Neural Networks Lecture 10: Fault Detection and Isolation (FDI) Using Neural Networks H.A. Talebi Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Winter 2011. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 1/37
An Actuator Additive FDI [1] Example:FDI on Robotic Manipulation An Actuator Gain FDI [1] Example Flexible Joint Manipulator A Sensor and Actuator FDI [1] Example of Satellite Attitude Control H. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 2/37
Additive FDI Consider the nonlinear dynamics: ẋ = f (x, u) + N 1 (x, u) + T F (x, u, t) (1) f : known nonlinear fcn. N 1 (x, u): Unmodeled dynamics and uncertainties of nominal model (fault free), estimated by NN1 TF (x, u, t): vector valued fcn. of unknown actuator fault Objective: Diagnose and Isolate Actuator Fault: (T F ) By defining a Hurwitz matrix A, (1) can be rewritten ẋ = Ax + g(x, u) + N 1 (x, u) + T F (x, u, t) g(x, u) = f (x, u) Ax M(s) = (si A) 1. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 3/37
Additive FDI H. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 4/37
NN2 T f is estimated by NN 2 : ˆTf (x, Ŵ, ˆV, t) = Ŵ σ( ˆV x) A three layer MLP Series Parallel identifier The estimated model ˆx = Aˆx + g(x, u) + N 1 (x, u) + Ŵ σ( ˆV x) It has been shown that by updating the weights of NN 2 : Ŵ = η 1 ( J ) ρ Ŵ 1 x Ŵ ˆV = η 2 ( J ˆV ) ρ 2 x ˆV x = x ˆx J = 1 2 ( x T x): objective fcn to be minimized then x, W = W Ŵ, Ṽ = V ˆV are ultimately bounded The main advantage of the given approach: fault detection and Isolation is provided in a unified fashion. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 5/37
Example:FDI on Robotic Manipulation Dynamic of a robot manipulator M(q) q + C(q, q) + G(q) = τ + τ F q: manipulator joints position q: joint velocity τ: applied torque τf : unknown fault Based[ on the defined ] model: [ 0 T f = M 1 ; f (x, u) = τ F q M 1 (q)(τ C(q, q) + G(q) + τ F ) ]. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 6/37
Experimental Results on Puma 560 H. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 7/37
Example Puma 560 It has 6 joints The unmodelled dynamics (for Ex. friction in joint which are not modeled precisely) are estimated by N 1 The modified fault free description of Puma: M(q) q + C(q, q) + G(q) + N 1 (q, q, τ) = τ Define θ = q At this stage, a PD controller is considered to control the robots behavior: τ C = K p (θ ref θ) + K v ( θ ref θ) + G Desired position: θ ref = θ H + 0.3sin(t) θ H = [0 π/2 π/2 0 0 0]. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 8/37
Fault Free Experimental Results (a)-(c) joint positions; (d)-(f) Joint velocities. Solid lines: actual measurements; Dashed lines: their estimates.. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 9/37
Faulty Experimental Results Consider a constant actuator fault 5Nm in the first three joints: τ F = [5 5 5 0 0 0] Estimate τ F by NN2: a three layer NN input layers: 12 neurons (θ, θ) hidden layer 10 neurons output layer 6 neurons ˆτ F. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 10/37
The outputs of the neural network estimating the faults H. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 11/37
Experimental Results with 5 Nm fault (a)-(c) Joint positions (d)-(f)joint velocities. Solid lines: actual measurements Dashed lines:their estimates. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 12/37
An Actuator Gain FDI [1] Consider the nonlinear fault free dynamics: ẋ = f (x, u) + η x (x, u, t) (2) y(t) = Cx(t) + η y (x, u, t) u R m : input; x R n : state;y R m : output f : known nonlinear fcn. η x (x, u, t) R n R m R R n : Plant unmodeled dynamics and noise η y (x, u, t) R n R m R R m : Sensor unmodeled dynamics and disturbances C: an m n conts. matrix Assume a multiplicative actuator fault occur: u(t) u f (t) = T A (x, u, t)u(t) TA (x, u, t) = diag{t Ai (x, u, t)}, i = 1,..., m unknown matrix. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 13/37
An Actuator Gain FDI The faulty dynamics: ẋ = f (x, u f ) + η x (x, u, t) (3) y(t) = Cx(t) + η y (x, u, t) Objective: Diagnose and Isolate Actuator Fault: (T F ) By defining a Hurwitz matrix A, (3) can be rewritten ẋ = Ax + g(x, u f ) + η x (x, u, t) y(t) = Cx(t) + η y (x, u, t) g(x, u) = f (x, u) Ax M(s) = (si A) 1. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 14/37
An Actuator Gain FDI H. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 15/37
