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