Fractional-order decoupled dual-mass mems gyroscope scheduled time fuzzy backstepping control method with delay constraint and actuator failure

Abstract

This invention relates to a predetermined-time fuzzy backstepping control method for a fractional-order decoupled dual-mass MEMS gyroscope with delay constraints and actuator faults, belonging to the field of MEMS gyroscope control technology. The method includes: S2: establishing a fractional-order motion equation for the decoupled dual-mass MEMS gyroscope with actuator faults based on the Lagrange equations to eliminate the influence of linear acceleration and vibration at the same frequency; S3: constructing a β-cut 2 type fuzzy logic system to address the uncertainties and nonlinear terms of the system model; superimposing a delay error function on a specified time function to ensure that the error does not violate the constraints after a finite time; using a fractional-order hyperbolic tangent tracking differentiator in the controller design to handle direct fractional derivatives and repeated derivatives; and finally, combining the β-cut 2 type fuzzy logic system to establish a control law with an adaptive law for fault parameters.

Description

Technical Field

[0001] This invention belongs to the field of MEMS gyroscope control technology, and relates to a predetermined time fuzzy backstepping control method for a fractional-order decoupled dual-mass MEMS gyroscope with delay constraints and actuator faults. Background Technology

[0002] MEMS (Micro-Electro-Mechanical Systems) micro-gyroscopes are primarily used to measure angular velocities in various applications, such as inertial navigation, intelligent vehicles, consumer electronics, and unmanned orbital satellites, due to their advantages such as small size, low power consumption, and high integration with integrated circuits. Traditional single-mass MEMS micro-gyroscopes are highly sensitive to any linear acceleration; along the sensing direction, linear acceleration can lead to common model errors and circuit saturation failures. Uncertainties in this model are unavoidable due to external interference, environmental changes, and manufacturing tolerances, all of which significantly affect the dynamic behavior of the MEMS gyroscope. Furthermore, operating a MEMS gyroscope under fault conditions or without fault tolerance mechanisms can lead to self-damage or shortened lifespan. Output constraints are a pressing issue due to physical limitations and safety requirements. In addition, inherent high-frequency chaotic oscillations cause significant damage to the system. Therefore, proposing a multi-mass MEMS gyroscope model is of great significance. This model will provide the possibility of analyzing its dynamic behavior and further address the corresponding uncertainties, constraints, actuator failures, and chaotic oscillation effects.

[0003] In recent years, research on the modeling and control of single MEMS gyroscopes has yielded many results. Fei and Zhou established a dynamic model of a three-axis MEMS gyroscope and proposed a robust adaptive control scheme to address its nonlinearity. Fazlyab et al. established the complete dynamic equations of a z-axis MEMS gyroscope and proposed an interval type-2 fuzzy sliding mode control method to reduce noise while addressing the problem of uncertain parameters. Chu et al. proposed a dynamic model of a MEMS gyroscope and designed an adaptive neural network dynamic global PID sliding mode controller. However, these works do not reflect the fractional-order characteristics of the dielectric and do not increase the degrees of freedom in controller design. These works were completed under the general condition of dynamic evolution and may be ineffective under high-frequency chaotic oscillations. Furthermore, these schemes will fail once the actuator fails. To improve bandwidth and vibration resistance, a few articles on multi-mass MEMS gyroscopes have been published in recent years. Efimovskaya et al. studied the dynamics of a dual-mass MEMS vibrating gyroscope with manufacturing defects and proposed an operating mode based on an electrostatic frequency tuning scheme. Yang et al. discussed the structural design, simulation, fabrication, signal measurement, and control of a symmetrical dual-mass micro-gyroscope. Hamed et al. investigated the dynamics, energy transfer, and vibration control of MEMS gyroscopes under linear and nonlinear parameter excitation. However, even with only model uncertainties, these control methods failed to achieve the aforementioned objectives.

[0004] Constraints of various forms are necessary in many engineering applications, and the corresponding problems have received considerable attention. Zhang et al. proposed a performance-defined barrier Lyapunov function for neural network adaptive control of chaotic permanent magnet synchronous motor systems within a backstepping framework. Zhao et al. implemented stability control of MEMS resonators using a cosine barrier function. Song et al. constructed a barrier Lyapunov function to realize neural network adaptive PI tracking control of nonlinear systems with unknown and non-stationary driving characteristics. These works presuppose that constraints appear immediately at the start of system operation. However, in many practical cases, relevant constraints are applied after the system has been running for a period of time. To address this issue, Zhao et al. proposed a delayed constraint function to implement event-triggered adaptive neural network control of nonlinear systems. However, without considering actuator failure mechanisms, single nonlinear systems in the integer order domain are completely different from fractional-order decoupled dual-mass MEMS gyroscopes. Due to the introduction of generalized function calculus into traditional backstepping control theory, some fractional-order backstepping control schemes for nonlinear systems have been proposed to improve the design freedom of the controller. Liu et al. discussed the adaptive neural network backstepping control problem for fractional-order nonlinear systems with actuator failures. Li et al. proposed an adaptive backstepping control scheme for fractional-order nonlinear systems with external disturbances and parameter uncertainties. However, ignoring performance constraints, the inherent "complexity term explosion" problem associated with traditional backstepping remains unresolved, and this problem becomes increasingly prominent with increasing system order. Furthermore, fractional-order decoupled dual-mass MEMS gyroscopes are physically completely different from the aforementioned single nonlinear systems. When such dual-mass MEMS gyroscopes face high-frequency chaotic oscillations and high-sensitivity parameters, these techniques become ineffective under certain conditions if delay constraints and actuator failures are not considered.

[0005] Therefore, there is an urgent need for a fractional-order decoupled dual-mass MEMS gyroscope control method that can simultaneously consider delay constraints and actuator failures. Summary of the Invention

[0006] In view of this, the purpose of this invention is to provide a fractional-order decoupled dual-mass MEMS gyroscope predetermined-time fuzzy backstepping control method with delay constraints and actuator failures, which can better solve problems such as constraints, actuator failures, uncertainties and chaotic oscillations.

[0007] To achieve the above objectives, the present invention provides the following technical solution:

[0008] A predetermined-time fuzzy backstepping control method for a fractional-order decoupled dual-mass MEMS gyroscope with delay constraints and actuator faults, specifically including the following steps:

[0009] S1: Based on the Lagrange equation, establish the fractional-order motion equation of a decoupled dual-mass MEMS gyroscope with actuator failure to eliminate the influence of linear acceleration and vibration at the same frequency.

