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Graphical symptom abstraction first draft after split

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Robin Clark 2010-08-14 08:12:34 +01:00
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*.bbl
*.blg
*paper.tex
*_paper.tex
*.txt

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#
# Make the propositional logic diagram a paper
#
SOURCE = symptom_abstraction.tex
all:
pdflatex symptom_abstraction.tex
okular symptom_abstraction.pdf
# abstract.tex algorithm.tex fmmd.tex introduction.tex paper.tex process.tex sys_abs.tex topbot.tex
#paper: paper.tex logic_diagram_paper.tex
# latex paper.tex
# dvipdf paper
# okular paper.pdf
paper: paper.tex symptom_abstraction_paper.tex $(SOURCE)
#cat introduction.tex | sed s/chapter/paper/ > introduction.tex
#latex paper.tex
#dvipdf paper pdflatex cannot use eps ffs
pdflatex paper.tex
mv paper.pdf symptom_ex_process_paper.pdf
okular symptom_ex_process_paper.pdf
# Remove the need for referncing graphics in subdirectories
#
#logic_diagram_paper.tex: logic_diagram.tex
# cat logic_diagram.tex | sed 's/logic_diagram\///' > logic_diagram_paper.tex
symptom_abstraction_paper.tex: symptom_abstraction_paper.tex paper.tex $(SOURCE)
cat symptom_abstraction.tex | sed 's/symptom_abstraction\///' > symptom_abstraction_paper.tex
bib: $(SOURCE)
bibtex paper

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\documentclass[a4paper,10pt]{article}
\usepackage{graphicx}
\usepackage{fancyhdr}
\usepackage{tikz}
\usepackage{amsfonts,amsmath,amsthm}
\usepackage{algorithm}
\usepackage{algorithmic}
\usepackage{ifthen}
\newboolean{paper}
\setboolean{paper}{true} % boolvar=true or false
%\input{../style}
%\newtheorem{definition}{Definition:}
\begin{document}
\pagestyle{fancy}
%\outerhead{{\small\bf Symptom Extraction Process}}
%\innerfoot{{\small\bf R.P. Clark } }
% numbers at outer edges
\pagenumbering{arabic} % Arabic page numbers hereafter
\author{R.P.Clark}
\title{Symptom Extraction Process}
\maketitle
\input{symptom_abstraction_paper}
\bibliographystyle{plain}
\bibliography{../vmgbibliography,../mybib}
\today
\end{document}

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symptom_abstraction/symptom_abstraction.tex
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\ifthenelse {\boolean{paper}}
{
\begin{abstract}
In modular systems design, it is often very useful to
know the failure modes of the sub-systems used.
This paper outlines a technique for determining the failure modes of a sub-system given
its component parts.
%, and the failure modes of those parts.
The technique uses a graphical notation, based on Euler\cite{eulerviz} and Constraint
diagrams\cite{constraint} to model failure modes and failure mode common symptom collection.
The technique is designed for making building blocks for a hierarchical fault model.
Once the failure modes have been determined for a sub-system,
that sub-system may be treated as a `component' or `black box' and used
in conjunction with other such analysed sub-systems, to model
higher level sub-systems. In this way a hierarchy to represent the fault behaviour
of a system can be built.
%FMMD hierarchy
The hierarchy is built from the bottom up.
Starting with component failure modes at the bottom.
Because the process is bottom-up, syntax checking and tracking can ensure that
no component failure mode can be overlooked.
Once a hierarchy is in place it can be converted into a fault data model.
From the fault data model, automatic generation
of FTA\cite{nasafta} (Fault Tree Analysis) and mimimal cuts sets\cite{nucfta} are possible.
Also statistical reliability\cite{en61508} and MTTF (Mean Time to Failure) calculations can be produced
automatically, where component failure mode statistics are available\cite{mil1991}.
This paper focuses on the process of building the blocks that are used in the hierarchy.
This paper applies the symptom extraction technique
using Propositional Logic diagrams, to identify
symptom groups in functional~groups under
analysis.
\end{abstract}
}
{}
{
\section{introduction}
This chapter applies the symptom extraction technique
using Propositional Logic diagrams, to identify
symptom groups in functional~groups under
analysis.
}
%\clearpage
\section{Introduction}
Fault finding is intinctively performed from the top-down.
