 


\section{Software and Hardware Failure Mode Concepts}
\label{sec:elecsw}
FMMD  can be applied to software, and thus we can 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 placed into a hierarchy, that of its call-tree.
A software function typically 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 functions that may call it.
%
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 can thus apply the $\derivec$ process to software functions, by viewing them in terms of their failure
mode behaviour. To simplify things as well, software already fits into a hierarchy.
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, that hierarchy is fixed.

%                    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 failure modes.}

A precondition, or requirement for a contract software function
defines the correct ranges of input conditions for the function
to operate successfully.
%
For a software function, a violation of a pre-condition is 
in effect a failure mode of `one of its components'.


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

A post condition is a definition of correct behaviour by a function.
A violated post condition is a symptom of failure, or derived failure mode, of a function.
Post conditions could  be either  actions performed (i.e. the state of hardware changed) or an output value of a function.

\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.

\subsection{Simple Software Example}


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 
 

\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}

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     
    
\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}:

$$ fm(RADC) = \{ VV\_ERR, HIGH, LOW \} .$$





\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 may read      &      $OUT\_OF\_$           \\  
                               &   wrong channel  &  $RANGE$                 \\ \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}



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

 



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

%\clearpage
\subsection{Conclusion: 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}.
%
%Detailing this however,  is beyond the scope %and page-count 
%of this paper. 



%Its solved. Hoooo-ray !!!!!!!!!!!!!!!!!!!!!!!!






\vspace{20pt}

%typeset in {\Huge \LaTeX} \today
