Journal of Optimization Theory and Applications

, Volume 179, Issue 3, pp 1025–1042 | Cite as

Optimal Control of Vibration-Based Micro-energy Harvesters

  • Thuy T. T. Le
  • Felix Jost
  • Sebastian SagerEmail author


We analyze the maximal output power that can be obtained from a vibration energy harvester. While recent work focused on the use of mechanical nonlinearities and on determining the optimal resistive load at steady-state operation of the transducers to increase extractable power, we propose an optimal control approach. We consider the open-circuit stiffness and the electrical time constant as control functions of linear two-port harvesters. We provide an analysis of optimal controls by means of Pontryagin’s maximum principle. By making use of geometric methods from optimal control theory, we are able to prove the bang–bang property of optimal controls. Numerical results illustrate our theoretical analysis and show potential for more than 200% improvement of harvested power compared to that of fixed controls.


Optimal control Pontryagin’s maximum principle Switching function Energy harvesting Power optimization 

Mathematics Subject Classification

49K15 49J30 93B40 



Financial support from the European Research Council via the Consolidator Grant MODEST-647573 is gratefully acknowledged. Thanks to Prof. Giovanni Colombo for his valuable suggestions and to Robert Rantz for proofreading the article.


  1. 1.
    Mitcheson, P.D., Yeatman, E.M., Rao, G.K., Holmes, A.S., Green, T.C.: Energy harvesting from human and machine motion for wireless electronic devices. Proc. IEEE 96(9), 1457–1486 (2008)CrossRefGoogle Scholar
  2. 2.
    Roundy, S., Wright, P.K., Rabaey, J.: A study of low level vibrations as a power source for wireless sensor nodes. Comput. Commun. 26(11), 1131–1144 (2003)CrossRefGoogle Scholar
  3. 3.
    Ottman, G., Hofmann, H., Bhatt, A., Lesieutre, G.: Adaptive piezoelectric energy harvesting circuit for wireless remote power supply. IEEE Trans. Power Electron. 17(5), 669–676 (2002)CrossRefGoogle Scholar
  4. 4.
    Renno, J.M., Daqaq, M.F., Inman, D.J.: On the optimal energy harvesting from a vibration source. J. Sound Vib. 320, 386–405 (2008)CrossRefGoogle Scholar
  5. 5.
    Mitcheson, P.D., Green, T.C., Yeatman, E.M., Holmes, A.S.: Architectures for vibration-driven micropower generators. J. Microelectromech. Syst. 13(3), 429–440 (2004)CrossRefGoogle Scholar
  6. 6.
    Halvorsen, E., Le, C.P., Mitcheson, P.D., Yeatman, E.M.: Architecture-independent power bound for vibration energy harvesters. J. Phys. Conf. Ser. 476(1), 012–026 (2013)Google Scholar
  7. 7.
    Heit, J., Roundy, S.: A framework to determine the upper bound on extractable power as a function of input vibration parameters. Energy Harvest. Syst. 3(1), 069–078 (2015)Google Scholar
  8. 8.
    Halvorsen, E.: Optimal load and stiffness for displacement-constrained vibration energy harvesters. ArXiv e-prints (2016)Google Scholar
  9. 9.
    Marboutin, C., Suzuki, Y., Kasagi, N.: Vibration-driven MEMS energy harvester with vertical electrets. In: Proceedings of the PowerMEMS, pp. 141–144. Freiburg, Germany (2007)Google Scholar
  10. 10.
    Erturk, A., Inman, D.J.: Parameter identification and optimization in piezoelectric energy harvesting: analytical relations, asymptotic analyses, and experimental validations. Proc. Inst. Mech. Eng. I J. Syst. Control Eng. 225(4), 485–496 (2011)Google Scholar
  11. 11.
    Le, C.P., Halvorsen, E., Søråsen, O., Yeatman, E.M.: Microscale electrostatic energy harvester using internal impacts. J. Intell. Mater. Syst. Struct. 23, 1409–1421 (2012)CrossRefGoogle Scholar
  12. 12.
    von Büren, T., Tröster, G.: Design and optimization of a linear vibration-driven electromagnetic micro-power generator. Sens. Actuators A 135(2), 765–775 (2007)CrossRefGoogle Scholar
  13. 13.
    Schättler, H., Ledzewicz, U.: Geometric Optimal Control: Theory, Methods and Examples, vol. 38. Springer, Berlin (2012)zbMATHGoogle Scholar
  14. 14.
    Bressan, A., Piccoli, B.: Introduction to the Mathematical Theory of Control, vol. 2. American Institute of Mathematical Sciences, Springfield (2007)zbMATHGoogle Scholar
