Adaptive Inversion Control Method for a Triaxial MEMS Gyroscope with Output Constraints

Abstract

The present invention relates to a three-axis MEMS gyroscope acceleration adaptive inversion control method with output constraints, belonging to the technical field of MEMS gyroscope control. The method includes: S1: constructing a three-axis MEMS gyroscope dynamic model and constructing a time-varying barrier Lyapunov function to limit the state variables within a specified range; S2: constructing a type-2 fuzzy wavelet neural network to approximate the nonlinear unknown function of the gyroscope; S3: constructing a three-axis MEMS gyroscope acceleration adaptive inversion control model, specifically integrating the type-2 fuzzy wavelet neural network, velocity function, time-varying barrier Lyapunov function, hyperbolic tangent tracking differentiator and adaptive law into the inversion control model. The controller designed by the present invention not only satisfies the constraints of the state variables, but also can suppress chaotic oscillations and has good tracking accuracy.

Description

Technical Field

[0001] The present invention belongs to the technical field of MEMS gyroscope control, and relates to a three-axis MEMS gyroscope acceleration adaptive inversion control method with output constraints. Background Art

[0002] MEMS is a device with advantages such as multi-functionality, miniaturization, and easy circuit integration. In recent years, MEMS has been increasingly widely used in engineering. With the development of MEMS technology, micro gyroscopes have become one of the hot topics in the MEMS research community. Due to the ability to measure angular velocity, MEMS gyroscopes are widely used in smartphones, medical services, north-seeking systems, and automobiles. However, MEMS gyroscopes have inherent chaotic oscillations, which will inevitably endanger their stability and even lead to intermittent failures. Therefore, analyzing its dynamic behavior and further designing an effective controller to handle this dynamic instability is crucial and challenging.

[0003] Due to the rich nonlinear dynamic behavior of MEMS gyroscopes, in order to better analyze and understand its motion process, scholars have studied its dynamic behavior. Fei and Zhou established the dynamic equation of a three-axis MEMS gyroscope. Lestev analyzed the nonlinear differential motion equation, which is helpful for studying the vibration analysis of MEMS gyroscopes. Hamed et al. studied the dynamics of MEMS gyroscopes with linear and nonlinear excitations, and determined the stability through Poincaré maps and bifurcation diagrams. Larkin et al. revealed the nonlinear dynamic behavior of MEMS gyroscopes and studied their vibration characteristics when they have cracks, which is helpful for further improving the manufacturing process. Ouakad analyzed the static and dynamic behaviors of MEMS gyroscopes through Galerkin's modal expansion method and multi-scale approximation method. Wei et al. studied a non-parametric method based on Hilbert transform to measure the nonlinear dynamic characteristics of MEMS gyroscopes, which can control the excitation within a certain range and avoid complex calculations. However, the parameters and states of three-axis MEMS gyroscopes are easily affected by various factors such as temperature changes, noise interference, mechanical shock, and manufacturing tolerances. Worse still, if these changes are ignored, faults may occur. However, none of the above studies mentioned the dynamic analysis under different conditions, let alone chaos suppression.

[0004] In the field of control of nonlinear electromechanical systems, many control strategies have been proposed by scholars. Zhou et al. proposed a new adaptive backstepping control scheme with hyperbolic tangent function for uncertain nonlinear systems, which can effectively compensate for unknown nonlinear interactions. Zhao et al. developed an adaptive backstepping controller to suppress chaotic oscillations in MEMS resonators. Sui et al. designed a fuzzy adaptive finite-time controller with event-triggering, which can not only ensure the stability of nonlinear systems with unknown functions and unmodeled dynamics, but also save communication resources. Luo et al. proposed an optimal synchronization scheme based on hierarchical neural network for fractional-order chaotic electromechanical devices. A large number of results have shown the boundedness of the whole system and the minimization of the cost function. In addition, for the control problem of MEMS gyroscopes, many control methods have been proposed by scholars to ensure their performance and stability. Li et al. controlled the frequency mismatch between the sensing and driving modes to improve the sensitivity stability of mismatched MEMS gyroscopes within a certain temperature range. Fei et al. combined the backstepping scheme with an adaptive sliding mode controller to ensure the stability of MEMS gyroscopes. Guo et al. designed a neural terminal sliding mode control scheme based on finite-time learning for MEMS gyroscopes and proved that this scheme has better tracking and learning abilities than traditional schemes. Rahmani proposed a fractional-order integral terminal sliding mode controller with proportional derivative for MEMS gyroscopes, which can not only track the reference trajectory but also eliminate the chattering phenomenon. In recent years, fuzzy logic and neural networks have been studied and widely applied in adaptive backstepping control methods to solve the unknown nonlinear terms and parameters of MEMS gyroscopes. Fei and Feng designed a double-loop neural network with fuzzy logic to approximate the uncertain function of MEMS gyroscopes and proved its high approximation accuracy. Yan et al. used FNN to estimate the unknown terms in the robust adaptive terminal sliding control scheme of MEMS gyroscopes. Shao and Shi developed an adaptive echo state network control scheme with minimum learning parameters, which can compensate for the matched and mismatched disturbances in the MEMS gyroscope system. Asad et al. proposed a recurrent type-2 fuzzy system to estimate the uncertainties in the backstepping sliding mode control scheme of MEMS gyroscopes. Vafaie et al. used a type-3 fuzzy logic system to estimate the unknown equations in MEMS gyroscopes. It is worth noting that there is a problem of "complexity explosion" in the backstepping process due to the repeated calculation of the derivative of the virtual control signal. To solve this problem, scholars have proposed effective methods, such as first-order filters, tracking differentiators, and neural networks. Zhou et al. used neural networks to approximate the nonlinear virtual control signal.Tognetti and de Oliveira applied a first-order filter in the derivation. Since only one parameter needs to be adjusted, it is easy to debug. However, compared with the tracking differentiator, its accuracy is not satisfactory. Wang et al. incorporated the tracking differentiator into the backstepping process, successfully reducing the computational complexity and having good filtering accuracy. However, the chaotic behavior and chaotic suppression of the three-axis MEMS gyroscope were not involved, which are extreme conditions brought about by its complex operating environment. Therefore, it is meaningful to design an accelerated adaptive backstepping controller to achieve chaotic suppression, accelerated convergence, nonlinear function approximation, and reduced computational complexity.

