Controlled systems, failure prediction and maintenance

IFAC Conference on Manufacturing Modelling, IFAC Conference Manufacturing Modelling, Management and on Control IFAC Conference Conference on Manufacturing Modelling, Modelling, IFAC Manufacturing Management and on Control Available online at www.sciencedirect.com June 28-30, 2016. Troyes, France Management and Control Management and Control June 28-30, 2016. Troyes, France June June 28-30, 28-30, 2016. 2016. Troyes, Troyes, France France

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IFAC-PapersOnLine 49-12 (2016) 805–808

Controlled Controlled Controlled

systems, failure prediction systems, systems, failure failure prediction prediction maintenance maintenance maintenance

and and and

∗∗∗ Yves Langeron ∗∗ Mitra Fouladirad ∗∗ ∗∗ Antoine Grall ∗∗∗ Yves Langeron ∗ Mitra Fouladirad ∗∗ ∗∗ Antoine Grall ∗∗∗ ∗∗∗ ∗ Yves Langeron Mitra Fouladirad Antoine Grall Yves Langeron Mitra Fouladirad Antoine Grall ∗ ∗ Institut Charles Delaunay, 12 rue Marie Curie Troyes 10010 France, Charles Delaunay, 12 rue Marie Curie Troyes 10010 France, ∗ ∗ Institut Institut 12 (e-mail: [email protected]). Institut Charles Charles Delaunay, Delaunay, 12 rue rue Marie Marie Curie Curie Troyes Troyes 10010 10010 France, France, (e-mail: [email protected]). ∗∗ (e-mail: [email protected]). Institut Charles Delaunay, 12 rue Marie Curie Troyes 10010 France (e-mail: [email protected]). ∗∗ ∗∗ Institut Charles Delaunay, 12 rue Marie Curie Troyes 10010 France ∗∗ Institut Charles Charles (e-mail: Delaunay, 12 rue rue Marie Marie Curie Curie Troyes Troyes 10010 10010 France France [email protected]). Institut Delaunay, 12 (e-mail: [email protected]). ∗∗∗ (e-mail: [email protected]). Delaunay, 12 rue Marie Curie Troyes 10010 (e-mail: [email protected]). ∗∗∗ Institut Charles Charles Delaunay, 12 rue Marie Curie Troyes 10010 ∗∗∗ ∗∗∗ Institut Institut 12 FranceDelaunay, (e-mail: [email protected]) Institut Charles Charles Delaunay, 12 rue rue Marie Marie Curie Curie Troyes Troyes 10010 10010 France (e-mail: [email protected]) France France (e-mail: (e-mail: [email protected]) [email protected])

