CN109412806A - Applied to the controlled Lorenz system of secret communication and the generalized synchronization method of Genesio-Tesi system - Google Patents

Abstract

A kind of generalized synchronization method of controlled Lorenz system applied to secret communication and Genesio-Tesi system, the described method comprises the following steps: 1) generalized chaotic synchronization problem describes；2) state transformation and feedback of response system；3) the state conversion of drive system；4) generalized synchronization.The present invention provides the generalized chaotic synchronization method of a kind of controlled Lorenz system applied to secret communication and Genesio-Tesi system, using Genesio-Tesi chaos system as drive system, using the controlled Lorenz system of single input as response system, a kind of Chaotic Synchronous algorithm is designed using the Lie derivatives method of vector field in Differential Geometry, realize generalized synchronization, Control platform is higher.

Description

Controlled Lorenz system and Genesio-Tesi system applied to secret communication it is wide Adopted synchronous method

Technical field

The invention belongs to can be applied to the Chaotic Synchronous technical field of secret communication more particularly to it is a kind of realization with Genesio-Tesi chaos system is drive system, using the controlled Lorenz system of single input as the Chaotic Synchronous side of response system Method.

Background technique

Chaotic motion is the branch of non-linear ambit, but its range being related to is well beyond traditional non-thread sexology Section territorial limit, develops into comprehensive, intercrossing, cross-cutting subject branch, and very big to have widened people's understanding non- The ken of linear science, it is more deep to the understanding of nonlinear science.

Chaos is also applied to laser secret communication.One typical application is chaotic modulation.Chaotic modulation is 1992 The it is proposeds such as Halle, Hasler solve the problems, such as a kind of method complicated in confidential corespondence, basic thought be by original signal with One chaotic signal modulation is sent together；And receiver is demodulated, and isolates original signal according to chaotic signal；It is right Third party is unaware of the dynamic characteristic of the chaotic signal due to it, can not decrypt.The advantages of chaotic laser light secret communication, has: 1) it is hardware encryption.It uses the structural parameters of sending and receiving laser as key, avoids the security risk of algorithm for encryption；2) add The speed of decryption quickly because it lean on be laser response speed；3) hidden due to coming by the chaotic waves of laser output Information is hidden, and is no longer single photon, transmission range is long；4) compatible with existing optical fiber telecommunications system, it can advantageously transplant existing All technologies such as amplification, wavelength-division multiplex in Fibre Optical Communication Technology.2005, European Union was in the 5th scientific and technological framework planning OCCULT Purpose subsidize under, the seven state researcher such as moral, method, English in the Metropolitan Area Network (MAN) of Athens city 120km rate under realize communication speed The chaotic laser light secret communication of rate 1Gb/s.2010, the scientific and technological framework planning PICASSO project of the 6th, European Union completed exocoel The integreted phontonics of feedback chaos semiconductor laser, and the mixed of 10Gb/s is completed in the Metropolitan Area Network (MAN) of Besancon, France 100km Ignorant secret communication experiment.

One so is led to the problem of, for transmitter and receiver, it is necessary to have almost consistent chaotic signal, this is needed Chaotic Synchronous technology is realized.Chaotic Synchronous refers to the different running tracks of two chaos systems, with the variation of time, simultaneously Identical value is converged to, the always consistent Chaotic Synchronous research work of the running track of the two systems can be divided into following (several synchronisation control means and its application study referring to Gu Baohua chaos system, Institutes Of Technology Of Nanjing are rich for several wheel synchronization types Bachelorship paper .2009.):

1) fully synchronized (Complete Synchronization) is that the running track of drive system and response system is complete It is complete consistent, it is the basis of Chaotic Synchronous research.

2) generalized synchronization (Generalized Synchronization) is the operation of drive system and response system output Track keeps functional relation, and generalized synchronization is fully synchronized and Projective Synchronization popularization.

3) Phase synchronization (Phase Synchronization) is that the chaos system of two couplings can enter a middle area Domain is able to maintain the synchronization of system running track phase.

4) late synchronous (Lag Synchronization) is that there are the same of time delay for the tracks of two chaos systems Step requires strictly, to require loosely than fully synchronized than Phase synchronization.

