基于不同混沌时延系统同步的混沌遥测系统参数估计

魏恒东

（中国西南电子技术研究所，成都 610036）

基于不同混沌时延系统同步的混沌遥测系统参数估计

魏恒东

（中国西南电子技术研究所，成都 610036）

针对混沌遥测系统的参数估计问题，研究了基于混沌同步的不同混沌时延系统的参数估计方法。构造了利用不同混沌系统进行参数估计的自适应方案，利用Krasovskii-Lyapunov理论推导出两个不同混沌时延系统同步和参数估计的充分条件。所提方案应用到混沌遥测系统参数估计可以有效估计系统参数。非合作者可用所构造方法利用不同混沌系统获取混沌遥测系统的有用信息。仿真分析验证了方案的有效性。

时延混沌系统；混沌遥测；参数估计；混沌同步

1 Introduction

Chaotic telemetry is a new technology in aircraft telemetry system.It completes telemetry system parameters transmission and measurement through utilizing chaotic sequence substitute traditional pseudo code sequence［1-2］.The basic principle of chaotic telemetry is that the transmitter uses high-speed chaotic sequence to modulate the transmitted message to spread its spectrum，then，the receiver uses synchronization circuit to generate same chaotic sequence to correlate with the received signal to recover telemetry data and estimate the telemetry parameters.The usually used chaotic telemetry modes are chaotic spread spectrum［3］and chaotic modulation［4］.As a non-cooperator，how to get useful information from intercepted chaotic telemetry signal is an important task. However，the report about non-cooperate chaotic telemetry signal analysis is too few.

One interesting application of chaos synchronization is estimation of the system parameters with the available time series from the driving system.During the past decade，an approach of using synchronization as a param-eter estimation method called auto-synchronization［5-6］has attracted much attentions［7-13］.This process is governed by additional update rules for the parameters that are controlled by the synchronization error.

The parameter estimation of chaotic telemetry system and getting useful information hidden in intercepted signal by chaotic synchronization is an important issue in chaotic telemetry signal analysis.However，the precondition of most of the parameters estimation methods is that the two systems which synchronize with each other should have the same structure，which，however，is not the case in practice.For instance，in chaotic telemetry signal reconnaissance，the non-cooperative detector doesn′t know the system structure.That is to say，the detector has to use a different system to estimate the parameters modulated by information.However，the synchronization and parameter estimation of different chaotic systems is far less understood because of their different structures and parameters.So，it is not only of theoretical interest but also of practical value to investigate the parameter estimation between two different chaotic systems.The immediate configuration for synchronization between two different systems is generalized synchronization.The basic idea of generalized synchronization is that there exists a transformation which is able to map asymptotically the trajectories of the driving attractor into the response attractor.So，if the generalized synchronization is used to parameter estimation，the first problem we must face is the estimation of the“transformation”. However，this transformation is so complex that we can not get any analytical results［14-15］.Therefore，designing a simple scheme to estimate parameters between two different chaotic systems is a significant issue.

In this paper，synchronization based parameters estimation between two different chaotic systems is proposed and applied to chaotic telemetry signal analysis.A control signal is designed based on Krasovskii-Lyapunov theory to make these two systems synchronize with each other.An analytical approach to the sufficient condition for synchronization and parameter estimation is also established. Chaotic telemetry is an active branch of TT＆C communication，because of the noise-like characteristic of chaotic signal.Here，as an eavesdropper，a different system is used to decode the message transferred between the cooperators.Firstly，a control signal is designed for parameter estimation between two different chaotic systems based on Krasovskii-Lyapunov theory.The sufficient condition for synchronization and parameter estimation is established analytically.Secondly，the method is verified by numerical examples.Then chaotic telemetry signal reconnaissance scheme is given，too.Finally，results are summarized.

2 Parameters Estimation Scheme

Consider the following time-delay system，

where p＝（p1，p2，…，pm）∈Rmare unknown parameters to be estimated；xτ＝x（t-τ）.This system is used to drive a response system，

where a，b are positive parameters；τis the delay time. The response system，called prototype model，is proposed as a chaos generator and is studied in Reference［16］. Figure 1（a）shows the chaotic attractor of this system，where a＝1.7，b＝1 andτ＝1.Our aim is to design a control signal u and parameter update rules gi（i＝1，2，…，m）such that the system（2）synchronize with system（1）and the unknown parameters are estimated correctly.

