Modern Control Engineering

472 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method

In the following, we shall summarize the correlation between the step transient response

and frequency response of the standard second-order system given by Equation (7–16):

1.The phase margin and the damping ratio are directly related. Figure 7–74 shows a plot

of the phase margin gas a function of the damping ratio z. It is noted that for the stan-

dard second-order system shown in Figure 7–73, the phase margin gand the damping

ratiozare related approximately by a straight line for 0z0.6, as follows:

Thus a phase margin of 60° corresponds to a damping ratio of 0.6. For higher-order

systems having a dominant pair of closed-loop poles, this relationship may be used

as a rule of thumb in estimating the relative stability in the transient response (that

is, the damping ratio) from the frequency response.

2.Referring to Equations (7–17) and (7–19), we see that the values of vrandvdare

almost the same for small values of z. Thus, for small values of z, the value of vris

indicative of the speed of the transient response of the system.

3.From Equations (7–18) and (7–20), we note that the smaller the value of zis, the

larger the values of MrandMpare. The correlation between MrandMpas a func-

tion of zis shown in Figure 7–75. A close relationship between MrandMpcan be

seen for z>0.4. For very small values of z,Mrbecomes very large AMr 1 B, while

the value of Mpdoes not exceed 1.

Correlation between Step Transient Response and Frequency Response in

General Systems. The design of control systems is very often carried out on the basis

of the frequency response. The main reason for this is the relative simplicity of this ap-

proach compared with others. Since in many applications it is the transient response of

the system to aperiodic inputs rather than the steady-state response to sinusoidal in-

puts that is of primary concern, the question of correlation between transient response

and frequency response arises.

z=

g

100°

90 °

60 °

30 °

0 °

0 0.4 0.8 1.2 1.6 2.0

z

g

Straight-line

approximation

Figure 7–74

Curveg(phase

margin) versus zfor

the system shown in

Figure 7–73.

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