A fuzzy adaptive backstepping control method for micro-electro-mechanical resonators

Abstract

The application discloses a fuzzy adaptive backstepping control method of a micro-electro-mechanical resonator, and steps include: fractional order micro-electro-mechanical resonator modeling; design of an inaccurate target trajectory fuzzy adaptive optimal backstepping controller with a disturbance compensation mechanism; stability analysis. In order to reduce the damage of matching disturbance to the control effect, the application proposes a disturbance compensation mechanism based on fractional order disturbance observation. Secondly, in order to approximate unknown functions in the fractional order MEMS resonator, an interval type-3 fuzzy logic system (IT3FLS) with an adaptive law is constructed. Then, the Fourier series and the IT3FLS are introduced to reconstruct the inaccurate target trajectory, and the fractional order hyperbolic tangent tracking differentiator is introduced to deal with the "complexity explosion" related to the traditional backstepping control. Finally, the optimal control input is embedded into the technical framework of the backstepping controller, so that the tracking accuracy of the controller is improved and the cost function is minimized.

Description

Technical Field

[0001] This invention relates to the field of microelectromechanical resonator (MEMS) control technology, and more specifically to a fuzzy adaptive back-calculation control method for MEMS resonators. Background Technology

[0002] Microelectromechanical systems (MEMS) resonators are multi-scale, highly integrated electromechanical coupling systems with broad application prospects in robotics, gyroscopes, probe microscopes, sensors, and medical monitoring. Nonlinear motion behavior, especially chaotic oscillations, severely compromises the safety, stability, and reliability of these systems. Therefore, studying the nonlinear motion behavior (especially chaotic oscillations) of MEMS resonators and proposing feasible control strategies to suppress them is highly challenging. The prerequisite for researching and designing a reliable controller to suppress chaotic oscillations is establishing an accurate model of the system, and parameter identification techniques are an effective tool for combining theory with experimental data to achieve accurate system modeling. Current research mainly focuses on parameter identification and dynamic behavior analysis of integer-order MEMS resonator models, which cannot fully reflect the operation of integer-order systems.

[0003] Fractional differential equations have attracted considerable attention from researchers due to their ability to comprehensively and accurately describe the dynamic behavior of research objects. In existing techniques, to fully understand the dynamic behavior of fractional hyperchaotic systems, He et al. used Lyapunov exponents, bifurcation diagrams, and multi-scale complexity to study the dynamic behavior of systems with varying parameters and derivative orders. They also studied the nonlinear dynamic behavior of a novel fractional dynamic system with fixed system parameters and varying initial values ​​using bifurcation diagrams, dynamic distribution diagrams, and phase diagrams. However, these studies are limited to the dynamic behavior analysis of fractional systems and do not address the problem of suppressing chaotic oscillations. To suppress chaotic oscillations in a class of variable-order fractional differential systems, Jiang et al. designed a novel variable-order fractional controller based on global sliding control, achieving chaotic oscillation suppression and trajectory tracking control for both variable-order and constant-order fractional systems. To track ideal currents and suppress harmonic distortion of active filters within a finite time, Fang et al. proposed an adaptive fuzzy neural fractional current control. Wei et al. designed a fractional back-calculation controller to achieve stability control of a class of fractional chaotic systems. Luo et al. proposed an adaptive back-calculation optimal control strategy to suppress oscillations caused by chaos and dead zones in fractional-order magneto-electro-mechanical transducer systems. However, the equivalent analog circuit and experimental analysis of the fractional-order MEMS resonator were not addressed. In summary, current work does not cover research on inaccurate target trajectories, optimal control, and disturbance compensation. Summary of the Invention

[0004] The purpose of this invention is to overcome the above-mentioned shortcomings and propose a fuzzy adaptive back-propagation control method for microelectromechanical resonators with a disturbance compensation mechanism when the fractional-order MEMS resonator is disturbed and the target trajectory is inaccurate.

[0005] The present invention provides a fuzzy adaptive back-mapping control method for a microelectromechanical resonator, wherein the method includes the following steps:

[0006] Step 1: Modeling a fractional-order microelectromechanical resonator:

[0007] The microelectromechanical resonator includes a movable microbeam, an AC drive voltage, a DC bias voltage, a drive electrode, and a sensing electrode. The AC drive voltage and DC bias voltage serve as drive sources, providing a driving force F to the microelectromechanical resonator. act It can be expressed as:

[0008]

[0009] Where C0 is the structural capacitance at rest, V AC Ω represents the AC amplitude and frequency, V DC d is the DC drive voltage, d is the initial gap width, and x is the displacement of the movable microbeam at the midpoint.

