CN112068427A - Chaotic synchronization control method - Google Patents

Chaotic synchronization control method Download PDF

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CN112068427A
CN112068427A CN202010875038.9A CN202010875038A CN112068427A CN 112068427 A CN112068427 A CN 112068427A CN 202010875038 A CN202010875038 A CN 202010875038A CN 112068427 A CN112068427 A CN 112068427A
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马少娟
李曼
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North Minzu University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
    • G—PHYSICS
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    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
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    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/25—Design optimisation, verification or simulation using particle-based methods
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/27—Design optimisation, verification or simulation using machine learning, e.g. artificial intelligence, neural networks, support vector machines [SVM] or training a model
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Abstract

The invention relates to a chaotic synchronization control method, and belongs to the field of nonlinear science. The chaotic synchronization control method comprises the steps of obtaining a training sample number set by solving a master system, combining a slave system, establishing fitness functions of the master system and the slave system under the method of an unbiased least square support vector machine, solving a mixed synchronization trajectory and a control input signal, optimizing parameters of a kernel function used in the unbiased least square through a particle swarm algorithm to obtain optimal parameters, and then using the designed control input signal to realize the mixed synchronization control of the master system and the slave system. The chaotic synchronization control method can simplify the calculation mode and effectively improve the calculation precision.

Description

Chaotic synchronization control method
Technical Field
The invention belongs to the field of nonlinear science, and particularly relates to a chaotic synchronization control method.
Background
Nonlinear science has gained vigorous development over the last half of the twentieth century. Because of the complexity of natural phenomena and social phenomena, linear kinetic equations are difficult to describe the dynamics of the natural phenomena and the social phenomena, so that the nonlinear kinetic equations are necessary to describe the dynamics. Another aspect provides a favorable condition for scientists to study nonlinear systems due to the miniaturization of computers. Non-linear equations generally do not have an analytical solution, except for a few. It has been found that a system described by a deterministic, non-linear evolution equation may, under conditions far from equilibrium, enter a bifurcation into a spatio-temporally ordered dissipative system with less symmetry or a more chaotic state when the system's parameters change. Chaotic states are a sort of "order" in disorder that is regular, but without apparent symmetry. The discovery of chaos is one of the greatest discoveries in physics after relativity and quantum mechanics, and a new example of scientific modeling is created. Chaos theory and experimental research have been the main research direction in nonlinear scientific research for half a century. The chaotic system has random motion and also follows a deterministic law. The mechanical movements at classical li-law can deduce from the value at a certain moment a definite value at any moment. The motion state of the chaotic system is unpredictable and random motion. The basic property of chaos is sensitivity to initial values. When the initial value changes slightly, the system changes significantly after a period of time. This is closely related to the randomness of the chaotic system. The "butterfly effect" is an example of an initial value sensitivity. A butterfly stunts several wings in amazon tropical rainforests in south america, resulting in tornadoes in texas, usa after several weeks, which is a widely known butterfly effect. It starts from a paper published by a meteorologist Loranz, and finds that a small change in initial conditions causes a huge chain reaction phenomenon of the system, which is a typical chaos phenomenon.
The chaotic dynamics of random motion has important breakthrough in both theoretical depth and application range. An important breakthrough is the proposal of chaotic control and synchronization since the last 90 th century, thereby initiating the research heat of chaotic synchronization control theory and method at home and abroad. The first step towards success of the synchronization theory is to control the chaos. The chaos control can not only eliminate chaos when the chaos motion is harmful, but also generate chaos by control when the chaos is beneficial, namely chaotic control. The chaotic synchronization and the chaotic control have uniformity, and a plurality of chaotic control methods can be applied to the chaotic synchronization, such as an external noise synchronization method, a self-adaptive control synchronization method and a mutual coupling synchronization method. At present, the research of chaos control and synchronization has obtained initial results in many fields such as human body life science, laser, secret communication, safety, strong current particle beams and the like and is paid more and more attention by scientists.
The existing chaotic synchronization control method is complex in calculation and insufficient in calculation precision.
Disclosure of Invention
In view of this, an object of the embodiments of the present invention is to provide a chaotic synchronization control method, which can implement hybrid synchronization control of a master system and a slave system, simplify a calculation manner, and effectively improve calculation accuracy.
The embodiment of the invention is realized by the following steps:
the embodiment of the invention provides a chaotic synchronization control method, which is used for realizing the synchronous control of a master system and a slave system, and comprises the following steps:
step S1, sampling the main system to obtain a training sample number set.
