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component_failure_modes_definition/component_failure_modes_definition.tex

@@ -62,8 +62,8 @@ Let us first define a component. This is anything which we use to build a
 product or system with. This could be something quite complicated 
 like an integrated microcontroller, or quite simple like the humble resistor.
 We can define a 
-component by its name, a manufacturers part number and perhaps
-a vendors reference number.
+component by its name, a manufacturers' part number and perhaps
+a vendors' reference number.
 What these components all have in common is that they can fail, and fail in
 a number of well defined ways. For common components
 there is established literature for the failure modes for the system designer to consider (often with accompanying statistical
@@ -120,7 +120,7 @@ When building from the bottom up, it is more meaningful to call them `derived~co
 %% Paragraph using failure modes to build from bottom up
 %%
 
-\section{Fault Mode Analysis, top down or bottom up?}
+\section{Fault Mode Analysis, \\ top down or bottom up?}
 
 Traditional static fault analysis methods work from the top down.
 They identify faults that can occur in a system, and then work down
@@ -167,16 +167,16 @@ all the failure modes of all the components in the group,
 and analysing it is called `symptom abstraction' and 
 is dealt with in detail in chapter \ref{symptom_abstraction}.
 
-In terms of our UML model the symptom abstraction process takes a functional~group,
-and creates a new derived component from it.
+In terms of our UML model, the symptom abstraction process takes a {\fg}
+and creates a new {\dc} from it.
 %To do this it first creates
 %a new set of failure modes, representing the fault behaviour
 %of the functional group. This is a human process and to do this the analyst
 %must consider all the failure modes of the components in the functional
 %group.
-The newly created derived~component requires a set of failure modes of its own.
-These failure modes are the failure mode behaviour of the functional group that it was derived from.
-Because these new failure modes were determined from a derived component we can call 
+The newly created {\dc} requires a set of failure modes of its own.
+These failure modes are the failure mode behaviour of the {\fg} that it was derived from.
+Because these new failure modes were derived from a {\fg} we can call 
 these `derived~failure~modes'.
 %It then creates a new derived~component object, and associates it to this new set of derived~failure~modes.
 We thus have a `new' component, or system building block, but with a known and traceable
@@ -185,7 +185,7 @@ fault behaviour.
 The UML representation shows a `functional group' having a one to one relationship with a derived~component.
 We can represent this using a UML diagram in figure \ref{fig:cfg}.
 
-Using the symbol $\bowtie$ to indicate the analysis process that takes a
+The symbol $\bowtie$ is used to indicate the analysis process that takes a
 functional group and converts it into a new component.
 
 \[ \bowtie ( FG ) \mapsto DerivedComponent \] 
@@ -258,7 +258,7 @@ $$  FM ( C ) = F $$
 For FMMD failure mode analysis we need to consider the failure modes
 from all the components in a functional~group as a flat set.
 Consider the components in a functional group to be $C$  indexed by j thus $C_j$.
-Thflat set of failure modes we are after can be found by applying function $FM$ to all the components
+The flat set of failure modes we are after can be found by applying function $FM$ to all the components
 in the functional~group and taking the union of them thus:
 
 $$ FunctionalGroupAllFailureModes = \bigcup_{j \in \{1...n\}} FM(C_j) $$
@@ -271,12 +271,12 @@ FM : FG \mapsto \mathcal{F}
 \end{equation}
 
 
-\section{Unitary State Component Failure Mode sets}
+\section{Unitary State Component \\ Failure Mode sets}
 
 \paragraph{Design Descision/Constraint}
 An important factor in defining a set of failure modes is that they
 should be as clearly defined as possible.
-It should not be possible for instance for
+It should not be possible, for instance for
 a component to have two or more failure modes active at once.
 Having a set of failure modes where $N$ modes could be active simultaneously
 would mean having to consider an additional $2^N-1$ failure mode scenarios.
@@ -329,7 +329,7 @@ For a given resistor R we can apply the
 the function $FM$ to find its set of failure modes thus  $ FM(R) =  \{R_{SHORTED},R_{OPEN}\} $.
 A resistor cannot fail with both conditions open and short active at the same time! The conditions
 OPEN and SHORT are thus mutually exclusive.
-Because of this the failure mode set $F=FM(R)$ is `unitary~state'.
+Because of this, the failure mode set $F=FM(R)$ is `unitary~state'.
 
 
 Thus because both fault modes cannot be active at the same time, the intersection of $ R_{SHORTED} $ and $ R_{OPEN} $ cannot exist.
@@ -365,9 +365,9 @@ Note where there are more than two failure~modes,
 by banning any pairs from being active at the same time,
 we have banned larger combinations as well.
 
-\section{Handling Simultaneous Component Faults}
+\section{Handling Simultaneous \\ Component Faults}
 
-For some integrity levels of static analysis there is a need to consider not only single
+For some integrity levels of static analysis, there is a need to consider not only single
 failure modes in isolation, but cases where more then one failure mode may occur
 simultaneously.
 It is an implied requirement of EN298\cite{en298} for instance to consider double simultaneous faults.
@@ -377,7 +377,7 @@ all combinations of failures modes of size $N$ and less.
 The Powerset concept from Set theory is useful to model this.
 The powerset, when applied to a set S is the set of all subsets of S, including the empty set 
 \footnote{The empty set ( $\emptyset$ ) is a special case for FMMD analysis, it simply means there
-is no fault active in the functional~group under analysis}
+is no fault active in the functional~group under analysis.}
 and S itself.
 In order to consider combinations for the set S where the number of elements in each sub-set of S is $N$ or less, a concept of the `cardinality constrained powerset'
 is proposed and described in the next section. 
@@ -388,7 +388,7 @@ is proposed and described in the next section.
 
 A Cardinality Constrained powerset is one where sub-sets of a cardinality greater than a threshold
 are not included. This threshold is called the cardinality constraint.
-To indicate this the cardinality constraint $cc$, is subscripted to the powerset symbol thus $\mathcal{P}_{cc}$.
+To indicate this, the cardinality constraint $cc$ is subscripted to the powerset symbol thus $\mathcal{P}_{cc}$.
 Consider the set $S = \{a,b,c\}$. 
 
 The powerset of S: 
@@ -537,7 +537,7 @@ associated with the test cases, complete coverage would be verified.
 
 
 \pagebreak[1]
-\section{Component Failure Modes and Statistical Sample Space}
+\section{Component Failure Modes \\ and Statistical Sample Space}
 %\paragraph{NOT WRITTEN YET PLEASE IGNORE}
 A sample space is defined as the set of all possible outcomes.
 For a component in FMMD analysis, this set of all possible outcomes is its normal correct