Handling parametric and non-parametric additive faults in LTV - - PowerPoint PPT Presentation

Introduction Proposed algorithms Conclusion

Handling parametric and non-parametric additive faults in LTV Systems

Qinghua Zhang & Michèle Basseville

INRIA & CNRS-IRISA, Rennes, France

9th IFAC SAFEPROCESS, Paris, France, Sept. 2-4, 2015

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Introduction Proposed algorithms Conclusion 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 algorithms Conclusion Overview Problem statement

Different fault types Parametric fault : (rare) changes in a parameter vector Non-parametric fault : arbitrary unknown function of time Most FDI methods for LTV systems address the non-parametric fault case, or the parametric one Contribution A statistical approach exists for constant parametric faults Extension to both TV parametric and non-parametric faults Two solutions : Assuming (piecewise) constant parametric fault and rejecting the non-parametric fault Adapting to the TV parametric fault

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Introduction Proposed algorithms Conclusion 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 θk + Ek fk Yk = Hk Xk + Jk Uk + Vk known fault profile matrix Ψk, unknown fault vector θk known fault incidence matrix Ek, unknown fault profile vector fk

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Introduction Proposed algorithms Conclusion Overview Problem statement

Different fault cases Ekfk and Ψkθk typically represent actuator faults fk : no a priori information; θk : constant or slowly varying Parametric faults in both state and output equations (sensor faults) can be handled This modeling framework encompasses multiple faults Non-additive faults are not handled Particular cases Actuator bias: Uk → Uk + θ; then Ψk = Gk Actuator gain loss: Uk → (I − diag(θ))Uk; then Ψk = −Gk diag(Uk) Ψk = δr,k+1 I: investigated by Willsky-Jones, Gustafsson with Fk assumed exponentially stable

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Introduction Proposed algorithms Conclusion Fault effect First solution: rejecting the non-parametric fault Second solution: adapting to the parametric fault

Fault effect on the innovation of a linear filter

State prediction error and innovation - Fault free case

• Xk

∆

= Xk −

• Xk|k−1

εk

∆

= Yk − Jk Uk − 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 θk + Ek fk εk = Hk Xk + Vk

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Introduction Proposed algorithms Conclusion Fault effect First solution: rejecting the non-parametric fault Second solution: adapting to the parametric fault

Introducing a matrix gain ηk

∆

=

• Xk − Γk θk

Γk+1

∆

= Fk (I − Kk Hk) Γk + Ψk , Γ0

∆

= 0 ηk+1 = Fk (I − Kk Hk) ηk − Fk Kk Vk + Wk − Γk+1 (θk+1 − θk) + Ek fk Distinguishing two cases for the parametric fault vector Constant θ TV θk

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Introduction Proposed algorithms Conclusion Fault effect First solution: rejecting the non-parametric fault Second solution: adapting to the parametric fault

Constant parametric fault vector θk

∆

= θ Let ζk+1

∆

= Fk (I − Kk Hk) ζk + Ek fk , ζ0 = 0 cf. Γk+1

∆

= Fk (I − Kk Hk) Γk + Ψk , Γ0

∆

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

• X 0

k + ζk

Additive fault effect εk = ε0

k + Hk Γk θ + Hk ζk

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Introduction Proposed algorithms Conclusion Fault effect First solution: rejecting the non-parametric fault Second solution: adapting to the parametric fault

Guaranteed properties of the recursive Γk and ζk 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. Similarly, if fk is bounded, ζk is bounded. 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 algorithms Conclusion Fault effect First solution: rejecting the non-parametric fault Second solution: adapting to the parametric fault

TV parametric fault vector θk+1 = θk + ek , |ek| ≤ δ Let δk+1

∆

= Fk (I − Kk Hk) δk − Γk+1 ek, δ0

∆

= 0 ηk =

• X 0

k + δk + ζk

Additive fault effect εk = ε0

k + Hk Γk θk + Hk δk + Hk ζk

Γk is bounded Fk (I − Kk Hk) defines an exponentially stable LTV system

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Introduction Proposed algorithms Conclusion Fault effect First solution: rejecting the non-parametric fault Second solution: adapting to the parametric fault

Kitanidis filter (UI-KF) for rejecting Ek fk

Xk+1 = Fk Xk + Gk Uk + Wk + Ek fk Yk = Hk Xk + Jk Uk + Vk

• Xk+1 = Fk

Xk + GkUk + Fk Lk (Yk − JkUk − Hk Xk) Lk = Kk +(I − KkHk)Ek−1(ET

k−1HT k Σ−1 k HkEk−1)−1ET k−1HT k Σ−1 k

Kk = Pk HT

k Σ−1 k

Pk+1 = Fk(I − LkHk)Pk (I − LkHk)TF T

k + FkLkRkLT k F T k + Qk

Σk = Hk Pk HT

k + Rk

Kk

∆

= Lk

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Introduction Proposed algorithms Conclusion Fault effect First solution: rejecting the non-parametric fault Second solution: adapting to the parametric fault

Monitoring a constant parametric fault

Fault effect on the Kitanidis filter innovation εk = ε0

k + Hk ∆k θ

∆k+1 = Fk (I − Lk Hk)∆k + Ψk , ∆0 = 0 The Kitanidis filter innovation is white Proof in the notes. Use the GLR algorithm

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Introduction Proposed algorithms Conclusion Fault effect First solution: rejecting the non-parametric fault Second solution: adapting to the parametric fault

MLE of θ under H1 - Known fault profile matrix 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 algorithms Conclusion Fault effect First solution: rejecting the non-parametric fault Second solution: adapting to the parametric fault

Monitoring the non-parametric fault

While the GLR does not detect anything Run a Kalman filter based on the fault-free model

• Xk+1

= Fk Xk + GkUk + FkKk(Yk − JkUk − Hk Xk) Kk = Pk HT

k Σ−1 k

Pk+1 = Fk(I − Kk Hk)Pk F T

k + Qk

Σk = Hk Pk HT

k + Rk

Monitor its energy OK when dim(fk) ≥ dim(Yk), i.e. testing a Gaussian white noise against an arbitrary signal. More sophisticated tests might be considered in the case where dim(fk) < dim(Yk).

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Introduction Proposed algorithms Conclusion Fault effect First solution: rejecting the non-parametric fault Second solution: adapting to the parametric fault

Tracking a slowly time-varying θk

εk = ε0

k + Hk Γk θk + Hk δk + Hk ζk

RLS

• θk

=

• θk−1 + Lk
• εk − Hk Γk

θk−1

• ,

θ0

∆

= 0 Sk =

• λ Σk + Hk Γk Pk−1 ΓT

k HT k

−1 Lk = Pk−1 ΓT

k HT k Sk

Pk = λ−1 Pk−1 − Pk−1 ΓT

k HT k Sk Hk Γk Pk−1

• , P0

∆

= I Ek

∆

= εk − Hk Γk θk ; monitor its energy to detect Ek fk

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Introduction Proposed algorithms Conclusion

FDI for LTV systems with TV additive faults Constant parametric 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

Handling both TV parametric and non-parametric faults Two solutions

Assuming constant parametric fault and rejecting the non-parametric fault Adapting to the TV parametric fault

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