Chaotic Motions in the Real Fuzzy Electronic Circuits

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.

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 Section 2, 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: IF𝑥₁(𝑡)is𝑀_𝑖1⋯and𝑥_𝑛(𝑡)is𝑀_𝑖𝑛THEN𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡), (1) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇,𝑢(𝑡)=𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡)^𝑇, (2)𝑖=1,2,…,𝑟 (𝑟 is the number of IF-THEN rules), 𝑀_𝑖𝑗 are fuzzy sets, and 𝑥(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(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: ̇∑𝑥=^𝑟𝑖=1𝜔_𝑖(𝑥(𝑡))𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡)∑^𝑟_𝑖=1𝜔_𝑖(𝑥(𝑡,)) (3) where 𝜔_𝑖(𝑥(𝑡))=^𝑛_𝑗=1𝑀_𝑖𝑗(𝑥(𝑡)), (4) for all 𝑡, and𝑀_𝑖𝑗(𝑥(𝑡))is the grade of membership  𝑥_𝑗(𝑡)of in𝑀_𝑖𝑗.

The open-loop system of (3) is ̇∑𝑥=^𝑟𝑖=1𝜔_𝑖(𝑥(𝑡))𝐴_𝑖𝑥(𝑡)∑^𝑟_𝑖=1𝜔_𝑖(𝑥(𝑡)), (5) where it is assumed that ^𝑟_𝑖=1𝜔_𝑖(𝑥(𝑡))>0,𝜔_𝑖(𝑥(𝑡))≥0,𝑖=1,2,…,𝑟. (6)By introducing ℎ_𝑖(𝑥(𝑡))=𝜔_𝑖(𝑥(𝑡))/∑^𝑟_𝑖=1𝜔_𝑖(𝑥(𝑡))instead of𝜔_𝑖(𝑥(𝑡)), (3) and (5) can be rewritten as 𝑥̇=^𝑟_𝑖=1ℎ_𝑖(𝑥(𝑡))𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡,̇)𝑥=^𝑟_𝑖=1ℎ_𝑖(𝑥(𝑡))𝐴_𝑖𝑥(𝑡). (7)

Note that ^𝑟_𝑖=1ℎ_𝑖(𝑥(𝑡))=1,ℎ_𝑖(𝑥(𝑡))≥0,𝑖=1,2,…,𝑟, (8)

for all 𝑡. ℎ_𝑖(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], 𝑥̇₁=𝑎𝑥₂−𝑥₁,𝑥̇₂=𝑐𝑥₁−𝑥₁𝑥₃−𝑥₂,𝑥̇₃=𝑥₁𝑥₂−𝑏𝑥₃, (9) 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: IF𝑥is𝑀₁,THEN𝑋̇(𝑡)=𝐴₁𝑋(𝑡). (10)

Rule 2: IF𝑥is𝑀₂,THEN𝑋̇(𝑡)=𝐴₂𝑋(𝑡), (11) where 𝑋=[𝑥₁,𝑥₂,𝑥₃]^𝑇,𝐴₁=−𝑎𝑎0𝑐−1−𝑑0𝑑−𝑏,𝐴₂=,𝑀−𝑎𝑎0𝑐−1𝑑0−𝑑−𝑏₁(𝑥)=12𝑥1+¹𝑑,𝑀₂(𝑥)=12𝑥1−¹𝑑. (12)

Choosing  𝑑=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: 𝑋̇(𝑡)=²_𝑖=1ℎ_𝑖𝐴_𝑖𝑋(𝑡)=ℎ₁ℎ₁ℎ₁^𝑇𝑎𝑥₂−𝑥₁𝑐𝑥₁−𝑑𝑥₃−𝑥₂𝑑𝑥₂−𝑏𝑥₃+ℎ₂ℎ₂ℎ₂^𝑇𝑎𝑥₂−𝑥₁𝑐𝑥₁+𝑑𝑥₃−𝑥₂−𝑑𝑥₂−𝑏𝑥₃, (13) where ℎ₁=𝑀₁𝑀₁+𝑀₂,ℎ₂=𝑀₂𝑀₁+𝑀₂. (14)

The configuration of electronic circuit in T-S fuzzy chaotic Lorenz system is shown in Figure 1 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.

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Figure 2 

Projection of phase portraits outputs in fuzzy electronic circuit for the Lorenz system.

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Figure 3 

Projection of phase portraits in MATLAB for fuzzy chaotic Lorenz system.

