Extremum seeking for analog, digital and interconnected systems

求极值控制的目的是稳定或跟踪动态控制系统中某个目标（或性能）函数达到极值时的状态。这个目标必须在不需要知道最优状态或关于目标函数梯度的信息的情况下完成。只有对目标函数值的实时测量才能用于反馈律。由于过去几十年的深入研究，无论是从理论还是从实践的角度来看，都取得了重大进展。然而，关于极值寻求控制的设计、分析和实现的许多重要问题，目前还没有得到充分的解决。例如，现有的大多数研究假设所提出的控制策略的理想实现，从而忽略了干扰的影响。此外，现代控制系统中的数字组件带来了新的挑战，如采样测量，量化输入或离散时间动力学。由于控制系统趋向于越来越互联，对于由数字和模拟组件组成的网络的极值搜索控制器的需求也在增加。对于连续时间和离散时间系统及其相互关系，目前还没有一种通用的鲁棒的求极值控制方法。本研究项目的目的是在各个层面上对上述问题做出贡献。为此，我们将遵循最近介绍的连续时间控制系统的极值寻求控制方法。该方法采用了几何控制理论的思想。从适当选择的Lie括号的近似值中提取梯度信息。该研究项目的第一个主要目标是通过使用李括号的流解释将该方法扩展到大型离散时间系统。第二个主要目标是研究极值寻求控制系统在连续和离散时间中对量化、采样和噪声损坏测量引起的误差的鲁棒性。为此，我们将使用公认的输入到状态稳定性（ISS）概念。我们的目标是在存在干扰的情况下，推导出与李氏括号方法的底层平均技术兼容的闭环系统的合适的ISS-like性质。我们还旨在为机械系统（如自主机器人）推导极值搜索控制器，这些控制器侵入性较小，对干扰的鲁棒性更强。第三个主要目标是研究分布式互连极值寻优控制系统。在这种情况下，我们的目标是获得小增益的ISS结果，以研究极值寻求控制系统网络中干扰的传播。

Extremum Seeking control has the purpose to stabilize or track those states of a dynamical control system at which a certain objective (or performance) function attains an extreme value. This goal has to be accomplished without requiring knowledge of the optimal states or information about the gradient of the objective function. Only real-time measurements of the values of the objective function can be utilized for a feedback law. Due to intensive research efforts over the past decades, significant advances have been made from a theoretical as well as from a practical point of view. However, many important problems concerning the design, analysis, and implementation of Extremum Seeking control are not sufficiently addressed so far. For example, most of the existing studies assume an ideal implementation of a proposed control strategy and thereby neglect the influence of disturbances. Moreover, digital components in modern control systems have lead to novel challenges such as sampled measurements, quantized inputs, or discrete-time dynamics. Since control systems tend to be more and more interconnected, there is also an increasing demand for Extremum Seeking controllers for networks consisting of both digital and analog components. A general and robust approach to Extremum Seeking control for continuous-time and discrete-time systems as well as their interconnections is not developed yet. The intention of the research project is to contribute to the above problems on various levels. For this purpose, we will follow a recently introduced approach to Extremum Seeking control for continuous-time control systems. The method uses ideas from geometric control theory. Gradient information is extracted from approximations of suitably chosen Lie brackets. The first main goal of the research project is to extend the approach to a large class of discrete-time systems by using the flow interpretation of Lie brackets. The second main goal is to study robustness properties of Extremum Seeking control systems with respect to errors induced by quantization, sampling, and noise corrupted measurements, both in continuous and discrete time. For this purpose, we will use the well-established concept of input-to-state stability (ISS). Our goal is to derive suitable ISS-like properties of closed-loopsystems in the presence of disturbances that are compatible with the underlying averaging techniques of the Lie bracket approach. We also aim to derive Extremum Seeking controllers for mechanical systems, such as autonomous robots, which are less invasive and more robust against disturbances. The third main objective is the study of distributed and interconnected Extremum Seeking control systems. In this context,we aim to derive small-gain ISS results in order to investigate the propagation of disturbances in networks of Extremum Seeking control systems.