Preset finite time backstepping control method of four-degree-of-freedom MEMS resonator small network
By constructing a small network of four-degree-of-freedom MEMS resonators and combining it with a pre-defined finite-time backstepping control method, the problem of insufficient bandwidth and sensitivity of traditional MEMS resonators is solved, achieving high-performance suppression of chaotic oscillations and fast convergence, thus improving the robustness and flexibility of the system.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-11
- Publication Date
- 2026-03-31
AI Technical Summary
Traditional single-degree-of-freedom MEMS resonators have limited bandwidth, low sensitivity, and weak fault tolerance. Multi-degree-of-freedom MEMS resonators are difficult to achieve fast convergence under robust control in high-frequency chaotic oscillations, and there is a differential term explosion problem in traditional backstepping control.
A small network for a four-degree-of-freedom MEMS resonator is constructed. Using a preset performance function and a type-II sequential fuzzy neural network, an accelerated exponential integral tracking differentiator is designed. Combined with a preset finite-time backstepping controller, the trajectory tracking of the resonator's position and velocity is realized.
It achieves high-performance control within a finite time, suppresses chaotic high-frequency oscillations, improves sensitivity and bandwidth, avoids the differential term explosion problem, and has greater design flexibility and anti-interference capability.
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Figure CN121763732A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of microelectromechanical systems (MEMS) control technology, and relates to a preset finite-time backstepping control method for a small network of four-degree-of-freedom MEMS resonators. Background Technology
[0002] MEMS has made rapid progress in materials, structure, control, and circuit integration in recent years. It is a miniaturized crystal oscillator with advantages such as small size, lightweight structure, high reliability, strong resistance to external interference, and seamless integration with application-specific integrated circuits (ASICs) and central processing units (CPUs). Driven by the ever-increasing demand for improved device performance, traditional single-degree-of-freedom MEMS resonators, due to their limited bandwidth, low sensitivity, and weak fault tolerance, are no longer adequate. To address these limitations, constructing an effective multi-degree-of-freedom MEMS resonator structure with interconnected resonators has become a hot topic. Meanwhile, ensuring robust control while suppressing high-frequency chaotic oscillations remains a challenge, particularly in avoiding state constraint violations and ensuring rapid convergence of tracking errors.
[0003] In recent years, significant progress has been made in the structural design and mathematical modeling of single-degree-of-freedom (DOF) MEMS resonators. Mestrom et al. introduced a modeling method capable of capturing the nonlinear dynamic phenomena of MEMS resonators; Kaajakari et al. established an analytical model based on the nonlinear engineering Young's modulus; and Ouakad et al. proposed a model of a bent micro-beam MEMS resonator fixed at both ends. However, the limited bandwidth and sensitivity of single-DOF MEMS resonators can no longer meet the high-performance requirements of current engineering applications. To overcome these shortcomings, recent research has shifted towards multi-DOF, multi-coupled MEMS resonators. Thiruvenkatanathan et al. reported the vibration mode localization phenomenon in electrostatically coupled MEMS resonators and demonstrated its potential to significantly improve sensitivity. Sathiya et al. proposed a mathematical model for a two-DOF resonator. Zhao et al. developed a force sensor based on a three-DOF weakly coupled resonator, achieving a sensitivity improvement of two orders of magnitude compared to traditional schemes. Wang et al. investigated the mass sensitivity of a three-DOF electrostatically coupled resonator system under ambient pressure. Despite these advances, two-degree-of-freedom resonators still lack the performance required for high-precision sensing and complex signal processing, while three-degree-of-freedom resonators are limited by their odd-numbered topologies, hindering their ability to achieve symmetrical filtering characteristics. To address these challenges, developing small-scale networks composed of multiple coupled MEMS resonators has become an effective approach.
[0004] As is well known, meticulous design of dynamic nonlinear systems is fundamental to achieving optimal performance. Therefore, many scholars have studied the dynamic behavior of MEMS resonators. For example, Miandoab et al. investigated the nonlinear dynamics of bifacial electrostatic MEMS resonators, revealing that the chaotic response of the resonator is caused by homoclinic and heteroclinic orbits. Luo et al. studied the transient chaotic characteristics of single-degree-of-freedom MEMS resonators. Siewe et al. explored the chaotic dynamics of MEMS resonators using numerical methods. Luo et al. further revealed the dynamic characteristics of fractional-order MEMS resonators. However, these studies did not simultaneously consider the variations in system parameters and applied AC voltage.
