
\abstract{ This chapter defines what is meant by the terms
components, component fault modes and `unitary~state' component fault modes.
%The application of Bayes theorem  in current methodologies, and
%the suitability of the `null hypothesis' or `P' value statistical approach
%are discussed.
Data types and their relationships are described using UML.
Mathematical constraints and definitions are made using set theory.
}

 
\section{Introduction}
This chapter describes the data types and concepts for the Failure Mode Modular De-composition (FMMD) method.
When analysing a safety critical system using the 
this technique, we need clearly defined failure modes for
all the components that are used to model the system.
These failure modes have a constraint such that
the component failure modes must be mutually exclusive.
When this constraint is complied with we can use the FMMD process to 
build hierarchical bottom-up models of failure mode behaviour.
%This and the definition of a component are 
%described in this chapter.
%When building a system from components, 
%we should be able to find all known failure modes for each component.
%For most common electrical and mechanical components, the failure modes
%for a given type of part can be obtained from standard literature\cite{mil1991}
%\cite{mech}. %The failure modes for a given component $K$ form a set $F$.


%%
%% Paragraph component and its relationship to its failure modes
%%

\section{ What is a Component ?}


\begin{figure}[h]
 \centering
 \includegraphics[width=400pt,bb=0 0 437 141,keepaspectratio=true]{component_failure_modes_definition/component.jpg}
 % component.jpg: 437x141 pixel, 72dpi, 15.42x4.97 cm, bb=0 0 437 141
 \caption{A Component and its Failure Modes}
 \label{fig:component}
\end{figure}

Let us first define a component. This is anything 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.
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 consider (with accompanying statistical
failure rates)\cite{mil1991}. For instance, a simple resistor is generally considered 
to fail in two ways, it can go open circuit or it can short.  
Thus we can associate a set of faults to this component $ResistorFaultModes=\{OPEN, SHORT\}$.
The UML diagram in figure 
\ref{fig:component} shows a component as a data
structure with its associated failure modes.
From this diagram we see that each component must have at least one failure mode.
Also to clearly show that the failure modes are unique events associated with one component,
each failure mode is referenced back to only one component.
% Perhaps talk here about the failure modes being shared, but by being referenced
% by the component ?



A product naturally consists of many components and these are traditionally
kept in a `parts list'. For safety critical product this is usually a formal document
and is used by quality inspectors to ensure the correct parts are being fitted.
For our UML diagram the parts list is simply a collection of components 
as shown in figure \ref{fig:componentpl}.
\begin{figure}[h]
 \centering
 \includegraphics[width=400pt,bb=0 0 712 68,keepaspectratio=true]{component_failure_modes_definition/componentpl.jpg}
 % componentpl.jpg: 712x68 pixel, 72dpi, 25.12x2.40 cm, bb=0 0 712 68
 \caption{Parts List of Components}
 \label{fig:componentpl}
\end{figure}


 

%%
%% Paragraph using failure modes to build from 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
to see how they could be caused. Some apply statistical tequniques to
determine the likelihood of component failures 
causing specific system level errors (see Bayes theorem \ref{bayes}).
Another top down technique is to apply cost benifit analysis 
to determine which faults are the highest priority to fix\cite{FMEA}.
The aim of FMMD analysis is to produce complete  failure
models of safety critical systems  from the bottom-up,
starting, where possible with known component failure modes.

In order to analyse from the bottom-up, we need to take
small groups of components from the parts~list that naturally
work together to perform a simple function.
The components to include in a  functional group are chosen by a human, the analyst.
We can term this a `Functional~Group' and represent it as a class. When we have a 
`Functional~Group' we can look at the failure modes of all the components
in it and determine a failure mode behaviour for that group.
Or in other words we can determine the failure modes of the functional
group. An advantage of working from the bottom up is that we can ensure that
all component failure modes must be considered. A top down approach
could miss individual failure modes of components.


\subsection{From functional group to newly derived component}

The process for taking a functional~group, considering
all the failure modes of all the components in it,
and analysing these 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.
%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 fungtional group that it was derived from.
Because these new failure modes were determined from a derived component 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
fault behaviour.

The UML representation shows a `functional group' having a one to one relationship with a derived~component.
We can represet this using an UML diagram in figure \ref{fig:cfg}

\begin{figure}[h]
 \centering
 \includegraphics[width=400pt,bb=0 0 712 286,keepaspectratio=true]{./cfg.jpg}
 % cfg.jpg: 712x286 pixel, 72dpi, 25.12x10.09 cm, bb=0 0 712 286
 \caption{UML Meta model for FMMD hierarchy}
 \label{fig:cfg}
\end{figure}

\subsection{Keeping track of the dereived components position in the hierarchy}

The UML meta model in figure \ref{fig:cfg}, will build a hierarchy of
objects, with derived~components forming functional~groups, and creating
derived components higher up in the structure.
The level variable in each Component,
indicates the position in the hierarchy. Base or parts~list components
have a `level' of 0. Derived~components take a level based on the highest level
component used to build the functional group it was derived from plus 1.
So a derived component built from base level or parts list components
would have a level of 1.
%\clearpage



\section{Set Theory Description}

$$ System \stackrel{has}{\longrightarrow} PartsList $$

$$ PartsList  \stackrel{has}{\longrightarrow} Components $$

$$ Component \stackrel{has}{\longrightarrow} FailureModes $$

$$ FunctionalGroup  \stackrel{has}{\longrightarrow} Components $$

Using the symbol $\bowtie$ to indicate an analysis process that takes a
functional group and converts it into a new component.

$$ \bowtie ( FG ) \mapsto DerivedComponent $$



\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
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.
Should a component be analysed and simultaneous failure mode cases exit,
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
within one package.





\begin{definition}
A set of failure modes where only one fault mode
can be active at a time is termed a `unitary~state' failure mode set.
%This is termed the $U$ set thoughout this study.
This corresponds to the `mutually exclusive' definition in 
probability theory\cite{probandstat}.
\end{definition}

We can define a function $FM()$ to 
take a given component $K$ and return its set of failure modes $F$.

$$  FM : K \mapsto F $$

We can further define a set $U$ which is a set of sets of failure modes, where
the component failure modes in each of its members are unitary~state.
Thus if the failure modes of $F$ are unitary~state, we can say $F \in U$.


\section{Component failure modes : Unitary State example}

A component with an obvious set of  ``unitary~state'' failure modes is the electrical resistor.

Electrical resistors can fail by going OPEN or SHORTED.

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'.


Thus

$$ R_{SHORTED} \cap R_{OPEN} = \emptyset $$
therefore
$$ FM(R) \in U $$


We can make this a general case by taking a set $F$ (where $f1, f2 \in F$) representing a collection
of component failure modes.
We can define a boolean function {\ensuremath{\mathcal{ACTIVE()}}} that returns 
whether a fault mode is active (true) or dormant (false).

We can say that if any pair of fault modes is active at the same time, then the failure mode set is not
unitary state:
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 U 
 \end{equation}


% 
% \begin{equation}
%  c1 \cap c2 \neq \emptyset  | c1 \neq c2 \wedge c1,c2 \in C \wedge C \not\in U
% \end{equation}

That is to say that it is impossible that any pair of failure modes can be active at the same time
for the failure mode set $F$ to exists in the family of sets $U$

Note where that are more than two failure~modes, 
by banning pairs from being active at the same time
we have banned larger combinations as well.



\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 this set of all possible outcomes are its normal correct
operating state and all its failure modes.
When dealing with failure modes, we are not interested in 
the state where the component is working perfectly or `OK' (i.e. operating with no error).
We are interested only in ways in which it can fail.
By definition while all components in a system are `working perfectly' 
that system will not exhibit faulty behaviour.
Thus the statistical sample space $\Omega$ for a component or derived~component $K$ is
%$$ \Omega = {OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N} $$
$$ \Omega(K) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, \ldots ,failure\_mode_{N}\} $$
The failure mode set $F$ for a given component or derived~component $K$
is therefore
$$ F = \Omega(K) \backslash OK $$

The $OK$ statistical case is the largest in probability, and is therefore
of interest when analysing systems that have failed using techniques
such as bayes theorem to determine the likelyhood of the failure source.


\vspace{40pt}
