A Note on the Use of Step Responses Matrix and Lambert W Function in the Dynamics Analysis of Time Delay Systems

For investigation of the dynamics of linear time-delay system, we propose to use the step responses matrix of the system and apply the Lambert W function method. The elements of this matrix in the first stage are being found on the interval [0, tau] (here t is a fixed delay). For this, it is sufficient to solve an ordinary matrix differential equation. After that, using obtained solution as initial vector-function, elements of the step responses matrix are being found on the interval (tau,+ infinity) by solving a linear homogeneous matrix delay differential equation. The well known Lambert W function method can be used for solution of this equation. The proposed method is illustrated on concrete time-delay systems.