Event-triggered neural network backstepping control method for dual-Duffing MEMS gyroscope with state constraint

Abstract

The invention relates to an event-triggered neural network backstepping control method for a double-Duffing MEMS gyroscope with state constraint, and belongs to the technical field of MEMS gyroscope control. The method comprises the following steps: carrying out mathematical modeling on the dual-Duffing MEMS gyroscope, and introducing a digital signal processing (DSP) method to carry out dynamic analysis on the dual-Duffing MEMS gyroscope; in an event-triggered neural network backstepping controller, a tracking error, an acceleration function and an acceleration error of the event-triggered neural network backstepping controller are defined; designing a plurality of integral obstacle Lyapunov candidate functions, the control input of a controller and the adaptive rate according to the tracking error, the acceleration function and the acceleration error, estimating a nonlinear unknown function through a type 2 sequence fuzzy neural network, and solving the problem of virtual control rate differential item explosion through an acceleration second-order tracking differentiator; and establishing an event triggering mechanism based on a switching threshold strategy to assist the controller in performing event triggering neural network backstepping control. According to the method, the problems of chaotic oscillation, state constraint, uncertain interference, communication burden and the like in the system are solved.

Description

Technical Field

[0001] The invention belongs to the technical field of MEMS gyroscope control, and relates to an event-triggered neural network backstepping control method of a double Duffing type MEMS gyroscope with state constraints. Background Art

[0002] As one of the key components used in specific applications such as drones, robots, and aerospace, MEMS gyroscopes play a vital role in achieving precise navigation and attitude control. Compared with optical gyroscopes and mechanical rotor gyroscopes, their accuracy and environmental adaptability still have obvious defects, and there are harmful chaotic oscillations in the system. In addition, temperature changes, magnetic field interference, and noise put forward higher requirements on their anti-disturbance performance and robustness. To solve these problems, the performance of MEMS gyroscopes can be improved through two different approaches. The first approach focuses on the hardware level and designs gyroscopes with larger bandwidth and higher mechanical sensitivity through multi-mass coupling. The second approach solves the problems at the software level by developing high-performance controllers using various algorithms to improve safety, robustness, and tracking accuracy. However, on the one hand, the current research on the dynamics of multi-mass coupling structures is mostly limited to certain simulations and lacks connection with actual engineering applications. On the other hand, for these sensitive microstructures, there is an urgent need to balance the reliability, accuracy, and complexity of the control algorithm.

[0003] In order to accurately describe the dynamic behavior of the double Duffing type MEMS gyroscope, mathematical modeling based on the physical structure is required. Wu et al. designed a dual-mass gyroscope and optimized its mechanical structure for the existing mechanical orthogonal error. Li et al. proposed a dimensionless model of the dual-mass MEMS gyroscope, preliminarily studied the chaotic phenomenon, and designed a controller. However, these studies did not mention or explore in depth the external interference and chaotic oscillation. The existence of these problems will seriously affect the stability and safety of the gyroscope. In addition, the study of these dynamic characteristics is limited to computer simulation and has not yet been implemented on an embedded platform.

[0004] Dynamic analysis based on embedded platforms is a core component of MEMS gyroscope research. Given the high requirements for accuracy and real-time processing, DSP is considered to be an ideal choice for performing this critical task. At present, the dynamic characteristics analysis of various DSP systems has become a research hotspot. Tayyab et al. implemented a dynamic analysis of a new four-dimensional storage resistor system based on the DSP platform. Liu et al. constructed a chaotic map using memristors and completed the experiment using DSP. Li et al. proposed an image encryption system by improving the algorithm, which was implemented by DSP and comprehensively analyzed. Leng et al. implemented a new fractional-order conservative system using DSP, focusing on its hyperchaotic phenomenon and the existence of dissipation. Ma et al. analyzed the nonlinear characteristics of a neural network model on DSP.

[0005] Considering the limitations of the communication bandwidth of the above-mentioned embedded systems and the service life of the controller's peripheral circuit components under extreme conditions, it is necessary to develop an event trigger mechanism to improve communication efficiency and ensure the reliability of the controller design. Liu et al. introduced fixed threshold event triggering and relative threshold event triggering into the control of the electromagnetic active suspension system, solving the problem of limited vehicle communication resources. The fixed threshold event triggering method has low design complexity, but it cannot guarantee high-precision tracking. The relative threshold event triggering can guarantee control performance, but when the system state fluctuates greatly, the control input may jump due to the increase in threshold and the accumulated difference. Therefore, a strategy to balance system stability and control accuracy is urgently needed.

[0006] The integration of various algorithms based on the inversion framework has become the mainstream method for designing control schemes. MEMS gyroscopes are extremely susceptible to external shocks, which can cause the amplitude peak of the mass block to exceed its physical structural limit, resulting in system damage. Therefore, in the design of the controller, it is crucial to constrain its position. Among them, the application of the barrier Lyapunov function can effectively prevent the system operation from violating the constraint boundary, thereby improving the safety of the system under complex working conditions. The use of tan-type barrier Lyapunov function and log-type barrier Lyapunov function to establish system constraints indirectly ensures that the system is within the constraint range by limiting the size of the tracking error. When the state variable approaches the constraint boundary, the controller will immediately react and produce a sharply increased control input, which will inevitably have a certain impact on the system. Therefore, the integral barrier Lyapunov function is developed to enhance the system stability while providing smoother state constraint processing. Therefore, it is a meaningful topic to apply the integral barrier Lyapunov function to the control of the double Duffing type MEMS gyroscope. Summary of the invention

[0007] In view of this, an object of the present invention is to provide an event-triggered neural network backstepping control method for a double Duffing type MEMS gyroscope with state constraints.

[0008] In order to achieve the above object, the present invention provides the following technical solutions:

[0009] An event-triggered neural network backstepping control method for a double Duffing type MEMS gyroscope with state constraints comprises the following steps:

[0010] S1. Design a double Duffing MEMS gyroscope and perform mathematical modeling on it.

