Multi-Class Fault Detection Using Simulated Data

This example shows how to use a Simulink model to generate fault and healthy data. The data is used to develop a multi-class classifier to detect different combinations of faults. The example uses a triplex reciprocating pump model and includes leak, blocking, and bearing faults.

Setup the Model

This example uses many supporting files that are stored in a zip file. Unzip the file to get access to the supporting files, load the model parameters, and create the reciprocating pump library.

if ~exist('+mech_hydro_forcesPS','dir')
    unzip('pdmRecipPump_supportingfiles.zip')
end

% Load Parameters
pdmRecipPump_Parameters %Pump
CAT_Pump_1051_DataFile_imported %CAD

% Create Simscape library if needed
if exist('mech_hydro_forcesPS_Lib','file')~=4
    ssc_build mech_hydro_forcesPS
end

Reciprocating Pump Model

The reciprocating pump consists of an electric motor, the pump housing, pump crank and pump plungers.

mdl = 'pdmRecipPump';
open_system(mdl)

open_system([mdl,'/Pump'])

The pump model is configured to model three types of faults; cylinder leaks, blocked inlet, and increased bearing friction. These faults are parameterized as workspace variables and configured through the pump block dialog.

Simulating Fault and Healthy Data

For each of the three fault types create an array of values that represent the fault severity, ranging from no fault to a significant fault.

% Define fault parameter variations
numParValues = 10;
leak_area_set_factor = linspace(0.00,0.036,numParValues);
leak_area_set = leak_area_set_factor*TRP_Par.Check_Valve.In.Max_Area;
leak_area_set = max(leak_area_set,1e-9); % Leakage area cannot be 0
blockinfactor_set = linspace(0.8,0.53,numParValues);
bearingfactor_set = linspace(0,6e-4,numParValues);

The pump model is configured to include noise, thus running the model with the same fault parameter values will result in different simulation outputs. This is useful for developing a classifier as it means there can be multiple simulation results for the same fault condition and severity. To configure simulations for such results, create vectors of fault parameter values where the values represent no faults, a single fault, combinations of two faults, and combinations of three faults. For each group (no fault, single fault, etc.) create 125 combinations of fault values from the fault parameter values defined above. This gives a total of 1000 combinations of fault parameter values.

nPerGroup = 125; % Number of elements in each fault group
    
% No fault simulations
leakArea = repmat(leak_area_set(1),nPerGroup,1);
blockingFactor = repmat(blockinfactor_set(1),nPerGroup,1);
bearingFactor = repmat(bearingfactor_set(1),nPerGroup,1);

% Single fault simulations
idx = ceil(10*rand(nPerGroup,1));
leakArea = [leakArea; leak_area_set(idx)'];
blockingFactor = [blockingFactor;repmat(blockinfactor_set(1),nPerGroup,1)];
bearingFactor = [bearingFactor;repmat(bearingfactor_set(1),nPerGroup,1)];
idx = ceil(10*rand(nPerGroup,1));
leakArea = [leakArea; repmat(leak_area_set(1),nPerGroup,1)];
blockingFactor = [blockingFactor;blockinfactor_set(idx)'];
bearingFactor = [bearingFactor;repmat(bearingfactor_set(1),nPerGroup,1)];
idx = ceil(10*rand(nPerGroup,1));
leakArea = [leakArea; repmat(leak_area_set(1),nPerGroup,1)];
blockingFactor = [blockingFactor;repmat(blockinfactor_set(1),nPerGroup,1)];
bearingFactor = [bearingFactor;bearingfactor_set(idx)'];

% Double fault simulations
idxA = ceil(10*rand(nPerGroup,1));
idxB = ceil(10*rand(nPerGroup,1));
leakArea = [leakArea; leak_area_set(idxA)'];
blockingFactor = [blockingFactor;blockinfactor_set(idxB)'];
bearingFactor = [bearingFactor;repmat(bearingfactor_set(1),nPerGroup,1)];
idxA = ceil(10*rand(nPerGroup,1));
idxB = ceil(10*rand(nPerGroup,1));
leakArea = [leakArea; leak_area_set(idxA)'];
blockingFactor = [blockingFactor;repmat(blockinfactor_set(1),nPerGroup,1)];
bearingFactor = [bearingFactor;bearingfactor_set(idxB)'];
idxA = ceil(10*rand(nPerGroup,1));
idxB = ceil(10*rand(nPerGroup,1));
leakArea = [leakArea; repmat(leak_area_set(1),nPerGroup,1)];
blockingFactor = [blockingFactor;blockinfactor_set(idxA)'];
bearingFactor = [bearingFactor;bearingfactor_set(idxB)'];

