Analysis of Jump Linear Systems Driven by Lumped Processes

Safety critical control systems such as flight control systems use fault-tolerant technology to minimize the effect of faults and increase the dependability of the system. In fault-tolerant systems, the system availability process indicates the current operational mode of an interconnection of digital logic devices. It is a process that results from the transformation of the stochastic processes characterizing the availability of the devices forming the system. To assess safety critical control systems, the following measures of performance will be considered: the steady-state mean output power, the mean output energy, the mean time to failure and the mean time to repair. For this assessment it is important to determine the characteristics of the system availability process since both stability and the aforementioned measure of performance are directly dependent on it. When the system availability process results from a transformation of a homogeneous Markov chain, it is well-known that the resulting process is not necessarily a homogeneous Markov chain. In particular, when the Markov chain characterizing the faults is a zeroth order Markov chain, it is shown that the availability process results in another zeroth order Markov chain. Thus, all the results which are known to analyze closed-loop systems driven by a homogeneous Markov chain can be applied to the zeroth order Markov chain. However, simpler formulas that do not trivially follow from these Markov chain results can be derived in this case. Part of this dissertation is dedicated to the derivation of these new formulas. On the other hand, when the system availability results in either a non-homogeneous Markov chain or a non-Markov chain, the existing Markov results can not be directly applied. This problem is addressed here. The necessity for better integration of the fault tolerant and the control designs for safety critical systems is also addressed. The dependability of current designs is primarily assessed with measures of the interconnection of fault tolerant devices: the reliability metrics that include the mean time to failure and the mean time to repair. These metrics do not directly take into account the interaction of the fault tolerant components with the dynamics of the system. In this dissertation, a first step to better integrate fault tolerant and the control designs for safety critical systems is made. These are the problems that motivated this work. Therefore, the goals of this dissertation are: to develop a suitable methodology to analyze a jump linear system when the driving process is characterized by a zeroth order Markov chain, a non-homogeneous Markov chain and a non-Markov chain; and to integrate the analysis of jump linear systems with the reliability theory for network architectures