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JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1491
* T
e = u ( z) − u( z) = φ ( z) θ + δ( z)
(17)
u
*
where θ = θ −θ
is the parameter estimation error vector.
According to the mean value theorem, there exist
constant 0< α < 1, h(, ξη , u)
can be described as
( )
h ξηu h ξη u h u z u z (18)
* *
(, , ) = (, , ) +
u
() − ()
λ
where
h =∂h(, ξη , u) ∂u|
uλ
u=
uλ
uλ = αu z + − α u z
*
( ) (1 ) ( )
By substituting (5) into the equation (18) and considering
(7), we get
T
bPe
*
e
⎛ ⎞
= Ac
e −bλtanh ⎜ ⎟−b⎡
⎣h( ξη , , u ) −ν⎤
⎦−
⎝ Ξ ⎠
*
−bhu
( u( z) −u ( z)
)
(19)
λ
T
⎛bPe⎞
*
= Ae c −bλ
tanh ⎜ ⎟−bhu
( u( z) −u ( z)
)
λ
⎝ Ξ ⎠
T
Considering Ac
= A0
− bK , then (19) can be rewritten as
T
( r) T ⎛bPe⎞
*
e + K e + λ tanh ⎜ ⎟ = h ( u ( z) − u( z)
u ) = h e (20)
λ
uλ
u
⎝ Ξ ⎠
We notice here that u
* ( z)
is an unknown quantity, so
the signal e u
defined in (17) is not available. Eq. (20) will
be used to overcome the difficulty. Indeed, from (20), we
see that even if the signal e is not available for
measurement, the quantity h uλ
e u
is measureable. This fact
will be exploited in the design of the parameters adaptive
law.
In order to obtain the update law ofθ , we consider a
quadratic cost function defined as
u
1 2 1 * T
J ( ( ) ( ) ) 2
θ
= eu
= u z − φ z θ (21)
2 2
By applying the gradient descent method, we obtain as an
adaptive law for the parameters θ
θ = - γ∇ θ
J ( θ) = γφ( ze ) u
(22)
From (22), we know e u is not available, the adaptive
law (22) cannot be implemented. In order to render (22)
computable, from Eq. (20), we select the design
parameter γ = γ θ
hu
λ
, where γ
θ
is a positive constant. We
have
θ = γ φ( zh ) e
θ
γφ
⎧⎪
uλ
u
λ
( r)
T
=
θ
( z) ⎨e + K e + tanh
⎪⎩
⎛
⎜
⎝
T
bPe
Ξ
⎞⎫⎪
⎟⎬
⎠⎪⎭
(23)
At the same time, in order to improve the robustness of
adaptive law in the presence of the approximation error,
we modify it by introducing a σ -modification term as
follows:
T
⎧⎪
( r)
T
⎛b Pe ⎞⎫⎪
θ = γθφ( z) ⎨e + K e + λ tanh⎜
⎟⎬−γ θσθ
(24)
⎪⎩
⎝ Ξ ⎠⎪⎭
whereσ is a small positive constant
Since the function of the σ -modification adaptive law
is to avoid parameter drift, it does not need to be active
when the estimated parameters are within some
acceptable bound. The system stability relies entirely on
the neural network because the proposed adaptive
controller in the paper is only composed of a neural
network part without additional control terms. The term
T
⎛b Pe ⎞
λ tanh ⎜ ⎟in the parameter adaptive law (24) plays,
⎝ Ξ ⎠
in some way, the role of a robustifying control term. Thus
by selecting a large positive value for the design
parameter λ and a small positive value for the
parameter Ξ , the robustness of the controller can be
improved.
V. STABILITY AND CONVERGENCE ANALYSIS OF
CONTROL SYSTEM
In order to analysis the convergence of neural network
weights, we firstly consider the following positive
function:
V θ
1
= T
θ θ
(25)
2 γ
Using (17), (20) and (24), the time derivative of (25)
can be written as
Considering the inequalities
θ
( z hu
eu
)
T
V
θ
= - θ φ( ) -σθ
λ
T
T
= - φ ( z)
θhu
e
λ u
+ σθ
θ
T
=− h e + h δ ( z)
e + σθ
θ
2
uλ
u uλ
u
T σ σ σ
σθ θ =− θ − θ + θ + θ
2 2 2
σ 2 σ *
2
≤− θ + θ
2 2
2 2
2
(26)
(27)
2 1 2 1 2 1
− e ( ) ( ) ( ( )) 2
u
+ δ z eu = − eu + δ z − eu
−δ
z
2 2 2
(28)
1 2 1 2
≤− eu
+ δ ( z))
2 2
Considering (27) and (28), Eq. (26) can be bounded as
1 2
2 1 2 *
2
V σ σ
θ
≤− hu eu + hu
δ ( z)
− θ + θ (29)
λ
λ
2 2 2 2
Because the functions δ ( z)
and hu
λ
are bounded in this
paper, and the parameters θ * are constants, so we can
define a positive constant bound ψ as
⎛1
2 ⎞ σ *
2
ψ = sup ⎜ hu
δ ( z)
θ
λ ⎟+
(30)
t ⎝2 ⎠ 2
Then
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