% Triple fault simulations
idxA = ceil(10*rand(nPerGroup,1));
idxB = ceil(10*rand(nPerGroup,1));
idxC = ceil(10*rand(nPerGroup,1));
leakArea = [leakArea; leak_area_set(idxA)'];
blockingFactor = [blockingFactor;blockinfactor_set(idxB)'];
bearingFactor = [bearingFactor;bearingfactor_set(idxC)'];

Use the fault parameter combinations to create Simulink.SimulationInput objects. For each simulation input ensure that the random seed is set differently to generate different results.

for ct = numel(leakArea):-1:1
    simInput(ct) = Simulink.SimulationInput(mdl);
    simInput(ct) = setVariable(simInput(ct),'leak_cyl_area_WKSP',leakArea(ct));
    simInput(ct) = setVariable(simInput(ct),'block_in_factor_WKSP',blockingFactor(ct));
    simInput(ct) = setVariable(simInput(ct),'bearing_fault_frict_WKSP',bearingFactor(ct));
    simInput(ct) = setVariable(simInput(ct),'noise_seed_offset_WKSP',ct-1);
end

Use the generateSimulationEnsemble function to run the simulations defined by the Simulink.SimulationInput objects defined above and store the results in a local sub-folder. Then create a simulationEnsembleDatastore from the stored results.

Note that running these 1000 simulations in parallel takes around an hour on a standard desktop and generates around 620MB of data. An option to only run the first 10 simulations is provided for convenience.

% Run the simulation and create an ensemble to manage the simulation results
runAll = true;
if runAll
    [ok,e] = generateSimulationEnsemble(simInput,fullfile('.','Data'),'UseParallel',true);
else
    [ok,e] = generateSimulationEnsemble(simInput(1:10),fullfile('.','Data')); %#ok<UNRCH>
end

[09-Apr-2018 09:01:38] Checking for availability of parallel pool...
[09-Apr-2018 09:01:38] Loading Simulink on parallel workers...
Analyzing and transferring files to the workers ...done.
[09-Apr-2018 09:01:38] Configuring simulation cache folder on parallel workers...
[09-Apr-2018 09:01:38] Running SetupFcn on parallel workers...
[09-Apr-2018 09:01:39] Loading model on parallel workers...
[09-Apr-2018 09:01:39] Transferring base workspace variables used in the model to parallel workers...
[09-Apr-2018 09:01:41] Running simulations...
[09-Apr-2018 09:02:28] Completed 1 of 1000 simulation runs
[09-Apr-2018 09:02:33] Completed 2 of 1000 simulation runs
[09-Apr-2018 09:02:37] Completed 3 of 1000 simulation runs
[09-Apr-2018 09:02:41] Completed 4 of 1000 simulation runs
[09-Apr-2018 09:02:46] Completed 5 of 1000 simulation runs
[09-Apr-2018 09:02:49] Completed 6 of 1000 simulation runs
[09-Apr-2018 09:02:54] Completed 7 of 1000 simulation runs
[09-Apr-2018 09:02:58] Completed 8 of 1000 simulation runs
[09-Apr-2018 09:03:01] Completed 9 of 1000 simulation runs...

ens = simulationEnsembleDatastore(fullfile('.','Data'));

Processing and Extracting Features from the Simulation Results

The model is configured to log the pump output pressure, output flow, motor speed and motor current.

ens.DataVariables

ans = 8×1 string array
    "SimulationInput"
    "SimulationMetadata"
    "iMotor_meas"
    "pIn_meas"
    "pOut_meas"
    "qIn_meas"
    "qOut_meas"
    "wMotor_meas"

For each member in the ensemble preprocess the pump output flow and compute condition indicators based on the pump output flow. The condition indicators are later used for fault classification. For preprocessing remove the first 0.8 seconds of the output flow as this contains transients from simulation and pump startup. As part of the preprocessing compute the power spectrum of the output flow, and use the SimulationInput to return the values of the fault variables.

