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age. However, this is unlikely as in that case the
prescribed force by the primary controller should
be zero and the algorithm ends up in a
0
0
position.
To avoid such a numerically unstable situation,
the supply voltage near the zero state condition
is redefined as
of
k
e
determines the stability of the controller and
its tracking efficiency. It should be noted that
Equation (21) contain the dynamics of the pri-
mary control force,
f
and the force provided by
the MR damper,
u
. Equation (19) provides
f
and
Equation (1) provides
u f
=
, which are given in
Equations (22) and (23), respectively
v
=
a
redefined
0
x
<
tol
and
otherwise
x
<
tol
f t
( )
= −
K X
g
(22)
mr
1
mr
2
v
a
(25)
(
)
u
=
c x
+
K x
+
α
z
0
a mr
0
a mr
a mr
(
)
ˆ
ˆ
−
c x
+
K x
+
α
z
η
v
Backstepping Control Design
0
b mr
0
b mr
b mr
c
(23)
(
)
+
c x
+
K x
+
α
z
v
0
a mr
0
a mr
a mr
c
The DI technique designed in the previous sec-
tion considers the input voltage dynamics of the
MR damper in its algorithm development. Nev-
ertheless it has a drawback in that one needs to
design an intermediate controller like H2/LQG
and then employ dynamic inversion to determine
the voltage required to be supplied to the MR
damper such that the control force prescribed by
the intermediate controller is supplied. The main
scope of this section is to design a stable semi-
active controller maintaining the good features of
the DI algorithms but eliminating the intermedi-
ate primary controller, and for this the integral
backstepping controller proposed by Krstic et al.,
(1995) is adopted in this study.
In recent adaptive and robust control literature,
the backstepping design provides a systematic
framework for the design of tracking and regula-
tion strategies, suitable for a large class of state
feedback linearizable nonlinear systems. Integra-
tor backstepping is used to systematically design
controllers for systems with known nonlinearities.
The approach can be extended to handle systems
with unknown parameters, via adaptive backstep-
ping. However, adaptive backstepping design
for nonlinear control may dramatically increase
the complexity of the controller. In this chapter,
integrator backstepping is applied to deduce the
(
)
c x
K x
z
v
+
+
+
α
0
b mr
0
b
mr
b mr
a
The voltage supplied to the MR damper is
represented by
v
a
whereas the voltage driving the
magnetic flux, i.e., at the damper magnetic coils
(also known as commanded voltage), is repre-
sented by
v
c
.
v
c
represents the measured value of
the commanded voltage obtained from on-line
integration using Simulink. Substituting
u
from
Equation (23) into Equation (21), the following
simplified form of the supply voltage is obtained:
v
=
a
k
(
)
−
(
)
f
e
u
f
c x
K x
z
+
−
+
+
α
a mr
a mr
a mr
0
0
2
(
)
−
c x
+
K x
+
α
z
η
v
ˆ
ˆ
b mr
b mr
b
mr
c
0
0
(
)
+
c x
+
K x
+
α
z
v
a mr
a mr
a mr
c
0
0
(
)
×
c x
+
K x
+
α
z
0
b mr
0
b mr
b mr
(24)
It is to be noted that when the system dynam-
ics at the damper location goes to zero (particu-
larly at steady state) or in any situation where the
states simultaneously go to zero, an unstable
situation may arise in the computed applied volt-
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