Passivity-Based Control of Mechanical Systems

  • Romeo OrtegaEmail author
  • Alejandro Donaire
  • Jose Guadalupe Romero
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 473)


Stabilization of mechanical systems by shaping their energy function is a well-established technique whose roots date back to the work of Lagrange and Dirichlet. Ortega and Spong in 1989 proved that passivity is the key property underlying the stabilization mechanism of energy shaping designs and the, now widely popular, term of passivity-based control (PBC) was coined. In this chapter, we briefly recall the history of PBC of mechanical systems and summarize its main recent developments. The latter includes: (i) an explicit formula for one of the free tuning gains that simplifies the computations, (ii) addition of PID controllers to robustify and make constructive the PBC design and to track ramp references, (iii) use of PBC to solve the position feedback global tracking problem, and (iv) design of robust and adaptive speed observers.


Inertia Matrix Globally Exponentially Stable Gyroscopic Force Underactuated System Energy Shaping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



R. Ortega is supported by Government of Russian Federation (grant 074-U01, GOSZADANIE 2014/190 (project 2118)), the Ministry of Education and Science of Russian Federation (project 14.Z50.31.0031). The work of J.G. Romero is supported by a public grant overseen by the French National Research Agency (ANR) as part of the Investissement d’Avenir program, through the iCODE Institute, research project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02.


  1. 1.
    Acosta, J.A., Ortega, R., Astolfi, A., Mahindrakar, A.M.: Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one. IEEE Trans. Autom. Control 50(12), 1936–1955 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ailon, A., Ortega, R.: An observer-based controller for robot manipulators with flexible joints. Syst. Control Lett. 21(4), 329–335 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Astolfi, A., Karagiannis, D., Ortega, R.: Nonlinear and Adaptive Control with Applications. Springer, Berlin (2007)zbMATHGoogle Scholar
  4. 4.
    Astolfi, A., Ortega, R., Venkataraman, A.: A globally exponentially convergent immersion and invariance speed observer for mechanical systems with non-holonomic constraints. Automatica 46(1), 182–189 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Auckly, D., Kapitanski, L.: On the \(\lambda \)-equations for matching control laws. SIAM J. Control Optim. 41(5), 1372–1388 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Auckly, D., Kapitanski, L., White, W.: Control of nonlinear underactuated systems. Commun. Pure Appl. Math. 53(3), 354–369 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Blankenstein, G., Ortega, R., van der Schaft, A.J.: The matching conditions of controlled Lagrangians and interconnection and damping assignment passivity based control. Int. J. Control 75(9), 645–665 (2002)Google Scholar
  8. 8.
    Bloch, A., Leonard, N., Marsden, J.: Controlled Lagrangians and the stabilization of mechanical systems I: the first matching theorem. IEEE Trans. Autom. Control 45(12), 2253–2270 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bloch, A.M., Chang, D.E., Leonard, N., Marsden, J.E.: Controlled Lagrangians and the stabilization of mechanical systems II: potential shaping. IEEE Trans. Autom. Control 46(10), 1556–1571 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Byrnes, C., Isisdori, A., Willems, J.C.: Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Trans. Autom. Control 36(11), 1228–1240 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chang, D.E.: Generalization of the IDA–PBC method for stabilization of mechanical systems. In: The 18th Mediterranean Conference on Control and Automation, pp. 226–230. Marrakech, Morocco (2010)Google Scholar
  12. 12.
    Chang, D.E.: On the method of interconnection and damping assignment passivity-based control for the stabilization of mechanical systems. Regular Chaotic Dyn. 19(5), 556–575 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chang, D.E., Bloch, A.M., Leonard, N.E., Marsden, J.E., Woolsey, C.A.: The equivalence of controlled Lagrangian and controlled Hamiltonian systems for simple mechanical systems. ESAIM: Control Optim. Calc. Var. 8, 393–422 (2002)Google Scholar
  14. 14.
    Cisneros, R., Mancilla-David, F., Ortega, R.: Passivity-based control of a grid-connected small-scale windmill with limited control authority. IEEE J. Emerg. Sel. Top. Power Electr. 1(4), 2168–6777 (2013)Google Scholar
  15. 15.
