From 1a8dc98b2110bf8f16ac7dc566271b72ac39434b Mon Sep 17 00:00:00 2001 From: Robin Clark Date: Sun, 21 Nov 2010 19:42:34 +0000 Subject: [PATCH] Just need full UML digram now. --- .../component_failure_modes_definition.tex | 91 +++++++++++-------- 1 file changed, 51 insertions(+), 40 deletions(-) diff --git a/component_failure_modes_definition/component_failure_modes_definition.tex b/component_failure_modes_definition/component_failure_modes_definition.tex index 51cef35..0e6dc44 100644 --- a/component_failure_modes_definition/component_failure_modes_definition.tex +++ b/component_failure_modes_definition/component_failure_modes_definition.tex @@ -366,12 +366,13 @@ can be active at one time is termed a {\textbf{unitary~state}} failure mode set. Let the set of all possible components be $ \mathcal{C}$ and let the set of all possible failure modes be $ \mathcal{F}$. The set of failure modes of a particular component are of interest -here. What is required is to define a property for -a set of failure modes where only one failure mode can be active at a time, -or borrowing from the terms of statistics, the failure mode is an event, and it is mutually exclusive -with the a specific set $F$. +here. +What is required is to define a property for +a set of failure modes where only one failure mode can be active at a time; +or borrowing from the terms of statistics, the failure mode being an event that is mutually exclusive +with a set $F$. We can define a set of failure mode sets called $\mathcal{U}$ to represent this -property. +property for a set of failure modes.. \begin{definition} We can define a set $\mathcal{U}$ which is a set of sets of failure modes, where @@ -475,36 +476,36 @@ to dealing with double simultaneous failure modes.}. To generalise, we may need to consider $N$ simultaneous failure modes when analysing a functional group. This involves finding all combinations of failures modes of size $N$ and less. -The Powerset concept from Set theory is useful to model this. +%The Powerset concept from Set theory is useful to model this. The powerset, when applied to a set S is the set of all subsets of S, including the empty set \footnote{The empty set ( $\emptyset$ ) is a special case for FMMD analysis, it simply means there is no fault active in the functional~group under analysis.} and S itself. -In order to consider combinations for the set S where the number of elements in each sub-set of S is $N$ or less, a concept of the `cardinality constrained powerset' +In order to consider combinations for the set S where the number of elements in each subset of S is $N$ or less, a concept of the `cardinality constrained powerset' is proposed and described in the next section. %\pagebreak[1] \subsection{Cardinality Constrained Powerset } \label{ccp} -A Cardinality Constrained powerset is one where sub-sets of a cardinality greater than a threshold +A Cardinality Constrained powerset is one where subsets of a cardinality greater than a threshold are not included. This threshold is called the cardinality constraint. To indicate this, the cardinality constraint $cc$ is subscripted to the powerset symbol thus $\mathcal{P}_{cc}$. Consider the set $S = \{a,b,c\}$. The powerset of S: -$$ \mathcal{P} S = \{ \emptyset, \{a,b,c\}, \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} $$ +$$ \mathcal{P} S = \{ \emptyset, \{a,b,c\}, \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} $$. -$\mathcal{P}_{2} S $ means all non-empty subsets of S where the cardinality of the subsets is +$\mathcal{P}_{\le 2} S $ means all non-empty subsets of S where the cardinality of the subsets is less than or equal to 2 or less. -$$ \mathcal{P}_{2} S = \{ \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} $$ +$$ \mathcal{P}_{\le 2} S = \{ \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} $$. Note that $\mathcal{P}_{1} S $ (non-empty subsets where cardinality $\leq 1$) for this example is: -$$ \mathcal{P}_{1} S = \{ \{a\},\{b\},\{c\} \} $$ +$$ \mathcal{P}_{1} S = \{ \{a\},\{b\},\{c\} \} $$. \paragraph{Calculating the number of elements in a cardinality constrained powerset} @@ -515,7 +516,7 @@ with $n$ elements (size $n$) is the binomial coefficient~\cite{probstat} shown i \begin{equation} C^n_k = {n \choose k} = \frac{n!}{k!(n-k)!} \label{bico} -\end{equation} +\end{equation} . To find the number of elements in a cardinality constrained subset S with up to $cc$ elements in each combination sub-set, @@ -531,7 +532,7 @@ from $1$ to $cc$ thus \begin{equation} |{\mathcal{P}_{cc}S}| = \sum^{cc}_{k=1} \frac{|{S}|!