Laplace transform - Continuous and Discrete Time Signals and Systems - page 299

In this section, we derive an equivalent condition to determine the stability of
an LTIC system modeled with a rational Laplace transfer function H ( s )given
in Eq. (6.35). Since we are mostly interested in causal systems, we assume
that the Laplace transfer function H ( s ) corresponds to a right-sided system.
The poles of a system with a transfer function as given in Eq. (6.35) can be
calculated by solving the characteristic equation, Eq. (6.36). Three types of
poles are possible. Out of the n possible poles, assume that there are L poles at
s

= 0, K real poles at s =−σ k ,1 ≤ k ≤ K , and M pairs of complex-conjugate

poles at s =−α m j ω m ,1 ≤ m ≤ M , such that L + K + 2 M = n . In terms

of its poles, the transfer function, Eq. (6.35), is given by

H ( s ) = N ( s )
D ( s )

N ( s )

= . (6.40)

K M

s 2 + 2 α m s + α m + ω m

s L ( s + σ k )

k = 1 m = 1

From Table 6.1, the repeated roots at s = 0 correspond to the following term

in the time domain:

1
n ! t n u ( t )

1
s n .

←→

(6.41)

Since term t n u ( t ) is unbounded as t →∞ , a stable LTIC system will not contain
such unstable terms. Therefore, we assume that L

= 0. The partial fraction

expansion of Eq. (6.40) with L = 0 results in the following expression:

A 1
( s + σ 1 )

A K
( s + σ K )

B 1 s + C 1

H ( s ) = ++ + +

α 1 + ω 1

s 2 + 2 α 1 s +

B M s + C M

+ , (6.42)

s 2 + 2 α M s + α 2 M + ω 2 M

where { A k , B m , C m } are the partial fraction coefficients. Calculating the inverse
Laplace transform of Eq. (6.42) and assuming a causal system, we obtain the
following expression for the impulse response h ( t ) of the LTIC system:

K M

−σ k t u ( t ) −α m t cos( ω m t + θ m ) u ( t )

h ( t ) = A k e + r m e , (6.43)

k = 1 m = 1

h k ( t ) h m ( t )

where we have expressed the terms with conjugate poles in the polar format.
Constants { r m , θ m } are determined from the values of the partial fraction coef-
ficients { B m , C m } and α m .
In Eq. (6.43), we have two types of terms on the right-hand side of the
equation. Summation I consists of K real exponential functions of the type
h k ( t ) = A k exp( −σ k t ) u ( t ). Depending upon the value of σ k , each of these func-
tions h k ( t ) may have a constant, decaying exponential or a rising exponential
waveform.
Summation II consists of exponentially modulated sinusoidal functions of
the type h m ( t ) = r m exp( −α m t ) cos( ω m t + θ m ) u ( t ). The stability characteristic