Disturbance fractional order chaotic system quasi-synchronization method and device and storage medium

Abstract

The invention discloses a disturbance fractional order chaotic system quasi-synchronization method and device and a storage medium, and the method comprises the steps: constructing a fractional order chaotic master system with external disturbance and a fractional order chaotic slave system with external disturbance based on a fractional order chaotic system; setting a synchronization error according to the fractional order chaos master system and the fractional order chaos slave system; constructing a synchronization error system according to the fractional order chaos master system, the fractional order chaos slave system and the synchronization error; designing an intermittent synchronous controller according to the synchronous error; and acting the intermittent synchronous controller on the synchronous error system, so that the fractional order chaos slave system is synchronized with the fractional order chaos slave system. According to the method, under the condition that disturbance exists in the chaotic system, the fractional calculus is adopted to describe the system, and the fractional order chaotic system under external disturbance is constructed, so that the robustness of the system is improved, and the method has a wider application prospect; by designing the intermittent synchronous controller, quasi-synchronization of the fractional order chaotic system is realized, and the control theory of the fractional order chaotic system is enriched.

Description

Technical Field

[0001] The present invention relates to the field of new generation information technology, and particularly to a method, device, terminal device, computer-readable storage medium and computer program product for pseudo-synchronization of a perturbed fractional-order chaotic system. Background Art

[0002] The fractional-order chaotic system is a generalization of the integer-order chaotic system and is a class of complex nonlinear dynamic systems. It can describe systems with memory effects, such as viscoelastic materials, ecological systems, financial markets, etc. In addition, the fractional-order chaotic system has the characteristics of the chaotic system's dependence on the initial value and unpredictability, as well as the genetic characteristics of the fractional-order system and the property closely related to the system order. Therefore, it has potential application prospects in secure communication, meteorology, fluid dynamics, and has become a current research hotspot.

[0003] Actual systems are usually in a dynamically changing environment, resulting in inevitable external perturbations in real life. For example, in the fields of robot control, automation systems, and UAV flight, external perturbations such as wind force, temperature changes, and power fluctuations are often faced. Studying the fractional-order chaotic system under external perturbations can improve the robustness and stability of the system.

[0004] Synchronization is a key research direction of the fractional-order chaotic system, and the convergence of its error system determines the synchronization effect of the chaotic system. Currently, many synchronization control methods are complete synchronization, that is, the states of the systems are exactly the same after a certain time, and the error must be exactly zero. However, in the presence of external perturbations, it is difficult for the fractional-order chaotic system to achieve complete synchronization. The performance of some systems (such as distributed control systems, social network systems, financial systems, etc.) does not depend on complete synchronization, but on the coordination degree between subsystems. Therefore, the synchronization error can be controlled to a small area to achieve the expected goal. Intermittent control is a discontinuous control method, and its control input is activated within some non-zero time intervals and turned off within other time intervals. Because it is practical and easy to implement in engineering control, it is widely used in engineering fields such as manufacturing, transportation, and communication. Compared with continuous control methods, intermittent control is more economical and efficient. Currently, there is less research on intermittent control of the fractional-order chaotic system. Therefore, studying the pseudo-synchronization of the fractional-order chaotic system based on intermittent control has great practical significance. Summary of the Invention

[0005] In view of this, the present invention provides a method, apparatus, terminal device, computer-readable storage medium and computer program product for quasi-synchronization of a perturbed fractional-order chaotic system. Considering the case where there is perturbation in the chaotic system, the system is described by fractional-order calculus and a fractional-order chaotic system under external perturbation is constructed, which improves the robustness of the system, so that the present invention has a wider application prospect.

[0006] The first object of the present invention is to provide a method for quasi-synchronization of a perturbed fractional-order chaotic system.

[0007] The second object of the present invention is to provide a quasi-synchronization system for a perturbed fractional-order chaotic system.

[0008] The third object of the present invention is to provide a terminal device.

[0009] The fourth object of the present invention is to provide a computer-readable storage medium.

[0010] The fifth object of the present invention is to provide a computer program product.

