A non-backstepping controller for non-homogeneous fractional order Chua circuit system and a design method thereof

Abstract

The application discloses a non-backstepping controller for a non-homogeneous fractional-order Chua circuit system and a design method thereof, and the non-backstepping controller comprises a state transformation module, an error observer module, a fuzzy logic system approximation module and a controller and projection operator module. The state transformation module transforms an original non-homogeneous fractional-order Chua circuit system into a controllable standard type; the error observer module constructs an observer suitable for the characteristics of the non-homogeneous fractional-order system in view of the engineering problem that the state transformation is unobservable; the fuzzy logic system approximation module performs online approximation on a system lumped unknown nonlinear function; and the controller and projection operator module constructs a non-backstepping adaptive fuzzy controller containing a hyperbolic tangent function and a fractional-order projection operator parameter updating law. The application can realize effective tracking of a reference trajectory by a system output, ensure that all signals of a closed-loop system are uniformly ultimately bounded, and be suitable for various nonlinear engineering systems with non-homogeneous fractional-order dynamic characteristics.

Description

Technical Field

[0001] This invention belongs to the field of control technology for non-uniform Chua's circuit systems, specifically relating to a non-backstep controller for non-uniform fractional Chua's circuit systems and its design method. Background Technology

[0002] Circuit systems are the core research object of nonlinear dynamics and analog circuit control engineering. Accurate modeling and high-performance control are the core prerequisites for ensuring the stable and safe operation of various circuits. With the rapid iteration of modern intelligent control technology, actual nonlinear circuits generally exhibit multiple complex characteristics such as dynamic hysteresis of energy storage elements, strong coupling of multiple variables, and piecewise nonlinearity. Traditional integer-order modeling methods are difficult to accurately characterize such complex dynamics, and the corresponding control strategies are also difficult to meet the stringent performance indicators of high-precision trajectory tracking and circuit disturbance suppression. Among many nonlinear circuit models, the Chua's circuit system, with its simple circuit topology and rich dynamic evolution behavior, is widely used in analog circuit simulation, nonlinear closed-loop control, and other fields, and is a highly representative benchmark research model in the field of nonlinear circuits. Currently, most existing research on this system is based on the same-element fractional-order framework, which forcibly stipulates that the fractional derivatives of all state variables are consistent in order. This fails to take into account the objective physical law that the dynamic orders of energy storage elements such as capacitors and inductors are not equal in engineering circuits, resulting in poor matching between the model and the actual circuit. In contrast, the non-same-element fractional-order Chua circuit system can truly reflect the physical characteristics of the differences in the dynamic order of different energy storage branches: the dynamic evolution order of the capacitor branch and the inductor circuit in the circuit is naturally different. This modeling method no longer uniformly limits the fractional order, and compared with the same-element modeling scheme, it is more in line with the actual circuit in engineering, with higher model fidelity and wider engineering application potential.

[0003] However, existing adaptive control methods for non-uniform fractional-order Chua's circuit systems still have the following problems:

[0004] First, most existing analyses and control studies of fractional-order circuit systems are based on fractional-order models of the same order, assuming that the fractional derivatives of all state variables in the system have the same order. However, in actual Chua's circuit systems, the dynamic elements of different branches naturally differ, and the fractional orders corresponding to each channel are often not entirely the same. More importantly, while the stability of the equilibrium point of an integer-order nonlinear system can be verified by a convex Lyapunov function, the corresponding fractional-order nonlinear system of the same order can directly adopt this stability conclusion. However, this criterion only applies to fractional-order systems of the same order and does not hold for fractional-order systems of different orders. This means that the stability analysis and theoretical derivation of the controller convergence proof for non-fractional-order Chua's circuit systems are significantly more difficult, lacking mature and systematic theoretical analysis tools, resulting in significant theoretical research challenges and relatively scarce achievements.

[0005] Second, current controller designs for fractional-order Chua's circuit systems commonly employ backstepping recursive control methods. This method requires iterative differentiation of the virtual control law order by order during the design process, easily leading to a "computational complexity explosion." While improved schemes such as dynamic surface control and command filtering control avoid repeated differentiation processes by introducing filters, mitigating the complexity explosion to some extent, the filters themselves introduce filtering errors, requiring additional filter compensation mechanisms, further increasing the complexity of the controller structure and derivation. Simultaneously, within the non-uniform fractional-order framework, multi-order coupling increases the difficulty of filter order matching and error boundary quantity derivation, further limiting the control effectiveness of such methods.

