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May  2021, 17(3): 1269-1287. doi: 10.3934/jimo.2020021

A goethite process modeling method by Asynchronous Fuzzy Cognitive Network based on an improved constrained chicken swarm optimization algorithm

Junjie Peng , Ning Chen , Jiayang Dai , and  Weihua Gui

School of Automation, Central South University, Changsha 410083, China

* Corresponding author: Jiayang Dai

Received  June 2019 Revised  August 2019 Published  May 2021 Early access  January 2020

Fund Project: This research is supported by the Program of National Science Foundation of China, grant number 61673339 and the Program of National Science Foundation of Hunan Province, grant number 2017JJ2329

In order to solve the problem that the mechanism model of nonlinear system with uncertainty is difficult to establish, a modeling method of nonlinear system based on Asynchronous Fuzzy Cognitive Network (AFCN) is proposed. This method combines fuzzy cognitive network with time-lag system, and extends the node state values and weights of fuzzy cognitive network to the time interval, which enhances the adaptability of the model. At the same time an improved constrained chicken swarm optimization algorithm(ICCSOA) is proposed to identify model parameters of AFCN. A lag matrix corresponding to the actual measured values of the system lag of the nodes in the AFCN model is introduced, and a correction term including the difference between the measured values and the predicted values of the system is added to the model parameter updating mechanism. The simulation experiment results of goethite process system shows this modeling method can be used to model complex systems with uncertainties or partial missing data. The control model based on the established system model can make correct control decisions. ICCSOA has the advantages of fast convergence speed and accurate learning results, whose global search ability and convergence accuracy are higher than those of CSO algorithm, which can be widely used to the modeling of uncertain systems.

Keywords: Fuzzy cognitive network, time-lag system, chicken swarm optimization algorithm, terminal constraint.
Mathematics Subject Classification: 90C26, 90C70, 90C59.
Citation: Junjie Peng, Ning Chen, Jiayang Dai, Weihua Gui. A goethite process modeling method by Asynchronous Fuzzy Cognitive Network based on an improved constrained chicken swarm optimization algorithm. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1269-1287. doi: 10.3934/jimo.2020021
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N. Chen, J. Y. Dai, X. J. Zhou, Q. Q. Yang and W. H. Gui, Distributed model predictive control of iron precipitation process by goethite based on dual iterative method, International Journal of Control Automation and Systems, 17 (2019), 1233-1245.  doi: 10.1007/s12555-017-0742-6.

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[20]

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[21]

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[22]

J. Solana-Gutiérrez, G. Rincón, C. Alonso and D. García-de-Jalón, Using fuzzy cognitive maps for predicting river managementresponses: A case study of the Esla River basin, Spain, Ecological Modelling, 360 (2017), 260-269. 

[23]

W. Stach, L. Kurgan, W. Pedrycz and M. Reformat, Genetic learning of fuzzy cognitive maps, Fuzzy Sets and Systems, 153 (2005), 371-401.  doi: 10.1016/j.fss.2005.01.009.

[24]

D. H. Wu, S. P. Xu and F. Kong, Convergence analysis and improvement of the chicken swarm optimization algorithm, IEEE Access, 4 (2019), 9400-9412.  doi: 10.1109/ACCESS.2016.2604738.

[25]

B. Wang, W. Li, X. H. Chen and H. H. Chen, Improved chicken swarm algorithms based on chaos theory and its application in wind power interval prediction, Mathematical Problems in Engineering, (2019), Art. ID 1240717, 10 pp. doi: 10.1155/2019/1240717.

[26]

Z. Q. Wu, D. Q. Yu and X. H. Kang, Application of improved chicken swarm optimization for MPPT in photovoltaic system, Optimal Control Applications and Method, 39 (2018), 1029-1042.  doi: 10.1002/oca.2394.

[27]

X. W. Yu, L. X. Zhou and X. Y. Li, A novel hybrid localization scheme for deep mine based on wheel graph and chicken swarm optimization, Computer Networks, 154 (2019), 73-78.  doi: 10.1016/j.comnet.2019.02.011.

