Pole placement for delay differential equations with time-periodic delays using Galerkin approximations

Abstract

In this work, a new methodology is proposed to obtain the feedback gains for the closed-loop control systems having time-periodic delays. A new pseudo-inverse method combined with Galerkin approximations is developed which approximates the delay differential equations (DDEs) with time-periodic delays to a system of time-periodic ordinary differential equations (ODEs). Floquet theory is applied to obtain the stability of the resulting time-periodic ODEs. Later, an optimization approach is used to find suitable feedback gains to stabilize the system. The gains so obtained result in the spectral radius of the Floquet transition matrix (FTM) to be less than unity. The proposed pseudo-inverse method is validated by comparing the results so obtained for the examples from literature for both first and second-order systems. The proposed optimization approach in combination with the pseudo-inverse method was found to stabilize the systems that were considered from the literature and satisfactory results were obtained.