Accelerated Adaptive Backstepping Funnel Control Method for a Dual-Mass MEMS Gyroscope with an Event-Triggering Mechanism

Abstract

The present invention relates to an acceleration adaptive backstepping funnel control method for a dual-mass MEMS gyroscope with an event-triggering mechanism, belonging to the field of gyroscope control, and includes the following steps: S1: Based on the mechanical coupling structure between two gyroscopes, establish a mathematical model of the dual-mass MEMS gyroscope; S2: Analyze the dynamic behavior characteristics of the dual-mass MEMS gyroscope through phase diagrams and Lyapunov exponent diagrams; S3: Integrate a type-2 sequential fuzzy neural network, a velocity function, an asymmetric funnel boundary, an acceleration exponential integral tracking differentiator, and an event-triggering mechanism to design an acceleration adaptive backstepping funnel controller. This solution well solves the comprehensive control problems including chaos suppression, uncertainty, "explosion terms", state constraints, system parameter perturbations, and saving computational resources.

Description

Technical Field

[0001] The present invention belongs to the field of gyroscope control, and relates to an acceleration adaptive backstepping funnel control method for a dual-mass MEMS gyroscope with an event-triggering mechanism. Background Technique

[0002] MEMS gyroscopes have the advantages of low cost, small size, high integration, etc., which have greatly revolutionized the inertial sensor industry and are increasingly widely used in fields such as inertial navigation systems, intelligent vehicles, and satellites. Dual-mass MEMS gyroscopes have a larger bandwidth and higher mechanical sensitivity, and have become a hot topic in the research field of MEMS gyroscopes. However, due to the more complex structure of the dual-mass gyroscope, its modeling and dynamic analysis are more challenging. At the same time, due to non-linear characteristics, large computational amounts, the existence of state constraints and uncertain factors, this will bring greater difficulties to the design of the controller.

[0003] System modeling can qualitatively and quantitatively describe the dynamic process of the entire system, which is a prerequisite for controller design. Moreover, in engineering applications, the dynamic characteristics of gyroscopes are of great significance to the reliability of gyroscopes. Efimovskaya et al. studied the dynamic characteristics of circular dual-mass MEMS gyroscopes and established their motion equations. However, the above work ignored the mutual interference between the driving mode and the sensing mode, resulting in system instability problems. In order to eliminate unnecessary interference phenomena, Dai et al. decoupled the structure of the dual-mass gyroscope. Yang et al. designed a dual-mass MEMS gyroscope with a symmetric decoupling structure and analyzed the lever support system inside the gyroscope. Li et al. studied two coupling methods of dual-mass MEMS gyroscopes and discussed their physical characteristics. However, the inherent chaotic oscillations that endanger system stability and reliability have not been analyzed in depth. Moreover, in actual operation, system parameters are affected by internal and external disturbances such as temperature changes and mechanical shocks, which will cause system uncertainties.

[0004] At present, the backstepping method is widely used in the control of electromechanical systems. Aiming at the adaptive control problem of MEMS gyroscopes, Shao and Shi proposed an adaptive control scheme based on neural networks to deal with the quantization input and full-state constraints of MEMS gyroscopes. Luo et al. established the analog circuit of the fractional-order MEMS gyroscope and designed an adaptive backstepping controller. For the double gimbal control moment gyroscope, Lungu proposed a backstepping control method based on a neural observer. Vafaie et al. fused the adaptive sliding mode method with type-3 fuzzy logic to achieve the precise control of MEMS gyroscopes. However, the derivative of the virtual control signal in the traditional backstepping method needs to be repeatedly calculated, which leads to the problem of "explosion terms". To solve this problem, a first-order filter and a tracking differentiator are used to approximate the virtual signal, thus avoiding repeated differentiation. At the same time, the tracking accuracy and convergence speed of the first-order filter and the tracking differentiator are not good enough. In addition, most of the existing research focuses on the control schemes of single-mass MEMS gyroscopes.

[0005] In practical engineering applications, due to the limitations of storage space and real-time communication capabilities, the computing resources of a single processor are limited. In view of this, the event-triggered mechanism is used to reduce the computational burden. In addition, due to the limitations of the physical structure and other specific working requirements, the output signal must be controlled within a given region. If these constraints are violated, the system will become unstable or even malfunction. Therefore, to ensure the performance of the system, many tools such as symmetric / asymmetric barrier Lyapunov functions and preset performance functions are often designed to constrain the output signal. Zirkohi designed an adaptive backstepping control scheme based on the barrier Lyapunov function to achieve the output constraint of MEMS gyroscopes. Mei et al. used the asymmetric barrier Lyapunov function to constrain the state variables and output of the target system. However, these constant constraint methods are not applicable in the case of time-varying. To solve the above disadvantages, time-varying constraints have emerged in engineering applications. Luo et al. designed a time-varying preset performance function to ensure that the state constraints are not violated. Wang et al. introduced a funnel function in the backstepping process to reconstruct the Lyapunov function to limit the tracking error. Li et al. fused the time-varying function into the asymmetric barrier Lyapunov function to ensure the constraints of the nonlinear system. However, due to the reduction of the update rate under event triggering, the control accuracy will also decrease. To sum up, how to save the communication resources of the double-mass MEMS gyroscope and ensure the control performance is very challenging. Summary of the Invention

[0006] In view of this, the purpose of the present invention is to provide an accelerated adaptive backstepping funnel control method for a double-mass MEMS gyroscope with an event-triggered mechanism.

