# Solved – 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?

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