[0006] Third, in engineering applications, the internal state of a system often cannot be fully measured, requiring error observers to achieve output feedback control. However, most existing observer schemes are suitable for fractional-order systems with the same parameters. The multi-order dynamic coupling characteristics of non-same-parallel systems make it difficult to analyze the convergence of observation errors, and there is no standardized solution for observer gain. Furthermore, the system contains unknown nonlinear terms, and the collaborative design of the observer and fuzzy approximation module, as well as the joint stability proof of observation and tracking errors, lack mature theoretical support, making it impossible to balance state observation accuracy with overall closed-loop stability. Summary of the Invention

[0007] To address the shortcomings of existing technologies, this invention proposes a non-backstepping controller and its design method for non-uniform fractional-order Chua's circuit systems. It abandons the traditional backstepping method, avoiding the complexity explosion problem caused by repeated differentiation. It overcomes the inherent filtering errors introduced by dynamic surfaces and command filtering methods, which require additional filter compensation design. Ultimately, it achieves trajectory tracking and ensures that all closed-loop signals are consistent and eventually bounded, adapting to various nonlinear engineering systems with non-uniform fractional-order dynamics, such as motors and chaotic circuits.

[0008] To achieve the above objectives, the present invention adopts the following technical solution:

[0009] A non-inverse controller for non-coordinate fractional-order Chua's circuit systems includes a state transition module, an error observer module, a fuzzy logic system approximation module, and a controller and projection operator module, wherein:

[0010] The output of the non-same-element fractional-order Chua's circuit system is connected to the input of the state transformation module and the input of the error observer module, respectively; the output of the state transformation module is connected to the input of the error observer module; the state transformation module is used to transform the original non-same-element fractional-order Chua's circuit system into a controllable canonical state vector.

[0011] The reference trajectory signal is connected to the input terminals of the error observer module, the fuzzy logic system approximation module, and the controller and projection operator module, respectively; the output terminal of the error observer module is connected to the input terminals of the fuzzy logic system approximation module and the controller and projection operator module, respectively; the error observer module is used to estimate the unmeasurable state after transformation;

[0012] The output of the fuzzy logic system approximation module is connected to the input of the controller and the projection operator module; the fuzzy logic system approximation module is used to approximate the lumped unknown nonlinear function of the system;

[0013] The control output terminal of the controller and projection operator module is connected to the control input terminal of the non-uniform fractional Chua's circuit system; the controller and projection operator module are used to construct a non-instep adaptive fuzzy controller containing a hyperbolic tangent function and a parameter update law for the fractional projection operator.

[0014] A design method for a non-backstep controller for a non-coordinate fractional-order Chua's circuit system includes the following steps:

[0015] Step A: Establish a model of a non-identical fractional-order Chua's circuit system;

[0016] Step B: Design a system state transformation module to transform the original system state into a controllable canonical form state vector;

[0017] Step C: Design an error observer module to estimate the unmeasurable state after the transformation;

[0018] Step D: Design a fuzzy logic system approximation module to approximate the lumped unknown nonlinear function of the system;

[0019] Step E: Design the controller and projection operator modules, and construct a non-instep adaptive fuzzy controller with a hyperbolic tangent function and a fractional-order projection operator parameter update law.

[0020] Furthermore, in step A, the non-coordinate fractional-order Chua's circuit system is described by the following formula:

[0021] (1)

[0022] in, It is a three-dimensional circuit state;

[0023] This represents the first-dimensional state of the Chua's circuit, corresponding to the voltage of the nonlinear resistive branch. This is the second-dimensional state, corresponding to the capacitor branch voltage; This is the third-dimensional state, corresponding to the inductor circuit current;

[0024] It is a Caputo-type fractional differential operator; It consists of three fractional orders that are not all equal;

[0025] parameter yes Acting on The linear coupling coefficient is obtained by dimensionless simplification of the inductance, capacitance, and resistance parameters of the main circuit.

[0026] parameter It is the current-voltage characteristic coefficient of the nonlinear resistor;

[0027] parameter yes Acting on The coupling coefficient;

[0028] parameter yes Acting on The positive coupling coefficient;

[0029] parameter It is the second-dimensional state. The self-damping attenuation coefficient characterizes the energy dissipation characteristics of the capacitor branch.

