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

  • * Corresponding author: Jiayang Dai

    * Corresponding author: Jiayang Dai 

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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  • 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.

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


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  • Figure 1.  Process flow chart of goethite process in a smelting enterprise

    Figure 2.  AFCN model of 1# reactor for goethite process

    Figure 3.  AFCN simulation after CSO learning

    Figure 4.  AFCN simulation after ICCSO learning

    Figure 5.  FCN simulation after ICCSO learning

    Figure 6.  Comparison of ICCSO, GA and PSOA

    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
     | Show Table
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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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
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