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JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1493
Supposing
selecting μ =
1
T
2
2 −ρt
ε = + 2 bP ψ0e + 2ψ1K c
Ξ , and
4ε
2
2
1⎛
1 ⎞
⎜σ 3
− σ4Lξ
ε1⎟
2⎝
2 ⎠
, then
( σε
4 2( Lb
ξ d
+ Lq)
)
1
2
V
⎛
⎞
( e, η) ≤−⎜λmin ( Q) −0.5− μσ4Lξ
⎟ e +
⎝
2 ε1
⎠
Adjusting Q to make
⎛ 1 ⎞
3 4 1
1
⎜σ − σ Lξ
ε ⎟
2
2
−
⎝
⎠
η + ε
2
4
( σε
4 2( Lb
ξ d
+ Lq)
)
1
λ ( ) −0.5− μσ > 0.
min
Q
4L ξ
2ε1
2
2
(43)
From the above equation, we can know that tracking
error and internal states η are all uniformly ultimately
bounded.
Besides,
since ξ ≤ e + y y y ≤ e + b , then
(1) ( r−1)
T
(
d d
d
)
d
the state ξ is uniformly ultimately bounded too. This
completes the proof.
VI. SIMULATION RESULTS
In this part, the following SISO nonaffine nonlinear
system with zero dynamics is simulated to illustrate the
effectiveness of the proposed adaptive neural network
tracking controller. The nonaffine nonlinear system is
described as follows:
ξ = ξ
1 2
2
(( ) )( )
ξ =−2 ξ −η −1 ξ −η −η − 0.2η
+
2 1 1 2 2 1 2
⎡ 1 3 ⎤
+ ( 2 + sin ([ ξ1−η1][ ξ2 − η2]
))
⎢u+ u + sin( u)
3 ⎥ (44)
⎣
⎦
η1 = η2
η =−2η − 0.2η + ξ
2 1 2 1
y = ξ
1
The control objective is to force the system output y
to track the desired trajectory yd
= 2sint+ cos ( 0.5t).The
simulation parameters are selected as follows:
15 5
Q = diag[10,10] , P = ⎡ ⎤
⎢
5 5 ⎥ , K = [ 1, 2]
T
, Ξ = 0.01 ,
⎣ ⎦
λ = 10 , γ = 11 , σ = 0.02 .
θ
T
The output of RBFNN controller is uz ( ) = φ ( z)
θ . The
basis function vector is φ( z) = ( φ1
( z) φ ( )) T
M
z , where
T
( z μ ) ( z μ )
⎡− −
i
− ⎤
i
φi
( z) = exp ⎢
⎥, i = 1, , M . M is the
2
⎢⎣
σ
i ⎥⎦
number of hidden layer nodes which is stable at the 33
nodes by training on-line using the GGAP-RBF
algorithm.
According to (23), the control law is
T
uz ( ) = φ ( z)
θ
θ =− 11 φ( z) e + e+ 2e + 10 tanh 100 × (5e+
5 e
)
− 11×
0.02θ
{ ( )}
The system initial conditions are ξ = [ ]
(0) 1 2 T
.The
simulation results using MATLAB are shown in Fig1, 2,
3, 4.
Figure 1. Plots of output tracking of system
Figure 1 shows the result of output tracking, and the
control input signal is shown in Figure 2. The growing
and pruning automatically of hidden layer nodes are
shown in Figure 3.
Figure 2. Plots of Control input
Figure 3. Node Number of Hidden Layer
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