    Crasta, N., Ortega, R., Pillai, H.: On the matching equations of energy shaping controllers for mechanical systems. Int. J. Control 88(9), 1757–1765 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dirksz, D., Scherpen, J.: Power-based control: Canonical coordinate transformations, integral and adaptive control. Automatica 48(6), 1045–1056 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Do, K.D., Pan, J.: Underactuated ships follow smooth paths with integral actions and without velocity measurements for feedback: theory and experiments. IEEE Trans. Control Syst. Technol. 14(2), 308–322 (2006)CrossRefGoogle Scholar
  18. 18.
    Do, K.D., Jiang, Z.P., Pan, J.: A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots. IEEE Trans. Robot. Autom. 20(3), 589–594 (2004)CrossRefGoogle Scholar
  19. 19.
    Donaire, A., Junco, S.: On the addition of integral action to port-controlled Hamiltonian systems. Automatica 45(8), 1910–1916 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Donaire, A., Mehra, R., Ortega, R., Satpute, S., Romero, J.G., Kazi, F., Singh, N.M.: Shaping the energy of mechanical systems without solving partial differential equations. IEEE Trans. Autom. Control 61(4), 1051–1056 (2016)Google Scholar
  21. 21.
    Donaire, A., Ortega, R., Romero, J.G.: Simultaneous interconnection and damping assignment passivity–based control of mechanical systems using dissipative forces. Syst. Control Lett. 94, 118–126 (2016)Google Scholar
  22. 22.
    Donaire, A., Romero, J.G., Ortega, R., Siciliano, B., Crespo, M.: Robust IDA–PBC for underactuated mechanical systems subject to matched disturbances. Int. J.Robust Nonlinear Control (to appear) (2017)Google Scholar
  23. 23.
    Duindam, V., Macchelli, A., Stramigioli, S., Bruyninckx, H. (eds.): Modeling and Control of Complex Physical System: The Port-Hamiltonian Approach. Springer, Berlin (2009)zbMATHGoogle Scholar
  24. 24.
    Fradkov, A.L.: Synthesis of an adaptive system for linear plant stabilization. Autom. Remote Control 35(12), 1960–1966 (1974)zbMATHGoogle Scholar
  25. 25.
    Fujimoto, K., Sugie, T.: Canonical transformations and stabilization of generalized Hamiltonian systems. Syst. Control Lett. 42(3), 217–227 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Grillo, S., Marsden, J.E., Nair, S.: Lyapunov constraints and global asymptotic stabilization. J. Geom. Mech. 3(2), 145–196 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hatanaka, T., Chopra, N., Fujita, M., Spong, M.W.: Passivity-Based Control and Estimation in Networked Robotics. Springer International, Cham, Switzerland (2015)CrossRefzbMATHGoogle Scholar
  28. 28.
    Hill, D., Moylan, P.: The stability of nonlinear dissipative systems. IEEE Trans. Autom. Control 25(5), 708–711 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jonckeere, E.: Lagrangian theory of large scale systems. In: European Conference on Circuit Theory and Design, The Hague, The Netherlands, pp. 626–629 (1981)Google Scholar
  30. 30.
    Kelly, R.: A simple set–point robot controller by using only position measurement. In: IFAC World Congress, Sydney, Australia, pp. 173–176 (1994)Google Scholar
  31. 31.
    Kelly, R., Ortega, R.: Adaptive control of robot manipulators: an input–output approach. In: IEEE International Conference on Robotics and Automation, PA, USA, pp. 699–703 (1988)Google Scholar
  32. 32.
    Kelly, R., Carelli, R., Ortega, R.: Adaptive motion control design of robot manipulators: an input-output approach. Int. J. Control 49(12), 2563–2581 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kelly, R., Ortega, R., Ailon, A., Loria, A.: Global regulation of flexible joints robots using approximate differentiation. IEEE Trans. Autom. Control 39(6), 1222–1224 (1994)CrossRefzbMATHGoogle Scholar
  34. 34.
