Method for adaptive neural network preset time control of electromechanical transducer with memristor

Abstract

The present application relates to a method for adaptive neural network preset time control of electromechanical transducer with memristor, and belongs to the technical field of adaptive control. The method comprises the following steps: S1: system modeling of electromechanical transducer with memristor; S2: dynamic analysis; S3: design of adaptive neural network preset time controller, including: applying delay constraint function to tracking error starting from any position; processing parameter disturbance and system uncertainty items by using type 2 fuzzy wavelet neural network; using second-order fractional-order tracking differentiator; constructing fault control input; S4: stability analysis. The present application not only fully represents the fractional characteristics of dielectric and the relationship between magnetic flux and charge, but also converts chaotic oscillation into regular motion, and achieves the given tracking accuracy under the premise of considering and solving the actuator fault and system uncertainty.

Description

Technical Field

[0001] This invention belongs to the field of adaptive control technology and relates to a preset time control method for an adaptive neural network of an electromechanical transducer with a memristor. Background Technology

[0002] Due to its more attractive properties than resistors, such as its unique nonlinear, memristor, and alternating mechanisms, the memristor is considered a nonlinear electromagnetic element that more realistically describes the relationship between magnetic flux and charge in electromagnetic circuits and device fields. Electromechanical transducers, as typical representatives of electromagnetic elements, are widely used in various equipment in fields such as mixers, vibratory hammers, and compactors. However, their inherent dynamic characteristics related to memristors and fractional-order properties are very complex and rich, leading to adverse effects on such electromechanical transducers. Meanwhile, to meet the ever-increasing demands for high performance, achieving optimal tracking accuracy control within a specified time is also crucial, even with actuator failures and uncertainties. Therefore, it is highly challenging and significant to perform dynamic modeling and analysis considering memristor and fractional-order characteristics, and then propose an adaptive neural network preset-time control scheme with fault tolerance, preset time, and preset tracking accuracy.

[0003] Dynamic modeling is the most fundamental and crucial step in subsequent complexity analysis and controller design, and many scholars have achieved certain results in the field of electromechanical transducers. Pérez-Molina and Perez Polo established their model and conducted dynamic analysis using harmonic basis oscillations. Ngeuteu et al. established a model of two distributed coupled electromechanical transducers generating high-order nonlinearities and studied the related synchronization control problem. However, these works neglected the fractional properties of the dielectric and the relationship between magnetic flux and charge, and none of them described the inherent characteristics of such systems more clearly. To address this issue, Ngeuteu et al., building on their previous results, conducted a synchronization analysis of coupled electromechanical transducers, incorporating fractional properties into the model. However, effectively suppressing high-frequency oscillations of such electromechanical transducers remains a challenging problem, let alone achieving a preset tracking accuracy within a set time. In view of this, Luo et al. analyzed the dynamic behavior of electromechanical transducers and proposed an adaptive backstepping optimal control method to suppress chaotic and dead-zone oscillations. Furthermore, they extended their research results to two electromechanical transducers with capacitive and resistive coupling, achieving optimal synchronization while obtaining the minimum cost function. However, these documents can only handle fractional properties and dynamic effects with single-parameter variations, and the convergence accuracy and convergence time cannot be preset by the developers.

[0004] Fractional calculus theory has become a new hot topic in applications. Numerous physical phenomena and engineering systems have verified that fractional-order models and control methods can describe the characteristics of real-world systems more accurately than traditional integer-order models and methods, and offer greater design flexibility. Backstepping control, through the integration of fuzzy logic systems or neural networks, is widely considered an effective solution for nonlinear systems. The combination of backstepping control and fractional calculus undoubtedly increases design freedom and yields greater potential benefits. To this end, Liu et al. studied the fuzzy backstepping control problem for a class of uncertain fractional-order nonlinear systems. Wei et al. studied the adaptive backstepping problem for trigonometric fractional-order nonlinear systems. Heydarinejad et al. developed a fuzzy type-2 backstepping controller with the help of a sliding mode observer. However, the "complexity explosion" of repeating derivatives inevitably occurs with increasing system order, and these solutions will fail if a fault occurs or control effectiveness is lost. Furthermore, achieving fast response and high convergence accuracy after a preset time is unknown and uncertain.

[0005] Preset-time control and finite-time control play a crucial role in improving the performance of engineering systems and have become hot topics in the control field. Wang et al. studied adaptive neural network tracking control for unconstrained feedback stochastic nonlinear systems. Cao et al. investigated the preset-time control problem of Euler-Lagrange systems under partial / full state constraints. Wang et al. realized fault-tolerant finite-time consistency for multiple uncertain mechanical systems with unidirectional directional communication interaction and driving faults. Li et al. developed a dual-mass MEMS gyroscope-accelerated adaptive backstepping funnel controller with event triggers. However, these are limited to integer-order predetermined-time control, a specific case of fractional calculus. Furthermore, the preset time and tracking accuracy depend on initial conditions and other design parameters. Summary of the Invention

[0006] In view of this, the purpose of the present invention is to provide a preset time control method for an adaptive neural network of an electromechanical transducer with a memristor, which solves the technical problem of how to achieve optimal tracking accuracy control of the system within a specified time when actuator failure and uncertainty occur.

[0007] To achieve the above objectives, the present invention provides the following technical solution:

[0008] A preset-time control method for an adaptive neural network of an electromechanical transducer with a memristor, the method comprising the following steps:

[0009] S1: System modeling of an electromechanical transducer with memristors;

[0010] S2: Dynamics Analysis: Based on the dynamics analysis of bifurcation diagram, sample entropy and C0 complexity, this study reveals the effects of different fractional orders, memristor parameter β and initial memristor values ​​on the internal evolution of system dynamics and complexity performance.

[0011] S3: Design an adaptive neural network controller with a preset time, including: applying a delay constraint function to the tracking error starting from an arbitrary position; using a type 2 fuzzy wavelet neural network to handle parameter disturbances and system uncertainties; using a second-order fractional tracking differentiator; and constructing a fault control input.

[0012] S4: Stability analysis.

[0013] Furthermore, in S1, the electromechanical transducer with memristor consists of two parts: mechanical and electrical. The mechanical part, which oscillates along the Z-axis, consists of a moving beam, a spring, and an operating rod. The operating rod is fixed on the rigid beam and enclosed in the spring. The electrical part consists of a memristor, a capacitor, and an inductor. These three devices are connected in series with a sinusoidal voltage power supply.

[0014] Furthermore, in S1, system modeling is performed on the electromechanical transducer with memristors, specifically including:

[0015] In an electromechanical transducer with a memristor, based on the inherited voltage-charge characteristics, the capacitor exhibits Duffing-type nonlinearity, and the voltage is expressed as:

[0016]

[0017] in, Let q represent voltage, and C represent charge. a A represents the capacitance, and a3 and a5 represent the system coefficients.

