Controllability of Linear Dynamical Systems Under Input Sparsity Constraints

In this article, we consider the controllability of a discrete-time linear dynamical system with sparse control inputs. Sparsity constraints on the input arises naturally in networked systems, where activating each input variable adds to the cost of control. We derive algebraic necessary and sufficient conditions for ensuring controllability of a system with an arbitrary transfer matrix. The derived conditions can be verified in polynomial time complexity, unlike the more traditional Kalman-type rank tests. Further, we characterize the minimum number of input vectors required to satisfy the derived conditions for controllability. Finally, we present a generalized Kalman decomposition-like procedure that separates the state-space into subspaces corresponding to sparse-controllable and sparse-uncontrollable parts. These results form a theoretical basis for designing networked linear control systems with sparse inputs.

Automated summary

In this article, the controllability of a discrete-time linear dynamical system with sparse control inputs is considered. Algebraic necessary and sufficient conditions for ensuring controllability are derived and can be verified in polynomial time complexity. A generalized Kalman decomposition-like procedure is presented to separate the state-space into subspaces corresponding to sparse-controllable and sparse-uncontrollable parts, providing a theoretical basis for designing networked linear control systems with sparse inputs.