Fractional order Chua's circuit system finite time control method and system based on aperiodic intermittence

Abstract

The invention provides a finite time control method and system for a fractional-order Chua's circuit system based on non-periodic intermittence, and the method comprises the steps: setting an initial state of the fractional-order Chua's circuit system, and measuring a state variable of the fractional-order Chua's circuit system through a sensor; a finite-time non-periodic intermittent controller is constructed by using a state variable measured by a sensor; a digital control signal output by a finite-time non-periodic intermittent controller is converted into a physical quantity through an actuator, the physical quantity is injected into a corresponding node of the fractional-order Chua's circuit system in a working interval, and finite-time stable control over the fractional-order Chua's circuit system is achieved. The invention provides a new solution for finite time control of the fractional-order Chua's circuit system, can be expanded and applied to the fields of chaos synchronization, secret communication and the like, and has theoretical value and engineering practicability.

Description

Technical Field

[0001] The present invention belongs to the technical field of fractional-order nonlinear system control, and in particular relates to a finite-time control method and system for a fractional-order Chua's circuit system based on non-periodic intermittence. Background Art

[0002] The Chua circuit is a chaotic oscillator circuit that incorporates several common energy storage elements, nonlinear elements, and resistors. It exhibits a rich array of dynamic characteristics, demonstrating its nonlinear properties and making it a typical chaotic circuit. However, the dynamic behavior of Chua circuit systems, traditionally modeled using integer-order differential equations, is limited by the local characteristics of integer-order calculus, making it difficult to accurately characterize the memory effects and frequency-domain response characteristics of energy storage elements in actual circuits, such as capacitors and inductors. The hereditary and memory properties of fractional-order calculus give fractional-order systems unique advantages in handling complex systems with both long and short memories.

[0003] Therefore, fractional-order Chua circuit systems achieve higher modeling accuracy by introducing fractional-order differential operators. Current research focuses on achieving asymptotic control of fractional-order Chua circuit systems. In practical engineering applications, it is often desirable for the system to stabilize within a finite time. Finite-time control aims to design and implement a system that achieves a certain performance metric within a finite time. It offers advantages such as fast convergence, strong anti-interference capabilities, and high control accuracy. It is widely used in automation, aerospace, industrial production processes, and other fields to meet the demands for rapid system response, efficient operation, and precise control.

[0004] Although some results have been published on the finite-time control of integer-order Chua's circuit systems, the Newton-Leibniz formula and chain rule used in classical calculus are not applicable to fractional-order systems, making the analysis of finite-time control of fractional-order Chua's circuit systems more difficult. Furthermore, the continuous application of control signals can cause mechanical components to operate for extended periods, accelerating wear and increasing equipment maintenance costs. To further conserve control resources and reduce costs, a finite-time control method and system for fractional-order Chua's circuit systems based on aperiodic intermittent operation is investigated, which has practical applications. Summary of the Invention

[0005] The purpose of the present invention is to provide a finite-time control method and system for a fractional-order Chua's circuit system based on non-periodic intervals, which ensures that the system state converges rapidly within a finite time by intelligently adjusting the activation interval of the control signal, while significantly reducing control energy consumption and cost.

[0006] To achieve the above object, the present invention provides the following solutions:

[0007] A finite-time control method for a fractional-order Chua's circuit system based on non-periodic intermittent operation, comprising:

[0008] Setting the initial state of the fractional-order Chua's circuit system and measuring the state variables of the fractional-order Chua's circuit system using sensors;

[0009] Using the state variables measured by sensors, a finite-time aperiodic intermittent controller is designed that is activated only during the working period.

[0010] According to the different dynamic behaviors of the fractional-order Chua circuit system between the working zone and the rest zone, 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 the fractional-order Chua circuit system to achieve finite-time stability under a finite-time aperiodic intermittent controller are obtained;

[0012] Adjusting the control parameters of the finite-time aperiodic intermittent controller by using preset fractional-order Chua 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 an actuator and injected into the corresponding node of the fractional-order Chua's circuit system within the working range, thereby achieving the finite-time stability of the fractional-order Chua's circuit system and accurate estimation of the dwell 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 of the original fractional-order Chua circuit system is as follows:

[0016]

[0017] in, is the voltage across capacitor C1, is the voltage across capacitor C2, i L is the current through the inductor L, the function Represents the nonlinear resistance R NL The vi characteristics shown, represents the Caputo fractional derivative with the initial time t0 and fractional order β∈(0,1);

[0018] in,

[0019]