Abstract: The paper treats the problem of the maintenance of controlled systems. The actuator Abstract: The treats the the maintenance of controlled systems. The Abstract: Theispaper paper treats the problem problem of ofdue theto maintenance of shocks controlled systems. The actuator actuator of the system subject to deterioration wear and to caused by environmental Abstract: The paper treats the problem of the maintenance of controlled systems. The actuator of the system is subject to deterioration due to wear and to shocks caused by environmental of the the system system is subject to deterioration deterioration due variables to wear wear and and to shocks shocks caused bydeterioration environmental changes. This is latter is modeled by random called covariates. Theby of of subject to due to to caused environmental changes. This latter is modeled by random variables called covariates. The deterioration of changes. This latter is modeled by random variables called covariates. The deterioration of the actuator impact the control system and requires more effort to balance the deterioration changes. This latter is modeled by random variables called covariates. The deterioration of the actuator impact the control system and requires more effort to balance the deterioration the actuator impact the control system and requires more effort to balance the deterioration impact and the controlled out put of the system. The main purpose of this paper is to propose the actuator impact the control system and requires more effort to balance the deterioration impact and controlled out of the The main purpose of is to impact and the the policy controlled outonput put ofresidual the system. system. The of main purpose of this this paper paper is to propose propose aimpact maintenance based theof lifetime the purpose system regulated by is a Markovien and the controlled out put the system. The main of this paper to propose a maintenance policy based on the residual lifetime of the system regulated by aa Markovien a maintenance policy based on the residual lifetime of the system regulated by Markovien command by taking into account the deterioration of the actuator. Theregulated originality ofathe paper is a maintenance policy based on the residual lifetime of the system by Markovien command by taking into account the deterioration of the actuator. The originality of the paper is command by taking into account of The originality of paper in one hand to the of a random deterioration of the in the framework command bydue taking intoconsideration account the the deterioration deterioration of the the actuator. actuator. Theactuator originality of the the paper is is in one hand due to the consideration of a random deterioration of the actuator in the framework in one hand due to the consideration of aahand random in the framework of a controlled system, and on the other thedeterioration considerationof ofthe theactuator remaining useful lifetime in one hand due to the consideration of random deterioration of the actuator in the framework of controlled system, and the of a athe controlled system, and on on the other other hand hand the the consideration consideration of of the the remaining remaining useful useful lifetime lifetime in maintenance decision rule. of a controlled system, and on the other hand the consideration of the remaining useful lifetime in the maintenance decision rule. in the maintenance decision rule. in the maintenance decision rule. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Controlled system, command, deterioration, Gamma process, shock process, Keywords: Controlled system, system, command, command, deterioration, deterioration, Gamma Gamma process, process, shock process, process, Keywords: Controlled covariate, maintenance policy, command, Residual lifetime estimation. Keywords: Controlled system, deterioration, Gamma process, shock shock process, covariate, maintenance policy, Residual lifetime estimation. covariate, covariate, maintenance maintenance policy, policy, Residual Residual lifetime lifetime estimation. estimation. 1. INTRODUCTION The system is monitored through periodic inspections and 1. INTRODUCTION INTRODUCTION The system is monitored through periodic inspections and 1. The system is periodic inspections and at each inspection time athrough replacement time is calculated 1. INTRODUCTION The system is monitored monitored through periodic inspections and at each inspection time a replacement time is calculated at each inspection time a replacement time is calculated with respect to the remaining useful lifetime estimation In industrial automation, a system is operating with a at each inspection time a replacement time is calculated respect to the remaining useful lifetime estimation In industrial industrial automation, automation, aa system system is is operating operating with with aa with with respect toThe the aim remaining useful lifetime estimation of therespect system.to of this useful paper lifetime is twofold, first to In closed-loop andagoverned byoperating a dedicated con-a with the remaining estimation In industrialmechanism automation, system is with of the system. The aim of this paper is twofold, first to closed-loop mechanism and governed by a dedicated conof the system. The aim of this