I'm not sure it is the right department. I try my chance
I am wondering if there is a way to make a hybrid formulation of a least-square problem and a Kalman filter. Let me explain what I mean:
The (linear) least-square formulation reads for a set of observation $y \in \mathbb{R}^{q}$
$$\hat x = \min_x {\epsilon^\top \epsilon}\quad \text{subject to} \quad y = C x + L \epsilon$$
with $\epsilon \in \mathbb{R}^q$ , $C \in \mathbb{R}^{q \times n}$, $L \in \mathbb R^{q\times q}$ (weighting matrix) and $x \in \mathbb{R}^n$ the state to be estimated.
Kalman filter makes use of a dynamical model for the state and a linear mapping from the observation to the state
\begin{align} x_k & = A_k x_{k-1} + B_k u_{k} + w_k \\ y_k & = C_k x_k + v_k \end{align}
Where $w_k$ and $v_k$ (plant and measurement noise) account for uncertainty in the model. $u_k \in \mathbb{R}^p$ some known imput.
Although I understand good both least-squares and Kalman filters I was wondering how to build a hybrid formulation of the both. I've been asked to create a model for a part only of the state (call it $\tilde x \in \mathbb{R}^m$, $m < n$). I now managed to find a linear model that I'm happy with.
To include my model I perform a Kalman filter on $\tilde x$ and then on the rest of the state $x \backslash \tilde x \in \mathbb{R}^{n-m}$ (not the best notation sorry) I perform the least-square problem (already developed by someone else). It doesn't seem to give me good results though... I also tried to use the estimate of the filter as an observation entering the least-squares (the result is better, still less good than simple least-square without KF).
Is there a way to formulate the whole thing only as a least-square problem? I'm imagining something like using the prediction part of the Kalman filter algorithm
$$\tilde x_{k/ k-1} = A_k \tilde x_{k-1/ k-1} + B_k u_k$$
and then include the update step somehow directly in the least-squares.
$$\tilde x_{k/ k} = \tilde x_{k/ k-1} + K_k (\tilde y - C_k \tilde x_{k/ k-1})$$
I think it should be possible because it also uses the information of the observations $y$. Otherwise any other ideas would be most welcome!
Thank you very much!