Continuous Control Systems: a review

Figure 1.6. Step response for a first-order system

5. 
   Frequency Response
   

The frequency response of a dynamic system is studied and characterized for periodic inputs of variable frequency but of constant magnitude. For continuous- time systems, the gain-frequency characteristic is represented on a double logarithmic scale and the phase frequency characteristic is represented on a logarithmic scale only for the frequency axis.

-

)

 _B
⁽ ƒ ⁾ B


Figure 1.7. Frequency responses

 ƒ

The gain G() = |H(j)| is expressed in dB (|H(j)| dB = 20 log |H(j)|) on the vertical axis and the frequency , expressed in rad/s ( = 2 f where f represents the frequency in Hz) is represented on the horizontal axis. Figure 1.7 gives some typical frequency response curves.

The characteristic elements of the frequency response are:

• 
  f_B(_B) (bandwidth): the frequency (radian frequency) from which the zero- frequency (steady-state) gain G(0) is attenuated more than 3 dB;
  

G(_B )  G(0)  3dB; (G(_B )  0.707 G(0)) ^.

• 
  f_C(_C) (cut-off frequency): the frequency (rad/s) from which the attenuation introduced with respect to the zero frequency is greater than N dB;
  

G( j_C )  G(0)  N dB ^.

• 
  Q (resonance factor): the ratio between the gain corresponding to the maximum of the frequency response curve and the value G(0).
  
• 
  Slope: it concerns the tangent to the gain frequency characteristic in a certain region. It depends on the number of poles and zeros and on their frequency distribution.
  

Consider, as an example, the first-order system characterized by the transfer function given by Equation 1.1.9. For s = j the transfer function of Equation 1.1.9 is rewritten as

H ( j ) 
G
1  jT
 H ( j ) e ^{j ()}  H ( j )  ( )

(1.1.16)

where |H(j represents the modulus (gain) of the transfer function and  ^
the phase deviation introduced by the transfer function. We then have

G( )  H ( j )  ^G

(1.1.17)

 
 ( )  tan^{1  ImG( j ) }  tan^1  T  (1.1.18)
Re G( j )
 

From Equation 1.1.17 and from the definition of the bandwidth _B, we obtain:

_B  1/ T



B
Using Equation 1.1.18, we deduce that for  = _B the system introduces a phase

deviation
 ^{  )} = -45^°. Also note that for  = 0, G(0) = G,
 ^{ 0)} = 0^° and

for    , G(   = 0,  ^{ }  ⁾ = -90°.
Figure 1.8 gives the exact and asymptotic frequency characteristics for a first- order system (gain and phase).
As a general rule, each stable pole introduces an asymptotic slope of -20 dB/dec (or 6 dB/octave) and an asymptotic phase lag of -90°. On the other hand, each stable zero introduces an asymptotic slope of +20 dB/dec and an asymptotic phase shift of +90°.
It follows that the asymptotic slope of the gain-frequency characteristic in dB, for high frequencies, is given by

^G  (n  m)  20 dB / dec


(1.1.19a)

where n is the number of poles and m is the number of zeros.

0
-10

BODE diagram


0
-22.5

[dB] [Deg]

-20
-30

-40
-2 -1 0
-45
-67.5
-90
1 2

10 10
10 10 10
 ^[rad/s]