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基于变换函数与填充函数的模糊粒子群优化算法

吕柏权 张静静 李占培 刘廷章

吕柏权, 张静静, 李占培, 刘廷章. 基于变换函数与填充函数的模糊粒子群优化算法. 自动化学报, 2018, 44(1): 74-86. doi: 10.16383/j.aas.2018.c160547
引用本文: 吕柏权, 张静静, 李占培, 刘廷章. 基于变换函数与填充函数的模糊粒子群优化算法. 自动化学报, 2018, 44(1): 74-86. doi: 10.16383/j.aas.2018.c160547
LV Bai-Quan, ZHANG Jing-Jing, LI Zhan-Pei, LIU Ting-Zhang. Fuzzy Partical Swarm Optimization Based on Filled Function and Transformation Function. ACTA AUTOMATICA SINICA, 2018, 44(1): 74-86. doi: 10.16383/j.aas.2018.c160547
Citation: LV Bai-Quan, ZHANG Jing-Jing, LI Zhan-Pei, LIU Ting-Zhang. Fuzzy Partical Swarm Optimization Based on Filled Function and Transformation Function. ACTA AUTOMATICA SINICA, 2018, 44(1): 74-86. doi: 10.16383/j.aas.2018.c160547

基于变换函数与填充函数的模糊粒子群优化算法

doi: 10.16383/j.aas.2018.c160547
基金项目: 

国家自然科学基金 61273190

详细信息
    作者简介:

    张静静  上海大学机电工程与自动化学院硕士研究生.主要研究方向为非线性控制理论, 智能优化算法.E-mail:jingjiangzhang25@163.com

    李占培  上海大学机电工程与自动化学院博士研究生.主要研究方向为复杂系统的建模和控制, 建筑系统节能和控制.E-mail:woshilizhanpei@126.com

    刘廷章  上海大学机电工程与自动化学院教授.1996年获得西安交通大学机械工程博士学位.主要研究方向为复杂系统的建模和控制, 建筑系统节能和控制.E-mail:liutzh@stafi.shu.edu.cn

    通讯作者:

    吕柏权  上海大学机电工程与自动化学院副教授.1997年获得清华大学热能工程系博士学位.主要研究方向为计算智能, 非线性系统控制.本文通信作者.E-mail:lbq123188@aliyun.com

Fuzzy Partical Swarm Optimization Based on Filled Function and Transformation Function

Funds: 

National Natural Science Foundation of China 61273190

More Information
    Author Bio:

     Master student at the School of Mechatronic Engineering and Automation, Shanghai University. Her research interest covers nonlinear control theory and intelligent optimization algorithm

     Ph. D. candidate at the School of Mechatronic Engineering and Automation, Shanghai University. His research interest covers modeling and control for complex system, energy-saving and control for building system

     Professor at the School of Mechatronic Engineering and Automation, Shanghai University. He received his Ph. D. degree in mechanical engineering from Xi0an Jiaotong University in 1996. His research interest covers modeling and control for complex system, energysaving and control for building system

    Corresponding author: LV Bai-Quan  Associate professor at the School of Mechatronic Engineering and Automation, Shanghai University. He received his Ph. D. degree in thermal engineering from Tsinghua University in 1997. His research interest covers computational intelligence and nonlinear system control. Corresponding author of this paper
  • 摘要: 本文提出了一种基于变换函数与填充函数的模糊粒子群优化算法(Fuzzy partical swarm optimization based on filled function and transformation function,FPSO-TF).以基于不同隶属度函数的多回路模糊控制系统为基础,进一步结合变换函数与填充函数,使该算法减少了陷入局部最优的可能,又可以跳出局部极小值点至更小的点,快速高效地搜索到全局最优解.最后采用基准函数对此算法进行测试,并与几种不同类型的改进算法进行对比分析,验证了此算法的有效性与优越性.
    1)  本文责任编委 张毅
  • 图  1  单回路控制系统

    Fig.  1  The single loop control system

    图  2  变换函数图

    Fig.  2  Transformation function curve

    图  3  目标函数平面示意图

    Fig.  3  Objective function diagram

    图  4  $F4$变换前后的曲线图即$F4$和$T(1,0,F4)$

    Fig.  4  The curves of $F4$ before and after transformation: $F4$ and $T(1,0,F4)$

    图  5  $F11$变换前后的曲线图即$F11$和$T(1,0,F11)$

    Fig.  5  The stereogram of $F11$ before and after transformation: $F11$ and $T(1,0,F11)$

