CN110007596A - A kind of second order chaos ratio projective synchronization method controlling input-bound - Google Patents
A kind of second order chaos ratio projective synchronization method controlling input-bound Download PDFInfo
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Abstract
The present invention proposes a kind of second order chaos ratio projective synchronization method for controlling input-bound, comprising: according to drive system and response system, establishes ratio Projective Synchronization error system；Design global sliding mode face and adaptive exponentially approaching rule；Design total-sliding-mode control device；Sign function is replaced using sinusoidal pattern saturation function, improves total-sliding-mode control device；Control is balanced using total-sliding-mode control device comparative example Projective Synchronization error system after the improvement under constraint of saturation, realizes the ratio Projective Synchronization control of drive system and response system；Pass through experimental verification, present invention total-sliding-mode control device under constraint of saturation realizes the ratio Projective Synchronization control of different original state drive systems and response system, the speed of ratio Projective Synchronization is very fast, and external interference signals uncertain to modeling have good robustness and very high reliability.
Description
Technical field
The invention belongs to automatic control technology fields, and in particular to a kind of second order chaos ratio projection for controlling input-bound
Synchronous method.
Background technique
Chaos is the tie for connecting regular motion and random motion, is widely present in nature and human society.
Since Pecora and Carroll realizes Chaotic Synchronous by electronic circuit, since Chaotic Synchronous is in secret communication and control
Field etc. has huge potential application foreground, it is made to have obtained extensive concern and in-depth study.Mainieri and
Rehacek proposes the concept of Projective Synchronization, has unified different types of phase synchronization of two coupled chaotic.Second order chaos system only needs single
Control input can be achieved with ratio Projective Synchronization, be with a wide range of applications in terms of secret communication.
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.Common sliding formwork control is divided into reaching mode and sliding mode,
And only there is robustness in sliding mode.Total-sliding-mode control is realized by design kinematic nonlinearity sliding-mode surface,
Reaching mode and sliding mode all have robustness.The ratio projection for carrying out second order chaos system using total-sliding-mode control device is same
When step control, the size for controlling input will have certain limitation, and excessive control input not only can make the control of system be difficult to reality
It is existing, in some instances it may even be possible to actuator to be damaged, the second order chaos ratio projective synchronization method of research control input-bound is very necessary.
Summary of the invention
Based on above technical problem, the present invention provides a kind of second order chaos ratio Projective Synchronization side for controlling input-bound
Method establishes ratio Projective Synchronization error system according to drive system and response system, using global sliding mode face and adaptive index
Reaching Law designs total-sliding-mode control device, total-sliding-mode control device comparative example Projective Synchronization error system under constraint of saturation into
Row balance control, realizes the ratio Projective Synchronization control of drive system and response system, and external disturbance uncertain to modeling is believed
Number have robustness.
A kind of second order chaos ratio projective synchronization method controlling input-bound, comprising the following steps:
Step 1: drive system is second order chaos system, and response system is not know with modeling and external interference signals
Second order chaos system establishes ratio Projective Synchronization error system according to drive system and response system；
Drive system is second order chaos system, state equation are as follows:
Wherein, x1And x2For the state variable of system, x=[x1,x2]T, f1(x, t) is continuous function, and t is the time.
Response system is second order chaos system, does not know the controlled response system with external interference signals with modeling,
State equation are as follows:
Wherein, y1And y2For the state variable of system, y=[y1,y2]T, f2(y, t) is continuous function, and t is the time.△f2
(y) uncertain for modeling, d (t) is external interference signals, u1For control input.Drive system and response system are isomorphism chaos
It or is isomery chaos.
Model uncertain △ f2(y) and the equal bounded of external interference signals d (t), it may be assumed that
|△f2(y)|+|d(t)|≤d1 (3)
Wherein, d1To model the uncertain upper bound with external interference signals, and d1>0。
The ratio Projective Synchronization error of drive system and response system are as follows:
Wherein, k is proportionality constant, and k ≠ 0.As k=1, drive system and response system be it is fully synchronized, work as k=-1
When, drive system is that reverse phase is synchronous with response system.The fully synchronized special circumstances for ratio Projective Synchronization synchronous with reverse phase.
