Implementation on Electronic Circuits
7.3 New Fuzzy model of complicated chaotic systems
whereM1andM2are diagonal matrices as following:
dia(M1)=
[
M11 M21 ... Mi1]
, dia(M2)=[
M12 M22 ... Mi2]
Note that for each equation i:
∑
The new model provides a much more convenient approach for fuzzy model research and fuzzy application. The simulation results of complicated chaotic systems are discussed in next Section.
7.3 New Fuzzy model of complicated chaotic systems
In this section, the new fuzzy models of two different chaotic systems, two-cell Quantum-CNN system and Qi system, are shown in Model 1 and Model 2. In order to investigate the convenience and effectiveness of the new fuzzy model, original T-S fuzzy model is given for comparison.
Model 1: New fuzzy model of Quantum-CNN system
For a two-cell Quantum-CNN, the following differential equations are obtained
[26]:
⎪⎪
⎪⎪
⎩
⎪⎪
⎪⎪
⎨
⎧
+ −
−
−
=
−
−
=
+ −
−
−
=
−
−
=
2 4 3 2 3
1 3 2 4
4 2 3 2
3
2 2 1 1 1
3 1 1 2
2 2 1 1
1
x x cos 1 a x 2 ) x x ( w x
x sin x 1 a 2 x
x x cos 1 a x 2 ) x x ( w x
x sin x 1 a 2 x
&
&
&
&
(7-3-1)
where x1 and x3 are polarizations, x2 and x4 are quantum phase displacements, a1 and a2 are proportional to the inter-dot energy inside each cell and ω1 and ω2 are parameters that weigh effects on the cell of the difference of the polarization of neighboring cells, like the cloning templates in traditional CNNs.
Whena1 =−0.83,a2 =−0.53,w1 =0.5andw2 =0.5(assume two balanced cells) and initial states chosen as (0.001, 0.005, 0.001, 0.005), the nano system is chaotic which is shown in Fig. 7-1.
If T-S fuzzy model is used for representing local linear models of Quantum-CNN nano system, there are going to be 16 fuzzy rules, 16 linear subsystems and 64 equations. The process of modeling is shown as follow:
T-S fuzzy model:
Assume that:
(1) 1−x12sinx2∈[−Z1,Z1] and Z1>0,
(2) cosx2 1−x12 ∈[1+Z2,1−Z2] and Z2 >0, (3) 1−x32sinx4∈[−Z3,Z3] and Z3 > , 0
(4) cosx4 1−x23 ∈[1+Z4,1−Z4] and Z4 >0 Then we have the following T-S fuzzy rules:
Rule 1: IF 1−x12sinx2isM11,cosx2 1−x12 isM21, 1−x23 sinx4isM and 31
2 3
4 1 x
x
cos − isM41, THEN X& =A1X.
Rule 2: IF 1−x12sinx2isM11,cosx2 1−x12 isM21, 1−x23 sinx4isM and 31
2 3
4 1 x
x
cos − isM42, THEN X& =A2X.
Rule 3: IF 1−x12sinx2isM11,cosx2 1−x12 isM21, 1−x23 sinx4isM and 32
2 3
4 1 x
x
cos − isM41, THEN X& =A3X.
Rule 4: IF 1−x12sinx2isM11,cosx2 1−x12 isM21, 1−x23 sinx4isM and 32
2 3
4 1 x
x
cos − isM42, THEN X& =A4X.
Rule 5: IF 1−x12sinx2isM11,cosx2 1−x12 isM22, 1−x23 sinx4isM and 31
2 3
4 1 x
x
cos − isM41, THEN X& =A5X.
Rule 6: IF 1−x12sinx2isM11,cosx2 1−x12 isM22, 1−x23 sinx4isM and 31
2 3
4 1 x
x
cos − isM42, THEN X& =A6X.
Rule 7: IF 1−x12sinx2isM11,cosx2 1−x12 isM22, 1−x23 sinx4isM and 32
2 3
4 1 x
x
cos − isM41, THEN X& =A7X.
Rule 8: IF 1−x12sinx2isM11,cosx2 1−x12 isM22, 1−x23 sinx4isM and 32
2 3
4 1 x
x
cos − isM42, THEN X& =A8X.
