Sparse identification method of extracting hybrid energy harvesting system from observed data

Ya-Hui Sun(孙亚辉) Yuan-Hui Zeng(曾远辉) and Yong-Ge Yang(杨勇歌)

1School of Mathematics and Statistics,Guangdong University of Technology,Guangzhou 510520,China

2State Key Laboratory for Strength and Vibration of Mechanical Structures,Xi’an Jiaotong University,Xi’an 710049,China

Keywords: data-driven,hybrid energy harvester,harmonic excitation,Gaussian white noise

1. Introduction

The vibration energy harvester(VEH)can produce electric energy from ambient vibrations[1–3]to provide wireless sensors with sustaining energy. There are many types of energy harvesting techniques, such as piezoelectric,[4,5]electromagnetic,[6]etc.[7,8]Among them, combining piezoelectric components and electromagnetic elements,hybrid energy harvesters were investigated and widely applied due to the advantage of improving the harvesting efficiency.[9]Panyam and Daqaq[10]studied a tri-stable hybrid energy harvester under harmonic excitation by using multi-scales method. Karami and Inman[11]proposed an approximation method to explore the hybrid energy harvester under harmonic excitation. Xiaet al.[12]investigated the performances of the energy harvesters with different boundary conditions. Moreover, noises exist in the real environment and result in different dynamical behaviors.[13,14]Therefore, hybrid energy harvesters under random excitation were studied in previous researches.[15–20]For example,Zhouet al.[15]showed that the performance of hybrid energy harvesters under Gaussian excitation is improved by adding nonlinear components. Senghaet al.[16]modeled a hybrid energy harvesting system driven by harmonic excitation and colored noise. Foupouapouognigniet al.[17]indicated that the performance of the hybrid VEH,which was influenced by Gaussian white noise (GWN) and harmonic excitation, was improved. Sunet al.[18]used the stochastic averaging technique to analyze the stochastic responses of a fractional-order hybrid VEH driven by GWN.The influences of colored noise excitation on hybrid VEH were investigated by Yang and Cao.[20]

Stochastic responses of hybrid VEH in the above articles were studied in the condition that the governing equations were firstly known. However,for a complex system,it is trouble to model the governing equations precisely in practical applications. To overcome this difficulty, in recent years, datadriven modeling[21,22]was presented and applied in different areas,such as fluid dynamics,meteorology,finance,etc.[23–26]With the developments of machine learning and data science,some methods were proposed to identify elusive dynamical systems. For deterministic differential equations, Bruntonet al.[27]proposed a data-driven method called sparse identification of nonlinear dynamics (SINDy), and demonstrated that the method is well agreement in the deterministic differential equations by using the iterative threshold algorithm. Boninsegnaet al.[28]modified the SINDy to avoid the adjustment of the threshold parameter in Ref.[27]. However,stochastic differential equations widely exist in practical applications. For stochastic differential equations (SDEs), the methods which were devised in Refs. [27,28] were also used to model SDEs driven by GWN. Rudyet al.[29]presented a deep neural network approach to estimate the coefficients and the measurement noise simultaneously, and showed good robustness of the method by increasing the noise level. Daiet al.[30]proposed the maximum likelihood estimation to learn the SDE under fractional Brownian motion, and the results showed a good accuracy for true values. Lu and Lermusiaux[31]devised a Bayesian learning technique to model stochastic dynamical systems. Huang and Li[32]used the SINDy to discover the equations of a four-dimensional stochastic projectile system. Wuet al.[33]obtained the mean residence time and escape probability of SDEs from data by using the devised approach.Additionally,based on the Koopman generator,Zhanget al.[34]developed an approach to extract SDEs influenced by L´evy noise from data on mean exit time, and indicated that this method can also apply to dynamical systems under GWN. Lu and Duan[35]discovered the SDEs from data with L´evy noise by utilizing extended dynamic mode decomposition, and acquired the transition probability density functions by solving the Fokker–Planck equations. Together with the Kramers–Moyal formulas and SINDy, governing laws under different L´evy noise were extracted from the observed data of stochastic dynamics equations.[36–38]Among these methods,SINDy has been widely applied in discovering governing equations from massive datasets. The method combines the least-squares and compressed-sensing to solve the sparse coefficients of the equations, so that the approximate governing equations can be obtained. It is useful to analyze the subsequent dynamical behavior of the system.