[0007] To achieve the above object, the present invention provides the following technical solutions:

[0008] An acceleration adaptive backstepping funnel control method for a dual - mass MEMS gyroscope with an event - trigger mechanism, comprising the following steps:

[0009] S1: Based on the mechanical coupling structure between two gyroscopes, establish a mathematical model of the dual - mass MEMS gyroscope;

[0010] S2: Analyze the dynamic behavior characteristics of the dual - mass MEMS gyroscope through phase diagrams and Lyapunov exponent diagrams;

[0011] S3: Integrate a type - 2 sequential fuzzy neural network, a velocity function, an asymmetric funnel boundary, an acceleration - exponential integral tracking differentiator, and an event - trigger mechanism to design an acceleration adaptive backstepping funnel controller.

[0012] Furthermore, the mathematical model of the dual - mass MEMS gyroscope described in step S1 specifically includes:

[0013] Considering the nonlinear term K 3 ξ 3 and the asymmetric term K a ξ, establish the dynamic equation of the dual - mass MEMS gyroscope:

[0014]

[0015] where

[0016] x 1 、x 2 、y 1 、y 2 represent the axes of displacement, Ω z is the angular velocity; k x 、k y are the linear spring coefficients; k xy 、k yx represent the asymmetric spring terms; k cx 、k cy are the coupling spring coefficients, d x 、d y represent the damping coefficients; represents the control force; m s 、m d 、m c represent the masses of the mass blocks;

[0017] Define χ 1 =x 1 / l 0 , χ 3 =x 2 / l 0 , χ 5 =y 1 / l 0 , χ 7 =y 2 / l 0 , The dimensionless equations of the dual-mass MEMS gyroscope are expressed as:

[0018]

[0019] where M s =m s +m c , M d =m d +m c , l 0 =1 μm is the reference length, ω 0 =1 kHz represents the natural resonance frequency.

[0020] Furthermore, the parameters of the dimensionless gyroscope system are set as κ x =0.6, κ cx =0.05, κ xy =0.03, κ y =0.9, κ cy =0.06, κ yx =0.02, δ x =0.006, δ x =0.003, ω = 0.01, and the initial state is taken as [χ 1 (0), …, χ 8 (0)] = [0.42, 0, 0.61, 0, 0.36, 0, 0.55, 0]. The phase diagram, time history diagram, and Lyapunov exponent diagram of the dual-mass MEMS gyroscope are plotted, and it is concluded that the dual-mass MEMS gyroscope exhibits complex nonlinear dynamic behavior under the selected parameters and initial state.

[0021] Furthermore, the type-2 sequential fuzzy neural network includes a fuzzification layer, a membership layer, and a rule layer;

[0022] When x mn is input into the fuzzification layer and the membership layer, it is converted to:

[0023]

[0024] where represents the upper and lower inputs after conversion, and are the membership functions respectively and are the upper and lower widths of is the center of the membership function;

[0025] In the membership layer, the upper and lower membership functions are expressed as:

[0026]

[0027] In the rule layer, the IF-THEN fuzzy rule is:

[0028] where is the i-th type-2 Gaussian membership function of the mn-th input, represents the upper and lower consequent parameters;

[0029] The upper and lower firing degrees are written as:

[0030]

[0031] The output of the type-2 sequential fuzzy neural network is:

[0032]

[0033] where represents a positive constant;

[0034] Finally, formula (6) is expressed as:

[0035]

[0036] where the weight vectors and

[0037] For any smooth function g(x), there is:

[0038]

[0039] where ε(x) represents a positive approximation error, and Ω x is a compact set of x; define the optimal weight vector w * as Ω w is a compact set of w; finally define where w * is an artificial term,

[0040] Furthermore, in step S3, define the system tracking error e i , i = 1, …, 8 as:

[0041]

[0042] where χ i , i = 1, …, 8 are state variables, χ id , i = 1, 3, 5, 7 are reference trajectories, υ i , i = 2, 4, 6, 8 are virtual control laws;

[0043] Introduce the velocity function

[0044]

[0045] where is a positive increasing function that satisfies and is bounded and continuously differentiable; T ∈ (0, ∞) is a certain moment; β is a design parameter that satisfies 0 < β << 1;

[0046] Based on formulas (9) and (10), the tracking error e i , i = 1, …, 8, is converted into an acceleration error:

[0047]

[0048] Constraining the state variables will ensure that the funnel boundary function of the state variable constraints is integrated into the asymmetric barrier Lyapunov function, and the boundary is designed as a decreasing function:

[0049] Ψ j (t) = (ψ j0 - ψ j∞ ) exp(-ψ jr t) + ψ j∞ , j = u, l, (12)

[0050] where ψ j0 represents the initial state, ψ j∞ represents the convergence boundary, and ψ jr represents the convergence speed.