[0010] The equation of motion for the microelectromechanical resonator is shown below:

[0011]

[0012] Where x , and x ,, These are the first and second derivatives of the displacement x. is the effective lumped mass of the resonator, μ represents the damping coefficient, l is the linear stiffness coefficient, and β is the cubic stiffness coefficient;

[0013] set up According to Taylor series expansion, the mathematical model of a fractional-order microelectromechanical resonator with matching disturbances caused by external systems and control input u can be written as:

[0014]

[0015] in The fractional derivatives of x1 and x2, and the matched perturbation. It is generated through an external system as shown below:

[0016]

[0017] in Represents the state parameters of the external system. These are known exogenous system parameters;

[0018] Step 2: Design process of an adaptive optimal back-mapping controller for inaccurate target trajectory with perturbation compensation mechanism:

[0019] Step 2.1: Interval Type 3 Fuzzy Logic System IT3FLS:

[0020] To approximate the unknown function in a fractional-order microelectromechanical resonator, an interval-type 3 fuzzy logic system, IT3FLS, with an adaptive law is constructed. The operating mechanism for obtaining the output of the interval-type 3 fuzzy logic system, IT3FLS, is as follows:

[0021] In the input layer, the input variable of IT3FLS is x. i ,i=1,2,…,n;

[0022] In the fuzzy layer, membership functions are taken into consideration. It is x i The j-th order fuzzy set, membership function Membership degree It can be calculated as

[0023] Where k = 1, 2, ..., K represents the number of horizontal cuts. It is a membership function The center, It is a membership function Top / bottom width;

[0024] In the rule layer, the j-th order fuzzy set rule is:

[0025] If x1 belongs to x2 belongs to And...x n belong , So j=1,2,…,m,

[0026] Where m is the total number of fuzzy set rules. These are the result parameters of the j-th order fuzzy set rule; the activation level of the rule is shown below.

[0027]

[0028] The output of the deblurring layer of IT3FLS is calculated as follows:

[0029]

[0030] For ease of subsequent expression, formula (13) is simplified to

[0031]

[0032] in, It is the adaptive rate.

[0033] Indicates weight,

[0034] and

[0035] It is a subvector of the weights;

[0036] Step 2.2: Approximate the unknown function using IT3FLS:

[0037] The unknown non-periodic function in the reference trajectory is reconstructed by approximation, and the unknown nonlinear function in the fractional-order microelectromechanical resonator is approximated.

[0038] For an unknown nonperiodic function h 1N The fuzzy approximation of f(t) and the unknown nonlinear function f2(t) is designed as follows, and the corresponding IT3FLS outputs are designed as follows:

[0039] in They represent the IT3FLS approximation of the unknown non-periodic function h, respectively. 1N The optimal approximation vector, weight vector, and approximation error of (t); These are the optimal approximation vector, weight vector, and approximation error for IT3FLS to approximate the unknown nonlinear function f2(t); approximation error. Meet the conditions ,in and It is a given arbitrarily small positive constant;

[0040] make:

[0041] ,

[0042] in It is the estimation error; the adaptive law is designed as follows:

[0043]

[0044] in All are positive numbers; Indicates satisfaction , for any smooth and bounded function, where It is a positive design constant, and e2(t) is the tracking error variable;

[0045] Step 2.3: Reconstruct the target trajectory

[0046] The process of reconstructing an inaccurate target trajectory using a target trajectory reconstruction method based on Fourier series and IT3FLS is as follows:

[0047] The inaccurate target trajectory x 1d (t) is defined as

[0048]

[0049] in The estimated value x represents 1d (t), h 1d (t) is x 1d (t) and The estimation error between them; to improve tracking accuracy, Fourier series and IT3FLS are used to approximate h. 1d (t); h 1d (t) is an unknown function, which can be written as h 1T (t)+h 1N (t), where h 1T (t) represents an unknown continuous periodic function T, h with a fundamental period. 1N (t) is an unknown non-periodic function;

[0050] According to the Dirichlet boundary conditions, h 1T The Fourier series expansion of (t) can be written as

[0051]

[0052] Where m0, m l and n l Here, l is an unknown constant, and l is a sampling point;

[0053] We introduce a known constant T0 to approximate the amplitude T, with an approximation error of ΔT. According to T = T0 + ΔT, we have:

[0054]

[0055] in Represents an unknown and bounded constant;

[0056] Substituting formula (21) into formula (20) yields

[0057]

[0058] in,

[0059]

[0060] Simplify formula (22) to

[0061]

[0062] in,

[0063] It is an unknown vector function with dimension 2E+1.

[0064] It is a computable vector function, and E is a positive design constant.

[0065] It is the truncation error;

[0066] h 1d (t) is simplified to

[0067]

[0068] in h 1d The estimated value of (t), It is an estimation error;

[0069] Substituting formula (24) into formula (19) yields

[0070]

[0071] in It is the target trajectory for reconstruction;

[0072] Step 2.4: Design of a fuzzy adaptive optimal back-calculation controller based on a fractional hyperbolic tangent tracking differentiator

[0073] Step 2.4.1: Define the computable tracking error variable e1(t) as... ,in It reconstructs the target trajectory; the fractional derivative of e1(t) can be written as

[0074] To reduce harmonic disturbances To mitigate the damage to the controller's performance, a fractional-order disturbance observer was constructed.

