The consensus control with the leader of delayed linear multi-agent systems under the relative state saturations is investigated in this article. Using an incidence matrix, the consensus problem of delayed multi-agent systems (time-varying input delays) with relative state saturation is converted into the stability problem of edge dynamics on the closed sets. Then, a new linear control protocol is designed using a saturation function that guarantees stability for all agents under relative state saturation constraints. With the appropriate Lyapunov–Krasovskii functional definition, sufficient conditions are created for the consensus of the agents. Meanwhile, the relative state saturations do not happen. Also, the agents track the leader’s path well, while bounded delays can be time-varying and arbitrarily fast. This consensus and stability are achieved by maintaining graph connectivity. Finally, the correctness of the above content is confirmed by simulating a practical example.

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