And step S2, under the method of the unbiased least square support vector machine, establishing fitness functions of the master system and the slave system, and designing a control input signal to be used as a controller.
And step S3, optimizing parameters of the unbiased least squares support vector machine based on the particle swarm optimization.
And step S4, carrying out numerical simulation on the control input signal and verifying the validity.
In step S5, the active control input signal is used as the hybrid synchronous control signal for the master system and the slave system.
As an alternative to the above embodiment, step S2 includes:
step S21, establishing nonlinear chaotic models of the master system and the slave system:
Figure BDA0002652384890000031
Figure BDA0002652384890000032
wherein x, y ∈ RnF, g is a differentiable function, u (t, x, y) is a control input, a hybrid synchronization error is defined as e-x-qy, and different initial values x0、y0The system meets the following requirements:
Figure BDA0002652384890000033
when q is 0, the master system and the slave system are synchronized, and when q is-1, the master system and the slave system are desynchronized.
Step S22, introducing a fixed value parameter based on the least square support vector machine, so that the original bias term and weight are combined.
As an alternative to the above embodiment, in step S3, the unbiased least squares support vector machine selects the radial basis kernel function.
As an alternative to the above embodiment, step S3 includes the steps of:
in step S31, a population of particles of population size N is initialized, including the random position and velocity of the particles, the learning factor, and the maximum number of iterations.
And step S32, respectively using the least square support vector machine corresponding to each particle vector to predict the sample, and obtaining the current position and prediction error of each particle.
And step S33, taking the result of the step S33 as the current fitness of the corresponding particle, then comparing the current fitness of each particle with the optimal fitness of the particle, and if the particle has no optimal fitness, taking the current position of the corresponding particle as the optimal position of the particle.
And step S34, comparing the fitness of each particle optimal position with the fitness of the population optimal position, and if the performance is better, taking the particle optimal position as the population optimal position.
And step S35, calculating inertia weight, and updating the speed and position of the particles.
And step S36, judging whether the optimal condition is met, if so, ending the optimization, otherwise, returning to the step S32 and continuing a new round of optimization process.
As an alternative to the above embodiment, in step S4, if the state variables of the master system and the slave system are synchronized by the control input signal, the next step is executed, and if the state variables of the master system and the slave system are not synchronized, the process returns to step S2.
The invention has the beneficial effects that:
the chaotic synchronization control method provided by the invention obtains a training sample number set by solving a master system, combines with a slave system, establishes fitness functions of the master system and the slave system under the method of an unbiased least square support vector machine, calculates a mixed synchronization trajectory and a control input signal, optimizes parameters of a kernel function used in unbiased least square through a particle swarm algorithm to obtain optimal parameters, and then uses the designed control input signal to realize the mixed synchronization control of the master system and the slave system. The chaotic synchronization control method can simplify the calculation mode and effectively improve the calculation precision.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without creative efforts. The above and other objects, features and advantages of the present invention will become more apparent from the accompanying drawings. Like reference numerals refer to like parts throughout the drawings. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention.
FIG. 1 shows a trajectory diagram of a Lorentz system;
fig. 2 is a flowchart illustrating a chaotic synchronization control method according to an embodiment of the present invention;
FIG. 3 shows a flow chart of step 3 of FIG. 2;
FIG. 4 shows an algorithm flow diagram for optimizing parameters in an unbiased LS-SVM using particle swarm optimization;
FIG. 5 shows a Y1-X1 error curve;
FIG. 6 shows a Y1-X1 synchronous trajectory;
FIG. 7 shows a Y2-X2 error curve;
FIG. 8 shows a Y2-X2 synchronous trajectory;
FIG. 9 shows a Y3-X3 error curve;
FIG. 10 shows a Y3-X3 synchronous trajectory;
FIG. 11 shows a Y4-X4 error curve;
FIG. 12 shows the Y4-X4 synchronous trajectory.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. The components of embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations.
Thus, the following detailed description of the embodiments of the present invention, presented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It should be noted that: like reference numbers and letters refer to like items in the following figures, and thus, once an item is defined in one figure, it need not be further defined and explained in subsequent figures.
Referring to FIG. 1, a trajectory diagram of a Lorentz system is shown.
Referring to fig. 2, an embodiment of the present invention provides a chaotic synchronization control method for implementing synchronous control of a master system and a slave system, in this embodiment, the master system and the slave system may be a sensor, an image signal processing, a secure communication, and the like.
