Random control method of three-stable-state piezoelectric cantilever beam structure energy harvester

Abstract

The invention discloses a random control method of a three-stable-state piezoelectric cantilever beam structure energy harvester, and belongs to the field of energy harvesters. The implementation method comprises the following steps: introducing white noise by considering random excitation in an environment, establishing an electromechanical coupled strong nonlinear three-stable state energy harvester model, and carrying out dimensionless processing on the energy harvester model to obtain an electromechanical coupled strong nonlinear three-stable state energy harvester dimensionless model; based on the acquired displacement data and voltage of the three-stable-state energy harvester, establishing a model equation of the three-stable-state energy harvester by using dimensionless coefficients zeta, theta, km (m = 1, 2, 3) and D in an electromechanical coupling equation of the strong nonlinear three-stable-state energy harvester by using a sparse identification method of a nonlinear power system; based on a strong nonlinear three-stable-state energy harvester electromechanical coupling equation established by using an SINDy method and system response, an optimal control law alpha is calculated by using a model prediction control method, a control signal is applied through a Buck-boost circuit according to the optimal control law alpha, random control of the three-stable-state piezoelectric cantilever beam structure energy harvester is realized, and the output power of the energy harvester is increased.

Description

Technical Field

[0001] The present invention relates to a random control method of a tristable piezoelectric cantilever beam structure energy harvester, and in particular to a circuit control method of a tristable energy harvester in a random environment based on impedance matching and power maximization principles, belonging to the field of energy harvesters. Background Art

[0002] Vibration is a common phenomenon in nature. Given its wide distribution and high energy density, research on how to convert vibration energy and collect and utilize it has gradually emerged in recent decades. Piezoelectric vibration energy harvesters can harvest mechanical energy from the environment. Cantilever beam structures are often used as mechanical structures for energy harvesters because of their simple structure and high stability. The conversion of mechanical energy to electrical energy is achieved through the positive piezoelectric effect of the piezoelectric film on the cantilever beam. Vibration energy harvesting has been widely studied in the literature. Combined with the basic configuration of the spring-mass-damper system, the vibration amplitude of the device at resonance can be significantly greater than the vibration amplitude of the base movement. Early energy harvester designs focused on linear energy conversion in narrowband frequencies. However, any deviation of the excitation frequency from resonance can lead to a sharp reduction in power generation, and the optimal performance of the device is limited to a very narrow bandwidth near the fundamental frequency. Its utility is significantly limited when faced with wide-spectrum vibration energy sources. In order to overcome this problem of traditional linear configuration, researchers have artificially introduced nonlinear restoring forces to broaden the energy harvesting frequency band, and proposed a series of innovative designs for nonlinear energy harvesters, covering monostable, bistable and even tristable nonlinear oscillator structures. Compared with linear harvesters, nonlinear designs have three significant advantages: first, the frequency response range is significantly broadened, allowing the harvester to more effectively capture vibration energy of a wider frequency; second, by enhancing the response amplitude, the energy conversion efficiency is further improved; third, the output power is systematically increased, providing a stronger energy supply for practical applications.

[0003] In addition to the energy harvester structure, the energy management circuit is also an important factor affecting the output of the energy harvester. The energy output of the energy harvester based on the piezoelectric cantilever beam structure is discontinuous and random. How to make the power management circuit follow the working mode of the energy harvester to achieve maximum power transmission is one of the key issues in the design of the power management circuit. The piezoelectric energy harvesting system circuit mainly consists of two parts: AC-DC and DC-AC parts. The AC-DC part has a rectifier circuit and an interface circuit. The DC-DC part is mainly composed of a switching converter. Researchers Ottman GK and others first proposed a standard energy storage circuit for optimal power transmission, which was later called a standard energy harvesting circuit. However, the circuit lacks impedance matching, resulting in low energy conversion efficiency. The researchers proposed a synchronous charge extraction interface circuit, a parallel synchronous inductor switch interface circuit, and a series synchronous inductor switch interface circuit to improve the efficiency of mechanical energy to electrical energy conversion, which is used to reduce the power loss of the piezoelectric energy harvester. In order to achieve maximum energy transmission, the design of the power management circuit must be consistent with the output characteristics and electromechanical equivalent model of the energy harvester. Due to the randomness of vibrations in the natural environment and the nonlinearity of the energy harvester, the internal impedance of the energy harvester changes all the time, so it is difficult to achieve maximum power transmission with fixed impedance matching technology. Researchers Kong Na and others used a microcontroller unit to execute a maximum power point tracking algorithm. Through dynamic resistance matching, the energy harvester can achieve a power efficiency of 72% near the 47Hz resonant frequency. However, this algorithm is designed for dynamic impedance matching of linear energy harvesters. For tri-stable nonlinear energy harvesters, a dynamic power maximization control circuit is also required to improve the collection performance. At present, there is no power maximization control method for tri-stable energy harvesters under random excitation. Summary of the invention

