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

  • Irma Jankauskienė Kaunas University of Technology
  • Jonas Rimas
Keywords: differential equations, delayed arguments, Lambert W function


For investigation of the dynamics  of linear time-delay system it is proposed 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 \tau   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 +\infty)  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.

DOI: http://dx.doi.org/10.5755/j01.itc.46.2.16208