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

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 1 / 19

Introduction Overview Proposed algorithm Problem statement Example 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 2 / 19

Introduction Overview Proposed algorithm Problem statement Example Model and assumptions MIMO LTV system ( H 0 ) � X k + 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 ) controllable, both uniformly k Additive faults ( H 1 ) � X k + 1 = F k X k + G k U k + W k + Ψ k θ Y k = H k X k + J k U k + V k θ : unknown fault vector Ψ k : known TV fault profile 3 / 19

Introduction Overview Proposed algorithm Problem statement Example Different fault 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 ) Sensor faults: use a similar term Ψ k θ on the output equation (not treated here) Different fault occurrence speeds: ex: step change Ψ k ( r ) ∆ = � Ψ k × 1 l { k ≥ r } (and θ constant) A particular case Ψ k = δ r , k + 1 I : investigated by Willsky-Jones, Gustafsson with F k assumed exponentially stable 4 / 19

Introduction Fault effect Proposed algorithm Detection algorithm Example Isolation algorithm Fault effect on the Kalman filter innovation State prediction error and innovation - Fault free case ∆ � � X k = X k − X k | k − 1 ∆ H k � ε k = Y k − X k | k − 1 � F k ( I − K k H k ) � X 0 X 0 = − F k K k V k + W k k + 1 k H k � ε 0 X 0 = + V k k k State prediction error and innovation - Faulty case � F k ( I − K k H k ) � X k + 1 = X k − F k K k V k + W k + Ψ k θ H k � ε k = X k + V k 5 / 19

Introduction Fault effect Proposed algorithm Detection algorithm Example Isolation algorithm Introducing a matrix gain ∆ � η k = X k − Γ k θ ∆ ∆ Γ k + 1 = F k ( 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 � X 0 η k = k Additive fault effect ε k = ε 0 k + H k Γ k θ 6 / 19

Introduction Fault effect Proposed algorithm Detection algorithm Example Isolation algorithm Guaranteed properties of the recursive gain 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. The persistent excitation condition: � k Σ − 1 k Γ T k H T k H k Γ 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) 7 / 19

Introduction Fault effect Proposed algorithm Detection algorithm Example Isolation algorithm Known fault profile matrix MLE of θ under H 1 H 0 : ε k ∼ N ( 0 , Σ k ) , H 1 : ε k ∼ N ( H k Γ k θ, Σ k ) k � θ ) T Σ − 1 � ( ε j − H j Γ j � ( ε j − H j Γ j � θ ) = C − 1 θ k = arg min d k j k � θ j = 1 k Σ − 1 C k − 1 + Γ T k H T C k = H k Γ k k k Σ − 1 d k − 1 + Γ T k H T d k = ε k k GLR test = 2 ln p ( ε 1 , . . . , ε k | θ = � θ k ) ∆ k C − 1 p ( ε 1 , . . . , ε k | θ = 0 ) = d T l k d k k 8 / 19

Introduction Fault effect Proposed algorithm Detection algorithm Example Isolation algorithm Unknown jump fault onset time Handling the transient behavior after a jump Ψ k ( r ) = � Ψ k × 1 l { k ≥ r } = ⇒ Γ k + 1 ( r ) = F k ( I − K k H k ) Γ k ( r ) + Ψ k ( r ) Full treatment of the transient � C − 1 θ k ( r ) = k ( r ) d k ( r ) 1 ≤ r ≤ k d T k ( r ) C − 1 l k = max k ( r ) d k ( r ) 1 ≤ r ≤ k d T k ( r ) C − 1 ˆ r k = arg max k ( r ) d k ( r ) In practice Γ k ( r ) , C k ( r ) , d k ( r ) computed for r ∈ { k − w + 1 , k − w + 2 , . . . , k } 9 / 19

Introduction Fault effect Proposed algorithm Detection algorithm Example Isolation algorithm Locating non zero components   ε ˆ r + 1   . Z ∆ .  ∼ N ( M θ, S ) =  . ε k   r + 1 (ˆ H ˆ r + 1 Γ ˆ r )   . . M =  , S = diag (Σ ˆ r + 1 , . . . , Σ k )  . H k Γ k (ˆ r ) = M T S − 1 Z , F ∆ ζ ∆ = M T S − 1 M � θ a � θ = − θ a = 0 against θ a � = 0 , θ b nuisance parameter θ b 10 / 19

