Page 163 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

APPC of Strict-Feedback Systems With Non-linear Dead-Zone 159

To facilitate the control design, the derivative of the transformed error
is calculated

∂S −1 ˙ 1
1 1 ˙ e e ˙μ
˙ z 1 = λ = − −
∂λ 2 λ+δ λ− ¯ δ μ μ 2 (10.8)

= r f 1 (x 1 ) + g 1 (x 1 )x 2 + h 1 (t,x 1 (t − τ 1 (t))) −¨y d − e ˙μ/μ

1 1 1
¯
where r = − is well-defined with −δμ(0)< e(0)< δμ(0) ful-
2μ λ+δ λ− ¯ δ
filling 0 < r ≤ r M for a positive constant r M . It is clear that r can be calculated
based on e(t), μ(t) and used in the control design. Incorporating the trans-
formed error dynamics (10.8) into original system (10.1)provides
⎧

⎪ ˙ z 1 = r f 1 (x 1 ) + g 1 (x 1 )x 2 + h 1 (t,x 1 (t − τ 1 (t))) −¨y d − e ˙μ/μ
⎪
⎪
˙ x i = f i (¯x i ) + g i (¯x i )x i+1 + h i (t, ¯x i (t − τ i (t))), 2 ≤ i ≤ n − 1
⎨
(10.9)

⎪ ˙x n = f n (x) + g n (x) d(t)v(t) + ρ(t) + h n (t,x(t − τ n (t)))
⎪
⎪
y(t) = x 1 (t)
⎩
It is shown that system (10.9) is still in a strict-feedback form, which is
invariant with the proposed error transformation [22], then the stabilization
of system (10.9) is sufficient to guarantee PPF condition (10.4), and to
achieve control objectives according to Lemma 10.1.
10.2.2 High-Order Neural Network and Nussbaum-Type
Function

It has been proved in [23] that, a high-order neural network (HONN) can
approximate a non-linear continuous function up to arbitrary accuracy on
acompactset
as

Q(Z) = W ∗T (Z) + ε, ∀Z ∈
⊂ R n (10.10)

∗ T
L
where W =[w ,w ···w ] ∈ R are ideal bounded weights and ε ∈ Ris
∗
∗
∗
L
2
1
the bounded error, i.e., W ≤ W N , |ε| ≤ ε N with W N , ε N being positive
∗
L
T
constants. The basis vector is set as (Z) =[ 1 (Z),··· L (Z)] ∈ R with
d k (j)

k (Z) = [σ(Z j )] ,k = 1,...,L,where J k are collections of L not
j∈J k
ordered subsets of {0,1,...,n}, d k (j) are non-negative integers, and σ(·) is
a sigmoid function.
To deal with unknown signs of control gains g i (¯x i ), the Nussbaum-type
function [27,28]isutilized: