CN104079402A - Parameter identification and projective synchronization method of sectional chaotic system - Google Patents

Abstract

The invention discloses a parameter identification and projective synchronization method of a sectional chaotic system. Firstly, a four-dimensional sectional Lorenz-stenflo chaotic system model is established to analyze the Lyapunov index property; secondly, a response system is established by adopting the Lorenz-stenflo chaotic system as the driving system; a projective synchronous controller is designed and feedback factors are automatically regulated according to the magnitude of errors and state changes; finally, a self-adaptive identification estimator and the parameter updating ruler are designed, so that a generalized projective synchronization of all state variables in varying proportions can be realized when multi-group of unknown parameters of a real-time driving system and the self time delay response system are identified. The parameter identification and projective synchronization method of the sectional chaotic system has the advantages that the method is independent of known parameters of the driving chaotic system, has a good robustness on time delay disturbance, and can be applied in the fields of secure communication and signal processing.

Description

A kind of unknown parameter identification of segmentation chaos system and Projective Synchronization method

Technical field

The unknown parameter identification and the Projective Synchronization method that the present invention relates to a kind of segmentation chaos system, belong to nonlinear Control field.

Background technology

In the Practical Project implementation procedure of nonlinear chaotic system, because this type systematic often has complicated nonlinear terms, if therefore realizing this chaos system with electronic circuit generally can have very large difficulty.Research discovery, piecewise linear function, due to the particularity of its structure, is therefore often used to simple structure electronic circuit to generate chaos attractor.Cai's formula system is exactly first segmentation chaos system of building out by actual electronic circuits.In recent years, revising L ü system, correction Chen system and correction Lorenz-Stenflo chaos system etc. is also suggested in succession.

Piecewise function structure due to segmentation chaos system, therefore it can be decomposed into some stabistor systems, but different initial condition will make chaos system in different fragmentation states, therefore its control research is compared continuous system and is wanted difficult, achievement in research is also relatively less, especially segmentation Lorenz-Stenflo chaos system.

Chaotic Synchronous is extensive application all in fields such as secure communication and signal processing, and the concept of Generalized Projective Synchronization, and object further requires to realize the in proportion Projective Synchronization of the different chaos system of two initial values to each corresponding states vector.The method has very strong antidecoding capability.Because the parameter of actual chaos system may be unknown or drift completely, and real response circuit system affects from delayed perturbation in Chaotic Synchronous suffered the unknown in the process of drive system, and under some condition, these impacts are fatal for system.

The concept of brief description Generalized Projective Synchronization:

Consider following two Kind of Nonlinear Dynamical Systems

$\stackrel{\·}{X}=F_1\left(X\right)$

$\stackrel{\·}{Y}=F_2\left(Y\right)+U(X,Y)$

Wherein X and Y are respectively the state variables of drive system and responding system.U is control inputs vector.Under the prerequisite of the scale factor α of given diagonal matrix form, if exist controller function U to make

$\underset{t\→\∞}{\lim }\left|\right|Y\left(t\right)-\mathrm{\αX}\left(t\right)\left|\right|=0$

Claim drive system and responding system to reach Generalized Projective Synchronization.By changing scale factor α, just can obtain arbitrary proportion in the chaotic signal of former drive system, every one dimension state can be presented as different synchronous regimes.The Complete Synchronization of chaos system (α=I) and antiphase synchronous (α=-I) are all the special case of Generalized Projective Synchronization.

Summary of the invention

The object of the invention is to solve a class and take the Generalized Projective Synchronization problem of unknown parameters segmentation Lorenz-Stenflo chaos system as driving, design responding system is being subject to more new law of unknown controller under delayed perturbation affects and Adaptive Identification, to realize multi-form Projective Synchronization.Can be used for secure communication, to improving communication security, there is important practical value.

According to technical scheme provided by the invention, unknown parameter identification and the Projective Synchronization method of described segmentation chaos system comprise the steps:

The first step: the 4 dimension segmentation Lorenz-Stenflo Chaotic Systems of setting up unknown parameters.Reconstruct obtains 4 new dimension segmentation Lorenz-Stenflo chaos systems:

$\begin{array}{c}\stackrel{\·}{x}=a(y-x)+\mathrm{dw}\\ \stackrel{\·}{y}=x(c-z)-y\\ \stackrel{\·}{z}=\mathrm{sign}\left(y\right)x-\mathrm{bz}\\ \stackrel{\·}{w}=-x-\mathrm{aw}\end{array}---\left(1\right)$

The linear symbol function that wherein sign () is standard, a, b, c, d is the unknown constant coefficient of system.

