Civil Engineering Reference
In-Depth Information
In most cases, a smooth response is acceptable and even desirable from a mechanics standpoint,
and thus responses to the high-frequency components in an input signal are not needed. Summarizing,
at low frequencies, the gain of K( jv)G( jv), called the open-loop gain, should be high, while at high fre-
quencies it may be low. Note that K( jv)G( jv) is a complex-valued function, and that it can have a value
equal to or close to
1. To assess the behavior of the closed loop, the frequency response of the open-
2
loop, K( jv)G( jv),
is studied near this point where the magnitude of the open-loop response,
j
, is 1 or 0 dB. This point is the cross-over point, and the corresponding frequency is
called the cross-over frequency.
The phase shift of the open-loop at the cross-over frequency determines what the gain of the closed-
loop system will be at the cross-over frequency. A phase shift near 180
K( jv)G( jv)
j
will cause the magnitude of
the denominator of the closed-loop system in Equation 12.14 to become very small (while the numer-
ator's magnitude is 1), and the closed-loop frequency response will have what is called an oscillatory
peak. The response of the closed-loop system to, for example, a step change in the reference signal
will show an oscillation with approximately the cross-over frequency. Investigation of the phase at
cross-over is important for the assessment of the stability of the closed loop. The difference between
the phase shift at cross-over and a phase of
8
1) is called the phase margin.For
stability of the closed-loop system, the phase margin must be positive, that is, the phase shift of the
system is less negative than
2
180
8
(i.e., the point
2
is chosen.
For any feedback system, where the open-loop transfer function is not unstable, a positive phase margin
is a guarantee for closed-loop stability. Stability for a system that is open-loop unstable, that is, has open-
loop poles in the right-half complex plane, can be studied by means of the Nyquist stability theorem, which
also constitutes more formal proof of stability by means of the frequency response. Proof of and expla-
nation on the Nyquist stability theorem can be found in the engineering textbooks already mentioned.
An exemplary open-loop system, which, when used in a closed-loop feedback,
2
180
8
. Usually, a phase margin larger than 40
8
is the “single
in-tegrator”. For a single integrator, K( jv)G( jv)
1
( jv). The single integrator has an infinitely high
¼
/
gain at v
0, thus the output of the closed-loop system will perfectly follow the input for low frequen-
cies. The phase margin is always 90
¼
8
, whatever gain is chosen for the controller.
20
20
|
H
|
[dB]
|
H
|
[dB]
10
10
1+
t
2
s
1+
t
1
s
1+
t
1
s
1+
t
2
s
0
0
H
(
s
)=
H
(
s
)=
−10
−10
−20
−20
1/
t
1
1/
t
2
ω
[rad/sec]
1/
t
1
1/
t
2
[rad/sec]
ω
90
90
∠
H
∠
H
[deg]
[deg]
45
45
0
0
−45
−45
−90
−90
1/
t
1
1/
t
2
ω
[rad/sec]
1/
t
1
1/
t
2
ω
[rad/sec]
FIGURE 12.9 Bode diagram of a lag-lead compensating network (left) and of a lead-lag compensating network
(right).
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