Modélsaon de la dééroraon basée sur les données de survellance condonnelle e esmaon de la durée de ve résduelle T. T. Le, C. Bérenguer, F. Chaelan Unv. Grenoble Alpes, GIPSA-lab, F-38000 Grenoble, France CNRS, GIPSA-lab, F-38000 Grenoble, France Journées de l Auomaque GdR MACS
Conex FP7 European projec: SUsanable PREdcve Manenance for manufacurng Equpmen Purpose: Developmen of new ools for predcve manenance o mprove producvy, reduce machne downmes and ncrease energy effcency. Applcaon case: Paper machne Man objecves: Deeroraon modelng Remanng Useful Lfe (RUL) esmaon SUPREME parners Predcve manenance 2
Problem saemen Deeroraon models Represen emporal evoluon of defecs,.e. of healh ndcaors. Healh ndcaors: unobservable Observaons: nosy condon monorng daa => Sae-space represenaon x = f ( x, ω ) : Hdden saes y = g( x, ν ) : Observaons Two ype of saes Connuous Dscree Connuous-sae modelng Dscree-sae modelng 3
Problem saemen Remanng Useful Lfe (RUL) Z ( ) RUL Condonal random varable: ( ) T T >, Z f f 0 Sar of operaon T f Z(): nformaon up o me ; : me o falure Unceranes assessmen: characerze he RUL by a probablsc dsrbuon RUL predcon example T f 4
Problem saemen Problems Co-exsence of mulple deeroraon modes n compeon Proposed soluon: Mul-branch modelng Dscree healh sae: Markov based => Mul-branch Hdden Markov model (MB-HMM) Sem-Markov based => Mul branch Hdden sem-markov model (MB-HsMM) Connuous healh sae : Jump Markov lnear sysems 5
Oulne Mul-branch dscree-sae model Dagnoscs and Prognoscs framework Numercal sudes Jump Markov lnear sysems Parameers learnng Healh assessmen and RUL esmaon Numercal sudy Concluson & Perspecves 6
Mul-branchdscree-sae models 7
Mul-branch dscree-sae modelng Mul-branch M deeroraon modes M branches Mode probably : Observaons: connuous Fnal sae: sysem falure Sae ranson probables are dfferen beween he branches => Dfferen deeroraon raes Assumpons π k M k= π k = Monoonc deeroraon: Lef-rgh opology Faul deecon s perfec: Inal and falure saes are non-emng saes Deeroraon modes are exclusve once naed => No branches swchng 8
Mul-branch dscree-sae modelng Mul-branch Hdden Markov Model Each branch ~ lef-rgh Markov chan Markovan propery Sae sojourn me: Exponenal or geomercal dsrbued May no be rue n pracce Mul-branch Hdden sem-markov Model Each branch ~ lef-rgh sem-markov chan Sem-Markov propery: relax he Markovan assumpon => allow arbrary dsrbuons for sojourn me: Gaussan, Webull, 9
Dagnoscs and prognoscs framework Two-phase mplemenaon: offlne & onlne 0
Off-lne phase Model ranng Tranng daa: Hgh-level feaures exraced from condon monorng daa Topology selecon: BIC creron Daa classfcaon => M groups Each group s used o ran a consuen branch: MB-HMM: Adapon of he Baum-Welch algorhm MB-HsMM: Adapon of he Forward-Backward procedure [Yu 2006] A prormode probables π ( λ ) /, = P = K K k = K M k k k K k s he number of ranng sequences correspondng o he mode k *[Yu06]: Praccal mplemenaon of an effcen forward-backward algorhm for an explc-duraon hdden Markov model. IEEE Transacons on Sgnal Processng, 54(5), 947-95.
