Page 237 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Neural Dynamic Surface Control for Pure-Feedback Systems With Input Saturation 237

Step i: Consider the definition

˙ z i = z i+1 (15.35)

and denote the intermediate error as

(15.36)
s i = z i − β i
which is called the i-th error surface. Then, we have its derivative as

s ˙ i = z i+1 − β i (15.37)
˙
Choose a virtual control ¯z i+1 as

˙ (15.38)
¯ z i+1 =−k is i − s i−1 + β i
where k i > 0 is a positive constant.
Introduce a new state variable β i+1 and let ¯z i+1 pass through a first-order
filter with time constant τ i+1 > 0, and we have

˙
τ i+1 β i+1 + β i+1 =¯z i+1 ,β i+1 (0) =¯ z i+1 (0) (15.39)
so that the filter error is given by

y i+1 = β i+1 −¯z i+1 . (15.40)

Substituting (15.40)into(15.39), we can obtain

¯ z i+1 − β i+1 y i+1
˙
β i+1 = =− . (15.41)
τ i+1 τ i+1
Step n: The final control will be derived in this step. Now, we consider
the system dynamics as

˙ z n = a(¯ x n ) + b(¯ x n ,v ) u. (15.42)
α n
Then, we define the final n-th error as

(15.43)
s n = z n − β n
From (15.42)and (15.43), we have

s ˙ n = a(¯ x n ) + b(¯ x n ,v )u − β n . (15.44)
α n
˙