CN107086916B - Fractional order adaptive sliding mode control-based chaotic system synchronization method - Google Patents

Fractional order adaptive sliding mode control-based chaotic system synchronization method Download PDF

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CN107086916B
CN107086916B CN201710368880.1A CN201710368880A CN107086916B CN 107086916 B CN107086916 B CN 107086916B CN 201710368880 A CN201710368880 A CN 201710368880A CN 107086916 B CN107086916 B CN 107086916B
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杨宁宁
吴朝俊
贾嵘
韩宇超
徐诚
程书灿
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Xian University of Technology
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    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
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Abstract

The invention discloses a chaotic system synchronization method based on fractional order self-adaptive sliding mode control, which is implemented according to the following steps: step 1, selecting a driving chaotic system and calculating the state information quantity y of the driving chaotic system1,y2,y3(ii) a Step 2, calculating the state information quantity x of the following system of the driving chaotic system1,x2,x3(ii) a Step 3, obtaining an error system e according to the state information quantity of the driving chaotic system and the following system thereof1,e2,e3(ii) a Step 4, setting a fractional order sliding mode surface s (t); step 5, establishing an error feedback synchronous control rate u1,u2,u3And adding an adaptive coefficient. The synchronization of the driving chaotic system and the following system can be effectively realized.

Description

Fractional order adaptive sliding mode control-based chaotic system synchronization method
Technical Field
The invention belongs to the technical field of automatic control methods, and relates to a chaotic system synchronization method based on fractional order self-adaptive sliding mode control.
Background
Chaos is an apparent random motion that occurs in a deterministic system and is ubiquitous in natural and social sciences. The research of chaos is to find out the ordered information with practical value existing behind the chaos behavior of the system, thereby realizing the utilization of chaos. Chaotic synchronization is a key issue in chaotic applications, and has attracted attention because of its great application value in the field of information security such as secure communication.
The chaos synchronization is a chaos control in a broad sense, and a controlled chaotic system track moves according to a target chaotic system track, which is a control problem. The research of chaotic synchronization is not limited to synchronization between two systems, and is expanded to synchronization between a plurality of systems, namely synchronization of complex network node dynamics. In addition, the fractional calculus is popularized as the integral calculus, so that the rich dynamic behavior in the original integral system can be expanded, and the actual system in the nature can be simulated more truly.
The research at the present stage is generally based on integral order calculus, and the synchronization of the chaotic system is realized slowly.
Disclosure of Invention
The invention aims to provide a chaotic system synchronization method based on fractional order self-adaptive sliding mode control, which can effectively realize the synchronization of a driving chaotic system and a following system thereof.
The invention adopts the technical scheme that a chaotic system synchronization method based on fractional order self-adaptive sliding mode control is implemented according to the following steps:
step 1, selecting a driving chaotic system and calculating the state information quantity y of the driving chaotic system1,y2,y3；
Step 2, calculating the state information quantity x of the following system of the driving chaotic system1,x2,x3；
Step 3, obtaining an error system e according to the state information quantity of the driving chaotic system and the following system thereof1,e2,e3；
Step 4, setting a fractional order sliding mode surface s (t) to ensure that the driving chaotic system is converged on the sliding mode surface;
step 5, establishing an error feedback synchronous control rate u1,u2,u3Adding adaptationsAnd (4) the coefficient.
The present invention is also characterized in that,
state information quantity y of driving chaotic system in step 11,y2,y3The method specifically comprises the following steps:
Figure GDA0002409066720000021
Figure GDA0002409066720000022
Figure GDA0002409066720000023
wherein, U1To be applied at y1The control amount of (1); τ is a normalized function of time; u shape2To be applied at y2The control amount of (1); u shape3To be applied at y3M is the nonlinear part of the system, and q is the actual order of the error system.
State information quantity x of following system in step 21,x2,x3The method specifically comprises the following steps:
Figure GDA0002409066720000024
Figure GDA0002409066720000025
Figure GDA0002409066720000026
error system e in step 31,e2,e3The method specifically comprises the following steps:
Figure GDA0002409066720000031
Figure GDA0002409066720000032
Figure GDA0002409066720000033
the specific method for setting the fractional order sliding mode surface s (t) in the step 4 comprises the following steps:
step 4.1, selecting the definition of the fractional calculus, which specifically comprises the following steps:
Figure GDA0002409066720000034
wherein q is a fractional calculus and satisfies l-1 < q ≦ l, a is an initial value, t is an upper bound of the integral, and Γ (·) is a Gamma function.
Step 4.2, selecting a designated sliding mode surface, specifically:
Figure GDA0002409066720000035
wherein, the value range of i-1, 2,3, α is 0-1, q is the actual order of the error system, sgn is the sign function, and D in the formulaq-1The fractional calculus operator is specifically calculated by using a formula in the step 4.1, and the specific numerical value of the fractional calculus in the step 4.1 is calculated by an Oustaloup filter method.
Constructing the error feedback synchronous control rate u in step 51,u2,u3The specific method for adding the adaptive coefficient comprises the following steps:
