diff --git a/submission_thesis/CH3_FMEA_criticism/copy.tex b/submission_thesis/CH3_FMEA_criticism/copy.tex index bd748f5..0ff5553 100644 --- a/submission_thesis/CH3_FMEA_criticism/copy.tex +++ b/submission_thesis/CH3_FMEA_criticism/copy.tex @@ -31,7 +31,7 @@ This problem is compounded by the fact that traditional FMEA cannot integrate so \section{Reasoning Distance used to measure Comparison Complexity} - +\label{sec:reasoningdistance} Traditional FMEA cannot ensure that each failure mode of all its components are checked against any other components in the system which it may affect, due to state explosion. diff --git a/submission_thesis/CH5_Examples/software.tex b/submission_thesis/CH5_Examples/software.tex index 1fc2164..273b871 100644 --- a/submission_thesis/CH5_Examples/software.tex +++ b/submission_thesis/CH5_Examples/software.tex @@ -225,9 +225,11 @@ Let us define any value outside the 4mA to 20mA range as an error condition. As we read a voltage, we use Ohms law~\cite{aoe} to determine the mA current detected: $V=IR$, $0.004A * \ohms{220} = 0.88V$ and $0.020A * \ohms{220} = 4.4V$. % -Our acceptable voltage range\footnote{For the purpose of clarity we are ignoring resistor tolerance +Our acceptable voltage +range\footnote{For the purpose of clarity we are ignoring resistor tolerance for this example. In a practical {\ft} reader we would factor in resistor tolerance to the limits, or -allow `deadbands' of $\approx \half mA$ at either end of the range.} is therefore +allow `deadbands' of $\approx \half mA$ at either end of the range.} +is therefore $$(V \ge 0.88) \wedge (V \le 4.4) \; .$$ diff --git a/submission_thesis/CH6_Evaluation/copy.tex b/submission_thesis/CH6_Evaluation/copy.tex index 2f3e283..f0bde23 100644 --- a/submission_thesis/CH6_Evaluation/copy.tex +++ b/submission_thesis/CH6_Evaluation/copy.tex @@ -130,24 +130,25 @@ We can represent this set of components as $G$, and the number of components in $ | G | $. %, %(an indexing and sub-scripting notation to identify particular {\fgs} %within an FMMD hierarchy is given in section~\ref{sec:indexsub}). - +% %\paragraph{Defining Components} $G$ is simply a sub-set of all possible components. -We define the set of all components as $\mathcal{C}$ and can state $G \subset \mathcal{C}$.. Individual components are denoted as $c$ +We define the set of all components as $\mathcal{C}$ and can state $G \subset \mathcal{C}$. Individual components are denoted as $c$ with additional indexing where appropriate. -\paragraph{Defining a function to return the failure modes of a component.} -The function $fm$ has a component as its domain and the components failure modes % , $fms$, -as its range. % (see equation~\ref{eqn:fm}). -Where $\mathcal{F}$ is the set of all failures, -$$ fm: \mathcal{C} \rightarrow \mathcal{F}.$$ -we can represent the number of potential failure modes of a component $c$, to be $ | fm(c) | .$ +%\paragraph{Defining a function to return the failure modes of a component.} +The function $fm$ returns the failure modes of a component, +its signature is %has a component as its domain and the components failure modes % , $fms$, +%as its range. % (see equation~\ref{eqn:fm}). +$ fm: \mathcal{C} \rightarrow \mathcal{F},$ where $\mathcal{F}$ is the set of all failures. +We can represent the number of potential failure modes of a component $c$, to be $ | fm(c) | .$ -\paragraph{Indexing components with the group $G$.} -If we index all the components in the system under investigation $ c_1, c_2 \ldots c_{|G|} $ we can express +%\paragraph{Indexing components with the group $G$.