Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 3, pp. 605-616, Warsaw 2015 DOI: 10.15632/jtam-pl.53.3.605 LINEAR MATRIX INEQUALITIES CONTROL DRIVEN FOR NON-IDEAL POWER SOURCE ENERGY HARVESTING Douglas C. Ferreira UFMT – Federal University of Mato Grosso, Mechanical Engineering Department, Rondonópolis, MT, Brazil e-mail: dcferreira@ufmt.br Fábio R. Chavarette UNESP – Universidade Estadual Paulista, Mathematical Sciences Department, Ilha Solteira, SP, Brazil e-mail: fabioch@mat.feis.unesp.br Nelson J. Peruzzi UNESP – Universidade Estadual Paulista, Mathematical Sciences Department, Jaboticabal, SP, Brazil e-mail: peruzzi@fcav.unesp.br The dynamic model of a linear energy harvester excited by a non-ideal power source is coupled to a controller to maximum vibration adjustment. Numerical analysis is taken to evaluate the energy harvested keeping the vibration optimized for themaximum interaction to the energy source using linear matrix inequalities for control driven. The dimensionless power output, actuation power and net output power is determined. As a result, it is po- ssible to verify that the total energy harvested via exogenous vibration using the proposed controller is increased up to 65 times when in comparison to the open loop system. Keywords: energy harvesting, efficiency, control, non-ideal excitation 1. Introduction Harnessing energy from environment to supply low power devices can be accomplished from sources as thermal gradients, solar radiation and vibration (Huesgen et al., 2008; Cepnik et al., 2011; Miller et al., 2011). Concerning the vibration there are three main groups of harvesters: electrostatic, electromagnetic and piezoelectric (Roundy et al., 2003). Although harvesting to be a sustainable source of energy, it is still not efficient enough and its use is limited to very little power devices, and enhancing its capacity represents a science frontier.Theenergyharvesting systemmainapplication in thepresent is for remote sensornodes, wireless systems and smart structures actuators (Miller et al., 2011; Roundy et al., 2003). To enhance energy harnessed, there are several project solutions focusing in matching the natural frequency according to the vibration source, nevertheless the resonance solution has restrictions for power source and scale. Regarding the power source, an important consideration is that the environmental vibration has low power andwide range, and normally is random resulting in difficulties for a design to be coincident with the resonance. Considering the scale, it is notable that recent studies concerning the enhancement of energy harvesting efficiency produced better piezoelectric materials (Yeager andTrolier-Mckinstry, 2012; Baek et al., 2011;Kim et al., 2012) andnow thedesigned harvesters are facingnewbarrier applications ofmicro sizing.Asmostpower sources fromambientvibration have low frequency, it is difficult to design a small harvester resulting in his natural frequency to be coincident with the resonance. 606 D.C. Ferreira et al. Some project solutions regard to tuning a device to the resonance (Challa et al., 2008; Eichhorn et al., 2009) that can be accomplished by mechanical, magnetic and piezoelectric ad- justments (Peters et al., 2009; Tang et al., 2013). The mechanical adjustment requires known excitation frequency and high cost energy to tuning, magnetic adjustment has limited tuning result and faces a limited micro sizing requirements, and the piezoelectric adjustment is a pro- mising frontier to enhance energy harvesting capability (Zhu et al., 2010). First studies using piezoelectric adjustment results in negative net output power (Roundy and Zhang, 2005) but latter studies from Zhu et al. (2010) found amistake in Roundy and Zhang (2005) formulation