Extremum Seeking Control for Dynamic Maps: A Lie Bracket Averaging Framework
Final Report Abstract
The main goal of the project was the development of novel extremum seeking algorithms for dynamic maps based on Lie bracket averaging (or Lie bracket approximation) methods. Extremum seeking control is a powerful methodology to solve various control problems without the need of a detailed system model. There exist different approaches to analyze and design extremum seeking algorithms for dynamic maps (dynamical systems). The most common approach is based on classical averaging theory. In this project, we developed an alternative and novel approach based on Lie bracket approximation ideas. In particular, the goal was to develop an analysis and design framework which can cope with dynamic maps (dynamical systems) and which allows to incorporate constraints and system models in order to improve the performance of extremum seeking control loops. In the course of the project, we established several novel results. One important achievement was to fully incorporate dynamic maps in the Lie bracket framework. Hereby, we developed two approaches. The first approach goes along the lines of classical averaging theory, while the second is based on the Chen-Fliess series expansion. Both approaches are model-free and can deal with unknown dynamic maps. In applications, often some knowledge about the to-be-controlled system is available. In the course of the project, we were able to integrate such partial knowledge of the system in terms of state space models into the analysis and design of extremum seeking algorithms. For example, with the methods developed in the project, it is now possible do design extremum seeking algorithms for systems which are composed by a feedback-linearizable system, where a state space model is known, in conjunction with an unknown dynamic map. Finally, we also improved the performance of extremum seeking algorithms in the project. We could significantly improve the performance of Lie bracket based extremum seeking algorithms, by developing a whole class of novel algorithms which exploit the generality (non-uniqueness) of approximating gradients in terms of Lie bracket vector fields. These algorithms showed a significant better performance, in simulation and in experiments, than the standard approach. Overall, we believe that the established results in the project contribute significantly to the area of extremum seeking control. The obtained results have turned the Lie bracket based extremum seeking approach into a powerful and flexible methodological toolkit that can be flexibly applied to solve a wide variety of control problems.
Publications
- Model-based extremum seeking for a class of nonlinear systems. In Proc. of the American Control Conference (ACC), Chicago, USA, pages 2026– 2031, 2015
S. Michalowsky and C. Ebenbauer
(See online at https://doi.org/10.1109/acc.2015.7171031) - Extremum control of linear systems based on output feedback. In Proc. of the 55th IEEE Conference on Decision and Control, Las Vegas, USA, pages 2963–2968, 2016
S. Michalowsky and C. Ebenbauer
(See online at https://doi.org/10.1109/cdc.2016.7798711) - Gradient approximation and extremum seeking via needle variations. In Proc. of the American Control Conference (ACC), Chicago, USA, pages 6091–6096, 2016
S. Michalowsky and C. Ebenbauer
(See online at https://doi.org/10.1109/acc.2016.7526626) - Extremum seeking for dynamic maps using Lie brackets and singular perturbations. Automatica, 83:91–99, September 2017
H.B. Dürr, M. Krstic, A. Scheinker, and C. Ebenbauer
(See online at https://doi.org/10.1016/j.automatica.2017.05.002) - Extremum seeking for timevarying functions using Lie bracket approximations. In Proc. of the 20th IFAC World Congress, Toulouse, France, pages 5687–5693, 2017
V. Grushkovskaya, H.B. Dürr, C. Ebenbauer, and A. Zuyev
(See online at https://doi.org/10.1016/j.ifacol.2017.08.1093) - A family of extremum seeking laws for a unicycle model with a moving target: theoretical and experimental studies. In Proc. of the European Control Conference (ECC), Limassol, Cyprus, pages 912–917, 2018
V. Grushkovskaya, S. Michalowsky, A. Zuyev, M. May, and C. Ebenbauer
(See online at https://doi.org/10.23919/ecc.2018.8550280) - On a class of generating vector fields for the extremum seeking problem: Lie bracket approximation and stability properties. Automatica, 94:151–160, August 2018
V. Grushkovskaya, A. Zuyev, and C. Ebenbauer
(See online at https://doi.org/10.1016/j.automatica.2018.04.024)
