State Update Equation

This page is going to be the shortest page of this tutorial. I've provided extensive description of the State Update Equation in the "\( \alpha -\beta -\gamma \) filter" section and "One-dimensional Kalman Filter section".

The State Update Equation in the matrix form is given by:

\[ \boldsymbol{\hat{x}_{n,n} = \hat{x}_{n,n-1} + K_{n} ( z_{n} - H \hat{x}_{n,n-1} )} \]
where:
\( \boldsymbol{\hat{x}_{n,n}} \) is a estimated system state vector at time step \( n \)
\( \boldsymbol{\hat{x}_{n,n-1}} \) is a predicted system state vector at time step \( n - 1 \)
\( \boldsymbol{K_{n}} \) is a Kalman Gain
\( \boldsymbol{z_{n}} \) is a measurement
\( \boldsymbol{H} \) is an observation matrix

You should be familiar with all components of the State Update Equation except the Kalman Gain in a matrix notation. We will derive the Kalman Gain a little bit later.

You should pay attention on the dimensions. If, for instance, the state vector has 5 dimensions, while only 3 dimensions are measurable (the first, third and fifth states):

\[ \boldsymbol{x(n)} = \left[ \begin{matrix} x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\ x_{5}\\ \end{matrix} \right] \boldsymbol{z(n)} = \left[ \begin{matrix} z_{1}\\ z_{3}\\ z_{5}\\ \end{matrix} \right] \]

The observation matrix would be \( 3 \times 5 \) matrix:

\[ \boldsymbol{H} = \left[ \begin{matrix} 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\\ \end{matrix} \right] \]

The innovation \( \left( \boldsymbol{ z_{n} - H \hat{x}_{n,n-1}} \right) \) yields:

\[ \left( \boldsymbol{ z_{n} - H \hat{x}_{n,n-1}} \right) = \left[ \begin{matrix} z_{1}\\ z_{3}\\ z_{5}\\ \end{matrix} \right] - \left[ \begin{matrix} 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\\ \end{matrix} \right] \left[ \begin{matrix} \hat{x}_{1}\\ \hat{x}_{2}\\ \hat{x}_{3}\\ \hat{x}_{4}\\ \hat{x}_{5}\\ \end{matrix} \right] = \left[ \begin{matrix} (z_{1} - \hat{x}_{1})\\ (z_{3} - \hat{x}_{3})\\ (z_{5} - \hat{x}_{5})\\ \end{matrix} \right] \]

The Kalman Gain dimensions shall be \( 5 \times 3 \).

State Update Equation dimensions

The following table specifies the matrix dimensions of the State Update Equation variables:

Variable Description Dimension
\( \boldsymbol{x} \) state vector \( n_{x} \times 1 \)
\( \boldsymbol{z} \) output vector \( n_{z} \times 1 \)
\( \boldsymbol{H} \) observation matrix \( n_{z} \times n_{x} \)
\( \boldsymbol{K} \) Kalman gain \( n_{x} \times n_{z} \)
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