Time-Delayed Feedback Control in the Multiple Attractors Wind-Induced Vibration Energy Harvesting System

This paper proposes a time-delayed feedback control to improve collection performance of the multiple attractors wind-induced vibration energy harvester system. By employing a conversion mechanism for decoupling the electromechanical equations and the stochastic averagingmethod, the roles of the time-delayed feedback and the system parameters have been systematically examined. In the deterministic case, the time delay can control efficiently the birhythmic properties; thus one can realize a dividing line in the parameter plane that separates the space into two subspaces of generically distinct nature. Further, it is exclusively demonstrated that implementing time-delayed feedback control serves as a very simple but highly efficient scheme to increase the harvested energy from vibrations in presence of noise disturbances. Additionally, the time constant ratio and coupling coefficients can also be properly tuned to optimize power generation.


Introduction
Growing demand of electrical energy and environmental issues has spurred considerable efforts to generate electricity from renewable energy sources [1].Energy harvesting systems which served as a promising solution for generating electricity are designed to transform available ambient energy (e.g., wind energy, solar energy, tidal energy, piezoelectricity, thermoelectricity, and kinematic energy) into electrical energy for small and/or portable devices [2].Although common transduction mechanisms are electromagnetic, piezoelectric, and electrostatic, the piezoelectric conversion mechanism is in general preferable due to its good power generation performance, simple structure, and no reliance on external magnetic field or initial DC voltage [3].
In the early stages, different types of linear vibration energy harvesters have been designed to generate electrical energy [4][5][6].These systems, however, were found to be effective within a limited bandwidth near their resonant frequencies, thus limiting their application to frequency variant, amplitude variant, broadband, and random excitation sources which are typical excitation types in real applications [2,7,8].In order to solve this critical challenge, a burst of research activities have been devoted to widen the working frequency of energy harvesters by using nonlinear phenomenon [9][10][11][12][13][14].
Compared to linear systems, nonlinear systems can generate electrical energy over a larger continuous bandwidth and are less sensitive to changes in frequency characteristics [14].Nonlinear monostable, bistable, and tristable energy harvesters have been successively explored [13,15,16].Sebald [13] experimentally proved that the operating bandwidth of the nonlinear monostable energy collector is several times that of its linear counterpart.One of the drawbacks of monostable design is its poor performance when the excitation is random excitation [17,18].Therefore, research interests have turned to collectors that utilize multiple steady states.The unique advantage of bistable energy harvesters (BEH) has been demonstrated by Cottone et al. [19] from the viewpoint of stochastic resonance.Daqaq et al. [20] compared the energy harvesting performance between the BEH and monostable energy harvesters influenced by Gaussian white noise.Utilization of high-energy orbits in the bistable structural configuration for electrostatic, electromagnetic, and magnetostrictive transduction mechanisms is summarized by Erturk et al. [21].More recently, Li et al. [22] demonstrated the better energy harvesting performance of the tristable 2 Complexity energy harvester (THE) over its bistable counterpart under random base excitations.A tristable energy harvesting system with fractional-order viscoelastic material has been meticulously elaborated in [23].Further work in nonlinearity can include internal resonance, coherence resonance [12,24], and parametric excitation [25][26][27][28][29].
From the perspective of vibration suppression, the reduced amplitude of the vibration is beneficial.For energy harvesting systems, however, amplitude reduction is undesirable because the resultant energy is reduced.Consequently, more advanced control methods (e.g.synchronized switching harvesting on inductor and switching controller) have been proposed to solve this critical problem [30,31].A simple and effective control strategy is time-delayed feedback control, which has been widely used to control bifurcation, chaos, resonance, and birhythmicity [32][33][34][35][36].In addition, as an important renewable energy source, wind energy has received extensive attention from scholars [2,31].In this paper, we will focus on the multiple attractors wind-induced vibration energy harvesting system [2] and unveil the optimization effect of time-delayed feedback control on this system with random excitation.
The rest of this paper is organized as follows.The next section makes a detailed description of multiple attractors energy harvesting system with time-delayed feedback control and Gaussian white noise excitation and carries on a detailed theoretical analysis.The main deterministic and random theoretical and numerical contributions are presented in Sections 3 and 4, respectively.Finally, the results are concluded and summarized in Section 5.

