State Update Equation
This page is the shortest page of this tutorial. I've provided an extensive description of the State Update Equation in the "\( \alpha -\beta -\gamma \) filter" section and the "One-dimensional Kalman Filter section".
The State Update Equation in the matrix form is given by:
| \( \boldsymbol{\hat{x}}_{n,n} \) | is an 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 derive the Kalman Gain in the following chapters.
You should pay attention to 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 a \( 3 \times 5 \) 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 \( \left( \boldsymbol{z}_{n} - \boldsymbol{H \hat{x}}_{n,n-1} \right) \) yields:
\[ \left( \boldsymbol{z}_{n} - \boldsymbol{H \hat{x}}_{n,n-1} \right) = \left[ \begin{matrix} z_{1}\\ z_{3}\\ z_{5}\\ \end{matrix} \right] - \left[ \begin{matrix} 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\\ \end{matrix} \right] \left[ \begin{matrix} \hat{x}_{1}\\ \hat{x}_{2}\\ \hat{x}_{3}\\ \hat{x}_{4}\\ \hat{x}_{5}\\ \end{matrix} \right] = \left[ \begin{matrix} (z_{1} - \hat{x}_{1})\\ (z_{3} - \hat{x}_{3})\\ (z_{5} - \hat{x}_{5})\\ \end{matrix} \right] \]
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 | \( n_{x} \times 1 \) |
| \( \boldsymbol{z} \) | measurements vector | \( n_{z} \times 1 \) |
| \( \boldsymbol{H} \) | observation matrix | \( n_{z} \times n_{x} \) |
| \( \boldsymbol{K} \) | Kalman gain | \( n_{x} \times n_{z} \) |