Page 232 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

From (15.7)and (15.8), the system (15.1) can be rewritten as:
⎧

⎪ ˙ x i = f i (¯ x i ,0) + g i ¯ x i ,x α i x i+1 , 1 ≤ i ≤ n − 1
i+1
⎨
˙ x n = f n (¯ x n ,0) + g n (¯ x n ,v )v (15.9)
α i
⎪
y = x 1
⎩
which is now in the strict-feedback form.
To further reformulate system (15.9), we define new system states as [13]
z 1 = y = x 1

(15.10)
z 2 =˙z 1 = f 1 (x 1 ) + g 1 x 1 ,x α 1
2 x 2
Then the time derivative of z 2 is calculated as

α 1 α 1
∂g 1 x 1 ,x ∂g 1 x 1 ,x
∂f 1 (x 1 ) 2 2
α 1
˙ z 2 = ˙ x 1 + ˙ x 1 + ˙ x 2 x 2 + g 1 x 1 ,x 2 ˙ x 2
∂x 1 ∂x 1 ∂x 2

∂f 1 ∂g 1
∂g 1
= + x 2 f 1 + g 1x 2 + x 2 + g 1 f 2 + g 2x 3
∂x 1 ∂x 1 ∂x 2
α 2
= a 2 (¯x 2 ) + b 2 (¯x 2 ,x )x 3 (15.11)
3

∂f 1 ∂g 1
∂g 1
α 2
where a 2 (¯ x 2 ) = + x 2 f 1 + g 1x 2 + x 2 + g 1 f 2 and b 2 ¯ x 2 ,x 3 =
∂x 1 ∂x 1 ∂x 2

∂g 1
x 2 + g 1 g 2.
∂x 2
Again, let the coordinate as z 3 = a 2 + b 2x 3, and then its time derivative
is calculated as
2 3
∂a 2 ∂b 2

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

(15.12)
∂a 2 ∂b 2 ∂b 2
= x 3 + f j + g jx j+1 + x 3 +b 2 f 3 + g 3x 4
∂x j ∂x j ∂x 3
j=1

= a 3 (¯ x 3 ) + b 3 ¯ x 3 ,x α 3 x 4
4
2

∂a 2 ∂b 2 ∂b 2

where a 3 (¯ x 3 ) = + x 3 f j + g jx j+1 + x 3 +b 2 f 3 and b 3 ¯ x 3 ,
∂x j ∂x j ∂x 3
j=1

x α 3 = ∂b 2 x 3 +b 2 g 3.

4 ∂x 3
Similarly, with the derived a i−1 and b i−1, i = 2,··· ,n, we can define

α i−1
z i =a i−1 (¯ x i−1 ) + b i−1 ¯ x i−1 ,x i x i (15.13)
Then, following the similar mathematical manipulations as shown in the
above steps, we can obtain that
˙ z i = a i (¯ x i ) + b i (¯ x i )x i+1 (15.14)