Identification of linear systems with time-delay operating in a closed loop in the presence of noise

The subject of this short paper is on-line identification of the parameter vector defining a linear dynamical system which operates in a closed loop in the presence of noise, and incorporates a time delay. The method is based on the equation error. Other known subsystems in the closed-loop system increase the dimension of the closed-loop parameter vector which tends to degrade the estimation convergence process. By means of "composite state variables," introduced in this short paper, this increase is prevented and the open-loop parameters are directly identified from closed-loop input-output data. The pure time delay in the closed loop causes a representation problem in the equation error formulation. This is overcome by "composite delayed state variables." The value of the time delay is determined by means of excess parameters provided by a "higher order model" and a simple on-line search procedure. The method is illustrated by simulated examples.