### Kalman Filter

```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 Estimation
Hidden Markov Model
Discrete Kalman filter cycle
B
wkk
Roles of Variables in Kalman Filter
: state
: state transition model
: control-input model
: control vector
: zero mean multivariate
normal distribution
: observation model
: observation noise
: noise covariance
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 Constant
• Measurements are corrupted by a 0.1 volt
RMS white measurement noise.
• State xk  Axk 1  Buk 1  wk
 xk 1  wk
• Measurement
zk  Hxk  vk
 xk  vk
• 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.
Kalman Filter Simulation with
R=0.01
Kalman Filter Simulation with R=1
and R=0.0001
Slower response to the
measurements
More quick response
to 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
functions.
• At each time step the Jacobian is
evaluated with current predicted states
Unscented Kalman filter (UKF) (1/2)
• When the state transition and
observation 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 technique
known as the unscented transform to
pick a minimal set of sample points
(called sigma points) around the mean.
• The result is a filter which more
accurately 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)
,
Where
describe a vector Brownian motion process
with covariance
.
• Fokker-Planck equation (also named as
Kolmogorov’s equation)
: probability density
of the model state
```