Configure the ensemble so that the read only returns the variables of interest and call the preprocess function that is defined at the end of this example.

ens.SelectedVariables = ["qOut_meas", "SimulationInput"];
reset(ens)
data = read(ens)

data=1×2 table
        qOut_meas                SimulationInput        
    __________________    ______________________________

    [2001×1 timetable]    [1×1 Simulink.SimulationInput]

[flow,flowP,flowF,faultValues] = preprocess(data);

Plot the flow and flow spectrum. The plotted data is for a fault free condition.

% Figure with nominal
subplot(211);
plot(flow.Time,flow.Data);
subplot(212);
semilogx(flowF,pow2db(flowP));
xlabel('Hz')

The flow spectrum reveals resonant peaks at expected frequencies. Specifically, the pump motor speed is 950 rpm, or 15.833 Hz, and since the pump has 3 cylinders the flow is expected to have a fundamental at 3*15.833 Hz, or 47.5 Hz, as well as harmonics at multiples of 47.5 Hz. The flow spectrum clearly shows the expected resonant peaks. Faults in one cylinder of the pump will result in resonances at the pump motor speed, 15.833 Hz and its harmonics.

The flow spectrum and slow signal gives some ideas of possible condition indicators. Specifically, common signal statistics such as mean, variance, etc. as well as spectrum characteristics. Spectrum condition indicators relating to the expected harmonics such as the frequency with the peak magnitude, energy around 15.833 Hz, energy around 47.5 Hz, energy above 100 Hz, are computed. The frequency of the spectral kurtosis peak is also computed.

Configure the ensemble with data variables for the condition indicators and condition variables for fault variable values. Then call the extractCI function to compute the features, and use the writeToLastMemberRead command to add the feature and fault variable values to the ensemble. The extractCI function is defined at the end of this example.

ens.DataVariables = [ens.DataVariables; ...
    "fPeak"; "pLow"; "pMid"; "pHigh"; "pKurtosis"; ...
    "qMean"; "qVar"; "qSkewness"; "qKurtosis"; ...
    "qPeak2Peak"; "qCrest"; "qRMS"; "qMAD"; "qCSRange"];
ens.ConditionVariables = ["LeakFault","BlockingFault","BearingFault"];

feat = extractCI(flow,flowP,flowF);
dataToWrite = [faultValues, feat];
writeToLastMemberRead(ens,dataToWrite{:})

The above code preprocesses and computes the condition indicators for the first member of the simulation ensemble. Repeat this for all the members in the ensemble using the ensemble hasdata command. To get an idea of the simulation results under different fault conditions plot every hundredth element of the ensemble.

%Figure with nominal and faults
figure,
subplot(211);
lFlow = plot(flow.Time,flow.Data,'LineWidth',2);
subplot(212);
lFlowP = semilogx(flowF,pow2db(flowP),'LineWidth',2);
xlabel('Hz')
ct = 1;
lColors = get(lFlow.Parent,'ColorOrder');
lIdx = 2;

% Loop over all members in the ensemble, preprocess 
% and compute the features for each member
while hasdata(ens)
    
    % Read member data
    data = read(ens);
    
    % Preprocess and extract features from the member data
    [flow,flowP,flowF,faultValues] = preprocess(data);
    feat = extractCI(flow,flowP,flowF);
    
    % Add the extracted feature values to the member data
    dataToWrite = [faultValues, feat];
    writeToLastMemberRead(ens,dataToWrite{:})
    
    % Plot member signal and spectrum for every 100th member
    if mod(ct,100) == 0
        line('Parent',lFlow.Parent,'XData',flow.Time,'YData',flow.Data, ...
            'Color', lColors(lIdx,:));
        line('Parent',lFlowP.Parent,'XData',flowF,'YData',pow2db(flowP), ...
            'Color', lColors(lIdx,:));
        if lIdx == size(lColors,1)
            lIdx = 1;
        else
            lIdx = lIdx+1;
        end
    end
    ct = ct + 1;
end

Note that under different fault conditions and severities the spectrum contains harmonics at the expected frequencies.