Derivation is carried out to formula (4), obtains ratio Projective Synchronization error system are as follows:
The control of the ratio Projective Synchronization of drive system and response system, is converted to the balance of ratio Projective Synchronization error system
Control, i.e.,
Step 2: design global sliding mode face and adaptive exponentially approaching rule；
In the design of total-sliding-mode control device, the global sliding mode face of use are as follows:
S=e2+ce1-p(t) (6)
Wherein, c > 0, p (t) are the functions in order to realize total-sliding-mode control design.As t=0, s (0)=0.When t →
When ∞, s → 0.Function p (t) needs to meet three following conditions:
(1) p (0)=e2(0)+ce1(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), is obtained:
In the design of total-sliding-mode control device, the adaptive exponentially approaching rule of use are as follows:
Wherein, k1, k2And k3For constant, and k1> 0, k2> 0, k3≥d1。
Step 3: according to ratio Projective Synchronization error formula (5), global sliding mode face formula (6) and adaptive exponential approach
Rule designs total-sliding-mode control device are as follows:
It is proved using stability of the Lyapunov Theory of Stability to system.Lyapunov function are as follows:
Wherein, s is global sliding mode face defined in formula (6).
Derivation is carried out to formula (11) to obtain after then bringing formula (5) and formula (10) into:
Due to V >=0,According to Lyapunov stability principle, total-sliding-mode control device can be realized ratio projection
The balance of synchronization error system controls, i.e.,To realize the ratio of drive system and response system
Projective Synchronization control, and external interference signals uncertain to modeling have robustness.
Step 4: sign function sgn (s) is replaced using sinusoidal pattern saturation function sat (s), improves total-sliding-mode control device,
It is specific as follows:
There are sign function sgn (s) in total-sliding-mode control device, and control input can be made discontinuous, chattering phenomenon occur.
The expression formula of sign function sgn (s) are as follows:
In order to weaken the influence of buffeting, sign function sgn (s) is replaced using sinusoidal pattern saturation function sat (s).Sinusoidal pattern
The expression formula of saturation function sat (s) are as follows:
Wherein, δ is constant, and δ > 0.
Improved total-sliding-mode control device are as follows:
Step 5: the constraint of saturation that the control input of total-sliding-mode control device is subject to are as follows:
Wherein, umaxInput value, and u are controlled for maximummax> 0, u1For the total-sliding-mode control device of formula (14), u is saturation
Total-sliding-mode control device under constraint.
Step 6: control is balanced using the total-sliding-mode control device comparative example Projective Synchronization error system under constraint of saturation
System, realizes the ratio Projective Synchronization control of drive system and response system, and external interference signals uncertain to modeling have Shandong
Stick.
Advantageous effects:
Total-sliding-mode control device is designed using global sliding mode face and adaptive exponentially approaching rule, is proposed using under constraint of saturation
Total-sliding-mode control device comparative example Projective Synchronization error system be balanced control.It is sliding in the overall situation in order to weaken chattering phenomenon
In mould controller, sign function is replaced using sinusoidal pattern saturation function.It is mixed to can be realized second order when controlling input-bound
The ratio Projective Synchronization of ignorant system controls, and the speed of ratio Projective Synchronization is very fast, and external interference signals uncertain to modeling
With good robustness and very high reliability.
Detailed description of the invention
Fig. 1 is general principles figure of the invention；
When Fig. 2 is symbolization function in specific embodiment 1 under constraint of saturation total-sliding-mode control device response curve；
Fig. 3 is in specific embodiment 1 using the response of total-sliding-mode control device under constraint of saturation when sinusoidal pattern saturation function
Curve；
Fig. 4 is the response curve of ratio Projective Synchronization error in specific embodiment 1；
When Fig. 5 is symbolization function in specific embodiment 2 under constraint of saturation total-sliding-mode control device response curve；
Fig. 6 is in specific embodiment 2 using the response of total-sliding-mode control device under constraint of saturation when sinusoidal pattern saturation function
Curve；
Fig. 7 is the response curve of ratio Projective Synchronization error in specific embodiment 2.
