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The Koopman and moment-sum-of-squares approach for control: computational methods and applications
Author: Vít Cibulka
This thesis uses Sum-of-squares (SOS) and Koopman operator frameworks for analysis and control synthesis of nonlinear control systems. Both techniques propose linearization and convexification of nonlinear control problems by casting the problems into infinite-dimensional space where they permit linear descriptions leading to convex programming. We investigate the current challenges of both techniques and propose solutions aimed at making them more applicable in practice.
The SOS framework provides tooling for solving nonconvex polynomial problems via convex programming. The tradeoff for the global optimality is reflected in the size of convex programs, which limits the framework to small or very sparse problems. In this work, we improve the scalability, resource demands,and accuracy of the SOS framework for control-related tasks by splitting the problem into several interconnected parts of lesser complexity while also providing a method for optimizing the splitting, thus mitigating the impact of increasing the number of parameters.
The Koopman framework provides tools for global representation of nonlinear dynamical systems by high-dimensional linear systems, allowing the use of linear control methods for analysis and control design for the underlying nonlinear system. The current methods for learning the Koopman operator assume some partial knowledge about the operator, which is usually challenging to obtain,thus transferring the problem of finding the Koopman operator to the problem of finding the right parametrization for the particular numerical method, which can be just as difficult.
This thesis presents a new method for approximating the Koopman operator for nonlinear control systems. The method does not assume any prior knowledge about the operator, successfully outperforms the current state-of-the-art,increases the class of systems which can be approximated by the methodology, and is capable of exploiting and replicating symmetries of the underlying nonlinear system, thus guaranteeing consistent controller behaviour when used for control synthesis.