This　paper　explores　the　leader-following　consensus　of　second-order　heterogeneous　linear　multi-agent　systems　with　the　directed　communication　topology.　Firstly,　we　propose　a　general　consensus　algorithm　which　composed　of　different　gain　matrices　and　generic　communication　matrices,　which　increases　the　system　freedom　degree.　Then,　we　adopt　the　state　linear　transformation　method　to　convert　the　consensus　study　to　a　partial　variable　stability　study　of　the　corresponding　auxiliary　system,　thereby　reducing　the　computation　complexity　of　the　research　problem.　On　this　basis,　we　get　a　sufficient　and　necessary　consensus　standard.　According　to　this　criterion,　by　solving　bilinear　matrix　inequality,　we　propose　an　algorithm　for　solving　the　protocol　gain　matriies.　Finally,　two　sets　of　simulations　prove　the　effectiveness　of　the　algorithm　proposed　in　this　paper.　©　2021　IEEE