Observerbased fault detection and isolation

1
Observer-based fault detection and isolation

• Michel Kinnaert
• Dpt of Control Engineering and System Analysis
• Université libre de Bruxelles
• Michel.Kinnaert_at_ulb.ac.be

2
Content

• Introduction and motivation
• Notion of redundancy
• Structure of an FDI system
• Preliminaries on geometric system theory
• Residual generation for linear systems
• Residual generation for nonlinear systems
• Decision system
• Conclusion

3
Introduction Motivation (1)

• Notion of fault
• Event that modifies the operation of the process
  in such a way that its performance is degraded
  and/or its missions cannot be achieved
• Classical monitoring system
• Comparison of measured signals to fixed
  thresholds and/or tests on the gradient of the
  signals
• Alarm operator chooses synoptic with detailed
  information on the signal ? action to be
  performed

4
Introduction Motivation (2)

• Detection of important faults (not incipient
  faults )
• Little help to analyse the origin of the faults
• System for fault detection and isolation
• Improved decision help for the operator
• Incipient faults
• Predictive maintenance
• Automatic controller reconfiguration
• (autonomous system)

5
Introduction Motivation (3)

• Consequences
• Less unexpected process shutdowns
• Higher product quality
• Reduction of maintenance costs
• Improved compliance with environmental
  constraints

6
Content

• Introduction and motivation
• Notion of redundancy
• Structure of an FDI system
• Preliminaries on geometric system theory
• Residual generation for linear systems
• Residual generation for nonlinear systems
• Decision system
• Conclusion

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Notion of redundancy

• Material redundancy
• drawbacks cost, space,
• safety critical applications (nuclear,
  aeronautics, )
• Analytical redundancy
• Check compatibility between different types of
  measurements and a mathematical model of the
  supervised process

8
Content

• Introduction and motivation
• Notion of redundancy
• Structure of an FDI system
• Preliminaries on geometric system theory
• Residual generation for linear systems
• Residual generation for nonlinear systems
• Decision system
• Conclusion

9
Structure of an FDI system
Faults
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Content

• Introduction and motivation
• Notion of redundancy
• Structure of an FDI system
• Preliminaries on geometric system theory
• Residual generation for linear systems
• Residual generation for nonlinear systems
• Decision system
• Conclusion

11
Preliminaries on geometric system theory (1)
(Wonham, 1985)

• A-invariant subspace
• Definition
• Interpretation for a dynamical system
• Examples 0, , the subspaces generated by
  the eigenvectors of A
• The intersection and the sum of two A-invariant
  subspaces is A-invariant

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Preliminaries on geometric system theory (1bis)

• Change of coordinates xTz such that Z is the set
  of vectors of the form

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Preliminaries on geometric system theory (2)

• Unobservable subspace

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Preliminaries on geometric system theory (3)

• Change of coordinate exhibiting unobservable
  state variables

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Preliminaries on geometric system theory (3)

• (C,A)-invariant subspace

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Preliminaries on geometric system theory (4)

• Set of all (C,A)-invariant subspaces containing a
• given subspace V

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Preliminaries on geometric system theory (5)

• (C,A)-unobservability subspace

18
Preliminaries on geometric system theory (6)
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Preliminaries on geometric system theory (7)

• Observation space and observability rank
  condition for bilinear system

20
Preliminaries on geometric system theory (8)
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Preliminaries on geometric system theory (9)

• Kabore, 1998Hammouri et al.,2001De Persis et
  al., 2000
• (C,A)-invariant subspace
• Define

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Preliminaries on geometric system theory (10)

• Set of all (C,A)-invariant subspaces containing a
• given subspace V

23
Preliminaries on geometric system theory (11)
24
Preliminaries on geometric system theory (12)
25
Preliminaries on geometric system theory (13)

• Generalization to control affine nonlinear
  systems of the form

(h, )-invariant coditribution Observability
codistribution (De Persis and Isidori, 2001)
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Content

• Introduction and motivation
• Notion of redundancy
• Structure of an FDI system
• Preliminaries on geometric system theory
• Residual generation for linear systems
• Residual generation for nonlinear systems
• Decision system
• Conclusion

27
Linear Systems - Model of faulty system

• Linear state space model
• Additive faults
• Examples
• Actuator faults EB, G0
• Sensor faults E0, GI
• Multiplicative faults
• Changes in the entries of A, B, C f0 (not
  considered
• here)

28
Linear systems Basic principle of observer
based residual generation(1)

• Process model
• Luenberger observer
• State estimation error

fault
Supervised system
State observer
Estimated state
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Linear systems Basic principle of observer
based residual generation(2)

