On the statistical convergence of bias in mode-based Kalman filter for switched systems

Abstract

Many physical and engineered systems (e.g., smart grid, autonomous vehicles, and robotic systems) that are observed and controlled over a communication/cyber infrastructure can be efficiently modeled as stochastic hybrid systems (SHS). This paper quantifies the bias of a mode-based Kalman filter commonly used for state estimation in SHS. The main approach involves modeling the bias dynamics as a transformed switched system and the transitions across modes are abstracted via arbitrary switching signals. This general model effectively captures a wide range of SHS systems where the modes may follow deterministic, Markovian, or guard condition based transitions. By leveraging techniques developed to analyze the stability of switched systems, we derive conditions for statistical convergence of the bias in a mode-based Kalman filter in the presence of mode mismatch errors. Developed upon the foundations of Lyapunov theory, we demonstrate a linear matrix inequality condition that guarantees asymptotic stability of the corresponding autonomous switched system irrespective of the choice of mode mismatch probability. Furthermore, we obtain the range of mode mismatch probabilities that assures bounded input bounded output stability of the bias dynamics for both stable and unstable SHS. Using numerical simulations of a smart grid with network topology errors, we verify and validate the theoretical results and demonstrate the potency of using the analysis in critical infrastructures.

Quantification of Mismatch Error in Randomly Switching Linear State-Space Models

“…In the case of switching dynamics, [23] explores the conditions under which the instantaneous mode detection is successful based on the statistics of the predicted and collected measurement residuals. Also, in [24]- [27] the convergence of a mode-based KF is studied and the conditions under which the steady state bias term would converge to zero are investigated. However, to the best of our knowledge, the work presented in this paper is the first to quantify the transient error in SLDS state estimation based on model parameters alone and without measurements.…”

Quantification of mismatch error in randomly switching linear state-space models

“…In the case of switching dynamics, [18] explores the conditions under which the instantaneous mode detection is successful or not based on the statistics of the residuals of the predicted and the collected measurements. Also, Zhang et al [19]- [22] study the convergence of a mode-based KF and argue the conditions under which the steady state's bias term will converge to zero in a switching mode dynamic system. However, to the best of our knowledge, estimation of the transient evolution of the error in an SLDS prior to running the experiment and collecting measurements has not been investigated.…”