3.2. Fuzzy Modeling of Chen-Lee System

For Chen-Lee system, 𝑦̇₁=−𝑦₂𝑦₃+𝑎₁𝑦₁,𝑦̇₂=𝑦₁𝑦₃+𝑏₁𝑦₂,𝑦̇₃=𝑦₁𝑦₂/3+𝑐₁𝑦₃, (15) 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: IF 𝑦₁ is𝑃₁and IF 𝑦₂is𝑄₁, THEN 𝑌̇(𝑡)=𝐵₁𝑌(𝑡). (16)

Rule 2: IF 𝑦₁ is𝑃₁and IF 𝑦₂is𝑄₂, THEN 𝑌̇(𝑡)=𝐵₂𝑌(𝑡). (17)

Rule 3: IF 𝑦₁ is𝑃₂  and IF  𝑦₂is𝑄₁, THEN 𝑌̇(𝑡)=𝐵₃𝑌(𝑡). (18)

Rule 4: IF 𝑦₁ is𝑃₂  and IF 𝑦₂ is𝑄₂, THEN 𝑌̇(𝑡)=𝐵₄𝑌(𝑡), (19) where 𝑌=𝑦₁,𝑦₂,𝑦₃^𝑇,𝐵₁=01𝑎0−𝑒0𝑏𝑑3𝑑𝑐,𝐵₂=01𝑎0𝑒0𝑏𝑑3,𝐵𝑑𝑐₁=1𝑎0−𝑒0𝑏−𝑑0−3𝑑𝑐,𝐵₂=1𝑎0𝑒0𝑏−𝑑0−3,𝑃𝑑𝑐₁𝑦=12𝑦1+¹𝑑₁,𝑃₂𝑦=12𝑦1−¹𝑑₁,𝑄₁𝑦=12𝑦1+²𝑒₁,𝑄₂𝑦=12𝑦1−²𝑒₁. (20)

Choose 𝑑₁=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:𝑍̇(𝑡)=²_𝑖=1𝐼_𝑖𝐵_𝑖𝑍(𝑡)=𝐼₁𝐼₁𝐼₁^𝑇−𝑒₁𝑦₃+𝑎₁𝑦₁𝑑₁𝑦₃+𝑏₁𝑦₂𝑑₁𝑦₂/3+𝑐₁𝑦₃+𝐼₂𝐼₂𝐼₂^𝑇𝑒₁𝑦₃+𝑎₁𝑦₁𝑑₁𝑦₃+𝑏₁𝑦₂𝑑₁𝑦₂/3+𝑐₁𝑦₃+𝐼₃𝐼₃𝐼₃^𝑇−𝑒₁𝑦₃+𝑎₁𝑦₁−𝑑₁𝑦₃+𝑏₁𝑦₂−𝑑₁𝑦₂/3+𝑐₁𝑦₃+𝐼₄𝐼₄𝐼₄^𝑇𝑒₁𝑦₃+𝑎₁𝑦₁−𝑑₁𝑦₃+𝑏₁𝑦₂−𝑑₁𝑦₂/3+𝑐₁𝑦₃, (21)

where 𝐼₁=𝑃₁×𝑄₁𝑃₁+𝑃₂+𝑄₁+𝑄₂,𝐼₂=𝑃₁×𝑄₂𝑃₁+𝑃₂+𝑄₁+𝑄₂,𝐼₃=𝑃₂×𝑄₁𝑃₁+𝑃₂+𝑄₁+𝑄₂,𝐼₄=𝑃₂×𝑄₂𝑃₁+𝑃₂+𝑄₁+𝑄₂. (22)

The configuration of electronic circuit in T-S fuzzy chaotic Chen-Lee system is shown in Figure 4 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.

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Figure 5 

Projection of phase portraits outputs in fuzzy electronic circuit for Chen-Lee system.

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Figure 6 

Projection of phase portraits in MATLAB for fuzzy chaotic Chen-Lee system.

Two illustrations given in Sections 3.1 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.

Acknowledgments

This work was supported in part by the UST-UCSD International Center of Excellence in Advanced Bioengineering sponsored by the Taiwan National Science Council I-RiCE Program under Grant no. NSC-101-2911-I-009-101. This work was also by the Aiming for the Top University Plan of National Chiao Tung University, the Ministry of Education, Taiwan, under Contract 101W963, and by the Army Research Laboratory and was accomplished under Cooperative Agreement no. W911NF-10-2-0022.