Koopman-based control using sum-of-squares optimization: Improved stability guarantees and data efficiency

Abstract: In this paper, we propose a novel controller design approach for unknown nonlinear systems using the Koopman operator. In particular, we use the recently proposed stability- and certificate-oriented extended dynamic mode decomposition (SafEDMD) architecture to generate a data-driven bilinear surrogate model with certified error bounds. Then, by accounting for the obtained error bounds in a controller design based on the bilinear system, one can guarantee closed-loop stability for the true nonlinear system. While existing approaches over-approximate the bilinearity of the surrogate model, thus introducing conservatism and providing only local guarantees, we explicitly account for the bilinearity by using sum-of-squares (SOS) optimization in the controller design. More precisely, we parametrize a rational controller stabilizing the error-affected bilinear surrogate model and, consequently, the underlying nonlinear system. The resulting SOS optimization problem provides explicit data-driven controller design conditions for unknown nonlinear systems based on semidefinite programming. Our approach significantly reduces conservatism by establishing a larger region of attraction and improved data efficiency. The proposed method is evaluated using numerical examples, demonstrating its advantages over existing approaches.