[0005] In the practical application of electromechanical systems, the output and state are restricted by the physical structure and performance requirements. To meet the constraints, Zhu et al. used the tracking performance function to meet the specified requirements of the power system. Luo et al. designed the prescribed performance function and the symmetric barrier Lyapunov function to achieve the tracking performance and constraints of the fractional-order seismograph system. For the safety of the vehicle system, Guo et al. proposed a fault-tolerant controller with symmetric and asymmetric barrier Lyapunov functions to limit the output in the vehicle row. Zirkohi designed an adaptive backstepping controller with a barrier Lyapunov function to limit the output of the three-axis MEMS gyroscope, thus achieving good tracking performance with less computational effort. Zhang et al. designed a preset performance function to limit the error of the MEMS gyroscope. Sun et al. integrated the time-varying barrier Lyapunov function based on logarithms and trigonometric functions into the backstepping controller, which helps to keep the output and error within the specified range. However, the three-axis MEMS gyroscope is a high-precision device operating under complex conditions, so the requirements for performance and stability should be more stringent. In addition, during the process of tracking the signal, the tracking error has a large fluctuation at first. Considering these two factors, it is necessary and meaningful to design a controller to meet the performance requirements and ensure that the time-varying constraints are not violated. Summary of the Invention

[0006] In view of this, the purpose of the present invention is to provide an acceleration adaptive inversion control method for a three-axis MEMS gyroscope with output constraints, establish a mathematical model of the three-axis MEMS gyroscope with output constraints, and reveal its dynamic behavior through phase diagrams, time history diagrams, and Lyapunov exponent diagrams. In the controller design, a type-2 fuzzy wavelet neural network is used to approximate the nonlinear unknown function of the system. The velocity function is utilized to achieve accelerated convergence with a small amplitude under the backstepping framework, and a time-varying barrier Lyapunov function is constructed to limit the state variables within a specified range. At the same time, a hyperbolic tangent tracking differentiator is adopted to reduce the computational complexity of the time-varying barrier Lyapunov function in the backstepping framework. The type-2 fuzzy wavelet neural network, velocity function, time-varying barrier Lyapunov function, hyperbolic tangent tracking differentiator, and adaptive law are integrated into the backstepping framework to design the entire controller. In addition, the stability analysis proves that all signals in the closed-loop system are ultimately bounded. The controller designed by the present invention not only satisfies the constraints of the state variables but also suppresses chaotic oscillations and has good tracking accuracy.

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

[0008] An acceleration adaptive inversion control method for a three-axis MEMS gyroscope with output constraints, specifically comprising the following steps:

[0009] S1: Construct a dynamic model of a three-axis MEMS gyroscope, and construct a time-varying barrier Lyapunov function to limit the state variables within a specified range;

[0010] S2: Construct a type-2 fuzzy wavelet neural network to approximate the nonlinear unknown function of the gyroscope;

[0011] S3: Construct an acceleration adaptive inversion control model for a three-axis MEMS gyroscope, specifically by integrating a type-2 fuzzy wavelet neural network, a velocity function, a time-varying barrier Lyapunov function, a hyperbolic tangent tracking differentiator, and an adaptive law into the inversion control model.

[0012] Further, in step S1, the constructed dynamic model of the three-axis MEMS gyroscope is:

[0013]

[0014] Wherein, x, y, z are displacement axes, d xy , d xz , d yz are the symmetry terms of the spring, d xx , d yy , d zz are damping coefficients, kxy , k xz , k yz is the asymmetry term of the spring, k xx , k yy , k zz is the linear spring coefficient, is the angular velocity, is the control force, m is the mass of the internal mass block, ω 0 = 1 kHz is the natural resonance frequency, l 0 = 1 μm represents the reference length; x 1 = x / l 0 , x 3 = y / l 0 , x 5 = z / l 0 ,

[0015] Furthermore, in step S1, the expression of the constructed time-varying obstacle Lyapunov function is:

[0016]

[0017] where, k i (t) is a positive time-varying decreasing function, e represents the state variable and satisfies k i (t) > |e|.

[0018] Furthermore, in step S2, a type-2 fuzzy wavelet neural network is constructed, specifically including: Assume the network has N inputs and M fuzzy rules, and the IF-THEN rule is: If x 1 is w m1 , x 2 is w m2 … x n is w mn , then

[0019]

[0020] where, x n is the nth input, w mn represents the mth fuzzy rule of the nth input, W m is the output of the fuzzy rule;

[0021] The upper and lower membership functions are written as:

[0022]

[0023] where, c mn , dmn respectively represent the center, upper width, and lower width of the membership function;

[0024] The calculations of the upper and lower activation functions for fuzzification are as follows:

[0025] f m = μ 1n · μ 2n ·...· μ mn

[0026] where, f m respectively represent the upper and lower activation functions for fuzzification, μ mn respectively represent the upper and lower membership functions;

[0027] Then the output F of the fuzzy neural network part m is:

[0028]

[0029] The wavelet function calculation is:

[0030]

[0031] where, a mn and b mn represent the dilation and translation parameters;

[0032] The output W of the wavelet network m is:

[0033]

[0034] Finally, the output O of the type-2 fuzzy wavelet neural network m is:

[0035]

[0036] where, θ = [θ 11 , …, θ 1n , …, θ m1 , …, θ mn T , σ = [σ 11 , …, σ 1n , …, σ m1 , …, σ mn T , σ mn = F m w​​mn , θ mn represents the weights of the type-2 fuzzy wavelet neural network, and σ mn represents the basis function of the type-2 fuzzy wavelet neural network.