step 5.1, establishing an error feedback synchronous control rate:
U(t)＝-By-g(y)+Ax+f(x)-sgn(ei)|ei|α
b and A are coefficient matrixes of the driving chaotic system and the following system thereof, g (y) and f (x) are nonlinear term parts in the driving chaotic system and the following system thereof respectively, and when i is 1,2 and 3, synchronous control rates u are obtained respectively1,u2,u3；
Step 5.2, the synchronous control rate obtained in the step 5.1 is givenSelf-adaptive coefficient is added in the method, so that the system can be defined according to faster convergence of error magnitude
Figure GDA0002409066720000041
Then the process of the first step is carried out,
Figure GDA0002409066720000042
in the formula, θ is an adaptive coefficient, and n is 3.
The method has the advantages that the synchronization of the chaotic system is realized, the real-time feedback can be realized according to the error information, the system is adaptively adjusted according to the error information, and the synchronization of the driving chaotic system and the following system is realized.
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FIG. 1 is a time domain waveform of a following system and a driving system x1-y1 in a chaotic system synchronization method based on fractional order adaptive sliding mode control according to the present invention;
FIG. 2 is a time domain waveform of a following system and a driving system x2-y2 of a chaotic system synchronization method based on fractional order adaptive sliding mode control in the invention;
FIG. 3 is a time domain waveform of a following system and a driving system x3-y3 of a chaotic system synchronization method based on fractional order adaptive sliding mode control in the invention;
FIG. 4 is a time domain waveform of an error system in the chaotic system synchronization method based on fractional order adaptive sliding mode control according to the present invention.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
The invention discloses a chaotic system synchronization method based on fractional order self-adaptive sliding mode control, which is implemented according to the following steps:
if q is 0.95 and m is 7, then:
step 1, selecting a driving chaotic system and calculating the state information quantity y of the driving chaotic system1,y2,y3(ii) a The method specifically comprises the following steps:
Figure GDA0002409066720000051
Figure GDA0002409066720000052
Figure GDA0002409066720000053
wherein, U1To be applied at y1The control amount of (1); τ is a normalized function of time; u shape2To be applied at y2The control amount of (1); u shape3To be applied at y3The control amount of (3).
Step 2, calculating the state information quantity x of the following system of the driving chaotic system1,x2,x3(ii) a The method specifically comprises the following steps:
Figure GDA0002409066720000054
Figure GDA0002409066720000055
Figure GDA0002409066720000056
step 3, obtaining an error system e according to the state information quantity of the driving chaotic system and the following system thereof1,e2,e3(ii) a The method specifically comprises the following steps:
Figure GDA0002409066720000057
Figure GDA0002409066720000058
Figure GDA0002409066720000059
step 4, setting a fractional order sliding mode surface s (t) to ensure that the driving chaotic system is converged on the sliding mode surface; the specific method comprises the following steps:
step 4.1, selecting the definition of the fractional calculus, which specifically comprises the following steps:
Figure GDA00024090667200000510
wherein q is a fractional calculus and satisfies l-1 < q ≦ l, a is an initial value, t is an upper bound of the integral, and Γ (·) is a Gamma function.
Step 4.2, selecting a designated sliding mode surface, specifically:
Figure GDA0002409066720000061
wherein, the value range of i-1, 2,3, α is 0-1, q is the actual order of the error system, sgn is the sign function, and D in the formulaq-1The fractional calculus operator is specifically calculated by using a formula in the step 4.1, and the specific numerical value of the fractional calculus in the step 4.1 is calculated by an Oustaloup filter method.
Step 5, establishing an error feedback synchronous control rate u1,u2,u3Adding adaptive coefficients; the specific method comprises the following steps:
step 5.1, establishing an error feedback synchronous control rate:
U(t)＝-By-g(y)+Ax+f(x)-sgn(ei)|ei|α
b and A are coefficient matrixes of the driving chaotic system and the following system thereof, g (y) and f (x) are nonlinear term parts in the driving chaotic system and the following system thereof respectively, and when i is 1,2 and 3, synchronous control rates u are obtained respectively1,u2,u3；
Step 5.2, adding an adaptive coefficient to the synchronous control rate obtained in the step 5.1, and defining the system according to faster convergence of error magnitude
Figure GDA0002409066720000062
Then the process of the first step is carried out,
Figure GDA0002409066720000063
in the formula, θ is an adaptive coefficient, and n is 3.
In step 4.2 of the method, q is the actual order of the error system, and q is 0.95.
According to the Lyapunov stability law, a Lyapunov function is always positive, and the control rate of the method can be positively fixed on an error system in the acquired sliding mode, which proves that the process is as follows:
Figure GDA0002409066720000071
b in the above formulaijFor the coefficient matrix of the error system, it can be seen that the lyapunov function satisfies the following relationship:
Figure GDA0002409066720000072
and (3) after simplification:
Figure GDA0002409066720000073
it can be seen that the simplified equation is always negative, and from the above proof, it can be seen that the derivative of the lyapunov function is always less than zero, so the system can be stable on the sliding mode surface.
After Matlab and Simulink simulation, the results are shown in FIGS. 1-4, all variables of the following system and the driving chaotic system realize chaotic synchronization, and when the system starts to intervene in the control rate within 20 seconds, the system converges within 2.5 seconds, so that the fractional order sliding mode control has better performance.