[0011] The first object of the present invention can be achieved by adopting the following technical solutions:

[0012] A method for quasi-synchronization of a perturbed fractional-order chaotic system, the method comprising:

[0013] Based on the fractional-order chaotic system, construct a fractional-order chaotic master system with external perturbation and a fractional-order chaotic slave system with external perturbation;

[0014] According to the fractional-order chaotic master system and the fractional-order chaotic slave system, set the synchronization error; according to the fractional-order chaotic master system, the fractional-order chaotic slave system and the synchronization error, construct a synchronization error system;

[0015] According to the synchronization error, design an intermittent synchronization controller; apply the intermittent synchronization controller to the synchronization error system to make the fractional-order chaotic slave system synchronize with the fractional-order chaotic master system.

[0016] Further, the intermittent synchronization controller is:

[0017]

[0018] where u ★ (t) is the intermittent synchronization controller at time t, is the synchronization error at time t, k is a positive control gain; in the sub-interval [t i , s i ), the intermittent synchronization controller is running, and the control time τ 1 = s i - t i; In the sub - interval [s i , t i+1 ), the intermittent synchronization controller is in the rest state, and the rest time δ = t i+1 - s i ; The time series satisfies

[0019] Furthermore, the intermittent synchronization controller is applied to the synchronization error system to update the synchronization error system as:

[0020]

[0021] where W ∈ R n×n is a constant matrix; is the synchronization error at time t; Λ(t) ∈ R n and are the state variables of the fractional - order chaotic master and slave systems at time t, respectively; D(t) ∈ R n and are the external disturbances existing in the fractional - order chaotic master and slave systems, respectively, and satisfy d ∈ R + ; k is a positive control gain; η: R n → R n is a Lipschitz - continuous non - linear function, where η(0) = 0. For any two different vectors Λ and n in R there exist constants l j > 0 (j = 1, 2, …, n) and satisfy If l max = max{l 1 , l 2 , …, l n}, it follows that

[0022] If there exists a constant k and satisfies the following conditions:

[0023] λ max (W - kI n ) ≤ - ξ 1

[0024] λ max (W) ≤ ξ 2

[0025]

[0026] where are all positive numbers, I n is the n - dimensional identity matrix, and λ max : R n → R is the function of taking the maximum eigenvalue of the matrix;

[0027] Under the action of the intermittent synchronization controller, quasi-synchronization of the fractional-order chaotic system under external disturbances is achieved;

[0028] Furthermore, the synchronization error trajectory remains within the range of the set Ω, where

[0029] Furthermore, the synchronization error system is:

[0030]

[0031] where u(t) ∈ R n is the synchronization controller to be designed at time t.

[0032] Furthermore, the synchronization error is:

[0033]

[0034] where is the synchronization error at time t, Λ(t) ∈ R n and are the state variables of the fractional-order chaotic master and slave systems at time t, respectively.

[0035] Furthermore, the fractional-order chaotic master system with external disturbances is:

[0036]

[0037] The fractional-order chaotic slave system with external disturbances is:

[0038]

[0039] where is the Caputo derivative of the fractional-order system at time t, q is the order of the fractional order in the system; Λ(t) ∈ R n is the state variable of the fractional-order chaotic master system, is the state variable of the fractional-order chaotic slave system; D(t) ∈ R n and are the external disturbances existing in the fractional-order chaotic master and slave systems, respectively, and satisfy d ∈ R + ; W ∈ R n×n is a constant matrix; u'(t) ∈ R n is the synchronization controller to be designed at time t; η: R n → R n is a Lipschitz continuous non-linear function, where η(0) = 0, and for any two different vectors Λ and n in R there exists a constant lj > 0 (j = 1, 2, …, n) and satisfy If l max = max{l 1 , l 2 , …, l n}}, it is obtained that

[0040] The second object of the present invention can be achieved by adopting the following technical solutions:

[0041] A quasi-synchronization device for a perturbed fractional-order chaotic system, the device comprising:

[0042] A master-slave system construction module, configured to construct a fractional-order chaotic master system with external perturbation and a fractional-order chaotic slave system with external perturbation based on a fractional-order chaotic system;

[0043] An error system construction module, configured to set a synchronization error according to the fractional-order chaotic master system and the fractional-order chaotic slave system; construct a synchronization error system according to the fractional-order chaotic master system, the fractional-order chaotic slave system, and the synchronization error;

[0044] A synchronization module, configured to design an intermittent synchronization controller according to the synchronization error; apply the intermittent synchronization controller to the synchronization error system to synchronize the fractional-order chaotic slave system to the fractional-order chaotic slave system.

