Modern Control Engineering

Example Problems and Solutions 547

Figure 7–143 shows the Bode diagram of the closed-loop transfer function. The magnitude

curve of the closed-loop frequency response shows two resonant peaks. Note that such a case

occurs when the closed-loop transfer function involves the product of two lightly damped second-

order terms and the two corresponding resonant frequencies are sufficiently separated from each

other. As a matter of fact, the closed-loop transfer function of this system can be written

=

9

As^2 +0.487s+ 1 BAs^2 +0.613s+ 9 B

C(s)

R(s)

=

G(s)

1 +G(s)

30

24

18

12

6

0

• 18


• 12
  
  
  • 6
    
    
    • 360  – 270  – 180  – 90 
    
    
    
    
  
  
  
  

M= 0.5 dB

M=–2 dB

M= 8 dB

M= 2 dB

M=–4 dB

0.1

0.3

0.5

1

2.5 (^2) 1.5
3
3.5
G
G
in dB
Figure 7–142
Log-magnitude-
versus-phase plot of
G(s)given by
Equation (7–32).
20
0

• 20


• 40

0 °

• 90 °


• 180 °


• 270 °


• 360 °
  0.1 0.2 0.4 0.6 1 2 4 6 10
  v in rad/sec

dB

Figure 7–143
Bode diagram of

whereG(s)is given
by Equation (7–32).

G(s)C 1 +G(s)D,