**7.3 Simulation results of the centralized indirect (SM) adaptive neural control with Iterm using L-M learning**

In this case the indirect adaptive I-term control is a sum of the I-term control signal and the SM control, computed using the state and parameter information issued from the RTNN-1 neural identifier. The control signals are shown on Fig. 17. The X1 control simulation results are given on Fig. 18-20. The simulation results of SMC are obtained on-line during 1000 days with a step of 0.1 day. The given on Fig. 18-20 graphical results of I-term SMC demonstrated smooth behavior. It could be seen also that the L-M learning converge fast and the I-term remove the constant noise Of, and the plant uncertainties.

Fig. 17. Plant input control signals generated by the I-term centralized indirect SMC: a) Sin1, and b) Sin2

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Identification and I-Term Control Using Recurrent Neural Network Model 197

Fig. 20. Three dimensional plot of the I-term indirect SM control results of the plant output

The Fig. 21 illustrated the behavior of the SMC system without I-term perturbed by a

Fig. 21. Graphical simulation results of the indirect SM control without I-term of the plant output X1 vs. system reference in four measurement points for the total time of L-M

X1 in four measurement points of L-M learning : z=0.2H, z=0.4H, z=0.6H, z=0.8H

constant noise Of, causing a big error of reference tracking.

learning: a) z=0.2H, b) z=0.4H, c) z=0.6H, d) z=0.8H

Fig. 18. Graphical simulation results of the I-term indirect SM control of the plant output X1 vs. system reference in four measurement points for the total time of L-M learning: a) z=0.2H, b) z=0.4H, c) z=0.6H, d) z=0.8H

Fig. 19. Graphical simulation results of the I-term indirect SM control of the plant output X1 vs. system reference in four measurement points for the beginning of L-M learning: a) z=0.2H, b) z=0.4H, c) z=0.6H, d) z=0.8H

Fig. 18. Graphical simulation results of the I-term indirect SM control of the plant output X1 vs. system reference in four measurement points for the total time of L-M learning: a)

Fig. 19. Graphical simulation results of the I-term indirect SM control of the plant output X1 vs. system reference in four measurement points for the beginning of L-M learning: a)

z=0.2H, b) z=0.4H, c) z=0.6H, d) z=0.8H

z=0.2H, b) z=0.4H, c) z=0.6H, d) z=0.8H

Fig. 20. Three dimensional plot of the I-term indirect SM control results of the plant output X1 in four measurement points of L-M learning : z=0.2H, z=0.4H, z=0.6H, z=0.8H

The Fig. 21 illustrated the behavior of the SMC system without I-term perturbed by a constant noise Of, causing a big error of reference tracking.

Fig. 21. Graphical simulation results of the indirect SM control without I-term of the plant output X1 vs. system reference in four measurement points for the total time of L-M learning: a) z=0.2H, b) z=0.4H, c) z=0.6H, d) z=0.8H

Centralized Distributed Parameter Bioprocess

b) z=0.4H, c) z=0.6H, d) z=0.8H

b) z=0.4H, c) z=0.6H, d) z=0.8H

Identification and I-Term Control Using Recurrent Neural Network Model 199

Fig. 23. Graphical simulation results of the I-term optimal control of the plant output X1 vs. system reference in four measurement points for the total time of L-M learning: a) z=0.2H,

Fig. 24. Graphical simulation results of the I-term optimal control of the plant output X1 vs. system reference in four measurement points for the beginning of L-M learning: a) z=0.2H,

The given on Fig. 18-20 graphical results of I-term SMC showed smooth exponential behavior, fast convergence and the removal of the constant noise terms. The Fig. 21 showed that the constant perturbation in the input of the plant caused a deviation of the plant output X1 with respect of the set point R1 and this occurred for all other plant output signals and measurement points. The final MSE for all point output variables are given in Table 5.


Table 5. MSE of the I-term indirect (SM) centralized control of the bioprocess plant variables in all measurement points

The final MSE given on Table 5 possessed small values (2.1345E-5 in the worse case).

