Error convergence to zero is not necessarily assured, Sujay.

A bit difficult to discuss details on the social media, but there are proofs and conditions of error convergence (some general and some system specific) that you may wish to go through...

-Sanjay 

I haven't looked properly into this topic, but to my knowledge, the Riccati equation yields the steady state covariance matrix, i.e. the covariance matrix converges to a steady state matrix if the system is observable (full rank observability matrix).

Stability of the system should not be an issue, the state estimate would of course grow unbounded too, but the error should stay bounded.

What you wrote down is the Lyapunov equation, which is used for the smoothing step in Kalman smoothing, to find the steady state filtering covariance you have to solve the (discrete) algebraic riccati equation.

Kalman filter convergence ( in stochastic sense – it  roughly means convergence of covariance matrices) seems not to  not require the stability of   the system – the boundedness of the solutions is (probably) sufficient.

  As a simplistic example – let us look at the aircraft and INS/GPS data  (rougly – a KF) – when the aircraft  gets out of control (and therefore unstable in most of the cases),   nothing  prevents the navigation system from getting a good estimate of the aircraft position.

 As for the   plots you provided in  KF.pdf  - it seems that the solution gets   unbounded ( 10 exp 20), estimates are following it and the computational errors (due to very big numbers the algo operates) grow in such a way that  Kalman fiter algo finally diverges as well.

 What I am about to say – nothing prevent form using the KF for an unstable system along the finite time period when the system behaves in acceptable way (even being unstable).  KF-stability  needs to be checked not only for theoretical reasons) but also to ensure that the estimations behave correctly.   

For linear system, detectability insures convergence.

The equation Pk+1|k = A Pk|k AT + Q is only the time update then comes the measurement update. If they are combined you get

Pk+1|k = (A-KkC) Pk|k-1 (A-KkC)^T+R+KkQkk^T

where Kk is the Kalman gain and the ABC state space notation is used. This means that not A but A-KC must have eigen values inside the unit circle which it very well can have even though A haven't

As you correctly wrote the error dynamics is given by the equation :

ek = (A-KkCA)ek-1

So the error will converge towards zero iff (A-KkCA) is Schur (eigen values inside the unit circle). (A,C) pair observable is sufficient but not necessary. And observability of often difficult to verify for large order systems. (A,C) pair detectable is not sufficient.

What we know is that :

if :

(A,C) pair detectable

Q factorized as Q = TT' with (A,T) stabilisable

R definite positive

Then (A-KkCA) is Schur and then the error converges towards zero

This is what I think ... hope this is correct ...

agree with Federico Wadehn      

"The convergence of P is guaranteed only is case where A has stable eigen values."

In a Kalman filter for a discrete-time linear stochastic system with white noise inputs, the steady state solution to the discrete-time matrix Riccati equation exists if the system matrix pair (F,H) is completely observable. I think to get a positive definite solution you also need complete controllability of (F,Q^{1/2}) where Q is the process noise covariance.

If the system is unstable, I think you are right. The KF covariance will converge (given the above conditions) but the error may diverge. As other respondents have said, you need to check the stability of the governing equation for the estimation error. The equation M. Malaterre has given is a good place to start. What you need to check is whether controllability and observability give stability of this equation. I don't know the answer to this.

Sujay Hello

Of course that Kalman filter may be applied for unstable systems. Ship and aircraft coordnate changing are examples of non-stable systems. In our work (file is attached) 

Bashirov, A.E., Eppelbaum, L.V. and Mishne, L.R., 1992. Improving Eötvös corrections by wide-band noise Kalman filtering. Geophysical Journal International, 108, 1, 193-197.

we considered improved version of Kalman filtering with wide-band noise and delay between the "useful signal" and noise.

Best

Lev

As long as the system is observable, the Kalman filter is applicable to an unstable system yearling stable error dynamics. However, for an unstable system, the Kalman filter will be unstable itself. 

Yes.

if the system is unstable, the state estimator won't converge. The system has to be closed loop stable or open loop stabilizable, and observable  in order for this to work. If the state estimation is not converging to the right states, what is the value of the Kalman filter at that point? 

I concur with my colleagues, this is the observability of the system that is the important element in this case.

Take a look at this document

http://robotics.eecs.berkeley.edu/~wlr/226F06/226a-ch11.pdf

Here is a link that will help you in your work

Introduction for Kalman Filter

oatao.univ-toulouse.fr/2248/1/Alazard_2248.pdf

From my point view the most valuable results in this aea you can find in the following paper:

S.W. Chan, G.C. Goodwin, K.S. Sin. Convergence Properties of the Riccati Difference Equation in Optimal Filtering of Nonstabilizable Systems.

IEEE Trans/ on Automatic Control,vol. AC-29, no. 2, February 1984.

I agree with the proposals of Anatol Tunik

Good luk

Many thanks to all of you for the answers and for suggesting the reading materials.

Kalman Filter needs the system is observable for error dynamics is stable. For an unstable but observable system, the intention of Kalman filter is to estimate the states and to design state feedback, so the closed loop control system with Kalman filter and state feedback control must be stable.

I agree with Yunhua Li

You can augment your original system model to become standardized!

Article Multi-site modeling and prediction of annual and monthly pre...