[0011] S2, introduce digital signal processing DSP method to perform dynamic analysis on double Duffing type MEMS gyroscope;

[0012] S3, defining the tracking error, acceleration function and acceleration error of the double Duffing type MEMS gyroscope in the event-triggered neural network backstepping controller;

[0013] S4. According to the tracking error, acceleration function and acceleration error, several integral barrier Lyapunov candidate functions, control input and adaptive rate of the controller are designed, and the nonlinear unknown function is estimated by type 2 sequential fuzzy neural network, and the explosion problem of virtual control rate differential term is solved by accelerating the second-order tracking differentiator;

[0014] S5. Establish an event triggering mechanism based on the switching threshold strategy to assist the controller in performing event-triggered neural network backstepping control.

[0015] The beneficial effects of the present invention are:

[0016] First, the research in the existing literature is limited to simulation calculation, completing the numerical calculation of the model, while ignoring the functional definition of various pins and registers, data storage and retrieval process, and communication protocols between devices, which are all necessary for the application of embedded platforms in engineering. In contrast, the DSP in the present invention solves these deficiencies and lays a theoretical foundation for the real-time control of MEMS gyroscopes. At the same time, it better solves the problems of time-consuming and poor anti-interference performance in circuit design.

[0017] Secondly, compared with the existing literature, the present invention not only avoids the signal jump caused by the relative threshold event trigger when the control input of the double Duffing MEMS gyroscope is too large, but also eliminates the problems of low accuracy and control input jitter of the fixed threshold event trigger in the steady state.

[0018] Finally, the integral barrier Lyapunov function designed in the present invention limits the maximum displacement of the mass block within the physical limit while ensuring the tracking accuracy, providing a smoother signal transition near the boundary. In addition, the proposed event-triggered neural network backstepping control scheme can not only effectively suppress chaotic oscillations, but also improve safety and stability, and save communication resources to a certain extent.

[0019] Other advantages, objectives and features of the present invention will be described in the following description to some extent, and to some extent, will be obvious to those skilled in the art based on the following examination and study, or can be taught from the practice of the present invention. The objectives and other advantages of the present invention can be realized and obtained through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below in conjunction with the accompanying drawings, wherein:

[0021] Figure 1 It is an overall framework diagram of the event-triggered neural network backstepping control method of the double Duffing type MEMS gyroscope with state constraints of the present invention;

[0022] Figure 2 It is a schematic structural diagram of a double Duffing type MEMS gyroscope of the present invention;

[0023] Figure 3 It is a schematic diagram of the principle of the double Duffing type MEMS gyroscope of the present invention;

[0024] Figure 4 is the phase diagram and time diagram of the double Duffing MEMS gyroscope under computer simulation, where, Figure 4 (a), 4(c), 4(e) and 4(g) are phase diagrams under different parameters. Figure 4 (b), 4(d), 4(f), and 4(h) are the corresponding time course diagrams;

[0025] Figure 5 Lyapunov index diagram of double Duffing type MEMS gyroscope;

[0026] Figure 6 is the Lyapunov index diagram of double Duffing MEMS gyroscope with different coefficients, where: Figure 6 (a) is the Lyapunov index diagram of the dual Duffing MEMS gyroscope in driving mode. Figure 6 (b) Lyapunov index diagram of the dual Duffing MEMS gyroscope in sensing mode;

[0027] Figure 7 The phase diagram and time diagram of the double Duffing MEMS gyroscope based on the DSP platform, where: Figure 7 (a) is the phase diagram of channel 1 and channel 2; Figure 7 (b) is the time course diagram of channel 1 and channel 2; Figure 7 (c) is the phase diagram of channel 3 and channel 4; Figure 7 (d) is the time course diagram of channel 3 and channel 4; Figure 7 (e) is the phase diagram of channel 5 and channel 6; Figure 7 (f) is the time course diagram of channel 5 and channel 6; Figure 7 (g) is the phase diagram of channel 7 and channel 8; Figure 7 (h) is the time course diagram of channel 7 and channel 8;

[0028] Figure 8 It is the principle diagram of the two-class sequential fuzzy neural network;

[0029] Fig. 9 Schematic diagram of position tracking performance comparison results of different control schemes;

[0030] Fig.10 This is a schematic diagram of the results of position tracking errors;

[0031] Fig.11 It is a schematic diagram of the phase diagram tracking performance analysis results of the controlled double Duffing type MEMS gyroscope;

[0032] Fig.12 It is a schematic diagram of control input under the event trigger mechanism;

[0033] Fig.13 It is a schematic diagram of the number of event triggers;

[0034] Fig.14 Schematic diagram of position tracking performance under different initial conditions and system parameters, where: Fig.14 (a) is a schematic diagram of different initial conditions. Fig.14 (b) is a schematic diagram under system parameters;

[0035] Fig.15 Schematic diagram of position tracking effect and controller response under system disturbance. DETAILED DESCRIPTION

[0036] The following describes the embodiments of the present invention by specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways 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 only illustrate the basic concept of the present invention in a schematic manner, and the following embodiments and features in the embodiments can be combined with each other without conflict.

[0037] Among them, the drawings are only used for illustrative explanations, and they only represent schematic diagrams rather than actual pictures, and should not be understood as limitations on the present invention. In order to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of actual products. For those skilled in the art, it is understandable that some well-known structures and their descriptions in the drawings may be omitted.

[0038] The same or similar numbers in the drawings of the embodiments of the present invention correspond to the same or similar parts; in the description of the present invention, it should be understood that if the terms "upper", "lower", "left", "right", "front", "rear", etc. indicate the orientation or position relationship, they are based on the orientation or position relationship shown in the drawings, which is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operate in a specific orientation. Therefore, the terms describing the position relationship in the drawings are only used for illustrative purposes and cannot be understood as limiting the present invention. For ordinary technicians in this field, the specific meanings of the above terms can be understood according to specific circumstances.

