Dynamic Neural Networks for Actuator Fault Diagnosis: Application to the DAMADICS Benchmark Problem

Transcription

1 Dynamic Neural Networks for Actuator Fault Diagnosis: Application to the DAMADICS Benchmark Problem Krzysztof PATAN and Thomas PARISINI University of Zielona Góra Poland Dept. of Electrical, Electronic and Computer Engineering DEEI-University of Trieste, Italy

2 OUTLINE 1. Introduction 2. Dynamic neural network 3. Stochastic approximation 4. Sugar actuator 5. Experiments 6. Concluding remarks

3 Motivations INTRODUCTION Artificial neural networks a useful and efficient tool for modelling of nonlinear dynamic processes Fault detection applications a high model quality required Back-propagation based methods suffer from entrapment in local minima of an error function Objectives Designing of an actuator FDI system using dynamic neural networks Application of a stochastic algorithm for dynamic neural network training Evaluation of the proposed FDI system using real process data (sugar evaporation process)

4 Dynamic model with IIR filter DYNAMIC NEURAL NETWORKS u k 1( ) u k 2( ) u k P( ) w 1 w 2... w P Σ xk ( ) yk ~ ( ) IIR filter g c F(.) yk ( ) weighted adder filter module activation function ỹ(k) = x(k) = P w p u p (k) p=1 n b i x(k i) i= n a i ỹ(k i) i=1 y(k) = F (g ỹ(k) + c)

5 LEARNING PROCESS A vector of all unknown network parameters θ = [w, a, b, g, c] Optimization task θ = min θ C J(θ); θ Rp C a set of constraint, and cost function J(k; θ) = 1 N N number of learning patterns N (y(k) ŷ(k; θ)) 2 i=1

6 STOCHASTIC APPROXIMATION Recursive form of Stochastic Approximation (SA) ˆθ k+1 = ˆθ k a k ĝ k (ˆθ k ) where ĝ k (ˆθ k ) gradient estimate J/ ˆθ, a k small positive number Simultaneous Perturbation Stochastic Approximation (SPSA) Gradient estimate ĝ ki (ˆθ k ) = L(ˆθ k + c k k ) L(ˆθ k c k k ) 2c k ki where L( ) cost function measurement, k random perturbation vector, c k small positive number

7 Random distribution 1. Symmetrically distributed about zero 2. Finite inverse moments E( ki 1 ) Bernoulli ±1 distribution satisfies these conditions popular normal and uniform distributions do not fulfill condition 2 Gain sequences (i) a k, c k > k; lim a k ; k ( ) 2 ak (ii) a k = ; < k= Gains calculation k= c k a k = lim c k k a (A + k) α, c k = c k γ

8 Advantages To calculate gradient estimate 2 measurements needed Low numerical complexity in contrast to gradient based methods and other stochastic algorithms Useful in the case of noisy data Global optimization property 1. Chin D.C.: A more efficient global optimization algorithm based on Styblinski and Tang, Neural Networks, 7, pp , Maryak J.L. and Chin D.C.: Global random optimization by Simultaneous Perturbation Stochastic Approximation, Proc American Control Conference, June 21, Arlington, VA, pp

9 SUGAR ACTUATOR CV POSITIONER X positioner P S servo-motor T 1 P 1 X P 2 F control valve V C V V C V B

10 Causal graph P Z CV CVI P S X P V F measured variables variables realistic to measure unmeasurable variables F VC F V3 X 3 P 1 P 2 T 1 Servo-motor rod displacement X = r 1 (CV, P 1, P 2, T 1, X) Flow through the valve F = r 2 (X, P 1, P 2, T 1 )

11 Examined faulty scenarios f 1 f 2 f 3 positioner supply pressure drop interpretation: oversized system air consumption, air leading pipes breaks, etc. fault nature: rapidly developing unexpected pressure change across the valve interpretation: media pump station failure, increased pipes resistance, external media leakage fault nature: rapidly developing fully opened by-pass valve interpretation: valve corrosion, seat sealing wear, etc. fault nature: abrupt

12 EXPERIMENT Neural models 2 4 X F 3 5 = NN(P 1, P 2, T 1, CV ) F - flow through the valve, X - servo-motor rod displacement P 1 - pressure on the valve inlet P 2 - pressure on the valve outlet T 1 - juice temperature on the valve inlet CV - control signal Data preprocessing 1. Inputs normalized to zero mean and standard deviation of one 2. Outputs transformed to fall in the range [-1,1]

13 Modelling Neural models chosen using information criteria: AIC and FPE Fault Structure Filter order Activation function f N4,5,2 2 2 hyperbolic tangent f 1 N4,7,2 2 1 hyperbolic tangent f 2 N4,7,2 2 1 hyperbolic tangent f 3 N4,5,2 2 1 hyperbolic tangent Data - the sugar campaign 21 Assumed accuracy -.1 Parameters of the training procedure: SPSA: A = 1, a =.1, c =.1, α =.62, γ =.11

14 Decision making Let us assume that residual r(k) is N (m, v) A significance level β corresponds to probability that a residual exceeds a random value t β with N (, 1) β = prob( r(k) > t β ) Fixed threshold T = t β v + m Significance level used β =.5

15 Fault detection model (a) fault f 1 (b) fault f f f (c) fault f f

16 Fault isolation fault f 2, output F Fault model f 1 (a), fault model f 2 (b), fault model f 3 (c) a) fault 1 f b).2.1 fault f c) fault f Discrete time

17 Fault isolation fault f 2, output X Fault model f 1 (a), fault model f 2 (b), fault model f 3 (c) a) fault 1 f b) fault f c) -.5 fault f Discrete time

18 Fault diagnosis results Fault detection results (X detectable, N non detectable) Fault f 1 f 2 f 3 flow output X X X rod displacement output X X X Fault isolation results (X isolable, N non isolable) Fault f 1 f 2 f 3 flow output N N N rod displacement output N X X

19 Comparative study Sum of squared errors SSE Detection time t dt False detection rate r fd Isolation time t it False isolation rate r fi

20 Modelling quality SSE DN - Dynamic Network ARX - Auto-Regressive with exogenous input NNARX - Neural Networks ARX Method f f 1 f 2 f 3 F X F X F X F X DN ARX NNARX ,

21 FDI properies T f threshold used for the flow output T x threshold used for the rod displacement output Index DN NNARX f 1 f 2 f 3 f 1 f 2 f 3 t d t i r fd r id T f T x

22 CONCLUDING REMARKS Dynamic neural networks can be easily and effectively applied to design model-based fault detection and isolation systems Impossibility to model all potential system faults Data for faulty scenarios can be simulated Simultaneous Perturbation Stochastic Approximation algorithm strong alternative to gradient based methods useful when the search direction can not be determined accurately property of the global optimization