CN108931917A - A kind of tight feedback chaos projective synchronization method of three ranks - Google Patents
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Abstract
The present invention proposes a kind of tight feedback chaos projective synchronization method of three ranks, and process includes: to establish drive system and response system according to the state equation of the tight feedback chaos system of three ranks, establishes Projective Synchronization error system according to drive system and response system；Design non-linear global sliding mode face and adaptive exponentially approaching rule；All there is robustness in the overall process that system responds using the total-sliding-mode control device in non-linear global sliding mode face, and traditional sliding mode controller does not have robustness in reaching mode.Total-sliding-mode control device and adaptive sliding mode controller are combined, propose adaptive total-sliding-mode control device, modeling is not known by adaptive rate and external interference signals are estimated, all there is robustness in reaching mode and sliding formwork mode, it only needs single control input to can be achieved with the Projective Synchronization control of the tight feedback chaos of three ranks, and overcomes modeling uncertain and the influence of external interference signals.
Description
Technical field
The invention belongs to automatic control technology fields, and in particular to a kind of tight feedback chaos projective synchronization method of three ranks.
Background technique
Chaos is the tie for connecting regular motion and random motion, is widely present in nature and human society.
Since the concept of Mainieri and Rehacek proposition Projective Synchronization, different types of phase synchronization of two coupled chaotic is unified.Three ranks are sternly anti-
Feedback chaos only needs single input to can be achieved with Projective Synchronization, is with a wide range of applications in terms of secret communication.Due to depositing
Uncertain and external interference signals are being modeled, it is very tired for the Projective Synchronization of the tight feedback chaos system of different three ranks of original state
It is difficult.
Sliding formwork control has very strong robustness for modeling uncertain and external interference signals, and has fast response time
And the advantages that easy to accomplish, it is widely used in the control of nonlinear system.Using the total-sliding-mode control in non-linear global sliding mode face
There is device global robustness to have very big advantage compared with linear sliding mode face.Adaptive sliding mode controller can be by adaptive
Rate does not know modeling and the upper bound of external interference signals is estimated.By non-linear global sliding mode face and adaptive sliding-mode observer
Device combines, and the Projective Synchronization for designing adaptive total-sliding-mode control device for the tight feedback chaos of three ranks is very necessary.
Summary of the invention
Based on above technical problem, the present invention provides a kind of three ranks tight feedback chaos projective synchronization method, proposes non-
Linear global sliding mode face and adaptive exponentially approaching rule, using adaptive rate to the upper bound for modeling uncertain and external interference signals
Estimated, using single adaptive total-sliding-mode control device, in the case where modeling is not known with external interference signals, really
Protect the Projective Synchronization of different original state isomorphisms or the tight feedback chaos of three rank of isomery.
The tight feedback chaos projective synchronization method of a kind of three rank, comprising the following steps:
Step 1: drive system and response system being established according to the state equation of the tight feedback chaos system of three ranks, according to driving
System and response system establish Projective Synchronization error system:
Drive system is the tight feedback chaos system of three ranks, and state equation is as follows:
Wherein, x1, x2And x3For the state variable of system, x=[x1,x2,x3]T, fx(x, t) is continuous function, and t is the time.
Using formula (1) as drive system；
Response system is the tight feedback chaos system of three ranks, and state equation is as follows:
Wherein, y1, y2And y3For the state variable of system, y=[y1,y2,y3]T, fy(y, t) is continuous function, and t is the time；
With the uncertain controlled response system with external interference signals of modeling, state equation is as follows:
Wherein, △ f (y) is that modeling is not known, and d (t) is external interference signals, and u is control input, using formula (3) as sound
System is answered, f is worked asx(x, t) and fyWhen (y, t) has identical structure, drive system and response system are isomorphism chaos；Work as fx(x,t)
And fyWhen (y, t) has different structure, drive system and response system are isomery chaos；
Model uncertain △ f (y) and the equal bounded of external interference signals d (t), it may be assumed that
Wherein, d1For the upper bound for modeling uncertain △ f (y), d2For the upper bound of external interference signals d (t), and d1>=0, d2
≥0。d1And d2For unknown parameter, estimated using adaptive rate；
The Projective Synchronization error of drive system and response system is ei=yi-kxi, wherein i=1,2,3, k be proportionality constant,
And it is as follows to establish Projective Synchronization error system according to drive system (1) and response system (3) for k ≠ 0:
Wherein e1, e2And e3For Projective Synchronization error system state variable；
Step 2: designing non-linear global sliding mode face and adaptive exponentially approaching rule；
The non-linear global sliding mode face are as follows:
S=e3+k1e2+k2e1-p(t) (6)
Wherein, k1> 0, k2> 0, p (t) is the function in order to realize total-sliding-mode control design, and function p (t) must satisfy down
Three conditions in face:
(1) p (0)=e3(0)+k1e2(0)+k2e1(0)；
(2) as t → ∞, p (t) → 0；
(3) p (t) has first derivative.
