Summary This article considers uncertain disturbed systems represented under initial and measurement errors in linear discrete‐time state‐space. A novel bias‐constrained a posteriori H2$$ {H}_2 $$ optimal unbiased finite impulse response (OUFIR) state observer is derived, in which the requirements of the initial state is removed by embedding the unbiasedness constraint. A suboptimal bias‐constrained H2$$ {H}_2 $$‐FIR filtering algorithms using linear matrix inequality is also obtained. Based on an example of uncertain system and error matrices, it is shown that the H2$$ {H}_2 $$‐OUFIR state observer offers a better tradeoff between the accuracy and robustness than the OFIR filter, maximum likelihood FIR filter, and Kalman filter. It loses in robustness to the unbiased FIR filter, but provides a better accuracy. The performance of the proposed filter is verified experimentally based on daily glucose measurements in diabetic persons with timing jitter.

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Finite impulse response (FIR) filtering is known to be more robust than Kalman filtering. In this article, we derive a discrete convolution-based H ∞ -FIR observer for disturbed systems under measurement and initial errors. The gain for the H ∞ -FIR observer is obtained numerically by solving a linear matrix inequality (LMI). Since the LMI has a term that is quadratic with respect to the filter gain, we modify and constrain LMI by introducing an additional variable and proving a theorem. It is shown numerically and experimentally that for disturbed systems operating under measurement and initial errors, the developed H ∞ -FIR observer surpasses the optimal FIR and Kalman filters in accuracy and has almost the same robustness as a robust unbiased FIR filter.

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