[0004] The purpose of the present invention is to improve the collection power of a tristable energy harvester under random excitation, and a random control method for a tristable piezoelectric cantilever beam structure energy harvester is provided; the method uses data drive to perform dynamic modeling on the tristable nonlinear piezoelectric cantilever beam energy harvester, and the system response calculation predicted power can be calculated according to the model. Based on the predicted power and the reference power, the circuit damping is dynamically adjusted using the model predictive control method to obtain the optimal matching resistance value, thereby further improving the power of the energy harvester to capture electrical energy from the random vibration source.

[0005] The objective of the present invention is achieved through the following technical solutions:

[0006] The present invention discloses a random control method for a tristable piezoelectric cantilever beam structure energy harvester. The random excitation in the environment is considered to introduce white noise excitation. Through the displacement data of the energy harvester, a random system model of the electromechanical coupling tristable energy harvester is established by using a sparse identification method of a data-driven nonlinear dynamic system. According to the predicted response of the random system, a reasonable loss function is set using a model predictive control method, and the optimal control rate of the tristable energy harvester under noise excitation of any intensity can be obtained. According to the optimal control law α, a control signal is applied through a Buck-boost circuit to realize the random control of the tristable piezoelectric cantilever beam structure energy harvester and increase the output power of the energy harvester.

[0007] The random control method of the tristable piezoelectric cantilever beam structure energy harvester disclosed in the present invention comprises the following steps:

[0008] Step 1: Considering the random excitation in the environment to introduce white noise, establish a strongly nonlinear tristable energy harvester model of electromechanical coupling, and perform dimensionless processing on the energy harvester model to obtain a dimensionless model of the strongly nonlinear tristable energy harvester of electromechanical coupling;

[0009] The strongly nonlinear tristable energy harvester model is shown in formula (1):

[0010]

[0011] In the formula, represents the displacement of the tip magnet, represents the piezoelectric voltage, M represents the mass of the tip magnet, represents time, C represents the damping coefficient, θ represents the electromechanical coupling coefficient, represents the random excitation from the outside world, C p represents the effective capacitance, Indicates the resistance of a circuit; The potential energy function of the system is expressed as a polynomial:

[0012]

[0013] In the formula is the stiffness coefficient;

[0014] The following transformation is introduced to make the system equation (1) dimensionless:

[0015]

[0016] in is the natural frequency of the cantilever beam, is the length scale of the cantilever beam. t, X, V represent the dimensionless time, displacement and voltage respectively; ζ, Θ, α represent the dimensionless damping coefficient, electromechanical coupling coefficient and electromechanical time constant ratio respectively; k m(m=1, 2, 3) represents the dimensionless stiffness coefficient.

