A New Approach to Modeling and Stabilizing 2-D Markovian FMII Systems

Unlike the existing works on 2-D Markovian systems, this work proposes a new approach to modeling and stabilizing such systems according to mode relationships and directional influences. First, 2-D Markovian Fornasini-Marchesini (FMII) system models are developed by establishing two new 2-D Markov chains with different classification criteria. This involves reclassifying all possible mode combinations at the two previous points to ensure that the current mode is jointly determined by two preceding modes. Both the initial mode space and total probability formula are developed, which are not only consistent with the classical structure of 2-D Markov chains, but also effectively capture the dynamic behavior of 2-D FMII systems. Subsequently, for autonomous 2-D Markovian FMII systems, sufficient conditions on asymptotic stability in the mean square are provided, by which transition probabilities can be designed. For controlled 2-D Markovian FMII systems, the stabilization problem is addressed by simultaneously designing the transition probabilities and asynchronous controllers. Finally, an example is given to demonstrate the proposed schemes.