According to three above condition, function p (t) is designed are as follows:
P (t)=p (0) e-βt (7)
Wherein, β is constant, and β > 0.Derivation is carried out to function p (t), available:
The adaptive exponentially approaching rule design are as follows:
Wherein, s is the non-linear global sliding mode face that formula (6) define,λ0For constant, and λ0≥0。
WithRespectively unknown parameter d1And d2Estimated value, obtained by adaptive rate, parameter lambda is according to the size of Projective Synchronization error
It is adaptively adjusted, with the reduction of Projective Synchronization error, parameter lambda levels off to λ0；
Step 3: according to Projective Synchronization error formula (5), non-linear global sliding mode face formula (6) and adaptive exponential approach
It restrains (9), designs adaptive total-sliding-mode control device, the single adaptive total-sliding-mode control device is to Projective Synchronization error system
It is controlled, forms closed-loop system, which can be realized the Projective Synchronization of drive system and response system, not to modeling
Determining and external interference signals have robustness；
According to formula (5), formula (6) and formula (9), adaptive total-sliding-mode control device is designed are as follows:
The unknown parameter d1And d2Adaptive rate are as follows:
Wherein, μ1And μ2For constant, and μ1> 0, μ2>0。d10And d20RespectivelyWithInitial value, and d10> 0, d20>0；
S is the non-linear global sliding mode face that formula (6) define.
There are sign function sgn (s) in the controller of formula (10),Controller can be made not connect
It is continuous, there is chattering phenomenon, in order to weaken the influence of buffeting, sign function sgn (s) is replaced using saturation function sat (s), finally
The adaptive total-sliding-mode control device are as follows:
Wherein, the expression formula of saturation function sat (s) isWherein, δ is constant, and δ > 0；
It is proved by stability of the Lyapunov Theory of Stability to closed-loop system, wherein Lyapunov function is
Wherein, s is the non-linear global sliding mode face that formula (6) define, μ1And μ2For constant, and μ1> 0, μ2> 0,WithPoint
It Wei not be by unknown parameter d that adaptive rate obtains1And d2Estimated value.
Derivation is carried out to formula (13), is then brought formula (6), formula (5) and formula (11) into available
Then formula (10) is brought into, it is available after abbreviation
It is stable for being demonstrated by Lyapunov Theory of Stability and forming closed-loop system by formula (5), formula (10) and formula (11)
, the Projective Synchronization error asymptotic convergence of drive system and response system to zero.Single adaptive total-sliding-mode control device energy
The Projective Synchronization for enough realizing different original state drive systems and response system does not know modeling and external interference signals has
Good robustness.
Advantageous effects:
The present invention proposes a kind of tight feedback chaos projective synchronization method of three ranks, sliding using the overall situation in non-linear global sliding mode face
Mould controller all has robustness in the overall process that system responds, and traditional sliding mode controller does not have robust in reaching mode
Property.Total-sliding-mode control device and adaptive sliding mode controller are combined, adaptive total-sliding-mode control device is proposed, by certainly
Adaptation rate is uncertain to modeling and external interference signals estimate all there is robustness in reaching mode and sliding formwork mode, only
It needs the single input that controls to can be achieved with the Projective Synchronization control of the tight feedback chaos of three ranks, and overcomes modeling uncertain and outside
The influence of interference signal.
