  Achieving average consensus without disclosing sensitive information can be a
critical concern for multi-agent coordination. This paper examines
privacy-preserving average consensus (PPAC) for vector-valued multi-agent
networks. In particular, a set of agents with vector-valued states aim to
collaboratively reach an exact average consensus of their initial states, while
each agent's initial state cannot be disclosed to other agents. We show that
the vector-valued PPAC problem can be solved via associated matrix-weighted
networks with the higher-dimensional agent state. Specifically, a novel
distributed vector-valued PPAC algorithm is proposed by lifting the agent-state
to higher-dimensional space and designing the associated matrix-weighted
network with dynamic, low-rank, positive semi-definite coupling matrices to
both conceal the vector-valued agent state and guarantee that the multi-agent
network asymptotically converges to the average consensus. Essentially, the
convergence analysis can be transformed into the average consensus problem on
switching matrix-weighted networks. We show that the exact average consensus
can be guaranteed and the initial agents' states can be kept private if each
agent has at least one "legitimate" neighbor. The algorithm, involving only
basic matrix operations, is computationally more efficient than
cryptography-based approaches and can be implemented in a fully distributed
manner without relying on a third party. Numerical simulation is provided to
illustrate the effectiveness of the proposed algorithm.