    Koditschek, D.E.: Natural motion of robot arms. In: IEEE Conference on Decision and Control, Las Vegas, USA, pp. 733–735 (1984)Google Scholar
  35. 35.
    Koditschek, D.E.: Robot planning and control via potential functions. In: Khatib, O., Craig, J.J., Lozano-Pérez, T. (eds.) The Robotics Review 1, pp. 349–367. The MIT Press, Cambridge (1989)Google Scholar
  36. 36.
    Landau, I.D.: Adaptive Control: The Model Reference Approach. Marcel Dekker, New York (1979)zbMATHGoogle Scholar
  37. 37.
    Lewis, A.: Notes on energy shaping. In: IEEE Conference on Decision and Control, Paradise Island, Bahamas, pp. 4818–4823 (2004)Google Scholar
  38. 38.
    Loria, A.: Observers are unnecessary for output-feedback control of Lagrangian systems. IEEE Trans. Autom. Control 61(4), 905–920 (2016)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Mahindrakar, A.D., Astolfi, A., Ortega, R., Viola, G.: Further constructive results on IDA-PBC of mechanical systems: the Acrobot example. Int. J. Robust Nonlinear Control 16, 671–685 (2006)CrossRefzbMATHGoogle Scholar
  40. 40.
    Monroy, A., Alvarez-Icaza, L., Espinosa-Perez, G.: Passivity-based control for variable speed constant frequency operation of a DFIG wind turbine. Int. J. Control 81(9), 1399–1407 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Moylan, P., Anderson, B.D.O.: Nonlinear regulator theory and an inverse optimal control problem. IEEE Trans. Autom. Control 18(5), 460–465 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Nunna, K., Sassano, M., Astolfi, A.: Constructive interconnection and damping assignment for port-controlled Hamiltonian systems. IEEE Trans. Autom. Control 60(9), 2350–2361 (2015)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Nuño, E., Ortega, R., Basañez, L.: An adaptive controller for nonlinear teleoperators. Automatica 46(2), 155–159 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Ortega, R.: Passivity properties for stabilization of cascaded nonlinear systems. Automatica 27(2), 423–424 (1991)Google Scholar
  45. 45.
    Ortega, R.: Applications of input–output techniques to control problems. In: European Control Conference, Grenoble, France, pp. 1307–1313 (1991)Google Scholar
  46. 46.
    Ortega, R., Borja, P.: New results on control by interconnection and energy–balancing passivity–based control of port–Hamiltonian systems. In: IEEE Conference on Decision and Control, Los Angeles, USA, pp. 2346–2351 (2014)Google Scholar
  47. 47.
    Ortega, R., García-Canseco, E.: Interconnection and damping assignment passivity-based control: a survey. Eur. J. Control 10, 432–450 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Ortega, R., Romero, J.G.: Robust integral control of port-Hamiltonian systems: the case of non-passive outputs with unmatched disturbances. Syst. Control Lett. 61(1), 11–17 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Ortega, R., Spong, M.W.: Adaptive motion control of rigid robots: a tutorial. Automatica 25(6), 877–888 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Ortega, R., Rodriguez, A., Espinosa, G.: Adaptive stabilization of non-linearizable systems under a matching condition. In: American Control Conference, San Diego, USA, pp. 67–72 (1990)Google Scholar
  51. 51.
    Ortega, R., Loria, A., Nicklasson, P., Sira-Ramirez, H.: Passivity-Based Control of Euler-Lagrange Systems: Mechanical, Electrical, and Electromechanical Applications. Springer, London (1998)CrossRefGoogle Scholar
  52. 52.
    Ortega, R., Spong, M.W., Gomez, F., Blankenstein, G.: Stabilization of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans. Autom. Control 47(8), 1218–1233 (2002)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Ortega, R., van der Schaft, A., Castaños, F., Astolfi, A.: Control by interconnection and standard passivity-based control of port-Hamiltonian systems. IEEE Trans. Autom. Control 53(11), 2527–2542 (2008)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Paden, B., Panja, R.: Globally asymptotically stable PD+ controller for robot manipulators. Int. J. Control 4, 1697–1712 (1988)CrossRefzbMATHGoogle Scholar
  55. 55.
