Advanced numerical computation of χ2 tests for fault detection

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					                                                                                                          Introduction
                Advanced numerical computation of χ2 tests                     • Typical fault detection and isolation (FDI) procedure:

                      for fault detection and isolation                               ◦ residual generation

                                                                                      ◦ residual evaluation
                                            `
                      Qinghua Zhang and Michele Basseville
                                                                               • Evaluation of Gaussian residuals
                             IRISA (INRIA & CNRS)
                                 Rennes, France
                                                                                      ◦ for parametric change in linear systems

                                                                                      ◦ for small parametric change in nonlinear systems
                                                                                        (local asymptotic approach to change detection)

                                                                               • Changes in the mean of a Gaussian vector — χ2-tests

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                                                                                              Basic formulas for the χ2-tests
                 Gaussian residual evaluation — χ2-tests                                                                      ηa                        Iaa Iab
                                                                             z ∼ N (M η, Σ) I = M T Σ−1M η =                       M = [Ma Mb] I =
                                                                                                                              ηb                        Iba Ibb
       Consider m-dimensional residual z ∼ N (M η, Σ)
       with M ∈ Rm×n, η ∈ Rn, Σ ∈ Rm×m, m ≥ n.                               Fault detection (global test)

       Fault detection
                                                                                                                        T
                                                                                              t = zT Σ−1M (M T Σ−1Ma)−1Ma Σ−1z
       H0 : η = 0 against H1 : η = 0
                                                                             Fault isolation by sensitivity test
                                                ηa
       Fault isolation for some partition η =                                                ˜
                                                ηb                                                          T          T
                                                                                             ta = zT Σ−1Ma(Ma Σ−1Ma)−1Ma Σ−1z
       H0 : ηa = 0 against H1 : ηa = 0
                                                                             or equivalently
                                                     2
       These hypothesis testing problems lead to χ -tests.                                                    ˜       T
                                                                                                              ζa = Ma Σ−1z
                                                                                                              ˜    ˜T     ˜
                                                                                                              ta = ζa I−1ζa
                                                                                                                       aa



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                Basic formulas of the χ2-tests (contd.)                                    Advanced numerical computation – Global test

       Fault isolation by min-max test                                                               t = zT Σ−1M (M T Σ−1M )−1M T Σ−1z
                                                                                         • Use pseudo-inverse for Σ−1 if badly conditioned
                             ˜
                             ζa =       T
                                    Ma Σ−1z                                                                                             1
                                                                                         • Compute t as a square : t = (M T Σ−1M )− 2 M T Σ−1z   2
                             ˜
                             ζb =       T
                                    Mb Σ−1z
                                                                                         • Avoid the inverse involving M .
                              ∗
                             ζa =   ˜
                                    ζa − IabI−1ζb  ˜
                                               bb                                        Proposed solution:
                             I∗ =
                              a     Iaa − IabI−1Iba
                                                bb                                                    1
                                                                                         • let Γ = Σ− 2 (with pseudo-inverse if necessary),
                             t∗ =
                              a     ζa T I∗ −1ζa
                                     ∗
                                          a
                                               ∗

                                                                                         • QR decomposition of ΓM : ΓM = QR with QT Q = I,
                                                                                         • Then
       Numerical difficulties: when the matrices to be inverted are badly                                  t = zT ΓQR(RT QT QR)−1RT QT Γz
       conditioned, these basic formulas can lead to large numerical er-                                    = zT ΓQR(RT R)−1RT QT Γz
       rors. It is thus important to develop advanced numerical algo-
       rithms.                                                                                              = QT Γz 2

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                                                                                          Advanced numerical computation – Minmax test

                                                                                       More computations are involved.
       Advanced numerical computation – Sensitivity test
                                                                                       QR decompositions:
                   ˜              T          T
                   ta = zT Σ−1Ma(Ma Σ−1Ma)−1Ma Σ−1z                                                                   ΓMa = QaRa
                                                                                                                      ΓMb = QbRb
       Same as global test (M being replaced by Ma).
                                                                                                                      T
                                                                                                              (I − QbQb )Qa = QcRc
       Advanced numerical computation:                                                 then
                                                                                                                  t∗ = QT Γz
                                                                                                                   a    c
                                                                                                                               2
       QR decomposition of ΓMa: ΓMa = QaRa with        QT Qa
                                                        a      = I,
       then
                                                                                       A non-trivial step for deriving the algorithm:
                                ˜
                                ta = QT Γz
                                      a
                                               2
                                                                                            (I − QT QbQT Qa) = QT (I − QbQT )(I − QbQT )Qa
                                                                                                     a      b           a           b b




                                                                                       Remark: SVD can be used instead of QR decomposition.
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                           A numerical example
        Gaussian vector: dim(z) = 9, dim(η) = 5, dim(ηa) = 2.
       Condition number of Σ: 3.4 × 1010.                                              0.06

       Histograms are based on 10000 random realizations.                              0.05

                                                                                       0.04
     0.06
                                                                                       0.03
     0.05
                                                                                       0.02
     0.04
                                                                                       0.01
     0.03
                                                                                          0
     0.02                                                                                     0   10   20   30   40    50   60    70    80    90   100

     0.01

        0
                                                                                         Advanced minmax test (solid line) and χ 2(2, 44.955) density
            0   10   20     30    40     50    60     70    80     90   100              function (dashed line).
       Basic minmax test (solid line) and χ 2(2, 44.955) density function
       (dashed line).

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                                 Conclusion

         • χ2-tests are frequently used for residual evaluation.


         • Advanced numerical algorithms can significantly improve the
           numerical accuracy of badly conditioned problems.




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