CN109412804A - Controlled Shimizu-Morioka system and Chen system generalized synchronization method applied to secret communication - Google Patents
Controlled Shimizu-Morioka system and Chen system generalized synchronization method applied to secret communication Download PDFInfo
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/001—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols using chaotic signals
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/12—Transmitting and receiving encryption devices synchronised or initially set up in a particular manner
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Abstract
A kind of controlled Shimizu-Morioka system applied to secret communication and Chen system generalized synchronization method, 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 Shimizu-Morioka system applied to secret communication and Chen system, using Chen system as drive system, using the controlled Shimizu-Morioka 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
Technical field
It is mixed with Chen the invention belongs to can be applied to the Chaotic Synchronous technical field of secret communication more particularly to a kind of realization
Ignorant system is drive system, using the controlled Shimizu-Morioka system of single input as the chaos synchronization of response system.
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 Shimizu-Morioka system and Chen system generalized synchronization method of communication, using Chen system as drive system, with list
The controlled Shimizu-Morioka system of input is response system, is designed using the Lie derivatives method of vector field in Differential Geometry
A kind of Chaotic Synchronous algorithm, realizes generalized synchronization, and Control platform is higher.
The technical solution adopted by the present invention to solve the technical problems is:
A kind of controlled Shimizu-Morioka system applied to secret communication and Chen system generalized synchronization method, packet
Include following steps:
1) generalized chaotic synchronization problem describes
Drive system is Chen system, and form is as follows:
Wherein x=(x1,x2,x3)TIt is state variable, a, b and c are known positive real number parameter, 2a-b ≠ 0；
Using controlled Shimizu-Morioka system as response system, form is as follows:
Wherein ξ=(ξ1,ξ2,ξ3)TIt is state variable, u is scalar input, and α and β are known positive real number parameter in system,
B in this β and formula (1) is equivalent, i.e. β=b；
The target to be realized of generalized chaotic synchronization is: being respectively x (t in drive system (1) and response system (2) initial value0)
With ξ (t0), response system track is fed back by state
U=u (x, ξ, t) (3)
Wherein t indicates the state transformation between time and phase space
ξ=T (x) (4)
It is intended to the track of drive system afterwards, i.e.,
Here | | | | it representsThe 2- norm of vector in space；
2) state transformation and feedback of response system
Following state transformation η=S (ξ) wherein η=(η is made to response system (2)1,η2,η3)T
So this is a linear transformation, MSFor 3 rank square matrixes, this linear transformation is inversely transformed into
Using η as state, system representation is
It feeds back
U=- (1- η1)η2+αη3+u0 (9)
Consider further that β=b, system is reduced to
The system belongs to controlled lower cam system, and the general type of cam system is under three ranks
Wherein w is input control quantity；Latter two equation of another aspect observing system (10) has been actually formed linear system
Form, so system (10) is to realize Partial Linear；
3) the state conversion of drive system
In order to look for the state transformation of drive system (1) to simplify system, first system becomes plus control amount thus
Wherein v is the input control quantity being added；
System (12) is fed back
V=(a-c) x1-cx2+x1x3+v1 (13)
System is reduced to
Consider by system (14) by state transformation and further feed back be converted to more simply with system (10) more
Similar form, in order to design generalized synchronization control method；Due to the controlled ordinary differential side that system (10) are lower triangular form
Journey, it is desirable to system (14) can be converted to same the latter's similar form,
For this purpose, the shift vectors field of note system (14) 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 (14) can not be real
Present condition feedback linearization；But as 2a-b ≠ 0,Only order is 2 in a null set, this collection
Order is 3 except conjunction, so system (14) can be converted to lower cam system (11) by state transformation equivalence, however, still needing to
The problem of probing into system (14) actually and can be converted to which kind of lower cam system, and intentionally get in form relatively simple lower three
Angle system, thus, it is noted that
Distribution in global scope at this timeOrder be 3 and pairing, enable
Take following distribution
Δ0=span { X0}；Δ1=span { X0,X1}；Δ2=span { X0,X1,X2, (23)
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 a form of lower cam system such as system (11), but from second formula of system (29)
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
Become known to comparison system (12) and system (32) by state change (30) system (1)
Preceding 2 equations of above system have been realized unanimously in form in the equivalent form (10) of drive system；
4) generalized synchronization
The stationary problem for considering system (33) and system (10) now, enabling the two state difference is e=η-y=(e1,e2,e3)T,
Then
Design of feedback
System is represented by
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 (37) 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, there is 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 (37)1In 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 (37)0Moment should meet
Since formula (40) and formula (41) provide 6 conditions altogether, so p (t0) it should be 5 order polynomials, recycle formula (40) to obtain
Wherein C0And C1For undetermined coefficient, obtained using the 1st formula of formula (41)
Again by the 2nd formula of formula (41)
Arrangement obtains
The e2(t) meet the requirements of formula (40) and formula (41), then
And
Obvious e2(t1)=e3(t1)=u1(t1)=0；
In time t1Later, first equation of system (36) becomesThis equation is obviously a wide range of progressive steady
Fixed, thus system (36) a wide range of asymptotically stability, illustrating system (10), the realization under this control law is synchronous with system (33)；
The generalized synchronization problem of system (1) Yu system (2) is returned to, system (2) becomes by feedback and state transformation (6) is
It unites (10), system (1) becomes system (33) after having made state transformation, and state transformation therebetween needs composite type (28) and formula (30)
And naming this state transformation is y=(y1,y2,y3)T=Y ((x1,x2,x3)T)=Y (x), and control law visible
(35), wherein u1Expression formula see formula (47).