Second step: the segmentation Lorenz-Stenflo chaos system of take is drive system, tectonic response system.The segmentation chaos drive system of rewriting formula (1) matrix form is

${\stackrel{\·}{x}}_m=f\left(x_m\right)+F\left(x_m\right)\mathrm{\θ}---\left(2\right)$

Wherein: $x_m^T=\left[\begin{array}{cccc}{x}^T& y^T& z^T& w^T\end{array}\right],$ $f\left(x_m\right)=\left[\begin{array}{c}0\\ -\mathrm{xz}-y\\ \mathrm{sign}\left(y\right)x\\ -x\end{array}\right],$ $\mathrm{\θ}=\left[\begin{array}{c}a\\ b\\ c\\ d\end{array}\right],$ $F\left(x_m\right)=\left[\begin{array}{cccc}y-x& 0& 0& w\\ 0& 0& x& 0\\ 0& -z& 0& 0\\ -w& 0& 0& 0\end{array}\right].$

For drive system (2), design the responding system of following form

$\begin{array}{c}{\stackrel{\·}{x}}_s=a(y_s-x_s)+d{w}_s+u_1\\ {\stackrel{\·}{y}}_s=x_s(c-z_s)-y_s+u_2\\ {\stackrel{\·}{z}}_s=\mathrm{sign}\left(y_s\right)x_s-b{z}_s+u_3\\ {\stackrel{\·}{w}}_s=-x_s-a{w}_s+u_4\end{array}---\left(3\right)$

U in formula _i(i=1,2,3,4) are the isochronous controller of design.

Because chaos system generally all can exist the phenomenons such as signal transmission delay in actual synchronization response process, so responding system will be subject to unknown delayed perturbation impact.Now, sync response system need be rewritten as:

Wherein ${\hat{x}}_s^T=\left[\begin{array}{cccc}{x}_s^T& y_s^T& z_s^T& w_s^T\end{array}\right],$ $D\left({\hat{x}}_s(t-\mathrm{\τ})\right)=\mathrm{diag}\{x_s(t-\mathrm{\τ}),y_s(t-\mathrm{\τ}),z_s(t-\mathrm{\τ}),w_s(t-\mathrm{\τ})\},$

$u^T=\left[\begin{array}{cccc}{u}_1^T& u_2^T& u_3^T& u_4^T\end{array}\right],$ $f\left({\hat{x}}_s\right)=\left[\begin{array}{c}0\\ -x_s{z}_s-y_s\\ \mathrm{sign}\left(y_s\right)x_s\\ -x_s\end{array}\right],$ $\widehat{\mathrm{\θ}}=\left[\begin{array}{c}\hat{a}\\ \hat{b}\\ \hat{c}\\ \hat{d}\end{array}\right],$

$F\left({\hat{x}}_s\right)=\left[\begin{array}{cccc}{y}_s-x_s& 0& 0& w_s\\ 0& 0& x_s& 0\\ 0& -z_s& 0& 0\\ -w_s& 0& 0& 0\end{array}\right]\text{,}$ And for nonlinear function vector, for delayed perturbation item, τ is lag time, the estimate vector of responding system to drive system unknowm coefficient θ, q ₁, q ₂, q ₃and q ₄unknowm coefficient for disturbance.

The 3rd step: design Projective Synchronization controller, regulates feedback factor automatically according to error size and state variation.Get error vector α is the scale factor of diagonal matrix form.Get simultaneously cONTROLLER DESIGN:

A wherein _mbe the matrix that all characteristic roots all have negative real part, or meet MATRIX INEQUALITIES p is symmetric positive definite matrix.

The 4th step: design Adaptive Identification estimator and parameter be new law more, is picking out Real Time Drive system and when time lag response system is organized unknown parameter, is realizing the different proportion Generalized Projective Synchronization of all state variables more.In formula (5), the adaptive updates of parameter vector rule is

K wherein ₁and K ₂for symmetric positive definite matrix.

Advantage of the present invention is: this method does not need to depend on and drives the parameter of chaos system known, and for there is good robust property from delayed perturbation; The Projective Synchronization problem that solves fractional order unification and isomery chaos system be can further promote, and secure communication and signal process field are applied to.

