Civil Engineering Reference
In-Depth Information
environment
inputs
u
1
u
2
u
3
outputs
y
1
y
2
system
FIGURE 12.1 Representation of a system, with inputs and outputs, and a system boundary that separates the
system from the environment.
This definition stresses the fact that the choice of what is considered to be the system and what belongs to
the environment (surroundings) is often a subjective one. However, once the system boundaries are
defined, the interaction between the system and its environment, by means of input and output
signals, also becomes clear, see Figure 12.1.
A system is linear with respect to the inputs and outputs, if the output to a sum of two or more input
signals, is the sum of the outputs each of these inputs would give individually. Most real-world systems
are not linear, but in many control situations, where there are only small excursions around an “operating
point,” they can be well approximated with linear models. If also the properties of the system do not
change over time, thus if we neglect processes such as wear and tear, the change of mass due to use of
fuel, etc., one obtains an LTI system. The behavior of an LTI system can be described with an ordinary
differential equation (ODE), with constant coefficients. For instance, consider the following ODE,
describing the relationship between the input signal of a system, u(t) and the output signal of that
system, y(t), see Figure 12.2:
a
2
d
2
y(t)
dt
2
a
1
dy(t)
dt
b
1
du(t)
dt
a
0
y(t)
þ
þ
¼
b
0
u(t)
þ
:
(12
:
1)
As in this example, the ODE generally consists of time-derivatives of the input and output signals,
characterizing the dynamic response of that system to the input signals.
12.2.2 Transfer Function
The calculation of solutions for ODEs, and the combination of systems described in ODE form, is usually
quite laborious. Therefore, in classical control theory, extensive use is made of the Laplace Transform, to
transform the ODEs to the Laplace domain, where manipulation such as the combination of systems
with other systems and signals, and the solution of the ODEs, is simpler, since all the equations
become algebraic. Textbooks on control theory contain tables of Laplace transforms that show the trans-
formation of signals to and from the Laplace domain, and Laplace transform theorems that can be used
to transform system descriptions to and from the Laplace domain. One can use the Laplace real differ-
entiation theorem, which states that:
¼
df (t)
dt
L
sF(s)
f (0)
(12
:
2)
Y
(
jw
)
y
(
t
)
U
(
jw
)
u
(
t
)
h
(
t
)
H
(
jw
)
or
FIGURE 12.2 LTI systems in the time domain (left) and the frequency domain (right).
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