Chaotic system with sinusoidal feedback control of finite-time synchronization model

Abstract

The application relates to a chaotic system with a sine feedback control limited time synchronization model, which is mainly applied to limited time synchronization of master-slave non-autonomous chaotic systems, and is characterized in that the limited time synchronization model comprises a master non-autonomous chaotic system, a slave non-autonomous chaotic system and a sine feedback controller; wherein the output end of the master non-autonomous chaotic system is connected with the input end of the sine feedback controller, the output end of the sine feedback controller is connected with the input end of the slave non-autonomous chaotic system, and the output end of the slave non-autonomous chaotic system is connected with the input end of the sine feedback controller. The sine feedback control can make two chaotic systems reach synchronization in the shortest possible limited time. Moreover, no matter how large the input master-slave system state variable is, the sine feedback control can always output a smooth bounded control signal, so that the adverse effect on the slave system due to the too large input of the slave system can be prevented.

Description

Technical Field

[0001] This application belongs to the field of chaotic synchronization and relates to a finite-time synchronization model for chaotic systems based on sinusoidal feedback control. Background Technology

[0002] Chaotic synchronization has demonstrated enormous application prospects and market potential in fields such as secure communication, laser control, human life sciences, ecosystems, and chemistry. The vast majority of theoretical achievements in chaotic synchronization focus on the asymptotic synchronization of two chaotic systems, where the synchronization time is infinite. In engineering and technological fields that apply chaotic synchronization, such as secure digital communication, target recognition based on dynamic target patterns, and underwater vibration and noise control, it is often desirable for two chaotic systems to achieve synchronization within a finite timeframe, and for this synchronization time to be estimable. Therefore, research on finite-time synchronization of chaotic systems has significant application value. Summary of the Invention

[0003] This application provides a finite-time synchronization model for chaotic systems based on sinusoidal feedback control, which is used to solve the problem that two chaotic systems are difficult to synchronize within a finite time.

[0004] This application provides a finite-time synchronization model for chaotic systems based on sinusoidal feedback control, mainly applied to the finite-time synchronization of master-slave non-autonomous chaotic systems. The finite-time synchronization model comprises:

[0005] Non-autonomous chaotic system

[0006] From non-autonomous chaotic systems,

[0007] Sine feedback controller

[0008] The output of the master non-autonomous chaotic system is connected to the input of the sinusoidal feedback controller, the output of the sinusoidal feedback controller is connected to the input of the slave non-autonomous chaotic system, and the output of the slave non-autonomous chaotic system is connected to the input of the sinusoidal feedback controller.

[0009] Furthermore, the finite-time synchronization model is expressed as:

[0010] (3)

[0011] in,

[0012] (4)

[0013] (5)

[0014] x and z are state variables, and t is a time variable. Let be a time-varying bounded coefficient matrix. For a continuous nonlinear function, K and S are constant coupling matrices. , It is a symbolic function.

[0015] Furthermore, define an error variable. The following error system is obtained:

[0016] (7)

[0017] in, , , ; Let be an assumed time-varying bounded matrix, and such that... The pause time of the error system is the synchronization time of the master-slave non-autonomous chaotic system.

[0018] Furthermore, if the constant coupling matrix K=0, then the finite-time synchronization model is expressed as:

[0019] (9).

[0020] Furthermore, define an error variable. The following error system is obtained:

[0021] (10)

[0022] The pause time of the error system is the synchronization time of the master-slave non-autonomous chaotic system.

[0023] Furthermore, for the finite-time synchronization model (9), a finite-time synchronization cost is introduced, and the calculation formula is as follows:

[0024] (52)

[0025] in, The estimated synchronization time refers to the time when... At that time, the error of the master-slave non-autonomous chaotic system .

[0026] Furthermore, the finite-time synchronization model is applied to the SMIB power system, which includes a master SMIB non-autonomous chaotic system and a slave SMIB non-autonomous chaotic system, and the sinusoidal feedback controller is introduced.