The chaotic synchronization control method comprises the following steps:
step S1, sampling the main system to obtain a training sample number set.
Model building
And step S2, under the method of the unbiased least square support vector machine, establishing fitness functions of the master system and the slave system, and designing a control input signal to be used as a controller.
Wherein, step S2 includes:
step S21, establishing nonlinear chaotic models of the master system and the slave system:
Figure BDA0002652384890000071
Figure BDA0002652384890000072
wherein x, y ∈ RnF, g is a differentiable function, u (t, x, y) is a control input, a hybrid synchronization error is defined as e-x-qy, and different initial values x0、y0The system meets the following requirements:
Figure BDA0002652384890000073
when q is 0, the master system and the slave system are synchronized, and when q is-1, the master system and the slave system are desynchronized.
Step S22, introducing a fixed value parameter based on the least square support vector machine, so that the original bias term and weight are combined.
Specifically, a least squares support vector machine (LS-SVM) is an improved algorithm of the Support Vector Machine (SVM), which changes an inequality constraint term in the support vector machine into an equality constraint, introduces an error variable to each sample in order to solve the situation that some special points exist, and adds an l2 regular term of the error variable to an original function. To simplify the form of the calculation, the bias term is eliminated. The unbiased LS-SVM method is to introduce a constant value parameter on the basis of LS-SVM, so that the original bias term and weight are combined. The basic principle of the method is as follows.
The known sample data can be marked as
Figure BDA0002652384890000081
Wherein xk∈RnFor the kth sample input value, yk∈RnIs the target value of the kth sample, and k is the number of training samples.
According to the unbiased LS-SVM model in the feature space, the slave system can be represented as
Figure BDA0002652384890000082
Is provided with
Figure BDA0002652384890000083
Wherein
Figure BDA0002652384890000084
Is a term of the weight, and the weight,
Figure BDA0002652384890000085
a non-linear mapping function maps the input space from a low-dimensional space to a high-dimensional space. λ is a constant parameter, and bias terms are incorporated into the weights to increase computation speed
Figure BDA0002652384890000086
In (1). The same method maps f (x) in the slave system into a high-dimensional space to obtain
Figure BDA0002652384890000087
Where l 1, 2., n, k ═ l 1, 2., n, θ (x) is a certain nonlinear mapping function, which maps the input space into a high-dimensional space.
The constraint problem can be expressed as:
Figure BDA0002652384890000088
s.t x(t1)＝qx0
x(ti)＝xd(ti)+ζi i＝1，2，...m
wherein the initial condition is x0＝[p1；p2；p3；...pn]Sample is xd＝[yd1 2；yd1 m；ydn 2；...；ydn m]Substituting the above formula according to the principle of minimizing the structural risk can obtain:
Figure BDA0002652384890000089
Figure BDA0002652384890000091
Figure BDA0002652384890000092
Figure BDA0002652384890000093
Figure BDA0002652384890000094
Figure BDA0002652384890000095
wherein
Figure BDA0002652384890000096
Is the error in the synchronization that is,
Figure BDA0002652384890000097
is the error in the system, and is,
Figure BDA0002652384890000098
is the systematic approximation error of (x), and a, b, c are penalty factors, in order to solve the problems of over-fitting and under-fitting.
Then, a Lagrange multiplier alpha, beta and gamma are introduced to be more than or equal to 0 for conversion, and the Lagrange function obtained from the system and substituted into the unbiased LS-SVM model in the characteristic space is as follows:
Figure BDA0002652384890000099
wherein, i is 2, 1,2, n, l is 1,2, m, j is 1, 2. According to the Karush-Kuhn-Tucker (KKT) optimization condition, derivation is carried out on each variable in the Lagrangian function:
Figure BDA0002652384890000101
Figure BDA0002652384890000102
the following matrix equation is obtained:
Figure BDA0002652384890000103
wherein:
Figure BDA0002652384890000104
Figure BDA0002652384890000105
is provided with
Figure BDA0002652384890000106
Figure BDA0002652384890000107
1, 2.. m, each partition matrix in the coefficient matrix may be:
Figure BDA0002652384890000108
Figure BDA0002652384890000109
according to the Mercer condition, the kernel function can be written as:
Figure BDA0002652384890000111
defining a kernel function matrix omega epsilon RN*NThe derivative of (c) is:
Figure BDA0002652384890000112
wherein when m and n are 1 and 2 respectively:
Figure BDA0002652384890000113
Figure BDA0002652384890000114
Figure BDA0002652384890000115
from the above equations, the synchronous trajectory of the slave system is determined as:
Figure BDA0002652384890000116
Figure BDA0002652384890000117
Figure BDA0002652384890000118
in the least square support vector machine, the support vector is related to the selection of the kernel function. The most commonly used at present are Polynomial kernels (multinomial Kernel), Linear kernels (Linear Kernel), Gaussian kernels (Gaussian Kernel), hyperbolic tangent Sigmoid kernels, Exponential kernels (exponentialkernel), laplace kernels (Laplacian Kernel). The radial basis kernel function is widely used, has a simple structure and strong generalization capability, needs few optimized parameters, and only has the kernel function and the normalization parameter, so the radial basis kernel function is selected.
And step S3, optimizing parameters of the unbiased least squares support vector machine based on the particle swarm optimization.
Particle swarm algorithm
Particle Swarm Optimization (PSO) is an evolutionary computing technique, introduced by doctor Eberhart and doctor kennedy in 1995, from behavioral studies on the predation of groups of birds. The algorithm is initially inspired by the regularity of bird cluster activities, and then a simplified model is established by utilizing group intelligence. The particle swarm optimization enables the movement of the whole colony to generate an evolution process from disorder to order in a problem solving space by utilizing the sharing of the individual pair information in the colony on the basis of observing the activity behaviors of the animal colony, thereby obtaining an optimal solution.
Particle swarm optimization simulates birds in a flock of birds by designing a particle without mass, which has only two attributes: speed and position, speed representing the speed and direction of movement. Birds are abstracted as particles (dots) without mass and volume, and extendExtending into the m-dimensional space, the current position of the particle i in the m-dimensional space is represented as a vector xi＝(xi1，xi2，...，xim) The flight speed is expressed as vector vi ═ (v)i1，vi2，...，vim)。
Each particle has an adaptation value (fixness value) determined by the objective function and knows its best position (pbest) found so far and the current position xi. This can be seen as the flight experience of the particles themselves. In addition to this, each particle also knows the best position (gbest) found so far for all particles in the whole population (gbest is the best value in pbest), which can be seen as an experience of particle companions. The particles determine the next step of movement through their own experience and the best experience among the partners.
The PSO is initialized to a population of random particles (random solution). The optimal solution is then found by iteration. In each iteration, the particle updates itself by tracking two "extrema" (pbest). After finding these two optimal values, the particle updates its velocity and position by the following formula:
vi＝vi+c1*rand()*(pbesti-xi)+c2*rand()*(gbesti-xi)
xi＝xi+vi
in the above formula, i is 1, 2. N is the total number of particles in the population. Rand () is a random number between 0 and 1. x is the number ofiIs the current position of the particle. c. C1，c2Is a scholar factor, usually taken as 2. v. ofiMaximum value of vmax(greater than 0) if vi＞vmaxThen v isi＝vmax. The above formula is a standard form of particle swarm. From the above equations, it can be seen that the PSO is defined in terms of velocity, and the particle also has memory characteristics.
PSO-based unbiased LS-SVM parameter optimization
The radial basis kernel function (RBF) has only the kernel function σ and the normalization parameter γ that have a large influence on the accuracy of the prediction. Therefore, in the present application, a particle swarm algorithm is used to optimize parameters in an unbiased LS-SVM, please refer to fig. 3, which shows the overall steps of the algorithm as follows:
in step S31, a population of particles of population size N is initialized, including the random position and velocity of the particles, the learning factor, and the maximum number of iterations.
And step S32, respectively using the least square support vector machine corresponding to each particle vector to predict the sample, and obtaining the current position and prediction error of each particle.
And step S33, taking the result of the step S33 as the current fitness of the corresponding particle, then comparing the current fitness of each particle with the optimal fitness of the particle, and if the particle has no optimal fitness, taking the current position of the corresponding particle as the optimal position of the particle.
And step S34, comparing the fitness of each particle optimal position with the fitness of the population optimal position, and if the performance is better, taking the particle optimal position as the population optimal position.
And step S35, calculating inertia weight, and updating the speed and position of the particles.
And step S36, judging whether the optimal condition is met, if so, ending the optimization, otherwise, returning to the step S32 and continuing a new round of optimization process.