[0045] The third object of the present invention can be achieved by adopting the following technical solutions:

[0046] A terminal device, comprising a processor and a memory for storing a program executable by the processor, and when the processor executes the program stored in the memory, the above-mentioned quasi-synchronization method for a perturbed fractional-order chaotic system is implemented.

[0047] The fourth object of the present invention can be achieved by adopting the following technical solutions:

[0048] A computer-readable storage medium, storing a program, and when the program is executed by a processor, the above-mentioned quasi-synchronization method for a perturbed fractional-order chaotic system is implemented.

[0049] The fifth object of the present invention can be achieved by adopting the following technical solutions:

[0050] A computer program product, comprising a computer program, and when the computer program is executed by a processor, the above-mentioned quasi-synchronization method for a perturbed fractional-order chaotic system is implemented.

[0051] The invention has the following beneficial effects compared with the prior art:

[0052] In the present invention, considering the case of fractional calculus and the existence of perturbations in chaotic systems simultaneously, a more general fractional-order chaotic system model, i.e., a fractional-order chaotic system under external perturbations, is constructed, which improves the robustness of the system, thus enabling the present invention to have a broader application prospect; the quasi-synchronization of the fractional-order chaotic system is realized by designing an intermittent synchronization controller, enriching the control theory of the fractional-order chaotic system. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on the structures shown in these drawings.

[0054] Figure 1 It is a flowchart of the quasi-synchronization method for the perturbed fractional-order chaotic system in Embodiment 1 of the present invention;

[0055] Figure 2 It is a curve graph showing the variation of each state quantity of the fractional-order chaotic main system with time in Embodiment 1 of the present invention;

[0056] Figure 3 It is a curve graph showing the variation of each state quantity of the fractional-order chaotic slave system with time in Embodiment 1 of the present invention;

[0057] Figure 4 It is a trajectory comparison graph of the error system without the action of a controller in Embodiment 1 of the present invention;

[0058] Figure 5 It is a trajectory comparison graph of the error system under the action of an intermittent controller in Embodiment 1 of the present invention;

[0059] Figure 6 It is that under the action of the intermittent controller in Embodiment 1 of the present invention, the synchronization error trajectory remains within the range of the set Ω;

[0060] Figure 7 It is a structural block diagram of the quasi-synchronization device for the perturbed fractional-order chaotic system in Embodiment 2 of the present invention;

[0061] Figure 8 It is a structural block diagram of the terminal device in Embodiment 3 of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0062] To make the objectives, 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 accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the scope of protection of the present invention. It should be understood that the specific embodiments described are only used to explain the present application and are not used to limit the present application.

[0063] Embodiment 1:

[0064] As Figure 1 shown, this embodiment provides a method for quasi-synchronization of a perturbed fractional-order chaotic system, which specifically includes the following steps:

[0065] Step S1: Based on the fractional-order chaotic system, construct a fractional-order chaotic master system with external perturbation and a fractional-order chaotic slave system with external perturbation.

[0066] Further, step S1 includes:

[0067] Step S11: Construct a fractional-order chaotic master system with external perturbation.

[0068] The master system is:

[0069]

[0070] Step S12: Construct a fractional-order chaotic slave system with external perturbation.

[0071] The slave system is:

[0072]

[0073] Wherein, represents the Caputo derivative of the fractional-order system at time t, t 0 represents the initial time, q represents the order of the fractional order in the system; Λ(t) ∈ R n represents the state variable of the master system, Λ(t 0 ) is the initial value of the master system; represents the state variable of the slave system, is the initial value of the slave system; D(t) ∈ R n and are respectively the external perturbations existing in the master and slave systems, and satisfy d ∈ R + ; W ∈ R n×n is a constant matrix; u'(t) ∈ R n represents the synchronization controller to be designed; η: R n → Rn is a Lipschitz continuous nonlinear function, where η(0) = 0, for any two different vectors Λ and n in R there exists a constant l j > 0 (j = 1, 2, …, n) that satisfies it can be obtained that where l max = max{l 1 , l 2 , …, l n}.