On-lne phase Dagnoss Mode deecon: ( ) kˆ = arg max P λk O Healh-sae assessmen: Verb algorhm Deermne of he bes sae sequence: Consder he las sae as he acual sae k ˆ ( ˆ ˆ ) Q = arg max P O, Q λ Q k k k 2
RUL esmaon One branch (HMM case) Suppose ha he sysem s followng he mode k RUL : dscree me assumpon => number of ranson seps o reach for he s me he falure sae: ( l) ( = = ) = (,,, ) RUL = P RUL l q S P q = S q S q S q = S + l N + l N + N Lef-rgh HMM: Gven he curren sae, he sysem can eher say n he same sae or jump o he nex one Recursve compuaon 3
RUL esmaon (con.) One branch (HMM case) A sae : A sae :...... A sae : RUL SN 2 () N RUL = a =... S () 2 ( N 2) N 0 ( l) ( l ) ( l ) N 2 = a( N 2)( N 2) RULN 2 + ( N 2)( N ) ULN RUL a R... SN () N = ( N ) N RUL a... ( l) ( l ) N = ( N )( N ) R N RUL a UL = a N ( l) ( l ) ( l ) = a + a( + ) + RUL RUL RUL 4
RUL esmaon (con.) One branch (HsMM case) Srcly lef-rgh model: RUL N = D + D j j= + : Sojourn me n saes j D j N D = D D D > D j= + D j ~ Normal dsrbuon ~ runcaed Normal dsrbuon RUL = sum of Normal dsrbuon + runcaed Normal dsrbuon Bayesan Model Averagng Take no accoun model uncerany: P( RUL O) = P( RUL λk, O) P( λk O) M k= 5
Numercal examples Smulaed deeroraon daa Fague Crack Growh (FCG) model o represen he evoluon of a crack deph Observaon model: Mul-mode: Two propagaon raes Crack deph s proporonal wh ( γ β ) e w b x = x + e C e x y = x + ξ C = 0.005, n =.3, σ w =.7 2 γ e = [ 0 0.75] T σ ξ = 0, L = 00, π = π = 0.5 2 γ e n Two-mode ranng daa 6
Numercal examples Onlne RUL esmaon (MB-HsMM model) 7
Numercal examples Mul-branch model vs. Average model Ams: Invesgae he advanages of he mul-branch models Mehod: Evaluae RUL esmaon resuls a dfferen mode dsances Mode dsance : proporonal o Creron: Roo mean squared error (RMSE) Resul: γ e Average model 8
Case sudy (MB-HsMM model): PHM08 compeon C-MAPSS: Modelng a large realsc commercal urbofan engne 2 daa se for ranng and es 28 dencal and ndependen uns Objecve: Consruc a prognosc mehod basng on ranng daa se Use o esmae he RUL of each un n es daa se Evaluaon creron: S 28 = = Where s penaly score for un : S = S S d /3 e d d /0 e d >, 0, 0 where = es real d RUL RUL *C-MAPSS: Commercal Modular Aero-Propulson Sysem Smulaon Smplfed dagram of engne smulaed n C-MAPSS 9
PHM2008 daa Healh ndcaor consrucon From [Le Son e al.]* Raw daa Consruced healh ndcaors - Clear endency of ndcaor emporal evoluon - Beer score han he wnners of he compeon: S = 5520 wh he Wener process based mehod S = 470 wh non-homogeneous Gamma based mehod * [Le Son e al.] Remanng useful lfe esmaon based on sochasc deeroraon models: A comparave sudy. Relably Engneerng & Sysem Safey 202 20
