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Input saturation, which brings input non-linearity to systems and limits the closed-loop systems' performances severely, is an ubiquitous problem in practice. Here, triggered finitetime consensus problem of disturbed leader-follower multi-agent systems is addressed under input saturation constraint. To overcome the coupling of the non-linearity and finitetime convergence together with input saturation, non-linear triggered control protocol has been proposed, which is confirmed through a two-step stabilization procedure. First, the saturated closed-loop system is designed to be asymptotically stable, under which the systems converge into a desaturated region in finite time. Then in the second step, with the proposed control protocol, the desaturated closed-loop multi-agent systems can achieve practical finite-time consensus. In addition, to get rid of continuous communication in the monitoring of event-triggered condition, a self-triggered approach is further proposed. With rigorous analyses, practical finite-time consensus can be achieved for the overall closed-loop systems and Zeno behavior is also excluded. Simulations are carried out in detail to illustrate the theoretical results in this study.

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