Optimal Filtering for the 2-D Markov Jump Linear System with Fading Channels

This paper is concerned with the optimal filtering for the two-dimensional (2-D) Markov jump linear system (MJLS) with fading channels, where the two-dimensional system is represented by the Fornasini-Marchesini's second (FMII) model and the fading channel is described by a stochastic diagonal matrix. The aim of this paper is to design an optimal recursive filter in the 2-D scheme, where the filter gain is subject to Markov jumping property. A probability constraint is required. When the sum of the changing steps in two directions remain the same, the corresponding transition probabilities keep the same. Based on this condition, an optimal recursive Markov jump linear filter under the 2-D framework is designed via the complete mean square method. And the filter gain is deduced based on a recursive coupled difference Riccati equation and a Lyapunov equation. The most challenge exists in this paper is the deduction of the recursive form of the 2-D coupled Riccati equation, where the correlation of the jumping parameter needs to be considered as well as the 2-D circumstances.