Page 336 - Mechanical Engineers' Handbook (Volume 2)

5 Approaches to Linear Systems Analysis 327

n
where (a) P(s) a s a n 1 a s a is the characteristic polynomial of the
n
1
0
system
(b) G(s) b s m b m 1 s m 1 b s b represents the numerator dynamics
1
0
m
of the system
(c) U(s) N(s)/D(s) is the transform of the input to the system, u(t), assumed to be
a rational function
dy (0) a
n 1 n 2
a
n n 1
(d) F(s) ay(0)s
n y(0) s
dt
d n 1 y d n 2 y (0) ay(0)
(0) a
a
n n 1 n 1 1
dt dt
reflects the initial system state [i.e., the initial conditions on y(t) and its
first n 1 derivatives]
The transformed response can be thought of as the sum of two components,
Y(s) Y (s) Y (s)
zi
zs
where (e) Y (s) [G(s)/P(s)][N(s)/D(s)] H(s)U(s) is the transform of the zero-state re-
zs
sponse, that is, the response of the system to the input alone
(f) Y (s) F(s)/P(s) is the transform of the zero-input response, that is, the response
zi
of the system to the initial state alone
The rational function
(g) H(s) Y (s)/U(s) G(s)/P(s)isthe transfer function of the system, defined as
zs
the Laplace transform of the ratio of the system response to the system
input, assuming zero initial conditions
The transfer function plays a crucial role in the analysis of fixed linear systems using
transforms and can be written directly from knowledge of the system I/O equation as
m
bs b
H(s) m 0
n
as a n 1 s n 1 as a 0
1
n
Impulse Response
Since U(s) 1 for a unit impulse function, the transform of the zero-state response to a unit
impulse input is given by the relation (g) as
Y (s) H(s)
zs
that is, the system transfer function. In the time domain, therefore, the unit impulse response
is
0 for t 0
L [H(s)] for t 0
h(t) 1
This simple relationship is profound for several reasons. First, this provides for a direct
characterization of time-domain response h(t) in terms of the properties (poles and zeros) of
the rational function H(s) in the complex-frequency domain. Second, applying the convo-
lution transform pair (Table 5) to relation (e) above yields
t
Y (t) h( )u(t ) d
zs
0