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 (35) and formula (47) are determinedThen
Due to the nonnegativity of norm
Above formula illustrates that the requirement formula (5) of generalized synchronization meets, and chooses in formula (5)
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
Under above-mentioned setting, generalized synchronization is realized in drive system (1) and response system (2).
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 Shimizu-Morioka chaos system Yu Chen 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, the speed that generalized synchronization is realized is adjusted.
Detailed description of the invention
Fig. 1 is 3 dimension phasors of Chen system, parameter a=35, b=3 and c=28, initial value x1(t0)=1, x2
(t0)=1, x3(t0)=20；
Fig. 2 be in response to the i.e. controlled Shimizu-Morioka system of system 3 dimension phasors, parameter alpha=0.75, β=3, just
Value is ξ1(t0)=1.0286, ξ2(t0)=1, ξ3(t0)=1.017；
Fig. 3 is the asymptotically stability of error system, initial value e1(t0)=1, e2(t0)=1, e3(t0)=1；
Fig. 4 is control amount u1, 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 Shimizu-Morioka system and Chen system applied to secret communication are wide
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 Chen system, and Chen system is big in houston, U.S.A at that time
Professor Chen Guanrong has found and the chaos system for proving that it had previously found with Lorenz system and Rossler system etc. cannot
Homeomorphic is a completely new chaos attractor, and concrete form is as follows:
Wherein x=(x1,x2,x3)TIt is state variable, a, b and c are known positive real number parameters, limit 2a-b in the present invention
≠0；
Shimizu-Morioka system was suggested in 1980 by T.Shimizu and N.Moriok, it has been found that it has
Chaos phenomenon, this system, which is also verified, to be realized with circuit.Using controlled Shimizu-Morioka system as response system, tool
Body form is as follows:
Wherein ξ=(ξ1,ξ2,ξ3)TIt is state variable, u is scalar input, and α and β are known positive real number parameter in system,
B in this β and formula (1) is equivalent, i.e. β=b；If inputting u is 0, when α=0.75, β=0.45, system has chaos locus；
The target to be realized of generalized chaotic synchronization is: being respectively x (t in drive system (1) and response system (2) initial value0)
With ξ (t0), response system track is fed back by state
U=u (x, ξ, t) (3)
Wherein t indicates the state transformation between time and phase space
ξ=T (x) (4)
It is intended to the track of drive system afterwards, i.e.,
Here | | | | it representsThe 2- norm of vector in space；
2) state transformation and feedback of response system
Following state transformation η=S (ξ) wherein η=(η is made to response system (2)1,η2,η3)T
So this is a linear transformation, MSFor 3 rank square matrixes, this linear transformation is inversely transformed into
Using η as state, system representation is
It feeds back
U=- (1- η1)η2+αη3+u0 (9)
Consider further that β=b, system is reduced to
The system belongs to controlled lower cam system, and the general type of cam system is under three ranks
Wherein w is input control quantity；Latter two equation of another aspect observing system (10) has been actually formed linear system
Form, so system (10) is to realize Partial Linear；
3) the state conversion of drive system
In order to look for the state transformation of drive system (1) to simplify system, first system becomes plus control amount thus
Wherein v is the input control quantity being added；
System (12) is fed back
V=(a-c) x1-cx2+x1x3+v1 (13)
System is reduced to
Consider by system (14) by state transformation and further feed back be converted to more simply with system (10) more
Similar form, in order to design generalized synchronization control method；Due to the controlled ordinary differential side that system (10) are lower triangular form
Journey, it is desirable to system (14) can be converted to same the latter's similar form,
For this purpose, the shift vectors field of note system (14) 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 Lee
Form equal Differential Manifold 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 (14) can not be real
Present condition 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,The only order in a null set
Be 2, this set except order be 3, so system (14) can be converted to by state transformation equivalence lower cam system (11) (see
Celikovsky S,Nijmeijer H.Equivalence of nonlinear systems to triangular form:
The singular case.Systems&Control Letters, 1996,27:135-144.), however, still needing to the system of probing into
(14) the problem of cam system under which kind of can be converted to actually, and lower cam system relatively simple in form is intentionally got,
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, (23)
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 old
Save the Beijing Differential Geometry handout (second edition) such as body, BJ University Press, 2001.)