Nonlinear Vibration Energy Harvesting System with Delayed Feedback.
A schematic of a multiple attractors windinduced vibration energy harvester system at low Reynolds number is shown in Figure 1 (see [2]).The physical model consists of a flexible beam with distributed piezoelectric patches and an electrical circuit having a load resistance   .The dimensionless form of the original model is dominated by the following set of equations [2,37]: Here  is the dimensionless transversal beam deflection function and the coefficients , ,  are related to wind velocity, fluid mass density, Reynolds number, etc. V is the dimensionless form of the voltage across an equivalent resistive load   and  is time constant ratio ( = 1/    ,   is inherent capacitance of the piezoelectric element). 0 is the piezoelectric coupling coefficient in the mechanical equation and  1 is the piezoelectric coupling coefficient in the electrical equation.A time-delayed feedback is known as an efficient tool [35,36] for continuous-time control of dynamical systems [38].A simple and efficient scheme of time-delayed feedback control is Pyragas control since it does not require detailed information of the system to be controlled [38].Of particular interest to us is such a form of feedback () = [ ẋ ( − ) − ẋ ()], where ẋ () is the signal coming from the system,  is the feedback strength, and  is the time delay.This control method by changing both the feedback strength and time delay has also been applied in a multicycle van der Pol oscillator [35].The object of this paper is to understand the dynamic responses of the energy harvester system toward such a delayed feedback control.On the other hand, the energy harvesting system is inevitably affected by random excitation.This paper introduces Gaussian white noise as a random noise source.Application of delayed feedback and consideration of Gaussian white noise excitation result in the following equation: Here () is Gaussian white noise which has the following properties ⟨()⟩ = 0, ⟨()( + )⟩ = 2().ẋ  stands for the delayed variable ẋ ( − ),  is the strength of the delayed feedback.And  defines the time delay ( > 0) characterizing the finite time in signal propagation, in measuring and estimating the system states, calculating and executing the control force, etc.
Further, substituting Eq. ( 4) into Eq.( 3) yields [40] V The steady part of ( 6) is Due to the amplitude of the steady-state voltage is Substitution of ( 5) and ( 7) into (2) yields a modified governing equation for a nonlinear energy harvester To facilitate the analysis, we use the following substitutions: The new form of the equations is thus On the hypothesis that damping and noise intensity  is small, introduce a transformation of variables [41]  () =  () cos , ẋ () = − ()  sin ,  =  + .(13) By applying ( 13) into (12), the differential functions versus (), () can be obtained By adopting the standard stochastic averaging method [41], the stochastic differential equations (in the It ǒ sense) for the instantaneous amplitude () and phase () are determined by Thereby,  1 () and  2 () are independent normalized Wiener processes and ,  is given by (11).Obviously, the amplitude  does not depend on the phase .Consequently, the Fokker-Planck-Kolmogorov (FPK) equation of amplitude yields with Considering the stationary case, the stationary probability density function (PDF) for amplitude can be solved as in which  is a normalization constant and ,  is given by (11).Thus, the expected value of the mean square voltage can be calculated as In this paper, the fixed parameters are  0 = 1,  = 0.01.In order to validate the effectiveness of the analytical results (see (18)), direct numerical simulations based on the secondorder Runge-Kutta algorithm (provided in the appendix) of ( 2), (3) are performed.
Letting  = 0.15,  = 0.005,  = 0.01,  = ,  0 = 0.01,  1 = 0.25, = 0.2, Figure 2 demonstrates the stationary PDFs of the amplitude for three different values of feedback strength, respectively.In Figure 2, the solid lines are the analytical results and the dots result from numerical simulations.The basic agreement between the analytical results and numerical simulations best confirm the effectiveness of the theoretical analyses.
Observations of Figure 2 also show that the negative time delay feedback control induces an increase in the probability of large amplitude motion of the energy harvesting system.

