A common approach to the design based on the Bode diagram is that we first adjust
the open-loop gain so that the requirement on the steady-state accuracy is met. Then the
magnitude and phase curves of the uncompensated open loop (with the open-loop gain
just adjusted) are plotted. If the specifications on the phase margin and gain margin are
not satisfied, then a suitable compensator that will reshape the open-loop transfer func-
tion is determined. Finally, if there are any other requirements to be met, we try to sat-
isfy them, unless some of them are mutually contradictory.
Information Obtainable from Open-Loop Frequency Response. The low-
frequency region (the region far below the gain crossover frequency) of the locus indi-
cates the steady-state behavior of the closed-loop system. The medium-frequency region
(the region near the gain crossover frequency) of the locus indicates relative stability.
The high-frequency region (the region far above the gain crossover frequency) indi-
cates the complexity of the system.
Requirements on Open-Loop Frequency Response. We might say that, in many
practical cases, compensation is essentially a compromise between steady-state accura-
cy and relative stability.
To have a high value of the velocity error constant and yet satisfactory relative sta-
bility, we find it necessary to reshape the open-loop frequency-response curve.
The gain in the low-frequency region should be large enough, and near the gain
crossover frequency, the slope of the log-magnitude curve in the Bode diagram should
be–20dBdecade. This slope should extend over a sufficiently wide frequency band to
assure a proper phase margin. For the high-frequency region, the gain should be atten-
uated as rapidly as possible to minimize the effects of noise.
Examples of generally desirable and undesirable open-loop and closed-loop
frequency-response curves are shown in Figure 7–89.
Referring to Figure 7–90, we see that the reshaping of the open-loop frequency-
response curve may be done if the high-frequency portion of the locus follows the G 1 (jv)
locus, while the low-frequency portion of the locus follows the G 2 (jv)locus. The reshaped
locusGc(jv)G(jv)should have reasonable phase and gain margins or should be tangent
to a proper Mcircle, as shown.
492 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response MethodIm- 10 Re
DesirableUndesirableIm dBRe– (^1) Logv
0
Desirable
Undesirable
(a) (b)
Desirable
Undesirable
Figure 7–89
(a) Examples of desirable and undesirable open-loop frequency-response curves;
(b) examples of desirable and undesirable closed-loop frequency-response curves.
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