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Kalman Filter (KF) 

Linear dynamical system (Linear RJ11 Phone to RJ45 Jack evolution functions)

Extended Kalman Filter (EKF) 

Non-linear dynamical system (Non-linear evolution functions)

Consider the following non-linear system:

Assume that we can somehow determine a reference trajectory 

Then:

where

For the measurement equation, we have:

We can then apply the standard Kalman filter to the linearized model

How to choose the reference trajectory?

Idea of the extended Kalman filter is to re-linearize the model around the most recent state estimate, i.e.

The Extended Kalman Filter (EKF) has become a standard    technique used in a number of 

# nonlinear estimation and 

# machine learning applications

#State estimation

#estimating the state of a nonlinear dynamic system

#Parameter estimation

#estimating parameters for nonlinear system identification

#e.g., learning the weights of a neural network

#dual estimation 

#both states and parameters are estimated simultaneously

#e.g., the Expectation Maximization (EM) algorithm

MATLAB CODE
########################################################################
function [x_next,P_next,x_dgr,P_dgr] = ekf(f,Q,h,y,R,del_f,del_h,x_hat,P_hat);
% Extended Kalman filter
%
% -------------------------------------------------------------------------
%
% State space model is
% X_k+1 = f_k(X_k) + V_k+1   -->  state update
% Y_k = h_k(X_k) + W_k       -->  measurement
%
% V_k+1 zero mean uncorrelated gaussian, cov(V_k) = Q_k
% W_k zero mean uncorrelated gaussian, cov(W_k) = R_k
% V_k & W_j are uncorrelated for every k,j
%
% -------------------------------------------------------------------------
%
% Inputs:
% f = f_k
% Q = Q_k+1
% h = h_k
% y = y_k
% R = R_k
% del_f = gradient of f_k
% del_h = gradient of h_k
% x_hat = current state prediction
% P_hat = current error covariance (predicted)
%
% -------------------------------------------------------------------------
%
% Outputs:
% x_next = next state prediction
% P_next = next error covariance (predicted)
% x_dgr = current state estimate
% P_dgr = current estimated error covariance
%
% -------------------------------------------------------------------------
%

if isa(f,'function_handle') & isa(h,'function_handle') & isa(del_f,'function_handle') & isa(del_h,'function_handle')
    y_hat = h(x_hat);
    y_tilde = y - y_hat;
    t = del_h(x_hat);
    tmp = P_hat*t;
    M = inv(t'*tmp+R+eps);
    K = tmp*M;
    p = del_f(x_hat);
    x_dgr = x_hat + K* y_tilde;
    x_next = f(x_dgr);
    P_dgr = P_hat - tmp*K';
    P_next = p* P_dgr* p' + Q;
else
    error('f, h, del_f, and del_h should be function handles')
    return
end

##############################################################################


For more

https://drive.google.com/folderview?id=0B2l8IvcdrC4oMzU3Z2NVXzQ0Y28&usp=sharing