European Journal of Scientific Research - EuroJournals

Model-Based Adaptive Chaos Control using Lyapunov Exponents 329
Figure 1: The Nonlinear observer.
Substituting x ( k + 1)
and x ˆ ( k + 1)
from (12) and (13) into (15), and using the gradient method
for computing the gain of observer we have:
L( k + 1)
= L(
k)
− α∇
( ( 1),
ˆ
L(
k ) J x k + x(
k + 1))
(16)
Where, α > 0 . Algorithm (16) makes the current observer gain correction
∆ L( k)
= L(
k + 1)
− L(
k)
in the descent direction of the current goal function J .
5. Simulation Results
+
u
k
+
+
Chaotic
Plant
Z -1
xˆ k + 1
k
fˆ
( ⋅)
Consider the Henon map as:
2
xk
+ 1 = yk
+ 1−
axk
yk
+ 1 = bxk
(17)
which behaves chaotically when a=1.4 and b=0.3 and periodically when a=1 and b=0.1. Figure 2
shows the Henon behavior when the parameters change occurs at 1500 from periodically to chaotically.
h (⋅)
Figure 2: Henon output whit parameters change at k=1500.
Lk
xˆ
y
k
yˆ
k
+
−