[0020] and E represent the internal slope, external slope and power-off voltage of the vi characteristic curve respectively;

[0021] The introduced preset variables include:

[0022] x3=Ri L E -1 ,t=s(RC2) -1 ,

[0023] r2=C2R 2 L -1 ,

[0024] s represents time;

[0025] The final fractional-order Chua circuit system expression is as follows:

[0026]

[0027] in,

[0028] Preferably, the controlled system expression of the fractional-order Chua circuit system is as follows:

[0029]

[0030] in, C and A both represent unknown parameter matrices, Δ represents the coupling strength, Γ is the internal coupling matrix, and If nodes i and j are connected, the coupling configuration matrix H = (h ij ) N×N Satisfy h ij =h ji > 0, otherwise, h ij =h ji =0,(i≠j).

[0031] Preferably, based on the controlled system expression of the fractional-order Chua circuit system, a finite-time aperiodic intermittent controller is constructed in combination with the start and end times of the control interval, the control width and the non-control width; the expression of the finite-time aperiodic intermittent controller is as follows:

[0032]

[0033] Where, 0=t0<θ0<t1<…<t m <θ m <t m+1 <…,d i >0α>0,ζ1>0,δ∈(0,1), m represents a non-negative integer, d i, α, ζ1 and δ are all control parameters; t m and θ m are the start and end time of the m+1th control interval, t m+1 -t m is the m+1th control width, θ m -t m is the m+1th control width, t m+1 -θ m Indicates the m+1th non-controlled width.

[0034] Preferably, in the process of controlling the fractional-order Chua circuit system using the finite-time aperiodic intermittent controller, the control width and the control period are dynamically adjusted so that the finite-time aperiodic intermittent controller is activated only within the working range.

[0035] Preferably, the rest period refers to the time required for the fractional-order Chua's circuit system to reach an equilibrium state from an initial moment.

[0036] The present invention also provides a finite-time control system of a fractional-order Chua's circuit system based on non-periodic intermittents, which is used to implement the method, comprising:

[0037] a state variable acquisition module, used to set the initial state of the fractional-order Chua's circuit system and measure the state variables of the fractional-order Chua's circuit system using sensors;

[0038] A controller building block for designing a finite-time aperiodic intermittent controller that is activated only during the working interval using state variables measured by sensors;

[0039] A model building module is used to construct a fractional-order aperiodic intermittent finite-time Lyapunov stability method based on the different dynamic behaviors of the fractional-order Chua circuit system between the working interval and the rest interval;

[0040] A finite-time stability condition acquisition module is used to obtain the finite-time stability condition of the fractional-order Chua circuit system under the control of the finite-time aperiodic intermittent controller based on the fractional-order aperiodic intermittent finite-time Lyapunov stability method;

[0041] a parameter adjustment module, configured to adjust control parameters of the finite-time aperiodic intermittent controller by using preset fractional-order Chua circuit system parameters and based on the sufficient condition for finite-time stability;

[0042] 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 through an actuator, and inject it into the corresponding node of the fractional-order Chua circuit system within the working range to achieve finite-time stable control of the fractional-order Chua circuit system and accurate estimation of the dwell time result.

[0043] Compared with the prior art, the present invention has the following beneficial effects: in response to the demand for fast convergence, low energy consumption and strong adaptability of fractional-order Chua's circuit systems in key application fields such as chaotic synchronization and secure communication, the present invention provides a finite-time control method for fractional-order Chua's circuit systems based on non-periodic intermittent control. By dynamically adjusting the control width and period, the finite-time controller is activated only in the non-periodic working interval, reducing the application time of the control signal and the overall energy consumption. Furthermore, based on the different dynamic behaviors of the system between the working interval and the resting interval, a fractional-order non-periodic intermittent finite-time Lyapunov stability method is designed. This method not only strictly guarantees that the system state can quickly converge to the equilibrium point within the estimated finite time, but also solves the problem of slow convergence and inability to predict the exact time of existing asymptotic stability methods. In addition, by changing the working interval and resting interval according to actual needs, there is no need to change the controller structure and design parameters, making this method flexible to adapt to changes in communication protocols or the dynamic adjustment requirements of work / rest modes proposed by different task requirements, enhancing the practicality of the control strategy in complex and changeable practical application scenarios (such as adaptive secure communication networks and multi-node chaotic synchronization). This invention provides an effective solution for the efficient, fast and low-power control of fractional-order Chua circuit systems, supporting their reliable application in cutting-edge engineering fields such as chaos synchronization and secure communications. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] In order to more clearly illustrate the technical solution of the present invention, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0045] Figure 1 This is a structural diagram of a fractional-order Chua's circuit system according to an embodiment of the present invention;

