Page 96 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Basic Concepts in Process Analysis 71

For example, in the development of Eq. (3-33) for the first-order
model
- g u
Y=---c
't'S + 1 S

there is a pole at s = -1 I -r and at s = 0. These poles correspond to an
exponential term e-1/r and a constant term.
In general, the Laplace transform can be written as a ratio of a
numerator N(s) to a denominator D(s)

(3-48)

showing that G(s) has m zeros, Z , z , ••• , zm and n poles, p , p , ••• , Pn'
1 2 1 2
any of which can be real or complex; however, complex poles and
zeros must appear as paired complex conjugates so that their product
will yield a real quantity.
The inversion of N(s)/D(s) will yield

II
Y(t) = L CkePk' (3-49)
k-1

Note that if some of the poles are complex they will occur as com-
plex conjugates and the associated exponential terms will contain
sinusoidal terms via Euler's formula. Finally, note that to find the
poles one usually sets the denominator of the Laplace transform,
D(s), to zero and solves for the roots.
The transfer function for the controlled system is

To see if this controlled system is stable one could find the values
of s (or the poles of G(s)) that cause

or

(3-50)

We will return to this equation many times in subsequent chapters.