Controllability of a Linear System With Nonnegative Sparse Controls

This article studies controllability of a discrete-time linear dynamical system using nonnegative and sparse inputs. These constraints on the control input arise naturally in many real-life systems, where the external influence on the system is unidirectional, and activating each input node adds to the cost of control. We derive the necessary and sufficient conditions for the controllability of the system, without imposing any constraints on the system matrices. Unlike the straightforward extension of the well-known Kalman-rank-based controllability criteria, the conditions presented in this article can be verified in polynomial time. The verification complexity of our conditions is independent of the sparsity level. Our results also provide a closed-form expression for the minimum number of control nodes to be activated at every time instant to ensure controllability of the system using nonnegative controls. Furthermore, for the systems satisfying the derived controllability conditions, we provide approximate algorithms that solve for the nonnegative inputs.