  A resilient distributed algorithm is proposed to solve the distributed
resource allocation problem of a first-order nonlinear multi-agent system who
is subject to false data injection (FDI) attacks. An intelligent attacker
injects false data into agents' actuators and sensors such that agents execute
the algorithm according to the compromised control inputs and interactive
information. The goal of the attacker is to make the multi-agent system to be
unstable and to cause the deviance of agents' decisions from the optimal
resource allocation. At first, we analyze the robustness of a distributed
resource allocation algorithm under FDI attacks. Then, the unknown nonlinear
term and the false data injected in agents are considered as extended states
which can be estimated by extended state observers. The estimation was used in
the feedback control to suppress the effect of the FDI attacks. A resilient
distributed resource allocation algorithm based on the extended state observer
is proposed to ensure that it can converge to the optimal allocation without
requiring any information about the nature of the attacker. An example is given
to illustrate the results.