[0046] Figure 2 This is a chaotic diagram of the fractional-order Chua's circuit system according to an embodiment of the present invention;

[0047] Figure 3 The control area and rest area of ​​the controller according to the embodiment of the present invention;

[0048] Figure 4 The embodiment of the present invention is to provide a limited time stability of the system under the controller;

[0049] Figure 5 Flowchart of a finite-time control method for a fractional-order Chua's circuit system based on non-periodic intermittent operation according to an embodiment of the present invention. DETAILED DESCRIPTION

[0050] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0051] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0052] Example 1

[0053] like Figure 5 As shown, a finite-time control method for a fractional-order Chua circuit system based on non-periodic intermittent operation includes:

[0054] 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 shown.

[0055] A further implementation method is to introduce preset variables into the original fractional-order Chua's circuit system expression to obtain the final fractional-order Chua's circuit system expression:

[0056] The expression of the original fractional-order Chua circuit system is as follows:

[0057]

[0058] in, is the voltage across capacitor C1, is the voltage across capacitor C2, i L is the current through the inductor L, the function Represents the nonlinear resistance R NL The vi characteristics shown, represents the Caputo fractional derivative with the initial time t0 and fractional order β∈(0,1); where,

[0059]

[0060] and E represent the internal slope, external slope and power-off voltage of the vi characteristic curve respectively;

[0061] The introduced preset variables include:

[0062] x3=Ri L E -1 ,t=s(RC2) -1 ,

[0063] r2=C2R 2 L -1 ,

[0064] In the formula, s represents time; variable replacement is to convert the physical quantities of the actual circuit into mathematical variables through dimensionless conversion and parameter ratio form, simplify the equation form, and retain the essence of the circuit dynamic behavior.

[0065] The final fractional-order Chua circuit system expression is as follows:

[0066]

[0067] in,

[0068] A further implementation method is that the controlled system expression of the fractional-order Chua circuit system is as follows:

[0069]

[0070] where x(t) = [x1(t), x2(t), x3(t)] T Represents system state variables C and A both represent unknown parameter matrices, Δ represents the coupling strength, Γ is the internal coupling matrix, and If nodes i and j are connected, the coupling configuration matrix H = (h ij ) N×N Satisfy h ij =h ji > 0, otherwise, h ij =h ji =0,(i≠j).

[0071] S2: Using the state variables measured by sensors, a finite-time aperiodic intermittent controller is constructed. Figure 3 、 Figure 4 shown.

[0072] A further implementation method is to construct a finite-time aperiodic intermittent controller based on the controlled system expression of the fractional-order Chua circuit system, combined with the start and end times of the control interval, the control width and the non-control width; the expression of the finite-time aperiodic intermittent controller is as follows:

[0073]

[0074] Where, 0=t0<θ0<t1<…<t m <θ m <t m+1 <…, d i >0, α>0, ζ1>0, δ∈(0,1), m represents a non-negative integer, d i , α, ζ1 and δ are all control parameters; t m and θ m are the start and end time of the m+1th control interval, t m+1 -t m is the m+1th control cycle, θ m -t m is the m+1th control width, t m+1 -θ m represents the m+1th non-controlled width. In addition, assume that T c is the lower bound of the control width, representing the minimum control width, T r is the maximum non-controlled width, and T is the maximum controlled cycle length.

[0075] S3: Based on the different dynamic behaviors of the fractional-order Chua circuit system between the working zone and the rest zone, a fractional-order aperiodic intermittent finite-time Lyapunov stability method is constructed;

[0076] S4: Based on the fractional-order aperiodic intermittent finite-time Lyapunov stability method, sufficient conditions for the fractional-order Chua circuit system to achieve finite-time stability under the control of a finite-time aperiodic intermittent controller are obtained;

[0077] This embodiment designs a non-periodic intermittent finite-time controller for a fractional-order Chua circuit system, and then limits the control parameters to certain conditions that are sufficient 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.

[0078] Specifically, the controller (4) is substituted into the system (3). Since the controller (4) has a sign function, the system (3) is a discontinuous system. The following constructs a set-valued mapping of the system state to deal with this discontinuity. Then the solution of the system (3) satisfies:

[0079]

[0080] Where SIGN(x i (t))=(SIGN(x i1 (t)),…,SIGN(x in(t))) T Moreover, for any 1≤i≤N,1≤k≤n, we have

[0081]

[0082] SIGN is a set-valued mapping constructed to handle the discontinuity of the sign function, which maps the discontinuity points in the system to another set to handle the discontinuity.

