State estimation of nonlinear dynamical systems by an asymptotic method (Q1282555)

Consider the finite-dimensional dynamic system with noises \[ {dx(t)\over dt}= f(x(t), t)+ G(x(t),t) w(t) \] with initial conditions \(E[x(0)]= \widehat x_0\), \(\text{Cov}[x(0)]= V_0\). The output signal is measured, \[ y(t)= m(x(t), t)+ n(t),\quad t\in [0,t_N]. \] The components \(w_j(t)\), \(j= 1,\dots, p\), and \(n_i(t)\), \(i= 1,\dots, r\) of noises are stationary stochastic processes with arbitrary probability distribution laws. The estimation problem of the phase vector \(x(t)\) is solved. By the Fourier series expansion of covariance functions of the processes \(w_j(t)\) and \(n_i(t)\) and by an asymptotic method of analyses of differential equations with small parameters, an estimation algorithm with minimum variance of the estimation error is obtained. As an example, the problem of determining the value of the heat flow with the help of the battery heat flow meter is investigated. The numerical results show that the proposed algorithm yields unbiased consistent estimators that have a rapid rate of convergence and a variance of estimation error that is less than in the recursive least square method and in the Kalman filter.