[0027] The present invention has at least the following beneficial effects: This application enables two chaotic systems to achieve synchronization within a finite timeframe as short as possible through sinusoidal feedback control. Furthermore, regardless of the magnitude of the input state variables of the master-slave chaotic system, the sinusoidal feedback controller always outputs a smooth and bounded control signal, thus preventing adverse effects on the slave system due to excessive output from other controllers. Attached Figure Description

[0028] The accompanying drawings in this application are for illustrating preferred embodiments and to facilitate a clear understanding by those skilled in the art of various other advantages and benefits, and should not be construed as limiting the scope of this application. Furthermore, the same reference numerals denote the same parts throughout the drawings.

[0029] Figure 1 This is a schematic diagram of a finite-time synchronization model of a chaotic system based on sinusoidal feedback control in the embodiment.

[0030] Figure 2 The diagram shows the chaotic trajectory of the master system in the master-slave SMIB power system in this embodiment.

[0031] Figure 3 This is a schematic diagram of the synchronization process of the master-slave SMIB power system in the embodiment.

[0032] Figure 4 This is a schematic diagram of the synchronization process of the master-slave SMIB power system in the embodiment.

[0033] Figure 5 This is a schematic diagram of the synchronization cost of the synchronization model (9) in the embodiment. Detailed Implementation

[0034] The technical solutions in the embodiments of this application will be clearly and completely described below with reference to specific examples. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of this application without creative effort are within the scope of protection of this application.

[0035] This embodiment demonstrates how a master-slave non-autonomous chaotic system achieves consistent finite-time synchronization under two sinusoidal feedback controllers, providing proof criteria and an application example in a SMIB (single-machine infinite-bus) power system. Furthermore, introducing the concept of synchronization cost in finite-time synchronization allows for the numerical calculation of the minimum synchronization cost and optimal coupling strength. Minimum synchronization cost implies minimum energy input, which is highly significant in engineering applications such as SMIB power systems.

[0036] I. Description of Chaotic Systems and Synchronization Models

[0037] Consider a general non-autonomous chaotic system (or simply a non-autonomous system):

[0038]

[0039] Where x is a state variable, t is the time quantity, and the coefficient matrix is... It is time-varying and bounded, that is: , , and It is a constant; It is a continuous nonlinear function.

[0040] Assume there exists a bounded matrix , making

[0041]

[0042] Where z is the state variable. It is time-varying and bounded, that is... , , and It is a constant.

[0043] Therefore, a finite-time synchronization model for chaotic systems based on sinusoidal feedback control mainly includes a master non-autonomous chaotic system and a slave non-autonomous chaotic system (referred to as the master system and slave system, or the driving system and the response system). The master system and slave system can be combined into the following finite-time synchronization model through a sinusoidal feedback controller (referred to as the controller):

[0044]

[0045] in, , Let be a constant coupling matrix to be determined, and

[0046]

[0047]

[0048] , It is a symbolic function.

[0049] By designing coupling matrices K and S, it is possible to achieve chaos in the main system under any initial state. and from any initial state of the system Trajectories of the master and slave systems and satisfy

[0050]

[0051] And when hour, ,in The Euclidean norm of a vector is denoted by .

[0052] Define an error variable We can obtain the following dynamic error system:

[0053]

[0054] in, , , .

[0055] If the error system (7) is at the origin If the local (global) consistent finite-time stability is achieved, then the synchronization model (3) achieves local (global) consistent finite-time synchronization under condition (6). Therefore, we call the chaotic synchronization under condition (6) local (global) chaotic finite-time synchronization. The resting time of the error system is the synchronization time of the master-slave system.

[0056] If the constant coupling matrix We can then obtain a simpler controller:

[0057]

[0058] Then through the sinusoidal feedback controller The following finite-time synchronization model can be constructed:

[0059]

[0060] The dynamic error system is represented as:

[0061]

[0062] A simpler finite-time synchronization model for chaotic systems based on sinusoidal feedback control is provided.

[0063] II. Finite-Time Synchronization Criterion

[0064] Here we will prove that the finite-time synchronization model (3) and the finite-time synchronization model (9) are sufficient criteria for achieving local (global) finite-time synchronization.

[0065] Lemma 1 If ,So .

[0066] Lemma 2 Consider the following non-autonomous system:

[0067]

[0068] in, It is continuous, and For any All of these hold true, meaning the origin is a non-autonomous system. A balance point.

