# Dynamics of a Nonlinear Energy Harvesting System in Time-Delayed Feedback Control under Stochastic Excitations

Authors: Shuling Zhang; Ying Zhang; Zhongkui Sun; Xiaxia Duan
Journal: Complexity
 (2020)

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## Abstract

A time-delayed feedback control is applied to a nonlinear piezoelectric energy harvesting system excited by additive and multiplicative Gaussian white noises to improve its energy harvesting performance. An equivalent decoupling system can be obtained by using a variable transformation. Based on the standard stochastic averaging method, the Fokker–Plank–Kolmogorov equation and the stationary probability density functions of the amplitude, displacement, and velocity of the harvester are obtained, respectively. In addition, the approximate expressions of mean square electric voltage and the mean extracted output power are derived. Finally, the paper explores the influences of parameters on the mean square electric voltage. The results show that noise intensity, time delay, feedback strength, time constant ratio, and coupling coefficients have great influences on the mean square electric voltage. The accuracy of the theoretical method is verified by the Monte Carlo simulation.


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## Body

## 1. Introduction

Due to the development of wireless sensors towards low energy consumption and miniaturization, conventional power supply devices are no longer adapted to technological innovation [1, 2]. Nowadays, how to improve the efficiency of an environmental vibration energy harvesting system is becoming a hot research field.In the recent years, vibration energy harvesters based on the piezoelectric effect have become one of the most effective energy harvesting methods due to their advantages of simple structure, no electromagnetic interference, easy manufacture, and miniaturization [3, 4]. As the first design scheme adopted, a linear vibration energy harvester relies on the resonance phenomenon and has a narrow effective frequency band. Only when the excitation frequency matches its basic natural frequency can it work effectively. A slight perturbation of the basic frequency of a linear harvester to the excitation frequency causes a sharp drop in the energy output [5–7]. Therefore, the linear harvesters based on the resonance principle are not suitable for obtaining energy from broadband and stochastic excitation sources, both of which are typical excitation types in the real world. To extend the harvester’s bandwidth, some recent solutions call for utilizing energy harvesters with stiffness-type nonlinearities. [8–13]. Cottone et al. [14] proposed that the performance of the nonlinear oscillators under stochastic excitation is superior to the linear oscillators through numerical and experimental results. McInnes et al. [15] established the theoretical model of an energy harvesting system based on stochastic resonance and found that its performance was significantly improved. Huang et al. [16] analyzed the dynamical characteristics of the nonlinear vibration energy harvester with an uncertain parameter. Erturk et al. [17, 18] investigated the stochastic response of a bistable energy harvester under harmonic excitation through numerical calculation and experiment. Daqaq [19] explored the stochastic response of the bistable energy harvesting system driven by white noise and exponential correlated Gaussian noise. However, when the bistable energy harvesters are driven by low level excitation, it will not be able to conduct high-energy interwell oscillation. Zhou et al. [20–23] designed a tristable energy harvester, and their results demonstrated that compared to the bistable energy harvester, tri-stable energy harvesters have a smaller high-energy motion threshold and can efficiently collect energy over a larger low-frequency range. Panyam and Daqaq [24] explored the responses of the tristable energy harvester to harmonic excitation, and the results showed that the tristable energy harvesters have higher output power than the bistable energy harvesters. Huang et al. [25] investigated the resonance mechanism of nonlinear vibrational multistable energy harvesters under narrow-band stochastic parametric excitations.In fact, vibration energy harvesters are inevitably affected by environmental excitation. Therefore, the researchers investigated the performance of the nonlinear energy harvesting system under stochastic excitation [26–28]. Sun et al. [26] motivated to study stochastic dynamics in a fractional-order system subjected to Gaussian noise. Jin et al. [27] established the semianalytical solution of random response of the nonlinear vibration energy harvester under Gaussian white noise excitation by using generalized harmonic transformation and equivalent nonlinearization. Xiao and Jin [29] explored the energy harvesting of a monostable duffing type harvester with piezoelectric coupling under correlated multiplicative and additive white noise.Some recent studies have shown that deliberately introducing delay into a continuous-time control dynamical system yields favorable results. In distributed generation systems, the introduction of time delays of a communication network is to ensure the precise voltage restoration and optimal load sharing. [30–32]. Belhaq and Hamdi [33] investigated a delayed van der Pol oscillator with time-varying delay amplitude coupled to an electromagnetic energy harvesting device. Zhang et al. [34] studied the fractional-delay system in the presence of both the colored noise and delayed feedback. Xiao et al. [35, 36] found that the coupling delay provides a strategy to control the oscillating dynamics in fractional-order oscillators. The robust distributed control algorithms consider time-delay communications and noise communication disturbances of a communication network for distributed generation systems.In general, due to the time required for signal propagation, each electronic equipment inevitably has time delay. However, time delays in both controllers and actuators may deteriorate the control performance or even cause the instability of the piezoelectric energy harvesting system. The appropriate choices of feedback strength and time delays may improve the performance of dynamical systems. However, there are still rarely nonlinear analyses into the energy harvesting enhancement with time-delayed feedback control. Guo et al. [37] proposed a time-delayed feedback control to improve collection performance of the multiple attractors wind-induced vibration energy harvester system. The results show that time-delayed feedback is a powerful tool for achieving a wide range of operating regimes and enhancing amplitude-frequency characteristics and controlling the stochastic or deterministic dynamics of nonlinear systems. However, time-delayed feedback control of the nonlinear piezoelectric energy harvesting system [38, 39] may influence the harvesting performance and the power generation under different excitation conditions, but it has not yet been explored. Therefore, it is necessary to deeply analyze the influences of time-delayed feedback control on the energy harvesting system.In this paper, the aimed system is a nonlinear piezoelectric energy harvesting system containing time-delayed feedback control in the presence of additive and multiplicative Gaussian white noise. We devote ourselves to studying the stochastic responses induced by time-delayed feedback in the energy harvesting system. It has been found that not only the noise time, constant ratio, and coupling coefficients but also the time delay and the feedback strength can increase the harvested energy from vibrations. The main work and the results are as follows. The piezoelectric energy harvesting system and its equivalent system are given in Section2. The stationary probability density functions (PDFs) of displacement and velocity are calculated by using the stochastic average method, and the approximate expressions of the mean square electric voltage (MSEV) are derived in Section 3. In addition, the influences of parameters on the MSEV are discussed in detail. Finally, conclusions are drawn in Section 5.


