Geology Reference
In-Depth Information
1
2
The error dynamics is given by
2
2
2
V
=
(
kx
+
mx
+
qx
)
(31)
1
1
2
3
e
= −
=
v
v
c
c
(35)
des
{
}
(
)
+
v
v
X
−
−
c
+
c
x
2
γ
q x x x
2
dum
c
a
0
2
2 3
3
des X
,
V
(
)
=
k x x
+
α
−
Aq x x
+
q x x
β
3
a
a
1
0
1 2
2 3
2 3
−
(
)
where
v
c
des X
,
is the derivative of
v
c
des
w.r.t.
state
X
.
Choosing a second Lyapunov function as
V
+
k
x x
+
c x
2
+
α
x x
v
b
b
b
c
0
1 2
0
2
2 3
(32)
= +
and the voltage variable
v
dum
as
given in Equation (36), it can be shown that the
system defined in Equation (30) becomes asymp-
totically stable (see Marquez, (2003); Krstic et
al., (1995)).
V
e
2
1
2
2
1
To design a stable closed loop system the
Lyapunov time-derivative
V
1
should be made
negative-definite. The first term in
V
1
, i.e.,
c
{
}
(
)
+
γ
, is free of the voltage
variable
v
c
and is negative-definite
∀
(
+
c
x
2
q x x x
a
0
2
2 3
3
)
, ,
q
is a positive constant given by
α
A
. Out of many
solutions, we select the designed commanded
voltage
v
c
dzs
to be
x x x
1
v
=
v
F t X
( ,
)
+
G t X v
( ,
)
2
3
dum
c X
,
1
1
c
(36)
des
−
V G t X
.
( ,
)
−
K v
(
−
v
)
1
,
X
1
c
c
des
with
F
1
and
G
1
defined in Equation (28);
K
>0
is any constant to be decided by the designer. For
our analysis
K=
1 is considered. The voltage ap-
plied to the MR damper can be obtained by sub-
stituting Equation (36) into equation (29).
2
3
k x
−
K x x
−
q x x
β
α
v
=
d
1
0
a
1 2
2 3
(33)
c
k x x
+
c x
2
+
x x
des
b
b
b
0
1 2
0
2
2 3
where
k
d
≥0
is a positive constant to be decided
by the designer. This simple form makes
V
NUMERICAL SIMULATION
OF NONLINEAR CONTROL
STRATEGIES
{
}
≤ ∀ ≠
(
)
+
γ
,
In the present analysis,
k
d
=1
is considered.
There can be a numerical stability problem, when
all
x
1
→
,
x
2
→
and
x
3
→
simultaneously.
Therefore, a tolerance is set for all the state vari-
ables, below which the damper input voltage is
kept at zero.
Nevertheless,
v
c
is a state variable, and perfect
tracking to
v
c
des
is desired and hardly achieved in
reality. Therefore, an error variable
e
(given in
Equation (34)) is defined as the error between the
target and the designed.
c
c
x
2
q x x x
2
k x
2
0
X
0
= −
+
+
1
0
a
2
2 3
3
d
1
Numerical simulations are carried out with eight
earthquake records. For the sake of brevity results
are reported for Big Bear earthquake excitation
with recorded magnitude 6.4 M on 28
th
June 1992
at San Bernardino Hospitality, California. Figure
25 shows the time history of the input excitation
and frequency spectrum. It is to be noted that
the frequency spectrum shows high peaks at low
frequency, which excites low frequency structures
like base isolated buildings. Mathematical model
of the three storey base isolated building shown in
Figure 4 is used for the study. A classical damp-
ing at the base is considered for the numerical
e
= −
v
v
(34)
c
c
des
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