Page 57 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

The first part σ 2 ω m +[f c + (f s − f c )e −(ω m /ω s ) 2 ]sgn(ω m ) is a static function of
1
the velocity. The second part σ 0 [1 − −(ωm /ωs ) 2 |ω m |] − J l ˙ω l is scaled by
f c +(f s −f c )e
the error due to the dynamic perturbation in the friction. Then one can
verify that

−(ω m /ω s ) 2
F ≤ 1 |ω m |+ 2 + f c + (f s − f c )e sgn(ω m ) (3.42)
where is bounded since σ 1 and σ 0 are bounded. 1 and 2 = σ 2 ω m − J l ˙ω l
are positive constants. Let f 1 = f c + (f s − f c )e −(ω m /ω s ) 2 ,and f 2 = 1 |ω m |+ 2.
Then (3.42) can be rewritten as

F ≤ f 1sgn(ω m ) + f 2 . (3.43)

Since the dynamics given in (3.43) are not a smooth function, it cannot be
directly approximated via ESNs. However, f 1 is a smooth function, which
is approximated by an ESN in a compact set as:

T
ˆ
f 1 = X(x) (3.44)
ˆ
where is the estimated NN weight, X(x) is the regressor. Then, the
ˆ
adaptive law for updating ˆ is given by

˙

= r 3z 3Xsgn(ω m ) − 1 (3.45)
ˆ
ˆ
where > 0and 1 > 0 are all positive constants. We denote the NN
∗
˜
weight error as = − .
ˆ
Moreover, theestimateof f 2 canalsobegivenby
ˆ ˆ
f 2 = 1 |ω m |+ 2 (3.46)
where 1 is theestimateof 1, which is online updated by
ˆ
˙
ˆ ˆ
1 = 1 (r 3z 3 |ω m |− 2 1 ) (3.47)
> 0and 2 > 0 are positive constants.
where 1
From (3.44)–(3.47), one can obtain the friction compensation term as
ˆ
ˆ F = f 1sgn(ω m ) + f 2 . (3.48)
ˆ
Then, the controller u givenin(3.40) with friction compensation (3.48)
is given by

˙ μ 3
ˆ ˆ
u = J m − k 3z 3 + ϑ 2,3 + s 3 +ˆ x 2 + f 1sgn(ω m ) + f 2 (3.49)
μ 3