Nonlinear Kalman filters for dynamic estimation over complex networks

Abstract

This thesis studies the applications of nonlinear Kalman filters in solving problems associated with the dynamic estimation of networked systems such as epidemic tracking over networks and monitoring the operation of power networks in real time. In particular, the epidemic spreading on networks is concerned with controlling morbidity. A compartmental model is in general utilized for describing epidemic transmission on networks. The compartmental model, however, is inadequate for describing the dynamics of epidemic spreading without considering measurements from transmission processes. A nonlinear Kalman filter can be utilized for solving this problem by considering the inherent dynamic model together with epidemic transmission processes. It is, however, non-trivial to choose appropriate nonlinear Kalman filters for epidemic tracking over various networks, such as the Erdös and Rényi (ER) network, the Newman and Watts (NW) network, and the Watts and Strogatz (WS) network. A guideline will be provided for choosing traditional nonlinear Kalman filters for studying epidemic spreading on commonly used complex networks. Specifically, epidemic spreading on networks is described by compartmental models, such as Susceptible-Infected-Recovered, Susceptible-Infected-Susceptible, and Susceptible-Infected-Recovered-Susceptible models. The dynamic study of epidemic spreading on various homogeneous networks is performed using nonlinear Kalman filters, including the extended Kalman filter (EKF), the unscented Kalman filter(UKF), and the cubature Kalman filter (CKF). Various traditional Kalman filters are compared in terms of accuracy, stability, and complexity. These traditional Kalman filters are, however, based on the optimization of the minimum mean square error. As a result, these nonlinear Kalman filters may have degraded filtering precision when available measurements are corrupted by non-Gaussian noise. For solving this issue, a novel generalized correntropy sparse Gauss-Hermite quadrature filter is proposed by combining the generalized correntropy with the sparse Gauss-Hermite quadrature filter. Dependent on the Susceptible-Infected-Recovered-Susceptible compartmental model, the proposed generalized correntropy sparse Gauss-Hermite quadrature filter is applied to tracking epidemic spreading on homogeneous networks in the presence of non-Gaussian noise. In addition, the dynamic estimation of power systems is studied with the aim of enhancing the operation of power distribution infrastructure. Since the available measurements may be corrupted by non-Gaussian noise, a robust mixed p-norm square root unscented Kalman filter is proposed for estimating the state of power systems in the presence of non-Gaussian noise. The mixed p-norm square root unscented Kalman filter applies a mixed p-norm for weighting measurement errors for robustness improvement. Furthermore, unlike the generalized correntropy sparse Gauss-Hermite quadrature filter, the mixed p-norm square root unscented Kalman filter utilizes a piecewise function, i.e., multiple p-norms for handling varying measurements. As a result, the mixed p-norm square root unscented Kalman filter is more flexible in dealing with corrupted measurements but may be confronted with the choice of multiple parameters in comparison with the generalized correntropy sparse Gauss-Hermite quadrature filter. Simulation results demonstrate the efficiency of the p-norm square root unscented Kalman filter in the WSCC (Western System Coordinating Council) 3-machine system and the NPCC (Northeastern Power Coordinating Council) 48-machine system.