Page 231 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Neural Dynamic Surface Control for Pure-Feedback Systems With Input Saturation 231

is the function of any fixed u ξ , d 1 (u) = sat(u)−g (u) is a bounded
where g u ξ
function with bound given by

d 1 (u) = sat(u) − g (u) ≤ v max [1 − tanh(1)] = D 1
(15.5)

where D 1 is a positive constant defining the upper bound of d 1 (u) .

The control design objective is to find an appropriate control u such
that the output y of system (15.1) can track a given trajectory y d .
To facilitate the control design, the following assumption is used in this
chapter:

Assumption 15.1. The non-linear functions f i (·),i = 1,··· ,nof (15.1)are
continuously differentiable to n-th order with respect to the state variables ¯x i and the
input v(u).

15.2.2 Coordinate Transformation
In the following, we will show that the original system (15.1) can be trans-
formed into the canonical form with respect to the newly defined state
variables [13], which is more suitable for control design.
Since the unknown functions f i (·), i = 1,··· ,n are continuously differ-
entiable with respect to ¯x i and v, we apply the first-order Taylor expansion
on f i (·), i = 1,··· ,n, such that:

0 ∂f i ¯ x i ,x i+1
0
f i (¯ x i ,x i+1 ) = f i ¯ x i ,x + α i · x i+1 − x ,1 ≤ i ≤ n − 1
i+1 ∂x i+1 x i+1 =x i+1 i+1

f n (¯ x n ,v) = f n ¯ x n ,v 0 + ∂f n (¯ x n ,v) | v=v n · v − v 0 (15.6)
α
∂u
α n
where x α i = α ix i+1 + (1 − α i )x 0 ,with 0 <α i < 1, 1 ≤ i ≤ n −1, and v =
i+1 i+1
0
0
α nv + (1 − α n )v ,with 0 <α n < 1. By choosing x 0 i+1 = 0and v = 0, then
Eq. (15.6) can be rewritten as:

f i (¯ x i ,x i+1 ) = f i (¯ x i ,0) + ∂f i ¯ x i ,x i+1 x i+1 =x α i · x i+1 ,1 ≤ i ≤ n − 1
∂x i+1 i+1 (15.7)
∂f n (¯ x n ,v)
f n (¯ x n ,v) = f n (¯ x n ,0) + | v=v αn · v.
∂v
For the convenience of notation, it is defined that

α i ∂f i ¯ x i ,x i+1
g i ¯ x i ,x = α i ,1 ≤ i ≤ n − 1
i+1 ∂x i+1 x i+1 =x i+1 (15.8)
∂f n (¯ x n ,v)
α n
g n (¯ x n ,v ) = | v=v n
α
∂v
which are also unknown non-linear functions.