Page 151 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

ANDSC of Strict-Feedback Systems With Non-linear Dead-Zone 147

∗2
∗2
∗
ϑ n = σ ng n0 θ /4 + 1/2g n0 + σ an ¯ε /2g n0 + 0.2875¯ε ω n .
n
n
n
Similar to previous Step i, the last term of (9.46) is bounded since ϕ nj (x) is
2
bounded on any compact set C n and −1 ≤ 1 − 2tanh (z n /ω n ) ≤ 1, which
can guarantee the boundedness of z n, θ n, ˜ε n for small enough ϑ n, c n1 and/or
˜
large γ n on a compact set C n. Moreover, as discussed in above Step 1,for
, the inequality
the case that the tracking error z n is large, i.e., z n /∈ z n
2
e nj

à nm ϕ (x)

2 z n
1 − 2tanh ( ) m n ≤ 0 holds, such that the Lyapunov function
τ
ω n j=1 1−¯ n
(9.46) can be rewritten as ˙ V n ≤−γ nV n + ϑ n, which improves the error
,the last term
convergence. On the other hand, for a small error z n ∈ z n
can be tuned to be small by decreasing ω n and c n1.
∗
Remark 9.3. By introducing a novel unknown positive constant θ =
i
W i ∗T W as the adaptive parameter of HONN (9.5), there are only two
∗
i
scalar parameters ˆ (independent of the number of NN nodes) and ˆε i to
θ i
be updated online at each step design of α i and v. Thus the “explosion of
learning parameters” is circumvented and the computational cost can be
reduced significantly. This is clearly different to the conventional NN con-
trollers where the NN weight ˆ W i to be updated are vectors or matrices.
9.3.2 Stability Analysis
From (9.26)–(9.27)and (9.29), it is easy to obtain

s
˙ e i =˙ i −¨α i =− e i + ∂α i ˙ z i + ∂α i ˙ ˆ + ∂α i ˙ ˆ ε i + ∂α i ˙
θ i
i
μ i ∂z i ∂ ˆ θ i ∂ ˆε i ∂ε i (9.47)
¯
e i ˆ ¯
=− + ξ i (¯z i , ¯e i ,θ i , ˆε i ,y d , ˙y d , ¨y d ), i = 1··· ,n − 1
μ i
¯
ˆ ¯
where ξ i (¯z i , ¯e i ,θ i , ˆε i ,y d , ˙y d , ¨y d ) is a continuous function of the vectors
T T ¯ ˆ ˆ T ˆ T ˆ T T ¯
¯ z i =[z 1 ,z 2 ,··· ,z i ] , ¯e i =[e 1 ,e 2 ,··· ,e i ] , θ i =[θ ,θ ,··· ,θ ] ,and ˆε i =
1 2 i
T
[ˆε 1 , ˆε 2 ,··· , ˆε i ] .
Then it follows that
e 2 i ¯
ˆ ¯

˙ e ie i ≤− + e i ξ i (¯z i , ¯e i ,θ i , ˆε i ,y d , ˙y d , ¨y d ) , i = 1··· ,n − 1. (9.48)
μ i
2

Define the compact set of the desired trajectory as d := (y d , ˙y d , ¨y d ) : y +
d

3
2
T
2
T ¯ T

ˆ
˙ y +¨y ≤ B 0 ⊂ R with B 0 being a positive constant i := [¯z , ¯e ,θ ,
d d i i i

¯T T n n−1 2 4i
ˆ ε ] : V i + e ≤ 2P 0 ⊂ R as the compact set of the ini-
i i=1 i=1 i
tial conditions with P 0 a positive constant. Then for any B 0 > 0,P 0 > 0,