[0075]

[0076] in It is an intermediate variable. The estimated value, It is the fractional-order perturbation observation gain. It is the design constant; the estimation error of the fractional-order perturbation observer is defined as... ;e do1 The fractional derivative can be calculated as

[0077]

[0078] Based on the computable tracking error variable e1(t) and the estimation error e of the fractional-order perturbation observer do1 The first Lyapunov candidate function can be designed as follows:

[0079] According to the fractional derivative rule, the fractional derivative of V1(t) is calculated as follows:

[0080]

[0081] The virtual control input α1(t) is designed as

[0082]

[0083] Where k1>0 is a positive design constant;

[0084] Define the tracking error variable e2(t) as x2(t)-α1(t), and substitute equations (28) and (31) into equation (30) to obtain

[0085]

[0086] Step 2.4.2: The fractional derivative of e²(t) is calculated as follows:

[0087] in,

[0088] It is an unknown time-varying function;

[0089] To compensate for the disturbance A fractional-order perturbation observer was established as shown below.

[0090] in It is an intermediate variable. It is the observer gain. It is a design constant. The estimated value; definition ,exist

[0091]

[0092] Based on the computable tracking error variable e2(t) and the estimation error e of the fractional-order perturbation observer do2 The second Lyapunov candidate function is constructed as follows:

[0093]

[0094] Where γ2 is a positive design constant;

[0095] The fractional derivative of V2(t) is calculated as

[0096]

[0097] Substituting formulas (32) and (33) into formula (37) yields

[0098]

[0099] To approximate the fractional derivative of α1(t), a fractional hyperbolic tangent tracking differentiator as shown below was designed.

[0100]

[0101] Where H1 and H2 are the state parameters of the fractional-order hyperbolic tangent tracking differentiator. These are positive design parameters;

[0102] Based on formulas (38) and (39), the control input u is designed as follows:

[0103]

[0104] Where k2 is a positive constant, H2 is the output of the fractional hyperbolic tangent tracking differentiator, and u o This represents the optimal control input used to compensate for tracking errors;

[0105] Substituting formula (40) into formula (38) yields

[0106]

[0107] The performance-cost function minimized by optimal control can be written as:

[0108]

[0109] in Represents positive design parameters; for arbitrarily small approximation errors of IT3FLS ,exist ;

[0110] The Ricardi algebraic equation for Q(e2) can be written as follows:

[0111]

[0112] The analytical solution of formula (43) is calculated as follows:

[0113]

[0114] Optimal control input design is

[0115]

[0116] Substituting formula (45) into formula (41) yields

[0117]

[0118] in

[0119]

[0120] Step 3: Stability Analysis

[0121] The stability of a fuzzy adaptive optimal back-calculation controller based on a fractional hyperbolic tangent tracking differentiator under an imprecise target trajectory is proved using Lyapunov stability theory. The proof is as follows:

[0122] Construct the global Lyapunov function as follows

[0123]

[0124] The fractional derivative of formula (47) is calculated as follows:

[0125]

[0126] The Laplace transform of formula (48) is calculated as follows:

[0127]

[0128] Where V(s) is the Laplace transform of V(t);

[0129] Formula (49) is rewritten as

[0130]

[0131] For t>0, there exists Makes the following inequalities true

[0132]

[0133] According to formula (51), we can obtain

[0134]

[0135] For any small positive number The following inequalities hold.

[0136]

[0137] In formula (53) It can be calculated as

[0138]

[0139] For any small positive number According to formulas (50) and (54), it can be known that

[0140]

[0141] By choosing appropriate parameters, the following inequality can be made to hold.

[0142]

[0143] Substituting formulas (53), (55), and (56) into formula (50), we can obtain...

[0144]

[0145] In summary, all signals of a fractional-order microelectromechanical resonator eventually become consistent, and chaotic oscillations are suppressed.

[0146] Compared with the prior art, the present invention has significant advantages, as can be seen from the above solutions:

[0147] (1) This invention establishes a fractional-order model and fractional-order circuit differential equations for MEMS resonators to accurately describe the dynamic behavior of MEMS resonators. Subsequently, based on the fractional-order model and fractional-order circuit differential equations, numerical simulation and fractional-order simulation circuit diagrams are constructed to reveal the relationship between chaotic oscillations and the order α and parameter Vac.

[0148] (2) This invention introduces Fourier series and IT3FLS to reconstruct inaccurate target trajectories and uses the universal approximation property of IT3FLS to solve the problem of unknown nonlinear functions.

[0149] (3) This invention proposes a perturbation compensation mechanism based on a fractional perturbation observer to estimate the modeling perturbation, establishes a fractional hyperbolic tangent tracking differentiator to replace the virtual control input and its derivative, in order to cope with the “complexity explosion” brought about by traditional back-reasoning, and designs an optimal control input to improve tracking accuracy and minimize the cost function.

[0150] In summary, this invention enables fuzzy adaptive optimal back-pushing control with a disturbance compensation mechanism when the fractional-order MEMS resonator is disturbed and the target trajectory is inaccurate.

