Page 171 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

APPC of Strict-Feedback Systems With Non-linear Dead-Zone 167

m n e à nm
c n1 2 z n 2
+ 1 − 2tanh ( ) k (x)
nj
2 ω n j=1 1 −¯τ n
˙
≤− γ nV n + ϑ n +[g n (x)dN(ξ n ) + 1]ξ n
m n e à nm

c n1 2 z n 2
+ 1 − 2tanh ( ) k (x) (10.46)
nj
2 ω n j=1 1 −¯τ n
where γ n and ϑ n are positive constants, which is given as
∗2 ∗2
γ n = min 2(k n − m n /2c n1 ), n σ n2 , an σ n3 , ,ϑ n = σ n1 +σ n2 θ /2+σ n3 ε /2.
n n
Similar to the analysis in the previous steps, the last term of (10.46)is
bounded since the functions k nj (x) are bounded on any compact set C n and
2
−1 ≤ 1 − 2tanh (z n /ω n ) ≤ 1 holds, which can guarantee the boundedness
of z n, θ n, ˜ε n for small enough ϑ n, c n1,orlarge γ n.
˜
10.3.2 Stability Analysis
In above analysis, Nussbaum functions N(ξ i ) are included in the Lyapunov
functions. Therefore extra efforts should be made to prove the system sta-
bility and guarantee the prescribed tracking control performance (10.4).
The following theorem states the main results of this chapter:

Theorem 10.1. Consider system (10.1) with unknown non-linear dead-zone
(10.2), the control is given by (10.37)–(10.40), then for any bounded initial condi-
θ i (0) ≥ 0, ˆε i (0) ≥ 0 and −δμ(0)< e(0)< δμ(0), there exist control feedback
tion ˆ ¯
gains k i fulfilling (10.47)suchthat

⎧ 2
m 1 r 1
⎪ k 1 ≥ M +
2c 11 4c 12
⎨
m i 1 2
k i ≥ + + c i−1,2g i−1,1 , i = 2,··· ,n − 1 (10.47)
2c i1 4c i2
⎪
⎩ m n 2
k n ≥ + c n−1,2g n−1,1
2c n1
i) All signals in the closed-loop system remain semi-globally bounded;
ii) The tracking control with prescribed performance condition (10.4)ispreserved.

˜
Proof. For any given initial condition compact set
0 = z i (0),θ i (0), ˜ε i (0),
!
i ≤ 1,···n , we can always construct a larger compact set
than
0 com-
,i = 1,··· ,n, in which the NN approximation is valid and
prising C i ,
z i
the functions k ij (¯x i ) are bounded. We denote G i (x) = g i (¯x i ),i = 1,··· ,n−1,
G n (x) = g n (x)d,and α = max{r M ,1}, which are also bounded functions