Classical and Resilient Filtering

Abstract

This chapter concerns the problem of \(H_{\infty }\) estimation for a class of Markov jump linear system (MJLS) with time-varying transition probabilities (TPs) in discrete-time domain. The time-varying character of TPs is also considered as finite piecewise homogeneous and the variations in the finite set are considered as two types: arbitrary variation and stochastic variation, respectively. The latter means that the variation is subject to a higher-level transition probability matrix (TPM). The mode-dependent and variation-dependent \(H_{\infty }\) filter is designed such that the resulting closed-loop systems are stochastically stable and have a guaranteed \(H_{\infty }\) filtering error performance index. Using the idea of partially unknown TPs for the traditional MJLS with homogeneous TPs, a generalized framework covering the two kinds of variation is proposed. Then, the derived results are extended to the study of the resilient \(H_{\infty }\) filtering problem for a class of discrete-time Markov jump neural networks (MJNNs) with time-varying delays, unideal measurements and multiplicative noises. The transitions of neural networks modes and desired mode-dependent filters are considered to be asynchronous, and a nonhomogeneous mode TPM of filters is used to model the asynchronous jumps to different degrees that are also mode-dependent. The unknown time-varying delays are also supposed to be mode-dependent with lower and upper bounds known a priori. The unideal measurements model includes the phenomena of randomly occurring quantization and missing measurements in a unified form. The desired resilient filters are designed such that the filtering error system is stochastically stable with a guaranteed \(H_{\infty }\) performance index. A monotonicity is disclosed in filtering performance index as the degree of asynchronous jumps changes. Numerical examples are provided to demonstrate the potential and validity of the theoretical results.