    图  6  多回路模糊控制系统框图

    Fig.  6  A multi-loop distributed fuzzy control system

    图  7  不同宽度的隶属度函数

    Fig.  7  The membership functions with different width

    图  8  多回路控制系统关系图

    Fig.  8  The relationship with all subsystems

    图  9  填充函数示意图

    Fig.  9  Graph of the filled function

    图  10  FPSO-TF算法流程图

    Fig.  10  Flowchart of FPSO-TF algorithm

    图  11  (a) $F$1, (b) $F$2, (c) $F$3, (d) $F$4, (e) $F$5, (f) $F$6 (这里曲线在每次迭代时都加78.33233140745), (g) $F$7, (h) $F$8, (i) $F$9, (j) $F$10, (k) $F$11, (l) $F$12 (这里曲线在每次迭代时都加0.000381827), (m) $F$13, (n) $F$14和(o) $F$15的收敛曲线

    Fig.  11  Convergence progress of the FPSO-TF on (a) $F$1, (b) $F$2, (c) $F$3, (d) $F$4, (e) $F$5, (f) $F$6 (where curves are obtained by subtracting 78.33233140745 from the true value of $F$6 for each iteration), (g) $F$7, (h) $F$8, (i) $F$9, (j) $F$10, (k) $F$11, (l) $F$12 (where curves are obtained by subtracting 0.000381827 from the true value of $F$12 for each iteration), (m) $F13$, (n) $F14$ and (o) $F15$

    表  1  测试函数

    Table  1  Test functions

    测试函数 维数 可行域 最优值/最优点
    $f_1=\sum\limits_{i=1}^n(x_i^2-10\cos(2\pi x_i)+10)$ 30 $[-5.12,5.12]^D$ 0.0/0, 0, $\cdots$, 0
    $f_2=-20\exp(-0.2\sqrt{\sum\limits_{i=1}^n\frac{x_i^2}{n}})-\exp(\sum\limits_{i=1}^n\frac{\cos(2\pi x_i)}{n})+20+\exp(1)$ 30 $[-32,32]^D$ 0.0/0, 0, $\cdots$, 0
    $f_3=\sum\limits_{i=1}^n\frac{x_i^2}{4\,000}-\prod\limits_{i=1}^n\cos(\frac{x_i}{\sqrt{i}})+1$ 30 $[-600,600]^D$ 0.0/0, 0, $\cdots$, 0
    $f_4=\frac{\pi}{n}(10\sin^2(\pi y_1)+(y_n-1)^2+\sum\limits_{i=1}^{n-1}(y_i-1)^2(1+10\sin^2(\pi y_{i+1}))) y_i=1+\frac{(1+x_i)}{4}$ 30 $[-50,50]^D$ $ 0.0/-1,\ -1, \cdots,\ -1 $
    $ f_5=\frac{1}{10}(\sin^2(3\pi x_1)+(x_n-1)^2(1+\sin^2(2\pi x_n))+ \quad \sum\limits_{i=1}^{n-1}(x_n-1)^2(1+\sin^2(3\pi x_{i+1})))$ 30 $[-50,50]^D$ 0.0/1, 1, $\cdots$, 1
    $f_6=\sum\limits_{i=1}^n{\frac{(x_i^4-16x_i^2+5x_i)}{n}}$ 30 $[-5,5]^D$ $-78.3323/-2.90353,-2.90353, \cdots,-2.90353$
    $f_7=\sum\limits_{i=1}^{n-1}\left({100(x_i^2-x_{i+1})^2+(x_i-1)^2}\right)$ 30 $[-5,10]^D$ 0.0/1, 1, $\cdots$, 1
    $f_8=\sum\limits_{i=1}^nx_i^2$ 30 $[-100,100]^D$ 0.0/0, 0, $\cdots$, 0
    $f_9=\sum\limits_{i=1}^nx_i^4$ 30 $[-1.28,1.28]^D$ 0.0/0, 0, $\cdots$, 0
    $f_{10}=\sum\limits_{i=1}^n\left[{\sum\limits_{j=1}^i x_j}\right]^2$ 30 $[-100,100]^D$ 0.0/0, 0, $\cdots$, 0
    $f_{11}=\sum\limits_{i=1}^n\left[\sum\limits_{k=0}^{k\max}\left(a^k\cos(2\pi b^k(x_i+0.5))\right)\right]- \quad n\sum\limits_{k=0}^{k\max}\left(a^k\cos(2\pi b^k0.5)\right) a=0.5,b=3,k\max=20$ 30 $[-0.5,0.5]^D$ 0.0/0, 0, $\cdots$, 0
    $f_{12}=418.9829\times n-\sum\limits_{i=1}^nx_i\sin(\sqrt{|x_i|})$ 30 $[-500,500]^D$ 0.000381827/420.97, 420.97, $\cdots$, 420.97
    $f_{13}=\sum\limits_{i=1}^ni\times x_i^4+{\rm Random}(0,1)$ 30 $[-1.28,1.28]^D$ 0.0/0, 0, $\cdots$, 0
    $f_{14}=\frac{\pi}{n}\Big(10\sin^2(\pi y_1)+(y_n-1)^2+\sum\limits_{i=1}^{n-1}(y_i-1)^2(1+ \quad 10\sin^2(\pi y_{i+1}))\Big) +\sum\limits_{i=1}^nu_i(x_i,10,100,4), \\ y_i=1+\frac{1+x_i}{4} u_i(x_i,a,k,m)=\left\{\!\!\!\begin{array}{ll} k(x_i-a)^m, & x_i>a \\ 0, &-a \leq x_i \leq a \\ k(-x_i-a)^m, & x_i <a \end{array}\right. $ 30 $[-50,50]^D$ $0.0/-1,-1,\cdots,-1 $
    $f_{15}=\frac{1}{10}(\sin^2(3\pi x_1)+(x_n-1)^2(1+\sin^2(2\pi x_n))+\quad \sum\limits_{i=1}^{n-1}(x_n-1)^2(1+\sin^2(3\pi x_{i+1})))+\sum\limits_{i=1}^nu_i(x_i,5,100,4)$ 30 $[-50,50]^D$ 0.0/1, 1, $\cdots$, 1
    下载: 导出CSV

