Page 185 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

11.3.2 Non-linear Extended State Observer Design
In this section, by treating F(¯x n ,u ,v) as an additional state variable, we
λ n
can employ the following ESO to estimate both the unknown dynamics
F(·) and state z 2 ,...,z n of the transformed system (11.14)as

⎧ γ
⎪ ˙ = ξ 2 − β 1 [η 1 ]
ξ 1
⎪
⎪
⎨ ˙ = ξ i+1 − β i [η 1 ] iγ −(i−1) , i = 2,··· ,n − 1
ξ i (11.15)
⎪ ˙ = ξ n+1 − β n [η 1 ] nγ −(n−1) + b 0v
ξ n
⎪
⎪ (n+1)γ −n
⎩ ˙
ξ n+1 =−β n+1 [η 1 ]
where ξ i , i = 1,··· ,n are the estimation of unknown states z i , i = 1,··· ,n,
ξ n+1 is the estimation of the lumped uncertainty a(¯x n ),and η 1 = ξ 1 − y =
i
i
ξ 1 − z 1 is the input of ESO, the operation [x] |x| sgn(x) for all x ∈ Rand
i > 0, b 0 is the estimated nominal value of b(x n ),and β i , i = 1,··· ,n +1are
positive constants selected by the designers.
From (11.14)and (11.15), the observer errors η i = ξ i −z i ,i = 1,··· ,n+1
of ESO can be given as
⎧
γ
⎪ ˙ η 1 = η 2 − β 1 [η 1 ]
⎨
iγ −(i−1)
˙ η i = η i+1 − β i [η 1 ] , i = 2,··· ,n (11.16)
⎪ (n+1)γ −n
˙ η n+1
⎩ =−β n+1 [η 1 ] .
The finite-time error convergence of ESO in (11.15) has been proved
in [13] by considering the error dynamics described by (11.16). It is sum-
marized as in the following lemma.
Lemma 11.1. Let the gains β 1 ,··· ,β n+1 of ESO be a Hurwitz vector. Then,
1
there exists ∈[1 − ,1) such that for all γ ∈ (1 − ,1),theerrorsystem
n−1
(11.16) is finite-time stable with a Lyapunov function chosen as
T
V ξ (γ,η) = ν Pν (11.17)
where

⎛ 1 ⎞
[η 1 ] q
1
⎜ ⎟
[η 2 ] γq
⎜ ⎟
⎜ ⎟
. ⎟
ν = ⎜ . . ⎟ (11.18)
⎜
1
⎜ ⎟
⎜ [nγ −(n−1)]q ⎟
⎝ [η n ] ⎠
1
[η n+1 ] [(n+1)γ −n]q