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; Weihua Gui

doi:10.3934/jimo.2020021

Article Contents

2021, Volume 17, Issue 3: 1269-1287. Doi: 10.3934/jimo.2020021

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

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

^* Corresponding author: Jiayang Dai
^* Corresponding author: Jiayang Dai

Received: June 2019

Revised: August 2019

Early access: January 2020

Published: May 2021

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

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Abstract

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.

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Mathematics Subject Classification: 90C26, 90C70, 90C59.

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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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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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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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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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