[0030] parameter yes Acting on The coupling coefficient;

[0031] The existence of unknowns , Make ;

[0032] parameter It is the feedback coupling coefficient of the inductor branch;

[0033] For external control input, This is the system's observable output.

[0034] Furthermore, in step B, the input signals of the system state transformation module are the original state vector and the measurable output signal of the fractional-order Chua's circuit system; the output signal is the transformed state vector and the transformed controllable canonical system. The state transformation rule of the system state transformation module is expressed as follows:

[0035] (2)

[0036] in, This is the transformed state vector;

[0037] and The relationship is There exists a smooth implicit function. Make This achieves the inverse mapping from the standard state to the original system state;

[0038] The transformed unknown smooth nonlinear function is derived from the original system's nonlinear functions of various orders through a composite derivation using the fractional chain rule.

[0039] Furthermore, in step C, the input signals of the error observer module are the system's measurable output tracking error, the third-order fractional derivative of the reference trajectory, and the adaptive control input signal; the output signals are the tracking error full-state estimation vector, the measurable output component of the observation error, and the observer gain configuration that satisfies the Hurwitz stability constraint.

[0040] The tracking error system model is represented in the following matrix form.

[0041] (3)

[0042] in, This is the full state vector of the tracking error;

[0043] The transformed unknown smooth nonlinear function;

[0044] It is a diagonal matrix composed of three non-identical fractional orders;

[0045] Here is the state matrix of the error dynamic system;

[0046] The input matrix of the error dynamic system

[0047] To track the reference trajectory;

[0048] It is a Caputo-type fractional differential operator;

[0049] The designed error observer module is represented as follows:

[0050] (4)

[0051] in The tracking error is the one that can be directly measured at the system output. The original system's measurable output, To track the reference trajectory;

[0052] The estimated vector for tracking the full state of the error corresponds to the state estimate of the 3D error dynamic system;

[0053] This is the observation error vector;

[0054] This is the input matrix of the error dynamic system;

[0055] The measurable output component of the observation error;

[0056] Here, represents the error state feedback gain vector, and represents the pre-designed controller parameters;

[0057] is the observer gain vector, and is the pre-designed observer parameters;

[0058] Furthermore, in step D, the input signals of the fuzzy logic system approximation module are the state estimation vector output by the error observer and the third-order fractional derivative information of the reference trajectory; the output signals are the ideal weight vector of the fuzzy logic system, the fuzzy basis function vector, and the bounded approximation error.

[0059] The approximation module of the fuzzy logic system is represented as follows:

[0060] (5)

[0061] in, The lumped unknown continuous nonlinear function to be approximated;

[0062] For the approximation output of an unknown function in a fuzzy logic system;

[0063] This is the transformed state vector;

[0064] , Let be the ideal weight vector of the fuzzy logic system and its adaptive estimate, and their corresponding transposes be respectively , ;

[0065] The fuzzy basis function vector is constructed from Gaussian membership functions through product inference and central average defuzzification, and consists of continuous and smooth basis functions.

[0066] The inherent approximation error of the fuzzy logic system is unknown but satisfies boundedness constraints.

[0067] Furthermore, in step E, the input signals of the controller and the projection operator module are the error state estimation vector output by the error observer, the basis function output and weight estimation value of the fuzzy logic system approximation module, and the third-order fractional derivative of the reference trajectory; the output signals are the system control input and the fractional adaptive update law of the fuzzy weight parameters.

[0068] The controller and projection operator modules are represented as follows:

[0069] (6)

[0070] (7)

[0071] in, For parameters The reciprocal of;

[0072] This is the shape parameter (a constant greater than 0) of the hyperbolic tangent function, which adjusts the steepness of the function.

[0073] As an intermediate virtual controller;

[0074] This is the actual controller.

[0075] Furthermore, in step E, to ensure the fuzzy logic system and unknown parameters... To ensure the boundedness of the estimate, the following parameter update law for the fractional projection operator is designed:

[0076] (8)

[0077] (9)

[0078] in, For parameter estimates The first derivative with respect to time;

[0079] For parameter estimates The first derivative with respect to time;

[0080] It is a 3D column vector constructed from observation errors. ;

[0081] It is a binary projection operator, with two inputs being the parameter update direction term and the current parameter estimate;

[0082] This is the adaptive gain coefficient, used to adjust the convergence speed of the parameters;

[0083] Let be the positive definite symmetric matrix corresponding to the Lyapunov function.

