Prognosis of Systems with Stochastic Inputs

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Prognosis of Systems with Stochastic Inputs Prognosis of Systems with Stochastic Inputs Prognosis of Systems with Stochastic Prognosis of Systems with Stochastic Inputs Inputs

∗ ∗∗ Radouane Radouane Ouladsine Ouladsine ∗∗∗ Rachid Rachid Outbib Outbib ∗∗ ∗∗ Radouane Ouladsine Rachid Outbib Radouane Ouladsine Rachid Outbib ∗∗ ∗ Universit´ ee Internationale ∗ Internationale de de Rabat Rabat Rocade Rocade Rabat-Sal´ Rabat-Sal´ee 11100 11100 Sala Sala El El jadida jadida ∗∗ Universit´ Universit´ e Internationale de Rabat Rocade Rabat-Sal´ e 11100 Sala [email protected]. Universit´e Internationale de Rabat Rocade Rabat-Sal´e 11100 Sala El El jadida jadida [email protected]. ∗∗ LSIS laboratory, [email protected]. Aix-Marseille University, Avenue Escadrille ∗∗ LSIS laboratory, [email protected]. Aix-Marseille University, Avenue Escadrille ∗∗ ∗∗ LSIS Aix-Marseille University, Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex LSIS laboratory, laboratory, Aix-Marseille Escadrille Normandie-Niemen, 13397University, Marseille Avenue Cedex 20, 20, Normandie-Niemen, 13397 Marseille Cedex 20, France.(e-mail:[email protected]). Normandie-Niemen, 13397 Marseille Cedex 20, France.(e-mail:[email protected]). France.(e-mail:[email protected]). France.(e-mail:[email protected]). Abstract: This paper focuses Abstract: This paper focuses on on the the problem problem of of the the systems systems prognostic. prognostic. More More precisely, precisely, it it is is about about Abstract: This paper focuses on the problem of the systems prognostic. More precisely, it is the prognostic of systems subject to stochastic inputs based on the expert’s knowledge. The proposed Abstract: This paper focuses on the problem of the systems prognostic. More precisely, it is about about the prognostic of systems subject to stochastic inputs based on the expert’s knowledge. The proposed the prognostic of systems subject to stochastic inputs based on the expert’s knowledge. The proposed approach consists in assessing the system availability during a mission. This mission is supposed the prognostic of systems subject to stochastic inputs based on the expert’s knowledge. The proposed approach consists in assessing the system availability during a mission. This mission is supposed to to be be approach consists in assessing system availability during aa mission. This is supposed to the user (the which system evolve). This is aa partial approach consists assessing the thein availability during mission. This mission mission supposed to be be the user profile profile (theinenvironment environment insystem which the the system will will evolve). This profile profile is given givenisthrough through partial the (the in which the system will evolve). This profile is through aadamage partial knowledge provided by the expert. aim this work to estimate trajectory system the user user profile profile (the environment environment in The which theof system willis evolve). This the profile is given givenof through partial knowledge provided by the expert. The aim of this work is to estimate the trajectory of system damage knowledge provided by expert. The of this is to the trajectory of system and analyze the success. the of the proposed methodology, the knowledge by the the expert.Finally, The aim aimto this work work to estimate estimate system damage damage and analyzeprovided the mission mission success. Finally, toofillustrate illustrate theisrelevance relevance of the thetrajectory proposed of methodology, the and analyze the mission success. Finally, to illustrate the relevance of the proposed methodology, the suspension damage trajectory is treated. and analyze the mission success. Finally, to illustrate the relevance of the proposed methodology, the suspension damage trajectory is treated. suspension damage trajectory is treated. suspension damage trajectory is treated. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Prognosis Prognosis of of system, system, Expert’s Expert’s knowledge, knowledge, Damage Damage trajectory, trajectory, Stochastic Stochastic system, system, Partial Partial Keywords: Prognosis of system, Expert’s knowledge, Damage trajectory, Stochastic system, information, Maximum of entropy principal. Keywords: Prognosis of system, Expert’s knowledge, Damage trajectory, Stochastic system, Partial Partial information, Maximum of entropy principal. information, information, Maximum Maximum of of entropy entropy principal. principal. 1. The 1. INTRODUCTION INTRODUCTION The paper paper is is organized organized as as follows. follows. In In section section 2, 2, adopted adopted nono1. The paper is organized as follows. In section 2, adopted notations and definitions are given. Section 3 is dedicated to the 1. INTRODUCTION INTRODUCTION The paper is organized as follows. In section 2, adopted tations and definitions are given. Section 3 is dedicated to nothe tations and definitions are given. Section 3 is dedicated to formulation of the considered issue and statement of a prelimtations and definitions are given. Section 3 is dedicated to the the formulation of the considered issue and statement of a prelimformulation of the considered issue and statement of inary result. In section 44 we give on probabilities formulation of the considered issueour andresults statement of aa prelimpreliminary result. In section we give our results on probabilities inary In section 4 we our on probabilities Nowadays, main on to inary result. result.and In the section we give give our results resultsIn probabilities Nowadays, the the problem problem of of prognosis prognosis becomes becomes an an important important estimation estimation and the main4result result on prognosis. prognosis. Inonorder order to illusillusNowadays, the problem of prognosis becomes an important estimation and the main result on prognosis. In order to topic in automatic control (see [1][2]). Generally,in the litertrate the relevance of the proposed methodology, in section 5 Nowadays, the problem of