# Parameter Identification and Synchronization of Dynamical System by Introducing an Auxiliary Subsystem

- Haipeng Peng
^{1, 2, 3}, - Lixiang Li
^{1, 2, 3}Email author, - Fei Sun
^{1, 2, 3}, - Yixian Yang
^{1, 2, 3}and - Xiaowen Li
^{1}

**2010**:808403

**DOI: **10.1155/2010/808403

© Haipeng Peng et al. 2010

**Received: **23 December 2009

**Accepted: **29 May 2010

**Published: **24 June 2010

## Abstract

We propose a novel approach of parameter identification using the adaptive synchronized observer by introducing an auxiliary subsystem, and some sufficient conditions are given to guarantee the convergence of synchronization and parameter identification. We also demonstrate the mean convergence of synchronization and parameters identification under the influence of noise. Furthermore, in order to suppress the influence of noise, we complement a filter in the output. Numerical simulations on Lorenz and Chen systems are presented to demonstrate the effectiveness of the proposed approach.

## 1. Introduction

Since the pioneering work of Pecora and Carroll [1], chaos synchronization has become an active research subject due to its potential applications in physics, chemical reactions, biological networks, secure communication, control theory, and so forth [2–12]. An important application of synchronization is in adaptive parameter estimation methods where parameters in a model are adjusted dynamically in order to minimize the synchronization error [13–15]. To achieve system synchronization and parameter convergence, there are two general approaches based on the typical Lyapunov's direct method [2–9] or LaSalle's principle [10]. When adaptive synchronization methods are applied to identify the uncertain parameters, some restricted conditions on dynamical systems, such as persistent excitation (PE) condition [11, 15] or linear independence (LI) conditions [10], should be matched to guarantee that the estimated parameters converge to the true values [12].

In the following, we explore a novel method for parameter estimation by introducing an auxiliary subsystem in adaptive synchronized observer instead of Lyapunov's direct method and LaSalle's principle. It will be shown that through harnessing the auxiliary subsystem, parameters can be well estimated from a time series of dynamical systems based on adaptive synchronized observer. Moreover, noise plays an important role in parameter identification. However, little attention has been given to this point. Here we demonstrate the mean convergence of synchronization and parameters identification under the influence of noise. Furthermore, we implement a filter to recover the performance of parameter identification suppressing the influence of the noise.

## 2. Parameter Identification Method

where is the state vector, is the unique unknown parameter to be identified, and are the nonlinear functions of the state vector in the th equation.

where is a positive constant.

Lemma 2.1.

If is bounded and does not converge to zero as , then the state of system (2.2) is bounded and does not converge to zero, when .

Proof.

If is bounded, we can easily know that is bounded [16]. We suppose that the state of system (2.2) converges to zero, when . According to LaSalle principle, we have the invariant set , then ; therefore, from system (2.2), we get as . This contradicts the condition that does not converge to zero as . Therefore, the state does not converge to zero, when .

Based on observer theory, the following response system is designed to synchronize the state vector and identify the unknown parameters.

Theorem 2.2.

where are the observed state and estimated parameter of and , respectively, and and are positive constants.

Proof.

Obviously, as .

The solution of system (2.8) is . From the lemma, we know that does not converge to zero. According to Barbalat theorem, we have as ; correspondingly, as , that is, the system is asymptotically stable.

Now from the exponential convergence of in system (2.6) and asymptotical convergence of in system (2.8), we obtain that in system (2.7) are asymptotical convergent to zero.

Finally, from , , and being bounded, we conclude that are global asymptotical convergence.

The proof of Theorem 2.2 is completed.

Note.

When and is the offset, in this condition no matter is in stable, periodic, or chaotic state, we could use system (2.3) to estimate and synchronize the system (2.1).

Note.

In doing so, synchronization of the system and parameters estimation can be achieved.

## 3. Application of the Above-Mentioned Scheme

where the parameters , , and are unknown, and all the states are measurable. When , , , Lorenz system is chaotic.

where the parameters , , and are unknown, and all the states are measurable. When , , and , Chen system is chaotic.

From the simulation results of Lorenz and Chen system above, we can see that the unknown parameters could be identified. It indicates that the proposed parameter identifier in this paper could be used as an effective parameter estimator.

## 4. Parameter Identification in the Presence of Noise

where is the zero mean, bounded noise.

Theorem 4.1.

If the above lemma is hold and is independent to , and , using the synchronized observer (2.3), then for any set of initial conditions, and converge to zero asymptotically as , where and are mean values of and , respectively.

Proof.

So similarly we have , , and therefore, as .

It is clear to see from Figure 7(b) that unknown parameters , , and can be identified with high accuracy even in the presence of large random noise.

## 5. Conclusions

In this paper, we propose a novel approach of identifying parameters by the adaptive synchronized observer, and a filter in the output is introduced to suppress the influence of noise. In our method, Lyapunov's direct method and LaSalle's principle are not needed. Considerable simulations on Lorenz and Chen systems are employed to verify the effectiveness and feasibility of our approach.

