1. Fault Detection and Diagnosis in Engineering Systems Basic concepts with simple examples Janos Gertler George Mason University Fairfax, Virginia

2. Outline What is a fault What is diagnosis Diagnostic approaches –Model - free methods –Principal component approach –Model - based methods –Systems identification Application example: car engine diagnosis

3. What is a fault Fault: malfunction of a system component - sensor fault- bias - actuator fault- parameter change - plant fault- leak, etc. Symptom: an observable effect of a fault Noise and disturbance: nuissances that may affect the symptoms

4. What is a fault actuator command actuator leak sensor faults fault sensor readings Sensor fault: reading is different from true value Actuator fault: valve position is different from command Plant fault: leak

5. What is fault diagnosis Fault detection: indicating if there is a fault Fault isolation: determining where the fault is Detection + Isolation = Diagnosis Fault identification: –Determining the size of the fault –Determining the time of onset of the fault

6. Model-free methods Fault-tree analysis - cause-effect trees analysed backwards Spectrum analysis - fault-specific frequencies in sound, vibration, etc Limit checking - checking measurements against preset limits

7. flow l1 l2 l3 s1 s2 s3 y 1 y 2 y 3 Limit checking y 1 y 2 y 3 S1 faultoffnormalnormal Leak3normalnormaloff Leak2normaloffoff Leak1offoffoff High/low flowoffoffoff

8. Limit checking Easy to implement Requires no design BUT To accommodate “normal” variations, must have limited fault sensitivity Has limited fault specificity (symptom explosion)

9. Principal Component Approach Modeling phase: based on normal data - determine the subspace where normal data exists (representation space, RepS) - determine the spread (variances) of data in the RepS Monotoring phase: compare observations to representation space - if outside RepS, there are faults - if inside RepS but outside thresholds, abnormal operating conditions

10. Principal Component Approach u flow y 1 = u y 2 = u y 1 y 2 y 2 Representation space Fault Normal spread y 1 u

11. Principal component modeling Centered normalised measurements x(t) = [x 1 (t) … x n (t)]’ Data matrix: X = [ x(1) x(2) … x(N)] Covariance matrix: R = XX’/N Compute eigenvalues 1 … n and eigenvectors q 1 … q n q 1 … q k, k  n, belonging to nonzero 1 … k,, span RepS 1 … k are the variances in the respective directions

12. Principal Components – Residual Space Residual Space (ResS): complement of Representation Space, spanned by the e-vectors q k+1 … q n, belonging to (near) - zero e-values Residual = (Observation) – (Its projection on RepS) Residuals exist in ResS ResS provides isolation information - directional property (fault-specific response directions) - structural property (fault-specific Boolean structures)

13. Residual Space – Directional Property u flow  u  y 1  y 2 y 1 y 2 y 2 residual observation Repres. Space q 1 y 1 u on  u q 3 q 2 on  y 1 on  y 2 Residual Space

14. Residual Space – Structural Property  u  u  u r 2 r 3 r 1  y 1  y 2  y 1  y 2  y 1  y 2 r 1, r 2, r 3 : residuals obtained by projection  u  y 1  y 2 Structure matrix r 1 0 1 1 r 2 1 1 0 Fault codes r 3 1 0 1

15. Model-Based Methods faults f(t) disturbances d(t) noise n(t) outputs y(t) inputs u(t) parameters  Complete model: y(t) = f[u(  ), f(  ), d(  ), n(  ),  ] Nominal model: y^(t) = f[u(  ),  ] Models are: static/dynamic linear/nonlinear

16. Obtaining Models First principle models Empirical models - “classical” systems identification - principal component approach - neuronets

17. Analytical Redundancy d(t) f(t) n(t) u(t) y(t) PLANT + e(t) RESIDUAL r(t) PROCESSING - MODEL y^(t) Primary residuals: e(t) = y(t) – y^(t) Processed residuals: r(t)

18. Analytical redundancy f(t) d(t) n(t) u(t) y(t) PLANT RESIDUAL GENERATOR r(t)

19. Residual Properties Detection properties - sensitive to faults - insensitive to disturbances (disturbance decoupling) - insensitive to model errors (model-error robustness)  perfect decoupling under limited circumstances  “optimal” decoupling - insensitive to noise  noise filtering  statistical testing

