Modern Control Engineering

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

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

0 0.2 0.4 0.6 0.8 1.0

z

vb

vn

Figure 7–134

Curve of vb/vn

versusz, where vbis

the bandwidth.

By dividing both sides of this last equation by v^4 n, we obtain

Solving this last equation for Avb/vnB^2 yields

SinceAvb/vnB^2 >0, we take the plus sign in this last equation. Then

or

Figure 7–134 shows a curve relating vb/vnversusz.

A–7–18. A Bode diagram of the open-loop transfer function G(s)of a unity-feedback control system is

shown in Figure 7–135. It is known that the open-loop transfer function is minimum phase. From

the diagram, it can be seen that there is a pair of complex-conjugate poles at v=2radsec.

Determine the damping ratio of the quadratic term involving these complex-conjugate poles.

Also, determine the transfer function G(s).

Solution.Referring to Figure 7–9 and examining the Bode diagram of Figure 7–135, we find the

damping ratio zand undamped natural frequency vnof the quadratic term to be

z=0.1, vn= 2 radsec

vb=vnA 1 - 2 z^2 + 24 z^4 - 4 z^2 + 2 B^1 ^2

v^2 b=v^2 nA 1 - 2 z^2 + 24 z^4 - 4 z^2 + 2 B

a

vb

vn

b

2

=- 2 z^2 + 1 ; 24 z^4 - 4 z^2 + 2

1 =0.5ec 1 - a

vb

vn

b

2

d

2

+ 4 z^2 a

vb

vn

b

2

f

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