Modern Control Engineering

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174 Chapter 5 / Transient and Steady-State Response Analyses

z

%

100

90

80

70

60

50

40

30

20

10

0 0.5 1.0 1.5

Mp

Mp: Maximum overshoot

C(s)

R(s)=

vn

s^2 + 2 zvns+vn

2

2

Figure 5–12

Mpversuszcurve.

2 T

3 T

4 T

T

5 T

6 T

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

z

Settling time,

ts

5% Tolerance band

2% Tolerance band

Figure 5–11

Settling time ts

versuszcurves.

is determined primarily by the undamped natural frequency vn. This means that the

duration of the transient period may be varied, without changing the maximum over-

shoot, by adjusting the undamped natural frequency vn.

From the preceding analysis, it is evident that for rapid response vnmust be large. To limit

the maximum overshoot Mpand to make the settling time small, the damping ratio zshould

not be too small. The relationship between the maximum percent overshoot Mpand the

damping ratio zis presented in Figure 5–12. Note that if the damping ratio is between 0.4

and 0.7, then the maximum percent overshoot for step response is between 25%and 4%.

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