Specific embodiment
Invention is described further with specific implementation example with reference to the accompanying drawing: as shown in Figure 1, drive system is second order
Chaos system, response system be with modeling is uncertain and the second order chaos system of external interference signals, according to drive system and
Response system establishes ratio Projective Synchronization error system；Global sliding mode face and adaptive exponentially approaching rule are designed, and using global
Sliding-mode surface and adaptive exponentially approaching rule design total-sliding-mode control device；Total-sliding-mode control device comparative example under constraint of saturation is thrown
Shadow synchronization error system is balanced control, realizes the ratio Projective Synchronization control of drive system and response system.
For a kind of more intuitive display second order chaos ratio Projective Synchronization for controlling input-bound proposed by the present invention
The validity of method carries out computer simulation experiment to this control program using MATLAB/Simulink software.In emulation experiment
In, using ode45 algorithm ,-five rank Runge-Kutta algorithm of ode45 algorithm, that is, quadravalence, is a kind of ordinary differential of adaptive step
Equation numerical solution, maximum step-length 0.0001s, simulation time 3s.The parameter setting in sinusoidal pattern saturation function sat (s)
For δ=0.001.
Specific embodiment 1:
Step 1: drive system is second order chaos system, and response system is not know with modeling and external interference signals
Second order chaos system establishes ratio Projective Synchronization error system according to drive system and response system；
Drive system and response system are second order Duffing chaos system.Drive system is Duffing chaos system,
State equation are as follows:
Wherein, x=[x1,x2]T, t is the time.The original state of drive system is set as x1(0)=- 0.5, x2(0)=
0.6。
Response system is second order Duffing chaos system.With the uncertain controlled response with external interference signals of modeling
System, state equation are as follows:
Wherein, y=[y1, y2]T, t is the timeModel uncertain △ f2
(y) it is set as △ f2(y)=0.5cos (y1+y2), external interference signals d (t) is set as d (t)=0.5sin (π t).Modeling is not
Determine △ f2(y) and the equal bounded of external interference signals d (t), and | △ f2(y)|+|d(t)|≤d1, then d1=1.Response system
Original state is set as y1(0)=1, y2(0)=1.
The ratio Projective Synchronization error of drive system and response system uses formula (4):
Wherein, parameter setting k=1.1.
Step 2: design global sliding mode face and adaptive exponentially approaching rule；
Global sliding mode face uses formula (6):
S=e2+ce1-p(t) (6)
Wherein, parameter setting c=7.
In global sliding mode face, function p (t) uses formula (7):
P (t)=p (0) e-βt (7)
Wherein, parameter setting is β=5.
In the design of total-sliding-mode control device, adaptive exponentially approaching rule uses formula (9):
Wherein, parameter setting k1=3, k2=0.5, k3=1.2, and k3≥d1。
Step 3: according to ratio Projective Synchronization error formula (5), global sliding mode face formula (6) and adaptive exponential approach
Rule designs total-sliding-mode control device are as follows:
Step 4: sign function sgn (s) is replaced using sinusoidal pattern saturation function sat (s), improves total-sliding-mode control device,
It is specific as follows:
There are sign function sgn (s) in total-sliding-mode control device, and control input can be made discontinuous, chattering phenomenon occur.
The expression formula of sign function sgn (s) are as follows:
In order to weaken the influence of buffeting, sign function sgn (s) is replaced using sinusoidal pattern saturation function sat (s).Sinusoidal pattern
The expression formula of saturation function sat (s) are as follows:
Wherein, δ is constant, and δ > 0.
Improved total-sliding-mode control device are as follows:
Step 5: the constraint of saturation that the control input of total-sliding-mode control device is subject to is using formula (16):
Wherein, parameter setting umax=10.
Step 6: being carried out using total-sliding-mode control device comparative example Projective Synchronization error system after the improvement under constraint of saturation
The ratio Projective Synchronization control of drive system and response system, and external interference signals uncertain to modeling are realized in balance control
With robustness.
