Page 103 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

tion f (x) can be represented by

N

f (x) = p 0 + p r σ r (0,x − α r ,β r − α r ) (6.2)
r=1
where α r and β r are the lower and upper boundaries of x confined in the
r-th subinterval, respectively. σ r (0,x−α r ,β r −α r ) is the basic function which
is given by

σ r (a,b,c) = max(a,min(b,c)) ∀a,b,c ∈ R (6.3)
and p r (r = 0,...,N) is the unknown coefficient that can be determined
by the least squares. It has been proven in [13] that any piecewise linear
function can be represented by (6.2).
The use of (6.2) to approximate any piecewise linear function is on the
precondition of partitioning the domain of x into finite closed subinter-
vals whose interiors do not overlap (see Lemma 1 in [13]). Moreover, the
boundaries α r , β r satisfy: 1) α r <β r ;2) β r = α r + 1. Then it is noted that
the basic function σ r (0,x − α r ,β r − α r ) is actually a special piecewise linear
function which can be decomposed as the difference of two simpler piece-
wise linear function, i.e., α r (0,x − α r ,β r − α r ) = max(0,x − α r ) − max(0,x −
max(α r ,β r )). In fact, the coefficient p r is the slope of the local linear func-
tion given by α r = x − α r when α r ≤ x <β r . Furthermore, it can be seen
that α r = 0when x <α r or α r = β r − α r which is a constant for x ≥ β r .

6.3.2 DPPR Modeling of Friction
The expression (6.2) provides a potential way for model-free parameteri-
zation of frictions since the friction dynamics appear sectionally and nearly
linear in the high speed operation region [6]. However, in the low veloc-
ity operation region, the Stribeck effect plus the Coulomb friction force
mainly contributes to the friction, which makes the friction highly non-
linear and non-smoothing in the low speed especially operation near zero
crossings. In addition, the static friction has a jumping behavior at zero
velocity where the direction of motion may change.
In order to solve the approximation of Stribeck effect and jumping be-
havior at zero velocity, two additional terms are introduced into (6.2)to
describe the friction dynamics:
1) A jump term h 1 (v) related to the maximum static force is introduced
into (6.2) to represent the reversal behavior of friction force when the
motion direction is changed;