20. Residual Properties Isolation properties - selectively sensitive to faults  structured residuals perfect  directional residuals decoupling  “optimal” residuals

21. Residual Generation u flow Model:  u y 1 = u +  u +  y 1  y 1  y 2 y 1 y 2 y 2 = u +  u +  y 2 Primary residuals: e 1 = y 1 – u =  u +  y 1  u  y 1  y 2 e 2 = y 2 – u =  u +  y 2 r 1 1 1 0 Processed residuals: r 2 1 0 1 r 1 = e 1 =  u +  y 1 r 3 0 1 1 r 2 = e 2 =  u +  y 2 r 3 = e 2 – e 1 =  y 2 –  y 1 Structured residuals

22. Residual Generation u flow Model:  u y 1 = u +  u +  y 1  y 1  y 2 y 1 y 2 y 2 = u +  u +  y 2 Primary residuals: e 1 = y 1 – u =  u +  y 1 e 2 = y 2 – u =  u +  y 2 Processed residuals: r 1 = e 1 =  u +  y 1 r 2 = e 2 =  u +  y 2 r 3 = e 1 – e 2 =  y 1 –  y 2 r 3 on  y 1 r 2 on  u r 1 on  y 2 Directional residuals

23. Linear Residual Generation Methods Perfect decoupling - direct consistency relations - parity relations from state-space model - Luenberger observer - unknown input observer Approximate decoupling - the above with singular value decomposition - constrained least-squares - H-infinity optimization

24. Linear Residual Generation Methods Under identical conditions (same plant, same response specification) the various methods lead to identical residual generators

25. Dynamic Consistency Relations System description: y(t) = M(q)u(t) + S f (q)f(t) + S d (q)d(t) q : shift operator Primary residuals: e(t) = y(t) – M(q)u(t) = S f (q)f(t) + S d (q)d(t) Residual transformation: r(t) = W(q)e(t) = W(q)[S f (q)f(t) + S d (q)d(t)]

26. Dynamic Consistency Relations Response specification: r(t) =  f (q)f(t) +  d (q)d(t)  f (q) : specified fault response (structured or directional)  d (q) : specified disturbance response (decoupling)  W(q)[S f (q) S d (q)] = [  f (q)  d (q)] Solution for square system: W(q) = [  f (q)  d (q)] [S f (q) S d (q)] -1

27. Dynamic Consistency Realtions Realization: The residual generator W(q) must be causal and stable; [S f (q) S d (q)] -1 is usually not so Modified specification: W(q) = [  f (q)  d (q)]  (q) [S f (q) S d (q)] -1  (q) : response modifier, to provide causality and stability without interfering with specification Implementation: inverse is computed via the fault system matrix

28. Diagnosis via Systems Identification Approach: - create reference model by identification - re-identify system on-line  discrepancy indicates parametric fault Difficulty: discrete-time model parameters are nonlinear functions of plant parameters  for small faults, fault-effect linearization  continuous-time model identification (noise sensitive or requires initialization)

29. Applications Very large systems - Principal Components are widely used in chemical plants - reliable numerical package is available An intermediate-size system: rain-gauge network in Barcelona, Spain (structured parity relations) Aerospace: traditionally Kalman filtering

30. Applications Mass-produced small systems: on-board car-engine diagnosis car-to-car variation (model variation robustness) - GM: parity relations - Ford: neuronets - Daimler: parity relations + identification Many published papers “with application to” are just simulation studies

31. GM – GMU On-Board Diagnosis Project OBD-II: any component fault causing emissions (CH, CO, NOX) go 50% over limit must be detected on-line Pilot project: intake manifold subsystem (THR, MAP, MAF, EGR) Structured parity relations based on direct identification After more in-house development, this is being gradually introduced on GM cars

32. Filtered and integrated residual with fault

33. On-board report – MAP fault

34. GM fleet experiment Fleet of “identical” vehicles (Chevy Blazer) available at GM Collect data from 25 vehicles Identify models from combined data from 5 vehicles Test on data from 25 vehicles Residual means and variances vary  increase thresholds (sacrifice sensitivity) Only a 50% increase is necessary

35. Fault sensitivities – GM fleet experiment Critical fault sizes for detection and diagnosis (fleet experiment) ThrIacEgrMapMaf detection 2%10%12% 5% 2% diagnosis6%20%17% 7% 8%