Page 62 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

CHAPTER 3

Basic Concepts in

Process Analysis

ince the basic tenet of this book is to analyze the process before
one attempts to control it, we will have to develop some process
S analysis concepts. Furthermore, since controlling a process
inherently deals with transient behavior we will have to deal with
process dynamics. Finally, since we need to keep the level of the mate-
rial in this book reasonable, that is, attractive to a busy manager, we
will start with the simplest of constructs-the first-order process. In
fact, we will beat it to death. During the beating, the widely used
proportional-integral (PI) control algorithm will be introduced. One
of the sophisticated tools of control engineering, the Laplace trans-
form, will also be introduced. For technical support, the reader may
want to consult App. A (introductory calculus), App. B (complex
numbers), App. E (first- and second-order differential equations),
and App. F (Laplace transforms).

3-1 The First-Order Process-an Introduction
Let's go back to the tank of water introduced in Chap. 1 (Fig. 3-1). It
will be our prototypical first-order process. The dynamic analysis of
this tank often consists of studying the step response, which is shown
in Fig. 3-2. Here the process input U, the valve, is given a step at time
t = 9, from an initial value of zero to unity. The process output Y, the
tank level, begins to rise and appears to line out at a value of 2.0. For
convenience, we have chosen the initial value of the valve and the
tank level to be zero. In general, these quantities could have almost
any initial value but this graph would still apply if the reader is will-
ing to allow us to subtract these nonzero initial values, that is, nor-
malizing the initial values of these quantities to zero.
To proceed we need some nomenclature. First, let the change in
the process input be signified as ~U. The symbol~ usually signifies a
change in the quantity following it (or upon which it operates). Sec-
ond, let the resulting change in the tank level be signified as ~ Y.