Resilience for Distributed Consensus With Constraints

This paper proposes a new approach that enables multi-agent systems to achieve resilient <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">constrained</i> consensus in the presence of Byzantine attacks, in contrast to existing literature that is only applicable to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unconstrained</i> resilient consensus problems. The key enabler for our approach is a new device called a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><inline-formula><tex-math notation="LaTeX">$(\gamma _{i},\alpha _{i})$</tex-math></inline-formula>-resilient convex combination</i>, which allows normal agents in the network to utilize their locally available information to automatically isolate the impact of the Byzantine agents. Such a resilient convex combination is computable through linear programming, whose complexity scales well with the size of the overall system. By applying this new device to multi-agent systems, we introduce network and constraint redundancy conditions under which resilient constrained consensus can be achieved with an exponential convergence rate. We also provide insights on the design of a network such that the redundancy conditions are satisfied. Finally, numerical simulations and an example of safe multi-agent learning are provided to demonstrate the effectiveness of the proposed results.