JMC PR

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.