A faulty piece of equipment is examined and will have a
symptom or specific fault. The area or sub-system within the
equipemnt will next be looked into. Secific measurements
and checks will be made, and finally a component or a low level sub-system
will be found to be faulty.
The technique here works the other way. It works from the bottom up.
Starting with a collection of compoents that form
a simple functional group, the effect of all component error modes are
examined, as to their effect on the functional group.
The effects on the functional group can then be collected as common symptoms,
and now we may treat the functional group as a component. It has a known set of failure modes.
By working from the bottom up, we can trace all possible sources
that could cause a particular mode of equipment failure.
This means that we can obtainm statistical estimates based on the known reliabilities
of the components.
It also means that every component failure mode must at the very least be considered.
\subsection{Static Analysis}
In the field of safety critical engineering; to comply with
European Law a product must be certified under the approriate `EN' standard.
Typically environmental stress, EMC, electrical stressing, endurance tests,
software~inspections and project~management quality reviews are applied\cite{sccs}.
Static testing is also applied. This is theoretical analysis of the design of the product from the safety
perspective.
Three main techniques are currenly used,
Statistical failure models, FMEA (Failure mode Effects Analysis) and FTA (Fault Tree Analysis).
The technique outlined here aims to provide a mathematical frame work
to assist in the production of these three results of static analysis.
{
The aims are
\begin{itemize}
\item To automate the process where possible
\item To apply a documented trail for each analysis phase (determination of functional groups, and analysis of component failure modes on those groups)
\item To use a modular approach so that analysed sub-systems can be re-used
\item Automatically ensure no failure mode is unhandled
\item To produce a data model from which FTA, FMEA and statistical failure models may be obtained automatically
\end{itemize}
}
\subsection{Systems, functional groups, sub-systems and failure modes}
It is helpful here to define some terms, `system', `functional~group', `component', `base~component' and `sub-system'.
A System, is really any coherent entity that would be sold as a safety critical product.
A sub-system is a system that is part of some larger system.
For instance a stereo amplifier separate is a sub-system. The
whole Sound System, consists perhaps of the following `sub-systems':
CD-player, tuner, amplifier~separate, loudspeakers and ipod~interface.
%Thinking like this is a top~down analysis approach
%and is the way in which FTA\cite{nucfta} analyses a System
%and breaks it down.
A sub-system will be composed of component parts, which
may themselves be sub-systems. However each `component part'
will have a fault/failure behaviour and it should
always be possible to obtain a set of failure modes
for each `component'.
If we look at the sound system again as an
example; the CD~player could fail in serveral distinct ways, no matter
what has happened to it or has gone wrong inside it.
Using the reasoning that working from the bottom up forces the consideration of all possible
component failures (which can be missed in a top~down approach)
we are presented with a problem. Which initial collections of base components should we choose ?
For instance in the CD~player example; to start at the bottom; we are presented with
a massive list of base~components, resistors, motors, user~switches, laser~diodes all sorts !
Clearly, working from the bottom~up we need to pick small
collections of components that work together in some way.
These are termed `functional~groups'. For instance the circuitry that powers the laser diode
to illuminate the CD might contain a handful of components, and as such would make a good candidate
to be one of the base level functional~groups.
In choosing the lowest level (base component) sub-systems we would look
for the smallest `functional~groups' of components within a system. A functional~group is a set of components that interact
to perform a specific function.
When we have analysed the fault behaviour of a functional group, we can treat it as a `black box'.
We can now call our functional~group a sub-system. The goal here is to know how will behave under fault conditions !
%Imagine buying one such `sub~system' from a very honest vendor.
%One of those sir, yes but be warned it may fail in these distinct ways, here
%in the honest data sheet the set of failure modes is listed!
This type of thinking is starting to become more commonplace in product literature, with the emergence
of reliability safety standards such as IOC1508\cite{sccs},EN61508\cite{en61508}.
FIT (Failure in Time - expected number of failures per billion hours of operation) values
are published for some micro-controllers. A micro~controller
is a complex sub-system in its self and could be considered a `black~box' with a given reliability.
\footnote{Microchip sources give an FIT of 4 for their PIC18 series micro~controllers\cite{microchip}, The DOD
1991 reliability manual\cite{mil1991} applies a FIT of 100 for this generic type of component}
As electrical components have detailed datasheets a useful extension of this would
be failure modes of the component, with environmental factors and MTTF statistics.
Currently this sort of information is generally only available for generic component types\cite{mil1991}.