[0017] The dimensionless electromechanical coupling equation of the strongly nonlinear tristable energy harvester is obtained:

[0018]

[0019] In the formula, represents the dimensionless potential energy function, and uses Gaussian white noise ξ(t) to describe the dimensionless external excitation. The mean and correlation function of its statistical characteristics are:

[0020] E[ξ(t)]=0, E[ξ(t)ξ(t+τ)]=Dδ(τ);

[0021] E[·] means averaging the function in brackets, δ(τ) means the Dirac function, τ means the correlation time interval, and D is the noise intensity of the dimensionless white noise;

[0022] Step 2: Based on the collected displacement data and voltage of the tristable energy harvester, the dimensionless coefficients ζ, Θ, and k in the electromechanical coupling equation of the strong nonlinear tristable energy harvester are identified using the sparse identification method of nonlinear dynamic systems. m (m=1,2,3), D, establish the model equation of the tri-stable energy harvester.

[0023] The electromechanical coupling equation (3) of the strongly nonlinear tristable energy harvester can be rewritten as Stochastic differential equations SDEs:

[0024]

[0025] Among them, Y(t) is the dimensionless velocity; B(t) is the standard Brownian motion;

[0026] Acquisition time [t1, t2, ..., t N+1 ] corresponds to the voltage [v1,v2,...,v N+1 ] and the cantilever tip displacement [x1,x2,...,x N+1 ], speed [y1,y2,...,y N ]Through (x i+1 -x i ) / (t i+1 -t i )(i=1,2,...,N) is calculated, x i ,y i and v i Indicates t i The displacement, velocity and voltage corresponding to the moment, the time interval is Δt, and N is the number of sampling points.

[0027] The non-zero diffusion term and drift term in formula (4) are expressed as:

[0028]

[0029] The drift term a1 only contains the velocity y i And the coefficient is 1, and there is no need to identify its coefficient. For the drift term a2, use the basis function Φ k and coefficient c k Approximate the product of

[0030]

[0031] a2(x i ,y i ,v i ) indicates t i The diffusion drift term corresponding to the moment, coefficient c k is the basis function Φ k The corresponding coefficients are: For using 14 basis functions, k = 1, ..., 14.

[0032] For a3, use the basis function M j and coefficient p j Approximate the product of

[0033]

[0034] a3(x i ,y i ,v i ) indicates t i The diffusion drift term corresponding to the time, coefficient p j is the basis function M j The corresponding coefficients are: For using 15 basis functions, j = 1, ..., 15.

[0035] For b 22 , using the basis function Ψ r and coefficient q r Approximate the product of

[0036]

[0037] b 22 (x i ,y i ) indicates t i The diffusion drift term corresponding to the moment, the coefficient q r is the basis function Ψ r The corresponding coefficients are: For using 6 basis functions, r = 1,...,6.

[0038] According to the Kramers–Moyal formula, the drift terms a2, a3 and the non-zero diffusion term b22 Expressed as

[0039]

[0040] According to the relationship between formula (7) and formula (12), the following iterative equation for solving the coefficient of drift term a2 is obtained:

[0041] Ac=B (15)

[0042] in

[0043]

[0044] c=[c1,...,c 14 ],

[0045] B=[y2-y1,...,y N -y N-1 ] / Δt.

[0046] A is composed of basis functions Φ k (x i ,y i ,v i ) composed of a (N-1) × 14-dimensional matrix, c is a matrix composed of coefficients c k is a vector composed of , and B is the drift term a2 calculated according to formula (12).

[0047] Similarly, the iterative equation of the drift term a3 is:

[0048]

[0049] in

[0050]

[0051] p=[p1,...,p 15 ],

[0052]

[0053] The basis function M j (x i ,y i ,v i ) composed of an (N-1) × 15-dimensional matrix, p is a matrix consisting of coefficients p j The vector composed of is the drift term a3 calculated according to formula (13).

[0054] Diffusion term b 22 Iterative equation of ;

[0055]

[0056] in

[0057]

[0058] q=[q1,...,q6],

[0059]

[0060] The basis function Ψ r (x i ,y i ,v i ) composed of a (N-1) × 6-dimensional matrix, q is a matrix consisting of coefficients q r The vector composed of is the drift term b calculated according to formula (14) 22 .

[0061] The SINDy algorithm is used to solve the system parameters; formula (15) is simplified using the least squares method to obtain Therefore, the solution of formula (15) is

[0062] c=(A T A) -1 (A T B) (18)

[0063] The solution with the optimal sparsity is selected from c by the stepwise sparse regression SSR algorithm.