[0018] The expression for fractional impedance is:

[0019]

[0020] Where j represents the imaginary part, α represents the fractional order, and ω represents the frequency;

[0021] In the fractional properties of a dielectric, current and voltage The relationship between time t and time t is expressed as:

[0022]

[0023] In electromechanical transducers, memristors are used to replace resistors. The configuration of a memristor is defined using a quadratic nonlinear smooth function, expressed as:

[0024] φ(q)=-βq+0.5γq 2sgn(q) (4)

[0025] Where φ is the magnetic flux, and β and γ represent positive constants;

[0026] The memristor function W(q) is expressed as:

[0027] W(q)=-β+γ|q| (5)

[0028] Using Newton's laws and Kirchhoff's law, the mathematical model of a fractional-order electromechanical transducer can be expressed as follows:

[0029]

[0030] Where m represents mass, h represents the coefficient of viscous friction, k represents the stiffness coefficient, l represents the length of the moving coil, B represents the magnetic flux density, L represents the inductance, v0 represents the amplitude, Ω represents the frequency, and z represents the displacement in the direction of gravity.

[0031] Define new variables t=ω e τ and in Q0 represents the reference charge of the capacitor;

[0032] By adding control inputs, the dimensionless mathematical model of an electromechanical transducer with a memristor can be expressed as follows:

[0033]

[0034] in, and There are three dimensionless parameters α, C, and ω, representing the fractional order, Caputo fractional derivative, and frequency, respectively, and u2 =

[11] [u1 u2] T and u4 = [1 1][u3 u4] T This indicates the control input when a fault occurs;

[0035] The definition of a stuck fault is:

[0036]

[0037] in, Let t represent a positive integer. The stuck failure occurs at t. j time;

[0038] Control failure is represented as

[0039]

[0040] Where, δ i Indicates at t i Once control effectiveness is lost, the effectiveness of the remaining actuators... and u i This represents the lower limit of the i-th control input;

[0041] Based on the two types of actuator failures mentioned above, the control input is expressed as follows:

[0042]

[0043] Among them, u=[u1,…,u n ] T , λ=diag{λ1,…,λ n} and δ=diag{δ1,…,δ n},λ i i = 1, ..., n have only two values: 0 or 1;

[0044] When λ i When the value is 1, the actuator is stuck; otherwise, the control is in failure. The number of actuators that fail is less than n-1.

[0045] Furthermore, by definition, for any differentiable function, its Caputo fractional derivative is derived as:

[0046]

[0047] in, This indicates that n-1 < a < n and The Euler Gamma function, where α represents the fractional value, n represents a natural number, and t represents time;

[0048] For any continuous functions F1(t) and F2(t), on [0, t... s Within the interval [0 < α < 1], where t s Representing time, the following equation exists:

[0049]

[0050] Assume that the desired trajectory and its fractional derivative are continuous and smooth.

[0051] Furthermore, in S3, the type 2 fuzzy wavelet neural network specifically includes:

[0052] Input layer: Each neuron receives the input signal vector and transmits the corresponding information directly to the next layer;

[0053] Membership layer: Each neuron converts the input signal into a membership function to represent fuzzy language terms. The upper and lower membership degrees based on the Mexican-hat wavelet function can be written as:

[0054]

[0055] Where, x i Let i be the vector of input signals received by the neuron, i = 1, ..., N S j = 1, ..., n s n s and N S This indicates the number of input signals and the number of rules. and These are called the translation and expansion parameters of the member functions;

[0056] Rule layer: There exists a series of fuzzy IF-THEN rules.

[0057] If x1 is and...and yes So It is ω j j = 1, ..., N s (14)

[0058] Where i = 1, ..., n s j = 1, ..., N s , This represents the j-th member function of the i-th input;

[0059] In this context, the inference engine of each neuron performs fuzzy inference as follows: The relevant information is passed down to the lower level, where ^ denotes the minimal operator. and Indicates membership degree;

[0060] Degradation: Incentive output is defined as

[0061]

[0062] in and ξ i Indicates the upper and lower values ​​of fuzzy inference;

[0063] Output layer: The entire output of the type 2 fuzzy wavelet neural network is defined as follows:

[0064]

[0065] Where w T ξ(x) represents the weight transpose and basis functions of a type 2 fuzzy wavelet neural network;

[0066] Therefore, it exists.

[0067]

[0068] Wherein, ε and D xThis indicates that x has an approximate error and a compact set with appropriate boundaries;

[0069] Introduce an optimal parameter w * This parameter equals The solution, where Ω w Represent the compact set of w; define the equation. And w * These are artificial variables; furthermore, and

[0070] Convert the type 2 fuzzy wavelet neural network to:

[0071] w T ξ(x)≤ζξ T (x)ξ(x) / 2b 2 +b 2 / 2, (18)

[0072] Where ζ=||w|| 2 and Established, and Let b represent the estimate and positive constant of ζ; And the Caputo derivative of the constant is zero.

[0073] Furthermore, in S3, the delay constraint function is specifically expressed as follows:

[0074] Define the time-varying constraint boundary as:

[0075]

[0076] Among them, T,s i And p means that the condition p > (n) is satisfied. s The settling time, tracking accuracy, and positive integer n are given by (+1) / 2. s As the order of the system; when t→0 It is infinitely large;

[0077] Throughout the process The delayed tracking constraint was followed, and the tracking target has been achieved, where e i (t) represents the tracking error; simultaneously, we obtain

[0078] Introduce a distance function between the tracking error and the constraint boundary:

[0079]

[0080] Where c i It is a positive constant; as can be seen from formula (20),

[0081] To better understand the value γ i (t) is mapped to the range (0,1], and the delay constraint function is set as follows:

[0082]

[0083] Where γ i and a i Indicates γ i (t) abbreviation and preset safety distance;

[0084] when At that time, there is ρ i (γ i ) = 1; the system is close to the constraint boundary, γ i (t)∈(0,a i This could lead to a collision and a violation of the rules;

[0085] When γ i (t)∈(a i When ,∞), the system remains at a safe point far from the constraint boundary, and there exists a i ≡c i σ i 2 make sure

[0086] The transformed tracking error is defined as:

[0087] s i (t)=e i (t) / ρ i (γ i ), i = 1, 3. (22)

[0088] In formula (22), when e i (0) = 0 when s i (0) = 0; s i (t)→∞ is represented as And s i (t)→-∞ is represented as

[0089] Furthermore, in S3, the controller for the preset time of the adaptive neural network is designed, specifically including:

[0090] The first tracking error is constructed as follows:

[0091] e1 = x1 - x d1 (twenty three)

[0092] Where x d1 Let x1 represent the ideal trajectory and x1 represent the state variable.