[0151] The following specific embodiments further illustrate the beneficial effects of the present invention. Attached Figure Description

[0152] Figure 1 This is a flowchart of the control scheme of the present invention;

[0153] Figure 2 This is a flowchart of the interval type 3 fuzzy logic system IT3FLS of the present invention;

[0154] Figure 3Different parameters α and V in the embodiments of the present invention AC The tracking error performance curve is shown below.

[0155] Figure 4 Different parameters α and V in the embodiments of the present invention AC Performance curves of the control input and optimal control input under the given conditions;

[0156] Figure 5 Different parameters α and V in the embodiments of the present invention AC The unknown vector function M m and Estimated value;

[0157] Figure 6 Different parameters α and V in the embodiments of the present invention AC Performance curves of a lower fractional hyperbolic tangent tracking differentiator. Detailed Implementation

[0158] The following detailed description, in conjunction with the accompanying drawings and preferred embodiments, details the specific implementation, features, and effects of a fuzzy adaptive back-propagation control method for a microelectromechanical resonator proposed according to the present invention.

[0159] The present invention provides a fuzzy adaptive back-mapping control method for a microelectromechanical resonator, wherein the method includes the following steps:

[0160] To address the control problem of a disturbed fractional-order MEMS resonator with an inaccurate target trajectory, a fuzzy adaptive optimal back-tracking controller design method with a disturbance compensation mechanism is proposed. The main design process is as follows: First, to reduce the impact of matching disturbances on control performance, a disturbance compensation mechanism based on fractional-order disturbance observation is proposed. Second, to approximate the unknown function in the fractional-order MEMS resonator, an interval type-3 fuzzy logic system (IT3FLS) with an adaptive law is constructed. Then, Fourier series and IT3FLS are introduced to reconstruct the inaccurate target trajectory, and a fractional-order hyperbolic tangent tracking differentiator is introduced to handle the "complexity explosion" associated with traditional back-tracking control. Finally, the optimal control input is embedded into the technical framework of the back-tracking controller to improve the tracking accuracy and ensure the minimization of the cost function. The control flow of the designed controller is as follows: Figure 1 As shown, the specific implementation steps are as follows:

[0161] Step 1: Modeling a fractional-order MEMS resonator

[0162] It consists of a movable microbeam, an AC driving voltage, a DC bias voltage, driving electrodes, and sensing electrodes. The AC driving voltage and DC bias voltage serve as the driving source, providing the driving force for the MEMS resonator. Driving force F act It can be expressed as:

[0163] Where C0 is the structural capacitance at rest, V AC Ω represents the AC amplitude and frequency, V DC d is the DC drive voltage, d is the initial gap width, and x is the displacement of the movable microbeam at the midpoint.

[0164] The equation of motion for the microelectromechanical resonator is shown below:

[0165]

[0166] Where x , and x ,, These are the first and second derivatives of the displacement x. is the effective lumped mass of the resonator, μ represents the damping coefficient, l is the linear stiffness coefficient, and β is the cubic stiffness coefficient;

[0167] set up According to Taylor series expansion, the mathematical model of a fractional-order microelectromechanical resonator with matching disturbances caused by external systems and control input u can be written as:

[0168]

[0169] in, The fractional derivatives of x1 and x2, and the matched perturbation. It is generated through an external system as shown below:

[0170]

[0171] in Represents the state parameters of the external system. These are known exogenous system parameters.

[0172] Step 2: Design process of an adaptive optimal back-mapping controller for inaccurate target trajectory with perturbation compensation mechanism

[0173] Compared to Type 1 fuzzy logic systems with explicit membership, Type 2 fuzzy logic systems with Type 1 fuzzy set membership, and Type 2 sequential fuzzy neural networks, IT3FLS improves the approximation capability of unknown nonlinear systems with more complex nonlinearity and uncertainty by adding fuzzy sets. Therefore, this invention introduces IT3FLS to approximate unknown functions. The flowchart of IT3FLS is as follows... Figure 2 As shown. The operational mechanism for obtaining the output of the proposed fuzzy system is summarized as follows:

[0174] In the input layer, the input variable of IT3FLS is x. i ,i=1,2,…,n;

[0175] In the fuzzy layer, considering that the membership function Aj i is x i The upper / lower membership degree of the membership function Aj for a j-th order fuzzy set. It can be calculated as

[0176]

[0177] Where k = 1, 2, ..., K represents the number of horizontal cuts. It is a membership function The center, It is a membership function Top / bottom width;

[0178] In the rule layer, the j-th order fuzzy set rule is:

[0179] If x1 belongs to x2 belongs to And...x n belong , So j=1,2,…,m,

[0180] Where m is the total number of fuzzy set rules. These are the result parameters of the j-th order fuzzy set rule; the activation level of the rule is shown below.