5) Projective Synchronization (Projective Synchronization) is that two chaos systems keep proportionate relationship, i.e. frequency Rate is identical, and amplitude keeps proportionate relationship, and Projective Synchronization is fully synchronized extension.

6) weighted array and response that synchronous (Combination Synchronization) is two drive systems are combined System is synchronous, and it is fully synchronized and Projective Synchronization popularization that combination, which synchronizes,.

7) composite sync (Compound Synchronization) is the composite system of three drive systems and response is System synchronizes.

In addition to this, there are also anti-synchronous, refer to that its running track frequency of the state variable of two chaos systems is identical, amplitude It is identical, contrary, i.e., the state variable of two chaos systems and for 0 synchronous situation；Similarly there are also reverse phase synchronizations, part The synchronias such as synchronize.These synchronous method are technologies with practical value in laser secret communication.

Summary of the invention

In order to overcome the lower deficiency of Control platform of existing chaos synchronization, the present invention provides a kind of applied to secrecy The controlled Lorenz system of communication and the generalized synchronization method of Genesio-Tesi system are with Genesio-Tesi chaos system Drive system, using the controlled Lorenz system of single input as response system, using the Lie derivatives method of vector field in Differential Geometry A kind of Chaotic Synchronous algorithm is designed, realizes generalized synchronization, Control platform is higher.

The technical solution adopted by the present invention to solve the technical problems is:

A kind of generalized synchronization method of controlled Lorenz system applied to secret communication and Genesio-Tesi system, packet Include following steps:

1) generalized chaotic synchronization problem describes

Drive system is Genesio-Tesi system, and form is as follows:

Wherein ξ=(ξ₁,ξ₂,ξ₃)^TIt is state variable, α, β and γ are known real parameter；System (1) also requires that there are L So that α-L > 0, while L is a real solution of following equation

L³-2αL²+(α²+ β) β=0 L+ γ-α (2)

Using controlled Lorenz system as response system, form is as follows:

Wherein x=(x₁,x₂,x₃)^TIt is state variable, u is scalar input, and a, b and c are known positive real number parameter, b=α-L And 2a-b ≠ 0；

The target to be realized of generalized chaotic synchronization is: response system (3) is being respectively x (t with drive system (1) initial value₀) With ξ (t₀), response system track is fed back by state

U=u (x, ξ, t) (4)

Wherein t indicates the state transformation between time and phase space

ξ=T (x) (5)

It is intended to the track of drive system afterwards, i.e.,

Here | | | | it representsThe 2- norm of vector in space；

2) state transformation of drive system

Following state transformation η=S (ξ) wherein η=(η is made to drive system (1)₁,η₂,η₃)^T

Here K and L is undetermined parameter, M_SFor 3 rank square matrixes, this linear transformation is inversely transformed into

Using η as state, system representation is

If met in above system

Then system (9) is reduced to

K=β-L (α-L) is obtained by second equation of formula (10) and substitutes into first equation and obtains formula (2)；

3) the state feedback equivalence conversion of response system

To response system (3), if fed back

U=- (cx₁-x₂-x₁x₃)+u₁ (12)

System is reduced to

Consider by system (13) by state transformation and further feed back be converted to more simply with system (11) more Similar form, in order to design generalized synchronization control method, observing system (11) is the ordinary differential side of lower triangular form Journey is namely to meet following form to 3 levels system

Wish that system (13) can be converted to controlled lower triangular form, i.e.,

Wherein v is input, for this purpose, the shift vectors field of note system (13) is

And input vector field is

Enable vector field

Calculate following vector field Lie bracket

Pay attention toOrder is 2 in global scope, andIllustrate this distribution pairing； It enables

Calculate following vector field Lie bracket

In x₁When=0 or 2a-b=0It is order is still 2, this also illustrates that system (13) can not Realize state feedback linearization；But as 2a-b ≠ 0,Only order is 2 in a null set, Order is 3 except this set, so system (13) can be converted to lower cam system (15) by state transformation equivalence, however, The problem of system (13) need to be probed into actually capable of being converted to which kind of lower cam system, and intentionally get it is relatively simple in form under Cam system, thus, it is noted that

