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This survey presents recent research on determining control-theoreticproperties and designing controllers with rigorous guarantees usingsemidefinite programming and for nonlinear systems for which no mathematicalmodels but measured trajectories are available. Data-driven control techniqueshave been developed to circumvent a time-consuming modelling by firstprinciples and because of the increasing availability of data. Recently, thisresearch field has gained increased attention by the application of Willems'fundamental lemma, which provides a fertile ground for the development ofdata-driven control schemes with guarantees for linear time-invariant systems.While the fundamental lemma can be generalized to further system classes, theredoes not exist a comparable data-based system representation for nonlinearsystems. At the same time, nonlinear systems constitute the majority ofpractical systems. Moreover, they include additional challenges such asdata-based surrogate models that prevent system analysis and controller designby convex optimization. Therefore, a variety of data-driven control approacheshas been developed with different required prior insights into the system toensure a guaranteed inference. In this survey, we will discuss developments inthe context of data-driven control for nonlinear systems. In particular, wewill focus on methods based on system representations providing guarantees fromfinite data, while the analysis and the controller design boil down to convexoptimization problems given as SDP. Thus, these approaches achieve reasonableadvances compared to the state-of-the-art system analysis and controller designby models from system identification. Specifically, the paper covers systemrepresentations based on extensions of Willems' fundamental lemma, setmembership, kernel techniques, the Koopman operator, and feedbacklinearization.

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