%At higher levels of analysis, functional~groups are pre-analysed sub-systems that interact to
%erform a given function.
\vspace{0.3cm}
%\begin{table}[p]
@ -209,6 +82,8 @@ in a region enclosed by the contours representing the failure modes it investiga
The effect on the sub-system of each test case is analysed.
%It is then represented on the diagram by an asterisk on the contour representing the failure mode.
The `test~case~results' are archived.
That is to say the results from each test case are now associated with the
asterisk point.
When all test cases have been analysed, we switch our attention to a higher abstraction level.
% We treat the sub-system as a black box, or as a component part itsself.
% We can now look at the test case results from the perspective of a `user'
@ -268,7 +143,7 @@ is represented as a contour. These contours represent the failure modes of the s
\section{A general Sub-System example}
Consider a functional group $FG$ with component parts $A$,$B$ and $C$.
Consider a functional group $FG$ with component parts $C_1$,$C_2$ and $C_3$.
Each part has a set of related fault modes (i.e. ways in which it can fail to operate correctly).
Let us define the following failure modes for each component part, defining a function $FM()$ where $K$
is a component part and $F$ is its set of failure modes\footnote{Base component failure modes are defined, often with
@ -281,9 +156,9 @@ $$
\\
For our example above
\\
$$ FM(A) = \{ a_1, a_2, a_3 \} $$
$$ FM(B) = \{ b_1, b_2 \} $$
$$ FM(C) = \{ c_1, c_2 \} $$
$$ FM(C_1) = \{ a_1, a_2, a_3 \} $$
$$ FM(C_2) = \{ b_1, b_2 \} $$
$$ FM(C_3) = \{ c_1, c_2 \} $$
We can now represent the sub-system as a set of component faulure modes $FG_{cfm}$,
thus
@ -371,7 +246,7 @@ This is performed on the diagram by linking the test cases with lines to form `s
For the sake of example let us consider the fault symptoms $SP1 = \{fs_2, fs_4, fs_5\}$ to be an identical
failure mode at the {\em sub-system} level. These can then be joined to form a spider. Likewise
let $SP2 = \{fs_1, fs_3, fs_7\}$ be an identical failure mode at the {\em sub-system} level.
Let $\{fs_6\}$ be a distinct failure mode at {\em sub-system} level.
Let $SP3 = \{fs_6\}$ be a distinct failure mode at {\em sub-system} level.
The diagram can now be drawn as in figure \ref{fig:gensubsys3}.
@ -399,7 +274,7 @@ The result of this will be, a set of failure modes for the sub-system, as thoug
or a {\em component} to be used in higher level designs.
We have now in $SP1$, $SP2$ and $fs_6$ the three ways in which this sub-system can fail.
We have now in $SP1$, $SP2$ and $SP_3$ the three ways in which this sub-system can fail.
In other words we have derived failure modes for this sub-system.
@ -447,7 +322,7 @@ In other words we have derived failure modes for this sub-system.
This sub-system may now therfore, be represented as three separate failure modes.
We may now treat this sub-system as we would a component with a known set of failure modes.
The failure modes of the Sub-system $SS$ are now the set $SS_{fm} = \{ SP1, Sp2, fs_6 \}$.
The failure modes of the Sub-system $SS$ are now the set $SS_{fm} = \{ SP1, SP2, SP_3 \}$.
Defining the function `$\bowtie$' to represent the {\em symptom abstraction} process, we may now
write
@ -491,222 +366,7 @@ with known failure modes.
\clearpage
\section{A Formal Algorithmic Description of `Symptom Abstraction'}
The algorithm for {\em symptom abstraction} is described in
this section
%describes the symptom abstraction process
using set theory.
The {\em symptom abstraction process} (given the symbol `$\bowtie$') takes a functional group $FG$
and converts it to a sub-system $SS$.
The sub-system $SS$ is a collection
of failure~modes of the sub-system.
The sub-system $SS$ may now be treated
as a component with a known set of failure modes.
Thus $SS$ can be used as a system building block at a higher
level of fault abstraction.
The algorithm has been broken down into five stages, each following on from the other.
\begin{algorithm}[h+]
~\label{alg:sympabs1}
\caption{Determine failure modes: $FG \mapsto FG_{cfm}$} \label{alg:sympabs11}
\begin{algorithmic}[1]
%\REQUIRE Obtain a list of components for the System $S$ under investigation.
