Chapter 2
MATHEMATICAL
MODELS IN CONTROL
In this chapter we present the basic properties of control and highlight significant
design and operating criteria ofmodel-basedcontrol theory. We discuss these
properties in the context of two very popular classical methods of control: state-
variable feedback control, and proportional-integral-derivative (PID) control.
This chapter serves as a platform for discussing the desirable properties of a
control system in the context of fuzzy and neural control in subsequent chapters.
It is not our intent to present a thorough treatment of classical control theory,
but rather, to present relevant material that provides the foundations for fuzzy
and neural control systems. The reader therefore is urged to refer to the many
excellent sources in classical control theory for further information.
Standard control theory consists of two tasks, analysis and synthesis.Analy-
sisrefers to the study of the plant and the control objectives.Synthesisrefers
to designing and building the controller to achieve the objectives. In standard
control theory, mathematical models are used in both the analysis and the syn-
thesis of controllers.
2.1 Introductoryexamples:pendulumproblems............
We present two simple, but detailed, examples to bring out the general frame-
work and techniques of standard control theory. Thefirst is a simple pendulum,
fixed at one end, controlled by a rotary force; and the second is an inverted
pendulum with one end on a moving cart. The concepts introduced in these ex-
amples are all discussed more formally, and in more detail, later in this chapter.
2.1.1 Example:fixed pendulum
We choose the problem of controlling a pendulum to provide an overview of
standard control techniques, following the analysis in [70]. In its simplified