Left invertibility and duality for linear systems

Left and right invertibilities, as well as inversion procedures and algorithms, have been widely studied for years. We consider here, for linear time invariant (LTI) systems, such types of one side invertibilities but in the time domain, which is more demanding than just in the frequency domain, i.e. for the Transfer Function Matrix (TFM). New geometric conditions are given for Time Domain (TD) Left Invertibility, as well as structural conditions which require not only full column rank but also the absence of some types of finite zeros, namely the "observable" ones. For general monic (full column rank) systems having "observable" zeros, we enhance a cascade decomposition which allows for the TD Left Inversion of the "pole" factor, and for the TFM Left Inversion of the "zero" factor. Duality in the classical TFM setting just relies on transposition. In the present TD context, this is not relevant. We give necessary and sufficient conditions for TD Right Invertibility, that show that the underlying duality is indeed the one between zeros and poles, which is a posteriori natural. Full row rank LTI systems can be TD Right Inverted if and only if they have no finite observable poles.