Introduction Fault effect Proposed algorithm Detection algorithm Example Isolation algorithm Minmax test − 1 ζ ∗ ∆ = ζ ∗ T F ∗ l a a a a ∆ ∆ ζ ∗ = ζ a − F ab F − 1 bb ζ b , F ∗ = F aa − F ab F − 1 bb F ba a a l a ∼ χ 2 ( dim ( θ a )) 11 / 19

Introduction A simulated example Proposed algorithm Numerical results Example 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  X k + 1 = ( F 0 + F p p k ) X k + ( G 0 + G p p k ) U k + K k e k   = ( H 0 + H p p k ) X k + ( J 0 + J p p k ) U k + Y k e k U k ∈ R : input mass flow, Y k ∈ R : output mass flow X k ∈ R 2 : mass flow and pressure drop within the first section p k ∈ R : scheduling parameter (pressure pattern) 12 / 19

Introduction A simulated example Proposed algorithm Numerical results Example Conclusion Simulated leakage Additive changes on G 0 , G p ∈ R 2 (actuator gain loss) - Monitoring the first component Nominal values: G 0 ( 1 ) = − 7 . 8297 e − 4 G p ( 1 ) = + 3 . 8290 e − 5 Changed values: G 0 ( 1 ) + 1 . 6 e − 5 Fault 1 G p ( 1 ) + 9 . 5 e − 6 Fault 2 Available data, simulated data U k provided by P . Lopes dos Santos et al. Y k simulated using the LPV model 13 / 19

Introduction A simulated example Proposed algorithm Numerical results Example Conclusion Fault detection - Fault 1 4 2 x 10 Fault ¡onset ¡*me ¡ Kalman ¡Innova*on ¡ 1 0 − 1 − 2 0 200 400 600 800 1000 1200 100 l k Alarm ¡*me ¡ Es*mated ¡fault ¡ 80 804 ¡min ¡ onset ¡*me ¡ 60 604 ¡min ¡ 40 20 0 0 200 400 600 800 1000 1200 Time ¡(min) ¡ 14 / 19

Introduction A simulated example Proposed algorithm Numerical results Example Conclusion Detection delay and onset time estimate - Fault 1 60 50 Detec%on ¡delays ¡ 40 30 20 10 0 0 50 100 150 200 250 200 Fault ¡onset ¡ 150 es%mate ¡errors ¡ 100 50 0 − 200 − 150 − 100 − 50 0 50 100 150 200 Time ¡(min) ¡ 15 / 19

Introduction A simulated example Proposed algorithm Numerical results Example Conclusion Detection delay and onset time estimate - Fault 2 120 100 Detec%on ¡delays ¡ 80 60 40 20 0 0 20 40 60 80 100 120 140 160 500 Fault ¡onset ¡ ¡ 400 es%mate ¡errors ¡ 300 200 100 0 − 200 − 150 − 100 − 50 0 50 100 150 200 Time ¡(min) ¡ 16 / 19

Introduction A simulated example Proposed algorithm Numerical results Example Conclusion Fault isolation with the minmax tests - Fault 1 8 Minmax ¡test ¡ focusing ¡on ¡fault ¡1 ¡ 6 Threshold ¡theore7c ¡ 4 1% ¡false ¡alarm ¡rate ¡ 2 0 0 20 40 60 80 100 120 140 160 180 30 Minmax ¡test ¡ 25 focusing ¡on ¡fault ¡2 ¡ 20 15 Threshold ¡theore7c ¡ 10 1% ¡false ¡alarm ¡rate ¡ 5 0 0 1 2 3 4 5 6 7 8 17 / 19

Introduction A simulated example Proposed algorithm Numerical results Example Conclusion Fault isolation with the minmax tests - Fault 2 500 Minmax ¡test ¡ Threshold ¡for ¡1% ¡ 400 focusing ¡on ¡fault ¡1 ¡ theore8c ¡false ¡alarm ¡rate ¡ ¡ ¡ 300 Empirical ¡false ¡ ¡ 200 alarm ¡rate ¡1.3% ¡ ¡ ¡ 100 0 0 5 10 15 20 25 100 Threshold ¡for ¡1% ¡ Minmax ¡test ¡ 80 theore8c ¡false ¡alarm ¡rate ¡ ¡ ¡ focusing ¡on ¡fault ¡2 ¡ 60 40 Empirical ¡miss-­‑ detec8on ¡rate ¡0.3% ¡ ¡ ¡ 20 0 0 50 100 150 200 250 300 350 400 18 / 19