Computation of the State Bias and Initial States for Stochastic State Space Systems in the General 2-D Roesser Model Form

Abstract

Recently \cite{Ramos2017a} presented a subspace system identification algorithm for 2-D purely stochastic state space models in the general Roesser form. However, since the exact problem requires an oblique projection of $Y_f^h$ projected onto $W_p^h$ along ${\hat{X}}_f^{vh}$, where $W_p^h=\left[\begin{array}{c}{\hat{X}}_p^{vh}\\ Y_p^h\end{array}\right]$, this presents a problem since $\{{\hat{X}}_p^{vh},{\hat{X}}_f^{vh}\}$ are unknown. In the above mentioned paper, the authors found that by doing an orthogonal projection $Y_f^h/Y_p^h$, one can identify the future horizontal state matrix ${\hat{X}}_f^h$ with a small bias due to the initial conditions that depend on $\{{\hat{X}}_p^{vh},{\hat{X}}_f^{vh}\}$. Nevertheless, the results on modeling 2-D images were very good despite lack of knowledge of $\{{\hat{X}}_p^{vh},{\hat{X}}_f^{vh}\}$. In this note we delve into the bias term and prove that it is insignificant, provided $i$ is chosen large enough and the vertical and horizontal states are uncorrelated. That is, the cross covariance of the state estimates $x_{r,s}^h$ and $x_{r,s}^v$ is zero, or $P_{hv}=0_{n_x\times n_x}$ and $P_{vh}=0_{n_x\times n_x}$. Our simulations use $i=30$. We also present a second iteration to improve the state estimates by including the vertical states computed from a vertical data processing step, i.e., by doing an orthogonal projection $Y_f^v/Y_p^v$. In this revised algorithm we include a step to compute the initial states. This new portion, in addition to the algorithm presented in \cite{Ramos2017a}, forms a complete 2-D stochastic subspace system identification algorithm.

Companion paper of "A Stochastic Subspace System Identification Algorithm for State Space Systems in the General 2-D Roesser Model Form" published in International Journal of Control as a regular paper (https://doi.org/10.1080/00207179.2017.1418983)