    表  2  参数初值

    Table  2  The initial values of the parameters

    测试函数 $d_1/d_2/d_3/d_4/d_5$ $\eta_1/\eta_2/\eta_3/\eta_4$ $a_1/R/N_\sigma$ $MN/TT/FMN$
    $F1 $ 1E-6/1E-18/30/1/12 2E-2/1E-4/1E-4/0.1 1E-3/-1/6 1 000/2 000/100
    $F2 $ 1E-4/1E-26/40/0.1/30 5E-3/1E-4/1E-5/0.1 1E-4/-1/5 5 000/8 000/100
    $F3 $ 1E-6/1E-15/25/0.3/18 1E-4/1E-3/1E-3/1E-2 1E-3/-1/3 15 000/9 000/60
    $F4$ 1E-4/1E-18/35/0.2/15 1.6/0.1/1E-5/0.1 1E-4/-1/5 12 000/20 000/100
    $F5 $ 1E-4/1E-18/35/0.05/18 1.4/1E-2/1E-5/0.1 1E-3/0/3 8 000/20 000/100
    $F6 $ 1E-5/1E-11/12/1/13 0.6/1E-4/1E-3/1E-2 1E-3/-1/5 1 000/2 000/60
    $F7 $ 1E-15/1E-16/19/0.2/10 1/1E-3/1E-4/1E-3 1E-3/-1/4 190 000/40 000/100
    $F8 $ 1E-5/1E-24/35/1/30 6E-2/1E-3/1E-4/6E-3 1E-4/-1/3 30 000/4 000/100
    $F9$ 1E-3/1E-18/25/1/25 3E-2/1E-4/1E-4/1E-3 1E-3/-1/5 8 000/5 000/60
    $F10$ 1E-9/1E-21/35/1/30 0.6/0.3/1E-4/0.8 1E-4/-1/5 50 000/250 000/130
    $F11$ 1E-9/1E-20/25/1/25 3E-3/1E-4/1E-3/1E-3 1E-4/-1/5 20 000/18 000/100
    $F12 $ 1E-6/1E-4/10/1/14 6E-4/1E-3/1E-5/0.1 1E-4/-1/5 300/1 000/60
    $F13$ 1E-15/1E-10/20/1/11 1/1E-3/1E-4/1E-2 1E-4/-1/10 80 000/6 000/60
    $F14 $ 1E-4/1E-18/35/0.2/15 1.5/0.1/1E-5/0.1 1E-4/-1/3 12 000/20 000/100
    $F15 $ 1E-4/1E-18/35/0.05/18 1.4/0.01/1E-5/0.1 1E-4/0/3 10 000/20 000/100
    下载: 导出CSV