[0037] Furthermore, in step S2, the constructed type-2 fuzzy wavelet neural network is used to approximate the nonlinear unknown function of the gyroscope, which specifically includes: for any smooth continuous function f(x), there exists where ε(x) represents the positive approximation error, and Ω x is a compact set of x; the optimal vector θ * is defined as where Ω θ is a compact set of θ; finally, define where θ * is a manually defined term, and the approximation error satisfies

[0038] Furthermore, in step S3, a three-axis MEMS gyroscope acceleration adaptive backstepping control model is constructed, which specifically includes the following steps:

[0039] S31: Calculate the derivative 1 of the first tracking error A The expression is:

[0040]

[0041] where, represents the first derivative of; represents the velocity function, and the expression is: where is a positive increasing function, and its initial value satisfies and is bounded and continuously differentiable; T ∈ (0, ∞) represents the specified time, b(t) represents the smoothing function and satisfies b(0) = 1 and is usually selected as b(t) = 1 + t 2 ; η is a design parameter that satisfies 0 < η << 1;

[0042] e i is the tracking error, and the expression is: where x i , i = 1, …, 6 are state variables, x id , i = 1, 3, 5 and α i , i = 2, 4, 6 are the reference trajectories and virtual controls of the X, Y, and Z axes respectively;

[0043] Select the first barrier Lyapunov function V 1 as:

[0044]

[0045] where k 1 (t) represents a positive definite time-varying attenuation function;

[0046] Differentiating V 1 yields:

[0047]

[0048] The first virtual control rate α 2 is designed as:

[0049]

[0050] where c 1 represents a positive design parameter;

[0051] Substituting (4) into (3) gives:

[0052]

[0053] where A 2 represents the second acceleration tracking error;

[0054] S32: Select the second Lyapunov function V 2 as:

[0055]

[0056] Calculating the derivative of V 2 yields:

[0057]

[0058] where f 2 is regarded as an unknown function, and the expression is

[0059] Due to the existence of uncertainties such as parameter variations and drifts, the motion of this gyroscope is always unknown. Considering this, the f 2 involved in the gyroscope dynamics model is regarded as an unknown function. To solve this problem, a type-2 fuzzy wavelet neural network is used to approximate f 2 , that is

[0060]

[0061] where represents the weights of the type-2 fuzzy wavelet neural network, σ 2 represents the basis function of the type-2 fuzzy wavelet neural network, ε 2Denote the approximation error of the type-2 fuzzy wavelet neural network;

[0062] In addition, to avoid "complexity explosion" and simplify the complex calculations of the traditional backstepping technique, a hyperbolic tangent tracking differentiator is designed to approximate the derivative of α i Then

[0063]

[0064] where τ i1 denotes a variable, and τ i2 denotes a variable; ρ i , c i1 and c i2 are adjustable parameters, which mainly affect the tracking speed, tracking effect and differentiation effect;

[0065] Remark 1: It is found during the parameter adjustment process that the larger ρ i , c i1 and the smaller c i2 can improve the tracking speed and accuracy of the hyperbolic tangent tracking differentiator. However, if the parameters are selected too large or too small, the output will fluctuate. Based on this, the above parameters should be appropriately selected to enhance the approximation ability of the hyperbolic tangent tracking differentiator.

[0066] In addition, for the convenience of calculation, define:

[0067]

[0068] where E τi denotes the positive approximation error of the hyperbolic tangent tracking differentiator;

[0069] Then, the control input u x and the adaptive law are derived as:

[0070]

[0071]

[0072] where c 2 , λ 2 and μ 2 are designed positive constants; γ 2 denotes a positive constant;

[0073] Substitute (5) and (11) into (7), and we get

[0074]

[0075] S33: The derivative of the third tracking error A 3 is

[0076]

[0077] Select the third barrier Lyapunov function V 3 as:

[0078]

[0079] where k 3 (t) represents a positive definite time-varying decay function;

[0080] Then the derivative of V 3 is

[0081]

[0082] Then, the second virtual control rate α 4 is designed as:

[0083]

[0084] where c 3 represents a designed positive parameter;

[0085] Based on (14) and (17), (16) can be rewritten as:

[0086]

[0087] S34: Select the fourth Lyapunov function V 4 as:

[0088]

[0089] where A 4 is the fourth acceleration tracking error;

[0090] Taking the derivative of V 4 gives

[0091]

[0092] Similarly, using τ 42 to replace Considering that is uncertain, the type-2 fuzzy wavelet neural network is used again to approximate it, defined as

[0093]

[0094] The control input u y and the adaptation rate can be designed as

[0095]

[0096]

[0097] Among them, c 4 , λ 4 and μ 4 represent positive design parameters; γ 4 represents a positive constant;

[0098] Based on (18) and (22), (20) is calculated as

[0099]

[0100] S35: The derivative of the fifth tracking error A 5 is

[0101]

[0102] Define the fifth barrier Lyapunov function V 5 as:

