Page 73 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

θ = θ − θ are parameter errors between the bounded
where ˜ε = ε − ε and ˜ ˆ
∗
T
constants ε = ε N /g, θ = and their estimations ˆε, θ.
ˆ
∗
The derivative of V can be obtained as
g g
T ˙

˜ ˜
˙ V ≤ sr + ε + gu + θθ + ˜ ε˜ε ˙
a
2 2
gr ˆ 2 grˆε s
θs
T 2 T
≤ sr + rε N |s| − k 1gs − − (4.22)
2η 2 ˆ ε|s|+ σ 1
s T
2
− gθr − σ 2 θ − g˜εr |s| − σ 3 ˆε
ˆ
˜
2η 2
Applying Young’s inequality with η> 0, one can obtain the following in-
equalities:
grθs 2 r M η 2
T T
sr ≤ + (4.23)
2η 2 2g
σ 2gr M θ ˜ 2 σ 2gr M θ 2
˜ ˆ
σ 2grθθ ≤− + (4.24)
2 2
σ 3gr M ˜ε 2 σ 3gr M ε 2 N
σ 3gr˜εˆε ≤− + (4.25)
2 2
Moreover, the fact θ(t), ˆε(t) ≥ 0,t ≥ 0 holds for any initial conditions
ˆ
ˆ
θ(0), ˆε(0) ≥ 0, and 0 ≤ ab ≤ a,∀a,b > 0 is true. Then one can rewrite
a+b
(4.22)as
2 2
r M η 2 rgˆε s
2 ∗
˙ V ≤−k 1gs + + rgε |s| − rg˜ε |s| − + σ 2gr ˜ ˆ + σ 3gr˜εˆε
θ 1 θ 1
2g ˆ ε|s|+ σ 1
σ 2gr M θ ˜ 2 σ 3gr M ˜ε 2 σ 2gr M θ 2 σ 3gr M ε 2 r M η 2
2 N
≤−k 1gs − − + + + + σ 1gr M
2 2 2 2 2g
≤−γV + ϑ, (4.26)
where γ and ϑ are positive constants

γ = min 2k 1g, r M σ 2 , ar M σ 3 ,
2
2
2
ϑ = σ 1gr M + σ 2gr M θ /2 + σ 3gr M ε /2 + r M η /2g.
N
Then according to Lyapunov theorem, V is uniformly ultimately bounded
and thus the errors s, θ, ˜ε are bounded. This further guarantees the bound-
˜
edness of the transformed error z 1 and ˙z 1 according to (4.13). Moreover,
T
∗
since θ = , ε = ε N /g are bounded, the adaptive parameters θ, ˆε are all
ˆ
bounded. Consequently, the control signal u is bounded.