Left-invertibility under sparse assumption: the linear case

This paper deals with the issue of dynamical left-invertibility for linear continuous-time dynamical systems. More precisely, we provide sufficient conditions in order to estimate unknown inputs, using known outputs, under sparse input assumptions for linear continuous-time dynamical systems that are not necessarily square. In fact, there exists an algorithm in the literature that allows verification of left-invertibility for linear square systems; that is, systems with p known outputs and p unknown inputs. However, a similar algorithm does not exist for the rectangular case, where the number of inputs is much larger than the number of outputs. In this paper, we first use the square case algorithm as a stepping stone in order to propose a new algorithm for the rectangular case. However, it shown that even if the proposed algorithm converges successfully, it is not sufficient to estimate the unknown inputs. Consequently, it has been deemed necessary to include a sparse input assumption and to verify the well-known Restrictive Isometric Property (RIP) conditions of a specific matrix. Finally, an academic example is given in order to highlight the feasibility of the proposed approach.