Linear systems with sparse inputs: Observability and input recovery

In this work, we introduce a new class of linear time-invariant systems for which, at each time instant, the input is sparse with respect to an overcomplete dictionary of inputs. Such systems may be appropriate for modeling a system which exhibits multiple discrete behaviors orchestrated by the sparse input. Although the input is assumed to be unknown, we show that the additional structure imposed on the input allows us to recover both the initial state and the sparse, but unknown, input from output measurements alone. For this purpose, we derive sufficient observability and sparse recovery conditions that integrate classical observability conditions for linear systems with incoherence conditions for sparse recovery. We also propose a convex optimization algorithm for jointly estimating the initial condition and recovering the sparse input.