Question

因此，这是符号计算和优化问题的组合。我有一个phi21，phi31和phi32的三个微分方程组。我最终想要针对k1，k2，f和s优化方程式中的四个参数。我在下面的代码中设置了方程式和雅可比矩阵的构造：

syms phi21 phi31 phi32 k1 k2 f s a

w1 = (2*pi/24)*0.99;
w2 = (2*pi/24)*1.01;
w3 = (2*pi/24)*1.02;

f21 = (w2 - w1) + (1/3)*(-2*k1*sin(phi21) - 2*k2*sin(2*phi21) - k1*sin(phi31) - k2*sin(2*phi31) + k1*sin(phi32) + k2*sin(2*phi32)) + 2*f*cos((2*s - phi21)/2)*sin(-phi21/2);
f31 = (w3 - w1) + (1/3)*(k1*sin(phi21) + k2*sin(2*phi21) - k1*sin(phi31) - k2*sin(2*phi31) - 2*k1*sin(phi32) - 2*k2*sin(2*phi32)) + 2*f*cos((2*s - phi31)/2)*sin(-phi31/2);
f32 = (w3 - w2) + (1/3)*(-k1*sin(phi21) - k2*sin(2*phi21) - 2*k1*sin(phi31) - 2*k2*sin(2*phi31) - k1*sin(phi32) - k2*sin(2*phi32)) + + 2*f*cos((2*s - phi32)/2)*sin(-phi32/2);

df21d21 = diff(f21, phi21);
df21d31 = diff(f21, phi31);
df21d32 = diff(f21, phi32);

df31d21 = diff(f31, phi21);
df31d31 = diff(f31, phi31);
df31d32 = diff(f31, phi32);

df32d21 = diff(f32, phi21);
df32d31 = diff(f32, phi31);
df32d32 = diff(f32, phi32);


J = [df21d21 df21d31 df21d32; df31d21 df31d31 df31d32; df32d21 df32d31 df32d32];
lambda = eig(J);
rlambda = real(lambda);

srlambda = subs(rlambda, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271]);
seq = [subs(f21, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271]), subs(f31, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271]), subs(f32, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271])];

完成此操作后，我希望进行优化，以使f21 = f31 = f32 = 0并且特征值均为负。但是，我不知道如何在某些非线性优化过程中使用我的符号表达式。我有一些看起来像这样的代码：

x0 = [];
lb = [];
ub = [];

[sol, fval, exitflag, output] = fmincon(@eq1, x0, A, b, Aeq, beq, lb, ub, @constraints)

function objfun = eq1(k)
objfun = ;
end

function [c, ceq] = constraints(k)
c = [];
ceq = [];
end

在这里我可以为我的ceq，f21，f31条件和向量指定初始搜索点，上限和下限以及向量f32 c为我的特征值条件。我已经知道了几个问题。首先，优化部分希望使用k(1)，k(2)，k(3)和k(4)形式的变量，而不是k1，k2，{{ 1}}和f。有没有一种方法可以轻松地做到这一点？其次，我是否需要将符号约束转换为MATLAB函数？可能还有其他问题，但是我不确定。任何帮助将不胜感激：）

Answer 1

您可以使用matlabFunction将所有必需的符号表达式（分别为f21，f31，f32或seq转换为可执行的Matlab函数。这将使它们可执行（并输出双精度值而不是符号值），并使它们采用按字母顺序排序的多个输入参数。

因此，matlabFunction(seq)将导致匿名函数使用(f,k1,k2,s)作为输入参数。您还可以使用'File'的{{1}}参数将函数存储在文件中。

要让此函数采用要优化的参数向量，可以编写一个小的“包装器”：

matlabFunction

由于对象函数具有自己的工作空间（即无法访问基本工作空间中的匿名函数句柄），因此我建议将所有函数保存在文件中（也包括包装器）。

Answer 2

使用rinkert所述的matlabFunction来转换syms 函数function handle
平等f21 = f31 = f32等同于f21 - f31 = 0 and f21 - f32 = 0
要仅获得约束，请将目标函数定义为 恒定函数

eq = @(k)0

请仔细阅读评论

syms phi21 phi31 phi32 k1 k2 f s a

w1 = (2*pi/24)*0.99;
w2 = (2*pi/24)*1.01;
w3 = (2*pi/24)*1.02;

f21 = (w2 - w1) + (1/3)*(-2*k1*sin(phi21) - 2*k2*sin(2*phi21) - k1*sin(phi31) - k2*sin(2*phi31) + k1*sin(phi32) + k2*sin(2*phi32)) + 2*f*cos((2*s - phi21)/2)*sin(-phi21/2);
f31 = (w3 - w1) + (1/3)*(k1*sin(phi21) + k2*sin(2*phi21) - k1*sin(phi31) - k2*sin(2*phi31) - 2*k1*sin(phi32) - 2*k2*sin(2*phi32)) + 2*f*cos((2*s - phi31)/2)*sin(-phi31/2);
f32 = (w3 - w2) + (1/3)*(-k1*sin(phi21) - k2*sin(2*phi21) - 2*k1*sin(phi31) - 2*k2*sin(2*phi31) - k1*sin(phi32) - k2*sin(2*phi32)) + + 2*f*cos((2*s - phi32)/2)*sin(-phi32/2);

df21d21 = diff(f21, phi21);
df21d31 = diff(f21, phi31);
df21d32 = diff(f21, phi32);

df31d21 = diff(f31, phi21);
df31d31 = diff(f31, phi31);
df31d32 = diff(f31, phi32);

df32d21 = diff(f32, phi21);
df32d31 = diff(f32, phi31);
df32d32 = diff(f32, phi32);


J = [df21d21 df21d31 df21d32; df31d21 df31d31 df31d32; df32d21 df32d31 df32d32];
lambda = eig(J);
rlambda = real(lambda);

srlambda = subs(rlambda, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271]);
seq = [subs(f21, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271]), subs(f31, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271]), subs(f32, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271])];

% Transform syms function to function handles

f21 = matlabFunction(seq(1));
f31 = matlabFunction(seq(2));
f32 = matlabFunction(seq(3));

lambda = matlabFunction(srlambda);

% Inequality constraint, input is passed as a vector 
c =  @(k)lambda(k(1), k(2), k(3), k(4));

% Equality constraint, input is passed as a vector 
% f21 = f31 = f32 --> f21 -f31 = 0 and f21 -f32 = 0

ceq = @(k)[f21(k(1), k(2), k(3), k(4))-f31(k(1), k(2), k(3), k(4));...
            f21(k(1), k(2), k(3), k(4))-f32(k(1), k(2), k(3), k(4))];

% Combine all the constraints to one function handle 
constraints = @(k)deal(c(k),ceq(k)); 

% Only need the constraints to be satisfied, define a constant objective
% function
eq1 = @(k)0;

% A random starting guess, lower bound, upper bound 
% You can change this part to what you want
x0 = ones(1,4);
lb = [-inf, -inf, -inf, -inf];
ub = [inf, inf, inf, inf];

% No linear constraints 
A = [];
b = [];
Aeq = [];
beq = [];
[sol, fval, exitflag, output] = fmincon(eq1, x0, A, b, Aeq, beq, lb, ub, constraints);

解决方案

sol = [0.0116    0.5946   -0.3432    1.0064]

MATLAB中具有优化功能的符号计算

2 个答案: