Saturday morning edit

component_failure_modes_definition/component_failure_modes_definition.tex

@@ -258,7 +258,8 @@ derived components higher up in the structure.
 To keep track of the level in the hierarchy (i.e. how many stages of component 
 derivation `$\bowtie$' have lead to the current derived component)
 we can add an attribute to the component data type. 
-This can be a natural number called the level variable $\alpha \in \mathbb{N}_0$.
+This can be a natural number called the level variable $\alpha \in \mathbb{N}$.
+% J. Howse says zero is a given in comp sci. This can be a natural number called the level variable $\alpha \in \mathbb{N}_0$.
 The $\alpha$ level variable in each component,
 indicates the position in the hierarchy. Base or parts~list components
 have a `level' of $\alpha=0$. 
@@ -290,7 +291,8 @@ would have an $\alpha$ value of 1.
 \subsection{Relationships between functional~groups and failure modes}
 
 Let the set of all possible components  be $\mathcal{C}$
-and let the set of all possible failure modes be $\mathcal{F}$.
+and let the set of all possible failure modes be $\mathcal{F}$ and $\mathcal{PF}$ is the powerset of
+all $\mathcal{F}$..
 
 We can define a function $fm$ as equation \ref{eqn:fmset}.
 
@@ -299,22 +301,30 @@ fm : \mathcal{C} \rightarrow \mathcal{P}\mathcal{F}
   \label{eqn:fmset}
 \end{equation}
 
-The is defined by equation \ref{eqn:fminstance}, where C is a component and F is a set of failure modes.
-
-\begin{equation}
-  fm ( C ) = F
-  \label{eqn:fminstance}
-\end{equation}
+%% 
+% Above def gives below anyway
+%
+%The is defined by equation \ref{eqn:fminstance}, where C is a component and F is a set of failure modes.
+%
+%\begin{equation}
+%  fm ( C ) = F
+%  \label{eqn:fminstance}
+%\end{equation}
 
 \paragraph{Finding all failure modes within the functional group}
 
 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 $C_1...C_N$.
+from all the components in a functional~group.
+In a functional group we have a collection of Components
+that hold failure mode sets.
+We need to collect these failure mode sets and place all the failure
+modes into a single set; this can be termed flattening the set of sets.
+%%Consider the components in a functional group to be $C_1...C_N$.
 The flat set of failure modes $FSF$ 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:
 
-$$ FSF = \bigcup_{j=1}^{N} FM(C_j) $$
+%%$$ FSF = \bigcup_{j=1}^{N} fm(C_j) $$
+$$ FSF = \bigcup_{c \in FG} fm(c) $$
 
 We can actually overload the notation for the function $fm$ % FM
 and define it for the set components within a functional group $\mathcal{FG}$ (i.e. where $\mathcal{FG} \subset \mathcal{C} $) 
@@ -337,7 +347,7 @@ Were this to be the case, we would have to consider additional combinations of
 failure modes within the component.
 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.
-Should a component be analysed and simultaneous failure mode cases exit,
+Should a component be analysed and simultaneous failure mode cases exist,
 the combinations could be represented by new failure modes, or
 the component should be considered from a fresh perspective,
 perhaps considering it as several smaller components
@@ -350,11 +360,18 @@ probability theory~\cite{probstat}.
 
 \begin{definition}
 A set of failure modes where only one failure mode
-can be active at one time is termed a `unitary~state' failure mode set.
+can be active at one time is termed a {\textbf{unitary~state}} failure mode set.
 \end{definition}
 
-Let the set of all possible components to be $ \mathcal{C}$
+Let the set of all possible components be $ \mathcal{C}$
 and let the set of all possible failure modes be $ \mathcal{F}$.
+The set of failure modes of a particular component are of interest 
+here. What is required is to define a property for
+a set of failure modes where only one failure mode can be active at a time,
+or borrowing from the terms of  statistics, the failure mode is an event, and it is mutually exclusive
+with the a specific set $F$.
+We can define a set of failure mode sets called $\mathcal{U}$ to represent this
+property.
 
 \begin{definition}
 We can define a set $\mathcal{U}$ which is a set of sets of failure modes, where
@@ -394,7 +411,7 @@ we state this formally
 
 
  \begin{equation}
-  \forall f_1,f_2 \in F \dot ( f_1 \neq f_2 \wedge \mathcal{ACTIVE}({f_1}) \wedge \mathcal{ACTIVE}({f_2}) ) \implies F \not\in \mathcal{U} 
+  \exists f_1,f_2 \in F \dot ( f_1 \neq f_2 \wedge \mathcal{ACTIVE}({f_1}) \wedge \mathcal{ACTIVE}({f_2}) ) \implies F \not\in \mathcal{U} 
  \end{equation}
 
 
@@ -411,18 +428,31 @@ we have banned larger combinations as well.
 
 \subsection{Design Rule: Unitary State}
 
-All components must have unitary state failure modes to be used with the FMMD methodology.
-Where a complex component is used, for instance a microcontroller
+
+
+
+All components must have unitary state failure modes to be used with the FMMD methodology,
+for base~components, this is usually the case. Most simple components fail in one
+clearly defined way and generally stay in that state.
+
+However, where a complex component is used, for instance a microcontroller
 with several modules that could all fail simultaneously, a process
-of reduction into smaller theoretical components will have to be made
-\footnote{A modern microcontroller will typically have several modules, which are configured to operate on
+of reduction into smaller theoretical components will have to be made.
+This is sometimes termed `heuristic~de-composition'.
+A modern microcontroller 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}}.
+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
 inside the microcontroller package.
-The microcontroller thuis becomes a collection of smaller components
-the can be analysed separately.
+The microcontroller 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
+component like a micro controller, the process of heuristic de-compostion
+applied to it}.
+
+
+
 \paragraph{Reason for Constraint} Were this constraint to not be applied
 each component could not have $N$ failure modes to consider but potentially
 $2^N$. This would make the job of analysing the failure modes