# Optimality of Kalman Filter for Process Noise dependent on magnitude of state

Consider I have a dynamical system $\dot{x} = Ax + w(t)$, $x \in \mathbb{R^2}$ where $w(t)$ is a Gaussian random variable with mean $E(w(t)) = C\|x\|^2$ where $C \in R^2$ is a constant and covariance matrix $Q(t) \leq \begin{bmatrix} f_{1}(\|x\|^2) & 0 \\ 0 & f_{2}(\|x\|^2) \end{bmatrix}$ where $f_{1}$ and $f_{2}$ decrease as $\|x\|^2$ decreases. Also $A$ is a stable system matrix. Assume noisy measurements $y = Hx + v(t)$ ( $v$ is standard white noise with zero mean and known fixed covariance). The aim is to generate estimates of $x(t)$ using a Kalman Filter.How do I begin to analyze the theoretical implications of process noise of this nature ? Will the standard Kalman Filter "work" in this case ?

• Welcome to DSP.SE! You've set up the problem but you've not finished it: what are you trying to achieve? Estimate $x(t)$? I assume so, based on your question as to whether the Kalman Filter will "work", but you haven't explicitly stated it. I expect the KF equations will be usable, but whether it'll gain you bounds on variance of the state estimate, I can't say quickly. You may need to assume monotonicity on the functions $f_1$ and $f_2$ or some other simplifying condition. – Peter K. Nov 3 '15 at 14:25
• Hi ! Yes I am trying to estimate $x(t)$ . Assuming $f_{1}(\|x\|^2)$ and $f_{2}(\|x\|^2)$ to just be constant scaling functions of $\|x\|^2$, and assuming that A is a stable matrix ( eigenvalues with negative real parts), intuitively it feels like the filter should give bounded covariance. This is essentially a situation where the noise statistics are bounded by the magnitude of the state variable itself. My question was mainly in relevance to whether the KF was theoretically designed for a situation where the process noise was independent of the state variable ? – sid Nov 3 '15 at 14:40
• I improved the question based on your suggestions. – sid Nov 3 '15 at 14:43
• as a Kalman recursion, in this case, you will get a fixed point equation. A straightforward approach is to iterate that fixed point equation until you see some convergence (in some norm). You can investigate the theoretical properties of that fixed point iteration and try to say something general for the Kalman filter. – odea Oct 6 '17 at 12:57