[0074] Step S2: According to the fractional-order chaotic master system and the fractional-order chaotic slave system with external disturbances, set the synchronization error and establish a synchronization error system.

[0075] Furthermore, Step S2 includes:

[0076] Step S21: Set the synchronization error between the master system and the slave system.

[0077] The synchronization error between the master system and the slave system is:

[0078]

[0079] Step S22: Construct a synchronization error system according to the master system, the slave system, and the synchronization error.

[0080] The synchronization error system is:

[0081]

[0082] where u(t) ∈ R n is the synchronization controller at time t to be designed.

[0083] Step S3: Design an intermittent controller according to the synchronization error; apply the controller to the slave system so that the slave system and the master system achieve quasi-synchronization.

[0084] Furthermore, Step S3 includes:

[0085] Step S31: Design an intermittent synchronization controller according to the synchronization error.

[0086] The intermittent synchronization controller is:

[0087]

[0088] where k is a positive control gain; in the sub-interval [t i , s i )), the controller is running, and the control time τ 1 = s i - ti ; In the sub - interval [s i , t i+1 ), the controller is in a rest state, and the rest time δ = t i+1 - s i ; The time series satisfies

[0089] Step S32: Apply the intermittent synchronization controller to the synchronization error system so that the slave system synchronizes with the master system.

[0090] Take the intermittent synchronization controller as the designed synchronization controller and apply it to the synchronization error system to update the synchronization error system as:

[0091]

[0092] If there exists a constant k and satisfies the following conditions:

[0093] λ max (W - kI n ) ≤ - ξ 1

[0094] λ max (W) ≤ ξ 2

[0095]

[0096] Wherein, are all positive numbers, I n is the n - dimensional identity matrix, and λ max : R n →R is the function of taking the maximum eigenvalue of the matrix.

[0097] Under the action of the intermittent synchronization controller, the quasi - synchronization of the fractional - order chaotic system in the presence of disturbances can be achieved. The synchronization error trajectory remains within the range of the set Ω, and its value is: β = (Ξ 1 + Ξ 2 )E q (h 2 δ q ).

[0098] Next, the principle of the quasi - synchronization method for the fractional - order chaotic system with external disturbances provided in this embodiment is explained:

[0099] Definition 1: The q - th order Caputo fractional - order derivative of the m - th order continuously differentiable function φ(t) in the Banach space is defined as:

[0100]

[0101] Among them, the integer m satisfies q < m < q + 1, t 0 is the initial time;

[0102] Definition 2: For the error system of the above fractional-order chaotic master system and fractional-order chaotic slave system, if there exists a compact set Ω such that the error function converges to the set Then the fractional-order chaotic master system and the fractional-order chaotic slave system can achieve quasi-synchronization, where the synchronization error ∈ * > 0.

[0103] Hypothesis 1: Assume that the nonlinear function η(·) is Lipschitz continuous, then for any there must exist a constant l j > 0 (j = 1, 2,..., n) such that η(·) = (η 1 (·), η 2 (·),..., η n (·)) T satisfies If l max = max{l 1 , l 2 ,..., l n}, then there is

[0104] Hypothesis 2: Assume that there exists a positive real constant d such that the external disturbance D(t) and satisfy

[0105] Lemma 1: When 0 < q ≤ 1, if the function V(t) is continuous and differentiable, then the inequality holds.

[0106] Lemma 2: When S and Q are real vectors of appropriate dimensions, there exists ∈ ∈ R + , such that the inequality holds. In particular, for ∈ = 1, the inequality simplifies to 2S T Q ≤ S T S + Q T Q.

[0107] Lemma 3: When 0 < q < 1 and ζ ∈ R, if the function V(t) is continuous on [t 0 , +∞) such that Then the inequality V(t) ≤ V(t 0 )E q (ζ(t - t 0 )) q , t ≥ t 0 is established, where is the one-parameter Mittag-Leffler function and Γ is the gamma function.

