Page 294 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Estimation and Control of Magneto-Rheological Damper for Semi-Active Suspensions 297

Figure 19.1 Work process of vehicle suspension system with MR.

Wen model [13], hyperbolic model [20]. We first review some of these
models in terms of the modeling complexity and dynamical behaviors.
1) Bingham model: this is the most commonly used MR damper model,
which can describe essential characteristics of MR fluid. We can obtain a
dynamic equation as [11]

F = f c sgn(˙x) + c 0 ˙x + f 0 (19.1)

where F is the generated damping force, ˙x is the piston velocity, sgn(·) is a
signum function. f c is the friction coefficient associated with the MR fluid,
c 0 is the viscosity damping coefficient, and f 0 is the damper force induced
by the internal pressure difference of the damper.
Bingham model is simple and easy for analysis. It can describe the force-
velocity relationship. However, this model assumes that the damper is rigid,
and the viscoelastic property of the damper force in the pre-yield region
is ignored. Thus, the force-velocity curve may be non-smooth when the
velocity is around zero.

2) Bouc-Wen model: The following Bouc-Wen model consists of a
spring, a viscous damper, and a Bouc-Wen hysteretic operator [12]. Bouc-
Wen model can be used to capture the hysteresis behavior of MR dampers,
where the damping force is given by

F = az + c 0 ˙x + k 0 (x − x 0 ) (19.2)
n−1 n
˙ z = A˙x − γ |˙x|z|z| − β ˙x|z| (19.3)

where c 0 is the viscosity damping coefficient, k 0 is the stiffness coefficient,
x 0 is the initial displacement, and a is a constant proportional to the current.
z is an auxiliary function that represents the hysteretic component of the
MR damper, and γ,β,A are the model parameters that can change the