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 directly 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 (differentiable) functions.

• At each time step the Jacobian is evalu-ated 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