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This paper considers the designated convergence rate (DCR) (or the designated convergence margin) problems of consensus or flocking of coupled double-integrator agents. The DCR problems are more valuable for systems design than just convergence or stability conditions. The system setting in this paper is general, i.e., the velocity coupling and position coupling (VCPC) between agents, respectively, are set to be general and nonequal (up to rescaling), together with distinct damping and stiffness gains for the VCPC, respectively. This paper has two primary contributions on consensus: 1) further necessary and sufficient conditions are established to guarantee the DCR problems of the system, which have enriched the previous results and 2) the patterns of the convergence rate contours for the DCR are characterized, in terms of the damping and stiffness gains, which are closely related to the characteristics of the spectra of the two Laplacian matrices of the VCPC. Additionally, this paper has a contribution on matrix theory, i.e., the sufficient conditions for the simultaneous upper-triangularization of two independent Laplacian matrices, particularly from an easily verifiable topological perspective on the corresponding digraphs of these Laplacian matrices.

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The Designated Convergence Rate Problems of Consensus or Flocking of Double-Integrator Agents With General Nonequal Velocity and Position Couplings: Further Results and Patterns of Convergence Rate Contours