paper is twofold, first propose the reliability and the remaining useful lifetime closed-loop and governed by aa dedicated controller. Thismechanism latter allows the system to track a given of the system. The aim of this paper is twofold, first to to closed-loop mechanism and governed by dedicated conpropose the reliability and the remaining useful lifetime troller. This latter allows the system to track a given propose the reliability and the useful lifetime calculation and to propose an remaining optimal condition-based troller. This latter allows to track aa given target value (i.e a set point)the by system generating a satisfactory propose the reliability and the remaining useful lifetime troller. This latter allows the system to track given and to propose an optimal condition-based target value value (i.e (i.e aa set set point) point) by by generating generating aa satisfactory satisfactory calculation calculation and to an policy. The originally of the condition-based paper is in one target control action. with the aaction of the maintenance calculation and to propose propose an optimal optimal condition-based target value (i.eThis a setmeans point) that by generating satisfactory maintenance policy. The originally of the paper is control action. This means that with the action of the maintenance policy. The originally of the paper is in in one one hand due to the presence of random actuator deterioration control action. This means that with the action of the actuator, the system is able to reach a given set point maintenance policy. The originally of the paper is in one control action. This means that with athegiven action ofpoint the hand due to the presence of random actuator deterioration actuator, the system is able to reach set hand due to the presence of random actuator deterioration and random covariate andofinrandom the other hand deterioration due to a new actuator, the system able to reach aa of given point within acceptable time.is The performance such set a system hand due to the presence actuator actuator, the system is able to reach given set point random covariate and in the other hand due to a new within acceptable acceptable time. time. The The performance performance of of such such aa system system and and random and hand due policy regulated by other command within depends on the setting theperformance controller that requests the maintenance and random covariate covariate and in in the the other handparameters. due to to a a new new within acceptable time. of The of such a system maintenance policy regulated by command parameters. depends on the setting of the controller that requests the maintenance depends on the setting of the controller requests actuator. Therefore, this setting has anthat impact on the maintenance policy policy regulated regulated by by command command parameters. parameters. depends on the setting of the controller that requests the actuator. Therefore, Therefore, this this setting has has an impact impact on the the actuator. actuator deterioration. actuator. Therefore, this setting setting has an an impact on on the actuator deterioration. 2. DETERIORATION MODEL actuator deterioration. actuator deterioration. 2. The safety of controlled systems is essential and this 2. DETERIORATION DETERIORATION MODEL MODEL 2. DETERIORATION MODEL The safety safety of of controlled controlled systems systems is is essential essential and and this this The problem has been largely addressed in the literature, The safety of controlled systems is essential and this In this paper, one considers a system to wear and problem has has been been largely largely addressed addressed in in the the literature, literature, In this paper, one considers a system subject problem subject to wear and see for instance Clarhaut al (2009); et al shocks. problem has been largely et addressed in Ghostine the literature, In this paper, one considers a system subject to wear The wear which can be considered as a monotone see for instance Clarhaut et al (2009); Ghostine et al In this paper, one which considers abesystem subject to wear and and see for instance Clarhaut et al (2009); Ghostine et al shocks. The wear can considered as a monotone (2011). In the controlled command literature, the random see for instance Clarhaut et al (2009); Ghostine et al shocks. The The wear which whichcan canbe be modeled consideredbyas as aaa stochastic monotone gradual deterioration (2011). In In the the controlled controlled command command literature, literature, the the random random shocks. wear can be considered monotone (2011). gradual deterioration can be modeled by a stochastic deterioration is scarcely considered, however, lately the (2011). In the iscontrolled literature, random deterioration can modeled by stochastic process such as a Gamma process. Several are deterioration scarcely command considered, however,the lately the gradual deterioration can be be modeled by aa papers stochastic deterioration lately the process such as aa Gamma process. Several papers are random pointis ofscarcely view isconsidered, attracting however, more attention see gradual deterioration is scarcely considered, however, lately the process such as Gamma process. Several papers are devoted to the deterioration modeling by Gamma process random point of