[0069] If there exists a continuously differentiable Lyapunov function In the neighborhood of the origin Internal satisfaction:

[0070]

[0071] in, It is continuous. And there exists a certain real number Make

[0072]

[0073] So, non-autonomous systems At the origin, it is stable for a finite time, and the resting time satisfies...

[0074]

[0075] if If it is gradually decreasing, then non-autonomous systems... At the origin, it is consistent and stable over a finite time, and the resting time satisfies...

[0076]

[0077] Furthermore, if non-autonomous systems Defined in the global scope, function It is radially unbounded, and Then the system At the origin, it is globally consistent and stable in a finite time.

[0078] Theorem 1 If there exists a symmetric positive definite matrix Two constant coupling matrices and This makes it possible for any In the neighborhood of the origin Inner, matrix

[0079]

[0080] Maximum eigenvalue satisfy

[0081]

[0082] and

[0083] Then the master-slave synchronization model (3) in the neighborhood Achieving localized, finite-time synchronization, wherein,

[0084]

[0085] And synchronize time function satisfy

[0086]

[0087] in,

[0088] .

[0089] Proof: Choose the following Lyapunov function

[0090] ,

[0091] So, Time along the trajectory of the error system (7) The derivative is:

[0092]

[0093] .

[0094] in, like As defined by the formula. Based on the assumption, It is bounded, therefore It always exists.

[0095] According to Lemma 1, when At that time, one can obtain Therefore, if the error variable , by the condition Then there is

[0096]

[0097] in, like Defined.

[0098] Let function

[0099]

[0100] Then, based on the conditions... It can be seen that, It is positive definite, and .

[0101] Choose a positive real number You can get

[0102]

[0103] Therefore, according to Lemma 2, the error system (7) is locally uniform and finite-time stable at the origin, that is, the master-slave synchronization model (3) is stable in... Achieving local finite-time synchronization within the time frame, and the synchronization time function satisfy

[0104] .

[0105] Below, we will prove the sufficient conditions for the simpler finite-time synchronization model (9) to achieve finite-time synchronization.

[0106] Theorem 2: Assume that for any There exists a constant , making

[0107]

[0108] Among them, the symmetric positive definite matrix If a constant coupling matrix exists... and real numbers This makes it possible for any ,

[0109]

[0110] in,

[0111]

[0112] Then the finite-time synchronization model (9) has an initial state error. Synchronization is achieved in a finite amount of time. And the synchronization time function... satisfy

[0113]

[0114] Proof: Choose the following Lyapunov function

[0115] , ,

[0116] So, by It can be seen that, Along the error system The trajectory of time The derivative is:

[0117] .

[0118] According to Lemma 1, when At that time, we can obtain:

[0119]

[0120] in, , .

[0121] Obviously, if

[0122]

[0123] If it is established, then .

[0124] From the above analysis, it can be seen that if the conditions are... Established, and initial values ​​selected. So there are ,therefore, Obviously, when the condition is met... At that time, under certain conditions It must also be true.

[0125] Let function , ,So, Choose a positive number. We can get

[0126]

[0127] From Lemma 2, we know that the error system (10) is uniformly finite-time stable at the origin, that is, the finite-time synchronization model (9) is stable at the origin. When finite-time synchronization is achieved, and the synchronization time function... satisfy .

[0128] from As can be seen from the formula, if If the value is small enough, then the synchronization region It can be large enough that Theorem 2 can be used as a criterion for global finite-time synchronization.

[0129] Compared to Theorem 2, Theorem 1 yields a locally finite-time criterion, but the criterion in Theorem 1 is independent of the initial state in the master-slave system, and the estimated synchronization time is also shorter.

[0130] III. Application Examples

[0131] The aforementioned finite-time synchronization model for chaotic systems based on sinusoidal feedback control can be widely applied to engineering chaotic systems. The following detailed explanation uses the SMIB power system as an example. SMIB power systems generally include master SMIB non-autonomous chaotic systems and slave SMIB non-autonomous chaotic systems. The aforementioned finite-time synchronization model for chaotic systems based on sinusoidal feedback control will be applied to the SMIB power system.

[0132] A typical SMIB power system can be represented as:

[0133] ,

[0134] in It is the moment of inertia. It is the damping constant. It is the generator's maximum power. It refers to the machine's power.