[0083] For a given set-valued mapping, there exists a measurable selection function satisfy:

[0084]

[0085] Construct the Lyapunov function:

[0086]

[0087] When t∈(t m ,θ m ]hour,

[0088]

[0089] Because h ij > 0 (i≠j) and Γ is a positive definite diagonal matrix, then according to the properties of the coupling configuration matrix, we can further obtain:

[0090]

[0091] in According to the Lipschitz condition,

[0092]

[0093] Where P>0 is the Lipschitz constant,

[0094] Therefore, system (6) can be written as follows:

[0095]

[0096] Where D = diag(d1, d2, ..., d N ).

[0097] Since δ∈(0,1), then we have the inequality If it holds, then formula (9) can be written as:

[0098]

[0099] Similarly, when t∈(θ m ,tm+1 ], there are:

[0100]

[0101] in ε is a definition variable, representing a constant greater than 0, and σ also represents a definition variable. represents the N-dimensional identity matrix; λ min represents the minimum eigenvalue of the matrix; γ k is the component of the internal coupling matrix Γ; c k are the component elements of matrix C.

[0102] According to the results in (10) and (11), we can get:

[0103]

[0104] By mathematical induction, we can get

[0105]

[0106] in,

[0107]

[0108] Based on the above conclusions, the parameters in system (3) meet the following conditions:

[0109]

[0110] Then the convergence of the synchronous error system can be achieved, that is, the fractional-order Chua circuit system can achieve finite-time stability under the finite-time intermittent control strategy (4), and the dwell time T can be obtained. ° satisfy:

[0111]

[0112] in,

[0113]

[0114] The settling time refers to the time required for the fractional-order Chua circuit system to reach the equilibrium state from the initial moment.

[0115] S5: adjusting the control parameters of the finite-time aperiodic intermittent controller by using preset fractional-order Chua circuit system parameters and based on the sufficient condition for finite-time stability.

[0116] S6: The digital control signal output by the finite-time non-periodic intermittent controller with adjusted control parameters is converted into a physical quantity through an actuator and injected into the corresponding node of the fractional-order Chua circuit system within the working range to achieve finite-time stable control of the fractional-order Chua circuit system and accurate estimation of the dwell time result.

[0117] A further implementation method is to dynamically adjust the control width and control period during the process of controlling the fractional-order Chua circuit system using the finite-time aperiodic intermittent controller so that the finite-time aperiodic intermittent controller is activated only within the working range.

[0118] Example 2

[0119] The present invention further provides a finite-time control system of a fractional-order Chua's circuit system based on non-periodic intermittents, which is used to implement the method of embodiment 1, including:

[0120] a state variable acquisition module, used to set the initial state of the fractional-order Chua's circuit system and measure the state variables of the fractional-order Chua's circuit system using sensors;

[0121] A controller building module for building a finite-time aperiodic intermittent controller using state variables measured by sensors;

[0122] A model building module is used to construct a fractional-order aperiodic intermittent finite-time Lyapunov stability method based on the different dynamic behaviors of the fractional-order Chua circuit system between the working interval and the rest interval;

[0123] A 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 the control of a finite-time aperiodic intermittent controller and an estimation result of the dwell time based on the fractional-order aperiodic intermittent finite-time Lyapunov stability method;

[0124] a parameter adjustment module, configured to adjust control parameters of the finite-time aperiodic intermittent controller by using preset fractional-order Chua circuit system parameters and based on the sufficient condition for finite-time stability;

[0125] 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 through an actuator, and inject it into the corresponding node of the fractional-order Chua circuit system within the working range to achieve finite-time stable control of the fractional-order Chua circuit system and accurate estimation of the dwell time result.

[0126] Example 3

[0127] This embodiment further illustrates the present invention with reference to specific examples. In practical applications, first, the state variables (such as capacitor voltage and inductor current) of the fractional-order Chua's circuit are measured by sensors, and these measured values ​​are transmitted to the controller. Based on the received state information, the controller executes the designed finite-time intermittent control algorithm - the core of the algorithm is: 1) executing a control law with finite-time convergence in the working range; 2) determining the opening (working) and closing (resting) of the control signal according to the intermittent strategy, and strictly proving that the system state can remain stable within a finite time through the Lyapunov function. 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 is injected into the corresponding node of the Chua's circuit only in the working range specified by the algorithm to achieve finite-time stable control of the system.

