This　paper　studies　the　leader-following　state　consensus　problem　of　high-order　linear　interconnected　multi-agent　systems　(LIMASs)　under　a　general　fixed　directed　communication　topology.　A　novel　consensus　protocol　is　firstly　proposed,　in　which　different　agents　can　possess　different　gain　matrices　and　different　communication　weight　matrices.　Then,　a　state　linear　transformation　method　is　presented　to　equivalently　transform　the　leader-following　state　consensus　problem　into　a　partial-variable　stability　problem　of　a　corresponding　auxiliary　system.　Next,　a　sufficient　and　necessary　consensus　criterion　is　deduced　in　terms　of　the　Hurwitz　stability　of　a　real　matrix.　Based　on　this　consensus　criterion,　an　iterative　algorithm　is　given　to　design　the　gain　matrices　in　the　consensus　protocol　by　solving　a　bilinear　matrix　inequality　(BMI).　Finally,　simulation　examples　are　provided　to　verify　the　effectiveness　of　the　proposed　method.　©　2022,　The　Author(s),　under　exclusive　license　to　Springer　Nature　Singapore　Pte　Ltd.