Koopman Observability of Nonlinear Systems and Data-Driven Koopman--Luenberger Observer

Abstract

While nonlinear systems can be generally described as infinite-dimensional linear systems via their Koopman semigroups, the direct synthesis of state observers on the premise of learned Koopman operator models, in an analogous manner to finite-dimensional linear systems, remains an open problem. In this paper, the concepts of Koopman observability (exact and approximate), Koopman detectability (exponential and strong), and Koopman--Luenberger observer of nonlinear systems are provided. By defining the Koopman semigroup on a reproducing kernel Hilbert space (RKHS), the conditions for Koopman exponential detectability are established, which involves solving an operator Lyapunov inequality that only implicitly involves the the infinitesimal generator of the Koopman semigroup. Through the use of a finite-horizon Yosida approximation of the Koopman generator and an empirical sampling operator, the data-driven solution of the observer gain can be reduced to a finite-dimensional subspace specified by the sample and hence fulfilled by convex semidefinite programming. The state observation error is guaranteed to be bounded in mean squares, under closed-loop dissipation conditions on the sampling operator and its empirical operator.