}{ k! ( |{S}| - k)!} \label{eqn:ccps} -\end{equation} +\end{equation} . @@ -584,14 +585,14 @@ $$ \mathcal{P}_{2}(fm(FG)) = \{ \} $$ -And % by inspection -$$ -| -\{ - \{R_o T_o\}, \{R_o T_s\}, \{R_o T_h\}, \{R_s T_o\}, \{R_s T_s\}, \{R_s T_h\}, \{R_o \}, \{R_s \}, \{T_o \}, \{T_s \}, \{T_h \} -\} -| = 11 -$$ +And whose cardinality is 11. % by inspection +%$$ +%| +%\{ +% \{R_o T_o\}, \{R_o T_s\}, \{R_o T_h\}, \{R_s T_o\}, \{R_s T_s\}, \{R_s T_h\}, \{R_o \}, \{R_s \}, \{T_o \}, \{T_s \}, \{T_h \} +%\} +%| = 11 +%$$ \pagebreak[1] @@ -600,29 +601,36 @@ cardinality calculation} The cardinality constrained powerset in equation \ref{eqn:ccps}, can be modified for % corrected for unitary state failure modes. -This is written as a general formula in equation \ref{eqn:correctedccps}. +%This is written as a general formula in equation \ref{eqn:correctedccps}. %\indent{ -To define terms : -\begin{itemize} -\item Let $C$ be a set of components (indexed by $j \in J$) +%To define terms : +%\begin{itemize} +%\item +Let $C$ be a set of components (indexed by $j \in J$) that are members of the functional group $FG$ -i.e. $ \forall j \in J | C_j \in FG $ -\item Let $|fm({C}_{j})|$ +i.e. $ \forall j \in J | C_j \in FG $. + +%\item +Let $|fm({C}_{j})|$ indicate the number of mutually exclusive fault modes of component $C_j$. -\item Let $fm(FG)$ be the collection of all failure modes +%\item + +Let $fm(FG)$ be the collection of all failure modes from all the components in the functional group. -\item Let $SU$ be the set of failure modes from the {\fg} where all $FG$ is such that +%\item + +Let $SU$ be the set of failure modes from the {\fg} where all $FG$ is such that components $C_j$ are in -`unitary state' i.e. $(SU = fm(FG)) \wedge (\forall j \in J | fm(C_j) \in \mathcal{U}) $ -\end{itemize} +`unitary state' i.e. $(SU = fm(FG)) \wedge (\forall j \in J | fm(C_j) \in \mathcal{U}) $, then +%\end{itemize} %} \begin{equation} |{\mathcal{P}_{cc}SU}| = {\sum^{cc}_{k=1} \frac{|{SU}|!}{k!(|{SU}| - k)!}} - {\sum_{j \in J} {|FM({C_{j})}| \choose 2}} \label{eqn:correctedccps} -\end{equation} +\end{equation} . Expanding the combination in equation \ref{eqn:correctedccps} @@ -631,7 +639,7 @@ Expanding the combination in equation \ref{eqn:correctedccps} |{\mathcal{P}_{cc}SU}| = {\sum^{cc}_{k=1} \frac{|{SU}|!}{k!(|{SU}| - k)!}} - {{\sum_{j \in J} \frac{|FM({C_j})|!}{2!(|FM({C_j})| - 2)!}} } \label{eqn:correctedccps2} -\end{equation} +\end{equation} . \paragraph{Use of Equation \ref{eqn:correctedccps2} } Equation \ref{eqn:correctedccps2} is useful for an automated tool that @@ -639,11 +647,12 @@ would verify that a single or double simultaneous failures model has complete fa By knowing how many test cases should be covered, and checking the cardinality associated with the test cases, complete coverage would be verified. -\paragraph{N Venn disallowed combinations} -The general case of equation \ref{eqn:correctedccps2}, involves not just dis-allowing pairs -of failure modes within components, but also ensuring that combinations across components -do not involve any pairs of failure modes within the same component. -A recursive algorithm and proof is described in appendix \ref{chap:vennccps}. +%\paragraph{Multiple simultaneous failure modes disallowed combinations} +%The general case of equation \ref{eqn:correctedccps2}, involves not just dis-allowing pairs +%of failure modes within components, but also ensuring that combinations across components +%do not involve any pairs of failure modes within the same component. +%%%%- NOT SURE ABOUT THAT !!!!! +%%%- A recursive algorithm and proof is described in appendix \ref{chap:vennccps}. %%\paragraph{Practicality} %%Functional Group may consist, typically of four or five components, which typically @@ -701,7 +710,9 @@ Thus the statistical sample space $\Omega$ for a component or derived~component $$ \Omega(C) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, \ldots ,failure\_mode_{N}\} $$ The failure mode set $F$ for a given component or derived~component $C$ is therefore -$$ F = \Omega(C) \backslash \{OK\} $$ +$ fm(C) = \Omega(C) \backslash \{OK\} $ +(or expressed as +$ \Omega(C) = fm(C) \cup \{OK\} $). The $OK$ statistical case is the largest in probability, and is therefore of interest when analysing systems from a statistical perspective.