Reference: H. Sadeghian, H. Salarieh, A. Alasty, A. Meghdari. "On the Fractional-Order Extended Kalman Filter and its Application to Chaotic Cryptography in Noisy Environment" Applied Mathematical Modelling, 2014, 38, pp. 961-973.

1. On the fractional-order extended
Kalman filter and its application
to chaotic cryptography in noisy
environment
By Hoda Sadeghian, Hassan Salarieh,
Aria Alasty, Ali Meghdari
Presentation by Mostafa Shokrian Zeini

2. 
Chaotic Systems
Chaotic systems
characteristics
Very complex and
nonlinear
behaviour
Dependence of the
trajectories to the
initial conditions
Impossible long-
time prediction

3. 
1990, Pecora & Carroll: synchronization of chaotic
systems
Chaos synchronization: the slave/response system
should track the master/drive system trajectories.
A synchronization system consists of a transmitter
module, a channel for communication and a receiver
module.
Chaotic Systems

4. 
-Application:
*secure
communication
Chaotic Cryptography

5. 
BUT
•Most of the works in chaotic communication have modeled the
chaotic systems in deterministic form.
IN THIS
CASE
•in real world applications due to random uncertainties such as
stochastic forces on physical systems and noisy measurements
caused by environmental uncertainties, a stochastic chaotic behavior
is produced instead of a deterministic one.
the deterministic differential equation of a system must be substituted by a
stochastic one.
Chaotic Communication

6. 
As to increase the complexity of the transmitter, one may
use a fractional-order stochastic chaotic system as
transmitter.
A chaotic fractional-order dynamical equation produces
a complex behavior which makes the masked signal
more encrypted and consequently hard to decipher.
The complexity of fractional-order systems is due to
dealing with integration and derivation of non-integer
orders.
Fractional-order Stochastic Chaotic
Systems

7. 
Kalman Filter for Linear Fractional-order
Stochastic Systems
Lemma

8. 
Then the Kalman filter can be obtained via following steps
Kalman Filter for Linear Fractional-order
Stochastic Systems

9. 
d. Updating equation
c. Kalman gain equation
b. Innovation equation
a. Output prediction equation
Kalman Filter for Linear Fractional-order
Stochastic Systems:
Design & Calculations

10. 
and
where
e. State prediction equation
Kalman Filter for Linear Fractional-order
Stochastic Systems:
Design & Calculations

11. 
Then the Kalman filter can be obtained via following steps
Extended Kalman Filter for Non-linear
Fractional-order Stochastic Systems

12. 
c. Kalman gain equation
b. Innovation equation
where
a. Output prediction equation
Extended Kalman Filter for Non-linear
Fractional-order Stochastic Systems:
Design & Calculations

13. 
where
where the initial condition assumed to be at time
e. State prediction equation
d. Updating equation
Extended Kalman Filter for Non-linear
Fractional-order Stochastic Systems:
Design & Calculations

14. 
Except 𝛼 = 1, 𝜔 𝑘 is not a Wiener process and thus the
Kalman filter presented here is dealing with non-Wiener
process.
The above theorems can be easily verified in the case of 𝛼
= 1 to be exactly equal to the classical Kalman and extended
Kalman filter.
FKF Design Remarks

15. 
while the output of the communication channel would be
The output/measurement of this system can be assumed as
The Caputo derivative of fractional order is defined as
Let assume the transmitter module as a fractional-order
stochastic chaotic system be
Cryptography and Synchronization

16. 
To decrypt the message in the receiver, assuming the
synchronization via nonlinear filtering has been done
The noise of communication channel would change this output
in the form of
Let now assume the transmitter module has got another output
with embedded message that should be encrypted.
Cryptography and Synchronization

17. 
Cryptography and Synchronization

18. 
Using fractional-order Chen system
for chaotic fractional-order
cryptography

19. 
Fractional-
order Chen
system
the output
of noisy
channel
Using fractional-order Chen system
for chaotic fractional-order
cryptography
where

20. 
Using fractional-order Chen system
for chaotic fractional-order
cryptography
The first synchronized state of the fractional-order stochastic Chen chaotic system
and its estimated error

21. 
Using fractional-order Chen system
for chaotic fractional-order
cryptography
The second synchronized state of the fractional-order stochastic Chen chaotic
system and its estimated error

22. 
Using fractional-order Chen system
for chaotic fractional-order
cryptography
The third synchronized state of the fractional-order stochastic Chen chaotic system
and its estimated error

23. 
The
embedded
message
signal
The output
for
encryption
Using fractional-order Chen system
for chaotic fractional-order
cryptography
where

24. 
Using fractional-order Chen system
for chaotic fractional-order
cryptography
Encryption/Decryption of signal via fractional-order synchronization in the
fractional-order stochastic Chen chaotic system and its estimated error