[0051] Furthermore, the constraining of the state variables specifically includes the following steps:

[0052] S31: The first derivative of the tracking error is:

[0053]

[0054] where

[0055] The first asymmetric barrier Lyapunov function V 1 is designed as:

[0056]

[0057] Derive with respect to V 1 Derivation:

[0058]

[0059] where z ui = E i / Ψ u (t), z li (t) = E i / Ψ l (t), i = 1, 3, 5, 7, Ψ u (t) and Ψ l (t) are the upper and lower boundary functions, Ψ l (t) < E i < Ψ u (t) and q satisfy:

[0060]

[0061] If E i ≥ 0, the tracking error will be limited within Ψ u (t). If E i < 0, Ψ l (t) starts to take effect;

[0062] The first virtual control law υ 2 is designed correspondingly as:

[0063]

[0064] where k is a positive number to ensure that the virtual control signal υ and are zero and the virtual control signal υ i is bounded, m 1 is a positive design parameter;

[0065] Define Substitute equation (17) into equation (15) to get:

[0066]

[0067] S32: Define the conversion weight error related to the type-2 sequential fuzzy neural network as Select the second Lyapunov function as:

[0068]

[0069] Calculate the derivative of V 2 as

[0070]

[0071] Among them

[0072] Regarding the g involved in the gyroscope dynamics model 2 function as an uncertain function, a type-2 sequential fuzzy neural network is used to estimate it, and its definition is:

[0073]

[0074] Design an accelerated exponential integral tracking differentiator to approximate the derivative of υ i to solve the "explosion term" problem in traditional backstepping. The expression is as follows:

[0075]

[0076] where the input is the virtual control signal υ i , θ i1 and θ i2 are the outputs of the accelerated exponential integral tracking differentiator; is the velocity function; ρ i is the main parameter, c i1 , c i2 and b < 1 affect the tracking performance, differential performance, and stability respectively, and their values are all adjustable positive numbers;

[0077] Let

[0078] where η i , i = 2, 4, 6, 8 represent the approximation errors;

[0079] The control input u 2 (t) and the adaptation law are designed as

[0080]

[0081]

[0082] where m 2 , r 2 and h 2 are positive constants;

[0083] Substitute equations (18), (24), and (25) into equation (20) to obtain:

[0084]

[0085] Integrate the event-triggered strategy into the control scheme, which is expressed as:

[0086]

[0087] where represents the update time of the controller, a i > 0, b i ∈(0, 1), is a positive number; once |u ei (t) - u i (t)| ≥ a i u i (t) + b i , the time will be recorded as the control signal will act on the system;

[0088] Based on the above formula, we get:

[0089]

[0090] where |λ i1 (t)| ≤ 1 and |λ i2 (t)| ≤ 1, i = 2, 4, 6, 8, are time-varying terms, and we have:

[0091]

[0092] When -1 ≤ λ i1 (t) ≤ 1 and -1 ≤ λ i2 (t) ≤ 1, we have the following derivation:

[0093]

[0094] Therefore, formula (20) is written as:

[0095]

[0096] According to formulas (27) and (32), formula (33) is derived as:

[0097]

[0098] According to formulas (26) and (34), it is simplified to:

[0099]

[0100] S33: Differentiate the third tracking error E 3 :

[0101]

[0102] Select the asymmetric barrier Lyapunov function V 3 as:

[0103]

[0104] Calculate V 3 The derivative of

[0105]

[0106] Design the second virtual control υ 4 as

[0107]

[0108] where m 3 is a positive constant;

[0109] Based on formulas (35) and (39), formula (38) is written as:

[0110]

[0111] S34: Select the fourth Lyapunov function

[0112]

[0113] Take the derivative of V 4 There is

[0114]

[0115] θ 42 is used to replace and the uncertain function is approximated by a type-2 sequential fuzzy neural network, that is

[0116]

[0117] Then, the control input u 4 (t) and the adaptation law are derived as:

[0118]

[0119] where m 4 , r 4 and h 4 are positive constants;

[0120] According to formula (30), the event-triggered input is given as:

[0121]

[0122] According to formulas (34), (40) and (45), is written as:

[0123]

[0124] S35: Take the derivative of the fifth tracking error E 5 Derivative:

[0125]

[0126] Define the fifth Lyapunov candidate function as:

[0127]

[0128] Calculate as

[0129]

[0130] Design the third virtual control signal υ 6 as

[0131]

[0132] where m 5 is a positive coefficient;

[0133] Combining formulas (47) and (51), we get

[0134]

[0135] S36: Design the sixth Lyapunov function V 6 as:

[0136]

[0137] Take the derivative of V 6 and we get

[0138]

[0139] where

[0140] θ 62 is used to replace and the uncertain term g 6 is estimated by a type-2 sequential fuzzy neural network:

[0141]

[0142] Design the third group of control inputs and adaptation laws as:

[0143]

[0144] where m 6 , r 6 and h 6denotes a positive constant;

[0145] According to formula (30), the third event-triggered control input is designed as:

[0146]

[0147] Combining formulas (34), (52), (57) and (58), formula (54) is derived as:

[0148]

[0149] S37: E 7 The derivative of is:

[0150]

[0151] Construct the last asymmetric barrier Lyapunov candidate function as:

[0152]

[0153] Find the derivative of V 7 which is:

[0154]

[0155] The last virtual control law υ 8 is:

[0156]

[0157] where m 7 denotes a positive parameter

[0158] Combining formulas (59) and (63), formula (62) is derived as:

[0159]

[0160] S38: Design the last Lyapunov function V 8 as:

[0161]

[0162] V 8 The derivative of is:

[0163]

[0164] where

[0165] θ 82 is used to estimate and the uncertain g 8 is approximated by a type-2 sequential fuzzy neural network:

[0166]

[0167] The last set of control inputs and adaptation laws are given as:

[0168]

[0169] where m 8 , r 8 and h 8 represent positive parameters;

[0170] The last event-triggered input is given as:

[0171]

[0172] Combining Eqs. (34), (64), (69) and (70), Eq. (66) is derived as:

[0173]

[0174] The beneficial effects of the present invention are as follows:

[0175] 1. Considering the high-order nonlinear terms and asymmetric terms, a dynamic model of the dual-mass MEMS gyroscope closer to actual engineering applications is established, and then the dynamic evolution laws of the dual-mass MEMS gyroscope under different conditions are revealed, which is of great significance for improving the system sensitivity.

