352 Practical MATLAB® Applications for Engineers
be a system transfer function.
Verify that multiple poles are present in H(s), and evaluate the MATLAB partial
fraction expansion of H(s).
MATLAB Solution
>> num = [5 7 3 5 -30];
>> den = [1 4 7 6 2];
>> [r,p,k] = residue(num,den)
r = % PFE coefficients
-6.5000 -20.5000I
-6.5000 +20.5000i
0.0000
-34.0000
p = % poles
-1.0000 + 1.0000i
-1.0000 - 1.0000i
-1.0000
-1.0000
k = % stand alone term
5
The partial fraction coeffi cients given by the column vector r are then matched
with the corresponding poles given by the column vector p, obtaining the follow-
ing expansion:
Hs
i
si
i
sis s
()
,...
()
5
6 5 20 5
1
6 5 20 5
1
0
1
34
12
Observe that H(s) has repeated poles at s = −1, then the PFE consists of two terms,
a linear and a quadratic as a consequence of the repeated pole, as well as, two other
terms as a consequence of the pair of complex poles, and the stand-alone term k.
R.4.98 Four examples of the evaluations of the direct and inverse LTs by hand calcula-
tions, using Table 4.2, are provided as follows, to gain practice and insight into the
process.
a. Example (#1)
Let
f(t) = 5 u(t) + 2 e−^3 t u(t) + 10 sin( 3 t) u(t) + 5 e −^2 t cos( 7 t)
Find F(s).
ANALYTICAL Solution
(From Table 4.2)
F(s) = £ [f(t)] = £^ [ 5 + 2 e−^3 t u(t) + 10 sin( 3 t) u(t) + 5 e−^2 t cos( 7 t)]
F(s) = £^ [ 5 ] + 2 £^ [e−^3 t u(t)] + 10 £^ [sin( 3 t) u(t)] + 5 £^ [e−^2 t cos( 7 t)]