Page 22 - Servo Motors and Industrial Control Theory -

14 1 Feedback Control Theory

Fig. 1.10 Simplifying two
cascaded system in a simple X(s) G (s)G 2 (s) Y(s)
block diagram 1

Fig. 1.11 A negative feed- X(s) + e Y(s)
back control system G(s)
–

H(s)

ys() Gs() (1.43)

xs() = 1 + Hs Gs() ()
Equation (1.43) can be shown in block diagram, (Fig. 1.12).
A positive feedback is used as shown in Fig. 1.11. With positive feedback instead
of negative sign; the reduction procedure is the same as before. The positive feed-
back of Fig. 1.11 is reduced to a single block diagram as in Fig. 1.13.

ys() = Gs() (1.44)
xs() 1 − Gs Hs() ()

The above procedure can be used for any complicated block diagram and helps to
reduce it to a single block.

1.10 Frequency Response

For a second order system with transfer function,

y = 1
x 1 s + 2ζ s + 1
2
ω 2 ω
n n
the frequency response is obtained by replacing s with iω. After some algebraic
manipulation, the amplitude ratio and phase angle are given by