[0103]

[0104] Among them, k 5 (t) represents a positive time-varying decay function;

[0105] Take the derivative of V 5 with respect to

[0106]

[0107] Then, given the third virtual control rate α 6 as:

[0108]

[0109] Among them, c 5 represents a positive parameter;

[0110] Combining (24) and (28), (27) can be derived as

[0111]

[0112] S36: Design the last Lyapunov function V 6 :

[0113]

[0114] Among them, A 6 represents the sixth tracking error;

[0115] Take the derivative of V 6 with respect to,

[0116]

[0117] Among them,

[0118] Similarly to steps S32 and 34, use τ 62 to replace the uncertain terms and use a type-2 fuzzy wavelet neural network to approximate the unknown term f 6 , that is

[0119]

[0120] The last group of control inputs u z and the adaptation law are

[0121]

[0122]

[0123] where c 6 , λ 6 and μ 6 represent positive parameters; γ 6 represents a positive number;

[0124] From (29) and (33), (31) can be further deduced as

[0125]

[0126] The beneficial effects of the present invention are as follows:

[0127] 1) In the practical application of a three-axis MEMS gyroscope, parameters and states are easily affected by internal and external disturbances. The present invention more detailedly analyzes the dynamic behaviors of the three-axis MEMS gyroscope under different initial states and system parameters through phase diagrams, time history diagrams, and Lyapunov exponent diagrams, which are closely related to the stability of the gyroscope and provide a basis for continuous controller design.

[0128] 2) For the "complexity explosion" of traditional backstepping control, the present invention designs a hyperbolic tangent tracking differentiator. Compared with a first-order filter and an ordinary tracking differentiator, the designed hyperbolic tangent tracking differentiator can not only eliminate the computational burden brought by repeated iterations, but also approximate the virtual control signal faster and more accurately. In addition, the time-varying barrier Lyapunov function designed by the present invention combines the advantages of a preset performance function and a traditional barrier Lyapunov function, can satisfy the time-varying constraints of state variables, and at the same time limit the tracking error within strict specified performance.

[0129] 3) The acceleration adaptive backstepping controller designed by the present invention not only solves the problems of chaotic oscillation, output constraint, "complexity explosion", and uncertainty existing in the three-axis MEMS gyroscope, but also achieves good tracking performance under various conditions.

[0130] 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

[0131] In order to make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be described in detail preferably with reference to the accompanying drawings, where:

[0132] Figure 1 is the structural diagram of the acceleration adaptive inversion controller for the three-axis MEMS gyroscope designed by the present invention;

[0133] Figure 2 is the two-dimensional schematic diagram of the three-axis MEMS gyroscope;

[0134] Figure 3 is the Lyapunov exponent diagram of the three-axis MEMS gyroscope;

[0135] Figure 4 is the phase diagram and time history diagram of the three-axis MEMS gyroscope;

[0136] Figure 5 is the Lyapunov exponent diagram under different system parameters;

[0137] Figure 6 is the phase diagram under different damping terms;

[0138] Figure 7 is the phase diagram under different spring coefficients;

[0139] Figure 8 is the phase diagram under different angular velocities;

[0140] Figure 9 is the phase diagram under different initial states;

[0141] Figure 10 is the structural diagram of the type-2 fuzzy wavelet neural network;

[0142] Figure 11 is the position tracking performance and control input;

[0143] Figure 12 is the tracking error with the time-varying obstacle Lyapunov function constraint;

[0144] Figure 13 is the comparison between controllers with and without a velocity function;

[0145] Figure 14 is the approximation performance of the hyperbolic tangent tracking differentiator;

[0146] Figure 15 is for different δ xx and the phase diagrams and tracking performance of the controller;

[0147] Figure 16 is the tracking performance of the controller under different initial states;

[0148] Figure 17 is the tracking performance of the controller at different angular velocities. Specific implementation manner

[0149] The following uses specific specific examples to illustrate the implementation manner of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific implementation manners. The details in this specification can also be variously modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the diagrams provided in the following embodiments only illustrate the basic concept of the present invention in a schematic manner. Without conflict, the following embodiments and the features in the embodiments can be combined with each other.

[0150] Please refer to Figures 1 to 10 , the present invention provides a three-axis MEMS gyroscope acceleration adaptive inversion control method with output constraints. As Figure 1 shown, it specifically includes the following parts:

[0151] 1. System modeling

[0152] The three-axis MEMS gyroscope consists of a base, a mass block with spring beams and dampers, an electrostatic actuator for generating a driving force, and an induction device for sensing the position and velocity of the mass block. Figure 2 shows a two-dimensional schematic diagram of the three-axis MEMS gyroscope.

[0153] In order to better establish the dynamic equation of the above gyroscope, two assumptions are proposed:

[0154] Assumption 1 The gyroscope moves at a linear velocity and rotates at a certain speed.

[0155] Assumption 2 Since the displacement is very small, the centrifugal force can be ignored. And the gyroscope moves along the X, Y, and Z axes.

[0156] Based on Newton's laws of motion, the dynamic equation of a three-axis MEMS gyroscope can be expressed as (1), and the parameters are shown in Table 1.

[0157]

[0158] Table 1 System parameters of a three-axis MEMS gyroscope

[0159]

[0160] Define x 1 = x / l 0 , x 3 = y / l 0 , x 5 = z / l 0 , and the dimensionless time t * = ω 0 t. The dimensionless model of the three-axis MEMS gyroscope is rewritten as

[0161]

[0162] where, l 0 = 1 μm represents the reference length, and ω 0 = 1 kHz is the natural resonance frequency.

