# Chaotic Motions in the Real Fuzzy Electronic Circuits

Authors: Shih-Yu Li; Cheng-Hsiung Yang; Chin-Teng Lin; Li-Wei Ko; Tien-Ting Chiu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Fuzzy electronic circuit (FEC) is firstly introduced, which is implementing Takagi-Sugeno (T-S) fuzzy chaotic systems on electronic circuit. In the research field of secure communications, the original source should be blended with other complex signals. Chaotic signals are one of the good sources to be applied to encrypt high confidential signals, because of its high complexity, sensitiveness of initial conditions, and unpredictability. Consequently, generating chaotic signals on electronic circuit to produce real electrical signals applied to secure communications is an exceedingly important issue. However, nonlinear systems are always composed of many complex equations and are hard to realize on electronic circuits. Takagi-Sugeno (T-S) fuzzy model is a powerful tool, which is described by fuzzy IF-THEN rules to express the local dynamics of each fuzzy rule by a linear system model. Accordingly, in this paper, we produce the chaotic signals via electronic circuits through T-S fuzzy model and the numerical simulation results provided by MATLAB are also proposed for comparison. T-S fuzzy chaotic Lorenz and Chen-Lee systems are used for examples and are given to demonstrate the effectiveness of the proposed electronic circuit.


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## Body

## 1. Introduction

Nonlinear dynamics, commonly called the chaos theory, changes the scientific way of looking at the dynamics of natural and social systems, which has been intensively studied over the past several decades [1–10]. The phenomenon of chaos has attracted widespread attention amongst mathematicians, physicists, and engineers and has also been extensively studied in many fields, such as chemical reactions [11, 12], biological systems [13, 14], information processing [15, 16], and secure communications [17–20].The mathematical meteorologist Lorenz discovered chaos in a simple system of three autonomous ordinary differential equations in order to describe the simplified Rayleigh-Bénard problem [21] in 1963 which is the most popular system for studying [22–26]. Chen and Lee reported a new chaotic system [27] in 2004, which is now called the Chen-Lee system [28]. The chaotic Chen-Lee system was developed based on the Euler equations for the motion of rigid body. It was proved that this system is the governing set of equations for gyromotion with feedback control. Recently, studies were conducted on this system to explore its dynamic behavior, including the fractional order behavior, the generation of hyperchaos and perturbation analysis, the control and anti-control of chaos, and the synchronization [29, 30].Since the fuzzy set theory [31] and the fuzzy logic [32] were initiated by Zadeh in 1965 and 1973, fuzzy logic has received much attention as a powerful tool for the nonlinear filed. Among various kinds of fuzzy methods, Takagi-Sugeno fuzzy system is widely accepted as a tool for design and analysis of fuzzy control system [33]. The T-S fuzzy model proposes a successful method to deal with certain complex nonlinear systems via some local linear subsystems. There are plenty of researches using the Takagi-Sugeno (T-S) fuzzy model to represent typical chaotic models and then apply some effective fuzzy techniques [34–42]. However, there are still no real experimental models in electronic circuit for Takagi-Sugeno (T-S) fuzzy-based chaotic systems. In this paper, we carry out the powerful tool, Takagi-Sugeno (T-S) fuzzy model, in electronic circuit and show good agreement between computer simulations in MATLAB and experimental results in our circuits.The layout of the rest of the paper is as follows. In Section2, the Takagi-Sugeno fuzzy model is introduced. In Section 3, experimental results and configurations in electronic circuits for T-S fuzzy chaotic Lorenz and Chen-Lee systems are presented. In Section 4, conclusions are given.


