CN107086916A - A kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer - Google Patents
A kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer Download PDFInfo
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- 230000000739 chaotic effect Effects 0.000 title claims abstract description 22
- 230000003044 adaptive effect Effects 0.000 title claims abstract description 17
- 230000006978 adaptation Effects 0.000 claims abstract description 12
- 238000000034 method Methods 0.000 claims description 14
- 239000011159 matrix material Substances 0.000 claims description 4
- 230000010354 integration Effects 0.000 claims description 3
- 230000001360 synchronised effect Effects 0.000 abstract description 6
- 238000009415 formwork Methods 0.000 description 2
- 230000009286 beneficial effect Effects 0.000 description 1
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- 238000005516 engineering process Methods 0.000 description 1
- 238000004088 simulation Methods 0.000 description 1
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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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- 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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Abstract
The invention discloses a kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer, specifically implement according to following steps：Step 1, selection drives chaos system and calculates the amount of state information y of the driving chaos system1,y2,y3；Step 2, the amount of state information x of the system for tracking of the driving chaos system is calculated1,x2,x3；Step 3, error system e is tried to achieve according to the amount of state information of the driving chaos system and its system for tracking1,e2,e3；Step 4, fractional order sliding-mode surface s (t) is set；Step 5, error feedback synchronization control rate u is built1,u2,u3, add adaptation coefficient.It can effectively realize that driving chaos system is synchronous with its system for tracking.
Description
Technical field
The invention belongs to autocontrol method technical field, it is related to a kind of chaos based on fractional order adaptive sliding-mode observer
System synchronization method.
Background technology
Chaos is that occur it is determined that random motion seemingly in system, is prevalent in natural science and social science
In.The research of chaos is the information orderly, with practical value existed behind for discovery chaotic systems behavior, so that real
Now to the utilization of chaos.Chaotic Synchronous as chaos applications emphasis problem, because it is in information security fields such as secret communications
Huge applications are worth and received much concern.
The synchronization of chaos allows controlled chaos system track according to target chaos system from will be broadly a kind of control of chaos
System track motion, is a class control problem.The research of Chaotic Synchronous is not limited to the synchronization between two systems, has been extended to multiple systems
Synchronization between system, the i.e. dynamic (dynamical) synchronization of complex network node.In addition, fractional calculus pushing away as integer rank calculus
Extensively, the abundant dynamic behavior during original integer level can be united is expanded, reality that can be in truer simulation nature
Border system.
Research at this stage is generally basede on integer rank calculus, realizes that Synchronization of Chaotic Systems is slower.
The content of the invention
It is an object of the invention to provide a kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer, can have
Effect realizes that driving chaos system is synchronous with its system for tracking.
The technical solution adopted in the present invention is, a kind of Synchronization of Chaotic Systems side based on fractional order adaptive sliding-mode observer
Method, specifically implements according to following steps：
Step 1, selection drives chaos system and calculates the amount of state information y of the driving chaos system1,y2,y3；
Step 2, the amount of state information x of the system for tracking of the driving chaos system is calculated1,x2,x3；
Step 3, error system e is tried to achieve according to the amount of state information of the driving chaos system and its system for tracking1,e2,e3；
Step 4, fractional order sliding-mode surface s (t) is set, it is ensured that driving chaos system is restrained on this sliding-mode surface；
Step 5, error feedback synchronization control rate u is built1,u2,u3, add adaptation coefficient.
The features of the present invention is also resided in,
The amount of state information y of driving chaos system in step 11,y2,y3Specially：
Wherein, U1To be applied to y1On controlled quentity controlled variable；τ is the function of time after a normalized；U2To be applied to y2
On controlled quentity controlled variable；U3To be applied to y3On controlled quentity controlled variable, m be the mission nonlinear part, q be error system actual exponent number.
The amount of state information x of system for tracking in step 21,x2,x3Specially：
Error system e in step 31,e2,e3Specially：
The setting fractional order sliding-mode surface s (t) of step 4 specific method is：
Step 4.1, the definition of fractional calculus is selected, is specially：
In formula, q is fractional calculus number and meets l-1 ＜ q≤l, and a is the lower bound that initial value t is integration, and Γ () is
Gamma functions.