Control parameter is for example preceding set, carries out the emulation of system.Global sliding mode under constraint of saturation when Fig. 2 is symbolization function
The response curve of controller u.Fig. 3 is bent using the response of total-sliding-mode control device u under constraint of saturation when sinusoidal pattern saturation function
Line.In Fig. 2, there is apparent chattering phenomenon in control input.In Fig. 3, there is not chattering phenomenon in control input.Scheming
In 2 and Fig. 3, there is constraint of saturation, u=-u in total-sliding-mode control devicemax=-10.Fig. 4 is drive system and response system ratio
The response curve of example Projective Synchronization error.Ratio Projective Synchronization error base in 1.9s can be intuitively observed from simulation curve
Originally zero is converged to, the speed of ratio Projective Synchronization is very fast.Total-sliding-mode control device realizes different initial shapes under constraint of saturation
The control of the ratio Projective Synchronization of state drive system and response system, the speed of ratio Projective Synchronization is very fast, uncertain to modeling
There is good robustness and very high reliability with external interference signals.
Specific embodiment 2:
Step 1: drive system is second order chaos system, and response system is not know with modeling and external interference signals
Second order chaos system establishes ratio Projective Synchronization error system according to drive system and response system；
Drive system is second order Duffing chaos system, and response system is second order van der Pol chaos system.Driving
System is Duffing chaos system, state equation are as follows:
Wherein, x=[x1,x2]T, t is the time.The original state of drive system is set as x1(0)=0.8, x2(0)=-
0.5。
Response system is van der Pol chaos system.With the uncertain controlled response with external interference signals of modeling
System, state equation are as follows:
Wherein, y=[y1,y2]T, t is time, f2(y, t)=- y1+3(1-y12)y2+ 5sin (1.788t) is modeled not true
Determine △ f2(y) it is set as △ f2(y)=0.4cos (y1+ 0.5)+0.1, external interference signals d (t) are set as d (t)=0.5sin
(4t).Model uncertain △ f2(y) and the equal bounded of external interference signals d (t), and | △ f2(y)|+|d(t)|≤d1, then d1=1.
The original state of response system is set as y1(0)=1.5, y2(0)=- 1.2.
The ratio Projective Synchronization error of drive system and response system uses formula (4):
Wherein, parameter setting k=-1.5.
Step 2: design global sliding mode face and adaptive exponentially approaching rule；
Global sliding mode face uses formula (6):
S=e2+ce1-p(t) (6)
Wherein, parameter setting c=7.
In global sliding mode face, function p (t) uses formula (7):
P (t)=p (0) e-βt (7)
Wherein, parameter setting is β=5.
In the design of total-sliding-mode control device, adaptive exponentially approaching rule uses formula (9):
Wherein, parameter setting k1=3, k2=0.5, k3=1.2, and k3≥d1。
Step 3: according to ratio Projective Synchronization error formula (5), global sliding mode face formula (6) and adaptive exponential approach
Rule designs total-sliding-mode control device are as follows:
Step 4: sign function sgn (s) is replaced using sinusoidal pattern saturation function sat (s), improves total-sliding-mode control device,
It is specific as follows:
There are sign function sgn (s) in total-sliding-mode control device, and control input can be made discontinuous, chattering phenomenon occur.
The expression formula of sign function sgn (s) are as follows:
In order to weaken the influence of buffeting, sign function sgn (s) is replaced using sinusoidal pattern saturation function sat (s).Sinusoidal pattern
The expression formula of saturation function sat (s) are as follows:
Wherein, δ is constant, and δ > 0.
Improved total-sliding-mode control device are as follows:
Step 5: the constraint of saturation that the control input of total-sliding-mode control device is subject to is using formula (16):
Wherein, parameter setting umax=40.
Step 6: being carried out using total-sliding-mode control device comparative example Projective Synchronization error system after the improvement under constraint of saturation
The ratio Projective Synchronization control of drive system and response system, and external interference signals uncertain to modeling are realized in balance control
With robustness.
Control parameter is for example preceding set, carries out the emulation of system.Global sliding mode under constraint of saturation when Fig. 5 is symbolization function
The response curve of controller u.Fig. 6 is bent using the response of total-sliding-mode control device u under constraint of saturation when sinusoidal pattern saturation function
Line.In Fig. 5, there is apparent chattering phenomenon in control input.In Fig. 6, there is not chattering phenomenon in control input.Scheming
In 5 and Fig. 6, there is constraint of saturation, u=-u in total-sliding-mode control devicemax=-40.Fig. 7 is drive system and response system ratio
The response curve of example Projective Synchronization error.Ratio Projective Synchronization error base in 1.9s can be intuitively observed from simulation curve
Originally zero is converged to, the speed of ratio Projective Synchronization is very fast.Total-sliding-mode control device realizes different initial shapes under constraint of saturation
The control of the ratio Projective Synchronization of state drive system and response system, the speed of ratio Projective Synchronization is very fast, uncertain to modeling
There is good robustness and very high reliability with external interference signals.
