0195136047.pdf

814 BASIC CONTROL SYSTEMS

Position

control

Variable

resistance

Potentiometer arm

Computer

dc source

Data inputs:

Dc source Dc

motor

Load

ω

ω

+

+

−

−

arm position

desired performance

load

Vm

Ia

Vm

Ra

Ia

Figure 16.3.12Adaptive control process as applied to a motor-speed control system.

Methods of Analysis

Just as differential equations are used to represent systems with analog signals,difference

equationsare used for systems with discrete or digital data. Difference equations are also used

to approximate differential equations, since the former are more easily programmed on a digital

computer, and are generally easier to solve.

One of the mathematical tools devised for the analysis and design of discrete-data systems

is thez-transformwithz=eTs. The role of thez-transform for digital systems is similar to that

of the Laplace transform for continuous-data systems. While the Laplace transform can be used

to solve linear ordinary differential equations, for linear difference equations and linear systems

with discrete or digital data, thez-transform becomes more appropriate to use. Since it is not a

simple matter to perform an inverse Laplace transform on transcendental functions which involve

terms likee−kT s, the need arises to convert transcendental functions insinto algebraic ones in

z. The development ofz-transform methods of analysis are considered to be outside the scope of

this book.

Various techniques and methods mentioned earlier, such as state-variable analysis, time-

domain analysis, frequency-domain analysis, root-locus techniques, and Bode diagrams, are

applied to the analysis of digital control systems. Details of these are obviously outside the

scope of this introductory text.

Finally, no one can be an expert in all areas discussed in this chapter, or indeed in the preceding

chapters. Therefore, it is always good advice toconsultwith those who are. The basics you have

been exposed to will help you to select such consultants, either in or out of house, who will provide

the knowledge to solve the problem confronting you.

16.4 Learning Objectives

Thelearning objectivesof this chapter are summarized here, so that the student can check whether

he or she has accomplished each of the following.