[0005] Dynamic behavior, especially high-frequency chaotic oscillations, can directly lead to a decrease in system accuracy, damage to stability, and even system collapse. To suppress these harmful oscillations, Qiao et al. proposed a stochastic optimal control scheme for electrostatic MEMS resonators, Song et al. developed a second-order fast terminal sliding mode controller, effectively eliminating chaotic oscillations in MEMS resonators, and Yau et al. proposed a robust fuzzy sliding mode controller to achieve a stable transition from chaotic oscillations to periodic motion in MEMS resonators under parameter uncertainty. However, these studies have not effectively solved the inherent "differential term explosion" problem in traditional backstepping control. To address this problem, Luo proposed a second-order tracking differentiator and applied it to an advanced control framework, while Li et al. designed an accelerated exponential integral tracking differentiator. In practical applications, electromechanical systems must operate within specified safety boundaries, and constraint control is considered an effective method for solving this problem. However, traditional asymptotic stability methods cannot guarantee a predetermined convergence time. To overcome this limitation, Moulay et al. explored the properties of the convergence time function and its Lyapunov method, proposing a finite-time control method. In the above works, the convergence time increases as the initial state deviates further from the equilibrium point. Therefore, their performance is inherently dependent on initial conditions and cannot be guaranteed to converge within a specified time. Summary of the Invention
[0006] In view of this, the purpose of this invention is to provide a preset finite-time backstepping control method for a small network of four-degree-of-freedom MEMS resonators. To achieve the above objectives, the present invention provides the following technical solution: A pre-defined finite-time backstepping control method for a small four-degree-of-freedom MEMS resonator network includes: A small network of four-degree-of-freedom MEMS resonators was constructed, and its dynamic model was established. Construct a preset performance function and define the constraint boundary of the position tracking error, requiring the tracking error to converge within a preset finite time and remain within the range defined by the preset performance function; Perform system state transitions by using constraint transition functions to transform error variables that satisfy preset performance function constraints into new unconstrained state variables; Design a type II sequential fuzzy neural network to approximate unknown nonlinear functions and external disturbances in a dynamic model; Design an accelerated exponential integral tracking differentiator to estimate the derivative of the virtual control law during backstep control design; A pre-defined finite-time controller is designed. Based on the system state transition, the output of the type II sequence fuzzy neural network, and the output of the accelerated exponential integral tracking differentiator, the control input applied to the small network of the four-degree-of-freedom MEMS resonator is derived to achieve trajectory tracking of the resonator's position and velocity.
[0007] The beneficial effects of this invention are as follows: (1) Compared with single-degree-of-freedom, two-degree-of-freedom and three-degree-of-freedom MEMS resonators, this invention proposes a small network structure composed of four coupled MEMS resonators. The proposed network structure has superior sensitivity and wider bandwidth through frequency response and 3dB bandwidth analysis. Based on this, a dynamic model of a small network of four-degree-of-freedom MEMS resonators (including electrostatic coupling and mechanical coupling) is established.
[0008] (2) This invention proposes a predefined finite-time backstepping control method for a small network of a four-degree-of-freedom MEMS resonator, which achieves high-performance control while suppressing chaotic high-frequency oscillations.
[0009] (3) The present invention can pre-set the convergence time and error range according to user needs, which has greater design flexibility, avoids directly processing state constraints by transforming the system, and eliminates the common problem of repeated differential term explosion in backstep design.
[0010] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description
[0011] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein: Figure 1 This is a schematic diagram of a small network for a four-degree-of-freedom MEMS resonator. Figure 2 (a) The frequency response of small MEMS resonator networks with two-coupling, three-coupling, and four degrees of freedom at 3dB; Figure 2 (b) Comparison of bandwidth at 3dB for small networks of two-coupled, three-coupled, and four-degree-of-freedom MEMS resonators; Figure 3 A schematic diagram of the mass-spring-damping model of a small network for a four-degree-of-freedom MEMS resonator. Figure 4 This is a schematic diagram of the structure of a type II sequence fuzzy neural network; Figure 5 (a) and Figure 5 (b) are schematic diagrams of position tracking performance; Figure 6 (a) and Figure 6 (b) is a schematic diagram of velocity tracking performance under different reference signals; Figure 7 (a) and Figure 7 (b) is a schematic diagram of position tracking error under different parameters; Figure 8 (a) and Figure 8 (b) is a schematic diagram of different control inputs; Figure 9 (a) and Figure 9 (b) Schematic diagrams of position tracking performance and error under different initial states; Figure 10 (a) and Figure 10 (b) Schematic diagrams comparing tracking performance and error under different control schemes; Figure 11 (a) and Figure 11 (b) are schematic diagrams comparing the tracking performance and error of the accelerated exponential integral tracking differentiator; Figure 12 This is a structural block diagram of the method provided by the present invention. Detailed Implementation
[0012] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of the present invention. Unless otherwise specified, the following embodiments and features can be combined with each other.
