A perturbation method for estimation of dynamic systems

A novel framework called the Perturbed Jth Moment Extended Kalman Filter (PJMEKF), based on a classical perturbation technique is proposed for estimating the states of a nonlinear dynamical system from sensor measurements. This method falls under a class of architectures under investigation primarily to study the interplay of major issues in nonlinear estimation such as nonlinearity, measurement sparsity, and initial condition uncertainty in an environment with low levels of process noise. Taylor series expansion of the departure motion dynamics about the best estimate is used to derive a series representation of the unforced motion. It is found that such series representation evolves as a set of differential equations that force each other in a cascade manner, adding up to give the unforced motion (in a so-called “triangular” structure). This formal perturbation solution for the departure motion dynamics is used in deriving the differential equations governing the time evolution of the high order statistical moments of the estimation error. These tensor differential equations are found to possess a similar high order triangular structure in addition to being symmetric (in N tensorial dimensions and we appropriately term the evolution equations as Tensor Lyapunov Equations of statistical moment perturbations). Elegance of the tensor differential equations thus derived is accompanied by the computational advantages due to symmetry in all tensorial dimensions. A vector matrix representation of tensors is proposed with which the representation and solution of the tensor differential equations can be carried out effectively. Approximations are introduced to incorporate low levels of process noise forcing function in the propagation phase of the moment equations. The statistics thus propagated are used in a filtering framework to estimate the state vector of a nonlinear system from noisy measurements, within the traditional Kalman update paradigm. The Kalman gain thus determined is utilized in updating all high order moments in preparation for the subsequent propagation phase leading to improved estimation accuracy. The filter developed is applied to an orbit estimation problem and comparisons are presented with classical extended Kalman filter.