Page 88 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

x d , ˙x d to replace the measured signals x 1 ,x 2 in the ESN approximation, and
introduce a new robust term to guarantee the asymptotic convergence. We
rewrite (5.11)as

r = ¯ F d + ρθu + S + (5.12)
˜
where the non-linear function ¯ F d including the unknown non-linearities
and frictions is defined as

¯ F d = F d (x d , ˙x d , ¨x d ,ρ) + ρT f (˙x d )/J (5.13)

and the auxiliary function S is defined as

S = k 1 ˙z 1 + k 2z 2 + E(e 1 ) + F d (x, ˙x d , ¨x d ,ρ) − F d (x d , ˙x d , ¨x d ,ρ)
(5.14)
+ ρ T f (˙x) − T f (˙x d ) /J

Hence, thetimederivativeof r can be given as

˙
˙
˙ r = ¯ F d + ρθ ˙u + S + (5.15)
˙
˜
Since the friction model (4.4) and the reference x d are all continuous,
the non-linear function in (5.15) can be approximated by an ESN as [10]
˙ ¯ F d = (Z) + ε (5.16)
T

T
where =[ 1 ,..., L ] is the bounded ESN weight, (Z) =[ 11 (Z),...,
T
L
1L (Z)] ∈ R is the regressor vector, and ε is a bounded approximation
error.
Assumption 5.1. [11] The function approximation error ε and its time deriva-
tives are bounded by |ε|≤ ε b1, |˙ε|≤ ε b2, |¨ε|≤ ε b3,where ε b1, ε b2, ε b3 are positive
constants.

Substituting (5.16)into(5.15), one can have

T
˙
˜
˙ r = (Z) + ρθ ˙u + S + + ε (5.17)
˙
In the following, an alternative control will be designed to retain the con-
vergence of r and thus z 1 and e 1.