利用fminconodeMatLab进行优化 [英] Optimization using fmincon ODE Matlab

问题描述

问题是通过改变kst、x1、x5和xo，在(-8e-4到2e-4)的范围内找到x3的最佳(最大)值)

x5=5    %Input 2 (Input 2 is a state variable and could vary in range of 4 to 15 while performing 
                  optimization)
kst=1   %Input 3 (Input 3 is in terms of rate constant, it could vary from 0.1 to 2)
xo=4    %Input 4 (Input 4 is a state variable and could vary in range of 4 to 10)
x1=1e-7 %Input 1 could vary from 1e-9 to 1e-6

1. 脚本文件

    function rest = Scrpt1(t,X)
    x2 = X(1); 
    x3 = X(2); 
    
    %Parameters
    
    if t<15
    
    x1 = 1e-7; %Input 1 could vary from 1e-9 to 1e-6
    
    else x1 = 0;
    
    end
    
    x5=5 %Input 2 (Input 2 is a state variable and could vary in range of 4 to 15 while performing optimization)
    
    kst=1 %Input 3 (Input 3 is in terms of rate constant, it could vary from 0.1 to 2)

    xo=4 %Input 4 (Input 4 is a state variable and could vary in range of 4 to 10)
    
    k1 = 6e7;

    km1 = 0.20;

    km4 = 0.003;

    k3 = 2500.00;

    k4 = km4/9;

    km3 = km1;

    LAP=1.5

% Differential equations

    dx2dt = km1*x3 + km3*LAP - k1*x1*x2 + km4*x3 - k4*x2;

    dx3dt = k1*x1*x2 - km1*(x3+x5+xo) - k3*x3*kst;

    rest = [dx2dt; dx3dt];

    end

2. ODE解决方案函数文件

    options = odeset('InitialStep',0.0001,'RelTol',1e-09);
    
    [T,Y]=ode15s(@Scrpt1,[0 60],[9e-13,0],options);
    
    X3= Y(:,2);
    
    plot(T,X3)

如何使用fmincon或任何其他优化求解器来解决前面提到的寻找x3的最大值的优化问题。对于x5、kst、x0、x1的哪个值，我们得到最大x3？

推荐答案

首先，您必须将要优化的值添加为耦合的微分方程式的参数：

function rest = Scrpt1(t,X,X_opt)
    
    x5=X_opt(1);
    kst=X_opt(2);
    xo=X_opt(3);
    x1=X_opt(4);  
    x2 = X(1); 
    x3 = X(2); 
    
    %Parameters
    
    if t>=15
    x1 = 0;  
    end
    
    k1 = 6e7;
    km1 = 0.20;
    km4 = 0.003;
    k3 = 2500.00;
    k4 = km4/9;
    km3 = km1;
    LAP=1.5;

% Differential equations

    dx2dt = km1*x3 + km3*LAP - k1*x1*x2 + km4*x3 - k4*x2;
    dx3dt = k1*x1*x2 - km1*(x3+x5+xo) - k3*x3*kst;
    rest = [dx2dt; dx3dt];
    end

然后，您必须编写一个要最小化的包装函数。因为你想最大化x3，你必须在你的目标值上加一个减号。

function max_X3=fun(X_opt) 
    tspan=[0 60];
    y0=[9e-13,0];
    options = odeset('InitialStep',0.0001,'RelTol',1e-09);
    
    [~,y] = ode15s(@(t,y) Scrpt1(t,y,X_opt), tspan, y0,options);
    
    max_X3=-max(y(:,2));
end

最后，您可以这样使用fmincon：

% x5, kst, xo, x1
initial_search_point=[5, 1, 4, 1e-7]
lower_bounds=[4, 0.1, 4, 1e-9]
upper_bounds=[15, 2, 10, 1e-6]

fmincon(@fun,initial_search_point,[],[],[],[], lower_bounds,upper_bounds)

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