[0163] Due to the constraints of the physical structure and the actual operating environment, the motion state of the MEMS gyroscope is usually restricted within a certain range. Without taking effective measures, the MEMS gyroscope may frequently malfunction. To meet the constraint conditions of the state variables and ensure the preset tracking performance, a time-varying barrier Lyapunov function is incorporated into the control scheme. The time-varying barrier Lyapunov function is designed as:

[0164]

[0165] where, k i (t) is a positive time-varying decreasing function, e represents the state variable and satisfies k i (t) > |e|.

[0166] 2. Dynamic analysis

[0167] The system parameters of the three-axis MEMS gyroscope are selected as δ xx = δ xy = 0.025, δ xz = δ yy = 0.04, δ xz = δ yy = 0.052, fxy = 98, f xz = 117, f yz = 136.5, ω x = 0.001, ω y = 0.0007, ω z = 0.0017, The initial state is selected as [x 1 (0), …, x 6 (0)] = [0.48, 0.01, 0.51, 0.01, 0.56, 0.01]. Figure 3 Shows the variation of Lyapunov exponents with time. From Figure 3 It can be seen that due to the positive and negative values of the exponents, the motion of the system will enter chaotic and periodic states. Figure 4 Shows the phase diagram of the above gyroscope system ( Figure 4 (a)) and the corresponding time history diagram ( Figure 4 (b)). From Figure 4 It can be found that under the selected parameters and initial state, the whole system exhibits chaotic phenomena. Considering that the chaotic state may only exist under specific conditions, the dynamic behaviors under different initial states and parameters are further studied. Figure 5 Depicts the Lyapunov exponent diagram under the variation of δ xy , f xy and ω x . From Figure 5 It can be found that the three-axis MEMS gyroscope will exhibit chaotic behavior because there are always positive values under different conditions. Figure 6 and Figure 7 Show the phase diagrams under different damping and spring terms. In addition, the phase diagrams under different angular velocities and initial states are as shown in Figure 8 and Figure 9 . From Figures 6 to 9 The phase diagram, it can be seen that the three-axis MEMS gyroscope has disordered and unpredictable motion states under different conditions. In addition, its dynamic behavior is sensitive to the changes of parameters and initial states.

[0168] 3. Design of Accelerated Adaptive Backstepping Controller

[0169] (1) Type-2 Fuzzy Wavelet Neural Network

[0170] Type-2 fuzzy wavelet neural network integrates fuzzy logic, neural network and wavelet transform, and is often used for the identification, control and prediction of unknown nonlinear systems. Due to the combination of wavelet function and fuzzy logic, the performance and computational ability of neural network can be improved. In view of this, a type-2 fuzzy wavelet neural network is constructed to approximate the unknown function of gyroscope and simplify the complex design process of controller. The structure of type-2 fuzzy wavelet neural network is as shown in Figure 10 shown.

[0171] The fuzzy part has N inputs and M fuzzy rules, and the IF-THEN rule is:

[0172] If x 1 is w m1 , x 2 is w m2 …x n is w mn , then

[0173]

[0174] where is the input, w mn represents the m-th fuzzy rule of the n-th input, is the output.

[0175] The upper and lower membership functions can be written as

[0176]

[0177] where c mn , d mn represent the center, upper width and lower width of the membership function.

[0178] The upper and lower activation functions of fuzzification can be calculated as

[0179] f m = μ 1n · μ 2n ·...· μ mn (6)

[0180] Then, the output of the fuzzy neural network part is

[0181]

[0182] The wavelet function is calculated as

[0183]

[0184] Among them, a mn and b mn represent the dilation and translation parameters.

[0185] The output of the wavelet network can be calculated as

[0186]

[0187] Finally, the output of the type-2 fuzzy wavelet neural network is

[0188]

[0189] Among them, θ = [θ 11 , …, θ 1n , …, θ m1 , …, θ mn T , σ = [σ 11 , …, σ 1n , …, σ m1 , …, σ mn T , σ mn = F m w mn .

[0190] For any smooth and continuous function f(x), there exists

[0191]

[0192] Among them, ε(x) represents the positive approximation error and Ω x is a compact set of x. Define the optimal vector θ * as where Ω θ is a compact set of θ. Finally, define where θ * is a manually defined term, and the approximation error satisfies

[0193] (2) Controller design

[0194] To achieve the accelerated convergence of the error in the gyroscope system, a velocity function is introduced.

[0195]

[0196] Among them, is a positive increasing function, whose initial value satisfies and is bounded and continuously differentiable. T ∈ (0, ∞) represents the specified time, b(t) represents the smoothing function and satisfies b(0) = 1 and ​​It is usually selected that b(t) = 1 + t 2 . η is a design parameter satisfying 0 < η << 1.

[0197] Tracking error e i , i = 1, …, 6 is defined as

[0198]

[0199] where x i , i = 1, …, 6 are state variables, x id , i = 1, 3, 5 and α i , i = 2, 4, 6 are the reference trajectories and virtual controls of the X, Y, and Z axes.

[0200] According to (12) and (13), the acceleration tracking error A i is

[0201]

[0202] Based on the backstepping control framework, the design process of the acceleration adaptive backstepping controller can be divided into the following six steps:

[0203] Step 1: The derivative of the first tracking error A 1 is

[0204]

[0205] The first barrier Lyapunov function V 1 is selected as:

[0206]

[0207] Taking the derivative of V 1 gives

[0208]

[0209] The first virtual control rate α 2 is designed as

[0210]

[0211] where c 1 represents a positive design parameter.