An Actuator Gain FDI The following assumptions should be made to facilitate the FDI design The nonlinear sys (3) is observable The nominal closed loop sys is asym. stable (ẋ = f (x, k(cx))) η x, η y are uniformly bounded η x η x, η y η y [ ] x ˆx g is Lipshtiz in x and u: g(x, u) g(ˆx, û) lg u û T f is estimated by NN 1 : ˆT f (x, Ŵ, ˆV, t) = Ŵ σ( ˆV x) A three layer MLP The estimated model ẋ = Aˆx + g(ˆx, û f ) ŷ(t) = C ˆx û f = Ŵ σ( ˆV x)u. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 16/37
An Actuator Gain FDI It has been shown that by updating the weights of NN 1 : Ŵ = η 1 ( J Ŵ ) ρ 1 ỹ Ŵ ˆV = η 2 ( J ˆV ) ρ 2 ỹ ˆV x = x ˆx ỹ = y ŷ W = W Ŵ Ṽ = V ˆV J = 1 2 (ỹ T ỹ): objective fcn to be minimized then x, ỹ, W, Ṽ are ultimately bounded This FDI can diagnose and isloate simultaneous fault. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 17/37
Example of Flexible Joint Manipulator [?] The dynamical model: D l (q 1 ) q 1 + C 1 (q 1, q 1 ) + g(q 1 ) + B 1 q 1 = τ s J q 2 + τ s + B 2 q 2 = τ q2 R n : motor shaft position g(q 1 ) R n : gravity force C 1 (q 1, q 1 ) R n : centrifugal and Coriolis force B1 R n n, B 2 R n n : viscus damping of input and output shaft D1 (q 1 ) R n n, J R n n robot and actuator inertia matrix τ input torque τ s = K(q 2 q 1 ) + β(q 1, q 2, q 1, q 2 ): reaction torque from rotational spring K: stiffness matrix of rotational spring represent the flexibility between the input and output shaft β(q1, q 2, q 1, q 2 ): a combination of nonlinear spring and friction at output shafts of the manipulator The output vector y = [q21 q 22 q 21 q 22 ] T. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 18/37 q1 R n : link position
Numerical Data of Two Link Flexible Joint Manipulator J = diag{1.16, 1.16}, m = diag{1, 1} l 1 = l 2 = 1m, K = diag{100, 100}, η = 0.3, ρ 1 = ρ 2 = 0.001 A = 2I 8 A PD controller is used to stabilize the open-loop system β is uncertainty in dynamics and unknown Sensory measurement is corrupted with a 10% Gaussian noise The NN FDI: A Three layer NN 10 neurons in input layer; 10 neurons in hidden layer; 2 neurons in output layer ( faults in actuator τ 1 and τ 2 The actuator { fault in both joints: diag{0.7 0.4} t [200 300]secs T A = diag{1 1} otherwise. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 19/37
Example Flexible Joint Manipulator (a) angular velocities of the first link (b) angular velocity of the second link Red lines: real signals; Blue lines: the estimated signals by NN. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 20/37
(a) Estimated fault in the first actuator (b) Estimated fault in the second actuator (c) Estimated fault in the first actuator; a closer look (d) Estimated fault in the second actuator; a closer look. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 21/37
A Sensor and Actuator FDI [1] Consider the nonlinear dynamics: ẋ = f (x, u) + η x (x, u, t) + T A (x, u) (4) y(t) = Cx(t) + η y (x, u, t) + T s (x, u) u R m : input; x R n : state;y R m : output f : known nonlinear fcn. η x (x, u, t) R n R m R R n : Plant unmodeled dynamics and noise ηy (x, u, t) R n R m R R m : Sensor unmodeled dynamics and disturbances C: an m n conts. matrix T s R n R m R m : unknown sensor fault T A R n R m R n : unknown actuator fault. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 22/37
A Sensor and Actuator FDI Objective: Diagnose and Isolate Actuator and Sensor Faults: (T A, T S ) By defining a Hurwitz matrix A, (4) can be rewritten ẋ = Ax + g(x, u) + η x (x, u, t) + T A (x, u) y(t) = Cx(t) + η y (x, u, t) + T s (x, u) g(x, u) = f (x, u) Ax M(s) = (si A) 1. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 23/37
A Sensor and Actuator FDI. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 24/37
A Sensor and Actuator FDI The following assumptions should be made Sensor and actuator fault will not occur simultaneously The nonlinear sys (4) is observable The nominal closed loop sys is asym. stable (ẋ = f (x, k(cx))) η x, η y are uniformly bounded η x η x, η y η y g is Lipshtiz in x: g(x, u) g(ˆx, u) l g x ˆx T A is estimated by NN 1 and T S is estimated by NN 2 : ˆT A (ˆx, Ŵ 1, ˆV 1, t) = Ŵ 1 σ( ˆV 1 ˆx) ˆT S (ˆx, Ŵ2, ˆV 2, t) = Ŵ2σ( ˆV 2 ˆx) Each is a three layer MLP The estimated model ˆx = Aˆx + g(ˆx, u) + Ŵ 1 σ( ˆV 1 ˆx) ŷ(t) = C ˆx + Ŵ 2 σ( ˆV 2 ˆx). A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 25/37