prognosis becomes an important estimation and the main result on prognosis. In order to illusillustopic in automatic control (see [1][2]). Generally,in the liter- trate the relevance of the proposed methodology, in section 5 topic automatic control (see Generally,in the litertrate the of proposed methodology, in 5 ature, proposed methodologies for prognosis renumerical example, concerning half-car topic in inthe (see [1][2]). [1][2]). thebe the relevance relevance of the the proposedthe methodology, in section section is ature, theautomatic proposedcontrol methodologies for Generally,in prognosis can can beliterre- aatrate numerical example, concerning the half-car suspension, suspension, is5 ature, the proposed methodologies for prognosis can be rea numerical example, concerning the half-car suspension, is grouped in three approaches ([3][4]). The first approach is treated. The last section is a conclusion. ature, theinproposed methodologies for The prognosis can be rea numerical example, the half-car suspension, is grouped three approaches ([3][4]). first approach is treated. The last sectionconcerning is aa conclusion. grouped in approaches ([3][4]). The first is model prognostic, in case damage evolution groupedbased in three three approaches first approach approach is treated. treated. The The last last section section is is a conclusion. conclusion. model based prognostic, in this this([3][4]). case the theThe damage evolution is model prognostic, this case damage evolution is defined by or stochastic law (see and model based based prognostic, in in case the the evolution is defined by aa deterministic deterministic or this stochastic lawdamage (see [3][4] [3][4] and [7]). [7]). defined by a deterministic or stochastic law (see [3][4] and [7]). The second approach is prognosis based data. The damage evodefined by aapproach deterministic or stochastic (see [3][4] and [7]). 2. The second is prognosis basedlaw data. The damage evo2. NOTATIONS NOTATIONS The approach is prognosis data. evolution, in this case, is by aa statistical analysis (for 2. The second second approach isdetermined prognosis based based data. The The damage damage evo2. NOTATIONS NOTATIONS lution, in this case, is determined by statistical analysis (for lution, in this case, is determined by a statistical analysis (for instance or using technical as lution, insee this[8]) case, is determined by aIntelligence statistical analysis instance see [8]) or by by using Artificial Artificial Intelligence technical(for as Let n, m ∈ N. < ., . >n and ∇n refer to standard inner product ∇n refer to standard inner product n, m ∈ N. < ., instance see [8]) or using Intelligence technical as in [9][10]. is based prognostic, n and n. > n×m instance seeThe [8])third or by byapproach using Artificial Artificial Intelligence as Let Let m < ., to inner product in [9][10]. The third approach is experience experience based technical prognostic, and gradient respectively. R denotes the set of nn and n×m Let n, n, m∈ ∈ N. N.in > and ∇ ∇nn refer refer to standard standard inner product and gradient in R respectively. R denotes the set of real real [9][10]. The third approach is experience based prognostic, in this case the methods are used when the physical model are n×m n×n [9][10]. third approach is experience based prognostic, n n×m in this caseThe the methods are used when the physical model are and gradient in R , respectively. R denotes the set of matrices of order n by m. I is the identity matrix in R n×n n and gradient in R , respectively. R denotes the set of.. real real matrices of order n by m. In is the identity matrix in Rn×n in this case the methods are used when the physical model are difficult to obtain, or the damage state is impossible to monitor in this case the methods are used when the physical model are n×n .. difficult to obtain, or the damage state is impossible to monitor matrices of order nnn by m. IInn is the identity matrix in R matrices of order by m. is the identity matrix in R 1 m m−1 difficult to obtain, or the damage state is impossible to monitor with the sensor [11]. = × difficult obtain, or the damage state is impossible to monitor For with the to sensor [11]. For SS aa finite finite set, set, SSnnn is is the the set set defined defined by by SS111 = = SS and and SSm =S Sm−1 × m−1 m S. m−1 × with SS m aa finite set, SSP(S) is the set defined by SS all = SSparts and SSofm = S SSFor for = 1 . . . n. denotes the set of with the the sensor sensor [11]. [11]. For finite set, is the set defined by = and = S × for m = 1 . . . n. P(S) denotes the set of all parts of S. SS for m = 11 .. .. .. n. P(S) denotes the set of all parts of S. for m = n. P(S) denotes the set of all parts of S. This This paper paper is is aa contribution contribution of of prognostic prognostic based based on on expert’s expert’s This paper is aa contribution of prognostic based on knowledge, and the proposed prognosis framework can This paper is contribution of prognostic based on expert’s expert’s knowledge, and the proposed prognosis framework can be be Γ U → → {0, {0, 1} 1} is is the the function function defined defined by by Γ ΓSS (u) (u) = =1 1 for for u u∈ ∈S S ΓSS :: U knowledge, and the proposed prognosis framework can U → {0, is described as follows. Given a system considered to perform and elsewhere. SS :: Γ knowledge,as and the proposed prognosis framework can be be Γ USS (u) →= {0,001} 1} is the the function function defined defined by by Γ ΓSS (u) (u) = =1 1 for for u u∈ ∈S S Γ described follows. Given a system considered to perform and Γ (u) = elsewhere. described as Given aa system considered to Γ = 00 elsewhere. mission aa series of the is the SS (u) described(i.e. as follows. follows. system considered to perform perform and Γ (u) = elsewhere. n mission (i.e. series Given of tasks), tasks), the goal goal is to to estimate estimate the and Let j≤n will denote the n × n matrix mission (i.e. of the is the Let vv ∈ ∈R Rnnn ,, Diag(v) Diag(v) = = (a (ai,i, jj ))1≤i, damage and the availability. Note 1≤i, j≤n will denote the n × n matrix mission trajectory (i.e. aa