Strong Left Inversion of Linear Systems and Input Reconstruction

Left inversion of linear time-invariant systems aims at identifying the input acting on a system, for the zero initial state, from partial information on the input and the state. In the present contribution, we propose to generalize such left inversion for linear systems in various directions that take arbitrary initial state into consideration to address both exact and asymptotic input reconstructions. Necessary and sufficient algebraic conditions are given to achieve such strong left inversion properties. Complete characterizations of introduced concepts in terms of system zeros are provided. Relationship between inversion and input reconstruction is therefore investigated, with emphasis on causal realization. Conditions for input observer existence are proposed, and a constructive causal design for an asymptotically convergent input observer is presented.