Numerical computation of cross‐covariance functionals for linear systems with multiple time delays

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Abstract
The problem of evaluating a cross‐covariance functional of the form 1/2πi?i8 ‐i8 F(s)G(‐s)ds arises, e.g., in parametric optimization of linear systems and in optimal linear filtering. In this paper, an algorithm is presented for computing cross‐covariance functionals for linear systems with multiple commensurate time delays. The algorithm is based on converting a cross‐covariance functional into an equivalent auto‐covariance one. Then by utilizing an existing partial‐fraction‐expansion scheme for a class of two‐dimensional rational transfer functions, the evaluation of a cross‐covariance functional is reduced to the determination of symbolic inversion of a polynomial matrix and the spectral factorization of a finite‐degree real‐coefficient polynomial. It is shown that the algorithm presented here can be extended to systems having non‐commensurate time delays by either introducing fictitious delays or adjusting time delays with rational approximations such that the resulting delays are commensurate. In order to illustrate the effectiveness of the algorithm, several examples are provided.

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