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Changed the notation FG now G, RD now CC

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and entered a general formula for finding the
Comparison Complexity of an FMMD hierarchy.

Sent it ( ref section 9.1) to Andrew to see
idf its a clear description.
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Robin Clark 2011-11-19 11:56:07 +00:00
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opamp_circuits_C_GARRETT/opamps.tex
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@ -19,7 +19,7 @@
\newcommand{\bcs}{\em base~components} \newcommand{\bcs}{\em base~components}
\newcommand{\irl}{in~real~life} \newcommand{\irl}{in~real~life}
\newcommand{\abslevel}{\ensuremath{\psi}} \newcommand{\abslevel}{\ensuremath{\psi}}
\newcommand{\FG}{\ensuremath{G}}
%\usepackage{glossary} %\usepackage{glossary}
%opening %opening
@ -880,20 +880,20 @@ of components.
%We thus define $FG$ as a set of chosen components defining %We thus define $FG$ as a set of chosen components defining
%a {\fg}; all functional groups %a {\fg}; all functional groups
We can state that We can state that
$FG$ is a member of the power set of all components, $ FG \in \mathcal{P} \mathcal{C}. $ {\FG} is a member of the power set of all components, $ \FG \in \mathcal{P} \mathcal{C}. $
We can overload the $fm$ function for a functional group $FG$ We can overload the $fm$ function for a functional group {\FG}
where it will return all the failure modes of the components in $FG$ where it will return all the failure modes of the components in {\FG}
given by given by
$$ fm (FG) = F. $$ $$ fm (FG) = F. $$
Generally, where $\mathcal{FG}$ is the set of all functional groups, Generally, where $\mathcal{{\FG}}$ is the set of all functional groups,
\begin{equation} \begin{equation}
fm : \mathcal{FG} \rightarrow \mathcal{P}\mathcal{F}. fm : \mathcal{{\FG}} \rightarrow \mathcal{P}\mathcal{F}.
\end{equation} \end{equation}
@ -914,7 +914,7 @@ We can further define the abstraction level of a {\fg}.
We can say that it is the maximum abstraction level of any of its We can say that it is the maximum abstraction level of any of its
components. Thus a functional group containing only base components components. Thus a functional group containing only base components
would have an abstraction level zero and could be represented with a superscript of zero thus would have an abstraction level zero and could be represented with a superscript of zero thus
`$FG^0$'. The functional group set may also be indexed. `${\FG}^0$'. % The functional group set may also be indexed.
We can apply symptom abstraction to a {\fg} to find We can apply symptom abstraction to a {\fg} to find
its symptoms. its symptoms.
@ -930,7 +930,12 @@ The symptom abstraction process must always raise the abstraction level
for the newly created {\dc}. for the newly created {\dc}.
Using $\abslevel$ to symbolise the fault abstraction level, we can now state: Using $\abslevel$ to symbolise the fault abstraction level, we can now state:
$$ \bowtie(FG^{\abslevel}) \rightarrow c^{{\abslevel}+N} | N \ge 1. $$ $$ \bowtie({\FG}^{\abslevel}) \rightarrow c^{{\abslevel}+N} | N \ge 1. $$
\paragraph{Functional Groups may be indexed}
We will typically have more than one {\fg} on each level of FMMD hierarchy ( expect the top level where there will only be one)
we could index the {\fgs} with a sub-script, and can then uniquely identify them using their level and their index.
For example ${\FG}^{3}_{2}$ would be the second {\fg} at the third level of abstraction in an FMMD hierarchy.
\paragraph{The symptom abstraction process in outline.} \paragraph{The symptom abstraction process in outline.}
The $\bowtie$ function processes each component in the {\fg} and The $\bowtie$ function processes each component in the {\fg} and
@ -978,6 +983,8 @@ Idea stage on this section
\end{itemize} \end{itemize}
\clearpage \clearpage
Two areas that cannot be automated. Choosing {\fgs} and the analysis/symptom collection process itself.
\section{Side Effects: A Problem for FMMD analysis} \section{Side Effects: A Problem for FMMD analysis}
A problem with modularising according to functionality is that we can have component failures that would A problem with modularising according to functionality is that we can have component failures that would
intuitively be associated with one {\fg} that may cause unintended side effects in other intuitively be associated with one {\fg} that may cause unintended side effects in other
@ -1060,7 +1067,7 @@ Our logic circuit may be able to cope with $LOW\_VOLTAGE$ and $NOISE\_LF$, but r
But in order to process these failure modes it must be at a higher stage in the FMMD hierarchy. But in order to process these failure modes it must be at a higher stage in the FMMD hierarchy.
