PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

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)]