[0093] Design the first Lyapunov function as:

[0094]

[0095] Taking the fractional derivative of formula (24), we get:

[0096]

[0097] in,

[0098]

[0099]

[0100] And s2(t) = x2 - α2·α2 represents the first virtual control law in the subsequent design;

[0101] The first virtual control law is designed as follows:

[0102]

[0103] Where k1 > 0;

[0104] Substituting formula (28) into formula (25), we get:

[0105]

[0106] Design the second Lyapunov function as follows:

[0107]

[0108] Where θ2>0;

[0109] Taking the fractional derivative of formula (30) yields:

[0110]

[0111] in,

[0112]

[0113] Using a type 2 fuzzy wavelet neural network, f2(x) can be approximated as:

[0114]

[0115] Design a fractional differentiator, represented as:

[0116]

[0117] Where, α ik κ represents the i-th input variable of the fractional-order tracking differentiator. i1 and κ i2All of these are its variables, υ i1 and υ i2 Indicates that the parameter is being adjusted;

[0118] Based on formulas (33) and (34), formula (31) can be rewritten as:

[0119]

[0120]

[0121] Where b2 > 0;

[0122] The fault control input and the associated adaptive law are expressed as follows:

[0123]

[0124]

[0125] Where k2>0, m2>0;

[0126] Substituting formulas (36) and (37) into formula (35), formula (35) is simplified to:

[0127]

[0128] The third tracking error is designed as e3 = x3 - x d3 , where x d3 Represents the ideal trajectory;

[0129] The third Lyapunov function is chosen as:

[0130]

[0131] The fractional derivative of formula (39) is then derived as follows:

[0132]

[0133] in,

[0134]

[0135]

[0136] and

[0137] s4(t)=x4-α4 (43)

[0138] α4 represents the second virtual control law;

[0139] The second virtual control is defined as:

[0140]

[0141] Where k3 > 0;

[0142] Substituting formula (44) into formula (40) yields:

[0143]

[0144] Design the fourth Lyapunov function as follows:

[0145]

[0146] Where θ4>0;

[0147] The fractional derivative of formula (46) is calculated as follows:

[0148]

[0149] in,

[0150] Using a type 2 fuzzy wavelet neural network, f4(x) can be approximated as:

[0151]

[0152] To avoid the complex fractional derivative of α4, the fractional-order tracking differentiator proposed in formula (34) is used, and according to formulas (34) and (48), formula (47) can be further written as:

[0153]

[0154] Where b4 > 0;

[0155] The fault control input and the corresponding adaptive law are then derived as follows:

[0156]

[0157]

[0158] Where k4 > 0, m4 > 0;

[0159] Substituting formulas (50) and (51) into formula (49), we get:

[0160]

[0161] The beneficial effects of this invention are as follows:

[0162] First, compared with integer-order modeling of electromechanical transducers, the fractional-order mathematical model constructed in this invention can better characterize the fractional properties of the dielectric and the relationship between magnetic flux and charge. Based on the constructed model, dynamic analysis provides a clear perspective on the dynamic evolution law and complexity performance of the electromechanical transducer.

[0163] Second, unlike predetermined time control results, the preset time and tracking accuracy in this invention are manually specified before the event occurs, without relying on initial conditions and other design parameters in the integer domain. Unlike cases where tracking accuracy is unknown and depends on too many parameters, this invention eliminates these limitations.

[0164] Third, the proposed solution not only ensures the boundedness of all signals in the closed-loop system, but also transforms the chaotic oscillation of the electromechanical transducer into normal motion, achieving the specified tracking accuracy within a set time, while solving the problems of actuator failure and system uncertainty.

[0165] 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

[0166] 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:

[0167] Figure 1 A schematic diagram of an electromechanical transducer with a memristor;

[0168] Figure 2 Lyapunov exponents and bifurcation diagrams for electromechanical transducers with memristors at different fractional orders;

[0169] Figure 3 The phase diagrams for an electromechanical transducer with memristors at α = 0.98, 0.998, 0.993, 0.997, 0.999 and 1 are shown. Figure 3 (a) is the phase diagram of an electromechanical transducer with memristors at α = 0.98; Figure 3 (b) is the phase diagram of an electromechanical transducer with memristors at α = 0.998; Figure 3 (c) is the phase diagram of an electromechanical transducer with memristors at α = 0.993; Figure 3 (d) is the phase diagram of an electromechanical transducer with memristors at α = 0.98; Figure 3 (e) is the phase diagram of an electromechanical transducer with memristors at α = 0.998; Figure 3(f) is the phase diagram of an electromechanical transducer with memristors at α = 0.993; Figure 3 (g) is the phase diagram of an electromechanical transducer with memristors at α = 0.997; Figure 3 (h) is the phase diagram of an electromechanical transducer with memristors at α = 0.999;

[0170] Figure 3 (i) is the phase diagram of an electromechanical transducer with memristors when α = 1; Figure 3 (j) is the phase diagram of an electromechanical transducer with memristors at α = 0.997; Figure 3 (k) is the phase diagram of an electromechanical transducer with memristors at α = 0.999; Figure 3 (l) is the phase diagram of an electromechanical transducer with memristors when α=1;

[0171] Figure 4 Lyapunov exponents and bifurcation diagrams for electromechanical transducers with different memristors β and α = 0.999;

[0172] Figure 5 Phase diagrams of an electromechanical transducer with memristors at β = 0.03, 0.06, 0.12 and α = 0.999; Figure 5 (a) is the phase diagram of an electromechanical transducer with memristors at β = 0.03; Figure 5 (b) is the phase diagram of an electromechanical transducer with memristors at β = 0.06; Figure 5 (c) is the phase diagram of an electromechanical transducer with memristors at β = 0.12; Figure 5 (d) is the phase diagram of an electromechanical transducer with memristors at β = 0.03; Figure 5 (e) is the phase diagram of an electromechanical transducer with memristors at β = 0.06; Figure 5 (f) is the phase diagram of an electromechanical transducer with memristors at β = 0.12;

[0173] Figure 6 Lyapunov exponents and bifurcation diagrams for electromechanical transducers with different memristor initial values ​​and α = 0.999;

