Page 146 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Substituting (9.14)–(9.22)into(9.12), one can obtain ˙ V 1 as

˜ 2
m 1 1 1 2 σ 1g 10 θ 1 σ a1 2
∗
˙ V 1 ≤− g 10k 1 − − − z − − (ε − g 10 ˆε 1 )
1 1
2c 11 4c 12 4c 13 4 2g 10
σ a1g 10 ˆε 2 1 2 2 2 2 σ 1g 10 θ 1 ∗2 1 σ a1 ε ∗2
1
− + c 12g z + c 13g e + + +
11 2
11 1
2 4 2g 10 2g 10
m 1 e à 1m

c 11 2 z 1 2
∗
+ 0.2785ω 1 ε + 1 − 2tanh ( ) ϕ (x 1 ) − V d1
1j
1
2 ω 1 j=1 1 −¯τ 1
2
2
2
≤− γ 1V 1 + ϑ 1 + c 12g z + c 13g e 2
11 2
11 1
m 1 e ϕ (x 1 )
à 1m 2
c 11 2 z 1 1j
+ 1 − 2tanh ( ) (9.23)
2 ω 1 j=1 1 −¯ 1
τ
where γ 1 and ϑ 1 are constants specified by

γ 1 = min 2(g 10k 1 − m 1 /2c 11 − 1/4c 12 − 1/4c 13 ),
1 σ 1 /2,
a1 σ a1 , ,
∗2
∗2
∗
ϑ 1 = σ 1 θ g 10 /4 + 1/2g 10 + σ a1 ε /2g 10 + 0.2785ω 1 ε .
1 1 1
The last term in (9.23) may be positive or negative, which depends on
thesizeof z 1. However, since the functions ϕ 1j (x 1 ) are bounded on any
2
compact set C 1 and −1 ≤ 1 − 2tanh (z 1 /ω 1 ) ≤ 1, the last term in (9.23)
is bounded. If z 2 and e 1 can be proven to be bounded (to be shown in
the next step), k 1 is large (then γ 1 is positive), and c 11, ϑ 1 are small, then
according to the extended Lyapunov Theorem, z 1, θ 1, ˜ε 1 are uniformly
˜
ultimately bounded (UUB) over the compact set C 1.
For a possible large tracking error in the initial control procedure, e.g.,
, then it is shown according to Lemma 9.2 that
z 1 /∈ z 1
m 1 à 1m 2
z 1 e ϕ (x 1 )
1j
2
1 − 2tanh ( ) ≤ 0. (9.24)
τ
ω 1 1 −¯ 1
j=1
In this case, the Lyapunov function (9.23) can be rewritten as
2
2
2
2
˙ V 1 ≤−γ 1V 1 + ϑ 1 + c 12g z + c 13g e . (9.25)
11 2 11 1
. On the other
This can improve the error convergence rate for z 1 /∈ z 1
, the system output x 1 is
hand, if the tracking error is small, i.e., z 1 ∈ z 1
bounded according to z 1 = x 1 −y d . For this case, the last term in (9.23)can
be tuned to be small by decreasing ω 1 and c 11.