[0013] The accompanying drawings are for illustrative purposes only and are schematic diagrams, not actual pictures. They should not be construed as limiting the invention. To better illustrate the embodiments of the invention, some parts in the drawings may be omitted, enlarged, or reduced, and do not represent the actual product dimensions. It is understandable to those skilled in the art that some well-known structures and their descriptions may be omitted in the drawings.
[0014] In the accompanying 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 terms such as "upper," "lower," "left," "right," "front," and "rear" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, they are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, the terms used to describe positional relationships in the drawings are only for illustrative purposes and should not be construed as limiting the present invention. For those skilled in the art, the specific meaning of the above terms can be understood according to the specific circumstances.
[0015] An embodiment of the present invention provides a preset finite-time backstepping control method for a small network of a four-degree-of-freedom MEMS resonator, the method mainly including: First, in order to improve sensitivity and bandwidth, a novel small network structure for coupled MEMS resonators is proposed, and a corresponding dynamic model is established.
[0016] Secondly, in order to suppress harmful chaotic oscillations and achieve high-performance control objectives, a pre-set finite-time backstepping control method is proposed. This method integrates a pre-set performance function, transforms the state constraint problem into a boundedness problem of newly defined variables, uses a type-two sequential fuzzy neural network to approximate the unknown nonlinear function, and employs an improved accelerated exponential integral tracking differentiator to solve the "differential term explosion" problem in backstepping control design.
[0017] Finally, the feasibility, robustness, and superiority of the proposed method are verified through simulation results.
[0018] Please see Figures 1 to 5 The preset finite-time backstepping control method for a small four-degree-of-freedom MEMS resonator network provided in this embodiment is as follows: 1. Design a small network for a four-degree-of-freedom MEMS resonator, such as... Figure 1 As shown, it includes four identical resonators, two inertial mass blocks, four drive combs, six anchor points, two sets of parallel capacitor plates, and a mechanical coupling beam connecting resonators 2 and 3.
[0019] Among them, resonators 1 and 2, as well as resonators 3 and 4, are electrostatically coupled through parallel capacitor plates, while resonators 2 and 3 are mechanically coupled through mechanical coupling beams.
[0020] The detection electrodes are located on the right side of resonator 1 and the left side of resonator 4 to achieve accurate measurement of the resonator dynamics. Its working principle is as follows: when an AC voltage is applied to the driving electrode and a DC bias voltage is applied to resonators 1 and 4, an electrostatic driving force is generated, inducing micro-oscillations in the resonators. The energy from these oscillations is transferred to resonators 2 and 3 through electrostatic coupling provided by the parallel plates. Simultaneously, resonators 2 and 3 interact through mechanical coupling.
[0021] To theoretically verify the feasibility of the proposed structure, the frequency response and 3dB bandwidth of the small network were analyzed. For example... Figure 2 As shown in (a), the two-coupled, three-coupled, and four-degree-of-freedom MEMS resonators have 4, 6, and 8 resonance peaks, respectively. Figure 2 (b) gives the corresponding 3dB bandwidth, and it is clear that the small network of four-degree-of-freedom MEMS resonators has a wider bandwidth than that of two-resonators and three-resonators.
[0022] 2. In order to perform system modeling of a small network of four-degree-of-freedom MEMS resonators, its equivalent mass-spring-damping model is established, such as... Figure 3 As shown.
[0023] Wherein, the spring force is expressed as (1) in, and Let these represent the first-order and third-order terms of the nonlinear stiffness, respectively. This represents the displacement.
[0024] Based on equation (1), the nonlinear stiffness can be expressed as: (2) in, This represents the nonlinear stiffness of each resonator.
[0025] The total kinetic energy of the four-degree-of-freedom MEMS resonator small network is: (3) in, This represents the total mass of each resonator.