Distribution in global scope at this timeOrder be 3 (in 2a-b ≠ 0) and pairing, enable

Take following distribution

Δ₀=span { X₀}；Δ₁=span { X₀,X₁}；Δ₂=span { X₀,X₁,X₂, (24)

Profile Δ₀,Δ₁,Δ₂And X₀,X₁,X₂It has the property that

1. can verify that [X₀,X₁]=0, [X₁,X₂]=0 and [X₀,X₂]=0

2. by 1., Δ₀,Δ₁,Δ₂It is involutive distribution；

3. existence converts h=(h by 1.₁(x),h₂(x),h₃(x))^T=H (x)=H (x₁,x₂,x₃) meet

4. due toIllustrative 3. in state transformation h under, system must still have lower three

Angular formula；

3. above-mentioned property also implies that following 3 partial differential equations of satisfaction, the 1st group is

Wherein h₁(x) be smooth function, symbol " L " expression do Lie derivatives, the 2nd group is

Wherein h₂It (x) is smooth function, the 3rd group is

Wherein h₃It (x) is smooth function, the feasible solution of above-mentioned 3 groups of partial differential equation is respectively

In h=(h₁,h₂,h₃)^TSystem becomes under state

The system has actually had a form of lower cam system such as system (15), but from second formula of system (30) It sees, above system can be also further simplified by following state transformation

Such as use y=(y₁,y₂,y₃)^TState indicates that x state is then

The write through system under y state

Design input

u₁=-ax₁+ax₂+u₂, (34)

Above system is further simplified as

Preceding 2 equations of above system have been realized unanimously in form in the equivalent form (9) of drive system；

4) generalized synchronization

The stationary problem for considering system (35) and system (9) now, enabling the two state difference is e=η-y=(e₁,e₂,e₃)^T, Then

Design of feedback

u₂=η₁-Kη₂-Lη₃-u₃ (37)

System representation is

For the subsystem of above system

Device control as follows can be designed according to the classical way of linear system:

Under the controller system (39) by finite-time control in, i.e. t₁Moment realizes e₂(t₁)=e₃(t₁)=0, if A kind of controller is counted from t₀Moment realizes e through finite time₂(t₁)=e₃(t₁)=0, and guarantee that control amount has in the process for this Continuous first derivative is simultaneously transitioned into 0；Firstly, the e of design anticipation₂(t) it is

Wherein p (t) is polynomial of one indeterminate, due to requiring t₁Moment reaches the origin and u of system (39)₃In t > t₀Range Inside there is continuous first derivative, it means that e₂(t) in t₁When have continuous three order derivative, actually e₂(t) and one, two, three Order derivative t again₁Moment is to guarantee continuously to be only 0, i.e.,

Consider further that the t of system (39)₀Moment should meet

Since formula (42) and formula (43) provide 6 conditions altogether, so p (t₀) it should be 5 order polynomials, recycle formula (42) to obtain

Wherein C₀And C₁For undetermined coefficient, obtained using the 1st formula of formula (43)

Again by the 2nd formula of formula (43)

Arrangement obtains

The e₂(t) meet the requirements of formula (42) and formula (43), then

And

Obvious e₂(t₁)=e₃(t₁)=u₃(t₁)=0；

In time t₁Later, first equation of system (38) becomesThis equation is obviously a wide range of progressive steady Fixed, thus system (38) a wide range of asymptotically stability, illustrating system (9), the realization under this control law is synchronous with system (35)；

The generalized synchronization problem of system (1) Yu system (3) is returned to, system (1) becomes system (9) by state transformation (7), System (3) becomes system (35) after having made feedback and state transformation, and wherein state transformation needs composite type (29) and formula (31) is

And naming this state transformation is y=(y₁,y₂,y₃)^T=Y ((x₁,x₂,x₃)^T)=Y (x), and composite type (12), formula (34), formula (37) and formula (40), control law are

Wherein u₃Expression formula see formula (49).