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 (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 (11), but from second formula of system (29)
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
Become known to comparison system (12) and system (32) by state change (30) system (1)
Preceding 2 equations of above system have been realized unanimously in form in the equivalent form (10) of drive system；
4) generalized synchronization
The stationary problem for considering system (33) and system (10) now, enabling the two state difference is e=η-y=(e1,e2,e3)T,
Then
Design of feedback
System is represented by
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 (37) 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, there is 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 (37)1In 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 (37)0Moment should meet
Since formula (40) and formula (41) provide 6 conditions altogether, so p (t0) it should be 5 order polynomials, recycle formula (40) to obtain
Wherein C0And C1For undetermined coefficient, obtained using the 1st formula of formula (41)
Again by the 2nd formula of formula (41)
Arrangement obtains
The e2(t) meet the requirements of formula (40) and formula (41).So
And
Obvious e2(t1)=e3(t1)=u1(t1)=0；
In time t1Later, first equation of system (36) becomesThis equation is obviously a wide range of progressive steady
Fixed, thus system (36) a wide range of asymptotically stability, illustrating system (10), the realization under this control law is synchronous with system (33)；
The generalized synchronization problem of system (1) Yu system (2) is returned to, system (2) becomes by feedback and state transformation (6) is
It unites (10), system (1) becomes system (33) after having made state transformation, and state transformation therebetween needs composite type (28) and formula (30)
And naming this state transformation is y=(y1,y2,y3)T=Y ((x1,x2,x3)T)=Y (x), and control law visible
(35), wherein u1Expression formula see formula (47)；
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 (35) and formula (47) are determinedThen
Due to the nonnegativity of norm
Above formula illustrates that the requirement formula (5) of generalized synchronization meets, and chooses in formula (5)
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
Under above-mentioned setting, generalized synchronization is realized in drive system (1) and response system (2).
To verify this generalized synchronization technology, phasor is tieed up using Matlab software emulation Chen system, that is, drive system 3
(see Fig. 1)；3 dimension phasors of the i.e. controlled Shimizu-Morioka system of response system are simulated, wherein the setting of controller u is abided by
Formula (52) (see Fig. 2) is followed；Error system (36) and its controller are simulated, wherein controller u1Also in compliance with formula (52) (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 (30) under x state in Fig. 1
Become the initial value under y state afterwards, the initial value of response system becomes under η state after state transformation (6) under ξ 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 u1
In t1Place has continuous first derivative (see Fig. 4), but without second dervative；If designing control amount u1In 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 controlled Shimizu-Morioka system applied to secret communication and Chen system generalized synchronization method, special
Sign is, comprising the following steps:
1) generalized chaotic synchronization problem describes
Drive system is Chen system, and form is as follows:
Wherein x=(x1,x2,x3)TIt is state variable, a, b and c are known positive real number parameter, 2a-b ≠ 0；
Using controlled Shimizu-Morioka system as response system, form is as follows:
Wherein ξ=(ξ1,ξ2,ξ3)TState variable, u is scalar input, and α and β are known positive real number parameter in system, this β with
B in formula (1) is equivalent, i.e. β=b；
The target to be realized of generalized chaotic synchronization is: being respectively x (t in drive system (1) and response system (2) initial value0) and ξ
(t0), response system track is fed back by state
U=u (x, ξ, t) (3)
Wherein t indicates the state transformation between time and phase space
ξ=T (x) (4)
It is intended to the track of drive system afterwards, i.e.,
Here | | | | it representsThe 2- norm of vector in space；
2) state transformation and feedback of response system
Following state transformation η=S (ξ) wherein η=(η is made to response system (2)1,η2,η3)T
So this is a linear transformation, MSFor 3 rank square matrixes, this linear transformation is inversely transformed into
Using η as state, system representation is
