Abstract: The dual use of covariance information enables filtering and smoothing algorithms to effectively capture statistical dependencies and mitigate the effects of system uncertainties and observation noise. In the fixed-point smoother and filter, the existing literature approximates covariance information using finite Fourier series expansions. Building upon this foundation, the present work originally introduces a newly designed robust recursive least-squares fixed-lag (FL) smoother tailored for linear continuous-time stochastic systems with uncertainties. In this framework, the autocovariance function of the degraded signal is modeled as a semidegenerate function, while the cross-covariance function between the signal and the observed measurements is represented as a degenerate function. This paper emphasizes approximating both covariance structures using polynomials and finite Fourier series, thereby enabling flexible, computationally efficient modeling of statistical dependencies essential for robust state estimation. As demonstrated in the numerical simulation example, the mean-square values (MSVs) of the approximation errors associated with both the autocovariance and cross-covariance functions are negligible, indicating high fidelity in the modeling process. Based on polynomial approximations, the estimation accuracy of both the robust filter and the FL smoother is evaluated, highlighting their effectiveness in capturing key statistical dependencies. Additionally, the simulation includes a comparative example in which signal estimation is performed using covariance data approximated via finite Fourier series expansions. The results indicate that both the polynomial- and Fourier-based approximations strategies reliably and consistently support the robustness of the proposed estimation framework.

Keywords: Robust RLS fixed-lag smoother, stochastic systems with uncertainties, continuous-time stochastic systems, covariance information, polynomial approximation, finite Fourier series expansion

Pages: 38-52

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