Fig.1 Phase portrait of the prototype model and Ikeda system图1 Prototype模型和Ikeda系统的相图

Introduce a control signal u to system（2），which can be written as

To determine the control signal and parameter update rules，we investigate the dynamics of the differencesΔ＝x-y.Then the dynamics of the error are

where giare the update rules to be determined to ensure the parameter estimation；qiare the estimation of pi；k is the coupled strength.

According to Krasovskii-Lyapunov theory，we define a continuous positive-definite Lyapunov function［14，17］

whereξ＞0，μ＞0 andδi＞0 for all i is the update rate.Then

In view of Krasovskii-Lyapunov theory，V˙ will be negative if and only ifμ＝a/2 and

On the other hand，

Then，we get the sufficient condition for synchronization and parameter estimation as

where sup（f（x））is the supremum of f（x）.

3 Numerical Simulation

To illustrate the suggested parameter estimation method，we first consider the Ikeda system with one unknown parameter，

where f（x）＝sin（x）.We choose the parameter values as c＝1，p＝4 andτ1＝2.Figure 1（b）shows the chaotic attractor of this system.Here p＝4 is the only estimated parameter.

Another chaotic system，referred in last section，whose structure is different from Ikeda system，

If we chooseδ＝10，from the condition（12），one can obtain the sufficient condition for synchronization and parameter estimation as k＞1.2.Figure 2 shows the synchronization error and parameter estimation of these two different systems when k＝2.5.Although，these two systems have different structure，it is easy to see from the figure that the synchronization error diminishes gradually. At last，the parameter is estimated correctly as shown in Fig.2（a）.Noting that，the update rateδcontrols the convergence speed of the parameter estimation.Improper chosen of update rate will lead to the too slow converge，serious fluctuate，or divergence.

Fig.2 Parameter estimation and synchronization error between Ikeda system（with one unknown parameter）and prototype model图2 Ikeda系统（具有一个未知参数）和Prototype模型之间的参数估计和同步误差

4 Applications

Owing to the random noise-like properties and the discovery of the synchronization of chaotic system，chaotic signals are applicable to secure communication systems［18］，including chaotic encryption and chaotic spread spectrum systems.Chaotic telemetry is a new technology in aircraft telemetry system which is an active branch in chaotic communication.In chaotic telemetry scheme with coherent receiver，the transmitter and the receiver use the same chaotic system to encoding and decoding the message，respectively.Here，as an eavesdropper，we use a different chaotic system to decode the useful information transmitted by the transmitter.The Ikeda system，as described in（13），is used to transmit information between the cooperators.The receiver uses the same system to recover the message as

The modulation parameter p is switched between 4 and 5 in the transmitter according to the digital message. That is to say，when the transmitted message bit is 0，the parameter p is set to be 4；when the transmitted message bit is 1，the parameter p is set to be 5.Figure 3 shows decoding by the cooperator and the eavesdropper.The eavesdropper use a different system as

Fig.3 Parameter estimation of chaotic telemetry system using same and different system图3 相同和不同混沌系统对混沌遥测系统参数估计结果

It should be noted that our scheme can be extended to parameter estimation of multi-parameter chaotic telemetry system.Every transmitted telemetry parameter exploits a chaotic system parameter to modulate it.

5 Conclusions

Parameter estimation between two different time-delayed chaotic systems and its application has been investigated.A control signal is designed to ensure synchronization and parameter estimation of different systems based on Krasovskii-Lyapunov theory.This scheme can be used to estimate multi-parameters of a chaotic telemetry system from a scalar observable signal.We exploited a different system to recover the message transmitted between the cooperators.The update rate which determines the convergence speed will decrease the fluctuant time till the correct parameter value is estimated，with the increase of it.So，in chaotic telemetry signal reconnaissance，we can increase the update rate to decrease the convergence time to satisfy the fast parameter switch.An immediate result of the increase of update rate is the serious fluctuation of the initial steps of parameter estimation.However，a filter can be used to smooth the estimated curve，and a decision device is used to decide the message transmitted from the transmitter.Therefore，our results can be used to practical engineering application.

It is worth noting that the construction of the parameter estimation scheme does not consider the effect of the noise，however，this is an important problem should be faced and solved in synchronization based parameter estimation between chaotic systems.Next，we will study the impact of the noise on our scheme.

［1］WEI Heng-dong.A study on the detection of chaotic direct sequence spread spectrum signal an chaos synchronization［D］.Chengdu：University of Electronic Science and Technology of China，2010.（in Chinese）魏恒东.混沌直扩信号检测与混沌同步研究［D］.成都：电子科技大学，2010.