    Rodriguez, A., Ortega, R.: Adaptive stabilization of nonlinear systems: the non–feedback–linearizable case. In: IFAC World Congress, Tallinn, USSR, pp. 121–124 (1990)Google Scholar
  56. 56.
    Romero, J.G., Ortega, R.: Two globally convergent adaptive speed observers for mechanical systems. Automatica 60, 7–11 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Romero, J.G., Donaire, A., Ortega, R.: Robust energy shaping control of mechanical systems. Syst. Control Lett. 62(9), 770–780 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Romero, J.G., Ortega, R., Sarras, I.: A globally exponentially stable tracking controller for mechanical systems using position feedback. IEEE Trans. Autom. Control 60(3), 818–823 (2015)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Romero, J.G., Ortega, R., Donaire, A.: Energy shaping of mechanical systems via PID control and extension to constant speed tracking. IEEE Trans. Autom. Control 61(11), 3551–3556 (2016)Google Scholar
  60. 60.
    Ryalat, M., Laila, D., Torbati, M.: Integral IDA–PBC and PID–like control for port–controlled Hamiltonian systems. In: American Control Conference, Chicago, USA, pp. 5365–5370 (2015)Google Scholar
  61. 61.
    Sarras, I., Acosta, J.A., Ortega, R., Mahindrakar, A.: Constructive immersion and invariance stabilization for a class of underactuated mechanical systems. Automatica 49(5), 1442–1448 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Slotine, J., Li, W.: Adaptive manipulator control: a case study. IEEE Trans. Autom. Control 33(11), 995–1003 (1988)CrossRefzbMATHGoogle Scholar
  63. 63.
    Spong, M.W.: Partial feedback linearization of underactuated mechanical systems. In: The IEEE/RSJ International Conference on Intelligent Robots and Systems, Munich, Germany, pp. 314–321 (1994)Google Scholar
  64. 64.
    Stramigioli, S., Maschke, B., van der Schaft, A.: Passive output feedback and port interconnection. In: IFAC Symposium on Nonlinear Control Systems, Enschede, The Netherlands, pp. 613–618 (1998)Google Scholar
  65. 65.
    Takegaki, M., Arimoto, S.: A new feedback for dynamic control of manipulators. Trans. ASME: J. Dyn. Syst. Meas. Control 12, 119–125 (1981)zbMATHGoogle Scholar
  66. 66.
    van der Schaft, A.: \(L_2\)-Gain and Passivity Techniques in Nonlinear Control. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  67. 67.
    Venkatraman, A., Ortega, R., Sarras, I., van der Schaft, A.: Speed observation and position feedback stabilization of partially linearizable mechanical systems. IEEE Trans. Autom. Control 55(5), 1059–1074 (2010)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Viola, G., Ortega, R., Banavar, R., Acosta, J.A., Astolfi, A.: Total energy shaping control of mechanical systems: simplifying the matching equations via coordinate changes. IEEE Trans. Autom. Control 52(6), 1093–1099 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Willems, J.C.: The behavioral approach to open and interconnected systems. IEEE Control Syst. Mag. 27(6), 46–99 (2007)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Woolsey, C., Bloch, A., Leonard, N., Marsden, J.: Physical dissipation and the method of controlled Lagrangians. In: European Control Conference, Porto, Portugal, pp. 2570–2575 (2001)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Romeo Ortega
    • 1
    Email author
  • Alejandro Donaire
    • 2
    • 3
  • Jose Guadalupe Romero
    • 4
  1. 1.Laboratoire des Signaux et Systèmes, CNRS–SUPELECGif–sur–YvetteFrance
  2. 2.PRISMA Lab, Dipartimento di Ingegneria Elettrica e Tecnologie dell’InformazioneUniversità di Napoli Federico IINaplesItaly
  3. 3.School of EngineeringThe University of NewcastleCallaghanAustralia
  4. 4.Departamento Académico de Sistemas DigitalesInstituto Tecnológico Autónomo de México-ITAMDistrito FederalMexico

Personalised recommendations