Civil Engineering Reference
In-Depth Information
Solving for E(s) yields:
(1
þ
G(s)K(s))E(s)
¼
R(s),
(12
:
10)
so
1
E(s)
¼
G(s)K(s)
R(s)
(12
:
11)
1
þ
and
G(s)K(s)
Z(s)
¼
G(s)K(s)
R(s)
:
(12
:
12)
1
þ
Thus,
Z(s)
R(s)
¼
K(s)G(s)
(12
:
13)
1
þ
K(s)G(s)
One can consider this solution in the frequency domain, by substituting jvfor s, which yields:
Z(jv)
R(jv)
¼
K(jv)G(jv)
K(jv)G(jv)
:
(12
:
14)
1
þ
The design requirement is that the system output Z( jv) equals the reference signal R( jv), and so
(Z(jv)
The solution to this problem would be to achieve a high “open-loop gain”
K( jv)G( jv). When K( jv)G( jv) is very large, one can see that Z( jv)
=
R(jv)
1
:
R( jv) [Equation (12.14) and
that E( jv)
0 Equation (12.11)].
However, one should bear in mind that K( jv)G( jv) is still a function of v, with a complex-valued
outcome. In general, it is not possible, and for many practical reasons not desirable to obtain a large
value for K( jv)G( jv) for all v. Essentially K( jv)G( jv) determines the “speed” of reaction to an error
signal. When time delays are small then a faster response will yield a lower tracking error. When time
delays are large, however, it is possible to respond too quickly and cause the system to become unstable.
Hence, as will be discussed in more detail, in the following paras, generally there is a trade-off between
response speed and accuracy.
Just as transfer functions can be considered in terms of their frequency response, that is, what (sine
signal) frequencies they pass and what frequencies they block, signals can be considered in terms of
their frequency content. A reference signal such as a block or sawtooth signal can be seen as constructed
from an infinitely large sum of sine signals (see Figure 12.8), a much smoother signal has less high-
frequency components.
(a)
(b)
(c)
3
3
3
0
0
0
−3
−3
−3
0
5
10
0
5
10
0
5
10
FIGURE 12.8 Triangular function approximated by sums of sine functions, base frequency 0.1 Hz, following sine
components at 0.3, 0.5, 0.7 Hz, etc. (a) one sine function; (b) three sine functions; (c) ten sine functions.
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