Control of Birhythmicity Using Delayed Feedback without Noise
Letting ()/ = 0, the extreme   of the distribution of ( 18) is given by the roots of the following equation: For  = 0, the amplitude equation is dominated by It should be noted that the theoretical method in our paper is more accurate than the theoretical method proposed by Debabrata Biswas and his collaborators [37].For the same parameters as in Figure 3 in [37], one can obtain that the Hopf bifurcation of the control parameter is approximately 0.0953 by using the continuation package XPPAUT.The theoretical value of the Hopf bifurcation of the control parameter in [2] is   =  = 0.1.However, the theoretical value of the Hopf bifurcation of the control parameter that we derived is  =  −  0  1 /( 2 +  2 ) ≈ 0.095192.
Based on (11) and ( 21), Figure 3 illustrates the bifurcation diagram in the  −  parameter domain for  0 = 0.01,  1 = 0.25,  = 0.2.In absence of delayed feedback, the birhythmic region possessing a bistability mode with coexistence of two stable limit cycles (LC) of the system consists of all colored regions without the dotted line.The birhythmic region, whereas, is split into two parameter zones corresponding to the points  1 (0.005, 0.15) for  = 0.005,  = 0.15 and  2 (0.001, 0.12) for  = 0.001,  = 0.12 charactering two distinct dynamical scenarios in presence of delayed feedback ( = 0.01), which are verified in Figures 4(a) and 4(b), respectively.Observing Figure 4(a), when the time delay increases gradually from zero, the system shows an interesting transition from three limit cycles (two are stable limit cycles and the other is unstable) to a stable limit cycle then to a stable fixed point and thence to oscillation recovery for  1 (0.005, 0.15).Four exemplary illustrations for  = 0.2, 1.16, 3, 6 are shown in Figure 5.For  = 0.02, the system exhibits birhythmic behavior possessing two stable limit cycles as shown in Figure 5(a).With the gradual increase of time delay, the outer cycle ultimately vanishes beyond a critical time delay ( > 1.5236), and the system becomes monorhythmic in nature.This can be verified by Figure 5

(b).
As is shown in Figure 5(c), a further increase in time delay will cause the loss of self-sustained oscillatory character.Again with further increase of time delay, the system reproduces oscillation (Figure 5(d)).
Then, the evolution of the electric voltage of the harvested energy with time delay for 1(0.005;0.15) is exhibited in Figure 6.When the time delay increases slowly from zero, the system will have two different voltages due to the birhythmic nature of the system.For example, for initial condition  = 0.1, ẋ = 0, V = 0.3, the system's stable voltage is in the lower branch; for initial condition  = 7.0, ẋ = 0,V = 0.3, the system's stable voltage will be in the upper branch.As the time delay increases further, the system has only one stable voltage regardless of initial conditions.Again with further increase of time delay, the system will reside in a stable state with the voltage of 0. An further increase of the time delay will cause the system to resume oscillation, thereby generating electrical energy.
Comparing Figure 6 and the red curve of Figure 10, one can see that when the system is subjected to random perturbations, it will randomly jump between two limit cycle oscillations with different amplitudes without being affected by the initial conditions.Further, we examine another dynamical scenario for  2 (0.001, 0.12) as presented in Figure 4(b).For  = 0.02, the system exhibits birhythmic behavior possessing two stable limit cycles as shown in Figure 7(a).As the gradual increase of time delay, the inner cycle becomes smaller in amplitude and ultimately vanishes for a threshold of time delay about 1.5236, and the system exhibits the bistable behavior with coexistence of a stable LC and stable steady state (SSS).This can be verified by Figures 7(b) and 7(c), respectively.Again with further increase of time delay, the inner cycle reappears implying the recovery of birhythmicity (Figure 7(d)).By inspecting Figures 4(b) and 7, it can be also found that the time delay can hardly affect the outer stable cycle and the unstable limit cycle.The outer stable limit cycle that remains unaffected by time delay has an amplitude about 13.17.So based on the above findings, we can found that the delayed feedback can be suitably tuned to control the nature of the attractors depending on the parameter space.For the region  in Figure 3 where the point  2 is located, the outer stable cycle is hardly affected by the time delay, which facilitates the energy harvester to be in a limit cycle of greater amplitude, thereby collecting more energy.So we will next focus on the region  and further explore the influence mechanism of time delay and noise on the points in the region to seek optimal control parameters.