## 2. Model and the Equivalent System

### 2.1. Model

A simplified model of a piezoelectric energy harvester is shown in Figure1 [38, 39]. Through theoretical analysis and experimental measurement, the control electromechanical model of the piezoelectric energy harvesting system is deduced:(1)mX¯¨+cX¯˙+δ1X¯+δ3X¯3+θV¯˙=−mX¯¨b,(2)CpV¯˙+V¯R=θX¯˙,where X¯ is the displacement of mass m and c, δ1, and δ3 are the damping coefficient and the linear and nonlinear stiffness coefficients, respectively. θ is the electromechanical coupling coefficient, Cp is the piezoelectric capacitance, and V is the voltage measured across the equivalent resistive load R.Figure 1
A simplified model of a piezoelectric energy harvester.For simplicity, a set of nondimensionless parameters is introduced as(3)X=X¯lc,t=ω1τ,ω1=δ1m,λ=δ3lc2δ1,V=CplcθV¯,ξ=c2δ1m,κ=θ2Cpk1,α=1RCpω1.The nonlinear piezoelectric energy harvesting system is dimensionless as follows:(4)X¨+2ξX˙+X+λX3+κV=−X¨b,(5)V˙+αV=X˙.On the other hand, time-delayed feedback controlFt=uX˙t−τ−X˙t is proposed [40]. Meanwhile, the energy harvesting system is inevitably affected by the external environment noise, and the additive and multiplicative Gaussian white noise excitation can more accurately replace the harmonic excitation as the stochastic noise source.Therefore, the corresponding model with time-delayed feedback control under additive and multiplicative Gaussian white noise is(6)X¨+2ξX˙+X+λX3+κV=ξ1t+Xξ2t+uX˙τ−X˙,(7)V˙+αV=X˙,where 2ξ is the nondimensional linear damping coefficient. κ is the piezoelectric coupling coefficient in the mechanical equation. λ is the nondimensional nonlinear stiffness coefficient. α is time constant ratio. u is the feedback strength, and τ is the time delay. ξ1t and ξ2t denote Gaussian white noise which have the following properties:(8)ξit=0,Rijτ=ξitξjt+τ=2Dijδτ,i=j,0,i≠j,i,j=1,2,where D11 and D22 are the Gaussian white noise intensities of ξ1t and ξ2t, respectively.


