Page 9 - Servo Motors and Industrial Control Theory -

Chapter 1
Feedback Control Theory

1.1 Linear System

In any system, if there exists a linear relationship between two variables, then it is
said that it is a linear system.
For example, the equation

y = Kx (1.1)

represents a linear system. It means that if K is constant then the relationship (1.1)
represents a linear relationship between two variables y and x. In general, any gov-
erning differential equations between two variables x and y in the form of
d n d n−1 d m
+
a ya y + ay = b x + bx (1.2)
n n n−1 n−1 m m
t d t d t d
is linear, where n and m represent the order of differential equations, and a , b are
n
m
constants. For real system n > m, any other form of equations that is not similar to
Eq. (1.2) is called nonlinear system.
There are extensive theories that deal with linear systems, but the theories on
nonlinear systems are very complex and little.
Example 1 The circuit diagram of equivalent DC servo motors is shown in Fig. 1.1.
The governing differential equation may be written as

V = RI + L dI + C ω (1.3)
i m m
dt
where V, I, ω are the input voltage, current, and angular speed. R and L are the
i
m
resistance and inductance, respectively. This represents a linear system, where ω
m
is the output variable and V represents the input voltage.
i
For DC servo motor, we can write
T = K I (1.4)
t
R. Firoozian, Servo Motors and Industrial Control Theory, Mechanical Engineering Series, 1
DOI 10.1007/978-3-319-07275-3_1, © Springer International Publishing Switzerland 2014