Page 331 - Mechanical Engineers' Handbook (Volume 2)

322 Mathematical Models of Dynamic Physical Systems

system I/O equations. Transform methods provide a particularly convenient algebra for com-
bining the component submodels of a system and form the basis of so-called classical control
theory. State-variable methods use the vector state and output equations directly. State-
variable methods permit the adaptation of important ideas from linear algebra and form the
basis for so-called modern control theory. Despite the deceiving names of ‘‘classical’’ and
‘‘modern,’’ the two approaches are complementary. Both approaches are widely used in
current practice and the control engineer must be conversant with both.

5.1 Transform Methods
A transformation converts a given mathematical problem into an equivalent problem, ac-
cording to some well-defined rule called a transform. Prudent selection of a transform fre-
quently results in an equivalent problem that is easier to solve than the original. If the solution
to the original problem can be recovered by an inverse transformation, the three-step process
of (1) transformation, (2) solution in the transform domain, and (3) inverse transformation
may prove more attractive than direct solution of the problem in the original problem domain.
This is true for fixed linear dynamic systems under the Laplace transform, which converts
differential equations into equivalent algebraic equations.

Laplace Transforms: Definition
The one-sided Laplace transform is defined as

F(s) L[ƒ(t)] ƒ(t)e st dt
0
and the inverse transform as
ƒ(t) L [F(s)]
j st
1
1
2 j
j F(s)e ds
The Laplace transform converts the function ƒ(t) into the transformed function F(s); the
inverse transform recovers ƒ(t) from F(s). The symbol L stands for the ‘‘Laplace transform
1
of’’; the symbol L stands for ‘‘the inverse Laplace transform of.’’
The Laplace transform takes a problem given in the time domain, where all physical
variables are functions of the real variable t, into the complex-frequency domain, where all
physical variables are functions of the complex frequency s
j , where j 1 is
the imaginary operator. Laplace transform pairs consist of the function ƒ(t) and its transform
F(s). Transform pairs can be calculated by substituting ƒ(t) into the defining equation and
then evaluating the integral with s held constant. For a transform pair to exist, the corre-
sponding integral must converge, that is,

ƒ(t) e
*t dt
0
for some real
* 0. Signals that are physically realizable always have a Laplace transform.

Tables of Transform Pairs and Transform Properties
Transform pairs for functions commonly encountered in the analysis of dynamic systems
rarely need to be calculated. Instead, pairs are determined by reference to a table of trans-
forms such as that given in Table 4. In addition, the Laplace transform has a number of
properties that are useful in determining the transforms and inverse transforms of functions