### 2.2. The Equivalent System

Supposing that the noise excitation intensity, damping coefficient, and cubic nonlinear term coefficient are small, the responses of equation (6) can be regarded as stochastic disturbance of periodic solutions about a conservative linear system. Therefore, assume that the solution of system (5) has the following form [41, 42]:(9)Xt=AtΦ,X˙t=−Atω0sinΦ,where Φt=ω0t+θt, At is the amplitude, and Φt is the phase angle.Substituting equation (9) into equation (6) yields [43](10)Vt=Cte−αt+Aω0α2+ω02ω0cosΦ−αsinΦ.The stationary part of equation (10) is(11)Vt=Aω0α2+ω02ω0cosΦ−αsinΦ=ω02α2+ω02Xt+αα2+ω02X˙t.As(12)ω0cosΦ−αsinΦ=α2+ω02cosΦ+tan−1αω0,the amplitude of the stationary voltage can be obtained as(13)L=Aω0α2+ω02.Then,(14)X˙τ=X˙t−τ=−At−τω0sinω0t−τ+θt−τ,=−Atω0sinΦcosω0τ+Atω0sinω0τcosΦ,=cosω0τX˙+ω0sinω0τX.Substitution of equations (11) and (14) into equation (6) yields a modified decoupling equation:(15)X¨+SX˙+ω2X+λX3=ξ1t+Xξ2t,where ω2=1+κω02/α2+ω02−uω0sinω0τ and S=2ξ+κα/α2+ω02−ucosω0τ+u.


## 2.1. Model

A simplified model of a piezoelectric energy harvester is shown in Figure1 [38, 39]. Through theoretical analysis and experimental measurement, the control electromechanical model of the piezoelectric energy harvesting system is deduced:(1)mX¯¨+cX¯˙+δ1X¯+δ3X¯3+θV¯˙=−mX¯¨b,(2)CpV¯˙+V¯R=θX¯˙,where X¯ is the displacement of mass m and c, δ1, and δ3 are the damping coefficient and the linear and nonlinear stiffness coefficients, respectively. θ is the electromechanical coupling coefficient, Cp is the piezoelectric capacitance, and V is the voltage measured across the equivalent resistive load R.Figure 1
A simplified model of a piezoelectric energy harvester.For simplicity, a set of nondimensionless parameters is introduced as(3)X=X¯lc,t=ω1τ,ω1=δ1m,λ=δ3lc2δ1,V=CplcθV¯,ξ=c2δ1m,κ=θ2Cpk1,α=1RCpω1.The nonlinear piezoelectric energy harvesting system is dimensionless as follows:(4)X¨+2ξX˙+X+λX3+κV=−X¨b,(5)V˙+αV=X˙.On the other hand, time-delayed feedback controlFt=uX˙t−τ−X˙t is proposed [40]. Meanwhile, the energy harvesting system is inevitably affected by the external environment noise, and the additive and multiplicative Gaussian white noise excitation can more accurately replace the harmonic excitation as the stochastic noise source.Therefore, the corresponding model with time-delayed feedback control under additive and multiplicative Gaussian white noise is(6)X¨+2ξX˙+X+λX3+κV=ξ1t+Xξ2t+uX˙τ−X˙,(7)V˙+αV=X˙,where 2ξ is the nondimensional linear damping coefficient. κ is the piezoelectric coupling coefficient in the mechanical equation. λ is the nondimensional nonlinear stiffness coefficient. α is time constant ratio. u is the feedback strength, and τ is the time delay. ξ1t and ξ2t denote Gaussian white noise which have the following properties:(8)ξit=0,Rijτ=ξitξjt+τ=2Dijδτ,i=j,0,i≠j,i,j=1,2,where D11 and D22 are the Gaussian white noise intensities of ξ1t and ξ2t, respectively.


