Page 92 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

√
T

where y =[z 1 ,z 2 ,r , Q, P],and U 1 and U 2 are defined as
2 2
U 1 (y) = λ 1 y ,U 2 (y) = λ 2 y (5.36)
with λ 1 and λ 2 being positive constants.
Thetimederivativeof V L can be expressed as

˙
˙ V L = z 1 ˙z 1 + z 2 ˙z 2 + r˙r + Q + ˙ P
= z 1 (z 2 − k 1z 1 ) + z 2 (r − k 2z 2 ) + r[ ˜ N + N − (k s + 1)r − β 1sgn(z 2 ) − z 2 ]
˜ T −1 ˙ ˜ 2
+ − r[N B + N d − β 1sgn(z 2 )]− Hsgn(z 2 ) − β 2z − z 2N B
2
2 2 2 2
=−k 1z − k 2z + z 1z 2 − (k s + 1)r − β 2z + r ˜ N − z 2N B
1
2
2
T
−1 ˙
˜
+ − Hsgn(z 2 ) (5.37)
˜
−1 ˙
˜ T
˜
Substituting the adaptive law (5.21)into , then one may verify that
T
T
−1 ˙
T
= (z)z 2 − σ (5.38)
ˆ
˜
˜
˜
˜
˜ T
˜ T
Then, we consider N B = (Z), and have (z)z 2 + z 2N B = 0. Con-
sequently, it follows
2
2
2
2
˙ V L =− k 1z − k 2z + z 1z 2 − (k s + 1)r + β 2z + r ˜ N
1 2 2 (5.39)
T
T
− σ + σ − Hsgn(z 2 )
˜
˜
˜
By using Young’s inequality, one has:
1 2 1 2
z 1z 2 ≤ z + z 2 (5.40)
1
2 2
2 2
T T σ
−σ ˜ ˜ ˜ + (5.41)
˜
+ σ ≤−σ −
2 4
Substituting (5.40)and (5.41)into(5.39)results in
1 2 1 2 2 2
˜
˙ V L ≤−(k 1 − )z − (k 2 − − β 2 )z − (k s + 1)r − σ − + r ˜ N
2
1
2 2 2
2

2 2
˜
≤−λ 3 z − σ − +[k sr − η( z ) r z ] (5.42)
2
1
1
where λ 3 = min{k 1 − ,k 2 − − β 2 ,1}.
2 2
After completing the squares for the third term in (5.42), it follows
2 2
η (z) z 2
2
˜
˙ V L ≤−λ 3 z + − σ − ≤−U(y) (5.43)
4k s 2