Step 4.2, choose and specify sliding-mode surface, be specially：
Wherein, i=1,2,3, α span is 0-1, and q is the actual exponent number of error system, and sgn is sign function, public
D in formulaq-1It is fractional calculus operator, is specifically obtained using the formula in step 4.1, step 4.1 mid-score rank calculus
Concrete numerical value is tried to achieve by Oustaloup filtered methods.
Structure error feedback synchronization control rate u described in step 51,u2,u3, adding adaptation coefficient specific method is：
Step 5.1, error feedback synchronization control rate is built：
U (t)=- Bx-g (y)+Ax+f (x)-sgn (ei)|ei|α
Wherein, B and A is driving chaos system and the coefficient matrix of its system for tracking, and g (y) and f (x) are driving chaos system
Nonlinear terms part in system, works as i=1, when 2,3, respectively obtains Synchronization Control rate u1,u2,u3；
Step 5.2, adaptation coefficient is added in the Synchronization Control rate tried to achieve to step 5.1, system is allowed according to error size
Convergence faster, definitionThen,
In formula, θ is adaptation coefficient, n=3.
The beneficial effects of the invention are as follows a kind of method of fractional order sliding formwork control realizes the synchronization of chaos system, Neng Gougen
According to control information Real-time Feedback, according to control information self-adaption regulation system, driving chaos system and its system for tracking are realized
It is synchronous.
Brief description of the drawings
Fig. 1 for system for tracking in a kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer of the present invention with
Drive system x1-y1 time domain waveforms；
Fig. 2 for the present invention in a kind of Synchronization of Chaotic Systems system for tracking based on fractional order adaptive sliding-mode observer with
Drive system x2-y2 time domain waveforms；
Fig. 3 for the present invention in a kind of Synchronization of Chaotic Systems system for tracking based on fractional order adaptive sliding-mode observer with
Drive system x3-y3 time domain waveforms；
Fig. 4 is in a kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer of the present invention during error system
Domain waveform.
Embodiment
The present invention is described in detail with reference to the accompanying drawings and detailed description.
A kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer of the present invention, specifically according to following step
It is rapid to implement：
Q=0.95, m=7 are taken, then：
Step 1, selection drives chaos system and calculates the amount of state information y of the driving chaos system1,y2,y3；Specially：
Wherein, U1To be applied to y1On controlled quentity controlled variable；τ is the function of time after a normalized；U2To be applied to y2
On controlled quentity controlled variable；U3To be applied to y3On controlled quentity controlled variable.
Step 2, the amount of state information x of the system for tracking of the driving chaos system is calculated1,x2,x3；Specially：
Step 3, error system e is tried to achieve according to the amount of state information of the driving chaos system and its system for tracking1,e2,e3；
Specially：
Step 4, fractional order sliding-mode surface s (t) is set, it is ensured that driving chaos system is restrained on this sliding-mode surface；Specific method
For：
Step 4.1, the definition of fractional calculus is selected, is specially：
In formula, q is fractional calculus number and meets l-1 ＜ q≤l, and a is the lower bound that initial value t is integration, and Γ () is
Gamma functions.
Step 4.2, choose and specify sliding-mode surface, be specially：
Wherein, i=1,2,3, α span is 0-1, and q is the actual exponent number of error system, and sgn is sign function, public
D in formulaq-1It is fractional calculus operator, is specifically obtained using the formula in step 4.1, step 4.1 mid-score rank calculus
Concrete numerical value is tried to achieve by Oustaloup filtered methods.
Step 5, error feedback synchronization control rate u is built1,u2,u3, add adaptation coefficient；Specific method is：
Step 5.1, error feedback synchronization control rate is built：
U (t)=- Bx-g (y)+Ax+f (x)-sgn (ei)|ei|α
Wherein, B and A is driving chaos system and the coefficient matrix of its system for tracking, and g (y) and f (x) are driving chaos system
Nonlinear terms part in system, works as i=1, when 2,3, respectively obtains Synchronization Control rate u1,u2,u3；
Step 5.2, adaptation coefficient is added in the Synchronization Control rate tried to achieve to step 5.1, system is allowed according to error size
Convergence faster, definitionThen,
In formula, θ is adaptation coefficient, n=3.
