Measurement Equation
Until now, we've dealt with the future. We've derived two Kalman Filter prediction equations:
- State Extrapolation Equation
- Covariance Extrapolation Equation
From now on, we are going to deal with the present. Let's start with the Measurement Equation.
In the "One-dimensional Kalman Filter" section, we denoted the measurement by \( z_{n} \).
The measurement value represents a true system state in addition to the random measurement noise \( v_{n} \), caused by the measurement device.
The measurement noise variance \( r_{n} \) can be constant for each measurement - for example, scales with a precision of 0.5kg (standard deviation). On the other hand, the measurement noise variance \( r_{n} \) can be different for each measurement - for example, a thermometer with a precision of 0.5% (standard deviation). In the latter case, the noise variance depends on the measured temperature.
The generalized measurement equation in matrix form is given by:
| \( \boldsymbol{z}_{n} \) | is a measurement vector |
| \( \boldsymbol{x}_{n} \) | is a true system state (hidden state) |
| \( \boldsymbol{v}_{n} \) | is a random noise vector |
| \( \boldsymbol{H} \) | is an observation matrix |
The observation matrix \( \boldsymbol{H} \)
In many cases, the measured value is not the desired system state. For example, a digital electric thermometer measures an electric current, while the system state is the temperature. There is a need for a transformation of the system state (input) to the measurement (output).
The purpose of the observation matrix \( \boldsymbol{H} \) is to convert the system state into outputs using linear transformations. The following chapters include examples of observation matrix usage.
Scaling
A range meter sends a signal toward a destination and receives a reflected echo. The measurement is the time delay between the transmission and reception of the signal. The system state is the range.
In this case, we need to perform a scaling:
\[ \boldsymbol{z}_{n} = \left[ \begin{matrix} \frac{2}{c}\\ \end{matrix} \right] \boldsymbol{x}_{n} + \boldsymbol{v}_{n} \]
\[ \boldsymbol{H} = \left[ \begin{matrix} \frac{2}{c}\\ \end{matrix} \right] \]
Where:
\( c \) is the speed of light
\( \boldsymbol{x}_{n} \) is the range
\( \boldsymbol{z}_{n} \) is the measured time delay
State selection
Sometimes certain states are measured while others are not. For example, the first, third, and fifth states of a five-dimensional state vector are measurable, while the second and fourth states are not measurable:
\[ \boldsymbol{z}_{n} = \boldsymbol{Hx}_{n} + \boldsymbol{v}_{n} = \left[ \begin{matrix} 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\\ \end{matrix} \right] \left[ \begin{matrix} {x}_{1}\\ {x}_{2}\\ {x}_{3}\\ {x}_{4}\\ {x}_{5}\\ \end{matrix} \right] + \boldsymbol{v}_{n} = \left[ \begin{matrix} {x}_{1}\\ {x}_{3}\\ {x}_{5}\\ \end{matrix} \right] + \boldsymbol{v}_{n} \]
Combination of states
Sometimes some combination of states can be measured instead of each separate state. For example, maybe the lengths of the sides of a triangle are the states, and only the total perimeter can be measured:
\[ \boldsymbol{z}_{n} = \boldsymbol{Hx}_{n} + \boldsymbol{v}_{n} = \left[ \begin{matrix} 1 & 1 & 1 \\ \end{matrix} \right] \left[ \begin{matrix} {x}_{1}\\ {x}_{2}\\ {x}_{3}\\ \end{matrix} \right] + \boldsymbol{v}_{n} = ({x}_{1} + {x}_{2} + {x}_{3}) + \boldsymbol{v}_{n} \]
Measurement equation dimensions
The following table specifies the matrix dimensions of the measurement 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 | \( n_{z} \times n_{x} \) |
| \( \boldsymbol{v} \) | measurement noise vector | \( n_{z} \times 1 \) |