To the best of our knowledge, few authors pay attention to discovering the equations from data for the hybrid energy harvester in present. In this paper,we develop a sparse identification approach to identify the equations for the hybrid VEH.The framework of this paper is organized as follows. In Section 2, the mathematical model of the hybrid VEH with nondimensional form is given.In Section 3,a sparse identification process is developed to solve the sparse coefficients for the hybrid VEH. In Sections 4 and 5, two examples of the hybrid VEH are taken to examine the validity of the devised method.In Section 6,some conclusions are remarked.

2. Hybrid VEH under noise excitation

A family of hybrid electromagnetic and piezoelectric energy harvesters are considered as shown in Fig.1. The model is simplified as a mass–spring–damper system in Fig.1(a)coupled with a piezoelectric circuit in Fig.1(b)and an electromagnetic circuit in Fig.1(c)under base acceleration.

Fig.1. (a)Simplified diagram of a hybrid VEH coupled with(b)a piezoelectric energy harvesting circuit and(c)an electromagnetic circuit.

The coupling equations of the hybrid energy harvesting system are given by

Here, ¯X, ˙¯Xand ¨¯Xbdenote the displacement, the velocity and the base acceleration of the massM,respectively. ¯Vis the electric voltage measured across the equivalent resistance loadRp.Cpand ¯ζ1are the the piezoelectric capacitance and the piezoelectric coupling coefficient.L, ¯Iand ¯ζ2denote the inductance of the coil, the output current and the electromagnetic coupling coefficient, respectively.RcandResuccessively denote the load resistances of the electromagnetic and the resistance of the coil.f(¯X)andg(¯X)represent the damping term and the stiffness term.

Then,we make the equations dimensionless by means of a transformation[18,39]Here,f(x) andg(x) are the non-dimensional damping force and stiffness term.ζ1,ζ2,λ1andλ2are the non-dimensional coupling coefficients in Eq. (1).µ1is the reciprocal of the product of resistance and capacitance.µ2represents the ratio of resistance and inductance.

As examples of different external excitation, harmonic excitation and GWN[18,40]are considered in Sections 4 and 5. With the method of stepwise sparse regressor(SSR),[28,41]we devise the method of identified sparse regression to discover the governing equations by learning the coefficients of the formula.

3. Sparse identification for hybrid VEH

Drawing on the ideas of machine learning, we combine the least-square sense, SSR algorithm and cross-validation(CV). A sparse identification process is developed to learn the unknown coefficients in the equations. Assume that we have observedNdata points of system state time series of displacement, voltage and output current denoted byX,VandI, respectively. Meanwhile, data pointsX,VandIhave been dimensionless to [x1,x2,x3,...,xN], [v1,v2,v3,...,vN]and[i1,i2,i3,...,iN]at[t1,t2,t3,...,tN]. We transform Eq.(3)into the following differential equations:

3.1. Learned process of drift term

In this subsection, we introduce the sparse identification of drift termb=[b1,b2,b3,b4]T. Firstly, we approximate the first-order derivative by using the first-order difference,i.e.,

The learned results will be better if we select an abundant type of basis functions, while the real workload is enormous and polynomial basis functions are enough precise for most cases.

By referring to Ref.[42],b2,b3andb4are estimated by using the Kramers–Moyal formula,i.e.,

where

However,Eq.(11)may have no solutions due to the equations more than variables. Based on this, we use the least square sense,i.e.,

whereρ>0 is called as the penalty factor applied in restricting the weight of the sparsity constraint. Due to the meaning of theL1regular,some terms in the solutions will be equal to zero.

The approach we used to solve Eq. (15) is iterative algorithm SSR together with CV. Compared with the iterative threshold algorithm,[21,27]Lasso,[43]elastic net[44]and matching pursuit,[45]SSR not only can adaptively select the number of iterations and the sparsity level,but also does not needlessly adjust the external parameters like threshold parameterλ. The purposes of CV[46]are the division of data and the selection of optimal parameter. The pipeline works are summarized in Table 1.

Table 1. The algorithm for sparse identification.