It feeds back
U=- (1- η1)η2+αη3+u0 (9)
Consider further that β=b, system is reduced to
The system belongs to controlled lower cam system, and the general type of cam system is under three ranks
Wherein w is input control quantity；Latter two equation of another aspect observing system (10) has been actually formed linear system form,
So system (10) is to realize Partial Linear；
3) the state conversion of drive system
In order to look for the state transformation of drive system (1) to simplify system, first system becomes plus control amount thus
Wherein v is the input control quantity being added；
System (12) is fed back
V=(a-c) x1-cx2+x1x3+v1 (13)
System is reduced to
Consider by system (14) by state transformation and further feed back be converted to more simply with system (10) it is increasingly similar
Form, in order to design generalized synchronization control method；Due to the controlled ODE that system (10) are lower triangular form, wish
Prestige system (14) can be converted to same the latter's similar form,
For this purpose, the shift vectors field of note system (14) 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 (14) can not be achieved shape
State feedback linearization；But as 2a-b ≠ 0,Only order is 2 in a null set, this set
Except order be 3, so system (14) can be converted to lower cam system (11) by state transformation equivalence, however, still needing to visit
The problem of cam system under which kind of can be converted to by studying carefully system (14) 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 and pairing, enable
Take following distribution
Δ0=span { X0}；Δ1=span { X0,X1}；Δ2=span { X0,X1,X2, (23)
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 (11), but in terms of second formula of system (29),
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
Become known to comparison system (12) and system (32) by state change (30) system (1)
Preceding 2 equations of above system have been realized unanimously in form in the equivalent form (10) of drive system；
4) generalized synchronization
The stationary problem for considering system (33) and system (10) now, enabling the two state difference is e=η-y=(e1,e2,e3)T, then
Design of feedback
System is represented by
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 (37) by finite-time control in, i.e. t1Moment realizes e2(t1)=e3(t1)=0, but should
The control amount of controller is in t1Moment is not still 0, it is easy to which control is excessive, there is certain defect；For this purpose, designing a kind of controller
From t0Moment realizes e through finite time2(t1)=e3(t1)=0, and guarantee that control amount has continuous single order to lead in the process for this
It counts and is 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 (37)1In 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 (37)0Moment should meet
Since formula (40) and formula (41) provide 6 conditions altogether, so p (t0) it should be 5 order polynomials, recycle formula (40) to obtain
Wherein C0And C1For undetermined coefficient, obtained using the 1st formula of formula (41)
Again by the 2nd formula of formula (41)
Arrangement obtains
The e2(t) meet the requirements of formula (40) and formula (41), then
And
Obvious e2(t1)=e3(t1)=u1(t1)=0；
In time t1Later, first equation of system (36) becomesThis equation is obviously a wide range of asymptotically stability,
To system (36) a wide range of asymptotically stability, illustrating system (10), the realization under this control law is synchronous with system (33)；
The generalized synchronization problem of system (1) Yu system (2) is returned to, system (2) becomes system by feedback and state transformation (6)
(10), system (1) becomes system (33) after having made state transformation, and state transformation therebetween needs composite type (28) and formula (30)
And naming this state transformation is y=(y1,y2,y3)T=Y ((x1,x2,x3)T)=Y (x), and control law visible (35),
Middle u1Expression formula see formula (47).
2. as described in claim 1 applied to the controlled Shimizu-Morioka system of secret communication and Chen system broad sense
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 (35) and formula (47) are determinedThen
Due to the nonnegativity of norm
Above formula illustrates that the requirement formula (5) of generalized synchronization meets, and chooses in formula (5)
3. wide applied to the controlled Shimizu-Morioka system of secret communication and Chen system as claimed in claim 1 or 2
Adopted synchronous 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
Under above-mentioned setting, generalized synchronization is realized in drive system (1) and response system (2).
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