Page 439 - Mathematical Techniques of Fractional Order Systems

424 Mathematical Techniques of Fractional Order Systems

sensitive to parameter values which makes it suitable for secure communica-
tion. The proposed master system at the transmitter side (Sheu et al., 2010):

D x 1 5 aðx 2 2 x 1 Þ; ð14:46aÞ
α 1
D x 2 52 x 1 x 3 1 bx 1 2 x 2 ; ð14:46bÞ
α 2
D x 3 5 x 1 x 2 2 cx 3 : ð14:46cÞ
α 3
where α 1 , α 2 , and α 3 are the fractional orders and a, b, and c are the system
parameters. The signal x 1 ðtÞ is chosen to be the synchronization signal used
to drive the slave system at the receiver side. The slave system is given by
(Sheu et al., 2010):
D y 1 5 aðy 2 2 y 1 Þ; ð14:47aÞ
α 1
D y 2 52 x 1 y 3 1 bx 1 2 y 2 ; ð14:47bÞ
α 2
D y 3 5 x 1 y 2 2 cy 3 : ð14:47cÞ
α 3
It was proved using Laplace transform that the master and slave systems
will synchronize. The proposed scheme of the communication system is
shown in Fig. 14.7. Two highly nonlinear functions are used to encrypt and
decrypt the message. The encryption and decryption operations are given as
(Sheu et al., 2010):
2
2
T 1 ðtÞ 5 x ðtÞ 1 ð1 1 x ðtÞÞSðtÞ; ð14:48aÞ
2
2
2
y ðtÞ T 1 ðtÞ
S d ðtÞ 52 2 2 1 2 ; ð14:48bÞ
1 1 y ðtÞ 1 1 y ðtÞ
2 2
where SðtÞ and S d ðtÞ are the message signals at the sender and receiver sides
respectively. It can be easily proved that when the two systems are synchro-
nized ðx 2 5 y 2 Þ then the received message is the same as the sent one.
As a numerical example, the systems were simulated at the following
fractional order, parameters, and initial conditions:
ðα 1 ; α 2 ; α 3 Þ 5 ð0:96; 0:98; 1:1Þ; ð14:49aÞ

ða; b; cÞ 5 ð10; 28; 8=3Þ; ð14:49bÞ

FIGURE 14.7 System block diagram of two channel secure communication system.