A Note on Least Square Approximated Linear Systems

Least square approximated linear systems are projection of linear systems on space of different dimensions, which are proposed to model dimension-varying systems. The projection from linear systems to least square approximated linear systems is invertible if the least square approximated linear system can be projected back to the original space to get exactly the same linear system as before. This paper presents a necessary and sufficient condition for the invertibility of the projection from linear systems to least square approximated linear systems. Based on this condition, we investigate the relation of the controllability matrix and the observability matrix between the linear system and its least square approximated linear system. Finally, we prove that the projection of linear systems onto quotient space is invertible, based on which dimension-varying systems can be modeled by quotient space approach. Some numerical examples are provided to illustrate the results in the paper.