[0026] The potential energy of a small network of a four-degree-of-freedom MEMS resonator can be expressed as: (4) in, This indicates the coupling stiffness between adjacent resonators.
[0027] The electrostatic driving force within a small four-degree-of-freedom MEMS resonator network is generated by the combined effect of the AC voltage applied to the driving electrodes and the DC bias voltage applied to the resonators. Based on the above working principle, only resonators 1 and 4 generate the electrostatic driving force, as expressed below: (5) in, , , and These represent capacitance, initial gap width, displacement, and frequency, respectively. and These represent DC voltage and AC voltage, respectively. For time.
[0028] Lagrange function and generalized force The derivation is as follows: (6) in, Indicates damping.
[0029] The Lagrange equation for a four-degree-of-freedom MEMS resonator is derived as follows: (7) in, , , , All are first-order terms of nonlinear stiffness. , , , All are cubic terms of nonlinear stiffness; The electrostatic coupling stiffness can be written as: (8) in, Indicates bias voltage and difference, Represents the vacuum permittivity. and These represent the effective cross-sectional area and initial gap between adjacent resonators, respectively.
[0030] definition , , , , , , , ,in This indicates the mechanical coupling stiffness.
[0031] To simplify the design, new variables are introduced: , , , , , , , , , Equation (7) can then be derived as: (9) make , , , , and By performing a Taylor expansion on equation (9) and ignoring higher-order nonlinear terms beyond the third order, the equation with control input is simplified. The mathematical model is as follows: (10) in:
[0032]
[0033]
[0034]
[0035] 3. Design a preset finite-time controller (1) Design a type II sequence fuzzy neural network, which consists of five layers: input layer, fuzzification layer, membership layer, fuzzy rule layer, and output layer, such as Figure 4 As shown.
[0036] In the input layer, external input variables are directly connected to the input of each neuron in the fuzzy layer.
[0037] In the fuzzing layer, input variables Through the following transformation: (11) in, and These represent the up-conversion input and the down-conversion input, respectively. and Membership functions The top and bottom widths and Membership functions The top and bottom widths It represents the center of the membership function. J , M , N All are integers.
[0038] In the membership hierarchy, the membership functions of the upper and lower layers are expressed as follows: (12) At the rule level, the IF-THEN fuzzy rule is: :like for , ...; for ,…;and for ;So , ,in Let jm be the nth Gaussian type 2 membership function of the jmth input. For input, and These represent the upper and lower weights, respectively.
[0039] According to the above design rules, the upper activation intensity and the lower activation intensity can be written as follows: (13) In the output layer, the linear combination of the inputs is as follows: (14) in, and This represents the weight vector.
[0040] The output of the neural network for: (15) in, , .
[0041] For any smooth function, it is continuous There always exists a sufficiently small constant. , so that: (16) in, Represents the approximation function The absolute value of the approximation error, and It is a vector A compact set. Additionally... and .
[0042] The optimal vector is defined as: (17) in, Representing variables A compact set. , This represents the ideal weight.
[0043] (2) Design the preset performance function To achieve fast convergence within a given time, a pre-defined performance function is constructed: (18) in, The given time set by the user .
[0044] about It has the following properties: ①When hour, Monotonically decreasing from 1 to ;when hour, equal .
[0045] ②For , and They are all continuous and bounded.
[0046] ③For , It is bounded.
[0047] Position tracking error is defined as: (19) in and These are the state variables and reference trajectory of a small network of a four-degree-of-freedom MEMS resonator.
[0048] To achieve tracking within a preset finite time, the position tracking error should meet the following constraints: (20) in It is a bounded constant.
[0049] The designed preset performance function mainly determines the convergence time of the controller and subsequent system transformations. It is the convergence time set in advance by the user. Related to tracking accuracy, i.e. The smaller the value, the higher the tracking accuracy.
[0050] (3) System conversion make Equation (20) is equivalent to: (twenty one) To handle error constraint issues, the constraint transformation function is designed as follows: (twenty two) Will use express, for The Kth derivative is obtained. , ,therefore It is a bounded, second-order continuously differentiable function.
[0051] To address the velocity constraint problem, a constraint transformation function is introduced. (twenty three) Will use Indicate, then It is a bounded, first-order, continuously differentiable function.
