diff --git a/submission_thesis/CH6_Evaluation/copy.tex b/submission_thesis/CH6_Evaluation/copy.tex index 75f1223..287c96f 100644 --- a/submission_thesis/CH6_Evaluation/copy.tex +++ b/submission_thesis/CH6_Evaluation/copy.tex @@ -32,6 +32,9 @@ check for in double failure analysis. % MOVE TO CH5 temperature measurement sensor circuit. This example is also used to show how component failure rate statistics can be % MOVE TO CH5 used with FMMD. % +% +% MIGHT MOVE TO CONCLUSIONS? +%FDefining a function that This is followed by some critiques of FMMD. % in use.%i.e. possible areas of difficulty when performing FMMD, and then %a general evaluation. % comparing it with traditional FMEA. @@ -65,7 +68,7 @@ We can view FMEA as a process, taking each component in the system and for each applying analysis with respect to the whole system. % This however entails a problem: which other components in the system must we -check, against %current failure mode. +check against %current failure mode. each particular failure mode? % Often a component failing will have obvious effects on functionally adjacent components. @@ -133,22 +136,22 @@ $G$ 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 $c$ with additional indexing where appropriate. -\paragraph{Defining a function that returns failure modes given a component.} +\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) | .$ +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 the number of checks required to rigorously examine every failure mode against all the other components in a system. - +% 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 a rigorous FMEA inspection. -Where $\mathcal{G}$ represents the set of all {\fgs}%, and $ \mathbb{Z}^{+} $, +Where $\mathcal{G}$ represents the set of all {\fgs} %, and $ \mathbb{Z}^{+} $, $CC$ is defined by, \begin{equation} %$$ @@ -158,7 +161,7 @@ $CC$ is defined by, % %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 +in component ${c_i}$. Comparison complexity, $CC$ for a group of components $G$, is given by \begin{equation} \label{eqn:CC} @@ -200,7 +203,7 @@ i.e. at the zeroth level of an FMMD hierarchy where $\alpha=0$, would have the s % \end{equation} \subsection{A general formula for counting Comparison Complexity in an FMMD hierarchy} -An FMMD Hierarchy will have reducing numbers of {\fgs} as we progress up the hierarchy. +An FMMD hierarchy will have reducing numbers of {\fgs} 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. We can define an FMMD hierarchy as a set of {\fgs}, $\hh$. @@ -255,13 +258,13 @@ Ensuring all component failure modes are checked against all other components in 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 +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 RFMEA for anything but trivial systems. % % Next statement needs alot of justification % -It is the authors belief that FMMD reduces the comparison complexity enough to make +It is the author's belief that FMMD reduces the comparison complexity enough to make rigorous checking feasible. @@ -364,7 +367,7 @@ 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 $|G|$ and $|fm(c)|$) +We can now use equation~\ref{eqn:anscen} (FMMD) and \ref{eqn:CC} (RFMEA) to compare (for fixed sizes of $|G|$ and $|fm(c)|$) the two approaches, for the work required to perform rigorous checking. @@ -389,7 +392,7 @@ $$ $$ %\clearpage -\subsection{Complexity Comparison applied to FMMD electroinc circuits analysed in chapter~\ref{sec:chap5}.} +\subsection{Complexity Comparison applied to FMMD electronic circuits analysed in chapter~\ref{sec:chap5}.} All the FMMD examples in chapters \ref{sec:chap5} and \ref{sec:chap6} showed a marked reduction in comparison @@ -467,7 +470,7 @@ are presented in table~\ref{tbl:firstcc}. The complexity comparison figures for the example circuits in chapter~\ref{sec:chap5} show that for the non trival examples, as we use more levels in the FMMD hierarchy, the performance -gains over RFMEA become apparent. %for increasing complexity the performance benefits from FMMD are apparent. +gain over RFMEA becomes apparent. %for increasing complexity the performance benefits from FMMD are apparent. @@ -476,7 +479,7 @@ gains over RFMEA become apparent. %for increasing complexity the performance ben \subsection{Comparison Complexity for the Bubba Oscillator Example.