Q is the actual exponent number of error system in the step 4.2 of the present invention, and q takes 0.95.
According to Liapunov stability law, there is liapunov function positive definite always, for the present invention's
Control rate can realize positive definite in taken sliding-mode surface for error system, it was demonstrated that process is as follows：
B in above-mentioned formulaijFor the coefficient matrix of error system, it can be seen that liapunov function meets following relation：
After abbreviation：
It can be seen that the equation perseverance after abbreviation is negative, the derivative of liapunov function is all the time it can be seen from above-mentioned proof
Less than zero, therefore system can be stablized on sliding-mode surface.
After Matlab and Simulink emulation, as a result as Figure 1-4, system for tracking and driving chaos system
All variables all realize Chaotic Synchronous, and when 20 seconds systems start intervention control rate, system restrained in 2.5 seconds, embodied
Fractional order sliding formwork control has preferable performance capabilities.
Claims (6)
1. a kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer, it is characterised in that specifically according to following
Step is implemented：
Step 1, selection, which is driven chaos system and calculated using the system dynamics equation, obtains state of chaotic system information content y1,
y2,y3；
Step 2, the amount of state information x of the system for tracking of the driving chaos system is calculated1,x2,x3；
Step 3, error system e is tried to achieve according to the amount of state information of the driving chaos system and its system for tracking1,e2,e3；
Step 4, fractional order sliding-mode surface s (t) is set, it is ensured that driving chaos system is restrained on this sliding-mode surface；
Step 5, error feedback synchronization control rate u is built1,u2,u3, add adaptation coefficient.
2. a kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer according to claim 1, it is special
Levy and be, the amount of state information y of the driving chaos system described in step 11,y2,y3Specially：
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Wherein, U1To be applied to y1On controlled quentity controlled variable；τ is the function of time after a normalized；U2To be applied to y2On
Controlled quentity controlled variable；U3To be applied to y3On controlled quentity controlled variable, q be error system actual exponent number, m be the mission nonlinear part.
3. a kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer according to claim 2, it is special
Levy and be, the amount of state information x of the system for tracking described in step 21,x2,x3Specially：
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4. a kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer according to claim 3, it is special
Levy and be, the error system e described in step 31,e2,e3Specially：
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5. a kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer according to claim 4, it is special
Levy and be, the specific method of the setting fractional order sliding-mode surface s (t) described in step 4 is：
Step 4.1, the definition of fractional calculus is selected, is specially：
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1
In formula, q is fractional calculus number and meets l-1 ＜ q≤l, and a is the lower bound that initial value t is integration, and Γ () is Gamma
Function；
Step 4.2, choose and specify sliding-mode surface, be specially：
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Wherein, i=1,2,3, α span is 0-1, and q is the actual exponent number of error system, during sgn is sign function, formula
Dq-1Be fractional calculus operator, specifically obtained using the formula in step 4.1, step 4.1 mid-score rank calculus it is specific
Numerical value is tried to achieve by Oustaloup filtered methods.
6. a kind of Synchronization of Chaotic Systems based on fractional order adaptive sliding-mode observer according to claim 5, it is special
Levy and be, the structure error feedback synchronization control rate u described in step 51,u2,u3, adding adaptation coefficient specific method is：
Step 5.1, error feedback synchronization control rate is built：
U (t)=- Bx-g (y)+Ax+f (x)-sgn (ei)|ei|α
Wherein, B and A is driving chaos system and the coefficient matrix of its system for tracking, and g (y) and f (x) is in driving chaos systems
Nonlinear terms part, work as i=1, when 2,3, respectively obtain Synchronization Control rate u1,u2,u3；
Step 5.2, add adaptation coefficient in the Synchronization Control rate tried to achieve to step 5.1, allow system according to error size faster
Convergence, definitionThen,
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In formula, θ is adaptation coefficient, n=3.
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