[0064] Formula (16) is simplified by the least squares method to obtain Therefore, the solution of formula (16) is

[0065]

[0066] The solution with the optimal sparsity is selected from p by the stepwise sparse regression SSR algorithm.

[0067] Formula (17) is simplified by the least squares method to obtain Therefore, the solution of formula (17) is

[0068]

[0069] The solution with the optimal sparsity is selected from q by the stepwise sparse regression SSR algorithm.

[0070] Solution with optimal sparsity The dimensionless coefficients ζ, Θ, k are included in m (m=1,2,3), contains the dimensionless coefficient α, contains the dimensionless coefficient D. After identifying the specific system parameters through this method, the model equation of the tristable energy harvester can be obtained by substituting them into equation (3).

[0071] Step 3: Based on the model equation of the tri-stable energy harvester established in step 2, solve the system response, use the model predictive control method to calculate the optimal control law α, apply the control signal through the Buck-boost circuit according to the optimal control law α, realize the random control of the tri-stable piezoelectric cantilever beam structure energy harvester, and increase the output power of the energy harvester.

[0072] The Monte Carlo method is used to simulate the energy harvester system output. The fourth-order Runge-Kutta method is used to numerically solve equation (3), and the system output is calculated multiple times to obtain the average value to obtain the mean square voltage E[V 2 ]; According to the characteristics of the energy harvester system, the dimensionless system output mean square power is expressed using mean square voltage as:

[0073] E[P]=ΘαE[V 2 ] (twenty one)

[0074] There is an optimal time constant ratio α to maximize the system output power; however, for the randomly excited tristable energy harvester, the optimal α value is not a fixed value, and the control method is used to obtain the time-varying control rate to achieve the maximum power effect;

[0075] Formula (21) is the system response formula.

[0076] Calculate 0~T c Optimal control law in time m c To control the number of time steps, T c To control the duration, α l (l=1,...,m c ) is the control parameter corresponding to the lth time step, α is 0~T c A collection of control parameters within a time period.

[0077] The optimal control law is obtained by minimizing the loss function within the computation time:

[0078]

[0079] Where m p is the number of prediction time steps; is the predicted power of the kth prediction step, Pref k is the reference power of the kth prediction step, and the coefficient Q is the weight coefficient of the difference between the two; α l is the control signal of the lth control step, Δα lis the control signal change rate of the lth control step, R α is the weight of the control signal, R Δα is the weight of the control signal change rate.

[0080] The power As a predicted response, by setting the reference power Pref k The optimal control rate that minimizes the loss function is obtained. The control signal is applied through the Buck-boost circuit to achieve random control of the tri-stable piezoelectric cantilever beam structure energy harvester to increase its output power.

[0081] Beneficial effects:

[0082] 1. The random control method of the tri-stable piezoelectric cantilever beam structure energy harvester disclosed in the present invention can intelligently identify system parameters and complete the modeling of electromechanical coupling equations using a data-driven method. It can still complete the digital modeling of the system under the condition of having control items. Compared with the existing modeling methods, the model can be adjusted accordingly as the system state changes, and control can be achieved more conveniently and accurately.

[0083] 2. The random control method of the tristable piezoelectric cantilever beam structure energy harvester disclosed in the present invention is based on the principle of impedance matching and power maximization. It achieves impedance matching and power maximization by adjusting the optimal duty cycle of the circuit. The present invention is implemented through model predictive control, which not only adapts to a variety of random environments, but also overcomes the problem that the internal resistance of nonlinear systems is difficult to match. Therefore, compared with the existing impedance matching energy circuit, it has stronger adaptability.

[0084] 3. The random control method of the tri-stable piezoelectric cantilever beam structure energy harvester disclosed in the present invention, in addition to having the beneficial effect of realizing the dynamic adjustment circuit to maximize the output power, realizes the performance optimization of the tri-stable energy harvester, controls the signal convergence and provides an important reference for the optimal design of the energy harvester.

