State-space models and matrix methods

Abstract

In many cases a two-state approach for reliability analysis of systems composed of multiple components is not appropriate. This is so especially when redundancy is applied or when some components have multiple failure modes having non-identical impacts on system performance. For example, consider a redundant rectifier system consisting of two diodes connected in series (Fig. 5.1(a)) and without repair. The diodes might suffer two distinct failure modes: a short-circuit failure D₂allowing current to pass in both directions and an open failure D₃. Figure 5.1(b) shows the states and the corresponding transitions for a diode. It will be obvious that the system is only redundant to short-circuit failures and that the redundancy is not active for open failures. Even worse, the probability of an open system failure has almost been doubled. This means that the two different failure modes of the diodes have different impacts on system performance and, therefore, cannot be treated as one single failure mode. For this simple system a combinatorial solution is still possible as:R = P _r{Da₁Db₁∪ Da₁Db₂∪ Da₂Db₁}, however, the logic expression for system success contains dependent events since the diode states D₁and D₂are mutually exclusive. In Chapter 7 it is shown how to deal with systems of multistate items more generally while Chapter 8 explains how to perform probability calculations on large logical expressions with dependent events.