Geology Reference
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voltage required by the MR damper to minimize
the structural responses.
This is also a two-stage control design. In
the first stage a Lyapunov control is designed to
stabilize the dynamics of the structural system.
Next, considering the MR damper input voltage
dynamics, a second Lyapunov based control is
developed to stabilize the full system, considering
both the structural system and the MR damper. It
is assumed that the three storey building behaves
as a SDOF system due to the presence of base iso-
lation (Chopra, 2005). The integral backstepping
based semi-active MR damper voltage monitoring
is developed for a SDOF system.
T
=
X
x x x
,
,
;
1
2
3
 
) + +
x
= =
x
x
;
x
= =
x
x
;
x
=
z
1
mr
2
mr
3
mr
{
}
(
(
) +
x
k
+
k
x
c
c
x
α
x
1
2
0
a
1
0
a
2
a
3
m
F
=
n
1
n
1
γ
x x x
β
x x
+
Ax
2
3
3
2
3
2
T
G
=
1
{
}
0
1
k x
+
c x
+
α
x
0
0
b
1
0
b
2
b
3
m
F
= −
η
i G
;
=
η
;
2
c
2
(28)
3 = is responsible for the
hysteretic behaviour of the MR damper and it
evolves with time. Therefore it is a hidden variable
and is considered as an additional state variable.
The variable x
z mr
System Model
Backstepping Controller Design
An SDOF model is considered with an MR damper
connected to it. The linear dynamics of SDOF
systems with an MR damper is given by
Equation (27) is in a second-order strict feedback
form. Let us define a dummy variable v dum such
that it satisfies the following relation:
mx
 
+ + +
cx
kx
u t
= −
mx g

(26)
( )
1
v
=
) (
v
F t X v
( ,
,
))
(29)
where m , c , and k are the mass, damping, and
stiffness of the SDOF system and (·) denotes the
derivative w.r.t. time, t . u t
a
dum
2
c
G t X v
( ,
,
2
c
( ) = is the MR
damper control force and x g is the external exci-
tation force. u ( ) is added as the system restoring
force as the MR damper acts as a passive device
in the absence of driver voltage. Substitute
u t
f c
The dummy variable v dum is defined to convert
the second-order strict feedback system to a sim-
plified form amenable for integrator backstepping
application. Combining Equations (27) and (29),
we reduce the strict feedback system to an integra-
tor backstepping form:
( ) ( = from Equation (1) to Equation (26).
Rewriting the closed loop system dynamics and
considering the MR damper dynamics (and ne-
glecting the external excitation) in state space
form, one gets
f t
c
X
=
F t X
( ,
)
+
G t X v
( ,
)
1
1
c
(30)
v
=
v
c
dum
The design objective is the state variable
X →0 as the time t → ∞ . The control law can
be synthesized in two steps. We regard the com-
manded voltage, v c , to the damper as the real
voltage driver, first. By choosing the Lyapunov
candidate function of the system as
X
=
F t X
( ,
)
+
G t X v
( ,
)
1
1
c
(27)
v
=
F t X i
( ,
,
)
+
G t X i v
( ,
,
)
c
2
c
2
c
a
where X, F1, G1, F2 and G2 are given in Equa-
tion (28).
 
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