[0084] Beneficial effects:

[0085] Compared with the prior art, the present invention has the following beneficial effects:

[0086] First, this invention constructs a control design framework for non-same-element fractional-order Chua's circuit systems, which are widely used in practical engineering, overcoming the limitation of existing technologies that are only applicable to same-element fractional-order systems. Existing mature stability determination conclusions can only be used for same-element systems with uniform differential orders, and cannot be directly applied to non-same-element fractional-order systems where the differential orders of each state are not entirely the same. However, this invention specifically designs observers and controllers for multi-order dynamically coupled non-same-element systems, and can adapt to real nonlinear systems with different dynamic orders.

[0087] Second, this invention employs a non-backstepping control method, avoiding the complexity explosion problem caused by the sequential differentiation of the virtual control law in traditional backstepping methods. Furthermore, unlike dynamic surface and command filtering control methods, it eliminates the need for filters, thus avoiding filtering errors and requiring no additional filtering compensation. This ensures stable system operation and tracking without increasing computational burden.

[0088] Third, existing observers are mostly developed for fractional-order systems with the same parameters, making it difficult to handle the convergence analysis of observation errors caused by multi-order coupling in non-same-order systems. This invention designs a fractional-order error observer adapted to the coupling characteristics of non-same-order systems, solving the problem that the system state cannot be directly measured after state transformation. It can be directly applied to various nonlinear engineering systems with non-same-order fractional-order dynamic characteristics, such as motor drive systems and fractional-order chaotic circuits, and has a wide range of engineering applications. Attached Figure Description

[0089] Figure 1 This is a schematic diagram of a non-uniform fractional Chua's circuit system.

[0090] Figure 2 This is a diagram illustrating the dynamic behavior of the fractional-order Chua's circuit system used in this invention.

[0091] Figure 3 This is a diagram showing the trajectory tracking effect;

[0092] Figure 4 It is a control state Renderings;

[0093] Figure 5 This is a diagram showing the effect of adaptive parameters;

[0094] Figure 6 It is a control input Line graph. Detailed Implementation

[0095] The invention will now be further explained with reference to the accompanying drawings.

[0096] The design of the non-coefficient fractional Chua circuit system of this invention is as follows: Figure 1 As shown, the present invention will be further illustrated using a fractional-order Chua's circuit system as an example. The control objective of this example is to achieve tracking control of the reference trajectory using the non-backstep controller designed for non-uniform fractional-order Chua's circuit systems proposed in this invention. Simultaneously, it utilizes a fuzzy logic system to approximate the unknown nonlinear terms of the system, avoids the computational complexity explosion problem of traditional backstepping methods by using a non-backstep control method, and ensures that all signals in the closed-loop system are consistent and ultimately bounded, thus achieving tracking control of the non-uniform fractional-order Chua's circuit system.

[0097] To address the aforementioned problems, this embodiment provides a non-backstep controller for non-coordinate fractional-order Chua's circuit systems, comprising a state transition module, an error observer module, a fuzzy logic system approximation module, and a controller and projection operator module, wherein:

[0098] The output of the non-same-element fractional-order Chua's circuit system is connected to the input of the state transformation module and the input of the error observer module, respectively; the output of the state transformation module is connected to the input of the error observer module; the state transformation module is used to transform the original non-same-element fractional-order Chua's circuit system into a controllable canonical form state vector.

[0099] The reference trajectory signal is connected to the input of the error observer module, the input of the fuzzy logic system approximation module, and the input of the controller and projection operator module, respectively; the output of the error observer module is connected to the input of the fuzzy logic system approximation module and the input of the controller and projection operator module, respectively; the error observer module is used to estimate the unmeasurable state after transformation;

[0100] The output of the fuzzy logic system approximation module is connected to the input of the controller and the projection operator module; the fuzzy logic system approximation module is used to approximate the lumped unknown nonlinear function of the system;

[0101] The control output of the controller and projection operator module is connected to the control input of the non-uniform fractional Chua's circuit system; the controller and projection operator module are used to construct a non-instep adaptive fuzzy controller containing a hyperbolic tangent function and a parameter update law for the fractional projection operator.

[0102] The design method of the non-backstep controller for a non-simultaneous fractional-order Chua's circuit system in this embodiment includes the following steps:

[0103] Step A: Establish a model of a non-coordinate fractional Chua's circuit system.

