Extremum seeking for analog, digital and interconnected systems

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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.