Recursive$\mathcal { \, L}_{2}$-to-$\mathcal {L}_\infty$ Filtering of Disturbed Models Based on the Backward Euler Method

When a system operates under impulsive disturbances and conventional state estimation methods become ineffective, robust peak error reduction is required. In this paper, the error-to-error transfer function approach is used to compute the bias correction gain $\mathbf {K}$ for a robust recursive $\mathcal {L}_{2}$-to-$\mathcal {L}_\infty$ 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 $\mathbf {K}$ is computed using the energy-to-peak lemma, a newly formulated theorem, and a linear matrix inequality. The performances of the $\mathcal {L}_{2}$-to-$\mathcal {L}_\infty$, $H_\infty$, 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 $\mathbf {K}$ of the $\mathcal {L}_{2}$-to-$\mathcal {L}_\infty$ 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 $\mathcal {L}_{2}$-to-$\mathcal {L}_\infty$ filter exhibited the smallest peak errors.