This　paper　deals　with　the　problem　of　leader-following　consensus　under　a　directed　communication　topology　for　a　class　of　discrete-time　multiagent　systems　with　large　delay　sequences　(LDSs).　More　specifically,　to　analyze　the　consensus　problem　of　such　a　time-delay　system　with　two　different　types　of　time-varying　delays,　new　Lyapunov　functionals　are　constructed.　By　imposing　some　constraints　on　the　length　and　frequency　of　LDSs,　a　sufficient　condition　for　ensuring　the　leader-following　consensus　is　provided　in　the　form　of　linear　matrix　inequalities　(LMIs).　A　simulation　example　is　provided　to　show　the　effectiveness　of　the　proposed　scheme.　(C)　2017　Elsevier　Inc.　All　rights　reserved.