The Extended Kalman Filter: An Interactive Tutorial for Non-Experts

Part 8: A More Realistic Model

Recall the two equations describing our system: \[x_k = a x_{k-1}\] \[z_k = x_k + v_k \] where $x_k$ is the current state of our system, $x_{k-1}$ is its previous state, $a$ is some constant, $z_k$ is our current observation of the system, and $v_k$ is the current noise (inaccuracy) associated with the observation.

Although these two equations apply well to many kinds of systems, they are sometimes not the whole story. For one thing, we have not accounted for the time-varying control that the pilot exercises over the airplane, by (for example) moving the control column forward and back. To account for the control we introduce another subscripted variable $u_k$, representing the current value of the control signal that the pilot is sending to the airplane. Just as the previous state $x_{k-1}$ was scaled by a constant amount $a$, this control signal can be scaled by a constant amount if we like; call it $b$. So our complete equation for the state becomes

$x_k = a x_{k-1}$ $~+~b u_k$

with the new component highlighted in blue.

In general, any signal other than noise can be scaled by some constant, so our equation for the observation $z_k$ can be rewritten thus: [11]

$z_k =$ $c$ $x_k + v_k$

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[11] For the sake of consistency with the original example I have chosen to use variables $a$, $b$, and $c$ here for the constants, instead of the more common $f$, $b$, and $h$.