The multi-innovation methods can be applied to adaptive filtering,parameter estimation,self-tuning control,adaptive fault detection and diagnosis,etc.for linear systems and nonlinear systems.Linear systems include two basic types:equation-error type systems and output-error type systems.This paper studies the multi-innovation (MI) extended stochastic gradient algorithm and the MI extended least squares identification algorithm for equation-error moving average (EEMA) systems (namely,CARMA systems),and presents the decomposition based MI generalized extended stochastic gradient (GESG) algorithm and the MI generalized extended least squares (GELS) algorithm,and the filtering based MI-GESG algorithm and the filtering based MI-GELS algorithm for equation-error autoregressive moving average (EEARMA) systems (namely,CARARMA systems).