Extremum seeking for analog, digital and interconnected systems

寻优控制的目的是稳定或跟踪动态控制系统的那些状态，在这些状态下，某个目标（或性能）函数达到极值。这个目标必须在不需要最优状态的知识或关于目标函数的梯度的信息的情况下实现。只有实时测量的目标函数的值可以用于反馈法。由于过去几十年的深入研究，从理论和实践角度都取得了重大进展。然而，许多重要的问题，有关的设计，分析和实施的搜索控件没有得到充分的解决。例如，大多数现有的研究假设一个理想的实施所提出的控制策略，从而忽略了干扰的影响。此外，现代控制系统中的数字元件带来了新的挑战，例如采样测量、量化输入或离散时间动态。由于控制系统的互联性越来越强，因此对用于数字和模拟组件组成的网络的寻迹控制器的需求也越来越大。对于连续时间和离散时间系统以及它们之间的关联，还没有一种通用的鲁棒寻优控制方法。该研究项目的目的是在各个层面上对上述问题作出贡献。为此，我们将遵循最近推出的连续时间控制系统的寻轨控制方法。该方法使用几何控制理论的思想。梯度信息提取适当选择的李括号的近似。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一大类离散时间系统。第二个主要目标是研究连续时间和离散时间下的混沌搜索控制系统对量化、采样和噪声干扰引起的误差的鲁棒性。为此，我们将使用输入到状态稳定性（ISS）的成熟概念。我们的目标是获得合适的ISS一样的性能的闭环系统中存在的干扰，是兼容的基本平均技术的李括号的方法。我们的目标还包括为机械系统（如自主机器人）推导出具有更小侵入性和更鲁棒性的控制器。第三个主要目标是研究分布式互联寻优控制系统。在这种情况下，我们的目标是获得小增益ISS的结果，以调查网络中的干扰传播的搜索控制系统。

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.