[0176] 2. To solve the inherent "explosion term" problem in the backstepping technique, an accelerated exponential integral tracking differentiator is designed, which overcomes the problem of repeated differentiation of virtual control and accelerates the convergence of the output signal. Compared with the first-order filter, the traditional tracking differentiator and the hyperbolic tangent tracking differentiator, the designed accelerated exponential integral tracking differentiator has higher tracking accuracy and faster convergence speed. In addition, the designed asymmetric funnel boundary not only ensures that the position tracking error is strictly limited within the specified region, but also makes the boundary design non-conservative.

[0177] 3. This scheme well solves the comprehensive control problems including chaos suppression, uncertainty, "explosion term", state constraints, system parameter perturbation and saving computational resources.

[0178] Other advantages, objectives and features of the present invention will be described to some extent in the subsequent specification, and to some extent, will be obvious to those skilled in the art based on the study of the following text, 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 specification. Brief Description of the Drawings

[0179] To make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be described in detail and preferably below with reference to the accompanying drawings, where:

[0180] Figure 1 It is a block diagram of the acceleration adaptive backstepping funnel control process for a dual-mass MEMS gyroscope with an event trigger mechanism;

[0181] Figure 2 It is a schematic diagram of a dual-mass MEMS gyroscope;

[0182] Figure 3 It is a phase diagram and time history diagram of a dual-mass MEMS;

[0183] Figure 4 It is a Lyapunov exponent diagram of a dual-mass MEMS gyroscope;

[0184] Figure 5 It is a schematic structural diagram of a type-2 sequential fuzzy neural network. Specific Embodiments

[0185] The following illustrates the embodiments of the present invention through specific examples. 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 different specific embodiments. Various details in this specification can also 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 drawings provided in the following embodiments only illustrate the basic concept of the present invention schematically. Without conflict, the following embodiments and the features in the embodiments can be combined with each other.

[0186] Among them, the drawings are only for illustrative purposes, showing only schematic diagrams rather than physical diagrams, and should not be construed as a limitation to the present invention; to better illustrate the embodiments of the present invention, some components in the drawings will be omitted, enlarged or reduced, which do not represent the dimensions 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.

[0187] In the drawings of the embodiments of the present invention, the same or similar reference numerals correspond to the same or similar components; in the description of the present invention, it should be understood that if there are terms such as "upper", "lower", "left", "right", "front", "rear", etc. indicating the orientation or position relationship, they are based on the orientation or position relationship shown in the drawings, and are 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 operated in a specific orientation. Therefore, the terms describing the position relationship in the drawings are only for illustrative purposes and should not be construed as a limitation to the present invention. For those of ordinary skill in the art, the specific meanings of the above terms can be understood according to specific circumstances.

[0188] The present invention provides an acceleration adaptive backstepping funnel control method for a dual-mass MEMS gyroscope with an event-triggering mechanism, and its control process is as follows Figure 1 shown.

[0189] As Figure 2 shown, the dual-mass MEMS gyroscope consists of a base, two mass blocks with spiral springs and dampers, an electrostatic actuator for generating a driving force, and a device for sensing the position and velocity of the mass blocks.

[0190] For the convenience of establishing mathematical equations, it is assumed that the mass blocks move and rotate at a certain linear velocity; for invisible displacements, the centrifugal force can be ignored; the two mass blocks move along the X and Y directions. The corresponding structure of the mass blocks is as shown on the right. The advantage of this form is to separate the driving mode and sensing mode of the mass blocks, avoiding mutual interference of the movements along the two axes. Figure 1 right side.

[0191] In an actual mass-spring system, there are nonlinear terms and inevitable manufacturing asymmetry factors in the springs. For the convenience of calculation, these asymmetry terms are usually ignored. However, for a more accurate establishment of the gyro model, the nonlinear terms and asymmetric spring terms are considered.

[0192] Based on this, considering the nonlinear term K 3 ξ 3 and the asymmetric term K a ξ, the dynamic equation of the dual-mass MEMS gyroscope is established

[0193]

[0194] where

[0195]

[0196] represents the axis of displacement, (m), Ω z is the angular velocity (rad / s), k x , k y is the linear spring coefficient (N / m), k xy , k yx represents the asymmetric spring term (N / m), k cx , k cy is the coupling spring coefficient (N / m), d x , d y represents the damping coefficient (N / m), represents the control force (N), m s , m d , m c represents the mass of the mass block (Kg).

[0197] For the convenience of subsequent design, define χ 1 = x 1 / l 0 , χ 3 = x 2 / l 0 , χ 5 = y 1 / l 0 , χ 7 = y 2 / l 0 , Then, the dimensionless equation of the dual - mass MEMS gyroscope can be written as:

[0198]

[0199] where M s = m s + m c , M d = m d + m c , l 0 = 1μm is the reference length, ω 0 = 1kHz represents the natural resonance frequency.

