Spectral Submanifolds of Delay Systems: Control in Continuous and Discrete Time

ABSTRACT This paper presents both continuous‐time and discretization‐based approaches for constructing spectral submanifolds (SSMs) for time delay systems. The continuous‐time approach built upon sun–star calculus, reformulates the governing delay differential equation (DDE) as an operator differential equation (OpDE). This way, the obtained SSM captures the inherent infinite‐dimensionality of the underlying time delay system, while the corresponding reduced dynamics is low‐dimensional. In contrast, by discretizing the infinitesimal generator, we approximate the original DDE with a large system of ordinary differential equations (ODEs) and then carry out the SSM calculation. We prove that as the discretization number approaches infinity, the eigenvalues, the eigenvectors, and the SSM obtained via the discretization‐based approach converge to those obtained via the continuous‐time approach. The accuracy of three different discretization methods is compared with the exact continuous‐time results through two case studies. First, we analyze a human–machine interaction model involving a human driver guided by an automated vehicle. Here, the SSM reduction is carried out corresponding to the dominant real eigenvalue. Then, a two‐degree‐of‐freedom oscillatory system is considered under delayed proportional‐derivative control, where a two‐dimensional SSM is constructed for the dominant pair of complex conjugate eigenvalues.