Detect and Classify Pump Faults

The previous section preprocessed and computed condition indicators from the flow signal for all the members of the simulation ensemble, which correspond to the simulation results for different fault combinations and severities. The condition indicators can be used to detect and classify pump faults from a pump flow signal.

Configure the simulation ensemble to read the condition indicators, and use the tall and gather commands to load all the condition indicators and fault variable values into memory

% Get data to design a classifier.
reset(ens)
ens.SelectedVariables = [...
    "fPeak","pLow","pMid","pHigh","pKurtosis",...
    "qMean","qVar","qSkewness","qKurtosis",...
    "qPeak2Peak","qCrest","qRMS","qMAD","qCSRange",...
    "LeakFault","BlockingFault","BearingFault"];
idxLastFeature = 14;

% Load the condition indicator data into memory
data = gather(tall(ens));

Starting parallel pool (parpool) using the 'local' profile ...
Preserving jobs with IDs: 1 2 because they contain crash dump files.
You can use 'delete(myCluster.Jobs)' to remove all jobs created with profile local. To create 'myCluster' use 'myCluster = parcluster('local')'.
connected to 6 workers.
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 45 sec
Evaluation completed in 45 sec

data(1:10,:)

ans=10×17 table
    fPeak      pLow       pMid     pHigh     pKurtosis    qMean      qVar     qSkewness    qKurtosis    qPeak2Peak    qCrest     qRMS      qMAD     qCSRange    LeakFault    BlockingFault    BearingFault
    ______    _______    ______    ______    _________    ______    ______    _________    _________    __________    ______    ______    ______    ________    _________    _____________    ____________

    43.909    0.86472    117.63    18.874     276.49      35.572    7.5242    -0.72832      2.7738        13.835      1.1494    35.677    2.2326     42690         1e-09          0.8              0      
    43.909    0.44477    125.92    18.899     12.417      35.576     7.869     -0.7094      2.6338        13.335      1.1449    35.686    2.3204     42697         1e-09          0.8              0      
    43.909     1.1782    137.99    17.526     11.589      35.573    7.4367    -0.72208      2.7136        12.641      1.1395    35.678    2.2407     42695         1e-09          0.8              0      
    44.151     156.74    173.84    21.073      199.5      33.768    12.466    -0.30256      2.4782        17.446      1.2138    33.952    2.8582     40518       2.4e-06          0.8              0      
    43.848    0.71756    110.92    18.579     197.02      35.563    7.5781    -0.72377       2.793         14.14      1.1504    35.669    2.2671     42682         1e-09          0.8              0      
    43.909    0.43673    119.56    20.003     11.589       35.57    7.5028    -0.74797      2.7913        13.833      1.1551    35.676    2.2442     42689         1e-09          0.8              0      
    43.788    0.31617     135.3    19.724     476.82      35.568    7.4406    -0.70964      2.6884        14.685      1.1473    35.673    2.2392     42687         1e-09          0.8              0      
    43.848    0.72747    121.63    19.733     11.589      35.523     7.791    -0.72736      2.7864        14.043      1.1469    35.633    2.2722     42633         1e-09          0.8              0      
    43.848    0.62777    128.85    19.244     11.589      35.541    7.5698     -0.6953      2.6942        13.451      1.1415    35.647    2.2603     42654         1e-09          0.8              0      
    43.848     0.4631    134.83    18.918     12.417      35.561    7.8607    -0.68417      2.6664        13.885      1.1504    35.671    2.3078     42681         1e-09          0.8              0      

The fault variable values for each ensemble member (row in the data table) can be converted to fault flags and the fault flags combined to single flag that captures the different fault status of each member.

% Convert the fault variable values to flags
data.LeakFlag = data.LeakFault > 1e-6;
data.BlockingFlag = data.BlockingFault < 0.8;
data.BearingFlag = data.BearingFault > 0; 
data.CombinedFlag = data.LeakFlag+2*data.BlockingFlag+4*data.BearingFlag;

Create a classifier that takes as input the condition indicators and returns the combined fault flag. Train a support vector machine that uses a 2nd order polynomial kernel. Use the cvpartition command to partition the ensemble members into a set for training and a set for validation.

rng('default') % for reproducibility
predictors = data(:,1:idxLastFeature); 
response = data.CombinedFlag;
cvp = cvpartition(size(predictors,1),'KFold',5);

