Editing Kalman filter (section)

== Example application, technical ==
[[File:kalman.png|class=skin-invert-image|thumb|
{{legend-line|solid light-dark(black,white)|Truth}}
{{legend-line|solid green|Filtered process}}
{{legend-line|dotted red|Observations}}
]]
Consider a truck on frictionless, straight rails. Initially, the truck is stationary at position 0, but it is buffeted this way and that by random uncontrolled forces. We measure the position of the truck every Δ''t'' seconds, but these measurements are imprecise; we want to maintain a model of the truck's position and [[velocity]]. We show here how we derive the model from which we create our Kalman filter.

Since <math>\mathbf{F}, \mathbf{H}, \mathbf{R}, \mathbf{Q}</math> are constant, their time indices are dropped.

The position and velocity of the truck are described by the linear state space
:<math>\mathbf{x}_k = \begin{bmatrix}
    x \\
    \dot{x}
  \end{bmatrix}
</math>

where <math>\dot{x}</math> is the velocity, that is, the derivative of position with respect to time.

We assume that between the (''k''&nbsp;−&nbsp;1) and ''k'' timestep, uncontrolled forces cause a constant acceleration of ''a''<sub>''k''</sub> that is [[normal distribution|normally distributed]] with mean 0 and standard deviation ''σ''<sub>''a''</sub>. From [[Newton's laws of motion]] we conclude that
:<math>\mathbf{x}_k = \mathbf{F} \mathbf{x}_{k-1} + \mathbf{G} a_k</math>

(there is no <math>\mathbf{B}u</math> term since there are no known control inputs. Instead, ''a''<sub>''k''</sub> is the effect of an unknown input and <math>\mathbf{G}</math> applies that effect to the state vector) where
:<math>\begin{align}
  \mathbf{F} &= \begin{bmatrix}
    1 & \Delta t \\
    0 & 1
  \end{bmatrix} \\[4pt]
  \mathbf{G} &= \begin{bmatrix}
    \frac{1}{2}{\Delta t}^2 \\[6pt]
    \Delta t
  \end{bmatrix}
\end{align}</math>

so that
:<math>\mathbf{x}_k = \mathbf{F} \mathbf{x}_{k-1} + \mathbf{w}_k</math>

where
:<math>\begin{align}
  \mathbf{w}_k &\sim N(0, \mathbf{Q}) \\
    \mathbf{Q} &=
  \mathbf{G}\mathbf{G}^\textsf{T}\sigma_a^2 =
  \begin{bmatrix}
    \frac{1}{4}{\Delta t}^4 & \frac{1}{2}{\Delta t}^3 \\[6pt]
    \frac{1}{2}{\Delta t}^3 & {\Delta t}^2
  \end{bmatrix}\sigma_a^2.
\end{align}</math>

The matrix <math>\mathbf{Q}</math> is not full rank (it is of rank one if <math>\Delta t \neq 0</math>). Hence, the distribution <math>N(0, \mathbf{Q})</math> is not absolutely continuous and has [[Multivariate normal distribution#Degenerate case|no probability density function]]. Another way to express this, avoiding explicit degenerate distributions is given by
:<math>\mathbf{w}_k \sim \mathbf{G} \cdot N\left(0, \sigma_a^2\right). </math>

At each time phase, a noisy measurement of the true position of the truck is made. Let us suppose the measurement noise ''v''<sub>''k''</sub> is also distributed normally, with mean 0 and standard deviation ''σ''<sub>''z''</sub>.
:<math>\mathbf{z}_k = \mathbf{H x}_k + \mathbf{v}_k</math>

where
:<math>\mathbf{H} = \begin{bmatrix} 1 & 0 \end{bmatrix} </math>

and
:<math>
  \mathbf{R} = \mathrm{E}\left[\mathbf{v}_k \mathbf{v}_k^\textsf{T}\right]
             = \begin{bmatrix} \sigma_z^2 \end{bmatrix}
</math>

We know the initial starting state of the truck with perfect precision, so we initialize
:<math>\hat{\mathbf{x}}_{0 \mid 0} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} </math>

and to tell the filter that we know the exact position and velocity, we give it a zero covariance matrix:
:<math>\mathbf{P}_{0 \mid 0} = \begin{bmatrix}
    0 & 0 \\
    0 & 0
  \end{bmatrix}
</math>

If the initial position and velocity are not known perfectly, the covariance matrix should be initialized with suitable variances on its diagonal:
:<math>\mathbf{P}_{0 \mid 0} = \begin{bmatrix}
    \sigma_x^2 & 0 \\
    0          & \sigma_\dot{x}^2
  \end{bmatrix}
</math>

The filter will then prefer the information from the first measurements over the information already in the model.