Optimal Linear Filtering for Discrete-Time Systems with Infinite-Dimensional Measurements
<|reference_start|>Optimal Linear Filtering for Discrete-Time Systems with Infinite-Dimensional Measurements: Systems equipped with modern sensing modalities such as vision and lidar gain access to increasingly high-dimensional measurements with which to enact estimation and control schemes. In this article, we examine the continuum limit of high-dimensional measurements and analyze state estimation in linear time-invariant systems with infinite-dimensional measurements but finite-dimensional states, both corrupted by additive noise. We propose a linear filter and derive the corresponding optimal gain functional in the sense of the minimum mean square error, analogous to the classic Kalman filter. By modeling the measurement noise as a wide-sense stationary random field, we are able to derive the optimal linear filter explicitly, in contrast to previous derivations of Kalman filters in distributed-parameter settings. Interestingly, we find that we need only impose conditions that are finite-dimensional in nature to ensure that the filter is asymptotically stable. The proposed filter is verified via simulation of a linearized system with a pinhole camera sensor.<|reference_end|>
@article{varley2024optimal,
    title={Optimal Linear Filtering for Discrete-Time Systems with
  Infinite-Dimensional Measurements},
    author={Maxwell Varley, Timothy L. Molloy, Girish N. Nair},
    journal={arXiv preprint arXiv:2409.12368},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.12368},
    primaryClass={eess.SY cs.SY}
}