Extremum Seeking Control for Dynamic Maps: A Lie Bracket Averaging Framework

极值搜索是一种控制方法，用于将动态系统导向与给定动态系统相关的部分或完全未知的输入输出映射的极值（最小或最大）。它有着悠久的历史，在汽车、飞机或过程控制中的实时优化和最优设定值调节等控制问题中得到了许多应用。极值查找格式的稳定性通常借助于经典的平均理论和奇异摄动理论进行分析。通常，这些方法导致局部稳定结果，并且分析涉及复杂的极值寻求任务，例如具有约束或不稳定植物动力学的问题。这些限制不仅是一般理论的缺点，而且也不利于将极值寻求控制的范围扩展到新的应用。最近，一组申请人和他们的同事建立了一个非常有前途的替代方法来分析极值搜索方案，基于李氏平均技术。这种新方法有几个优点。该方法可以为极值搜索方案建立强稳定性结果，并且可以系统地应用于广泛的问题，包括具有多种约束的极值搜索问题，例如在机械系统和同步问题中出现的问题。另一方面，目前关于李括号平均的结果主要局限于静态的输入输出映射。因此，极值搜索方案的一个关键优势，即动态映射的分析，到目前为止还不能在李括号平均方法中得到解决。因此，新的李氏平均方法在设计高级极值搜索方案方面的潜在优势，如可以处理复杂约束和动态模型的方案，迄今尚未得到充分利用。这个项目的动机是为了解决上述缺点。目标是建立一个基于李氏平均技术的极值搜索框架，该框架可以以各种方式处理动态映射和约束。

Extremum seeking is a control method to steer a dynamical system to an extremum (minimum ormaximum) of a partially or completely unknown input-output map associated to a given dynamicalsystem. It has a long history and has found many applications in control problems such as real-time optimization and optimal setpoint regulation in cars, airplanes or in process control.Stability of extremum seeking schemes is often analyzed with the help of classical averaging and singular perturbation theory. Very often, these methods lead to local stability results and the analysis becomes involved in complex extremum seeking tasks such as problems with constraints or unstable plant dynamics. These limitations are not only a drawback for a general theory but also for extending the scope of extremum seeking control to novel applications.Recently, a very promising alternative approach for analyzing extremum seeking schemes basedon Lie bracket averaging techniques has been established by the group of the applicant and their coworkers. The new approach has several advantages. The approach allows to establish strongstability results for extremum seeking schemes and it can be systematically applied to a wide range of problems including extremum seeking problems with manifold constraints as they appear for example in mechanical systems and in synchronization problems. On the other hand, the current results on Lie bracket averaging are mainly limited to static input-output maps. Thus a key advantage of extremum seeking schemes, namely the analysis of dynamic maps, cannot be addressed so far in the Lie bracket averaging approach. Consequently the potential advantages of the new Lie bracket averaging approach in designing advanced extremum seeking schemes, like schemes which can deal with complex constraints and dynamic models, have not been exploited so far. The motivation of this project is to address the above mentioned shortcomings. The goal is to establish a framework for extremum seeking schemes which is based on Lie bracket averaging techniques and which can deal in various ways with dynamic maps and constraints.