  Consensus in multi-agent dynamical systems is prone to be sabotaged by the
adversary, which has attracted much attention due to its key role in broad
applications. In this paper, we study a new false data injection (FDI) attack
design problem, where the adversary with limited capability aims to select a
subset of agents and manipulate their local multi-dimensional states to
maximize the consensus convergence error. We first formulate the FDI attack
design problem as a combinatorial optimization problem and prove it is NP-hard.
Then, based on the submodularity optimization theory, we show the convergence
error is a submodular function of the set of the compromised agents, which
satisfies the property of diminishing marginal returns. In other words, the
benefit of adding an extra agent to the compromised set decreases as that set
becomes larger. With this property, we exploit the greedy scheme to find the
optimal compromised agent set that can produce the maximum convergence error
when adding one extra agent to that set each time. Thus, the FDI attack set
selection algorithms are developed to obtain the near-optimal subset of the
compromised agents. Furthermore, we derive the analytical suboptimality bounds
and the worst-case running time under the proposed algorithms. Extensive
simulation results are conducted to show the effectiveness of the proposed
algorithm.