[0085] 4. The random control method of the tri-stable piezoelectric cantilever beam structure energy harvester disclosed in the present invention considers the external environment excitation as random excitation, which more realistically simulates the actual random environment, making the effect of improving the output power of the present invention in practical applications more accurate and real. BRIEF DESCRIPTION OF THE DRAWINGS

[0086] Figure 1 It is a structural schematic diagram of the tri-stable energy harvester in a random environment of the present invention;

[0087] Figure 2 It is a schematic diagram of the random control method of the tri-stable piezoelectric cantilever beam structure energy harvester;

[0088] Figure 3 It is a comparison diagram of the potential energy function of the original system of the energy harvester and the potential energy function of the identified system;

[0089] Figure 4 is a graph showing the variation of mean square power output of controlled and uncontrolled energy harvesters with noise intensity;

[0090] Figure 5 is a graph showing the control signal of the controlled energy harvester changing with the noise intensity;

[0091] Figure 6 is a graph showing the variation of the mean square power output of the controlled and uncontrolled energy harvesters with the damping coefficient;

[0092] Figure 7 It is a graph showing the change of control signal of the controlled energy harvester with the damping coefficient. DETAILED DESCRIPTION

[0093] In order to better illustrate the purpose and advantages of the present invention, the invention is further described below with reference to the accompanying drawings and examples.

[0094] Embodiment 1:

[0095] The object of this embodiment is a strongly nonlinear tristable energy harvester coupled with electromechanical coupling. Its basic structure is shown in the accompanying drawings of the specification. Figure 1 A magnet A is installed at the tip of the cantilever beam. This magnet is subjected to a nonlinear force from two magnets B and C at the fixed end. The force effect can be adjusted by adjusting the distance d between the two magnets at the fixed end. v and the distance d from the tip to the fixed end h . The placement position and magnetic pole direction in the figure construct the potential well function of the system's tristable state. Considering the randomness of external excitation in the real environment, Gaussian white noise excitation is applied to the base. The cantilever beam adopts a dual crystal structure, and a piezoelectric sheet is attached to the upper and lower parts of the cantilever beam and connected in series. A displacement sensor is used to monitor the tip displacement and collect data, and the signal is connected to the computer for modeling and giving the optimal control rate, and the control rate is adjusted to the circuit through the switch control signal. When the collector works under the condition of stable noise intensity, the optimal control parameters can be selected, so the performance of the energy harvester is not affected by the power consumption of the displacement sensor, controller, etc. The performance optimization method of the tristable piezoelectric cantilever beam energy harvester proposed by the present invention can directly adjust the control circuit according to the expected power, so as to improve the energy harvester collection power. In addition, the random control method of the tristable piezoelectric cantilever beam structure energy harvester, the present invention also provides a data-driven modeling method for a tristable energy harvester in a random environment, by processing the output data of the energy harvester of the tristable energy harvesting system, a system model containing control items is established, and accurate modeling ensures the optimization of system performance.

[0096] like Figure 2 As shown, the random control method of the tristable piezoelectric cantilever beam structure energy harvester disclosed in this embodiment has the following specific implementation steps:

[0097] Step 1: Establish the electromechanical coupling tristable energy harvester system model under white excitation. Figure 1 The basic structure diagram of the three-state energy harvester and Figure 1 A simplified schematic diagram of the three-stable energy harvester is shown in formulas (1)-(2). For ease of analysis, the system model is dimensionless, and the specific expression is shown in formula (3). In this embodiment, considering the position of the magnet at the end of the cantilever beam and the piezoelectric effect of the piezoelectric ceramic, we assume that the dimensionless parameters are: ζ=0.1, Θ=0.3, k1=1, k2=-4.5, k3=3.5. At this time, we consider the white noise excitation intensity D=0.1.