[0108] Lemma 4: When 0 < q < 1 and t ≥ t 0 the Mittag-Leffler function E q (ζ(t - t 0 )) q satisfies the following property: If ζ ≥ 0, then E q (ζ(t - t 0 )) q ) ≥ 1 is non-decreasing; if ζ ≤ 0, then 0 ≤ E q (ζ(t - t 0 )) q ) ≤ 1 is non-increasing.

[0109] Synchronization error system updated according to step S32:

[0110]

[0111] First, construct the Lyapunov function as: According to Lemma 1, the q-order fractional derivative of V(t) can be obtained. When t ∈ [t i , s i ),

[0112]

[0113] Then, according to Assumption 1, Assumption 2, and Lemma 2, it can be obtained that the non-linear term in (1.1) satisfies:

[0114]

[0115] and

[0116]

[0117] Substituting inequalities (1.2) and (1.3) into (1.1) again, we get:

[0118]

[0119] Similarly, when t ∈ [s i , t i+1 ), we can obtain:

[0120]

[0121] Immediately afterwards, combining inequalities (1.4) and (1.5), we can obtain:

[0122]

[0123] Based on the fractional differential property, the inequality group (1.6) can be rewritten as:

[0124]

[0125] Then, by applying Lemma 3, the inequality group (1.7) can be transformed into

[0126]

[0127] Furthermore, with the aid of Lemma 4, we can obtain:

[0128]

[0129] Next, based on the inequality group (1.9), we prove by mathematical induction that for all \(t\in[t 0 ,t i+1 \), the inequality always holds.

[0130] When \(t\in[t 0 ,s 0 \), we can get \(V(t)\leq V(t 0 )E q (-h 1 (t - t 0 ) q )+\Xi 1 . In particular,

[0131]

[0132] When \(t\in[s 0 ,t 1 \), \(V(t)\leq(V(s 0 )+\Xi 2 )E q (h 2 (t - t 0,1 ) q ). According to the result of the above inequality (1.10) at \(s 0 , we can obtain:

[0133]

[0134] When \(t\in[t 1 ,s 1 \), we can get:

[0135] V(t)\leq V(t 1 )E q (-h 1 (t - t 1 ) q )+\Xi 1

[0136] ≤V(t 0 )ωE q (-h 1 (t - t 1 ) q ) + βE q (-h 1 (t - t 1 ) q ) + Ξ 1

[0137] It can be concluded that:

[0138]

[0139] When t ∈ [s 1 , t 2 ), the following can be obtained:

[0140]

[0141] Thus, it can be deduced that:

[0142]

[0143] By mathematical induction, the following results can be summarized:

[0144]

[0145] For t ∈ [t l , s l ), according to Lemma 4, it is known that 0 < E q (-h 1 (t - t l ) q ) < 1 and E q (h 2 δ q ) > 1. In addition, β > Ξ 1 , from which the following can be obtained:

[0146]

[0147] Then, when t ∈ [s l , t l+1 ), the following inequality can be obtained:

[0148]

[0149] In summary, for any t ∈ [t 0 , t i+1 ), the following holds:

[0150]

[0151] Therefore, as time approaches infinity (t → +∞), the synchronization error system will converge to the set . Based on Definition 2, it can be known that the fractional-order chaotic master system and the fractional-order chaotic slave system can achieve quasi-synchronization, and the synchronization error is the set Ω.

[0152] Next, numerical simulations are carried out to verify that the fractional-order chaotic master system and the fractional-order chaotic slave system can achieve quasi-synchronization:

[0153] The fractional-order chaotic master system with external disturbances is described as follows:

[0154]

[0155] where Λ(t) = [Λ 1 (t), Λ 2 (t), Λ 3 (t)] T , the non-linear term η(Λ(t)) = [0, -Λ 1 (t)Λ 3 (t), Λ 1 (t)Λ 2 (t)] T , the disturbance term D(t) = [D 1 (t), D 2 (t), D 3 (t)] T ,

[0156] The fractional-order chaotic slave system with external disturbances is described as follows:

[0157]

