A non-periodic intermittent-based fractional-order Chua circuit system finite time control method and system

Abstract

The application provides a non-periodic intermittent-based fractional-order Chua circuit system finite time control method and system, comprising: setting an initial state of the fractional-order Chua circuit system, and measuring a state variable of the fractional-order Chua circuit system by using a sensor; constructing a finite time non-periodic intermittent controller by using the state variable measured by the sensor; converting a digital control signal output by the finite time non-periodic intermittent controller into a physical quantity by using an executor, and injecting the physical quantity into a corresponding node of the fractional-order Chua circuit system in a working interval, so as to realize finite time stable control of the fractional-order Chua circuit system. The application provides a new solution for finite time control of the fractional-order Chua circuit system, can be applied to fields such as chaos synchronization and secret communication, and has theoretical value and engineering practicability.

Description

Technical Field

[0001] This invention belongs to the field of fractional nonlinear system control technology, specifically relating to a finite-time control method and system for a fractional Chua's circuit system based on non-periodic intermittent operation. Background Technology

[0002] Chua's circuit is a chaotic oscillating circuit containing several common energy storage elements, nonlinear elements, and resistors, exhibiting rich dynamic characteristics. It is nonlinear and a typical chaotic circuit. However, the dynamic behavior of traditional Chua's circuit systems, modeled based on integer-order differential equations, is limited by the local properties of integer-order calculus, making it difficult to accurately characterize the memory effect and frequency response of energy storage elements in real circuits, such as capacitors and inductors. The inheritance and memory properties of fractional-order calculus give fractional-order systems a unique advantage in handling complex systems with long or short memories.

[0003] Therefore, fractional-order Chua's circuit systems can be modeled with higher accuracy by introducing fractional-order differential operators. Current research largely focuses on implementing asymptotic control of fractional-order Chua's circuit systems, as practical engineering requirements often favor system stability within a finite time. Finite-time control aims to design and implement systems that achieve certain performance indicators within a finite time. It offers advantages such as fast convergence, strong anti-interference capability, and high control accuracy, and is widely used in automation, aerospace, and industrial production processes to meet the demands for rapid system response, efficient operation, and precise control.

[0004] Although some results have been achieved in the finite-time control of integer-order Chua's circuits, the Newton-Leibniz formula and chain rule in classical calculus are not applicable to fractional-order systems, making the finite-time control analysis of fractional-order Chua's circuits more difficult. Furthermore, continuously applying control signals may cause mechanical components to operate for extended periods, accelerating device wear and increasing equipment maintenance costs. To further conserve control resources and reduce costs, this paper studies a finite-time control method and system for fractional-order Chua's circuits based on non-periodic intermittent operation, which has practical application value. Summary of the Invention

[0005] The purpose of this invention is to provide a finite-time control method and system for fractional-order Chua's circuit systems based on non-periodic intermittent operation. By intelligently adjusting the activation interval of the control signal, the system state is ensured to converge rapidly within a finite time, while significantly reducing control energy consumption and cost.

[0006] To achieve the above objectives, the present invention provides the following solution:

[0007] A finite-time control method for fractional-order Chua's circuit system based on aperiodic intermittent operation includes:

[0008] Set the initial state of the fractional-order Chua's circuit system and use sensors to measure the state variables of the fractional-order Chua's circuit system;

[0009] Design a finite-time aperiodic intermittent controller that is active only within the operating range by utilizing state variables measured by sensors;

[0010] Based on the different dynamic behaviors of fractional-order Chua's circuit system between the working and resting regions, a fractional-order non-periodic intermittent finite-time Lyapunov stability method is constructed.

[0011] Based on the fractional-order aperiodic intermittent finite-time Lyapunov stability method, sufficient conditions for fractional-order Chua's circuit system to achieve finite-time stability under a finite-time aperiodic intermittent controller are obtained.