} +%If we index all +Indexing the components in the system under investigation $ c_1, c_2 \ldots c_{|G|} $ allows us to express the number of checks required to exhaustively % rigorously examine every -failure mode against all the other components in a system. +failure mode against all the other components in a system in equation~\ref{eqn:CC}. % Comparison Complexity can be represented by a function $CC$, with its domain as $G$, and its range as the number of checks---or reasoning stages---to perform to satisfy an XFMEA inspection. @@ -156,24 +157,33 @@ Where $\mathcal{G}$ represents the set of all {\fgs} %, and $ \mathbb{Z}^{+} $, $CC$ is defined by, \begin{equation} %$$ - CC:\mathcal{G} \rightarrow \mathbb{Z}^{+}, + CC:\mathcal{G} \rightarrow \mathbb{Z}^{ }. % could be zero, one component like an op-amp used as a NIBUFF %$$ \end{equation} % %and, where n is the number of components in the system/{\fg}, -and $|fm(c_i)|$ is the number of failure modes -in component ${c_i}$. Comparison complexity, $CC$ for a group of components $G$, is given by +%and $|fm(c_i)|$ is the number of failure modes +%in component ${c_i}$. +Comparison complexity, $CC$ for a group of components $G$, is given by \begin{equation} \label{eqn:CC} %$$ %%% when it was called reasoning distance -- 19NOV2011 -- RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1) - CC(G) = (n-1) \sum_{1 \le i \le n} fm(c_i). + CC(G) = (n-1) \sum_{1 \le i \le n} |fm(c_i)|. %$$ \end{equation} - -This can be simplified if we can determine the total number of failure modes in the system $K$, (i.e. $ K = \sum_{n=1}^{|G|} {|fm(c_n)|}$); -equation~\ref{eqn:CC} becomes +% +% J Howse requires justification for the CC equation above 10MAR2013. +% +Equation~\ref{eqn:CC} says that for every failure mode in the group $G$, we must check it against all other +components in the group (except its-self). This gives us a count of the number of reasoning paths to perform {\XFMEA}. +These reasoning distance concepts are discussed in section~\ref{sec:reasoningdistance}. % from CH3 +% +Equation~\ref{eqn:CC} can be simplified if we can determine the total number of +failure modes in the system $K$, (i.e. $ K = \sum_{n=1}^{|G|} {|fm(c_n)|}$); +%equation~\ref{eqn:CC} +the equation becomes %$$ \begin{equation} \label{eqn:rd2} @@ -182,17 +192,26 @@ equation~\ref{eqn:CC} becomes An FMMD hierarchy consists of many {\fgs} which are subsets of $G$. -We define the set of all {\fgs} as $\mathcal{FG}$. -Using $FG$ to represent individual {\fgs} we %can therefore -state $$ \forall FG \in \mathcal{FG} | FG \subset \mathcal{G} .$$ - -FMMD analysis creates a hierarchy $\hh$ of {\fgs} where $\hh \subset \mathcal{FG}$. +%We define the set of all {\fgs} as $\mathcal{FG}$. +%Using $FG$ to represent individual {\fgs} +%i.e. FG \subset G. +%we %can therefore +%state +%$$ \forall FG \in \mathcal{FG} | FG \subset \mathcal{G} .$$ +% +FMMD analysis creates a hierarchy $\hh$ of {\fgs}. % where $\hh \subset \mathcal{FG}$. +% +We can define individual {\fgs} using $FG^{\alpha}_{i}$ with an index, +$i$ for identification and a superscript for the $\alpha$~level (see section~\ref{sec:alpha}). % -We can define individual {\fgs} using $FG^{\alpha}_{i}$ with an index, $i$ for identification and a superscript for the $\alpha$~level (see section~\ref{sec:alpha}). %--- %o identify the hierarchy. -For instance the first {\fg} in a hierarchy, containing base components only +For example the first {\fg} in a hierarchy, containing base components only i.e. at the zeroth level of an FMMD hierarchy where $\alpha=0$, would have the superscript 0 and a subscript of 1: $FG^{0}_{1}$. +% +The {\fg} representing the potential divider in section~\ref{sec:pd} +has an $\alpha$ level of 0 (as it contains base components). The {\fg} +with the potential divider and the operational amplifier has an $\alpha$ level of 1. %$$ %Equation~\ref{eqn:rd} can also be expressed as % @@ -213,7 +232,8 @@ We can define an FMMD hierarchy as a set of {\fgs}, $\hh$. % that returns % the sum of all complexity comparison % applied to {\fgs} at a particular hierarchy level in \hh, -We define a helper function, g, that applies $CC$ to all {\fgs} at a particular level, $\xi$ in an FMMD hierarchy {\hh} +We define a helper function, %g, +that applies $CC$ to all {\fgs} at a particular level, $\xi$ in an FMMD hierarchy {\hh} and returns the sum of the comparison complexities, \begin{equation} g: \hh \times \mathbb{N} \rightarrow \mathbb{N} . @@ -239,18 +259,33 @@ we overload the comparison complexity function $CC$, to obtain the comparison co \subsection{Complexity