Page 21 - Servo Motors and Industrial Control Theory

1.9 Block Diagram Representation 13

Fig. 1.8 Block diagram
representation of a control X(s) G(s) Y(s)
system

Fig. 1.9 Two cascaded con- X(s) Y (s) Y (s)
2
1
trol system G 1 (s) G 2 (s)

The frequency response can be obtained similar to the first order system. The
amplitude ratio reaches a peak value when approximately ω = ω . For ς < 1, there is
n
a resonance frequency showing the fact that the response is oscillatory. For ς > 1,
the amplitude ratio always is less than 1 showing the fact that the system is over-
damped. For exercise, the reader is recommended to obtain the frequency response
by replacing s with iω. The reader is also encouraged to obtain the response to a
ramp input. The frequency response will be studied at the end of this chapter.

1.9 Block Diagram Representation

Complicated system may have several elements in cascade or in the feedback. It is
useful to represent this kind of system in block diagram form. A single transfer func-
tion may be shown as a single block, where the input of the system x( s) entering the
block diagram and after transformation leaves the block diagram as y( s).
This is shown in Fig. 1.8, where G( s) is the transfer function. If two block dia-
grams appear in cascade as shown in Fig. 1.9, the result is obtained by multiplying
the two transfer functions together. This is shown in Fig. 1.10.
When two or more variables have to be added or subtracted, a circle is used to
show this. The summation or subtraction is shown by + or − sign on the circle.
A feedback is obtained when the subtraction is performed. This is shown in
Fig. 1.11.
For simple multivariable control system, principle of superposition can be used
and a single output variable with respect to a single input variable can be studied.
Any simple or complicated block diagram can be reduced to a single block. For
example, the block diagram in Fig. 1.11 can be reduced to a single block as

e = x s −() y sH s() () (1.41)
and

ys() = eG s() (1.42)

eliminating e from Eqs. (1.41) and (1.42) yields