Detailed description of the invention
Fig. 1 is the general structure schematic diagram of the embodiment of the present invention；
The response curve of control input when Fig. 2 is symbolization function in the embodiment of the present invention 1；
Fig. 3 is the response curve of control input when using saturation function in the embodiment of the present invention 1；
Fig. 4 is the response curve of Projective Synchronization error in the embodiment of the present invention 1；
The response curve of control input when Fig. 5 is symbolization function in the embodiment of the present invention 2；
Fig. 6 is the response curve of control input when using saturation function in the embodiment of the present invention 2；
Fig. 7 is the response curve of Projective Synchronization error in the embodiment of the present invention 2；
Specific embodiment
Invention is described further with specific implementation example with reference to the accompanying drawing:
As shown in Figure 1, drive system and response system are established according to the state equation of the tight feedback chaos system of three ranks, and
Projective Synchronization error system is established, non-linear global sliding mode face and adaptive exponentially approaching rule are designed, designs adaptive rate and oneself
Total-sliding-mode control device is adapted to, which controls Projective Synchronization error system, forms closed loop
Control system, the closed-loop control system realize the Projective Synchronization of drive system and response system.
A kind of validity of the tight feedback chaos projective synchronization method of three ranks proposed by the present invention is shown in order to more intuitive,
Computer simulation experiment is carried out to this control program using MATLAB/Simulink software.In emulation experiment, using ode45
Algorithm ,-five rank Runge-Kutta algorithm of ode45 algorithm, that is, quadravalence, is a kind of numerical solution of ordinary differential equations of adaptive step
Method, maximum step-length 0.0001s, simulation time 15s.In saturation functionIn, parameter setting is
δ=0.001.
Specific embodiment 1:
Drive system and response system are isomorphism system, are Arneodo chaos system.The state side of Arneodo system
Journey are as follows:
As parameter a1=-1, a2=-5.5, a3=3.5, a4When=1, Arneodo system will appear chaos phenomenon.Formula
It (14) is drive system.The original state of drive system is set as x1(0)=2, x2(0)=- 2, x3(0)=2.5.
Response system is also Arneodo chaos system, state equation are as follows:
It is indicated with the uncertain controlled response system with external interference signals of modeling are as follows:
Wherein, it models uncertain △ f (y) and is set as △ f (y)=1.5sin (2y2), external interference signals d (t) is set as
D (t)=1.5sin (3t).To be in response with the uncertain controlled response system (16) with external interference signals of modeling
System.The original state of response system is set as y1(0)=1, y2(0)=2, y2(0)=0.5.
Projective Synchronization error system is formula (5)
Wherein, the state variable of parameter setting k=0.5, i.e. drive system and response system level off to yi=0.5xi,
Middle i=1,2,3.
Non-linear global sliding mode face uses formula (6):
S=e3+k1e2+k2e1-p(t) (6)
Wherein, parameter setting k1=2, k2=2.
Function p (t) uses formula (7):
P (t)=p (0) e-βt (7)
Wherein, parameter setting is β=4.
Adaptive exponentially approaching rule uses formula (9):
Wherein,Parameter setting is λ0=0.2.WithRespectively unknown parameter d1And d2Estimation
Value, is obtained by adaptive rate.
Unknown parameter d1And d2Adaptive rate use formula (11):
Wherein, parameter setting μ1=50, μ2=50, d10=1.2, d20=1.3.
Control parameter is for example preceding set, carries out the emulation of system.It is adaptive global when Fig. 2 is symbolization function sgn (s)
The control input curve of sliding mode controller.When Fig. 3 is using saturation function sat (s), the control of adaptive total-sliding-mode control device
Input curve.In Fig. 2, there is apparent chattering phenomenon in control input.In Fig. 3, control input is existing without occurring buffeting
As smoother.Fig. 4 is the response curve of Projective Synchronization error.It can intuitively observe that Projective Synchronization misses from simulation curve
Difference converges to zero in 7s substantially, and the speed of Projective Synchronization is very fast.
Adaptive total-sliding-mode control device formula (12) and adaptive rate formula (11) are to Projective Synchronization error system formula
(5) it is controlled, forms closed-loop control system, which realizes the Projective Synchronization of drive system and response system.
It does not know in modeling under external interference signals, different original state drive systems and response system realize Projective Synchronization, tool
There are good robustness and very high reliability.
Specific embodiment 2:
Drive system and response system are heterogeneous system, and drive system is Arneodo chaos system, and response system is
Genesio-Tesi chaos system.The state equation of Arneodo system uses formula (14):
As parameter a1=-1, a2=-5.5, a3=3.5, a4When=1, Arneodo system will appear chaos phenomenon.Formula
It (14) is drive system.The original state of drive system is set as x1(0)=2.5, x2(0)=- 2.2, x3(0)=2.6.