series series of tasks), tasks), the goal goal is to to estimate estimate the defined damage trajectory and predict predict the process process availability. Note that that Let vv ∈ R ,,aDiag(v) = (a will denote the n× matrix by vvi and aai,i, jj ))= 00 j≤n for ii = Also, we the 1≤i, i,i = Let ∈ R Diag(v) = (a denote ×n n use matrix 1≤i, j≤n trajectory and predict the process availability. Note that defined by a = and = for will = j. j. Also,the wennwill will use the aadamage task is defined by a specific environment (internal solicitation; i,i i i, j damage predictenvironment the process availability. Note that abuse task is trajectory defined byand a specific (internal solicitation; defined by a = v and a = 0 for i =  j. Also, we will use the of notation exp(v) to denote the vector in R defined by i,i i i, j n defined by a = v and a = 0 for i =  j. Also, we will use the i,i i i, j aa task is defined by a specific environment (internal solicitation; abuse of notation exp(v) to denote the vector in R defined by i.e speed, charge, or external disturbances; i.e. wind, road, n task is defined by aorspecific environment (internal solicitation; v1 , . .of vn )T . i.e speed, charge, external disturbances; i.e. wind, road, n defined abuse notation exp(v) to denote the vector in R by (e . , e v v T n abuse of notation exp(v) to denote the vector in R defined by 1 i.e speed, charge, or external disturbances; i.e. wind, road, (e , . . . , e ) . whether). More precisely, it is about the prognostic of systems v v T i.e speed, charge, or external disturbances; i.e. wind, road, n 1 , . . . , evn )T . whether). More precisely, it is about the prognostic of systems (e v 1 (e , . . . , e ) . 2 whether). More precisely, it is about the prognostic of systems subject to stochastic inputs based on the expert’s whether). precisely, about of systems For σ > 0 we define the function f on R by f (x) = √e−x 2 subject to More stochastic inputsit is based onthe theprognostic expert’s knowledge. knowledge. e−x 2 . subject to stochastic inputs based the expert’s knowledge. 2 . σ > 0 we define the function fσσ on R by fσσ (x) = √ The aim to the system damage 2πσ ee−x subject based on on of thethe −x The aimtois isstochastic to estimate estimateinputs the trajectory trajectory of theexpert’s systemknowledge. damage in in For √ 2πσ . on R by f (x) = For σ > 0 we define the function f σ σ √ on R by f (x) = . For σ > 0 we define the function f The aim is to estimate the trajectory of the system damage in σ σ 2πσ order to evaluate the possible success mission. The aim is to estimate the trajectory of the system damage in Let m, σ ∈ R+ . N (m, σ ) denotes the gaussian random variable 2πσ order to evaluate the possible success of mission. order to evaluate the possible success of mission. Let m, σ ∈ R+ . N (m, σ ) denotes the gaussian random variable order to evaluate the possible success of mission. Let ∈ R . N (m, σ the random where deviation are m and σ ,, variable respecIn Let m, m, σ σthe ∈ mean R+ (m,standard σ )) denotes denotes the gaussian gaussian random variable + . Nand the mean and standard deviation are m and σ respecIn this this framework, framework, the the information information about about environment environment is is supsup- where where the mean and standard deviation are m and σ , respecIn this framework, the information about environment is suptively. posed to be provided by expert. However, because of the comwhere the mean and standard deviation are m and σ , respecIn thistoframework, thebyinformation about because environment iscomsup- tively. posed be provided expert. However, of the tively. posed to be provided by expert. However, because of the complexity of systems under consideration and the possible random tively. posed to be provided by expert. However, because of the complexity of systems under consideration and the possible random Note aa sub set on P (U ). The damage is aa continue that, E i is set on P (U ). The damage is Note that, E plexity under and the possible random i is phenomena this is to Hence, in plexity of of systems systems under consideration consideration thepartial. possible random aa sub sub set on P (U ). The damage is aa continue continue Note that, E phenomena this knowledge knowledge is supposed supposedand to be be partial. Hence, in variable which means degradation. ii is is sub set on P (U ). The damage is continue Note that, E variable which means degradation. phenomena this knowledge is supposed to be partial. Hence, in this context, we present a probabilistic approach, based on the phenomena is supposed approach, to be partial. Hence, in variable which means degradation. this context,this we knowledge present a probabilistic based on the variable which means degradation. T this we aa probabilistic approach, based on is variable describing the theoretical damage disinformation technical [12], able to quantify the uncerthis context, context, theory we present present probabilistic approach, based on the the X tT X is aaa random random variable describing the theoretical damage disinformation theory technical [12], able to quantify the uncertT R is the random T is information theory technical [12], able to quantify the uncerX random variable describing the theoretical damage distribution at time point t. X variable describing tainties and then predict accurately and precisely the damage R t information theory technical [12], able to quantify the uncerX is a random variable describing the theoretical damage dist at time point t. Xt RR is the random variable describing tainties and then predict accurately and precisely the damage tribution t tainties and then predict accurately and precisely the damage tribution at time point t. X is the random variable describing the real damage distribution at time point t. state. t tainties and then predict accurately and precisely the damage tribution at time point t. X is the random variable describing the real damage distributiont at time point t. state. state. the state. the real real damage damage distribution distribution at at time time point point t. t.