## 2.2. The Equivalent System

Supposing that the noise excitation intensity, damping coefficient, and cubic nonlinear term coefficient are small, the responses of equation (6) can be regarded as stochastic disturbance of periodic solutions about a conservative linear system. Therefore, assume that the solution of system (5) has the following form [41, 42]:(9)Xt=AtΦ,X˙t=−Atω0sinΦ,where Φt=ω0t+θt, At is the amplitude, and Φt is the phase angle.Substituting equation (9) into equation (6) yields [43](10)Vt=Cte−αt+Aω0α2+ω02ω0cosΦ−αsinΦ.The stationary part of equation (10) is(11)Vt=Aω0α2+ω02ω0cosΦ−αsinΦ=ω02α2+ω02Xt+αα2+ω02X˙t.As(12)ω0cosΦ−αsinΦ=α2+ω02cosΦ+tan−1αω0,the amplitude of the stationary voltage can be obtained as(13)L=Aω0α2+ω02.Then,(14)X˙τ=X˙t−τ=−At−τω0sinω0t−τ+θt−τ,=−Atω0sinΦcosω0τ+Atω0sinω0τcosΦ,=cosω0τX˙+ω0sinω0τX.Substitution of equations (11) and (14) into equation (6) yields a modified decoupling equation:(15)X¨+SX˙+ω2X+λX3=ξ1t+Xξ2t,where ω2=1+κω02/α2+ω02−uω0sinω0τ and S=2ξ+κα/α2+ω02−ucosω0τ+u.


## 3. Theoretical Analysis

Using the stochastic average method, suppose the responses of equation (15) can be expressed as follows [44]:(16)Xt=AtcosΦ,X˙t=−AtωsinΦ,Φt=ωt+Θt.Substituting equation (16) into equation (15) yields(17)A˙=1ω−AωSsinΦ+λA3cos3ΦsinΦ−1ωξ1tsinΦ−1ωξ2tAcosΦsinΦ,(18)Θ˙=cosΦAω−AωSsinΦ+λA3cos3Φ−1Aωξ1tsinΦ−1Aωξ2tAcosΦsinΦ.By adopting the standard stochastic averaging method, the stochastic differential equation (in the Ito sense) for the instantaneous amplitudeAt is determined by(19)dA=mAdt+σAdWt,where Wtis an independent normalized Wiener process. A,Θ can be approximated as the two-dimensional stochastic diffusion process. The drift coefficient and diffusion coefficient of the Fokker–Plank–Kolmogorov (FPK) equation can be obtained as follows:(20)mA=−ASsin2Φ+λω2A3cos3ΦsinΦ+1Aω2D11cos2Φ−I11ωcosΦsinΦ+D22A2cos4Φ−A2cos3ΦsinΦI222ωΦ,(21)σ2A=2D11ω2sin2Φ+2D22A2cos2Φω2sin2ΦΦ,where Φ=limT⟶∞1/T∫0Tdt=1/2π∫02πdΦ. Sω and Iω have the following form:(22)Sω=1π∫−∞0cosωtRτdτ,Iω=1π∫−∞0sinωtRτdτ.Obviously, the amplitudeA does not depend on the phase Θ. In conclusion, the FPK equation of amplitude A yields(23)∂p∂t=−∂∂AmAp+12∂2∂A2σ2Ap.In the stationary sense, equation (23) can be solved to yield the stationary PDFs for amplitude(24)pA=Cσ2Aexp∫0Amuσ2udu,in which C is a normalization factor.The joint PDF of the amplitude and phase can be retrieved as(25)pA,Θ=pApΘ=12πpA.According to the transformation fromA,Θ to the X,X˙, we can obtain the expression of the joint PDF of X and X˙ as follows:(26)pX,X˙=1ω1ApA,Θ=12πω1ApA.The corresponding marginal stationary PDFs of displacementX and velocity X˙ are shown in equations (27) and (28), respectively.(27)pX=∫−∞+∞pX,X˙dX˙,(28)pX˙=∫−∞+∞pX,X˙dX.Thus, the expected value of the MSEV and the mean extracted output power are(29)V2t=ω02α2+ω02A2=ω02α2+ω02∫0+∞A2pAdA,(30)P=αkV2t.


