Nonlinear Dynamics

, Volume 92, Issue 3, pp 1431–1451 | Cite as

Online estimation and adaptive control for a class of history dependent functional differential equations

  • Shirin Dadashi
  • Parag Bobade
  • Andrew J. Kurdila
Original Paper


This paper presents sufficient conditions for the convergence of online estimation methods and the stability of adaptive control strategies for a class of history-dependent, functional differential equations. The study is motivated by the increasing interest in estimation and control techniques for robotic systems whose governing equations include history-dependent nonlinearities. The functional differential equations in this paper are constructed using integral operators that depend on distributed parameters. As a consequence, the resulting estimation and control equations are examples of distributed parameter systems whose states and distributed parameters evolve in finite and infinite dimensional spaces, respectively. Well-posedness, existence, and uniqueness are discussed for the class of fully actuated robotic systems with history-dependent forces in their governing equation of motion. By deriving rates of approximation for the class of history-dependent operators in this paper, sufficient conditions are derived that guarantee that finite dimensional approximations of the online estimation equations converge to the solution of the infinite dimensional, distributed parameter system. The convergence and stability of a sliding mode adaptive control strategy for the history-dependent, functional differential equations is established using Barbalat’s lemma.


Online estimation Adaptive control Functional differential equations 



We would like to thank the anonymous referees for their constructive input and valuable suggestions that helped us improve the manuscript.


  1. 1.
    Bahlman, J.W., Swartz, S.M., Breuer, K.S.: Design and characterization of a multi-articulated robotic bat wing. Bioinspir. Biomim. 8(1), 1–17 (2013)CrossRefGoogle Scholar
  2. 2.
    Banks, H.T., Kunisch, K.: Estimation Techniques for Distributed Parameter Systems. Birkhauser, Boston (1989)CrossRefzbMATHGoogle Scholar
  3. 3.
    Baumeister, J., Scondo, W., Demetriou, M.A., Rosen, I.G.: On-line parameter estimation for infinite dimensional dynamical systems. SIAM J. Control Optim. 35(2), 678–713 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  5. 5.
    Corduneaunu, C.: Integral Equations and Applications. Cambridge University Press, Cambridge (2008)Google Scholar
  6. 6.
    Dahmen, W.: Stability of multiscale transformations. J. Fourier Anal. Appl. 4, 341–362 (1996)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Dadashi, S., Feaster, J., Bayandor, J., Battaglia, F.: Kurdila, Andrew J.: Identification and adaptive control of history dependent unsteady aerodynamics for a flapping insect wing. Nonlinear Dyn. 85, 1405 (2016)CrossRefzbMATHGoogle Scholar
  8. 8.
    Dadashi, S., Bobade, P., Kurdila, A.J.: Error estimates for multiwavelet approximations of a class of history dependent operators. In: 2016 IEEE 55th Conference on Decision and Control (CDC) (2016)Google Scholar
  9. 9.
    Demetriou, M.A.: Adaptive parameter estimation of abstract parabolic and hyperbolic distributed parameter systems. Ph.D. thesis, Departments of Electrical-Systems and Mathematics, University of Southern California, Los Angeles, CA (1993)Google Scholar
  10. 10.
    Demetriou, M.A., Rosen, I.G.: Adaptive identification of second order distributed parameter systems. Inverse Probl. 10, 261–294 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Demetriou, M.A., Rosen, I.G.: On the persistence of excitation in the adaptive identification of distributed parameter systems. IEEE Trans. Autom. Control 39, 1117–1123 (1994)CrossRefzbMATHGoogle Scholar
  12. 12.
    DeVore, R.A.: Nonlinear approximation. Acta Numer. 7, 51–150 (1998)CrossRefzbMATHGoogle Scholar
  13. 13.
    DeVore, R.A., Lorentz, G.: Constructive Approximation. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  14. 14.
    Driver, R.D.: Existence and stability of solutions of a delay-differential system. Arch. Ration. Mech. Anal. 10(1), 401–426 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ilchmann, A., Ryan, E.P., Sangwin, C.J.: Systems of controlled functional differential equations and adaptive tracking. SIAM J. Control Optim. 40(6), 1746–1764 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ilchmann, A., Logemann, H., Ryan, E.P.: Tracking with prescribed transient performance for hysteretic systems. SIAM J. Control Optim. 48(7), 4731–4752 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ioannou, P., Sun, J.: Robust Adaptive Control. Dover, New York (2012)zbMATHGoogle Scholar
  18. 18.
    Keinert, F.: Wavelets and Multiwavelets. Chapman & Hall, London (2004)zbMATHGoogle Scholar
  19. 19.
    Khalil, H.: Nonlinear Systems, 3rd edn. Prentice-Hall, Upper Saddle River (2003)Google Scholar
  20. 20.
    Krasovskii, N.N.: On the application of the second method of A.M. Lyapunov to equations with time delays. Prikl. Mat. Mekh. 20, 315–327 (1956)Google Scholar
  21. 21.
    Krasovskii, N.N.: On the asymptotic stability of systems with after-effect. Prikl. Mat. Mekh. 20, 513–518 (1956)MathSciNetGoogle Scholar
  22. 22.
    Krasnoselskii, M.A., Pokrovskii, A.V.: Systems with Hysteresis. Springer, Berlin (1989)CrossRefGoogle Scholar
  23. 23.
    Kurdila, A., Li, J., Strganac, T., Webb, G.: Nonlinear control methodologies for hysteresis in PZT actuated on-blade elevons. J. Aerosp. Eng. 16(4), 167–176 (2003)CrossRefGoogle Scholar
  24. 24.
    Lewis, F.L., Dawson, D., Abdallah, C.: Robot Manipulator Control: Theory and Practice. Marcel Dekker Inc, New York (2004)Google Scholar
  25. 25.
    Myshkis, A.D.: General theory of differential equations with a retarded argument. Uspekhi Mat. Nauk 4(5), 99–141 (1949)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Narendra, K., Annaswamy, A.: Stable Adaptive Systems. Dover, New York (2005)zbMATHGoogle Scholar
  27. 27.
    Rudakov, V.P.: Qualitative theory in a Banach space, Lyapunov–Krasovskii functionals, and generalization of certain problems. Ukr. Mat. Zh. 30(1), 130–133 (1978)Google Scholar
  28. 28.
    Rudakov, V.P.: On necessary and sufficient conditions for the extendability of solutions of functional-differential equations of the retarded type. Ukr. Mat. Zh. 26(6), 822–827 (1974)Google Scholar
  29. 29.
    Ryan, E., Sangwin, C.: Controlled functional differential equations and adaptive tracking. Syst. Control Lett. 47, 365–374 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sastry, S., Bodson, M.: Adaptive Control: Stability. Convergence and Robustness. Dover, New York (2011)zbMATHGoogle Scholar
  31. 31.
    Siciliano, B., Sciavicco, L., Villani, L., Oriolo, G.: Robotics: Modeling, Planning, and Control. Springer, London (2010)Google Scholar
  32. 32.
    Spong, M., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control. Wiley, New York (2005)Google Scholar
  33. 33.
    Tavernini, L.: Linear multistep methods for the numerical solution of Volterra functional differential equations. Appl. Anal. 3, 169–185 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Visintin, A.: Differential Models of Hysteresis. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringVirginia TechBlacksburgUSA
  2. 2.Department of Engineering Science and MechanicsVirginia TechBlacksburgUSA

Personalised recommendations