Nonlinear Dynamics

, Volume 92, Issue 3, pp 885–903 | Cite as

Two-scale command shaping for feedforward control of nonlinear systems

  • J. Justin Wilbanks
  • Christopher J. Adams
  • Michael J. Leamy
Original Paper


This paper presents a critical analysis of the two-scale command shaping (TSCS) feedforward control technique as applied to nonlinear Duffing oscillators. TSCS is an approach for tailoring a flexible system’s applied control input to reduce undesirable residual vibrations using multiple problem scales, command shaping of a linear subproblem, and cancelation of a remaining nonlinear subproblem. As shown herein, TSCS proves to be an effective method for feedforward control of nonlinear systems. The strategy outperforms conventional and nonlinearly informed command shaping strategies in traditional and non-traditional Duffing systems (e.g., Duffing systems with quadratic nonlinearity and Coulomb damping). The TSCS approach is also extended herein to nonlinear systems with uncertain parameters through the implementation of robust command shaping strategies and the extended Kalman filtering parameter estimation technique.


Command shaping Perturbation Multiple scale Feedforward control Nonlinear Kalman filtering 



This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1148903.


  1. 1.
    Daqaq, M.F., Masoud, Z.N.: Nonlinear input-shaping controller for quay-side container cranes. Nonlinear Dyn. 45(1–2), 149–170 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Singhose, W.: Command shaping for flexible systems: a review of the first 50 years. Int. J. Precis. Eng. Manuf. 10(4), 153–168 (2009)CrossRefGoogle Scholar
  3. 3.
    Daqaq, M., Reddy, C., Nayfeh, A.: Input-shaping control of nonlinear MEMS. Nonlinear Dyn. 54(1–2), 167–179 (2008)CrossRefzbMATHGoogle Scholar
  4. 4.
    Daqaq, M.F.: Adaptation of nontraditional control techniques to nonlinear micro and macro mechanical systems. Virginia Polytechnic Institute and State University (2006)Google Scholar
  5. 5.
    Canova, M., Guezennec, Y., Yurkovich, S.: On the control of engine start/stop dynamics in a hybrid electric vehicle. J. Dyn. Syst. Meas. Control Trans. ASME 131(6), 1–12 (2009)CrossRefGoogle Scholar
  6. 6.
    Park, K., Lee, J., Park, J.: Torque control of a vehicle with electronic throttle control using an input shaping method. Int. J. Autom. Technol. 14(2), 189–194 (2013)CrossRefGoogle Scholar
  7. 7.
    Wilbanks, J.J., Leamy, M.J.: Two-scale command shaping for reducing powertrain vibration during engine restart. J. Dyn. Syst. Meas. Control 139, 091004 (2017)CrossRefGoogle Scholar
  8. 8.
    Singer, N.C., Seering, W.P.: Preshaping command inputs to reduce system vibration. J. Dyn. Syst. Meas. Control 112(1), 76–82 (1990)CrossRefGoogle Scholar
  9. 9.
    Vaughan, J., Yano, A., Singhose, W.: Comparison of robust input shapers. J. Sound Vib. 315(4), 797–815 (2008)CrossRefGoogle Scholar
  10. 10.
    Singhose, W., Seering, W., Singer, N.: Residual vibration reduction using vector diagrams to generate shaped inputs. J. Mech. Des. 116(2), 654–659 (1994)CrossRefGoogle Scholar
  11. 11.
    Vaughan, J., Singhose, W.: Reducing multiple modes of vibration by digital filtering and input shaping. In: ASME 2010 Dynamic Systems and Control Conference. American Society of Mechanical Engineers (2010)Google Scholar
  12. 12.
    Smith, O.J.: Posicast control of damped oscillatory systems. Proc. IRE 45(9), 1249–1255 (1957)CrossRefGoogle Scholar
  13. 13.
    Smith, O.J.: Feedback control systems. McGraw-Hill, New York, NY (1958)Google Scholar
  14. 14.
    Singer, N.C.: Residual vibration reduction in computer controlled machines. (1989)
  15. 15.
    Singhose, W., Seering, W., Singer, N.C.: Time-optimal negative input shapers. J. Dyn. Syst. Meas. Control 119(2), 198–205 (1997)CrossRefzbMATHGoogle Scholar
  16. 16.
    Pao, L.Y., Singhose, W.E.: On the equivalence of minimum time input shaping with traditional time-optimal control. In: Proceedings of the 4th IEEE Conference on Control Applications. IEEE (1995)Google Scholar
  17. 17.
    Alsop, C., Forster, G., Holmes, F.: Ore unloader automation—a feasibility study. In: Proceedings of IFAC on Systems Engineering for Control Systems, pp. 295–305 (1965)Google Scholar
