CN111736458B - Adaptive synchronization method of fractional order heterogeneous structure chaotic system - Google Patents

Abstract

The invention discloses an adaptive synchronization method for a fractional-order heterogeneous structure chaotic system. The main steps are: (1) selecting two fractional-order chaotic systems with different structures and determined parameters as the driving system and the response system; (2) finding Obtain fractional error systems e ₁ , e ₂ , e ₃ , e ₄ ; (3) In fractional error systems e ₁ , e ₂ , e ₃ , e ₄ , add u ₁ (t), u ₂ (t ), u ₃ (t), u ₄ (t) controllers; (4) Design u ₁ (t), u ₂ (t), u ₃ (t), u ₄ (t) controllers and adaptive laws; (5) Construct the Lyapunov control function, and use the properties of the Caputo derivative operator and the Mittag-Leffler stability theory to determine that the fractional error systems e ₁ , e ₂ , e ₃ , and e ₄ are asymptotically stable, and realize different parameters under certain parameters. Synchronous control of structural chaotic systems. The invention can effectively control the fractional-order chaotic system with deterministic parameters. By designing an adaptive controller, the constructed Lyapunov function is combined with the fractional-order Mittag-Leffler stability theory, which effectively reduces the complexity of control and shortens the synchronization period. Time is a universal control method.

Description

An Adaptive Synchronization Method for Fractional Heterogeneous Structure Chaotic System

technical field

The invention relates to a synchronization method of a fractional-order heterogeneous structure chaotic system, and belongs to the technical field of automatic control methods.

Background technique

Because nonlinear chaotic systems have very rich dynamic characteristics, they have been applied in many fields, such as meteorology, mechanics and secure communication. The application of chaotic secure communication and related scientific fields has a very broad prospect, and chaotic synchronization is the key technology. Since the concept of chaotic synchronization was proposed by Pecora and Carroll of the US Naval Laboratory in the 1990s, many scholars have proposed many effective synchronization methods, such as drive-response synchronization, active-passive synchronization, coupled synchronization, adaptive synchronization, projection synchronization. , Sliding mode control synchronization.

The adaptive synchronization method is mostly used in integer-order chaotic systems, because the Lyapunov function constructed for the synchronization error system is easy to find the integer-order derivative, and for the fractional-order system, because the synchronization error function is also a fractional-order system. However, after calculating the integer derivative of the constructed Lyapunov function, it is difficult to calculate the fractional term. There are two ways to deal with it. The literature "Adaptive feedback control and synchronization of non-identicalchaotic fractional order systems" realizes the adaptive synchronization control of fractional order systems by finding an inequality substitution method. The literature "Finite-time synchronization between two complex-variablechaotic systems with unknown parameters via nonsingular terminal" Sliding modecontrol" and the literature "Synchronization of Chaotic Fractional-Order Systems via Fractional-Order Adaptive Controller" are converted by constructing sliding mode control surfaces. These two methods are complicated in mathematical operation process, and currently lack more generalization for fractional-order chaotic systems. A unique adaptive synchronization control method.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide an adaptive synchronization method for fractional-order heterogeneous chaotic systems, which can effectively simplify the synchronization between different fractional-order chaotic systems.

The technical scheme adopted in the present invention is:

An adaptive synchronization method for a fractional-order heterostructure chaotic system, characterized in that it comprises the following steps:

(1) Select two fractional-order chaotic systems with different structures, the parameters of both systems are determined, and the state information of the driving system is determined to be x ₁ , x ₂ , x ₃ , x ₄ , and the state information of the response system is y ₁ , y ₂ , y ₃ , y ₄ ;

(2) Obtain fractional-order error systems e ₁ , e ₂ , e ₃ , e ₄ according to the state information of the chaotic drive system and the chaotic response system;

(3) Add u ₁ (t), u ₂ (t), u ₃ (t), and u ₄ (t) controllers to the fractional-order error systems e ₁ , e ₂ , e ₃ , and e ₄ respectively;

(4) Design u ₁ (t), u ₂ (t), u ₃ (t), u ₄ (t) controllers and adaptive rates;

(5) According to the Mittag-Leffler stability theory, the Lyapunov control function is constructed, and the properties of the Caputo derivative operator are used to determine that the derivative of the Lyapunov function is not positive, and then the fractional error system e ₁ is obtained from the Mittag-Leffler stability theory, e ₂ , e ₃ , e ₄ are globally asymptotically stable, and it is concluded that two chaotic systems with different structures are synchronized under certain parameters.

In step (1), first determine the fractional calculus as the fractional calculus defined by Caputo, which is as follows:

In the formula, C indicates that the definition method is the Caputo fractional order definition, q is the order of the differential operator, n is the smallest integer greater than q, and n-1<q<n, t, a are the upper and lower limits of the definite integral, respectively, Γ(·) is a Gamma function.

If q is a fractional order, a ₁ is a definite parameter of the system, and the drive system is selected according to the definition of fractional calculus:

If q is a fractional order and a ₂ is a definite parameter of the system, the response system selected according to the definition of fractional calculus is:

In step (2), the fractional-order error system is:

In step (3), in the fractional-order error system, add the controller input u ₁ (t), u ₂ (t), u ₃ (t), u ₄ (t) respectively:

The self-adaptive synchronization method of a fractional-order heterostructure chaotic system according to claim 1, characterized in that, in step (4), the designed controller is:

是对参数a₁的估计，若λ为参数，估计参数的自适应律为：where parameters

is the estimation of the parameter a _1. If λ is a parameter, the adaptive law of the estimated parameter is:

In step (5), (a) Lemma 1: According to the Mittag-Leffler stability theory:

Denote the equilibrium point of the nonlinear fractional-order dynamical system as x _eq =0, D is the region including the far point, V(t,x(t)):[0,∞)×D→R ⁺ is a continuously differentiable function and Satisfy:

In the formula, γ(·) is a K-type function, x∈D and 0<α<1, then the equilibrium point x _eq =0 is globally stable;

(b) Construct the Lyapunov control function from Lemma 1:

参数

是参数a₁的估计值where e=[e ₁ , e ₂ , e ₃ , e ₄ ] ^T ,

parameter

is the estimated value _of parameter a1

(c) Lemma 2: Properties of Caputo derivative operator:

If x(t)∈R is a continuously differentiable function, then for any t≥b, the following relation holds:

则分数阶误差系统e₁，e₂，e₃，e₄是全局渐近稳定的，得出确定参数下两不同结构混沌系统同步。(d) When λ≤0, D ^q V≤0, then Lemma 1 shows that the error system (iv) has equilibrium points e=0 and

Then the fractional-order error systems e ₁ , e ₂ , e ₃ , e ₄ are globally asymptotically stable, and it is concluded that two chaotic systems with different structures are synchronized under certain parameters.

Compared with the existing method, the present invention has significant advantages and beneficial effects, and the details are as follows:

The invention designs the self-adaptive law for estimating parameters in the control function according to self-adaptive theory. For the fractional-order chaotic system whose parameters are determined, the Lyapunov function is simply constructed by using the fractional-order Mittag-Leffler stability theory, and the fractional-order error function is avoided. The derivation of , uses the properties of the Caputo derivative operator to realize the heterostructure synchronization of two fractional-order chaotic systems; the existing adaptive synchronization methods are mainly aimed at integer-order chaotic systems; the existing adaptive synchronization methods for fractional-order chaotic systems It is necessary to find sliding mode surface or inequality substitution to realize synchronization, the realization process is complicated, and the control precision is low; the fractional-order adaptive synchronization controller designed by the present invention has strong universality, simple control steps, and is a fractional-order chaotic system. Lay a foundation for deeper theoretical research and practical engineering techniques.

Description of drawings

FIG. 1 is a Matlab two-dimensional attractor projection phase diagram of an adaptive synchronization method driving system (ii) of a fractional-order heterostructure chaotic system of the present invention.

FIG. 2 is a Matlab two-dimensional attractor projection phase diagram of the adaptive synchronization method response system (iii) of a fractional-order heterostructure chaotic system of the present invention.

FIG. 3 is a Matlab dynamic modeling analog function acting on the error system (v) at time t=0 according to an adaptive synchronization method of a fractional-order heterostructure chaotic system of the present invention.

Figure 1(a) is the x ₁ -x ₂ plane phase diagram, and Figure 1(b) is the x ₂ -x ₃ plane phase diagram.

Fig. 2(a) is the y ₁ -y ₃ plane phase diagram, and Fig. 2(b) is the y ₁ -y ₄ plane phase diagram.

Figure 3(a) is the time domain response curve of the adaptive synchronous controller e ₁ acting on the error system at time t=0.

Figure 3(b) is the time domain response curve of the adaptive synchronous controller e ₂ acting on the error system at time t=0.

Fig. 3(c) is the time domain response curve of the adaptive synchronous controller e ₃ acting on the error system at time t=0.

Fig. 3(d) is the time domain response curve of the adaptive synchronous controller e ₄ acting on the error system at time t=0.