Further, in the step 4), whether verifying generalized synchronization be may be implemented, process are as follows:

WhereinFor matrix2- norm, it is clear that at the control law u that formula (51) and formula (49) are determinedThen

Due to the nonnegativity of norm

Above formula illustrates that the requirement formula (6) of generalized synchronization meets, and chooses in formula (6)

Further, the method also includes following steps:

5) according to the requirement of generalized synchronization, when the input of response system is set as

Wherein parameter t₁It can be used for adjusting the speed of generalized synchronization realization, the state transformation between phase space is set as

Wherein K and L is to meet the following formula

K (α-L)=γ

L (α-L)=β-K (57)

α-L > 0

Real number, under above-mentioned setting, generalized synchronization is realized in drive system (1) and response system (3).

Beneficial effects of the present invention are mainly manifested in: first, using the control of the State space transition based on Differential Geometry Method discloses the internal association and uniformity of Genesio-Tesi chaos system Yu Lorenz chaos system from profound level； Second, a kind of technology of directly design asymptotically stability track is proposed, wherein also containing a kind of raising finite-time control device light The method of slippery, relative to the control method of the design Lyapunov function generallyd use, (see Hong Yiguang, Chen Daizhan is non-linear Network analysis and control Beijing, Science Press, 2005.), be conducive to improve Control platform；Third, using the same of single input Step, controller are simply easy to circuit realization；4th, by changing parameter t₁-t₀, the speed that generalized synchronization is realized is adjusted.

Detailed description of the invention

Fig. 1 is 3 dimension phasors of the i.e. drive system of Genesio-Tesi chaos system, parameter alpha=0.44, β=1.1, γ =1, so K=1.3328114068120458 and L=-0.31029369863505435, initial value ξ₁(t₀)=0.2, ξ₂ (t₀)=0.1, ξ₃(t₀)=- 0.2；

Fig. 2 is in response to 3 dimension phasors of system, parameter a=10, b=α-L=0.75029369863505435 and c= 28, initial value x₁(t₀)=1, x₂(t₀)=1, x₃(t₀)=1；

Fig. 3 is the asymptotically stability of error system, initial value e₁(t₀)=0.025663, e₂(t₀)=0.1, e₃(t₀)= 0.1；

Fig. 4 is control amount u3, and wherein parameter setting is as follows: t₁-t₀=1, i.e., finite-time control when it is 1 second a length of.

Specific embodiment

The invention will be further described below in conjunction with the accompanying drawings.

Referring to Fig.1~Fig. 4, a kind of controlled Lorenz system applied to secret communication are wide with Genesio-Tesi system Adopted synchronous method, comprising the following steps:

1) generalized chaotic synchronization problem describes

The drive system that generalized chaotic synchronization technology is related to is Genesio-Tesi system, and the system is initially in quilt in 1992 It is proposed (Genesio R, Tesi A.Harmonic balance methods for the analysis of chaotic Dynamics in nonlinear systems.Automatica 1992,28:531-548.), it has been found that it is with chaos Phenomenon, this system, which is also verified, to be realized with circuit, and concrete form is as follows:

Wherein ξ=(ξ₁,ξ₂,ξ₃)^TIt is state variable, α, β and γ are known real parameter, suitable parameter is chosen, such as In the case of α=0.44, β=1.1, γ=1, chaotic characteristic is presented in system.System (1) also requires that there are L to make α-L > 0, simultaneously L is a real solution of following equation

L³-2αL²+(α²+ β) β=0 L+ γ-α (2)

Lorenz system is considered as allowing it was recognized that there are first dynamical systems of chaos phenomenon, here with controlled Lorenz system is response system, and concrete form is as follows:

Wherein x=(x₁,x₂,x₃)^TIt is state variable, u is scalar input, and a, b and c are known positive real number parameters, the present invention Middle limitation b=α-L and 2a-b ≠ 0；

The target to be realized of generalized chaotic synchronization is: response system (3) is being respectively x (t with drive system (1) initial value₀) With ξ (t₀), response system track is fed back by state

U=u (x, ξ, t) (4)

Wherein t indicates the state transformation between time and phase space

ξ=T (x) (5)

It is intended to the track of drive system afterwards, i.e.,

Here | | | | it representsThe 2- norm of vector in space；

2) state transformation of drive system

Following state transformation η=S (ξ) wherein η=(η is made to drive system (1)₁,η₂,η₃)^T

