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

Abstract

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. INDEX TERMS Controllability, computational complexity, networked control systems, optimization.

Minimal controllability problems on linear structural descriptor systems

“…where l denotes an integer l ≤ n. That is, in contrast to the "structural" controllability problem in [19] and this paper, [22] addressed usual controllability problems. Moreover, the column size of B in Problem ( 7) is given in advance, whereas MCP0 is the problem of minimizing the column size of B.…”

“…Unlike the problems in [19] and this study, [22] considered the input selection problems for non-structural descriptor system (2) formulated as…”

“…Moreover, the column size of B in Problem ( 7) is given in advance, whereas MCP0 is the problem of minimizing the column size of B. In [22], the authors showed that Problems ( 6) and ( 7) are NPhard. These results are analogous to the NP hardness of MCPs for non-structural LTI systems [6].…”