# 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
Please house, I need help to output the inverse Laplace transform of the 4 by 4 matrix attached below. I will be grateful for your assistance. Thanks

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)
Gsol = struct with fields:
x: C1*(exp(-(t*(880740210851004153290091*((3^(1/2)*43740438401693126157487504478370365822531694805937937204624469331174935258419763369366273851571079352664988310139012389846237400821611425481… w: C1*(exp(-(t*(880740210851004153290091*((3^(1/2)*43740438401693126157487504478370365822531694805937937204624469331174935258419763369366273851571079352664988310139012389846237400821611425481… y: C1*(exp(-(t*(880740210851004153290091*((3^(1/2)*43740438401693126157487504478370365822531694805937937204624469331174935258419763369366273851571079352664988310139012389846237400821611425481… z: C3*exp(-(t*(880740210851004153290091*((3^(1/2)*437404384016931261574875044783703658225316948059379372046244693311749352584197633693662738515710793526649883101390123898462374008216114254819…
%% 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)
Psol = struct with fields:
x: (2245501681194912713180821901898316369833105182787823237505398159820541372242590057761026213554694888135433837660368384056135068619771591528250850247291290758676480000000000000000000000000… w: (2245501681194912713180821901898316369833105182787823237505398159820541372242590057761026213554694888135433837660368384056135068619771591528250850247291290758676480000000000000000000000000… y: (18429039118821823205910363600473469*(exp(-(t*(880740210851004153290091*((3^(1/2)*4374043840169312615748750447837036582253169480593793720462446933117493525841976336936627385157107935266498… z: (2245501681194912713180821901898316369833105182787823237505398159820541372242590057761026213554694888135433837660368384056135068619771591528250850247291290758676480000000000000000000000000…
Sam Chak on 23 Nov 2023
Edited: Sam Chak on 23 Nov 2023
Hmm... This is strange. It looks complete on my machine. That's why I requested you to post exactly the same code and then hit the Run button . Let me check if I can the display completely from your side.
Walter Roberson on 23 Nov 2023
You might need to use the scroll-bars under the output expressions.

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
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.
Sunday Aloke on 22 Nov 2023
Yes, it's a state space matrix Please I am totally ignorant of the code to use. Sample of code will really help

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