[0181]

[0182] The output of the deblurring layer of IT3FLS is calculated as follows:

[0183]

[0184] For ease of subsequent expression, formula (13) is simplified to

[0185]

[0186] in It is the adaptive rate.

[0187] Indicates weight,

[0188] and It is a subvector of the weights.

[0189] Controllers operating in extreme environments cannot obtain accurate target trajectories due to environmental interference and signal attenuation and loss during transmission. Furthermore, the parameters in the fractional-order MEMS resonator are unknown and time-varying due to environmental interference. Therefore, IT3FLS is used to reconstruct the reference trajectory, approximating the unknown aperiodic functions in the reconstructed trajectory and approximating the unknown nonlinear functions in the fractional-order MEMS resonator.

[0190] For an unknown nonperiodic function h 1N The fuzzy approximation of f(t) and the unknown nonlinear function f2(t) is designed as follows, and the corresponding IT3FLS outputs are designed as follows:

[0191] in They represent the IT3FLS approximation of the unknown non-periodic function h, respectively. 1N The optimal approximation vector, weight vector, and approximation error of (t); These are the optimal approximation vector, weight vector, and approximation error for IT3FLS to approximate the unknown nonlinear function f2(t); approximation error. Meet the conditions ,in and It is a given arbitrarily small positive constant;

[0192] make:

[0193] ,

[0194] in It is the estimation error; the adaptive law is designed as follows:

[0195]

[0196] in All are positive numbers; Indicates satisfaction , for any smooth and bounded function, where It is the positive design constant, and e2(t) is the tracking error variable.

[0197] In the field of remote control, the target trajectory x 1d (t) may be affected by various interferences and transmission characteristics, leading to inaccurate target trajectories, and therefore cannot be directly used in controller design. In view of this, a target trajectory reconstruction method based on Fourier series and IT3FLS is proposed. The reconstruction process of inaccurate target trajectories can be summarized as follows:

[0198] The inaccurate target trajectory x 1d (t) is defined as

[0199]

[0200] in The estimated value x represents 1d (t), h 1d (t) is x 1d (t) and The estimation error between them; to improve tracking accuracy, Fourier series and IT3FLS are used to approximate h. 1d (t); h 1d (t) is an unknown function, which can be written as h 1T (t)+h 1N (t), where h 1T (t) represents an unknown continuous periodic function T, h with a fundamental period. 1N (t) is an unknown non-periodic function;

[0201] According to the Dirichlet boundary conditions, h 1T The Fourier series expansion of (t) can be written as

[0202]

[0203] Where m0, m l and n l Here, l is an unknown constant, and l is a sampling point;

[0204] We introduce a known constant T0 to approximate the amplitude T, with an approximation error of ΔT. According to T = T0 + ΔT, we have:

[0205]

[0206] in Represents an unknown and bounded constant;

[0207] Substituting formula (21) into formula (20) yields

[0208]

[0209] in,

[0210]

[0211] Simplify formula (22) to

[0212]

[0213] in,

[0214] It is an unknown vector function with dimension 2E+1.

[0215] It is a computable vector function, and E is a positive design constant.

[0216] It is the truncation error;

[0217] h 1d (t) is simplified to

[0218]

[0219] in h 1d The estimated value of (t), It is an estimation error;

[0220] Substituting formula (24) into formula (19) yields

[0221]

[0222] in It is the target trajectory for reconstruction.

[0223] Note 1: Unknown vector function M m (t), Both and their derivatives satisfy , A bounded smooth function, where M l M d , It is a normal number.

[0224] estimation error Defined as

[0225]

[0226] in It is an unknown vector function M m The estimated value of (t), It is the estimation error.

[0227] The adaptive rate is designed as follows:

[0228] in It is a positive design constant. Indicates satisfaction , for any smooth and bounded function, where It is a positive number.

[0229] In conclusion, it can be concluded that It is semi-globally consistent and eventually bounded. The detailed proof is as follows:

[0230] Proof: According to The Lyapunov function was designed as

[0231] V m The derivation of the fractional integral is as follows:

[0232] Substituting formulas (17) and (27) into formula (28) yields

[0233] According to Young's inequality, there exists

[0234] Where λ mmin and λ mmax yes The minimum and maximum eigenvalues, λ nmin and λ nmax yes The minimum and maximum eigenvalues.

[0235] Substituting formula (31) into formula (30) yields

[0236] in In summary, the estimation error can be derived. It is semi-globally consistent and eventually bounded, and M can be approximated with extremely high precision. m (t) and .

[0237] The computable tracking error variable e1(t) is defined as ,in It reconstructs the target trajectory; the fractional derivative of e1(t) can be written as

[0238] To reduce harmonic disturbances To mitigate the damage to the controller's performance, a fractional-order disturbance observer was constructed.

[0239] in It is an intermediate variable. The estimated value, It is the fractional-order perturbation observation gain. It is the design constant; the estimation error of the fractional-order perturbation observer is defined as... ;e do1 The fractional derivative can be calculated as

[0240] Based on the computable tracking error variable e1(t) and the estimation error e of the fractional-order perturbation observer do1 The first Lyapunov candidate function can be designed as follows:

[0241] According to the fractional derivative rule, the fractional derivative of V1(t) is calculated as follows:

[0242] The virtual control input α1(t) is designed as

[0243]

[0244] Where k1>0 is a positive design constant.