[0010] S2: Design a pre-defined time fuzzy backstepping controller: To better address issues such as constraints, actuator failures, uncertainties, and chaotic oscillations, the following steps are included:

[0011] S21: Construct a β-cut type 2 fuzzy logic system to solve the uncertainty and nonlinearity of the system model;

[0012] S22: Controller Design: A delay error function is superimposed on the specified time function to ensure that the error will not violate the constraints after a finite time; a fractional hyperbolic tangent tracking differentiator is used in the controller design to handle the problems of direct fractional derivatives and repeated derivatives; finally, a β-cut 2 type fuzzy logic system is combined to establish a control law with an adaptive law of fault parameters to solve the actuator fault problem including bias faults and failure faults.

[0013] Furthermore, in step S1, the decoupled dual-mass MEMS gyroscope includes two equivalent MEMS gyroscopes, a coupling beam, a lever support beam, and a drive coupling suspension beam; each MEMS gyroscope includes an anchor, an inertial mass block, a drive suspension beam, an inductive coupling suspension beam, a feedback electrode, an inductive decoupling beam, a drive decoupling beam, a drive electrode, a drive-inductive electrode, a drive frame, an inductive electrode, and an inductive suspension beam.

[0014] When voltage is applied to the driving electrodes on both sides of the dual-mass MEMS gyroscope, the left and right inertial mass blocks oscillate in opposite directions on the driving axis; simultaneously, the upper and lower sensing frames are stationary along the sensing axis; the driving coupling beam is used to adjust the in-phase and out-of-phase resonant frequencies, playing a key role in synchronizing the motion of the two inertial mass blocks; when Coriolis acceleration from angular velocity is generated, the left inertial mass block and sensing frame vibrate relative to the right inertial mass block and sensing frame along the sensing axis; the complex coupling system of the lever consists of the inductive coupling beam, the coupling crossbeam, and the lever support beam, which ensures the synchronization of the motion of the two inertial mass blocks by adjusting the in-phase and out-of-phase resonant frequencies in the sensing direction.

[0015] Further, in step S1, the fractional-order equation of motion for the decoupled dual-mass MEMS gyroscope with actuator failure is established. Specifically, it is assumed that the decoupled dual-mass MEMS gyroscope rotates around the z-axis with a slowly varying angular velocity, and the centrifugal force is negligible, where the z-axis is the direction axis perpendicular to the plane containing the sensing axis and the driving axis. Using the Lagrange equation, the fractional-order equation of motion for the decoupled dual-mass MEMS gyroscope can be written as:

[0016]

[0017] Where α and Ω z These represent the fractional order and angular velocity on the z-axis, respectively. m1 and m2 represent the effective mass for mass verification in drive and sensing modes, respectively. and These are the damping coefficients along the driving and sensing directions, respectively. and These are the spring stiffnesses in the driving and sensing directions, respectively. and It is the damping cross-coupling coefficient caused by manufacturing tolerances. The stiffness cross-coupling coefficient caused by manufacturing tolerances. It refers to the spring stiffness; -2m1Ω z d α Y1 / dt α 2m1Ω z d α X1 / dt α -2m2Ω z d α Y2 / dt α and 2m2Ω z d α X2 / dt α This represents the Coriolis force caused by rotation. and This represents the control force along the driving and sensing directions; X1 and X2 represent the drive shafts, and Y1 and Y2 represent the sensing shafts.

[0018] Using dimensionless variables and And the scaling frequency ω0 and scaling length l0, rewritten as:

[0019]

[0020] in, ω=Ω z / ω0, It is the Caputo derivative originating from the origin;

[0021] Note 1: Compared to a single MEMS gyroscope, a decoupled dual-mass MEMS gyroscope can better eliminate the effects of linear acceleration in the sensing direction and vibrations of the same frequency from the internal resonator. If α = 1 and This fractional-order equation of motion simplifies to two integer-order equations of motion.

[0022] Introducing variables and Then the fractional-order equation of motion for the decoupled dual-mass MEMS gyroscope is defined as:

[0023]

[0024] In engineering applications, actuator failures frequently lead to decreased control performance and even accidents. The mathematical model for actuator failure, including bias faults and malfunction faults, can be written as:

[0025]

[0026] Among them, v i (t), and Let γ represent the unknown time-varying function of the control input, the bias fault, and the start time of the actuator fault, respectively. i This implies an unknown control failure rate, and satisfies 0≤γ i <1.

[0027] Furthermore, in step S2, when designing the predetermined time fuzzy backstepping controller,

[0028] Definition 1: For a differentiable function, the Caputo fractional derivative can be written as

[0029]

[0030] Among them, Named the Euler Gamma function and satisfying n-1 < α < n and

[0031] Performing a Laplace transform on equation (5) yields...

[0032]

[0033] For interval Given two continuous functions F1(t) and F2(t) in the equation, under the condition 0 < α < 1, the following equation is derived.

[0034]

[0035] Assumption 1: Desired signal x d (t) and its time derivative are continuous and exist; at the same time, |x d (t)|≤A0<k c Where A0 and k c Represents positive numbers;

[0036] Lemma 1: For Υ and any Θ > 0, there exists an inequality

[0037] 0<|Υ|-Υtanh(Υ / Θ)≤0.2785Θ (8).

[0038] Furthermore, in step S21, a β-cut type fuzzy logic system is constructed, specifically including the following steps:

[0039] 1) Acquire the input signal;

[0040] 2) Calculate the upper and lower membership functions of the quadratic membership degree β.

[0041]

[0042]

[0043] have and in, This means the input signal x i The j-th membership function at level β = 0, δ A and Represents a small positive constant and and The center is in the upper and lower parts of the standard division when β = 0;

[0044] 3) Calculate the up-down activation rule

[0045]

[0046] Where j = 1, ..., N, N represents the rule and membership function number; ξ j,β These represent the trigger strength under the upper and lower activation rules, respectively;

[0047] 4) Derive the anti-fuzzy output based on price reduction.

