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.
Highlights
Parameter estimation deals with the problem of obtaining mathematical models that represent dynamic systems based on observed data
For ordinary dynamic systems, various methods are available in the literature, such as linear least-squares gradient-weighted least-squares, Kalman filtering and other robust techniques (Ljung, 1999; Panich, 2010)
They have been widely used in signal processing, robotics, civil engineering
Summary
Parameter estimation deals with the problem of obtaining mathematical models that represent dynamic systems based on observed data. The technique is based on analytical solutions to ODEs in terms of exponential functions. Estimation of delay from free responses: The Lambert W function has been used to solve DDEs (Asl and Ulsoy, 2003).
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