[0174] Figure 7 The phase diagram of the electromechanical transducer is given when α = 0.999 and the initial values ​​of the memristor are 0.2, 4, and 7. Figure 7 (a) is the phase diagram of the magneto-electro-mechanical transducer when α = 0.999 and the initial value of the memristor is 0.2; Figure 7 (b) is the phase diagram of the magneto-electro-mechanical transducer when α = 0.999 and the initial value of the memristor is 4; Figure 7 (c) is the phase diagram of the magneto-electro-mechanical transducer when α = 0.999 and the initial value of the memristor is 7; Figure 7(d) is the phase diagram of the magneto-electro-mechanical transducer when α = 0.999 and the initial value of the memristor is 0.2; Figure 7 (e) is the phase diagram of the magneto-electro-mechanical transducer when α = 0.999 and the initial value of the memristor is 4; Figure 7 (f) is the phase diagram of the magneto-electro-mechanical transducer when α = 0.999 and the initial value of the memristor is 7;

[0175] Figure 8 The sample entropy and C0 complexity plots of electromechanical transducers with different fractional orders, memristor parameters and initial values; Figure 8 (a) is a sample entropy diagram of electromechanical transducers with different fractional orders; Figure 8 (b) is a sample entropy diagram of electromechanical transducers with different memristor parameters; Figure 8 (c) is a sample entropy diagram of electromechanical transducers with different initial values; Figure 8 (d) is a C0 complexity diagram of electromechanical transducers with different fractional orders; Figure 8 (e) is a C0 complexity diagram of electromechanical transducers with different memristor parameters; Figure 8 (f) is the C0 complexity diagram of electromechanical transducers with different initial values;

[0176] Figure 9 Equivalent cloud plots of sample entropy and C0 complexity for electromechanical transducers with memristors; Figure 9 (a), (b), and (c) are sample entropy contour maps based on electromechanical transducers with memristors; Figure 9 (d), (e), and (f) are equivalent cloud maps of C0 complexity based on an electromechanical transducer with memristors;

[0177] Figure 10 Figures showing the tracking performance and tracking error of an electromechanical transducer with memristors at different fractional orders; Figure 10 (a) shows the tracking performance of an electromechanical transducer with memristors at different fractional orders; Figure 10 (b) shows the tracking error of an electromechanical transducer with memristors at different fractional orders;

[0178] Figure 11 The tracking performance and tracking error of electromechanical transducers with different memristor β are shown in the figure. Figure 11 (a) is a tracking performance diagram of electromechanical transducers with different memristor β; Figure 11 (b) is a tracking error diagram of electromechanical transducers with different memristor β;

[0179] Figure 12 Figures showing the tracking performance and tracking error of electromechanical transducers with different memristor initial values; Figure 12 (a) Tracking performance diagram of electromechanical transducers with different memristor initial values; Figure 12 (b) Tracking error diagram of electromechanical transducers with different memristor initial values;

[0180] Figure 13 An adaptive law graph for a type 2 fuzzy wavelet neural network with different β and memristor initial values; Figure 13 (a) is the adaptive law graph of a type 2 fuzzy wavelet neural network with different β; Figure 13 (b) is the adaptive law graph of a type 2 fuzzy wavelet neural network with different memristor initial values;

[0181] Figure 14 The predicted performance diagram of the proposed second-order fractional TD is shown. Figure 14 (a) is the predicted performance plot of the proposed second-order fractional TD when α = 0.98; Figure 14 (b) is the predicted performance plot of the proposed second-order fractional TD when β = 0.12; Figure 14 (c) is the predicted performance plot of the proposed second-order fractional TD when x5(0)=4;

[0182] Figure 15 The fault control input diagrams for electromechanical transducers with memristors at different fractional orders are shown. Figure 15 (a) and (b) are fault control input diagrams for electromechanical transducers with memristors at different fractional orders;

[0183] Figure 16 The fault control input diagram for electromechanical transducers with different memristor β is shown. Figure 16 (a) and (b) are fault control input diagrams for electromechanical transducers with different memristor β;

[0184] Figure 17 This is a schematic diagram of the method flow of the present invention. Detailed Implementation

[0185] 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.

[0186] 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.

[0187] 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.

[0188] Please see Figures 1 to 17 A preset time control method for an adaptive neural network of an electromechanical transducer with memristors includes the following steps:

[0189] S1: System modeling of an electromechanical transducer with memristors;

[0190] S2: Dynamics Analysis: Based on the analysis of bifurcation diagrams, sample entropy, and C0 complexity dynamics, this study reveals the effects of different fractional orders, system parameters β, and initial values ​​of memristors on the internal evolution of system dynamics and complexity performance.

[0191] S3: Design an adaptive neural network controller with a preset time, including: applying a delay constraint function to the tracking error starting from an arbitrary position; using a type 2 fuzzy wavelet neural network to handle parameter disturbances and system uncertainties; using a second-order fractional tracking differentiator; and constructing a fault control input.

[0192] S4: Stability analysis.

[0193] 1. System modeling of an electromechanical transducer with memristors:

[0194] 1.1 Electromechanical transducers with memristors:

[0195] A schematic diagram of an electromechanical transducer with a memristor is shown below. Figure 1As shown, this transducer typically consists of both electrical and mechanical parts. The mechanical part, which reciprocates along the Z-axis, comprises a moving beam, springs, and an operating lever, while the electrical part consists of a memristor, capacitor, and inductor. The operating lever is fixed to a rigid beam and enclosed within a spring. An external sinusoidal voltage applied between the operating lever, electrical, and mechanical components enables the transducer to operate normally.

[0196] Based on the inherited voltage-charge characteristics, capacitors exhibit Duffing-type nonlinearity, which is expressed as follows:

[0197]

[0198] in and C a a3 and a5 represent voltage, charge, and capacitance, respectively, and represent system coefficients.

[0199] The expression for fractional impedance is: Where j, α, and ω represent the imaginary part, fractional order, and frequency, respectively. In the fractional characteristic of a dielectric, the current... and voltage Relationship with time t

[0200]

[0201] To better characterize the relationship between magnetic flux and charge, memristors are used instead of resistors in electromechanical transducers. The relationship between a memristor and its components is defined using a quadratic nonlinear smooth function, written as...

[0202] φ(q)=-βq+0.5γq 2 sgn(q), (3)

[0203] Where φ is the magnetic flux, and β and γ represent positive constants. The memristor function W(q) is then given as W(q) = -β + γ|q|.

[0204] Considering the fractional order and memristor characteristics, and utilizing Newton's laws and Kirchhoff's law, the mathematical model of the fractional-order electromechanical transducer is written as follows:

[0205]

[0206] Where m represents mass, h represents the coefficient of viscous friction, k represents the stiffness coefficient, l represents the length of the moving coil, B represents the magnetic flux density, L represents the inductance, v0 represents the amplitude, Ω represents the frequency, and z represents the displacement in the direction of gravity.

[0207] Define new variables t=ω e τ and in Q0 represents the reference charge of the capacitor;

[0208] By adding control inputs, a dimensionless mathematical model of an electromechanical transducer with a memristor is given.