[0048]

[0049] make

[0050] Therefore, for any uncertain and bounded function f, the β-cut type 2 fuzzy logic system can achieve high-precision approximation in a compact set, given Ω. X

[0051]

[0052] Where ε(X) represents the positive approximation error, Let Ω w / Ω X It is the set of boundary conditions with respect to w / X; the required term w * Having the definition Furthermore, the approximation error satisfies the constraints therein. in Then, there was

[0053] Furthermore, in step S22, the controller design specifically includes: reducing the delay error equal Where e i It is the tracking error between the actual signal and the desired signal; the delay error function is designed as follows:

[0054]

[0055] Where, k b It is a positive constant that satisfies the condition 0 < k b <k c and 0 < k b <k c -A0;

[0056] From the description above, we can see and To ensure that the tracking error converges within a given time, the predetermined time function is constructed as follows:

[0057]

[0058] Where T and n r This represents the given time and associated degrees of freedom of system (3);

[0059] Using the delay error correlation function and the predetermined time function, the following barrier function is established to ensure that the system output delay constraint is strictly adhered to after a given time, as follows:

[0060]

[0061] Among them, Γ i and β A It is Γ i (e c ) and β A (t) is an abbreviation;

[0062] It can be clearly seen from equations (14) and (15) that when t = 0, |η i (0)|=|Γ(e c (0))|<1;When i When the barrier function S approaches ±1, i It tends towards infinity;

[0063] The tracking error is defined as follows:

[0064]

[0065] Where, αi This indicates the virtual control law designed subsequently;

[0066] Step 1: Consider the first candidate Lyapunov function;

[0067] Step 2: Define the weight error after transformation with the β-cut type 2 fuzzy logic system as... set up in and express and φ i An approximation of the value; using the properties defined by Caputo, it can be obtained immediately. exist Then, the second candidate Lyapunov function is selected;

[0068] In the calculation, the decoupled dual-mass MEMS gyroscope is susceptible to external disturbances, environmental changes, and manufacturing errors, all of which can lead to model uncertainty. To address these issues and improve system performance, an approximation is performed on a specified set using the β-cut 2 type fuzzy logic system described in equation (13).

[0069] A fractional-order hyperbolic tangent tracking differentiator was introduced into the controller design.

[0070] Step 3: Consider the third Lyapunov function;

[0071] Step 4: Introduce a fourth candidate Lyapunov function and use the β-cut 2 type fuzzy logic system described in equation (13) for estimation of the specified set. To solve the problem caused by... The resulting "complexity explosion" led to the use of the aforementioned fractional hyperbolic tangent tracking differentiator in the controller design.

[0072] Step 5: Introduce the fifth candidate Pnov function;

[0073] Step 6: Consider the sixth candidate Lyapunov function; to overcome the resulting "explosion of complex terms", the fractional hyperbolic tangent tracking differentiator described above is used here.

[0074] Step 7: Select the seventh candidate Lyapunov function;

[0075] Step 8: Select the eighth candidate Lyapunov function; similarly, use the β-cut 2 type fuzzy logic system proposed in Equation (13) to approximate the nonlinear function. At the same time, in order to better solve the "complexity term explosion" from the function, a fractional hyperbolic tangent tracking differentiator is used.

[0076] The beneficial effects of this invention are as follows:

[0077] 1) This invention proposes a delay error correlation function integrated with a deterministic time function so as to retain the system output within a bounded region after a finite time without violating the delay constraint.

[0078] 2) This invention proposes a fractional-order decoupled dual-mass MEMS gyroscope predetermined-time fuzzy backstepping control scheme with delay constraints and actuator faults, which makes all signals of the closed-loop system bounded, and its output achieves high-precision trajectory tracking, and achieves control objectives such as suppressing chaotic oscillations and solving actuator faults.

[0079] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0080] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0081] Figure 1 This is the overall flowchart of the fractional-order decoupled dual-mass MEMS gyroscope predetermined time fuzzy backstepping control method of the present invention;

[0082] Figure 2 A schematic diagram of a decoupled dual-mass MEMS gyroscope;

[0083] Figure 3 To simplify the motion model of the dual-mass MEMS gyroscope;

[0084] Figure 4 To decouple the bifurcation diagram of a dual-mass MEMS gyroscope and the relationship between the Lyapunov exponent and the fractional order;

[0085] Figure 5 To decouple the phase diagram and time history of the dual-mass MEMS gyroscope at α = 0.992 (a), α = 0.999 (b), and α = 1.0 (c);

[0086] Figure 6 To decouple the bifurcation diagram of a dual-mass MEMS gyroscope and the relationship between the Lyapunov exponent and the stiffness cross-coupling coefficient;

[0087] Figure 7 When α = 0.999 and Phase diagram and time history of a decoupled dual-mass MEMS gyroscope;

[0088] Figure 8 To decouple the bifurcation diagram of a dual-mass MEMS gyroscope and the relationship between the Lyapunov exponent and the fractional order;

[0089] Figure 9 When α = 0.999, m² = 1.6 × 10⁻⁶. -7 (a) m2=1.9×10 -7 (b) and m2 = 2.2 × 10 -7 (c) Phase diagram and time history of the decoupled dual-mass MEMS gyroscope;

[0090] Figure 10 To improve the trajectory tracking performance of decoupled dual-mass MEMS gyroscopes of different fractional orders;

[0091] Figure 11 To decouple the trajectory tracking error of dual-mass MEMS gyroscopes of different fractional orders;

[0092] Figure 12 The trajectory tracking error of decoupled dual-mass MEMS gyroscopes under different stiffness cross-coupling coefficients;

[0093] Figure 13 Weights for β-cut type 2 fuzzy logic systems that decouple dual-mass MEMS gyroscopes of different fractional orders;

[0094] Figure 14 β-cut type 2 fuzzy logic system for decoupling dual-mass MEMS gyroscopes with different stiffness cross-coupling coefficients and

[0095] Figure 15 For the control input of decoupled dual-mass MEMS gyroscopes of different fractional orders;

[0096] Figure 16 The trajectory tracking error of decoupled dual-mass MEMS gyroscopes of different fractional order under actuator failure;

[0097] Figure 17 The law governing the cross-coupling coefficient of a decoupled dual-mass MEMS gyroscope with different stiffnesses under actuator failure, based on the controller update rate;

[0098] Figure 18 In the event of actuator failure, the controller updates different fractional-order laws of the decoupled dual-mass MEMS gyroscope;

[0099] Figure 19 For the control input of decoupled dual-mass MEMS gyroscopes of different orders under actuator failure;

[0100] Figure 20 Performance of a fractional-order hyperbolic tangent tracking differentiator in the event of actuator failure. Detailed Implementation

[0101] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of the present invention. Unless otherwise specified, the following embodiments and features can be combined with each other.