[28]

Y. L. Zhang, Modeling and Control of Dynamic System Based on Fuzzy Cognitive Maps, Dalian University of Technology, 2012.

show all references

References:
[1]

A. P. Antigoni and P. P. Groumpos, Modeling of parkinson's disease using fuzzy cognitive maps and non-linear hebbian learning, International Journal on Artificial Intelligence Tools, 23 (2014), 1450010. 

[2]

N. Chen, J. Y. Dai, X. J. Zhou, Q. Q. Yang and W. H. Gui, Distributed model predictive control of iron precipitation process by goethite based on dual iterative method, International Journal of Control Automation and Systems, 17 (2019), 1233-1245.  doi: 10.1007/s12555-017-0742-6.

[3]

N. Chen, J. Y. Dai, W. H. Gui, Y. Q. Guo and J. Q. Zhou, A hybrid prediction model with a selectively updating strategy for iron removal process in zinc hydrometallurgy, Science China Information Sciences, 63 (2020), 119205.  doi: 10.1007/s11432-018-9711-2.

[4]

N. Chen, Y. Fan, W. H. Gui, C. H. Yang and Z. H. Jiang, Hybrid modeling and control of iron precipitation by goethite process, Chinese Journal of Nonferrous Metals, 24 (2014), 254-261. 

[5]

B. Christen, C. Kjeldsen, T. Dalgaard and J. Martin-Ortega, Can fuzzy cognitive mapping help in agricultural policy design and communication?, Land Use Policy, 45 (2015), 64-75.  doi: 10.1016/j.landusepol.2015.01.001.

[6]

N. Chen, J. Q. Zhou, J. J. Peng, W. H. Gui and J. Y. Dai, Modeling of goethite iron precipitation process based on time-delay fuzzy gray cognitive network, Journal of Central South University, 26 (2019), 63-74.  doi: 10.1007/s11771-019-3982-1.

[7]

N. Chen, J. J. Peng, L. Wang, Y. Q. Guo and W. H. Gui, Fuzzy grey cognitive networks modeling and its application, Acta Automatica Sinica, 44 (2018), 1227-1236. 

[8]

N. Chen, L. Wang, J. J. Peng, B. Liu and W. H. Gui, Improved nonlinear Hebbian learning algorithm based on fuzzy cognitive networks model, Control Theory and Applications, 33 (2016), 1273-1280. 

[9]

Y. G. Deng, Q. Y. Chen, Z. L. Yin and P. M. Zhang, Iron removal from zine leaching solution by goethite method, Non-ferrous Metal, 62 (2014), 80-84. 

[10]

Z. Djaafar, A. Yahia and N. Farid, Multi-objective chicken swarm optimization: A novel algorithm for solving multi-objective optimization problems, Computers and Industrial Engineering, 129 (2019), 377-391. 

[11]

S. Fatahi and H. Moradi, A fuzzy cognitive map model to calculate a user's desirability based on personality in e-learning environments, Computers in Human Behavior, 63 (2016), 272-281.  doi: 10.1016/j.chb.2016.05.041.

[12]

B. Kosko, Fuzzy cognitive maps, International Journal of Man-Machine Studie, 24 (1986), 65-75.  doi: 10.1016/S0020-7373(86)80040-2.

[13]

V. Kreinovich and C. D. Stylios, Why fuzzy cognitive maps are efficient, International Journal of Computers Communications & Control, 10 (2015), 825-833.  doi: 10.15837/ijccc.2015.6.2073.

[14]

T. Kottas, D. Stimoniaris and D. Tsiamitros, New operation scheme and control of Smart Grids using Fuzzy Cognitive Networks, PowerTech, 2015 IEEE Eindhoven, 63 (2015), 1-5.  doi: 10.1109/PTC.2015.7232563.

[15]

D. B. Li and J. M. Jiang, Present situation and development trend of zinc smelting technology at home and abroad, China Metal Bulletin, 6 (2015), 41-44. 

[16]

P. C. Marchal, J. G. García and J. G. Ortega, Application of fuzzy cognitive maps and run-to-run control to a decision support system for global set-point determination, IEEE Transactions on Systems Man & Cybernetics Systems, 47 (2017), 2256-2267.  doi: 10.1109/TSMC.2016.2646762.

