Generalized Markov Models for Real-Time Modeling of Continuous Systems

This paper presents a modeling framework based on finite-state space Markov chains (MCs) and fuzzy subsets to represent signals that vary in a continuous range. Our special attention to this extension of finite-state space MC modeling is motivated by numerous opportunities in applying MC models to represent physical variables in automotive and aerospace systems and, subsequently, using these models for fault detection, estimation, prediction, stochastic dynamic programming, and stochastic model predictive control. Our generalized MC modeling framework synergistically combines the notion of transition probabilities with information granulation based on fuzzy partitioning. As compared with the case of more familiar interval partitioning, the transition probabilities in our model are defined for transitions between fuzzy subsets rather than intervals/rectangular cells. Our framework is first introduced for scalar-valued signals and then extended to vector-valued signals. A real-time capable recursive algorithm for learning transition probabilities from measured signal data is derived. Formulas that characterize the possibility distribution of the next signal value and predict the next signal value are given. It is shown that the introduced modeling framework based on MC models defined over fuzzy partitioning inherits all properties and represents a natural extension of MC models defined over interval partitioning, while providing interpolation ability and improved prediction accuracy. In addition, we derive an alternative formulation of the Chapman–Kolmogorov equation that applies to models in possibilistic/fuzzy environment. Examples are given to illustrate the key notions and results based on modeling of the vehicle speed and road grade signals.