Controllability of Linear Dynamical Systems Under Input Sparsity Constraints

TL;DR

This paper investigates the controllability of discrete-time linear systems with sparse inputs, providing algebraic conditions, complexity analysis, and a decomposition method to facilitate the design of networked control systems with input sparsity constraints.

Contribution

It introduces polynomial-time verifiable controllability conditions and a Kalman-like decomposition for systems with sparse control inputs, advancing control design under sparsity constraints.

Findings

Derived algebraic necessary and sufficient controllability conditions.

Characterized the minimum number of inputs needed for controllability.

Developed a generalized Kalman decomposition for sparse control systems.

Abstract

In this work, 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.