Rule 9: IF 1−x12sinx2isM12,cosx2 1−x12 isM21, 1−x23 sinx4isM and 31
2 3
4 1 x
x
cos − isM41, THEN X& =A9X.
Rule 10: IF 1−x12 sinx2isM12,cosx2 1−x12 isM21, 1−x23 sinx4isM and 31
2 3
4 1 x
x
cos − isM42, THEN X& =A10X.
Rule 11: IF 1−x12sinx2isM12,cosx2 1−x12 isM21, 1−x23 sinx4isM and 32
2 3
4 1 x
x
cos − isM41, THEN X& =A11X.
Rule 12: IF 1−x12 sinx2isM12,cosx2 1−x12 isM21, 1−x23 sinx4isM and 32
2 3
4 1 x
x
cos − isM42, THEN X& =A12X.
Rule 13: IF 1−x12sinx2isM12,cosx2 1−x12 isM22, 1−x23 sinx4isM and 31
2 3
4 1 x
x
cos − isM41, THEN X& =A13X.
Rule 14: IF 1−x12sinx2isM12,cosx2 1−x12 isM22, 1−x23 sinx4isM and 31
2 3
4 1 x
x
cos − isM42, THEN X& =A14X.
Rule 15: IF 1−x12sinx2isM12,cosx2 1−x12 isM22, 1−x23 sinx4isM and 32
2 3
4 1 x
x
cos − isM41, THEN X& =A15X.
Rule 16: IF 1−x12sinx2isM12,cosx2 1−x12 isM22, 1−x23 sinx4isM and 32
2 3
4 1 x
x
cos − isM42, THEN X& =A16X.
Then the final output of the two cells Quantum-CNN system can be composed by fuzzy linear subsystems mentioned above. It is obviously an inefficient and complicated work.
New fuzzy model:
By using the new fuzzy model, Quantum-CNN system can be linearized as simple linear equations. The steps of fuzzy modeling are shown as follow:
Step of fuzzy modeling:
Step 1:
Assume that 1−x12sinx2∈[−Z1,Z1]and Z1 >0, then the first equation of (7-3-1) can be exactly represented by new fuzzy model as following:
Rule 1: IF 1−x12sinx2isM11, THEN x&1=−2a1Z1, (7-3-2)
Rule 2: IF 1−x12sinx2isM12, THEN x&1=2a1Z1 (7-3-3) where
)
Z x sin x 1 1
2( M 1
1 2 2 1 11
+ −
= , )
Z x sin x 1 1
2( M 1
1 2 2 1 12
− −
= ,
andZ1 =0.01.M11andM12 are fuzzy sets of the first equation of (7-3-1) and 1
M
M11+ 12 = . Step 2:
Assume that cosx2 1−x12 ∈[1−Z2,1+Z2] and Z2 >0 , then the second equation of (8-3-1) can be exactly represented by new fuzzy model as following:
Rule 1: IFcosx2 1−x12 isM21, THEN
2 1 1 3 1 1
2 w (x x ) 2a x Z
x& =− − + (7-3-4) Rule 2: IFcosx2 1−x12 isM22, THEN
2 1 1 3 1 1
2 w (x x ) 2a x Z
x& =− − − (7-3-5) where
)
Z x 1 x 1 cos 2( M 1
2 12 2
21
+ −
= , )
Z x 1 x 1 cos 2( M 1
2 12 2
22
− −
= ,
andZ2 =0.01. M21andM22are fuzzy sets of the second equation of (8-3-1) and 1
M
M21 + 22 = . Step 3:
Assume that 1−x23sinx4∈[−Z3,Z3]andZ3 > , then the third equation of 0
(7-3-1) can be exactly represented by new fuzzy model as following:
Rule 1: IF 1−x23 sinx4 isM , THEN 31 x&3 =−2a2Z3, (7-3-6)
Rule 2: IF 1−x32sinx4 isM , THEN 32 x&3=2a2Z3 (7-3-7) where
)
Z x sin x 1 1
2( M 1
3 4 2 3 31
+ −
= , )
Z x sin x 1 1
2( M 1
3 4 2 3 32
− −
= ,
andZ3 =0.01. M and31 M are fuzzy sets of the third equation of (7-3-1) and 32 1
M
M31+ 32 = . Step 4:
Assume that cosx4 1−x23 ∈[1−Z4,1+Z4] and Z4 >0 , then the fourth equation of (7-3-1) can be exactly represented by new fuzzy model as following:
Rule 1: IFcosx4 1−x32 isM41, THEN
4 3 2 1 3 2
4 w (x x ) 2a x Z
x& =− − + (7-3-8) Rule 2: IFcosx4 1−x32 isM42, THEN
4 3 2 1 3 2
4 w (x x ) 2a x Z
x& =− − − (7-3-9) where
)
Z x 1 x 1 cos 2( M 1
4 2 3 4
41
+ −
= , )
Z x 1 x 1 cos 2( M 1
4 2 3 4
42
− −
= ,
andZ4 =0.01. M41andM42are fuzzy sets of the fourth equation of (7-3-1) and 1
M
M41 + 42 = .