% Create and train the classifier
template = templateSVM(...
    'KernelFunction', 'polynomial', ...
    'PolynomialOrder', 2, ...
    'KernelScale', 'auto', ...
    'BoxConstraint', 1, ...
    'Standardize', true);
combinedClassifier = fitcecoc(...
    predictors(cvp.training(1),:), ...
    response(cvp.training(1),:), ...
    'Learners', template, ...
    'Coding', 'onevsone', ...
    'ClassNames', [0; 1; 2; 3; 4; 5; 6; 7]);

Check the performance of the trained classifier using the validation data and plot the results on a confusion plot.

% Check performance by computing and plotting the confusion matrix
actualValue = response(cvp.test(1),:);
predictedValue = predict(combinedClassifier, predictors(cvp.test(1),:));
confdata = confusionmat(actualValue,predictedValue);
figure,
labels = {'None', 'Leak','Blocking', 'Leak & Blocking', 'Bearing', ...
    'Bearing & Leak', 'Bearing & Blocking', 'All'};
h = heatmap(confdata, ...
    'YLabel', 'Actual leak fault', ...
    'YDisplayLabels', labels, ...
    'XLabel', 'Predicted fault', ...
    'XDisplayLabels', labels, ...
    'ColorbarVisible','off');

The confusion plot shows for each combination of faults the number of times the fault combination was correctly predicted (the diagonal entries of the plot) and the number of times the fault combination was incorrectly predicted (the off-diagonal entries).

The confusion plot shows that the classifier did not correctly classify some fault conditions (the off diagonal terms). However, the no fault condition was correctly predicted. In a couple of places a no fault condition was predicted when there was a fault (the first column), otherwise a fault was predicted although it may not be exactly the correct fault condition. Overall the validation accuracy was 84% and the accuracy at predicting that there is a fault 98%.

% Compute the overall accuracy of the classifier
sum(diag(confdata))/sum(confdata(:))

ans = 0.6150

% Compute the accuracy of the classifier at predicting 
% that there is a fault
1-sum(confdata(2:end,1))/sum(confdata(:))

ans = 0.9450

Examine the cases where no fault was predicted but a fault did exist. First find cases in the validation data where the actual fault was a blocking fault but a no fault was predicted.

vData = data(cvp.test(1),:);
b1 = (actualValue==2) & (predictedValue==0);
fData = vData(b1,15:17)

fData=11×3 table
    LeakFault    BlockingFault    BearingFault
    _________    _____________    ____________

      1e-09          0.77              0      
      1e-09          0.77              0      
      1e-09          0.71              0      
      1e-09          0.77              0      
      1e-09          0.77              0      
      1e-09          0.62              0      
      1e-09          0.77              0      
      1e-09          0.77              0      
      1e-09          0.71              0      
      8e-07          0.74              0      
      1e-09          0.74              0      

Find cases in the validation data where the actual fault was a bearing fault but a no fault was predicted.

b2 = (actualValue==4) & (predictedValue==0);
vData(b2,15:17)

ans =

  0×3 empty table

Examining the cases where no fault was predictive but a fault did exist reveals that they occur when the blocking fault value of 0.77 is close to its nominal value of 0.8, or the bearing fault value of 6.6e-5 is close to its nominal value of 0. Plotting the spectrum for the case with a small blocking fault value and comparing with a fault free condition reveals that spectra are very similar making detection difficult. Re-training the classifier but including a blocking value of 0.77 as a non fault condition would significantly improve the performance of the fault detector. Alternatively, using additional pump measurements could provide more information and improve the ability to detect small blocking faults.