• Output estimation error
• Generally does not decay to zero in the presence
  of a fault or at least exhibits significant
  transient upon occurrence of a fault
• can be used as a residual vector
  

30
Linear systems Basic principle of observer
based residual generation(3)

• Structured residuals
• Coding set
• Incidence matrix

31
Linear systems Basic principle of observer
based residual generation(4)

• - No need to estimate the whole state vector
• - Assure sensitivity to certain faults and
  unsensitivity to others
• Fundamental problem of residual generation
  (FPRG)

32
Linear systems Statement of FPRG (1)

• Supervised system
• Fundamental problem of residual generation (FPRG)
• Determine a filter of the form
• Such that 1) when , as
  for all u and d
• 2) when , r is affected by the
  fault

33
Linear systems - Statement of FPRG (2)

• Concatenation of supervised system and filter

34
Linear systems Existence of a solution to FPRG
(1)

• Equivalent necessary and sufficient conditions
  for the existence of a solution to FPRG
  (Massoumnia et al.,1989 Isidori et al., 2000)
  f scalar
• 1)
• 2) Existence of suitable changes of
• coordinates on the state space and the
  output
• space
• 3)

35
Linear systems - Existence of a solution to FPRG
(2)

• 2) Existence of suitable changes of coordinates
  on the state space the output space

36
Linear systems - Existence of a solution to FPRG
(3)
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Linear systems Design of residual generator (1)

• 3 step solution
• 1)
• 2)

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Linear systems Design of residual generator (2)

• 2) continued

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Linear systems Design of residual generator (3)

• 3) Design a Luenberger observer for system
• (Eq1), (Eq2) residual output estimation
  error
• Remark Steps 1 and 2 ? extraction of an
  observable
• subsystem which is not directly affected by the
  unknown
• Inputs (d only enters in the equations via y).

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Content

• Introduction and motivation
• Notion of redundancy
• Structure of an FDI system
• Preliminaries on geometric system theory
• Residual generation for linear systems
• Residual generation for nonlinear systems
• Decision system
• Conclusion

41
Nonlinear system - sensor faults (1)

• Dedicated observer scheme
• Model of the class of processes
• f(t) sensor fault (bias, drift, )
• Assume that an exponential observer can be
  constructed for each single output

42
Nonlinear system - sensor faults (2)

• Structure of the
• dedicated observer
• scheme

43
Nonlinear system - sensor faults (3)
44
Bilinear system Model Class

• Process model

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Bilinear system - Residual generation

• Considered structure for residual generator

46
Bilinear system Statement of BFPRG(1)

• BFPRG for system (bil1), (bil2)
• Consider the concatenated system (bil1)-(bil5)
• (input u,d,f state x, xr,g output r)
• Determine a filter of the form (bil3)-(bil5) for
• which there exists a subset U of the set of
• admissible inputs s.t.
• when f0, r is not affected by d and it
  asymptotically decays to zero for all u in U and
  for all initial conditions
• r is affected by f

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Bilinear system Statement of BFPRG (2)

• Definition
• The output y of

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Bilinear system Existence of a solution to
BFPRG (1)

• Equivalent necessary and sufficient conditions
  for
• the existence of a solution to BFPRG (Hammouri et
  al.,
• 2001 Kinnaert, 1999)
• 1)
• 2) Existence of suitable changes of coordinates
  on the state space and the output space
• 3) Existence of a solution to a set of algebraic
  equations observability condition

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Bilinear system Existence of a solution to
BFPRG (2)

• Remark1) Condition (1) written for ,
  see
• (Hammouri et al., 2001) for
• 2) In comparison with linear case,
• convergence of residual only for a particular
• class of inputs
• 3) Solution for a linear time invariant
• residual generator up to output injection, see
• (Yu and Shields, 1996)

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Bilinear systems Design of residual generator
(1)
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Bilinear systems Design of residual generator
(2)
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Bilinear systems Design of residual generator
(3)
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Comparison linear/ bilinear (1)

• Simulation study
• -Control levels
• in tanks R1 and
• R3 by acting on
• speed of pump
• P1 and aperture
• of valve V5
• -Faults
• 1)Leak in tank R1
• 2)Clog in branch P2
• 3)Bias on level in R3

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Comparison linear/ bilinear (2)

• Linearized model obtained by computing
  analytically linear model around set point
• Bilinear model obtained by least square
  identification from data resulting from the
  complete nonlinear simulator
• Synthesis of linear and bilinear FDI systems for
  detection and isolation of the three faults
• Illustration with residual sensitive to fault 3