  18. 18.
    Singhose, W., Kim, D., Kenison, M.: Input shaping control of double-pendulum bridge crane oscillations. J. Dyn. Syst. Meas. Control 130(3), 034504 (2008)CrossRefGoogle Scholar
  19. 19.
    Singhose, W., et al.: Effects of hoisting on the input shaping control of gantry cranes. Control Eng. Pract. 8(10), 1159–1165 (2000)CrossRefGoogle Scholar
  20. 20.
    Vaughan, J., Kim, D., Singhose, W.: Control of tower cranes with double-pendulum payload dynamics. IEEE Trans. Control Syst. Technol. 18(6), 1345–1358 (2010)Google Scholar
  21. 21.
    Popa, D.O. et al.: Dynamic modeling and input shaping of thermal bimorph MEMS actuators. In: IEEE International Conference on Robotics and Automation. Proceedings, ICRA’03. IEEE (2003)Google Scholar
  22. 22.
    Togai, K., Platten, M.: Input torque shaping for driveline NVH improvement and torque profile approximation problem with combustion pressure. In: Proceedings of the FISITA 2012 World Automotive Congress. Springer (2013)Google Scholar
  23. 23.
    Singhose, W., Derezinski, S., Singer, N.: Extra-insensitive input shapers for controlling flexible spacecraft. J. Guid. Control Dyn. 19(2), 385–391 (1996)CrossRefGoogle Scholar
  24. 24.
    Sorensen, K.L.: Operational Performance Enhancement of Human Operated Flexible Systems. Georgia Institute of Technology, Atlanta (2008)Google Scholar
  25. 25.
    Bradley, T.H., et al.: Command shaping under nonsymmetrical acceleration and braking dynamics. J. Vib. Acoust. 130(5), 054503 (2008)CrossRefGoogle Scholar
  26. 26.
    Blackburn, D., et al.: Command shaping for nonlinear crane dynamics. J. Vib. Control 16(4), 477–501 (2010)CrossRefzbMATHGoogle Scholar
  27. 27.
    Smith, J.Y., Kozak, K., Singhose, W.E.: Input shaping for a simple nonlinear system. In: American Control Conference, 2002. Proceedings of the 2002. IEEE (2002)Google Scholar
  28. 28.
    Duffing, G.: Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung. Vieweg & Sohn, Braunschweig (1918)zbMATHGoogle Scholar
  29. 29.
    Kovacic, I., Brennan, M.J.: The Duffing Equation: Nonlinear Oscillators and Their Behaviour. Wiley, Hoboken (2011)CrossRefzbMATHGoogle Scholar
  30. 30.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, Hoboken (2008)zbMATHGoogle Scholar
  31. 31.
    Kevorkian, J., Cole, J.D.: Multiple Scale and Singular Perturbation Methods, vol. 114. Springer, Berlin (2012)zbMATHGoogle Scholar
  32. 32.
    Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, Hoboken (2011)zbMATHGoogle Scholar
  33. 33.
    Singh, T., Vadali, S.: Robust time-delay control. J. Dyn. Syst. Meas. Control 115(2A), 303–306 (1993)CrossRefzbMATHGoogle Scholar
  34. 34.
    Singhose, W.E., Porter, L.J., Singer, N.C.: Vibration reduction using multi-hump extra-insensitive input shapers. In: Proceedings of the American Control Conference. American Automatic Control Council (1995)Google Scholar
  35. 35.
    Chowdhary, G., Jategaonkar, R.: Aerodynamic parameter estimation from flight data applying extended and unscented Kalman filter. Aerosp. Sci. Technol. 14(2), 106–117 (2010)CrossRefGoogle Scholar
  36. 36.
    McGee, L.A., Schmidt, S.F.: Discovery of the Kalman filter as a practical tool for aerospace and industry. NASA, NASA Ames Research Center; Moffett Field, CA, United States. (1985)
  37. 37.
    Gelb, A.: Applied Optimal Estimation. MIT Press, Cambridge (1974)Google Scholar
  38. 38.
    Jategaonkar, R., Plaetschke, E.: Estimation of aircraft parameters using filter error methods and extended Kalman filter. Forschungsbericht- Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt (1988)Google Scholar
  39. 39.
    Grewal, M., Andrews, A.: Kalman Filtering: Theory and Practice Using MATLAB. Wiley, New York (2001)zbMATHGoogle Scholar
  40. 40.
    Julier, S.J., Uhlmann, J.K.: Unscented filtering and nonlinear estimation. Proc. IEEE 92(3), 401–422 (2004)CrossRefGoogle Scholar
  41. 41.
    Leonard, A.: Vehicle Tracking Using Ultra-Wideband Radar. Georgia Institute of Technology, Atlanta (2016)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • J. Justin Wilbanks
    • 1
  • Christopher J. Adams
    • 1
  • Michael J. Leamy
    • 1
  1. 1.George W. Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations