 

%%% CHAPTER 6
\label{sec:chap6}

\section{Software and Hardware Failure Mode Concepts}
\label{sec:elecsw}
In this chapter we show that FMMD  can be applied to software enabling us to build build complete failure models
of typical modern safety critical systems.
With modular FMEA i.e. FMMD %(FMMD) 
we have the concepts of failure~modes
of components, {\fgs} and symptoms of failure for a functional group.
%
A  programmatic function has similarities with a {\fg} as defined by the FMMD process.
%
An FMMD {\fg} is placed into a hierarchy.
A software function is typically placed into the  hierarchy of its call-tree.
A software function calls other functions and uses data sources via hardware interaction, which could be viewed as its `components':
it has outputs, i.e. it can perform actions  on data or hardware.
%which will be used by other functions that may call it.
%
We show that we can map a software function to a {\fg} in FMMD: its failure modes
are the failure modes of the software components (other functions it calls)
and the hardware from which it reads values.
Its outputs are the data it changes, or the hardware actions it performs.
%%
%% Talk about how software specification will often say how hardware
%% will react and how to interpret readings---but they do not
%% always cover the failure modes of the hardware being interfaced too.
%
When we have analysed a software function---using failure conditions 
of its inputs as failure modes---we can 
determine its symptoms of failure (i.e. how calling functions will see its failure mode behaviour).

%
We   apply the FMMD process to software functions by viewing them in terms of their failure mode behaviour. 
%
As software already fits into a hierarchy we have one less analysis decision to make, compared
to analysing electronics. 
%
For Electronics and Mechanical systems, although we may be guided by the original designers
concepts of modularity and sub-systems in design, applying FMMD means deciding on the members for {\fgs}
and the subsequent hierarchy. 
%
With software already written, the hierarchies are given.
%
To apply FMMD to software, we collect the elements used by a software function, along with the function its-self
to form a {\fg}. When we have analysed the failure mode behaviour of this {\fg}
and have its failure mode symptoms, we can create a {\dc}. That {\dc} can be 
used by functions that call the function we have just analysed, until
we form a complete failure mode hierarchy of the system under investigation.
%                    map   the FMMD concepts of {\fms}, {\fgs} and {\dcs}
%to software functions.
%
%However, we need to map a the FMMD concepts of {\fms}, {\fgs} and {\dcs}
%to software functions.
% failure modes of a function in order to
%map FMMD to software.



%                    map   the FMMD concepts of {\fms}, {\fgs} and {\dcs}
%to software functions.
%
%However, we need to map a the FMMD concepts of {\fms}, {\fgs} and {\dcs}
%to software functions.
% failure modes of a function in order to
%map FMMD to software.

\subsection{Software, a natural hierarchy}

Software written for safety critical systems is usually constrained to
be modular~\cite{en61508}[3] and non recursive~\cite{misra}[15.2]. %{iec61511}.
Because of this we can assume direct call trees~\footnote{A typical embedded system
will have a run time call tree, and (possibly multiple) interrupt sourced call tress.}.  Functions call functions
from the top down and eventually call the lowest level library or IO
functions that interact with hardware/electronics.

What is potentially difficult with a software function, is deciding  what
its failure modes and symptoms are.
With electronic components, we can use literature to point us to suitable sets of
{\fms}~\cite{fmd91}~\cite{mil1991}~\cite{en298}. %~\cite{en61508}~\cite{en298}.
With software, only some library functions are well known and rigorously documented
enough to have the equivalent of known failure modes.
Most software is `bespoke'. 
%
We need a different strategy to
describe the failure mode behaviour of software functions.
We can use definitions from contract programming to assist here.

\subsection{Contract programming description}

Contract programming is a discipline~\cite{dbcbe} for building software functions in a controlled
and traceable way. Each function is subject to pre-conditions (constraints on its inputs),
post-conditions (constraints on its outputs) and function wide invariants (rules).


\paragraph{Mapping contract `pre-condition' violations to component failure modes.}

A precondition, or requirement for a contract software function
defines the correct ranges of input conditions for the function
to operate successfully.
%
% C Garret said this was unclear so I have added the following two sentences.
%
%If we consider a software function to be a {\fg} in the FMMD sense, i.e.
We can consider a software function to be 
a collection of code, functions called and values/variables used.
In this way it is similar to an electronic circuit, which is a collection
of components connected in a specific way.
Using this analogy for software, the connections are the functions code, and the 
called functions and variables are the components.
%
Erroneous behaviour from called functions and variables/inputs has the same effect as component failure modes
on an electronic {\fg}.
%
%
If we consider the 
called functions and variables/inputs to be components of a function,
we can build a modular and hierarchical failure mode model
from existing software.
%
Thus for FMMD applied to software, we consider a violation of a pre-condition to be
equivalent to failure mode of `one of its components'.


\paragraph{Mapping contract `post-condition' violations to symptoms.}

A post condition is a definition of correct behaviour of a function.
%
A violated post condition is a symptom of failure, or derived failure mode, from a function.
%
Post conditions could  be either  actions performed (i.e. the state of hardware changed) or an output value of a function.
In pure contract programming, a violation of a pre-condition would not cause the function to
be executed.
%
In implementation code, a pre-condition violation should cause
an error to be generated, and thus a post condition to fail.
% 
A function can fail for reasons other than the 
a failure of one the variables/inputs or functions that it calls.
Variables can become corrupted, by radiation affecting RAM or 
by another software function erroneously overwriting variables.
Current work on software FMEA generally focuses on mapping
variable corruption to failure modes~\cite{procsfmea,procsfmeadb,sfmeaauto,sfmea}.
However, errors other than variable corruption can occur,
for instance a microprocessor may have subtle bugs in its instruction set or
incorrectly handled
interrupt contention could cause side effects in software.
For the failure mode model of any software function
we must consider all failure modes of post condition
violations as well as those caused by `components'.


\paragraph{Mapping contract `invariant' violations to symptoms and failure modes.}

Invariants in contract programming may apply to inputs to the function (where violations can be considered {\fms} in FMMD terminology),
and to outputs (where violations can be considered {failure symptoms} in FMMD terminology).


\subsection{Combined Hardware/Software FMMD}

For the purpose of example, we chose a simple common safety critical industrial circuit
that is nearly always used in conjunction with a programmatic element.
A common method for delivering a quantitative value in analogue electronics is
to supply a current signal to represent the value to be sent~\cite{aoe}[p.934].
Usually, $4mA$ represents a zero or starting value and $20mA$ represents the full scale,
and this is referred to as {\ft} signalling.
%
{\ft} signalling has  intrinsic electrical  safety advantages.
%
Because the current in a loop is constant~\cite{aoe}[p.20]
resistance in the wires between the source and receiving end is not an issue
that can alter the accuracy of the signal.
%
%This circuit has many advantages for safety. 
If the signal becomes disconnected
it reads $0mA$ at the receiving end: as this is outside the {\ft} range,
it is easily detectable as an error condition rather than an incorrect value.
%
Should the driving electronics go wrong at the source end, it will usually
supply far too little or far too much current, also making error conditions easy to detect.
%
At the receiving end, we only require one simple component to convert the 
current signal into a voltage that we can read with an AD---a resistor---given
its properties defined by Ohms law. % the humble resistor!


%BLOCK DIAGRAM HERE WITH FT CIRCUIT LOOP

\begin{figure}[h]
 \centering
 \includegraphics[width=230pt]{./CH5_Examples/ftcontext.png}
 % ftcontext.png: 767x385 pixel, 72dpi, 27.06x13.58 cm, bb=0 0 767 385
 \caption{Context Diagram for {\ft} loop}
 \label{fig:ftcontext}
\end{figure}


The diagram in figure~\ref{fig:ftcontext}, shows some equipment which is sending a {\ft}
signal to a micro-controller system.
The signal is locally driven over a load resistor, and then read into the micro-controller via
an ADC and its multiplexer.
With the voltage determined at the ADC we read the intended quantitative
value from the external equipment.

\section{Simple Software Example: Reading a \ft input into software}


Consider a software function that reads a {\ft} input, and returns a value between 0 and 999 (i.e. per mil $\permil$)
representing the current detected with an additional error indication flag .
%
Let us assume the {\ft} detection is via a \ohms{220} resistor, and that we read a voltage
from an ADC into the software.
Let us define any value outside the 4mA to 20mA range as an error condition.
%
As we read a voltage voltage, we use Ohms law~\cite{aoe} to determine the mA current detected: $V=IR$, $0.004A * \ohms{220} = 0.88V$
and $0.020A * \ohms{220} = 4.4V$. 
%
Our acceptable voltage range is therefore 

$$(V \ge  0.88) \wedge (V \le 4.4) \; .$$ 

This voltage range forms our input requirement and can be considered as an invariant condition.
%
We can now examine a software function that performs a conversion from the voltage read to 
a per~mil representation of the {\ft} input current.
%
For the purpose of example the `C' programming language~\cite{DBLP:books/ph/KernighanR88} is used.
We initially  assume a function \textbf{read\_ADC} which returns a floating point %double precision 
value which represents the voltage read (see code sample in figure~\ref{fig:code_read_4_20_input}).


%%{\vbox{
\begin{figure}[h+]

\footnotesize
\begin{verbatim}
/***********************************************/
/* read_4_20_input()                           */
/***********************************************/
/* Software function to read 4mA to 20mA input */
/* returns a value from 0-999 proportional     */
/* to the current input.                       */
/***********************************************/
int  read_4_20_input ( int * value ) {
  double input_volts;
  int error_flag;

  /* require: input from ADC to be 
              between 0.88 and 4.4 volts */


  input_volts = read_ADC(INPUT_4_20_mA);

  if ( input_volts < 0.88 || input_volts > 4.4 ) {
    error_flag = 1; /* Error flag set to TRUE */
  }
  else {
    *value = (input_volts - 0.88) * ( 4.4 - 0.88 ) * 999.0;
    error_flag = 0; /* indicate current input in range */
  }

  /* ensure: value is proportional (0-999) to the 
             4 to 20mA input                      */

  return error_flag;
}
\end{verbatim}
%}
%}\clearpage

\caption{Software Function:  \textbf{read\_4\_20\_input}}
\label{fig:code_read_4_20_input}
%\label{fig:420i}
\end{figure}
\clearpage
We now look at the function called by \textbf{read\_4\_20\_input}, \textbf{read\_ADC}, which returns a 
voltage for a given ADC channel.
%
This function
deals directly with the hardware in the micro-controller on which the software is running. %software on.
%
The software's job  is to select the correct channel (ADC multiplexer) and then to initiate a 
conversion by setting an ADC 'go' bit (see code sample in figure~\ref{fig:code_read_ADC}).
%
It takes the raw ADC reading and converts it into a
floating point\footnote{the type, `double' or `double precision', is a 
standard C language floating point type~\cite{DBLP:books/ph/KernighanR88}.}
voltage value.





%{\vbox{
\begin{figure}[h+]

\footnotesize
\begin{verbatim}
/***********************************************/
/* read_ADC()                                  */
/***********************************************/
/* Software function to read voltage from a    */
/* specified ADC MUX channel                   */
/* Assume 10 ADC MUX channels 0..9             */
/* ADC_CHAN_RANGE = 9                          */
/* Assume ADC is 12 bit and ADCRANGE = 4096    */
/* returns voltage read as double precision    */
/***********************************************/
double  read_ADC( int  channel ) {
  int timeout = 0;
  /* require: a) input channel from ADC to be 
              in valid ADC range 
              b) voltage ref is 0.1% of 5V     */

  /* return out of range result  */
  /* if invalid channel selected */
  if ( channnel > ADC_CHAN_RANGE )
     return -2.0; 

  /* set the multiplexer to the desired channel */  
  ADCMUX = channel; 

  ADCGO = 1; /* initiate ADC conversion hardware */

  /* wait for ADC conversion with timeout */
  while ( ADCGO == 1 || timeout < 100 ) 
     timeout++;

  if ( timeout < 100 )
       dval = (double) ADCOUT * 5.0 / ADCRANGE;
  else
       dval = -1.0; /* indicate invalid reading */ 

  /* return voltage as a floating point value */

  /* ensure: value is voltage input to within 0.1% */

  return dval; 
}
\end{verbatim}
\caption{Software Function: \textbf{read\_ADC}}
\label{fig:code_read_ADC}
\end{figure}
%}
%}
\clearpage

We now have a very simple software structure, a call tree, shown in figure~\ref{fig:ct1}.

\begin{figure}[h]
 \centering
 \includegraphics[width=100pt]{./CH5_Examples/ct1.png}
 % ct1.png: 151x224 pixel, 72dpi, 5.33x7.90 cm, bb=0 0 151 224
 \caption{Call tree for software example}
 \label{fig:ct1}
\end{figure}

This software is above the hardware in the conceptual call tree---from a programmatic perspective---%in software terms---the
the software is reading values from the `lower~level' electronics.
%
FMEA is always a bottom-up process and so we must begin with this hardware.
%
The hardware is simply a load resistor, connected across an ADC input
pin on the micro-controller and ground.
%
We can identify the resistor and the ADC module of the micro-controller as 
the base components in this design.
%
We now apply FMMD starting with the hardware.


\subsection{FMMD Process}

\paragraph{Functional Group - Convert mA to Voltage - CMATV}

This functional group contains the load resistor 
and the physical Analogue to Digital Converter (ADC).
Our functional group, $G_1$ is thus the set of base components: $G_1 = \{R, ADC\}$.
We now determine the {\fms} of all the components in $G_1$.
For the resistor we can use a failure mode set from the literature~\cite{en298}.
Where the function $fm$ returns a set of failure modes for a given component we can state:

$$ fm(R) = \{OPEN,SHORT\}. $$
\vbox{
For the ADC we can determine the following failure modes:

\begin{itemize}
 \item STUCKAT --- The ADC outputs a constant value,
 \item MUXFAIL --- The ADC cannot select its input channel correctly,
 \item LOW --- The ADC output is always LOW, or zero ADC counts,
 \item HIGH --- The ADC output is always HIGH, or max ADC counts.
\end{itemize}
}
We can use the function $fm$ to define the {\fms} of an ADC thus:
$$ fm(ADC) = \{ STUCKAT, MUXFAIL,LOW, HIGH \}. $$

With these failure modes, we can analyse our first functional group, see table~\ref{tbl:cmatv}.

{
\tiny
\begin{table}[h+]
\center
\caption{$G_1$: Failure Mode Effects Analysis} % title of Table
\label{tbl:cmatv}

\begin{tabular}{|| l   | c |   l ||} \hline
 %\textbf{Failure}   &  \textbf{failure}     & \textbf{Symptom}          \\ 
 %\textbf{Scenario}  &  \textbf{effect}      &     \textbf{ADC }                     \\ \hline 
 %                          & &                          &   &                             \\

 \textbf{Failure} &  \textbf{Failure }         & \textbf{Derived Component}          \\ 
 \textbf{cause}   &   \textbf{Effect}        & \textbf{Failure Mode}          \\
  
 
               \hline \hline
    1: $R_{OPEN}$           &  resistor open,               &      $HIGH$       \\  
                                &   voltage on pin high         &                 \\ \hline 

     2: $R_{SHORT}$               &  resistor shorted,         &    $LOW$          \\ 
                                   &   voltage on pin low         &                   \\   \hline \hline 


 
     3: $ADC_{STUCKAT}$            &  ADC reads out         &      $V\_ERR$          \\   
                                      &  fixed value          &                 \\ \hline 



     4: $ADC_{MUXFAIL}$        & ADC may read      &         $V\_ERR$           \\  
                                  &   wrong channel   &                  \\ \hline 

     5: $ADC_{LOW}$        &  output low   &       $LOW$              \\       
     6: $ADC_{HIGH}$      &  output high   &       $HIGH$                \\  \hline 
     7:  post condition fails                 &  software fails              &          $V\_ERR$                  \\ \hline
 

\hline


\hline

\end{tabular}
\end{table} 
}


We now collect the symptoms for the hardware functional group, $\{ HIGH , LOW, V\_ERR \} $.
We now create a {\dc} to represent this called $CMATV$.

%We can express this using the `$\derivec$' function thus:
%$$ CMATV = \; \derivec (G_1) .$$

As its failure modes, are the symptoms of failure from the functional group we can now state:
$$fm ( CMATV ) =  \{ HIGH , LOW, V\_ERR \} .$$


\paragraph{Functional Group - Software -  Read\_ADC - RADC}
\label{readADC}
The software function $Read\_ADC$ uses the ADC hardware analysed
as the {\dc} CMATV above.


The code fragment in figure~\ref{fig:code_read_ADC} states pre-conditions, as
{\em/* require: a) input channel from ADC to be 
              in valid ADC range 
              b) voltage ref is 0.1\% of 5V     */}.
%
From the above contractual programming requirements, we see that
the function must be sent the correct channel number.
%
A violation  of this  can be considered a {\fm} of the function,
which we can call $ CHAN\_NO $.
%
The reference voltage for the ADC has a 0.1\% accuracy requirement.
%
If the reference value is outside of this, it is also a {\fm}
of this function, which we can call $V\_REF$.

Taken as a component for use in FMEA/FMMD our function has
two failure modes. We can therefore treat it as a generic component, $Read\_ADC$,
by stating:

$$ fm(Read\_ADC) = \{ CHAN\_NO, VREF \} $$

As we have a failure mode model for our function, we can now use it in conjunction with
with the ADC hardware {\dc} CMATV, to form a {\fg} $G_2$, where $G_2 =\{ CMSTV, Read\_ADC \}$.

We now analyse this hardware/software combined {\fg}.



{
\tiny
\begin{table}[h+]
\caption{$G_2$: Failure Mode Effects Analysis} % title of Table
\label{tbl:radc}

\begin{tabular}{|| l   | c |   l ||} \hline
% \textbf{Failure}   &  \textbf{failure}     & \textbf{Symptom}          \\ 
% \textbf{Scenario}  &  \textbf{effect}      &     \textbf{RADC }                     \\ \hline 
 
 \textbf{Failure} &  \textbf{Failure }        & \textbf{Derived Component}          \\ 
 \textbf{cause}   &   \textbf{Effect}               & \textbf{Failure Mode}          \\
  
 
               \hline
    1: ${CHAN\_NO}$           &  wrong voltage               &      $VV\_ERR$       \\  
                                &   read         &                 \\ \hline 

     2: ${VREF}$          &  ADC volt-ref         &    $VV\_ERR$          \\ 
                             &  incorrect         &                   \\    \hline  \hline 


 
     3: $CMATV_{V\_ERR}$      &  voltage value     &      $VV\_ERR$          \\   
                             &  incorrect          &                 \\ \hline 



     4: $CMATV_{HIGH}$        & ADC may read      &         $HIGH$           \\  
                               &   wrong channel  &                  \\ \hline 

     5: $CMATV_{LOW}$        &  output low   &       $LOW$              \\  \hline
     
     6:  post condition fails                 &  software fails              &          $VV\_ERR$                  \\ \hline
 
\hline


\hline

\end{tabular}
\end{table} 
}



We  now collect the symptoms of failure for the {\fg} analysed (see table~\ref{tbl:radc}) 
as $\{ VV\_ERR, HIGH, LOW \}$. We can add as well the violation of the postcondition
for the function.
This postcondition, {\em /* ensure: value is voltage input to within 0.1\% */ },
corresponds to $VV\_ERR$, and is already in the {\fm} set for this {\fg}.

%We can now create a {\dc} called $RADC$ thus: $$RADC = \; \derivec(G_2)$$ which has the following
%{\fms}:
We can now create a {\dc} called $RADC$ thus:
$$ fm(RADC) = \{ VV\_ERR, HIGH, LOW \} .$$





\paragraph{Functional Group - Software - voltage to per mil - VTPM }

This function sits on top of the $RADC$ {\dc} determined above.
We look at the pre-conditions for the function $read\_4\_20\_input$  , % which we can call $RI$
to determine its {\fms}.
Its pre-condition is, {\em  /* require: input from ADC to be between 0.88 and 4.4 volts */}.
We can map this violation of the pre-condition, to the {\fm} VRNGE; %As this function has one pre-condition
we can state,

$$ fm(read\_4\_20\_input) = \{ VRNGE \} .$$

We can now form a functional group with the {\dc} $RADC$ and the 
software component $read\_4\_20\_input$, i.e. $G_3 = \{read\_4\_20\_input, RADC\} $.



{
\tiny
\begin{table}[h+]
\caption{$G_3$: Read\_4\_20: Failure Mode Effects Analysis} % title of Table
\label{tbl:r420i}

\begin{tabular}{|| l   | c |   l ||} \hline
% \textbf{Failure}   &  \textbf{failure}     & \textbf{Symptom}          \\ 
% \textbf{Scenario}  &  \textbf{effect}      &     \textbf{RADC }                     \\ \hline 
 \hline 
 \textbf{Failure} &  \textbf{Failure }         & \textbf{Derived Component}          \\ 
 \textbf{cause}   &   \textbf{Effect}                & \textbf{Failure Mode}          \\
  
 
               \hline
    1: $RI_{VRGE}$           &  voltage     &      $OUT\_OF\_$       \\  
                                & outside range    &    $RANGE$             \\ \hline 

     2: $RADC_{VV_ERR}$          &  voltage           &    $VAL\_ERR$          \\ 
                             &  incorrect         &                   \\    \hline  \hline 


 
     3: $RADC_{HIGH}$      &  voltage value     &      $VAL\_ERR$          \\   
                             &  incorrect          &                 \\ \hline 



     4: $RADC_{LOW}$        & ADC  low voltage      &      $OUT\_OF\_$           \\  
                               &  so out of range  &  $RANGE$                 \\ 
                               &  i.e. < 0.88V  &                   \\                          
                               \hline
    
\hline


\hline

\end{tabular}
\end{table} 
}

The failure symptoms for the {\fg} are $\{OUT\_OF\_RANGE, VAL\_ERR\}$.
The postcondition for the function $read\_4\_20\_input$, {\em /* ensure: value is proportional (0-999) to the 
             4 to 20mA input                      */} corresponds to the $VAL\_ERR$ and is already in the set of failure modes.
% \paragraph{Final Functional Group}
For single failures these are the two ways in which this function
can fail. An $OUT\_OF\_RANGE$ will be flagged by the error flag variable.
The $VAL\_ERR$ will simply mean that the value read is incorrect.

We can finally make a {\dc} to represent a failure mode model for our function $read\_4\_20\_input$. %thus:

% $$ R420I = \; \derivec(G_3) .$$

This new {\dc} has the following {\fms}:
$$fm(R420I) = \{OUT\_OF\_RANGE, VAL\_ERR\} .$$

% 
% Using the derived components, CMATV and VTPM we create
% a new functional group. This
% integrates FMEA's from software and eletronics
% into the same failure mode model.



We can now represent the software/hardware FMMD analysis 
as a hierarchical diagram, see figure~\ref{fig:eulerswhw}. % see figure~\ref{fig:hd}.

% HTR 27OCT2012 % \begin{figure}[h]
% HTR 27OCT2012 %  \centering
% HTR 27OCT2012 %  \includegraphics[width=200pt]{./CH5_Examples/hd.png}
% HTR 27OCT2012 %  % hd.png: 363x520 pixel, 72dpi, 12.81x18.34 cm, bb=0 0 363 520
% HTR 27OCT2012 %  \caption{FMMD hierarchy with hardware and software elements}
% HTR 27OCT2012 %  \label{fig:hd}
% HTR 27OCT2012 % \end{figure}

\begin{figure}[h]
 \centering
 \includegraphics[width=300pt]{./CH5_Examples/eulerswhw.png}
 % eulerswhw.png: 510x344 pixel, 72dpi, 17.99x12.14 cm, bb=0 0 510 344
 \caption{Electronics and Software shown in an integrated failure mode
 model---an Euler diagram showing relationship between {\dcs} determined from electronics and software---the two outermost contours are software functions,
 and the inner two are electronic {\dcs}.}
 \label{fig:eulerswhw}
\end{figure}

\subsection{Conclusion: {\ft} Reader Software/Hardware FMMD Model}

The {\dc} representing the {\ft} reader
in software shows that by FMMD, we can integrate
software and electro-mechanical FMMD models.
With this analysis
we have a complete `reasoning~path' linking the failures modes from the 
electronics to those in the software.
Each functional group to {\dc} transition represents a 
reasoning stage. 
%
Each reasoning stage will have an associated analysis report.
%
With traditional FMEA methods the reasoning~distance is large, because
it stretches from the component failure mode to the top---or---system level failure.
For this reason applying traditional FMEA to software stretches
the reasoning distance even further. This is exacerbated by the fact that traditional SFMEA is 
performed separately from HFMEA~\cite{sfmea,sfmeaa}, additionally even the software/hardware 
interfacing is treated as a separate FMEA task~\cite{sfmeainterface,embedsfmea,procsfmea}


We now have a {\dc} for a {\ft} input in software.
Typically, more than one such input could be present in a real-world system.
Not only have we integrated electronics and software in an FMEA, we can also
re-use the analysis for each {\ft} input in the system.

The unsolved symptoms, or unobservable errors, i.e. $VAL\_ERR$ could be addressed
by another software function to read other known signals
via the MUX (i.e. voltage references). This strategy would
detect ADC\_STUCK\_AT and MUX\_FAIL failure modes.

A software specification for a hardware interface will concentrate on
how to interpret raw readings, or what signals to apply for actuators.
Using FMMD we can determine an accurate failure model for the interface as well~\cite{sfmeainterface}.


% HTR == HATE TO REMOVE
%HTR 18NOV2012 We can represent %the hierarchy in figure~\ref{fig:hd} algebraically, 
%HTR 18NOV2012 the analysis hierarchy algebraically using the `$\derivec$' function:
%HTR 18NOV2012 %using the groups as intermediate stages:
%HTR 18NOV2012 \begin{eqnarray*}
%HTR 18NOV2012 G_1 &=&  \{R,ADC\} \\
%HTR 18NOV2012 CMATV &=&  \;\derivec (G_1) \\
%HTR 18NOV2012 G_2 &=& \{CMATV, read\_ADC \} \\
%HTR 18NOV2012 RADC &=&  \; \derivec (G_2) \\
%HTR 18NOV2012 G_3 &=& \{ RADC, read\_4\_20\_input \} \\
%HTR 18NOV2012 R420I &=&  \; \derivec (G_3) \\
%HTR 18NOV2012 \end{eqnarray*}
%HTR 18NOV2012 or, a nested definition,
%HTR 18NOV2012 $$ \derivec \Big( \derivec \big( \derivec(R,ADC), read\_4\_20\_input \big), read\_4\_20\_input \Big). $$ 

 
%\section


%HTR 18NOV2012 This nested structure means that we have multiple traceable
%HTR 18NOV2012 stages of failure mode reasoning in our analysis. Traditional FMEA would have only one stage 
%HTR 18NOV2012 of reasoning for each component failure mode.



\section{Closed Loop Control Hardware/Software Hybrid Example}

It is desirable to model a complete standalone system with FMMD.
Not only a standalone system, but ideally a hybrid software/hardware system.
Temperature control is a first order differential problem, and is often 
addressed using the Proportional Integral differential (PID) algorithm~\cite{dcods}[p.66].
Traditionally this was performed in analogue electronics
with trimmer potentiometers providing the P and I parameters.
Since the introduction of micro-processors, it has been possible to
implement PID programmatic-ally.
An FMMD analysis of a PID temperature controller would mean an
analysis of a standalone system without being un-wieldingly large.
\paragraph{PID Temperature Control.}
PID control starts with a setpoint, or desired value for a process
(here the temperature). It reads the process value and determines an error value for it.
The aim of the PID controller is to minimise this error term, by setting an output value,
which is fed back into the process (in this example the amount of power to supply the heater).
The error value is integrated and multiplied by an I constant.
A differential of the error value is calculated and multiplied by a D constant.
The error value its self is multiplied by a P constant, and all three of these are added
to obtain the output required.
\subsection{Design Stage: Implementation on a micro-controller.}
When designing  a computer program it is often useful to 
produce a structured analysis `Yourdon' context diagram~\cite{Yourdon:1989:MSA:62004}, see figure~\ref{fig:context_diagram_PID}.

\begin{figure}[h]+
 \centering
 \includegraphics[width=300pt]{./CH5_Examples/context_diagram_PID.png}
 % context_diagram_PID.png: 818x324 pixel, 72dpi, 28.86x11.43 cm, bb=0 0 818 324
 \caption{Yourdon Context Diagram for PID Temperature Controller.}
 \label{fig:context_diagram_PID}
\end{figure}
We have two voltage inputs (see section~\ref{sec:Pt100}) from the Pt100 temperature sensor.
For the Pt100 sensor, we will need to read the voltages it outputs and for this
we will need an ADC and MUX. 
%
For the output, we can use a Pulse Width Modulator (PWM) (this is a common module found on micro-controllers
allowing a variable power output~\cite{pwm}). PWM's ADC's and MUX's are commonly built into cheap micro-controllers~\cite{pic18f2523}.
We can now build more detail into the Yourdon diagram, with the afferent data flow coming through the MUX and ADC on the micro-controller, and the efferent 
channelled through a PWM module, %again built into the micro-controller, 
see figure~\ref{fig:context_diagram2_PID}.
\begin{figure}[h]+
 \centering
 \includegraphics[width=300pt]{./CH5_Examples/context_diagram2_PID.png}
 % context_diagram_PID.png: 818x324 pixel, 72dpi, 28.86x11.43 cm, bb=0 0 818 324
 \caption{Yourdon Context Diagram for PID Temperature Controller.}
 \label{fig:context_diagram2_PID}
\end{figure}
The Yourdon methodology allows us to zoom into data transform bubbles and analyse them in more
detail.
%
We define the controlling software, by looking at or zooming into its transform bubble.
We have the inputs and outputs from the software.
We refine the data flow within the software and thus define software functions.
%, and
%this in terms of software functions.
%
We follow the data streams through the process, creating transform bubbles as required.
In all `bare~metal' software architectures, we need a rudimentary operating system, often referred to as the monitor.
%
We bare in mind that PID, because the algorithm depends heavily on integral calculus, is time sensitive
and we therefore need to call at precise intervals specific to its integration and differential coefficients.
%
Most micro-controllers feature several general purpose timers~\cite{pic18f2523}.
We can use an internal timer in conjunction with the monitor function
to call the PID algorithm at a specified interval.
%
\paragraph{Data flow model to programmatic call tree.}
The Yourdon methodology also gives us a guide as to which software
functions should be called to control the process, or in `C' terms be the main function.
%
\begin{figure}[h]
 \centering
 \includegraphics[width=300pt]{./CH5_Examples/context_software.png}
 % context_software.png: 1023x500 pixel, 72dpi, 36.09x17.64 cm, bb=0 0 1023 500
 \caption{Context diagram of the software in the PID temperature controller}
 \label{fig:contextsoftware}
\end{figure}
Using figure~\ref{fig:contextsoftware} we can now pick the transform bubble we 
want to be the `main' or controlling function in the software.
This can be thought of as picking one bubble and holding it up. The other bubbles hang underneath
forming the software call tree hierarchy, see figure~\ref{fig:context_calltree}.
From is clearly going to be the monitor function.
\begin{figure}[h]+
 \centering
 \includegraphics[width=300pt]{./CH5_Examples/context_calltree.png}
 % context_calltree.png: 800x783 pixel, 72dpi, 28.22x27.62 cm, bb=0 0 800 783
 \caption{Software yourdon diagram converted to programatic call tree.}
 \label{fig:context_calltree}
\end{figure}
%

\paragraph{Software Algorithm.}
The monitor function will orchestrate the control process.
Firstly it will examine the timer value, and when appropriate, call the PID function, which will call first
the determine\_set\_point\_error function with that calling convert\_ADC\_to\_T
which calls Read\_ADC (the function developed in the earlier example).
%
With the set point error value the PID function will return 
 output control value to its calling
function (i.e. the PID 
demand which will be returned to the monitor function).
% 
On returning to the monitor function, it will return the PID demand value.
The PID demand value will be applied via the PWM.
We now have a rudimentary closed loop control system incorporating both hardware and software.
%
Using the Yourdon methodology we have the system design: we have all the components, i.e. hardware elements and software functions
that will be used in the temperature controller.
We list these, and begin, from the bottom-up, to apply FMMD analysis.

\clearpage
\subsection{FMMD Analysis of PID temperature Controller}

To summarise from the design stage,
Identified electronic components:
\begin{itemize}
 \item ADCMUX --- Electronics, analysed in previous example.
 \item TIMER --- Internal micro controller timer
 \item HEATER --- Heating element, essentially a resistor.
 \item Pt100 --- Pt100 Temperature sensor, as analysed in section~\ref{sec:Pt100}.
 \item PWM --- Internal micro controller pulse width modulation module
 \item micro-controller --- the medium for running the software
 \item setLEDs --- Set status indication LEDs via GPIO
\end{itemize}






\subsection{Temperature Controller Hardware Elements FMMD}

\paragraph{ACDMUX and Read\_ADC}
We re-use this derived component from section~\ref{readADC}.
$$ fm(RADC) = \{ VV\_ERR, HIGH, LOW \} .$$


\paragraph{TIMER}
The internal timer in use is a register which when read 
returns an incremented time value.
Using two's complement mathematics, by subtracting
the time we last read it, we can calculate the interval
between readings (assuming the timer has not completely wrapped around).
We can say that a timer can fail by
incrementing its value at an incorrect rate, or can stop incrementing.
$$ fm(TIMER) = \{ STOPPED, INCORRECT\_INTERVAL \}$$

\paragraph{HEATER}
A heating element is typically some configuration of resistive wire.
It therefore has the same failure modes as a resistor and we can state
$$fm(HEATER) = \{ OPEN, SHORT \}$$

\paragraph{Pt100 Platinum Temperature Sensor}
The Pt100 four wire configuration is analysed in section~\ref{sec:Pt100}
$$ fm(Pt100) = \{ OUT\_OF\_RANGE \} $$


\paragraph{PWM}
The PWM, in use, is a hardware register written to with an integer value.
It then applies a mark space ratio proportional to that value providing
a means of applying varying amounts of power. When the PWM
action is halted the digital output pin associated with it will typically be held in a high or low state.
We therefore state:
$$ fm(PWM) = \{ HIGH, LOW \}.$$

\paragraph{Micro-Controller}
The Micro controller is a complex piece of highly integrated electronics.
Typically, along with a micro-processor with PROM and RAM, they have many I/O modules including UARTS, PWM, ADCMUX, CAN
General I/O and interrupt lines to name but a few.
In this project we are using the ADCMUX, TIMER, PWM and the general purpose computing facilities.
We have to therefore consider the general~computing, CLOCK, PROM and RAM failure modes.
$$fm (micro-controller) =\{ PROM\_FAULT, RAM\_FAULT, CPU\_FAULT, ALU\_FAULT, CLOCK\_STOPPED \}.$$

\subsection{Temperature Controller Software Elements FMMD}
Identified Software Components:
\begin{itemize}
 \item --- Monitor (which calls PID algorithm and sets status LEDS)
 \item --- PID (which calls determine\_set\_point\_error and output\_control)
 \item --- determine\_set\_point\_error (which calls convert\_ADC\_to\_T)
 \item --- convert\_ADC\_to\_T (which calls read\_ADC which we can re-use from the last example)
 \item --- read\_ADC
 \item --- output\_control (which sets the PWM hardware according to the PID demand value)
\end{itemize}
With the call tree structure defined (see figure~\ref{fig:context_calltree}), we can now analyse these 
components from the bottom-up, starting with the afferent flow, the reading in of the temperature and its conversion
to a PID calculated heater output demand.

\subsubsection{Afferent flow FMMD analysis , Pt100, temperature, set point error, PID output demand.}
We start with the afferent flow from the Pt100.
%with the software, and consider the hardware elements
%used (if any) by each software function.
Starting at the bottom we form a {\fg} with
the function read\_ADC and the Pt100.
This gives us a {\dc} we shall call ReadPt100.

{
\tiny
\begin{table}[h+]
\caption{ Read\_Pt100: Failure Mode Effects Analysis} % title of Table
\label{tbl:readPt100}

\begin{tabular}{|| l   | c |   l ||} \hline
% \textbf{Failure}   &  \textbf{failure}     & \textbf{Symptom}          \\ 
% \textbf{Scenario}  &  \textbf{effect}      &     \textbf{RADC }                     \\ \hline 
 \hline 
 \textbf{Failure} &  \textbf{Failure }         & \textbf{Derived Component}          \\ 
 \textbf{cause}   &   \textbf{Effect}                & \textbf{Failure Mode}          \\
  
 
               \hline
     FC1: $RI_{VRGE}$           &  voltage     &      $VOLTAGE\_HIGH$       \\  
                                & outside range    &                 \\ \hline 

     FC2: $RADC_{VV_ERR}$          &  voltage           &    $VAL\_ERR$          \\ 
                             &  incorrect         &                   \\    \hline  \hline 


 
     FC3: $RADC_{HIGH}$      &  voltage value     &      $VAL\_ERR$          \\   
                             &  incorrect          &                 \\ \hline 



     FC4: $RADC_{LOW}$        & ADC may read      &      $VOLTAGE\_LOW$           \\  \hline
\end{tabular}
\end{table}
}
                 
The {\dc} Read\_Pt100 is a failure mode model of the Read\_ADC function and the Pt100
hardware, and has the following failure modes:

$$ fm (Read\_Pt100) = \{ VOLTAGE\_HIGH, VAL\_ERR, VOLTAGE\_LOW \}. $$


We can now move along in the afferent flow, and we come to the convert\_ADC\_to\_T function.
This will call Read\_ADC thwice, one for the high Pt100 value, again for the lower. % and once for to read a current sense.
We  then, calculate the resistance of the Pt100 element, and with this---using a 
polynomial or a lookup table~\cite{eutothermtables}---and calculate the temperature.
The pre-conditions for the function are that:
\begin{itemize}
% \item The current calculated is within pre-defined bounds i.e. Pt100\_current,
 \item The lower Pt100 value is within an acceptable voltage range i.e. Pt100\_lower\_voltage,
 \item The higher Pt100 value is within an acceptable voltage range i.e. Pt100\_higher\_voltage,
 \item The Lower and higher values agree to within a given tolerance i.e. Pt100\_high\_low\_mismatch.
\end{itemize}
Any violation of these pre-conditions is equivalent to a failure mode.
Note that a temperature outside the pre-defined range will also cause these errors.
The postcondition is that it returns a temperature within a given tolerance to the temperature at the sensor.
A failure of this post-condition can be termed temp\_incorrect.
\clearpage
We now apply FMMD to the {\fg} formed by Read\_Pt100 and the function convert\_ADC\_to\_T.
We can call the resulting {\dc} Get\_Temperature.

{
\tiny
\begin{table}[h+]
\caption{ Get\_Temperature: Failure Mode Effects Analysis} % title of Table
\label{tbl:gettemperature}

\begin{tabular}{|| l   | c |   l ||} \hline
% \textbf{Failure}   &  \textbf{failure}     & \textbf{Symptom}          \\ 
% \textbf{Scenario}  &  \textbf{effect}      &     \textbf{RADC }                     \\ \hline 
 \hline 
 \textbf{Failure} &  \textbf{Failure }         & \textbf{Derived Component}          \\ 
 \textbf{cause}   &   \textbf{Effect}                & \textbf{Failure Mode}          \\
  
 
               \hline
     FC1: $Pt100:Voltage\_High$           &  Pt100 voltage too high     & Pt100\_out\_of\_range            \\  
                                         & Pt100\_higher\_voltage    &                 \\ 
                                         & OR Pt100\_current         &    \\ \hline
                                         
       FC2: $Pt100:Voltage\_Low$           &  Pt100 voltage too low     & Pt100\_out\_of\_range            \\  
                                         & Pt100\_lower\_voltage    &                 \\ 
                                         & OR Pt100\_current         &        \\ \hline     
                                         
                                         
                                         
                                            
       FC3: $Pt100\_high\_low\_mismatch$ & temperature can be calculated     & Pt100\_out\_of\_range            \\  
                                         & from either high or low           &                 \\ 
                                         & reading, but should correlate     &        \\ \hline                                       
                                         
                                  
%        FC4: $Pt100\_current$             & the current applied is     & Pt100\_out\_of\_range            \\  
%                                          & necessary to calculate resistance,        &                 \\ 
%                                          &  but should be within given bounds     &        \\ \hline                                       
%                                          
%                                                                   
                                         
       FC4: $Pt100:VAL\_ERR$              & could cause an out of     & temp\_incorrect\\
                                          & range error, but may also & \\
                                          & cause us to read an       &       \\
                                          & incorrect temperature     &    \\ \hline
                                         \hline
     
\end{tabular}
\end{table}
}


We now collect the failure symptoms for the {\dc} Get\_Temperature and can state:

$$fm(Get\_Temperature) = \{  Pt100\_out\_of\_range, temp\_incorrect \}$$
\clearpage

Following the afferent flow further, we come to a function to determine the control error value.
The is simply the target temperature subtracted from the measured.
We thus form a {\fg} with our newly {\dc} Get\_Temperature
and the function determine\_set\_point\_error.

The pre-condition for determine\_set\_point\_error is that the temperature read by it
is accurate, and its post condition is to return the correct control error value.
Most failure modes from a Pt100 are observable.
we can divide the post condition into two variants, a known incorrect error value, KnownIncorrectErrorValue
where we can detect the Pt100 value is suspect, and IncorrectErrorValue where we simply have
an incorrect error value.


{
\tiny
\begin{table}[h+]
\caption{ GetError: Failure Mode Effects Analysis} % title of Table
\label{tbl:geterror}

\begin{tabular}{|| l   | c |   l ||} \hline
% \textbf{Failure}   &  \textbf{failure}     & \textbf{Symptom}          \\ 
% \textbf{Scenario}  &  \textbf{effect}      &     \textbf{RADC }                     \\ \hline 
 \hline 
 \textbf{Failure} &  \textbf{Failure }         & \textbf{Derived Component}          \\ 
 \textbf{cause}   &   \textbf{Effect}                & \textbf{Failure Mode}          \\
  
 
               \hline
     FC1: $ Pt100\_out\_of\_range $      &   pre-condition violated        & KnownIncorrectErrorValue          \\  
                                         &   observable/detectable        &                 \\ 
                                         &   failure mode        &    \\ \hline
                                         
       FC2: $temp\_incorrect$           &  pre-condition violated          &  IncorrectErrorValue               \\  
                                        &  unobservable                    &                 \\ 
                                        &  undetectable failure mode          &        \\ \hline     
                                         
                                         
     
\end{tabular}
\end{table}
}


We collect failure mode symptoms, and can create a new {\dc} GetError
where
$$fm(GetError) = \{ KnownIncorrectErrorValue, IncorrectErrorValue \}.$$


We now follow the afferent path to the PID algorithm.
Here we assume that the PID constants are fixed (i.e. are not parameters).
We use the $GetError$ {\dc} and the PID function to form a {\fg}.
The pre-condition for the PID function is that % are that it is called 
%iat the correct frequency and that 
it receives the correct error value.
The post-condition is that it outputs correct control values.
% RESP FOR TIMEING IS ON CALLING FUNCTION AND IS A SEPARATE ERROR- TGHINK ABOUT JITTER.....
% and controll values..... Jitter might not matter, wrong int times would
% controlling function provdes context of use.
Those familiar with the PID algorithm may here notice raise the point of calling frequency.
were this function to be called at an incorrect rate its output
would be wrong (the differential and integral parameters would effectively have been changed).
However this problem  is a failure mode for the function calling it.
The calling function sets the context for the PID algorithm (i.e. what it is used for).
If this PID were to be used, say as some form of low pass filter, we could consider jitter
for instance. In a control environment with PID jitter would not be a significant factor.
This harks back to the context of use (see section~\ref{sec:subjectiveobjective}) discussion, the subjective
being the context the {\dc} is used for/in, and the objective
being the logic and process  of the failure mode analysis. 

{
\tiny
\begin{table}[h+]
\caption{ PID: Failure Mode Effects Analysis} % title of Table
\label{tbl:pidfunction}

\begin{tabular}{|| l   | c |   l ||} \hline
% \textbf{Failure}   &  \textbf{failure}     & \textbf{Symptom}          \\ 
% \textbf{Scenario}  &  \textbf{effect}      &     \textbf{RADC }                     \\ \hline 
 \hline 
 \textbf{Failure} &  \textbf{Failure }         & \textbf{Derived Component}          \\ 
 \textbf{cause}   &   \textbf{Effect}                & \textbf{Failure Mode}          \\
  
 
               \hline
     FC1: $ KnownIncorrectErrorValue $      &   pre-condition violated        & KnownControlValueErrorV         \\  
                                         &   observable/detectable        &                 \\ 
                                         &   failure mode        &    \\ \hline
                                         
       FC2: $ IncorrectErrorValue $           &  pre-condition violated          &  IncorrectControlErrorV               \\  
                                        &  unobservable                    &                 \\ 
                                        &  undetectable failure mode          &        \\ \hline     
                                         
                                         
     
\end{tabular}
\end{table}
}

We now create a PID {\dc}, with the following failure modes:

$$ fm(PID) = \{  KnownControlValueErrorV, IncorrectControlErrorV \} .$$

\begin{figure}[h]
 \centering
 \includegraphics[width=400pt]{./CH5_Examples/euler_afferent_PID.png}
 % euler_afferent_PID.png: 1002x342 pixel, 72dpi, 35.35x12.06 cm, bb=0 0 1002 342
 \caption{Euler diagram representing the hierarchy of FMMD analysis applied to the afferent branch of call tree for the PID temperature controller example.}
 \label{fig:euler_afferent_PID}
\end{figure}



We have now modelled the the software call tree for the afferent  flow, we represent this as an Euler diagram in figure~\ref{fig:euler_afferent_PID}.
Two call tree branches remain. The LED indication branch and the 
PWM/heater output.

