Page 191 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

1) All signals in the closed-loop system are UUB for any given initial conditions
x i (0),ξ j (0),ˆι i (0),i = 1,··· ,n, j = 1,··· ,n + 1.
2) The tracking errors e i ,i = 1,··· ,n, converge to a compact set around zero
defined by (11.53).

Proof. Defined the following Lyapunov function as

n
n ˜ι i (t) 2

V(t) = V i (t) + V e (t) + (11.45)
i=1 i=1 2θ i
T
T
where V e (t) = η Pη,for η =[η 1 ,...,η n+1 ] .
From (11.29), (11.36), and (11.44), the derivative of V satisfies
n n ι
2
˙ V ≤− c ie + e n (F − ξ n+1 ) + ι i (e i − e i tanh( 0.2785ˆ ie i )) + ˙ V e
i
i=1 i=1 ω
n n
ι
0.2785ˆ ie i
ι
ι
− σ i ˜ i ˆ i + ˆ ι ie i tanh( )
i=1 i=1 ω (11.46)
n n ι
2 0.2785ˆ ie i
≤− c ie + e n (F − ξ n+1 ) + ι i (e i − e i tanh( ))
i
i=1 i=1 ω
n n ι
T T 0.2785ˆ ie i
ι
ι
+η (A P + PA o )η − σ i ˜ i ˆ i + ˆ ι ie i tanh( )
o
i=1 i=1 ω
Then, by using the fact that
ι
0.2785ˆ ie i
|e i |− e i tanh( ) ≤ 0.2785ω
ω 2 2 (11.47)
σ i ˜ ι i σ i ι i
−σ i ˜ι i ˆι i ≤− + .
2 2
The inequality (11.46) can be rewritten as
n n 2 n 2
T
2
T
˙ V ≤− c ie + η (A P + PA o )η − σ i ˜ι 2 i + σ i ι 2 i + 0.2785ωnι
o
i
i=1 i=1 i=1
n
e n 2 α e ι
ι
+ |ˆ ie i |+ + (11.48)
2α e 2
i=1
n−1
2 2
1 1
≤− c ie − (c n − )e − V e + ς
i n λ max (P)
i=1 2α e
n n
σ i ι α e ι
2
ι
where ς = + 0.2785ωnι + |ˆ ie i |+ .
2 2
i=1 i=1
Letting
1 1 1
, θ i σ i }
ρ v = min 1≤i≤n {c i ,c n − , λ max (P) 2 (11.49)
2α e