How to solve the matrix P in the Lyapunov equation "A'P+PA=- Q" in Simulink

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LIULIYAN 2023-7-12

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编辑： Bruno Luong 2023-7-12

For the solution of matrix P in Lyapunov equation, I can call the ‘lyap’ function in the command line window or m file to solve. The method is as follows:

A=[0,1;-1,-2]';Q=[1,0;0,1];P=lyap(A,Q)

However, now I need to solve the matrix P in 'A'P+PA=- Q' in Simulink (matrices A and Q are known). At this point, the ‘lyap’ function cannot be called. Is there any good way to solve this problem? I look forward to your reply very much.

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LIULIYAN 2023-7-12

Hi@Sam Chak

In fact, matrix A is a Hurwitz matrix about the control gains Kp and Kv, which means it is variable. Q is a Positive-definite matrix. The control output of the controller I designed contains matrix P, so I need to solve it.

Sam Chak 2023-7-12

Hi @LIULIYAN

If

where

and

are the known control gains in Simulink, why is

a variable?

If

is Hurwitz and

is positive-definite, then

can be found, and you want to use it to compute the control gain matrix

? But this

is usually found by solving the Algebraic Riccati Equation (ARE).

In other words, the Lyapunov equation is used to show stability, while the solution to the ARE is used to show optimality.

By the way, the lyap() function has not been supported for standalone code generation for many years. However, if you simply want to find

via solving the Lyapunov equation, consider @Bruno Luong's two methods.

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Answer 1

Divyajyoti Nayak 2023-7-12

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Hi LIULYAN,

You can use a MATLAB function block and call the lyap function inside it to calculate P.

Implement MATLAB Functions in Simulink with MATLAB Function Blocks - MATLAB & Simulink (mathworks.com)

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Sam Chak 2023-7-12

在 MATLAB Online 中打开

Hi @Divyajyoti Nayak

Can you test if this simple Simulink model works on your machine?

P = lyap(-1, 1)
P = 0.5000

LIULIYAN 2023-7-12

Hi@Sam Chak

When running the above Simulink model on my computer, it still reported an error. The version I am using is MATLAB 2016b.

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Answer 2

Bruno Luong 2023-7-12

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编辑：Bruno Luong 2023-7-12

在 MATLAB Online 中打开

I don't know simulink nor lyap function but just to tell you you can solve with standard algebra, so I hope it could be incorporated easier to simulink

A=[0,1;-1,-2]';
Q=[1,0;0,1];
P = (kron(eye(2),A') + kron(A.',eye(2))) \ -Q(:);
P = reshape(P,[2 2]),
P = 2×2
    1.5000   -0.5000
   -0.5000    0.5000
% Check
A'*P + P*A + Q
ans = 2×2
     0     0
     0     0

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Bruno Luong 2023-7-12

编辑：Bruno Luong 2023-7-12

在 MATLAB Online 中打开

A variation to avoid using kron is solving by one of the linear solver where function-handle user supply is possible (instead of matrix representation)

A=[0,1;-1,-2]';
Q=[1,0;0,1];
P = lsqr(@(varargin) LyaProd(A, varargin{:}), -Q(:));
lsqr converged at iteration 3 to a solution with relative residual 2.5e-15.
P = reshape(P,[2 2]);
disp(P)
    1.5000   -0.5000
   -0.5000    0.5000
%%
function AP = LyaProd(A, P, opt)
P = reshape(P, [2 2]);
if strcmp(opt,'notransp')
    AP = A'*P + P*A;
else
    AP = A*P + P*A';
end
AP = AP(:);
end