    表  3  仿真结果

    Table  3  Results of simulation

    Functions Average value Best value Worst value Confidence interval N/30 CPU times (s) SA
    $F1 $ 0 0 0 0±0 30/30 12.7 1E-15
    $F2$ 2.37E-16 0 3.55E-15 2.37E-16±2.90724E-31 30/30 58.2 1E-10
    $F3 $ 0 0 0 0±0 30/30 42.7 1E-15
    $F4$ 1.57E-32 1.57E-32 1.60E-32 1.57E-32±1.5E-69 30/30 119 1E-15
    $F5$ 1.35E-32 1.35E-32 1.35E-32 1.35E-32±2.5E-95 30/30 118 1E-15
    $F6$ -78.33233141 -78.33233141 -78.33233141 -78.33233141±1.9E-24 30/30 7.3 1E-7
    $F7 $ 3.23E-22 2.61E-26 4.39E-21 3.23E-22±3.66E-43 30/30 205 1E-15
    $F8 $ 1.21E-78 2.91E-221 3.60E-77 1.21E-78±1.54E-155 30/30 42.8 1E-15
    $F9$ 1.64E-52 1.94E-53 4.59E-52 1.64E-52±5.11E-105 30/30 18.2 1E-15
    $F10$ 2.74E-22 3.35E-32 1.11E-21 2.74E-22±5.12E-44 30/30 219 1E-15
    $F11$ 2.75E-07 0 7.87E-06 2.75E-07±7.39168E-13 30/30 1 816 1E-5
    $F12$ 3.82E-04 3.82E-04 3.82E-04 3.82E-04±2.71969E-38 30/30 4.1 1E-8
    $F13$ 6.36E-04 3.03E-05 2.54E-03 6.36E-04±1.7E-07 30/30 30.9 1E-2
    $F14$ 1.57E-32 1.57E-32 1.6E-32 1.57E-32±1.55E-69 30/30 126 1E-15
    $F15$ 1.35E-32 1.35E-32 1.35E-32 1.35E-32±2.5E-95 30/30 119 1E-15
    下载: 导出CSV

    表  4  与现有算法的结果比较

    Table  4  Comparison with other algorithms

    Functions FPSO-TF FEP OGA/Q CMA-ES JADE OLPSO-L OLPSO-G
    $F1$ Mean 0 4.6E-2 0 1.76E+2 0 0 1.07
    SD 0 1.2E-2 0 13.89 0 0 0.99
    Rank 1 2 1 4 1 1 3
    t-test - -19.17 0 -63.36 0 0 -8.5
    $F2$ Mean 2.37E-16 1.8E-2 4.4E-16 12.124 4.4E-15 4.14E-15 7.98E-015
    SD 2.90724E-31 2.1E-3 3.99E-17 9.28 0 0 2.03E-15
    Rank 1 6 2 7 4 3 5
    t-test - -42.8 -5 -6.5 -7.2E+16 -6.5+16 -19
    $F3$ Mean 0 1.6E-2 0 9.59E-16 2.E-4 0 4.8E-3
    SD 0 2.2E-2 0 3.5E-16 1.4E-3 0 8.63E-3
    Rank 1 5 1 2 3 1 4
    t-test - -3.64 0 -13.7 -0.7 0 -2.7
    $F7$ Mean 3.23E-22 5.06 0.75 2.33E-15 0.32 1.26 21.52
    SD 3.66E-43 5.87 0.11 7.7E-16 1.1 1.4 29.92
    Rank 1 6 4 2 3 5 7
    t-test - -4.26 -34.1 -15.1 -1.45 -4.5 -1.2
    $F8$ Mean 1.21E-78 5.7E-4 0 4.56E-16 1.3E-54 1.11E-38 4.1E-54
    SD 1.54E-155 1.3E-4 0 1.13E-16 9.2E-54 1.3E-38 6.32E-54
    Rank 2 7 1 5 3 6 4
    t-test - -21.9 0 -20.4 -0.7 --4.3 -3.2
    $F12$ Mean 3.82E-4 14.98 3.03E-2 3.15E+3 7.1 3.82E-4 3.84E+2
    SD 2.71969E-38 52.6 6.45E-4 5.79E+2 28 0 2.17E+2
    Rank 1 4 2 6 3 1 5
    t-test - -1.4 -234.8 -27.2 -1.3 0 -8.8
    $F13$ Mean 6.36E-4 7.6E-3 6.3E-3 5.92E-2 6.8E-4 1.64E-2 1.16E-2
    SD 1.7E-7 2.6E-3 4.07E-4 1.73E-2 2.5E-4 3.25E-3 4.1E-3
    Rank 1 4 3 7 2 6 5
    t-test - -13.5 -71.3 -16.9 -0.88 -26.3 -12.8
    $F14$ Mean 1.57E-32 9.2E-6 6.02E-6 1.63E-15 1.6E-32 1.57E-32 1.57E-32
    SD 1.55E-69 3.6E-6 1.16E-6 4.93E-16 5.5E-48 2.79E-48 1.01E-33
    Rank 1 5 4 3 2 1 1
    t-test - -12.8 -1.65 -16 -2.7E+18 0 0
    $F15$ Mean 1.35E-32 1.6E-4 1.87E-4 1.71E-15 1.4E-32 1.35E-32 4.39E-4
    SD 2.5E-95 7.3E-5 2.62E-5 3.7E-16 1.1E-47 5.6E-48 2.2E-3
    Rank 1 4 5 3 2 1 6
    t-test - -10 -35.6 -23 -0.23E+15 0 -0.99
    Ave.rank 1.1 4.7 2.5 4 2.6 4.9 5.6
    Final rank 1 5 2 4 3 6 7
    下载: 导出CSV
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出版历程
  • 收稿日期:  2016-07-24
  • 录用日期:  2016-12-18
  • 刊出日期:  2018-01-20

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