that not consideredmean voltage to active power determination. A piezoelectric tuning solution withpositive net outputpowerwasnumerically and experimentally provenbyLallart and Inman (2010) and Lallart andGuyomar (2010). According toWang and Inman (2012), the state of the art of enhancement of energy harvesting capability relies in control projects formulation which characterizes the mean objective of this study. The proposed solution considers that vibration excitation has low energy and the vibration generated by the harvester can influence the sourcewhich characterizes a non-ideal power source (Balthazar et al., 2003; Balthazar and Dantas, 2004; Chavarette, 2012). For a limited power in controller systems, the input vibration control influences the own controller which characterizes a non-ideal system (Souza et al., 2008). Thus the model of movement sums the feedback term increasing thenumberof degrees of freedom(Piccirillo et al., 2008). According toBalthazar et al. (2003), when the vibration source is near natural frequency there appears a jump phenomenon which it is not possible while arriving the resonance as the maximum vibration response. In this study, the proposed controller is based on Linear Matrix Inequalities (LMI) theory. The dynamic systems control for vibrationmaximization is performed by optimum controlH∞ as a convex optimization problem involving Linear Matrix Inequalities – LMI (Chilali and Ga- hinet, 1996). The LMI can be in form of linear inequalities, linear convex inequalities or matrix inequalities. Several restriction formats of control theory as Lyapunov and Riccati inequalities can bedescribed via LMI (Antwerp andBraatz, 2000). TheLMI has large use in control, mainly because sustains the stability of a system based in restrictions and can incorporated in dynami- cal systems with not singular parameters that vary within a known range (Wan and Kothare, 2003; Andrea et al., 2008). To explore controller efficiency, this study is conducted according to a bimorph energy ha- rvester (Erturk and Inman, 2011) and the use of non-ideal power source (Balthazar et al., 2003; Balthazar and Dantas, 2004; Chavarette, 2012). The definition of a Linear Matrix Inequali- ties controller (LMI) to set up a harvester to optimize interaction to the power source to take advantage of full range of vibration is the main propose of this study. Considering the paper organization: Section 2 presents the harvester model, Section 3 pre- sents the controller definition, Section 4 presents the efficiency analysis and Section 5 presents the final remarks from this investigation and acknowledgements. 2. Harvester model Anon-ideal excitation can bemodeled for an unbalancedmass receiving torque by aDCmotor as shown in Fig. 1 (Balthazar et al., 2003; Balthazar and Dantas, 2004; Chavarette, 2012). This arrangement is based in Kononenkomodel and can bemathematically modeled accor- ding to equations (2.1) wherem1 is DCmotor mass,m0 is unbalanced mass, l is dumping, k is rigidity, r is the eccentricity distance from the unbalancedmass to the torque source, (I+m0r 2) is moment of inertia of the unbalanced mass, and the state variables are X for the beam tip position andϕ for the unbalancedmass angular position.The net torque is a function of angular Linear matrix inequalities control driven... 