Design of observers for Takagi-Sugeno fuzzy models for Fault Detection and Isolation

Design of observers for Takagi-Sugeno fuzzy models for Fault Detection and Isolation Abdelkader Akhenak, Mohammed Chadli, José Ragot and Didier Maquin Institut de Recherche en Systèmes Electroniques Embarqués Rouen France Laboratoire de Modélisation, Information et systèmes Amiens France Centre de Recherche en Automatique de Nancy Nancy France 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, Barcelona, Spain, 30th June 3rd July, 2009 Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 1 / 26

Motivations Goal To develop an observer design method for Takagi-Sugeno models subjected to unknown inputs To use the proposed observer in a bank of observers for fault detection and isolation Context To take into consideration the complexity of the system in the whole operating range (nonlinear models are needed) Observer design problem for generic nonlinear models is very delicate Proposed strategy Multiple model representation of the nonlinear system Design an unknown input observer on the basis of that model Convergence conditions are obtained using the Lyapunov method Conditions are given under a LMI form Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 2 / 26

Outline 1 Multiple model approach for modeling 2 Unknown input observer design Structure of the unknown input observer Unknown input observer existence conditions: main result 3 Model of an automatic steering 4 Fault detection and isolation 5 Conclusions Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 3 / 26

Multiple model approach for modeling Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 4 / 26

Multiple model approach Operating range decomposition into several local zones. A local model represents the behavior of the system in each zone. The overall behavior of the system is obtained by the aggregation of the sub-models with adequate weighting functions. ξ 1(t) ξ 1(t) zone 1 Operating space zone 2 zone 3 Nonlinear system ξ 2(t) zone 4 ξ 2(t) Multiple model representation Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 5 / 26

Multiple model approach Why using a multiple model? Appropriate tool for modelling complex systems (e.g. black box modelling) Tools for linear systems can partially be extended to nonlinear systems Specific analysis of the system nonlinearity is avoided Takagi-Sugeno model The considered model is described by the following equations 8 >< NX ẋ(t) = µ i (ξ(t))(a i x(t) + B i u(t) + R i ū(t) + D i ) i=1 >: y(t) = Cx(t) Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 6 / 26

Multiple model approach Why using a multiple model? Appropriate tool for modelling complex systems (e.g. black box modelling) Tools for linear systems can partially be extended to nonlinear systems Specific analysis of the system nonlinearity is avoided Takagi-Sugeno model The considered model is described by the following equations 8 >< NX ẋ(t) = µ i (ξ(t))(a i x(t) + B i u(t) + R i ū(t) + D i ) i=1 >: y(t) = Cx(t) Interpolation mechanism : NX µ i (ξ(t)) = 1 and 0 µ i (ξ(t)) 1, t, i {1,..., N} i=1 The premise variable ξ(t) is assumed to be measurable (input u(t) or output y(t)). Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 6 / 26

Modeling using a Takagi-Sugeno structure Two main different approaches Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 7 / 26

Modeling using a Takagi-Sugeno structure Two main different approaches Identification approach Choice of premise variables Choice of the number of modalities of each premise variable Choice of the structure of the local models Parameter identification Transformation of a nonlinear model into a multiple model 8 8 P < ẋ(t) = f(x(t), u(t)) >< ẋ(t) = N µ i (ξ(t))`a i x(t) + B i u(t) + D i i=1 : P y(t) = Cx(t) >: y(t) = N µ i (ξ(t))c i x(t) Exact representation: sector nonlinearity approach (the premise variables depend on the state of the system) Linearization of the model around different operating points i=1 Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 7 / 26

Unknown input observer design Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 8 / 26

Structure of the unknown input observer Model 8 >< >: ẋ(t) = y = Cx NX µ i (ξ) (A i x + B i u + R i ū + D i ) i=1 Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 9 / 26

Structure of the unknown input observer Model 8 >< >: ẋ(t) = y = Cx NX µ i (ξ) (A i x + B i u + R i ū + D i ) i=1 Assumption: ū(t) is bounded ū < ρ where ρ is a positive scalar Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 9 / 26

