Page 10 - Servo Motors and Industrial Control Theory

2 1 Feedback Control Theory

Fig. 1.1 Equivalent circuit R
diagram of a DC servo motor

L
V i
M C m W m

dω (1.5)
T = J m
dt

where K , J are the torque constant and rotor moment of inertia.
t
Eliminating T, from Eqs. (1.4) and (1.5) and substituting for I in Eq. (1.3) yields

2
RJ dω LJ d ω
V = m + m + C ω m (1.6)
i
m
K t dt K t dt 2
Equation (1.6) now represents a linear differential system, and in control terminol-
ogy, V is called the input variable and ω is called the output variable. The Eq. (1.6)
m
i
can be solved for ω in terms of the input variable. In deriving Eq. (1.6), we ignore
m
the external torque acting on the motor. If we consider the external torque, the gov-
erning differential equation would have two input variables and one output variable.
For linear systems, the principle of superposition holds. It means that if input x
1
causes output y and input x causes output y , then input x + x causes output y + y .
2
1
1
2
1
2
2
This is a powerful principle, and we will use it throughout this book.
1.2 Nonlinear Systems
There are different kinds of nonlinearities. For example, on–off control systems are
inherently nonlinear. Transport lag, saturation, and transport lag are other kinds of
nonlinearities. These kinds of nonlinearities cannot be solved with linear control
theory. This is shown in Fig. 1.2. There is complicated theory that covers discontin-
uous nonlinearities, but they are beyond the scope of this book. Most nonlinearities
that exist in servo control systems are shown in Fig. 1.2.
For linearized equation, it is better to use Laplace Transform. In this way, the dif-
ferential equations become algebraic equation in s. Throughout this book, the lower
case s represents Laplace Transform.
Some nonlinearity is continuous, and they can be solved by the linearization
technique. One example of this kind of nonlinearity is

y = Kx 2 (1.7)