## 2. Takagi-Sugeno Fuzzy Model

In system analysis and design, it is important to select an appropriate model representing a real system. As an expression model of a real plant, we use the fuzzy implications and the fuzzy reasoning method suggested by Takagi and Sugeno. The Takagi-Sugeno (T-S) fuzzy model is described by fuzzy IF-THEN rules which represent local linear input-output relations of a nonlinear system. The main feature of the T-S fuzzy model is to express the local dynamics of each fuzzy rule by a linear system model.The overall fuzzy model of the system is achieved by fuzzy blending of the linear system models. Consider a continuous-time nonlinear dynamic system as follows.Rule  i:(1)IFx1(t)isMi1⋯andxn(t)isMinTHENx˙(t)=Aix(t)+Biu(t),
where
(2)x(t)=[x1(t),x2(t),…,xn(t)]T,u(t)=[u1(t),u2(t),…,un(t)]T,i=1,2,…,r (r is the number of IF-THEN rules), Mij are fuzzy sets, and x(t)=Aix(t)+Biu(t) is the output from the ith IF-THEN rule. Given a pair of (x(t),u(t)), the final output of the fuzzy system is inferred as follows:
(3)x˙=∑i=1rωi(x(t)){Aix(t)+Biu(t)}∑i=1rωi(x(t)),
where
(4)ωi(x(t))=∏j=1nMij(x(t)),
for all t, andMij(x(t))is the grade of membership  xj(t)of inMij.The open-loop system of (3) is
(5)x˙=∑i=1rωi(x(t))Aix(t)∑i=1rωi(x(t)),
where it is assumed that
(6)∑i=1rωi(x(t))>0,ωi(x(t))≥0,i=1,2,…,r.By introducing hi(x(t))=ωi(x(t))/∑i=1rωi(x(t))instead ofωi(x(t)), (3) and (5) can be rewritten as
(7)x˙=∑i=1rhi(x(t)){Aix(t)+Biu(t)},x˙=∑i=1rhi(x(t))Aix(t).Note that(8)∑i=1rhi(x(t))=1,hi(x(t))≥0,i=1,2,…,r,for allt. hi(x(t))can be regarded as the normalized weight of the IF-THEN rules.


## 3. Implementation of T-S Fuzzy Systems on Electronic Circuit

This section shows the electronic circuit implementations of the T-S fuzzy model of classical Lorenz system and Chen-Lee system. The experimental results are going to be compared with the simulation results given by MATLAB.


### 3.1. Fuzzy Modeling of Lorenz System

For Lorenz system [21],
(9)x˙1=a(x2-x1),x˙2=cx1-x1x3-x2,x˙3=x1x2-bx3,
where a, b,c are the parameters. When a=10, b=8/3,  c=28, and initial states are (−0.1, 0.2, 0.3), the dynamic behavior is chaotic. Assume thatx1∈[-d,d]andd>0, then Lorenz system can be exactly represented by T-S fuzzy model as follows [43].Rule 1:(10)IFxisM1,THENX˙(t)=A1X(t).Rule 2:(11)IFxisM2,THENX˙(t)=A2X(t),
where
(12)X=[x1,x2,x3]T,A1=[-aa0c-1-d0d-b],A2=[-aa0c-1d0-d-b],M1(x)=12(1+x1d),M2(x)=12(1-x1d).Choosingd=30,M1 andM2  are fuzzy sets of Lorenz system. Here, we call (10) the first linear subsystem under the fuzzy rule and (11) the second linear subsystem under the fuzzy rule. The final output of the fuzzy Lorenz system is inferred as follows:
(13)X˙(t)=∑i=12hiAiX(t)=[h1h1h1]T[a(x2-x1)cx1-dx3-x2dx2-bx3]+[h2h2h2]T[a(x2-x1)cx1+dx3-x2-dx2-bx3],
where
(14)h1=M1M1+M2,h2=M2M1+M2.The configuration of electronic circuit in T-S fuzzy chaotic Lorenz system is shown in Figure1 and the chaotic behaviors in circuit and MATLAB are shown in Figures 2 and 3. It can be found out that the experimental result in our circuit is actually effective. It means that the nonlinear chaotic systems can be represented by real operations in electronic circuits, not just existing in simulation results.Figure 1
The fuzzy electronic circuit for chaotic Lorenz system.Projection of phase portraits outputs in fuzzy electronic circuit for the Lorenz system.
(a)
(b)Projection of phase portraits in MATLAB for fuzzy chaotic Lorenz system.
(a)
(b)


