Generalizing the Letov formula for the discrete-time case

In this brief note, the generalization of the Letov formula to the case of not-strictly proper discrete-time systems is proposed. Recently, the Letov formula for continuous-time not-strictly proper system has been derived by using an approach based on Hamiltonian matrices. This strategy cannot be directly transferred in the discrete-time domain, and therefore, an appropriate approach to derive and prove the generalized Letov formula for not-strictly proper discrete-time systems is outlined. Numerical examples, including technologically relevant systems, regarding both single-input single-output (SISO) and multi-input multi-output (MIMO) systems, are reported and discussed. The results presented in this note fall within a wider investigation scope related to revisiting classic control theory to establish further general results about unexplored details of well-posed problems.