Civil Engineering Reference
In-Depth Information
to transform a differential equation to the Laplace domain. Assuming that the system is at rest at t
0, so
¼
f (0)
0 and df (0)
dt
0, one obtains:
¼
/
¼
a
2
s
2
Y(s)
a
0
Y(s)
þ
a
1
sY(s)
þ
¼
b
0
U(s)
þ
b
1
sU(s)
(12
:
3)
This yields a constant relation between the input U(s) and the output Y(s):
Y(s)
U(s)
¼
b
0
þ
b
1
s
H(s)
¼
a
2
s
2
,
(12
:
4)
a
0
þ
a
1
s
þ
which is known as the system's transfer function. For an LTI system with one input and one output,
the transfer function completely defines the system. The transfer function is a convenient format for
manipulation of system models. For example, the transfer function for two systems placed in series,
that is, the output of the first system is the input of the second, is simply the product of the two transfer
functions, and, likewise, the transfer function for the total of the two systems placed in parallel is the sum
of the transfer functions.
The stability of an LTI system can be determined from its transfer function by determining the poles
of the transfer function, which are the (complex) numbers for which the denominator equals zero (these
solutions are referred to as the “roots” of the denominator). Any poles with a positive real part are
associated with an unstable response of the system to an input signal. This response may be oscillating
for a pair of complex poles with positive real part, or it may be a-periodic when the corresponding poles
are on the positive real axis. A system with such poles is unstable. Poles with negative real part produce
responses with exponentially decreasing amplitude. Pole pairs on the imaginary axis produce an
undamped oscillatory response of the system, a pole in the origin produces an integration. An experi-
enced control engineer can interpret the location of the poles and zeros of a transfer function in
terms of a system's dynamical behavior.
12.2.3 Frequency Response
Another method of characterizing an LTI system is by means of its frequency response. If a sine wave input
signal is applied to an LTI system, and the system is stable, then after the initial response to the start of the
sine wave has faded, the output of the system will be a sine wave with the same frequency as the input
signal, but with a different amplitude, and a different phase from the input signal. Figure 12.3 shows the
sine input and the output signal for a system. The ratio of the output sine to the input sine signal, as a
function of frequency, is called the gain. The difference in phase between the two sine signals, is called the
phase shift. Together they form the systems' frequency response. For physically implemented systems, the
frequency response can be determined with a Frequency Analyzer, a device that can generate sinusoid test
signals and measure the response of the system.
1
amplitude response
0.5
0
phase response
−0.5
−1
0
10
20
30
40
50
60
70
80
90
100
FIGURE 12.3
Input sinusoid (black) and system output (grey) as a function of time.
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