Statistical fault detection and isolation for linear time-varying - - PowerPoint PPT Presentation

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Statistical fault detection and isolation for linear time-varying - - PowerPoint PPT Presentation

Jan 16, 2023 292 likes •488 views

Introduction Proposed algorithm Example Statistical fault detection and isolation for linear time-varying systems Qinghua Zhang & Michle Basseville INRIA & CNRS-IRISA, Rennes, France 16th IFAC SYSID, Brussels, Belgium, July 11-13,

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Introduction Proposed algorithm Example

Statistical fault detection and isolation for linear time-varying systems

Qinghua Zhang & Michèle Basseville

INRIA & CNRS-IRISA, Rennes, France

16th IFAC SYSID, Brussels, Belgium, July 11-13, 2012

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Introduction Proposed algorithm Example Overview Problem statement

Problem and approaches

FDI for LTV systems Relevant approach to FDI of NL systems (linearization along the actual or nominal trajectory) LTV systems more general than widely used LPV systems Three main approaches Detection filter, game theoretic approach to filter design, unknown input decoupled filter, UIO, finite horizon fault detection filter

Keviczky, Edelmayer, Chung-Speyer, Chen-Patton, Hou-Muller, Zhong-Ding, ...

Adaptive observers, set-valued observers, time domain solutions to different H−/H∞ problems

Zhang-Xu, Rosa-Shamma-Athans, Li-Zhou, ...

Parity-based fault estimation Zhong-Ding

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Introduction Proposed algorithm Example Overview Problem statement

Model and assumptions

MIMO LTV system (H0) Xk+1 = Fk Xk + Gk Uk + Wk Yk = Hk Xk + Jk Uk + Vk Fk, Gk, Hk, Jk: bounded TV matrices Wk, Vk: independent white Gaussian noises, TV cov. Qk, Rk (Hk, Fk) observable & (Fk, Q1/2

k

) controllable, both uniformly Additive faults (H1) Xk+1 = Fk Xk + Gk Uk + Wk + Ψk θ Yk = Hk Xk + Jk Uk + Vk θ: unknown fault vector Ψk: known TV fault profile

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Introduction Proposed algorithm Example Overview Problem statement

Different fault cases Actuator bias: Uk → Uk + θ; then Ψk = Gk Actuator gain loss: Uk → (I − diag(θ))Uk; then Ψk = − Gk diag(Uk) Sensor faults: use a similar term Ψk θ on the output equation (not treated here) Different fault occurrence speeds: ex: step change Ψk(r) ∆ = Ψk × 1 l{k≥r} (and θ constant) A particular case Ψk = δr,k+1 I: investigated by Willsky-Jones, Gustafsson with Fk assumed exponentially stable

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Introduction Proposed algorithm Example Fault effect Detection algorithm Isolation algorithm

Fault effect on the Kalman filter innovation

State prediction error and innovation - Fault free case

  • Xk

∆

= Xk −

  • Xk|k−1

εk

∆

= Yk − Hk Xk|k−1

  • X 0

k+1

= Fk (I − Kk Hk) X 0

k

− Fk Kk Vk + Wk ε0

k

= Hk X 0

k

+ Vk State prediction error and innovation - Faulty case

  • Xk+1

= Fk (I − Kk Hk) Xk − Fk Kk Vk + Wk + Ψk θ εk = Hk Xk + Vk

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Introduction Proposed algorithm Example Fault effect Detection algorithm Isolation algorithm

Introducing a matrix gain ηk

∆

=

  • Xk − Γk θ

Γk+1

∆

= Fk (I − Kk Hk) Γk + Ψk , Γ0

∆

= 0 ηk+1 = Fk (I − Kk Hk) ηk − Fk Kk Vk + Wk ηk =

  • X 0

k

Additive fault effect εk = ε0

k + Hk Γk θ

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Introduction Proposed algorithm Example Fault effect Detection algorithm Isolation algorithm

Guaranteed properties of the recursive gain Γk depends on the fault gain Ψk, not on the fault vector θ. The matrix gain Γk computed from the bounded Ψk is bounded even when the system is not stable. The persistent excitation condition:

  • k ΓT

k HT k Σ−1 k HkΓk strictly positive definite

is satisfied even when the number of sensors is smaller than the number of faults. Difference with the Willsky-Jones algorithm Computations based on recursive formulas involving Fk (thus required to be stable)

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Introduction Proposed algorithm Example Fault effect Detection algorithm Isolation algorithm

Known fault profile matrix

MLE of θ under H1 H0 : εk ∼ N (0, Σk) , H1 : εk ∼ N (Hk Γk θ, Σk)

  • θk = arg min
  • θ

k

  • j=1

(εj − Hj Γj θ)T Σ−1

j

(εj − Hj Γj θ) = C−1

k

dk Ck = Ck−1 + ΓT

k HT k Σ−1 k

Hk Γk dk = dk−1 + ΓT

k HT k Σ−1 k

εk GLR test lk

∆

= 2 ln p(ε1, . . . , εk | θ = θk) p(ε1, . . . , εk | θ = 0) = dT

k C−1 k

dk

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Introduction Proposed algorithm Example Fault effect Detection algorithm Isolation algorithm

Unknown jump fault onset time

Handling the transient behavior after a jump Ψk(r) = Ψk × 1 l{k≥r} = ⇒ Γk+1(r) = Fk (I − KkHk) Γk(r) + Ψk(r) Full treatment of the transient

  • θk(r)

= C−1

k (r) dk(r)

lk = max

1≤r≤k dT k (r) C−1 k (r) dk(r)

