Parameter Estimation in a System of Integro-Differential Equations with Time-Delay

• Published in: IEEE transactions on circuits and systems. II, Express briefs Vol. 70; no. 11; p. 1
• Main Authors: Kumar, Abhimanyu, Saxena, Sahaj
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.11.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Circuits and systems; Delays; Differential calculus; Differential equations; Error minimization; Functions (mathematics); Integro-differential equations with time-delay; Linear equations; Mathematical models; Numerical stability; Optimization; Orthogonal function; Parameter estimation; Polynomials; Stability analysis; Upper bound; Upper bounds
• ISSN: 1549-7747; 1558-3791
• DOI: 10.1109/TCSII.2023.3276080

This brief presents a technique for estimating the parameters of a system of integro-differential equations with a time-delay component using some given samples. The considered system is reduced to a set of linear equations by expanding the involved functions in the triangular orthogonal polynomials. While the unknowns (system parameters) can now be obtained by solving the linear equations, however, it is an overdetermined system that is usually inconsistent. In addition, there is no guarantee that the estimated parameters will predict the states nicely outside the time range from which the samples are given. Hence, the upper bound on the error (in the state prediction) is minimized, which depends only on the system parameters and is obtained using the Parseval-Plancherel identity, Cauchy-Schwarz inequality, and the Mean Value theorem. Consequently, an optimization problem is set up to minimize the accumulated error subject to the set of linear equations. The proposed technique is validated numerically where the estimated parameters are found to be close to the true values.