[0200] To reveal the dynamic characteristics of the dual - mass MEMS gyroscope and clarify the necessity of the designed controller, its dynamic behavior is analyzed through time - history diagrams, phase diagrams, and Lyapunov exponent diagrams.

[0201] The parameters of the dimensionless gyroscope system are set as κ x = 0.6, κ cx = 0.05, κ xy = 0.03, κ y = 0.9, κ cy = 0.06, κ yx = 0.02, δ x = 0.006, δ x = 0.003, ω = 0.01, and the initial state is taken as [χ 1 (0), …, χ 8 (0)] = [0.42, 0, 0.61, 0, 0.36, 0, 0.55, 0]. It can be seen from Figure 3 the phase diagrams and time - history diagrams of the four axes that the whole system exhibits chaotic behavior. Figure 4Depicts the Lyapunov exponent diagram on the time scale. It can be found that the exponents have both positive and negative values, indicating that the gyroscope enters chaotic motion over time. From the above two diagrams, it can be seen that under the selected parameters and initial states, the entire system exhibits complex nonlinear dynamic behaviors.

[0202] For any Ψ l <E i <Ψ u , there is

[0203]

[0204] where Ψ j (t), j = u, l, are time-varying functions, and E i , i = 1, 3, 5, 7 are state variables.

[0205] For and any tanh(·), the following inequality holds

[0206] 0 ≤ |X| - tanh(X / φ i ) ≤ 0.2785φ i , i = 2, 4, 6, 8, (4)

[0207] where φ i , i = 2, 4, 6, 8 are positive constants.

[0208] The type-2 sequential fuzzy neural network combines the advantages of fuzzy logic systems and neural networks and can handle the identification, control, and prediction problems of uncertainties in nonlinear systems. Its structure is as Figure 5 shown.

[0209] When x mn is input into the fuzzification layer and membership layer, it will be converted to:

[0210]

[0211] where represents the upper and lower inputs after conversion, and are the upper and lower widths of the membership functions and (MFs) respectively, and is the center of the membership function.

[0212] In the membership layer, the upper and lower membership functions are expressed as:

[0213]

[0214] In the rule layer, the IF-THEN fuzzy rule is

[0215] wherein is the i-th type-2 Gaussian membership function of the mn-th input, representing the upper and lower consequent parameters.

[0216] The upper and lower firing degrees can be written as:

[0217]

[0218] The output of the type-2 sequential fuzzy neural network is:

[0219]

[0220] wherein represents a positive constant.

[0221] Finally, (8) can be further expressed as:

[0222]

[0223] where the weight vector and

[0224] For any smooth function g(x), there is

[0225]

[0226] where ε(x) represents a positive approximation error, and Ω x is a compact set of x. Define the optimal weight vector w * as Ω w is a compact set of w. Finally, define where w * is an artificial term,

[0227] Define the system tracking error e i , i = 1, …, 8 as:

[0228]

[0229] where χ i , i = 1, …, 8, χ id , i = 1, 3, 5, 7 and υ i , i = 2, 4, 6, 8 are the state variables, reference trajectory and virtual control law, respectively.

[0230] To achieve the accelerated convergence of the actual signal, the velocity function

[0231]

[0232] where is a positive increasing function satisfying and is bounded and continuously differentiable. \(T\in(0,\infty)\) is a certain moment. \(\beta\) is a design parameter satisfying \(0 < \beta\ll1\).

[0233] Based on (11) and (12), the tracking error \(e\) i , \(i = 1,\ldots,8\), is converted into the acceleration error:

[0234]

[0235] Considering that the stability of the gyroscope is affected by internal and external disturbances such as temperature changes, magnetic field interference, and mechanical shocks, in order to ensure performance, the state variables should be constrained. In addition, there are more or less asymmetric factors in manufacturing. To meet the performance requirements, the funnel boundary function that guarantees the state variable constraints is incorporated into the asymmetric barrier Lyapunov function. The boundary is designed as a decreasing function:

[0236] \(\varPsi\) j (t)=( \(\psi\) j0 -\(\psi\) j∞ )\(\exp(-\psi\) jr t)+\(\psi\) j∞ , \(j = u,l\), (14)

[0237] where \(\psi\) j0 \(\psi\) j∞ and \(\psi\) jr represent the initial state, the convergence boundary, and the convergence rate, respectively.

[0238] Based on the backstepping framework, the design of the acceleration adaptive backstepping funnel control method with an event-triggered mechanism is divided into 8 steps.

[0239] Step 1: The first derivative of the tracking error is

[0240]

[0241] where

[0242] To ensure the limitation of the tracking error \(E\) 1 , the first asymmetric barrier Lyapunov function \(V\) 1 is designed as:

[0243]

[0244] Differentiate \(V\) 1 :

[0245]

[0246] where z ui = E i / Ψ u (t), z li (t) = E i / Ψ l (t), i = 1, 3, 5, 7, Ψ u (t) and Ψ l (t) are the upper and lower boundary functions, Ψ l (t) < E i < Ψ u (t) and q satisfy:

[0247]

[0248] If E i ≥ 0, the tracking error will be limited within Ψ u (t). If E i < 0, Ψ l (t) starts to take effect.

[0249] Then, the first virtual control law υ 2 is correspondingly designed as

[0250]

[0251] where k is a positive number to ensure that the virtual control signal υ and are zero, and m i is a positive design parameter. 1 is a positive design parameter.