## **7.4 Simulation results of the centralized I-term optimal control using neural identifier and L-M RTNN learning**

The integral term extended the identified local linear plant model so it is part of the indirect optimal control algorithm. The generated by the optimal control plant input signals are given on Fig. 22. The Fig. 23-25 illustrated the X1 I-term optimal control results. The MSE numerical results for all final process variable and measurement points control results, given on Table 6 possessed small values (1.4949E-5 in the worse case).

Fig. 22. Plant input control signals generated by the I-term centralized optimal control: a) Sin1, and b) Sin2


Table 6. MSE of the I-term opt. control of the bioprocess plant variables in all meas. points

The given on Fig. 18-20 graphical results of I-term SMC showed smooth exponential behavior, fast convergence and the removal of the constant noise terms. The Fig. 21 showed that the constant perturbation in the input of the plant caused a deviation of the plant output X1 with respect of the set point R1 and this occurred for all other plant output signals and measurement points. The final MSE for all point output variables are given in Table 5.

*Collocation point X1 X2 S1 / S1T S2 / S2T* z=0.2 2.6969E-8 1.7122E-7 9.9526E-6 2.1347E-5 z=0.4 1.3226E-8 1.2511E-7 5.2323E-6 1.2903E-5 z=0.6 1.0873E-8 6.5339E-8 3.2234E-6 7.0511E-6 z=0.8 5.9589E-9 4.4750E-8 1.6759E-6 4.4548E-6 Recirculation tank 1.1842E-6 2.5147E-6 Table 5. MSE of the I-term indirect (SM) centralized control of the bioprocess plant variables

The final MSE given on Table 5 possessed small values (2.1345E-5 in the worse case).

on Table 6 possessed small values (1.4949E-5 in the worse case).

**7.4 Simulation results of the centralized I-term optimal control using neural identifier** 

The integral term extended the identified local linear plant model so it is part of the indirect optimal control algorithm. The generated by the optimal control plant input signals are given on Fig. 22. The Fig. 23-25 illustrated the X1 I-term optimal control results. The MSE numerical results for all final process variable and measurement points control results, given

Fig. 22. Plant input control signals generated by the I-term centralized optimal control: a)

*Collocation point X1 X2 S1 / S1T S2 / S2T*  z=0.2 2.06772E-8 1.5262E-7 9.3626E-6 1.4949E-5 z=0.4 1.3819E-8 7.5575E-8 5.6917E-6 1.0197E-5 z=0.6 1.8115E-8 4.7505E-8 2.8872E-6 6.1763E-6 z=0.8 1.5273E-8 5.9744E-8 1.6295E-6 4.2868E-6 Recirculation tank 1.3042E-6 2.5136E-6 Table 6. MSE of the I-term opt. control of the bioprocess plant variables in all meas. points

in all measurement points

**and L-M RTNN learning** 

Sin1, and b) Sin2

Fig. 23. Graphical simulation results of the I-term optimal control of the plant output X1 vs. system reference in four measurement points for the total time of L-M learning: a) z=0.2H, b) z=0.4H, c) z=0.6H, d) z=0.8H

Fig. 24. Graphical simulation results of the I-term optimal control of the plant output X1 vs. system reference in four measurement points for the beginning of L-M learning: a) z=0.2H, b) z=0.4H, c) z=0.6H, d) z=0.8H

Centralized Distributed Parameter Bioprocess

September 2006, CINVESTAV-IPN, Mexico

21459-3, Berlin Heidelberg New York

numbers 21-32, ISSN 1311-9702

953-7619-08-4, Vienna, Austria

2009) page numbers (1094-1114), ISSN 0884-8173

(37-52), ISSN 1897-8645 (print), ISSN 2080-2145 (on-line)

0884-8173

0654

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Fig. 25. Three dimensional plot of the I-term optimal control results of the plant output X1 in four measurement points of L-M learning : z=0.2H, z=0.4H, z=0.6H, z=0.8H

The given on Fig. 23-25 graphical results of I-term optimal control showed smooth exponential behavior, fast convergence and the removal of the constant noise terms.