Claims (2)
1. a kind of second order chaos ratio projective synchronization method for controlling input-bound, which is characterized in that specific step is as follows:
Step 1: drive system is second order chaos system, and response system is with the uncertain second order with external interference signals of modeling
Chaos system establishes ratio Projective Synchronization error system according to drive system and response system；
Drive system is second order chaos system, state equation are as follows:
Wherein, x1And x2For the state variable of system, x=[x1,x2]T, f1(x, t) is continuous function, and t is the time；
Response system is second order chaos system, with the uncertain controlled response system with external interference signals of modeling, state
Equation are as follows:
Wherein, y1And y2For the state variable of system, y=[y1,y2]T, f2(y, t) is continuous function, and t is the time；△f2(y) it is
Modeling is uncertain, and d (t) is external interference signals, u1For control input, drive system and response system are isomorphism chaos or are
Isomery chaos；
Model uncertain △ f2(y) and the equal bounded of external interference signals d (t), it may be assumed that
|△f2(y)|+|d(t)|≤d1 (3)
Wherein, d1To model the uncertain upper bound with external interference signals, and d1>0；
The ratio Projective Synchronization error of drive system and response system are as follows:
Wherein, k is proportionality constant, and k ≠ 0, as k=1, drive system and response system be it is fully synchronized, as k=-1,
Drive system is that reverse phase is synchronous with response system；
Derivation is carried out to formula (4), obtains ratio Projective Synchronization error system are as follows:
The control of the ratio Projective Synchronization of drive system and response system, is converted to the balance control of ratio Projective Synchronization error system
System, i.e.,
Step 2: design global sliding mode face and adaptive exponentially approaching rule；
In the design of total-sliding-mode control device, the global sliding mode face of use are as follows:
S=e2+ce1-p(t) (6)
Wherein, c > 0, p (t) they are the functions in order to realize total-sliding-mode control design, as t=0, s (0)=0, and as t → ∞,
S → 0, function p (t) need to meet three following conditions:
(1) p (0)=e2(0)+ce1(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 obtains function p (t) progress derivation:
In the design of total-sliding-mode control device, the adaptive exponentially approaching rule of use are as follows:
Wherein, k1, k2And k3For constant, and k1> 0, k2> 0, k3≥d1；
Step 3: according to ratio Projective Synchronization error formula (5), global sliding mode face formula (6) and adaptive exponentially approaching rule, if
Count total-sliding-mode control device are as follows:
Step 4: sign function sgn (s) being replaced using sinusoidal pattern saturation function sat (s), improves total-sliding-mode control device, specifically
It is as follows:
There are sign function sgn (s) in total-sliding-mode control device, and control input can be made discontinuous, chattering phenomenon occur, described
The expression formula of sign function sgn (s) are as follows:
In order to weaken the influence of buffeting, sign function sgn (s), the sinusoidal pattern are replaced using sinusoidal pattern saturation function sat (s)
The expression formula of saturation function sat (s) are as follows:
Wherein, δ is constant, and δ > 0；
Improved total-sliding-mode control device are as follows:
Step 5: the constraint of saturation that the control input of total-sliding-mode control device is subject to are as follows:
Wherein, umaxInput value, and u are controlled for maximummax> 0, u1For the total-sliding-mode control device of formula (14), u is constraint of saturation
Under total-sliding-mode control device；
Step 6: being balanced using total-sliding-mode control device comparative example Projective Synchronization error system after the improvement under constraint of saturation
The ratio Projective Synchronization control of drive system and response system is realized in control.
2. a kind of second order chaos ratio projective synchronization method for controlling input-bound according to claim 1, which is characterized in that
It is proved using stability of the Lyapunov Theory of Stability to system, Lyapunov function are as follows:
Wherein, s is global sliding mode face defined in formula (6).
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