Algebraic Dynamic Fault Tree Analysis for Avionic Systems
RESUME Les analyses par arbres de defaillance se basent sur une representation graphique des evene ments probabilistes qui peuvent engendrer la panne d'un systeme quelconque. Dans l'analyse traditionnelle, les evenements sont consideres comme etant statistiquement independants et sont relies entre eux par la logique combinatoire. L'arbre de defaillance permet de determiner les relations existantes entre les differents evenements et leurs differents modes de fonctionnement, de facon a deduire analytiquement une probabilite de panne du systeme. A chaque evenement est associe un taux de panne. Les portes representant les relations de la logique combinatoire AND, OR,.etc, sont incapables de representer convenablement la relation entre differents evenements dont la sequence d'apparition est importante. Nous avons alors affaire a un arbre de defaillance dit "dynamique" qui ne peut etre modelise sans le recours de portes logiques dynamiques. L'arbre de defaillance dynamique est alors modelise a partir de portes dynamiques et de portes logiques standards ou statiques. Parmi les portes dynamiques les plus utilisees, on retrouve la porte priorite AND (PAND) la porte sequentielle (SEQ) la porte de rechange ou en attente (SPARE) et la porte de dependance fonctionnelle (FDEP) Ces portes permettent de conceptualiser un arbre de defaillance entre des evenements ayant une certaine causalite. Au cours des dernieres decennies, plusieurs accidents d'avion ont ete rapportes dont la cause etait reliee au systeme de pilotage. La modelisation du systeme de pilotage par un arbre de defaillance dynamique est un bon moyen d'evaluer la probabilite de panne et d'ameliorer la robustesse du systeme. Ce memoire porte sur la conception d'un arbre de defaillance dynamique pour un systeme avionique simple. Une methode algebrique est utilisee pour calculer la probabilite de panne du systeme global. Des theoremes mathematiques sur les portes dynamiques sont utilises pour developper un modele Matlab qui calcule les differentes probabilites de panne des sous systemes et la probabilite globale. Compare aux methodes traditionnelles de resolution des arbres de defaillance dynamique, soient les methodes Monte Carlo et les chaines de Markov, la methode algebrique proposee dans ce travail est beaucoup plus simple a implementer et demande moins de temps de calcul, tout en conservant une precision semblable sur les resultats. ABSTRACT Fault Tree Analysis (FTA) is a method mainly based on a graphical representation of dif ferent combinations of basic events which result in a Top Event (TE) In the traditional analysis, the TE occurrences are logically investigated by considering all possible events sta tistically independent implementing logical Boolean gates. A fault tree is used to determine the operational relationship among different components under different modes to derive an analytical expression giving the probability of failure. Since traditional logical gates, such as AND, OR, etc. are not capable to demonstrate the dynamic behavior of failure mecha nisms in a system with dependent events and failures, a Dynamic FTA (DFTA) is used as a promising method. A DFTA employs dynamic gates in its fault tree to represent a sequential notation for the system failures by considering the order of occurrence and different com binations of basic events. The dynamic relationships between basic events can be analyzed using DFT, whereas the traditional static FT expresses only the probability of failure as a function of basic event failure rates. A DFT has at least one dynamic gate along with a combination of static gates such as AND OR and voting gates. The most commonly used dynamic gates in DFTA are; the priority AND (PAND) gate, the sequence enforcing (SEQ) gate, the standby or spare (Spare) gate, and the functional dependency (FDEP) gate. Dynamic gates in fault trees provide a conceptually simple modeling framework to represent system level Failure in terms of interactions between component failures through basic events relationships. Some fatal risks in flight control are typically due to unforeseen failures in different sections of the systems, particularly, in flight control hardware and software systems. Modeling them necessitates a reliable quantitative analysis such as FTA that dynamically predicts faults in the related components. The use of dynamic gates enables the analysis to be more dependable and accurate compared to conventional FTA. This thesis investigates the design of a reliable DFT for an avionic system by taking into account the least reliable flight components. An algebraic method for solving a DFT with a promising algorithm is demonstrated to determine the failure probability of the aircraft which is referred to as the top event (TE) based on the relationships between many basic events. To this end, using temporal operators and some related theorems, the behavioral model of the dynamic gates and the subtrees in the DFT are determined. Using the obtained behavioral models, the probabilistic models of the dynamic gates and subtrees in the DFT are determined. Finally, the total probability of failure is calculated through algebraic analysis of the subtrees in the structure and their individual failure probabilities. Compared to traditional methods, such as Monte Carlo simulations and Markov analysis, our proposed algebraic method is simpler to implement with lower computational work load and less computer intensive to solve a DFT rapidly and accurately.