[17]

A. Mourhir, E. I. Papageorgiou, K. Kokkinos and T. Rachidi, Exploring precision farming scenarios using Fuzzy Cognitive Maps, Sustainability, 9 7 (2017), 1241. doi: 10.3390/su9071241.

[18]

X. B. Meng, Y. Liu and X. Z. Gao, A new bio-inspired algorism: Chicken swarm optimization, Proc of International Conference in Swarm of Intelligence, Cham: Springer, (2014), 86-94. 

[19]

M. Obiedat and S. Samarasinghe, A novel semi-quantitative Fuzzy Cognitive Map model for complex systems for addressing challenging participatory real life problems, Applied Soft Computing, 48 (2016), 91-110.  doi: 10.1016/j.asoc.2016.06.001.

[20]

E. I. Papageorgiou, K. D. Aggelopoulou and T. A. Gemtos, Yield prediction in apples using Fuzzy Cognitive Map learning approach, Computers & Electronics in Agriculture, 91 (2013), 19-29.  doi: 10.1016/j.compag.2012.11.008.

[21]

K. E. Parsopoulos, E. I. Papagergiou, P. P. Groumpos and M. N. Vrahatis, A first study of fuzzy cognitive maps learning using particle swarm optimization, Proceedings of IEEE Congress on Evolutionary Computation 2003, (2003), 1440-1447.  doi: 10.1109/CEC.2003.1299840.

[22]

J. Solana-Gutiérrez, G. Rincón, C. Alonso and D. García-de-Jalón, Using fuzzy cognitive maps for predicting river managementresponses: A case study of the Esla River basin, Spain, Ecological Modelling, 360 (2017), 260-269. 

[23]

W. Stach, L. Kurgan, W. Pedrycz and M. Reformat, Genetic learning of fuzzy cognitive maps, Fuzzy Sets and Systems, 153 (2005), 371-401.  doi: 10.1016/j.fss.2005.01.009.

[24]

D. H. Wu, S. P. Xu and F. Kong, Convergence analysis and improvement of the chicken swarm optimization algorithm, IEEE Access, 4 (2019), 9400-9412.  doi: 10.1109/ACCESS.2016.2604738.

[25]

B. Wang, W. Li, X. H. Chen and H. H. Chen, Improved chicken swarm algorithms based on chaos theory and its application in wind power interval prediction, Mathematical Problems in Engineering, (2019), Art. ID 1240717, 10 pp. doi: 10.1155/2019/1240717.

[26]

Z. Q. Wu, D. Q. Yu and X. H. Kang, Application of improved chicken swarm optimization for MPPT in photovoltaic system, Optimal Control Applications and Method, 39 (2018), 1029-1042.  doi: 10.1002/oca.2394.

[27]

X. W. Yu, L. X. Zhou and X. Y. Li, A novel hybrid localization scheme for deep mine based on wheel graph and chicken swarm optimization, Computer Networks, 154 (2019), 73-78.  doi: 10.1016/j.comnet.2019.02.011.

[28]

Y. L. Zhang, Modeling and Control of Dynamic System Based on Fuzzy Cognitive Maps, Dalian University of Technology, 2012.

