Page 265 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

with

ˆ
ˆ
2ρ|˙u d | ε(˙ u d )|˙u d |(−d 1 + d 2 )
˙ W(t) =˙u d − W(t) + (17.29)
ˆ ˆ ˆ ˆ
d 1 + d 2 d 1 + d 2
where ρ> 0 is a constant gain, and ε(˙u d ) is given by

0, if ˙u d = 0
ε(˙u d ) = (17.30)
1, otherwise

From (17.29), when ˙ W(t) = 0, we have
1 1
ˆ
ˆ
W(t) = (sgn(˙u d ) − ε)d 1 + (sgn(˙u d ) + ε)d 2 (17.31)
2ρ 2ρ
Considering (17.27)–(17.31), we know that ρW(t) satisfies the first
and third conditions of (17.27); and when ˙u d = 0, the second condition
of (17.27) can be satisfied. Hence, Eq. (17.31) shows that the conversion
ˆ d 1 ˆ d 2
time between the two states − and − can be decreased by increasing
p p
the value of gain ρ, which will help eliminate the effect of backlash.
In fact, we have the following results:
Lemma 17.1. [11]Let = BI(BIV(u d )) − u d,thenfor any t > 0, is a
ˆ
ˆ
bounded variable and | |≤|d 1 |+|d 2 |.
Proof. The proof of this lemma can be conducted by considering three
different cases described in (17.27) are discussed.
1) When the first condition of (17.27) is true, then we can verify from
ˆ
(17.27)and (17.28)that ρW(t) = d 2 holds, thus it follows
1
ˆ
ˆ
ˆ
ˆ
ˆ
w(t) = l(u(t) − d 2 ) = l ˆ u d (t) + ρW(t) − d 2 = u d (t) + l ρW(t) − d 2
ˆ l
(17.32)
According to (17.27), we have
ˆ
= w(t) − u d (t) = ρW(t) − d 2 = 0 (17.33)
2) When the third condition of (17.27)istrue, thenwe canverifyfrom
ˆ
(17.27)and (17.28)that ρW(t) = d 1 and thus

ˆ ˆ ˆ ˆ
w(t) = l(u(t) + d 1 ) = u d (t) + l ρW(t) − d 1 (17.34)