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Mainly J Howse comments on CH7.

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submission_thesis/CH3_FMEA_criticism/copy.tex
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@ -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.

6
submission_thesis/CH5_Examples/software.tex
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@ -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
$$(V \ge 0.88) \wedge (V \le 4.4) \; .$$

119
submission_thesis/CH6_Evaluation/copy.tex
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@ -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}.}