State estimation for bilinear systems through minimizing the covariance matrix of the state estimation errors

Abstract

This paper considers the state estimation problem of bilinear systems in the presence of disturbances. The standard Kalman filter is recognized as the best state estimator for linear systems, but it is not applicable for bilinear systems.It is well known that the extended Kalman filter (EKF) is proposed based on the Taylor expansion to linearize the nonlinear model. In this paper, we show that the EKF method is not suitable for bilinear systems because the linearization method for bilinear systems cannot describe the behavior of the considered system. Therefore, this paper proposes a state filtering method for the single-input-single-output bilinear systems by minimizing the covariance matrix of the state estimation errors. Moreover, the state estimation algorithm is extended to multiple-input-multiple-output bilinear systems. The performance analysis indicates that the state estimates can track the true states. Finally, the numerical examples illustrate the specific performance of the proposed method.

Recursive parameter estimation methods and convergence analysis for a special class of nonlinear systems

“…Lemma 3. For the system in (7) and the BSO-HSG algorithm in (16)- (29), the following equalities hold:…”

“…Theorem 1. For the bilinear system in (7) and the BSO-HSG algorithm in (16)- (29), assume that v t is a white noise sequence with zero mean and variance 2 , r 1,t → ∞ and there exists an integer sequence {t 0 , t 1 , t 2 , … , t s , … , t s+1 , … }, t 0 = 0 and t * s = t s+1 − t s ⩾ n 2 + n and two positive constants c 1 and c 2 such that the following persistent excitation conditions hold. a.s., for s = 1, 2, 3, … Remark 5.…”

Moving data window gradient‐based iterative algorithm of combined parameter and state estimation for bilinear systems

“…Remark Recent work about identifying the bilinear state‐space systems exists. For example, the state estimation method by minimizing the covariance matrix of the state estimation errors was proposed in Reference , the method is using the extremum principle differing from the previous linearization method like Taylor expansion. The state filter‐based on the delta operator was proposed in Reference .…”

Adaptive parameter estimation for a general dynamical system with unknown states

“…Parameter estimation and state filtering are basic for system control and system analysis . Many parameter estimation methods, such as hierarchical identification methods, Newton identification methods, and coupled identification methods, have been widely studied.…”

“…10,11 Parameter estimation and state filtering are basic for system control and system analysis. 12,13 Many parameter estimation methods, such as hierarchical identification methods, 14,15 Newton identification methods, [16][17][18] and coupled identification methods, 19,20 have been widely studied. In the literature, Waschburger and Galvão investigated a method to estimate the input delays of a discrete-time state-space model by utilizing the standard least squares methods to minimize a quadratic cost function of the prediction error of the system states within a given time range.…”