On the Performance of Kalman Filter for Markov Jump Linear Systems with Mode Mismatch
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On the Performance of Kalman Filter for Markov Jump Linear Systems with Mode Mismatch Wenji Zhang1
· Balasubramaniam Natarajan1
Received: 19 January 2020 / Revised: 30 August 2020 / Accepted: 6 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Markov jump linear system (MJLS) is a class of hybrid systems where the continuous states evolve linearly and the discrete state transitions are modeled via a Markov chain. It has been shown that the optimal estimator for MJLS is a mode-based Kalman filter when the discrete mode is given. However, in practical applications, there are situations where the mode information is inaccurate and mode mismatches result in a biased estimate from the mode-based Kalman filter. This paper, for the first time, studies the impact of time correlated mode mismatch errors on MJLS state estimation using a mode-based Kalman filter. Unlike prior efforts that (1) assume mode mismatches are independent and identically distributed and (2) are limited to bi-modal systems, this paper is built on a general MJLS setup with a Markovian model for mode mismatch errors. The main contribution of this work lies in deriving the statistics of the bias term from a mode-based Kalman filter estimation with the aforementioned system setup. The sufficient conditions based on the probabilities of mode mismatch errors for the bias to be statistically convergent are derived. Since time correlated mode mismatch errors can effectively capture communication link impairments in a cyberphysical system, this new fundamental result provides guidance on the design of estimation strategies for MJLS and sheds light on their resilience in the presence of mode mismatch errors. Keywords Markov jump linear systems · Kalman filter · Error analysis · Mode mismatch error
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Wenji Zhang [email protected] Balasubramaniam Natarajan [email protected]
1
Department of Electrical and Computer Engineering, Kansas State University, Manhattan, KS 66506, USA
Circuits, Systems, and Signal Processing
1 Introduction Stochastic hybrid system (SHS) refers to a system that involves the interaction of continuous and discrete dynamics with uncertainties. SHS models have been widely used to model cyber-physical systems (CPS) such as smart grid [35], communication networks [18] and air traffic control systems [14,26]. One important category of SHS is Markov jump linear system (MJLS). MJLS models are applicable to systems that can be represented by a set of linear systems with modal transitions governed by a Markov chain. MJLS has attracted significant attention in the research community due to its analytical tractability as well as applicability to many practical systems. The applications of MJLS include, but are not limited to, power systems [28], networked control systems (NCS) [29], etc. State estimation in MJLS is critical for both situational awareness and implementation of control actions. Due to the interaction between continuous states and discrete states, there exists some challenges in MJLS state e
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