%ENSURE Decomposition of $S$ into atomic components where each component $c$ has a know set of $fm$ failure modes.
%\STATE Determine functional groups $fg_n \subset S$ of components, where n is an index number and the number of functional groups found.
\STATE { Let $FG$ be a set of components } \COMMENT{ The functional group should be chosen to be minimally sized collections of components that perform a specific function}
\STATE { Let $c$ represent a component}
\STATE { Let $CFM$ represent a set of failure modes }
\STATE { $FM(c) \mapsto CFM $} \COMMENT {Let the function $FM$ take a component and return a set of all its failure modes}
%\ENSURE { $ \forall c | c \in FG \wedge FM(c) \neq \emptyset $}
%\ENSURE { $ c | c \in FG \wedge FM(c) \neq \emptyset $}
\ENSURE{ Each component $c \in FG $ has a known set of failure modes i.e. $FM(c) \neq \emptyset$ }
%\REQUIRE{ Ensure that all components belong to at least one functional group $\bigcup_{i=1...n} fg_i = S $ }
%symptom_abstraction
% okular
\STATE {let $FG_{cfm}$ be a set of failure modes}
\STATE {Collect all failure modes from the components into the set $FM_{cfm}$}
%\FORALL { $c \in FG $ }
%\STATE { $ FM(c) \in FG_{cfm} $ } \COMMENT {Collect all failure modes from the components into the set $FM_{cfm}$}
%\ENDFOR
%\hline
Algorthim \ref{alg:sympabs11} has taken a functional group $FG$ and returned a set of failure~modes $FG_{cfm}$.
The next task is to formulate `test cases'. These are the collections of failure~modes that will be used
in the analysis stages.
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[h+]
~\label{alg:sympabs2}
\caption{Determine Test Cases: $FM_{cfm} \mapsto TC $} \label{alg:sympabs22}
\begin{algorithmic}[1]
\REQUIRE {Determine the test cases to be applied}
\STATE { All test cases are now chosen by the investigating engineer(s). Typically all single
component failures are investigated
with some specially selected combination faults}
\STATE { Let $TC$ be a set of test cases }
\STATE { Let $tc_j$ be set of component failure modes where $j$ is an index of $J$}
\COMMENT { Each set $tc_j$ is a `test case' }
\STATE { $ \forall j \in J | tc_j \in TC $ }
%\STATE { $ \bigcup_{j=1...N} tc_j = \bigcup TC $ }
%\COMMENT { All $tc_j$ test cases sets belong to $TC$ }
%\REQUIRE { $ TC \subset \bigcup (FM_{cfm}) $ }
%\COMMENT { $TC$ is the set of all test_cases
% Let TC be a subset of the powerset of the failure modes $ FG_{cfm} $,
%i.e. only failure modes present in $ FG_{cfm} $ are present in sets belonging to $ TC $}
\COMMENT { Ensure the test cases are complete and unique }
\FORALL { $tc_j \in TC$ }
%\ENSURE {$ tc_j \in \bigcap FG_{cfm} $}
\ENSURE {$ tc_j \in \mathcal{P} FG_{cfm} $}
\COMMENT { require that the test case is a member of the powerset of $FM_{cfm}$ }
\ENSURE { $ \forall \; j2 \; \in J ( \forall \; j1 \; \in J | tc_{j1} \neq tc_{j2} \; \wedge \; j1 \neq j2 ) $}
\COMMENT { Test cases must be unique }
\ENDFOR
\STATE { let $f$ represet a component failure mode }
\REQUIRE { That all failure modes are represented in at least one test case }
\ENSURE { $ \forall f | (f \in FM_{cfm}) \wedge (f \in \bigcup TC) $ }
\COMMENT { This corresponds to checking that at least each failure mode is considered at least once in the analysis; some european standards
imply checking all double fault combinations\cite{en298} }
%\hline
Algorithm \ref{alg:sympabs22} has taken the set of failure modes $FM_{cfm}$ and returned a set of test cases $TC$.
The next stages is to analyse the effect of each test case on the functional group.
