

Function	Kernel
Radial basis function kernel	$kr(z_i,z_j)=e^{-\frac{\parallel z_i-z_j{\parallel }^2}{2{\sigma }^2}}$
Polynomial kernel	$kr(z_i,z_j)=(a.z_i.z_j+b{)}^n$
Linear kernel	$kr(z_i,z_j)=z_i.z_j$
Sigmoid kernel	$kr(z_i,z_j)=tanh(a.z_i.z_j+b)$

Figure 4. $a_1(k)$ and $a_2(k)$ trajectories.

Table 2. The comparison results of the used kernel function in the identification error.

Kernel function	$MS{E}_e$
RBF kernel, $\sigma =6$	$4.6252{10}^{-7}$
Polynomial kernel, $a=1;b=1;n=3$	$4.6854{10}^{-7}$
Linear kernel	$4.6829{10}^{-7}$
Sigmoid kernel, $a=1;b=1$	$4.0955{10}^{-7}$

Table 3. The influence of the dimensionality reduction in the identification error.

	NN model	KPCA NN model
$\eta$	Variable	Variable
$MS{E}_e$	$4,{6252.10}^{-7}$	$4,{0955.10}^{-7}$
$MA{E}_e$	$5,{4126.10}^{-4}$	${\mathrm{4.9423.10}}^{-4}$
$max(e)$	${\mathrm{9.9953.10}}^{-4}$	$9,{9666.10}^{-4}$
time (s)	$16.2345$	$1.0987$

Table 4. The influence of the dimensionality reduction in the control error.

	NN controller	KPCA NN controller
${\eta }_c$	Variable	Variable
$MS{E}_{e_c}$	$0.0074$	$0.0027$
$MA{E}_{e_c}$	$0.0225$	$0.0166$
$max(e_c)$	$0.5000$	$0.3000$
time (s)	$47.4502$	$3.1092$

Figure 5. The pre-processing control system output and the desired values.

Figure 6. The control law.

Figure 7. The control error.

Figure 8. The pre-processing control system output and the desired values.

Figure 9. The control law.

Figure 10. The control error.

Table 5. The influence of the dimensionality reduction in the identification error.

	NN model	KPCA NN model
$\eta$	Variable	Variable
$MS{E}_e$	$4,{4236.10}^{-7}$	$4,{2343.10}^{-7}$
$max(e)$	$9,{9904.10}^{-4}$	$9,{9793.10}^{-4}$
$MA{E}_e$	$5,{1255.10}^{-4}$	$4,{9632.10}^{-4}$
time (s)	$41.3576$	$1.8007$

Table 6. The influence of the dimensionality reduction in the control error.

	NN model	KPCA NN model
${\eta }_c$	Variable	Variable
$MS{E}_{e_c}$	$0.0073$	$0.0028$
$MA{E}_{e_c}$	$0.0221$	$0.0166$
$max(e_c)$	$0.5000$	$0.3073$
time (s)	$22.3142$	$3.1241$

Table 7. The influence of the dimensionality reduction in the identification error.

	NN model	KPCA NN model
$\eta$	Variable	Variable
$MS{E}_e$	$4,{2044.10}^{-8}$	$3,{5275.10}^{-8}$
$max(e)$	$9,{8520.10}^{-4}$	$9,{7021.10}^{-4}$
$MA{E}_e$	$1,{2436.10}^{-4}$	$1,{1753.10}^{-4}$
time (s)	$58.2112$	$3.2024$

Table 8. The influence of the dimensionality reduction in the control error.

	NN Controller	KPCA NN controller
${\eta }_c$	Variable	Variable
$MS{E}_{e_c}$	$0.0074$	$0.0027$
$MA{E}_{e_c}$	$0.0223$	$0.0165$
$max(e_c)$	$0.5000$	$0.3000$
time (s)	$62.2231$	$2.3321$

Table 9. The influence of the dimensionality reduction in the identification error.

	NN model	KPCA NN model
$\eta$	Variable	Variable
$MS{E}_e$	$2,{8501.10}^{-8}$	$1,{6541.10}^{-8}$
$max(e)$	$9,{4522.10}^{-4}$	$8,{9819.10}^{-4}$
$MA{E}_e$	$1,{1140.10}^{-4}$	$1,{0016.10}^{-4}$
time (s)	$17.5536$	$2.0092$

Table 10. The influence of the dimensionality reduction in the control error.

	NN controller	KPCA NN controller
${\eta }_c$	Variable	Variable
$MS{E}_{e_c}$	$0.0074$	$0.0028$
$MA{E}_{e_c}$	$0.0233$	$0.0175$
$max(e_c)$	$0.5000$	$0.3073$
time (s)	$38.4532$	$3.00941$

Figure 11. The control system output, the desired values and the control error.

Figure 12. The control law $u_1$ and $u_2$ trajectories.

Table 11. The influence of the dimensionality reduction in the model error.

	KPCA NN model
${\eta }_i$ $i=1:2$	Variable
$MS{E}_{e_1}$	$5,{4430.10}^{-7}$
$MS{E}_{e_2}$	$8,{1510.10}^{-7}$
$MA{E}_{e_1}$	$2,{4000.10}^{-4}$
$MA{E}_{e_2}$	$1,{3635.10}^{-4}$
$max(e_1)$	$0.0023$
$max(e_2)$	$0.0053$
time (s)	$13,8162$

Table 12. The influence of the dimensionality reduction in the control error.

	KPCA NN controller
${\eta }_{ic}$	Variable
$MS{E}_{e_{1c}}$	$5,{4430.10}^{-7}$
$MS{E}_{e_{2c}}$	$8,{1510.10}^{-7}$
$MA{E}_{e_{1c}}$	$0,0057$
$MA{E}_{e_{2c}}$	$0,0091$
$max(e_{1c})$	$0,4251$
$max(e_{2c})$	$0,5000$
time (s)	$34,7886$