Page 267 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

to satisfy the following condition
n

r(u r + ˜ a jx n−j+1 ) ≤ v (17.41)
j=1

where ˜a j =ˆa j − a j is the identification error, v is a small constant set by the
designers. Hence, inspired by the adaptive robust control [5], we can set
1
2
u r =− τ r, for any τ ≥||˜a||||¯x||,with ˜a =[˜a 1 ,··· , ˜a n ] and ¯x =[x 1 ,··· ,x n ].
4v
17.4.3 Stability Analysis
The main results of this paper can be summarized as follows:

Theorem 17.2. For the system (17.26) with the feedback control (17.40)and
the inverse compensation (17.28), then the following properties can be obtained:
1) The closed-loop system is stable, and the positive definite function V r (t) =
1 2
2 r (t) of the control error r has an upper bound given by
v e
V r (t) ≤ exp(−k pt)V r (0) + [1 − exp(−k pt)] (17.42)
k p
2) The tracking error e = y − y d converges to a small set around zero.

Proof. 1) From (17.39)and (17.40), the derivative of V r can be obtained
2
˙ V r =−k 0r + r(
˜ a ix n−i+1 + u r + ) (17.43)
where = BI(BIV(u d )) − u d .From(17.41), it can be seen that

2
˙ V r ≤−k pr + v + r (17.44)
k p 2 2 2
Then by applying Young’s inequality r ≤ r + ,wehave
2 k p
1 2 2 2
˙ V r ≤− k pr + v + (17.45)
2 k p

From Lemma 17.1,itisknownthat | |≤|d 1 |+|d 2 |, thus we can obtain
1 2 2 2
˙ V r ≤− k pr + v + (|d 1 |+|d 2 |) =−k pV r + v e (17.46)
2 k p
2 2
where v e = v + (|d 1 |+|d 2 |) . Hence, by deriving the solution of (17.46),
k p
the Lyapunov function V r (t) is bounded as given in (17.42), where the