Page 57 - Servo Motors and Industrial Control Theory -

Chapter 3
State Variable Feedback Control Theory

3.1 Introduction

In the previous two chapters, the classical control theory was discussed and the
concept of transfer function was used to describe the dynamic behavior of various
systems. In practice, the transfer function of higher order than three or four becomes
tedious and it is better to use state variable control theory. It was stated that with PID
control, three parameters can be adjusted to design control systems. With three pa-
rameters, all roots of characteristic equation can not be adjusted to achieve desirable
transient and steady state behavior. In practice, it is only possible to make one or
two roots to become dominant in response and a compromise between the transient
response and steady state error has to be made. The derivative term always ampli-
fies the noise in the practical systems and is not recommended. Instead a lead-lag
network produces a better response.
State variable feedback control theory makes use of matrices to describe con-
trol systems. This makes it possible to write the governing differential equation in
compact form. The use of state variable feedback makes it possible to adjust the
location of all roots to desirable position in the s-plane. The practical limitations of
measuring all state variables, and the nonlinearities of transducers and saturation of
amplifiers limits as where the roots can be located in the s-plane.
Observer may be used to predict state variables from the measurement of a single
or few state variables. This makes it possible to control practical systems with many
state variables.
In practice, if high performance is not required it is better to model servo systems
with second or third order transfer function by making simplifying assumptions.
When high performance is required, many factors such as inductance and the com-
pliance of the transmission mechanism must be considered. These considerations
make the model very complicated and state variable control theory is recommended
as the concept of transfer function becomes very tedious.
In this chapter, the concept of state variables is described from the transfer func-
tion but in the later chapters it will be shown that state variables can be defined right
from the beginning when the governing equations for each element is written and
there is no need to find the overall transfer function.

R. Firoozian, Servo Motors and Industrial Control Theory, Mechanical Engineering Series, 49
DOI 10.1007/978-3-319-07275-3_3, © Springer International Publishing Switzerland 2014