This　article　proposes　a　novel　framework　for　studying　the　consensus　problem　for　a　class　of　nonlinear　multiagent　systems　(MASs)　based　on　decentralized　output-feedback　stabilization　of　derived　reduced-order　fundamental　edge　states　dynamic　systems.　First,　by　introducing　edge　states　dynamics　with　respect　to　a　directed　spanning　tree　of　the　communication　topology,　we　equivalently　transform　the　state　consensus　problem　into　a　decentralized　output　feedback　stabilization　problem.　Then,　for　the　case　of　linear　MASs,　we　derive　a　new　sufficient　and　necessary　consensus　criterion　in　terms　of　the　concept　of　decentralized　fixed　modes　and　also　present　a　design　procedure　of　the　protocol　gains　in　terms　of　linear　matrix　inequalities　(LMIs).　Next,　a　sufficient　consensus　condition　is　derived　in　terms　of　LMIs　for　the　nonlinear　MAS　with　local　proportional-integral　(PI)　observers.　Furthermore,　to　reduce　the　complexity　of　the　condition　verification,　we　introduce　a　graph　representation　of　the　fundamental　edge　states　dynamic　system　and　decompose　it　into　strongly　connected　components　and,　thus,　reorganize　the　condition　in　a　decentralized　form.　On　this　basis,　we　present　a　hierarchical　method　to　design　the　consensus　protocols　and　PI　observers　based　on　the　information　of　the　subsystem　itself　only.　Finally,　several　numerical　examples　are　given　to　illustrate　the　effectiveness　of　the　proposed　method.