For the standard second-order system shown in Figure 7–73, mathematical rela-
tionships correlating the step transient response and frequency response can be obtained
easily. The time response of the standard second-order system can be predicted exactly
from a knowledge of the Mrandvrof its closed-loop frequency response.
For nonstandard second-order systems and higher-order systems, the correlation is
more complex, and the transient response may not be predicted easily from the fre-
quency response because additional zeros and/or poles may change the correlation be-
tween the step transient response and the frequency response existing for the standard
second-order system. Mathematical techniques for obtaining the exact correlation are
available, but they are very laborious and of little practical value.
The applicability of the transient-response–frequency-response correlation existing for
the standard second-order system shown in Figure 7–73 to higher-order systems depends on
the presence of a dominant pair of complex-conjugate closed-loop poles in the latter systems.
Clearly, if the frequency response of a higher-order system is dominated by a pair of com-
plex-conjugate closed-loop poles, the transient-response– frequency-response correlation
existing for the standard second-order system can be extended to the higher-order system.
For linear, time-invariant, higher-order systems having a dominant pair of complex-
conjugate closed-loop poles, the following relationships generally exist between the step
transient response and frequency response:
1.The value of Mris indicative of the relative stability. Satisfactory transient per-
formance is usually obtained if the value of Mris in the range 1.0<Mr<1.4
A0dB<Mr<3dBB, which corresponds to an effective damping ratio of
0.4<z<0.7. For values of Mrgreater than 1.5, the step transient response may
exhibit several overshoots. (Note that, in general, a large value of Mrcorresponds
to a large overshoot in the step transient response. If the system is subjected to
noise signals whose frequencies are near the resonant frequency vr, the noise will
be amplified in the output and will present serious problems.)
2.The magnitude of the resonant frequency vris indicative of the speed of the tran-
sient response. The larger the value of vr, the faster the time response is. In other
words, the rise time varies inversely with vr. In terms of the open-loop frequency
Section 7–7 / Relative Stability Analysis 4733Mr21Mp0
0.2 0.4 0.6 0.8 1.0
zMr=^1
2 z 1 – z^2Mp=c(tp)– 1
[Equation (5-21)]
Figure 7–75
CurvesMrversusz
andMpversuszfor
the system shown in
Figure 7–73.