[0209]

[0210] in and There are three dimensionless parameters α, C, and ω, representing the fractional order, Caputo fractional derivative, and frequency, respectively, and u2 =

[11] [u1 u2] T and u4 = [1 1][u3 u4] T This indicates the control input when a fault occurs.

[0211] Memristors better characterize the physical properties between magnetic flux and charge in electromechanical transducers. The associated fractional-order model accurately describes the internal characteristics of the electromechanical transducer, including current, voltage, memristor, viscosity, and carbon nanotube percentage, increasing the freedom of controller design. When α = 1 and a resistor replaces the memristor, the proposed mathematical model degenerates from existing models.

[0212] 1.2 Actuator Failure Model:

[0213] The definition of a stuck fault is:

[0214]

[0215] in This represents a positive integer, meaning the jamming failure occurred at time t. j time.

[0216] Control failure is represented as

[0217]

[0218] Where, δ i Indicates at t i Once control effectiveness is lost, the effectiveness of the remaining actuators... and u i This represents the lower limit of the i-th control input.

[0219] Considering the two types of actuator failures mentioned above, the control input is written as follows:

[0220]

[0221] where u=[u1,…,u n ] T , λ=diag{λ1,…,λ n} and δ=diag{δ1,…,δ n},λ i i = 1, ..., n have only two values: 0 or 1;

[0222] When λ i When the value is 1, the actuator is stuck; otherwise, the control fails. It is important to note that to ensure the control objective, the number of malfunctioning actuators should be less than n-1.

[0223] 1.3 Preparatory Work

[0224] Definition 1: For any differentiable function, its Caputo fractional derivative is derived as follows:

[0225]

[0226] in, This indicates that n-1 < α < n and The Euler Gamma function, where α represents the fractional value, n represents a natural number, and t represents time;

[0227] For any continuous functions F1(t) and F2(t), on [0, t... s Within the interval [0 < α < 1], where t s Representing time, the following equation exists:

[0228]

[0229] Assumption 1: The expected trajectory and its fractional derivative are continuous and smooth.

[0230] 2. Dynamic Analysis

[0231] To further reveal the evolution of system dynamics and facilitate subsequent controller design, dynamic analysis was conducted under different fractional orders, system parameters, and memristors.

[0232] The system parameters of the fractional-order electromechanical transducer, including the memristor, are γ1=0.2, γ2=0.1, β1=0.9, β2=0.1, ζ1=0.01, ζ2=0.05, ω2=1.2, ω=0.85, E0=23.5, ζ3=0.01, β=0.03, γ=0.02, and α=0.99.

[0233] The fractional order, β, and initial values, which are closely related to the system model in Equation (5), greatly influence the dynamics of the electromechanical transducer. From... Figure 2 It can be seen that this electromechanical transducer exhibits rich nonlinear dynamic characteristics in the fractional order [0.981] range, and alternates between periodic and chaotic states. When this electromechanical transducer is in a multi-periodic state, such as Figure 3 (a) and Figure 3The phase diagram of (d) is shown. As the fractional order increases to 0.9925, the system immediately falls into a chaotic state, as shown... Figure 3 (b) and Figure 3 As shown in (e). In [0.9925 0.9935], the system instantaneously returns to as... Figure 3 (c) and Figure 3 (f) shows the multi-period state. When α is greater than 0.9935 and less than 0.9975, this electromechanical transducer always remains in a chaotic state. As a representative example, Figure 3 (g) and Figure 3 (j) provides the corresponding phase diagram. When the fractional order continues to increase to around 1, the system returns to the state shown in Figure (j). Figure 3 (h) and Figure 3 The multi-periodic state is shown in (k). Finally, this electromechanical transducer produces chaotic oscillations near 1, such as... Figure 3 (i) and Figure 3 As shown in (l).

[0234] To study the effects of changes in the memristor parameters and initial values ​​on the system dynamics of the electromechanical transducer under consideration, this study investigated the influence of such changes on the system dynamics. Figure 4 The Lyapunov exponents and bifurcation diagrams of magneto-electromagnetic transducers with different memristors β and α = 0.999 are shown. Figure 5 (a) and Figure 5 As shown in (d), if β is less than 0.055, the system will exhibit multi-period motion. When β is in the interval [0.055-0.15], the system will fall into chaotic motion, as shown in (d). Figure 5 As shown in (b), (c), (e) and (f).

[0235] Figure 6 The Lyapunov exponent and bifurcation diagrams of electromechanical transducers with different initial memristor values ​​are shown. When the initial charge values ​​are at [0, 0, 4) and [6, 4, 8], the system produces high-frequency chaotic oscillations, such as... Figure 7 As shown in (a), (c), (d), and (f). In the remaining images of

[08] , this electromechanical transducer exhibits a multi-period phenomenon, such as... Figure 7 As shown in (b) and (d).

[0236] Complexity analysis of electromechanical transducers with memristors was performed using sample entropy and C0 complexity.

[0237] Suppose there exists a time series {x(n), n=0,…,N-1}, then the relevant discrete Fourier transform can be written as:

[0238]

[0239] in It is the rotation factor.

[0240] Define time series S N and frequency sequence The square value is

[0241]

[0242] Where γ a This indicates the control parameters.

[0243] The C0 complexity is then expressed as

[0244]

[0245] The sample entropy and C0 complexity of fractional-order changes of electromechanical transducers are as follows: Figure 8 (a) and Figure 8 As shown in (d), its sample entropy and C0 complexity with different memristor β are as follows. Figure 8 (b) and Figure 8 As shown in (e). Figure 8 (c) and Figure 8 (f) shows the sample entropy and C0 complexity of the system with memristor initial values. Clearly, the results of the complexity analysis are consistent with those of the previous Lyapunov exponent and bifurcation analysis. Figure 1 To. The equivalent contour plot of sample entropy and C0 complexity for electromechanical transducers with memristors is shown below. Figure 9 As shown. In Figure 9 (a) and Figure 9 The complex dynamics of the system in the β-α plane (d) are shown in the figure. Figure 9 (b) and Figure 9 In the β-x5(0) plane of (e), the complex dynamics of the system occur in the lower right corner. Figure 9 (c) and Figure 9 In the x5(0)-α plane of (f), the contour plot shows that the electromechanical transducer exhibits rich dynamic behavior in the upper half of the figure. In summary, the order, parameters, and initial values ​​of the memristor directly determine the complexity performance of the electromechanical transducer.

[0246] 3. Design a controller for an adaptive neural network with a preset time.

[0247] Type 3.1 Fuzzy Wavelet Neural Network

[0248] Fuzzy neural networks possess powerful approximation capabilities in areas such as model recognition, intelligent control, and prediction. To further enhance the computational and mapping capabilities of advanced control methods, a five-layer type-2 fuzzy wavelet neural network was introduced.