[0212] Substituting (18) into (17) gives

[0213]

[0214] Step 2: Select the second Lyapunov function

[0215]

[0216] Calculate V 2 derivative of

[0217]

[0218] wherein, and

[0219] Due to the existence of uncertainties such as parameter variations and drifts, the motion of this gyroscope is always unknown. Considering this, the f involved in the gyroscope dynamics model 2 is regarded as an unknown function. To solve this problem, a type-2 fuzzy wavelet neural network is used to approximate it, that is

[0220]

[0221] In addition, to avoid "complexity explosion" and simplify the complex calculations of the traditional backstepping technique, a hyperbolic tangent tracking differentiator is designed to approximate the derivative of α i then

[0222]

[0223] wherein, ρ i , c i1 and c i2 are adjustable parameters, which mainly affect the tracking speed, tracking effect and differentiation effect.

[0224] Remark 1: It is found during the parameter adjustment process that the larger ρ i , c i1 and the smaller c i2 can improve the tracking speed and accuracy of the hyperbolic tangent tracking differentiator. However, if the parameters are selected too large or too small, the output will fluctuate. Based on this, the above parameters should be appropriately selected to enhance the approximation ability of the hyperbolic tangent tracking differentiator.

[0225] In addition, for the convenience of calculation, it is defined that:

[0226]

[0227] wherein, E τi represents the positive approximation error of the hyperbolic tangent tracking differentiator.

[0228] Then, the control input u x and the adaptive law are derived as

[0229]

[0230]

[0231] Among them, c 2 , λ 2 and μ 2 are positive constants of the design.

[0232] Substituting (19) and (25) into (21), we get

[0233]

[0234] Step 3: The derivative of the third tracking error A 3 is

[0235]

[0236] Select the third barrier Lyapunov function V 3 :

[0237]

[0238] V 3 The derivative of can be calculated as

[0239]

[0240] Then, the second virtual control rate α 4 is designed as

[0241]

[0242] Among them, c 3 represents a positive parameter of the design.

[0243] Based on (28) and (31), (30) can be rewritten as

[0244]

[0245] Step 4: Select the fourth Lyapunov function

[0246]

[0247] Taking the derivative of V 4 , we have

[0248]

[0249] Similarly, using τ 42 to replace Considering uncertain, the type-2 fuzzy wavelet neural network is used to approximate it again, defined as

[0250]

[0251] Control input u y and the adaptation rate can be designed as

[0252]

[0253]

[0254] where c 4 , λ 4 and μ 4 represent positive design parameters.

[0255] Based on (32) and (36), (34) is calculated as

[0256]

[0257] Step 5: The derivative of the fifth tracking error A 5 is

[0258]

[0259] Define the fifth barrier Lyapunov function:

[0260]

[0261] Take the derivative of V 5 with respect to

[0262]

[0263] Then, given the third virtual control rate α 6

[0264]

[0265] where c 5 represents a positive parameter.

[0266] Combining (38) and (42), (41) can be derived as

[0267]

[0268] Step 6: Design the last Lyapunov function V 6 :

[0269]

[0270] Take the derivative of V 6 with respect to, and we get

[0271]

[0272] where

[0273] Inspired by Steps 2 and 4, replace the uncertain terms with τ 62 and approximate the unknown term f with a type-2 fuzzy wavelet neural network i.e., 6 That is

[0274]

[0275] The last set of control inputs and adaptation laws are

[0276]

[0277]

[0278] where c 6 , λ 6 and μ 6 represent positive parameters.

[0279] From (43) and (47), (45) can be further derived as

[0280]

[0281] 4. Stability analysis

[0282] Theorem 1: For the three-axis MEMS gyroscope (2), under the assumptions 1 and 2, the control inputs of the accelerated adaptive backstepping controller are constructed as (25), (36) and (47), with the adaptation laws (26), (37) and (48). Then, if the control parameters are properly selected, the following conclusions hold.

[0283] All signals in the closed-loop system are bounded, and the state variables are restricted within the given ranges.

[0284] The inherent chaotic oscillations that may cause system failures are completely eliminated.

[0285] Proof: Select the global Lyapunov function as

[0286]

[0287] The derivative of Equation (50) can be calculated as

[0288]

[0289] According to Young's inequality, we have

[0290]

[0291] Then Equation (52) can be rewritten as

[0292]

[0293] In addition, there exists

[0294]

[0295] Equation (51) can be further derived as

[0296]

[0297] where r 0 = min{2c i , γ i} and

[0298] Based on the general solution of the first-order linear differential equation, the solution of Equation (55) can be further expressed as

[0299]

[0300] where t 0 represents the initial time.

[0301] In addition, a compact set is defined as

[0302]

[0303] It can be concluded that all signals of the closed-loop system are bounded, that is The proof of Theorem 1 is completed.

[0304] In addition, when |A i | → |k i (t)| This proves that when k i (t) ≠ A i and are both ultimately bounded. From this, it can be concluded that if |A i | < |k i (t)|, the constraints on the system can be satisfied.