[0012] The control parameters of the finite-time aperiodic intermittent controller are adjusted by using preset fractional-order Chua's circuit system parameters and based on the sufficient condition for finite-time stability.

[0013] The digital control signal output by the finite-time non-periodic intermittent controller with adjusted control parameters is converted into a physical quantity by the actuator and injected into the corresponding node of the fractional-order Chua's circuit system within the working interval, thereby achieving finite-time stability of the fractional-order Chua's circuit system and accurate estimation of the resting time result.

[0014] Preferably, the preset variables are introduced into the original fractional-order Chua's circuit system expression to obtain the final fractional-order Chua's circuit system expression:

[0015] The expression for the original fractional-order Chua's circuit system is as follows:

[0016]

[0017] in, It is a capacitor The voltage at both ends, It is a capacitor The voltage at both ends, Through inductor The current, function Represents nonlinear resistance shown characteristic, Indicates the initial time is fractional order Caputo fractional derivative;

[0018] in,

[0019] ,

[0020] , and Represent The internal slope, external slope, and power-off voltage of the characteristic curve;

[0021] The predefined variables introduced include:

[0022]

[0023] 's' represents time;

[0024] The final expression for the fractional-order Chua's circuit system is as follows:

[0025]

[0026] in, .

[0027] Preferably, the controlled system expression of the fractional-order Chua's circuit system is as follows:

[0028]

[0029] in, , C and A both represent unknown parameter matrices. Indicates coupling strength. It is the internal coupling matrix, for nodes ,exist When, if node and They are connected, and the coupling configuration matrix is ​​used. satisfy ,otherwise, .

[0030] Preferably, a finite-time aperiodic intermittent controller is constructed based on the controlled system expression of the fractional-order Chua's circuit system, combined with the start and end times of the control interval, and the control width and non-control width; the expression of the finite-time aperiodic intermittent controller is as follows:

[0031]

[0032] In the formula, , , , , , m represents a non-negative integer. , , as well as All are control parameters; (Note: The last part is a typo and can be left as is.) and They are the first The start and end times of each control interval. For the first One control cycle, It is the first One control width, Indicates the first One uncontrolled width.

[0033] Preferably, during the process of controlling the fractional-order Chua's circuit system using a finite-time aperiodic intermittent controller, the control width and control period are dynamically adjusted so that the finite-time aperiodic intermittent controller is activated only within the working interval.

[0034] Preferably, the rest interval refers to the time required for a fractional-order Chua's circuit system to reach an equilibrium state from its initial moment.

[0035] The present invention also provides a finite-time control system for a fractional-order Chua's circuit system based on non-periodic intermittent circuits, for implementing the method, comprising:

[0036] The state variable acquisition module is used to set the initial state of the fractional-order Chua's circuit system and to measure the state variables of the fractional-order Chua's circuit system using sensors.

[0037] The controller building block is used to design a finite-time non-periodic intermittent controller that is active only within the operating range, utilizing state variables measured by sensors.

[0038] The model building module is used to construct a fractional-order non-periodic intermittent finite-time Lyapunov stability method based on the different dynamic behaviors of fractional-order Chua's circuit system between the working and resting regions.

[0039] The finite-time stability condition acquisition module is used to obtain the finite-time stability condition of the fractional-order Chua's circuit system under the control of the finite-time aperiodic intermittent controller based on the fractional-order aperiodic intermittent finite-time Lyapunov stability method.

[0040] The parameter adjustment module is used to adjust the control parameters of the finite-time aperiodic intermittent controller by using preset fractional-order Chua's circuit system parameters and based on the sufficient condition of finite-time stability.

[0041] The system control module is used to convert the digital control signal output by the finite-time non-periodic intermittent controller with adjusted control parameters into a physical quantity via an actuator, and inject it into the corresponding node of the fractional-order Chua's circuit system within the working interval, so as to realize the finite-time stable control of the fractional-order Chua's circuit system and the accurate estimation of the rest time result.

