Nonlinear FDI based on state derivatives, as provided by inertial measurement units Julien Marzat 8th IFAC Symposium on Nonlinear Control Systems Bologna, Italy, September 01-03 2010

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Outline Introduction Related work Objectives Principles of the approach Illustration of the procedure Aeronautical case study FDI algorithm description Simulation results Simulation set-up Robustness Summary and future work

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Nonlinear Fault Detection and Isolation - related work Fault Detection and Isolation of actuator faults for Nonlinear control-affine systems Differential-geometric approach (De Persis & Isidori) Transform coordinates to design nonlinear residual filters sensitive to faults and decoupled from disturbances. Differential-algebraic approach (Diop, Bokor, Shumsky...) Transform the system into a set of differential polynomials, functions of inputs, outputs and their successive derivatives. Use elimination theory to extract fault information. Inversion-based FDI (Edelmayer, Szigeti...) Compute the left-inverse to obtain a new dynamical model, outputs = faults, inputs = original inputs, outputs and their successive derivatives. NOLCOS 2010 - J.Marzat - 02/09/2010 - 3/14

Objectives Known drawbacks of those nonlinear methods Design of coordinate transforms, tuning of inner parameters Successive time derivatives of noisy and disturbed measurements Integration of dynamical filters Objectives of the present work Avoid numerical differentiation of measured variables Avoid dynamical integration, to reduce computational cost Assess robustness to model & measurement uncertainty Means of synthesis Take advantage of systems involving measured state derivatives (e.g., autonomous vehicles equipped with IMUs) Design a completely nonlinear actuator fault diagnosis method

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Principles of the approach

Residuals : discrepancy between computed and reconstructed inputs

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Planar aeronautical model (longitudinal) State vector : x = [x, z, vbx , vbz , q, θ]T , position, speed, orientation, Input vector : u = [δm , η]T , rudder and propulsion Measurements : y = [abx , abz , x, z, vbx , vbz , q, θ]T , acceleration Nonlinear model   x˙ = cos(θ)vbx + sin(θ)vbz      z ˙ = cos(θ)vbz − sin(θ)vbx    1  [fmin + (fmax − fmin )η]  abx = − QsMref [cx0 + cxa α + cxδm δm ] + M Qsref abz = −  M [cz0 + cza α + czδm δm ]      lref Qsref  √  c + c α + c δ + c q q ˙ = m0 ma mδm m mq  2 +v 2 b vbx  bz    θ˙ = q fmin , fmax , M, sref , lref constant parameters. Q, α, c(·) nonlinear functions of x

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Preliminary step Extract state equations containing control inputs, and involving only measured (or estimated) state variables and their measured derivatives ( 1 abx = − QsMref [cx0 + cxa α + cxδm δm ] + M [fmin + (fmax − fmin )η] Qsref abz = − M [cz0 + cza α + czδm δm ] Model reformulation:

"

e f1 e f2

#

 =

ge11 ge12

 e  f1 = abx + QsMref [cx0 + cxa α] − fmin  M    e    f2 = abz + QsMref [cz0 + cza α]  ge21 δ · m where ge11 = − QsMref cxδm 0 η   −fmin  ge12 = fmaxM     ge21 = − QsMref czδm

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Direct Residual Generation Substitute inputs by their computed values δmc and ηc  −e f1 +e g11 δmc +e g21 ηc   e11 g  r11 = e g21 ηc mc +e r21 = −f1 +eg11geδ21   e g12 δmc  r12 = −f2 +e e12 g If denominator too close to zero → residual rij not taken into account

Sensitivity to faults - example Inject expression of e f2 in residual r12 =

−e g12 δm + ge12 δmc = δmc − δm ge12

→ identification of the rudder fault NOLCOS 2010 - J.Marzat - 02/09/2010 - 8/14

Additional Residual Generation Further combinations between equations Here, δbm = e f2 /e g12 is used in r11 and r21 to get  e g11 (e g21 ηc f2 /e g12 )+e  e 1 = −f1 +e r11 e11 g e g11 (e g21 ηc f2 /e g12 )+e  e 1 = −f1 +e r21 e21 g Sensitivity to faults Inject expression of e f1 and e f2 in residual 1 e r21 =

− (e g11 δm + ge21 η) + ge11 (e g12 δm /e g12 ) + ge21 ηc = ηc − η ge21 → identification of the propulsion fault

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Simulation set-up Multiple actuator faults considered

Propulsion loss: ∀t > 10s, η(t) = 0.5ηcalc (t) Rudder locking-in-place: ∀t > 15s, δm (t) = δm (15s) IMU uncertainty

Measurement of q is q˜ = kq q + bq + wq kq : scale factor, bq : bias, wq : Gaussian white noise Extreme values considered for each measurement Delay of 2 time steps Multiplicative model uncertainty

Randomly, each aerodynamic coefficient value is csim = 0.95cmodel or csim = 1.05cmodel

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Trajectories

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Residuals

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Robustness of the residuals Model error ge11−sim = ge11−model + ε, ε small and bounded 1 e r21 =

ge11 1 [e g11 (−δm + δm − ε) + ge21 (−η + ηc )] = − εδm +ηc −η ge21 ge21

1 Zoom on residual e r21

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Summary and future work Summary Nonlinear FDI scheme applied to a realistic aeronautical model Multiple faults detectable, isolable and identifiable Static residuals : hard-coding possible, no tuning required Acceptable robustness to model and measurement uncertainty Formal description of the procedure in the paper MAPLE implementation Future work Extend to the 3D case (to be presented at IEEE SYSTOL 2010) Enhance residual analysis (statistical tests) Compare systematically with other FDI approaches

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