Parameter Estimation and Identification of Nonlinear Systems with the Koopman Operator

We report on a novel framework for parameter estimation and nonlinear system identification. This framework is based on the key idea that nonlinear system identification in the state space is equivalent to linear identification of the Koopman operator in the space of observables. The proposed technique is divided into three steps. First, data is lifted in a higher dimensional space spanned by properly chosen basis functions. Second, a matrix representation of the Koopman operator is identified in this lifted space. This is equivalent to a linear identification problem. Finally, the infinitesimal generator of the Koopman semigroup is computed and used to identify the unknown vector field. Two methods (main and dual) are proposed in this framework and complemented with theoretical convergence results. The main method is applied to parameter estimation in the case of polynomial vector fields and both autonomous and input–output systems are considered. It is shown to be efficient with a general class of systems, including chaotic systems, and well suited to low sampling rate datasets. The other (dual) method is well suited to high-dimensional systems and is used in the context of network reconstruction.