Recursive identification methods for general stochastic systems with colored noises by using the hierarchical identification principle and the filtering identification idea

This article reviews and investigates several basic recursive parameter identification methods for a general stochastic system with colored noise (i.e., output-error autoregressive moving average system or Box–Jenkins system). These recursive identification methods are derived by means of the hierarchical identification principle and the filtering identification idea, including a filtered auxiliary-model hierarchical generalized extended stochastic gradient algorithm, a filtered auxiliary-model hierarchical multi-innovation generalized extended stochastic gradient algorithm, a filtered auxiliary-model hierarchical recursive generalized extended gradient algorithm, a filtered auxiliary-model hierarchical multi-innovation recursive generalized extended gradient algorithm, a filtered auxiliary-model hierarchical generalized extended least squares algorithm, and a filtered auxiliary-model hierarchical multi-innovation generalized extended least squares algorithm by using the auxiliary-model identification idea. The presented filtered auxiliary-model hierarchical generalized extended identification algorithms can be extended to other linear and nonlinear systems with colored noises.