Deutsch Intern
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Dynamics and Control

Dr. Victoriia Grushkovska

Academic Staff
Professorship at the Chair of Mathematics II
Emil-Fischer-Straße 40
97074 Würzburg
Building: 40 (Mathematik Ost)
Room: 01.015
Portrait Viktoriia Grushkovska

Gradient Flow Approximation for Stabilization and Motion Planning in Dynamic Environments

Project management: Dr. Victoriia Grushkovska

Project period: 2018 - 2020

Funding institution: DFG

Funding sum: 190.600,00 €

Funding code: GR 5293/1-1

Project description

Motion planning for nonlinear control systems is one of the most important problems of mathematical control theory because of its theoretical challenges and many practical applications. Many applied problems are related to dynamic environments, where the locations of targets and obstacles change with time, e.g., when an autonomous vehicle should follow a moving target avoiding collisions with other moving objects. From a theoretical point of view, the complexity of control design for such tasks increases significantly, and many classical methods developed for static environments are no longer applicable.

The goal of this project is to develop a general framework for stabilization and motion planning of nonholonomic systems governed by driftless control-affine systems. The main idea is to guarantee that the motion of the system steers along the approximated gradient flow of a certain potential function. Depending on the approximation method, two types of controls will be constructed: gradient-based and gradient-free. Gradient-based controls may explicitly depend on the derivatives of a potential function and can be used in situations where its analytical expression is known, for example, when complete information on the target and obstacles is available. However, in a variety of practical scenarios an explicit analytical expression of the potential function is partially or completely unknown, for example, when the target trajectory is unknown, or in extremum seeking problems. For such cases, it is planned to construct gradient-free controls.

Team Work Dynamical Systems and Control


Prof. Dr. Sergey Dashkovskiy
Dr. Victoriia Grushkovska

Wednesday,  12:00 - 16:00 (SE40, Mathematik Ost), Hubland Nord


WueCampus course
Class Schedule of the Faculty of Mathematics and Computer Science


  1. V. Grushkovskaya, A. Zuyev, C. Ebenbauer
    On a class of generating vector fields for the extremum seeking problem: Lie bracket approximation and stability properties
    Automatica, 2018, Vol. 94, pp. 151-160.
    DOI: 10.1016/j.automatica.2018.04.024
  2. V. Grushkovskaya,  S. Michalowsky, A. Zuyev, M. May,  C. Ebenbauer
    A family of extremum seeking laws for a unicycle model with a moving target: theoretical and experimental studies
    Proc. 17th European Control Conference (ECC 2018), 2018, pp. 1-6.
    DOI: 10.23919/ECC.2018.8550280
  3. V. Grushkovskaya, A. Zuyev
    Obstacle Avoidance Problem for Second Degree Nonholonomic Systems
     Proc. 57th IEEE Conf. on Decision and Control, 2018, pp. 1500-1505.
  4. V. Grushkovskaya, A. Zuyev, C. Ebenbauer
    On extremum seeking controllers based on the Lie bracket approximation in domains with obstacles
    PAMM - Proceedings in Applied Mathematics and Mechanics, 2018, Vol. 18, Issue 1, pp. 1-2.
    DOI: 10.1002/pamm.201800298
  5. V. Grushkovskaya, H.-B. Dürr, C. Ebenbauer, A. Zuyev
    Extremum Seeking for Time-Varying Functions using Lie Bracket Approximations
    IFAC-PapersOnLine (Proc. 20th IFAC World Congress),  2017, Vol. 50, No.1, pp. 5522–5528.
    DOI: 10.1016/j.ifacol.2017.08.1093
  6. A. Zuyev, V. Grushkovskaya
    Obstacle Avoidance Problem for Driftless Nonlinear Systems with Oscillating Controls
    IFAC-PapersOnLine (Proc. 20th IFAC World Congress),  2017, Vol. 50, No.1, pp. 15343–15348.
    DOI: 10.1016/j.ifacol.2017.08.1979 
  7. C. Ebenbauer, S. Michalowsky, V. Grushkovskaya, B. Gharesifard
    Distributed Optimization Over Directed Graphs with the Help of Lie Brackets
    IFAC-PapersOnLine (Proc. 20th IFAC World Congress), 2017, Vol. 50, No. 1, 10476–10481.
    DOI: 10.1016/j.ifacol.2017.08.2456 
  8. A. Zuyev, V. Grushkovskaya
    Motion Planning for Control-Affine Systems Satisfying Low-Order Controllability Conditions
    International Journal of Control, 2017, Vol. 90, No.11, pp. 2517-2537.
    DOI: 10.1080/00207179.2016.1257157 
  9. V. Grushkovskaya, A. Zuyev
    Two-Point Problem for Systems Satisfying the Controllability Condition with Lie Brackets of the Second Order 
    Journal of Mathematical Sciences, 2017, Vol. 220, No. 3, P. 301-317 (Translation).
    DOI: 10.1007/s10958-016-3185-3 
  10. V. Grushkovskaya
    On the influence of resonances on the asymptotic behavior of trajectories of nonlinear systems in critical cases
    Nonlinear Dynamics, 2016, Vol. 86, No. 1, P.587-603.
    DOI:  10.1007/s11071-016-2909-8 
  11. V. Grushkovskaya
    Asymptotic Decay of Solutions to an Essentially Nonlinear System with Two-Frequency Resonances
    Applicable Analysis, 2016, Vol. 95, No. 11, P. 2501-2516. 
    DOI: 10.1080/00036811.2015.1094798 
  12. V. Grushkovskaya, C. Ebenbauer
    Multi-Agent Coordination with Lagrangian Measurements
    IFAC-PapersOnLine, 2016, Vol. 49, No. 22, P.115-120.
    DOI: 10.1016/j.ifacol.2016.10.382 
  13. A. Zuyev, V. Grushkovskaya, P. Benner
    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.
    DOI:  10.1109/ECC.2016.7810346 
  14. V. Grushkovskaya
    Asymptotic behavior of solutions of nonlinear systems with multiple imaginary eigenvalues
    PAMM - Proceedings in Applied Mathematics and Mechanics, 2016, Vol. 16, No.1, P. 271-272.  
    DOI: 10.1002/pamm.201610124 
  15. V. Grushkovskaya, A. Zuyev
    Attractors of Nonlinear Dynamical Systems with a Weakly Monotonic Measure
    Journal of Mathematical Analysis and Applications, 2015, Vol. 422, P.559-570.
    DOI: 10.1016/j.jmaa.2014.08.046 
  16. V. Grushkovskaya, A. Zuyev
    Optimal Stabilization Problem with Minimax Cost in a Critical Case 
    IEEE Transactions on Automatic Control, 2014, Vol. 59, N.9,  P.2512-2517.
    DOI: 10.1109/TAC.2014.2304399 
  17. V. Grushkovskaya, A. Zuyev
    Optimal Stabilization of Nonlinear Systems by an Output Feedback Law in a Critical Case
    Proc. of the 52 nd IEEE Conference on Decision and Control, 2013, P.4607-4612.
    DOI 10.1109/CDC.2013.6760610 
  18. V. Grushkovskaya, A. Zuyev
    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, Vol. 80, P.156-178.
    DOI 10.1016/j.na.2012.10.007