Home /Archive /2018 /2018 №8 /2018_8_7

Visn. Nac. Akad. Nauk Ukr. 2018. (8):66-75  
https://doi.org/10.15407/visn2018.08.066 

V.V. Grushkovska
Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Sloviansk
https://orcid.org/0000-0003-0439-6834

GRADIENT-FREE CONTROL ALGORITHMS FOR DYNAMIC OPTIMIZATION PROBLEMS
According to the materials of scientific report at the meeting of the Presidium of NAS of Ukraine, May 30, 2018

The paper focuses on the study of dynamic optimization problems with cost function, whose analytical expression is partially or completely unknown. This limitation leads to inefficiency of classical control methods, for which the gradient of a cost functions has to be computed explicitly. This paper presents a novel gradient-free control design approach for dynamic optimization problems. It unifies and generalizes some known results and, moreover, allows constructing new controls with favourable properties. In contrast to many existing gradient-free control algorithms which imply only the practical asymptotic stability, we propose conditions for asymptotic (and even exponential) stability in the sense of Lyapunov. The results obtained are illustrated by numerical simulations and experiments with a mobile robot.
Keywords: dynamic optimization problems, extremum seeking, asymptotic stability, gradient-free control algorithms, Lie bracket approximation.

Language of article: ukrainian

 

REFERENCES

  1. Liu S.J., Krstić M. Stochastic averaging and stochastic extremum seeking. (Springer Science & Business Media, 2012)
  2. Ariyur K.B., Krstić M. Real-time optimization by extremum-seeking control. (Wiley-Blackwell, 2003)
  3. Tan Y., Moase W.H., Manzie C., Nešić D., Mareels I.M.Y. Extremum seeking from 1922 to 2010. In: Proc. 29th IEEE Chinese Control Conference (July 28-31, 2010, Beijing, China).
  4. Grushkovskaya V., Zuyev A., Ebenbauer C. On a class of generating vector fields for the extremum seeking problem: Lie bracket approximation and stability properties. Automatica. 2018. 94: 151-160. https://doi.org/10.1016/j.automatica.2018.04.024
  5. Grushkovskaya V., Michalowsky S., Zuyev A., May M., Ebenbauer C. A family of extremum seeking laws for a unicycle model with a moving target: theoretical and experimental studies. In: Proc. European Control Conference’18. 2018.
  6. Grushkovskaya V., Dürr H.-B., Ebenbauer C., Zuyev A. Extremum Seeking for Time-Varying Functions using Lie Bracket Approximations. IFAC-PapersOnLine. 2017. 50 (1): 5522-5528. https://doi.org/10.1016/j.ifacol.2017.08.1093
  7. Dürr H.-B., Stankovic M.S., Ebenbauer C., Johansson K. Lie bracket approximation of extremum seeking systems. Automatica. 2013. 49: 1538-1552. https://doi.org/10.1016/j.automatica.2013.02.016
  8. Scheinker A., Krstić M. Extremum seeking with bounded update rates. Systems & Control Letters. 2014. 63: 25-31. https://doi.org/10.1016/j.sysconle.2013.10.004
  9. Scheinker A., Krstić M. Non-C2 Lie bracket averaging for nonsmooth extremum seekers. J. of Dynamic Systems, Measurement, and Control. 2014. 136 (1): 011010-1–011010–10. https://doi.org/10.1115/1.4025457
  10. Suttner R., Dashkovskiy S. Exponential stability for extremum seeking control systems. IFAC-PapersOnLine. 2017. 50 (1): 15464-15470. https://doi.org/10.1016/j.ifacol.2017.08.2106
  11. Zuyev A. Exponential stabilization of nonholonomic systems by means of oscillating controls. SIAM Journal on Control and Optimization. 2016. 54: 1678-1696. https://doi.org/10.1137/140999955
  12. Zuyev A., Grushkovskaya V. Motion Planning for Control-Affine Systems Satisfying Low-Order Controllability Conditions. International Journal of Control. 2017. 90 (11): 2517-2537. https://doi.org/10.1080/00207179.2016.1257157
  13.  Zuyev A., Grushkovskaya V., Benner P. Time-varying stabilization of a class of driftless systems satisfying second-order controllability conditions. Proc. of the European Control Conference’16, 2016, P. 575-580. https://doi.org/10.1109/ECC.2016.7810346
  14. Grushkovskaya V., Zuyev A. Asymptotic Behavior of Solutions of a Nonlinear System in the Critical Case of q Pairs of Purely Imaginary Eigenvalues. Nonlinear Analysis: Theory, Methods & Applications. 2013. 80: 156-178. https://doi.org/10.1016/j.na.2012.10.007
  15. Grushkovskaya V. On the influence of resonances on the asymptotic behavior of trajectories of nonlinear systems in critical cases. Nonlinear Dynamics. 2016. 86 (1): 587-603. https://doi.org/10.1007/s11071-016-2909-8
  16. Grushkovskaya V. Gradient-free control algorithms for motion planning with obstacle avoidance. Proceedings of IAMM NASU. 2017. 31.
  17. Grushkovskaya V., Ebenbauer C. Multi-Agent Coordination with Lagrangian Measurements. IFAC-PapersOnLine. 2016. 49 (22): 115-120. https://dx.doi.org/10.1016/j.ifacol.2016.10.382