# Kalman filter without model versus RLS

As I found out in my previous question Where to get transtion matrix for Kalman filter?, I need a model for correct usage of Kalman filter.

In paper On the Intrinsic Relationship Between the Least Mean Square and Kalman Filters there is model-less Kalman filter - The Kalman filter for deterministic states:

\begin{align} \textbf{g}(k) &= \textbf{P}(k)\textbf{x}(k) / (\textbf{x}^T(k)\textbf{P}(k)\textbf{x}(k) + r)\\ y(k) &= \textbf{x}^T(k)\textbf{w}(k)\\ e(k) &= d(k) - y(k)\\ \textbf{w}(k) &= \textbf{w}(k-1) + \textbf{g}(k)e(k)\\ \textbf{P}(k) &= \textbf{P}(k-1) - \textbf{g}(k)\textbf{x}(k)\textbf{P}(k-1) \end{align}

where $\textbf{g}(k)$ is gain vector $\textbf{P}(k)$ is covariance matrix, $e(k)$ is error, $\textbf{w}(k)$ adaptive filter weights and $\textbf{x}(k)$ are inputs (not really a states here), $d(k)$ is target, $y(k)$ is filter output.

Nice thing is that $\textbf{x}^T(k)\textbf{P}(k)\textbf{x}(k)$ is scalar, so no matrix inversion is needed.

• What is difference of this in comparison with RLS filter?
• What should be better?
• Is it case specific or is there some obvious difference what I am missing?
• Is it even a good idea to use this Kalman filter over RLS or NLMS?