### 3.2. Fuzzy Modeling of Chen-Lee System

For Chen-Lee system,(15)y˙1=-y2y3+a1y1,y˙2=y1y3+b1y2,y˙3=y1y2/3+c1y3,
where a1, b1, and c are the parameters. When a1=5, b1=-10, c1=-38, and initial states are (0.2,0.2,0.2), the dynamic behavior is chaotic. Assume thaty1∈[-d1,d1], y2∈[-e1,e1],  andd1>0,e1>0, then Chen-Lee system can be exactly represented by T-S fuzzy model as follows.Rule 1: IFy1 isP1and IF y2isQ1, THEN
(16)Y˙(t)=B1Y(t).Rule 2: IFy1 isP1and IF y2isQ2, THEN
(17)Y˙(t)=B2Y(t).Rule 3: IFy1 isP2  and IF  y2isQ1, THEN
(18)Y˙(t)=B3Y(t).Rule 4: IFy1 isP2  and IF y2 isQ2, THEN
(19)Y˙(t)=B4Y(t),
where
(20)Y=[y1,y2,y3]T,B1=[a0-e0bd013dc],B2=[a0e0bd013dc],B1=[a0-e0b-d0-13dc],B2=[a0e0b-d0-13dc],P1(y)=12(1+y1d1),P2(y)=12(1-y1d1),Q1(y)=12(1+y2e1),Q2(y)=12(1-y2e1).Choosed1=40and e1=30.  N1 andN2are fuzzy sets of Chen-Lee system. Here, we call (16) the first linear subsystem under the fuzzy rule and (17) the second linear subsystem under the fuzzy rule. The final output of the fuzzy Chen-Lee system is inferred as follows:(21)Z˙(t)=∑i=12IiBiZ(t)=[I1I1I1]T[-e1y3+a1y1d1y3+b1y2d1y2/3+c1y3]+[I2I2I2]T[e1y3+a1y1d1y3+b1y2d1y2/3+c1y3]+[I3I3I3]T[-e1y3+a1y1-d1y3+b1y2-d1y2/3+c1y3]+[I4I4I4]T[e1y3+a1y1-d1y3+b1y2-d1y2/3+c1y3],where(22)I1=P1×Q1P1+P2+Q1+Q2,I2=P1×Q2P1+P2+Q1+Q2,I3=P2×Q1P1+P2+Q1+Q2,I4=P2×Q2P1+P2+Q1+Q2.The configuration of electronic circuit in T-S fuzzy chaotic Chen-Lee system is shown in Figure4 and the chaotic behaviors in circuit and MATLAB are shown in Figures 5 and 6. This experimental result in T-S fuzzy chaotic Chen-Lee system is exactly effective as well.Figure 4
The fuzzy electronic circuit for chaotic Chen-Lee system.Projection of phase portraits outputs in fuzzy electronic circuit for Chen-Lee system.
(a)
(b)
(c)Projection of phase portraits in MATLAB for fuzzy chaotic Chen-Lee system.
(a)
(b)
(c)Two illustrations given in Sections3.1 and 3.2 all show the agreement between our experimental and MATLAB simulation results. It means the T-S fuzzy model would no longer be just a mathematical tool, it can be applied to electronic circuits for various kinds of applications in practice.


## 3.1. Fuzzy Modeling of Lorenz System

For Lorenz system [21],
(9)x˙1=a(x2-x1),x˙2=cx1-x1x3-x2,x˙3=x1x2-bx3,
where a, b,c are the parameters. When a=10, b=8/3,  c=28, and initial states are (−0.1, 0.2, 0.3), the dynamic behavior is chaotic. Assume thatx1∈[-d,d]andd>0, then Lorenz system can be exactly represented by T-S fuzzy model as follows [43].Rule 1:(10)IFxisM1,THENX˙(t)=A1X(t).Rule 2:(11)IFxisM2,THENX˙(t)=A2X(t),
where
(12)X=[x1,x2,x3]T,A1=[-aa0c-1-d0d-b],A2=[-aa0c-1d0-d-b],M1(x)=12(1+x1d),M2(x)=12(1-x1d).Choosingd=30,M1 andM2  are fuzzy sets of Lorenz system. Here, we call (10) the first linear subsystem under the fuzzy rule and (11) the second linear subsystem under the fuzzy rule. The final output of the fuzzy Lorenz system is inferred as follows:
(13)X˙(t)=∑i=12hiAiX(t)=[h1h1h1]T[a(x2-x1)cx1-dx3-x2dx2-bx3]+[h2h2h2]T[a(x2-x1)cx1+dx3-x2-dx2-bx3],
where
(14)h1=M1M1+M2,h2=M2M1+M2.The configuration of electronic circuit in T-S fuzzy chaotic Lorenz system is shown in Figure1 and the chaotic behaviors in circuit and MATLAB are shown in Figures 2 and 3. It can be found out that the experimental result in our circuit is actually effective. It means that the nonlinear chaotic systems can be represented by real operations in electronic circuits, not just existing in simulation results.Figure 1
The fuzzy electronic circuit for chaotic Lorenz system.Projection of phase portraits outputs in fuzzy electronic circuit for the Lorenz system.
(a)
(b)Projection of phase portraits in MATLAB for fuzzy chaotic Lorenz system.
(a)
(b)


