Two-stage gradient-based iterative algorithm for bilinear stochastic systems over the moving data window

For the bilinear stochastic system, the difficulty of identification lies in the product of the state vector and input in the system. This paper studies the iterative estimation of the parameters and states for the bilinear state-space systems in the observer canonical form. The standard Kalman filter is recognized as the best state estimator for linear systems, but it is not applicable for bilinear systems. Therefore, this paper proposes a state filter (SF) for the bilinear systems based on the extremum principle. By means of the hierarchical principle, we decompose the identification model into two sub-identification models by introducing two fictitious output variables. Then an SF two-stage gradient-based iterative algorithm is proposed to achieve the combined parameter and state estimation according to the gradient search. For the purpose of improving the identification performance, an SF two-stage moving data window gradient-based iterative algorithm is derived by increasing the data utilization. The numerical example demonstrates the validity of the proposed algorithms.