[0245] The tracking error variable e2(t) is defined as x2(t) - α1(t).

[0246]

[0247] The fractional derivative of e²(t) is calculated as

[0248]

[0249] in It is an unknown time-varying function;

[0250] To compensate for the disturbance A fractional-order perturbation observer was established as shown below.

[0251]

[0252] in It is an intermediate variable. It is the observer gain. It is a design constant. The estimated value; definition ,exist

[0253] Based on the computable tracking error variable e2(t) and the estimation error e of the fractional-order perturbation observer do2 The second Lyapunov candidate function is constructed as follows:

[0254] Where γ2 is the positive design constant.

[0255] The fractional derivative of V2(t) is calculated as

[0256] Substituting formulas (39) and (40) into formula (44) yields

[0257] As is well known, the fractional derivative of α1(t) is difficult to obtain directly due to the complexity of its calculation. More importantly, the multiple FO derivatives of α1(t) lead to a "complexity term explosion" associated with traditional inverse derivation. Therefore, a fractional hyperbolic tangent pursuing differentiator is used to address this problem. The fractional hyperbolic tangent pursuing differentiator shown below is designed to approximate the fractional derivative of α1(t).

[0258] Where H1 and H2 are the state parameters of the fractional-order hyperbolic tangent tracking differentiator, the parameters are... These are positive design parameters;

[0259] Based on formulas (45) and (46), the control input u is designed as follows:

[0260]

[0261] Where k2 is a positive constant, H2 is the output of the fractional hyperbolic tangent tracking differentiator, and u o This represents the optimal control input used to compensate for tracking errors;

[0262] Substituting formula (47) into formula (45) yields

[0263]

[0264] The performance-cost function minimized by optimal control can be written as:

[0265]

[0266] in Represents positive design parameters; for arbitrarily small approximation errors of IT3FLS ,exist .

[0267] The Ricardi algebraic equation for Q(e2) can be written as follows:

[0268]

[0269] The analytical solution of formula (50) is calculated as follows:

[0270]

[0271] Optimal control input design is

[0272] Substituting formula (52) into formula (48) yields

[0273] in .

[0274] Theorem 1: For the optimal control problem of a fractional-order MEMS resonator under an inaccurate target trajectory, the control input is designed as formula (47) using an adaptive law and optimal control input. By selecting appropriate design parameters, the following conclusions can be drawn:

[0275] 1) All signals in a fractional-order MEMS resonator are eventually uniformly bounded.

[0276] 2) It completely suppresses chaotic oscillations that disrupt the stability of fractional-order MEMS resonators and minimizes the cost function.

[0277] Step 3: Stability Analysis

[0278] The stability of a fuzzy adaptive optimal back-calculation controller based on a fractional hyperbolic tangent tracking differentiator under an imprecise target trajectory is proved using Lyapunov stability theory. The proof is as follows:

[0279] The global Lyapunov function is constructed as follows:

[0280] The fractional derivative of formula (54) is calculated as follows:

[0281]

[0282] The Laplace transform of equation (55) is calculated as follows:

[0283]

[0284] Where V(s) is the Laplace transform of V(t).

[0285] Formula (56) is rewritten as

[0286]

[0287] For t>0, there exists Makes the following inequalities true

[0288]

[0289] According to formula (58), we can obtain

[0290] For any small positive number The following inequalities hold.

[0291] In formula (60) It can be calculated as

[0292]

[0293] For any small positive number According to formulas (57) and (61), it can be seen that

[0294]

[0295] By choosing appropriate parameters, the following inequality can be made to hold.

[0296]

[0297] Substituting formulas (60), (62), and (63) into formula (57), we can obtain...

[0298]

[0299] In summary, all signals of a fractional-order microelectromechanical resonator eventually become consistent, and chaotic oscillations are suppressed.

[0300] Performance Analysis:

[0301] Extensive simulation results are presented to verify the effectiveness of the designed control scheme. The initial parameters of the fractional-order MEMS resonator are selected as x. i =0, i=1, 2. Target trajectory set The estimated target trajectory is The parameter values ​​in the fractional-order perturbation compensation observer are set to... k f1 =32 and k f2 =24. The relevant parameters in the Fourier series are set to T0=5 and E=4. The parameters in the adaptive rate of the reconstructed reference trajectory are selected as follows: The proposed controller parameters are chosen as k1=64 and k2=35. The optimal controller parameters are chosen as follows: and The adjustment parameters for the fractional hyperbolic tangent tracking differentiator are selected as follows: The parameters in formula (18) are designed to be The parameters of IT3FLS are set to... .