Structure of the unknown input observer Model 8 >< >: ẋ(t) = y = Cx NX µ i (ξ) (A i x + B i u + R i ū + D i ) i=1 Assumption: ū(t) is bounded ū < ρ where ρ is a positive scalar Observer 8 >< NX ˆx = µ i (ξ) A iˆx + B i u + R iˆū i + D i + G i (y Cˆx) i=1 ˆū i = γw i`y Cˆx >: ˆū = NX µ i (ξ) ˆū i i=1 Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 9 / 26

Unknown input observer existence conditions: main result Theorem The unknown input observer estimates asymptotically with any desired degree of accuracy ε > 0, the state of the T-S model, if there exists symmetric positive definite matrices P and Q and gain matrices G i and W i which satisfies the following constraints: 8 < (A i G i C) T P + P(A i G i C) < Q i [1, N] : W i C = Ri T P The observer is then completely defined by choosing: γ 1 2 and the unknown input estimation is given by λ min (P 1 Q)λ min (P) ε 2 1 ρ 2 ˆū = γ NX µ i (ξ) W i`y Cˆx i=1 Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 10 / 26

Sketch of the proof Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 11 / 26

Sketch of the proof (i) Consider the following quadratic Lyapunov function: V = e T P e (ii) Express the state estimation error e(t) = x ˆx and its derivative w.r.t. time ė = NX i=1 µ i (ξ) (A i G i C)e + R i ū γr i W i Ce (iii) Express the derivative of the Lyapunov function using ė V = NX i=1 µ i (ξ) e T ((A i G i C) T P + P(A i G i C))e + 2e T PR i ū 2γe T PR i W i Ce (iv) Use some upper bound properties and guarantee that V < 0 (v) See the proceedings for a detailed proof Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 11 / 26

Application to automatic steering of vehicle Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 12 / 26

Representation of the vehicle model by a T-S model Coupling model of longitudinal and lateral motions of a vehicle Variables in red u v r longitudinal velocity, lateral velocity, yaw rate, u = vr fg + (fk 1 k 2 ) M v = ur (c f + c r) Mu u 2 v + ar + c f Mu δ + T M v + (bcr ac f) r + c fδ + Tδ Mu M ṙ = (bcr ac f) `b2 c r + a 2 c f v r + atδ + ac fδ I zu I zu I z δ T steering angle, traction and/or braking force Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 13 / 26

Representation of the vehicle model by a T-S model Model parameters M Mass of the full vehicle 1480 kg I z Moment of inertia 2350 kg.m 2 g Acceleration of gravity force 9.81 m/s 2 f Rotating friction coefficient 0.02 a Distance from front axle to CG a 1.05 m b Distance from rear axle to CG 1.63 m c f Cornering stiffness of front tyres 135000 N/rad c r Cornering stiffness of rear tyres 95000 N/rad k 1 Lift parameter from aerodynamics 0.005 Ns 2 /m 2 k 2 Drag parameter from aerodynamics 0.41 Ns 2 /m 2 a CG: Center of gravity Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 14 / 26

Representation of the vehicle model by a T-S model General form of the nonlinear model 0 1 u where x = @ va, w = δ! r T Takagi-Sugeno model ẋ = A i = F x, B x=x (i) i = w=w (i) ẋ = F(x, w) y = Cx and C = 1 0 0!, 0 0 1 NX µ i (y 1 )(A i x + B i w + D i ) i=1 F w x=x (i) w=w (i), D i = F(x (i), w (i) ) A i x (i) B i w (i) Membership functions µ i (y 1 ) are chosen as triangular functions The number N of local models is chosen by the user (here N = 3) The operating points around which the linearization is done are optimized. Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 15 / 26