[0039] See also Figure 1 to Figure 15 , an event-triggered neural network backstepping control method for a double Duffing type MEMS gyroscope with state constraints.

[0040] Example

[0041] This embodiment proposes an event-triggered neural network backstepping control method for a double Duffing MEMS gyroscope with state constraints, such as Figure 1 As shown, it at least includes the following steps:

[0042] S1. Design a double Duffing MEMS gyroscope and perform mathematical modeling on it.

[0043] S2, introduce digital signal processing DSP method to perform dynamic analysis on double Duffing type MEMS gyroscope;

[0044] S3, defining the tracking error, acceleration function and acceleration error of the double Duffing type MEMS gyroscope in the event-triggered neural network backstepping controller;

[0045] S4. According to the tracking error, acceleration function and acceleration error, several integral barrier Lyapunov candidate functions, control input and adaptive rate of the controller are designed, and the nonlinear unknown function is estimated by type 2 sequential fuzzy neural network, and the explosion problem of virtual control rate differential term is solved by accelerating the second-order tracking differentiator;

[0046] S5. Establish an event triggering mechanism based on the switching threshold strategy to assist the controller in performing event-triggered neural network backstepping control.

[0047] In step S1, Figure 2 The schematic diagram of the structure of the double Duffing type MEMS gyroscope of the present invention is shown, which consists of two equal and symmetrical parts, each of which consists of a mass block, a sensing and driving frame, four fixed piles, seven beams and four electrodes. The above two parts are connected by a coupling spring to ensure synchronous vibration. According to their structure and function, the above seven beams can be divided into a driving decoupling beam, a driving suspension beam, a driving coupling suspension beam, a sensing decoupling beam, a sensing suspension beam, a sensing coupling suspension beam and a coupling beam. Similarly, according to their role in the gyroscope, the types of electrodes can be divided into driving electrodes, driving sensing electrodes, sensing electrodes and orthogonal zero electrodes.

[0048] The designed MEMS gyroscope has a fully decoupled structure. The working principle is summarized as follows: The independent folded beams of the drive / sensing part effectively reduce the mechanical coupling between the drive / sensing modes. A DC bias voltage is applied to the left and right drive electrodes to make the left and right blocks vibrate along the drive direction with the drive frame. The drive decoupling beam keeps the upper and lower beams stationary in the sensing direction to achieve drive decoupling. When the angular velocity is input along the z-axis, the Coriolis acceleration causes the left and right mass blocks and the detection frame to vibrate in the detection direction. The sensing decoupling beam prevents interference with the drive frame in the sensing direction and ensures sensing decoupling. The sensing electrode then measures the vibration displacement in the sensing direction and indirectly provides the input angular rate.

[0049] In order to conveniently and clearly carry out mathematical modeling and dynamic analysis of the double Duffing MEMS gyroscope, please refer to Figure 3 The schematic diagram of the double Duffing type MEMS gyroscope shown.

[0050] In the actual manufacturing process of MEMS gyroscopes, their structure involves delicate components such as drive / sensing frames, coupling and decoupling beams. Obviously, there are inevitable manufacturing tolerances, which lead to nonlinear terms in the system. In addition, this kind of gyroscope is very susceptible to transient or long-term interference caused by external factors. Therefore, asymmetric spring terms, nonlinear terms and external interference should be considered when establishing mathematical models.

[0051] According to Newton's second law, the mathematical model of the double Duffing MEMS gyroscope is established:

[0052]

[0053] in:

[0054]

[0055] According to formula (1), the mathematical model of the above-mentioned MEMS gyroscope can be further derived as follows:

[0056]

[0057] in:

[0058]

[0059] Introducing new variables Then the dimensionless mathematical model of the MEMS gyroscope is:

[0060]

[0061] Where: M S =m S +m C , M D =m D +m C , χ 1 ,x 2 is the length of the X axis, γ 1 ,γ 2 is the length of the Y axis, m S is the perceived frame quality, m C is the mass of the mass block, m D is the driving frame mass, k X ,k Y is the linear spring constant, d X ,d Y Damping coefficient, k XY ,k YX Spring constant of the asymmetric term, k CX ,k CY Coupling spring constant, Ω Z Angular velocity, Input system force in the driving direction, System force in the input sensing direction, reference length l 1 is 1 micron and the natural resonant frequency ξ 0 Serves 1 kHz , χ 1 ,x 2 Indicates the length of the X axis; D 2 and D 6 It is a interference term.

[0062] Lemma 1: There exists an inequality In the hyperbolic tangent function tanh(·):

[0063]

[0064] where h j Represents a positive constant.

[0065] The integral barrier Lyapunov function is constructed as follows:

[0066]

[0067] where e j =x j -x jd ;x j ,j=1,3,5,7 are state variables; k aj , j = 1, 3, 5, 7 are all positive numbers and x jd ,j=1,3,5,7; continuous differentiable function satisfies |x jd |≤Z j <k aj , Z j is a positive constant. j is continuously differentiable and positive definite and satisfies the decreasing condition:

[0068]

[0069] In the set |x j | <k aj whereσ=Ge j , G is a positive constant, which is useful for stability analysis.

[0070] Lemma 2: If |x j | <k aj And k aj >0, then the function V j satisfy:

[0071]

[0072] In step S2, due to the use of registers, storage and reading of data, memory allocation and device communication, the method based on computer simulation and analog circuit cannot be directly applied to industrial production. For the double Duffing type MEMS gyroscope, the converted analog circuit system becomes more complicated. Compared with the analog circuit, the DSP-based method has the advantages of high calculation accuracy, fast processing speed, strong anti-interference ability, and modular operation. Compared with computer simulation, the DSP-based method is more in line with the application scenario of the gyroscope. Computer simulation often ignores many critical underlying processes in embedded platform engineering applications, such as the call and definition of various registers, the communication protocol between modules, and the storage and retrieval of data. Therefore, in order to enhance program compatibility, DSP is introduced and compared with computer simulation to complete the dynamic analysis and verification of the gyroscope on the embedded platform.

