Solved – Kalman filter with input control noise

The noise in the input can be embedded into the process noise, so that the new system, too, is a linear state-space model with additive Gaussian noise. Thus, Kalman filter can be applied, and only the process noise covariance must be changed to take into account the noise in the input. Details follow.

Substitute $mathbf{c}_k = mathbf{u}_k + mathbf{m}_k$ into the dynamic part of the new model to obtain begin{equation} mathbf{x}_k = mathbf{F}_{k},mathbf{x}_{k-1} + mathbf{B}_k,mathbf{u}_k + mathbf{B}_k,mathbf{m}_k + mathbf{w}_k. end{equation} Let us define a new random variables begin{equation} tilde{mathbf{w}}_k = mathbf{B}_k,mathbf{m}_k + mathbf{w}_k. end{equation} Gaussianity is preserved under affine transformations, thus $mathbf{B}_k,mathbf{m}_k$ is Gaussian. Furthermore, $mathbf{tilde{w}}_k$ is the sum of two Gaussian random variables and thus it is Gaussian with begin{equation} mathbb{E}(mathbf{tilde{w}}_k) = mathbb{E}(mathbf{B}_k,mathbf{m}_k + mathbf{w}_k) = mathbf{B}_k,mathbb{E}(mathbf{m}_k) + mathbb{E}(mathbf{w}_k) = mathbf{0} end{equation} and begin{equation} mathrm{Cov}(mathbf{tilde{w}}_k) = mathrm{Cov}(mathbf{B}_k,mathbf{m}_k + mathbf{w}_k) = mathrm{Cov}(mathbf{B}_k,mathbf{m}_k) + mathrm{Cov}(mathbf{w}_k) = mathbf{B}_k,mathbf{M}_k,mathbf{B}^mathrm{T}_k + mathbf{Q}_k, end{equation} where the second equality follows from assuming that the noises $mathbf{m}_k$ and $mathbf{w}_k$ are independent. Thus, the system with input noise can be expressed as begin{equation} begin{array}{ll} mathbf{x}_k &= mathbf{F}_k,mathbf{x}_{k-1} + mathbf{B}_k,mathbf{u}_k + mathbf{tilde{w}}_k, \ mathbf{z}_k &= mathbf{H}_k,mathbf{x}_k + mathbf{v}_k end{array} end{equation} where $mathbf{tilde{w}}_k sim mathrm{N}(mathbf{0},mathbf{tilde{Q}}_k),~mathbf{v}_k sim mathrm{N}(mathbf{0}, mathbf{R}_k)$ with begin{equation} mathbf{tilde{Q}}_k = mathbf{B}_k,mathbf{M}_k,mathbf{B}^mathrm{T}_k + mathbf{Q}_k. end{equation}