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.
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 [
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 [
Since the fuzzy set theory [
The layout of the rest of the paper is as follows. In Section
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:
The open-loop system of (
Note that
for all
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.
For Lorenz system [
Rule 1:
Rule 2:
Choosing
The configuration of electronic circuit in T-S fuzzy chaotic Lorenz system is shown in Figure
The fuzzy electronic circuit for chaotic Lorenz system.
Projection of phase portraits outputs in fuzzy electronic circuit for the Lorenz system.
Projection of phase portraits in MATLAB for fuzzy chaotic Lorenz system.
For Chen-Lee system,
Rule 1: IF
Rule 2: IF
Rule 3: IF
Rule 4: IF
Choose
where
The configuration of electronic circuit in T-S fuzzy chaotic Chen-Lee system is shown in Figure
The fuzzy electronic circuit for chaotic Chen-Lee system.
Projection of phase portraits outputs in fuzzy electronic circuit for Chen-Lee system.
Projection of phase portraits in MATLAB for fuzzy chaotic Chen-Lee system.
Two illustrations given in Sections
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.
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.