\pagebreak[4] \pagebreak[4]
\section{Defining the concept of `reasoning distance' in FMEA} \section{Defining the concept of `comparison~complexity' in FMEA}
% %
% DOMAIN == INPUTS % DOMAIN == INPUTS
@ -1088,73 +1095,99 @@ Obviously, for a small number of components and failure modes we have a smaller
of checks to make than for a complicated larger system. of checks to make than for a complicated larger system.
% %
We can consider the system as a large {\fg} of components. We can consider the system as a large {\fg} of components.
We represent the number of components in the {\fg} by We represent the number of components in the {\fg} $G$, by
$ | fg | .$ $ | G | .$
The function $fm$ has a component as its domain and the components failure modes as its range. The function $fm$ has a component as its domain and the components failure modes as its range.
We can represent the number of failure modes in a component $c$, to be $ | fm(c) | .$ We can represent the number of failure modes in a component $c$, to be $ | fm(c) | .$
If we index all the components in the system under investigation $ c_1, c_2 \ldots c_{|fg|} $ we can express If we index all the components in the system under investigation $ c_1, c_2 \ldots c_{|G|} $ we can express
the number of checks required to rigorously examine every the number of checks required to rigorously examine every
failure mode against all the other components in the system. failure mode against all the other components in the system.
We can define this as a function, $RD$, with its domain as the system We can define this as a function, Comparison Complexity, $CC$, with its domain as the system
or {\fg}, $fg$, and or {\fg}, $G$, and
its range as the number of checks to perform to satisfy a rigorous FMEA inspection. its range as the number of checks to perform to satisfy a rigorous FMEA inspection.
\begin{equation} \begin{equation}
\label{eqn:rd} \label{eqn:CC}
%$$ %$$
RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1) %%% 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)
%$$ %$$
\end{equation} \end{equation}
This can be simplified if we can determine the total number of failure modes in the system $fT$, (i.e. $ fT = \sum_{n=1}^{|fg|} {|fm(c_n)|}$); 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:rd} becomes $$ RD(fg) = fT.(|fg|-1).$$ equation~\ref{eqn:CC} becomes $$ CC(G) = K.(|G|-1).$$
Equation~\ref{eqn:rd} can also be expressed as %Equation~\ref{eqn:rd} can also be expressed as
%
% \begin{equation}
% \label{eqn:rd2}
% %$$
% CC(G) = {|G|}.{|fm(c_n)|}.{(|fg|-1)} .
% %$$
% \end{equation}
\subsection{A general formula for counting Comparison Complexity in an FMMD hierarchy}
\begin{equation} An FMMD Hierarchy will have reducing numbers of functional groups as we progress up the hierarchy.
\label{eqn:rd2} In order to calculate its comparison~complexity we need to apply equation~\ref{eqn:CC} to
%$$ all {\fgs} on each level.
RD(fg) = {|fg|}.{|fm(c_n)|}.{(|fg|-1)} .
%$$ Where $L$ represents the number of levels in the FMMD hierarchy,
\end{equation} $|\xi|$ represents the number of functional groups on the level
and $H$ represents an FMMD hierarchy,
we overload the comparison complexity equation thus:
$$
\label{eqn:gf}
CC(H) = \sum_{\xi=0}^{L} \sum_{j=1}^{|\xi|} CC({G}_{j}^{\xi}).
$$
\pagebreak[4] \pagebreak[4]
\subsection{Reasoning Distance Examples} \subsection{Complexity Comparison Examples}
The potential divider discussed in section~\ref{potdivfmmd} has a four failure modes and two components and therefore has an $RD$ of 4. The potential divider discussed in section~\ref{potdivfmmd} has a four failure modes and two components and therefore has an $CC$ of 4.
$$RD(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$ $$CC(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$
Were we to consider a $fictitious$ system with 81 components, with these components Were we to consider a $fictitious$ system with 81 components, with these components
having 3 failure modes each, we would have an $RD$ of having 3 failure modes each, we would have an $CC$ of
$$RD(fictitious) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$ $$CC(fictitious) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$
This would be the polynomial ($O(N^2)$) result of applying FMEA rigorously (we could term this Ensuring all component failure modes are checked against all other components in a system
Rigorous FMEA (RFMEA). -- applying FMEA rigorously -- could be termed
Rigorous FMEA (RFMEA).
The computational order for RFMEA would be polynomial ($O(N^2.K)$) (where $K$ is the variable number of failure modes).
This order may be acceptable in a computational environment: However, the choosing of {\fgs} and the analysis
process are human activities. It can be seen that it is practically impossible to achieve
RFMEA for anything but trival systems. FMMD reduces the comparison complexity enough to make
rigorous checking feasible.
