Modern Control Engineering

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

Nonminimum-Phase Transfer Functions. If, at the high-frequency end, the com-

puted phase lag is 180° less than the experimentally obtained phase lag, then one of the

zeros of the transfer function should have been in the right-half splane instead of the

left-halfsplane.

If the computed phase lag differed from the experimentally obtained phase lag by a

constant rate of change of phase, then transport lag, or dead time, is present. If we assume

the transfer function to be of the form

whereG(s)is a ratio of two polynomials in s, then

where we used the fact that constant. Thus, from this last equation, we

can evaluate the magnitude of the transport lag T.

lim

vSq

/G(jv)=

= 0 - T=-T

= lim

vSq

d

dv

C/G(jv)-vTD

lim

vSq

d

dv

/G(jv)e-jvT= lim

vSq

d

dv

C/G(jv)+ /e-jvTD

G(s)e-Ts

(a)

0

• 20

20 log K

• 40
  
  
  • 40
  
  
  
  

dB

v =K v =K

v in log scale

(b)

(c)

0

• 20
  
  
  • 20
    
    
    • 20
    
    
    
    
  
  
  
  


• 20


• 40


• 40


• 40


• 40

dB

v in log scale

0

dB

v in log scale

0

dB

v in log scale

0

dB

v in log scale

v = K v = K

Figure 7–87

(a) Log-magnitude

curve of a type 0

system; (b) log-

magnitude curves of

type 1 systems;

(c) log-magnitude

curves of type 2

systems. (The slopes

shown are in

dBdecade.)

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