Page 183 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

0
By choosing x 0 i+1 = 0and u = 0, then Eq. (11.4) can be rewritten as
∂f i (¯x i ,x i+1 )
f i (¯x i ,x i+1 ) =f i (¯x i ,0) + | λ i × x i+1 ,1 ≤ i ≤ n − 1,
∂x i+1 x i+1 =x i+1 (11.5)
∂f n (¯x n ,u)
f n (¯x n ,u) =f n (¯x n ,0) + | u=u n × u.
λ
∂u
For analysis convenience, it is defined that
g i (¯x i ,x λ i ) = ∂f i (¯x i ,x i+1 ) | x i+1 =x λ i
i+1
∂x i+1
∂f n (¯x n ,u) i+1 (11.6)
λ n
g n (¯x n ,u ) = | u=u n ,
λ
∂u
which are unknown non-linear functions.
Based on (11.5), the original system (11.1)isreformulatedasastrict-
feedback system. Hence, we can further represent this system into the
Brunovsky form with respect to the newly defined state variables as [8,9].
Hence, we define a set of new coordinates as
z 1 = y
(11.7)
z 2 = ˙ z 1 = f 1 + g 1x 2 .
Thetimederivativeof z 2 is derived as

∂f 1 ∂g 1 ∂f 1 ∂g 1
˙ z 2 = ˙ x 1 + ˙ x 1x 2 + g 1 ˙x 2 = ( + x 2 )(f 1 + g 1x 2 ) + g 1f 2 + g 1g 2x 3
∂x 1 ∂x 1 ∂x 1 ∂x 1
a 2 (¯x 2 ) + b 2 (¯x 2 )x 3 (11.8)
where a 2 (¯x 2 ) = ( ∂f 1 + ∂g 1 x 2 )(f 1 + g 1x 2 ) + g 1f 2 and b n (¯x 2 ) = g 1g 2.
∂x 1 ∂x 1
Again, let z 3 = a 2 + b 2x 3, and its time derivative is

2 ∂a 2 2 ∂b 2
˙ z 3 = ˙ x j + ˙ x jx 3 + b 2 ˙x 3
j=1 ∂x j j=1 ∂x j
2

= ( ∂a 2 + ∂b 2 x 3 )(f j + g jx j+1 ) + b 2 (f 3 + g 3x 4 ) (11.9)
j=1 ∂x j ∂x j
a 3 (¯x 3 ) + b 3 (¯x 3 )x 4
2

where a 3 (¯x 3 ) = ( ∂a 2 + ∂b 2 x 3 )(f j + g jx j+1 ) + b 2f 3 and b 3 (¯x 3 ) = b 2g 3 =
j=1 ∂x j ∂x j
g 1g 2g 3.
Similar to the above derivations, for i = 2,...,n,wecanhave

z i a i−1 (¯x i−1 ) + b i−1 (¯x i−1 )x i
(11.10)
˙ z i = a i (¯x i ) + b i (¯x i )x i+1
where x n+1 = u and
a i (¯x i ) i−1 ∂a i−1 + ∂b i−1
(
j=1 ∂x j ∂x j x i )(f j + g jx j+1 ) + b i−1f i
i (11.11)
g

b i (¯x i ) b i−1g i = j=1 j .