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JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1489
system is given. In Section VI, simulation results are
presented to confirm the effectiveness and applicability of
the proposed method. Finally, conclusions are included.
A. Notations and Preliminaries
The following notations and definitions will
extensively be used throughout the paper. Let be the
real number, n
n×
m
and represent the real n-vectors
and the real n× m matrices, respectively. i denotes the
usual Euclidean norm of a vector. In the case where y is
a scalar, y denotes its absolute value and if Y is a
matrix, Y means Frobenious norm defined as
Y
T
{ }
= tr Y Y .where tr{ i}
stands for trace operator.
Implicit Function Theorem: Assume that
n m n
h : × → is continuously differentiable at each
n m
ab of an open set S ⊂ × a , b be a
point ( , )
. Let ( 0 0)
point in S for which ( , )
h a b and for which the
0 0
Jacobian matrix ⎡∂h
⎤( a , b )
⎣
∂a⎦
0 0
is nonsingular. Then
n
m
there exist neighborhoods U ⊂ of a0
and V ⊂ of
b0
such that for each b∈V
the equation h( a, b ) = 0has a
unique solution a∈
U . Moreover, the solution can be
given as a = g( b)
where g is continuously differentiable
at b= b0
.
II. PROBLEM FORMALATION
Consider the following SISO nonaffine nonlinear
system [8]:
where [ ]
⎧ dξi
⎪ = ξi+
1
i =1, , r−1
dt
⎪
dξr
⎪ = h( ξη , , u)
⎨ dt
⎪ dη
⎪ = q(, ξη, u)
⎪ dt
⎪ ⎩ y = ξ1
, ,
T r
ξ = ξ1 ξ r
∈ Rξ
⊂ ,
(1)
n−r
η∈R η
⊂ are
system states and u ∈Ωu
⊂ , y ∈ are system input
and output respectively. h(, ξη, u)
is a smooth partially
known function, and q(, ξη , u)
is a smooth partially
known vector field.
The control objective is to design an adaptive neural
network controller for a class of SISO nonaffine
nonlinear systems (1) such that the system output follows
a desired trajectory while all signals in the closed-loop
system remain bounded.
∂h(, ξη, u)
Assumption 1: The function hu
(, ξη , u)
=
∂u
is nonzero and bounded for all (, ξη, u) ∈ × ×
R
R
.
This implies that h (, ξη , u)
is strictly either positive or
u
ξ
η
negative for all ( ξη , , u) ∈ × ×
R
R
.Without loss of
generality, it is assumed that it exists a constant c such
that hu
(, ξη , u) ≥ c > 0.
Define the reference vector
( r−1)
T r
y = ( y y y ) ∈R
ξ
d d d d
The reference signal y d
and its time derivative are
assumed to be smooth and bounded. We also define the
tracking error as
e= yd
− y
and corresponding error vector as
( r−1)
T r
e = (,, e e e ) ∈ R .
Assumption 2: When the desired output y d
and its r-
order derivative are of known bound, there exists a
positive constant bd
to satisfy
(1) ( r−1)
T
( yd yd yd ) ≤ bd
Then the error equation is as follows:
η
(2)
( r)
e = A0 e + b⎡
⎣yd
−h(, ξη, u)
⎤
⎦ (3)
⎡ 0 1 0 0⎤
⎡0⎤
⎢
0 0 1 0
⎥
⎢
⎢
⎥
0
⎥
⎢ ⎥
where A0
= ⎢ ⎥
, b = ⎢0⎥
.
⎢ ⎥
⎢ ⎥
⎢ 0 0 0 1⎥
⎢
⎥
⎢
⎣ 0 0 0 0 0⎥
⎦
⎢
r × r ⎣1⎥
⎦ r × 1
A b is controllable, then there will exist a
Obviously, ( , 0 )
constant matrix [ , , ]
T
0 1 r 1
T
c
K = k k k −
which makes
eigenvalues of matrix A = A0
− bK all have negative
real part. Thus, for any given positive definite symmetric
matrix Q , there exists a unique positive definite
symmetric solution P to the following Lyapunov
algebraic equation:
A P+ PA = − Q
(4)
T
c
Define a signal
T
( r)
T
⎛b Pe ⎞
ν= yd
+ K e + λtanh⎜ ⎟
⎝ Ξ ⎠
where tanh( •) ∈− ( 1,1) is the hyperbolic tangent function,
Ξ,λ
are the positive design parameters, when error
T
⎛bPe⎞ e →+
∞ , the value of tanh ⎜ ⎟ → + ∞ , and when
⎝ Ξ ⎠
T
⎛b Pe ⎞
error e →-∞ the value of tanh ⎜ ⎟ → - ∞ .
⎝ Ξ ⎠
T
⎛b Pe ⎞
When e → 0 , tanh ⎜ ⎟ → 0 .The term
⎝ Ξ ⎠
T
⎛b Pe ⎞
λ tanh ⎜ ⎟ is a smooth approximation of the
⎝ Ξ ⎠
T
discontinuous term sign( b Pe )
c
λ usually used in robust
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