3.2. Learned process of diffusion term

For GWN,the diffusion terma22in Eq.(4)is calculated as 2D. Analogously, we construct the basis functionΨ(X).Then,the diffusion term is approximated as

According to the steps in Table 1, we strengthen the sparse level of the solutionqand the sparse solution ^qis obtained.

4. Discover equations for a hybrid VEH under harmonic excitation

4.1. A hybrid VEH under harmonic excitation

Letf(x) =c4x4+c2x2+c0,g(x) =δ1x+δ3x3+δ5x5[47,48]andD=0. The equations of the system can be expressed as

4.2. Sparse learning of the system under harmonic excitation

We set parameters asc0=−0.5,c2=0.5,c4=−0.1,δ1=1,δ3=−3,δ5=1,ζ1=0.5,ζ2=0.5,F=1,µ1=1,λ1=1,µ2=0.5,λ2=1.

According to Eq.(22),five independent system state trajectories ofNh=105steps each are generated as the observed data by using the method of the fourth order Runge–Kutta.We compute the derivatives ˙X, ˙Y, ˙Vand ˙Iby applying Eq.(6)with the time step ∆t=0.001,where the smaller ∆t,the higher the accuracy. Then,Kh=27 basis function dictionaryφh(t,X,˙X,V,I)is considered. The specific composition of the dictionary reads as

The matricesAandBcan be obtained by Eqs.(9),(10)and (12). To avoid under-fitting and over-fitting, 7-fold CV is utilized to select the specific number of iterations on which SSR needs to be run. We have a family of models with the number of iterations as parameter(SSR(0),...,SSR(27)). By performing the algorithm in Table 1,we select the one model that is best to fit the observed datasets. Figures 2(a), 2(b)and 2(c) respectively demonstrate MSE of coefficients learning from Eqs. (22b), (22c) and (22d), wherendenotes the number of non-zero coefficients.

From Fig.2(a),we can see that whenn<8,the MSE vibrates firstly and then decreases slowly.Subsequently,the part ofn ≥8 is magnified in the upper right of Fig.2(a).The results show that whennchanges from 8 to 9,the mean square error plummets, and then slowly decreases untiln=11. We hold the opinion that the model is under-fitting inn<8 and overfitting inn>11.[28]Analogously,Figs.2(b)and 2(c)show that the number of non-zero coefficients in Eqs.(22c)and(22d)is 2. Thus, for Eq. (22b), we focus onn=9 andn=10. For Eqs.(22c)and(22d),n=2 is the best parameter value.

Fig.2. MSE from applying SSR algorithm is plotted as a function of solution size n. (a)Eq.(22b);(b)Eq.(22c);(c)Eq.(22d).

Table 2. Identified coefficients for Eq.(22b).

The sparse solutions ˆuhcan be learned as listed in Tables 2–4. The results indicate that the coefficients learned from the algorithm are similar to the true values,andn=9 is more suitable for Eq.(22b).Furthermore,Figs.3(a)and 3(b)demonstrate the comparisons of learned values and true values with regard to damping forcef(x)and stiffness termg(x),respectively. Figure 4 shows the comparison of time-varying displacement and time-varying voltage from the original system and the learned system.The results show that the true system and the learning system are in good agreement.

Table 3. Identified coefficients for Eq.(22c).

Table 4. Identified coefficients for Eq.(22d).

Fig.3. Comparisons between the learned values and true values of(a)f(x)and(b)g(x).

Fig. 4. Comparisons of (a) time-varying displacement and (b) timevarying voltage.

Fig.5. Comparisons between true values and learned values of the b2h. (a)and(b)y=0.5,I=0.5;(c)and(d)y=0.5,V =0.5.

Fig.6. Comparisons between true and learned functions. (a)and(b)b3h;(c)and(d)b4h.

Table 5. The learned results of 10 sets data with different ∆t and different Nh.

Since Eq. (23b) is four-dimensional, the figure of equation is impossible to plot intuitively. Thus, we plot the equation as two-dimensional by fitting two state variables. In Figs.5 and 6,comparisons of the true and learnedb2h,b3handb4hare shown to indicate the learning results. Here,Figs.5(a)and 5(c) represent the true functionb2h; Figs. 5(b) and 5(d)represent the learning functionb2h. The two rows of the figures denote the case with(i)y=0.5 andI=0.5; (ii)y=0.5 andV=0.5, respectively. Figures 6(a) and 6(b) are the true functions ofb3handb4h, respectively. Figures 6(c) and 6(d)are the learning functions ofb3handb4h,respectively. The results show that the learning functions agree well with the true functions.

