10 5 transient response specifications - KFUPM Open Courseware

ME 413 Systems Dynamics & Control
Chapter 10: Time‐Domain Analysis and Design of Control Systems
T s
4
ζω
= ( 2%Criterion)
(10‐19)
n
Similarly for 5% we can get
3
T s =
ζω
( 5%Criterion)
(10‐20)
n
REVIEW AND SUMMARY
TRANSIENT RESPONSE SPECIFICATIONS OF A SECOND ORDER SYSTEM
TABLE 1.
Useful Formulas and Step Response Specifications for the Linear
Second‐Order Model m & x&+ c x&
+ k x = f (t)
where m, c, k constants
2
− c ± c − 4mk
1. Roots
s1,2
=
2m
2. Damping ratio or ζ = c / 2 mk
3. Undamped natural frequency
ω
n
=
2
4. Damped natural frequency
ω d
= ω n
1−ζ
5. Time constant τ = 2 m / c = 1/
ζω
n
if ζ ≤ 1
k
m
6. Stability Property Stable if, and only if, both roots have negative real parts, this occurs if
and only if , m, c, and k have the same sign.
7. Maximum Percent Overshoot: The maximum % overshoot M
p
is the maximum peak value of the
response curve.
M p
= 100e
2
−πζ / 1−ζ
8. Peak time: Time needed for the response to reach the first peak of the overshoot
2
T = π / ω 1−
ζ
9. Delay time: Time needed for the response to reach 50% of its final value the first time
1+
0.
7ζ
Td
≈
ω
p
n
10. Settling time: Time needed for the response curve to reach and stay within 2% of the final value
4
Ts
=
ζω
n
11. Rise time: Time needed for the response to rise from (10% to 90%) or (0% to 100%) or (5% to 95%) of
π − β
its final value Tr
= (See Figure 10‐25)
ω
d
n
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