Bias analysis in mode-based Kalman filters for stochastic hybrid systems

Abstract

Stochastic hybrid system (SHS) is a class of dynamical systems that experience interaction of both discrete mode and continuous dynamics with uncertainty. State estimation for SHS has attracted research interests for decades with Kalman filter based solutions dominating the area. Mode-based Kalman filter is an extended version of the traditional Kalman filter for SHS. In general, as Kalman filter is unbiased for non-hybrid system estimation, prior research efforts primarily focus on the behavior of error covariance. In SHS state estimate, mode mismatch errors could result in a bias in the mode-based Kalman filter and have impacts on the continuous state estimation quality. The relationship between mode mismatch errors and estimation stability is an open problem that this dissertation attempts to address. Specifically, the probabilistic model of mode mismatch errors can be independent and identically distributed (i.i.d.), correlated across different modes and correlated across time. The proposed approach builds on the idea of modeling the bias evolution as a transformed system. The statistical convergence of the bias dynamics is then mapped to the stability of the transformed system. For each specific model of the mode mismatch error, the system matrix of the transformed system varies which results in challenges for the stability analysis. For the first time, the dissertation derives convergence conditions that provide tolerance regions for the mode mismatch error for three mode mismatch situations. The convergence conditions are derived based on generalized spectral radius theorem, Lyapunov theorem, Schur stability of a matrix polytope and interval matrix method. This research is fundamental in nature and its application is widespread. For example, the spatially and timely correlated mode mismatch errors can effectively capture cyber-attacks and communication link impairments in a cyber-physical system. Therefore, the theory and techniques developed in this dissertation can be used to analyze topology errors in any networked system such as smart grid, smart home, transportation, flight management system etc. The main results provide new insights on the fidelity in discrete state knowledge needed to maintain the performance of a mode-based Kalman filter and provide guidance on design of estimation strategies for SHS.