[0305] 5. Analysis of Simulation Experiment Results

[0306] The reference trajectory is selected as x 1d = 0.5sin(1.5t) + 0.5sin(2t + π / 3), x 3d = -0.3sin(2t) + 0.5cos(1.2t) and x 5d = 0.5cos(1.5t) + 0.3sin(2t). In the type-2 fuzzy wavelet neural network part, the parameters are set as c 1n = [0, 15, 0.2, 0.25]c 2n ​= [0, 2, 0.25, 0.3], c 3n = [0, 16, 0.21, 0.26], c 4n = [1.95, 2, 2.05], c 5n = [0.95, 1, 1.05], c 6n = [13.95, 14, 14.05] and c mn = 0.4.b mn with c mn set to be the same, the expansion coefficient and n = 1, …, 6. Then, the parameters of the velocity function are designed as T = 1 and η = 0.5. Considering the different motions and initial parameters of the three axes, three time-varying obstacle Lyapunov functions are designed: k 1 (t) = (0.05 - 0.006)e -5t + 0.005, k 3 (t) = (0.04 - 0.007)e -4t + 0.005 and k 5 (t) = (0.06 - 0.007)e -3t + 0.005. In addition, the parameters of the hyperbolic tangent tracking differentiator can be designed as ρ i = 12, c i1 = 30 and c i2 = 5. During the backstepping process, the controller parameters are set as [c 1 , …, c 6 = [700, 800, 600, 700, 600, 800], and the parameters of the adaptive law are set as λ i = 1.1 and γ i = 2.5, i = 2, 4, 6.

[0307] Figure 11 shows the position tracking performance and control input of the designed controller. Among them, Figure 11 (a) is the tracking performance of x 1 , Figure 11 (b) is the tracking performance of x 3 , Figure 11 (c) is the tracking performance of x 5 , Figure 11 (d) is the control input of the X, Y, and Z axes. It can be seen from Figure 11 that the actual signals (dashed lines) of the three axes can quickly and accurately track their reference trajectories (curves). At the same time, Figure 11 (d) shows that the control input of the controller is smooth and periodic. The tracking error of the designed controller is as Figure 12As shown, it can be seen that the three tracking errors can be restricted to the interval [-0.005, 0.005] by their associated time-varying barrier Lyapunov functions.

[0308] To verify the effectiveness of the designed speed function, a controller without the speed function was redesigned while other conditions were the same as those of the acceleration controller. As Figure 13 shown, the fluctuation amplitude (curve) of the tracking error of the acceleration controller is less than that of the non-acceleration controller (dashed line). From Figure 13 it can also be seen that within the first few seconds, the three acceleration errors converge faster and oscillate less. In addition, Figure 14 shows the approximate performance of the designed hyperbolic tangent tracking differentiator, where Figure 14 (a) is the virtual control rate αi and its approximation τ i1 , Figure 14 (b) is the approximation performance at different angular velocities. From Figure 14 it can be found that the hyperbolic tangent tracking differentiator (dashed line) not only reduces the computational complexity but also can approximate the virtual control input (curve) with high precision. At the same time, the hyperbolic tangent tracking differentiator is proven to have anti-interference ability, as Figure 14 (b) shows.

[0309] To verify the robustness of the proposed scheme, factors such as the initial state and system parameters were changed. Figure 15 shows the phase diagram ([[]] and δ xx under different [[[]] Figure 15 (a)) and the tracking error ([[]] Figure 15 (b)) of the controlled three-axis MEMS gyroscope. Compared with [[[]] Figure 4 , it can be seen that the proposed controller suppresses the chaotic oscillation of the gyroscope. In addition, the tracking performance can be restricted within the specified constraints and the fluctuation is very small. Figure 16 shows that with the change of the initial state, the tracking error is still restricted within the specified performance range and the fluctuation is very small. From Figure 17 it can be seen that although the angular velocities ω x , ω y and ω z are increased by 30 times, the tracking performance of the proposed controller can still achieve good results. It can be summarized from the above figures that the proposed control scheme not only has the ability to resist the interference caused by parameter and state changes but also meets the strict specified performance requirements.

