# Properties of a Kalman filter of a non controllable system

Let's say I have a standard system \begin{align} x(t+1)&=Ax\\ y(t) &=Cx(t) \end{align}

As you can see $B=0$, so the system is not controllable.

For the steady state Kalman filter I'd say that if $A$ is Hurwitz then it converges.

But what if $A$ is say, $\mathbf I_n$ (i.e. eye(n)), what can I say about the optimality of the SS. filter varying $C$?

In this setting you are considering an observation problem. Controllability does not play a role. In geenral, the condition under which you can obtain a convergent estimate is that the couple $(A,C)$ is detectable, i.e. that you can design a gain matrix $L$ so that $A+LC$ is Hurwitz. Under this hypothesis the eigenvalues of $A+LC$ will be with negative real-part and they are assigned in a optimal way, in the kalman sense. The kalman gains depend on $C$, therefore as you vary $C$ you should vary the gains accordingly to obtain optimality
• Well, if A is not Hurwitz but $A+LC$ is, then your estimate converges. If the state diverges then obviously so does your estimate, infact I dont know how much sense does it make to estimate unstable systems. – LJSilver Jun 8 '16 at 8:28