Response system is Genesio-Tesi chaos system, state equation are as follows:
Wherein, parameter a>0, b>0, c>0, and ab<c.As parameter a=1.2, b=2.92, c=6, Genesio-
Tesi system will appear chaos phenomenon.It is indicated with the uncertain controlled response system with external interference signals of modeling are as follows:
Wherein, it models uncertain △ f (y) and is set as △ f (y)=3sin (2y2)sin(y1), external interference signals d (t) is set
It is set to d (t)=2sin (3t).It will controlled system (18) uncertain with modeling and external interference signals system in response.
The original state of response system is set as y1(0)=1, y2(0)=- 2, y3(0)=- 1.5.
Projective Synchronization error system is formula (5):
Wherein, the state variable of parameter setting k=-0.5, i.e. drive system and response system level off to yi=-0.5xi,
Wherein i=1,2,3.
Non-linear global sliding mode face uses formula (6):
S=e3+k1e2+k2e1-p(t) (6)
Wherein, parameter setting k1=2, k2=2.
Function p (t) uses formula (7):
P (t)=p (0) e-βt (7)
Wherein, parameter setting is β=4.
Adaptive exponentially approaching rule uses formula (9):
Wherein,Parameter setting is λ0=0.2.WithRespectively unknown parameter d1And d2Estimated value,
It is obtained by adaptive rate.
Unknown parameter d1And d2Adaptive rate use formula (11):
Wherein, parameter setting μ1=60, μ2=60, d10=2.6, d20=1.8.
Control parameter is for example preceding set, carries out the emulation of system.It is adaptive global when Fig. 5 is symbolization function sgn (s)
The control input curve of sliding mode controller.When Fig. 6 is using saturation function sat (s), the control of adaptive total-sliding-mode control device
Input curve.In Fig. 5, there is apparent chattering phenomenon in control input.In Fig. 6, control input is existing without occurring buffeting
As smoother.Fig. 7 is the response curve of Projective Synchronization error.It can intuitively observe that Projective Synchronization misses from simulation curve
Difference converges to zero in 7s substantially, and the speed of Projective Synchronization is very fast.Adaptive total-sliding-mode control device formula (12) and adaptive
Should rate formula (11) Projective Synchronization error system formula (5) is controlled, formed closed-loop control system, the closed-loop control system
Realize the Projective Synchronization of drive system and response system.Uncertain under external interference signals in modeling, different original states are driven
Dynamic system and response system realize Projective Synchronization, have good robustness and very high reliability.
Claims (2)
1. a kind of tight feedback chaos projective synchronization method of three ranks, which comprises the following steps:
Step 1: drive system and response system being established according to the state equation of the tight feedback chaos system of three ranks, according to drive system
Projective Synchronization error system is established with response system:
Drive system is the tight feedback chaos system of three ranks, and state equation is as follows:
Wherein, x1, x2And x3For the state variable of system, x=[x1,x2,x3]T, fx(x, t) is continuous function, and t is the time；With formula
(1) it is used as drive system；
Response system is the tight feedback chaos system of three ranks, and state equation is as follows:
Wherein, y1, y2And y3For the state variable of system, y=[y1,y2,y3]T, fy(y, t) is continuous function, and t is the time；It has
The uncertain controlled response system with external interference signals of modeling, state equation are as follows:
Wherein, △ f (y) is that modeling is uncertain, and d (t) is external interference signals, and u is control input, is in response with formula (3)
System, works as fx(x, t) and fyWhen (y, t) has identical structure, drive system and response system are isomorphism chaos；Work as fx(x, t) and fy
When (y, t) has different structure, drive system and response system are isomery chaos；
Model uncertain △ f (y) and the equal bounded of external interference signals d (t), it may be assumed that
Wherein, d1For the upper bound for modeling uncertain △ f (y), d2For the upper bound of external interference signals d (t), and d1>=0, d2>=0, d1
And d2For unknown parameter, estimated using adaptive rate；
The Projective Synchronization error of drive system and response system is ei=yi-kxi, wherein i=1,2,3, k be proportionality constant, and k
