Koopman-Theoretic Modeling of Quasiperiodically Driven Systems: Example of Signalized Traffic Corridor

Abstract

This article presents a novel approach to analyzing quasiperiodically driven dynamical systems. It presents a holistic data-driven framework for reconstructing such a system and gaining insights into its dynamical behavior. A quasiperiodically driven dynamics has two components: 1) the quasiperiodic driving source with generating frequencies and 2) the driven nonlinear dynamics. The driving dynamics are reconstructed by accurately computing a large number of Koopman eigenfrequencies, using techniques based on the theory of reproducing kernel Hilbert space (RKHS) and from ergodic theory. Unlike Fourier analysis and dynamic mode decomposition (DMD), the proposed framework can analyze systems/signals with nondominant or absent periodic components. The driven system is next reconstructed by continuing to use the RKHS framework. As a case study, we investigate data collected from a corridor of nine consecutive signalized traffic intersections. The proposed framework provides insights into generating frequencies and associated modes, accurately reconstructs the queue lengths at the signalized intersections, and makes stable long-term forecasts. The unique contributions of this article are spread across multiple domains. It includes the novel modeling framework for quasiperiodically driven systems, a computationally efficient approach able to handle a large amount of data, utilization of true Koopman eigenfrequencies, and an illustration of the proposed approach on a corridor of signalized traffic intersections.