The states evolves continuously in time and the measurements are also continuous in time. In this implementation only the covariance P(t) is solved for through the Differential Riccati equation and that only works for linear systems. A basic numerical scheme is used so the code should only be used for small state count and care taken in setting timestep. The code example is available in matlab_implementation/continuous_continuous/examples/cc_example.m and the code for the filter is in matlab_implementation/continuous_continuous of https://github.com/mannyray/KalmanFilter.

clear all;
addpath('../');
C = [1, 0.1]; %observation matrix to obtain measurement y (y = C*x + noise)
A = [-1,0.2;-0.1,-1]; %(d/dt)x(t) = A*x(t)
R_c = [0.1^2]; %continuous sensor noise
Q_c = [0.001^2,0;0,0.001^2]; %continuous process noise
P_0 = eye(2);%initial covariance
%next three variables dictate time step size
t_start = 0;
t_final = 20;
outputs = 10000;
covariances = cckf(t_start,t_final,outputs,P_0,A,C,R_c,Q_c);
covariances{1}%at t = 0
covariances{2000}%at t = 1.999 
format long;
covariances{end}%at t = 19.999

produces

ans =
   8.2441e-05   1.1739e-04
   1.1739e-04   1.8665e-04
ans =
   5.04888842493018e-07   2.45085608502966e-08
   2.45085608502966e-08   4.97548868164108e-07