[0042] Compared with existing technologies, the beneficial effects of this invention are as follows: Addressing the requirements of rapid convergence, low energy consumption, and strong adaptability for fractional-order Chua's circuit systems in key applications such as chaotic synchronization and secure communication, this invention provides a finite-time control method for fractional-order Chua's circuit systems based on aperiodic intermittent intervals. By dynamically adjusting the control width and period, the finite-time controller is activated only within the aperiodic working interval, reducing the application time of the control signal and overall energy consumption. Furthermore, based on the different dynamic behaviors exhibited by the system between the working and rest intervals, a fractional-order aperiodic intermittent finite-time Lyapunov stability method is designed. This method not only strictly guarantees that the system state can rapidly converge to the equilibrium point within the estimated finite time, but also solves the problems of slow convergence and inability to predict the exact time in existing asymptotic stability methods. In addition, by changing the working and rest intervals according to actual needs, the controller structure and design parameters do not need to be changed. This allows the method to flexibly adapt to changes in communication protocols or the dynamic adjustment requirements of different task demands on the working / rest modes, enhancing the practicality of the control strategy in complex and variable practical application scenarios (such as adaptive secure communication networks and multi-node chaotic synchronization). This invention provides an effective solution for efficient, fast, and low-power control of fractional-order Chua's circuit systems, supporting their reliable application in cutting-edge engineering fields such as chaotic synchronization and secure communication. Attached Figure Description

[0043] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0044] Figure 1 This is a structural diagram of a fractional-order Chua's circuit system according to an embodiment of the present invention;

[0045] Figure 2 This is a chaotic diagram of a fractional-order Chua's circuit system according to an embodiment of the present invention.

[0046] Figure 3 This refers to the control area and rest area of ​​the controller in this embodiment of the invention;

[0047] Figure 4 This invention provides a finite-time stability of the system under the controller in an embodiment of the invention.

[0048] Figure 5 This is a flowchart of a finite-time control method for a fractional-order Chua's circuit system based on non-periodic intermittent circuits, according to an embodiment of the present invention. Detailed Implementation

[0049] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0050] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0051] Example 1

[0052] like Figure 5 As shown, a finite-time control method for a fractional-order Chua's circuit system based on non-periodic intermittent circuits includes:

[0053] S1: Set the initial state of the fractional-order Chua's circuit system and use sensors to measure the state variables of the fractional-order Chua's circuit system (such as capacitor voltage and inductor current). Figure 1 , Figure 2 As shown.

[0054] A further implementation involves introducing preset variables into the original fractional-order Chua's circuit system expression to obtain the final fractional-order Chua's circuit system expression:

[0055] The expression for the original fractional-order Chua's circuit system is as follows:

[0056] (1)

[0057] in, It is a capacitor The voltage at both ends, It is a capacitor The voltage at both ends, Through inductor The current, function Represents nonlinear resistance shown characteristic, Indicates the initial time is fractional order The Caputo fractional derivative; where,

[0058] ,

[0059] , and Represent The internal slope, external slope, and power-off voltage of the characteristic curve;

[0060] The predefined variables introduced include:

[0061]

[0062] In the formula, s represents time; variable substitution transforms the physical quantities of the actual circuit into mathematical variables through dimensionless transformation and parameter ratios, simplifying the equation form while preserving the essence of the circuit's dynamic behavior.

[0063] The final expression for the fractional-order Chua's circuit system is as follows:

[0064] (2)

[0065] in, .

[0066] A further implementation method is that the controlled system expression of the fractional-order Chua's circuit system is as follows:

[0067] (3)

[0068] in Represents system state variables , , C and A Both represent unknown parameter matrices. Indicates coupling strength. It is the internal coupling matrix, for nodes When When, if node and They are connected, and the coupling configuration matrix is ​​used. satisfy ,otherwise, .

[0069] S2: Construct a finite-time aperiodic intermittent controller using state variables measured by sensors. For example... Figure 3 , Figure 4 As shown.

