Guarantees for data-driven control of nonlinear systems using semidefinite programming: A survey

This survey presents recent research on determining control-theoretic properties and designing controllers with rigorous guarantees using semidefinite programming and for nonlinear systems for which no mathematical models but measured trajectories are available. Data-driven control techniques have been developed to circumvent a time-consuming modelling by first principles and because of the increasing availability of data. Recently, this research field has gained increased attention by the application of Willems' fundamental lemma, which provides a fertile ground for the development of data-driven control schemes with guarantees for linear time-invariant systems. While the fundamental lemma can be generalized to further system classes, there does not exist a comparable data-based system representation for nonlinear systems. At the same time, nonlinear systems constitute the majority of practical systems. Moreover, they include additional challenges such as data-based surrogate models that prevent system analysis and controller design by convex optimization. Therefore, a variety of data-driven control approaches has been developed with different required prior insights into the system to ensure a guaranteed inference. In this survey, we will discuss developments in the context of data-driven control for nonlinear systems. In particular, we will focus on methods based on system representations providing guarantees from finite data, while the analysis and the controller design boil down to convex optimization problems given as semidefinite programming. Thus, these approaches achieve reasonable advances compared to the state-of-the-art system analysis and controller design by models from system identification. Specifically, the paper covers system representations based on extensions of Willems' fundamental lemma, set membership, kernel techniques, the Koopman operator, and feedback linearization.