## 4. Numerical Simulation

In this section, we mainly focus on the influences of the parameters on the stochastic responses of the piezoelectric energy harvesting system. To verify the validity of the analysis results, equations (6) and (7) will be directly simulated by Monte Carlo simulations. Let α=0.05, λ=0.05, and the initial conditions are set as the static equilibrium position X0=0,X˙0=0,V0=0. The Monte Carlo simulations are selected as the reference to evaluate the accuracy of the analytical results.


### 4.1. The Influences of Parameters on PDFs

Whenτ=0.5, k=0.3, and u=0.05, the stationary PDFs of amplitude, displacement, and velocity under different noise intensities are investigated, as in Figure 2. Specifically, when Gaussian white noise intensity increases continuously, the stationary PDFs peak values decrease and shift to the right, which indicates that the system leads to a large response and gives out a high harvested energy. It can be concluded that the larger excitation intensity plays an important role in the stationary responses of the nonlinear piezoelectric energy harvester. The results are consistent with those of numerical simulation, which can prove the validity of theoretical analysis.Figure 2
The stationary PDFs under different noise intensities: (a) amplitude, (b) displacement, and (c) velocity. Theoretical analysis results (-) and Monte Carlo simulation results (∗).
(a)(b)(c)While fixing parametersD11=0.02, D22=0.02, and τ=0.5, the stationary PDFs of the amplitude, displacement, and the velocity for different feedback strengths are demonstrated in Figure 3. As the feedback strength increases from negative to positive, the shape of the stationary PDFs become steeper gradually, which results in the probability of the system stabilizing at the equilibrium position increasing. It can be clearly explained that under this set of parameters, negative feedback intensity is more conducive to the energy harvester. According to Figures 2 and 3, we can conclude that the results obtained by the theoretical method are consistent well with those from Monte Carlo simulations.Figure 3
The stationary PDFs under feedback strength. (a) Amplitude, (b) displacement, and (c) velocity. Theoretical analysis results (-) and Monte Carlo simulation results (∗).
(a)(b)(c)


