PLAT: An Automated Fault and Behavioural Anomaly Detection Tool for PLC Controlled Manufacturing Systems

Algorithm 1

Transition time clustering algorithm (CS-means algorithm with HAC).

() Inputs: Time_Array⁢⁢, ⁢,

   % The transition time instances are stored in Time_Array⁢; (>1) is the

   initial number of clusters (set by the user and will be the same for all the

   transitions); and is the total number of transition time instances %

() Outputs: Clus_Avg⁢ , Clus_Max⁢⁢, Clus_Prob⁢⁢,

   % Clus_Avg⁢ stores the mean values of the clusters (average transition

   times); Clus_Max⁢ stores the highest-valued instance of each cluster

   (maximum transition times); Clus_Prob⁢ stores the support of each cluster

   (transition time probabilities); and is the final number of clusters %

() declare and initialize Clus_Avg, Clus_Max and Clus_Prob

   % all the arrays are filled with zeros %

() call function  Introsort(Time_Array)    % apply Introsort algorithm to sort Time_Array⁢  %

   % code for single cluster %

() if (Time_Array⁢  −  Time_Array⁢) < (user-defined) threshold  value

   then

()   for   to   do

()     Clus_Avg⁢ = Clus_Avg⁢ + Time_Array⁢;

    end for

()   if  Clus_Avg⁢  then

()     Clus_Avg⁢ = Clus_Avg⁢; Clus_Max⁢ = Time_Array⁢;

()   Clus_Prob⁢; ;

   % code for (probable) multiple clusters %

() else

()   declare and initialize Gap_Val[], Pos and

   % Gap_Val⁢ and Pos⁢ are filled with −1’s; Gap_Val⁢ keeps track of

   the () largest “GAP”s in the sorted number line (or Time_Array⁢);

   Pos⁢ stores the corresponding array positions (ending points of clusters or “breaks”);

   and (>0) is a small integer number (user-defined) %

()   for   to   do

()     var gap = (Time_Array⁢  −  Time_Array⁢/(Time_Array⁢;

   % calculate GAP between two instances; var implies Temporary Variable %

()     var min = Gap_Val⁢; var minIndex = 0;

()     for   to   do

()       if  Gap_Val⁢ < min  then

()         min = Gap_Val⁢; minIndex = ;

      end for

()     if  gap > min  then

()       Gap_Val⁢minIndex⁢ = gap; Pos⁢minIndex⁢ = ;

     end for

()   call function  Introsort(Pos)  % apply Introsort() to sort Pos⁢  %

   % split the array into partitions (based on sorted () positions in Pos⁢);

   then, calculate the mean of each partition and store them in Clus_Avg⁢  %

()   for   to   do

()     var temp1 = −1; var temp2 = −1;

()     if    then

()      temp1 = 0; temp2 = Pos⁢;

()     else if    then

()      temp1 = Pos⁢; temp2 = Pos⁢;

()     else

()      temp1 = Pos⁢; temp2 = ;

()     for   = temp1 to temp2  do

()       Clus_Avg⁢ = Clus_Avg⁢ + Time_Array⁢;

      end for

()     if  Clus_Avg⁢  then

()      Clus_Avg⁢ = Clus_Avg⁢/(temp2 − temp1 + 1);

     end for

   % pass the newly calculated mean values (stored in Clus_Avg⁢) as the

   initial cluster centers to the (traditional) -means algorithm %

()   call function  ⁢_Means_Algorithm(Time_Array, , , Clus_Avg, Pos);

     % Apply HAC algorithm to merge the clusters whose centers are not far away

   (in a user-defined way) from each other %

()   call function  HAC_Algorithm(Time_Array, , , Pos, Clus_Avg, Clus_Max, Clus_Prob);

   % the average and maximum transition times of the clusters and their corresponding

   probabilities can be found in Clus_Avg⁢, Clus_Max⁢ and

   Clus_Prob⁢, respectively from 0th to (⁢)⁢th  ⁢position⁢  %

() Save the clustering results into the file and Exit the procedure