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Unscented Kalman filter

The `trackingUKF`

class creates
a discrete-time unscented Kalman filter used for tracking positions and velocities of
target platforms. An unscented Kalman filter is a recursive algorithm for estimating the
evolving state of a process when measurements are made on the process. The unscented
Kalman filter can model the evolution of a state that obeys a nonlinear motion model.
The measurements can also be nonlinear functions of the state. In addition, the process
and the measurements can have noise. Use an unscented Kalman filter when the current
state is a nonlinear function of the previous state or when the measurements are
nonlinear functions of the state or when both conditions apply. The unscented Kalman
filter estimates the uncertainty about the state, and its propagation through the
nonlinear state and measurement equations, using a fixed number of sigma points. Sigma
points are chosen using the unscented transformation as parameterized by the
`Alpha`

, `Beta`

, and
`Kappa`

properties.

`filter = trackingUKF`

creates an unscented Kalman filter object for a
discrete-time system using default values for the
`StateTransitionFcn`

, `MeasurementFcn`

, and
`State`

properties. The process and measurement noises are
assumed to be additive.

specifies
the state transition function, `filter`

= trackingUKF(`transitionfcn`

,`measurementfcn`

,`state`

)`transitionfcn`

,
the measurement function, `measurementfcn`

, and
the initial state of the system, `state`

.

configures
the properties of the unscented Kalman filter object using one or
more `filter`

= trackingUKF(___,`Name,Value`

)`Name,Value`

pair arguments. Any unspecified
properties have default values.

clone | Create unscented Kalman filter object with identical property values |

correct | Correct Kalman state vector and state error covariance matrix |

correctjpda | Correct state and state estimation error covariance using JPDA |

distance | Distance from measurements to predicted measurement |

initialize | Initialize unscented Kalman filter |

likelihood | Measurement likelihood |

predict | Predict unscented Kalman state vector and state error covariance matrix |

residual | Measurement residual and residual covariance |

The unscented Kalman filter estimates the state of a process governed by a nonlinear stochastic equation

$${x}_{k+1}=f({x}_{k},{u}_{k},{w}_{k},t)$$

where *x _{k}* is
the state at step

$${x}_{k+1}=f({x}_{k},{u}_{k},t)+{w}_{k}$$

To use the simplified
form, set `HasAdditiveProcessNoise`

to `true`

.

In the unscented Kalman filter, the measurements are also general functions of the state,

$${z}_{k}=h({x}_{k},{v}_{k},t)$$

where
*h(x _{k},v_{k},t)* is the
measurement function that determines the measurements as functions of the state. Typical
measurements are position and velocity or some function of these. The measurements can
include noise as well, represented by

$${z}_{k}=h({x}_{k},t)+{v}_{k}$$

To use the simplified form, set
`HasAdditiveMeasurmentNoise`

to `true`

.

These equations represent the actual motion of the object and the actual measurements. However, the noise contribution at each step is unknown and cannot be modeled exactly. Only statistical properties of the noise are known.

[1] Brown, R.G. and P.Y.C. Wang. *Introduction
to Random Signal Analysis and Applied Kalman Filtering*.
3rd Edition. New York: John Wiley & Sons, 1997.

[2] Kalman, R. E. “A New Approach to Linear Filtering
and Prediction Problems.” *Transactions of the ASME–Journal
of Basic Engineering*, Vol. 82, Series D, March 1960, pp.
35–45.

[3] Wan, Eric A. and R. van der Merwe. “The Unscented
Kalman Filter for Nonlinear Estimation”. *Adaptive
Systems for Signal Processing, Communications, and Control*.
AS-SPCC, IEEE, 2000, pp.153–158.

[4] Wan, Merle. “The Unscented Kalman Filter.”
In *Kalman Filtering and Neural Networks*, edited
by Simon Haykin. John Wiley & Sons, Inc., 2001.

[5] Sarkka S. “Recursive Bayesian Inference on Stochastic Differential Equations.” Doctoral Dissertation. Helsinki University of Technology, Finland. 2006.

[6] Blackman, Samuel. *Multiple-Target Tracking
with Radar Applications*. Artech House, 1986.

`cameas`

|`cameasjac`

|`constacc`

|`constaccjac`

|`constturn`

|`constturnjac`

|`constvel`

|`constveljac`

|`ctmeas`

|`ctmeasjac`

|`cvmeas`

|`cvmeasjac`

|`initcaukf`

|`initctukf`

|`initcvukf`

`trackingABF`

|`trackingCKF`

|`trackingEKF`

|`trackingGSF`

|`trackingIMM`

|`trackingKF`

|`trackingMSCEKF`

|`trackingPF`