[0102] Please see Figures 1 to 20 This invention provides a predetermined-time fuzzy backstepping control method for a fractional-order decoupled dual-mass MEMS gyroscope with delay constraints and actuator faults, such as... Figure 1 As shown, within the framework of backstepping control, a fractional-order decoupled dual-mass MEMS gyroscope predetermined-time fuzzy backstepping control method with delay constraints and actuator faults is proposed by integrating a delay error function, a β-cut type fuzzy logic system, a fractional-order hyperbolic tangent tracking differentiator, and a control law with an adaptive law for fault parameters.

[0103] 1. System Modeling and Preparation

[0104] A. Modeling of a fractional-order decoupled dual-mass MEMS gyroscope

[0105] like Figure 2 As shown, the decoupled dual-mass MEMS gyroscope consists of two equivalent MEMS gyroscopes, a coupling beam, a lever support beam, and a drive coupling suspension beam. Each MEMS gyroscope comprises an anchor, an inertial mass block, a drive suspension beam, an inductive coupling suspension beam, a feedback electrode, an inductive decoupling beam, a drive decoupling beam, a drive electrode, a drive-inductive electrode, a drive frame, an inductive electrode, and an inductive suspension beam.

[0106] The working principle is summarized as follows. When voltage is applied to the drive electrodes on both sides of the dual-mass MEMS gyroscope, the left and right inertial mass blocks oscillate in opposite directions along the drive axis. Simultaneously, the upper and lower sensing frames remain stationary along the sensing axis. The drive coupling beam, used to adjust the in-phase and out-of-phase resonant frequencies, plays a crucial role in synchronizing the motion of the two mass blocks. When Coriolis acceleration from angular velocity is generated, the left inertial mass block and sensing frame vibrate relative to the right inertial mass block and sensing frame along the sensing axis. The complex coupling system of the lever consists of the inductive coupling beam, the coupling crossbeam, and the lever support beam, which ensures the synchronization of the motion of the two inertial mass blocks by adjusting the in-phase and out-of-phase resonant frequencies in the sensing direction.

[0107] Assume the decoupled dual-mass MEMS gyroscope rotates about the z-axis with a slowly varying angular velocity, and the centrifugal force is negligible. The simplified motion model is as follows: Figure 3 As shown, Westerlund and Ekstam used extensive experimental results to confirm the inherent fractional-order characteristics of capacitors with different dielectrics. Using the Lagrange equation, the fractional-order equation of motion for decoupling a dual-mass MEMS gyroscope can be written as follows:

[0108]

[0109] Where α and λ z These represent the fractional order and angular velocity on the z-axis, respectively. m1 and m2 represent the effective mass for mass verification in drive and sensing modes, respectively. and These are the damping coefficients along the driving and sensing directions, respectively. and These are the spring stiffnesses in the driving and sensing directions, respectively. and It is the damping cross-coupling coefficient caused by manufacturing tolerances. The stiffness cross-coupling coefficient caused by manufacturing tolerances. It refers to the spring stiffness; -2m1Ω z d α Y1 / dt α 2m1Ω z d α X1 / dt α -2m2Ω z d α Y2 / dt α and 2m2Ω z d α X2 / dt α This represents the Coriolis force caused by rotation. and This indicates the control force along the driving and sensing directions; X1 and X2 represent the driving shafts, and Y1 and Y2 represent the sensing shafts.

[0110] Using dimensionless variables and And the scaling frequency ω0 and scaling length l0, rewritten as:

[0111]

[0112] in, ω=Ω z / ω0, It is the Caputo derivative originating from the origin.

[0113] Note 1: Compared to a single MEMS gyroscope, a decoupled dual-mass MEMS gyroscope can better eliminate the effects of linear acceleration in the sensing direction and vibrations of the same frequency from the internal resonator. If α = 1 and This fractional-order equation of motion simplifies to two integer-order equations of motion.

[0114] Introducing variables and Then the fractional-order equation of motion for the decoupled dual-mass MEMS gyroscope is defined as:

[0115]

[0116] In engineering applications, actuator failures frequently lead to decreased control performance and even accidents. The mathematical model for actuator failure, including bias faults and malfunction faults, can be written as:

[0117]

[0118] in, and Let γ represent the unknown time-varying function of the control input, the bias fault, and the start time of the actuator fault, respectively. i This implies an unknown control failure rate, and satisfies 0≤γ i <1.

[0119] B. Preparations

[0120] Definition 1: For a differentiable function, the Caputo fractional derivative can be written as

[0121]

[0122] Among them, Named the Euler Gamma function and satisfying n-1 < α < n and

[0123] Performing a Laplace transform on equation (5) yields...

[0124]

[0125] For interval Given two continuous functions F1(t) and F2(t) in the equation, under the condition 0 < α < 1, the following equation is derived.

[0126]

[0127] Assumption 1: Desired signal x d (t) and its time derivative are continuous and exist; at the same time, |x d (t)|≤A0<kc Where A0 and k c Represents positive numbers;

[0128] Lemma 1: For Υ and any Θ > 0, there exists an inequality

[0129] 0<|Υ|-Υtanh(Υ / Θ)≤0.2785Θ (8)

[0130] 2. Dynamics Analysis and Problem Statement

[0131] To fully reveal the evolution of the system's dynamic characteristics and facilitate subsequent design, a large amount of dynamic analysis was performed on the dual-mass MEMS gyroscope.

[0132] The system parameters for decoupling the dual-mass MEMS gyroscope are defined as ω0 = 1.5 × 10⁻⁶. 3 Hz, l0=1μm, Ω z =200 rad / s, m1 = 1.8 × 10 -7 kg, m2 = 1.7 × 10 -7 kg, and All initial values ​​for the system state are set to 0.01.

[0133] Figure 4 The bifurcation diagram and the relationship between the Lyapunov exponent and fractional order are shown for the decoupled dual-mass MEMS gyroscope in the interval [0.992 1]. When all LEs are less than 0, the decoupled dual-mass MEMS gyroscope operates in a periodic state, which is... Figure 5 This was verified in subgraph (a). When 0.998 < α < 0.9998, this MEMS gyroscope can generate chaotic oscillations. Figure 5 Subgraph (b) reflects this oscillation. With further increases in fractional order, the decoupled dual-mass MEMS gyroscope exhibits hyperchaotic phenomena, such as... Figure 5 The subgraph (c) is shown.