[0249] Input layer: Each neuron receives an input signal vector, for example... And so on, and transmit the corresponding information directly to the next layer.

[0250] Membership layer: Each neuron converts the input signal into a membership function to represent fuzzy language terms. Then, the upper and lower membership degrees based on the Mexican-hat wavelet function are written as...

[0251]

[0252] Where i = 1, ..., N S j = 1, ..., n s ,n s and N S This indicates the number of input signals and the number of rules. and These are called translation and expansion parameters of the member functions.

[0253] Rule layer: There exists a series of fuzzy IF-THEN rules.

[0254] If x1 is and...and yes So It is ω j j = 1, ..., N s (15)

[0255] Where i = 1, ..., n s j = 1, ..., N s , This represents the j-th member function of the i-th input;

[0256] In this context, the inference engine of each neuron performs fuzzy inference as follows: The relevant information is passed down to the lower level, where ^ denotes the minimal operator. and Indicates membership degree;

[0257] Degradation: Incentive output is defined as

[0258]

[0259] in and ξ i Indicates the upper and lower values ​​of fuzzy inference;

[0260] Output layer: The entire output of the type 2 fuzzy wavelet neural network is defined as follows:

[0261]

[0262] Where w T ξ(x) represents the weight transpose and basis functions of a type 2 fuzzy wavelet neural network;

[0263] Obviously, there is

[0264]

[0265] Where ε and D x This represents the approximate error and compact set of x with appropriate boundaries.

[0266] Introduce an optimal parameter w * This parameter equals The solution, where Ω w Let w be a compact set. Define the equation. And w * These are artificial variables. Furthermore, and

[0267] To accelerate the solution process and save computational resources, the type 2 fuzzy wavelet neural network was converted to...

[0268] w T ξ(x)≤ζξ T (x)ξ(x) / 2b 2 +b 2 / 2, (19)

[0269] Where ζ=||w|| 2 and Established, and Let b represent the estimate of ζ and its positive constant. From the preceding introduction, we know that... Considering that the derivative of the constant Caputo is zero.

[0270] In typical forward or backward fuzzy neural networks, the negative error gradient for weight correction is obtained by calculating numerous partial differential equations, and the offline training process takes considerable time. Therefore, an update law is constructed to adjust the weights online. Furthermore, compared to Type I fuzzy neural networks, the established Type II fuzzy wavelet neural network exhibits more flexible modeling and nonlinear function handling capabilities.

[0271] 3.2 Delay Constraint Function

[0272] To obtain a pre-set stable time and tracking accuracy in advance, the time-varying constraint boundary is defined as follows:

[0273]

[0274] Where, T,σ i And p means that the condition p > (n) is satisfied. s The settling time, tracking accuracy, and positive integer n are given by (+1) / 2. s As the order of the system. It is worth noting that as t→0... It is infinitely large.

[0275] Due to the entire process The delayed tracking constraint was followed, and the tracking target has been achieved, where e i (t) is the tracking error given later. Meanwhile, we obtain...

[0276] A distance function between tracking error and constraint boundary is introduced. Where c i It is a positive constant. Clearly...

[0277] To better understand the value γ i (t) is mapped to the range (0,1], and the delay constraint function is set to

[0278]

[0279] Where γ i and a i Indicates γ i (t) is an abbreviation for and a preset safe distance.

[0280] when At that time, there is ρ i (γ i ) = 1. The system is close to the constraint boundary, γ i (t)∈(0,a i [This could lead to collisions and rule violations. Conversely, γ] i (t)∈(a i When ∞), the system remains at a safe point far from the constraint boundary. There exists a i ≡c i s i 2 make sure

[0281] To facilitate subsequent controller design, the transformed tracking error is defined as follows:

[0282] s i (t)=e i (t) / ρ i (γ i ), i = 1, 3. (22)

[0283] Regarding (22), it should also be pointed out that only when e i (0) = 0 when s i (0) = 0; s i (t)→∞ is represented as And s i (t)→-∞ is represented as in this case, It is impossible to reach infinity at the beginning.

[0284] 3.3 Controller Design

[0285] The entire controller design consists of four steps.

[0286] Step 1: Construct the first tracking error as e1 = x1 - x d1 , where x d1 Let x1 represent the ideal trajectory and x1 represent the state variable. Design the first Lyapunov function as...

[0287]

[0288] Taking the fractional derivative of (23) yields

[0289]

[0290] in,

[0291]

[0292]

[0293] And s2(t)=x2-α2·α2 represents the first virtual control law in the subsequent design.

[0294] The first virtual control law is designed as

[0295]

[0296] Where k1 > 0.

[0297] Substituting (25) into (24) gives

[0298]

[0299] Step 2: Design the second Lyapunov function as follows

[0300]

[0301] Where θ2>0.

[0302] Its fractional derivative is as follows

[0303]

[0304] in

[0305] In practical applications, electromechanical transducers are inevitably affected by factors such as load variations, temperature, electromagnetic interference, and manufacturing errors. This fact, related to parameter disturbances and system uncertainties, significantly impacts the performance of electromechanical transducers to a certain extent. To address this issue, a type-2 fuzzy wavelet neural network is used to approximate f2(x).

[0306]

[0307] Meanwhile, the first virtual control law contains time-varying constraint functions, making it difficult to directly derive the fractional derivative of α². To address this issue, a fractional differentiator is designed:

[0308]

[0309] Where α ik κ represents the i-th input variable of the fractional-order tracking differentiator. i1 and κ i2 All of these are its variables, υ i1 and υ i2 This indicates that the parameters are being adjusted.

[0310] Based on (29) and (30), (28) is rewritten as

[0311]

[0312]

[0313] Where b2 > 0.

[0314] The fault control input and the associated adaptive law are expressed as follows:

[0315]

[0316]

[0317] Where k2>0, m2>0.

[0318] Substituting (32) and (33) into (31), (31) simplifies to

[0319]

[0320] Step 3: The third tracking error is designed as e3 = x3 - x d3 , where x d3 Represents the ideal trajectory. The third Lyapunov function is chosen as:

[0321]

[0322] Obviously, the fractional derivative of (35) is derived as follows:

[0323]

[0324] in,

[0325]

[0326]

[0327] And s4(t) = x4 - α4. α4 represents the second virtual control law designed in the following content.

[0328] Accordingly, the second virtual control is defined as

[0329]

[0330] Where k3 > 0.

[0331] Substituting (37) into (36) gives

[0332]

[0333] Step 4: Design the fourth Lyapunov function as follows

[0334]

[0335] Where θ4>0.

[0336] The fractional derivative of formula (39) is:

[0337]

[0338] in

[0339] To replicate parameter perturbations and system uncertainties, a type 2 fuzzy wavelet neural network is used to approximate f4(x).