Applcaon of he MB-HsMM model Number of deeroraon modes Dfferen faul propagaon rajecores dependng on he decrease raes of he flow rae (f) and effcency (e) parameers Consder 3 modes of deeroraon: 2 modes 3 modes 4 modes Mode f< e f< e f<< e Mode 2 f > e f e f< e Mode 3 f > e f> e Faul propagaon rajecores Mode 4 Nbof branches f>> e 2 3 4 2
Applcaon of he MB-HsMM model Observaons model Mxure of Gaussan: K ( x) = N ( x;, Σ ) b c µ j jk jk jk k= K: number of mxure componens Topology selecon BIC creron: N = 7; K = 2 RUL esmaon resul RSE MSE = 28 d 2 = 28 = = d 2 28 = es real d RUL RUL Mehod Score RSE MSE -branch HsMM 2246 502 57 2-branch HsMM 6456 45 936 3-branch HsMM 5458 40 773 4-branch HsMM 379 389 694 Wener-based mehod 5575 423 823 Gamma-based mehod 407 434 864 22
Mul-branchconnuous-sae models 23
Movaons Healh saes are connuous Sae-space represenaon Deeroraon modes n compeon => mode swchng: x y = = f ( x, ω, s ) : Hdden saes g( x, ν, s ) : Observaons : realzaon a me of dscree varable S s S ~ dscree-me Markov chan Graphcal represenaon Connuous-sae modelng Swchng sae-space model 24
Jump Markov Lnear Sysem Assumpon Deeroraon dynamc can be approxmaed by lnear model M deeroraon modes:,2, K, M Model formulaon x y = A x s = C x s + ω + ν where Transon probably marx π π2 K πm π2 π22 K π2m Π = M M O M π M π M 2 K π MM Inal sae dsrbuon: s ω ν x where { } ~ N 0, ~ N 0, ( Qs ) ( Rs ) ( µ Σ ) ~ N, 0 0 0 25 ( s s j) π j + ( ) P( s ) π = = = Ρ = =
Parameers learnng problem Model parameers Incomplee daa => Expecaon-Maxmzaon algorhm: E sep: M sep: Problem: Presence of swchng dynamc Compued over all possble sequences of dscree saes => Inracable => Approxmaed EM algorhm { A, C, Q, R, µ,,, π } Θ = Σ Π 0 0 =,, M ( k ) E P ( ) ( ) ( k) log T, T, T T, Q Θ Θ = X S Y Θ Y Θ ( k ) ( k+ ) ( ) Θ = arg maxq Θ Θ Θ ( k) ( k) ( k) ( Θ Θ ) = P (, Θ ) p (,, Θ ) log P (,, Θ) S T ( ) T T T T T T T T d T Q S Y X S Y X S Y X S T 26
Approxmaed EM algorhm Prunng echnque Approxmaon: Calculae he sum over he mos lkely sae sequence Adapon of he Verb algorhm Do no guaranee he convergence, bu sll suffcen n several praccal cases Mos mporan: he algorhm s lnear n number of me seps. From he radonal Verb algorhm Defne he bes paral cos a me : ( X Y S ) J ( ) = max log P,,, s = S, X Then, for each sae ranson j ->, assgn an nnovaon cos : j, J, + = y C x CΣ C + R y C x 2 log CΣ +, j, C + R + log Π ( j, ) 2 ( +, j, ) ( +, j, ) ( +, j, ) 27
Recurson A me +: Termnaon A me T: Backrackng +, + j j, ψ, + j * T T, For = T, T 2, K,: Approxmaed Q funcon J s = max J * Denoe he mos lkely sae sequence: S T ( j, ) J ( ) = max J + J ( j ) * T ( ) ( ) = arg max J + J ( j ) = arg max J T, ( s ) * * =ψ + + => Calculaed by Rauch-Tung-Sreber(RTS) smooher s ( k) * ( ) (,, ( k) * Θ Θ p Θ ) log P (,, Θ) Q X S Y X S Y dx T T T T T T T Illusraon of paral cos calculaon 28
