Finite-Time Scaled Consensus in Hybrid Multi-Agent Systems via Conjugate Gradient Methods

This paper addresses the finite-time scaled consensus problem for hybrid multi-agent systems (HMASs) comprising both continuous-time and discrete-time agents. Motivated by the limitations of traditional consensus models that neglect scaling effects and hybrid dynamics, we propose a unified control framework under two sampling-based protocols. By modeling the system as a directed communication graph and formulating the consensus condition as a linear system, we employ the conjugate gradient method (CGM) to achieve exact convergence within at most \(N\) steps, where \(N\) is the number of agents. Sufficient and necessary conditions are derived under each protocol to guarantee scaled consensus, accounting for heterogeneous agent dynamics and non-uniform scaling parameters. Numerical simulations on scale-free networks validate the theoretical results. The key innovation lies in integrating CGM with hybrid protocols to realize fast and scalable consensus in finite time for complex distributed systems.