Detailed ways

The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

The present invention is an adaptive synchronization method for a fractional-order heterostructure chaotic system, which is specifically implemented according to the following steps:

Taking the fractional order q=0.8 and the system parameter a ₁ =a ₂ =1, then:

Step (1), select the driving fractional-order chaotic system as:

Select the response system as:

The system (ii) and system (iii) are numerically simulated by matlab software to verify that the system has chaotic behavior. The simulation algorithm adopts the prediction-correction method. The step size is h=0.01, and the number of simulation points is N=4000.

Figure 1(a) shows the attractor projected phase diagram of system (ii) in the x ₁ -x ₂ plane, in the x ₁ -x ₂ plane (-25, 25) and (-40, 40) phase space range, The motion trajectory of system (ii) is a symmetrical double scroll attractor, indicating that system (ii) is a chaotic system.

Figure 1(b) shows the attractor projected phase diagram of system (ii) in the x ₂ -x ₃ plane. In the x ₂ -x ₃ planes (-40, 40) and (20, 60) phase space range, the system The motion trajectory of (ii) is a symmetrical butterfly-shaped attractor, which proves that the system (ii) is a chaotic system.

Figure 2(a) shows the attractor projected phase diagram of the system (iii) in the y ₁ -y ₃ plane. In the y ₁ -y ₃ plane (-20, 25) and (6, 24) phase space range, the system The trajectory of (iii) is a typical butterfly-shaped chaotic attractor, indicating that the system (iii) has chaotic behavior.

Figure 2(b) shows the attractor projected phase diagram of system (iii) on the y ₁ -y ₄ plane, and in the y ₁ -y ₄ plane (-20, 20) and (-10, 10) finite space range, the system The motion trajectory of (iii) has a self-similar shape but never intersects, indicating that the system (iii) is a chaotic system.

Step (2), the fractional order error system is:

Step (3), in the fractional error system, add controllers respectively:

Step (4), Theorem 1: The designed controller is

是对参数a₁的估计，且估计参数的自适应律为：where parameters

is an estimate of the parameter a ₁ , and the adaptive law of the estimated parameter is:

响应系统与驱动系统全局渐近同步，即对于任意初始值有

If the control parameter λ≤0, the error system has equilibrium point e=0 and

The response system is globally asymptotically synchronized with the drive system, that is, for any initial value, there is

Proof: Substitute system (vi) into system (v), we can get:

The error dynamic system formula (I) has nothing to do with the response system parameter a ₂ , as long as the error system is stabilized, the synchronization of two heterogeneous chaotic systems with different parameters can be realized.

Step (5), (a) Lemma 1: According to the Mittag-Leffler stability theory:

Denote the equilibrium point of the nonlinear fractional-order dynamical system as x _eq =0, D is the region including the far point,

V(t,x(t)):[0,∞)×D→R ⁺ is a continuously differentiable function and satisfies:

In the formula, γ(·) is a K-type function, x∈D and 0<α<1, then the equilibrium point x _eq =0 is globally stable.

为变量构造Lyapunov控制函数为：(b) with e ₁ , e ₂ , e ₃ , e ₄ and

The Lyapunov control function is constructed for the variables as:

参数

是参数a₁的估计值.where e=[e ₁ , e ₂ , e ₃ , e ₄ ] ^T ,

parameter

is _an estimate of parameter a1.

(c) Lemma 2: Properties of Caputo derivative operator:

If x(t)∈R is a continuously differentiable function, then for any t≥b, the following relation holds:

According to Lemma 2, the derivative of the Lyapunov control function (ix) is:

误差系统(iv)是渐近稳定的，所以两不同参数异结构混沌系统(ii)与(iii)同步，该非线性系统是全局渐近稳定的。According to the stability theory of Lemma 1, when λ≤0, D ^q V≤0, and Lemma 1 shows that the error system has equilibrium points e=0 and

The error system (iv) is asymptotically stable, so the two different parameter heterostructure chaotic systems (ii) and (iii) are synchronized, and the nonlinear system is globally asymptotically stable.

Certificate completed.

MATLAB R2016a is used for dynamic modeling and simulation of the synchronization method, and the simulation time is T _sim =10s.

分数阶阶次q＝0.8，驱动系统(ii)和响应系统(iii)的初值均为(1,2,2,3)。The simulation parameters are set as: variable step size, absolute error and relative error are both set to 10 ^-3 , λ = -1000, the solver uses ode15s, the initial value of the adaptive law

The fractional order q=0.8, and the initial values of the drive system (ii) and the response system (iii) are both (1, 2, 2, 3).

Fig. 3(a) shows the synchronization error curve of e ₁ under the action of the synchronization controller (vi). Under the ordinate with the minimum scale of ^10-4 , the synchronization establishment time is approximately 0s.