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3. DEFINITIONS AND PROBLEM FORMULATION

where |M| denotes the number of tasks that compose the mission and ti ∈ T for i = 1, . . . , |M| with ti−1 < ti .

3.1 Generalities and definitions In this paper, a dynamical system Σ (see Fig. 1) will be represented by Σ = (T , U , Y , F , D) where T = R+ is time set. U is the input values space which is assumed to be a finite set and u ∈ U will represent a damage mode. Y is the output values (damage values space) and F is a set of smooth functions used to describe the damage behaviors. D : U −→ F associates each damage mode u ∈ U to a specific damage behavior fu ∈ F . We suppose that Y = Yn ∪ Y f where Yn and Y f denote the sets of normal functioning and the functioning in presence of faults, respectively.   we assume that For seek of simplicity, Y = R+ , Yn = 0, d F and Y f = d F , +∞ where d F > 0 is used to denote an acceptable damage threshold defining the normal functioning. Hence, F is composed by smooth scalar non-decreasing functions defined from R+ to R+ that denote abaques determined by experts on the considered system. The operation of the system is supposed to be done in a framework of missions. A mission is defined as a series of tasks where each task is characterized by a duration. Furthermore, in this work, during a task the damage mode is assumed to be constant. However, the damage mode during a task is not perfectly known upstream of mission.

