Geology Reference
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first stage contains a primary controller, which
provides the force required to obtain a desired
closed loop response of the system. Then, DI
maps the required force to required voltage to be
supplied to the MR damper. Therefore the overall
control scheme forms a new two-stage stabilizing
state feedback control design approach.
To formulate the proposed two-stage control-
ler let us consider a system in state space form
as given by
gives a state feedback form of the control force
required (Ali and Ramaswamy 2009c).
f t
( )
= −
K X
g
(19)
where
K
g
is the feedback gain matrix and
X
the
states. The feedback gain (
K
g
) can be calculated or
can be obtained using the '
lqr
' function available
with the Control Toolbox in MATLAB. Once the
state feedback form of the optimal control force
has been obtained, it is necessary to compute the
voltage to be supplied to the MR damper such that
the MR damper provides similar control force.
Dynamic inversion is used to obtain a closed
form solution of the input voltage to be supplied
to the MR damper in order to obtain the desired
optimal force.
X
=
AX Bu Ex
g
+
+
(16)
where
X R
∈
is the state of the system, u ∈ R1
is the damper force, and
x
g
is the input excitation
to the system.
A R
n n
∈
×1
, and
E R
∈
×1
are the system state matrix, controller location
vector, and influence vector for support excitation,
respectively.
×
,
B R
n
∈
Secondary Controller Design
The secondary controller is designed with a goal
to minimize the error between the primary con-
troller and the control force supplied by the MR
damper in
L
2
norm sense. Let us define an error
term as follows
f
=
c x
+
k x
+
α
z
(17)
c
0
mr
0
mr
mr
where
f
c
is the MR damper force. For simplifica-
tion, we assume a perfectly observable and con-
trollable system, and the all states are measurable.
1
2
(
)
2
e
=
u
−
f
(20)
Primary Controller Design
An LQR (linear quadratic regulator) is considered
as the first-stage or primary controller. LQR is
designed to obtain the optimal force required to
minimize the cost function defined as
The idea is to minimize the error,
e
in an ex-
ponential decay fashion. Therefore a first order
dynamics is considered for the error variable.
e
+
k e
=
0
e
1
τ
{
}
(21)
∫
k
J
=
lim
τ
X QX u Ru dt
T
+
T
(18)
(
)
−
(
)
+
(
)
=
2
u
−
f u
f
e
u
−
f
1
0
τ
→∞
0
2
where Q and R are weighting matrices used to
appropriately weight the states and calculate the
controller force required. Minimization of the
performance index in Equation (27) with the
system dynamics Equation (25) as a constraint
In Equation (21),
k
e
>0
serves as a gain. One
=
1
τ
may choose it as
k
e
, where
τ
c
>0
serves
c
as a 'time constant' for the error
e
to decay. Choice
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