[0158] where, non-linear term disturbance term controller

[0159] Given the system parameters σ 1 = 10, σ 2 = 28, σ 3 = 8 / 3, let the orders of the fractional-order chaotic master system and the fractional-order chaotic slave system be q = 0.993, and the initial values be Λ(0) = [8; 8; 8] and The specific expression of the disturbance term is D 1 (t) = -0.26sin(3t), D 2 (t) = 18cos(6t), D 3 (t) = -0.6sin(8t), Since If it is assumed that Then When the control gain k = 341.87 and the control time τ 1 = 0.15, and the rest time δ = 0.1, the following parameter values of ξ 2 = λ max (W) = 11.8277 can be obtained. It can be known through calculation that λ max (W - kI n ) = -330.0423, and ξ 1 = 330.0423 can be obtained. Since l max = 25, Then β = (Ξ 1 + Ξ 2 )E q (h 2 δ q ) = 0.1107, Through these calculations, the conditions in step 3 are satisfied. When external disturbances exist, the fractional-order chaotic master system and the fractional-order chaotic slave system can achieve quasi-synchronization under the action of intermittent control, and the range of the synchronization error trajectory verifies the effectiveness of the proposed control strategy.

[0160] Figure 2 And Figure 3 are the curve graphs of the state variables of the fractional-order chaotic master system and the fractional-order chaotic slave system changing with time under disturbance respectively. It can be seen that the system has chaotic behavior at this time. Figure 4 is the trajectory comparison graph of the error system without the action of the controller. When there is no controller, the synchronization error trajectory oscillates continuously, indicating that the master system and the slave system cannot achieve quasi-synchronization without the action of the controller; Figure 5 is the trajectory comparison graph of the error system under the action of intermittent control, Figure 6 and the synchronization error trajectory under the action of the controller remains within the range of the set Ω. The above results show that the master system and the slave system achieve quasi-synchronization under the action of the synchronization controller designed in this embodiment.

[0161] In this embodiment, by selecting the fractional-order Lorenz system for numerical simulation, the accuracy and effectiveness of the invention are verified.

[0162] Those skilled in the art can understand that all or part of the steps in the method of implementing the above embodiment can be completed by instructing relevant hardware through a program, and the corresponding program can be stored in a computer-readable storage medium.

[0163] It should be noted that although the method operations of the above embodiments are described in a specific order in the accompanying drawings, this does not require or imply that these operations must be performed in that specific order, or that all the operations shown must be performed to achieve the desired result. On the contrary, the depicted steps can be executed in a changed order. Additionally or alternatively, certain steps can be omitted, multiple steps can be combined into one step for execution, and / or one step can be decomposed into multiple steps for execution.

[0164] Embodiment 2:

[0165] As Figure 7 shown, this embodiment provides a quasi-synchronization system for a perturbed fractional-order chaotic system, which system includes a master-slave system construction module 701, an error system construction module 702, and a synchronization module 703, where:

[0166] The master-slave system construction module 701 is configured to construct a fractional-order chaotic master system with external perturbation and a fractional-order chaotic slave system with external perturbation based on the fractional-order chaotic system;

[0167] The error system construction module 702 is configured to set a synchronization error according to the fractional-order chaotic master system and the fractional-order chaotic slave system; and construct a synchronization error system according to the fractional-order chaotic master system, the fractional-order chaotic slave system, and the synchronization error;

[0168] The synchronization module 703 is configured to design an intermittent synchronization controller according to the synchronization error; and apply the intermittent synchronization controller to the synchronization error system to synchronize the fractional-order chaotic slave system to the fractional-order chaotic slave system.

[0169] For the specific implementation of each module in this embodiment, reference can be made to Embodiment 1 above, which will not be elaborated here one by one; it should be noted that the system provided in this embodiment is only illustrated by the above division of each functional module. In practical applications, the above functions can be allocated to different functional modules as needed, that is, the internal structure can be divided into different functional modules to complete all or part of the functions described above.

