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The linear dynamic model

The Kalman filter was orginally developed to estimate time-varying quantities (state variables) from a set of noisy measurements (signals). The set of state variables is usually represented by a vector denoted as tex2html_wrap_inline3952 . For example, we could define the state of a satellite to be its position in Cartesian coordinates relative to the center of the earth. In this case, the state vector would look like the following:

equation408

where the (x y z) triplet represents the time-varying coordinates of the satellite's position (note the distinction between tex2html_wrap_inline3956 , the state vector, and x, the x-component of the state vector). gif In general, it is not possible to determine through direct measurements the precise state of such a system. Instead, we must rely on measurements that provide information about the state variables. In this example, we may have to rely on external radar signals or telemetry signals sent by the satellite itself. In other words, we must infer the state of the system based on the set of measurements we have about the system. In an ideal world, the state of the system can be determined precisely given an appropriate set of measurements. In practice, this is difficult because these measurements are corrupted by noise. In this example, if we had an onboard instrument that could measure the satellite's (x,y,z) coordinates in Cartesian using a gyroscopic system, the relationship between the state vector tex2html_wrap_inline3956 and the measurement vector tex2html_wrap_inline3968 can be written as

  equation421

where tex2html_wrap_inline3968 is the measured position vector and tex2html_wrap_inline3972 is the noise vector associated with the measurement. The matrix H is referred to as the transfer function because it transfers the input, which is the actual state, to the output, which is the set of measurements. Equation gif expresses the measurement vector as a function of the state vector. What we really need is the state vector expressed as a function of the measurement vector. If we know the value of tex2html_wrap_inline3972 , this would reduce to a simple matrix inversion ( tex2html_wrap_inline3978 ). However, the addition of the unknown tex2html_wrap_inline3972 term implies that we can no longer determine tex2html_wrap_inline3968 precisely. Instead, we must find an estimate of tex2html_wrap_inline3968 based on our knowledge of how the state variables are correlated to one another, and on the prior distribution of tex2html_wrap_inline3972 . A Kalman filter is a function that provides an optimal estimate of the state vector x given such knowledge. This is an example of a linear dynamic system because the input and output are described by a linear equation.


next up previous
Next: The non-linear dynamic model Up: The Kalman-filter model Previous: The Kalman-filter model

Ramani Pichumani
Mon Jul 7 10:34:23 PDT 1997