SymPy - 在反拉普拉斯变换时生成巨大的方程

Question

When I do a really simple inverse laplace transform with Sympy, I got a huge equation back.

For example:

from sympy import *
s = symbols ('s')
t = symbols ('t', positive=True) # Just to remove the Heaviside(t) equations
k, m = symbols ('k m', const=True) 
A = Matrix([[0, 1], [-k/m, 0]])
I = eye(2) # Diagonalmatrix 
Fi = inverse_laplace_transform((s*I-A).inv(), s,  t)
print(pretty(simplify(Fi)))

Now a get an enormous huge equation from Fi . Why? Is something wrong with inverse_laplace_transform() function from Sympy?

Answer 1

Inverse lapace transform is used to turn transfer functiuons into discrete form instead of time continuous.

Instead of using inverse laplace transform, we can use this code instead, where h is the sample time. Image that we have a transfer function G(s), and you want to find the discrete equivalent model of G(s). Find the state space model of G(s) and then run this Octave/MATLAB code:

% Compute sizes
a1 = size(A,2) + size(B,2) - size(A,1);
b1 = size(A,2);
a2 = size(A,2) + size(B,2) - size(B,1);
b2 = size(B,2);
% Compute square matrix
M = [A B; zeros(a1, b1)  zeros(a2, b2)];
M = expm(M*h);
 % Find the discrete matrecies
Ad = M(1:size(A,1), 1:size(A,2));
Bd = M(1:size(B,1), (size(A,2) + 1):(size(A,2) + size(B,2)));

Code source: https://github.com/DanielMartensson/Matavecontrol/blob/master/sourcecode/c2d.m