Page 211 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Theorem 13.1. Considering the servo system (13.3), thesliding surface(13.17),
the adaptive sliding mode controller (13.20), and the parametric adaptive law
(13.21). Then, the state variables x 1 and x 2 can track the desired signals x 1d
and x 2d .

Proof. The following Lyapunov function is used
1 1
2
V = s + k ˜ 2 (13.22)
2 2k m
˜
∗
where k = k(t) − k is the parameter estimation error.
Then, the derivative of V is given by
1
1 ˙
∗ ˙
˙ V = s˙s + ˜ ˜ (k − k )k.
kk = s[x 3 + a 0 + b 0u −¨x 2d + λ 1 (x 2 −¨x 1d )]+
k m k m
(13.23)
Substituting (13.20)into(13.23)yields
∗ ˙
˙ V = s[(x 3 − z 3 ) + λ 1 (x 2 − z 2 ) − ksg(s)]+ 1 (k − k )k
k m
∗ ∗ 1 ∗ ˙
= s( 3 + λ 1 2 ) − ks · sg(s) + k s · sg(s) − k s · sg(s) + (k − k )k
k m
1 ˙
∗
∗
=−s[k sg(s) − ( 3 + λ 1 2 )]+ (k − k )[ k − s · sg(s)].
k m
(13.24)
Next, we will provide the stability analysis according to the following
two cases. 1) When |s|≥ μ, it can be concluded that sg(s) = sgn(s),thenwe
have
1
∗ ∗ ˙
˙ V ≤−[k −| 3 + λ 1 2 |]|s|+ (k − k )[ k − s · sg(s)]. (13.25)
k m
∗
From (13.21), we can conclude ˙ V ≤−[k −| 3 + λ 1 2 |]|s|≤ 0. Hence,
the sliding mode variable s will eventually converge to set |s|≤ μ.
2|s|
2) When |s| <μ,wecanobtain sg(s) = sgn(s) and
|s|+ μ
1 ˙
˙ V ≤−sk · sg(s) + (k − k )[ k − s · sg(s)]
∗
k m
1 ˙
∗
≤−sk ( 2|s| )sgn(s) + (k − k )[ k − s · sg(s)] (13.26)
|s|+μ k m
1 ˙
∗
≤−|s|k ( 2|s| ) + (k − k )[ k − s · sg(s)]
|s|+μ k m
∗
with k = k −| 3 + λ 1 2 |≥ 0. Then, substituting (13.21)into(13.26), it
can be easily concluded that ˙ V ≤ 0. Therefore, the sliding mode variable s