% Configure the ensemble to only read the flow and fault variable values
ens.SelectedVariables = ["qOut_meas","LeakFault","BlockingFault","BearingFault"];
reset(ens)

% Load the ensemble member data into memory
data = gather(tall(ens));

Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 38 sec
Evaluation completed in 38 sec

% Look for the member that was incorrectly predicted, and 
% compute its power spectrum
idx = ...
    data.BlockingFault == fData.BlockingFault(1) & ...
    data.LeakFault == fData.LeakFault(1) & ...
    data.BearingFault == fData.BearingFault(1);
flow1 = data(idx,1);
flow1 = flow1.qOut_meas{1};
[flow1P,flow1F] = pspectrum(flow1);

% Look for a member that does not have any faults
idx = ...
    data.BlockingFault == 0.8 & ...
    data.LeakFault == 1e-9 & ...
    data.BearingFault == 0;
flow2 = data(idx,1);
flow2 = flow2.qOut_meas{1};
[flow2P,flow2F] = pspectrum(flow2);

% Plot the power spectra
semilogx(...
    flow1F,pow2db(flow1P),...
    flow2F,pow2db(flow2P));
xlabel('Hz')
legend('Small blocking fault','No fault')

Conclusion

This example showed how to use a Simulink model to model faults in a reciprocating pump, simulate the model under different fault combinations and severities, extract condition indicators from the pump output flow and use the condition indicators to train a classifier to detect pump faults. The example examined the performance of fault detection using the classifier and noted that small blocking faults are very similar to the no fault condition and cannot be reliably detected.

Supporting Functions

function [flow,flowSpectrum,flowFrequencies,faultValues] = preprocess(data)
% Helper function to preprocess the logged reciprocating pump data.

% Remove the 1st 0.8 seconds of the flow signal
tMin = seconds(0.8);
flow = data.qOut_meas{1};
flow = flow(flow.Time >= tMin,:);
flow.Time = flow.Time - flow.Time(1);

% Ensure the flow is sampled at a uniform sample rate
flow = retime(flow,'regular','linear','TimeStep',seconds(1e-3));

% Remove the mean from the flow and compute the flow spectrum
fA = flow;
fA.Data = fA.Data - mean(fA.Data);
[flowSpectrum,flowFrequencies] = pspectrum(fA,'FrequencyLimits',[2 250]);

% Find the values of the fault variables from the SimulationInput
simin = data.SimulationInput{1};
vars = {simin.Variables.Name};
idx = strcmp(vars,'leak_cyl_area_WKSP');
LeakFault = simin.Variables(idx).Value;
idx = strcmp(vars,'block_in_factor_WKSP');
BlockingFault = simin.Variables(idx).Value;
idx = strcmp(vars,'bearing_fault_frict_WKSP');
BearingFault = simin.Variables(idx).Value;

% Collect the fault values in a cell array
faultValues = {...
    'LeakFault', LeakFault, ...
    'BlockingFault', BlockingFault, ...
    'BearingFault', BearingFault};
end

function ci = extractCI(flow,flowP,flowF)
% Helper function to extract condition indicators from the flow signal 
% and spectrum.

% Find the frequency of the peak magnitude in the power spectrum.
pMax = max(flowP);
fPeak = flowF(flowP==pMax);

% Compute the power in the low frequency range 10-20 Hz.
fRange = flowF >= 10 & flowF <= 20;
pLow = sum(flowP(fRange));

% Compute the power in the mid frequency range 40-60 Hz.
fRange = flowF >= 40 & flowF <= 60;
pMid = sum(flowP(fRange));

% Compute the power in the high frequency range >100 Hz.
fRange = flowF >= 100;
pHigh = sum(flowP(fRange));

% Find the frequency of the spectral kurtosis peak
[pKur,fKur] = pkurtosis(flow);
pKur = fKur(pKur == max(pKur));

% Compute the flow cumulative sum range.
csFlow = cumsum(flow.Data);
csFlowRange = max(csFlow)-min(csFlow);

% Collect the feature and feature values in a cell array. 
% Add flow statistic (mean, variance, etc.) and common signal 
% characteristics (rms, peak2peak, etc.) to the cell array.
ci = {...
    'qMean', mean(flow.Data), ...
    'qVar',  var(flow.Data), ...
    'qSkewness', skewness(flow.Data), ...
    'qKurtosis', kurtosis(flow.Data), ...
    'qPeak2Peak', peak2peak(flow.Data), ...
    'qCrest', peak2rms(flow.Data), ...
    'qRMS', rms(flow.Data), ...
    'qMAD', mad(flow.Data), ...
    'qCSRange',csFlowRange, ...
    'fPeak', fPeak, ...
    'pLow', pLow, ...
    'pMid', pMid, ...
    'pHigh', pHigh, ...
    'pKurtosis', pKur(1)};
end