H∞ Filtering for 2-D Discrete-Time Periodic Markov Jump Systems With Multiplicative Noise: A Periodic HMM Approach

This article presents the implementation of an asynchronous ${\mathcal {H}}_{\infty }$ filter for 2-D discrete-time periodic Markov jump systems with multiplicative noise. The study addresses the issue of missing measurements, which is treated as a stochastic variable following the Bernoulli random distribution. To account for the nonsynchronous phenomenon between the system and filter due to the loss of mode information, the periodic hidden Markov model (HMM) is introduced. Moreover, the transition rate matrix of the system and the conditional probability matrix of the filter are general, allowing for transition probabilities in fully known, partly known or fully unknown cases. The objective is to implement an asynchronous ${\mathcal {H}}_{\infty }$ filter based on periodic HMM that guarantees the filtering error system is mean-square asymptotically stable while maintaining a specified ${\mathcal {H}}_{\infty }$ disturbance attenuation performance. In the light of linear matrix inequalities, the article offers sufficient conditions for filter to exist and provides a solution for the parameters of the filter. Ultimately, a demonstration of the validity of the presented design technique is provided through the Darboux equation.