## 3.2. Fuzzy Modeling of Chen-Lee System

For Chen-Lee system,(15)y˙1=-y2y3+a1y1,y˙2=y1y3+b1y2,y˙3=y1y2/3+c1y3,
where a1, b1, and c are the parameters. When a1=5, b1=-10, c1=-38, and initial states are (0.2,0.2,0.2), the dynamic behavior is chaotic. Assume thaty1∈[-d1,d1], y2∈[-e1,e1],  andd1>0,e1>0, then Chen-Lee system can be exactly represented by T-S fuzzy model as follows.Rule 1: IFy1 isP1and IF y2isQ1, THEN
(16)Y˙(t)=B1Y(t).Rule 2: IFy1 isP1and IF y2isQ2, THEN
(17)Y˙(t)=B2Y(t).Rule 3: IFy1 isP2  and IF  y2isQ1, THEN
(18)Y˙(t)=B3Y(t).Rule 4: IFy1 isP2  and IF y2 isQ2, THEN
(19)Y˙(t)=B4Y(t),
where
(20)Y=[y1,y2,y3]T,B1=[a0-e0bd013dc],B2=[a0e0bd013dc],B1=[a0-e0b-d0-13dc],B2=[a0e0b-d0-13dc],P1(y)=12(1+y1d1),P2(y)=12(1-y1d1),Q1(y)=12(1+y2e1),Q2(y)=12(1-y2e1).Choosed1=40and e1=30.  N1 andN2are fuzzy sets of Chen-Lee system. Here, we call (16) the first linear subsystem under the fuzzy rule and (17) the second linear subsystem under the fuzzy rule. The final output of the fuzzy Chen-Lee system is inferred as follows:(21)Z˙(t)=∑i=12IiBiZ(t)=[I1I1I1]T[-e1y3+a1y1d1y3+b1y2d1y2/3+c1y3]+[I2I2I2]T[e1y3+a1y1d1y3+b1y2d1y2/3+c1y3]+[I3I3I3]T[-e1y3+a1y1-d1y3+b1y2-d1y2/3+c1y3]+[I4I4I4]T[e1y3+a1y1-d1y3+b1y2-d1y2/3+c1y3],where(22)I1=P1×Q1P1+P2+Q1+Q2,I2=P1×Q2P1+P2+Q1+Q2,I3=P2×Q1P1+P2+Q1+Q2,I4=P2×Q2P1+P2+Q1+Q2.The configuration of electronic circuit in T-S fuzzy chaotic Chen-Lee system is shown in Figure4 and the chaotic behaviors in circuit and MATLAB are shown in Figures 5 and 6. This experimental result in T-S fuzzy chaotic Chen-Lee system is exactly effective as well.Figure 4
The fuzzy electronic circuit for chaotic Chen-Lee system.Projection of phase portraits outputs in fuzzy electronic circuit for Chen-Lee system.
(a)
(b)
(c)Projection of phase portraits in MATLAB for fuzzy chaotic Chen-Lee system.
(a)
(b)
(c)Two illustrations given in Sections3.1 and 3.2 all show the agreement between our experimental and MATLAB simulation results. It means the T-S fuzzy model would no longer be just a mathematical tool, it can be applied to electronic circuits for various kinds of applications in practice.


## 4. Conclusions

The implementations of Takagi-Sugeno (T-S) fuzzy chaotic systems on electronic circuits are proposed in this paper. We construct the powerful tool, Takagi-Sugeno (T-S) fuzzy model, on electronic circuit and show good agreement between computer simulations in MATLAB and experimental results in our circuits. Through our effort, the powerful Takagi-Sugeno (T-S) method is more than just a numerical strategy, it can be applied to electronic circuits for various kinds of applications in practice. Implementations of electronic circuits for Takagi-Sugeno (T-S) fuzzy chaotic systems are only the beginning for secure communication and other kinds of applications, this paper also creates both opportunities and challenges. Implementations of novel synchronization or control approaches on electronic circuits in nonlinear research filed would be definitely our future directions to achieve.



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