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. We will start from 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, if we have scales that have 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, if we have a thermometer that has 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:

\[ \boldsymbol{z_{n} = Hx_{n} + v_{n}} \]
Where:
\( \boldsymbol{z_{n}} \) 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 system state into outputs using linear transformations. The following chapters include examples of the observation matrix usage.

Scaling

A range meter sends a signal towards a destination and receives a reflected echo. The measurement is the time delay between 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

Consider

\( \boldsymbol{x_{n}} \) 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 second and fourth states are not measurable:

\[ \boldsymbol{z_{n} = Hx_{n} + v_{n}} = \left[ \begin{matrix} 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\\ \end{matrix} \right] \left[ \begin{matrix} {x}_{1}\\ {x}_{2}\\ {x}_{3}\\ {x}_{4}\\ {x}_{5}\\ \end{matrix} \right] + \boldsymbol{v_{n}} = \left[ \begin{matrix} {x}_{1}\\ {x}_{3}\\ {x}_{5}\\ \end{matrix} \right] + \boldsymbol{v_{n}} \]

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, only the total perimeter can be measured:

\[ \boldsymbol{z_{n} = Hx_{n} + v_{n}} = \left[ \begin{matrix} 1 & 1 & 1 \\ \end{matrix} \right] \left[ \begin{matrix} {x}_{1}\\ {x}_{2}\\ {x}_{3}\\ \end{matrix} \right] + \boldsymbol{v_{n}} = ({x}_{1} + {x}_{2} + {x}_{3}) + \boldsymbol{v_{n}} \]

Measurement equation dimensions

The following table specifies the matrix dimensions of the measurement equation variables:

Variable Description Dimension
\( \boldsymbol{x} \) state vector \( n_{x} \times 1 \)
\( \boldsymbol{z} \) output vector \( n_{z} \times 1 \)
\( \boldsymbol{H} \) observation matrix \( n_{z} \times n_{x} \)
\( \boldsymbol{v} \) measurement noise vector \( n_{z} \times 1 \)
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