This　paper　considers　the　problem　of　leader-following　consensus　and　fault　detection　for　a　class　of　multi-agent　systems　with　Lipschitz　nonlinear　dynamics.　To　reduce　the　amount　of　redundant　information　and　avoid　checking　triggering　conditions　continually,　this　paper　proposes　an　efficient　network　framework　with　a　double　periodic　event-triggered　mechanism.　Based　on　the　proposed　framework,　an　improved　fault　detection　observer　and　a　consensus　controller　are　designed.　Then,　the　original　problem　is　converted　into　a　set　of　stability　problems　with　constraints.　According　to　Lyapunov–Krasovskii　theorem　and　the　free-weighting　matrix　technique,　sufficient　conditions　for　solving　these　stability　problems　are　derived　in　the　form　of　bilinear　matrix　inequalities　(BMIs).　Further,　to　eliminate　the　nonlinear　terms　of　BMI　and　obtain　optimal　performance,　two　iterative　algorithms　based　on　linear　matrix　inequalities　(LMIs)　are　developed.　Two　simulation　examples　are　provided　to　verify　the　practicality　and　validity　of　the　theoretical　results.　©　2023,　The　Author(s),　under　exclusive　licence　to　Springer　Nature　B.V.