A Sensor and Actuator FDI It has been shown that by updating the weights of NN 1 and NN 2 : Ŵ 1 = η 1 ( J ) ρ 1 ỹ Ŵ 1 Ŵ1 ˆV 1 = η 2 ( J ˆV ) ρ 2 ỹ ˆV 1 1 Ŵ 2 = η 3 ( J ) ρ Ŵ 3 ỹ Ŵ2 2 ˆV 3 = η 4 ( J ) ρ ˆV 4 ỹ ˆV 4 3 x = x ˆx ỹ = y ŷ Wi = W i Ŵi, i = 1, 2 Ṽ i = V i ˆV i, i = 1, 2 J = 1 2 (ỹ T ỹ): objective fcn to be minimized then x, ỹ, W i, Ṽ i, i = 1, 2 are ultimately bounded. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 26/37
Example of Satellite Attitude Control Dynamical equation of satellite attitude model with magnetorquer actuator H ω = S(ω)Hω + BM ω = [ω 1 ω 2 ω 3 ] T : angular velocity of satellite 0 B z (t) B y (t) B(t) = B z (t) 0 B x (t) : magnetic field matrix B y (t) B x (t) 0 H: Symmetric P.D. inertia matrix S(ω) = 0 ω 3 ω 2 ω 3 0 ω 1 : cross product matrix ω 2 ω 1 0 M: magnetic dipole of the coil. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 27/37
Example of Satellite Attitude Control To define it in standard state space representation: u = M; f (x, u) = H 1 (B(t)u S(ω)Hω); x = ω the unmodeled dynamics: η x = H 1 (B(t) B)u; B = 1 T T 0 B(t)dt Magnetorquer fault (actuator fault) is considered as an additive term τ a : ẋ = f (x, u) + η x +T A ; T A = H 1 τ a Magnetometer fault (sensor fault): B m = B + EB + B + ν Bm : measured magnetic field B: true magnetic field B: magnetic bias ν measurement noise E = 0 E z E y E z 0 E x : magnetormeter misalignment matrix E y E x 0 any fault or misalignment can be considered as uncertainty vector ηy or additive fault T S. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 28/37
Numerical Data of φrsted Satellite H = 181.78 0 0 0 181.25 0 0 0 1.28 kg.m 2 All sensory measurements are assumed to be corrupted with 10% Gaussian noise The actuator fault is estimated by NN 1 and sensor fault by NN 2. They have: 3 layers 6 neurons in input layer; 5 neurons in hidden layer with sigmoidal activation fcn; 3 neurons in output layer with linear activation fcn. η1 = η 2 = 20; η 3 = η 4 = 0.1; ρ i = 1e 6, i = 1, 2, 3, 4 A = 2I 3 ˆTA = [ ˆT A1 ˆTA2 ˆTA3 ] T : output of NN 1 ˆTS = [ ˆT S1 ˆTS2 ] T : output of NN 2. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 29/37
Example of Satellite Attitude Control Note that: the output of the networks do not approach to zero due to presence of η y and η x Therefore a threshold should be considered for fault detection The threshold can be chosen by by adaptive, fuzzy or...approached In this simulation it has been chosen by trial and error A fault is detected if: ˆT Ai > th ai, i = 1,..., 3 (actuator fault) ˆT Si > th si, i = 1,..., 2 (sensor fault) t [t f, t f + t r ] th ai = [ 0.02 0.02] threshold of i th actuator th si = [ 0.02 0.02] threshold of i th sensor tr = 30sec.. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 30/37
Fault Free angular velocities, trajectory: 0.1sin(0.001t) Solid lines: actual velocities Dashed lines: Estimated velocities H. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 31/37
NN 1 and NN 2 responses to fault free operation Solid lines: estimated faults; Dashed lines: thresholds (a)-(c): Estimated actuator faults (d)-(e): Estimate sensor faults H. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 32/37
Angular velocities with bias fault: T A = [0.1 0 0] T Solid lines: actual velocities Dashed lines: Estimated velocities H. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 33/37
NN 1 and NN 2 responses to bias fault: T A = [0.1 0 0] T Solid lines: estimated faults; Dashed lines: thresholds (a)-(c): Estimated actuator faults (d)-(e): Estimate sensor faults H. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 34/37
NN 1 and NN 2 responses; traj: 0.1rad/s, fault: T S = [0.03 0.05] T Solid lines: estimated faults; Dashed lines: thresholds (a)-(c): Estimated actuator faults (d)-(e): Estimate sensor faults H. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 35/37
NN 1 and NN 2 responses; traj: 0.1sin(0.001t), fault free, with sensor and state uncertainties Solid lines: estimated faults; Dashed lines: thresholds (a)-(c): Estimated actuator faults (d)-(e): Estimate sensor faults H. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 36/37
References H.A.Talebi, F. Abdollahi, R.V.Patel, and K.Khorasani, Neural Network-Based State Estimation of Nonlinear Systems: Application to Fault Detection and Isolation. Springer, 2009.. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 10 37/37