Fig. 3(b) shows the synchronization error curve of e ₂ under the action of the synchronization controller (vi). The synchronization establishment time is approximately 0.2s under the ordinate with the minimum scale of 2×10 ^-4 .

Figure 3(c) shows the synchronization error curve of _e3 under the action of the synchronization controller (vi). Under the ordinate with the minimum scale of 0.05, the synchronization establishment time is approximately 0.2s.

Fig. 3(d) shows the synchronization error curve of e ₄ under the action of the synchronization controller (vi). Under the ordinate with a minimum scale of 0.5×10 ⁻⁵ , the synchronization establishment time is approximately 0.2s.

It can be seen from the time-domain curve diagram of the error system in Fig. 3 that the chaotic system based on fractional order with different structures, under the action method of the proposed adaptive law, the system converges to 0 within 0.2s, which proves that the present invention Effectiveness of the proposed adaptive synchronization method for fractional heterostructure chaotic systems.

The self-adaptive synchronization method of a fractional-order-differentiated structure chaotic system provided by the implementation of the present invention has been described in detail above. The above description is only a preferred example of the present invention, not a limitation of the invention, and the present invention is not limited to the above example. Changes, modifications, additions or substitutions made by those skilled in the art within the essential scope of the present invention also belong to the protection scope of the present invention.

Claims (3)

1. A self-adaptive synchronization method of a fractional order heterogeneous structure chaotic system is characterized by comprising the following steps:

(1) selecting two fractional order chaotic systems with different structures, determining the parameters of the two systems, and determining the state information quantity of the driving system to be x₁，x₂，x₃，x₄And the amount of response system state information is y₁，y₂，y₃，y₄；

(2) Obtaining a fractional order error system e according to the state information quantity of the chaotic driving system and the chaotic response system₁，e₂，e₃，e₄；

(3) In a fractional order error system e₁，e₂，e₃，e₄In (1), u is added separately₁(t)、u₂(t)、u₃(t)、u₄(t) a controller:

wherein q is the order of the differential operator, a₁ a₂Determining parameters for a system

(4) Design u₁(t)、u₂(t)、u₃(t)、u₄(t) controller and adaptation rate:

wherein the parameters

Is to the parameter a₁And the adaptive law of the estimated parameters is as follows:

wherein λ is a parameter;

(5) constructing a Lyapunov control function according to the Mittag-Leffler stability theory, judging that the derivative of the Lyapunov function is less than zero by using the property of a Caputo derivative operator, and judging a fractional order error system e according to the Mittag-Leffler stability theory₁，e₂，e₃，e₄The overall asymptotic stability is achieved, and synchronization of two chaotic systems with different structures under determined parameters is obtained;

(5) in (a), lemma 1: according to Mittag-Leffler stability theory:

recording the balance point of the nonlinear fractional order power system as x_eq0, D is a region including a far point, V (t, x (t)) [0, ∞) xD → R⁺Is a continuously differentiable function and satisfies:

wherein gamma (. cndot.) is a function of class K, x ∈ D and 0 < alpha < 1, then the equilibrium point x is_eq0 is globally stable;

(b) constructing a Lyapunov control function:

wherein e ═ e₁,e₂,e₃,e₄]^T，

Parameter(s)

Is a parameter a₁An estimated value of (d);

(c) 2, introduction: properties of the Caputo derivative operator:

if x (t) ε R is a continuous differentiable function, then for any t ≧ b, the following relationship holds:

wherein, C represents that the definition mode is Caputo fractional order definition, q is the order of a differential operator, and a is the upper and lower limits of definite integral respectively;

(d) when λ is less than or equal to 0, D^qV is less than or equal to 0, and the error system (iv) has the balance point e equal to 0 and

fractional order error system e₁，e₂，e₃，e₄The overall asymptotic stability is achieved, and synchronization of two chaotic systems with different structures under determined parameters is obtained.

2. The adaptive synchronization method of the fractional order heterogeneous structure chaotic system according to claim 1, wherein in the step (1), the cpu uto definition is adopted herein, and specifically:

c represents that the definition mode is Caputo fractional order definition, q is the order of a differential operator, n is the minimum integer larger than q, n-1 < q < n, t, a are the upper limit and the lower limit of definite integral respectively, and Gamma (·) is a Gamma function;

and then selecting a driving system according to the fractional calculus as follows:

wherein q is a fractional order, a₁Determining parameters for the system;

the selection response system is as follows:

wherein q is a fractional order, a₂Is a determined parameter of the system.

3. The adaptive synchronization method of the fractional order heterogeneous structure chaotic system according to claim 1, wherein in the step (2), the fractional order error system is obtained by:

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