607 velocity ϕ̇ and described forS(ϕ̇)= a−bϕ̇ (Palácios et al., 2003; Tusset et al., 2013), where a is the net torque applied by the DCmotor and b is the resistive net torque constant (m1+m0)Ẍ+ lẊ+kX =m0r(ϕ̇ 2cosϕ+ ϕ̈sinϕ) (I+m0r 2)ϕ̈−m0rẌ sinϕ=S(ϕ̇) (2.1) Fig. 1. Non-ideal power source (Chavarette, 2012) Fig. 2. Energy harvester system (Erturk and Inman, 2011) coupled to non-ideal power source (Chavarette, 2012) An energy harvester defined by Erturk and Inman (2011) with two layers of a piezoelectric material and an output voltage was coupled to the non-ideal power source as shown in Fig. 2. It is dimensionlessly modeled according to the resonance as given by equations (2.2), where ζ is damping factor, χ is piezoelectric mechanical coupling coefficient, Λ is reciprocal of time constant, κ is piezoelectric electric coupling coefficient, µ is unbalanced mass eccentricity, ξ is unbalancedmass eccentricity formoment of inertia,α is the net torque applied by theDCmotor and β is the resistive net torque constant. The dimensionless state variables are x for tip beam position, z for the angular position of the unbalanced mass and ν for the output voltage ẍ+2ζẋ+ 1 2 x−χν =µ(ż2cosz+ z̈ sinz) z̈= ξẍsinz+α−βż ν̇+Λν+κẋ=0 (2.2) Isolating ẍ, ν̇ and z̈, equation (2.2) can be presented by ẍ= −1 2 x−2ζẋ+χν+µż2cosz+(α−βż)µsinz 1−µξ(sinz)2 z̈= ( −1 2 x−2ζẋ+χν ) ξ sinz+µξż2cosz sinz+α−βż 1−µξ(sinz)2 ν̇ =−κẋ−Λν (2.3) 608 D.C. Ferreira et al. Adopting x = x1, z = x3 and ν = x5 and rearranging the terms, the space-state form of equations (2.3) is given by ẋ1 =x2 ẋ2 = −1 2 x1−2ζx2+χx5+µx 2 4cosx3+(α−βx4)µsinx3 1−µξ(sinx3) 2 ẋ3 =x4 ẋ4 = ( −1 2 x1−2ζx2+χx5 ) ξ sinx3+µξx 2 4cosx3 sinx3+α−βx4 1−µξ(sinx3)2 ẋ5 =−κx2−Λx5 (2.4) Applying Jacobian, the matrix form of equations (2.4) is given as              ẋ1 ẋ2 ẋ3 ẋ4 ẋ5              =        0 1 0 0 0 −1 2 −2ζ µα 0 χ 0 0 0 1 0 0 0 −1 2 ξ −β 0 0 −κ 0 0 −Λ                     x1 x2 x3 x4 x5              (2.5) The main propose of this study is to design a controller to optimize vibration response and, consequently, energy the harvesting performance. In this case, a LinearMatrix Inequalities controller is arranged to maximize the displacement and velocity. As the output voltage has directly influenced by velocity and displacement, it is expected to increase the output voltage and, then, the efficiency. 3. Controller definition To ensure the maximum energy harvested, the controller it is coupled using LinearMatrix Ine- qualities (LMI) according to Optimum Control H∞. An LMI Control Driven utilizes feedback state to optimize the interaction between exogenous the excitation w(t) and the resulting si- gnal y(t) (Andrea et al., 2008). A sufficient condition for LMI control is the existence of a matrix X=X′ ∈Rn×n andY∈Rm×n satisfying equations minµ ∣ ∣ ∣ ∣ ∣ ∣ ∣ AX+XA′−B2Y−Y ′B′2 XC ′+Y′D′ B1 CX+DY −I 0 B′1 0 −µI ∣ ∣ ∣ ∣ ∣ ∣ ∣ <0 |X|> 0 (3.1) The space-statemodel for the proposed controlled system is given by equations (3.2), and expla- ined in a schematic flow chart in Fig. 3 ẋ=Ax+B1w+B2u y=Cx (3.2) where A is the state matrix, B1 is the excitation vector, B2 is the feedback state vector and C is the actuation vector and control signal, in this case, a singular control signal. Theprevious study conductedbyErturkand Inman (2011) definedparameters for abimorph cantilever according to the resonance ζ = 0.01, χ = 0.05, κ = 0.5, Λ = 0.05, and the studies conducted by Balthazar et al. (2003), Balthazar and Dantas (2004) and Chavarette (2012) Linear matrix inequalities control driven... 609 Fig. 3. Dynamic control scheme defined the parameters for non-ideal excitation µ = 0.2, ξ = 0.3 and β = 1.5. Replacing the parameters in the state matrix according to equation (2.5) the matrices for equation (3.2) are shown as in the following A=        0 1 0 0 0 −0.5 −0.02 0.2α 0 0.05 0 0 0 1 0 0 0 −0.15 −1.5 0 0 −0.5 0 0 −0.05        B1 =        1 1 1 1 1        B)2 =        1 1 1 1 1        C= [ 1 0 0 0 0 ] (3.3) 4. Efficiency analysis For the designed controller the initial parameters are used to excite the beam, and then a LMI controller is set up to the maximum interaction between the excited beam and the external exogenous excitation, in this case, a non-ideal power source.When LMIs are feasible, there is a matrixX and a feedbackmatrixL for the space-state that optimizes the systembehavior, given by equation (3.4) L=YX−1 (4.1) OnceL is determined, it is possible to determine a feedback space-statematrix for the system (Af), given by equation (3.5) Af =A−B2L (4.2) The variable parameter is the dimensionless net torque applied by theDCmotorα that will assume values ranging from 0.4 to 5.0. 