Electronic – How to calculate this inverse Laplace Transform

control systemlaplace transformpid controllertransfer function

I have this transfer function and input, and I have to manually calculate what the systems response will be.
The transfer function is:

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and the input is:

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I multiplied the two functions and used fraction decomposition by partial fraction expansion and calculated the inverse laplace transform and this is what I got:

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But the response I get from that function does not correspond to the one I get using matlab. This is the matlab code I used:

t=0:0.1:100;
FT6=tf(0.1,[1 1 0.1])
r=4*sin(10*t);
V=lsim(FT6,r,t);
plot(t,V,'-')

How do I calculate it properly by hand?

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Best Answer

I understand you have \begin{equation} R(s) = \frac{40}{s^2+10^2} \end{equation} So \begin{equation} Y(s) = FT_6(s)R(s)=\frac{4}{(s^2+10^2)(s^2+s+0.1)} \end{equation} Then you proceed by taking partial fractions of Y(s) \begin{equation} Y(s) = \frac{A+sB}{(s^2+10^2)} +\frac{C+sD}{(s^2+s+0.1)} \end{equation} Then you expand the term \begin{equation} \frac{C+sD}{s^2+s+0.1} = \frac{C+sD}{(s+\frac{1}{2})^2-0.15} = D\frac{s+\frac{1}{2}}{(s+\frac{1}{2})^2-0.15} + (C-\frac{D}{2})\frac{1}{(s+\frac{1}{2})^2-0.15} \end{equation} Now you can use a table (keeping in mind the table items 7,8,21 and 22).

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