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This study is devoted to the finite-time consensus problem of multi-agent systems with higher-order dynamics, and presents a framework for effectively constructing distributed protocols which incorporate iterative learning control actions into output feedbacks. Using a terminal updating law, the feedback iterative learning protocols are shown with the ability to enable all agents to achieve the consensus at a finite time that can be prescribed. Furthermore, a model reference approach is employed to improve the feedback iterative learning protocols such that all agents can be guaranteed to achieve the consensus at any given desired terminal output. In both cases, necessary and sufficient conditions are provided which can also offer design criteria for the learning gains to ensure consensus. Simulation results are included to verify the effectiveness of the proposed theoretical results.