### 4.2. The Influences of Parameters on the MSEV

In order to completely clarify the influences of the parameters on the performance of the piezoelectric energy harvesters, noise intensity, feedback strength, and time delay are successively introduced to investigate their influences on MSEVs in the following analysis. By takingu=−0.05, k=0.5, and τ=0.5, the effects of additive and multiplicative Gaussian white noise intensity on the MSEV are illustrated in Figures 4(a) and 4(b), respectively. It is found that with the increase of the additive and multiplicative noise intensity, MSEVs increase monotonically. The results show that additive noise plays a leading role in energy harvesting. The energy extracted from multiplicative noise by the piezoelectric energy harvester is relatively small, but it still makes sense when the noise intensity is large. Therefore, appropriately increasing the additive and multiplicative excitation intensity is a feasible method to increase the output voltage.Figure 4
The influences of noise intensity on the MSEV. Theoretical analysis results (solid round) and Monte Carlo simulation results (∗).
(a)(b)The variation of the MSEV with the time delay and feedback strength is illustrated in Figure5. It is seen that for any time delay τ, the MSEV V2 increases monotonically with the decrease of the feedback strength u. Nevertheless, the influences of time delay on the MSEV are nonmonotone.Figure 5
The influences of time delay and feedback strength on the MSEV.To illustrate the influences of the feedback strength on MSEV, Figure6(a) shows the changing of the MSEV with the feedback strength u=0.01 and u=−0.01. For D11=0.02, D22=0.02, and k=0.5, the variation of the MSEV V2 over the time delay τ is investigated. The results show that MSEV of the energy harvesting system without delay feedback system can be stabilized at 0.51. When u=−0.01, the time delay τ reaches the critical value of 3.14, and the MSEV of the system reaches the maximum and then decreases monotonically. However, when the feedback strength u=0.01, the curve of the MSEV changing with time delay τ seems to be completely opposite. For τ<3.14, the MSEV decreases monotonically as the time delay increases. When τ>3.14, the MSEV increases with the continuous increase of time delay. Furthermore, negative feedback strength is more favorable to the energy harvesting system than positive feedback strength. Therefore, the harvesting efficiency of the energy harvesting system can be improved by selecting the optimal control parameters u and τ.Figure 6
The influences of time delay and feedback strength on the MSEV. Theoretical analysis results (solid line) and Monte Carlo simulation results (∗).
(a)(b)By takingD11=0.02, D22=0.02, and τ=1, the dependence of the MSEV V2on feedback strength u and time constant ratio α is depicted in Figure 7(a). Specifically, the MSEV decreases rapidly as the time constant ratio α increases for a negative feedback strength u=−0.02. However, the MSEV decreases slowly with the increase of the time constant ratio for a positive feedback strength u=0.02. As the time constant ratio continues to increase, both the positive and negative feedback strengths keep the MSEV in a small range. Through the abovementioned analysis, a smaller time constant ratio and the negative feedback control will make the energy harvesting system induce larger MSEV.Figure 7
The influences of time delay and feedback strength on the MSEV. Theoretical analysis results (solid line) and Monte Carlo simulation results (∗).
(a)(b)The influences of the MSEVV2on feedback strength u and the piezoelectric coupling coefficient κ are shown in Figure 8. The MSEV almost decreases proportionally with the increases of κ at any feedback strength u in Figure 8(b). This means a smaller κ and the negative feedback control will increase the production of output power.Figure 8
The influences of piezoelectric coupling coefficient and feedback strength on the MSEV. Theoretical analysis results (solid line) and Monte Carlo simulation results (∗).
(a)(b)


## 4.1. The Influences of Parameters on PDFs

Whenτ=0.5, k=0.3, and u=0.05, the stationary PDFs of amplitude, displacement, and velocity under different noise intensities are investigated, as in Figure 2. Specifically, when Gaussian white noise intensity increases continuously, the stationary PDFs peak values decrease and shift to the right, which indicates that the system leads to a large response and gives out a high harvested energy. It can be concluded that the larger excitation intensity plays an important role in the stationary responses of the nonlinear piezoelectric energy harvester. The results are consistent with those of numerical simulation, which can prove the validity of theoretical analysis.Figure 2
The stationary PDFs under different noise intensities: (a) amplitude, (b) displacement, and (c) velocity. Theoretical analysis results (-) and Monte Carlo simulation results (∗).
(a)(b)(c)While fixing parametersD11=0.02, D22=0.02, and τ=0.5, the stationary PDFs of the amplitude, displacement, and the velocity for different feedback strengths are demonstrated in Figure 3. As the feedback strength increases from negative to positive, the shape of the stationary PDFs become steeper gradually, which results in the probability of the system stabilizing at the equilibrium position increasing. It can be clearly explained that under this set of parameters, negative feedback intensity is more conducive to the energy harvester. According to Figures 2 and 3, we can conclude that the results obtained by the theoretical method are consistent well with those from Monte Carlo simulations.Figure 3
The stationary PDFs under feedback strength. (a) Amplitude, (b) displacement, and (c) velocity. Theoretical analysis results (-) and Monte Carlo simulation results (∗).
(a)(b)(c)


