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┃-J Tr ┃ - Tp Ts ┃
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Fig. 5.5 Characteristics ofsecond-order system response.
to plot y(c.ont) VS tont, which has the effect of normalizing the time with respect
to the system natural frequency COn. We observe that the lower the value of C , the
more oscillatory is the response and the larger is the overshoot.
The parameters that characterize the response of a second-order system (see
Fig. 5.5) are as follows:
D Peak time Tp. It is the time required to reach the first or maximum peak
y(max).
2) Percent overshoot Os. It is the maximum overshoot above the final or steady-
state value and expressed as a percentage of steady-state value y(oo).
3) Settlin,g time Ts. It is the time required for the transient response to come and
stay within +2% of the steady-state value.
4) Rise trme Tr. This is the time required for the response to rise from 0.1 to
0.9 of the final or steady-state value at its first occurrence.
Notice that the seffling time and rise time are basically the same as those defined
for first-order systems. The above defirutions are generalin nature and as such apply
to systems of any order.
In the following, we will derive analytical expressions for Tp, O.v, and Ts for
second-order systems. However, for the rise time Tr, it is not possible to obtain a
simple analytical expression.
The peak time Tp can be obtained by differentiating Eq. (5.69) with respect to
time t and finding the first zero crossing for t > 0 as
Tp= i
7r
eon,
(5.71)
(5.72)
r.!
'.1
'}
*1:
d
tl
/<,
~:
..
..
7!
i/d
r.,
~.
fq
c
,/r~/
452 PERFORMANCE, STABIL17Y, DYNAMICS, AND CONTROL
The percent overshoot Os is given by
Os - y nax) - y(oo) x 100
->
y(oo)
(5.73)
where y(rnax) is the value of y(t) at t - Tp. For the unit-step input, y(oo) : 1.
Then,
-<'r
Os = eh x ]OO
(5.74)
This relation states that the percent overshoot depends uniquely on the damping
ratio <. The value of the damping ratio corresponding to a given percent overshoot
is given by
>s]l00)
< = j~,,re0 (/
2( Os/l00)
(5.75)
The settling time Ts is the value of time t when the amplitude of the damped
response comes within 1:0.02 for the first occurrence. Using Eq. (5.69),
Solving, we get
e-<~" r, _ 0.02
- t,, (0.02 1)
Ts ~ --
< to,l
(5.76)
(5.77)
However, this expression is somewhat complex for frequent use. Instead, the fol-
lowing simple approximation is used to evaluate Ts. The numerator of the above
equation varies from 3.91 to 4.74 as < varies from O to 0.9. For typical under-
damped second-order systems, the numerator is usually close to 4. In view of this,
the following approximation is often used:
Ts=4 -4
(5.78)
5.4.3 Nonminimum Phase Systems
If all the poles and zeros of a system lie in the Ieft half of the s-plane, then the
given system is called a minimum phase system.lf a system has atleast one pole or
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