[0073] Firstly, the double Duffing MEMS gyroscope is simulated by computer. In order to reveal the dynamic characteristics of the double Duffing MEMS gyroscope, its dynamic behavior is explained through phase diagram, time diagram and Lyapunov index diagram.

[0074] Defining the dimensionless parameter λ of the double Duffing MEMS gyroscope X =0.52,λ CX =0.063,λ XY =0.018, λ Y =0.81,λ CY =0.069,λ YX =0.025, μ X =0.0031,μ Y =0.0058,ω Z =0.02 and initial state [-0.47,0,0.237,0,-0.42,0,0.21,0].

[0075] Figure 4 shows the computer simulation results of a MEMS gyroscope, where Figure 4 (a), 4(c), 4(e) and 4(g) are phase diagrams under different parameters. Figure 4 (b), 4(d), 4(f), and 4(h) are the corresponding time histories. These phase diagrams and time histories confirm that the MEMS gyroscope exhibits chaotic oscillations and complex nonlinear characteristics in both the driving and sensing directions.

[0076] Different models of double Duffing MEMS gyroscopes have different performance parameters, and further research is needed to study the overall impact of different parameters on the system. Figure 5This is the Lyapunov exponent diagram. It can be clearly seen that the exponent has both positive and negative values ​​on the time scale, indicating that the system has been in a state of chaotic motion since the initial state. Figure 6 (a) and Figure 6 (b) shows the Lyapunov exponent diagram under different spring coefficients (CY) and damping coefficients (Xd). It can be seen that different system parameters have a significant impact on the dynamic characteristics, but within the normal range of their selection, there is always a positive exponent in the system, indicating that the gyroscope has chaotic characteristics.

[0077] Due to the limitations of computer simulation, the compatibility of the dynamic analysis program is poor and cannot meet the programming requirements of the DSP platform. In order to solve this problem, it is imperative to complete dynamic analysis verification on the DSP platform.

[0078] In order to expand program compatibility and reveal the nonlinear dynamic characteristics of the MEMS gyroscope on the embedded platform, this embodiment is implemented in hardware on the DSPTMS320F28335 platform.

[0079] The hardware platform consists of a 3.3v DC power supply, a monitoring computer, a DSP development board, an XDS100V1 emulator, a DAC8552 and an oscilloscope. The implementation process is as follows:

[0080] First, considering that DSP can only process discrete data, the dimensionless mathematical model of the double Duffing MEMS gyroscope is discretized using the Runge-Kutta method. The discretized mathematical model is shown in equation (8):

[0081]

[0082] in,

[0083]

[0084] Secondly, due to the large amount of data, an external SRAM (IS61LV25616AL chip) is used to complete the storage operation. Thirdly, considering the actual needs, DAC8552 is used as the output signal, which can provide a two-dimensional array through dual-channel output while ensuring 16-bit high precision.

[0085] Finally, the oscilloscope shows Figure 7 The phase diagram and timing diagram shown, where Figure 7 (a) is the phase diagram of channel 1 and channel 2; Figure 7 (b) is the time course diagram of channel 1 and channel 2; Figure 7 (c) is the phase diagram of channel 3 and channel 4; Figure 7 (d) is the time course diagram of channel 3 and channel 4; Figure 7 (e) is the phase diagram of channel 5 and channel 6; Figure 7 (f) is the time course diagram of channel 5 and channel 6; Figure 7 (g) is the phase diagram of channel 7 and channel 8; Figure 7 (h) is the time history diagram of channel 7 and channel 8. These results are in good agreement with the computer simulation results, thus confirming the accuracy and feasibility of the DSP platform. Obviously, the double Duffing gyroscopes have a large number of chaotic oscillations, which reduces their detection accuracy and stability. However, Figure 7 As shown in the figure, compared with the computer simulation results, the DSP simulation results are obviously affected by the external circuit and data transmission, and there is a certain amount of noise interference. Therefore, the obtained model more accurately reflects the dynamic behavior of the double Duffing MEMS gyroscope in engineering applications.

[0086] Therefore, it is necessary to design a high-performance controller that can effectively suppress chaotic oscillations while ensuring low energy consumption and high precision of the gyroscope.

[0087] In step S3, based on the traditional backstepping theory, an event-triggered neural network backstepping control scheme is proposed by fully integrating the integral barrier Lyapunov function, type 2 sequential fuzzy neural network, acceleration function, accelerated second-order tracking differentiator and switching threshold event triggering.

[0088] The gyro tracking error is defined as follows:

[0089]

[0090] where x jd and α j are the expected trajectory and the virtual control rate.

[0091] The acceleration function is conducive to the rapid convergence of the tracking error. In view of this, the corresponding acceleration function is proposed as:

[0092]

[0093] where υ is a design parameter satisfying 0<υ<<1, t and T d Meet T d ∈(0,∞) represents time and a positive constant. Note that ψ(t) is a monotonically increasing, bounded, continuously differentiable function, when 0≤t <T d .Specific value T d It depends on the time required for the system to transition from the initial state to the steady state under the action of the controller.

[0094] Based on formula (9) and formula (10), the acceleration error is rewritten as:

[0095] S j =ψ(t)e j,j=1,…,8 (11)

[0096] MEMS gyroscopes often work in various extreme environments, and excessive mechanical shock and noise interference can easily damage their microstructures. Therefore, it is necessary to constrain the state variables, especially the displacement of the mass block, to ensure the safe and stable operation of the system. In view of this, the integral barrier Lyapunov function is used to achieve state constraints more smoothly by limiting the cumulative impact of variables. Unlike the tan-type barrier Lyapunov function and the log-type barrier Lyapunov function that constrain the system state error, the integral barrier Lyapunov function directly constrains the system state variables. Therefore, the application of the integral barrier Lyapunov function not only improves the operating safety of the gyroscope, but also reduces the conservatism of the controller design.