[0302] Figure 3 Different parameters α and V AC The performance curve of the tracking error is shown below. From... Figure 3 As can be seen from this, the tracking error can be controlled within... This further demonstrates that the proposed controller enables the fractional-order MEMS resonator system to achieve good tracking accuracy. Different parameters α and V... AC The performance curves of the control input and the optimal control input are as follows: Figure 4 As shown. The four enlarged sub-figures in the figure. Figure 4 Different parameters α and V are shown. AC The subtle difference between the control input under given conditions and the optimal control input. From Figure 3 and 4 It can be seen that regardless of changing the order α or the parameter V AC The position tracking trajectory, tracking error, control input, and optimal control input are almost unaffected. In other words, the controller designed in this invention has good robustness.

[0303] Unknown vector function With different parameters α and V AC The estimated value is as follows Figure 5 As shown. From this, we can see that for different parameters α and V... AC After the intervention of the scheme, all performance curves are trapped in a stable state. To solve the "complexity explosion" problem associated with the repeated differentiation of , a fractional hyperbolic tangent pursuing differentiator is introduced to approximate it. The approximate performance of the fractional hyperbolic tangent pursuing differentiator is as follows: Figure 6 As shown. From this, we can see that with different parameters α and V AC Below is a performance curve fitting of the fractional-order hyperbolic tangent tracking differentiator. It is worth noting that the fractional-order hyperbolic tangent tracking differentiator solves the "complexity explosion" problem. From... Figure 6 It can be seen that regardless of changing the order α or the parameter V AC The perturbation compensation mechanism based on the fractional-order perturbation observer and the performance of the fractional-order hyperbolic tangent tracking differentiator are almost unaffected. In other words, the components in the controller also exhibit excellent robustness.

[0304] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Any simple modifications, equivalent changes, and alterations made to the above embodiments without departing from the technical essence of the present invention shall still fall within the scope of the present invention.