[0170] Embodiment 3:

[0171] This embodiment provides a terminal device, which terminal device can be a computer, such as Figure 8As shown in the figure, it includes a processor 802, a memory, an input device 803, a display 804, and a network interface 805 connected through a system bus 801. The processor is used to provide computing and control capabilities. The memory includes a non-volatile storage medium 806 and an internal memory 807. The non-volatile storage medium 806 stores an operating system, a computer program, and a database. The internal memory 807 provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. When the processor 802 executes the computer program stored in the memory, the method for quasi-synchronization of the perturbed fractional-order chaotic system in Embodiment 1 above is implemented as follows:

[0172] Based on the fractional-order chaotic system, construct a fractional-order chaotic master system with external perturbation and a fractional-order chaotic slave system with external perturbation;

[0173] According to the fractional-order chaotic master system and the fractional-order chaotic slave system, set the synchronization error; according to the fractional-order chaotic master system, the fractional-order chaotic slave system, and the synchronization error, construct a synchronization error system;

[0174] According to the synchronization error, design an intermittent synchronization controller; apply the intermittent synchronization controller to the synchronization error system to make the fractional-order chaotic slave system synchronize with the fractional-order chaotic master system.

[0175] Embodiment 4:

[0176] This embodiment provides a computer-readable storage medium that stores a computer program. When the computer program is executed by a processor, the method for quasi-synchronization of the perturbed fractional-order chaotic system in Embodiment 1 above is implemented as follows:

[0177] Based on the fractional-order chaotic system, construct a fractional-order chaotic master system with external perturbation and a fractional-order chaotic slave system with external perturbation;

[0178] According to the fractional-order chaotic master system and the fractional-order chaotic slave system, set the synchronization error; according to the fractional-order chaotic master system, the fractional-order chaotic slave system, and the synchronization error, construct a synchronization error system;

[0179] According to the synchronization error, design an intermittent synchronization controller; apply the intermittent synchronization controller to the synchronization error system to make the fractional-order chaotic slave system synchronize with the fractional-order chaotic master system.

[0180] It should be noted that the computer-readable storage medium of this embodiment can be a computer-readable signal medium, a computer-readable storage medium, or any combination of the two. The computer-readable storage medium can be, for example, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination of the above. More specific examples of the computer-readable storage medium can include, but are not limited to: an electrical connection with one or more wires, a portable computer disk, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), an optical fiber, a portable compact disk read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the above.

[0181] Embodiment 5:

[0182] This embodiment provides a computer program product, including a computer program, which when executed by a processor implements the perturbation fractional-order chaotic system quasi-synchronization method of the above Embodiment 1, as follows:

[0183] Based on the fractional-order chaotic system, construct a fractional-order chaotic master system with external perturbation and a fractional-order chaotic slave system with external perturbation;

[0184] According to the fractional-order chaotic master system and the fractional-order chaotic slave system, set the synchronization error; according to the fractional-order chaotic master system, the fractional-order chaotic slave system, and the synchronization error, construct a synchronization error system;

[0185] According to the synchronization error, design an intermittent synchronization controller; apply the intermittent synchronization controller to the synchronization error system to make the fractional-order chaotic slave system synchronize with the fractional-order chaotic slave system.

[0186] In summary, the present invention describes the chaotic system by introducing fractional-order calculus, which is more applicable to the modeling and simulation in practical problems; by considering external perturbations, the stability and robustness of the system are improved; by adopting intermittent control, which only operates within the control interval, the control cost can be effectively saved.

[0187] The above is only a preferred embodiment of the present invention patent, but the protection scope of the present invention patent is not limited thereto. Any person skilled in the art within the scope disclosed by the present invention patent, according to the technical solution and inventive concept of the present invention patent, makes equivalent substitutions or changes, all belong to the protection scope of the present invention patent.

Claims

1. A quasi-synchronization method for a perturbed fractional-order chaotic system, characterized in that: The method comprises: Based on the fractional-order chaotic system, a fractional-order chaotic master system with external disturbance and a fractional-order chaotic slave system with external disturbance are constructed; According to the fractional-order chaos master system and the fractional-order chaos slave system, a synchronization error is set; according to the fractional-order chaos master system, the fractional-order chaos slave system and the synchronization error, a synchronization error system is constructed; According to the synchronization error, an intermittent synchronization controller is designed; the intermittent synchronization controller is applied to the synchronization error system to synchronize the fractional-order chaotic slave system with the fractional-order chaotic slave system.