[0340]

[0341] Meanwhile, to avoid the complex fractional derivative of α4, the fractional-order tracking differentiator proposed in (30) is used. From (30) and (41), it can be seen that (40) is further written as...

[0342]

[0343] Where b4 > 0.

[0344] Therefore, the fault control input and the corresponding adaptive law are derived as follows:

[0345]

[0346]

[0347] Where k4 > 0, m4 > 0.

[0348] Substituting (43) and (44) into (42), it has the following inequality.

[0349]

[0350] 4. Stability Analysis

[0351] Theorem 1: Consider a fractional-order electromechanical transducer (5) with a memristor under Assumption 1. If an adaptive neural network preset time controller composed of (32) and (43) is used, the following objectives can be achieved by reasonably selecting the design parameters:

[0352] 1) All signals of the closed system, such as tracking error e i The transformed tracking error s i and transformed weight error All converge to any small value near the origin.

[0353] 2) Solved the problems of high-frequency chaotic oscillation and unknown dynamics.

[0354] 3) Tracking error and tracking accuracy are achieved under preset stable time and time-varying constraints.

[0355] Proof: The entire Lyapunov function is designed as follows

[0356]

[0357] The fractional derivative of (46) is obtained.

[0358]

[0359] in γ v =min{2k i ,i=1,…,4,m2,m4},

[0360] The fractional integral of (47) exists.

[0361]

[0362] in This represents the Caputo fraction integral.

[0363] 5. Simulation Results and Analysis

[0364] The simulation experiment was analyzed to verify the effectiveness and feasibility of the proposed solution. First, the parameters of the delay constraint function were chosen as T = 1, s1 = 0.04, s3 = 0.03, c1 = 2, c3 = 2, and p = 3. The ideal trajectory is x. d1 = sint and x d3 =0.02cost. Control parameters are k1=k2=40, k3=k4=25, m2=m4=4, b2=0.2, b4=0.08, θ2=θ4=0.5. The parameters of the second-order fractional-order tracking differentiator are adjusted to υ. 21 =υ 22 =υ 41 =υ 42 =6. Meanwhile, the translation and dilation parameters of the member functions associated with the type 2 fuzzy wavelet neural network are set to [-1 -0.50 0.51] and 1, respectively. Furthermore, the corresponding actuator fault when t ≥ 12s is given as... and δ2=δ4=0.8.

[0365] Figure 10-12 The tracking performance and tracking error of the electromechanical transducer under different fractional orders, β, and memristor initial values ​​are shown. Based on the preceding description, the order, parameters, and initial values ​​of the memristor directly determine the complexity performance of the electromechanical transducer. Energy from chaotic oscillations can lead to system performance degradation or even system collapse. Clearly, this problem is effectively solved with the intervention of the adaptive neural network preset time control scheme. The system output signal tracks the ideal trajectory within the set time and achieves the preset accuracy under different operating conditions. Simultaneously, the tracking error never violates the delay constraint boundary throughout the entire time, and the system always operates within the safe and collision-avoidance regions thanks to the delay constraint function. Furthermore, even when there is a jamming fault or loss of control when t ≥ 12s, this scheme exhibits good effectiveness and robustness.

[0366] The adaptive law of the type 2 fuzzy wavelet neural network plays an important role in achieving excellent transient and steady-state performance and strong approximation capability in electromechanical transducers with memristors. Figure 13 The adaptive law of the type-2 fuzzy wavelet neural network was plotted under different initial values ​​of β and memristor. Clearly, throughout the process, the three curves almost overlap, and their values ​​converge rapidly to near zero within a specified time. Furthermore, it effectively avoids internal and external disturbances caused by load variations, temperature, electromagnetic interference, and manufacturing errors.

[0367] Directly calculating the fractional derivative of the virtual control is difficult and impractical. A second-order fractional tracking differentiator is used to address this problem. Figure 14The estimation performance of the proposed fractional-order tracking differentiator is shown in different cases. As can be seen from the figure, this fractional-order tracking differentiator still has a very strong approximation capability even when an actuator failure occurs at a certain moment.

[0368] Figure 15-16 The fault control input of the electromechanical transducer under different fractional orders and memristor β is described. The three curves essentially overlap after a very short time. Clearly, the proposed scheme provides a stable control input when the tracking error remains within the constraint boundaries. Meanwhile, actuator faults (including jamming and control failure) occur at twelfth second, but the fault control input recovers stability after a small jump. Furthermore, the fault control input is insensitive to variations in the fractional order and β of the memristor.

[0369] 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 preset time control method for an adaptive neural network of an electromechanical transducer with a memristor, characterized in that: The method includes the following steps: S1: System modeling of an electromechanical transducer with memristors; S2: Dynamic Analysis: Based on bifurcation diagram, sample entropy, and Complexity dynamics analysis reveals the parameters of different fractional orders and memristors. The impact of initial values ​​of the memristor on the internal evolution of system dynamics and complexity performance; S3: Design a controller for the preset time of an adaptive neural network, including: Apply a delay constraint function to the tracking error starting from an arbitrary position; Type II fuzzy wavelet neural network is used to process parameter disturbances and system uncertainties; The Type 2 fuzzy wavelet neural network comprises an input layer, a membership layer, a rule layer, a descent layer, and an output layer. In the input layer, each neuron receives an input signal vector and directly transmits the information to the next layer. In the membership layer, each neuron converts the input signal into a membership function and calculates upper and lower membership degrees based on the Mexican-hat wavelet function to represent fuzzy language terms. The rule layer performs fuzzy IF-THEN rule inference. The descent layer calculates the excitation output. The output layer outputs the entire output of the Type 2 fuzzy wavelet neural network. Use a second-order fractional-order tracking differentiator; Construct fault control inputs; In the fault control input, a stuck fault is defined as: （8） in, Representing a positive integer, the freezing failure occurs when... time; Control failure is represented as （9） in, Indicates in Once control effectiveness is lost, the effectiveness of the remaining actuators... and This represents the lower limit of the i-th control input; Based on the two types of actuator failures mentioned above, the control input is expressed as follows: （10） in, , , and , It has only two values: 0 or 1; when The actuator is stuck, otherwise it is a control failure; the number of actuators that have failed is less than n-1. S4: Stability analysis.

2. The method for preset time control of an adaptive neural network for an electromechanical transducer with a memristor according to claim 1, characterized in that: In S1, the electromechanical transducer with memristor consists of two parts: mechanical and electrical. The mechanical part, which oscillates along the Z-axis, consists of a moving beam, a spring, and an operating rod. The operating rod is fixed on the rigid beam and enclosed in the spring. The electrical part consists of a memristor, a capacitor, and an inductor. These three devices are connected in series with a sinusoidal voltage power supply.