view is attracting more attention see process such as a Gamma process. Several papers are random point of view is to the deterioration modeling by Gamma process for instance, Langeron et attracting al (2013, more 2015);attention Lefebvre see et devoted random point of view is attracting more attention see devoted to the deterioration modeling by Gamma process such as Abdel-Hameed (1975); C ¸ inlar (1980); Wenocur for instance, Langeron et al (2013, 2015); Lefebvre et devoted to the deterioration modeling by Gamma process for instance, Langeron et al (2013, 2015); Lefebvre et such as Abdel-Hameed (1975); C ¸ inlar (1980); Wenocur al (1996); Pereira et al Rishel This for Langeron et al(2010); (2013, 2015);(1991). Lefebvre et such such as as van Abdel-Hameed (1975); C C inlar (1980); (1980); Wenocur Wenocur (1989); Noortwijk (2009). al instance, (1996); Pereira Pereira et al al (2010); Rishel (1991). This Abdel-Hameed (1975); ¸¸ inlar al (1996); et (2010); This (1989); van Noortwijk (2009). paper considers a controlled system Rishel subject (1991). to a random al (1996); Pereira et al (2010); Rishel (1991). This (1989); van Noortwijk (2009). paper considers a controlled system subject to a random (1989); van Noortwijk (2009). paper considers aa controlled X(t) denote the deterioration due to wear at time t, deterioration of actuator. paper considers controlled system system subject subject to to aa random random Let Let X(t) denote deterioration due to at t, deterioration of actuator. actuator. Let X(t) denote the the deterioration due X(t) to wear wear at time time to t, deterioration of t ≥ X(t) 0. Hereafter it isdeterioration supposed that is subject Let denote the due to wear at time t, deterioration of actuator. ≥ 0. Hereafter it is supposed that X(t) is subject to The usage profile or environmental condition called co- ttGamma ≥ 0. Hereafter it is supposed that X(t) is subject to distribution with shape function α(t) and scale The usage profile or environmental condition called cot ≥ 0. Hereafter it is supposed that X(t) is subject to The usage profile or condition called Gamma distribution with shape function α(t) and scale variate is also considered random and is regulated bycoThe usage profile or environmental environmental condition calledby co-aa parameter Gamma with function α(t) and scale β. The probability density function then: variate is also also considered random and and is regulated regulated Gamma distribution distribution with shape shape function α(t)is and scale variate is considered random is β. The probability density function is then: Markovien process. Covariates are and random variablesby de-aa parameter variate is also considered random is regulated by parameter β. β. The Theβ probability probability density density function function is is then: then: Markovien process. Covariates are random variables deparameter Markovien process. Covariates are random variables describing theprocess. usage orCovariates the environmental conditions. The Markovien are random variables The deβ ·(βx)α(t)−1 ·exp (−βx), for x ≥ 0 (1) gα(t),β (x) = scribing the the usage usage or or the the environmental environmental conditions. β = ·(βx)α(t)−1 (−βx), for x≥ 0 (1) ggα(t),β (x) scribing The α(t)−1 β command modifies thethe usage profile of theconditions. system and it Γ(α(t)) scribing the usage or environmental conditions. The α(t)−1 ·exp ·(βx) (x) = ·exp α(t),β command modifies the usage profile of the system and it Γ(α(t)) (x) = ·(βx) ·exp (−βx), (−βx), for for x x≥ ≥0 0 (1) (1) g α(t),β command modifies the usage profile of the and it Γ(α(t)) also impacts the actuator deterioration rate.system command modifies the usage profile of the system and it Γ(α(t)) where α(t) is a non-decreasing, right-continuous, realalso impacts the actuator deterioration rate. also impacts the actuator deterioration rate. where α(t) right-continuous, realalso impacts the actuator deterioration rate. where function α(t) is is aaafornon-decreasing, non-decreasing, right-continuous, real valued t ≥ 0 with α(0) right-continuous, = 0, β > 0 and Γ isrealthe where α(t) is non-decreasing, and financial support acknowledgment goes here. Paper  Sponsor valued function for tt ≥ 00 with α(0) = 0, β > 00 and Γ is the Sponsor and financial support acknowledgment goes here. Paper  valued function for ≥ with α(0) = 0, β > and Γ is the Euler’s Gamma function. The expectation and variance titles should be written in uppercase and lowercase letters, not all Sponsor and financial support acknowledgment goes here. Paper  valued function for t ≥ 0 with α(0) = 0, β > 0 and Γ is the Sponsor and support acknowledgment goes here. not Paper Euler’s Gamma function. The expectation and variance titles should be financial written in uppercase and lowercase letters, all 2 Euler’s Gamma function. The expectation and variance are E(X(t)) = α(t)/β, var(X(t)) = α(t)/β respectively. uppercase. titles should be written in uppercase and lowercase letters, not all Euler’s Gamma function. The expectation 2 and variance titles should be written in uppercase and lowercase letters, not all are E(X(t)) = α(t)/β, var(X(t)) = α(t)/β uppercase. 2 respectively. are uppercase. are E(X(t)) E(X(t)) = = α(t)/β, α(t)/β, var(X(t)) var(X(t)) = = α(t)/β α(t)/β 2 respectively. respectively. uppercase.