Figure 1.  Process flow chart of goethite process in a smelting enterprise
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Figure 2.  AFCN model of 1# reactor for goethite process
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Figure 3.  AFCN simulation after CSO learning
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Figure 4.  AFCN simulation after ICCSO learning
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Figure 5.  FCN simulation after ICCSO learning
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Figure 6.  Comparison of ICCSO, GA and PSOA
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Table 1.  Standard test functions for testing algorithm performance
Test Function Expression Symbol Range of values Value of optimal solution
Sphere $ f(x)=\sum\limits_{i=1}^n {x_i^2 } $ F1 [-100, 100] 0
Rosenbrock $ \begin{array}{l} f(x)=-a\cdot \exp (-b\cdot \sqrt {\frac{1}{n}\sum\limits_{i=1}^n {x_i^2 } } ) \\ -\exp [\frac{1}{n}\sum\limits_{i=1}^n {\cos (cx_i } )]+a+e \end{array} $ F2 [-30, 30] 0
High Conditioned Elliptic $ f(x)=\sum\limits_{i=1}^n a ^{\frac{i-1}{n-1}x_i^2 } $ F3 [-100, 100] 0
Bent Cigar $ f(x)=x_1^2 +\sum\limits_{i=2}^n {ax_i^2 } $ F4 [-100, 100] 0
Discus $ f(x)=ax_1^2 +\sum\limits_{i=2}^n {x_i^2 } $ F5 [-100, 100] 0
Rotated hyper-ellipsoid $ f(x)=\sum\limits_{i=1}^n {\sum\limits_{j=1}^i {x_j^2 } } $ F6 [-100, 100] 0
Rotated rastrigin $ f(x)=\sum\limits_{i=2}^n {(x_i^2 } -a\cdot \cos 2\pi x_i +a) $ F7 [-100, 100] 0
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Test Function Expression Symbol Range of values Value of optimal solution
Sphere $ f(x)=\sum\limits_{i=1}^n {x_i^2 } $ F1 [-100, 100] 0
Rosenbrock $ \begin{array}{l} f(x)=-a\cdot \exp (-b\cdot \sqrt {\frac{1}{n}\sum\limits_{i=1}^n {x_i^2 } } ) \\ -\exp [\frac{1}{n}\sum\limits_{i=1}^n {\cos (cx_i } )]+a+e \end{array} $ F2 [-30, 30] 0
High Conditioned Elliptic $ f(x)=\sum\limits_{i=1}^n a ^{\frac{i-1}{n-1}x_i^2 } $ F3 [-100, 100] 0
Bent Cigar $ f(x)=x_1^2 +\sum\limits_{i=2}^n {ax_i^2 } $ F4 [-100, 100] 0
Discus $ f(x)=ax_1^2 +\sum\limits_{i=2}^n {x_i^2 } $ F5 [-100, 100] 0
Rotated hyper-ellipsoid $ f(x)=\sum\limits_{i=1}^n {\sum\limits_{j=1}^i {x_j^2 } } $ F6 [-100, 100] 0
Rotated rastrigin $ f(x)=\sum\limits_{i=2}^n {(x_i^2 } -a\cdot \cos 2\pi x_i +a) $ F7 [-100, 100] 0
Table 2.  Comparison of test results of algorithms
Title Symbol Algorithms Optimal value Worst value Average value Standard deviation Stable step
F1 CSO 1.8488e-133 2.8082e-123 6.0362e-125 3.9681e-124 30
ICSO 7.0233e-133 6.4356e-125 3.1959e-126 1.1357e-125 26
ICCSO 6.9244e-182 2.0424e-163 5.0084e-165 3.2075e-164 24
F2 CSO 6.1449 7.97 6.9651 0.3031 33
ICSO 5.9715 7.2163 6.7664 0.3324 29
ICCSO 2.1398e-07 4.9115e-05 6.6865e-06 8.2512e-06 27
F3 CSO 6.7415e-127 2.1961e-117 7.328e-119 3.3125e-118 32
ICSO 9.006e-128 1.4093e-118 3.3872e-120 1.9906e-119 28
ICCSO 8.9815e-177 1.2078e-160 2.4737e-162 1.7073e-161 25
F4 CSO 3.1473e-127 1.0419e-117 2.9859e-119 1.4796e-118 36
ICSO 9.4786e-127 5.5614e-119 2.5308e-120 8.8336e-120 28
ICCSO 5.148e-178 5.2831e-161 1.1419e-162 7.372e-162 25