[0134] Due to manufacturing errors, there is a coupling elastic stiffness between the left and right inertial masses. This elastic stiffness can affect the dynamic characteristics of the system. Figure 6 The bifurcation diagram and the relationship between the Lyapunov exponent and the stiffness cross-coupling coefficient are shown for a decoupled dual-mass MEMS gyroscope in the range of [80 100]. Clearly, the decoupled dual-mass MEMS gyroscope falls into chaotic oscillations throughout the process. Figure 7 The phase diagram and time history diagram further reveal the unpredictable, random, and disordered motion characteristics of the above system.

[0135] Due to the influence of internal and external environments, the mass of a MEMS gyroscope changes over time. It is necessary to study its impact on the dynamic behavior of a decoupled dual-mass MEMS gyroscope. Figures 8-9 It can be seen that for decoupled dual-mass MEMS gyroscopes, mass plays an important role in the formation of chaotic oscillations.

[0136] 3. Design of a fuzzy backstepping controller for a predetermined time

[0137] A. β-cut type 2 fuzzy logic system

[0138] Type-2 fuzzy logic systems offer significant advantages over Type-1 fuzzy logic systems in learning, fault detection, and function approximation. However, their feasibility in engineering applications is limited by substantial computational costs. To address this issue, a β-cut Type-2 fuzzy logic system is employed, where the value of β-cut is adjusted in real-time by an adaptive law. Its working principle is as follows:

[0139] 1) Acquire the input signal;

[0140] 2) Calculate the upper and lower membership functions of the quadratic membership degree β.

[0141]

[0142]

[0143] have and in, This means the input signal x i The j-th membership function at level β = 0, δ A and Represents a small positive constant and and The center is in the upper and lower parts of the standard division when β = 0;

[0144] 3) Calculate the up-down activation rule

[0145]

[0146] Where j = 1,,N, and N represents the rule and membership function number; ξ j,β These represent the trigger strength under the upper and lower activation rules, respectively;

[0147] 4) Derive the anti-fuzzy output based on price reduction.

[0148]

[0149] make

[0150] Therefore, for any uncertain and bounded function f, the β-cut type 2 fuzzy logic system can achieve high-precision approximation in a compact set, given Ω. X

[0151]

[0152] Where ε(X) represents the positive approximation error, Let Ω w / Ω X It is the set of boundary conditions with respect to w / X; the required term w * Having the definition Furthermore, the approximation error satisfies the constraints therein. in Then, there was

[0153] B. Controller Design

[0154] Cause delay error equal Where e i It is the tracking error between the actual signal and the desired signal; the delay error function is designed as follows:

[0155]

[0156] Where, k b It is a positive constant that satisfies the condition 0 < k b <k c and 0 < k b <k c -A0;

[0157] From the description above, we can see and

[0158] To ensure that the tracking error converges within a given time, the predetermined time function is constructed as follows:

[0159]

[0160] Where T and n r The given time and associated degrees of freedom of system (3) are represented.

[0161] Using the delay error correlation function and the predetermined time function, the following barrier function is established to ensure that the system output delay constraint is strictly adhered to after a given time, as follows:

[0162]

[0163] Among them, Γ i and β A It is Γ i (e c ) and β A (t) is an abbreviation.

[0164] It can be clearly seen from equations (14) and (15) that when t = 0, |η i (0)|=|Γ(e c (0))|<1;When η i When the barrier function S approaches ±1, i It tends towards infinity.

[0165] The tracking error is defined as follows:

[0166]

[0167] Where, α i This indicates the virtual control law designed subsequently.

[0168] Step 1: Consider the first candidate Lyapunov function

[0169]

[0170] The fractional derivative of formula (27) can be derived as follows:

[0171]

[0172] in, r1=ρ1μ1β A and

[0173] Then the virtual control law is designed as follows:

[0174]

[0175] Where k1 is a positive constant;

[0176] Substituting equation (20) into equation (19), we can obtain

[0177]

[0178] Step 2: Define the weight error after transformation with the β-cut type 2 fuzzy logic system as... set up in and express and φ i An approximation of the value; using the properties defined by Caputo, it can be obtained immediately. exist Then, select the second candidate Lyapunov function.

[0179]

[0180] The fractional derivative of V2 is calculated as follows:

[0181]

[0182] in,

[0183] In computation, decoupled dual-mass MEMS gyroscopes are susceptible to external disturbances, environmental changes, and manufacturing errors, all of which contribute to model uncertainty. To address these issues and improve system performance, an approximation is performed on a specified set using the β-cut 2 type fuzzy logic system described in equation (13), i.e.

[0184]

[0185] In this context, (·) is an abbreviation for (x1,x2,x3,x4,x5,x6,x7,x8).

[0186] To reduce computational burden and work with maximum efficiency, the formula is converted to...

[0187]

[0188] Where ζ2=||w2|| 2 This represents the transformed weights and b2 > 0.

[0189] Note 2: From equations (19) and (20), it is impractical and cumbersome to directly obtain the fractional derivative of the virtual control due to delay error correlation, barriers, and predetermined time functions. Direct use of the virtual control law will inevitably lead to the traditional "complexity term explosion". To solve this problem, a fractional hyperbolic tangent tracking differentiator is introduced into the controller design.

[0190] The fractional hyperbolic tangent tracking differentiator can be written as

[0191]

[0192] in, This represents the output of the hyperbolic tangent tracking differentiator. It is the input of the hyperbolic tangent tracking-differentiator, parameters λ i and λ i+1 Used to adjust tracking speed and tracking performance;

[0193] According to equations (24), (25) and Young's inequality, equation (23) simplifies to

[0194]

[0195] The control input with update rate can be expressed as

[0196]

[0197]

[0198]

[0199]

[0200]

[0201]

[0202] Where, k2,Θ2, c2 and l2 represent positive constants.

[0203] The following situations exist

[0204]

[0205]

[0206] Using Lemma 1 and equations (28) to (33), (27) can be derived as follows:

[0207]

[0208] Step 3: Consider the third Lyapunov function

[0209]

[0210] Calculation of the fractional derivative V3 with respect to time

[0211]

[0212] Where r3=ρ3μ3β A , and

[0213] Virtual control law design is

[0214]

[0215] Here, k3 represents a positive constant.