2. The quasi-synchronization method of a disturbed fractional-order chaotic system according to claim 1, characterized in that: The intermittent synchronous controller is: Among them, u ＊ (t) is the intermittent synchronous controller at time t, is the synchronization error at time t, k is the positive control gain; in the sub-interval [t i ,s i ), the intermittent synchronous controller is running, and the control time τ1 = s i -t i ; In the subinterval [s i ,t i+1 ), the intermittent synchronous controller is in a rest state, and the rest time δ = t i+1 -s i ; The time series satisfies 0=t0<s0<t1<s1…<t i <s i <t i+1 <…, t0 represents the initial time.

3. The quasi-synchronization method of a disturbed fractional-order chaotic system according to claim 1, characterized in that: The intermittent synchronous controller acts on the synchronous error system to update the synchronous error system as follows: Where W is a constant matrix; is the synchronization error at time t; Λ(t)∈R n and are the state variables of the fractional-order chaotic master and slave systems at time t; D(t)∈R n and are external disturbances existing in the fractional-order chaotic master and slave systems respectively, and satisfy d∈R + ; k is a positive control gain; η:R n →R n is a Lipschitz continuous nonlinear function, where η(0) = 0, for R n Any two different vectors Λ and There is a constant l j >0(j=1,2,…,n) and satisfies If l max =max{l1,l2,…,l n }, we get If there exists a constant k and the following conditions are met: λ max (W-kI n )≤-ξ1 l max (W)≤ξ2 Among them, ξ1,ξ2, All are positive numbers, I n is the n-dimensional identity matrix, λ max :R n →R is the function to take the maximum eigenvalue of the matrix; Under the action of the intermittent synchronization controller, the quasi-synchronization of the fractional-order chaotic system under external disturbance is achieved.

4. The quasi-synchronization method of a disturbed fractional-order chaotic system according to claim 3, characterized in that: The synchronization error trajectory remains within the range of the set Ω, where β=(Ξ1+Ξ2)E q (h2δ q ).

5. The quasi-synchronization method of a disturbed fractional-order chaotic system according to claim 3, characterized in that: The synchronization error system is: Where u(t)∈R n is the synchronous controller to be designed at time t.

6. The quasi-synchronization method of a disturbed fractional-order chaotic system according to any one of claims 1 to 5, characterized in that: The synchronization error is: In the formula, is the synchronization error at time t, Λ(t)∈R n and are the state variables of the fractional-order chaotic master and slave systems at time t respectively.

7. The quasi-synchronization method for a disturbed fractional-order chaotic system according to any one of claims 1 to 5, characterized in that: The fractional-order chaotic main system with external disturbance is: The fractional-order chaotic slave system with external disturbance is: In the formula, is the Caputo differential of the fractional-order system at time t, q is the number of fractional orders in the system; Λ(t)∈R n is the state variable of the fractional-order chaotic main system, is the state variable of the fractional-order chaotic slave system; D(t)∈R n and are external disturbances existing in the fractional-order chaotic master and slave systems respectively, and satisfy d∈R + ; W∈R n×n is a constant matrix; u'(t)∈R n is the synchronous controller to be designed at time t; η:R n →R n is a Lipschitz continuous nonlinear function, where η(0) = 0, for R n Any two different vectors Λ and There is a constant l j >0(j=1,2,…,n) and satisfies If l max =max{l1,l2,…,l n }, we get 8. A quasi-synchronization device for a perturbed fractional-order chaotic system, characterized in that: The device comprises: A master-slave system building module is used to build a fractional-order chaotic master system with external disturbance and a fractional-order chaotic slave system with external disturbance based on the fractional-order chaotic system; The error system construction module is used to set the synchronization error according to the fractional-order chaos master system and the fractional-order chaos slave system; and to construct the synchronization error system according to the fractional-order chaos master system, the fractional-order chaos slave system and the synchronization error; The synchronization module is used to design an intermittent synchronization controller according to the synchronization error; the intermittent synchronization controller is applied to the synchronization error system to synchronize the fractional-order chaotic slave system with the fractional-order chaotic slave system.

9. A terminal device, comprising a processor and a memory for storing a program executable by the processor, characterized in that: When the processor executes the program stored in the memory, the quasi-synchronization method of a disturbed fractional-order chaotic system according to any one of claims 1 to 7 is implemented.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the quasi-synchronization method of a disturbed fractional-order chaotic system according to any one of claims 1 to 7 is implemented.

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