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A T-S Fuzzy Model-Based Adaptive Exponential Synchronization Method for Uncertain Delayed Chaotic Systems: An LMI Approach

Abstract

This paper proposes a new fuzzy adaptive exponential synchronization controller for uncertain time-delayed chaotic systems based on Takagi-Sugeno (T-S) fuzzy model. This synchronization controller is designed based on Lyapunov-Krasovskii stability theory, linear matrix inequality (LMI), and Jesen's inequality. An analytic expression of the controller with its adaptive laws of parameters is shown. The proposed controller can be obtained by solving the LMI problem. A numerical example for time-delayed Lorenz system is presented to demonstrate the validity of the proposed method.

1. Introduction

Chaos synchronization is an important subject both theoretically and practically, for applications requiring oscillations out of chaos, machine and building structural stability analysis, chaos generators design and so on. Chaos synchronization, first described by Fujisaka and Yamada [1] in 1983, did not received great attention until 1990 [2]. From then on, chaos synchronization has been developed extensively due to its various applications [3]. During the last decade, several techniques for handling chaos synchronization have been developed, such as variable structure control [4], OGY method [5], observer-based control [6], active control [7], backstepping design technique [8], approach [9], and passivity based method [10].

Time delay inevitably appears in many physical systems such as aircraft, chemical, and biological systems. Unlike ordinary differential equations, time delayed systems are infinite dimensional in nature and time-delay is, in many cases, a source of instability. The stability issue and the performance of time delayed systems are, therefore, both of theoretical and practical importance. Since Mackey and Glass [11] first found chaos in time delayed system, there has been increasing interest in time delayed chaotic systems [12, 13]. The synchronization problem for time delayed chaotic systems is also investigated by several researchers [1420].

In recent years, fuzzy logic methodology has been proven effective in dealing with complex nonlinear systems containing certainties that are otherwise difficult to model. Among various kinds of fuzzy methods, Takagi-Sugeno (T-S) fuzzy model provides a successful method to describe certain complex nonlinear systems using some local linear subsystems [21, 22]. In [23], a fuzzy feedback control method was proposed for chaotic synchronization and chaotic model following control. The authors in [24, 25] proposed fuzzy observer-based chaotic synchronization and secure communication. In [26, 27], fuzzy adaptive synchronization methods for chaotic systems with unknown parameters were proposed. In spite of these advances in T-S fuzzy model-based chaos control and synchronization, most works were restricted to chaotic systems without time-delay. Due to finite signal transmission times, switching speeds and memory effects, time delayed systems are ubiquitous in nature, technology, and society [28, 29]. Time delayed chaotic systems are also interesting because the dimension of their chaotic dynamics can be increased by increasing the delay time sufficiently [30]. For this reason, the time delayed chaotic system has been suggested as a good candidate for secure communication [31]. The dimension of solution space of time delayed chaotic systems is infinite and so more than one positive Lyapunov exponents could be produced just by some low-dimension delayed chaotic systems. Therefore, communication system with a higher security level can be designed by means of time delayed chaotic systems. In addition, the time delayed system can be considered as a special case of spatiotemporal system [32]. From the above point of view, we can see that the study of fuzzy synchronization of time delayed chaotic systems is of high practical importance. To the best of our knowledge, however, for the fuzzy synchronization problem of time delayed chaotic systems, there is no result in the literature so far, which still remains open and challenging. This situation motivates our present investigation.

Motivated by the above discussions, the aim of this paper is to investigate the fuzzy adaptive exponential synchronization problem for time delayed chaotic systems with unknown parameters. T-S fuzzy model is adopted for the modeling of time delayed chaotic drive and response systems. Based on this fuzzy model, a new fuzzy synchronization controller is designed and an analytic expression of the controller with its adaptive laws of parameters is shown. By the proposed scheme, the closed-loop error system is adaptively exponentially synchronized. By virtue of Lyapunov-Krasovskii stability theory, linear matrix inequality (LMI) approach, and Jesen's inequality, an existence criterion for the proposed controller is represented in terms of the LMI, that can be readily checked by using some standard numerical packages [33].

This paper is organized as follows. In Section 2, we formulate the problem. In Section 3, a fuzzy adaptive exponential synchronization controller is proposed for time delayed chaotic systems with unknown parameters. In Section 4, an application example for time delayed Lorenz system is given, and finally, conclusions are presented in Section 5.

2. Problem Formulation

Consider a class of uncertain time delayed chaotic systems described by the following.

(2.1)

where is the state vector, is the time-delay of the chaotic system (2.1), and are known constant matrices, denotes a bias term which is generated by the fuzzy modeling procedure, and are activation function matrices, and represent the uncertain constant parameter vectors, is the premise variable, is the fuzzy set that is characterized by membership function, is the number of the IF-THEN rules, and is the number of the premise variables.