Environment

Damage

Process

Fig. 1. Operation of process considered as input/output system For all fu ∈ F , we define F u : R+ −→ R+ by   for all τ ≥ 0, d ≥ 0 Fτu (d) = fu τ + fu−1 (d)

We also use



(1)

to denote the operator defined by  u   Fτ22 (d) = Fτu22 Fτu11 (d) Fτu11

This operator allow to cumulate the damage caused by the environment u1 and u2 during τ1 and τ2 respectively. By recurrence, for n ≥ 3 and k ≤ n,   n  ui i=k

Fτi (d) =

n−1  ui i=k

Fτi



Fτunn (d)

for all τi ∈ R+ and ui ∈ U (i = 1, . . . , n).

We will use the convention that if n < k then identity function.

Let d0 ≥ 0 be the initial damage. A sample reasoning shows that the mission M will succeed if and only if |M|  uj j=1

Ft j −t j−1 (d0 ) ∈ Yn .

(3)

In rest part of the paper, we consider the situation when the inputs; i.e. the damage modes are not perfectly known upstream. First, let us consider the case with complete information. In this case, the mission can be described simply by M = (Pi , [ti−1 ,ti ])i=1,...,|M| (4) Pi is a probability distribution defined on Ei . We use Pl to denote the probability measurement defined on U |M| by |M|

P(u) = ∏ Pk (uk )

for all

k=1

u = (u1 , . . . u|M| ) ∈ U |M|

(5)

Now, we consider the case when during a mission, the damage modes are not known perfectly upstream. However, we suppose that during a task, the associated damage mode is constant a similar problem was treated in [15] and [16]. In this work, the stochastic concept is introduced to model the random aspect due to the lack of information. To do so, we assume that the expert’s knowledge, for each task i of the mission, can be described by a probability space (Ei , Pi ) where, for i = 1, . . . , |M|, Ei ⊂ P(U ) and Pi : Ei → [0, 1] is a probability measurement. In one hand, the complete information is considered to be obtained when Ei = U . Indeed, in this case all the probabilities, for each damage mode, are defined. Besides, by using for all Ω ∈ P(U ), Pi (Ω) = ∑ω∈Ω P(ω), the probability is defined over all P(U ). In another hand, the information is considered to be partial when Ei ⊂ P(U ) and Pi is not known for all elements of U . In contrast, the worst situation is concidered when Ei = {0, / U } with Pi (0) / = 0 and Pi (U ) = 1. This means that we have no information about the damage mode during the task. In the general case, the mission will be modeled as follows:    M = Ei , Pi , [ti−1 ,ti ] (6) i=1,...,|M|

n  i=k

Fτui i is the

where Ei ⊂ P(U ) for i = 1, . . . , |M|. 3.3 Damage behavior

3.2 Mission modeling The first situation concerns the case where the effect of the environment is assumed to be perfectly known (14). In this case a mission can be represented by (2) M = (ui , [ti−1 ,ti ])i=1,...,|M|

We use D to describe the damage behavior caused by the scenario u = (u1 , . . . , u|M| ) ∈ U |M| at each t ∈ [t0 ,t|M| ] as

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D(u,t, d) =



i−1  uj j=1

Ft j −t j−1



 u i

Ft−ti−1 (d0 )

for

d ≥ O (7)

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4.1 Probabilities estimation

where i is such that t ∈ [ti−1 ,ti ]. The damage is assumed to evolve in a random way by respecting the probabilities (Pi )i=1,...,|M| given in the model (4). Hence, for t ∈ [t0 ,t|M| ], we consider the stochastic process Xt T : U |M| −→ It ⊂ R+ to describe its behavior regardless the model error, with

P(Xt T = D(u,t, d)) = P(u). Therefore, the mean is given by µXt T (d0 ) = ∑ P(u) D(u,t, d0 )

σXt T (d0 ) =

∑

u∈U |M|

∑ Pˆi (u)ΓΩ (u) = Pi (Ω)

(8)

∑ Pˆi (u) = 1

u∈U

(9)

Xt R = Xt T + α Bt−tl for t ∈ [t0 ,t|M| ] where Bt is a Brownian motion and α > 0.

