Special Bilevel Quadratic Problems for Construction of Worst-Case Feedback Control in Linear-Quadratic Optimal Control Problems under Uncertainties

Almost all mathematical models that describe processes, for instance in industry, engineering or natural sciences, contain uncertainties which arise from different sources. We have to take these uncertainties into account when solving optimal control problems for such processes. There are two popula...

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Bibliographic Details
Main Author: Stibbe, Hilke Isabell
Contributors: Kostina, Ekaterina (Prof. Dr.) (Thesis advisor)
Format: Dissertation
Language:English
Published: Philipps-Universität Marburg 2019
Mathematik und Informatik
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Summary:Almost all mathematical models that describe processes, for instance in industry, engineering or natural sciences, contain uncertainties which arise from different sources. We have to take these uncertainties into account when solving optimal control problems for such processes. There are two popular approaches : On the one hand the so-called closed-loop feedback controls, where the nominal optimal control is updated as soon as the actual state and parameter estimates of the process are available and on the other hand robust optimization, for example worst-case optimization, where it is searched for an optimal solution that is good for all possible realizations of uncertain parameters. For the optimal control problems of dynamic processes with unknown but bounded uncertainties we are interested in a combination of feedback controls and robust optimization. The computation of such a closed-loop worst-case feedback optimal control is rather difficult because of high dimensional complexity and it is often too expensive or too slow for complex optimal control problems, especially for solving problems in real-time. Another difficulty is that the process trajectory corresponding to the worst-case optimal control might be infeasible. That is why we suggest to solve the problems successively by dividing the time interval and determining intermediate time points, computing the feedback controls of the smaller intervals and allowing to correct controls at these fixed intermediate time points. With this approach we can guarantee that for all admissible uncertainties the terminal state lies in a given prescribed neighborhood of a given state at a given final moment. We can also guarantee that the value of the cost function does not exceed a given estimate. In this thesis we introduce special bilevel programming problems with solutions of which we may construct the feedback controls. These bilevel problems can be solved explicitly. We present, based on these bilevel problems, efficient methods and approximations for different control policies for the combination of feedback control and robust optimization methods which can be implemented online, compare these approaches and show their application on linear-quadratic control problems.
Physical Description:168 Pages
DOI:https://doi.org/10.17192/z2019.0526