Page 221 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics
P. 221
220 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics
Differentiating s 2 along (14.14)and (14.15), we have
s
s ˙ 2 =¨ 1 + α˙ 1
s
s (14.17)
= F φ F (h(x,t) + b 0u −¨y d ) + H 1 (x,e,t) + α˙ 1
s
= F φ F (ξ(y d ,s 1 ,s 2 ) + b 0u −¨y d ) + α˙ 1
where the non-linear function ξ(y d ,s 1 ,s 2 ) is derived as
H 1 (x,e,t)
ξ(y d ,s 1 ,s 2 ) = h(x,t) + . (14.18)
F φ F
Since ξ(y d ,s 1 ,s 2 ) is not easy to calculate precisely, an NN will be used
to cope with this unknown function. Hence, there exists an ideal weight
vector W so that the non-linear function ξ(y d ,s 1 ,s 2 ) can be expressed as
∗
ξ(y d ,s 1 ,s 2 ) = W ∗T φ(X) + ε (14.19)
T 5
where the input vector of NN is X = y d , ˙y d , ¨y d ,s 1 ,s 2 ∈ R .
In the following, a non-singular terminal sliding mode funnel control
approach is developed for tracking control of the motor servo system (14.7).
To make s 2 converge to zero within a finite time, the non-singular terminal
sliding mode manifold is designed as
q/p
s
β|˙ 2 | sgn(s 2 ) + s 2 = 0 (14.20)
where β> 0, p and q arepositiveodd integers with p < q.
Substituting (14.17)into(14.20) and using (14.19), the controller is
designed as
1 T 1 1 p/q
u =− −¨ y d + ˆ W φ(X) + μsgn(s 2 ) + α˙ s 1 + |s 2 | sgn(s 2 )
b 0 F φ F β
(14.21)
where ˆ W is the estimate of the unknown NN weight W and μ is the
∗
T
upper bound of the NN approximation error ε and ˜ W φ(X),where ˜ W =
W − ˆ W is the NN weight estimation error.
∗
The adaptive law for updating ˆ W is given by
˙
ˆ W = è( X)s 2 (14.22)
where is a positive definite and diagonal matrix.