Convolutional 2D Kalman filter & LQG control
Abstract
This paper extends the solution of the discrete-time Kalman filtering and Linear Quadratic Gaussian (LQG) control problems to stochastic systems with spatially invariant linear dynamics over infinite 2-dimensional (2D) grids. Solutions of the corresponding deterministic LQ problem have been available in the literature for over two decades. However, the extension to stochastic problems (Kalman filtering/LQG) is so far restricted to random fields with almost-surely bounded realizations, and hence bounded spatial domains. Conversely, this extension is no longer valid when the state, inputs and outputs are vectors of unbounded random fields. In this paper, constructive solutions are provided for spatially invariant systems driven by Gaussian white noises with spatially invariant distributions over infinite grids. The optimal gains are obtained from Algebraic Riccati Equations (AREs) involving matrices of 2D operators. These convolutional AREs are then solved directly in the spatial domain. The resulting procedures enable massive parallelization of off-line and on-line computations. The convolutional approach also enables to better quantify and limit the impact of spatial and frequency-domain truncations. The construction of convolutional discrete-time models from systems of coupled ordinary differential equations and of spatially invariant stochastic Autoregressive (AR) disturbance models from spatial and temporal statistics is discussed. An illustrative application – state-feedback and output-feedback convolutional LQG control of a large deformable mirror – is presented.
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BibTeX
@article{raynaud2026convolutional,
title = {{Convolutional 2D Kalman filter & LQG control}},
author = {Henri-François Raynaud and Caroline Kulcsár},
journal = {European Journal of Control},
year = {2026},
month = oct,
doi = {10.1016/j.ejcon.2026.101543},
url = {https://iogs.hal.science/hal-05836768v1},
note = {CC BY},
}
RIS
TY - JOUR
TI - Convolutional 2D Kalman filter & LQG control
AU - Henri-François Raynaud
AU - Caroline Kulcsár
PY - 2026
DA - 2026/10/23
JO - European Journal of Control
DO - 10.1016/j.ejcon.2026.101543
UR - https://iogs.hal.science/hal-05836768v1
AB - This paper extends the solution of the discrete-time Kalman filtering and Linear Quadratic Gaussian (LQG) control problems to stochastic systems with spatially invariant linear dynamics over infinite 2-dimensional (2D) grids. Solutions of the corresponding deterministic LQ problem have been available in the literature for over two decades. However, the extension to stochastic problems (Kalman filtering/LQG) is so far restricted to random fields with almost-surely bounded realizations, and hence bounded spatial domains. Conversely, this extension is no longer valid when the state, inputs and outputs are vectors of unbounded random fields. In this paper, constructive solutions are provided for spatially invariant systems driven by Gaussian white noises with spatially invariant distributions over infinite grids. The optimal gains are obtained from Algebraic Riccati Equations (AREs) involving matrices of 2D operators. These convolutional AREs are then solved directly in the spatial domain. The resulting procedures enable massive parallelization of off-line and on-line computations. The convolutional approach also enables to better quantify and limit the impact of spatial and frequency-domain truncations. The construction of convolutional discrete-time models from systems of coupled ordinary differential equations and of spatially invariant stochastic Autoregressive (AR) disturbance models from spatial and temporal statistics is discussed. An illustrative application – state-feedback and output-feedback convolutional LQG control of a large deformable mirror – is presented.
SN - 0947-3580
ER -