\subsubsection{Efferent flow, PID demand value to PWM output}

The monitor function calls the output\_control function with the PID demand.
The output\_control function then sets the PWM hardware register, which causes the mark space output of the PWM module to
apply the demanded power. We form a {\fg} with the  Heating element, a PWM module and the output\_control function to model this branch
of the efferent flow. We apply FMMD analysis to this {\fg} in table~\ref{tbl:heateroutput}.
For the output\_control function, we have a pre-condition that the PWM module is 
configured and working, and has the correct clock frequency.
 A second pre-condition is that the heating element is connected and working.
The post condition is that is sets the correct value into the PWM register
to implement the PWM demand.

{
\tiny
\begin{table}[h+]
\caption{ HeaterOutput: Failure Mode Effects Analysis} % title of Table
\label{tbl:heateroutput}

\begin{tabular}{|| l   | c |   l ||} \hline
% \textbf{Failure}   &  \textbf{failure}     & \textbf{Symptom}          \\ 
% \textbf{Scenario}  &  \textbf{effect}      &     \textbf{RADC }                     \\ \hline 
 \hline 
 \textbf{Failure} &  \textbf{Failure }         & \textbf{Derived Component}          \\ 
 \textbf{cause}   &   \textbf{Effect}                & \textbf{Failure Mode}          \\
  
 
               \hline
     FC1: $ PWM stuck HIGH $      &   pre-condition violated        & HeaterOnFull         \\  
                                         &  PWM module not working        &                 \\  \hline    
                                       
                                         
       FC2: $ PWM stuck LOW $           &  pre-condition violated          &  HeaterOff               \\  
                                        &  PWM module not working                   &                 \\  \hline    
                                        
          FC3: $ output\_control$ wrong value            & The software supplies the wrong          &  HeaterOutputIncorrect               \\  
                                        &  value to the PWM register                   &                 \\  \hline 
                                        
                                        
      FC4:  HEATER $SHORT$               &  heating element resistor         & HeaterOff        \\  
                                         &  SHORT no heating effect      &                 \\  \hline    
                                       
                                         
       FC5:  HEATER $OPEN  $           &  heating element resistor          &  HeaterOff               \\  
                                        & OPEN no heating effect                  &                 \\  \hline                             
     
\end{tabular}
\end{table}
}

We now create a {\dc} called HeaterOutput
with the following failure modes:
$$fm(HeaterOutput) = \{ HeaterOnFull, HeaterOff, HeaterOutputIncorrect \}$$



\begin{figure}[h]
 \centering
 \includegraphics[width=300pt]{./CH5_Examples/euler_heater_output.png}
 % euler_heater_output.png: 392x141 pixel, 72dpi, 13.83x4.97 cm, bb=0 0 392 141
 \caption{Euler diagram showing HeaterOutput with its two hardware components, PWM and HEATER, and its software component output\_control.}
 \label{fig:eulerheateroutput}
\end{figure}



























\subsubsection{Efferent flow: LED status LEDs}

The status LEDS will be controlled by general purpose (GPIO) I/O pins.
We could have say, three LEDS one flashing with a human readable mark 
space ratio representing the heater output, one flashing at a regular interval to
indicate the processor is alive and another flashing at an interval related to the temperature,
(to indicate if the temperature readings are within expected ranges).
Each LED should flash in normal operation, and any LED being permanently on or off
would indicate to the operator that an error had occurred.
The pre condition for this function is that the GPIO
is connected to working LEDS.
The post condition is that the function setLEDS,  will supply correct indication by flashing the LEDs.
We form a {\fg} from the GPIO, the LEDs and the software function setLEDs.
We apply FMMD analysis to this {\fg} in table~\ref{tbl:ledoutput}.

{
\tiny
\begin{table}[h+]
\caption{ LEDOutput: Failure Mode Effects Analysis} % title of Table
\label{tbl:ledoutput}

\begin{tabular}{|| l   | c |   l ||} \hline
% \textbf{Failure}   &  \textbf{failure}     & \textbf{Symptom}          \\ 
% \textbf{Scenario}  &  \textbf{effect}      &     \textbf{RADC }                     \\ \hline 
 \hline 
 \textbf{Failure} &  \textbf{Failure }         & \textbf{Derived Component}          \\ 
 \textbf{cause}   &   \textbf{Effect}                & \textbf{Failure Mode}          \\
  
 
               \hline
     FC1: $ Temp LED fails $      &   LED will not light       & FailureIndicated         \\  
                                  &                            &                 \\  \hline    
                                       
                                         
       FC2: $ Processor LED fails $           &   LED will not light          &  FailureIndicated               \\  
                                              &                               &                 \\  \hline    
                                        
          FC3: $ PWM LED fails $           &  LED will not light         &  FailureIndicated               \\  
                                          &                              &                 \\  \hline
                                          
           FC4:  GPIO stuck HIGH & LED permanently OFF &   FailureIndicated    \\ \hline
           
 
           FC5: GPIO stuck Low  & LED permanently ON &   FailureIndicated    \\ \hline
           
           
            FC6:  Software SetLEDs                 &  Incorrect Indication        &  IndicationError               \\  
                  fails to set outputs correctly   &   Post condition failure     &                 \\  \hline                                             
  
                                        
                                        
\end{tabular}
\end{table}
}




\begin{figure}[h]
 \centering
 \includegraphics[width=300pt]{./CH5_Examples/euler_led_output.png}
 % euler_heater_output.png: 392x141 pixel, 72dpi, 13.83x4.97 cm, bb=0 0 392 141
 \caption{Euler diagram showing LEDOutput with its three LEDs and GPIO hardware elements, and its 
 and its software component setLEDS.}
 \label{fig:eulerheateroutput}
\end{figure}





















%OK STOP AT PID and follow the other data flows until we are ready to bring them to the top: i.e. 
%
%the monitor program.......



%\clearpage