Comparison Examples} %\pagebreak[4] -The potential divider discussed in section~\ref{subsec:potdiv} has four failure modes and two components and therefore has $CC$ of 4. -$$CC(potdiv) = \sum_{n=1}^{2} |2| \times (|1|) = 4 $$ -We combine the potential divider with an op-amp which has four failure modes +We initially work though the chapter ~\ref{sec:chap4} amplifier example, which has two +stages, the potential divider and then the amplifier. We add the complexities from +both these stages to determine how many reasoning paths there were to perform FMMD analysis on the +non-inverting amplifier. + +The potential divider discussed in section~\ref{subsec:potdiv} has +four failure modes and two components and therefore has $CC$ of 4. +We calculate this using equation~\ref{eqn:CC} thus, +$$CC(potdiv) = \sum_{n=1}^{2} \big( |2| \times (|1|) \big) = 4. $$ +% +We next combine the potential divider with an op-amp which has four failure modes to form a {\fg} with two components, one with four failure modes and the other (the potential divider) with two. $$CC(invamp) = 2 \times 1 + 4 \times 1 = 6 $$ + +We now add the two calculated complexities to determine the +amount of reasoning paths to analyse the amplifier using FMMD. +% +The potential divider has a $CC$ of four and the amplifier section a $CC$ of six. To analyse the inverting amplifier with FMMD we required 10 reasoning stages. -Using {\XFMEA} we obtain $ 2 \times (3-1) + 2 \times (3-1) + 4 \times (3-1)$ = 16. +Using traditional FMEA employing exhaustive checking ({\XFMEA}) +we obtain $ 2 \times (3-1) + 2 \times (3-1) + 4 \times (3-1)$ = 16. +Even with this very trivial example, we begin to see benefits of taking a modular approach to FMEA. \paragraph{Complexity Comparison for an hypothetical 81 component system.} %Even considering a $example$ A system, $example$, with just 81 components (with these components -having 3 failure modes each) we would have an $CC$ of +having 3 failure modes each) would, using equation~\ref{eqn:rd2} have an $CC$ of $$CC(example) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$ @@ -259,7 +294,8 @@ Ensuring all component failure modes are checked against all other components in %rigorously -- could be termed exhaustive FMEA ({\XFMEA}). -The computational order for {\XFMEA} would be polynomial ($O(N^2.K)$) (where $K$ is the variable number of failure modes). +The computational order for {\XFMEA} would be polynomial ($O((N)(N-1)K) \approx O(N^2.K)$) (where $K$ is the variable number of failure modes) +as discussed in section~\ref{eqn:fmea_single}. % This order may be acceptable in a computational environment. However, the choosing of {\fgs} and the analysis process are by-hand/human activities. It can be seen that it is practically impossible to achieve @@ -271,7 +307,8 @@ It is the author's belief that FMMD reduces the comparison complexity enough to exhaustive checking (within {\fgs}) entirely feasible. -\pagebreak[4] +%\pagebreak[4] +\clearpage %\subsection{Using the concept of Complexity Comparison to compare {\XFMEA} with FMMD} % \begin{figure} @@ -358,7 +395,7 @@ Thus we re-write equation~\ref{eqn:CC} as: \begin{equation} \label{eqn:fmea_state_exp21} - \sum_{n=1}^{k^{L+1}}.(k^{L+1}-1).f \; , % \\ + \sum_{n=1}^{k^{L+1}} (k^{L+1}-1).f \; , % \\ %(N^2 - N).f \end{equation} @@ -379,13 +416,16 @@ of FMMD analysis, with these fixed numbers, %(in addition to the top zeroth level) will require 81 base level components. -$$ -%\begin{equation} +%$$ +\begin{equation} \label{eqn:fmea_state_exp22} 3^4.(3^4-1).3 = 81.(81-1).3 = 19440 % \\ %(N^2 - N).f -%\end{equation} -$$ +\end{equation} +%$$ + +Equation \ref{eqn:fmea_state_exp22} shows that applying XFMEA where components all have three failure modes +and there are 81 components, would involve 19,440 reasoning paths. $$ %\begin{equation} @@ -394,6 +434,9 @@ $$ %\end{equation} $$ +For FMMD (where within {\fgs} the analysis \textbf{is exhaustive}) we only require +720 reasoning paths. + %\clearpage \subsection{Complexity Comparison applied to FMMD electronic circuits analysed in chapter~\ref{sec:chap5}.}