## Declarations

### Acknowledgments

Thanks are presented for all the anonymous reviewers for their helpful advices. Professor Lixiang Li is supported by the National Natural Science Foundation of China (Grant no. 60805043), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (FANEDD) (Grant no. 200951), and the Program for New Century Excellent Talents in University of the Ministry of Education of China (Grant no. NCET-10-0239); Professor Yixian Yang is supported by the National Basic Research Program of China (973 Program) (Grant no. 2007CB310704) and the National Natural Science Foundation of China (Grant no. 60821001).

## Authors’ Affiliations

## References

- Pecora LM, Carroll TL:
**Synchronization in chaotic systems.***Physical Review Letters*1990,**64**(8):821-824. 10.1103/PhysRevLett.64.821MATHMathSciNetView ArticleGoogle Scholar - Parlitz U:
**Estimating model parameters from time series by autosynchronization.***Physical Review Letters*1996,**76**(8):1232-1235. 10.1103/PhysRevLett.76.1232View ArticleGoogle Scholar - Maybhate A, Amritkar RE:
**Dynamic algorithm for parameter estimation and its applications.***Physical Review E*2000,**61:**6461-6470. 10.1103/PhysRevE.61.6461View ArticleGoogle Scholar - Tao C, Zhang Y, Du G, Jiang JJ:
**Estimating model parameters by chaos synchronization.***Physical Review E*2004,**69:**-5.Google Scholar - Abarbanel HDI, Creveling DR, Jeanne JM:
**Estimation of parameters in nonlinear systems using balanced synchronization.***Physical Review E*2008,**77**(1):-14.MathSciNetView ArticleGoogle Scholar - Ghosh D, Banerjee S:
**Adaptive scheme for synchronization-based multiparameter estimation from a single chaotic time series and its applications.***Physical Review E*2008,**78:**-5.Google Scholar - Li L, Peng H, Wang X, Yang Y:
**Comment on two papers of chaotic synchronization.***Physics Letters A*2004,**333**(3-4):269-270. 10.1016/j.physleta.2004.10.039MATHView ArticleGoogle Scholar - Chen M, Kurths J:
**Chaos synchronization and parameter estimation from a scalar output signal.***Physical Review E*2007,**76:**-4.Google Scholar - Li L, Yang Y, Peng H:
**Comment on Adaptive Q-S (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks.***Chaos*2007,**17**(3):-2.View ArticleGoogle Scholar - Yu W, Chen G, Cao J,
*et al*.:**Parameter identification of dynamical systems from time series.***Physical Review E*2007,**75:**-4.Google Scholar - Slotine J, Li W:
*Applied Nonlinear Control*. Prentice-Hall, Upper Saddle River, NJ, USA; 1991.MATHGoogle Scholar - Sun F, Peng H, Luo Q, Li L, Yang Y:
**Parameter identification and projective synchronization between different chaotic systems.***Chaos*2009.,**19**(2):Article ID 023109 - Yu D, Wu A:
**Comment on "Estimating Model Parameters from Time Series by Autosynchronization".***Physical Review Letters*2005,**94:**-1.Google Scholar - Wang S, Luo H, Yue C, Liao X:
**Parameter identification of chaos system based on unknown parameter observer.***Physics Letters A*2008,**372**(15):2603-2607. 10.1016/j.physleta.2007.12.025MATHMathSciNetView ArticleGoogle Scholar - Chen M, Min W:
**Unknown input observer based chaotic secure communication.***Physics Letters A*2008,**372**(10):1595-1600. 10.1016/j.physleta.2007.10.012MATHMathSciNetView ArticleGoogle Scholar - Liao XX:
*Theory and Application of Stability for Dynamical Systems*. National Defense Industry Press, Beijingm, China; 2000:xx+542.MATHGoogle Scholar - Lorenz E:
**Deterministic non-periodic flows.***Journal of the Atmospheric Sciences*1963,**20**(1):130-141. 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2MathSciNetView ArticleGoogle Scholar - Chen G, Ueta T:
**Yet another chaotic attractor.***International Journal of Bifurcation and Chaos*1999,**9**(7):1465-1466. 10.1142/S0218127499001024MATHMathSciNetView ArticleGoogle Scholar - Lü J, Chen G, Cheng D:
**A new chaotic system and beyond: the generalized Lorenz-like system.***International Journal of Bifurcation and Chaos*2004,**14**(5):1507-1537. 10.1142/S021812740401014XMATHMathSciNetView ArticleGoogle Scholar - Liu C, Liu T, Liu L, Liu K:
**A new chaotic attractor.***Chaos, Solitons and Fractals*2004,**22**(5):1031-1038. 10.1016/j.chaos.2004.02.060MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.