Here K and L is undetermined parameter, M_SFor 3 rank square matrixes.This linear transformation is inversely transformed into

Using η as state, system representation is

If met in above system

Then system (9) is reduced to

K=β-L (α-L) is obtained by second equation of formula (10) and substitutes into first equation and obtains formula (2)；Here it is formulas (2) cause can also directly solve K and L by equation group (10)；

The case where for α=0.44, β=1.1, γ=1, equation group (10) have unique one group of real solution K= 1.3328114068120458 and L=-0.31029369863505435, should have b=α-L=at this time 0.75029369863505435 > 0；

3) the state feedback equivalence conversion of response system

To response system (3), if fed back

U=- (cx₁-x₂-x₁x₃)+u₁ (12)

System can be reduced to

Consider by system (13) by state transformation and further feed back be converted to more simply with system (11) more Similar form, in order to design generalized synchronization control method.Observing system (11) is the ordinary differential side of lower triangular form Journey is namely to meet following form to 3 levels system

Wish that system (13) can be converted to controlled lower triangular form, i.e.,

Wherein v is input.For this purpose, the shift vectors field of note system (13) is

And input vector field is

Enable vector field

Calculate following vector field Lie bracket (see the Differential Manifold such as Li Yangcheng basis Beijing, Science Press, 2011.)

Pay attention toOrder is 2 in global scope, andIllustrate this distribution pairing (see the Differential Manifold such as Li Yangcheng basis Beijing, Science Press, 2011.).It enables

Calculate following vector field Lie bracket

In x₁When=0 or 2a-b=0It is order is still 2, this also illustrates that system (13) can not Realize state feedback linearization (see Isidori, A.Nonlinear Control Systems.3rd edition, Communications and Control Engineering Series,Springer-Verlag,New York- Heidelberg-Berlin,1995)；But as 2a-b ≠ 0,Only in a null set Order is 2, and order is 3 except this set, so system (13) can be converted to lower cam system (15) by state transformation equivalence (see Celikovsky S, Nijmeijer H.Equivalence of nonlinear systems to triangular Form:the singular case.Systems&Control Letters, 1996,27:135-144.), however, need to probe into System (13) can be converted to the problem of cam system under which kind of actually, and intentionally get lower triangle system relatively simple in form System, thus, it is noted that

Distribution in global scope at this timeOrder be 3 (in 2a-b ≠ 0) and pairing, enable

Take following distribution

Δ₀=span { X₀}；Δ₁=span { X₀,X₁}；Δ₂=span { X₀,X₁,X₂, (24)

Profile Δ₀,Δ₁,Δ₂And X₀,X₁,X₂It has the property that

1. can verify that [X₀,X₁]=0, [X₁,X₂]=0 and [X₀,X₂]=0

2. by 1., Δ₀,Δ₁,Δ₂It is involutive distribution；

3. existence converts h=(h by 1.₁(x),h₂(x),h₃(x))^T=H (x)=H (x₁,x₂,x₃) meet (see

The Beijing Differential Geometry handout (second edition) such as Shiing-Shen Chern, BJ University Press, 2001.)

4. due toIllustrative 3. in state transformation h under, system must still have lower three

Angular formula；

3. above-mentioned property also implies that following 3 partial differential equations of satisfaction, the 1st group is

Wherein h₁(x) be smooth function, symbol " L " expression do Lie derivatives (see the Differential Manifold such as Li Yangcheng basis Beijing, Science Press, 2011.), the 2nd group is

Wherein h₂It (x) is smooth function, the 3rd group is

Wherein h₃It (x) is smooth function, the feasible solution of above-mentioned 3 groups of partial differential equation is respectively

In h=(h₁,h₂,h₃)^TSystem becomes under state

The system has actually had a form of lower cam system such as system (15), but from system (30)

Such as use y=(y₁,y₂,y₃)^TState indicates that x state is then

The write through system under y state

Design input

u₁=-ax₁+ax₂+u₂, (34)

Above system is further simplified as

Preceding 2 equations of above system have been realized unanimously in form in the equivalent form (9) of drive system；

4) generalized synchronization

The stationary problem for considering system (35) and system (9) now, enabling the two state difference is e=η-y=(e₁,e₂,e₃)^T, Then

Design of feedback

u₂=η₁-Kη₂-Lη₃-u₃ (37)

System representation is

For the subsystem of above system

Device control as follows can be designed according to the classical way of linear system (to appear to be prosperous and receive nurse linear multivariable control: is a kind of The Beijing method of geometry, Science Press, 1984.)