[0070] A further implementation involves constructing a finite-time aperiodic intermittent controller based on the controlled system expression of a fractional-order Chua's circuit system, combined with the start and end times of the control interval, and the control and non-control widths; the expression of the finite-time aperiodic intermittent controller is as follows:

[0071] (4)

[0072] In the formula, , , , , , m represents a non-negative integer. , , as well as All are control parameters; (Note: The last part is a typo and can be left as is.) and They are the first The start and end times of each control interval. For the first One control cycle, It is the first One control width, Indicates the first A non-controlled width. Furthermore, assume... , , , The lower bound of the control width represents the minimum control width. T represents the maximum non-control width, and T represents the maximum control cycle length.

[0073] S3: Based on the different dynamic behaviors of fractional-order Chua's circuit system between the working and resting regions, a fractional-order non-periodic intermittent finite-time Lyapunov stability method is constructed.

[0074] S4: Based on the fractional-order aperiodic intermittent finite-time Lyapunov stability method, obtain sufficient conditions for the fractional-order Chua's circuit system to achieve finite-time stability under the control of a finite-time aperiodic intermittent controller.

[0075] This embodiment designs a non-periodic intermittent finite-time controller for a fractional-order Chua's circuit system. It then defines sufficient conditions for the control parameters to achieve finite-time stability. As long as the controller is designed according to this specification and the control parameters meet the above conditions, stability is guaranteed.

[0076] Specifically, substituting the controller (4) into the system (3), since the controller (4) has a sign function that makes the system (3) a discontinuous system, we will construct a set-valued mapping of the system state to handle this discontinuity. Then the solution of the system (3) satisfies:

[0077]

[0078] in Furthermore, for any have

[0079]

[0080] SIGN is a set-valued mapping constructed to handle discontinuities in symbolic functions, mapping discontinuous points in the system to another set to resolve discontinuities.

[0081] For a given set-valued mapping, there exists a measurable choice function. satisfy:

[0082] (5)

[0083] Constructing Lyapunov functions:

[0084]

[0085] when hour,

[0086] (6)

[0087] because and Since it is a positive definite diagonal matrix, then according to the properties of coupled configuration matrices, we can further obtain... :

[0088] (7)

[0089] in According to the Lipschitz condition,

[0090] (8)

[0091] in It is the Lipschitz constant. .

[0092] Therefore, system equation (6) can be written in the following form:

[0093] (9)

[0094] in .

[0095] because Then we have the inequality If true, then equation (9) can be written as:

[0096] (10)

[0097] Similarly, when At that time, there were:

[0098] (11)

[0099] in , To define a variable, representing a constant greater than 0, It also represents defining a variable. Represents an N-dimensional identity matrix; Represents the smallest eigenvalue of a matrix; It is the internal coupling matrix The amount; These are the component elements of matrix C.

[0100] Based on the results in (10) and (11), we can obtain:

[0101] (12)

[0102] By mathematical induction, we can obtain that

[0103] (13)

[0104] in, .

[0105] Based on the above conclusions, the parameters in system (3) satisfy the following conditions:

[0106]

[0107] The convergence of the synchronization error system can then be achieved, that is, the fractional-order Chua's circuit system can achieve finite-time stability under the finite-time intermittent control strategy (4), and the rest time can be obtained. satisfy:

[0108]

[0109] in,

[0110] , , .

[0111] The resting time refers to the time required for a fractional-order Chua's circuit system to reach equilibrium from the initial moment.

[0112] S5: Adjust the control parameters of the finite-time aperiodic intermittent controller by using preset fractional-order Chua's circuit system parameters and based on the sufficient condition of finite-time stability.

[0113] S6: The digital control signal output by the finite-time non-periodic intermittent controller with adjusted control parameters is converted into a physical quantity by the actuator and injected into the corresponding node of the fractional-order Chua's circuit system within the working range, thereby realizing the finite-time stable control of the fractional-order Chua's circuit system and the accurate estimation of the rest time result.

