1.算法描述

遗传算法的原理

遗传算法GA把问题的解表示成“染色体”，在算法中也即是以二进制编码的串。并且，在执行遗传算法之前，给出一群“染色体”，也即是假设解。然后，把这些假设解置于问题的“环境”中，并按适者生存的原则，从中选择出较适应环境的“染色体”进行复制，再通过交叉，变异过程产生更适应环境的新一代“染色体”群。这样，一代一代地进化，最后就会收敛到最适应环境的一个“染色体”上，它就是问题的最优解。

其主要步骤如下：

1．初始化

选择一个群体，即选择一个串或个体的集合bi，i=1，2，...n。这个初始的群体也就是问题假设解的集合。一般取n＝30-160。

通常以随机方法产生串或个体的集合bi,i＝1，2，...n。问题的最优解将通过这些初始假设解进化而求出。

2．选择

根据适者生存原则选择下一代的个体。在选择时，以适应度为选择原则。适应度准则体现了适者生存，不适应者淘汰的自然法则。

给出目标函数f，则f(bi)称为个体bi的适应度。以

为选中bi为下一代个体的次数。

显然．从式(3—86)可知：

(1)适应度较高的个体，繁殖下一代的数目较多。

(2)适应度较小的个体，繁殖下一代的数目较少；甚至被淘汰。

这样，就产生了对环境适应能力较强的后代。对于问题求解角度来讲，就是选择出和最优解较接近的中间解。

3．交叉

对于选中用于繁殖下一代的个体，随机地选择两个个体的相同位置，按交叉概率P。在选中的位置实行交换。这个过程反映了随机信息交换；目的在于产生新的基因组合，也即产生新的个体。交叉时，可实行单点交叉或多点交叉。

拟合公式：

该公式经过化简实部、虚部分离得：

公式化简

令：

所以：

需要拟合的参数有：

定义GA优化目标函数如下所示：

2.仿真效果预览

matlab2022a仿真结果如下：

3.MATLAB核心程序

%根据遗传算法进行参数的拟合
MAXGEN = 2000;
NIND   = 400;
Chrom  = crtbp(NIND,14*10);
%14个变量的区间
Areas  = [0   ,0   ,0   ,0   ,0   ,0    ,0   ,0   ,0   ,0    ,0   ,0   ,0   ,0;
          10  ,1   ,100 ,500 ,100 ,5e8  ,1   ,1   ,1   ,1    ,2e11,2e8 ,1e4 ,1e2];
 
FieldD = [rep([10],[1,14]);Areas;rep([0;0;0;0],[1,14])];
 
epls_inf_NIND    = zeros(NIND,1);
deltas_NIND      = zeros(NIND,1);
delta_epls1_NIND = zeros(NIND,1);
delta_epls2_NIND = zeros(NIND,1);
delta_epls3_NIND = zeros(NIND,1);
delta_epls4_NIND = zeros(NIND,1);
beta1_NIND       = zeros(NIND,1);
beta2_NIND       = zeros(NIND,1);
beta3_NIND       = zeros(NIND,1);
beta4_NIND       = zeros(NIND,1);
fc1_NIND         = zeros(NIND,1);
fc2_NIND         = zeros(NIND,1);
fc3_NIND         = zeros(NIND,1);
fc4_NIND         = zeros(NIND,1);
 
epls_inf         = zeros(MAXGEN,1);
deltas           = zeros(MAXGEN,1);
delta_epls1      = zeros(MAXGEN,1);
delta_epls2      = zeros(MAXGEN,1);
delta_epls3      = zeros(MAXGEN,1);
delta_epls4      = zeros(MAXGEN,1);
beta1            = zeros(MAXGEN,1);
beta2            = zeros(MAXGEN,1);
beta3            = zeros(MAXGEN,1);
beta4            = zeros(MAXGEN,1);
fc1              = zeros(MAXGEN,1);
fc2              = zeros(MAXGEN,1);
fc3              = zeros(MAXGEN,1);
fc4              = zeros(MAXGEN,1);
Error            = zeros(MAXGEN,1);
 
gen              = 0;
 
 
 
for a=1:1:NIND 
    epls_inf_NIND(a)    = epls_inf_0;      
    deltas_NIND(a)      = deltas_0;
    delta_epls1_NIND(a) = delta_epls1_0;
    delta_epls2_NIND(a) = delta_epls2_0;
    delta_epls3_NIND(a) = delta_epls3_0;      
    delta_epls4_NIND(a) = delta_epls4_0;
    beta1_NIND(a)       = beta1_0;
    beta2_NIND(a)       = beta2_0;          
    beta3_NIND(a)       = beta3_0;
    beta4_NIND(a)       = beta4_0;
    fc1_NIND(a)         = fc1_0;      
    fc2_NIND(a)         = fc2_0;
    fc3_NIND(a)         = fc3_0;
    fc4_NIND(a)         = fc4_0;    
    %计算对应的目标值
    [epls_1,epls_2] = func_obj(f,...
                               epls_inf_NIND(a),...
                               deltas_NIND(a),...
                               delta_epls1_NIND(a),delta_epls2_NIND(a),delta_epls3_NIND(a),delta_epls4_NIND(a),...
                               beta1_NIND(a),beta2_NIND(a),beta3_NIND(a),beta4_NIND(a),...
                               fc1_NIND(a),fc2_NIND(a),fc3_NIND(a),fc4_NIND(a));
    for m = 1:length(f)
        tmps1(m) = ((e1(m)-epls_1(m))^2)/(e1(m)^2);               
        tmps2(m) = ((e2(m)-epls_2(m))^2)/(e2(m)^2);    
    end
    E = sum(tmps1)+sum(tmps2);
    J(a,1)  = E;
end
 
