# Kalman filter with input control noise?

assume we have a standard Kalman filter with input controls, following wikipedia notation (http://en.wikipedia.org/wiki/Kalman_filter) 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?

Substitute $\mathbf{c}_k = \mathbf{u}_k + \mathbf{m}_k$ into the dynamic part of the new model to obtain $$\mathbf{x}_k = \mathbf{F}_{k}\,\mathbf{x}_{k-1} + \mathbf{B}_k\,\mathbf{u}_k + \mathbf{B}_k\,\mathbf{m}_k + \mathbf{w}_k.$$ Let us define a new random variables $$\tilde{\mathbf{w}}_k = \mathbf{B}_k\,\mathbf{m}_k + \mathbf{w}_k.$$ 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 $$\mathbb{E}(\mathbf{\tilde{w}}_k) = \mathbb{E}(\mathbf{B}_k\,\mathbf{m}_k + \mathbf{w}_k) = \mathbf{B}_k\,\mathbb{E}(\mathbf{m}_k) + \mathbb{E}(\mathbf{w}_k) = \mathbf{0}$$ and $$\mathrm{Cov}(\mathbf{\tilde{w}}_k) = \mathrm{Cov}(\mathbf{B}_k\,\mathbf{m}_k + \mathbf{w}_k) = \mathrm{Cov}(\mathbf{B}_k\,\mathbf{m}_k) + \mathrm{Cov}(\mathbf{w}_k) = \mathbf{B}_k\,\mathbf{M}_k\,\mathbf{B}^\mathrm{T}_k + \mathbf{Q}_k,$$ 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 $$\begin{array}{ll} \mathbf{x}_k &= \mathbf{F}_k\,\mathbf{x}_{k-1} + \mathbf{B}_k\,\mathbf{u}_k + \mathbf{\tilde{w}}_k, \\ \mathbf{z}_k &= \mathbf{H}_k\,\mathbf{x}_k + \mathbf{v}_k \end{array}$$ 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 $$\mathbf{\tilde{Q}}_k = \mathbf{B}_k\,\mathbf{M}_k\,\mathbf{B}^\mathrm{T}_k + \mathbf{Q}_k.$$