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

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LIULIYAN
LIULIYAN 2023-7-12
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
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
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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回答（2 个）

Divyajyoti Nayak
Divyajyoti Nayak 2023-7-12
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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LIULIYAN
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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Bruno Luong
Bruno Luong 2023-7-12
编辑：Bruno Luong 2023-7-12
Ran in:
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
Bruno Luong 2023-7-12
编辑：Bruno Luong 2023-7-12
Ran in:
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

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