[0104] The non-coefficient fractional Chua's circuit system described in this invention is expressed by the following formula:

[0105] (1)

[0106] in, It is a three-dimensional circuit state;

[0107] This represents the first-dimensional state of Chua's circuit, typically corresponding to the voltage of a nonlinear resistive branch. This is the second-dimensional state, corresponding to the capacitor branch voltage; This is the third-dimensional state, corresponding to the inductor circuit current;

[0108] It is a Caputo-type fractional differential operator; For three fractional orders that are not all equal, ;

[0109] parameter yes Acting on The linear coupling coefficient is obtained by dimensionless simplification of the inductance, capacitance, and resistance parameters of the main circuit.

[0110] parameter It is the current-voltage characteristic coefficient of the nonlinear resistor;

[0111] parameter yes Acting on The coupling coefficient;

[0112] parameter yes Acting on The positive coupling coefficient;

[0113] parameter It is the second-dimensional state. The self-damping attenuation coefficient characterizes the energy dissipation characteristics of the capacitor branch.

[0114] parameter yes Acting on The coupling coefficient;

[0115] parameter It is the feedback coupling coefficient of the inductor branch;

[0116] For external control input, This is the system's observable output.

[0117] Step B: Design of the State Transition Module

[0118] The input signals of the system state transformation module are the original state vector and the measurable output signal of the fractional-order Chua's circuit system; the output signal is the transformed state vector and the transformed controllable canonical system. The designed state transformation rule is expressed as follows:

[0119] (2)

[0120] in, This is the transformed state vector;

[0121] and The relationship is There exists a smooth implicit function. Make This achieves the inverse mapping from the standard state to the original system state;

[0122] The transformed unknown smooth nonlinear function is derived from the original system's nonlinear functions of various orders through a composite derivation using the fractional chain rule.

[0123] Step C: Design of the Error Observer Module

[0124] The input signals of the error observer module are the system's measurable output tracking error, the third-order fractional derivative of the reference trajectory, and the adaptive control input signal; the output signals are the tracking error full-state estimation vector, the measurable output component of the observation error, and the observer gain configuration that satisfies the Hurwitz stability constraint.

[0125] The tracking error system model is represented in the following matrix form.

[0126] (3)

[0127] in, This is the full state vector of the tracking error;

[0128] The transformed unknown smooth nonlinear function;

[0129] It is a diagonal matrix composed of three non-identical fractional orders;

[0130] Here is the state matrix of the error dynamic system;

[0131] The input matrix of the error dynamic system

[0132] To track the reference trajectory;

[0133] It is a Caputo-type fractional differential operator;

[0134] The designed error observer module is represented as follows:

[0135] (4)

[0136] in The tracking error is the one that can be directly measured at the system output. The original system's measurable output, To track the reference trajectory;

[0137] The estimated vector for tracking the full state of the error corresponds to the state estimate of the 3D error dynamic system;

[0138] This is the observation error vector;

[0139] This is the input matrix of the error dynamic system;

[0140] The measurable output component of the observation error;

[0141] Here, represents the error state feedback gain vector, and represents the pre-designed controller parameters;

[0142] is the observer gain vector, and is the pre-designed observer parameters.

[0143] Step D: Design of the fuzzy logic system approximation module

[0144] The input signals of the fuzzy logic system approximation module are the state estimation vector output by the error observer and the third-order fractional derivative information of the reference trajectory; the output signals are the ideal weight vector of the fuzzy logic system, the fuzzy basis function vector, and the bounded approximation error.

[0145] (5)

[0146] in, The lumped unknown continuous nonlinear function to be approximated;

[0147] For the approximation output of an unknown function in a fuzzy logic system;

[0148] This is the transformed state vector;

[0149] , Let be the ideal weight vector of the fuzzy logic system and its adaptive estimate, and their corresponding transposes be respectively , ;

[0150] The fuzzy basis function vector is constructed from Gaussian membership functions through product inference and central average defuzzification, and consists of continuous and smooth basis functions.

[0151] The inherent approximation error of the fuzzy logic system is unknown but satisfies boundedness constraints.

[0152] Step E: Design of the controller and projection operator module

[0153] The input signals of the controller and projection operator modules are the error state estimation vector output by the error observer, the basis function output and weight estimation value of the fuzzy logic system approximation module, and the third-order fractional derivative of the reference trajectory; the output signals are the system control input and the fractional adaptive update law of the fuzzy weight parameters.