Under the controller system (39) by finite-time control in, i.e. t₁Moment realizes e₂(t₁)=e₃(t₁)=0, but It is the control amount of the controller in t₁Moment is not still 0, it is easy to which control is excessive, has certain defect, for this purpose, designing a kind of control Device processed is from t₀Moment realizes e through finite time₂(t₁)=e₃(t₁)=0, and guarantee that control amount has continuous one in the process for this Order derivative is simultaneously transitioned into 0；Firstly, the e of design anticipation₂(t) it is

Wherein p (t) is polynomial of one indeterminate, due to requiring t₁Moment reaches the origin and u of system (39)₃In t > t₀Range Inside there is continuous first derivative, it means that e₂(t) in t₁When have continuous three order derivative, actually e₂(t) and one, two, three Order derivative t again₁Moment is to guarantee continuously to be only 0, i.e.,

Consider further that the t of system (39)₀Moment should meet

Since formula (42) and formula (43) provide 6 conditions altogether, so p (t₀) it should be 5 order polynomials, recycle formula (42) to obtain

Wherein C₀And C₁For undetermined coefficient, obtained using the 1st formula of formula (43)

Again by the 2nd formula of formula (43)

Arrangement obtains

The e₂(t) meet the requirements of formula (42) and formula (43), then

And

Obvious e₂(t₁)=e₃(t₁)=u₃(t₁)=0；

In time t₁Later, first equation of system (38) becomesThis equation is obviously a wide range of progressive steady Fixed, thus system (38) a wide range of asymptotically stability, illustrating system (9), the realization under this control law is synchronous with system (35)；

The generalized synchronization problem of system (1) Yu system (3) is returned to, system (1) becomes system (9) by state transformation (7), System (3) becomes system (35) after having made feedback and state transformation, and wherein state transformation needs composite type (29) and formula (31) is

And naming this state transformation is y=(y₁,y₂,y₃)^T=Y ((x₁,x₂,x₃)^T)=Y (x), and composite type (12), formula (34), formula (37) and formula (40), control law are

Wherein u₃Expression formula see formula (49).

Further, whether verifying generalized synchronization may be implemented, process are as follows:

WhereinFor matrix2- norm, it is clear that at the control law u that formula (51) and formula (49) are determinedThen

Due to the nonnegativity of norm

Above formula illustrates that the requirement formula (6) of generalized synchronization meets, and chooses in formula (6)

5) according to the requirement of generalized synchronization, when the input of response system is set as

Wherein parameter t₁It can be used for adjusting the speed of generalized synchronization realization, the state transformation between phase space is set as

Wherein K and L is to meet the following formula

K (α-L)=γ

L (α-L)=β-K (57)

α-L > 0

Real number, under above-mentioned setting, generalized synchronization is realized in drive system (1) and response system (3).

To verify this generalized synchronization technology, system is driven using Matlab software emulation Genesio-Tesi chaos system 3 dimensions phasor (see Fig. 1) of system (1)；3 dimension phasors of the i.e. controlled Lorenz system (3) of response system are simulated, wherein controller u Setting have followed formula (55) (see Fig. 2)；Error system (38) and its controller are simulated, wherein controller u₃Also in compliance with formula (55) (see Fig. 3 and Fig. 4).

Initial value in FIG. 1 to FIG. 3 is related.The initial value of drive system passes through state transformation (7) under ξ state in Fig. 1 Become the initial value under η state afterwards, the initial value of response system becomes under y state after state transformation (31) under x state in Fig. 2 Initial value, the initial value under η state subtract the initial value under y state and obtain the initial value under error system e state.

Have in Fig. 1 and Fig. 2 phasor form it is similar to a certain degree but not quite identical, this is because the two is generalized synchronization, Only it could become progressive track by state transformation.