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Comparison linear/ bilinear (3)
Measured inputs
Measured outputs
Residuals obtained from linear model and
bilinear model
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Bilinear system stochastic framework (1)

• Kinnaert and El Bahir (2000)
• Similar model as above but discrete time and with
  measurement and process noise (Gaussian white
  noise)
• Extracted subsystem independent of unknown input
  can be seen as a linear time-varying system up to
  output injection
• Kalman filter
• Residual innovation of Kalman filter

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Bilinear system stochastic framework (2)

• Form of the residual
• Distribution of the residual

58
Control affine nonlinear systems (1)

• Process model
• NLFPRG stated as a generalization of FPRG and
  BFPRG
• (De Persi and Isidori, 2001) Resort to notion of
  observability
• codistribution to extract a subsystem not
  directly affected
• by d. Extra hypotheses needed to assure the
  existence of
• an asymptotic observer for that system
• (Hammouri et al. 1999a,b) Less general
  transformation for the
• extraction of a subsystem not directly affected
  by d, but
• guarantee of existence of high gain observer for
  that
• subsystem

59
Control affine nonlinear systems (2)

• An example of a design by inspection
• Fault detection and isolation in a hydraulic
  system (Hammouri et al., 1999b)
• Aim detect and isolate the following two faults
• - drop of the spool control force
• - increase in the internal leakage of the piston

60
Control affine nonlinear systems (3)

• Schematic of the hydraulic system

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Control affine nonlinear systems (4)

• State space model of the hydraulic system

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Control affine nonlinear systems (5)
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Control affine nonlinear systems (6)

• Residual generator to detect fault 2
• Consider the last three equations of the model
  with
• and measurement equation

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Control affine nonlinear systems (7)
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Control affine nonlinear systems (8)

• Simulation results
• Residual as a function of time
• Fault 1 step-like fault between t10s and t20s
• Fault 2 step-like fault between t30s and t40s

66
Content

• Introduction and motivation
• Notion of redundancy
• Structure of an FDI system
• Preliminaries on geometric system theory
• Residual generation for linear systems
• Residual generation for nonlinear systems
• Decision system
• Conclusion

67
Decision system Deterministic framework

• Compare absolute value of the residual to a
  fixed threshold
• Compare integral of the absolute value (or the
  square) of the residual over a moving window to a
  fixed threshold
• Time varying threshold to handle modelling
  uncertainties (Emami-Naeini et al., 1998)

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Decision system Stochastic framework (1)

• Use on-line algorithm to detect changes in
• the mean of the residual
• 1) cumulative sum algorithm
• 2) generalized likelihood ratio algorithm
• Algorithms based on the log-likelihood ratio
• of the residual under fault-free and faulty
  situations (Basseville and Nikiforov, 1993
  Blanke et al., 2006)

69
Decision system Syochastic framework (2)

• Principle of CUSUM algorithm
• Log-likelihood ratio
• Fundamental property

70
Decision system Stochastic framework (3)

• Cumulative Sum (CUSUM)

Decision function
Alarm time
71
Content

• Introduction and motivation
• Notion of redundancy
• Structure of an FDI system
• Preliminaries on geometric system theory
• Residual generation for linear systems
• Residual generation for nonlinear systems
• Decision system
• Conclusion

72
Conclusion (1)

• Fundamental problem of residual generation solved
  for different classes of nonlinear system
• Resort to geometric system theory and nonlinear
  observers
• Related problem dealing with multiplicative
  faults ? adaptive observer approach,
• asymptotic local approach
• (Zhang et al., 1994)

73
Conclusion (2)

• Open issue proper handling of modelling
  uncertainties in a nonlinear framework
• Most work for linear systems Fault detection
  problem stated as an optimization problem to
  achieve trade-off between sensitivity to fault
  and insensitivity to unknown inputs
• Observer with disturbance attenuation (Besançon,
  2003), Bayesian approach (particle filter), set
  membership approach.

74
Conclusion (3)

• Towards fault tolerant control
• Design of FDI system and reconfiguration
  mechanism to assure specified closed-loop
  performance
• Interplay between FDI and reconfiguration

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Conclusion (4)

• Multi-controller scheme
• How many controllers to reach a specific
  performance level
• Proper handling of modelling uncertainties
• Keep in mind physical nature of faults

76
References (1)

• P. Alexandre and M. Kinnaert (1993) Numerically
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77
References (2)

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