Page 301 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

T

where W 1 = c 0 − b e , k 0 − k s ,k sn /m s is the parameter vector to be
T
3
estimated, φ 1 (Z 1 ) = x 1 − x 3 ,x 2 − x 4 ,(x 1 − x 3 ) is the regressor with
4
Z 1 = [x 1 ,x 2 ,x 3 ,x 4 ] ∈ R .
Substituting (19.24)into(19.23), then ˙s 1 can be written as
T
s ˙ 1 = x 2 +
(19.25)
T T T
where
=[W ,W ] is the augmented parameter vector with W 2 =
1 2
T
T
T
−θ I /m s,and =[φ (Z 1 ),φ (Z 2 )I] is the augmented regression vector
1
2
with φ 2 (Z 2 ) = tanh[c 1 (x 2 − x 4 ) + k 1 (x 1 − x 3 )].
T
T T
We denote
=[ ˆ W , ˆ W ] as the estimation of the unknown param-
ˆ
2
1
eter vector
and then the input current I of MR damper can be designed
as
1
T
I = − ˆ W φ 1 (Z 1 ) − k ss − x 2 (19.26)
1
T
ˆ W φ 2 (Z 2 )
2
where k s > 0 is the feedback gain, ˆ W 1 , ˆ W 2 are the estimation of W 1 ,W 2,
which will be updated based on the adaptive law given in (19.31).
To design a new adaptive law with guaranteed convergence, we define
the filtered variables s 1f , f ,x 2f of s 1 , ,x 2 givenin(19.25)as
⎧
⎪ k˙s 1f + s 1f = s 1 , s f (0) = 0
⎨
˙
k f + f = , f (0) = 0 (19.27)
⎪
k˙x 2f + x 2f = x 2 ,
⎩ x 2f (0) = 0
where k > 0 is a constant filter parameter.
According to (19.25)and (19.27), one can obtain that
s 1 − s 1f T
s ˙ 1f = = x 2f +
f (19.28)
k
Moreover, we define the auxiliary matrix M 1 and vector N 1 in terms of
the following filter operations:
T
˙ M 1 =− M 1 + f , M 1 (0) = 0
f

(19.29)
s 1 −s 1f
˙ N 1 =− N 1 + f − x 2f , N 1 (0) = 0
k
where > 0 is a positive constant.
Then another vector H 1 can be obtained based on M 1 ,N 1 as
ˆ (19.30)
H 1 = M 1
− N 1