Filter of $kalman

Examples of applications

The filter of $kalman is used in a broad range of technological fields (radar, electronic vision, communication…). It is a major topic of the Automatique and Treatment of the signal . An example of use can be the provision, uninterrupted, information such as the position or the speed of an object starting from a series of observations relative to its position, possibly including errors of measurement.

For example, for the case of the radars where one wishes to follow a target, data on its position, its speed and its acceleration are measured at every moment but with enormously disturbances due to the noise or the errors of measurement. The filter of $kalman calls upon the dynamics of the target which defines its evolution in time to obtain the best given, thus eliminating the effect of the noise. These data can be calculated for the moment present (filtering), in the past (smoothing), or on a future horizon (prediction).

The Kalman filter is used also more and more apart from the field of the treatment of the signal, for example in Météorologie and Océanographie, for the Assimilation of data in a digital model, in Finance or Navigation.

Paternity

The filter of $kalman owes his name with Rudolf $kalman although Thorvald Nicolai Thiele and Peter Swerling developed a similar algorithm before him. The paternity of the filter is the subject of a small controversy in the scientific community. The filter was described in various publications by Swerling (1958), $kalman (1960) and $kalman-Bucy (1961).

Stanley Schmidt is recognized like having carried out the first implementation of the filter. It was at the time of a visit of Rudolf $kalman to NASA Ames Research Center that he saw the potential of his filter for the estimate of the trajectory for the Programme Apollo. This led to the use of the filter in the computer of navigation.

A large variety of filters of $kalman, since, were developed starting from original the formulation known as filters of simple $kalman . Schmidt developed the filter of wide $kalman , Bierman, Thornton and well of others developed a whole range of filters root square . The filter more used is probably the '' phase-locked loop '', largely widespread in the radios, computers, equipment of communication, etc

The filter of $kalman

The filter of $kalman is a recursive estimator. That means that to consider the state running, only the preceding state and current measurements are necessary. The history of the observations and the estimates is thus not necessary.

The state of the filter is represented by 2 variables:

$\ hat {\ textbf {X}} _ {K|K}$ , the estimate of the state at the moment K ;
$\ textbf {P} _ {K|K}$ , the matrix of covariance of the error (a measurement of the precision of the estimated state).

The filter of $kalman has two distinct phases: Prediction and Updated . The phase of prediction uses the state estimated of the previous moment to produce an estimate of the current state. In the stage of update, the observations of the moment running are used to correct the state predicted with an aim of obtaining a more precise estimate.

Prediction

\ hat {\ textbf {X}} _ {K|k-1} = \ textbf {F} _ {K} \ hat {\ textbf {X}} _ {k-1|k-1} + \ textbf {B} _ {K} \ textbf {U} _ {k-1}

(predicted state)

\ textbf {P} _ {K|k-1} = \ textbf {F} _ {K} \ textbf {P} _ {k-1|k-1} \ textbf {F} _ {K} ^ {T} + \ textbf {Q} _ {K}

(predicted estimate of covariance)

Update

\ tilde {\ textbf {there}} _ {K} = \ textbf {Z} _ {K} - \ textbf {H} _ {K} \ hat {\ textbf {X}} _ {K|k-1}

(innovation)

\ textbf {S} _ {K} = \ textbf {H} _ {K} \ textbf {P} _ {K|k-1} \ textbf {H} _ {K} ^ {T} + \ textbf {R} _ {K}

(covariance of the innovation)

\ textbf {K} _ {K} = \ textbf {P} _ {K|k-1} \ textbf {H} _ {K} ^ {T} \ textbf {S} _ {K} ^ {- 1}

(profit of optimal $kalman )

\ hat {\ textbf {X}} _ {K|K} = \ hat {\ textbf {X}} _ {K|k-1} + \ textbf {K} _ {K} \ tilde {\ textbf {there}} _ {K}

(updated state)

\ textbf {P} _ {K|K} = (I - \ textbf {K} _ {K} \ textbf {H} _ {K}) \ textbf {P} _ {K|k-1}

(updated covariance)

The formula of the update of covariance is valid only for one profit of optimal $kalman. The use of other values of profits requires more complex formulas.