4.1. System response for α=0.4 Consideringα=0.4 and substitutingmatrices A,B1,B2 andC given in equations (3.3) in LMIs given in equations (3.1) and solving the inequalities, the resulting feedback matrix L for a feasible system is L=103 [ 6.1648 0.0004 −0.0000 −0.0007 −0.0000 ] Taking into consideration the feedback state matrixAf, it is possible to calculate the feedback parameters, comparing to the original matrixA in (3.3) as shown in Table 3. 610 D.C. Ferreira et al. Table 3. Feedback parameters for α=0.4 Feedback ζ χ κ Λ µ ξ β parameters Value 0.2308 0.0749 0.9416 0.0251 0.2988 0.2210 0.8244 Table 4. System eigenvalues (103) for α=0.4 λ1 −6.1635+0.0000i λ2 −0.0015+0.0000i λ3 −0.0005+0.0002i λ4 −0.0005−0.0002i λ5 −0.0001+0.0000i The controlled system exhibits stable behavior since all the eigenvalues have the real part negative as shown in Table 4. Performing Runge-Kutta fourth order algorithm for solving the ordinary differential equ- ations in equation (2.4) for open loop (without control) parameters given in Section 3 and for controlled system parameters given in Table 3, it is possible to visualize the total energy from the system to compare the efficiency from the open loop to the controlled system as shown in Fig. 4. Th following initial conditions are considered x(0) = 1, ẋ(0) = 0, ν(0) = 0, z(0) = 0 and ż(0) = 0 and time samples from 0 to 2500 in the interval of 0.1 totalizing 25000 samples in Figs. 4a-4d and samples from 2000 to 2500 in the interval of 0.1 totalizing 5000 samples in Fig. 4d to exclude transient behavior. The controlled system presents greater displacements, velocity and output voltage in com- parison to the open loop system. To explore control performance, the dimensionless net torque applied by theDCmotor assumes the valuesα=0.5,α=1.3 andα=5.0, considering parame- ters below the resonance, at the resonance and beyond it, the feedback vector L and feedback parameters are given in Table 5. Table 5. Feedback parameters α=0.5, α=1.3 and α=4.0 α L (103) Feedback parameters 0.5 [4.7795,0.0004,−0.0000,−0.0007,−0.0000] ζ =0.2304, χ=0.0742, κ=0.9408 Λ=0.0258, µ=0.2689, ξ=0.2311 β=0.8449 1.3 [2.2552,0.0005,−0.0000,−0.0007,−0.0000] ζ =0.2537, χ=0.0691, κ=0.9873 Λ=0.0309, µ=0.2173, ξ=0.2551 β=0.8212 5.0 [2.4482,0.0006,0.0000,−0.0007,−0.0000] ζ =0.3096, χ=0.0639, κ=1.0991 Λ=0.0361, µ=0.1957, ξ=0.3425 β=0.8231 Making use of the Runge-Kutta fourth order method for solving ordinary differential equ- ations in (2.4) foropen loop (without control) parameters given inSection3and for thecontrolled system parameters given in Table 5we find analogous results for the initial conditions x(0)= 1, ẋ(0) = 0, ν(0) = 0, z(0) = 0 and ż(0) = 0 and time samples from 0 to 2500 in the interval of 0.1 totalizing 25000 samples in Figs. 5a, 5c, 5e and samples from 2000 to 2500 in the interval of 0.1 totalizing 5000 samples to exclude transient behavior in Figs. 5b, 5d, 5f. It is possible to verify that the system energy is greater for the control system when out of the resonance forα=0.5 andα=5.0.When the system is at the resonant behavior atα=1.3, Linear matrix inequalities control driven... 