[0098] Step 2: Based on the collected system displacement data and voltage, the drift term and diffusion term are regressed and fitted using the basis function summation method. According to the Kramers–Moyal formula, the drift term and diffusion term can be expressed as (12)-(14). Use the expressions of formulas (15)-(17) to calculate the matrix A in the iterative equation respectively, and vector B, In sparse regression, cross-validation is used to calculate the error value of the solution under different sparsity levels. The specific steps of the cross-validation method are: randomly divide the sample data into ten groups, select one group as the test group, use the other nine groups of data to calculate the coefficient vector c according to formula (18), set the minimum coefficient to 0 in each cycle, and calculate is the error of different sparse solutions. After twenty cycles, the errors of different sparse degrees are averaged, and the optimal sparse solution is determined according to the minimum error value. The sparse solutions are shown in Table 1. The maximum error of the identification coefficient is 9.64% and the minimum is 0.17%. The potential energy function of the tristable system after parameter identification modeling is shown in Figure 3 As shown, the steady-state point distribution is consistent with the original system, and the error is within the acceptable range in engineering practice.

[0099]

[0100]

[0101] Table 1 Parameter values ​​identified by data-driven modeling

[0102] Step 3: From the perspective of improving the acquisition performance of the tristable energy harvester in a random environment, the output mean square power of the tristable energy harvester is calculated as formula (21) as the performance indicator, and formula (22) is used as the loss function. Set the time step Δt = 0.5s, and control the time T c =m c Δt=5s, prediction time T p =m pΔt = 10s, simulation time is 150s. According to the Monte Carlo simulation, the response results within the prediction time can be calculated, and the optimal control rate can be obtained by the minimum loss function J within the prediction time. Let the weight coefficient Q, R α and R Δα The values ​​are 0.9, 0.1, and 0.1 respectively, and the reference power Pref is 1.2 times that of the uncontrolled system, that is, 1.2E[P uncontrol ].

[0103] Figure 4 and Figure 5 The output power E[P] of the tristable energy harvester system changes when the noise intensity is different. Figure 4 As shown in the figure, the output voltage of the system without the control strategy is the red curve. As the noise intensity D increases, the output power also gradually increases. After the model predictive control is applied to the system, the yellow dotted line is used as the reference trajectory, and the blue circle is the output power of the controlled system. It can be observed that the control doubles the system output. Figure 5 The change of the control signal within 150 seconds can be observed in the figure. When the noise intensity D is 0.9, the control signal converges to about 2.5. When the noise intensity decreases to 0.6, the control signal converges to 2. As the noise intensity increases, the control signal gradually increases. Figure 6 and Figure 7 The output power E[P] of the tristable energy harvester system changes when different damping coefficients are taken. Figure 6 As shown in the figure, the output voltage of the system without the control strategy is the red curve. As the damping system increases, the output power gradually decreases. After applying model predictive control to the system, it can be observed that the control doubles the system output. Figure 7 The control signal changes with the damping coefficient, and Figure 5 Compared with the influence of noise intensity, the control signal is more affected by the damping coefficient. The tri-stable cantilever beam energy harvester system can increase the output power and improve the collection performance by implementing the control signal.

[0104] In summary, the random control method of the tristable piezoelectric cantilever beam structure energy harvester disclosed in this embodiment can intelligently identify system parameters using a data-driven method to complete the modeling of the electromechanical coupling equation. In the case of a control term, the digital modeling of the system can still be completed. Compared with the existing modeling method, the corresponding model adjustment can be made as the system state changes, and the control is more convenient and accurate. Based on the impedance matching and power maximization principles, the impedance matching is achieved by adjusting the optimal duty cycle of the circuit to achieve power maximization. This embodiment is implemented by model predictive control, which not only adapts to a variety of random environments, but also overcomes the problem that the internal resistance of the nonlinear system is difficult to match. Therefore, it is more adaptable than the existing impedance matching energy circuit. The influence of noise intensity and system damping coefficient on the control effect is analyzed in detail. In addition to the beneficial effect of realizing the dynamic adjustment circuit to maximize the output power, the performance optimization of the tristable energy harvester is achieved. The control signal converges and provides an important reference for the optimal design of the energy harvester, filling the gap in the circuit control method of the nonlinear energy harvester under random excitation.