[0114] A further implementation involves dynamically adjusting the control width and control period during the control of the fractional-order Chua's circuit system using a finite-time aperiodic intermittent controller, so that the finite-time aperiodic intermittent controller is activated only within the operating range.

[0115] Example 2

[0116] The present invention also provides a finite-time control system for a fractional-order Chua's circuit system based on non-periodic intermittent circuits, for implementing the method of Embodiment 1, comprising:

[0117] The state variable acquisition module is used to set the initial state of the fractional-order Chua's circuit system and to measure the state variables of the fractional-order Chua's circuit system using sensors.

[0118] The controller building module is used to construct a finite-time non-periodic intermittent controller using state variables measured by sensors.

[0119] The model building module is used to construct a fractional-order non-periodic intermittent finite-time Lyapunov stability method based on the different dynamic behaviors of fractional-order Chua's circuit system between the working and resting regions.

[0120] The finite-time stability condition acquisition module is used to obtain, based on the fractional-order aperiodic intermittent finite-time Lyapunov stability method, the sufficient conditions for the fractional-order Chua's circuit system to achieve finite-time stability under the control of a finite-time aperiodic intermittent controller, as well as the estimation results of the rest time.

[0121] The parameter adjustment module is used to adjust the control parameters of the finite-time aperiodic intermittent controller by using preset fractional-order Chua's circuit system parameters and based on the sufficient condition of finite-time stability.

[0122] The system control module is used to convert the digital control signal output by the finite-time non-periodic intermittent controller with adjusted control parameters into a physical quantity via an actuator, and inject it into the corresponding node of the fractional-order Chua's circuit system within the working interval, so as to realize the finite-time stable control of the fractional-order Chua's circuit system and the accurate estimation of the rest time result.

[0123] Example 3

[0124] This embodiment further illustrates the invention with specific examples. In practical applications, firstly, sensors measure the state variables (such as capacitor voltage and inductor current) of the fractional-order Chua's circuit, and these measurements are transmitted to the controller. Based on the received state information, the controller executes a designed finite-time intermittent control algorithm—the core of which lies in: 1) executing a control law with finite-time convergence within the operating interval; 2) determining the on (working) and off (resting) of the control signal according to the intermittent strategy, and rigorously proving through Lyapunov functions that the system state can remain stable within a finite time. Finally, the digital control signal output by the controller is converted into a physical quantity by an actuator (such as a controlled voltage / current source) and injected only into the corresponding nodes of the Chua's circuit within the operating interval specified by the algorithm, achieving finite-time stable control of the system.

[0125] like Figure 1 As shown, a finite-time control method for a fractional-order Chua's circuit system based on non-periodic intermittent circuits includes the following steps:

[0126] Step 1: Set the initial state variables of the fractional-order Chua's circuit system;

[0127] Step 2: Sensor Measurement and Transmission;

[0128] Step 3: Transmit the measured status to the controller;

[0129] Step 4: Design a finite-time intermittent control strategy;

[0130] Step 5: Verify whether the proposed method stabilizes the system state to an equilibrium point within a finite time.

[0131] For the established fractional-order Chua's circuit system, consider the following parameters:

[0132]

[0133] .