611 Fig. 4. System behavior for α=0.4 (open loop system – grey line, controlled system – black line); (a) displacement, (b) velocity, (c) output voltage, (d) phase portrait time sample and (e) phase portrait without transient it is not possible for the controller to get more energy than the resonance, and the controlled system gives less energy than the open loop system. According Tang and Zuo (2011), the system output power (P) is given by equation (4.3) related to theoutputvoltage rootmean square (Vrms) squared,dividedby the load resistance (R) P = V 2rms R (4.3) As the model is dimensionless, a dimensionless output power (Φ) given for the root mean square of the dimensionless output voltage (νrms) squared, divided by the dimensionless load resistance (Ψ), as shown in (4.4), is considered Φ= ν2rms Ψ (4.4) For the propose of numerical simulation, a value of the load resistance Ψ = 0.1 and for α ranging from0.5 to 5.0 the dimensionless output powerΦ values given inTable 6 are assumed. 612 D.C. Ferreira et al. Fig. 5. System behavior (open loop system – grey line, controlled system – black line); (a), (c), (e) output voltage, (b), (d), (f) phase portrait without transient Table 6.Dimensionless power Φ for α ranging from 0.4 to 5.0 Net torque (α) Φ – Open loop Φ – Controlled 0.4 0.0247 0.1695 0.5 0.0250 0.3040 1.3 16.0565 0.4015 5.0 0.0953 0.2995 Expanding the investigation for other values of thenet torque (α), the resultingdimensionless power (Φ) is given in Fig. 6. The LMI control show the efficiency of enhancing the system dimensionless output power as shown inFig. 6 andnot considering the resonance effectwhere the dimensionless output power is greater in the open loop system. This fact is explained because the higher energy orbit available in the system is the resonance behavior. To better visualize the control efficiency, the resonance net torque is shown in Fig. 7. Linear matrix inequalities control driven... 613 Fig. 6. Dimensionless output power (Φ) of the open loop compared to the controlled system; open loop system – grey line, controlled system – black line Fig. 7. Dimensionless output power (Φ) of the open loop compared to the controlled system; open loop system – grey line, controlled system – black line. Suppressing the resonance behavior To evaluate the dimensionless net output power (Φnet), Roundy and Zang (2005) compared the frequency of the open loop system to the controlled system, and later Lallart and Inman (2010) used the same evaluation to compare the output power from the open loop and controlled systems. In both cases, the power used for control (action power) is referred to the amount of change of the system parameters, in other words, it is the energetic cost to change the system behavior. Based on experimentally proven evaluations, the action power (Φact) is determined by applying the Runge-Kutta fourth order method to solve the ordinary differential equations in (2.4) for the absolute difference of the open loop and controlled parameters of each net torque (α). The results are shown in Fig. 8. Fig. 8. Output power for the open loop system (grey line); output power for the controlled system (black line); actuation power (dotted line); net output power (solid dark grey line) To better visualize the control efficiency, the resonance net torque is depicted in Fig. 9. The summary of power gains resulting from the control is shown in Table 7. 