Path: Common/Estimation
% Implement an Unscented Kalman Filter.
The filter uses numerical integration to propagate the state.
The filter propagates sigma points, points computed from the
state plus a function of the covariance matrix. For each state
there are two sigma states.
The default integrator is RK4, or you cab specify an integrator
with the field
d.integrator = @myIntegratorName
To initialize UKF call
d = UKF( 'initialize', d );
To update
d = UKF( 'update', d, y );
The filter appends internal data to the data structure.
d.x gives the current estimated state and d.pXX the state covariance.
The measurement function is of the form
meas = MeasFun( x, dMeasFun );
The state function is of the form
xDot = StateFun( x, t, dStateFun );
You can use either the Cholesky transform or matrix square root to
form the sigma points.
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Form:
d = UKF( action, d, y )
d = UKF( 'datastructure', nStates, nMeas )
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Inputs
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action (1,:) 'initialize' or 'update' or 'datastructure'
d (.) UKF data structure
.rHSFun (1,:) Name or handle of RHS function
.integrator (1,:) Handle of integrator function
.rHSFunData (.) RHS function data structure
.measFun (1,:) Handle of measurement function
.measFunData (.) Measurement function data
.x (n,1) Initial state vector
.p (n,n) Covariance matrix for x
.pXX (n,n) Covariance after state update
.dY (1,1) Number of measurements
.dX (1,1) Number of states
.rP (n,n) Process noise covariance
.rM (n,n) Measurement noise covariance
.alpha (1,1) Scaling 1e-4 <= alpha <= 1
.kappa (1,1) Secondary scaling usually 0
.beta (1,1) Prior knowledge of distribution = 2
for Gaussian distributed noise
.dT (1,1) Time step
.t (1,1) Time
.sigmaPtAlg (1,1) 1 = sqrtm, 2 = chol
y (m,1) Measurement vector
verbose (1,1) Flag for verbose output. 1 = warn, 2 = error.
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Outputs
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d (.) UKF data structure with the following appended
.dX (1,1) Number of states
.n (1,1) Number of sigma points
.y (m,1) Measurements based on the states
.xA {n} Sigma points
.pP (n,n) Covariance matrix prior to
measurement
.xP (n,1) State vector prior to measurement
incorporation
.res (:,1) Residuals
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References: Voss, H. U., Timmer, J., Kurths, J., "Nonlinear Dynamical
System Identification from Uncertain and Indirect
Measurements," International Journal of Bifurcation
and Chaos, Vol. 14, No. 6, 2005, pp. 1905-1933.
Van der Merwe, R. and Wan, E., "Sigma-Point Kalman Filters
for Probabilistic Inference in Dynamic State-Space Models",
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Common: DemoFuns/GXUKF Common: DemoFuns/RHSUKF Common: Estimation/SigmaPointsChol Math: Integration/RK4
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