From cbb1ce41a969dba799e49757553617ba00651a27 Mon Sep 17 00:00:00 2001 From: Robin Clark Date: Sat, 19 Nov 2011 11:56:07 +0000 Subject: [PATCH] Changed the notation FG now G, RD now CC 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. --- opamp_circuits_C_GARRETT/opamps.tex | 123 ++++++++++++++++++---------- 1 file changed, 78 insertions(+), 45 deletions(-) diff --git a/opamp_circuits_C_GARRETT/opamps.tex b/opamp_circuits_C_GARRETT/opamps.tex index e5472fe..e507c49 100644 --- a/opamp_circuits_C_GARRETT/opamps.tex +++ b/opamp_circuits_C_GARRETT/opamps.tex @@ -19,7 +19,7 @@ \newcommand{\bcs}{\em base~components} \newcommand{\irl}{in~real~life} \newcommand{\abslevel}{\ensuremath{\psi}} - +\newcommand{\FG}{\ensuremath{G}} %\usepackage{glossary} %opening @@ -880,20 +880,20 @@ of components. %We thus define $FG$ as a set of chosen components defining %a {\fg}; all functional groups 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 $FG$ -where it will return all the failure modes of the components in $FG$ +We can overload the $fm$ function for a functional group {\FG} +where it will return all the failure modes of the components in {\FG} given by $$ fm (FG) = F. $$ -Generally, where $\mathcal{FG}$ is the set of all functional groups, +Generally, where $\mathcal{{\FG}}$ is the set of all functional groups, \begin{equation} -fm : \mathcal{FG} \rightarrow \mathcal{P}\mathcal{F}. +fm : \mathcal{{\FG}} \rightarrow \mathcal{P}\mathcal{F}. \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 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 -`$FG^0$'. 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 its symptoms. @@ -930,7 +930,12 @@ The symptom abstraction process must always raise the abstraction level for the newly created {\dc}. 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.} The $\bowtie$ function processes each component in the {\fg} and @@ -978,6 +983,8 @@ Idea stage on this section \end{itemize} \clearpage +Two areas that cannot be automated. Choosing {\fgs} and the analysis/symptom collection process itself. + \section{Side Effects: A Problem for FMMD analysis} 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 @@ -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. \pagebreak[4] -\section{Defining the concept of `reasoning distance' in FMEA} +\section{Defining the concept of `comparison~complexity' in FMEA} % % 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. % We can consider the system as a large {\fg} of components. -We represent the number of components in the {\fg} by -$ | fg | .$ +We represent the number of components in the {\fg} $G$, by +$ | G | .$ 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) | .$ -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 failure mode against all the other components in the system. -We can define this as a function, $RD$, with its domain as the system -or {\fg}, $fg$, and +We can define this as a function, Comparison Complexity, $CC$, with its domain as the system +or {\fg}, $G$, and its range as the number of checks to perform to satisfy a rigorous FMEA inspection. \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} -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)|}$); -equation~\ref{eqn:rd} becomes $$ RD(fg) = fT.(|fg|-1).$$ -Equation~\ref{eqn:rd} can also be expressed as +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 $$ CC(G) = K.(|G|-1).$$ +%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} -\label{eqn:rd2} -%$$ - RD(fg) = {|fg|}.{|fm(c_n)|}.{(|fg|-1)} . -%$$ -\end{equation} +An FMMD Hierarchy will have reducing numbers of functional groups as we progress up the hierarchy. +In order to calculate its comparison~complexity we need to apply equation~\ref{eqn:CC} to +all {\fgs} on each level. + +Where $L$ represents the number of levels in the FMMD hierarchy, +$|\xi|$ 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] -\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. -$$RD(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$ +The potential divider discussed in section~\ref{potdivfmmd} has a four failure modes and two components and therefore has an $CC$ of 4. +$$CC(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$ 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 -Rigorous FMEA (RFMEA). +Ensuring all component failure modes are checked against all other components in a system +-- 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] -\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} \centering \includegraphics[width=400pt,keepaspectratio=true]{./three_tree.png} % 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} \end{figure} + + +\subsection{Comparing FMMD and RFMEA comparison complexity} + Because components have variable numbers of failure modes, - 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 +and {\fgs} have variable numbers of components it is difficult to +use the general formula for comparing the number of checks to make for RFMEA and FMMMD. 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 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 $L$ 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 @@ -1162,7 +1195,7 @@ with equation~\ref{eqn:anscen}. \begin{equation} \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} 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}). 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}, -$ RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1)$, where $|fg|$ is 27, $fm(c_n)$ is 3 -and $(|fg|-1)$ is 26. +apply RFMEA on it as a whole system, we can use equation~\ref{eqn:CC}, +$CC(G) = \sum_{n=1}^{|G|} |fm(c_n)|.(|G|-1)$, where $|G|$ is 27, $fm(c_n)$ is 3 +and $(|G|-1)$ is 26. 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 -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. % The number of components in the system, is number of components 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} @@ -1216,7 +1249,7 @@ or %(N^2 - N).f \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.