Control studies of time-delayed dynamical systems with the method of continuous time approximation

This paper presents control studies of delayed dynamical systems with the help of the method of continuous time approximation (CTA). The CTA method proposes a continuous time approximation of the delayed portion of the response leading to a high and finite dimensional state space formulation of the time-delayed system. Various controls of the system such as LQR and output feedback controls are readily designed with the existing design tools. The properties of the method in frequency domain are also discussed. We have found that time-domain methods such as semi-discretization and CTA, and other numerical integration algorithms can produce highly accurate temporal responses and dominant poles of the system, while missing all the fast and high frequency poles, which explains why many numerical methods can be applied to study the stability of time-delayed systems, and may not be a good tool for control design. Optimal feedback controls for a linear oscillator, collocated and non-collocated feedback controls of an Euler beam, and an experimental demonstration are presented in the paper.