Van der pol equation

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Chike
Chike 2024-8-22
编辑： Sam Chak 2024-8-26
Given the Van der pol equation
+ 𝝁 ( – 1) x + kx =0
Where 𝝁 = k = -1
Obtain the non-linear stage space representation of the system
Find the Jacobian matrix at the equilibrium point
Use the Lyaponov direct method to check for the stability of the system
(Hint V(x)= Px : PA + P = -1)

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回答（1 个）

Rahul
Rahul 2024-8-22
Hi @Chike,
I understand that you are trying to obtain the non-linear stage space representation of the Van der pol system. You require the Jacobian matrix at the equilibrium point and want to use Lyaponov direct method to check stability of the system.
Note: I am using ‘ (single quote) notation to demonstrate derivative of a variable in equations below.
The Van der Pol equation is given by: x'' + 𝝁 (x^2 - 1) x' + kx = 0
Substituting 𝝁 = -1 and k = -1, we have: x'' - (x^2 - 1) x' - x = 0 ]
To convert this into a state-space representation, define the state variables:
  • x1 = x
  • x2 = x'
The equilibrium point is where x1' = 0 and x2' = 0.
For this system, the equilibrium point is (x1, x2) = (0, 0).
Hence on calculating the partial derevatives we get the Jacobian Matrix "A" as: A = [0 1; 1 -1];
To check the stability using the Lyapunov direct method, the function "lyap" can be used.
Further, the eigen values of the matrix obtained from the "lyap" function can be used to check the stability of the system.
So you can follow this code to achieve the desired result:
% Jacobian matrix
A = [0 1; 1 -1];
I = eye(2);
% Lyapunov equation
P = lyap(A', I);
disp('Matrix P:');
disp(P);
% Check for stability
isPositiveDefinite = all(eig(P) > 0);
if isPositiveDefinite
disp('The system is stable.');
else
disp('The system is not stable.');
end
You can refer to the following documentations to know more about these functions:
"lyap": https://www.mathworks.com/help/releases/R2024a/control/ref/lyap.html?searchHighlight=lyap&s_tid=doc_srchtitle
"eig": https://www.mathworks.com/help/releases/R2024a/matlab/ref/eig.html?searchHighlight=eig&s_tid=doc_srchtitle
"jacobian": https://www.mathworks.com/help/releases/R2024a/symbolic/sym.jacobian.html?searchHighlight=jacobian&s_tid=doc_srchtitle
Hope this helps! Thanks.
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Torsten
Torsten 2024-8-26
编辑：Torsten 2024-8-26
Nobody spends one week on "just a question" . Especially if there is no progress in any of the three subproblems.
Sam Chak
Sam Chak 2024-8-26
编辑：Sam Chak 2024-8-26
i have been trying to resolve for a week now but i wasnt sure if i was right or wrong.
@Chike, could you please clarify which specific part of your attempted MATLAB-based solution you were uncertain about? We'd be happy to review the relevant section(s) and provide any feedback.

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