# Why discretize a continuous transition matrix in Kalman Filter?

In the Kalman filter toolbox at http://becs.aalto.fi/en/research/bayes/ekfukf/cwpa_demo.html the example code shows that a function lti_disc is called, which is essentially a matrix exponential function. What is the purpose of such matrix exponentiation? Given a really simple transition model such as, e.g.

F = [ 1 0 ;
0 1 ] ;


and doing the matrix exponentiation with lti_disc will always give a different transition matrix A, depending on the values of the Q process noise covariance, which is also an input to the lti_disc function.

• not quite sure what integration has to do with discretization? Mar 17, 2019 at 9:38

Have you taken a look at the documentation - section 2.2.4 discusses a Linear Kalman filter model that is very similar to the one you described. From that example you see that:

1. The resulting $$\bf{A}$$ matrix does not depend on the process noise
2. The $$\bf{A}$$ matrix only depends on the size of the time step.
3. The $$\bf{Q}$$ matrix only depends on the size of the time step, and the value of the continuous noise covariance, ie. diag(q) in the example

You can use discrete noise kinematic models, they are probably a bit easier to understand. The continuous noise model assume that the noise is applied throughout the time step interval i.e. from time $$t_0$$ to $$t_1$$. The discrete model assumes the noise is applied instantaneously at times $$t_0$$ and $$t_1$$ (and so on).

See chapter 6 -"Estimation for Kinematic Models" in the book - "Estimation with Applications to Tracking and Navigation" (Bar-Shalom, Rong Li, Kirubarajan) for a derivation of the continuous and discrete time versions of the state transition models.