A Kalman filter estimates a state from a viewpoint of statistics. let us consider a system

where $\bm{x}$, $\bm{y}$, and $\bm{u}$ are a state, output, and input; $\bm{A}$, $\bm{B}$, and $\bm{C}$ stand for a system matrix, input matrix, and observation matrix; $\bm{v}$ and $\bm{w}$ are a process noise and observation noise that satisfy

The filtering characteristic of a Kalman filter is determined by the variances of a process noise and observation noise. A filtering process is composed of two steps:

where a subscription $_{\rm m}$ stands for a model parameter, and subscriptions $_{\rm p}$ and $_{\rm K}$ denote a predicted value and filter output.

where $\bm{K}$ is a Kalman gain. This gain is uniquely sought by system information.

A Kalmain gain is selected to minimize a cost function

Such a gain is uniquely determined by focusing on the transition of an estimation error covariance.

A prediction error is calculated as

Then, an error covariance of prediction state is calculated as

An estimated error is calculated as

Then, an error covariance of a filtered state is calculated as

To minimize the cost function, the filtering system should satisfies a following condition

This condition derives

Assuming that a Kalman gain is appropriately set, the prediction process of an error covariance $\bm{P}_{\rm K}$ can be simplified as

A Kalmain filter for a linear time-invariant system has a constant error covariance matrix in a steady-state. The prediction process of error covariance $\bm{P}_{\rm p}$ can be expanded as

In a steady state, an error covariance is invariant and a following equation holds.

Here, a Kalman gain also get a constant value, and a observer is called a steady-state Kalman filter.

An output is composed of a low-pass filtered state and predicted state.

where

The filter gain is determined by the norms of process/observation noises. $\bm{F}[k]$ indicates an internal division point of these norms.

A predicted state $\bm{x}_{\rm p}$ is an output of a Luenberger observer and a filtered state $\bm{x}_{\rm K}$ is a smoothed signal of $\bm{x}_{\rm p}$.