Parameters in the unbiased LS-SVM are optimized by using a particle swarm algorithm, and a flowchart of the algorithm is shown in fig. 4.
Numerical simulation
And step S4, carrying out numerical simulation on the control input signal and verifying the validity.
If the state variables of the master system and the slave system are synchronized by the control input signal, the next step is executed, and if the state variables of the master system and the slave system are not synchronized, the process returns to the step S2.
To verify the effectiveness of the PSO-unbiased LS-SVM method, the following experiment was used for verification. The main system, namely the driving system, selects a new four-dimensional chaotic system, and the kinetic equation of the driving system is as follows:
Figure BDA0002652384890000141
wherein a, b, c, p, e are system parameters, which are respectively: 30,2.5,35,20,0.2,0.5. Initial state is y1(0)＝0.02,y2(0)＝0.01,y3(0)＝0.02,y4(0)＝0.01。
The slave system, i.e. the response system, is:
Figure BDA0002652384890000142
here, u ═ u (u1, u2, u3, u4)TIs a nonlinear control vector, in which a, b, c, p, e, l are system parameters, which are: 30,2.5,35,20,0.2,0.5. Initial state is y1(0)＝1.2,y2(0)＝-1,y3(0)＝1.5,y4(0)＝2。
Synchronous control trajectories for the drive system and response system referring to fig. 5-12, it can be seen that the master and slave systems are synchronized.
As can be seen from fig. 5, 7, 9 and 11, after the drive system is added to the controller, the error system e1，e2，e3，e4All converge to zero and the error system reaches a steady state. As can be seen from fig. 6, 8, 10 and 12, the state variables of the driving system and the response system are controlled to coincide with each other, and a completely synchronous state is achieved. From the numerical experiment result, the designed controller has obvious effect.
In step S5, the active control input signal is used as the hybrid synchronous control signal for the master system and the slave system.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (5)

1. A chaotic synchronization control method for realizing synchronous control of a master system and a slave system is characterized by comprising the following steps:
step S1, sampling the main system to obtain a training sample number set;
step S2, under the method of the unbiased least square support vector machine, the fitness function of the master system and the slave system is established, and a control input signal is designed to be used as a controller;
step S3, optimizing parameters of the unbiased least square support vector machine based on the particle swarm optimization;
step S4, carrying out numerical simulation on the control input signal, and verifying the validity;
in step S5, the active control input signal is used as the hybrid synchronous control signal for the master system and the slave system.
2. The chaotic synchronization control method according to claim 1, wherein the step S2 includes:
step S21, establishing nonlinear chaotic models of the master system and the slave system:
Figure FDA0002652384880000011
Figure FDA0002652384880000012
wherein x, y ∈ RnF, g is a differentiable function, u (t, x, y) is a control input, a hybrid synchronization error is defined as e-x-qy, and different initial values x0、y0The system meets the following requirements:
Figure FDA0002652384880000013
when q is 0, the master system and the slave system are synchronized, and when q is-1, the master system and the slave system are desynchronized;
step S22, introducing a fixed value parameter based on the least square support vector machine, so that the original bias term and weight are combined.
3. The chaotic synchronization control method according to claim 1, wherein in step S3, the unbiased least squares support vector machine selects the radial basis kernel function.
4. The chaotic synchronization control method according to claim 3, wherein the step S3 includes the steps of:
step S31, initializing a group of particles with the group size N, wherein the group of particles comprises the random positions and speeds of the particles, learning factors and the maximum iteration times;
step S32, respectively using the least square support vector machine corresponding to each particle vector to predict the sample, and obtaining the current position and prediction error of each particle;
step S33, taking the result of step S33 as the current fitness of the corresponding particle, then comparing the current fitness of each particle with the optimal fitness of the particle, and if the particle has no optimal fitness, taking the current position of the corresponding particle as the optimal position of the particle;
step S34, comparing the fitness of the optimal position of each particle with the fitness of the optimal position of the population, and if the performance is better, taking the optimal position of the particle as the optimal position of the population;
step S35, calculating inertia weight, and updating the speed and position of the particles;
and step S36, judging whether the optimal condition is met, if so, ending the optimization, otherwise, returning to the step S32 and continuing a new round of optimization process.
5. The chaotic synchronization control method according to claim 1, wherein in step S4, if the state variables of the master system and the slave system are synchronized by the control input signal, the next step is performed, and if the state variables of the master system and the slave system are not synchronized, the procedure returns to step S2.
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Application publication date: 20201211