A Three-Term Conjugate Gradient Framework for Matrix-Scaled Consensus in Hybrid Multi-Agent Systems

Abstract

This paper studies matrix-scaled consensus in hybrid multi-agent systems with both continuous-time and discrete-time dynamics. By introducing a coordinate transformation, the problem is reformulated as an agreement problem in a transformed space, leading to a regularized linear system. An optimization-based framework is developed, and a three-term conjugate-gradient (TTCG) algorithm is proposed to compute the consensus state efficiently. It is shown that the associated operator is symmetric positive definite, ensuring existence and uniqueness of the solution, while the TTCG iteration generates a strict descent sequence and converges to the projected agreement state. The framework is further extended to directed networks via a symmetrized formulation. Numerical results validate the theoretical findings and reveal that, although agreement is achieved in the transformed coordinates, the original states exhibit a clustered structure determined by the scaling matrices. The proposed approach provides a unified and computationally efficient solution for matrix-scaled consensus.