ˆ rk = arg max

1≤r≤k dT k (r) C−1 k (r) dk(r)

In practice Γk(r), Ck(r), dk(r) computed for r ∈ {k − w + 1, k − w + 2, . . . , k}

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Introduction Proposed algorithm Example Fault effect Detection algorithm Isolation algorithm

Locating non zero components Z ∆ =    εˆ

r+1

. . . εk    ∼ N (M θ, S) M =    Hˆ

r+1 Γˆ r+1(ˆ

r) . . . Hk Γk(ˆ r)    , S = diag(Σˆ

r+1, . . . , Σk)

ζ ∆ = MTS−1 Z , F ∆ = MTS−1M θ = θa θb

  • − θa = 0 against θa = 0, θb nuisance parameter

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Introduction Proposed algorithm Example Fault effect Detection algorithm Isolation algorithm

Minmax test la

∆

= ζ∗T

a

F∗

a −1 ζ∗ a

ζ∗

a ∆

= ζa − Fab F−1

bb ζb , F∗ a ∆

= Faa − Fab F−1

bb Fba

la ∼ χ2(dim(θa))

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Introduction Proposed algorithm Example A simulated example Numerical results Conclusion

Leakage detection in gas transportation networks

Thanks to Paulo Lopes dos Santos et al., IEEE CST, Jan. 2011

Gas dynamics as a LPV model Hyperbolic model linking edge pressure drop and mass flow Discrete time LPV model    Xk+1 = (F0 + Fp pk) Xk + (G0 + Gp pk) Uk + Kk ek Yk = (H0 + Hp pk) Xk + (J0 + Jp pk) Uk + ek Uk ∈ R: input mass flow, Yk ∈ R: output mass flow Xk ∈ R2: mass flow and pressure drop within the first section pk ∈ R: scheduling parameter (pressure pattern)

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Introduction Proposed algorithm Example A simulated example Numerical results Conclusion

Simulated leakage Additive changes on G0, Gp ∈ R2 (actuator gain loss) - Monitoring the first component Nominal values: G0(1) = −7.8297e − 4 Gp(1) = +3.8290e − 5 Changed values: G0(1) + 1.6e − 5 Fault 1 Gp(1) + 9.5e − 6 Fault 2 Available data, simulated data Uk provided by P . Lopes dos Santos et al. Yk simulated using the LPV model

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Introduction Proposed algorithm Example A simulated example Numerical results Conclusion

Fault detection - Fault 1

200 400 600 800 1000 1200 −2 −1 1 2 x 10

4

200 400 600 800 1000 1200 20 40 60 80 100

Kalman ¡Innova*on ¡ Es*mated ¡fault ¡

  • nset ¡*me ¡

604 ¡min ¡ Time ¡(min) ¡ Fault ¡onset ¡*me ¡

lk

Alarm ¡*me ¡ 804 ¡min ¡

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Introduction Proposed algorithm Example A simulated example Numerical results Conclusion

Detection delay and onset time estimate - Fault 1

50 100 150 200 250 10 20 30 40 50 60 −200 −150 −100 −50 50 100 150 200 50 100 150 200

Detec%on ¡delays ¡ Fault ¡onset ¡ es%mate ¡errors ¡ Time ¡(min) ¡

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Introduction Proposed algorithm Example A simulated example Numerical results Conclusion

Detection delay and onset time estimate - Fault 2

20 40 60 80 100 120 140 160 20 40 60 80 100 120 −200 −150 −100 −50 50 100 150 200 100 200 300 400 500

Detec%on ¡delays ¡ Fault ¡onset ¡ ¡ es%mate ¡errors ¡ Time ¡(min) ¡

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Introduction Proposed algorithm Example A simulated example Numerical results Conclusion

Fault isolation with the minmax tests - Fault 1

20 40 60 80 100 120 140 160 180 2 4 6 8 1 2 3 4 5 6 7 8 5 10 15 20 25 30

Minmax ¡test ¡ focusing ¡on ¡fault ¡1 ¡ Minmax ¡test ¡ focusing ¡on ¡fault ¡2 ¡ Threshold ¡theore7c ¡ 1% ¡false ¡alarm ¡rate ¡ Threshold ¡theore7c ¡ 1% ¡false ¡alarm ¡rate ¡

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Introduction Proposed algorithm Example A simulated example Numerical results Conclusion

Fault isolation with the minmax tests - Fault 2

5 10 15 20 25 100 200 300 400 500 50 100 150 200 250 300 350 400 20 40 60 80 100

Minmax ¡test ¡ focusing ¡on ¡fault ¡1 ¡ Minmax ¡test ¡ focusing ¡on ¡fault ¡2 ¡ Threshold ¡for ¡1% ¡ theore8c ¡false ¡alarm ¡rate ¡ ¡ ¡ Empirical ¡miss-­‑ detec8on ¡rate ¡0.3% ¡ ¡ ¡ Threshold ¡for ¡1% ¡ theore8c ¡false ¡alarm ¡rate ¡ ¡ ¡ Empirical ¡false ¡ ¡ alarm ¡rate ¡1.3% ¡ ¡ ¡

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Introduction Proposed algorithm Example A simulated example Numerical results Conclusion

FDI for LTV systems with TV additive faults Combining a recursive and stable filter that cancels out the fault dynamics and a GLR test Handling additive parametric faults with weaker assumptions than usual on the system stability and the number of required sensors Simulations confirm the stability of the proposed filter and suggest the relevance of the proposed approach Future investigations include numerical experiments on simulated and real cases to assess the quality of the full and simplified treatments of the transient

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