[0052] Construct a new function and convert it to an unconstrained form: (twenty four) Obviously, and It has the following properties: ① , .
[0053] ②When hour, ;when hour, .
[0054] Based on the above properties, the following theory can be derived: For any initial condition, , ,if , If it is bounded, then and It always holds true.
[0055] Differentiating equation (24) yields: (25) in, , , .
[0056] Furthermore, the original system (10) can be rewritten as: (26) (4) Design the controller definition and for: (27) in This is a virtual control law.
[0057] The first derivative is: (28) right Taking the derivative, we get: (29) Design an accelerated exponential integral-differential tracker to approximate the derivative of the virtual control law to solve the "differential term explosion" problem in traditional backstepping control. Its expression is as follows: (30) The input to the accelerated exponential integral-differential tracker is a virtual control signal. , and This represents the output of the accelerated exponential integral-differential tracker; , and All numbers represent positive integers. It should be noted that... and Determines tracking speed and accuracy. The rate at which the derivative of the approximation virtual control law is determined.
[0058] The inequality is defined as: (31) in .
[0059] Based on the above backstepping control framework, the design process of a predefined finite-time controller is decomposed into eight steps: Step 1: Consider the first Lyapunov function as: (32) right The derivative is: (33) The first virtual control law was designed as follows: (34) in Represents a positive integer.
[0060] Further, we can conclude that: (35) Step 2: The second Lyapunov function is defined as follows: (36) in For positive integers, These are the weights of a type II sequence fuzzy neural network.
[0061] right Differentiating, we get: (37) In practical applications, the small network of a four-degree-of-freedom MEMS resonator is affected by various unknown disturbances and component aging. These disturbances may lead to uncertainties in the mathematical model, making it difficult to obtain accurate physical parameters. Therefore, a type-two sequential fuzzy neural network is used to estimate... : (38) Substitute (38) into (37) to control the input. And adaptive law They are respectively: (39) (40) in and It is a positive number.
[0062] Substituting (39) and (40) into (37), we get (41) in .
[0063] Step 3: Consider the third Lyapunov function as follows: (42) The first derivative is: (43) The second virtual control law is designed as follows: (44) in It is a positive number.
[0064] Substituting (44) into (43) gives: (45) Step 4: The fourth Lyapunov function is defined as follows: (46) in It is a positive number.
[0065] right Differentiating, we get: (47) Similarly, using to replace A type II sequence fuzzy neural network is used to approximate the unknown function. ,Right now: (48) Subsequently, the corresponding control inputs and adaptive rates were designed: (49) (50) in and All are normal numbers.
[0066] Substituting (49) and (50) into (47) yields: (51) in .
[0067] As can be seen from equation (10), the mathematical models of resonators 3 and 4 exhibit a similar form to those of resonators 1 and 2. Therefore, based on the derivation process of steps 1-4 above, the following are obtained: the fifth, sixth, seventh, and eighth Lyapunov functions are: (52) (53) in It is a positive number.
[0068] Design virtual control law and for: (54) in It is a positive number.
[0069] The control inputs and adaptive laws for resonators 3 and 4 can be directly written as: (55) (56) in , For positive integers, .
[0070] Similar to equations (41) and (51), we can obtain: (57) in .
[0071] Stability analysis was performed on the method provided in this embodiment: A theorem is given: For a small network of a four-degree-of-freedom MEMS resonator, a pre-defined finite-time backstepping controller (composed of control inputs (39), (49), (55) and type II sequence fuzzy neural network adaptive laws (40), (50), (56)) is designed. By appropriately selecting parameters and initial conditions, the following conclusion is drawn: ① All signals in a closed-loop system are bounded.
[0072] ② All system states are confined to a predetermined range within a finite time.
[0073] Proof of the above theorem: Construct the Lyapunov function as follows: (58) Differentiating (58) yields (59) By applying Young's inequality, the following results are obtained: (60) Substituting (59) into (58) yields (61) in , .
[0074] make ,when At that time, there exists .therefore, It always holds true.
[0075] Integrating equation (60) yields: (62) It can be proven that all signals in a closed-loop system are bounded.
[0076] Q.E.D.
[0077] The method provided in this embodiment is analyzed through simulation experiments, as follows: Select reference trajectory , .