By referring to Refs.[49–51],we know that as the fluctuations of the estimated parameters are small with the increase of data length,the estimated parameters converge approximately to the true value of the parameters. Then, 10 sets data with different ∆tandNhare learned by the sparse identification algorithm. The results from Table 5 demonstrate that for the same time step ∆t, the longer the data length, the smaller the standard (Std.) deviation. For the same data lengthNh, the standard deviation decreases as the ∆tdecreases. Thus, with an increase of data length and a small time step,the identified coefficients can be close to the true values of the coefficients.

5. Discover equations for a hybrid VEH under GWN and harmonic excitation

5.1. A hybrid VEH under GWN and harmonic excitation

In this section,the parameter values are the same as those in Subsection 4.2, except for the parameter 2D=0.01. We consider a hybrid energy harvester under both GWN and harmonic excitation,i.e.,

Accordingly, the diffusion terma22=2D, and the drift termbof Eq.(26)can be obtained as

5.2. Pre-processing of drift term and diffusion term

For the drift termb,KG=18 basis function dictionaryφG(t,X,˙X,V,I)is given by

By using the sparse identification algorithm in Table 1,the drift coefficients ˆuGcan be solved by

whereAandBare obtained by Eqs.(9),(10)and(12).

Analogously,for the diffusion terma22=2D,L=4 basis function dictionaryψ(X)is constructed as

According to Subsection 3.2,the sparse solution ^qcan be calculated.

5.3. Sparse learning of the system under GWN and harmonic excitation

On the basis of Eq. (26), the fourth-order Runge–Kutta[52]is utilized to generate one hundred independent system state trajectories ofNG=105steps. Then, ˙X, ˙Y, ˙Vand ˙Iare approximated numerically by Eq.(6).Through Eqs.(28)and (29),AandBin Eq. (29) are established. Similarly, ^Aand ^Bare obtained via Eqs.(30)and(19).

Fig.7. The mean MSE of CV for fifty trajectories and each non-zero coefficients n. (a)Function b2G;(b)function b3G;(c)function b4G;(d)function a22.

Likewise,we use 10-fold CV to find the optimal number of iterations of SSR.For the one hundred trajectories and each the number of non-zero coefficientsn, the mean MSE of 10-fold CV is calculated by performing the algorithm in Table 1.Here,Figs.7(a),7(b),7(c)and 7(d)show the mean MSE of the functionb2G,b3G,b4Ganda22with every trajectories and the number of iterations. Then,to judge the number of iterations,we choose the optimal trajectory which has a minimum MSE in all average MSE. Figures 8(a), 8(b), 8(c) and 8(d) are the MSE of the functionsb2G,b3G,b4Ganda22. From Fig.8(a),the MSE has large fluctuations inn<4 and small fluctuations inn ≥4. The part ofn ≥7 is amplified in the upper right of Fig.8(a). It can be seen that the MSE decreases sharply fromn=8 ton=9 and fluctuates aftern>10. Meanwhile, the minimum MSE of the trajectory is acquired inn=10. Thus,we conclude that the best number of iterations isKG−9 orKG−10 in drift functionb2G,i.e.,n=9 orn=10. Due to the feature ofb2G,we only considern=9. Similarly according to the above analysis,because of the minimum MSE corresponding to then, the number of optimal non-zeros coefficients of the functionsb3G,b4Ganda22aren=2,n=2 andn=1,respectively. Differently,in Fig.8(d),we choosen=1 becausen=0 does not satisfy the existence of white noise in Eq.(25).

In the following,from the sparse identification algorithm in Table 1, we learn the coefficients ofb2G,b3G,b4Ganda22usingKG−9,KG−2,KG−2 andL−1 iterations,respectively.With the application of the optimal trajectory and 10-fold CV,the learned results are shown in Tables 6,7,8 and 9 correspond tob2G,b3G,b4Ganda22,respectively.

Fig.8. The MSE of optimal trajectory with non-zero coefficients n. (a)Function b2G;(b)function b3G;(c)function b4G;(d)function a22.

