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5 Problem A2.15 OGATA 4ed (Inverse Laplace transform, partial-fraction expansion)

Let's get the inverse Laplace transform using Scilab in the following transfer function:

\begin{displaymath}F(s)=\frac{(s^{5}+8\cdot s^{4}+23\cdot s^{3}+35\cdot s^{2}+28\cdot s+3)}{(s^{3}+6\cdot s^{2}+8\cdot s)}\end{displaymath}


Program in Scilab 
s=%s
num=s^5+8*s^4+23*s^3+35*s^2+28*s+3;
den=s^3+6*s^2+8*s;
g=syslin('c',num/den);
gf=tf2ss(g);
se=pfss(gf);
for k=1:size(se),
df(k)=clean(se(k)),
end;

Solution: 
 df  =
 
 
       df(1)
 
    0.375   
    -----   
      s     
 
       df(2)
 
    0.375   
    -----   
    4 + s   
 
       df(3)
 
    0.25    
    ----    
    2 + s   
 
       df(4)
 
              2  
    3 + 2s + s

 

 

 

Partial-fraction expansion would be:

\begin{displaymath}F(s)=\frac{0.375}{s}+\frac{0.375}{(s+4)}+\frac{0.25}{(s+2)}+3+2\cdot s+s^{2}\end{displaymath}


The inverse Laplace transform:

\begin{displaymath}f(t)=L^{-1}(F(s))=0.375+0.375\cdot e^{-4\cdot t}+0.25\cdot e^... ...a(t)+2\cdot \frac{d}{dt}(\delta(t))+\frac{d^{2}}{dt}(\delta(t))\end{displaymath}