[0216] Substituting equation (39) into equation (38), we can obtain

[0217]

[0218] Step 4: Introduce the fourth candidate Lyapunov function

[0219]

[0220] The fractional derivative of V4 can be obtained directly.

[0221]

[0222] in,

[0223] In real-world environments, decoupled dual-mass MEMS gyroscopes can be affected by external disturbances, environmental changes, and manufacturing errors. This undoubtedly leads to system uncertainty. To address this issue and improve system performance, the β-cut 2 type fuzzy logic system described in equation (13) is used to estimate f4 in a specified set, i.e.

[0224]

[0225] To improve computational efficiency, the corresponding formulas are converted and rewritten as follows:

[0226]

[0227] Where, ζ4=||w4|| 2 It is the transformed weight and b4 > 0.

[0228] In order to solve the problem The resulting "explosion of complex terms" led to the use of the aforementioned fractional hyperbolic tangent tracking differentiator in the controller design. Equation (42) then simplifies to...

[0229]

[0230] A control input with an update law was designed.

[0231]

[0232]

[0233]

[0234]

[0235]

[0236]

[0237] Where, k4,Θ4, c4 and l4 are positive constants.

[0238] Clearly, the following conditions hold true.

[0239]

[0240]

[0241] Using equations (46) to (51), equation (45) can be further calculated.

[0242]

[0243] Step 5: Introduce the fifth candidate Pnov function

[0244]

[0245] Taking the fractional derivative of V5, there exists

[0246]

[0247] Where r5=ρ5μ5β A , and

[0248] Virtual control law design is

[0249]

[0250] Here, k5 represents a positive constant.

[0251] Then equation (56) simplifies to

[0252]

[0253] Step 6: Consider the sixth candidate Lyapunov function

[0254]

[0255] The fractional derivative of V6 can be derived

[0256]

[0257] in,

[0258] Based on the preceding description, the β-cut 2 type fuzzy logic system given in equation (13) is used to estimate the nonlinear function, i.e.

[0259]

[0260] To improve calculation speed, the corresponding formulas were converted to...

[0261]

[0262] Where ζ6=||w6|| 2 This represents the transformed weights, and b6 > 0;

[0263] To overcome the resulting "explosion of complex terms," ​​the aforementioned fractional hyperbolic tangent tracking differentiator is used here. Equation (60) can then be further written as...

[0264]

[0265] The control input with update rate can be designed as

[0266]

[0267]

[0268]

[0269]

[0270]

[0271]

[0272] Where, k6,Θ6, c6 and l6 represent positive constants.

[0273] The following conditions are met

[0274]

[0275]

[0276] Substituting equations (64) to (69) into equation (63) yields

[0277]

[0278] Step 7: Select the seventh candidate Lyapunov function

[0279]

[0280] The fractional derivative of V7 is

[0281]

[0282] Where r7=ρ7μ7β A , and The virtual control law is designed to be

[0283]

[0284] Here, k7 is a positive constant.

[0285] Equation (74) rewritten as

[0286]

[0287] Step 8: Select the eighth candidate Lyapunov function

[0288]

[0289] Taking the derivative of the above equation, we have

[0290]

[0291] in, Similarly, the β-cut 2 type fuzzy logic system proposed in equation (13) is used to approximate the nonlinear function, i.e.

[0292]

[0293] To speed up the calculation process, the formula is transformed as follows:

[0294]

[0295] Where, ζ8=||w8|| 2 This represents the transformed weights and b8 > 0.

[0296] Meanwhile, to better address the "explosion of complex terms" from the equation, a fractional hyperbolic tangent tracking differentiator is used; then equation (78) can be rewritten as

[0297]

[0298] The control input with the update law is as follows:

[0299]

[0300]

[0301]

[0302]

[0303]

[0304]

[0305] Among them, k8,Θ8, c8 and l8 represent positive constants.

[0306] exist

[0307]

[0308]

[0309] Substituting equations (82) to (87) into equation (81), we get

[0310]

[0311] 4. Stability Analysis

[0312] Theorem 1: For the fractional-order decoupled dual-mass MEMS gyroscope predetermined-time fuzzy backstepping control problem with delay constraints and actuator failure under Assumption 1, control inputs with adaptive laws are designed as (28), (46), (64), and (82). Therefore, by reasonably selecting the design parameters, the following conclusions are drawn:

[0313] 1) All signals of a closed-loop fractional-order decoupled dual-mass MEMS gyroscope are always bounded.

[0314] 2) The tracking error can converge to a very small area after a specified time, thereby achieving the control objective.

[0315] Evidence: Select the entire Lyapunov function

[0316]

[0317] By using Young's inequality and Then we obtain the derivative of (91).

[0318]

[0319] in, and

[0320]

[0321] Taking the fractional integral of (92), we have

[0322]

[0323] The proof is complete.

[0324] 5. Simulation Experiment Results and Analysis

[0325] In this experiment, numerous results validated the feasibility and effectiveness of the proposed scheme. The required signal selection is... and The parameter of the function that relies on the delay error is set to k.b =0.4 and k c =2 and A0=1. The parameters of the fractional hyperbolic tangent tracking differentiator are selected as follows: λ i =32, i=1,3,5,7 and λ j =14, j=2,4,6,8. The parameters of the predetermined time fuzzy backstepping controller are selected as k. i =10, i=1,,8, b i =0.01, i=2,4,6,8, l2=l6=60, l4=l8=150, c2=c6=20, c4=c8=30 and Θ i =2, i=2,4,6,8. The parameter for actuator failure can be defined as γ. i =0.8 and The actuator failure time is between the 4th and 6th second. Furthermore, for the β-cut 2 type fuzzy logic system, δ A =0.8, β=0.5, Furthermore, its center is uniformly distributed within the interval [-5 5].

[0326] A. Normal operation without actuator malfunction.

[0327] Under normal operating conditions, decoupled dual-mass MEMS gyroscopes do not exhibit actuator failure. Therefore, Figures 10-12 The trajectory tracking performance and corresponding tracking error are shown. It is evident that each state variable curve can overlap with the desired signal throughout the process. The corresponding tracking error converges to a small interval within a defined time. Compared to the results of the dynamic analysis, the proposed scheme completely suppresses chaotic motion with disordered, random, and high-frequency oscillatory characteristics within a predetermined time.

