Page 12 - Servo Motors and Industrial Control Theory

4 1 Feedback Control Theory

In Eqs. (1.10) and (1.9) x, y represent small perturbation from the equilibrium point.
Equation (1.10) can be written as

y = Kx (1.11)

where

d Y
K = (1.12)
d X

K is constant at an operating point. Throughout this book, the lower case variable
represents small perturbation from equilibrium point. This is shown in Fig. 1.3.
Equation (1.8) represents one variable system. For a multivariable system, simi-
lar linearized equation can be obtained.
The solution of the governing equation simplifies if Laplace Transform is used.

1.4 Laplace Transform

By the definition, the Laplace Transform is defined as
∞
]
()e
F ( )s = L [ ()ft = ∫ ft − st dt (1.13)
0
By taking the Laplace Transform, the variable t is eliminated and the result is only
function of s.
Equation (1.13) appears to be very complicated, and indeed for complicated
transformation, the integral becomes very complex. Fortunately, for control systems
only a few functions are needed.
Example 2 Constant A.
∞
L ( )=A ∫ Ae − st dt (1.14)
0
This is a simple integration, and the integral becomes
A
LA() = (1.15)
s
The transformation of some common functions that are used in control are shown
in Table 1.1. There are a few important Laplace Transform that are often used in
defining performance of servo control systems. These are constant values which