European Journal of Scientific Research - EuroJournals

328 Amin Yazdanpanah Goharrizi and Mehdi Semin
3. Design Methodology
l
Let the controlled dynamical system (1), with the input sequence u( k)
∈ R , be as follows:
x ( k + 1)
= f [ x(
k)]
+ u(
k)
(8)
For this system finding a feedback control law u ( k)
= g[
x(
k)]
, where g: R R
n → is an
appropriate function such that the resulting closed loop system exhibits appropriate behavior, is
desired. So, consider the linear desired model is as follows:
x ( k + 1)
= Ax(
k)
(9)
n
n×
n
Where x ∈ R , A∈
R is a constant matrix and k = 0,
1,
2,...
is the discrete time index. So, the
controlled system orbit can behave like the system model "(9)," depends on how the eigenvalues of the
matrix A is chosen. If the eigenvalues lie in the unit disc then the control action is like a regulation
problem but if the eigenvalues are on the unit disc the behavior of the closed loop system becomes
periodic. So, it's sufficient to choose the appropriate control input as follows:
u ( k)
= − f [ x(
k)]
+ Ax(
k)
Then, the methodology is as follows:
(10)
⎧0
u(
k)
= ⎨
⎩−
f [ x(
k)]
+ Ax(
k)
if λmax
< 0
if λmax
> 0
(11)
The control action can be applied by adaptive calculation of Lyapunov exponents of the system,
as discussed in the pervious section.
4. Nonlinear Observer Design Methodology
Consider the controlled nonlinear chaotic system as follows:
x(
k + 1)
= f [ x(
k),
u(
k)]
y(
k)
= h[
x(
k)]
(12)
n
where f : ℜ
p ≤ n .
n
n
→ ℜ , and h : ℜ
p
→ ℜ are assumed to be differentiable and smooth functions and
An observer is a dynamic system driven by the observations which is shown in figure (1):
xˆ ( k + 1)
= f [ xˆ
( k),
u(
k)]
+ L(
k)[
h(
x)
− h(
xˆ
)]
In the case of state estimation, we have
(13)
lim[ xˆ
( k)
− x(
k)]
→ 0
k →∞
An approach to the observer design for system (12) is:
(14)
J ( x(
k + 1),
xˆ
( k + 1))
≤ ∆ when k > k ∗
(15)
With some nonnegative function J , and threshold value ∆ > 0 .