F5 CSO 1.9478e-131 9.9848e-123 5.2391e-124 1.7265-123 35
ICSO 9.8894e-132 2.2241e-123 4.6208e-125 3.1431e-124 27
ICCSO 4.9026e-181 1.0179e-166 5.8305e-168 2.5381e-167 23
F6 CSO 5.4415e-127 1.5173e-109 6.328e-129 3.3225e-117 37
ICSO 8.016e-138 1.9214e-140 4.6672e-110 2.1066e-119 31
ICCSO 9.148e-165 1.3078e-187 2.6728e-154 1.7953e-151 28
F7 CSO 4.1923e-172 1.0419e-117 2.1659e-139 1.4707e-118 36
ICSO 9.4554e-125 4.6634e-122 2.5325e-113 7.7543e-122 32
ICCSO 6.1579e-148 5.2635e-177 1.9719e-172 6.3823e-165 27
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Title Symbol Algorithms Optimal value Worst value Average value Standard deviation Stable step
F1 CSO 1.8488e-133 2.8082e-123 6.0362e-125 3.9681e-124 30
ICSO 7.0233e-133 6.4356e-125 3.1959e-126 1.1357e-125 26
ICCSO 6.9244e-182 2.0424e-163 5.0084e-165 3.2075e-164 24
F2 CSO 6.1449 7.97 6.9651 0.3031 33
ICSO 5.9715 7.2163 6.7664 0.3324 29
ICCSO 2.1398e-07 4.9115e-05 6.6865e-06 8.2512e-06 27
F3 CSO 6.7415e-127 2.1961e-117 7.328e-119 3.3125e-118 32
ICSO 9.006e-128 1.4093e-118 3.3872e-120 1.9906e-119 28
ICCSO 8.9815e-177 1.2078e-160 2.4737e-162 1.7073e-161 25
F4 CSO 3.1473e-127 1.0419e-117 2.9859e-119 1.4796e-118 36
ICSO 9.4786e-127 5.5614e-119 2.5308e-120 8.8336e-120 28
ICCSO 5.148e-178 5.2831e-161 1.1419e-162 7.372e-162 25
F5 CSO 1.9478e-131 9.9848e-123 5.2391e-124 1.7265-123 35
ICSO 9.8894e-132 2.2241e-123 4.6208e-125 3.1431e-124 27
ICCSO 4.9026e-181 1.0179e-166 5.8305e-168 2.5381e-167 23
F6 CSO 5.4415e-127 1.5173e-109 6.328e-129 3.3225e-117 37
ICSO 8.016e-138 1.9214e-140 4.6672e-110 2.1066e-119 31
ICCSO 9.148e-165 1.3078e-187 2.6728e-154 1.7953e-151 28
F7 CSO 4.1923e-172 1.0419e-117 2.1659e-139 1.4707e-118 36
ICSO 9.4554e-125 4.6634e-122 2.5325e-113 7.7543e-122 32
ICCSO 6.1579e-148 5.2635e-177 1.9719e-172 6.3823e-165 27
Table 3.  Algorithm analysis
Algorithm ICCSO GA PSO
$ A_3^{destination} $ 0.3600 0.3600 0.3600
$ A_3 $ 0.3559 0.3591 0.3490
Steady step 9 10 13
RMSE 0.0017 0.0022 0.0019
MAE 0.0011 0.0019 0.0015
MAX 0.0026 0.0028 0.0029
SD 0.0009 0.0017 0.0020
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Algorithm ICCSO GA PSO
$ A_3^{destination} $ 0.3600 0.3600 0.3600
$ A_3 $ 0.3559 0.3591 0.3490
Steady step 9 10 13
RMSE 0.0017 0.0022 0.0019
MAE 0.0011 0.0019 0.0015
MAX 0.0026 0.0028 0.0029
SD 0.0009 0.0017 0.0020
Table 4.  Errors analysis
Working conditions $ A_3^{destination} $ $ A_3 $ RMSE MAE MAX SD
SSS 0.1400 0.1421 0.0013 0.0011 0.0023 0.0011
SSB 0.1500 0.1490
SBS 0.1518 0.1533
SBB 0.1177 0.1199
MSS 0.2900 0.2906
MSB 0.2694 0.2689
MBS 0.3659 0.3659
MBB 0.0100 0.0120
BSS 0.1620 0.1627
BSB 0.1997 0.2002
BBS 0.3400 0.3407
BBB 0.3200 0.3189
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Working conditions $ A_3^{destination} $ $ A_3 $ RMSE MAE MAX SD
SSS 0.1400 0.1421 0.0013 0.0011 0.0023 0.0011
SSB 0.1500 0.1490
SBS 0.1518 0.1533
SBB 0.1177 0.1199
MSS 0.2900 0.2906
MSB 0.2694 0.2689
MBS 0.3659 0.3659
MBB 0.0100 0.0120
BSS 0.1620 0.1627
BSB 0.1997 0.2002
BBS 0.3400 0.3407
BBB 0.3200 0.3189
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