REGULATING STEADYSTATE BEHAVIOR
12
REGULATING STEADYSTATE
BEHAVIOR
This section treats the regulation problem associated with
a two degreeoffreedom system acted on by a persistent
excitation. For simplicity we’ll assume that the persistent
excitation is harmonic, although the material presented in
this chapter can be extended to systems acted on by
general persistent excitations. The material covered in
this chapter can be extended to general persistent
excitations by representing the general persistent
excitation as a linear superposition of harmonic
excitations. Also, we’ll assume in this section that the
control force is continuouslyacting.
When a system is acted on by a persistent excitation, the
dominant part of the response is a steadystate response,
at least after the transient response has damped out. The
regulation problem is essentially about changing steady
state behavior. This is clearly very different than the
regulation problems treated earlier in the notes where the
focus was on the transient response.
MAE 461: DYNAMICS AND CONTROLS
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1.
The Single DegreeofFreedom System
Consider a damped single degreeoffreedom system
whose base is excited harmonically, as shown in Fig. 12 
1.
Figure 12  1
The equation of motion is
(12  1)
f
x
x
c
x
x
k
x
m
+
−
−
−
−
=
)
(
)
(
0
0
where
is the base excitation and
f
is the regulation
force. Equation (12  1) is rewritten as
0
x
(12  2)
f
x
c
kx
kx
x
c
x
m
+
+
=
+
+
0
0
in which the right side of the equation contains the non
homogeneous terms. Assume that the system is subject to
the harmonic base excitation
(12  3)
t
i
e
X
t
x
ω
0
0
)
(
=
where
X
0
is the amplitude of the base excitation. We now
develop a regulator that prescribes a harmonic force
f
that
opposes the response in such a way as to reduce the
amplitude of the response. Accordingly, the regulating
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 Fall '08
 Silverberg
 Equations, Controls, k2, SteadyState Behavior

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