## 4.2. The Influences of Parameters on the MSEV

In order to completely clarify the influences of the parameters on the performance of the piezoelectric energy harvesters, noise intensity, feedback strength, and time delay are successively introduced to investigate their influences on MSEVs in the following analysis. By takingu=−0.05, k=0.5, and τ=0.5, the effects of additive and multiplicative Gaussian white noise intensity on the MSEV are illustrated in Figures 4(a) and 4(b), respectively. It is found that with the increase of the additive and multiplicative noise intensity, MSEVs increase monotonically. The results show that additive noise plays a leading role in energy harvesting. The energy extracted from multiplicative noise by the piezoelectric energy harvester is relatively small, but it still makes sense when the noise intensity is large. Therefore, appropriately increasing the additive and multiplicative excitation intensity is a feasible method to increase the output voltage.Figure 4
The influences of noise intensity on the MSEV. Theoretical analysis results (solid round) and Monte Carlo simulation results (∗).
(a)(b)The variation of the MSEV with the time delay and feedback strength is illustrated in Figure5. It is seen that for any time delay τ, the MSEV V2 increases monotonically with the decrease of the feedback strength u. Nevertheless, the influences of time delay on the MSEV are nonmonotone.Figure 5
The influences of time delay and feedback strength on the MSEV.To illustrate the influences of the feedback strength on MSEV, Figure6(a) shows the changing of the MSEV with the feedback strength u=0.01 and u=−0.01. For D11=0.02, D22=0.02, and k=0.5, the variation of the MSEV V2 over the time delay τ is investigated. The results show that MSEV of the energy harvesting system without delay feedback system can be stabilized at 0.51. When u=−0.01, the time delay τ reaches the critical value of 3.14, and the MSEV of the system reaches the maximum and then decreases monotonically. However, when the feedback strength u=0.01, the curve of the MSEV changing with time delay τ seems to be completely opposite. For τ<3.14, the MSEV decreases monotonically as the time delay increases. When τ>3.14, the MSEV increases with the continuous increase of time delay. Furthermore, negative feedback strength is more favorable to the energy harvesting system than positive feedback strength. Therefore, the harvesting efficiency of the energy harvesting system can be improved by selecting the optimal control parameters u and τ.Figure 6
The influences of time delay and feedback strength on the MSEV. Theoretical analysis results (solid line) and Monte Carlo simulation results (∗).
(a)(b)By takingD11=0.02, D22=0.02, and τ=1, the dependence of the MSEV V2on feedback strength u and time constant ratio α is depicted in Figure 7(a). Specifically, the MSEV decreases rapidly as the time constant ratio α increases for a negative feedback strength u=−0.02. However, the MSEV decreases slowly with the increase of the time constant ratio for a positive feedback strength u=0.02. As the time constant ratio continues to increase, both the positive and negative feedback strengths keep the MSEV in a small range. Through the abovementioned analysis, a smaller time constant ratio and the negative feedback control will make the energy harvesting system induce larger MSEV.Figure 7
The influences of time delay and feedback strength on the MSEV. Theoretical analysis results (solid line) and Monte Carlo simulation results (∗).
(a)(b)The influences of the MSEVV2on feedback strength u and the piezoelectric coupling coefficient κ are shown in Figure 8. The MSEV almost decreases proportionally with the increases of κ at any feedback strength u in Figure 8(b). This means a smaller κ and the negative feedback control will increase the production of output power.Figure 8
The influences of piezoelectric coupling coefficient and feedback strength on the MSEV. Theoretical analysis results (solid line) and Monte Carlo simulation results (∗).
(a)(b)


## 5. Conclusions

In this paper, the time-delayed feedback control is utilized to improve the efficiency of a nonlinear piezoelectric energy harvesting system under the influences of the additive and multiplicative Gaussian white noise excitation. The standard stochastic averaging method has been applied to derive the stationary PDFs of amplitude, displacement, and velocity. Then, the expression of MSEV is deduced via the approximate relationship between voltage and mechanical states. Furthermore, the influences of the parameters on the MSEV are investigated. The results show that the efficiency of the harvester can be improved by selecting the optimal time delay and negative feedback strength when the other parameters unaltered. Moreover, the increases of noise intensity will lead to the increases of MSEV. Besides, we find that time constant ratio and coupling coefficients will reduce the MSEV. The analysis results are almost consistent with the Monte Carlo simulation results of the original system, which verifies the rationality and validity of the theoretical method.Compared with the nonlinear monostable, the multistable piezoelectric energy harvesting systems have attracted much attention due to the activation of the large-orbit interwell snap-through mechanism. Therefore, we will focus on the bistable energy harvesting system with the time-delayed feedback control next steps.



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