[0128] like Figure 1 As shown, a finite-time control method for a fractional-order Chua circuit system based on non-periodic intermittent operation includes the following steps:

[0129] Step 1: Set the initial state variables of the fractional-order Chua circuit system;

[0130] Step 2: sensor measurement and transmission;

[0131] Step 3: Transmitting the measured state to the controller;

[0132] Step 4: Design a finite time intermittent control strategy;

[0133] Step 5: Verify whether the proposed method can stabilize the system state to the equilibrium point within a finite time.

[0134] For the established fractional-order Chua circuit system, the following parameters are considered:

[0135] x i (t) = [x i1 (t),x i2 (t),x i3 (t)] T ,

[0136]

[0137] The embodiments described above are merely descriptions of preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Without departing from the spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by persons skilled in the art should fall within the scope of protection defined by the claims of the present invention.

Claims

1. A finite-time control method for fractional-order Chua's circuit system based on non-periodic intermittent operation, characterized in that: include: Setting the initial state of the fractional-order Chua's circuit system and measuring the state variables of the fractional-order Chua's circuit system using sensors; Using the state variables measured by sensors, a finite-time aperiodic intermittent controller is designed that is activated only during the working period. According to the different dynamic behaviors of the fractional-order Chua circuit system between the working zone and the rest zone, 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 the fractional-order Chua circuit system to achieve finite-time stability under a finite-time aperiodic intermittent controller are obtained; Adjusting the control parameters of the finite-time aperiodic intermittent controller by using preset fractional-order Chua 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 an actuator and injected into the corresponding node of the fractional-order Chua's circuit system within the working range, thereby achieving the finite-time stability of the fractional-order Chua's circuit system and accurate estimation of the dwell time result.

2. The method according to claim 1, characterized in that Introduce the 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 of the original fractional-order Chua circuit system is as follows: in, is the voltage across capacitor C1, is the voltage across capacitor C2, i L is the current through the inductor L, the function Represents the nonlinear resistance R NL The vi characteristics shown, represents the Caputo fractional derivative with the initial time t0 and fractional order β∈(0,1); in, and E represent the internal slope, external slope and power-off voltage of the vi characteristic curve respectively; The introduced preset variables include: x3=Ri L E -1 ,t=s(RC2) -1 , r2=C2R 2 L -1 , s represents time; The final fractional-order Chua circuit system expression is as follows: in, 3. The method according to claim 2, characterized in that The controlled system expression of the fractional-order Chua circuit system is as follows: in, C and A both represent unknown parameter matrices, Δ represents the coupling strength, Γ is the internal coupling matrix, and If nodes i and j are connected, the coupling configuration matrix H = (h ij ) N×N Satisfy h ij =h ji > 0, otherwise, h ij =h ji =0,(i≠j).

4. The method according to claim 3, characterized in that Based on the controlled system expression of the fractional-order Chua circuit system, a finite-time aperiodic intermittent controller is constructed by combining the start and end times of the control interval, the control width, and the non-control width. The expression of the finite-time aperiodic intermittent controller is as follows: Where, 0=t0<θ0<t1<…<t m <θ m <t m+1 <…,d i >0α>0,ζ1>0,δ∈(0,1), m represents a non-negative integer, d i , α, ζ1 and δ are all control parameters; t m and θ m are the start and end time of the m+1th control interval, t m+1 -t m is the m+1th control width, θ m -t m is the m+1th control width, t m+1 -θ m Indicates the m+1th non-controlled width.

5. The method according to claim 4, 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 the control period are dynamically adjusted so that the finite-time aperiodic intermittent controller is activated only within a working range.

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

7. A finite-time control system based on a non-periodic intermittent fractional-order Chua's circuit system, used to implement the method according to any one of claims 1 to 6, characterized in that: include: a state variable acquisition module, used to set the initial state of the fractional-order Chua's circuit system and measure the state variables of the fractional-order Chua's circuit system using sensors; A controller building block for designing a finite-time aperiodic intermittent controller that is activated only during the working interval using state variables measured by sensors; A model building module is used to construct a fractional-order aperiodic intermittent finite-time Lyapunov stability method based on the different dynamic behaviors of the fractional-order Chua circuit system between the working interval and the rest interval; A 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; a parameter adjustment module, configured to adjust control parameters of the finite-time aperiodic intermittent controller by using preset fractional-order Chua circuit system parameters and based on the sufficient condition for 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 through an actuator, and inject it into the corresponding node of the fractional-order Chua circuit system within the working range to achieve finite-time stable control of the fractional-order Chua circuit system and accurate estimation of the dwell time result.

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