Fig. 3 error system energy asymptotically stability is to origin, but track has with the turnover relatively to smoothly transit at one, this is because Using finite-time control, until turning point i.e. t₁E near moment₂And e₃Have arrived at 0.After turning point, error system Control amount return 0, and e₁Tend to 0 by the dynamic characteristic of error system itself, so it is reasonable for there is turnover.Control amount u₃ In t₁Place has continuous first derivative (see Fig. 4), but without second dervative；If designing control amount u₃In t₁Place is only continuous but without single order Derivative, t₁Locate smoothness will decline, benefit be at this time without requiringCorrespondingly e₂(t) polynomial order It reduces, controller can be relatively simple.

Claims (3)

1. a kind of generalized synchronization method of controlled Lorenz system applied to secret communication and Genesio-Tesi system, special Sign is, comprising the following steps:

1) generalized chaotic synchronization problem describes

Drive system is Genesio-Tesi system, and form is as follows:

Wherein ξ=(ξ₁,ξ₂,ξ₃)^TIt is state variable, α, β and γ are known real parameter；System (1) also requires that there are L to make α- L > 0, while L is a real solution of following equation

L³-2αL²+(α²+ β) β=0 L+ γ-α (2)

Using controlled Lorenz system as response system, form is as follows:

Wherein x=(x₁,x₂,x₃)^TState variable, u is scalar input, and a, b and c are known positive real number parameters, b=α-L and 2a-b≠0；

The target to be realized of generalized chaotic synchronization is: response system (3) is being respectively x (t with drive system (1) initial value₀) and ξ (t₀), response system track is fed back by state

U=u (x, ξ, t) (4)

Wherein t indicates the state transformation between time and phase space

ξ=T (x) (5)

It is intended to the track of drive system afterwards, i.e.,

Here | | | | it representsThe 2- norm of vector in space；

2) state transformation of drive system

Following state transformation η=S (ξ) wherein η=(η is made to drive system (1)₁,η₂,η₃)^T

Here K and L is undetermined parameter, M_SFor 3 rank square matrixes, this linear transformation is inversely transformed into

Using η as state, system representation is

If met in above system

Then system (9) is reduced to

K=β-L (α-L) is obtained by second equation of formula (10) and substitutes into first equation and obtains formula (2)；

3) the state feedback equivalence conversion of response system

To response system (3), if fed back

U=- (cx₁-x₂-x₁x₃)+u₁ (12)

System is reduced to

Consider by system (13) by state transformation and further feed back be converted to more simply with system (11) it is increasingly similar Form, in order to design generalized synchronization control method, observing system (11) is the ODE of lower triangular form, to 3 Level system is namely to meet following form

Wish that system (13) can be converted to controlled lower triangular form, i.e.,

Wherein v is input, for this purpose, the shift vectors field of note system (13) is

And input vector field is

Enable vector field

Calculate following vector field Lie bracket

Pay attention toOrder is 2 in global scope, andIllustrate this distribution pairing；It enables

Calculate following vector field Lie bracket

In x₁When=0 or 2a-b=0It is order is still 2, this also illustrates that system (13) can not be achieved State feedback linearization；But as 2a-b ≠ 0,Only order is 2 in a null set, this collection Order is 3 except conjunction, so system (13) can be converted to lower cam system (15) by state transformation equivalence, however, needing to visit The problem of cam system under which kind of can be converted to by studying carefully system (13) actually, and intentionally get lower triangle relatively simple in form System, thus, it is noted that

Distribution in global scope at this timeOrder be 3 (in 2a-b ≠ 0) and pairing, enable

Take following distribution

Δ₀=span { X₀}；Δ₁=span { X₀,X₁}；Δ₂=span { X₀,X₁,X₂, (24)

Profile Δ₀,Δ₁,Δ₂And X₀,X₁,X₂It has the property that

1. can verify that [X₀,X₁]=0, [X₁,X₂]=0 and [X₀,X₂]=0

2. by 1., Δ₀,Δ₁,Δ₂It is involutive distribution；

3. existence converts h=(h by 1.₁(x),h₂(x),h₃(x))^T=H (x)=H (x₁,x₂,x₃) meet

4. due toIllustrative 3. in state transformation h under, system must still have lower triangular form；