[0097] (1) The process of obtaining the first integral barrier Lyapunov function is:

[0098] Derivative of the first error-integrated acceleration function:

[0099]

[0100] in,

[0101] The first integration barrier Lyapunov candidate function is defined as:

[0102]

[0103] Derivative of V1:

[0104]

[0105] in:

[0106]

[0107] According to L'Hôpital's law, we can infer that:

[0108]

[0109] where Φ 1 (·) represents Φ 1 (S 1 ,x 1d ).

[0110] Further simplifying formula (14) yields:

[0111]

[0112] Virtual control rate α 1 Designed to:

[0113]

[0114] where q 1 is a positive control gain.

[0115] The first integral barrier Lyapunov candidate function can be obtained:

[0116]

[0117] (2) The process of obtaining the second integral barrier Lyapunov function is:

[0118] Set the second Lyapunov candidate function:

[0119]

[0120] where p 2 is a positive constant.

[0121] V 2 Derivative:

[0122]

[0123] in

[0124] During the manufacturing and operation phases of the double Duffing MEMS gyroscope, the system is affected by many factors, including component manufacturing tolerances, external shocks, magnetic interference, and temperature fluctuations. Due to the existence of nonlinearity and uncertainty caused by the above factors, the mathematical model will not be able to accurately describe the dynamic behavior of the double Duffing MEMS gyroscope, resulting in performance degradation or even collapse of the backstepping controller. Therefore, a type-2 sequential fuzzy neural network is used to closely approximate the unknown function f 2 :

[0125]

[0126] In the traditional backstepping control derivation process, there is a problem of repeated derivation of complex virtual control rate, which further increases the amount of calculation and leads to "project explosion". In order to better solve this problem, an accelerated second-order tracking differentiator is provided here, and its expression is:

[0127]

[0128] Where R a ,b j1 ,b j2 and b j3 is a positive adjustable constant. j1 a and θ j2 is the variable that speeds up the second-order tracking differentiator.

[0129] In order to ensure the boundedness of the tracking error, the following inequality is introduced:

[0130]

[0131] in It is a myopic error.

[0132] Combining formulas (16), (20), (21) and (22), the control input u 2 and adaptive rate Designed for:

[0133]

[0134] where q 2 ,c 2 and d 2 is a positive constant.

[0135] Substituting equations (24) and (25) into equation (20), we can obtain:

[0136]

[0137] (3) The process of obtaining the third integral barrier Lyapunov function is as follows: The derivative of the third acceleration tracking error is:

[0138]

[0139] The third Lyapunov function is designed as:

[0140]

[0141] V 3 The derivative is:

[0142]

[0143] in:

[0144]

[0145] have:

[0146]

[0147] Further derivation and simplification of V 3 You can get:

[0148]

[0149] Design virtual control rate α 3

[0150]

[0151] where q 3 is a positive adjustable control gain.

[0152] Then formula (31) can be written as:

[0153]

[0154] Based on equations (26) and (33), It can be written as:

[0155]

[0156] (4) The process of obtaining the fourth integral barrier Lyapunov function is:

[0157] The fourth Lyapunov function is:

[0158]

[0159] where p 4 is a positive adjustable constant.

[0160] V 4 Derivative:

[0161]

[0162] in

[0163] Similarly, by accelerating the second-order tracking differentiator θ 42 To approximate When f 4 It is considered to be a nonlinear unknown function, estimated by a type 2 sequential fuzzy neural network, which is:

[0164]

[0165] The control input is selected as u 4 And the adaptive rate for:

[0166]

[0167] where q 4 ,c 4 and d 4 is a positive constant.

[0168] According to equations (34), (36), (38) and (39), Calculated as:

[0169]

[0170] (5) Furthermore, the following can be deduced.

[0171] Virtual control rate α 5 and α 7 yes:

[0172]

[0173] in

[0174] Use a sequential fuzzy neural network to handle subsequent unknowns:

[0175]

[0176] in: and

[0177] From equations (26), (34) and (40), we can get:

[0178]

[0179] Control input u j and adaptive rate Designed to:

[0180]

[0181] where q j ,c j and d j ,j=6,8 is a positive constant.

[0182] More specifically, the type 2 sequential fuzzy neural network is designed to solve the nonlinear unknown term problem in the backstepping design caused by the uncertainty and ambiguity of the double Duffing type MEMS gyroscope. Figure 8 As shown, the type 2 sequential fuzzy neural network includes an input layer, a fuzzification layer, a membership layer, a rule layer and an output layer, where:

[0183] In the input layer, the neural network input x ij It is determined by the system variable value. ij After the fuzzification layer and the membership layer, it is transformed into:

[0184]

[0185] Wherein, i=1,…,P;j=1,…,Q;n=1,…,N; and Represents the upper and lower input layers of the neural network, and Indicates the upper and lower widths of the membership function and The center of the membership function is

[0186] In the membership layer, the expression of the membership function is as follows:

[0187]

[0188] Each IF-THEN fuzzy rule is designed as:

[0189] If x 11 yes and x 1P yes And x PQ yes So in is the i-th Gaussian type 2 MF for the ij-th input, and are the up- and down-vectors.

[0190] The upper and lower emission intensities can be expressed as:

[0191]

[0192] In the output layer, the output of the type 2 sequential fuzzy neural network can be calculated as:

[0193]

[0194] in:

[0195]

[0196] Using the approximation characteristics, the output of the type 2 sequential fuzzy neural network It is possible to estimate any nonlinear unknown function with a small approximation error. Therefore, for any smooth continuous function h(x), there exists a sufficiently small constant η n So that:

[0197]

[0198] where η n Represents the absolute value of the approximation error, Ω x is a compact set of variables x.