Copyright 2016 IFAC 805 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 805 Copyright 2016 IFAC 805 Peer review© of International Federation of Automatic Copyright ©under 2016 responsibility IFAC 805Control. 10.1016/j.ifacol.2016.07.873

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The stochastic process {X(t), t ≥ 0} is considered to be a Gamma process satisfying the following properties: • X(0) = 0, • X(s + t) − X(t) ∼ Γ(α(s + t) − α(t), β) for s ≥ 0, t ≥ 0, ; • X(t) has independent increments.

When the shape function is linear, α(t) = αt, the process is a stationary Gamma process with shape parameter α and scale parameter β. At the inspection times tn , where t0 = 0, α(t0 ) = 0 and t0 < t1 < · · · < tn < · · · , the discrete process X(tn ) (n ≥ 1), according to the additivity of Gamma distribution Xn , has the following distribution: n  Xn ∼ Γ( (α(ti ) − α(ti−1 )), β) = Γ(α(tn ), β) (2) i=1

The choice of α and β allows to model various deterioration behaviors from almost deterministic to highly variable. The covariate is changed randomly and it impacts the deterioration rate and also implies sudden jumps considered as shocks. Let us denote Zt the covariate at time t and the time series {Zt , t ≥ 0} take values in E = {1, 2, · · · , r}. Suppose {Zt , t ≥ 0} be a continuous-time, time-homogeneous and state-discrete Markov process with the transition matrixP is as follows:  p11 p12 . . . p1r  p21 p22 . . . p2r  P = , ... ... ... ...  pr1 pr2 . . . prr where pij is the transition probability from state i to j. The covariate impacts the parameter α of the wear as follows: α(Zt ) ∈ Eα = {α1 , · · · , αr }, α(zt = i) = αi A wear model is not able to capture large deterioration fluctuations and sudden jumps occurrence. These jumps can be associated to extreme events, extreme loads or external shocks. Since these jumps can cause a high increase of the deterioration, they are major causes leading to failure. Therefore, in reliability engineering, shock modeling has drawn lots of attention (see in van der Weide et al (2011); Wang et al (2011); Noorossana et al (2015); Pham (2016)). In this paper, each time the covariate is changed the deterioration undergoes a shock. The shocks are assumed to occur at random times according to a stochastic point process. The shock magnitudes are random depending on the command setting and follow an exponential distribution with parameter θu . This assumption is supported by asymptotic extreme-value theory, see van Noortwijk et al (2007) and Hsing et al (1988) for more details. The deterioration due to shocks at time t is defined as follows: N (t)  S(t) = yk , (3)

θu (τk ) ∈ Eθ = {θu11 , · · · , θu1r , · · · θur1 , · · · , θurr }, and for zτk = j, zτk−1 = i, θu (τk ) = θuij . The total deterioration of the system up to time t, conditionally to the covariate Z(t) = zt , is the sum of the deterioration due to wear and shocks given by the following: D(t, zt ) = X(t, zt ) + S(t, zt ), t ≥ 0. (4) Considering the two deterioration mechanisms independent, the cumulative distribution function (CDF) of the total deterioration conditionally to covariate is as follows: FD(t,zt ) (x) = P(D(t, zt ) < x) = P(X(t, zt ) + S(t, zt ) < x)  x y gα(t,zt ),β (y − u)fS(t,zt ) (u)dudy (5) = 0

0

where gα(t,zt ),β (·) and fS(t,zt ) (x) = dP(S(t, zt ) ≤ x)/dx are respectively, the PDF of the Gamma process increments and of the sum of exponential shock increments (a hypo-exponential distribution) conditionally to covariate at time t. In the same way, FD(t,zt )−D(s,zs ) , the cumulative distribution function of the total deterioration increments D(t, zt ) − D(s, zs ) can be calculated. 3. RESIDUAL USEFUL LIFETIME ESTIMATION The system is considered to be failed when D(t) exceeds a safety threshold L. At the inspection time tj , given the covariate state zj and the deterioration level dj the the remaining useful lifetime at time t > tj is calculated as follows: P(D(t) < LF |Ztj = zj , D(tj ) = dj )

(6)

Regarding to equation (5), to derive the equation (6), first we need to calculate P(X(t) − X(tj ) < d|zj , X(tj )) and P(S(t) − S(tj ) < d|zj , S(tj )). 3.1 Wear process with covariate changes Let be Ent−s the event of exactly n covariate state transitions between t and s. First we develop the distribution function of the wear in the case of exactly n covariate changes between t and tj . Consider Ztj = ztj , assume a set of realizations of n change state time and intermediate states τ1 , . . . , τn , zτ1 , · · · , zτn , therefore ∆X(t, tj ) = ∆X(t, τn ) + . . . + ∆X(τ1 , tj ) where each ∆X(t, t ) = X(t, zt )−X(t , zt ) ∼ Ga(α(t, zt )−α(t , zt ), β) with Gα(t,zt )−α(t ,zt ),β as CDF and gα(t,zt )−α(t ,zt ),β as PDF. Hence, given the covariate trajectory, the PDF of ∆X(tj , tj−1 ) is a n-fold convolution of Gamma distribution functions. 3.2 Shock process with covariate changes

k=0

where N (t) is the number of shocks in (0, t], yk is the magnitude of the kth shock. Let be τ1 , τ2 · · · , the consecutive times of covariate changes. The parameter θu is time dependent and θu (t) = θu (τk ), for t ∈ [τk , τk+1 ], θu (τk ) depends on the state of Zτk and of Zτk−1 and for a command setting:

806

Suppose exactly n state transitions occur between two successive inspections. Therefore (7) ∆S(t, tj ) = ∆S(t, τn ) + . . . + ∆S(τ1 , tj ) Since for the given value of covariates each S(t, Zt ) − S(t , Zt ) has a probability distribution fS(t,zt )−S(t ,zt )

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defined in equation (5), given the covariate values the probability distribution function of ∆S(t, tj ) = S(t, Zt ) − S(tj , Ztj ) is a n-fold convolution. 3.3 Covariate state transitions within a time interval t−t