[0252] Define Substituting (19) into (17), we can obtain

[0253]

[0254] Step 2: Define the conversion weight error related to the type-2 sequential fuzzy neural network as Select the second Lyapunov function as:

[0255]

[0256] Calculate the derivative of V 2 as

[0257]

[0258] where

[0259] Considering that the actual working state of the dual-mass MEMS gyroscope is affected by various uncertain factors such as temperature changes and drifts, the g 2 function involved in the gyroscope dynamic model is regarded as an uncertain function. To handle this uncertainty, a type-2 sequential fuzzy neural network is used to estimate it, and its definition is as follows:

[0260]

[0261] Design an accelerated exponential integral tracking differentiator to approximate the derivative of υ i to solve the problem of "explosion terms" in traditional backstepping. The expression is as follows:

[0262]

[0263] where the inputs are the virtual control signal υ i , θ i1 and θ i2 are the outputs of the accelerated exponential integral tracking differentiator. is the velocity function. ρ i is the main parameter, c i1 , c i2 and b < 1 affect the tracking performance, differential performance, and stability respectively. Their values are all adjustable positive numbers.

[0264] Note 2: During the process of adjusting these parameters, larger ρ i , c i1 , c i2 and b can improve the tracking speed and accuracy of the accelerated exponential integral tracking differentiator. However, selecting overly large parameters will cause output fluctuations and lead to computational burdens. To ensure the approximation ability of the accelerated exponential integral tracking differentiator, appropriate parameters need to be selected.

[0265] For the convenience of subsequent calculations, there is:

[0266] Let

[0267] where η i , i = 2, 4, 6, 8 represent the approximation errors.

[0268] The control input u 2 (t) and the adaptation law are designed as

[0269]

[0270] where m 2 , r 2 and h 2 are positive constants.

[0271] Substituting (20), (26) and (27) into (22), we get

[0272]

[0273] In practical applications such as navigation and aerospace, due to the limitations of the storage space of hardware and the real-time communication capabilities of signal processors, the computing power is limited. Therefore, in order to reduce unnecessary waste of communication resources and improve economic efficiency at the same time, we incorporate an event-triggered strategy into the control scheme, which can be expressed as:

[0274]

[0275] where represents the update time of the controller, a i > 0, b i ∈(0, 1), is a positive number. Once |u ei (t) - u i (t)| ≥ a i u i (t) + b i , the time will be recorded as and the control signal will act on the system.

[0276] Based on the above formula, we can obtain:

[0277]

[0278] where |λ i1 (t)| ≤ 1 and |λ i2 (t)| ≤ 1, i = 2, 4, 6, 8, are time-varying terms. There is:

[0279]

[0280] When -1 ≤ λ i1 (t) ≤ 1 and -1 ≤ λ i2 (t) ≤ 1, it can be deduced that:

[0281]

[0282] Therefore, (22) is written as

[0283]

[0284] According to Lemma 2, (29) and (34), (35) can be deduced as:

[0285]

[0286] Thus, according to (28) and (36), Can be simplified to:

[0287]

[0288] Step 3: For the third tracking error E 3 Take the derivative

[0289]

[0290] Select the asymmetric barrier Lyapunov function V 3 as:

[0291]

[0292] Calculate the derivative of V 3 is

[0293]

[0294] Then design the second virtual control υ 4 as

[0295]

[0296] where m 3 is a positive constant.

[0297] Based on (37) and (41), (40) can be written as:

[0298]

[0299] Step 4: Select the fourth Lyapunov function

[0300]

[0301] Take the derivative of V 4 There is

[0302]

[0303] Similarly, θ 42 is used to replace and the uncertain function is approximated by a type-2 sequential fuzzy neural network, that is

[0304]

[0305] Then, the control input u 4 (t) and the adaptation law can be derived as

[0306]

[0307] where m4 , r 4 and h 4 are positive constants.

[0308] According to (32), the event-triggered input is given as:

[0309]

[0310] According to (36), (42) and (47), it can be written as:

[0311]

[0312] Step 5: Differentiate the fifth tracking error E 5 to obtain:

[0313]

[0314] Define the fifth Lyapunov candidate function as:

[0315]

[0316] Then, calculate which is

[0317]

[0318] Design the third virtual control signal υ 6 to be

[0319]

[0320] where m 5 is a positive coefficient.

[0321] Combining (49) and (53), we can obtain

[0322]

[0323] Step 6: Design the sixth Lyapunov function V 6 as:

[0324]

[0325] Differentiate V 6 to obtain

[0326]

[0327] where

[0328] Similar to Step 4, θ 62 is used to replace and the uncertain term g 6 Estimated by a type-2 sequential fuzzy neural network:

[0329]

[0330] Design the third group of control inputs and adaptation laws as:

[0331]

[0332] where m 6 , r 6 and h 6 represent positive constants.

[0333] According to (32), the third event-triggered control input is designed as:

[0334]

[0335] Combining (36), (54), (59) and (60), (56) can be further derived as

[0336]

[0337] Step 7: The derivative of E 7 is:

[0338]

[0339] Construct the last asymmetric barrier Lyapunov candidate function as:

[0340]

[0341] Find the derivative of V 7 is:

[0342]

[0343] Then, the last virtual control law υ 8 is:

[0344]

[0345] where m 7 represents a positive parameter.

