Page 23 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Lorentzian model
In [9], an improved model has been reported by replacing exponential term,
such that:
1
T f = f c sgn(˙x) + (f s − f c ) + f v ˙x (1.9)
1 + (˙x/˙x s ) 2
which shows a systematic dependence of ˙x s.

Dahl model
The Dahl model was introduced in [10], which is given by
α

T f
˙
T f = σ 1 − sgn(˙x) ˙ x (1.10)
f c
where σ is the stiffness coefficient and α is a parameter that determines the
shape of the stress-strain curve.

LuGre model
The LuGre model is a dynamic model which can capture the dynamic
behaviors of the contacting surfaces [3], where friction is related to the de-
flection of bristles. In this model, the rate-dependent friction phenomenon
and the Stribeck effect are all considered. The LuGre model is given by

T f = σ 0z + σ 1 ˙z + σ 2 ˙x
|˙ x| (1.11)
˙ z =˙x − z
g(˙x)
where σ 0 is the stiffness of the bristles, σ 1 is the damping coefficient and
σ 2 is the viscous coefficient, respectively. z represents the average bristle
deflection, and g(˙x) can be selected to model different friction effects.
A reasonable selection of g(˙x) which can characterize the Stribeck effect
is set as
g(˙x) = f c + (f s − f c )e −(˙x/˙x s ) 2 (1.12)

1.2.3 Continuously Differentiable Friction Model
The above conventional friction models (e.g., [3], [11], [12]and [13]) are
discontinuous or piecewise continuous, which may be problematic for de-
riving smooth control actions [6] when they are used in the control designs.
Moreover, the identification of such friction models with non-smooth dy-
namics is not a trivial task. In [6], a new continuously differentiable friction