614 D.C. Ferreira et al. Fig. 9. Output power for the open loop system (grey line); output power for the controlled system (black line); actuation power (dotted line); net output power (solid dark grey line). Suppressing the resonance behavior Table 7. Summary of dimensionless net power gains for the controlled system α 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Net power gain 40.69 59.67 65.85 31.70 14.99 3.54 0.77 0.13 0.04 0.02 α 1.4 1.5 1.6 1.7 1.8 2.0 3.0 4.0 5.0 Net power gain 0.68 0.99 0.82 1.74 1.64 2.03 2.28 2.65 2.59 5. Final remarks Various studies regarding the energy harvesting have been carried out for the last few years to enhance the capability of the energy harvesters. In the samedirection, this investigation presents a study of control of the dynamic behavior based on Linear Matrix Inequalities Optimum H∞ method and a non-ideal excitation as the power source. As themain result, it has been possible to verify a significant increase in the system energy available for harvesting when applying a controller to the system. Velocity and displacement have direct effect on the output voltage and, consequently, on the output power for energy harvesting. The proposed controller is able to increase the net output power up to 65 times. Table 8 shows the comparison of other studies with the present research. Table 8.Comparison of the obtained results with other works Author Resulting frequency band [Hz] Roundy and Zhang (2005) 61.8-67.0 Wu et al. (2006) 91.0-94.5 Peters et al. (2009) 66.0-89.0 Lallart and Inman (2010) 8.1-112.0 This work all the band Whereas other energy harvestingmethods, suchas piezoelectric tuning, increase the resonant band, this control approach enhances energy not regarding the band but focusing on the output power optimization with considerable efficiency for small power excitations. In this study, the parameters are considered constant, however in a real situation, damping and electrical coupling parameters depend on environmental conditions, for example humidity and temperature. Such variation in the parameters results in polytopic uncertainties. Thus, instead of an optimum method, a robust LMI method is required (Andrea et al., 2008) which characterizes possible research continuation. The energy sink due to controller circuit is result Linear matrix inequalities control driven... 615 of the electronic architecture and is not explored in this study as another possible research exploration area. Acknowledgments The authors would like to thank CNPq (Proc. No. 301769/2012-5) and Federal University of Mato Grosso (UFMT) for funding this research. References 1. Andrea C.Q., Pinto J.O.P., Assunção E., Teixeira M.C.M., Galotto L.J., 2008, Opti- mumcontrolH∞ of non-linear systemswith fuzzyTakagi-Sugenomodels (inPortuguese),Controle and Automação, 19, 3 2. Antwerp J.G.,BraatzR.D., 2000,A tutorial on linear andbilinearmatrix inequalities, Journal of Process Control, 10, 363-385 3. Baek S.H., Park J., Kim D.M., Aksyuk V.A., Das R.R., Bu S.D., Felker D.A., et al., 2011, Giant piezoelectricity on Si for hyperactiveMEMS, Science, 334 4. Balthazar J.M., Dantas M.J.H., 2004, On local analysis of oscillations of a non-ideal and non-linear mechanical model,Meccanica, 39, 4, 313-330 5. Balthazar J.M., Mook D.T., Weber H.I., Brasil R.M.L.R.F., Fenili A., Belato D., Felix P.J.L., 2003, An overview on non-ideal vibrations,Meccanica, 38, 613-621 6. Cepnik C., Radler O., Rosenbaum S., Ströhla T., Wallrabe U., 2011, Effective opti- mization of electromagnetic energy harvesters through direct computation of the electromagnetic coupling, Sensors and Actuators, 167, 416-421 7. Challa V.R., Prasad M.G., Shi Y., Fisher F.T., 2008, A vibration energy harvesting device with bidirectional resonance frequency tenability, Smart Materials and Structures, 17 8. Chavarette F.R., 2012, On an optimal linear control of a chaotic non-ideal Duffing system, Advances in Mechanical Engineering, 2, 3 9. Chilali M., Gahinet P., 1996, H∞ design with pole placement constrains: an LMI approach, IEEE Transactions on Automatic Control, 41, 3 10. Eichhorn C., Goldschmidtboeing F., Porro Y., Woias P., 2009, A piezoelectric harvester with an integrated frequency tuning mechanism, PowerMEMS, Washington DC, USA, December 1-4 11. Erturk A., Inman D., 