The filter of Information

In the filter of information, estimated covariance and the state are respectively replaced by the matrix of information and the vector of information. They are defined by:

$\ textbf {Y} _ {K|K} \ equiv \ textbf {P} _ {K|K} ^ {- 1}$

\ hat {\ textbf {there}} _ {K|K} \ equiv \ textbf {P} _ {K|K} ^ {- 1} \ hat {\ textbf {X}} _ {K|K}

In the same way, predicted covariance and the state have the equivalent forms of information, defined by:

$\ textbf {Y} _ {K|k-1} = \ textbf {P} _ {K|k-1} ^ {- 1}$

\ hat {\ textbf {there}} _ {K|k-1} = \ textbf {P} _ {K|k-1} ^ {- 1} \ hat {\ textbf {X}} _ {K|k-1}

The covariance and the vector of measurement are defined by:

$\ textbf {I} _ {K} = \ textbf {H} _ {K} ^ {T} \ textbf {R} _ {K} ^ {- 1} \ textbf {H} _ {K}$

\ textbf {I} _ {K} = \ textbf {H} _ {K} ^ {T} \ textbf {R} _ {K} ^ {- 1} \ textbf {Z} _ {K}

The update of information becomes now a commonplace sum:

$\ textbf {Y} _ {K|K} = \ textbf {Y} _ {K|k-1} + \ textbf {I} _ {K}$

\ hat {\ textbf {there}} _ {K|K} = \ hat {\ textbf {there}} _ {K|k-1} + \ textbf {I} _ {K}

The principal advantage of the filter of information is that NR measurements can be filtered at every moment simply by adding their matrices and vectors with information.

$\ textbf {Y} _ {K|K} = \ textbf {Y} _ {K|k-1} + \ sum_ {j=1} ^N \ textbf {I} _ {K, J}$

\ hat {\ textbf {there}} _ {K|K} = \ hat {\ textbf {there}} _ {K|k-1} + \ sum_ {j=1} ^N \ textbf {I} _ {K, J}

To predict the filter of information, the matrix and the vector of information can be converted again with their equivalents of the space of state or, alternatively, the prediction of the space of information can be used.

$\ textbf {M} _ {K} = ^ {T} \ textbf {Y} _ {K|K} \ textbf {F} _ {K} ^ {- 1}$

$\ textbf {C} _ {K} = \ textbf {M} _ {K} ^ {- 1}$

\ textbf {L} _ {K} = I - \ textbf {C} _ {K}

\ textbf {Y} _ {K|k-1} = \ textbf {L} _ {K} \ textbf {M} _ {K} \ textbf {L} _ {K} ^ {T} +

\ textbf {C} _ {K} \ textbf {Q} _ {K} ^ {- 1} \ textbf {C} _ {K} ^ {T}

$\ hat {\ textbf {there}} _ {K|k-1} = \ textbf {L} _ {K} ^ {T} \ hat {\ textbf {there}} _ {K|K}$

To also note that F and Q must be invertible.

Non-linear filters

The filter of $kalman is limited to the linear systems. However, the majority of the physical systems are nonlinear. Non-linearity can be associated with the model with the process, the model of observation or with both.

Filter of wide $kalman

In the filter of wide $kalman (FKE), the models of evolution and observation do not need to be linear functions of the state but can in the place being of the functions (differentiable).

$\ textbf {X} _ {K} = F (\ textbf {X} _ {k-1}, \ textbf {U} _ {K}, \ textbf {W} _ {K})$

$\ textbf {Z} _ {K} = H (\ textbf {X} _ {K}, \ textbf {v} _ {K})$

The function F can be used to calculate the state predicted starting from the state considered preceding and, similarly, the function H can be employed to calculate the predicted observation of the predicted state. However, F and H cannot be applied directly to the calculation of covariance: a matrix of the derivative partial, the Jacobienne, is calculated.

