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

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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.