Table 6. Identified coefficients for b2G.

Table 7. Identified coefficients for b3G.

Table 8. Identified coefficients for b4G.

Table 9. Identified coefficients for a22.

We show intuitively the learned results in Figs.9–11. Figure 9 demonstrates the fitting situation of the functionsf(x)andg(x). Figure 10 represents the comparison of the original system and the learned system in time-varying displacementx(t)and voltageV(t). It can be seen that compared with the true coefficients, the learned coefficients have slight deviation within an acceptable range.

Fig.9. Comparisons between the learned values and true values of(a) f(x)and(b)g(x).

Fig.10. Comparisons of(a)time-varying displacement and(b)time-varying voltage.

Fig.11. Comparisons between true and learned functions. (a)and(b)b3G;(c)and(d)b4G.

Then, the results of comparison between true values and learned values with respect tob3Gandb4Gare shown in Fig.11.Figures 11(a) and 11(b) are the true functions ofb3Gandb4G, respectively. Figures 11(c) and 11(d) are the learning functions ofb3Gandb4G, respectively. The true and learned results of functionb2Gare displayed in three partsx–V,x–Iandx–t. We demonstrate the equation in two-dimensional due to the dimension ofb2Gmore than two. Figures 12(a)–12(e)represent the true results. Figures 12(b)–12(f)represent the learned results. The three rows of the figures describe the case with(i)y=0.5,I=0.5 andt=10;(ii)y=0.5,V=0.5 andt=10;(iii)y=0.5,V=0.5 andI=0.5,respectively. It can be seen that the coefficients learned from the sparse identification algorithm have good enough accuracy.

Fig. 12. Comparisons between true values and learned values of the b2G. (a) and (b) y=0.5, I =0.5,t =10; (c) and (d) y=0.5,V =0.5,t=10;(e)and(f)y=0.5,V =0.5,I=0.5.

Above all,the hybrid energy harvesting system identified by sparse identification is consistent enough with the real system.

6. Conclusion

In this paper,a sparse identification approach was developed to acquire the governing equations of the hybrid energy harvesting system from the simulated sample state data. Two examples were taken to verify the feasibility and effectiveness of the method.

To begin with,a hybrid energy harvester under harmonic excitation was the first example. Through approximating derivatives by the first-order difference and constructing the basis functions dictionary,we obtained the expressions of differential equations of the system,which are equal to the linear combination of basis functions. Then,for the number of nonzero coefficientsnand each sample trajectory, 7-fold crossvalidation(CV)was applied to prevent under-fitting and overfitting by observing the variations of MSE.Removing the situations of under-fitting and over-fitting,we selectedn=9,n=2 andn=2 for the differential equations ˙y, ˙Vand ˙I,respectively.After solvingAu=Bby using the sparse identification algorithm,we learned the unknown coefficients,and discussed the degree of fitting.The results showed that the method is applied to solve the coefficients which are sufficiently accurate to the true functions,and all the learned coefficients are greatly converge to the true coefficients with the increase of data length under a small time step.Thus,this method can be well utilized to the deterministic hybrid energy harvesting system.

A hybrid energy harvester under both harmonic excitation and Gaussian white noise was the second example. Firstly,we received the approximated equations of the drift term and diffusion term based on the Kramers–Moyal formulas. According to the basis function dictionary,we calculated the iterative expressions of the drift term and diffusion term, respectively.Together with the 10-fold CV, the sparse identification algorithm was used to obtain the number of optimal non-zero coefficientsn=9,n=2,n=2 andn=1 corresponding to the functions ofb2G,b3G,b4Ganda22, respectively. According to the comparison of learned functions and true functions,the results demonstrated that the method is well applied to the hybrid energy harvesting system with an acceptable deviation.Meanwhile, compared with the first example, the second example depends on more sufficient data to reduce the effect of the noise.

Thus,measuring the time-series data of the system state,we can build the equations for the hybrid energy harvester.Then,the learned system can be applied to explore the subsequent dynamical behavior with the aim of the improvement of performance of the energy harvester.

Acknowledgements

Project supported by the National Natural Science Foundation of China (Grant Nos. 12002089 and 11902081) and Project of Science and Technology of Guangzhou (Grant No.202201010326).