[0328] β-cut type 2 fuzzy logic systems play an important role in handling unknown functions and parameter disturbances. Figures 13-14 The switching weights and matrix norms of the rule-excited β-cut type 2 fuzzy logic system are demonstrated. The resulting fuzzy logic system can achieve the predetermined objectives. Furthermore, the proposed scheme exhibits good robustness regardless of variations in order, stiffness, and cross-coupling coefficients. The control input directly determines the overall control performance. Figure 15 As can be seen, the four control inputs reach steady state in a very short time. Despite fractional-order changes, the control inputs remain stable.

[0329] B. Operation in case of actuator failure

[0330] In practical applications, increasingly complex industrial processes can lead to unexpected equipment failures. Ignoring this issue can result in potential failures that typically degrade the performance of decoupled dual-mass MEMS gyroscopes. Figure 16 This shows the trajectory tracking errors of a decoupled dual-mass MEMS gyroscope at different fractional orders under actuator failure. As can be seen, jitter only occurs at the beginning and end of the failure interval; at other times, all trajectory tracking errors remain stable within a very small range.

[0331] The update rate has a greater impact on the control performance of actuator failures (including deviations and failures). Figures 17-18 The update laws of decoupled dual-mass MEMS gyroscope controllers with different stiffness cross-coupling coefficients and fractional orders are shown. It can be seen that their amplitudes increase significantly during the fault interval. Regardless of the stiffness cross-coupling coefficient or fractional order, the three curves overlap. Figure 19 The control inputs of decoupled dual-mass MEMS gyroscopes with different fractional orders are shown. This is consistent with... Figures 17-18 The update rate differs, and the control input only changes at the 4th and 6th seconds. At other times, the results with actuator failure are the same as those without.

[0332] Figure 20 The approximate performance of the fractional hyperbolic tangent pursuing differentiator under actuator failure is demonstrated, and related approximate results are obtained. Clearly, the fractional hyperbolic tangent pursuing differentiator effectively addresses the "complexity term explosion" problem in the traditional backstepping framework.

[0333] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A fractional order decoupled dual-mass MEMS gyroscope scheduled time fuzzy backstepping control method with delay constraint and actuator failure, characterized in that, The method specifically includes the following steps: S1: Based on the Lagrange equation, establish the fractional-order motion equation of a decoupled dual-mass MEMS gyroscope with actuator failure to eliminate the influence of linear acceleration and vibration at the same frequency. S2: Design a predetermined time fuzzy backstepping controller, specifically including the following steps: S21: construct β a type-2 fuzzy logic system to address the uncertainty and non-linear terms of the system model; S22: Controller design: superimpose a delay error function on the specified time function to ensure that the error does not violate the constraint condition after a limited time; and use a fractional order hyperbolic tangent tracking differentiator in the controller design to handle direct fractional order derivatives and repeated derivatives; finally combine β a type-2 fuzzy logic system to establish a control law with a fault parameter adaptive law to solve the actuator fault problem containing bias faults and failure faults.

2. The fractional-order decoupled dual-mass MEMS gyroscope pre-defined time blurring backstepping control method according to claim 1, characterized in that, In step S1, the decoupled dual-mass MEMS gyroscope includes two equivalent MEMS gyroscopes, a coupling beam, a lever support beam, and a drive coupling suspension beam; each MEMS gyroscope includes an anchor, an inertial mass block, a drive suspension beam, an inductive coupling suspension beam, a feedback electrode, an inductive decoupling beam, a drive decoupling beam, a drive electrode, a drive-inductive electrode, a drive frame, an inductive electrode, and an inductive suspension beam. When a voltage is applied to the drive electrodes on both sides of the dual-mass MEMS gyroscope, the left and right inertial mass blocks oscillate in opposite directions on the drive axis; at the same time, the upper and lower sensing frames are stationary along the sensing axis; the drive coupling beam is used to adjust the in-phase and out-of-phase resonant frequencies; when Coriolis acceleration from angular velocity is generated, the left inertial mass block and sensing frame vibrate along the sensing axis relative to the right inertial mass block and sensing frame. The complex coupling system of the lever consists of an inductive coupling cantilever beam, a coupling crossbeam, and a lever support beam. The synchronous motion of the two inertial mass blocks is ensured by adjusting the in-phase and out-of-phase resonant frequencies in the inductive direction.

3. The fractional-order decoupled dual-mass MEMS gyroscope pre-defined time blurring backstepping control method according to claim 1, wherein, In step S1, the fractional-order equation of motion for a decoupled dual-mass MEMS gyroscope with actuator failure is established. Specifically, it is assumed that the decoupled dual-mass MEMS gyroscope rotates around the z-axis with a slowly varying angular velocity, and the centrifugal force is negligible, where the z-axis is the direction axis perpendicular to the plane containing the sensing axis and the driving axis. Using the Lagrange equation, the fractional-order equation of motion for the decoupled dual-mass MEMS gyroscope is written as follows: （1） in, and These are the fractional order and angular velocity on the z-axis, respectively. and Indicates the effective quality for quality verification under both drive and sensing modes. , , and These are the damping coefficients along the driving and sensing directions, respectively. , , , , and These are the spring stiffnesses in the driving and sensing directions, respectively. and It is the damping cross-coupling coefficient caused by manufacturing tolerances. The stiffness cross-coupling coefficient caused by manufacturing tolerances. It refers to the spring stiffness; , , and This represents the Coriolis force caused by rotation. , , and This indicates the control force along the driving and sensing directions; and Indicates the drive shaft. , Indicates the induction axis; Using dimensionless variables , , , and and scaling frequency and scaling length Rewritten as: （2） in, , , , , , , , , , , , , , , , , , , , , , It is the Caputo derivative originating from the origin; if and This fractional-order equation of motion simplifies to two integer-order equations of motion; Introducing variables , , , , , , and Then, the fractional-order equation of motion for the decoupled dual-mass MEMS gyroscope is defined as: （3） The mathematical model for actuator failure, including bias faults and failure faults, is written as: （4） in, , and These represent the unknown time-varying functions of the control input, bias fault, and actuator fault, respectively. This implies an unknown control failure rate, and satisfies... .