≠ 0, according to drive system (1) and response system (3), it is as follows to establish Projective Synchronization error system:
Wherein e1, e2And e3For Projective Synchronization error system state variable；
Step 2: designing non-linear global sliding mode face and adaptive exponentially approaching rule；
The non-linear global sliding mode face are as follows:
S=e3+k1e2+k2e1-p(t) (6)
Wherein, k1> 0, k2> 0, p (t) is the function in order to realize total-sliding-mode control design, and function p (t) must satisfy following
Three conditions:
(1) p (0)=e3(0)+k1e2(0)+k2e1(0)；
(2) when t approaches ∞, p (t) approach 0；
(3) p (t) has first derivative；
According to three above condition, function p (t) is designed are as follows:
P (t)=p (0) e-βt (7)
Wherein, β is constant, and β > 0, carries out derivation to function p (t), available:
The adaptive exponentially approaching rule design are as follows:
Wherein, s is non-linear global sliding mode face defined in formula (6),λ0For constant, and λ0>=0,WithRespectively unknown parameter d1And d2Estimated value；
Step 3: according to Projective Synchronization error formula (5), non-linear global sliding mode face formula (6) and adaptive exponentially approaching rule
(9), design adaptive total-sliding-mode control device, the single adaptive total-sliding-mode control device to Projective Synchronization error system into
Row control, forms closed-loop system, which can be realized the Projective Synchronization of drive system and response system, not true to modeling
Fixed and external interference signals have robustness；
According to formula (5), formula (6) and formula (9), adaptive total-sliding-mode control device is designed are as follows:
The unknown parameter d1And d2Adaptive rate are as follows:
Wherein, μ1And μ2For constant, and μ1> 0, μ2> 0, d10And d20RespectivelyWithInitial value, and d10> 0, d20>0；S is public affairs
Non-linear global sliding mode face defined in formula (6)；
There are sign function sgn (s) in the controller of formula (10),Controller can be made discontinuous,
There is chattering phenomenon, in order to weaken the influence of buffeting, sign function sgn (s) is replaced using saturation function sat (s), it is final described
Adaptive total-sliding-mode control device are as follows:
Wherein, the expression formula of saturation function sat (s) isWherein, δ is constant, and δ > 0.
2. the tight feedback chaos projective synchronization method of a kind of three ranks according to claim 1, which is characterized in that pass through Lyapunov
Theory of Stability proves the stability of the closed-loop system, wherein Lyapunov function are as follows:
Wherein, s is non-linear global sliding mode face, μ defined in formula (6)1And μ2For constant, and μ1> 0, μ2> 0,WithRespectively
For the unknown parameter d obtained by adaptive rate1And d2Estimated value.
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| CN109557817A (en) * | 2019-01-07 | 2019-04-02 | 东北大学 | A kind of improved total-sliding-mode control method |
| CN109557817B (en) * | 2019-01-07 | 2021-11-16 | 东北大学 | Improved global sliding mode control method |
| CN109742941A (en) * | 2019-01-16 | 2019-05-10 | 武汉工程大学 | DC-DC converter chaotic control method, system and medium based on supertwist control |
| CN109742941B (en) * | 2019-01-16 | 2020-05-22 | 武汉工程大学 | DC-DC converter chaotic control method, system and medium based on super-distortion control |
| CN110007596A (en) * | 2019-03-29 | 2019-07-12 | 东北大学 | A kind of second order chaos ratio projective synchronization method controlling input-bound |
| CN110020405A (en) * | 2019-04-12 | 2019-07-16 | 东北大学 | A kind of Jacobian matrix projective synchronization method of difference dimension chaos |
| CN110109349A (en) * | 2019-05-16 | 2019-08-09 | 东北大学 | The tight feedback chaos trace tracking method of three ranks under a kind of constraint of saturation |
| CN110568759A (en) * | 2019-09-26 | 2019-12-13 | 南京理工大学 | robust synchronization control method of fractional order chaotic system |
| CN110568759B (en) * | 2019-09-26 | 2022-06-28 | 南京理工大学 | Robust synchronization control method of fractional order chaotic system |
| CN111624881A (en) * | 2020-05-27 | 2020-09-04 | 中国地质大学（武汉） | Synchronous control method and system based on gene network |
| CN113285641A (en) * | 2021-04-12 | 2021-08-20 | 山东理工大学 | Motor speed tracking control method |
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