[0199] Optimal Vector is defined as:

[0200]

[0201] where θ nis the weight vector, Ω θ is a compact set of variables where θ * Ideal weight.

[0202] In step S5, unlike computer platforms, hardware platforms for dual Duffing MEMS gyroscopes are usually limited by bandwidth, energy consumption, and circuit component life. To address this problem, a switching threshold event trigger mechanism is introduced to optimize the resource utilization of the controller within the system. This method reduces controller energy consumption, increases the service life of peripheral circuits, and reduces communication pressure while ensuring fast and high-precision control goals. In addition, compared with fixed threshold event triggering and relative threshold event triggering, switching threshold event triggering solves the defects of low accuracy and control input jump. The specific method and derivation process are as follows:

[0203]

[0204]

[0205] in and g j >0 is an adjustable parameter and Represents the difference between the control input at the current time and the trigger time.

[0206] Case I: For the fixed threshold strategy, the compensation term is added to the control input to compensate for errors before and after the update, where Then the control output is updated as:

[0207]

[0208] When the time interval belongs to t z ,t z+1 ),exist Here we introduce a time-varying parameter R(t) satisfying |R(t)|≤1, then we have According to Lemma 1, we can infer that:

[0209]

[0210] Condition II: For the relative threshold strategy, the relevant control input is designed as:

[0211]

[0212] in and is a constant. is u j (t) is the update time. Similarly, when When the control input uj (t) is updated to u j (t z ) at t z time a and it will remain unchanged u j (t z ) until the next time it happens

[0213] From equation (52), equation (53) and equation (56), we can infer:

[0214]

[0215] Where R 1j (t) and R 1j (t) is a time-varying constant satisfying |R 1j (t)|≤1 and |R 2j (t)|≤1. Then we can conclude

[0216]

[0217] Where R 1j (t)∈[-1,1] and R 2j (t)∈[-1,1], t has:

[0218]

[0219]

[0220] According to Lemma 1, equations (59) and (60), we obtain:

[0221]

[0222] This embodiment also performs stability analysis on the control method proposed in the present invention.

[0223] Theorem 1: Aiming at the chaotic oscillation, complex nonlinear behavior and uncertainty of MEMS gyroscope, an event-triggered neural network backstepping controller is designed by using adaptive rate and corresponding switch threshold event-triggered control input. Under appropriate initial conditions and system parameter selection, all signals in the closed-loop system are bounded, do not violate any constraints, and avoid Zeno phenomenon.

[0224] Proof: Construct the global Lyapunov function as:

[0225]

[0226] Derivative with respect to V:

[0227]

[0228] From Young's inequality we can conclude that:

[0229]

[0230] Formula (63) can be further derived to obtain:

[0231]

[0232] have:

[0233]

[0234] Through equations (65) and (66), Described as:

[0235]

[0236] in And g 0 =min{k aj ,d j}.

[0237] Formula (67) can be further simplified as:

[0238]

[0239] A compact set is defined as follows:

[0240] Ω 0 ={t|V(t)≤E 0} (69)

[0241] in

[0242] According to Lemma 1 and the definition of V, all signals in the closed-loop system consisting of the dual Duffing MEMS gyroscope and the designed controller are bounded, that is,

[0243] In the design process of event trigger mechanism, event trigger control input is continuously differentiable in both formats, All variables are continuous and bounded. There are constants satisfy because and Existence * Make Therefore, the occurrence of Zeno phenomenon can be effectively prevented. The proof of Theorem 1 is completed.

[0244] This embodiment also tests the above scheme. In the experiment, the duration is set to 30s and the step size is set to 0.001s. The following reference trajectory composed of composite signals is selected as: 1d =0.7sin(1.6t-0.2π)+0.3sin(0.4t),x 3d =0.7sin(1.3t)+0.5sin(1.7t+0.85π),x 5d =0.5sin(1.6t-0.4π)+0.4sin(1.2t) and x 7d =0.6sin(0.8t)+0.5sin(1.7t+0.85π). In T2SFN-N, the upper and lower widths of the membership function are set to and, the membership function is the center The speed function parameter T is: [-1,-0.5,0,0.5,1]. d and υ are set to 2 and 0.5. The parameters involved in the controller derivation process are as follows: [q 2 ,q 4 ,q 6 ,q 8 ]=[42,30,40,34] and [k a1 ,k a3 ,k a5 ,k a7 ]=[3.3,3.4,3.6,3.9]. The parameters of the event trigger part are designed as follows: l j =0.01,g j =0.3, L = 2. In the accelerating second-order tracking differentiator, there is R a =4,b j1 =0.5,b j2 =0.6,b j3 =0.45,j=2,4,6,8. Positive parameter c of the adaptation rate j and d j ,j=2,4,6,8 are selected as 2 and 1.5. The parameters and initial conditions of the MEMS gyroscope are consistent with those in the dynamic analysis part.

[0245] from Fig. 9It can be seen that due to the intervention of the proposed control scheme, it can be clearly seen that the displacement signals corresponding to the four degrees of freedom completed the convergence process within 0.05 while maintaining a high accuracy. This fully proves the effectiveness of the proposed method. Then, in order to verify the superiority of the designed control scheme, it was compared with the RBFNN control based on the integral barrier Lyapunov function (Scheme A) and the accelerated adaptive inversion funnel control (Scheme B). Compared with the above two schemes, the proposed scheme has a faster convergence speed and higher steady-state accuracy in the initial stage.