[0134] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A finite-time control method for a fractional-order Chua's circuit system based on non-periodic intermittent circuits, characterized in that, include: Set the initial state of the fractional-order Chua's circuit system and use sensors to measure the state variables of the fractional-order Chua's circuit system; Design a finite-time aperiodic intermittent controller that is active only within the operating range by utilizing state variables measured by sensors; Based on the different dynamic behaviors of fractional-order Chua's circuit system between the working and resting regions, a fractional-order non-periodic intermittent finite-time Lyapunov stability method is constructed. Based on the fractional-order aperiodic intermittent finite-time Lyapunov stability method, sufficient conditions for fractional-order Chua's circuit system to achieve finite-time stability under a finite-time aperiodic intermittent controller are obtained. The control parameters of the finite-time aperiodic intermittent controller are adjusted by using preset fractional-order Chua's circuit system parameters and based on the sufficient condition for finite-time stability. The digital control signal output by the finite-time non-periodic intermittent controller with adjusted control parameters is converted into a physical quantity by the actuator and injected into the corresponding node of the fractional-order Chua's circuit system within the working range, thereby achieving finite-time stability of the fractional-order Chua's circuit system and accurate estimation of the resting time result. Based on the controlled system expression of the fractional-order Chua's circuit system, and combining the start and end times of the control interval, the control width and the non-control width, a finite-time aperiodic intermittent controller is constructed; the expression of the finite-time aperiodic intermittent controller is as follows: In the formula, , , , , , m represents a non-negative integer. , , as well as All are control parameters; (Note: The last part is a typo and can be left as is.) and They are the first The start and end times of each control interval For the first One control cycle, It is the first One control width, Indicates the first Uncontrolled width.

2. The method according to claim 1, characterized in that, Setting the initial state of the fractional-order Chua's circuit system and measuring its state variables using sensors includes: introducing preset variables into the original fractional-order Chua's circuit system expression to obtain the final fractional-order Chua's circuit system expression. The expression for the original fractional-order Chua's circuit system is as follows: in, It is a capacitor The voltage at both ends, It is a capacitor The voltage at both ends, Through inductor The current, function Represents nonlinear resistance shown characteristic, Indicates the initial time is fractional order Caputo fractional derivative; in, ， , and Represent The internal slope, external slope, and power-off voltage of the characteristic curve; The predefined variables introduced include: 's' represents time; The final expression for the fractional-order Chua's circuit system is as follows: in, .

3. The method according to claim 2, characterized in that, The controlled system expression of the fractional-order Chua's circuit system is as follows: in, , C and A both represent unknown parameter matrices. Indicates coupling strength. It is the internal coupling matrix, for nodes ,exist When, if node and They are connected, and the coupling configuration matrix is ​​used. satisfy ,otherwise, .

4. The method according to claim 1, characterized in that, In the process of controlling the fractional-order Chua's circuit system using a finite-time aperiodic intermittent controller, the control width and control period are dynamically adjusted so that the finite-time aperiodic intermittent controller is activated only within the working range.

5. The method according to claim 1, characterized in that, The pause time refers to the time required for a fractional-order Chua's circuit system to reach equilibrium from the initial moment.

6. A finite-time control system for a fractional-order Chua's circuit system based on non-periodic intermittent circuits, used to implement the method described in any one of claims 1-5, characterized in that, include: The state variable acquisition module is used to set the initial state of the fractional-order Chua's circuit system and to measure the state variables of the fractional-order Chua's circuit system using sensors. The controller building block is used to design a finite-time non-periodic intermittent controller that is active only within the operating range, utilizing state variables measured by sensors. The model building module is used to construct a fractional-order non-periodic intermittent finite-time Lyapunov stability method based on the different dynamic behaviors of fractional-order Chua's circuit system between the working and resting regions. The finite-time stability condition acquisition module is used to obtain sufficient conditions for the fractional-order Chua's circuit system to achieve finite-time stability under a finite-time aperiodic intermittent controller based on the fractional-order aperiodic intermittent finite-time Lyapunov stability method. The parameter adjustment module is used to adjust the control parameters of the finite-time aperiodic intermittent controller by using preset fractional-order Chua's circuit system parameters and based on the sufficient condition of finite-time stability. The system control module is used to convert the digital control signal output by the finite-time non-periodic intermittent controller with adjusted control parameters into a physical quantity via an actuator, and inject it into the corresponding node of the fractional-order Chua's circuit system within the working interval, so as to realize the finite-time stable control of the fractional-order Chua's circuit system and the accurate estimation of the rest time result.

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