[0078] For a type-II sequence fuzzy neural network, the upper and lower widths of the membership function are set as follows: =0.05, =0.1, =0.5 and =1, the center of the membership function is set as: For the accelerated exponential integral-differential tracker, the design parameters are selected as follows: , , , Meanwhile, the controller parameters are defined as follows: , , , , , , and The parameters of the adaptive law are selected as follows: , , Set the parameters of the predefined finite-time performance function to... , , and .
[0079] Position tracking and velocity tracking performance are respectively as follows Figure 5 and Figure 6 As shown, the actual signal achieves high-precision tracking of its reference trajectory, and the state of the four-degree-of-freedom MEMS resonator network is strictly limited within the specified constraints within a finite time. The position tracking error is as follows: Figure 7 As shown, its position tracking error stabilizes near zero within 0.5 s and is confined within the preset constraint boundaries. The control input is as follows: Figure 8 As shown, it can be seen that all control inputs of the controller quickly converge to a stable range. Figure 9 The position tracking performance under different initial states is demonstrated, showing that the control scheme proposed in this embodiment has strong anti-interference ability.
[0080] To verify the superiority of the control scheme proposed in this embodiment, it was compared with schemes A and B. From... Figure 10 It is evident that the solution presented in this embodiment has smaller tracking errors and faster convergence speed. For example... Figure 11 As shown, compared with the accelerated exponential integral tracking differentiator, the scheme in this embodiment has a faster response time and higher tracking accuracy.
[0081] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.
Claims
1. A pre-set finite-time backstepping control method for a four-degree-of-freedom MEMS resonator small network, characterized in that, The method comprises: A four-degree-of-freedom MEMS resonator small network is constructed, and a dynamic model thereof is established; A preset performance function is constructed, a position tracking error constraint boundary is defined, and the tracking error is required to converge within a preset limited time and to be kept within a range defined by the preset performance function; System state conversion is performed, and an error variable satisfying the preset performance function constraint is converted into a new state variable without constraint by using a constraint conversion function; A two-type sequence fuzzy neural network is designed for approximating an unknown nonlinear function in the dynamic model and external disturbance; An accelerated exponential integral tracking differentiator is designed for estimating a derivative of a virtual control law in a backstepping control design process; A predefined limited time controller is designed, and a control input applied to the four-degree-of-freedom MEMS resonator small network is derived according to the system state conversion, an output of the two-type sequence fuzzy neural network and an output of the accelerated exponential integral tracking differentiator, so that trajectory tracking of resonator positions and velocities is realized.
2. The method of claim 1, wherein, The four-degree-of-freedom MEMS resonator small network comprises four resonators, wherein the first resonator and the second resonator, and the third resonator and the fourth resonator are coupled by electrostatic coupling, and the second resonator and the third resonator are coupled by mechanical coupling; The dynamic model established for the four-degree-of-freedom MEMS resonator small network is an equivalent mass-spring-damper model, wherein a spring force is represented as wherein, and denote a linear and a cubic term of the non-linear stiffness, respectively, is the displacement amount; based on the spring force, the non-linear stiffness is expressed as: wherein, denotes the nonlinear stiffness of each resonator; The total kinetic energy of the four-degree-of-freedom MEMS resonator small network is: wherein, Mtotai represents the total mass of each resonator; The potential energy of the four-degree-of-freedom MEMS resonator small network is: wherein denotes the coupling stiffness between adjacent resonators; The electrostatic driving force in the four-degree-of-freedom MEMS resonator small network is generated by the combined effect of an alternating voltage applied to a driving electrode and a direct current bias voltage applied to the resonator, and only the resonator 1 and the resonator 4 generate the electrostatic driving force, and the electrostatic driving force is represented as: wherein, , , and respectively represent a capacitance, an initial gap width, a displacement amount, and a frequency; and respectively represent a direct current bias voltage and an alternating current voltage, is time; Lagrangian function and generalized forces is derived as follows: wherein represents the damping; The Lagrange equation of the four-degree-of-freedom MEMS resonator is derived as follows: wherein , , , are linear terms of non-linear stiffness, , , , are cubic terms of non-linear stiffness; The electrostatic coupling stiffness is represented as: wherein represents a bias voltage and the difference between represents the vacuum permittivity, and respectively represent the effective cross-sectional area and the initial gap between adjacent resonators; Definitions , , , , , , , where represents the mechanical coupling stiffness; while introducing the variable: , , , , , , , , , ; the Lagrange equation of the four-degree-of-freedom MEMS resonator is re-derived as: Let , , , , and , the Lagrangian equation of the counterweight derivation is Taylor expanded and the high-order nonlinear terms beyond the third order are ignored, and a mathematical model with control input is obtained: wherein, and is the output of the mathematical model; ； ；; ； 。 3. The method of claim 1, wherein, The preset performance function is represented as: In the formula, a given time set for the user, ; The position tracking error is defined as: where, and are the state variables and reference trajectories of the four-degree-of-freedom MEMS resonator mini-network, respectively.
