How can I express a set of nonlinear differential equations in the form 𝑥 ˙ = 𝐴 ( 𝑥 ) ⋅ 𝑥 + 𝐵 using MATLAB code ?

dXdt = @(X) [
    2*(x(1) + 0.005)^2 + 3*(x(1) + 0.005)*(x(2) + 0.003); 
    2*(x(1) + 0.005) + 4*(x(2) + 0.003)^2;
    (x(1) + 0.005)^3 + (x(2) + 0.003)*(x(3) + 0.05);
    2*(x(3) + 0.005) + (x(4) + 0.025)^2;
    (x(1) + 0.005)*(x(5) + 0.0236) + (x(2) + 0.003) + (x(4) + 0.0125)
];

Given a set of nonlinear differential equations for dxdt , how can I express the system in the form x_dot = A(X)*X+B using MATLAB code?

X=[x(1), x(2)....x(5)] is the states

[Torsten]

Given a set of nonlinear differential equations for dxdt , how can I express the system in the form x_dot = A(X)*X+B using MATLAB code?

Why do you want to do that ? MATLAB can solve the original nonlinear system directly.

[Sam Chak]

@Manish Kumar, If you are not referring to Jacobian linearization within the System Dynamics course, you must be implying Carleman linearization, which is typically not covered in undergraduate courses.

Answer 1

[Divyam]

Hi @Manish Kumar,

Assuming that you are referring to Jacobian linearization, for expressing a set of non-linear differential equations in the form , you can use the 'jacobian' function to find the linear part . Then you can identify any constant terms or terms that do not depend linearly on the state vector to form B.

syms x1 x2 x3 x4 x5 real

% Define the state vector
X = [x1; x2; x3; x4; x5];

% Define the nonlinear system
dXdt = [
    2*(x1 + 0.005)^2 + 3*(x1 + 0.005)*(x2 + 0.003); 
    2*(x1 + 0.005) + 4*(x2 + 0.003)^2;
    (x1 + 0.005)^3 + (x2 + 0.003)*(x3 + 0.05);
    2*(x3 + 0.005) + (x4 + 0.025)^2;
    (x1 + 0.005)*(x5 + 0.0236) + (x2 + 0.003) + (x4 + 0.0125)
];

% Compute the Jacobian matrix A(X)
A = jacobian(dXdt, X);

% Compute the constant term B by substituting X = 0
B = simplify(subs(dXdt, X, zeros(size(X))) - A * zeros(size(X)));

disp('A(X) = ');

A(X) = 

disp(A);

●

disp('B = ');

B = 

disp(B);

For more information regarding the 'jacobian' function, refer to the following documentation: https://www.mathworks.com/help/symbolic/sym.jacobian.html

[Sam Chak]

Hi @Divyam,

But if you make as the Jacobian matrix , then this product term is simply untrue according to the principle of linearization or approximation via Taylor series.

You can test performing the Jacobian on . Does approximate at every point where is differentiable?