Partial Fractions - Lecture 7: The Partial Fraction Expansion
Partial Fractions
Matthew M. Peet
Illinois Institute of Technology
Lecture 7: The Partial Fraction Expansion
Introduction
In this Lecture, you will learn: The Inverse Laplace Transform ? Simple Forms
The Partial Fraction Expansion ? How poles relate to dominant modes ? Expansion using single poles ? Repeated Poles ? Complex Pairs of Poles Inverse Laplace
M. Peet
Lecture 7: Control Systems
2 / 27
Recall: The Inverse Laplace Transform of a Signal
To go from a frequency domain signal, u^(s), to the time-domain signal, u(t), we use the Inverse Laplace Transform.
Definition 1.
The Inverse Laplace Transform of a signal u^(s) is denoted u(t) = -1u^.
u(t) = -1u^ = eitu^(i)d
0
? Like , the inverse Laplace Transform -1 is also a Linear system. ? Identity: -1u = u. ? Calculating the Inverse Laplace Transform can be tricky. e.g.
s3 + s2 + 2s - 1 y^ = s4 + 3s3 - 2s2 + s + 1
M. Peet
Lecture 7: Control Systems
3 / 27
Poles and Rational Functions
Definition 2.
A Rational Function is the ratio of two polynomials: n(s)
u^(s) = d(s)
Most transfer functions are rational.
Definition 3.
The
point
sp
is
a
Pole
of
the
rational
function
u^(s)
=
n(s) d(s)
if
d(sp)
=
0.
? It is convenient to write a rational function using its poles
n(s)
n(s)
=
d(s) (s - p1)(s - p2) ? ? ? (s - pn)
? The Inverse Laplace Transform of an isolated pole is easy:
1 u^(s) =
s+p
means
u(t) = e-pt
M. Peet
Lecture 7: Control Systems
4 / 27
Partial Fraction Expansion
Definition 4.
The Degree of a polynomial n(s), is the highest power of s with a nonzero coefficient. Example: The degree of n(s) is 4
n(s) = s4 + .5s2 + 1
Definition 5.
A
rational
function
u^(s)
=
n(s) d(s)
is
Strictly
Proper
if
the
degree
of
n(s)
is
less
than the degree of d(s).
? We assume that n(s) has lower degree than d(s) ? Otherwise, perform long division until we have a strictly proper remainder
s3 + 2s2 + 6s + 7
2
s2 + s + 5 = s + 1 + s2 + s + 5
M. Peet
Lecture 7: Control Systems
5 / 27
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