Handling parametric and non-parametric additive faults in LTV Systems

Transcription

1 1 / 16 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

2 2 / 16 Introduction Problem and approaches Overview Problem statement 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

3 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 3 / 16

4 Model and assumptions Overview Problem statement MIMO LTV system (H 0 ) { Xk+1 = F k X k + G k U k + W k Y k = H k X k + J k U k + V k F k, G k, H k, J k : bounded TV matrices W k, V k : independent white Gaussian noises, TV cov. Q k, R k (H k, F k ) observable & (F k, Q 1/2 k ) controllable, both uniformly Additive faults (H 1 ) { Xk+1 = F k X k + G k U k + W k + Ψ k θ k + E k f k Y k = H k X k + J k U k + V k known fault profile matrix Ψ k, unknown fault vector θ k known fault incidence matrix E k, unknown fault profile vector f k 4 / 16

5 Overview Problem statement Different fault cases E k f k and Ψ k θ k typically represent actuator faults f k : no a priori information; θ k : 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: U k U k + θ; then Ψ k = G k Actuator gain loss: U k (I diag(θ))u k ; then Ψ k = G k diag(u k ) Ψ k = δ r,k+1 I: investigated by Willsky-Jones, Gustafsson with F k assumed exponentially stable 5 / 16

6 6 / 16 Introduction on the innovation of a linear filter State prediction error and innovation - Fault free case X k = Xk X k k 1 ε k = Yk J k U k H k Xk k 1 X 0 k+1 = F k (I K k H k ) X 0 k F k K k V k + W k ε 0 k = H k X 0 k + V k State prediction error and innovation - Faulty case X k+1 = F k (I K k H k ) X k F k K k V k + W k + Ψ k θ k + E k f k ε k = H k Xk + V k

7 7 / 16 Introducing a matrix gain η k = Xk Γ k θ k Γ k+1 = Fk (I K k H k ) Γ k + Ψ k, Γ 0 = 0 η k+1 = F k (I K k H k ) η k F k K k V k + W k Γ k+1 (θ k+1 θ k ) + E k f k Distinguishing two cases for the parametric fault vector Constant θ TV θ k

8 8 / 16 Constant parametric fault vector θ k = θ Let ζ k+1 = Fk (I K k H k ) ζ k + E k f k, ζ 0 = 0 cf. Γ k+1 = Fk (I K k H k ) Γ k + Ψ k, Γ 0 = 0 η k+1 = F k (I K k H k ) η k F k K k V k + W k + E k f k η k = X 0 k + ζ k Additive fault effect ε k = ε 0 k + H k Γ k θ + H k ζ k

9 9 / 16 Guaranteed properties of the recursive Γ k and ζ k Γ k depends on the fault gain Ψ k, not on the fault vector θ. The matrix gain Γ k computed from the bounded Ψ k is bounded even when the system is not stable. Similarly, if f k is bounded, ζ k is bounded. The persistent excitation condition: k ΓT k HT k Σ 1 k H kγ k 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 F k (thus required to be stable)

10 10 / 16 TV parametric fault vector θ k+1 = θ k + e k, e k δ Let δ k+1 = Fk (I K k H k ) δ k Γ k+1 e k, δ 0 = 0 Additive fault effect η k = X 0 k + δ k + ζ k ε k = ε 0 k + H k Γ k θ k + H k δ k + H k ζ k Γ k is bounded F k (I K k H k ) defines an exponentially stable LTV system

11 11 / 16 Kitanidis filter (UI-KF) for rejecting E k f k { Xk+1 = F k X k + G k U k + W k + E k f k Y k = H k X k + J k U k + V k X k+1 = F k Xk + G k U k + F k L k (Y k J k U k H k Xk ) L k = K k +(I K k H k )E k 1 (Ek 1 T HT k Σ 1 k H ke k 1 ) 1 Ek 1 T HT k Σ 1 k K k = P k Hk T Σ 1 k P k+1 = F k (I L k H k )P k (I L k H k ) T F T k + F kl k R k L T k F T k + Q k Σ k = H k P k H T k + R k K k = Lk

12 12 / 16 Introduction Monitoring a constant parametric fault on the Kitanidis filter innovation ε k = ε 0 k + H k k θ k+1 = F k (I L k H k ) k + Ψ k, 0 = 0 The Kitanidis filter innovation is white Proof in the notes. Use the GLR algorithm

13 13 / 16 MLE of θ under H 1 - Known fault profile matrix H 0 : ε k N (0, Σ k ), H 1 : ε k N (H k Γ k θ, Σ k ) θ k = arg min θ GLR test k j=1 (ε j H j Γ j θ) T Σ 1 j C k = C k 1 + Γ T k HT k Σ 1 k d k = d k 1 + Γ T k HT k Σ 1 k (ε j H j Γ j θ) = C 1 k H k Γ k ε k d k l k = 2 ln p(ε 1,..., ε k θ = θ k ) p(ε 1,..., ε k θ = 0) = d T k C 1 k d k

14 Monitoring the non-parametric fault While the GLR does not detect anything Run a Kalman filter based on the fault-free model X k+1 = F k Xk + G k U k + F k K k (Y k J k U k H k Xk ) K k = P k H T k Σ 1 k P k+1 = F k (I K k H k )P k F T k + Q k Σ k = H k P k H T k + R k Monitor its energy OK when dim(f k ) dim(y k ), i.e. testing a Gaussian white noise against an arbitrary signal. More sophisticated tests might be considered in the case where dim(f k ) < dim(y k ). 14 / 16

15 15 / 16 Tracking a slowly time-varying θ k RLS ε k = ε 0 k + H k Γ k θ k + H k δ k + H k ζ k θ k = θ ( ) k 1 + L k ε k H k Γ k θk 1, θ0 = 0 ( ) 1 S k = λ Σ k + H k Γ k P k 1 Γ T k HT k L k = P k 1 Γ T k HT k S k P k = λ 1 ( P k 1 P k 1 Γ T k HT k S k H k Γ k P k 1 ), P 0 = I E k = εk H k Γ k θk ; monitor its energy to detect E k f k

16 16 / 16 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