3. above-mentioned property also implies that following 3 partial differential equations of satisfaction, the 1st group is

Wherein h₁(x) be smooth function, symbol " L " expression do Lie derivatives, the 2nd group is

Wherein h₂It (x) is smooth function, the 3rd group is

Wherein h₃It (x) is smooth function, the feasible solution of above-mentioned 3 groups of partial differential equation is respectively

In h=(h₁,h₂,h₃)^TSystem becomes under state

The system has actually had the form of lower cam system such as system (15), but in terms of second formula of system (30), Above system can be also further simplified by following state transformation

Such as use y=(y₁,y₂,y₃)^TState indicates that x state is then

The write through system under y state

Design input

u₁=-ax₁+ax₂+u₂, (34)

Above system is further simplified as

Preceding 2 equations of above system have been realized unanimously in form in the equivalent form (9) of drive system；

4) generalized synchronization

The stationary problem for considering system (35) and system (9) now, enabling the two state difference is e=η-y=(e₁,e₂,e₃)^T, then

Design of feedback

u₂=η₁-Kη₂-Lη₃-u₃ (37)

System representation is

For the subsystem of above system

Device control as follows can be designed according to the classical way of linear system:

Under the controller system (39) by finite-time control in, i.e. t₁Moment realizes e₂(t₁)=e₃(t₁)=0, design one Controller is planted from t₀Moment realizes e through finite time₂(t₁)=e₃(t₁)=0, and guarantee that control amount has continuously in the process for this First derivative and be transitioned into 0；Firstly, the e of design anticipation₂(t) it is

Wherein p (t) is polynomial of one indeterminate, due to requiring t₁Moment reaches the origin and u of system (39)₃In t > t₀Have in range Continuous first derivative, it means that e₂(t) in t₁When have continuous three order derivative, actually e₂(t) it is led with one, two, three ranks Count t again₁Moment is to guarantee continuously to be only 0, i.e.,

Consider further that the t of system (39)₀Moment should meet

Since formula (42) and formula (43) provide 6 conditions altogether, so p (t₀) it should be 5 order polynomials, recycle formula (42) to obtain

Wherein C₀And C₁For undetermined coefficient, obtained using the 1st formula of formula (43)

Again by the 2nd formula of formula (43)

Arrangement obtains

The e₂(t) meet the requirements of formula (42) and formula (43), then

And

Obvious e₂(t₁)=e₃(t₁)=u₃(t₁)=0；

In time t₁Later, first equation of system (38) becomesThis equation is obviously a wide range of asymptotically stability , thus system (38) a wide range of asymptotically stability, illustrating system (9), the realization under this control law is synchronous with system (35)；

The generalized synchronization problem of system (1) Yu system (3) is returned to, system (1) becomes system (9) by state transformation (7), system (3) become system (35) after having made feedback and state transformation, wherein state transformation needs composite type (29) and formula (31) is

And naming this state transformation is y=(y₁,y₂,y₃)^T=Y ((x₁,x₂,x₃)^T)=Y (x), and composite type (12), formula (34), Formula (37) and formula (40), control law are

Wherein u₃Expression formula see formula (49).

2. the broad sense applied to controlled the Lorenz system and Genesio-Tesi system of secret communication as described in claim 1 Synchronous method, which is characterized in that in the step 4), whether verifying generalized synchronization be may be implemented, process are as follows:

WhereinFor matrix2- norm, it is clear that at the control law u that formula (51) and formula (49) are determinedThen

Due to the nonnegativity of norm

Above formula illustrates that the requirement formula (6) of generalized synchronization meets, and chooses in formula (6)

3. as claimed in claim 1 or 2 applied to the controlled Lorenz system of secret communication and Genesio-Tesi system Generalized synchronization method, which is characterized in that the method also includes following steps:

5) according to the requirement of generalized synchronization, when the input of response system is set as

Wherein parameter t₁It can be used for adjusting the speed of generalized synchronization realization, the state transformation between phase space is set as

Wherein K and L is to meet the following formula

Real number, under above-mentioned setting, generalized synchronization is realized in drive system (1) and response system (3).