# Kalman Filtering with Unknown State Transition Matrix

I'm currently studying the use of Kalman filters for estimating linear systems. My current State Transition Matrix (STM) is the identity since so far I've been dealing with non time-varying systems. However my next step is to extend to time-varying systems and in this case there is no information on how the system varies and so the STM is unknown. And as of my testing so far a wrong STM causes a ton of errors on the Kalman Filter estimations. So my question is if there is a method for circumventing this, be it by updating the STM on the fly or some kind of way of updating the Process Noise Covariance Matrix in order to account for that?

• What do you mean by my next step is to extend to time-varying systems and in this case there is no information on how the system varies and so the STM is unknown What information do you mean? As long as a system is characterized by a differential equation you know how the system varies (although you may not solve analytically the equation for a closed form answer) – Fat32 Jul 8 '17 at 20:18

Unfortunately, in Classical Kalman filter applications you need to have a guess of your state transition model governing the states. Classical choices for such models for time-varying systems are the random walk model, or its generalization, a first-order Markov model. If do not have a clue about which state model you can assume, you can try to extract a trend component from your state time series (via EMD or a moving average filter, for example), and plot it to have an idea of how the first-order moment of your states vary with time, for example. If it decays as a power law with $n$ (i.e. $E[x(n)] \approx \alpha^n$) until a final value, it may be an indication that a first-order Markov model with pole equal to $\alpha$ can serve as a state transition model.