Transformations of Markov processes in fault tolerant interconnected systems

Safety-critical control systems use fault tolerant interconnections of components to minimize the effect of randomly triggered faults. The system availability process indicates whether or not the interconnection is operating correctly at each time instant. It is a 2-state process that results from the transformation of the stochastic processes characterizing the availability processes of the interconnected components. To analyze closed-loop systems controlled by these fault tolerant interconnected components, it is important to determine the characteristics of the system availability process. When the availability processes of the interconnected components are independent homogeneous Markov chains, the statistical nature of the system availability process is characterized. In particular, it is shown that the system availability process is not necessarily Markov, but has a well-defined one-step transition probability matrix that approaches a constant stochastic matrix at steady-state. Since it is simpler to analyze switched closed-loop systems when the switching process is Markov, conditions for the system availability process to be a Markov chain for all initial distributions are determined. A sufficient stability condition is given when the system availability process is a non-homogeneous Markov chain for a class of initial distributions.