[0135] make , , , , Therefore, the SMIB power system can be rewritten as follows:

[0136] ,

[0137] in ,and

[0138]

[0139] All parameters , , and All of these are positive numbers. When these parameters take certain values, the system may exhibit complex chaotic behavior.

[0140] Obviously,

[0141] ,

[0142] in,

[0143] ,

[0144] ,

[0145] The SMIB power system is a non-autonomous system. One such case. The following algebraic synchronization criterion can be proved by Theorem 1:

[0146] Theorem 3 If a constant coupling matrix is ​​chosen and This makes it possible for any ,

[0147]

[0148] and

[0149] .

[0150] So regarding the synchronization model (3) of the SMIB power system Achieving finite-time synchronization within the time frame. And the synchronization time function... satisfy

[0151] ,

[0152] in,

[0153] ,

[0154] .

[0155] Proof: Choose a diagonal positive definite matrix ,in ,So

[0156] .

[0157] By calculating the eigenvalues ​​of the above matrix, we can obtain...

[0158]

[0159] Depend on It can be seen that,

[0160]

[0161] Therefore, according to the conditions You can get

[0162]

[0163] when hour, It has a minimum value Therefore, if the conditions are met

[0164]

[0165] condition Established.

[0166] If we further restrict the state variables to sub-regions Inside, there is

[0167] .

[0168] Therefore, according to the inequality A simple criterion for achieving local finite-time synchronization in the synchronization model (3) of the SMIB power system can be obtained as follows: As shown, the synchronization time can be estimated as .

[0169] When the SMIB power system adopts a controller Then, using Theorem 2, we can obtain the following criterion:

[0170] Theorem 4 If there exists a symmetric positive definite matrix constant coupling matrix and real numbers This makes it possible for any ,

[0171]

[0172] in, So, regarding the synchronization model (9) of the SMIB power system in the region Achieving finite-time synchronization within a given time, and the synchronization time function satisfy

[0173]

[0174] Proof: Let ,So

[0175] ,

[0176] After simple calculation, the largest eigenvalue of the above matrix is:

[0177] .

[0178] Depend on It can be known

[0179] .

[0180] therefore,

[0181]

[0182] Satisfying the assumption .

[0183] According to Theorem 2, regarding the synchronization model (9) of the SMIB power system in the region... Conditions for achieving finite-time synchronization include: As shown.

[0184] The following are the criteria. Optimize.

[0185] when At that time, the judgment It can be rewritten as

[0186]

[0187] when At that time, the judgment It can be rewritten as

[0188] ,

[0189] Depend on and It can be seen that, yes The necessary condition for obtaining the minimum value is that, therefore, we choose [the appropriate value here]. ,in Then we can get

[0190]

[0191] when hour, The minimum value is .

[0192] The above analysis can be summarized into the following theorem:

[0193] Theorem 5 If a constant coupling matrix exists and real numbers This makes it possible for any ,

[0194]

[0195] in, , So, regarding the synchronization model (9) of the SMIB power system in the region Achieving finite-time synchronization within a given time, and the synchronization time function satisfy

[0196]

[0197] To verify the above algebraic criterion, the parameters of the SMIB power system are selected as follows: , , and The initial value of the master-slave system is set to , .

[0198] The chaotic trajectory of the main system of the SMIB power system is as follows: Figure 2 As shown. Figure 3 For the reason and The synchronization process of coupling, in which , . Figure 4 For the reason The synchronization process of coupling, in which .

[0199] Clearly, the initial values ​​of the SMIB power system satisfy... Using the controller defined in (3), the synchronization condition can be obtained from Theorem 3 as follows: , .Pick , , The synchronization process of the SMIB power system is as follows: Figure 3 As shown. By The estimated synchronization time is .

[0200] The initial values ​​of the SMIB power system also satisfy the following conditions. ,in Using the controller defined in (9), the synchronization condition can be obtained from Theorem 5 as follows: Select , The synchronization process of the SMIB power system is as follows: Figure 4 As shown. By The estimated synchronization time is .

