Page 295 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

298 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics

shape of the hysteresis loop and the smoothness within the pre-yield and
post-yield regions. The model can be reduced to a common damper if
a = 0. When a = 0 the hysteresis characteristics can be described.
Compared to the Bingham model, the curve of the Bouc-Wen model
is smooth, which can also reflect the non-linear behavior of MR damper at
the low speed regime. However, there are many parameters, which should
be calibrated based on the experiment data, i.e., the potential modeling
complexity makes it inefficient for application.

3) Modified Bouc-Wen model: To address the above mentioned issues
of Bouc-Wen model, Spencer et al. [13] proposed a modified Bouc-Wen
model described by

F = c 1 ˙y + k 1 (x − x 0 ) (19.4)

˙ y = 1 (c 0 + c 1 )[az + c 0 ˙x + k 0 (x − y)] (19.5)
n−1 n

˙ z = A(˙x −¨y) − γ ˙x −¨y |z| z − β(˙x −¨y)|z| (19.6)

where c 1 and k 1 are the viscosity coefficient and stiffness coefficient of
damper and spring, respectively. y and z are the auxiliary dynamic variables.
This modified Bouc-Wen model further improves the accuracy for mod-
eling the exact MR damper behaviors. However, there are two variables y
and z that cannot be directly observed, and their physical meaning is not
clearly justified. Moreover, the complexity of this model is also significant,
which may create difficulties in the modeling phase.
4) Hyperbolic model: to develop a simple, smooth MR model, which
is capable to describe hysteretic dynamics, a hyperbolic tangent function
can be used to represent the hysteresis characteristics embedded in the MR
damper. This is possible by considering the shape and mathematical expres-
sions of tangent functions. Thus linear functions representing the viscous
and stiffness together with a tangent function can lead to the following
hyperbolic MR model [20]

F = F yz + c 0 ˙x + k 0x + f 0 , (19.7)

z = tanh β ˙x + δ,sgn(x) (19.8)
where F y is the dynamic force coefficient associated with the current. z is
a hysteretic variable of the hyperbolic tangent function (19.8), β and δ are
the scale factors of hysteretic slope and bandwidth, respectively.
Compared to other models, the hyperbolic model contains a simple hy-
perbolic tangent function, which can be incorporated into the regressor