Representation of the vehicle model by a T-S model Takagi-Sugeno model A 11i = 2(fk 1 k 2 ) u i M 0 A 11i A 12i 1 A 13i A i = @ A 21i A 22i A 23i A A 31i A 32i A 33i c f (v i + ar i ) δ Mui 2 i, A 12i = r i + c fδ i Mu i, A 13i = v i + ac fδ i Mu i A 21i = (c f + c r) Mu 2 i A 31i = `b2 c r + a 2 c f I zu 2 i v i (bcr ac f) r Mui 2 i r i, A 22i = (c f + c r), A 23i = (bcr ac f) u i Mu i Mu i 0 B i = B @ r i (bcr ac f) v I zui 2 i, A 32i = (bcr ac f) `b2 c r + a 2 c f, A 33i = I zu i I zu i v c i +ar i 1 f Mu i M c f +T i M at i +ac f I z δ i M aδ i I z 1 C A,» D δi i = F(x i, δ i, T i ) A i x i B i T i Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 16 / 26

Representation of the vehicle model by a T-S model Takagi-Sugeno model validation 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0 5 10 15 20 25 30 Figure: δ: steering angle 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 T : traction force Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 17 / 26

Representation of the vehicle model by a T-S model Takagi-Sugeno model validation 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 µ 1 (u(t)) µ 2 (u(t)) µ 3 (u(t)) 18.5 18 17.5 17 16.5 16 15.5 15 14.5 0 14 15 16 17 18 19 Figure: Membership functions 14 0 5 10 15 20 u issued from nonlinear and TS models Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 18 / 26

Representation of the vehicle model by a T-S model Takagi-Sugeno model validation 18.5 18 17.5 17 16.5 16 15.5 15 14.5 14 0 5 10 15 20 Figure: v issued from nonlinear and TS models 1.5 1 0.5 0 0.5 1 1.5 0 5 10 15 20 r issued from nonlinear and TS models Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 19 / 26

Fault detection and isolation Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 20 / 26

Fault detection and isolation System model with additive faults with m = m1 m 2 ẋ = F(x, w) + Rm y = Cx + η «0 1 1 0, R = @ 0 1A, η N(0, V) 1 0 m 1 is a piece-wise constant signal with random amplitude. m 2 is a sinusoïdal signal Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 21 / 26

Fault detection and isolation Proposed unknow input observer 8 3X ˆx = µ i (ξ) A iˆx + B i u + R ˆm i + D i + G i (y Cˆx) i=1 >< ˆm i = γw i`y Cˆx >: ˆm = NX µ i (ξ) ˆm i i=1 Observer parameters 0 1 0 1 0 1 9.22 3.88 10.78 4.10 8.49 4.83 G 1 = @ 0.45 1.02 A, G 2 = @ 6.64 0.55 A, G 3 = @ 4.40 1.36 A 22.51 11.92 27.38 16.19 20.27 17.07 «34.14 0 γ = 78.12, W 1 = W 2 = W 3 = 0 10 Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 22 / 26

Fault detection and isolation State estimates K1 K2 F1 F2 19 1.5 18 17 1 16 15 0.5 14 0 13 12 0.5 11 10 0 5 10 15 20 1 0 5 10 15 20 Figure: longitudinal velocity u and lateral velocity v and their estimates Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 23 / 26

Fault detection and isolation Unknown input estimates u1 u2 H1 H2 0.6 0.8 0.6 0.4 0.2 0.4 0.2 0 0.2 0.4 0 0 5 10 15 20 0.6 0.8 0 5 10 15 20 Figure: fault m 1 and its estimate fault m 2 and its estimate Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 24 / 26

Conclusions & Prospects Conclusions A method for the estimation of the state and the unknown input of a nonlinear system has been proposed The proposed method uses a Takagi-Sugeno model to represent the behaviour of the nonlinear system Feasibility of sensor fault detection on a complex nonlinear model has been shown Futures prospects On a theoretical point of view Try to use another Lyapunov function in order to enlarge the stability region More precise analysis of the noise influence (particularly on the unknown input estimation) On a practical point of view Apply the proposed method on real data Implement the method on a vehicule Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 25 / 26

Thank you! Didier Maquin (CRAN) Observers for TS models Safeprocess 2009 26 / 26