Here, we call (7-3-2), (7-3-4), (7-3-6) and (7-3-8) the first liner subsystem under the fuzzy rules and (7-3-3), (7-3-5), (7-3-7) and (7-3-9) the second liner subsystem under the fuzzy rules.
The first linear subsystem is
⎪⎪
The second linear subsystem is
The final output of the fuzzy Quantum-CNN system is inferred as follows and the chaotic behavior of fuzzy system is shown in Fig. 7-2.
⎥⎥
Eq. (7-3-12) can be rewritten as a simple mathematical expression:
) whereΨiare diagonal matrices as follows:
[
11 21 31 41]
Via using new fuzzy model, the number of fuzzy rules in fuzzy Quantum-CNN system can be reduced from24to2× and only two subsystems can express such 4 complex chaotic behaviors. The simulation results are perfectly the same to the original chaotic behavior of the Quantum-CNN system.
Model 2: New fuzzy model of Qi system The four-order autonomous Qi system:
positive real parameters. This Qi system in Eq. (7-3-14) was recently introduced by Qi et al. [20] and it has been shown complex dynamical behavior including the familiar period-doubling route to chaos as well as Hopf bifurcations. For the system parameters: a3=35, b3=10, c3=1, d3=10 and initial conditions (y10, y20, y30, y40) = (2, 5, 2, 5), the Qi model exhibits chaotic motion which is shown in Fig. 7-3
First of all, T-S fuzzy model is used for representing local linear models of Qi system. The process of modeling is shown as follow:
T-S fuzzy model:
Assume that:
(1) ]y3y4∈[−Z5,Z5 andZ5 >0, (2) ]y y ∈[−Z ,Z andZ >0,
Then we have the following T-S fuzzy rules:
Rule 1: IFy3y4isN11andy1y2isN21,THEN YY& =C1 . Rule 2: IFy3y4isN11andy1y2isN22,THEN YY& =C2 . Rule 3: IFy3y4isN12andy1y2isN21,THEN YY& =C3 .
Rule 4: IFy3y4isN12andy1y2isN22,THEN YY& =C4 .
Then the final output of the Qi system can be composed by fuzzy linear subsystems mentioned above. There are 4 linear subsystems and 16 equations in this T-S fuzzy Qi system.
Novel fuzzy model:
Assume that:
(1) ]y3y4∈[−Z5,Z5 andZ5 >0, (2) ]y1y2∈[−Z6,Z6 andZ6 >0, then we have the following T-S fuzzy rules:
Rule 1: IFy3y4isN11,THEN
)
Here, we call (7-3-15) and (7-3-17) the first liner subsystem under the fuzzy rules and (7-3-16) and (7-3-18) the second liner subsystem under the fuzzy rules.
The first linear subsystem is
⎪⎪
The second linear subsystem is
The final output of the fuzzy Qi system is inferred as follows and the chaotic behavior of fuzzy system is shown in Fig. 7-4.
⎥⎥
Eq. (7-3-21) can be rewritten as a simple mathematical expression:
)
[
11 11 21 21]
Via using new fuzzy model, two linear subsystems are enough to express such complex chaotic behaviors. The simulation results are perfectly the same to the original chaotic behavior of the Qi system.