[0246] also, Fig.10 The tracking performance of the controller is given, and the position tracking error is constrained to be within the minimum range, denoted as [-0.0006, 0.0006]. Fig.11 It is the tracking effect of the phase diagram under controlled conditions. As can be seen from the figure, the controlled phase diagram is almost consistent with the ideal phase diagram, indicating that the proposed control scheme effectively suppresses the inherent chaotic oscillation of the gyroscope and improves the stability and safety of the system.

[0247] Fig.12 The control input is shown with the switching threshold event triggering the controller to engage. The waveform (shown as a solid green line) shows a square wave characteristic, indicating that the control input is updated when the difference between the control input at the current and previous time steps meets the set threshold. Fig.13 It can be seen that the update times of the control input only account for 10.13%, 9.72%, 8.65% and 9.39% of the total number of steps. The introduction of the switch threshold event trigger not only significantly reduces the update frequency of the control input, but also alleviates the high-frequency oscillation of the control input to a certain extent. This effectively reduces the communication bandwidth requirements and improves the stability of the controller.

[0248]

[0249] Fig.14 (a) and (b) show two sets of images showing the controller's tracking capability for the dual Duffing MEMS gyro under different initial conditions and system parameters, with all controller settings remaining unchanged. The example shows that changes in parameters or initial conditions have little effect on the control effect. To further verify the anti-disturbance capability of the controller, perturbations (70) and (71) are used. Fig.15 As shown in Figure 3, after the disturbance occurs, the controller promptly increases the control input to adjust the system so that the system maintains high-precision tracking. The negative impact of the disturbance on the tracking performance can be ignored, which confirms that the proposed scheme has good robustness.

[0250] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solution, which should be included in the scope of the claims of the present invention.

Claims

1. An event-triggered neural network backstepping control method for a double Duffing type MEMS gyroscope with state constraints, characterized by: S1. Design a double Duffing MEMS gyroscope and perform mathematical modeling on it. S2, introduce digital signal processing DSP method to perform dynamic analysis on double Duffing type MEMS gyroscope; S3, defining the tracking error, acceleration function and acceleration error of the double Duffing type MEMS gyroscope in the event-triggered neural network backstepping controller; S4. According to the tracking error, acceleration function and acceleration error, several integral barrier Lyapunov candidate functions, control input and adaptive rate of the controller are designed, and the nonlinear unknown function is estimated by type 2 sequential fuzzy neural network, and the explosion problem of virtual control rate differential term is solved by accelerating the second-order tracking differentiator; S5. Establish an event triggering mechanism based on the switching threshold strategy to assist the controller in performing event-triggered neural network backstepping control.

2. The event-triggered neural network backstepping control method of a double Duffing type MEMS gyroscope with state constraints according to claim 1, characterized in that: In step S1, the double Duffing type MEMS gyroscope includes two equal and symmetrical parts, which are connected by coupling springs, wherein each part includes a mass block, a sensing frame, a driving frame, a plurality of beam structures, a plurality of electrodes and a plurality of fixed piles, the beam structures include a driving decoupling beam, a driving suspension beam, a driving coupling suspension beam, a sensing decoupling beam, a sensing suspension beam, a sensing coupling suspension beam and a coupling cross beam; the electrode types include a driving electrode, a driving sensing electrode, a sensing electrode and an orthogonal zero electrode; the MEMS gyroscope has a completely decoupled structure, and its working principle is: The independent folded beams of the driving / sensing parts reduce the mechanical coupling between the driving / sensing modes, and a DC bias voltage is applied to the left and right driving electrodes to make the left and right blocks vibrate along the driving direction with the driving frame; The drive decoupling beam keeps the upper and lower beams stationary in the sensing direction, achieving drive decoupling; When the angular velocity is input along the z-axis, the Coriolis acceleration causes the left and right mass blocks and the detection frame to vibrate along the detection direction; the sensing decoupling beam is used to prevent interference with the drive frame in the sensing direction; The sensing electrodes then measure the vibration displacement in the sensing direction, indirectly providing the input angular rate.

3. The event-triggered neural network backstepping control method of a double Duffing type MEMS gyroscope with state constraints according to claim 2, characterized in that: According to Newton's second law, the mathematical model of the double Duffing MEMS gyroscope is established: in: M=diag{m C ,m C ,m D ,m D }+diag{m C ,m S ,m C ,m C } The mathematical model of the above-mentioned MEMS gyroscope is further derived as follows: in: Introducing new variables Then the dimensionless mathematical model of the MEMS gyroscope is: Where: M S =m S +m C , M D =m D +m C , χ1, χ2 are the lengths of the X axis, γ1, γ2 are the lengths of the Y axis, m S is the perceived frame quality, m C is the mass of the mass block, m D is the driving frame mass, k X ,k Y is the linear spring constant, d X ,d Y is the damping coefficient, k XY ,k YX is the spring constant of the asymmetric term, k CX ,k CY is the coupling spring constant, Ω Z is the angular velocity, is the system force in the input driving direction, is the system force in the input sensing direction, l1 represents the reference length, ξ0 represents the natural resonant frequency, χ1, χ2 represent the X-axis length; D2 and D6 are interference terms; In the above mathematical model, the following exists: There is an inequality In the hyperbolic tangent function tanh(·): where h j represents a positive constant; The integral barrier Lyapunov function is constructed as follows: where e j =x j -x jd ;x j ,j=1,3,5,7 are state variables; k aj , j = 1, 3, 5, 7 are all positive numbers and x jd ,j=1,3,5,7; continuous differentiable function satisfies |x jd |≤Z j <k aj , Z j is a positive constant; V j is continuously differentiable and positive definite and satisfies the decreasing condition: In the set |x j | <k aj whereσ=Ge j , G is a normal number; If |x j | <k aj And k aj >0, then the function V j satisfy:

4. The event-triggered neural network backstepping control method of a double Duffing MEMS gyroscope with state constraints according to claim 3, characterized in that: In step S2, a dynamic analysis of the double Duffing MEMS gyroscope is performed using a DSP platform, which includes the following steps: The dimensionless mathematical model of the double Duffing MEMS gyroscope is discretized using the Runge-Kutta method. The discretized mathematical model is expressed as: in, Use external SRAM to complete storage operations; Using DAC8552 as the output signal, a two-dimensional array is provided through dual-channel output while ensuring high accuracy; Finally, the phase diagram and time diagram of the double Duffing MEMS gyroscope are obtained through an oscilloscope.