Claims (3)

1. A chaotic system synchronization method based on fractional order adaptive sliding mode control is characterized by comprising the following steps:
step 1, selecting a driving chaotic system and calculating by utilizing a system dynamics equation to obtain a chaotic system state information quantity y1,y2,y3；
Step 2, calculating the state information quantity x of the following system of the driving chaotic system1,x2,x3；
Step 3, obtaining an error system e according to the state information quantity of the driving chaotic system and the following system thereof1,e2,e3；
Step 4, setting a fractional order sliding mode surface s (t) to ensure that the driving chaotic system is converged on the sliding mode surface;
step 5, establishing an error feedback synchronous control rate u1,u2,u3Adding adaptive coefficients;
the state information quantity y of the driving chaotic system in the step 11,y2,y3The method specifically comprises the following steps:
Figure FDA0002409066710000011
Figure FDA0002409066710000012
Figure FDA0002409066710000013
wherein, U1To be applied at y1The control amount of (1); τ is a normalized function of time; u shape2To be applied at y2The control amount of (1); u shape3To be applied at y3Q is the actual order of the error system, and m is the nonlinear part of the system;
the specific method for setting the fractional order sliding mode surface s (t) in the step 4 comprises the following steps:
step 4.1, selecting the definition of the fractional calculus, which specifically comprises the following steps:
Figure FDA0002409066710000014
wherein q is a fractional calculus and satisfies l-1 < q ≦ l, a is an initial value, t is an upper bound of the integral, and Γ (·) is a Gamma function;
step 4.2, selecting a designated sliding mode surface, specifically:
Figure FDA0002409066710000021
wherein, the value range of i-1, 2,3, α is 0-1, q is the actual order of the error system, sgn is the sign function, and D in the formulaq-1The fractional calculus operator is specifically calculated by using a formula in the step 4.1, and the specific numerical value of the fractional calculus in the step 4.1 is calculated by an Oustaloup filter method;
constructing the error feedback synchronous control rate u in step 51,u2,u3The specific method for adding the adaptive coefficient comprises the following steps:
step 5.1, establishing an error feedback synchronous control rate:
U(t)＝-By-g(y)+Ax+f(x)-sgn(ei)|ei|α
b and A are coefficient matrixes of the driving chaotic system and the following system thereof, g (y) and f (x) are nonlinear term parts in the driving chaotic system and the following system thereof respectively, and when i is 1,2 and 3, synchronous control rates u are obtained respectively1,u2,u3；
Step 5.2, adding an adaptive coefficient to the synchronous control rate obtained in the step 5.1, and defining the system according to faster convergence of error magnitude
Figure FDA0002409066710000022
Then the process of the first step is carried out,
Figure FDA0002409066710000023
in the formula, θ is an adaptive coefficient, and n is 3.
2. The chaotic system synchronization method based on fractional order adaptive sliding mode control as claimed in claim 1, wherein the state information quantity x of the following system in step 21,x2,x3The method specifically comprises the following steps:
Figure FDA0002409066710000024
Figure FDA0002409066710000025
Figure FDA0002409066710000026
3. the chaotic system synchronization method based on fractional order adaptive sliding mode control according to claim 2, characterized in that the error system e in step 31,e2,e3The method specifically comprises the following steps:
Figure FDA0002409066710000031
Figure FDA0002409066710000032
Figure FDA0002409066710000033
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