Using a standard fuzzy inference method (using a singleton fuzzifier, product fuzzy inference, and weighted average defuzzifier), the system (2.1) is inferred as follows:

(2.2)

where , , is the membership function of the system with respect to the fuzzy rule . can be regarded as the normalized weight of each IF-THEN rule and it satisfies

(2.3)

The system (2.2) is considered as a drive system. The synchronization problem of system (2.2) is considered by using the drive-response configuration. According to the drive-response concept, the controlled fuzzy response system is described by the following rules.

(2.4)

where is the state vector of the response system and is the control input. The fuzzy response system can be inferred as

(2.5)

Define the synchronization error . Then we obtain the synchronization error system

(2.6)

Throughout this paper, we define that and are the estimate values of and , respectively.

Definition 2.1 (Adaptive exponential synchronization).

With nonzero initial conditions, the error system (2.6) is adaptively exponentially synchronized if the synchronization error satisfies

(2.7)

where and are positive constants, under the control with the adaptive laws and , .

The purpose of this paper is to design the controller with the adaptive laws and , guaranteeing the adaptive exponential synchronization for time delayed chaotic systems with unknown parameters.

3. An LMI-Based Fuzzy Adaptive Exponential Synchronization

In this section, we present the LMI problem for achieving the fuzzy adaptive exponential synchronization of time delayed chaotic systems with unknown parameters.

Theorem 3.1.

If there exist , , , , , and such that

(3.1)

for , where is an enough small real number properly selected, then the fuzzy adaptive exponential synchronization is achieved under the control

(3.2)

and the adaptive laws

(3.3)

where and are positive definite matrices for design.

Proof.

The fuzzy adaptive exponential synchronization controller can be constructed via the parallel distributed compensation. The controller is described by the following rules.

(3.4)

where is the gain matrix of the controller for the fuzzy rule . The fuzzy controller can be inferred as

(3.5)

The closed-loop error system with the control input (3.5) can be written as

(3.6)

where and . Consider the following Lyapunov-Krasovskii functional:

(3.7)

The time derivative of along the trajectory of (3.6) is

(3.8)

Using the Jesen's inequality [34], we have

(3.9)

Finally, using (3.9), the time derivative of can be obtained as

(3.10)

where

(3.11)

If the adaptive laws (3.3) are used and the following matrix inequality is satisfied:

(3.12)

for , then we have

(3.13)

That is, for all . Thus, it implies that for any . In addition, from (3.7), one has

(3.14)

Also, we have

(3.15)

where is the minimum eigenvalue of the matrix . It follows immediately from (3.14) and (3.15) that

(3.16)

If we let

(3.17)

we obtain (2.7). If we let , (3.12) is equivalently changed into the LMI (3.1), then the gain matrix of the control input is given by . This completes the proof.

Remark 3.2.

Various efficient convex optimization algorithms can be used to check whether the LMI (3.1) is feasible. In this paper, in order to solve the LMI, we utilize MATLAB LMI Control Toolbox [35], which implements state-of- the-art interior-point algorithms.

4. Numerical Example

Consider the following time delayed Lorenz system [36]:

(4.1)

The parameter is assumed unknown. By defining two fuzzy sets, we can obtain the following fuzzy drive system that exactly represents the nonlinear equation of the time delayed Lorenz system under the assumption that with :

(4.2)

where

(4.3)

The membership functions are

(4.4)

For the numerical simulation, we use parameters , , and . Applying Theorem 3.1 to the fuzzy system (4.2) yields

(4.5)

Figure 1 shows state trajectories when the initial conditions are given by , , and . From Figure 1, it can be seen that drive and response systems are indeed achieving chaos synchronization. Figure 2 plots the time responses of synchronization errors. The estimate of the uncertain parameter is illustrated at Figure 3, which shows that the estimate approaches rapidly to target value . Simulation results reveal that the response system controlled using the proposed synchronization method performs well. The effectiveness and accuracy of the proposed method is demonstrated.

Figure 1
figure 1

State trajectories.

Figure 2
figure 2

Synchronization errors.

Figure 3
figure 3

The estimate value of parameter .

5. Conclusion

In this paper, a new fuzzy adaptive exponential synchronization scheme, which consists of time delayed fuzzy drive and response systems, is proposed for time delayed chaotic systems with unknown parameters. Based on Lyapunov-Krasovskii stability theory and LMI formulation, the proposed scheme can guarantee the adaptive exponential synchronization. The synchronization problem for the time delayed Lorenz system is given to illustrate the effectiveness of the proposed scheme. Finally, the proposed synchronization method has the advantage that it can be effectively used to adaptive exponential control and synchronization of other uncertain time delayed nonlinear systems described by a T-S fuzzy model.