Accompanying drawing explanation

Fig. 1 is unknown parameter identification and the Projective Synchronization structure chart of segmentation chaos system

Fig. 2 is that segmentation Lorenz-Stenflo chaos system is at the attractor figure of (x, y, z) and (x, y, w) plane

The movement locus of driving and responding system when Fig. 3 is α=I

The error curve of driving and responding system when Fig. 4 is α=I

The unknown parameter identification of drive system when Fig. 5 is α=I

The unknown parameter identification of time lag responding system when Fig. 6 is α=I

The movement locus of driving and responding system when Fig. 7 is α=-I

The error curve of driving and responding system when Fig. 8 is α=-I

The unknown parameter identification of drive system when Fig. 9 is α=-I

The unknown parameter identification of time lag responding system when Figure 10 is α=-I

Figure 11 is α=diag{-0.5 ,-1 ,-0.5, and the movement locus of driving and responding system during-0.5}

Figure 12 is α=diag{-0.5 ,-1 ,-0.5, and the error curve of driving and responding system during-0.5}

Figure 13 is α=diag{-0.5 ,-1 ,-0.5, and the unknown parameter identification of drive system during-0.5}

Figure 14 is α=diag{-0.5 ,-1 ,-0.5, and the unknown parameter identification of time lag responding system during-0.5}

Embodiment

Below in conjunction with drawings and Examples, the present invention will be further described.

Parameter identification and the Projective Synchronization control structure figure of segmentation Lorenz-Stenflo chaos system as shown in Figure 1.

The first step: according to set up the Projective Synchronization drive system of Vector-Matrix Form suc as formula the dimension segmentation of 4 shown in (1) Lorenz-Stenflo chaos system.As system parameters a=1, b=0.7, c=26, during d=1.5, the largest Lyapunov exponent of segmentation chaos system is 13.1.As shown in Figure 2, the variable amplitude of its chaotic characteristic is larger for the attractor shape of segmentation Lorenz-Stenflo chaos system, is easier to the structure of side circuit, and therefore the application prospect in fields such as secure communications also will be more extensive.

Second step: for chaos system (1) and time lag response system (4), design Adaptive synchronization controller (5) and identification more meet MATRIX INEQUALITIES in new law (6) p is the negative definite matrix A of symmetric positive definite matrix condition _m=diag{-10 ,-10 ,-10 ,-10}, symmetric positive definite matrix P=diag{0.01,0.01,0.01,0.01}.

The 3rd step: according to the adaptive updates rule formula of parameter vector get symmetric positive definite matrix K ₁=diag{10,10,10,10}, K ₂=diag{5,1,5,5}, substitution formula (5) completes the design of Projective Synchronization controller.

The 4th step: responding system controller (5) will be asked for different scale factors according to projection, at drive system arbitrary initial state $x_m^T\left(0\right)=\left[\begin{array}{cccc}0.1& 0.3& -0.1& -0.2\end{array}\right],$ The arbitrary initial state of responding system ${\hat{x}}_s^T=\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]$ Situation under, the result of the various Projective Synchronizations of drive system and responding system as shown in the figure:

(1) when scale factor α=I, Fig. 3 and Fig. 4 are respectively movement locus and the error curves of the driving of segmentation Lorenz-Stenflo chaos and responding system, and Fig. 5 and Fig. 6 are respectively the unknown parameter identification curves of driving and responding system.From synchronized result, can find out, the time lag synchronous error of system reduces fast, and adaptive controller has finally also accurately picked out 8 unknown parameters, thereby makes driving and responding system reach Complete Synchronization.

(2) when scale factor α=-I, the antiphase synchronizing process of segmentation Lorenz-Stenflo chaos system is as shown in Figure 7 to 10.From synchronized result, can find out, the now same asymptotic convergence of error system, and adaptive nonlinear control device almost remains unchanged to the identification precision of 8 unknown parameters in driving and responding system.

(3) as α=Λ=diag{-0.5 ,-1 ,-0.5, during-0.5}, segmentation Lorenz-Stenflo chaos system requires to reach Generalized Anti Phase synchronization, i.e. x _s=-0.5x, y _s=-y, z _s=-0.5z, w _s=-0.5w.Synchronized result respectively as shown in Figure 11 to 14.Can find out, now the state ratio of responding system is in the chaotic signal of former drive system, and every one dimension state is all presented as different synchronous regimes, is more conducive to increase the confidentiality of communication.

Above-described embodiment is only for example of the present invention is clearly described, and be not the restriction to embodiments of the present invention, for those of ordinary skill in the field, can also make other changes in different forms on the basis of the above description.

Claims (1)

1. the unknown parameter identification of segmentation chaos system and a Projective Synchronization method, its feature comprises: set up 4 dimension segmentation Lorenz-Stenflo Chaotic Systems of unknown parameters, analyze lyapunov index characteristic; Take Lorenz-Stenflo chaos system as drive system, tectonic response system; Design Projective Synchronization controller, regulates feedback factor automatically according to error size and state variation; Design Adaptive Identification estimator and parameter be new law more, is picking out Real Time Drive system and when time lag response system is organized unknown parameter, is realizing the different proportion Generalized Projective Synchronization of all state variables more.