[0154] (6)

[0155] (7)

[0156] in, For parameters The reciprocal of;

[0157] This is the shape parameter of the hyperbolic tangent function, used to adjust the steepness of the function;

[0158] As an intermediate virtual controller;

[0159] For the actual controller;

[0160] To ensure the fuzzy logic system and unknown parameters To ensure the boundedness of the estimate, the following parameter update law for the fractional projection operator is designed:

[0161] (8)

[0162] (9)

[0163] in, For parameter estimates The first derivative with respect to time;

[0164] For parameter estimates The first derivative with respect to time;

[0165] It is a 3D column vector constructed from observation errors. ;

[0166] It is a binary projection operator, with two inputs being the parameter update direction term and the current parameter estimate;

[0167] This is the adaptive gain coefficient, used to adjust the convergence speed of the parameters;

[0168] Let be the positive definite symmetric matrix corresponding to the Lyapunov function.

[0169] Simulation results are as follows Figure 2-6 As shown. Figure 2 This demonstrates the fractional-order Chua's circuit system used in this invention. The dynamic behavior diagram shows that the state trajectory is confined to a finite phase space and continuously nested and entangled in multiple layers. The trajectory does not converge to the equilibrium point and does not form a periodic trajectory, exhibiting a typical chaotic attractor geometry. Therefore, it can be determined that the system has chaotic dynamic effects. Figure 3 As can be observed from the tracking effect of the output signal on the reference trajectory, the tracking of the reference trajectory can be achieved relatively well. Figure 4 The system's control state is described. The response curves show that all state variables remain stable and do not exhibit divergence, indicating that the closed-loop system has good dynamic performance. Figure 5 For adaptive parameters The component graph has parameters that can be adaptively adjusted online and eventually stabilize. Figure 6 The control input is given. The change curve shows that the control input is continuous and bounded, which meets the engineering application requirements of practical systems.

[0170] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A non-backstep controller for a non-uniform fractional-order Chua's circuit system, characterized in that, It includes a state transition module, an error observer module, a fuzzy logic system approximation module, and a controller and projection operator module, wherein: The output of the non-same-element fractional-order Chua's circuit system is connected to the input of the state transformation module and the input of the error observer module, respectively; the output of the state transformation module is connected to the input of the error observer module; the state transformation module is used to transform the original non-same-element fractional-order Chua's circuit system into a controllable canonical state vector. The reference trajectory signal is connected to the input terminals of the error observer module, the fuzzy logic system approximation module, and the controller and projection operator module, respectively; the output terminal of the error observer module is connected to the input terminals of the fuzzy logic system approximation module and the controller and projection operator module, respectively; the error observer module is used to estimate the unmeasurable state after transformation; The output of the fuzzy logic system approximation module is connected to the input of the controller and the projection operator module; the fuzzy logic system approximation module is used to approximate the lumped unknown nonlinear function of the system; The control output terminal of the controller and projection operator module is connected to the control input terminal of the non-uniform fractional Chua's circuit system; the controller and projection operator module are used to construct a non-instep adaptive fuzzy controller containing a hyperbolic tangent function and a parameter update law for the fractional projection operator.

2. A design method for a non-backstep controller for a non-similarity fractional-order Chua's circuit system, characterized in that, Includes the following steps: Step A: Establish a model of a non-identical fractional-order Chua's circuit system; Step B: Design a system state transformation module to transform the original system state into a controllable canonical form state vector; Step C: Design an error observer module to estimate the unmeasurable state after the transformation; Step D: Design a fuzzy logic system approximation module to approximate the lumped unknown nonlinear function of the system; Step E: Design the controller and projection operator modules, and construct a non-instep adaptive fuzzy controller with a hyperbolic tangent function and a fractional-order projection operator parameter update law.