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[h+]
~\label{alg:sympabs3}
\caption{Analyse Test Cases: $ TC \mapsto R $} \label{alg:sympabs33}
\begin{algorithmic}[1]
\STATE { let r be a `test case result'}
\STATE { Let the function $Analyse : tc \mapsto r $ } \COMMENT { This analysis is a human activity, examining the failure~modes in the test case and determining how the functional~group will fail under those conditions}
\STATE { $ R $ is a set of test case results $r_j \in R$ where the index $j$ corresponds to $tc_j \in TC$}
\FORALL { $tc_j \in TC$ }
\STATE { $ rc_j = Analyse(tc_j) $} \COMMENT {this is Fault Mode Effects Analysis (FMEA) applied in the context of the functional group}
\STATE { $ rc_j \in R $ }
\ENDFOR
%\hline
Algorithm \ref{alg:sympabs33} has built the set $R$, the sub-system/functional group results for each test case.
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[h+]
~\label{alg:sympabs4}
\caption{Find Common Symptoms: $ R \mapsto SP $} \label{alg:sympabs44}
\begin{algorithmic}[1]
%\REQUIRE {All failure modes for the components in $fm_i = FM(fg_i)$}
\STATE {Let $sp_l$ be a set of `test cases results' where $l$ is an index set $L$}
\STATE {Let $SP$ be a set whose members are sets $sp_l$}
\COMMENT{ $SP$ is the set of `fault symptoms' for the sub-system}
%
%\COMMENT{This corresponds to a fault symptom of the functional group $FG$}
%\COMMENT{where double failure modes are required the cardinality constrained powerset of two must be applied to each failure mode}
\FORALL { $ r_j \in R$ }
\STATE { $sp_l \in \mathcal{P} R \wedge sp_l \in SP$ }
\STATE { $sp_l \in \bigcap R \wedge sp_l \in SP$ } \COMMENT{ Collect common symptoms.
Analyse the sub-system's fault behaviour under the failure modes in $tc_j$ and determine the symptoms $sp_l$ that it
causes in the functional group $FG$}
%\ENSURE { $ \forall l2 \in L ( \forall l1 \in L | \exists a \in sp_{l1} \neq \exists b \in sp_{l2} \wedge l1 \neq l2 ) $}
\ENSURE {$ \forall a \in sp_l | \forall sp_i \in \bigcap_{i=1..L} SP ( sp_i = sp_l \implies a \in sp_i)$}
\COMMENT { Ensure that the elements in each $sp_l$ are not present in any other $sp_l$ set }
\ENDFOR
\STATE { The Set $SP$ can now be considered to be the set of fault modes for the sub-system that $FG$ represents}
%\hline
Algorithm \ref{alg:sympabs44} raises the failure~mode abstraction level.
The failures have now been considered not from the component level, but from the sub-system or
functional group level.
We now have a set $SP$ of the symptoms of failure.
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[h+]
~\label{alg:sympabs5}
\caption{Treat the symptoms as failure modes of the Sub-System: $ SP \mapsto SS $} \label{alg:sympabs55}
\begin{algorithmic}[1]
\STATE { Let $SS$ be a set of faliure modes with failure modes $f$ indexed by $l$ }
\FORALL { $sp_l \in SP$ }
\STATE { $ f_l = ConvertToFaultMode(sp_l) $}
\STATE { $ f_l \in SS $}
\ENDFOR
%\hline
Algorithm \ref{alg:sympabs55} is the final stage in the process. We now have a
sub-system $SS$, which has its own set of failure~modes. This can now be treated
as a component, and used to form functional~groups at a higher level of failure~mode~abstraction.
\end{algorithmic}
\end{algorithm}
\clearpage
\section{To conclude}
The technique provides a methodology for bottom-up analysis of the fault behaviour of complex safety critical systems.
\subsection{Hierarchical Simplification}
Because symptom abstraction collects fault modes, the number of faults to handle decreases
as the hierarchy progresses upwards.
This is seen in real life Systems. At the highest levels the number of faults
reduces. A Sound system might have, for instance only four faults at its highest or System level,
\small
$$ SoundSystemFaults = \{TUNER\_FAULT, CD\_FAULT, SOUND\_OUT\_FAULT, IPOD\_FAULT\}$$
\normalsize
The number of causes for any of these faults is very large !
It does not matter which combination of causes caused the fault to the user.
But as the hierarchy goes up in abstraction level the number of faults goes down.
\subsection{Tracable Fault Modes}
Because the fault modes are determined from the bottom-up, the causes
for all high level faults naturally form trees.
Minimal cut sets \cite{nasafta} can be determined from these, and by
analysing the statistical likely hood of the component failures
the MTTF and SIL\cite{en61508} levels can be automatically calculated.