On the Observability of a Linear System With a Sparse Initial State

In this letter, we address the problem of observability of a linear dynamical system from compressive measurements and the knowledge of its external inputs. Observability of a high dimensional system state may require a large number of measurements in general, but we show that if the initial state vector admits a sparse representation, the number of measurements can be significantly reduced by using random projections for obtaining the measurements. We derive guarantees for the observability of the system using tools from probability theory and compressed sensing. Our analysis uses properties of the transfer matrix and random measurement matrices to derive concentration of measure bounds, which lead to sufficient conditions for the restricted isometry property of the observability matrix to hold. Hence, under the derived conditions, the initial state can be recovered by solving a computationally tractable convex optimization problem.