References

  1. Fujisaka H, Yamada T: Stability theory of synchronized motion in coupled-oscillator systems. Progress of Theoretical Physics 1983, 69(1):32–47. 10.1143/PTP.69.32

    MathSciNet  Article  MATH  Google Scholar 

  2. Pecora LM, Carroll TL: Synchronization in chaotic systems. Physical Review Letters 1990, 64(8):821–824. 10.1103/PhysRevLett.64.821

    MathSciNet  Article  MATH  Google Scholar 

  3. Chen G, Dong X: From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. Volume 24. World Scientific, River Edge, NJ, USA; 1998:xxii+753.

    Google Scholar 

  4. Wang C-C, Su J-P: A new adaptive variable structure control for chaotic synchronization and secure communication. Chaos, Solitons and Fractals 2004, 20(5):967–977. 10.1016/j.chaos.2003.10.026

    MathSciNet  Article  MATH  Google Scholar 

  5. Ott E, Grebogi C, Yorke JA: Controlling chaos. Physical Review Letters 1990, 64(11):1196–1199. 10.1103/PhysRevLett.64.1196

    MathSciNet  Article  MATH  Google Scholar 

  6. Yang X-S, Chen G: Some observer-based criteria for discrete-time generalized chaos synchronization. Chaos, Solitons and Fractals 2002, 13(6):1303–1308. 10.1016/S0960-0779(01)00127-8

    MathSciNet  Article  MATH  Google Scholar 

  7. Bai E-W, Lonngren KE: Synchronization of two Lorenz systems using active control. Chaos, Solitons and Fractals 1997, 8(1):51–58. 10.1016/S0960-0779(96)00060-4

    Article  MATH  Google Scholar 

  8. Hu J, Chen S, Chen L: Adaptive control for anti-synchronization of Chua's chaotic system. Physics Letters A 2005, 339(6):455–460. 10.1016/j.physleta.2005.04.002

    Article  MATH  Google Scholar 

  9. Ahn CK: An approach to anti-synchronization for chaotic systems. Physics Letters A 2009, 373(20):1729–1733. 10.1016/j.physleta.2009.03.032

    MathSciNet  Article  MATH  Google Scholar 

  10. Ahn CK: A passivity approach to synchronization for time-delayed chaotic systems. Modern Physics Letters B 2009, 23(29):3531–3541. 10.1142/S0217984909021594

    MathSciNet  Article  MATH  Google Scholar 

  11. Mackey MC, Glass L: Oscillation and chaos in physiological control systems. Science 1977, 197(4300):287–289. 10.1126/science.267326

    Article  Google Scholar 

  12. Farmer JD: Chaotic attractors of an infinite-dimensional dynamical system. Physica D 1981/82, 4(3):366–393.

    MathSciNet  Article  MATH  Google Scholar 

  13. Lu H: Chaotic attractors in delayed neural networks. Physics Letters A 2002, 298(2–3):109–116. 10.1016/S0375-9601(02)00538-8

    Article  MATH  Google Scholar 

  14. Tian Y-C, Gao F: Adaptive control of chaotic continuous-time systems with delay. Physica D 1998, 117(1–4):1–12.

    MathSciNet  Article  MATH  Google Scholar 

  15. Park JH, Kwon OM: Guaranteed cost control of time-delay chaotic systems. Chaos, Solitons and Fractals 2006, 27(4):1011–1018. 10.1016/j.chaos.2005.04.076

    MathSciNet  Article  MATH  Google Scholar 

  16. Chen B, Liu X, Tong S: Guaranteed cost control of time-delay chaotic systems via memoryless state feedback. Chaos, Solitons and Fractals 2007, 34(5):1683–1688. 10.1016/j.chaos.2006.05.009

    MathSciNet  Article  MATH  Google Scholar 

  17. Guan X, Feng G, Chen C, Chen G: A full delayed feedback controller design method for time-delay chaotic systems. Physica D 2007, 227(1):36–42. 10.1016/j.physd.2006.12.009

    MathSciNet  Article  MATH  Google Scholar 

  18. Chen M, Chen W-H: Robust adaptive neural network synchronization controller design for a class of time delay uncertain chaotic systems. Chaos, Solitons and Fractals 2009, 41(5):2716–2724. 10.1016/j.chaos.2008.10.003

    Article  MATH  Google Scholar 

  19. Zhu W, Xu D, Huang Y: Global impulsive exponential synchronization of time-delayed coupled chaotic systems. Chaos, Solitons and Fractals 2008, 35(5):904–912. 10.1016/j.chaos.2006.05.078

