Koopman Modeling and Stabilization of Discrete-Time Nonlinear Control Systems: Bilinearity on a Reproducing Kernel Hilbert Space
Despite the popularity of Koopman modeling for nonlinear systems, in the presence of input variables, the evident nonexistence of a fully linear time-invariant model even in infinite dimensions makes Koopman-based control largely an open problem to date. Focusing on discrete-time systems in this paper, which eschews from using operator semigroup and infinitesimal generator notions, it is proven that nonlinear systems, if satisfying appropriate smoothness and regularity conditions, can be expressed exactly as bilinear dynamics, when the state variables and input variables are separately lifted into their reproducing kernel Hilbert spaces (RKHSs). To account for the knowledge of an equilibrium point at the origin, the RKHS is defined by a linear--radial product kernel, and hence the functions belonging to this RKHS are spanned by the multiplications of component functions and Sobolev functions. The stabilization problem, namely the determination of a feedback law that causes a Lyapunov function (expressed as a kernel sum-of-squares form) to decrease, is then posed as an infinite-dimensional optimization problem over state-dependent conditional probability measures over the input space, solved via a discretization scheme.