To derive the probability of occurrence of En j , the event of exactly n covariate state transitions between t and tj , we proceed as follows. Let us denote Zτ1 = zτ1 , . . . , Zτn = zτn the unknown n states at state transition times τ1 , . . . , τn in interval [tj , t] with known Ztj = ztj . Define Wt = inf{s > 0, Zt+s = Zt } for all t > 0. Denote by τi the time moment of the ith state transition in [tj , t]. Based on the definition and according to CocozzaThivent (1997), we have Wτi ∼ exp(qzτi ) where qi is the ith diagonal element of the generator matrix of Z, and denote by vWτi its probability density function. According to these notations, if only two jumps happen, we have τ1 = tj + Wtj and τ2 = τ1 + Wτ1 . According to previous notations the following equivalence is satisfied: n n+1   Ent−tj ≡ Wtj + Wτk < t − tj , Wtj + W τ k > t − tj . k=1

k=1

Therefore the probability that n state transitions occur between tj and t for the given n state transition is defined as: P(Ent−tj | zτ1 , . . . , zτn )  t−tj  ∞ = vWτn+1 (y)VWt t−tj −x

0

where V Wt  

j

x 0

n

+



i=1

x−u0

0

Wτi (x)

= n−2

 x− ...  0 n

i=0

(8) j

n

+

i=1

Wτi (x)dydx.

ui

vWtj (u0 )vWτ1 (u1 ) . . .

 vWτn−2 (un−2 )vWτn−1 (x −

n−2 

the inspection cost is much lower than the replacement cost. The corrective replacement is carried out as soon as a failure is detected. To evaluate the maintenance policies, we focus on the asymptotic expected maintenance cost per unit over an infinite time span considered as a cost criterion: C(t) , C ∞ = lim t→∞ t C(t) = Ci Ni (t) + Cp Np (t) + Cc Nc (t) + Cd dd (t). where C(t) is the cumulated maintenance cost at time t with Np (t) the number of preventive replacements before t, Nc (t) the number of corrective replacements before t, dd (t) the cumulative unavailability duration of the system before t and N i (t)  the number of inspections before t. Note that Ni (t) = Tt where [x] denotes the integer part of the real number x. Let be RU L(t) the random variable such that

P(RU L(t) < h) = P(D(t + h) > LF |D(t) < LF )

In the framework of the proposed maintenance policy a fixed threshold  is defined and at each inspection time tk : • if D(tk ) < L and P(RU L(tk )) ≤ , the system is preventively replaced with a cost Cp . • if D(tk ) < L and P(RU L(tk )) > , the estimated mean RUL is still higher than the threshold , the decision is postponed until the next inspection tk+1 . • if D(tk ) ≥ L, the system has already failed, a corrective replacement is carried out with a cost Cc . Let us recall that at each inspection time the covariate state is known. The maintenance parameters are

(9)



807

• the inspection period ∆ • the preventive threshold  • θuij ∈ Eθ for i = 1, · · · r, i = 1, · · · , r.

Therefore, there are 2 + r2 parameters to optimise. The command setting is the key idea of the paper. Each command setting impacts the maintenance cost and its proper regulation will lead to a cost efficient maintenance policy and longstanding system. The latter requires a multi-objective optimisation which should be carried out by identifying the Pareto front, which is the set of all Pareto solutions and which represents the optimisation problem trade-offs. Being able to identify this set is a very useful aid in decision making.

ui )du0 du1 . . . dun−2

i=0

Furthermore, the probability that the n unknown intermediate states be zτ1 , . . . , zτn is calculated as follows: (10) P(Zτ1 = zτ1 , . . . , Zτn = zτn )  t−tj  t−tj −n−2 wi i=0 ... Pw0 (ztj , zτ1 )vWtj (w0 )Pw1(zτ1 , zτ2 ) In Figure 1, the average long run maintenance costs asso= 0 0 ciated to the proposed maintenance policy are depicted. vWτ1(w1 ). . .Pwn−1(zτn−1 , zτn )vWτn−1(wn−2 )dw0 dw1. . . dwn−1 The inspection interval and the preventive threshold are (11) optimized and their best values are pointed out. Finally the probability that exactly n state transitions The maintenance optimization results depends basically occur between tj−1 and tj can be easily deduced. on the priority that the user has on its decision rule. The optimal policy leads to a balance between control 4. MAINTENANCE DECISION RULE performances and maintenance efficiency. This latter can be noticed in Figure 2 where unit costs Ci = 5, Cp = 50, In this section, a prognostic based maintenance policy Cc = 100 are considered. As it can be noted the optimal is proposed. The maintenance decision is based on the maintenance policy does not lead necessarily to the best remaining useful life of the system (or component) at the control performances. The best policy is a compromise inspection time. At each inspection time two maintenance between these two factors. By reducing a little bit the operations are possible: a preventive replacement with cost maintenance efficiency it is possible in this case to increase Cp and a corrective replacement with cost Cc . We suppose significantly the control performances and obtain an effithat after replacements the system is as good as new and cient decision rule. 807

