Page 238 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Since the functions a(¯ x n ),b(¯ x n ,v ) are unknown, we will use an NN
α n
n
(15.18) in the final control. Given a compact set zn ∈ R , such that for
any (x 1 ,...,x n ) ∈ zn, then the following function approximation can be
used:

˙
a(¯ x n ) − β n ∗T
H(¯x n ) = = W φ (¯x n ) + ε (15.45)
α
b(¯ x n ,v n )
with W and ε are the ideal NN weight and approximation error, which
∗
∗
are bounded by W ≤ W N and |ε| ≤ ε N .
Then, the control u is designed as
s n

T
u =−k ns n − s n−1 − ˆ W φ (¯x n ) − ε N tanh (15.46)
δ
∗
where ˆ W is the estimation of W and ˆε N is the estimation of the upper
bound for ε,and δ> 0 is a small constant.
The adaptive laws of ˆ W and ˆε N are given by

˙ ˆ W = φ (¯x n )s n − σ ˆ W
(15.47)

˙ ˆ ε N = ε s n tanh
s n
δ
T
where = > 0, ε > 0 are constant learning gains, σ is a positive small
constant.
15.3.2 Stability Analysis

In this section, the stability of the closed-loop system and the convergence
of tracking error e and sliding mode variable s are all proved. The main
results of this chapter can be given as:

Theorem 15.1. Consider the non-linear system (15.1) with unknown input sat-
uration (15.4), the feedback control (15.24), (15.38), and (15.46), and adaptive
law (15.47) are applied. Given any positive constant p, for all initial condi-

T
2
1 2
tions satisfying n−1
s + y 2 + s + ˜ W W + 1 ˜ ε 2 ≤ 2p, then all the
−1 ˜

i i+1 b n v εN N
i=1
closed-loop system signals are semi-global uniformly ultimately bounded, and the
tracking error can be made arbitrarily small by properly choosing the design parame-
ters.
Proof. We define the estimation error as ˜ W = ˆ W − W , ˜ε =ˆε − ε N . Then,
∗
the closed-loop system with the new coordinates s i, β i , ˜ W i , can be ex-