Time-Delay Estimation using the Characteristic Roots of Delay Differential Equations

Abstract

Problem statement: For ordinary dynamic systems (i.e., non-delayed), various methods such as linear least-squares, gradient-weighted least-squares, Kalman filtering and other robust techniques have been widely used in signal processing, robotics, civil engineering. On the other hand, time-delay estimation of systems with unknown time-delay is still a challenging problem due to difficulty in formulation caused. Approach: The presented method makes use of the Lambert W function and analytical solutions of scalar first-order Delay Differential Equations (DDEs). The Lambert W function has been known to be useful in solving delay differential equations. From the solutions in terms of the Lambert W function, the dominant characteristic roots can be obtained and used to estimate time-delays. The function is already embedded in various software packages (e.g., MATLAB) and thus, the presented method can be readily used for time-delay systems. Results: The presented method and the provided examples show ease of formulation and accuracy of time-delay estimation. Conclusion: Estimation of time-delays can be conducted in an analytical way. The presented method will be extended to general systems of DDEs and application to physical systems.

Adaptive Control for Quadrotor UAVs Considering Time Delay: Study with Flight Payload

“…In this paper, the constant delay applied in the drone's altitude control system was estimated using an approach based on analytical solutions to DDEs [19,20]. The time delay has been estimated as 0.36s in [21,22].…”

“…, a first order plant, used in [19,20] for the estimation of the delay, then this time-delay system will be neutral type, see [21,22] for analysis of such control system. Also, if 1 ( ) G s s = and then we have a P-control, retarded type time-delay, system.…”