[0105] The specific description above further illustrates the purpose, technical solutions and beneficial effects of the invention in detail. It should be understood that the above is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A random control method for a tri-stable piezoelectric cantilever beam structure energy harvester, characterized in that: The following steps are included: Step 1: Considering the random excitation in the environment to introduce white noise, establish a strongly nonlinear tristable energy harvester model of electromechanical coupling, and perform dimensionless processing on the energy harvester model to obtain a dimensionless model of the strongly nonlinear tristable energy harvester of electromechanical coupling; Step 2: Based on the collected displacement data and voltage of the tristable energy harvester, the dimensionless coefficients ζ, Θ, and k in the electromechanical coupling equation of the strong nonlinear tristable energy harvester are identified using the sparse identification method of nonlinear dynamic systems. m (m=1,2,3), D, establish the model equation of the tristable energy harvester; Step 3: Based on the model equation of the tri-stable energy harvester established in step 2, solve the system response, use the model predictive control method to calculate the optimal control law α, apply the control signal through the Buck-boost circuit according to the optimal control law α, realize the random control of the tri-stable piezoelectric cantilever beam structure energy harvester, and increase the output power of the energy harvester.

2. The random control method of the tristable piezoelectric cantilever beam structure energy harvester according to claim 1, characterized in that: In step one, The strongly nonlinear tristable energy harvester model is shown in formula (1): In the formula, represents the displacement of the tip magnet, represents the piezoelectric voltage, M represents the mass of the tip magnet, represents time, C represents the damping coefficient, θ represents the electromechanical coupling coefficient, represents the random excitation from the outside world, C p represents the effective capacitance, Indicates the resistance of a circuit; The potential energy function of the system is expressed as a polynomial: In the formula is the stiffness coefficient; The following transformation is introduced to make the system equation (1) dimensionless: in is the natural frequency of the cantilever beam, is the length scale of the cantilever beam; t, X, V represent the dimensionless time, displacement and voltage respectively; ζ, Θ, α represent the dimensionless damping coefficient, electromechanical coupling coefficient and electromechanical time constant ratio respectively; k m (m=1,2,3) represents the dimensionless stiffness coefficient; The dimensionless electromechanical coupling equation of the strongly nonlinear tristable energy harvester is obtained: In the formula, represents the dimensionless potential energy function, and uses Gaussian white noise ξ(t) to describe the dimensionless external excitation. The mean and correlation function of its statistical characteristics are: E[ξ(t)]=0, E[ξ(t)ξ(t+τ)]=Dδ(τ); E[·] represents the mean of the function in the brackets, δ(τ) represents the Dirac function, τ represents the correlation time interval, and D is the noise intensity of the dimensionless white noise.