[0346] Combining (61) and (65), (64) can be derived as:

[0347]

[0348] Step 8: Design the last Lyapunov function V 8 as:

[0349]

[0350] V 8 The derivative of is:

[0351]

[0352] where

[0353] θ 82 is used to estimate and the uncertain g 8 is approximated by a type-2 sequential fuzzy neural network:

[0354]

[0355] The last group of control inputs and adaptation laws are given as:

[0356]

[0357]

[0358] where m 8 , r 8 and h 8 represent positive parameters.

[0359] Similarly, the last event-triggered input is given as:

[0360]

[0361] Combining (36), (66), (71) and (72), (68) is further derived as:

[0362]

[0363] A global Lyapunov function is constructed as:

[0364]

[0365] Calculating the derivative of (74), we can obtain:

[0366]

[0367] According to Young's inequality, we have

[0368]

[0369] Therefore, (75) can be rewritten as

[0370]

[0371] In addition, the following inequality holds

[0372]

[0373] Therefore, (77) can be further deduced as

[0374]

[0375] where H 0 = min{2m i , h i}.

[0376] Then the solution of (79) can be further expressed as

[0377]

[0378] From equations (74) to (80), it can be seen that for the dual-mass MEMS gyroscope (2) with uncertainty and chaotic oscillation, the control inputs (32), (48), (60) and (72) with an event-triggering mechanism are designed, and the adaptation laws are selected as (27), (47), (59) and (71). If the parameters are reasonably selected, then it can be ensured that all signals of the closed-loop system are bounded and all constraints are not violated. In addition, the inherent chaotic oscillation that may endanger the stability of the gyroscope system is completely eliminated. Furthermore, if E i → Ψ u (t) or Ψ l (t), j = u, l. That is, when Ψ l (t) < E i < Ψ u (t), E i and are ultimately uniformly bounded. According to the definition of the designed controller, it is proved that all signals are bounded.

[0379] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the spirit and scope of the present technical solution, and they should all be covered within the scope of the claims of the present invention.

Claims

1. An accelerated adaptive backstepping funnel control method for a dual-mass MEMS gyroscope with an event-triggering mechanism, Characterized in that: It includes the following steps: S1: Based on the mechanical coupling structure between two gyroscopes, establish a mathematical model of the dual-mass MEMS gyroscope; S2: Analyze the dynamic behavior characteristics of the dual-mass MEMS gyroscope through phase diagrams and Lyapunov exponent diagrams; S3: Integrate a type-2 sequential fuzzy neural network, a velocity function, an asymmetric funnel boundary, an accelerated exponential integral tracking differentiator, and an event-triggering mechanism to design an accelerated adaptive backstepping funnel controller; The mathematical model of the dual-mass MEMS gyroscope described in step S1 specifically includes: Consider the nonlinear term K 3 ξ 3 and the asymmetric term K a ξ, and establish the dynamic equation of the dual-mass MEMS gyroscope: wherein x 1 、x 2 、y 1 、y 2 represent the axes of displacement, and Ω z is the angular velocity; k x 、k y are the linear spring coefficients; k xy 、k yx represent the asymmetric spring terms; k cx 、k cy are the coupling spring coefficients, d x 、d y represent the damping coefficients; represents the control force; m s 、m d 、m c represent the masses of the mass blocks; Define χ 1 = x 1 / l 0 , χ 3 = x 2 / l 0 , χ 5 = y 1 / l 0 , χ 7 = y 2 / l 0 , The dimensionless equations of the dual-mass MEMS gyroscope are expressed as: where M s = m s + m c , M d = m d + m c , l 0 = 1 μm is the reference length, ω 0 = 1 kHz represents the natural resonance frequency; Set the parameters of the dimensionless gyroscope system to κ x = 0.6, κ cx = 0.05, κ xy = 0.03, κ y = 0.9, κ cy = 0.06, κ yx = 0.02, δ x = 0.006, δ x = 0.003, ω = 0.01, and the initial state is taken as [χ1(0), …,, χ 8 (0)] = [0.42, 0, 0.61, 0, 0.36, 0, 0.55, 0]. Plot the phase diagram, time history diagram, and Lyapunov exponent diagram of the dual-mass MEMS gyroscope. It is concluded that the dual-mass MEMS gyroscope exhibits complex nonlinear dynamic behavior under the selected parameters and initial state; The type-2 sequential fuzzy neural network includes a fuzzification layer, a membership layer, and a rule layer; When x mn is input into the fuzzification layer and the membership layer, it is converted to: Among them represent the upper and lower inputs after conversion, and are the upper and lower widths of the membership function and respectively, is the center of the membership function; In the membership layer, the upper and lower membership functions are expressed as: In the rule layer, the IF-THEN fuzzy rule is: wherein is the i-th type-2 Gaussian membership function of the mn-th input, representing the upper and lower consequent parameters; The upper and lower triggering degrees are written as: The output of the type-2 sequential fuzzy neural network is: wherein represent positive constants; Finally, formula (6) is expressed as: where the weight vector and For any smooth function g(x), there is: where ε(x) represents the positive approximation error, and Ω x is a compact set of x; define the optimal weight vector w * as Ω w is a compact set of w; finally, define where w * is an artificial term, In step S3, the system tracking error e is defined i , where i = 1, …, 8 is as follows: where χ i , i = 1, …, 8 are state variables, χ id , i = 1, 3, 5, 7 are reference trajectories, υ i , i = 2, 4, 6, 8 are virtual control laws; Introduce velocity function wherein is a positive increasing function satisfying and is bounded and continuously differentiable; T ∈ (0, ∞) is a certain moment; β is a design parameter satisfying 0 < β << 1; Based on formulas (9) and (10), the tracking error e i , i = 1, …, 8, is converted into an acceleration error: Constrain the state variables, and fuse the funnel boundary function that guarantees the state variable constraints into the asymmetric barrier Lyapunov function. The boundary is designed as a decreasing function: Ψ j (t) = (ψ j0 - ψ j∞ ) exp(-ψ j rt) + ψ j∞ , j = u, l, (12) where ψ j0 represents the initial state, ψ j∞ represents the convergence boundary, ψ jr represents the convergence rate.