4. The method of claim 1, wherein, The constraint conversion function comprises an error constraint conversion function and a velocity constraint conversion function; the error constraint conversion function is represented as: wherein, , is a position tracking error, is a preset performance function; is a bounded constant; The velocity constraint conversion function is represented as: wherein is a bounded constant; A new function is constructed to convert into a non-constrained form: For and The derivative is: wherein , , .
5. The method of claim 1, wherein, The two-type sequence fuzzy neural network comprises an input layer, a fuzzification layer, a membership degree layer, a fuzzy rule layer and an output layer; the membership degree layer adopts a Gaussian 2-type membership degree function, and the network weight is updated online by using an adaptive law to approximate the uncertain term in the system.
6. The method of claim 5, wherein, In the fuzzification layer, the input variables by the following transformation: wherein and represent upconversion input and downconversion input, respectively, and are upper and lower widths of the membership function , and are upper and lower widths of the membership function , represents the center of the membership function; J , M , N are integers; In the membership degree layer, the upper and lower membership degree functions are represented as: In the rule layer, the IF-THEN fuzzy rule is: : if is , …; is , …; and is ; then , where is the nth Gaussian 2 membership function of the j m input, is the input, and denote the upper and lower weights, respectively; According to the above design rules, the upper activation strength and the lower activation strength are represented as: In the output layer, the input linear combination is as follows: wherein and denotes a weight vector; Output of the neural network is: wherein , ; For any smooth function continuous , there exists a constant , small enough, such that: wherein represents an approximation function the absolute value of the approximation error, is a vector of a compact set; and ; The optimal vector is defined as: wherein denotes a variable of a compact set; , denotes an ideal weight.
7. The method of claim 1, wherein, The accelerated exponential integral tracking differentiator is represented as: where the input to the acceleration-exponential-integral-derivative tracker is the virtual control law , and denote the output of the acceleration-exponential-integral-derivative tracker; , and all denote a normal number.
8. The method of any one of claims 1-7, wherein, The design of a pre-specified finite-time controller includes defining the controller input as and : wherein , is the unconstrained state variable output by the system state transition, is the virtual control law; To take the derivative: To take the derivative: Integrating the output of an acceleration index proportional-derivative tracker Define inequality: wherein ; Based on the above backstepping control framework, the design process of the predefined limited time controller is divided into eight steps, including: Step 1: considering the first Lyapunov function as: For the derivative is: The first virtual control law is designed as: wherein represents a normal number; further obtained: Step 2: the second Lyapunov function is defined as: wherein is a normal number, is a weight of the bivariate sequence fuzzy neural network; For Taking the derivative gives: In the actual work process, the four degrees of freedom MEMS resonator small network is disturbed to cause the uncertainty of mathematical model, it is difficult to obtain accurate physical parameters, and the two type sequence fuzzy neural network is used to estimate unknown function : Substituting into , the control input and the adaptive law are respectively given by: wherein and is a normal number; Substituting and into yields wherein ; Step 3: considering the third Lyapunov function as: The first derivative is: The second virtual control law is designed as: wherein is a normal number; Substituting into gives: Step 4: the fourth Lyapunov function is defined as: wherein is a normal number; For Taking the derivative gives: Likewise, the unknown function is estimated using a type-2 fuzzy neural network : Obtaining control input and adaptive law : wherein and are both normal numbers; Substituting , into gives: wherein ; According to the mathematical model of the four-degree-of-freedom MEMS resonator small network, the mathematical model of the resonator 3 and the resonator 4 exhibits a similar form to the resonator 1 and the resonator 2; based on the derivation process of steps 1 to 4, the fifth, sixth, seventh and eighth Lyapunov functions are obtained as follows: wherein is a normal number; Designing virtual control laws and are: wherein is a normal number; The control input and adaptive law of the resonator 3 and the resonator 4 are as follows: wherein , is a normal number, ; Similarly, it is obtained that: wherein .