Page 152 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

4i
3
the sets d and i are compact in R and R , respectively. Thus

¯
ˆ ¯
ξ i (¯z i , ¯e i ,θ i , ˆε i ,y d , ˙y d , ¨y d ) has a maximum value M i ,i = 1,··· ,n − 1on

d × i.
Then the following theorem states the main results:
Theorem 9.1. Consider system (9.1) with unknown non-linear dead-zone un-
der Assumptions 9.1 and 9.2, and the adaptive control (9.36)–(9.38), then for

T T ¯ T ¯T T n n−1 2 4i
ˆ
any initial condition [¯ z , ¯e ,θ , ˆε ] : V i + e ≤ 2P 0 ⊂ R and
i i i i i=1 i=1 i
non-negative initial parameters θ i (0) ≥ 0, ˆε i (0) ≥ 0, there exist control feedback
ˆ
gains k i and filter parameters μ i fulfilling (9.49), such that the closed-loop control
system is semi-globally stable in the sense that all signals in the closed-loop system
remain ultimately bounded.
⎧ 1 1
m 1
⎪ k 1 ≥ + +
⎪ 2g 10 c 11 4g 10 c 12 4g 10 c 13
⎪ 2
⎪ 1 1 c i−1,2 g
⎪ m i i−1,1
⎨ k i ≥ + + + , i = 2,··· ,n − 1
2g i0 c i1 4g i0 c i2 4g i0 c i3 g i0 (9.49)
c n−1,2 g 2
m n n−1,1
⎪ k n ≥ +
⎪
2g n0 c n1 g n0
⎪
⎪
⎪ k e
μ i ≤ , i = 1,··· ,n − 1, k e > 0
⎩ 2
c i3 g +1
i1
Proof. For initial conditions in any given compact set i for P 0 > 0, it is
always possible to construct a compact set larger than i comprising the
,i = 1,··· ,n, in which the NN approximation (9.5)is
set d , C i and z i
valid and ϕ ij (¯x i ) is bounded. Consider the Lyapunov function candidate as
n n−1 2
V = V i + k e e (9.50)
i=1 2 i=1 i
where k e > 0 is a design parameter.
Recalling the previous recursive design procedures from Step 1 to Step n,
one can obtain

m 1 1 1 2
˙ V ≤− g 10k 1 − − − z
1
2c 11 4c 12 4c 13
n−1
m i 1 1 2 2
− g i0k i − − − − c i−1,2g z (9.51)
i−1,1 i
2c i1 4c i2 4c i3
i=2

m n 2 2
− k ng n0 − − c n−1,2g n−1,1 z n
2c n1
with gamma as:

2
2 g 10 k 1 −m 1 /2c 11 −1/4c 12 −1/4c 13 ,2(g i0 k i −m i /2c i1 −1/4c i2 −1/4c i3 −c i−1,2 g ),
γ =min

i−1,1
2
2 k n g n0 −m n /2c n1 −c n−1,2 g 2 n−1,1 ,2 1/μ i −(c i3 g −1)/k e ,
i σ i /2,
n σ n /2,
ai σ ai ,
an σ an ,
i1