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

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.