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1.8 Obtain partial-fraction expansion

Problem: Given the continuous time S transfer function defined by

\[ H(s)=\frac {s^{4}+8s^{3}+16s^{2}+9s+9}{s^{3}+6s^{2}+11s+6}\]
obtain the partial-fractions decomposition.

Comment: Mathematica result is easier to see visually since the partial-fraction decomposition returned in a symbolic form.

Mathematica 

Remove["Global`*"]; 
expr = (s^4+8 s^3+16 s^2+9 s+6)/ 
            (s^3+6 s^2+11 s+6); 
Apart[expr]
 


2 + s +3/(1+s) -4/(2+s) -6/(3+s)
 

Matlab 

clear all; 
s=tf('s'); 
tf_sys = (s^4+8*s^3+16*s^2+9*s+6)/... 
          (s^3+6*s^2+11*s+6); 
[num,den] = tfdata(tf_sys,'v'); 
[r,p,k] = residue(num,den)
 


r = 
   -6.0000 
   -4.0000 
    3.0000 
 
p = 
   -3.0000 
   -2.0000 
   -1.0000 
 
k = 
     1     2
 

Maple 

p:=(s^4+8*s^3+16*s^2+9*s+9)/(s^3+6*s^2+11*s+6); 
p0:=convert(p,parfrac);

\[ s+2- \frac {7}{s+2}-{\frac {9}{2\,s+6}}+{\frac {9}{2\,s+2 }} \]
 
[op(p0)];

\[ [s,2,\frac {7}{s+2},-{\frac {9}{2\,s+6}},{\frac {9}{2\,s+2}}] \]
 

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