Stochastic Responses
In the remaining part, we take the point  1 (0.005, 0.15) for  = 0.005,  = 0.15 in region  for further analysis.The effects of noise level, time delay and its feedback, time constant ratio, and coupling coefficients on mean square electric voltage will be explored in detail.In order to compare with the analytical results, direct numerical simulations of ( 2) and (3) are carried out, which verify the effectiveness of our theoretical predictions.

Effect of Noise
Level. Figure 8 exhibits the dependence of the mean square electric voltage ⟨V 2 ⟩ on the excitation intensity  and feedback strength ; the other parameters are  0 = 0.01,  1 = 0.25,  = 0.2,  = 1.0.The solid lines stand for the analytic results and the symbols denote the numerical simulations of original system.For positive feedback, an increase in noise intensity causes the mean square voltage to increase sharply and then slowly increase.Surprisingly, for negative feedback strength, the mean square voltage will decrease rapidly as the noise level increases and then slowly increase (see a partial enlargement of Figure 8(b)).In general, the Gaussian white noise with high noise level helps to increase the collection capacity of the energy harvesting system, and the negative feedback strength is more favorable for the energy harvesting system to collect more energy than the positive feedback strength.

Effect of Time Delay and Its
Feedback.The dependence of the mean square electric voltage ⟨V 2 ⟩ on the time delay  and feedback strength  is depicted in Figure 9; the parameters are  0 = 0.01,  1 = 0.25,  = 0.2,  = 0.01.This figure illustrates that the feedback strength makes the mean square voltage monotonously reduced, but the effect of time delay is nonmonotonic.
For an illustration of the impact of feedback strength on the mean square voltage, Figure 10(a) shows the change of the mean square voltage with feedback strength under two different time delay values  = 1.0 and  = , which reveals that the mean square voltage decreases monotonically with the increase of the feedback strength and rapidly decays when the feedback strength increases to near zero.voltage initially decreases monotonously, and it will show an upward trend after a gradual period.Surprisingly, with the monotonous increase of time delay, the mean square voltage initially monotonically increases, and it will undergo a monotonous decrease after exceeding the critical value  = 3.141 for  = −0.01.Based on the above analysis, one can get the optimal values of time delay and feedback strength are  = −0.01, = 3.141 for this set of parameters, with which the system has the largest mean square voltage.Thus, the optimal control parameters ,  can be selected to improve collection performance of the energy harvesting system.

Effect of Time Constant
Ratio.The dependence of the mean square electric voltage ⟨V 2 ⟩ on feedback strength  and time constant ratio  is depicted in Figure 11; the parameters are  0 = 0.01,  1 = 0.25, = 1.0,  = 0.01.For positive feedback strength, the mean square voltage decreases slowly with increasing time constant ratio (see the red curve in Figure 11(b)).However, for negative feedback strength, the mean square voltage decreases rapidly with increasing time constant ratio (see the blue curve in Figure 11(b)).And for a smaller time constant ratio, the mean square voltage at negative feedback strength is about 3-4 times the positive feedback strength.But for a larger time constant ratio, both the positive feedback strength and the negative feedback strength make the mean square voltage in a small range.That is, the negative feedback control and a smaller time constant ratio will induce larger mean square electric voltage.

Effect of Mechanical Coefficient.
The dependence of the mean square electric voltage ⟨V 2 ⟩ on  0 and feedback strength  is depicted in Figure 12; the parameters are  1 = 0.25,  = 1.0,  = 0.2,  = 0.01.For smaller  0 , the mean square voltage will drop sharply under negative feedback strength  = −0.01 and will slowly decrease under positive feedback strength.For larger  0 , the two curves almost coincide in the vicinity of the horizontal axis, implying very little mean square voltage.The observations suggest that the smaller  0 and the bigger of the absolute value of the negative feedback help the energy harvesting system produce higher mean square voltage.