1) first step: set up 4 dimension segmentation Lorenz-Stenflo Chaotic Systems of unknown parameters, analyze lyapunov index characteristic.Reconstruct chaos system structure obtains 4 new dimension segmentation Lorenz-Stenflo chaos systems:

$\begin{array}{c}\stackrel{\·}{x}=a(y-x)+\mathrm{dw}\\ \stackrel{\·}{y}=x(c-z)-y\\ \stackrel{\·}{z}=\mathrm{sign}\left(y\right)x-\mathrm{bz}\\ \stackrel{\·}{w}=-x-\mathrm{aw}\end{array}---\left(1\right)$

The equivalent matrice model that foundation obtains from the segmentation chaos drive system of first step formula (1) form

${\stackrel{\·}{x}}_m=f\left(x_m\right)+F\left(x_m\right)\mathrm{\θ}---\left(2\right)$

Wherein: $x_m^T=\left[\begin{array}{cccc}{x}^T& y^T& z^T& w^T\end{array}\right],$ $f\left(x_m\right)=\left[\begin{array}{c}0\\ -\mathrm{xz}-y\\ \mathrm{sign}\left(y\right)x\\ -x\end{array}\right],$ $\mathrm{\θ}=\left[\begin{array}{c}a\\ b\\ c\\ d\end{array}\right],$ $F\left(x_m\right)=\left[\begin{array}{cccc}y-x& 0& 0& w\\ 0& 0& x& 0\\ 0& -z& 0& 0\\ -w& 0& 0& 0\end{array}\right].$

2) second step: formula (2) the Lorenz-Stenflo chaos system of take is drive system, design responding system

$\begin{array}{c}{\stackrel{\·}{x}}_s=a(y_s-x_s)+d{w}_s+u_1\\ {\stackrel{\·}{y}}_s=x_s(c-z_s)-y_s+u_2\\ {\stackrel{\·}{z}}_s=\mathrm{sign}\left(y_s\right)x_s-b{z}_s+u_3\\ {\stackrel{\·}{w}}_s=-x_s-a{w}_s+u_4\end{array}---\left(3\right)$

U in formula _i(i=1,2,3,4) are the isochronous controller of design.

Because chaos system generally all can exist the phenomenons such as signal transmission delay in actual synchronization response process, so responding system will be subject to unknown delayed perturbation impact.Now, sync response system is rewritten as:

Wherein ${\hat{x}}_s^T=\left[\begin{array}{cccc}{x}_s^T& y_s^T& z_s^T& w_s^T\end{array}\right],$ $D\left({\hat{x}}_s(t-\mathrm{\τ})\right)=\mathrm{diag}\{x_s(t-\mathrm{\τ}),y_s(t-\mathrm{\τ}),z_s(t-\mathrm{\τ}),w_s(t-\mathrm{\τ})\},$ $u^T=\left[\begin{array}{cccc}{u}_1^T& u_2^T& u_3^T& u_4^T\end{array}\right],$ $f\left({\hat{x}}_s\right)=\left[\begin{array}{c}0\\ -x_s{z}_s-y_s\\ \mathrm{sign}\left(y_s\right)x_s\\ -x_s\end{array}\right],$ $\widehat{\mathrm{\θ}}=\left[\begin{array}{c}\hat{a}\\ \hat{b}\\ \hat{c}\\ \hat{d}\end{array}\right],$ $F\left({\hat{x}}_s\right)=\left[\begin{array}{cccc}{y}_s-x_s& 0& 0& w_s\\ 0& 0& x_s& 0\\ 0& -z_s& 0& 0\\ -w_s& 0& 0& 0\end{array}\right]\text{,}$ And for nonlinear function vector, for delayed perturbation item, τ is lag time, the estimate vector of responding system to drive system unknowm coefficient θ, q ₁, q ₂, q ₃and q ₄unknowm coefficient for disturbance.

3) the 3rd step: design Projective Synchronization controller, regulates feedback factor automatically according to error size and state variation.Get error vector α is the scale factor of diagonal matrix form.Get simultaneously cONTROLLER DESIGN:

A wherein _mbe the matrix that all characteristic roots all have negative real part, or meet MATRIX INEQUALITIES p is symmetric positive definite matrix.

4) the 4th step: design Adaptive Identification estimator and parameter be new law more, is picking out Real Time Drive system and when time lag response system is organized unknown parameter, is realizing the different proportion Generalized Projective Synchronization of all state variables more.The adaptive updates rule of design formula (5) controller parameter vector is

K wherein ₁and K ₂for symmetric positive definite matrix.The final Projective Synchronization controller of 4 dimension segmentation Lorenz-Stenflo chaos systems will can be obtained after formula (6) substitution formula (5).