Lie bracket approximation of extremum seeking systems
Lie bracket approximation of extremum seeking systems
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DOI：
10.1016/j.automatica.2013.02.016
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发表时间：
2011-09
期刊：
影响因子：
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通讯作者：
Hans-Bernd Dürr;M. Stanković;C. Ebenbauer;K. Johansson
中科院分区：
文献类型：
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作者：
Hans-Bernd Dürr;M. Stanković;C. Ebenbauer;K. Johansson
极值寻求反馈是将动力系统引导到部分或完全未知映射的极值的一种有效方法。它通常需要先进的系统论工具来理解极值求解系统的定性行为。本文对极值求法提出了一种新的解释。我们证明了极值搜索系统的轨迹可以用包含极值搜索系统向量场的某些李括号的系统的轨迹来近似。结果表明，李括号系统直接揭示了极值搜索系统的寻优行为。进一步，我们建立了理论基础，并证明了李括号系统的一致渐近稳定性蕴含了相应极值搜索系统的实用一致渐近稳定性。利用已有的结果，证明了多智能体极值求解系统某一映射极值的局部和半全局实用一致渐近稳定性。
Extremum seeking feedback is a powerful method to steer a dynamical system to an extremum of a partially or completely unknown map. It often requires advanced system-theoretic tools to understand the qualitative behavior of extremum seeking systems. In this paper, a novel interpretation of extremum seeking is introduced. We show that the trajectories of an extremum seeking system can be approximated by the trajectories of a system which involves certain Lie brackets of the vector fields of the extremum seeking system. It turns out that the Lie bracket system directly reveals the optimizing behavior of the extremum seeking system. Furthermore, we establish a theoretical foundation and prove that uniform asymptotic stability of the Lie bracket system implies practical uniform asymptotic stability of the corresponding extremum seeking system. We use the established results in order to prove local and semi-global practical uniform asymptotic stability of the extrema of a certain map for multi-agent extremum seeking systems.