5. The event-triggered neural network backstepping control method of a double Duffing MEMS gyroscope with state constraints according to claim 4, characterized in that: In step S3, the gyro tracking error is defined as follows: where x jd and α j are the expected trajectory and the virtual control rate; The acceleration function is defined as: where υ is a design parameter satisfying 0<υ<<1, t and T d Meet T d ∈(0,∞) represents time and a positive constant; when 0≤t <T d When ψ(t) is a monotonically increasing, bounded, continuously differentiable function; the value T d Depends on the time required for the system to transition from the initial state to the steady state under the action of the controller; Then the acceleration error can be rewritten as: S j =ψ(t)e j ,j=1,…,8。 6. The event-triggered neural network backstepping control method of a double Duffing MEMS gyroscope with state constraints according to claim 5, characterized in that: The state constraint is achieved by limiting the cumulative effect of variables using the integral barrier Lyapunov function, where: (1) The first integral barrier Lyapunov candidate function is: Derivative of the first error-integrated acceleration function: in, The first integration barrier Lyapunov candidate function is defined as: Derivative of V1: in: According to L'Hôpital's law: where Φ1(·) represents Φ1(S1,x 1d ) Further: The virtual control rate α1 is designed as: Where q1 is a positive control gain; Get the first integration barrier Lyapunov function: (2) The process of obtaining the second integral barrier Lyapunov function is: Set the second Lyapunov candidate function: Where p2 is a positive constant; Taking the derivative of V2: in Use a type 2 sequential fuzzy neural network to approximate the unknown function f2: Design an accelerated second-order tracking differentiator, whose expression is: Where R a ,b j1 ,b j2 and b j3 is the positive adjustable constant, θ j1 a and θ j2 is the variable that speeds up the second-order tracking differentiator; The following inequality is introduced to ensure the boundedness of the tracking error: in It is myopia error; Then the control input u2 and the adaptive rate Designed for: Where q2, c2 and d2 are positive constants; According to the above, the second integral barrier Lyapunov function is expressed as: (3) The process of obtaining the third integral barrier Lyapunov function is: The derivative of the third acceleration tracking error is: The third Lyapunov function is designed as: Taking the derivative of V3, we get: in: have: Further derivation and simplification have to: Design virtual control rate α3: Where q3 is a positive adjustable control gain; Then we have: Furthermore, the third integral barrier Lyapunov function is expressed as: (4) The process of obtaining the fourth integral barrier Lyapunov function is: The fourth Lyapunov function is: Where p4 is a positive adjustable constant; Derivative of V4: in By accelerating the θ in the second-order tracking differentiator 42 To approximate The type 2 sequential fuzzy neural network is used to estimate f4, which is: The control input is selected as u4 and the adaptive rate for: Where q4, c4 and d4 are positive constants; Then the fourth integral barrier Lyapunov function is expressed as: (5) Derive the following: Virtual control rates α5 and α7: in Use a sequential fuzzy neural network to handle subsequent unknowns: in: and Then we have: Control input u j and adaptive rate Designed for: where q j ,c j and d j ,j=6,8 is a positive constant.

7. The event-triggered neural network backstepping control method of a double Duffing MEMS gyroscope with state constraints according to claim 6, characterized in that: The type 2 sequential fuzzy neural network includes input layer, fuzzification layer, membership layer, rule layer and output layer, among which, In the input layer, the neural network input x ij Determined by the system variable value, x ij After the fuzzification layer and the membership layer, it is transformed into: Wherein, i=1,…,P;j=1,…,Q;n=1,…,N; and Represents the upper and lower input layers of the neural network, and Indicates the upper and lower widths of the membership function and The center of the membership function is In the membership layer, the expression of the membership function is as follows: Each IF-THEN fuzzy rule is designed as: If x 11 yes yes And x PQ yes So in is the i-th Gaussian type 2 MF for the ij-th input, and are the inverse and descending vectors; The upper and lower emission intensities are expressed as: In the output layer, the output of the type 2 sequential fuzzy neural network is expressed as: in: For any smooth continuous function h(x), there exists a sufficiently small constant η n So that: where η n Represents the absolute value of the approximation error, Ω x is a compact set of variables x; Optimal Vector is defined as: where θ n is the weight vector, Ω θ is a compact set of variables where θ * Ideal weight.

8. The event-triggered neural network backstepping control method of a double Duffing type MEMS gyroscope with state constraints according to claim 7, characterized in that: In step S5, a switching threshold event trigger mechanism is introduced to optimize the resource utilization of the controller in the system. The method and derivation process are as follows: in and g j >0 is an adjustable parameter and Represents the difference between the control input at the current time and the trigger time; Case I: For the fixed threshold strategy, the compensation term is added to the control input to compensate for errors before and after the update, where Then the control output is updated as: When the time interval belongs to t z ,t z+1 ),exist Introducing a time-varying parameter R(t) satisfying |R(t)|≤1, then we have According to Lemma 1, we can infer that: Condition II: For the relative threshold strategy, the relevant control input is designed as: in and is a constant, is u j (t) the update time; when When the control input u j (t) is updated to u j (t z ) at t z time a and it will remain unchanged u j (t z ) until the next occurrence: It can be inferred from this that: Where R 1j (t) and R 2j (t) is a time-varying constant satisfying |R 1j (t)|≤1 and |R 2j (t)|≤1, then Where R 1j (t)∈[-1,1] and R 2j (t)∈[-1,1]; have: From this we get:

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