How to solve the Lyapunov equation with unknowns

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Abdelkader Hd 2022-7-25

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1767545-how-to-solve-the-lyapunov-equation-with-unknowns

编辑： Torsten 2022-7-26

采纳的回答： Ivo Houtzager

I'd like to ask those with unknowns Lyapunov What function does the equation use

The original equation is MV+VM‘=-D,

The matrix code is like this , There is only one unknown ：

M=[0 w1 0 0 0 0;-q1 -r1 0 0 -g1u -g1v;0 0 0 w2 0 0;0 0 -q2 -r2 -g2u -g2v;g1v 0 g2v 0 -k x;-g1u 0 -g2u 0 -x -k];

d=[0 r1*(2n1+1) 0 r2(2*n2+1) k k];

D=diag(d);

V=lyap(M,D)

The following error occurred after running , Only numeric matrices can be calculated ：

Misuse lyap (line 35)

The input arguments of the "lyap" command must be numeric arrays.

error Untitled (line 38)

V=lyap(M,D)

Thank you

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Sam Chak 2022-7-26

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@Abdelkader Hd

What exactly do you gain from symbolically solving the linear matrix equation that is named after Lyapunov?

Do the symbolic parameters vary from

to ∞ in reality?

What are you trying to prove? Have you looked deeper into the problem such that it may be possible to decouple a few states to reduce the matrix size down to where it can be solved symbolically?

Do the reviewers suggest that you to solve the Lyapunov equation in order to get your manuscript accepted?

If you tell us your story, perhap we can find a workaround.

w1 = 1;
q1 = 1;
r1 = 1;
g1 = 1;
w2 = 1;
q2 = 1;
r2 = 1;
g2 = 1;
n1 = 1;
n2 = 1;
k  = 1;
u  = 1;
v  = 1;
x  = 1;
M  = [0 w1 0 0 0 0; -q1 -r1 0 0 -g1*u -g1*v; 0 0 0 w2 0 0; 0 0 -q2 -r2 -g2*u -g2*v; g1*v 0 g2*v 0 -k x; -g1*u 0 -g2*u 0 -x -k];
d  = [0 r1*(2*n1+1) 0 r2*(2*n2+1) k k];
D  = diag(d);
V  = lyap(M, D)
V = 6×6
    0.4032    0.0000   -1.0968    0.0000    0.0645    0.9032
    0.0000    1.3710   -0.0000   -0.1290   -0.1452    0.2742
   -1.0968   -0.0000    0.4032    0.0000    0.0645    0.9032
    0.0000   -0.1290    0.0000    1.3710   -0.1452    0.2742
    0.0645   -0.1452    0.0645   -0.1452    0.5645   -0.0645
    0.9032    0.2742    0.9032    0.2742   -0.0645   -1.2419

Abdelkader Hd 2022-7-26

编辑：Abdelkader Hd 2022-7-26

@Sam Chak well Thanks Sir for your response, I look to calculate logarithmic negativity "EN

" and it's defined as a function of V (solution of Lyapunov equation). IF I give all numerical values of the parameters I don't find the variable for plot EN as a function of some parameters such as temperature coupling ..

Please see the picture below to understand what I'm looking for it

https://arxiv.org/pdf/1807.07158.pdf "link of the paper"

Thank you.

Sam Chak 2022-7-26

编辑：Sam Chak 2022-7-26

I'm not good at magnomechanics, but it can clearly says that the elements of 𝒱are fully defined as function of u

and in Eq. (3), it is given that

So, maybe you can use ode45() to solve for

. Then, you can find a steady-state 𝒱 when

no longer change in time..

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采纳的回答

Ivo Houtzager 2022-7-25

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One inefficient way is too convert the Lyaponuv Matrix equation to linear system using the vectorization rule, and solve the linear system.

A = kron(eye(6),M) + kron(M,eye(6));
B = D(:);
X = A\B;
V = reshape(X,6,6);

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Abdelkader Hd 2022-7-26

@Sam Chak Thanks for your response,

Torsten 2022-7-26

编辑：Torsten 2022-7-26

@Sam Chak

If "\" and "kron" can deal with symbolic parameters, then in principle the problem can be solved using Ivo Houtzager 's suggestion. The inversion of a matrix does only use subdeterminants of the matrix itself - no roots are needed.

But even if it theoretically works: for a 36x36 or even 144x144 matrix, the expressions involved will become useless in my opinion.

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