486 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method7–9 Experimental Determination of Transfer Functions
The first step in the analysis and design of a control system is to derive a mathematical
model of the plant under consideration. Obtaining a model analytically may be quite dif-
ficult. We may have to obtain it by means of experimental analysis. The importance of the
frequency-response methods is that the transfer function of the plant, or any other com-
ponent of a system, may be determined by simple frequency-response measurements.
If the amplitude ratio and phase shift have been measured at a sufficient number of
frequencies within the frequency range of interest, they may be plotted on the Bode di-
agram. Then the transfer function can be determined by asymptotic approximations. We
build up asymptotic log-magnitude curves consisting of several segments. With some
trial-and-error juggling of the corner frequencies, it is usually possible to find a very
close fit to the curve. (Note that if the frequency is plotted in cycles per second rather
than radians per second, the corner frequencies must be converted to radians per sec-
ond before computing the time constants.)
Sinusoidal-Signal Generators. In performing a frequency-response test, suitable
sinusoidal-signal generators must be available. The signal may have to be in mechani-
cal, electrical, or pneumatic form. The frequency ranges needed for the test are ap-
proximately 0.001 to 10 Hz for large-time-constant systems and 0.1 to 1000 Hz for
small-time-constant systems. The sinusoidal signal must be reasonably free from har-
monics or distortion.
For very low frequency ranges (below 0.01 Hz), a mechanical signal generator
(together with a suitable pneumatic or electrical transducer if necessary) may be used.
For the frequency range from 0.01 to 1000 Hz, a suitable electrical-signal generator
(together with a suitable transducer if necessary) may be used.
Determination of Minimum-Phase Transfer Functions from Bode Diagrams.
As stated previously, whether a system is minimum phase can be determined from the
frequency-response curves by examining the high-frequency characteristics.
To determine the transfer function, we first draw asymptotes to the experimental-
ly obtained log-magnitude curve. The asymptotes must have slopes of multiples of
;20 dBdecade. If the slope of the experimentally obtained log-magnitude curve
changes from –20to–40dBdecade at v=v 1 , it is clear that a factor 1/C1+jAv/v 1 BD
exists in the transfer function. If the slope changes by –40dBdecade at v=v 2 , there
must be a quadratic factor of the form
in the transfer function. The undamped natural frequency of this quadratic factor is
equal to the corner frequency v 2. The damping ratio zcan be determined from the
experimentally obtained log-magnitude curve by measuring the amount of resonant
peak near the corner frequency v 2 and comparing this with the curves shown in
Figure 7–9.
Once the factors of the transfer function G(jv)have been determined, the gain can
be determined from the low-frequency portion of the log-magnitude curve. Since such
1
1 + 2 zaj
v
v 2
b + aj
v
v 2
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