} The Bubba oscillator example (see section~\ref{sec:bubba}) was chosen because it had a circular signal path. It was also analysed twice, once by -{na\"{\i}vely} using the first {\fgs} identified, and secondly be de-composing +{na\"{\i}vely} using the first {\fgs} identified, and secondly by de-composing the circuit further. We use these two analyses to compare the effect on comparison complexity (see table~\ref{tbl:bubbacc}) with that of RFMEA. % @@ -549,7 +552,7 @@ by more than a factor of ten. -\subsection{Sigma delta Example: Comparison Complexity Results} +\subsection{Sigma Delta Example: Comparison Complexity Results} \label{sec:bubbaCC} @@ -608,7 +611,7 @@ are level shifted, adding to the complication of analysing it for failures. % % can I say that ? % -\section{Unitary State Component Failure Mode sets} +\section{Unitary State Component Failure Mode Sets} \label{sec:unitarystate} \paragraph{Design Decision/Constraint} An important factor in defining a set of failure modes is that they @@ -720,9 +723,11 @@ We can term this `heuristic~de-composition'. A modern micro-controller will typically have several modules, which are configured to operate on pre-assigned pins on the device. Typically voltage inputs (\adcten / \adctw), digital input and outputs, PWM (pulse width modulation), UARTs and other modules will be found on simple cheap microcontrollers~\cite{pic18f2523}. -For instance the voltage reading functions which consist -of an ADC multiplexer and ADC can be considered to be components +% +For instance, the voltage reading functions which consist +of a multiplexer and ADC---which must work together to channel readings--- could be considered to be components inside the micro-controller package. +% The micro-controller thus becomes a collection of smaller components that can be analysed separately~\footnote{It is common for the signal paths in a safety critical product to be traced, and when entering a complex @@ -752,7 +757,7 @@ This does not preclude the possibility of two or more components failing simulta % %The scenarios presented deal with possibility of two or more components failing simultaneously. % -It is an implied requirement of EN298~\cite{en298} for instance to +It is an implied requirement of EN298~\cite{en298} for instance, to consider double simultaneous faults\footnote{Under the conditions of LOCKOUT~\cite{en298} in an industrial burner controller that has detected one fault already. However, from the perspective of static failure mode analysis, this amounts @@ -1088,9 +1093,9 @@ $ \Omega(C) = fm(C) \cup \{OK\} $). The $OK$ statistical case is the (usually) largest in probability, and is therefore of interest when analysing systems from a statistical perspective. -For these examples, the OK state is not represented area proportionately, but included +For these examples, the OK state is not represented area proportionately, but is included in the diagrams. -This is of interest for the application of conditional probability calculations +This type of diagram is germane to the application of conditional probability calculations such as Bayes theorem~\cite{probstat}. The current failure modelling methodologies (FMEA, FMECA, FTA, FMEDA) all use Bayesian @@ -1247,9 +1252,13 @@ Some logic chips are more susceptible to $INTERFERENCE$ than others. A logic chip with de-coupling capacitor failing, may operate correctly but interfere with other chips in the circuit. % -There is no reason why the de-coupling capacitors -could not be included % {\em in the {\fg} they would intuitively be associated with as well}.% poss split infinitive -in {\fgs} that they would not intuitively be associated with. +%%% There is no reason why the de-coupling capacitors +%%% could not be included % {\em in the {\fg} they would intuitively be associated with as well}.% poss split infinitive +%%% in {\fgs} that they would not intuitively be associated with. +% +There is no reason why we cannot include the de-coupling capacitors in each {\fg} +that could be affected by $INTERFERENCE$, meaning that the same +de-coupling capacitors can be members of different {\fgs}. % This allows for the general principle of a component failure affecting more than one {\fg} in a circuit. This allows functional groups to share components where necessary.