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Kalman Filter
12.03.22.(Thu)Joon Shik Kim
Computational Models of Intelligence
Application of Kalman Filter
NASA Apollo navigation and U.S. Navy’s Tomahawk missile
Recursive Bayesian Estima-tion
Hidden Markov Model Discrete Kalman filter cycle
Roles of Variables in Kalman FilterkB
: state : state transition model
: control-input model : control vector
: zero mean multi-variate normal dis-tribution
: observation model
: observation noise : noise covariance
kw
Predict Phase
- Predicted (a priori) state estimate
- predicted (a priori) estimate error covariance
Update Phase
- Innovation of measurement residual
- Innovation of residual covariance
- Optimal Kalman gain is chosen by minimizing the error covariance Pk
- Updated (a posteriori) state estimate
Estimating a Random Con-stant
• Measurements are corrupted by a 0.1 volt RMS white measurement noise.
• State
• Measurement
• The state does not change from step to step so A=1. There is no control input so u=0. Our measurement is of the state di-rectly so H=1.
1 1k k k kx Ax Bu w
1k kx w
k k kz Hx v
k kx v
Kalman Filter Simulation with R=0.01
Kalman Filter Simulation with R=1 and R=0.0001
Slower response to themeasurements
More quick responseto the measurements
Extended Kalman Filter (EKF)
• In the extended Kalman filter, (EKF) the state transition and observation models need not be linear functions of the state but may instead be (differen-tiable) functions.
• At each time step the Jacobian is eval-uated with current predicted states
Unscented Kalman filter (UKF) (1/2)
• When the state transition and obser-vation models – that is, the predict and update functions f and h– are highly non-linear, the extended Kalman filter can give particularly poor performance. This is because the covariance is propagated through linearization of the underlying non-linear model.
Unscented Kalman filter (UKF) (2/2)
• The unscented Kalman filter (UKF) uses a deterministic sampling tech-nique known as the unscented trans-form to pick a minimal set of sample points (called sigma points) around the mean.
• The result is a filter which more accu-rately captures the true mean and covariance.
Ensemble Kalman Filter (EnKF) (1/2)
• EnKF is a Monte Carlo approximation of the Kalman filter, which avoids evolving the covariance matrix of the probability density function (pdf) of the state vector.
• Instead, the pdf is represented by an ensemble
Ensemble Kalman Filter (EnKF) (2/2)
• Markov Chain Monte Carlo (MCMC)
• Fokker-Planck equation (also named as Kolmogorov’s equation)
Where describe a vector Brownian motion process with covariance .
,
: probability density of the model state