Solved – Kalman filter with input control noise

assume we have a standard Kalman filter with input controls, following wikipedia notation ( where the latent state is $x_{t}$ and the observation is $z_{t}$, following the equations:

$mathbf{x}_{k} = mathbf{F}_{k} mathbf{x}_{k-1} + mathbf{B}_{k} mathbf{u}_{k} + mathbf{w}_{k}$

$mathbf{z}_k = mathbf{H}_{k} mathbf{x}_k + mathbf{v}_k$

where $mathbf{u}_{k}$ and $mathbf{v}_k$ are Gaussian noise terms:

$mathbf{w}_k sim N(0, mathbf{Q}_k)$

$mathbf{v}_k sim N(0, mathbf{R}_k)$

assume now that the input controls $u_{k}$ are not given to the system perfectly. instead the system only senses what the control input corrupted by some additive Gaussian noise, which is denoted $c_{k}$:

$mathbf{c}_{k} = mathbf{u}_{k} + mathbf{m}_{k}$

where $mathbf{m}_{k} sim N(0, mathbf{M}_{k})$, so the full model is:

$mathbf{c}_{k} = mathbf{u}_{k} + mathbf{m}_{k}$

$mathbf{x}_{k} = mathbf{F}_{k} mathbf{x}_{k-1} + mathbf{B}_{k} mathbf{c}_{k} + mathbf{w}_{k}$

$mathbf{z}_k = mathbf{H}_{k} mathbf{x}_k + mathbf{v}_k$

is it still a Kalman filter? if so do the filtering equations significantly change or is it still as tractable as original Kalman filter?

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}

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