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In recent years data-driven analysis of dynamical systems has attracted a lotof attention and transfer operator techniques, namely, Perron-Frobenius andKoopman operators are being used almost ubiquitously. Since data is alwaysobtained in discrete-time, in this paper, we propose a purely data-drivenapproach for the design of a stabilizing feedback control law for a generalclass of discrete-time control-affine non-linear systems. In particular, we usethe Koopman operator to lift a control-affine system to a higher-dimensionalspace, where the control system's evolution is bilinear. We analyze thecontrollability of the lifted bilinear system and relate it to thecontrollability of the underlying non-linear system. We then leverage theconcept of Control Lyapunov Function (CLF) to design a state feedback law thatstabilizes the origin. Furthermore, we demonstrate the efficacy of the proposedmethod to stabilize the origin of the Van der Pol oscillator and the chaoticHenon map from the time-series data.

Data-driven Stabilization of Discrete-time Control-affine Nonlinear Systems: A Koopman Operator Approach