Objv  = (J+eps);
gen   = 0; 
 
while gen < MAXGEN;   
      gen
      FitnV=ranking(Objv);    
      Selch=select('sus',Chrom,FitnV);    
      Selch=recombin('xovsp', Selch,0.9);   
      Selch=mut( Selch,0.01);   
      phen1=bs2rv(Selch,FieldD);   
      for a=1:1:NIND  
          if  gen == 1
              epls_inf_NIND(a)    = epls_inf_0;      
              deltas_NIND(a)      = deltas_0;
              delta_epls1_NIND(a) = delta_epls1_0;
              delta_epls2_NIND(a) = delta_epls2_0;
              delta_epls3_NIND(a) = delta_epls3_0;      
              delta_epls4_NIND(a) = delta_epls4_0;
              beta1_NIND(a)       = beta1_0;
              beta2_NIND(a)       = beta2_0;          
              beta3_NIND(a)       = beta3_0;
              beta4_NIND(a)       = beta4_0;
              fc1_NIND(a)         = fc1_0;      
              fc2_NIND(a)         = fc2_0;
              fc3_NIND(a)         = fc3_0;
              fc4_NIND(a)         = fc4_0;           
          else
              epls_inf_NIND(a)    = phen1(a,1);      
              deltas_NIND(a)      = phen1(a,2);
              delta_epls1_NIND(a) = phen1(a,3);
              delta_epls2_NIND(a) = phen1(a,4);
              delta_epls3_NIND(a) = phen1(a,5);      
              delta_epls4_NIND(a) = phen1(a,6);
              beta1_NIND(a)       = phen1(a,7);
              beta2_NIND(a)       = phen1(a,8);          
              beta3_NIND(a)       = phen1(a,9);
              beta4_NIND(a)       = phen1(a,10);
              fc1_NIND(a)         = phen1(a,11);      
              fc2_NIND(a)         = phen1(a,12);
              fc3_NIND(a)         = phen1(a,13);
              fc4_NIND(a)         = phen1(a,14);  
          end
          
          %计算对应的目标值
          [epls_1,epls_2] = func_obj(f,...
                                     epls_inf_NIND(a),...
                                     deltas_NIND(a),...
                                     delta_epls1_NIND(a),delta_epls2_NIND(a),delta_epls3_NIND(a),delta_epls4_NIND(a),...
                                     beta1_NIND(a),beta2_NIND(a),beta3_NIND(a),beta4_NIND(a),...
                                     fc1_NIND(a),fc2_NIND(a),fc3_NIND(a),fc4_NIND(a));
          for m = 1:length(f)
              tmps1(m) = ((e1(m)-epls_1(m))^2)/(e1(m)^2);               
              tmps2(m) = ((e2(m)-epls_2(m))^2)/(e2(m)^2);    
          end
          E = sum(tmps1)+sum(tmps2);
          JJ(a,1)  = E;
      end 
      Objvsel=(JJ+eps);    
      [Chrom,Objv]=reins(Chrom,Selch,1,1,Objv,Objvsel);   
      gen=gen+1; 
 
      %保存参数收敛过程和误差收敛过程以及函数值拟合结论
      epls_inf(gen)         = mean(epls_inf_NIND);
      deltas(gen)           = mean(deltas_NIND);
      delta_epls1(gen)      = mean(delta_epls1_NIND);
      delta_epls2(gen)      = mean(delta_epls2_NIND);
      delta_epls3(gen)      = mean(delta_epls3_NIND);
      delta_epls4(gen)      = mean(delta_epls4_NIND);
      beta1(gen)            = mean(beta1_NIND);
      beta2(gen)            = mean(beta2_NIND);
      beta3(gen)            = mean(beta3_NIND);
      beta4(gen)            = mean(beta4_NIND);
      fc1(gen)              = mean(fc1_NIND);
      fc2(gen)              = mean(fc2_NIND);
      fc3(gen)              = mean(fc3_NIND);
      fc4(gen)              = mean(fc4_NIND);
      Error(gen)            = mean(JJ);
end 
 
 
MIN=min(Objv); 
for ttt=1:1:size(Objv)     
    if Objv(ttt)<=MIN         
       tt=ttt;         
       break;     
    end
end
 
 
epls_inf_best    = epls_inf_NIND(tt);      
deltas_best      = deltas_NIND(tt);
delta_epls1_best = delta_epls1_NIND(tt);
delta_epls2_best = delta_epls2_NIND(tt);
delta_epls3_best = delta_epls3_NIND(tt);      
delta_epls4_best = delta_epls4_NIND(tt);
beta1_best       = beta1_NIND(tt);
beta2_best       = beta2_NIND(tt);          
beta3_best       = beta3_NIND(tt);
beta4_best       = beta4_NIND(tt);
fc1_best         = fc1_NIND(tt);      
fc2_best         = fc2_NIND(tt);
fc3_best         = fc3_NIND(tt);
fc4_best         = fc4_NIND(tt);  
 
%计算对应的目标值
[epls_best1,epls_best2] = func_obj(f,...
                                 epls_inf_best,...
                                 deltas_best,...
                                 delta_epls1_best,delta_epls2_best,delta_epls3_best,delta_epls4_best,...
                                 beta1_best,beta2_best,beta3_best,beta4_best,...
                                 fc1_best,fc2_best,fc3_best,fc4_best);
%画图
 
 
figure;
subplot(211);
loglog(e1,'b','linewidth',2);
hold on
loglog(epls_best1,'r','linewidth',2);
legend('原始数据','拟合数据');
subplot(212);
loglog(e2,'b','linewidth',2);
hold on
loglog(epls_best2,'r','linewidth',2);
legend('原始数据','拟合数据');
02_016m