Page 234 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

by |ε| ≤ ε N for any positive constant ε N > 0, and the ideal NN weight is
bounded by W ∗T ≤ W N for positive constant W N . φ i (X) can be chosen
as the following sigmoid function

r 1
φ (X) = + r 4 (15.19)
r 2 + exp(−X/r 3 )
where r 1, r 2, r 3, r 4 are appropriate parameters, and exp(·) is an exponential
function.
The unknown system state z i will be addressed by using the following
high-order sliding mode (HOSM) differentiator.

15.2.3 High-Order Sliding Mode (HOSM) Differentiator
It is shownin(15.17) that the unknown system states z 2 ,··· ,z n are the high
order derivatives of the measurable system output y = x 1 = z 1.Inviewing
this fact, we can use a high-order sliding mode (HOSM) differentiator [16]
with finite-time convergence to estimate z 2 ,··· ,z n by using the system
output only. The generic form of HOSM observer can be given as

⎧
˙
⎪ ˆ z 1 = ω 1
n
⎪
⎪
⎪
⎪ ω 1 =−μ 1 |ˆz 1 − z 1 | n+1 sgn(ˆz 1 − z 1 ) +ˆz 2
⎪
⎪
⎪
⎪ ···
⎪
⎪
⎨ ˙
⎪
ˆ z i = ω i
n+1−i (15.20)
⎪ ω i =−μ i |ˆz i − ω i−1 | n+2−i sgn(ˆz i − ω i−1 ) +ˆz i+1
⎪
⎪
···
⎪
⎪
⎪
⎪ 1
⎪ ˙
⎪
⎪ ˆ z n =−μ n |ˆz n − ω n−1 | 2 sgn(ˆz n − ω n−1 ) +ˆz n+1
⎪
⎪
⎩ ˙ =−μ n+1sgn(ˆz n+1 − ω n )
ˆ z n+1
where sgn(·) is the signum function, μ i, i = 1,...,n + 1 are positive param-
eters, and z 1 is the measurement of system output x 1.
Lemma 15.1. [16] If a bounded noise is included in the input z 1 of differentiator
(15.20), i.e., |z 1 − y|≤ χ with χ being a positive constant, then for some positive
constants γ i and ¯μ i, the following inequalities hold in finite time:
n+2−i
|ˆz i − z i |≤ γ i χ n+1 , i = 1,...,n
n+1−i (15.21)
|ω i − z i+1 |≤ ¯μ i χ n+1 , i = 1,...,n − 1
Moreover, the corresponding solutions of the dynamic system (15.20) are finite-time
stable.