[0310] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not 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. A three-axis MEMS gyroscope acceleration adaptive inversion control method with output constraints, Characterized in that, In the controller design, a type-2 fuzzy wavelet neural network is used to approximate the nonlinear unknown function of the system. Under the backstepping framework, a velocity function is used to achieve accelerated convergence with a small amplitude, and a time-varying barrier Lyapunov function is constructed to limit the state variables within a specified range. At the same time, a hyperbolic tangent tracking differentiator is used to reduce the computational complexity of the time-varying barrier Lyapunov function in the backstepping framework. The type-2 fuzzy wavelet neural network, velocity function, time-varying barrier Lyapunov function, hyperbolic tangent tracking differentiator and adaptive law are integrated into the backstepping framework to design the entire controller. The method specifically includes the following steps: S1: Construct a three-axis MEMS gyroscope dynamic model, and construct a time-varying barrier Lyapunov function to limit the state variables within a specified range; S2: Construct a type-2 fuzzy wavelet neural network to approximate the nonlinear unknown function of the gyroscope; S3: Construct a three-axis MEMS gyroscope acceleration adaptive inversion control model, specifically by integrating the type-2 fuzzy wavelet neural network, velocity function, time-varying barrier Lyapunov function, hyperbolic tangent tracking differentiator and adaptive law into the inversion control model; In step S1, the constructed three-axis MEMS gyroscope dynamic model is: Among them, x, y, and z are displacement axes, d xy , d xz , d yz are the symmetry terms of the spring, d xx , d yy , d zz are damping coefficients, k xy , k xz , k yz are the asymmetry terms of the spring, k xx , k yy , k zz are linear spring coefficients, is the angular velocity, is the control force, m is the mass of the internal mass block, ω 0 = 1 kHz is the natural resonance frequency, l 0 = 1 μm represents the reference length; x 1 = x / l 0 , x 3 = y / l 0 , x 5 = z / l 0 , The expression of the constructed time-varying barrier Lyapunov function is: where k i (t) is a positive time-varying decreasing function, e represents the state variable and satisfies k i (t) > |e|; In step S3, constructing a three-axis MEMS gyroscope acceleration adaptive inversion control model specifically includes the following steps: S31: Calculate the derivative of the first tracking error A 1 The derivative The expression is: Among them, denotes the first derivative of; denotes the velocity function, and the expression is: where is a positive increasing function, and its initial value satisfies and is bounded and continuously differentiable; T ∈ (0, ∞) denotes the specified time, b(t) denotes the smoothing function and satisfies b(0) = 1 and η is a design parameter satisfying 0 < η << 1; e i is the tracking error, and the expression is: where x i , i = 1, …, 6 are state variables, x id , i = 1, 3, 5 and α i , i = 2, 4, 6 are the reference trajectories and virtual controls of the X, Y, and Z axes respectively; Select the first barrier Lyapunov function V 1 as follows: where k 1 (t) represents a positive definite time-varying attenuation function; Derive with respect to V 1 The derivative is obtained as follows: The first virtual control rate α 2 is designed as: where c 1 represents a positive design parameter; Substituting (4) into (3) gives: Among them, A 2 represents the second acceleration tracking error; S32: Select the second Lyapunov function V 2 as follows: Calculate V 2 The derivative of is: where f 2 is regarded as an unknown function, and the expression is Use a type-2 fuzzy wavelet neural network to approximate f 2 , that is Among them, represents the weights of the type-2 fuzzy wavelet neural network, and σ 2 represents the basis function of the type-2 fuzzy wavelet neural network, and ε 2 represents the approximation error of the type-2 fuzzy wavelet neural network; Design a hyperbolic tangent tracking differentiator to approximate the derivative of α i then where τ i1 represents a variable, and τ i2 also represents a variable; ρ i , c i1 and c i2 are adjustable parameters Define: Among them, E τi represents the positive approximation error of the hyperbolic tangent tracking differentiator; Then, the control input u x and the adaptation law are derived as follows: where c 2 , λ 2 and μ 2 are positive constants of the design; γ 2 represents a positive constant; Substituting (5) and (11) into (7) gives S33: The third tracking error A 3 has a derivative of Select the third barrier Lyapunov function V 3 It is: where k 3 (t) represents a positive definite time-varying attenuation function; Then V 3 has a derivative of Then, the second virtual control rate α 4 is designed as: where c 3 represents a positive parameter of the design; Based on (14) and (17), (16) is rewritten as: S34: Select the fourth Lyapunov function V 4 as follows: Among them, A 4 is the fourth acceleration tracking error; Derive with respect to V 4 The derivative is Similarly, using τ 42 to replace Considering uncertainty, the type-2 fuzzy wavelet neural network is used again to approximate it, defined as Control input u y and adaptation rate can be designed as where c 4 , λ 4 and μ 4 represent positive design parameters; γ 4 represents a positive constant; Based on (18) and (22), (20) is calculated as S35: The fifth tracking error A 5 The derivative of Define the fifth barrier Lyapunov function V 5 as follows: where k 5 (t) represents a positive definite time-varying attenuation function; Derivative with respect to V 5 Derivative Then, given a third virtual control rate α 6 is: where c 5 represents a positive parameter; Combining (24) and (28), (27) is derived as S36: Design the last Lyapunov function V 6 : Among them, A 6 represents the sixth tracking error; Derive with respect to V 6 and obtain the derivative where f 6 = -δ xz x 2 -δ yz x 4 -δ zz x 6 -f xz x 1 -f yz x 3 -f z 2 x 5 +2ω y x 2 -2ω x x 4 ; Similarly to steps S32 and 34, use τ 62 to replace the uncertain terms and use a type-2 fuzzy wavelet neural network to approximate the unknown term f 6 , that is The last set of control inputs u z and the adaptation law are where c 6 , λ 6 and μ 6 represent positive parameters; γ 6 denotes a positive number; From (29) and (33), (31) is further derived as 2. The three-axis MEMS gyroscope acceleration adaptive inversion control method according to claim 1, Characterized in that, In step S2, a type-2 fuzzy wavelet neural network is constructed, which specifically includes: Assuming that the network has N inputs and M fuzzy rules, the IF-THEN rule is: If x 1 is w m1 , x 2 is w m2 … x n is w mn , then where x n is the nth input, w mn represents the mth fuzzy rule of the nth input, and W m is the output of the fuzzy rule; The upper and lower membership functions are written as: Among them, c mn , d mn represent the center, upper width, and lower width of the membership function, respectively; The calculation of the upper and lower activation functions of fuzzification is: f m = μ 1n · μ 2n ·...· μ mn Among them, f m respectively represent the upper and lower activation functions for fuzzification, μ mn respectively represent the upper and lower membership functions; Then the output F of the fuzzy neural network part m is as follows: The wavelet function is calculated as: where a mn and b mn represent the dilation and translation parameters; The output W of the wavelet network m is as follows: Finally, the output O of the type-2 fuzzy wavelet neural network m is as follows: where θ = [θ 11 , …, θ 1n , …, θ m1 , …, θ mn T , σ = [σ 11 , …, σ 1n , …, σ m1 , …, σ mn T , σ mn = F m w mn , θ mn represents the weights of the type-2 fuzzy wavelet neural network, and σ mn represents the basis functions of the type-2 fuzzy wavelet neural network.​​ 3. The three-axis MEMS gyroscope acceleration adaptive inversion control method according to claim 2, Characterized in that, In step S2, a type-2 fuzzy wavelet neural network is constructed to approximate the nonlinear unknown function of the gyroscope, specifically including: for any smooth continuous function f(x), there exists where ε(x) represents a positive approximation error, and Ω x is a compact set of x; the optimal vector θ * is defined as where Ω θ is a compact set of θ; finally, define where θ * is an artificially defined term, and the approximation error satisfies

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