State-estimation

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State-estimation

The basic framework for the EKF involves estimation of the state of a discrete-time nonlinear dynamic system,

$\displaystyle {\bf x}_{k+1}$	$\displaystyle =$	$\displaystyle F({\bf x}_k,{\bf v}_k)$	(1)
$\displaystyle {\bf y}_{k}$	$\displaystyle =$	$\displaystyle H({\bf x}_k,{\bf n}_k),\vspace{-.07in}$	(2)

where ${\bf x}_k$ represent the unobserved state of the system and ${\bf y}_k$ is the only observed signal. The process noise ${\bf v}_k$ drives the dynamic system, and the observation noise is given by ${\bf n}_k$ . Note that we are not assuming additivity of the noise sources. The system dynamic model

and

are assumed known.

In state-estimation, the EKF is the standard method of choice to achieve a recursive (approximate) maximum-likelihood estimation of the state ${\bf x}_k$ . We will review the EKF itself in this context in Section 2 to help motivate the Unscented Kalman Filter (UKF).