［2］LIU Jia-xin.Spacecraft TT＆C and Information Transmission Technology［M］.Beijing：National Defense Industry Press，2011.（in Chinese）刘嘉兴.飞行器测控与信息传输技术［M］.北京：国防工业出版社，2011.

［3］Parlitz U，Ergezinger S.Robust communication based on chaotic spreading sequences［J］.Physics Letters A，1994，188（2）：146-150.

［4］Stavroulakis P.Chaos applications in telecommunications［M］.New York：CRC Press，2006.

［5］Parlitz U.Estimating model parameters from time series by autosynchronization［J］.Physical Review Letters，1996，76：1232-1235.

［6］Parlitz U，Junge L，Kocarev L.Synchronization-based parameter estimation from time series［J］.Physical Review E，1996，54（6）：6253-6259.

［7］Wei H，Li L.Estimating parameters by anticipating chaotic synchronization［J］.Chaos，2010，20（2）：023112.

［8］Li Z，Zhao X.Generalized function projective synchronization of two different hyperchaotic systems with unknown parameters［J］.Nonlinear Analysis：Real World Applications，2011，12（5）：2607-2615.

［9］Zhou P，Ding R.Modified Function Projective Synchronization between Different Dimension Fractional-Order Chaotic Systems［J］.Abstract and Applied Analysis，2012（2012）：1-12.

［10］Wu X，Li S.Dynamics analysis and hybrid function projective synchronization of a new chaotic system［J］.Nonlinear Dynamics，2012，69（4）：1979-1994.

［11］Parlitz U，YU D.Synchronization and control based parameter identification in Intelligent Computing based on Chaos［M］.Berlin：Springer，2009：227-249.

［12］Wang S，Yu Y.Generalized Projective Synchronization of Fractional Order Chaotic Systems with Different Dimensions［J］.Chinese Physics Letters，2012，29（2）：020505.

［13］Wei W，Li D H，Wang J.Synchronization of hyperchaotic Chen systems：a class of the adaptive control［J］.Chinese Physics B，2010，19（4）：1-11.

［14］Ghosh D，Banerjee S.Adaptive scheme for synchronizationbased multiparameter estimation from a single chaotic time series and its applications［J］.Physical Review E，2008，78（5）：056211.

［15］Pecora L，Carroll T，Johnson G，et al.Fundamentals of synchronization in chaotic systems，concepts，and applications［J］.Chaos，1997，7（4）：520.

［16］Ucar A.A prototype model for chaos studies［J］.International Journal of Engineering Science，2002，40（3）：251-258.

［17］Senthilkumar D V，Kurths J，Lakshmanan M.Stability of synchronization in coupled time-delay systems using Krasovskii-Lyapunov theory［J］.Physical Review E，2009，79（6）：1-4.

［18］Ren Hai-peng，Baptista M，Grebogi C.Wireless communication with chaos［J］.Physical Review Letter，2013，110（18）：1-5.

Biography：

WEI Heng-dong was born in Xichang，Sichuan Province，in 1980.He received the Ph.D.degree in 2010.He is now an engineer.His research concerns chaos theory，chaos synchronization and signal analysis.

魏恒东（1980—），男，四川西昌人，2010年获博士学位，现为工程师，主要研究方向为混沌理论、混沌同步和信号分析。Email：hdwei＠foxmail.com

Parameters Estimation of Chaotic Telemetry System between Two Different Time-delayed Chaotic Systems❋

WEI Heng-dong❋❋
（Southwest China Institute of Electronic Technology，Chengdu 610036，China）

For parameter estimation of chaotic telemetry system，chaos synchronization based parameter estimation between two different time-delayed chaotic systems is investigated.An adaptive scheme is proposed to estimate all the parameters of the interested system using a different system.Sufficient conditions for synchronization and parameter estimation are shown analytically according to Krasovskii-Lyapunov theory.By using the parameter estimation scheme in chaotic telemetry signal reconnaissance，the message transmitted can be recovered effectively.The proposed scheme can be used to obtain useful information of chaotic telemetry system between two different chaotic systems by non-cooperator.Numerical simulations illustrate the validity of the scheme.

time-delay chaotic systems；chaotic telemetry；parameter estimation；chaos synchronization

hdwei＠foxmail.com

hdwei＠foxmail.com

10.3969/j.issn.1001-893x.2013.05.007

TN911.6

A

1001-893X（2013）05-0565-05

❋Received date：2013-03-19；Revised date：2013-05-06

2013-03-19；修回日期：2013-05-06

❋❋