2011, Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling, Journal of Sound and Vibration, 330, 2339-2353 12. Huesgen T., Woias P., Kockmann N., 2008, Design and fabrication of MEMS thermoelectric generators with high temperature efficiency, Sensors and Actuators A, 423-429 13. Kim S., Leung A., Koo C.Y., Kuhn L., Jiang W., Kim D., Kingon A.I., 2012, Lead- free (Na0.5K0.5)(Nb0.95Ta0.05)O3-BiFeO3 thin films for MEMS piezoelectric vibration energy harvesting devices,Materials Letters, 69, 24-26 14. Lallart M., Guyomar D., 2010, Piezoelectric conversion and energy harvesting enhancement by initial energy injection,Applied Physics Letters, 97 15. Lallart M., Inman D.J., 2010, Frequency self-tuning scheme for broadband vibration energy harvesting, Journal of Intelligent Material Systems and Structures, 21, 897-906 16. Miller L.M., Halvorsen E., Dong T., Wright P.K., 2011,Modeling and experimental veri- fication of low-frequencyMEMS energy harvesting from ambient vibrations, Journal of Microme- chanics and Microengineering, 21, 045029, pp.13 616 D.C. Ferreira et al. 17. Palácios J.L.,BalthazarJ.M.,BrasilR.M.L.F.R., 2003,A shortnote onanonlinear system vibrations under two non-ideal excitations, Journal of the Brazilian Society ofMechanical Sciences and Engineering,XXV, 391-395 18. Peters C., Maurath D., Schock W., Mezger F., Manoli Y., 2009, A closed-loop wide- range tunablemechanical resonator for energy harvesting systems, Journal of Micromechanics and Microengineering, 19, pp.9 19. Piccirillo V., Balthazar J.M., Pontes Jr. B.R., Felix J.L.P., 2008, On a nonlinear and chaotic non-ideal vibrating system with shape memory alloy (SMA), Journal Of Theoretical and Applied Mechanics, 46, 597-620 20. Roundy S., Wright P.K., Rabaey J., 2003, A study of low level vibrations as a power source for wireless sensor nodes,Computer Communications, 26, 1131-1144 21. Roundy S., Zhang Y., 2005, Toward self-tuning adaptive vibration-basedmicrogenerators,Pro- ceedings of Smart Structures, Devices and Systems, 5649, 373-384 22. Souza S.L.T., Caldas I.L., Viana R.L., Balthazar J.M., 2008, control and chaos for vibro- impact and non-ideal oscillators, Journal of Theoretical and Applied Mechanics, 46, 641-664 23. Tang L., Yaowen Y., Soh C.K., 2013, Broadband vibration energy harvesting techniques, [In:] Advances in Energy Harvesting Methods (Chapter Two), N. Elvin and A. Erturk (Eds.), Springer Science &BusinessMedia, NewYork 24. Tang X., Zuo L., 2011, Enhanced vibration energy harvesting using dual-mass systems, Journal of Sound and Vibration, 330, 5199-5209 25. Tusset A.M., Balthazar J.M., Felix J.L.P., 2013, On elimination of chaotic behavior in a non-ideal portal frame structural system, using both passive and active controls, Journal of Vibration and Control, 19, 6, 803-813 26. WanZ.,KothareM.V., 2003,An efficient off-line formulation of robustmodel predictive control using linear matrix inequalities,Automatica, 39, 837-846 27. WangY., Inman J.D., 2012,A surveyof control strategies for simultaneous vibration suppression and energy harvesting via piezoceramics, Journal of Intelligent Material Systems and Structures, 23, 18, 2021-2037 28. Wu W.-J., Chen Y.-Y., Lee B.-S., He J.-J., Peng Y.-T., 2006, Tunable resonant frequency power harvesting devices,Proceedings of Smart Structures and Materials, 6169, 55-62 29. Yeager C.B., Trolier-Mckinstry S., 2012, Epitaxial Pb(Zrx,Ti1-x)O3 (0.30?x?0.63) films on (100)MgO substrates for energy harvesting applications, Journal of Applied Physics, 112 30. Zhu, D., Tudor, M. J., Beeby, S. P., 2010, Strategies for increasing the operating frequency range of vibration energy harvesters: a review,Measurement Science and Technology, 21, pp.29 Manuscript received February 5, 2014; accepted for print January 9, 2015