  15. 15.
    Ledzewicz, U., Schättler, H.: Antiangiogenic therapy in cancer treatment as an optimal control problem. SIAM J. Control Optim. 46(3), 1052–1079 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    d’Onofrio, A., Ledzewicz, U., Maurer, H., Schättler, H.: On optimal delivery of combination therapy for tumors. Math. Biosci. 222(1), 13–26 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ledzewicz, U., Mosalman, M.S.F., Schättler, H.: Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discrete Contin. Dyn. Syst. Ser. B 18(4), 1031–1051 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Sager, S.: Sampling decisions in optimum experimental design in the light of Pontryagin’s maximum principle. SIAM J. Control Optim. 51(4), 3181–3207 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Grigorieva, E., Khailov, E.: Optimal control of a nonlinear model of economic growth. In: Discrete and Continuous Dynamical Systems, pp. 456–466 (2007)Google Scholar
  20. 20.
    Grigorieva, E.V., Khailov, E.N.: Hierarchical differential game between manufacturer, retailer, and bank. J. Dyn. Control Syst. 15(3), 359–391 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tilmans, H.A.C.: Equivalent circuit representation of electromechanical transducers: I. Lumped-parameter systems. J. Micromech. Microeng. 6, 157–176 (1996)CrossRefGoogle Scholar
  22. 22.
    Andersson, J., Åkesson, J., Diehl, M.: CasADI: a symbolic package for automatic differentiation and optimal control. In: Barth, T.J., Griebel, M., Keyes, D.E., Nieminen, R.M., Roose, D., Schlick, T. (eds.) Recent Advances in Algorithmic Differentiation, pp. 297–307. Springer, Berlin (2012)CrossRefGoogle Scholar
  23. 23.
    Hindmarsh, A.C., Brown, P.N., Grant, K.E., Lee, S.L., Serban, R., Shumaker, D.E., Woodward, C.S.: SUNDIALS: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. (TOMS) 31(3), 363–396 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Truong, B.D., Le, C.P., Halvorsen, E.: Power optimization and effective stiffness for a vibration energy harvester with displacement constraints. J. Micromech. Microeng. 26(12), 124,006 (2016)CrossRefGoogle Scholar
  26. 26.
    Le, C.P., Halvorsen, E.: MEMS electrostatic energy harvesters with end-stop effects. J. Micromech. Microeng. 22(7), 74013–74024 (2012)CrossRefGoogle Scholar
  27. 27.
    Williams, C.B., Yates, R.B.: Analysis of a micro-electric generator for microsystems. Sens. Actuators A 52(1–3), 8–11 (1996)CrossRefGoogle Scholar
  28. 28.
    Lankarani, H.M., Nikravesh, P.E.: Continuous contact force models for impact analysis in multibody systems. Nonlinear Dyn. 5(2), 193–207 (1994)Google Scholar
  29. 29.
    Babitsky, V., Krupenin, V.: Vibration of Strongly Nonlinear Discontinuous Systems Engineering Online Library. Springer, Berlin (2001)CrossRefGoogle Scholar
  30. 30.
    Manevitch, L.I., Gendelman, O.V.: Oscillatory models of vibro-impact type for essentially non-linear systems. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 222(10), 2007–2043 (2008)CrossRefGoogle Scholar
  31. 31.
    Soliman, M., Abdel-Rahman, E., El-Saadany, E., Mansour, R.: A design procedure for wideband micropower generators. J. Microelectromech. Syst. 18(6), 1288–1299 (2009)CrossRefGoogle Scholar
  32. 32.
    Hoffmann, D., Folkmer, B., Manoli, Y.: Fabrication, characterization and modelling of electrostatic micro-generators. J. Micromech. Microeng. 19(9), 094001 (2009)CrossRefGoogle Scholar
  33. 33.
    Liu, H., Tay, C.J., Quan, C., Kobayashi, T., Lee, C.: Piezoelectric mems energy harvester for low-frequency vibrations with wideband operation range and steadily increased output power. J. Microelectromech. Syst. 20(5), 1131–1142 (2011)CrossRefGoogle Scholar
  34. 34.
    Le, C.P., Halvorsen, E.: An electrostatic energy harvester with end-stop effects. In: Proceedings of the 22nd Micromechanics and Microsystem Technology Europe Workshop. Tønsberg, Norway (2011)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematical Optimization, Mathematical Algorithmic OptimizationOtto-von-Guericke UniversityMagdeburgGermany

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