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