# Inverse Laplace transform of a 4 by 4 matrix

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Sunday Aloke
on 22 Nov 2023

Commented: Walter Roberson
on 23 Nov 2023

##### 0 Comments

### Accepted Answer

Sam Chak
on 22 Nov 2023

Since this is a linear dynamic system described in state-space, the output can be found using the dsolve() command.

syms w(t) x(t) y(t) z(t)

%% State equations

eqns = [diff(w,t) == - 0.00022074*w + 0*x - 0.321386*y + 0.0000004924*z,

diff(x,t) == 0.00012074*w - 0.187744*x + 0.321385*y + 0*z,

diff(y,t) == 0*w + 0.187644*x - 0.321215*y + 0*z,

diff(z,t) == 0*w + 0*x + 0.318725*y - 0.0001005*z];

%% General solution (if initial condition is unknown)

Gsol = dsolve(eqns, 'MaxDegree', 4)

%% Particular solution (if initial condition is known)

cond = [w(0) == 1,

x(0) == 0,

y(0) == 0,

z(0) == 0];

Psol = dsolve(eqns, cond, 'MaxDegree', 4)

##### 17 Comments

### More Answers (1)

Walter Roberson
on 22 Nov 2023

inverse laplace of a constant is dirac delta times the constant, so just multiply the matrix by dirac delta function.

Note that if your A is the A of a state space representation then you would have a completely different answer, and would need the other state space matrices as well

##### 2 Comments

Walter Roberson
on 22 Nov 2023

https://lpsa.swarthmore.edu/Representations/SysRepTransformations/TF2SS.html

talks about converting state space to transfer function. You would then take the inverse laplace of the transfer function.

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