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Gain on average renewal cycle 45 40 35 30 25 20 15 10 5 0 −5 200

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Fig. 1. Maintenance gain on a renewal cycle. 200

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Fig. 2. Maintenance gain on a renewal cycle. 5. CONCLUSION In this paper we have considered a random actuator deterioration subject to random usage profile. The system fails if its deterioration is higher than a safety level. The lifetime and the remaining useful time distribution are derived. We have proposed a maintenance policy considering the remaining useful time distribution of the system. The maintenance optimisation will be principally based on the command setting optimisation. This latter is possible by a multi-objective optimisation and by identifying the Pareto front. The work will be completed by numerical examples and implementations. REFERENCES Abdel-Hameed, M., A gamma wear process. IEEE Transactions on Reliability volume24, pp.152–153, 1975. Clarhaut, J. Cocquempot, V. Conrard, B. and Hayat, S. Optimal design of dependable control system architectures using temporal sequences of failures. IEEE Transaction on Reliability, 58(3), 2009. C ¸ inlar, E.. On a generalization of gamma processes. journalJournal of Applied Probability volume17,pp.467-480, 1980. Cocozza-Thivent, C., Processus stochastiques et fiabilit´e des syst`emes. volume 28. Springer, 1997. Ghostine, R. Thiriet, J.M. and Aubry, J.F. Variable delays and message losses: influence on the reliability of a control loop. Reliability Engeniering Sytem and Safety, 96(1):160-171, 2011. Hsing, T., H¨ usler, J., Leadbetter, M.. On the exceedance point process for a stationary sequence. Probability Theory and Related Fields volume78, pp.97–112, 1988. Langeron, Y.,Grall, A. and Barros,A. Actuator health prognosis for designing lqr control in feedback systems. Chemical engineering, 33:979-984, 2013. 808

Langeron, Y.,Grall, A. and Barros,A. A modeling framework for deteriorating control system and predictive maintenance of actuators Reliability Engineering and System Safety, ol. 140, pp. 2236, 2015 Lefebvre, M. and Gaspo, J. Optimal control of wear processes. IEEE Transactions on Automatic Control, 41(112-115), 1996. Noorossana, R. and Sabri-Laghaie, K. Reliability and maintenance models for a dependent competing-risk system with multiple time-scales Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability April 2015 vol. 229 no. 2 131-142 Pereira, E. Kawakami, R. and Yoneyama,T. “model predictive control using prognosis and health monitoring of actuators. In IEEE Intl Symposium on Industrial Electronics, Bari, Italy, pp. 237-243, 2010. Pham, Hoang (Ed.) Quality and Reliability Management and Its Applications Springer Series in Reliability Engineering 2016 Rishel, R. Controlled wear process : modeling optimal control. IEEE Transactions on Automatic Control, 36:1100-1102, 1991. van Noortwijk, J.M., van der Weide, J.A., Kallen, M.J., Pandey, M.D., 2007. Gamma processes and peaksover-threshold distributions for time-dependent reliability. Reliability Engineering & System Safety volume92, pp.1651-1658. Van Noortwijk, J.M., 2009. A survey of the application of gamma processes in maintenance. Reliability Engineering & System Safety volume94, pp2–21. van der Weide, J.A., Pandey, M.D., 2011. Stochastic analysis of shock process and modeling of conditionbased maintenance. Reliability Engineering & System Safety volume96, pp. 619–626. Wang, Y. ,Pham, H., Dependent competing-risk degradation systems, Safety and Risk Modeling and Its Applications. Springer London, pp. 197–218,2011. Wenocur, M., 1989. A reliability model based on the gamma process and its analytic theory. Advances in applied probability volume21, pp.899–918.