Page 223 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Differentiating (14.25) with respect to time t and using (14.23), we have

−1 ˙
T
˙ V = 1 s 2 ˙ 2 − ˜ W W
s
ˆ
k m
˜ T
1 p/q T −1 ˙
ˆ
≤ s 2 W φ(X) + ε − μsgn(s 2 ) − |s 2 | sgn(s 2 )) − ˜ W W
β
T −1 ˙ 1 (p+q)/q
ˆ
= ˜ W [s 2 φ(X) − W]+ εs 2 − μ|s 2 |− |s 2 | sgn(s 2 )
β
(14.26)
Substituting (14.22)into(14.26)yields
1 (p+q)/q
˙ V ≤− |s 2 | ≤ 0 (14.27)
k m β
Inequality (14.27) implies that both s 2 and ˜ W are bounded. Moreover,
considering (14.16) and the boundedness of W , we can conclude s 1, ˙s 1,
∗
and ˆ W are bounded, and thus the control u is bounded from (14.21), e, ˙e
are bounded from (14.13). Furthermore, the boundedness of y d , ˙y d ,and ¨y d
implies the boundedness of s 2 according to (14.16). Therefore, all signals of
the closed loop system are bounded. From (14.25)–(14.27), the stability of
the system (14.6) with control (14.21) and adaptive law (14.22) has been
proved. Now, we need to further prove the finite-time convergence of
terminal sliding manifold s 2.
2) We know that the sigmoid function of NN φ i (X) is bounded by 0 <
φ i (X)< n 0, i = 1,··· ,n with n 0 being a positive constant. φ(X) is bounded
by
√

φ(X)
ð n 0 n

T
where φ(X) =[φ 1 (X),φ 2 (X),··· ,φ n (X)] .
Select another Lyapunov function as

1 2
s
V 1 = 2k m 2 (14.28)
Differentiating (14.28)byusing (14.23), we have

1
s
˙ V 1 = s 2 ˙ 2
k m (14.29)
T 1 p/q
≤ s 2 [ ˜ W φ(X) + ε − μsgn(s 2 ) − |s 2 | sgn(s 2 ))]
k m β