Effect of Electrical Coefficient.
The dependence of the mean square electric voltage ⟨V 2 ⟩ on  1 and feedback strength  is depicted in Figure 13; the parameters are  0 = 0.01,  = 1.0,  = 0.2,  = 0.01.As shown in Figure 13(b), for smaller values of  1 , the two curves corresponding to  = −0.01,0.01 are somewhat coincident.As  1 increases, the two curves will gradually separate and the distance will be further and further, which reveals that the negative feedback strength induces a larger mean square voltage, approximately 5.8 times the positive feedback for  1 = 0.6.In addition, these phenomena indicate that a larger  1 helps to get a higher mean square electric voltage.

Conclusion and Discussions
In the multiple attractors wind-induced vibration energy harvester system, the birhythmic properties of the oscillator can be efficiently controlled by a time-delayed feedback.How the time-delayed feedback modify the bifurcation scenario in a general way is examined in detail, such that one may realize in the bifurcation diagram a dividing line that separates the space into two subspaces for which the dynamical scenarios are generically distinct.Depending on the time delay and its feedback strength, different regions may exist in the bifurcation diagram.The time delay feedback may play an Complexity important role in adjusting the position of the dividing line(s) between the regions.For the region  in the bifurcation diagram, the outer stable cycle is hardly affected by the time delay, which facilitates the energy harvester to collect more energy.So in the analysis of random responses, we focus on the region  and seek optimal control parameters.The effects of noise level, time delay and feedback intensity, time constant ratio, and coupling coefficients on the mean square voltage are systematically analyzed.All the results show that negative feedback strength can effectively improve the collection performance of the energy harvester.And when other parameters are unchanged, an optimal time delay value can be sought to maximize the system's mean square voltage.Gaussian white noise with higher noise intensity and larger coupling coefficient in the electrical equation, smaller time constant ratio, and smaller coupling coefficient in the mechanical equation can also enable the energy harvesting system to collect more energy.These findings may deepen our general understanding of the underlying mechanism of the multiple attractors wind-induced vibration energy harvester and provide a possible recipe to design energy harvesting systems with super performances in practical applications.

Figure 1 :
Figure 1: Schematic representation of the energy harvesting system under wind flow excitation with low Reynolds number.

Figure 2 :
Figure 2: Stationary PDFs of the amplitude P (a) for various values of feedback strength .The solid lines stand for the analytic results and the dots denote the numerical simulations of original system.

Figure 3 :
Figure 3: Parameter domain (, ) for the existence of the three limit cycles (area denoted by the colored area without the dotted line in absence of delayed feedback) and a single limit cycle (area denoted by white region).The yellow dotted line denotes the bifurcation line characterizing the two parameter zones corresponding to the points  1 (0.005, 0.15) and  2 (0.001, 0.12) for two distinct dynamical scenarios in presence of delayed feedback corresponding to Figures 4(a) and 4(b), respectively.

Figure 6 :Figure 7 :
Figure 6: The dependence of the electric voltage V 2 on .

Figure 8 :
Figure 8: The dependence of the mean square electric voltage ⟨V 2 ⟩ on  and .Solid lines are the analytical results; symbols result from numerical simulations.

2 ⟩Figure 9 :
Figure 9: The dependence of the mean square electric voltage ⟨V 2 ⟩ on  and .

Figure 10 (Figure 10 :Figure 11 :
Figure 10: (a) The dependence of the mean square electric voltage ⟨V 2 ⟩ on .(b) The dependence of the mean square electric voltage ⟨V 2 ⟩ on .Solid lines are the analytical results; symbols result from numerical simulations.

Figure 12 :Figure 13 :
Figure 12: The dependence of the mean square electric voltage ⟨V 2 ⟩ on  and  0 .Solid lines are the analytical results; symbols result from numerical simulations.