MATLAB: Strange ilaplace answer????

inverse laplace

Obviously I'm doing something wrong here
Why am i getting such a strange answer for an inverse laplace??
syms s
tf = 10/(s^4 + 6*s^3 + 8*s^2 + 10*s)
>> ilaplace(tf)
ans =
1 - 8*symsum(exp(root(s3^3 + 6*s3^2 + 8*s3 + 10, s3, k)*t)/(3*root(s3^3 + 6*s3^2 + 8*s3 + 10, s3, k)^2 + 12*root(s3^3 + 6*s3^2 + 8*s3 + 10, s3, k) + 8), k, 1, 3) - 6*symsum((root(s3^3 + 6*s3^2 + 8*s3 + 10, s3, k)*exp(root(s3^3 + 6*s3^2 + 8*s3 + 10, s3, k)*t))/(12*root(s3^3 + 6*s3^2 + 8*s3 + 10, s3, k) + 3*root(s3^3 + 6*s3^2 + 8*s3 + 10, s3, k)^2 + 8), k, 1, 3) - symsum((exp(root(s3^3 + 6*s3^2 + 8*s3 + 10, s3, k)*t)*root(s3^3 + 6*s3^2 + 8*s3 + 10, s3, k)^2)/(3*root(s3^3 + 6*s3^2 + 8*s3 + 10, s3, k)^2 + 12*root(s3^3 + 6*s3^2 + 8*s3 + 10, s3, k) + 8), k, 1, 3)
The answer is still in terms of S???
It did not even do the inverse? Its only a 4th order function? I don't get it
This TF happens to be the the step response of a system.
My goal was to transfer it back to the time domain so i could plot it in time.
I wanted to compare the manual plot i made to the internal step function of matlab
 step(tf_closed_loop)
Obviously they should of been the same
but i can't get a reasonable answer from ilapalce???

Best Answer

The output you are seeing is derived from the fact that the solution involves the sum of a polynomial times an exponential, with the sum taken over the three roots of a third degree polynomial. If you let the k'th root of the polynomial be designated by alpha(k) then the expression looks like
 1 - 8*(symsum(exp(alpha(k)*t)/(3*alpha(k)^2+12*alpha(k)+8), k, 1, 3)) - 6*(symsum(alpha(k)*exp(alpha(k)*t)/(3*alpha(k)^2+12*alpha(k)+8), k, 1, 3))) - symsum(exp(alpha(k)*t)*alpha(k)^2/(3*alpha(k)^2+12*alpha(k)+8), k, 1, 3)
Maple's invplace transforms it to a different form:
1 + (1/611) * (symsum((53*alpha(k)^2+258*alpha(k)-41)*exp(alpha*t), k, 1, 3))
where again alpha(k) represents the k'th root of s3^3 + 6*s3^2 + 8*s3 + 10
MATLAB is not expanding the polynomial roots, because the expression gets out of hand. A simplified version of Maple's invlaplace is
(1/10998)*(((-(1656*I)*3^(1/2)+1656)*(135+3*1833^(1/2))^(1/3)+((414*I)*3^(1/2)+414)*1833^(1/2)+((53*I)*(135+3*1833^(1/2))^(4/3)+10998*I)*3^(1/2)-53*(135+3*1833^(1/2))^(4/3)-3666*(135+3*1833^(1/2))^(2/3)+10998)*exp(-(1/6)*((12*I)*3^(1/2)/(135+3*1833^(1/2))^(1/3)-I*3^(1/2)*(135+3*1833^(1/2))^(1/3)-12/(135+3*1833^(1/2))^(1/3)-(135+3*1833^(1/2))^(1/3)+12)*t)+(((1656*I)*3^(1/2)+1656)*(135+3*1833^(1/2))^(1/3)+(-(414*I)*3^(1/2)+414)*1833^(1/2)+(-(53*I)*(135+3*1833^(1/2))^(4/3)-10998*I)*3^(1/2)-53*(135+3*1833^(1/2))^(4/3)-3666*(135+3*1833^(1/2))^(2/3)+10998)*exp((1/6)*((12*I)*3^(1/2)/(135+3*1833^(1/2))^(1/3)-I*3^(1/2)*(135+3*1833^(1/2))^(1/3)+12/(135+3*1833^(1/2))^(1/3)+(135+3*1833^(1/2))^(1/3)-12)*t)+(106*(135+3*1833^(1/2))^(4/3)-3666*(135+3*1833^(1/2))^(2/3)-828*1833^(1/2)-3312*(135+3*1833^(1/2))^(1/3)-21996)*exp(-(1/3)*((135+3*1833^(1/2))^(1/3)+12/(135+3*1833^(1/2))^(1/3)+6)*t)+10998*(135+3*1833^(1/2))^(2/3))/(135+3*1833^(1/2))^(2/3)
where I is sqrt(-1).
The MATLAB generated expression, expanded, can be shown to be exactly the same.
At the moment I do not know how to get MATLAB to expand the root() into explicit values.
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