3. The random control method of the tristable piezoelectric cantilever beam structure energy harvester according to claim 2, characterized in that: The implementation method of step 2 is: The electromechanical coupling equation (3) of the strongly nonlinear tristable energy harvester can be rewritten as Stochastic differential equations SDEs: Among them, Y(t) is the dimensionless velocity; B(t) is the standard Brownian motion; Acquisition time [t1, t2, ..., t N+1 ] corresponds to the voltage [v1,v2,...,v N+1 ] and the cantilever tip displacement [x1,x2,...,x N+1 ], speed [y1,y2,...,y N ]Through (x i+1 -x i ) / (t i+1 -t i )(i=1,2,...,N) is calculated, x i ,y i and v i Indicates t i The displacement, velocity and voltage corresponding to the moment, the time interval is Δt, and N is the number of sampling points; The non-zero diffusion term and drift term in formula (4) are expressed as: The drift term a1 only contains the velocity y i And the coefficient is 1, and there is no need to identify its coefficient; for the drift term a2, use the basis function Φ k and coefficient c k Approximate the product of a2(x i ,y i ,v i ) indicates t i The diffusion drift term corresponding to the moment, coefficient c k is the basis function Φ k The corresponding coefficients; For a3, use the basis function M j and coefficient p j Approximate the product of a3(x i ,y i ,v i ) indicates t i The diffusion drift term corresponding to the time, coefficient p j is the basis function M j The corresponding coefficients; For b 22 , using the basis function Ψ r and coefficient q r Approximate the product of b 22 (x i ,y i ) indicates t i The diffusion drift term corresponding to the moment, the coefficient q r is the basis function Ψ r The corresponding coefficients; According to the Kramers–Moyal formula, the drift terms a2, a3 and the non-zero diffusion term b 22 Expressed as According to the relationship between formula (7) and formula (12), the following iterative equation for solving the coefficient of drift term a2 is obtained: Ac=B (15) in c=[c1,...,c 14 ], B=[y2-y1,...,y N -y N-1 ] / Δt. A is composed of basis functions Φ k (x i ,y i ,v i ) composed of a (N-1) × 14-dimensional matrix, c is a matrix composed of coefficients c k The vector composed of , B is the drift term a2 calculated according to formula (12); Similarly, the iterative equation of the drift term a3 is: in p=[p1,...,p 15 ], The basis function M j (x i ,y i ,v i ) composed of an (N-1) × 15-dimensional matrix, p is a matrix consisting of coefficients p j The vector composed of is the drift term a3 calculated according to formula (13); Diffusion term b 22 Iteration equation of ; in q=[q1,...,q6], The basis function Ψ r (x i ,y i ,v i ) composed of a (N-1) × 6-dimensional matrix, q is a matrix consisting of coefficients q r The vector composed of is the drift term b calculated according to formula (14) 22 ; The SINDy algorithm is used to solve the system parameters; formula (15) is simplified using the least squares method to obtain Therefore, the solution of formula (15) is c=(A T A) -1 (A T B) (18) The solution with the optimal sparsity is selected from c by the stepwise sparse regression SSR algorithm. Formula (16) is simplified by the least squares method to obtain Therefore, the solution of formula (16) is The solution with the optimal sparsity is selected from p by the stepwise sparse regression SSR algorithm. Formula (17) is simplified by the least squares method to obtain Therefore, the solution of formula (17) is The solution q with the optimal sparsity is selected from q by the stepwise sparse regression SSR algorithm; Solution with optimal sparsity The dimensionless coefficients are included in Θ, k m (m=1,2,3), contains the dimensionless coefficient α, contains the dimensionless coefficient D. After identifying the specific system parameters through this method, the model equation of the tristable energy harvester can be obtained by substituting them into equation (3).

4. The random control method of the tristable piezoelectric cantilever beam structure energy harvester according to claim 3, characterized in that: The implementation method of step three is: The Monte Carlo method is used to simulate the energy harvester system output. The fourth-order Runge-Kutta method is used to numerically solve equation (3), and the system output is calculated multiple times to obtain the average value to obtain the mean square voltage E[V 2 ]; According to the characteristics of the energy harvester system, the dimensionless system output mean square power is expressed using mean square voltage as: E[P]=ΘαE[V 2 ] (21) There is an optimal time constant ratio α to maximize the system output power; however, for the randomly excited tristable energy harvester, the optimal α value is not a fixed value, and the control method is used to obtain the time-varying control rate to achieve the maximum power effect; Formula (21) is the system response formula. Calculate 0~T c Optimal control law in time m c To control the number of time steps, T c To control the duration, α l (l=1,...,m c ) is the control parameter corresponding to the lth time step, α is 0~T c A collection of control parameters in time; The optimal control law is obtained by minimizing the loss function within the computation time: Where m p is the number of prediction time steps; is the prediction power of the kth prediction step, Pref k is the reference power of the kth prediction step, and the coefficient Q is the weight coefficient of the difference between the two; α l is the control signal of the lth control step, Δα l is the control signal change rate of the lth control step, R α is the weight of the control signal, R Δα is the weight of the control signal change rate; The power As a predicted response, by setting the reference power Pref k The optimal control rate that minimizes the loss function is obtained; the control signal is applied through the Buck-boost circuit to achieve random control of the tri-stable piezoelectric cantilever beam structure energy harvester to increase its output power.

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