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In this paper, the problem of determining the sparsest input matrices to ensure controllability of linear singular systems is investigated. Firstly, it is shown that, determining the sparsest input matrices to ensure reachable controllability or complete controllability is NP-hard, even when the system `singularity' is arbitrarily large. Secondly, submodular functions for singular systems are built, upon which greedy algorithms are developed to approximate the sparsest input matrices with guaranteed performance bounds for the case where there is no restriction on the number of independent inputs. Thirdly, a two-step greedy algorithm is proposed for determining the sparsest input matrices with a given number of inputs to ensure controllability. Compared with the existing algorithms for sparsest input selections, the proposed algorithm achieves better trade-off between the approximation performances and computation efficiency, which are demonstrated by two simulation examples.

Sparsest Input Selection for Controllability of Singular Systems via a Two-Step Greedy Algorithm