M-sep: Parameer re-esmaon Connuous par ˆ = S0, S00, Α Cˆ Qˆ Rˆ = S20, S, ' = S, ΑS K 0, ( k) T k= ˆ ˆ ' = S22, CS K 20, ( k) T k= K ( k) 0 = ˆ 0 K k= K ˆ ( k) ( k) ( k) 0 P0 µ 0 µ 0 µ 0 µ 0 K k= ˆ µ µ Σ ˆ = + ˆ ˆ ˆ ˆ ( ' ) ( ) where ' S S S S K ( k) ( ( k) ) ˆ ( k), = xˆ ˆ T x T + Σ T k= F ( S ) * T K ( k) ( ( k) ) ˆ 0, = xˆ ˆ T x T + Σ T k= F ( S ) * T K ( k) ( ( k) ) ˆ 00, = xˆ ˆ T x T + Σ T k= F ( S ) * T K ( k) ( ) 22, = y k= F ( S ) * T ( k y ) K ( k) S ( ( k) ) 20, = y xˆ * T k= F ( S ) T 29
M-sep: Parameer re-esmaon Dscree par Smlar o he Hdden Markov model: ˆ π ( ) = number of mes n sae a me = π =, j number of ransons from sae o sae number of ransons from sae j We oban: where ˆ π K = e K *( k ) S ( ) T = k= ( k ) ( k ) K T ( ) ( ( ) ) ' T ˆ k k ( k) Π = ξ ξ dag ξ K k = = = ( k) S ( ) ξ = e *( k ) T [ 0 0 0 0] e = K K h elemen 30
JMLS based dagnoscs Mode probables Gven es daa * S, Y Denoe he bes sae sequence a me ha ends n Healh sae assessmen { y, y,, y } = K 2 ( ) M xˆ = µ ( ) xˆ, = M M Σ ˆ = µ ( ) Σ, + µ ( ) xˆ, xˆ xˆ, xˆ = = * where and are he mean value and varance gven S, x ˆ, Σ, = ( * S ), P µ ( ) = P s = = = M P ( * S ), + exp J J j, j, ( )( ) T 3
RUL esmaon Dscree me: RUL = mn k : x + k L x < L If fuure mode changes are deermnsc => x + k can be esmaed by mulsep ahead Kalman predcon In JMLS case: M fold ncrease n number of Gaussan dsrbuons o consder Inracable compuaon Approxmaon: merge all one-sep predced Gaussan dsrbuons no one Gaussan dsrbuon M ( + ) µ + ( ) N ( +,, Σ +, ) p x x = Mode probably updae µ ( ) = π µ ( j) + M j= j, ( ) 32
Numercal resuls Mode ranng Two deeroraon modes: quck-rae (mode ) and normal-rae (mode 2) Real parameers A A µ =.0 C = Q = 0.05 R = 2.5 =.002 C = Q = 0.005 R = 0.25 2 2 2 2 = 2 Σ = 0.2 0 0 0.99 0.0 0.3 Π = π = 0.7 0.02 0.98 Esmaed parameers Aˆ =.0 Cˆ =.0 Qˆ = 0.02 Rˆ = 2.59 Aˆ =.002 Cˆ = Qˆ = 0.0 Rˆ = 0.25 2 2 2 2 ˆ µ = 2.9 Σ ˆ = 0.285 0 0 0.3 ˆ 0.994 0.006 Π = ˆ π = 0.7 0.0 0.989 Convergence of he EM algorhm 33
Dagnoss resul Mode deecon and healh assessmen 34
RUL esmaon resul Tme o falure: Tf = 420[ h] Mean esmaed RUL a dfferen mes Evoluon of esmaed RUL pdf 35
Concluson & Perspecves Concluson Developmen of mul-branch models o deal wh he co-exsence problem of mulple deeroraon modes Dscree healh saes: MB-HMM & MB-HsMM Connuous healh saes: JMLS Proposon of mul-branch models based framework for dagnoscs & prognoscs Deecon of acual deeroraon mode Assessmen of curren healh saus Esmaon of he RUL Perspecves Exenson of he JMLS model o he non-lnear case Applcaon wh real-lfe daa 36
Thank you for your aenon! LE Thanh Trung PhD suden GIPSA-lab, Grenoble Insu of Technology rue des Mahémaques 38402 San Marn d Hères cedex Phone: +33 4 76 82 7 59 - Fax: +33 4 76 82 63 88 Emal: hanh-rung.le@gpsa-lab.grenoble-np.fr