Recursive$\mathcal { \, L}_{2}$-to-$\mathcal {L}_\infty$ Filtering of Disturbed Models Based on the Backward Euler Method
Abstract
When a system operates under impulsive disturbances and conventional state estimation methods become ineffective, robust peak error reduction is required. In this paper, the <italic>error-to-error transfer function</italic> approach is used to compute the bias correction gain <inline-formula><tex-math notation="LaTeX">$\mathbf {K}$</tex-math></inline-formula> for a robust recursive <inline-formula><tex-math notation="LaTeX">$\mathcal {L}_{2}$</tex-math></inline-formula>-to-<inline-formula><tex-math notation="LaTeX">$\mathcal {L}_\infty$</tex-math></inline-formula> filter applied to discrete-time state-space models based on the backward Euler method. The disturbance is treated as a Gauss-Markov sequence, and the gain <inline-formula><tex-math notation="LaTeX">$\mathbf {K}$</tex-math></inline-formula> is computed using the <italic>energy-to-peak</italic> lemma, a newly formulated theorem, and a linear matrix inequality. The performances of the <inline-formula><tex-math notation="LaTeX">$\mathcal {L}_{2}$</tex-math></inline-formula>-to-<inline-formula><tex-math notation="LaTeX">$\mathcal {L}_\infty$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula>, unbiased finite impulse response (UFIR), and Kalman filters are compared numerically based on the quasi periodic harmonic model in terms of mean square error, robustness, and estimation quality. An experimental verification is provided for air quality monitoring in a significantly polluted city area. It is shown that the gain <inline-formula><tex-math notation="LaTeX">$\mathbf {K}$</tex-math></inline-formula> of the <inline-formula><tex-math notation="LaTeX">$\mathcal {L}_{2}$</tex-math></inline-formula>-to-<inline-formula><tex-math notation="LaTeX">$\mathcal {L}_\infty$</tex-math></inline-formula> filter obeys the previously formulated rule of thumb, i.e. it ranges between the gains of the UFIR and Kalman filters. In both numerical and experimental examples, the <inline-formula><tex-math notation="LaTeX">$\mathcal {L}_{2}$</tex-math></inline-formula>-to-<inline-formula><tex-math notation="LaTeX">$\mathcal {L}_\infty$</tex-math></inline-formula> filter exhibited the smallest peak errors.