Parameter estimation for the input–output representation from the Kalman innovation filter

The Kalman innovation state-space model is a typical state filtering algorithm. This paper derives its corresponding input–output representation. Based on this representation, we propose a hierarchical extended stochastic gradient (HESG) parameter estimation algorithm and its variant. The convergence performance of the HESG algorithm is investigated in detail; in particular, conditions under which the parameter estimation errors converge to zero are established. These conditions include persistent excitation of the extended information vectors and strict positive realness of the noise models. Finally, the proposed algorithms are tested on a numerical example to demonstrate their advantages and effectiveness.