Page 397 - Mechanical Engineers' Handbook (Volume 2)

388 Basic Control Systems Design

The object to be controlled is the plant. The manipulated variable is generated by the
final control elements for this purpose. The disturbance input also acts on the plant. This is
an input over which the designer has no influence and perhaps for which little information
is available as to the magnitude, functional form, or time of occurrence. The disturbance can
be a random input, such as wind gust on a radar antenna, or deterministic, such as Coulomb
friction effects. In the latter case, we can include the friction force in the system model by
using a nominal value for the coefficient of friction. The disturbance input would then be
the deviation of the friction force from this estimated value and would represent the uncer-
tainty in our estimate.
Several control system classifications can be made with reference to Fig. 7. A regulator
is a control system in which the controlled variable is to be kept constant in spite of distur-
bances. The command input for a regulator is its set point. A follow-up system is supposed
to keep the control variable near a command value that is changing with time. An example
of a follow-up system is a machine tool in which a cutting head must trace a specific path
in order to shape the product properly. This is also an example of a servomechanism, which
is a control system whose controlled variable is a mechanical position, velocity, or acceler-
ation. A thermostat system is not a servomechanism, but a process control system, where
the controlled variable describes a thermodynamic process. Typically, such variables are
temperature, pressure, flow rate, liquid level, chemical concentration, and so on.

2.2 Transfer Functions
A transfer function is defined for each input–output pair of the system. A specific transfer
function is found by setting all other inputs to zero and reducing the block diagram. The
primary or command transfer function for Fig. 7 is
C(s) A(s)G (s)G (s)G (s)
a
m
p
(1)
V(s) 1 G (s)G (s)G (s)H(s)
p
m
a
The disturbance transfer function is
C(s) Q(s)G (s)
p
(2)
D(s) 1 G (s)G (s)G (s)H(s)
m
a
p
The transfer functions of a given system all have the same denominator.
2.3 System-Type Number and Error Coefficients
The error signal in Fig. 4 is related to the input as
1
E(s) R(s) (3)
1 G(s)H(s)
If the final-value theorem can be applied, the steady-state error is

sR(s)
e lim (4)
ss
s→0 1 G(s)H(s)
The static error coefficient c is defined as
i
i
c lim sG(s)H(s) (5)
i
s→0