\pagebreak[4] \pagebreak[4]
\subsection{Using the concept of Reasoning Distance to compare RFMEA with FMMD} \subsection{Using the concept of Complexity Comparison to compare RFMEA with FMMD}
\begin{figure} \begin{figure}
\centering \centering
\includegraphics[width=400pt,keepaspectratio=true]{./three_tree.png} \includegraphics[width=400pt,keepaspectratio=true]{./three_tree.png}
% three_tree.png: 851x385 pixel, 72dpi, 30.02x13.58 cm, bb=0 0 851 385 % three_tree.png: 851x385 pixel, 72dpi, 30.02x13.58 cm, bb=0 0 851 385
\caption{FMMD Hierarchy with $(|fg| = 3)$ } % \wedge (|fm(c)| = 3)$} \caption{FMMD Hierarchy with number of components in {\fg} fixed to 3 $(|G| = 3)$ } % \wedge (|fm(c)| = 3)$}
\label{fig:three_tree} \label{fig:three_tree}
\end{figure} \end{figure}
\subsection{Comparing FMMD and RFMEA comparison complexity}
Because components have variable numbers of failure modes, Because components have variable numbers of failure modes,
and {\fgs} have variable numbers of components it is difficult to and {\fgs} have variable numbers of components it is difficult to
come up with a general formula for comparing the number of checks to make for use the general formula for comparing the number of checks to make for
RFMEA and FMMMD. RFMEA and FMMMD.
If we were to create an example by fixing the number of components in a {\fg} If we were to create an example by fixing the number of components in a {\fg}
and the number of failure modes per component, we can derive formulae and the number of failure modes per component, we can derive formulae
to represent the number of checks to make. to represent the number of checks to make.
Consider $k$ to be the number of components in a {\fg} (i.e. $k=|fg|$), Consider $k$ to be the number of components in a {\fg} (i.e. $k=|{\FG}|$),
$f$ is the number of failure modes per component (i.e. $f=|fm(c)|$), and $f$ is the number of failure modes per component (i.e. $f=|fm(c)|$), and
$L$ to be the number of levels in the hierarchy of an FMMD analysis. $L$ to be the number of levels in the hierarchy of an FMMD analysis.
We can represent the number of failure scenarios to check in an FMMD We can represent the number of failure scenarios to check in an FMMD
@ -1162,7 +1195,7 @@ with equation~\ref{eqn:anscen}.
\begin{equation} \begin{equation}
\label{eqn:anscen} \label{eqn:anscen}
\sum_{n=0}^{L} {k}^{n}.k.f.(k-1) \sum_{n=0}^{L} {k}^{n}.k.f.(k-1)
\end{equation} \end{equation}
The thinking behind equation~\ref{eqn:anscen}, is that for each level of analysis -- counting down from the top -- The thinking behind equation~\ref{eqn:anscen}, is that for each level of analysis -- counting down from the top --
@ -1187,19 +1220,19 @@ Adding these together gives $242$ checks to make to perform FMMD (i.e. RFMEA \te
{\fgs}). {\fgs}).
If we were to take the system represented in figure~\ref{fig:three_tree}, and If we were to take the system represented in figure~\ref{fig:three_tree}, and
apply RFMEA on it as a whole system, we can use equation~\ref{eqn:rd}, apply RFMEA on it as a whole system, we can use equation~\ref{eqn:CC},
$ RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1)$, where $|fg|$ is 27, $fm(c_n)$ is 3 $CC(G) = \sum_{n=1}^{|G|} |fm(c_n)|.(|G|-1)$, where $|G|$ is 27, $fm(c_n)$ is 3
and $(|fg|-1)$ is 26. and $(|G|-1)$ is 26.
This gives: This gives:
$RD(fg) = \sum_{n=1}^{27} |3|.(|27|-1) = 2106$. $CC(G) = \sum_{n=1}^{27} |3|.(|27|-1) = 2106$.
In order to get general equations with which to compare RFMEA with FMMD In order to get general equations with which to compare RFMEA with FMMD
we can re-write equation~\ref{eqn:rd} in terms of the number of levels we can re-write equation~\ref{eqn:CC} in terms of the number of levels
in an FMMD hierarchy. in an FMMD hierarchy.
% %
The number of components in the system, is number of components The number of components in the system, is number of components
in a {\fg} raised to the power of the level plus one. in a {\fg} raised to the power of the level plus one.
Thus we re-write equation~\ref{eqn:rd} as: Thus we re-write equation~\ref{eqn:CC} as:
\begin{equation} \begin{equation}
@ -1216,7 +1249,7 @@ or
%(N^2 - N).f %(N^2 - N).f
\end{equation} \end{equation}
We can now use equation~\ref{eqn:anscen} and \ref{eqn:fmea_state_exp22} to compare (for fixed sizes of $|fg|$ and $|fm(c)|$) We can now use equation~\ref{eqn:anscen} and \ref{eqn:fmea_state_exp22} to compare (for fixed sizes of $|G|$ and $|fm(c)|$)
the two approaches, for the work required to perform rigorous checking. the two approaches, for the work required to perform rigorous checking.