    Article  MATH  Google Scholar 

  20. Liu X: Impulsive synchronization of chaotic systems subject to time delay. Nonlinear Analysis: Theory, Methods and Applications 2009, 71(12):e1320-e1327. 10.1016/j.na.2009.01.162

    Article  MathSciNet  MATH  Google Scholar 

  21. Takagi T, Sugeno M: Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man and Cybernetics 1985, 15(1):116–132.

    Article  MATH  Google Scholar 

  22. Tanaka K, Sugeno M: Stability analysis and design of fuzzy control systems. Fuzzy Sets and Systems 1992, 45(2):135–156. 10.1016/0165-0114(92)90113-I

    MathSciNet  Article  MATH  Google Scholar 

  23. Tanaka K, Ikeda T, Wang HO: A unified approach to controlling chaos via an LMI-based fuzzy control system design. IEEE Transactions on Circuits and Systems. I 1998, 45(10):1021–1040. 10.1109/81.728857

    MathSciNet  Article  MATH  Google Scholar 

  24. Lian K-Y, Chiu C-S, Chiang T-S, Liu P: LMI-based fuzzy chaotic synchronization and communications. IEEE Transactions on Fuzzy Systems 2001, 9(4):539–553. 10.1109/91.940967

    Article  Google Scholar 

  25. Lian K-Y, Chiang T-S, Chiu C-S, Liu P: Synthesis of fuzzy model-based designs to synchronization and secure communications for chaotic systems. IEEE Transactions on Systems, Man, and Cybernetics Part B 2001, 31(1):66–83.

    Article  Google Scholar 

  26. Kim J-H, Park C-W, Kim E, Park M: Adaptive synchronization of T-S fuzzy chaotic systems with unknown parameters. Chaos, Solitons and Fractals 2005, 24(5):1353–1361. 10.1016/j.chaos.2004.09.082

    MathSciNet  Article  MATH  Google Scholar 

  27. Kim J-H, Hyun C-H, Kim E, Park M: Adaptive synchronization of uncertain chaotic systems based on T-S fuzzy model. IEEE Transactions on Fuzzy Systems 2007, 15(3):359–369.

    Article  Google Scholar 

  28. Traub RD, Miles R, Wong RKS: Model of the origin of rhythmic population oscillations in the hippocampal slice. Science 1989, 243(4896):1319–1325. 10.1126/science.2646715

    Article  Google Scholar 

  29. Foss J, Longtin A, Mensour B, Milton J: Multistability and delayed recurrent loops. Physical Review Letters 1996, 76(4):708–711. 10.1103/PhysRevLett.76.708

    Article  Google Scholar 

  30. Pyragas K: Synchronization of coupled time-delay systems: analytical estimations. Physical Review E 1998, 58(3):3067–3071. 10.1103/PhysRevE.58.3067

    MathSciNet  Article  Google Scholar 

  31. Pyragas K: Transmission of signals via synchronization of chaotic time-delay systems. International Journal of Bifurcation and Chaos 1998, 8: 1839–1842. 10.1142/S0218127498001558

    Article  Google Scholar 

  32. Masoller C: Spatiotemporal dynamics in the coherence collapsed regime of semiconductor lasers with optical feedback. Chaos 1997, 7(3):455–462. 10.1063/1.166253

    MathSciNet  Article  MATH  Google Scholar 

  33. Boyd S, El Ghaoui L, Feron E, Balakrishnan V: Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics. Volume 15. SIAM, Philadelphia, Pa, USA; 1994:xii+193.

    Book  MATH  Google Scholar 

  34. Noldus E: Stabilization of a class of distributional convolution equations. International Journal of Control 1985, 41(4):947–960. 10.1080/0020718508961174

    MathSciNet  Article  MATH  Google Scholar 

  35. Gahinet P, Nemirovski A, Laub AJ, Chilali M: LMI Control Toolbox. The Mathworks; 1995.

    Google Scholar 

  36. Li L, Peng H, Yang Y, Wang X: On the chaotic synchronization of Lorenz systems with time-varying lags. Chaos, Solitons and Fractals 2009, 41(2):783–794. 10.1016/j.chaos.2008.03.014

    Article  MATH  Google Scholar 

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Acknowledgment

This work was supported by the grant of the Korean Ministry of Education, Science and Technology (The Regional Core Research Program/Center for Healthcare Technology Development).

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Ahn, C. A T-S Fuzzy Model-Based Adaptive Exponential Synchronization Method for Uncertain Delayed Chaotic Systems: An LMI Approach. J Inequal Appl 2010, 168962 (2010). https://doi.org/10.1155/2010/168962

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Keywords

  • Chaotic System
  • Linear Matrix Inequality
  • Fuzzy Model
  • Time Delayed System
  • Synchronization Error