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1490 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013
control. So, λ is selected larger than the magnitude of the
uncertainty and it will affect the convergence rate of the
tracking error, and Ξ is chosen very small to best
approximate the sign function and it will affect the size
of the residual set to which the tracking error will
converge. The sign function is not used here to avoid
problems associated with it as chattering and solutions
existence.
By adding and subtracting ν in (3), we obtain
T
T
bPe
e ⎛ ⎞
= ( A0 −bK ) e −bλ
tanh ⎜ ⎟−b[ h( ξη , , u)
−v]
(5)
⎝ Ξ ⎠
From the fact that the signal v does not explicitly
depend upon the control input u and Assumption 1, the
partial derivative of h(, ξη , u)
− v with respect to the
input u satisfies
( ξη )
∂ h(, , u) −v ∂h(, ξη, u)
= > 0
∂u
∂u
Thus according to the implicit function theorem, there
exists some ideal controller u * ( ξην , , ) satisfying the
following equality for all (, ξη,) v ∈ × ×
R
R
:
ξ
η
(6)
*
h(, ξη, u (, ξη ,)) v − v = 0
(7)
Therefore, if the control input u is chosen as the ideal
controller u
* (, ξη ,) v , the closed-loop error dynamic (5) is
reduced to
T
T
b Pe
e ⎛ ⎞
= ( A0 −bK ) e −bλ
tanh ⎜ ⎟ (8)
⎝ Ξ ⎠
Considering the following positive function to the
error dynamic:
V
T
= e Pe
(9)
Using (4) and (8), the time derivative of (9) becomes
T
T T b Pe
V ⎛ ⎞
=−e Qe −2λb Pe tanh⎜ ⎟ (10)
⎝ Ξ ⎠
T
⎛b Pe ⎞
Because the term b T Pe and tanh ⎜ ⎟ always
⎝ Ξ ⎠
have same sign, we conclude that V
≤ 0 , and only
when e = 0 , V = 0 , which means lim | e | = 0 .
t→∞
III. ZERO DYNAMICS
If system (1) is controlled by the input u, the state
vector η is completely unobservable from the output,
then the subsystem
dη
q(0, , u(0, , v(0, )))
dt = η η η (11)
is addressed as the zero dynamic.
Assumption 3: Zero dynamics (11) is exponentially
stable, and the function q( ξη , , u)
is Lipschitz inξ . There
exists Lipschitz constants L ξ
and L
q
such that
q(, ξη, u) − q(0, η, u ≤ L ξ + Lq
(12)
where u = u(0, η η
, v(0, η ))) .
By Lyapunov converse theorem, there is a Lyapunov
function V ( ) 0
η which satisfies
η
2 2
1
V0( )
2
ξ
σ η ≤ η ≤ σ η (13)
∂V0
( η)
q(0, η)
≤−σ3
η
∂η
∂V ( η)
0
∂η
≤ σ
Where σ , i = 1,2,3,4 are positive constant.
i
4
η
IV. DESIGN OF CONTROLLER
2
(14)
(15)
In control engineering, radial basis function (RBF)
NNs are usually used as a tool for modeling nonlinear
functions because of their good capabilities in function
approximation. RBFNN represents a class of linearly
parameterized approximations and can be replaced by any
other linearly parameterized approximations such as
spline functions or fuzzy systems. Moreover, nonlinearly
parameterized approximations, such as multilayer neural
network (MNN), can be linearized as linearly
parameterized approximations, with the higher order
terms of Taylor series expansions being taken as part of
the modeling error.
In this paper, the following RBF NN based on GGAP-
RBF [20] algorithm which can avoid to select initial
neural network parameters and nodes number of hidden
T
layer artificially uz ( ) = φ ( z)
θ is used to approximate
the ideal controller
1
* T T
u ( z ) , where z = ⎡ ⎣ ξ , η , v⎤
⎦ ,
φ( z) = ( φ ( z), , φ ( z)) T is the basic function vector, and
M
θ = ( θ1, , θ ) T
M
is the adjustable parameter. It has been
proven that neural network can approximate any smooth
q
function over a compact set ΩZ
⊂ R to arbitrarily any
accuracy as
( ) = φ ( ) θ + δ( ) (16)
* T *
u z z z
with bounded function approximation error δ ( z)
satisfying δ ( z)
≤ δ .Where
vector which minimizes the function δ ( z)
*
θ is an ideal parameter
T
. In this paper,
we assume that the used neural network does not violate
the universal approximation property on the compact set
Ω
Z
, which is assumed large enough so that the variable
z remains inside it under closed-loop control.
Let us define the control error between the controllers
uz ( ) and u
* ( z ) as
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