3. The design method according to claim 2, characterized in that, In step A, the non-coordinate fractional Chua's circuit system is described by the following formula: （1） in, It is a three-dimensional circuit state; This represents the first-dimensional state of the Chua's circuit, corresponding to the voltage of the nonlinear resistive branch. This is the second-dimensional state, corresponding to the capacitor branch voltage; This is the third-dimensional state, corresponding to the inductor circuit current; It is a Caputo-type fractional differential operator; It consists of three fractional orders that are not all equal; parameter yes Acting on The linear coupling coefficient is obtained by dimensionless simplification of the inductance, capacitance, and resistance parameters of the main circuit. parameter It is the current-voltage characteristic coefficient of the nonlinear resistor; parameter yes Acting on The coupling coefficient; parameter yes Acting on The positive coupling coefficient; parameter It is the second-dimensional state. The self-damping attenuation coefficient characterizes the energy dissipation characteristics of the capacitor branch. parameter yes Acting on The coupling coefficient; The existence of unknowns , Make ; parameter It is the feedback coupling coefficient of the inductor branch; For external control input, This is the system's observable output.

4. The design method according to claim 3, characterized in that, In step B, the input signals of the system state transformation module are the original state vector and the measurable output signal of the fractional-order Chua's circuit system; the output signal is the transformed state vector and the transformed controllable canonical system. The state transformation rules of the system state transformation module are expressed as follows: （2） in, This is the transformed state vector; and The relationship is There exists a smooth implicit function. Make This achieves the inverse mapping from the standard state to the original system state; The transformed unknown smooth nonlinear function is derived from the original system's nonlinear functions of various orders through a composite derivation using the fractional chain rule.

5. The design method according to claim 4, characterized in that, In step C, the input signals of the error observer module are the system's measurable output tracking error, the third-order fractional derivative of the reference trajectory, and the adaptive control input signal; the output signals are the tracking error full-state estimation vector, the measurable output component of the observation error, and the observer gain configuration that satisfies the Hurwitz stability constraint. The tracking error system model is represented in the following matrix form. （3） in, This is the full state vector of the tracking error; The transformed unknown smooth nonlinear function; It is a diagonal matrix composed of three non-identical fractional orders; Here is the state matrix of the error dynamic system; The input matrix of the error dynamic system To track the reference trajectory; It is a Caputo-type fractional differential operator; The designed error observer module is represented as follows: （4） in The tracking error is the one that can be directly measured at the system output. The original system's measurable output, To track the reference trajectory; The estimated vector for tracking the full state of the error corresponds to the state estimate of the 3D error dynamic system; This is the observation error vector; This is the input matrix of the error dynamic system; The measurable output component of the observation error; Here, represents the error state feedback gain vector, and represents the pre-designed controller parameters; is the observer gain vector, and is the pre-designed observer parameters.

6. The design method according to claim 5, characterized in that, In step D, the input signals of the fuzzy logic system approximation module are the state estimation vector output by the error observer and the third-order fractional derivative information of the reference trajectory; the output signals are the ideal weight vector of the fuzzy logic system, the fuzzy basis function vector, and the bounded approximation error. The approximation module of the fuzzy logic system is represented as follows: （5） in, The lumped unknown continuous nonlinear function to be approximated; For the approximation output of an unknown function in a fuzzy logic system; This is the transformed state vector; , Let be the ideal weight vector of the fuzzy logic system and its adaptive estimate, and their corresponding transposes be respectively , ; The fuzzy basis function vector is constructed from Gaussian membership functions through product inference and central average defuzzification, and consists of continuous and smooth basis functions. The inherent approximation error of the fuzzy logic system is unknown but satisfies boundedness constraints.

7. The design method according to claim 6, characterized in that, In step E, the input signals of the controller and the projection operator module are the error state estimation vector output by the error observer, the basis function output and weight estimation value of the fuzzy logic system approximation module, and the third-order fractional derivative of the reference trajectory; the output signals are the system control input and the fractional adaptive update law of the fuzzy weight parameters. The controller and projection operator modules are represented as follows: （6） （7） in, For parameters The reciprocal of; This is the shape parameter (a constant greater than 0) of the hyperbolic tangent function, which adjusts the steepness of the function. As an intermediate virtual controller; This is the actual controller.

8. The design method according to claim 7, characterized in that, In step E, to ensure the fuzzy logic system and unknown parameters To ensure the boundedness of the estimate, the following parameter update law for the fractional projection operator is designed: （8） （9） in, For parameter estimates The first derivative with respect to time; For parameter estimates The first derivative with respect to time; It is a 3D column vector constructed from observation errors. ; It is a binary projection operator, with two inputs being the parameter update direction term and the current parameter estimate; This is the adaptive gain coefficient, used to adjust the convergence speed of the parameters; Let be the positive definite symmetric matrix corresponding to the Lyapunov function.

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