Claims

1. A fuzzy adaptive back-mapping control method for a microelectromechanical resonator, characterized in that: The method includes the following steps: Step 1: Modeling a fractional-order microelectromechanical resonator: The microelectromechanical resonator includes a movable microbeam, an AC drive voltage, a DC bias voltage, a drive electrode, and a sensing electrode. The AC drive voltage and DC bias voltage serve as drive sources, providing a driving force F to the microelectromechanical resonator. act It can be expressed as: Where C0 is the structural capacitance at rest, V AC Ω represents the AC amplitude and frequency, V DC d is the DC drive voltage, d is the initial gap width, and x is the displacement of the movable microbeam at the midpoint. The equation of motion for the microelectromechanical resonator is shown below: Where x , and x ,, These are the first and second derivatives of the displacement x. is the effective lumped mass of the resonator, μ represents the damping coefficient, l is the linear stiffness coefficient, and β is the cubic stiffness coefficient; set up According to Taylor series expansion, the mathematical model of a fractional-order microelectromechanical resonator with matching disturbances caused by external systems and control input u can be written as: in, The fractional derivatives of x1 and x2 are the matched perturbations. It is generated through an external system as shown below: in Represents the state parameters of the external system. These are known exogenous system parameters; Step 2: Design of an adaptive optimal back-pushing controller for inaccurate target trajectory fuzziness with disturbance compensation mechanism. The design process is as follows: Step 2.1: Interval Type 3 Fuzzy Logic System IT3FLS: To approximate the unknown function in a fractional-order microelectromechanical resonator, an interval-type 3 fuzzy logic system, IT3FLS, with an adaptive law is constructed. The operating mechanism for obtaining the output of the interval-type 3 fuzzy logic system, IT3FLS, is as follows: In the input layer, the input variable of IT3FLS is x. i i = 1, 2, ..., n; In the fuzzy layer, membership functions are taken into consideration. It is x i The membership function of a j-th order fuzzy set. Membership degree It can be calculated as Where k = 1, 2, ..., K represents the number of horizontal cuts. It is a membership function The center It is a membership function Top / bottom width; In the rule layer, the j-th order fuzzy set rule is: If x1 belongs to x2 belongs to And...x n belong , So j=1,2,…,m, Where m is the total number of fuzzy set rules, These are the result parameters of the j-th order fuzzy set rule; the activation level of the rule is shown below. The output of the deblurring layer of IT3FLS is calculated as follows: For ease of subsequent expression, formula (13) is simplified to in It is the adaptive rate. Indicates weight, and It is a subvector of the weights; Step 2.2: Approximate the unknown function using IT3FLS: The unknown non-periodic function in the reference trajectory is reconstructed by approximation, and the unknown nonlinear function in the fractional-order microelectromechanical resonator is approximated. For an unknown non-periodic function h 1N The fuzzy approximation of f(t) and the unknown nonlinear function f2(t) is designed as follows, and the corresponding IT3FLS outputs are designed as follows: in, They represent the IT3FLS approximation of the unknown aperiodic function h, respectively. 1N The optimal approximation vector, weight vector, and approximation error of (t); These are the optimal approximation vector, weight vector, and approximation error for IT3FLS to approximate the unknown nonlinear function f2(t); approximation error. Meet the conditions ,in and It is a given arbitrarily small positive constant; make: ， in It is the estimation error; the adaptive law is designed as follows: in All are positive numbers; Indicates satisfaction , for any smooth and bounded function, where It is a positive design constant, and e2(t) is the tracking error variable; Step 2.3: Reconstruct the target trajectory The process of reconstructing an inaccurate target trajectory using a target trajectory reconstruction method based on Fourier series and IT3FLS is as follows: The inaccurate target trajectory x 1d (t) is defined as in The estimated value x represents 1d (t), h 1d (t) is x 1d (t) and The estimation error between them; to improve tracking accuracy, Fourier series and IT3FLS are used to approximate h. 1d (t); h 1d (t) is an unknown function, which can be written as h 1T (t)+h 1N (t), where h 1T (t) represents an unknown continuous periodic function T, h with a fundamental period. 1N (t) is an unknown non-periodic function; According to the Dirichlet boundary conditions, h 1T The Fourier series expansion of (t) can be written as Where m0, m l and n l Here, l is an unknown constant, and l is a sampling point; We introduce a known constant T0 to approximate the amplitude T, with an approximation error of ΔT. According to T = T0 + ΔT, we have: in Represents an unknown and bounded constant; Substituting formula (21) into formula (20) yields in, Simplify formula (22) to in, It is an unknown vector function with dimension 2E+1. It is a computable vector function, and E is a positive design constant. It is the truncation error; h 1d (t) is simplified to in h 1d The estimated value of (t), It is an estimation error; Substituting formula (24) into formula (19) yields in It is the target trajectory for reconstruction; Step 2.4: Design of a fuzzy adaptive optimal back-calculation controller based on a fractional hyperbolic tangent tracking differentiator Step 2.4.1: Define the computable tracking error variable e1(t) as... ,in It reconstructs the target trajectory; the fractional derivative of e1(t) can be written as To reduce harmonic disturbances To mitigate the damage to the controller's performance, a fractional-order disturbance observer was constructed. in, It is an intermediate variable. The estimated value, It is the fractional-order perturbation observation gain. It is the design constant; the estimation error of the fractional-order perturbation observer is defined as... ;e do1 The fractional derivative can be calculated as Based on the computable tracking error variable e1(t) and the estimation error e of the fractional-order perturbation observer do1 The first Lyapunov candidate function can be designed as follows: According to the fractional derivative rule, the fractional derivative of V1(t) is calculated as follows: The virtual control input α1(t) is designed as Where k1>0 is a positive design constant; Define the tracking error variable e2(t) as x2(t)-α1(t), and substitute equations (28) and (31) into equation (30) to obtain Step 2.4.2: The fractional derivative of e²(t) is calculated as follows: in, It is an unknown time-varying function; To compensate for the disturbance A fractional-order perturbation observer was established as shown below. in It is an intermediate variable. It is the observer gain. It is a design constant. The estimated value; definition ,exist Based on the computable tracking error variable e2(t) and the estimation error e of the fractional-order perturbation observer do2 The second Lyapunov candidate function is constructed as follows: Where γ2 is a positive design constant; The fractional derivative of V2(t) is calculated as Substituting formulas (32) and (33) into formula (37) yields To approximate the fractional derivative of α1(t), a fractional hyperbolic tangent tracking differentiator as shown below was designed. Where H1 and H2 are the state parameters of the fractional-order hyperbolic tangent tracking differentiator. These are positive design parameters; Based on formulas (38) and (39), the control input u is designed as follows: Where k2 is a positive constant, H2 is the output of the fractional hyperbolic tangent tracking differentiator, and u o This represents the optimal control input used to compensate for tracking errors; Substituting formula (40) into formula (38) yields The performance-cost function minimized by optimal control can be written as: in Represents positive design parameters; for arbitrarily small approximation errors of IT3FLS ,exist ; The Ricardi algebraic equation for Q(e2) can be written as follows: The analytical solution of formula (43) is calculated as follows: Optimal control input design is Substituting formula (45) into formula (41) yields in Step 3: Stability Analysis The stability of a fuzzy adaptive optimal back-calculation controller based on a fractional hyperbolic tangent tracking differentiator under an imprecise target trajectory is proved using Lyapunov stability theory. The proof is as follows: Construct the global Lyapunov function as follows The fractional derivative of formula (47) is calculated as follows: The Laplace transform of formula (48) is calculated as follows: Where V(s) is the Laplace transform of V(t); Formula (49) is rewritten as For t>0, there exists Makes the following inequalities true According to formula (51), we can obtain For any small positive number The following inequalities hold. In formula (53) It can be calculated as For any small positive number According to formulas (50) and (54), it can be known that By choosing appropriate parameters, the following inequality can be made to hold. Substituting formulas (53), (55), and (56) into formula (50), we can obtain... In summary, all signals of a fractional-order microelectromechanical resonator eventually become consistent, and chaotic oscillations are suppressed.

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