2. The accelerated adaptive backstepping funnel control method for a dual-mass MEMS gyroscope with an event-triggering mechanism according to claim 1, Characterized in that: The constraint on the state variables specifically includes the following steps: S31: The first derivative of the tracking error is: Among them The first asymmetric-barrier Lyapunov function V 1 is designed as: Derivative with respect to V 1 Derivation: where z ui = E i / Ψ u (t), z li (t) = E i / Ψ l (t), i = 1, 3, 5, 7, Ψ u (t) and Ψ l (t) are upper and lower boundary functions, Ψ l (t) < E i < Ψ u (t) and q satisfy: If E i ≥ 0, the tracking error will be limited within Ψ u (t). If E i < 0, Ψ l (t) starts to take effect; The first virtual control law υ 2 The corresponding design is as follows: wherein k is a positive number to ensure that and when it is zero, the virtual control signal υi is bounded, and m 1 is a positive design parameter; Definition Substituting formula (17) into formula (15), we get: S32: Define the conversion weight error related to the type-2 sequential fuzzy neural network as Select the second Lyapunov function as: Calculate V 2 The derivative of is Among them Regarding the g involved in the gyroscope dynamics model 2 as an uncertain function, a type-2 sequential fuzzy neural network is used to estimate it, and its definition is as follows: Design an accelerated exponential integral tracking differentiator to approximate the derivative of υ i to solve the problem of "explosion terms" in traditional backstepping. The expression is as follows: where the input is the virtual control signal υ i , θ i1 and θ i2 are the outputs of the acceleration exponential integral tracking differentiator; is the velocity function; ρ i is the main parameter, c i1 , c i2 and b < 1 affect the tracking performance, the differentiating performance and the stability respectively, and their values are all adjustable positive numbers; Let where η i , i = 2, 4, 6, 8 represents the approximation error; Control input u 2 (t) and adaptation law Designed as where m 2 , r 2 and h 2 are positive constants; Substitute formulas (18), (24), and (25) into formula (20) to obtain: Integrate the event-triggering strategy into the control scheme, expressed as: Among them represents the update time of the controller, a i > 0, b i ∈(0, 1), is a positive number; once |u ei (t) - u i (t)| ≥ a i u i (t) + b i , the time will be recorded as the control signal will act on the system; Based on the above formulas, obtain: where |λ i1 (t)| ≤ 1 and |λ i2 (t)| ≤ 1, i = 2, 4, 6, 8, are time-varying terms, and there is: When -1 ≤ λ i1 (t) ≤ 1 and -1 ≤ λ i2 (t) ≤ 1, the derivation is as follows: Therefore, formula (20) is written as: According to (27) and (32), formula (33) is deduced as: According to (26) and (34), Simplify to: S33: Derive the third tracking error E 3 with respect to: Select the asymmetric-barrier Lyapunov function V 3 as follows: Calculate V 3 The derivative of is Design the second virtual control υ 4 For where m 3 is a positive constant; Based on formulas (35) and (39), formula (38) is written as: S34: Select the fourth Lyapunov function Derive with respect to V 4 We get θ 42 used to replace and the uncertainty function approximated by a type-2 sequential fuzzy neural network, that is Thus, the control input u 4 (t) and the adaptation law are derived as follows: where m 4 , r 4 and h 4 are positive constants; According to formula (30), the event-triggering input is given as: According to formulas (34), (40) and (45), It is written as: S35: Derive the fifth tracking error E 5 with respect to: Define the fifth Lyapunov candidate function as: Calculation For Design the third virtual control signal υ 6 For where m 5 is a positive coefficient; Combining formulas (47) and (51), we get S36: Design the sixth Lyapunov function V 6 It is as follows: Derive with respect to V 6 and obtain Among them θ 62 used to replace υ 6 and the uncertainty term g 6 estimated by a type-2 sequential fuzzy neural network: Design the third set of control inputs and adaptation laws as: where m 6 , r 6 and h 6 represent positive constants; According to formula (30), the third event-triggering control input is designed as: Combining formulas (34), (52), (57), and (58), formula (54) is deduced as: S37: E 7 The derivative of: Construct the last asymmetric barrier Lyapunov candidate function as: Find V 7 The derivative of which is: The last virtual control law υ 8 is as follows: where m 7 represents a positive parameter Combining formulas (59) and (63), formula (62) is deduced as: S38: Design the last Lyapunov function V 8 as follows: V 8 The derivative of is: Among them θ 82 used to estimate and the uncertain g 8 approximated by a type-2 sequential fuzzy neural network: Give the last set of control inputs and adaptation laws as: where m 8 , r 8 and h 8 represent positive parameters; Give the last event-triggering input as: Combining formulas (34), (64), (69), and (70), formula (66) is deduced as:

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