[0201] IV. Synchronization Time and Synchronization Cost

[0202] Synchronizing two chaotic systems is not without cost. To measure the cost of achieving synchronization, we introduce the concept of finite-time synchronization cost for the synchronization model (9), and the calculation formula is as follows:

[0203]

[0204] in, This refers to the estimated synchronization time, which is the synchronization time defined here. It means when At that time, the error of the master-slave system .

[0205] from and It can be seen that with the optional parameters As the input energy increases, the coupling strength increases, but the estimated synchronization time decreases. To avoid an unnecessary increase in coupling strength, i.e., a wasteful use of input energy, we will calculate the minimum synchronization cost and the optimal coupling strength below.

[0206] Select , ,Depend on Calculation of synchronization costs and parameters Relationship such as Figure 5 As shown in the figure, the synchronization cost initially decreases rapidly, and then decreases again with the parameter. Increase with the increase, when At that time, the synchronization cost reaches a minimum. At this point, the corresponding coupling strength The coupling strength corresponding to the minimum synchronization cost can be chosen as the optimal coupling strength in terms of energy consumption.

[0207] In summary, the main beneficial effects of this application include:

[0208] 1. This application proposes a master-slave synchronization model consisting of a sinusoidal feedback controller and two identical non-autonomous chaotic systems connected by the sinusoidal feedback controller, as follows:

[0209]

[0210] and

[0211]

[0212] This allows two chaotic systems to synchronize in the shortest possible time.

[0213] 2. This application verifies the consistent finite-time synchronization criterion obtained from the proofs of Theorem 1 and Theorem 2 for a specific SMIB power system, proving that the finite-time synchronization model designed in this application can enable two chaotic systems to achieve synchronization in the shortest possible finite time.

[0214] 3. Introduce the concept of synchronization cost in finite-time synchronization, and calculate the minimum synchronization cost and the optimal coupling strength numerically.

[0215] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of this application, and not to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of this application, and they should all be covered within the scope of the claims and specification of this application. In particular, as long as there is no contradiction or conflict, the various technical features mentioned in the various embodiments can be combined in any way. This application is not limited to the specific embodiments disclosed herein, but includes all technical solutions falling within the scope of the claims.

Claims

1. A chaotic system with sinusoidal feedback control and a finite-time synchronization model, mainly applied to finite-time synchronization of master-slave non-autonomous chaotic systems, characterized in that... The finite-time synchronization model includes: Non-autonomous chaotic system From non-autonomous chaotic systems, Sine feedback controller Wherein, the output terminal of the master non-autonomous chaotic system is connected to the input terminal of the sinusoidal feedback controller, the output terminal of the sinusoidal feedback controller is connected to the input terminal of the slave non-autonomous chaotic system, and the output terminal of the slave non-autonomous chaotic system is connected to the input terminal of the sinusoidal feedback controller; The finite-time synchronization model is represented as follows: （3） in, （4） （5） x and z are state variables, and t is a time variable. It is a time-varying bounded coefficient matrix. For a continuous nonlinear function, K and S are constant coupling matrices. , It is a symbolic function; Define an error variable The following error system is obtained: （7） in, , , ; Let be an assumed time-varying bounded matrix, and such that... The pause time of the error system is the synchronization time of the master-slave non-autonomous chaotic system.

2. The chaotic system with sinusoidal feedback control and a finite-time synchronization model according to claim 1, characterized in that, If the constant coupling matrix K=0, then the finite-time synchronization model is expressed as: （9）。 3. The chaotic system with sinusoidal feedback control and a finite-time synchronization model according to claim 2, characterized in that, Define an error variable The following error system is obtained: （10） The pause time of the error system is the synchronization time of the master-slave non-autonomous chaotic system.

4. The chaotic system with sinusoidal feedback control and a finite-time synchronization model according to claim 2, characterized in that, For the finite-time synchronization model (9), the finite-time synchronization cost is introduced, and the calculation formula is as follows: （52） in, The estimated synchronization time refers to the time when... At that time, the error of the master-slave non-autonomous chaotic system .

5. The chaotic system with sinusoidal feedback control having a finite-time synchronization model according to claim 1 or 2, characterized in that, The invention is applied to an SMIB power system, which includes a master SMIB non-autonomous chaotic system and a slave SMIB non-autonomous chaotic system, and incorporates the sinusoidal feedback controller.

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