Advertisement

Novel Model Reference Adaptive Control Designed by a Lyapunov Function That is Kept at Low Value by Fixed Point Iteration

  • Bertalan Csanádi
  • József K. TarEmail author
  • János F. Bitó
Chapter
Part of the Topics in Intelligent Engineering and Informatics book series (TIEI, volume 14)

Abstract

Even in our days the generally prevailing methodology used for designing adaptive controllers is based on Lyapunov’s “direct method”. This statement holds for the special “Model Reference Adaptive Controllers (MRAC)”, too, that have to satisfy two typical requirements. Besides providing precise trajectory tracking they have to generate an illusion to an external control loop working on purely kinematic/kinetic basis that it controls a system the dynamic properties of which are identical to that of a “Reference System”. Based on the structure of the more or less plausible Lyapunov functions the Reference System used to be stable, linear, time-invariant. To evade the mathematical difficulties of the Lyapunov function-based design, as an alternative approach, “Fixed Point Iteration (FPI)”-based design technique was introduced that had a special MRAC variant, too. In this approach the usual restrictions for the dynamics of the Reference System were naturally released. Later it was shown that this adaptive approach can be interpreted from the point of view of the Lyapunov functions in which non-conventional feedback terms are used for driving the Lyapunov function near zero and keeping it in its vicinity. This approach naturally gave rise to a novel MRAC design idea that is akin to that of the “Backstepping Controllers”. In this paper this idea is presented and illustrated via simulation investigations. Simulation results are provided regarding a completely driven 2 degree of freedom physical system, that consists of two nonlinearly coupled and nonlinearly damped mass-points.

Notes

Acknowledgements

This work has been partially supported by the Doctoral School of Applied Informatics and Applied Mathematics of Óbuda University.

References

  1. 1.
    A. Lyapunov, Stability of Motion (Academic Press, New York, 1966)zbMATHGoogle Scholar
  2. 2.
    J.J.E. Slotine, W. Li, Applied Nonlinear Control (Prentice Hall International Inc, New Jersey, 1991)zbMATHGoogle Scholar
  3. 3.
    C. Nguyen, S. Antrazi, Z.L. Zhou, C. Campbell Jr., Adaptive control of a stewart platform-based manipulator. J. Robot. Syst. 10, 657–687 (1993)CrossRefGoogle Scholar
  4. 4.
    J. Somló, B. Lantos, P. Cát, Advanced Robot Control (Akadémiai Kiadó, Budapest, 2002)zbMATHGoogle Scholar
  5. 5.
    J. Tar, J. Bitó, L. Nádai, J. Tenreiro Machado, Robust fixed point transformations in adaptive control using local basin of attraction. Acta Polytech. Hung. 6, 21–37 (2009)Google Scholar
  6. 6.
    J. Tar, J. Bitó, I. Rudas, Replacement of Lyapunov’s direct method in model reference adaptive control with robust fixed point transformations, in Proceeding of the 14th IEEE Intelligent Conference on Intelligent Engineering Systems (Las Palmas of Gran Canaria, Spain 2010), pp. 231–235Google Scholar
  7. 7.
    B. Csanádi, P. Galambos, J. Tar, G. Györök, A. Serester, Revisiting Lyapunov’s technique in the fixed point transformation-based adaptive control, in Proceeding of the 22nd IEEE International Conference on Intelligent Engineering Systems, Las Palmas de Gran Canaria, Spain, 21–23 June 2018, pp. 329–334Google Scholar
  8. 8.
    J. Wang, B. Deng, X. Fei, Synchronizing two coupled chaotic neurons in external electrical stimulation using backstepping control. Chaos Solitons Fractals 29, 182–189 (2006)CrossRefGoogle Scholar
  9. 9.
    S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (About the operations in the abstract sets and their application to integral equations). Fund. Math. 3, 133–181 (1922)CrossRefGoogle Scholar
  10. 10.
    A. Dineva, J. Tar, A. Várkonyi-Kóczy, V. Piuri, Generalization of a sigmoid generated fixed point transformation from SISO to MIMO systems, in Proceeding of the IEEE 19th International Conference on Intelligent Engineering Systems (INES 2015), Bratislava, Slovakia 3–5 September 2015, pp. 135–140Google Scholar
  11. 11.
    A. Dineva, Non-conventional data representation and control (PhD Thesis, Supervisors: A.R. Várkonyi-Kóczy & J.K. Tar). Óbuda University, Budapest, Hungary (2016)Google Scholar
  12. 12.
    H. Redjimi, J. Tar, On the effects of time-delay on precision degradation in fixed point transformation-based adaptive control, in Proceeding of the IEEE 30th Jubilee Neumann Colloquium, Budapest, Hungary, 24–25 Nov 2017, pp. 125–130Google Scholar
  13. 13.
    B. Csanádi, P. Galambos, J. Tar, G. Györök, A. Serester, A novel, abstract rotation-based fixed point transformation in adaptive control, in Accepted for Publication at the 2018 IEEE International Conference on Systems, Man, and Cybernetics (SMC2018), Miyazaki, Japan, 7–10 Oct 2018, pp. 1–6Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Bertalan Csanádi
    • 1
  • József K. Tar
    • 2
    Email author
  • János F. Bitó
    • 2
  1. 1.Óbuda UniversityDoctoral School of Applied Informatics and Applied MathematicsBudapestHungary
  2. 2.University Research, Innovation, and Service CenterAntal Bejczy Center of Intelligent RoboticsBudapestHungary

Personalised recommendations