4. The fractional-order decoupled dual-mass MEMS gyroscope predetermined-time fuzzy backstepping control method according to claim 3, characterized in that, In step S2, when designing the predetermined time fuzzy backstepping controller, Definition 1: For a differentiable function, the Caputo fractional derivative is written as （5） Among them, Named the Euler Gamma function and satisfies and ; Performing a Laplace transform on equation (5) yields... （6） For interval Two continuous functions in and ,exist Under the given conditions, the following equation is derived. （7） Assumption 1: Desired signal Its time derivative is continuous and exists; at the same time, ,in and Represents positive numbers; Lemma 1: For and any For example, there exists an inequality （8）。 5. The fractional-order decoupled dual-mass MEMS gyroscope predetermined-time fuzzy backstepping control method according to claim 4, characterized in that, In step S21, construct β -The type 2 fuzzy logic system includes the following steps: 1) Acquire the input signal; 2) Calculate the upper and lower membership functions of the quadratic membership degree β. （9） （10） have and ,in, This means the input signal At level The first Membership functions, and Represents a small positive constant and , and The center is The upper and lower parts of standard division; 3) Calculate the up-down activation rule （11） in, , Representation rules and membership function numbers; , These represent the trigger strength under the upper and lower activation rules, respectively; 4) Derive the anti-fuzzy output based on price reduction. （12） make , , Therefore, for any uncertain and bounded function , β -cut type 2 fuzzy logic systems can achieve high-precision approximation in compact sets, setting （13） in, Indicates positive approximation error. ;set up / It is about / The set of boundary conditions; required items Having the definition Furthermore, the approximation error satisfies the constraints therein. in Then, there was .

6. The fractional-order decoupled dual-mass MEMS gyroscope predetermined-time fuzzy backstepping control method according to claim 5, characterized in that, In step S22, the controller design specifically includes: minimizing the delay error. equal ,in It is the tracking error between the actual signal and the desired signal; the delay error function is designed as follows: ， （14） in, It is a positive constant that satisfies the condition, such that and ; The predetermined time function is constructed as follows （15） in, and Represents the given time and associated degrees of freedom of system (3); Using the delay error correlation function and the predetermined time function, the following barrier function is established to ensure that the system output delay constraint is strictly adhered to after a given time, as follows: ， （16） in, and yes and Abbreviation; It is clear from equations (14) and (15) that when hour, ;when When close to ±1, the barrier function It tends towards infinity; The tracking error is defined as follows: （17） in, This indicates the virtual control law designed subsequently; Step 1: Consider the first candidate Lyapunov function （18） The fractional derivative of formula (18) is derived as follows: （19） in, , , and ; Then the virtual control law is designed as follows: （20） in, It is a positive constant; Substituting equation (20) into equation (19), we get （21） Step 2: Define the weight error after transformation with the β-cut type 2 fuzzy logic system as... ;set up , ,in and express and An approximation of the value; using the properties defined by Caputo, we obtain... , , , ;exist Then, select the second candidate Lyapunov function. （22） The fractional derivative is calculated as follows: （23） in, ; Using the β-cut 2 type fuzzy logic system described in equation (13) to approximate a specified set, i.e. （24） in, yes Abbreviation; To reduce computational burden and work with maximum efficiency, the formula is converted to... （25） in, Represents the transformed weights and ; A fractional hyperbolic tangent tracking differentiator was introduced into the controller design; The fractional hyperbolic tangent tracking differentiator is written as follows （26） in, This represents the output of the hyperbolic tangent tracking differentiator. It is the input of the hyperbolic tangent tracking-differentiator, parameters , and Used to adjust tracking speed and tracking performance; According to equations (24), (25) and Young's inequality, equation (23) simplifies to （27） The control input with update rate is expressed as: ,（28） ,（29） ,（30） ,（31） ,（32） ,（33） in, , , , and Represents positive numbers; The following situations exist （34） （35） Using Lemma 1 and equations (28)~(33), (27) are derived as follows: （36） Step 3: Consider the third Lyapunov function （37） Fractional derivative with respect to time Calculation （38） in, , , and ; Virtual control law design is （39） in, Represent a positive integer; Substituting equation (39) into equation (38), we get （40） Step 4: Introduce the fourth candidate Lyapunov function （41） The fractional derivative is obtained directly. （42） in, ; Using the β-cut type fuzzy logic system described in equation (13) for specified set estimation ,Right now （43） To improve computational efficiency, the corresponding formulas are converted and rewritten as follows: （44） in, It is the transformed weight and ; In order to solve the problem The resulting "explosion of complex terms" led to the use of the aforementioned fractional hyperbolic tangent tracking differentiator in the controller design; thus, equation (42) simplifies to （45） A control input with an update law was designed. ,（46） ,（47） ,（48） ,（49） ,（50） ,（51） in, , , , and It is a positive number; Clearly, the following conditions hold true. （52） （53） Using equations (46) to (51), we can further calculate equation (45). （54） Step 5: Introduce the fifth candidate Lyapunov function （55） Pick The fractional derivative exists （56） in, , , and ; Virtual control law design is （57） in, Represent a positive integer; Then equation (56) simplifies to （58） Step 6: Consider the sixth candidate Lyapunov function （59） The fractional derivative is derived （60） in, ; Based on the preceding description, the β-cut 2 type fuzzy logic system given in equation (13) is used to estimate the nonlinear function, i.e. （61） To improve calculation speed, the corresponding formulas were converted to... （62） in, Indicates the transformed weights and ; To overcome the resulting "explosion of complex terms", the aforementioned fractional hyperbolic tangent tracking differentiator is used here; then equation (60) is further written as （63） The control input with update rate can be designed as ,（64） ,（65） ,（66） ,（67） ,（68） ,（69） in, , , , and Represents positive numbers; The following conditions are met （70） （71） Substituting equations (64) to (69) into equation (63) yields （72） Step 7: Select the seventh candidate Lyapunov function （73） The fractional derivative is （74） in, , , and ; The virtual control law is designed to be （75） in, It is a positive constant; Equation (74) rewritten as （76） Step 8: Select the eighth candidate Lyapunov function （77） Taking the derivative of the above equation, we have （78） in, ; Similarly, the β-cut 2 type fuzzy logic system proposed in equation (13) is used to approximate the nonlinear function, i.e. （79） To speed up the calculation process, the formula is transformed as follows: （80） in, Indicates the transformed weights and ; Meanwhile, to better address the "explosion of complex terms" from the equation, a fractional hyperbolic tangent tracking differentiator is used; then equation (78) is rewritten as （81） The control input with the update law is as follows: ,（82） ,（83） ,（84） ,（85） ,（86） ,（87） in, , , , and Represents positive numbers; exist （88） （89） Substituting equations (82) to (87) into equation (81), we get （90）。

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