3. The method for preset time control of an electromechanical transducer with a memristor according to claim 2, characterized in that: In S1, system modeling is performed on the electromechanical transducer with memristors, specifically including: In an electromechanical transducer with a memristor, based on the inherited voltage-charge characteristics, the capacitor exhibits Duffing-type nonlinearity, and the voltage is expressed as: （1） in, Indicates voltage. Represents electric charge, Indicates capacitance. and Represents system coefficients; The expression for fractional impedance is: （2） in, Indicates the imaginary part. Indicates fractional order. Indicates frequency; In the fractional properties of a dielectric, current and voltage Over time The relation is expressed as: （3） In electromechanical transducers, memristors are used to replace resistors. The configuration of a memristor is defined using a quadratic nonlinear smooth function, expressed as: （4） in, It is magnetic flux. and Represents positive numbers; Then memristor function , represented as: （5） Using Newton's laws and Kirchhoff's law, the mathematical model of a fractional-order electromechanical transducer can be expressed as follows: （6） in, Indicates quality, Indicates the coefficient of viscous friction. Indicates the stiffness coefficient. Indicates the length of the moving coil. Represents magnetic flux density, Indicates inductance. Indicates amplitude, Indicates frequency, Indicates displacement in the direction of gravity; Define new variables , , and ,in , The reference charge of the capacitor; By adding control inputs, the dimensionless mathematical model of an electromechanical transducer with a memristor can be expressed as follows: （7） in, , , , , , , , , and There are three dimensionless parameters α , and , representing the fractional order, Caputo fractional derivative, and frequency, respectively. and This indicates the control input when a fault occurs.

4. The method for preset time control of an adaptive neural network for an electromechanical transducer with a memristor according to claim 3, characterized in that: by definition, for any differentiable function, its Caputo fractional derivative is derived as: （11） in, Indicates having and Euler's Gamma function, Represents the fractional order value. Represents natural numbers, Indicates time; For any continuous function and ,exist and Within the interval, where Representing time, the following equation exists: （12） Assume that the desired trajectory and its fractional derivative are continuous and smooth.

5. The method for preset time control of an adaptive neural network for an electromechanical transducer with a memristor according to claim 4, characterized in that: In S3, the type 2 fuzzy wavelet neural network specifically includes: Input layer: Each neuron receives the input signal vector and transmits the corresponding information directly to the next layer; Membership layer: Each neuron converts the input signal into a membership function to represent fuzzy language terms. The upper and lower membership degrees based on the Mexican-hat wavelet function can be written as: (13) in, For neurons to receive input signal vectors, , , and This indicates the number of input signals and the number of rules. and These are called the translation and expansion parameters of the member functions; Rule layer: There exists a series of fuzzy IF-THEN rules. if yes and and yes ; So yes , (14) in, , , This represents the j-th member function of the i-th input; In this context, the inference engine of each neuron performs fuzzy inference as follows: The relevant information was transmitted to the lower levels, among which... Represents the smallest operator. and Indicates membership degree; Degradation: Incentive output is defined as .(15) in and Indicates the upper and lower values ​​of fuzzy inference; Output layer: The entire output of the type 2 fuzzy wavelet neural network is defined as follows: ,(16) in and Represent the weight transpose and basis functions of a type 2 fuzzy wavelet neural network; Therefore, it exists. ,(17) in, and express Approximate errors and compact sets with appropriate boundaries; Introduce an optimal parameter This parameter equals The solution, where, express Compact set; Define equation and These are artificial variables; furthermore, and ; Convert the type 2 fuzzy wavelet neural network to: ,(18) in, and Established, and and Indicates to Estimates and normals; And the Caputo derivative of the constant is zero.

6. The method for preset time control of an electromechanical transducer with a memristor according to claim 5, characterized in that: In S3, the delay constraint function is specifically expressed as follows: Define the time-varying constraint boundary as: （19） in, , and This indicates that the condition is met. The settling time, tracking accuracy, and positive integers, As the order of the system; when hour It is infinitely large; Throughout the process The delayed tracking constraint was followed, and the tracking target has been achieved. For tracking error; at the same time, obtain ; Introduce a distance function between the tracking error and the constraint boundary: （20） in It is a positive constant; as can be seen from formula (20), , ; In order to better understand the value Mapping to the range (0,1], set the delay constraint function as follows: （21） in and express Abbreviations and preset safety distances; when Sometimes, The system is approaching the constraint boundary. This could lead to collisions and rule violations; when At this time, the system remains at a safe point far from the constraint boundary, which indicates that... make sure , ; The transformed tracking error is defined as: .（22） In formula (22), when hour ; Represented as and Represented as .

7. The method for preset time control of an electromechanical transducer with a memristor according to claim 6, characterized in that: In step S3, the controller for the preset time of the adaptive neural network is designed, specifically including: The first tracking error is constructed as follows: （23） in Represents the ideal trajectory. Represents state variables; Design the first Lyapunov function as: .（24） Taking the fractional derivative of formula (24), we get: （25） in, （26） （27） and . This represents the first virtual control law in the subsequent design; The first virtual control law is designed as follows: ,（28） in ; Substituting formula (28) into formula (25), we get: .（29） Design the second Lyapunov function as follows: ,（30） in, ; Taking the fractional derivative of formula (30) yields: （31） in, （32） A type 2 fuzzy wavelet neural network is used to approximate... Represented as: （33） Design a fractional differentiator, represented as: （34） in, This represents the i-th input variable of the fractional-order tracking differentiator. and They are all its variables. and This indicates that the parameter is being adjusted. Based on formulas (33) and (34), formula (31) can be rewritten as: （35） in, ; The fault control input and the associated adaptive law are expressed as follows: （36） （37） in, ; Substituting formulas (36) and (37) into formula (35), formula (35) is simplified to: .（38） The third tracking error is designed to be ,in Represents the ideal trajectory; The third Lyapunov function is chosen as: .（39） The fractional derivative of formula (39) is then derived as follows: （40） in, （41） （42） and （43） This represents the second virtual control law; The second virtual control law is defined as: ,（44） in, ; Substituting formula (44) into formula (40) yields: .（45） Design the fourth Lyapunov function as follows: ,（46） in, ; The fractional derivative of formula (46) is calculated as follows: ,（47） in, :; A type 2 fuzzy wavelet neural network is used to approximate... Represented as: .（48） In order to avoid The complex fractional derivative is obtained by using the fractional tracking differentiator proposed in formula (34), and according to formulas (34) and (48), formula (47) can be further written as: (49) in, ; The fault control input and the corresponding adaptive law are then derived as follows: （50） ,（51） in, ; Substituting formulas (50) and (51) into formula (49), we get: .（52）。

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