At every moment, Jacobienne is evaluated with the states considered current. These matrices can be employed in the equations of the filter of $kalman. This process linearizes primarily the nonlinear function around the current estimate.

This gives the equations of the filter of following wide $kalmans:

Prediction

$\ hat {\ textbf {X}} _ {K|k-1} = F (\ hat {\ textbf {X}} _ {k-1|k-1}, \ textbf {U} _ {K}, 0)$

$\ textbf {P} _ {K|k-1} = \ textbf {F} _ {K} \ textbf {P} _ {k-1|k-1} \ textbf {F} _ {K} ^ {T} + \ textbf {Q} _ {K}$

Update

$\ tilde {\ textbf {there}} _ {K} = \ textbf {Z} _ {K} - H (\ hat {\ textbf {X}} _ {K|k-1}, 0)$

$\ textbf {S} _ {K} = \ textbf {H} _ {K} \ textbf {P} _ {K|k-1} \ textbf {H} _ {K} ^ {T} + \ textbf {R} _ {K}$

$\ textbf {K} _ {K} = \ textbf {P} _ {K|k-1} \ textbf {H} _ {K} ^ {T} \ textbf {S} _ {K} ^ {- 1}$

$\ hat {\ textbf {X}} _ {K|K} = \ hat {\ textbf {X}} _ {K|k-1} + \ textbf {K} _ {K} \ tilde {\ textbf {there}} _ {K}$

$\ textbf {P} _ {K|K} = (I - \ textbf {K} _ {K} \ textbf {H} _ {K}) \ textbf {P} _ {K|k-1}$

Where the matrices of transition and observation are defined as being the following Jacobiennes:

$\ textbf {F} _ {K} = \ left. \ frac {\ partial F} {\ partial \ textbf {X}} \ right \ green _ {\ hat {\ textbf {X}} _ {k-1|k-1}, \ textbf {U} _ {K}}$

$\ textbf {H} _ {K} = \ left. \ frac {\ partial H} {\ partial \ textbf {X}} \ right \ green _ {\ hat {\ textbf {X}} _ {K|k-1}}$

Note: the convergence of this filter is not assured at all because it is about a local convergence. In fact, there exist many examples for which the convergence of the filter depends on the initialization of the state at the initial moment.

Filter of not-scented $kalman

Applications

Autopilot
Systems of dynamic localization
simultaneous Localization and construction of charts (SLAM)
System of positioning by satellites

See Too

particulate Filter

References

General

Year introduction to the $kalman filter, (16 pages) Greg Welch and Gary Bishop, 2006

Year introduction to the $kalman filter, (81 pages) Greg Welch and Gary Bishop, 2001

Gelb A., editor. optimal Applied estimate. MIT Near, 1974.

Harvey, A.C. Forecasting, Structural Time Series Models and the $kalman Filter. Cambridge University Near, Cambridge, 1989.

Stengel, RF, Optimal Control and Dover Estimate, 1994.

Julier, Simon J. and Jeffery K. Uhlmann. " In New Extension off the $kalman Filter to nonlinear Systems." In The Proceedings off AeroSense: The 11th International Symposium one Aerospace/Sensing Defense, Simulation and Controls, Multi Sensor Fusion, Alignment and Resource Management II, SPIE, 1997.

Roweis, S, and Z Ghahramani. " In Unifying Review off Linear Gaussian Models." Neural Computation Flight 11 No 2 (1999).

Simon, Optimal D. state estimate: $kalman, H-infinity, and nonlinear approaches. John Wiley & Sounds, 2006. (Book web site At http://academic.csuohio.edu/simond/estimation/)

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