(10)

Notice that Xt R is characterized by its distribution function defined by, for all δ ∈ R,

=

∑

P(αBt−t0 < δ − D(u,t, d0 ))

∑

P(u)

u∈|M|

u∈|M|



δ −D(u,t,d0 ) α

−∞

ft−t0 (τ) dτ

(11)

Let µXt R and σXt R be the mean and the standard deviation of Xt R , respectively. We have for all t ∈ [t0 ,t|M| ] : µXt R = µXt T

2 2 2 and σX R = σX T + α (t − t0 )]. t

t

(12)

3.4 Problem formulation In this work the concept of prognosis is defined as the ability to forecast the success of a mission in the sense that no fault will occur during the mission; i.e. X R (t|M| ) ∈ Y n . However, in the stochastic case, which is here the main issue, the prediction of mission success will be given with a confident rate; i.e. for θ > 0 a confident rate, we will ensure that P(X R (t|M| ) ∈ Y n ) > 1 − θ . Obviously, the greater is the likelihood, better is the prediction. 4. MAIN RESULTS The first part of this section  is dedicated to the probabilities measurements estimation Pˆi i=1,...,|M| by using the expert’s knowledge as a basic information. To do so, we use the maximum of entropy principal. Then, in the second part of this section, we give a result on system health prognosis.

i = 1, . . . |M|

and

Ω ∈ Ei

(C2) Normalization condition

u∈U |M|

2  P(u) µXt T (t, d) − D(u,t, d0 )

for all

u∈U

Now, and taking account of modeling error, we consider the stochastic process Xt R : U |M| −→ R to describe the real behavior of damage defined by

P(Xt R < δ ) =

The   goal of this subsection is to obtain a series of probabilities Pˆi i=1,...,|M| that are defined on U which respect and complete the information contained in the initial model   (6). To this end, we introduce the following constraints on Pˆi i=1,...,|M| . (C1) Available information from expert

It = {D(u,t, d) with u ∈ U |M| } In the case of complete information, Xt T is characterized by

and the standard deviation

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for all

i = 1, . . . , |M|

  Second, and in order to get Pˆi i=1,...|M| we use the maximum of entropy. The design is done by steps where Pˆi (for i = 1, . . . , |M|) is obtained taking into account the value Pˆi−1 . The goal, at each stage, is to keep the maximum of information from Pˆi−1 . To this end, we consider the relative entropy, introduced in [13], given as follows :     Pˆi (u) ˆ ˆ ˆ for i = 1, . . . , |M| H1 Pi , Pi−1 = ∑ Pi (u) ln Pˆi−1 (u) u∈U (13) where Pˆ0 is the initial values of probabilities which will be discussed hereafter. In the sequel, and for a seek of simplicity, we assume that U = {u¯1 , u¯2 , . . . , u¯|U | } where |U | designates the cardinal of U . We use Πi ∈ R|U | (i = 1, . . . , |M|) to denote the vector where the jth component is Πij = Pˆi (u¯ j ) ( j = 1, . . . , |U |). For suitable matrices Ai ∈ R(|Ei |+1)×|U | and vectors ai ∈ R|Ei |+1 the conditions (C1) and (C2) can be expressed by Ai Πi = ai (14) Finally, the probabilities estimation (Pˆi )i=1,...,|M| can be obtained by solving the following optimization problem:    max H1 Πi , Πi−1 ss Ai Πi = ai for i = 1, . . . , |M| (15)  i A¯ Let A¯ i be the matrix so that Ai = with In = (1 . . . 1). Let I  i n a¯ also a¯i be the vector so that ai = 1 We assume that (H1) For i = 1 . . . |M|, the matrices A¯ i are full row rank. Let η ∈ Rn . We define Ci (η) ∈ R|Ei |×|U | (i = 1, . . . |M|) by   i Ck,l (η) = ηk Aik,l − aik for k = 1, . . . , |Ei | and l = 1, . . . , |U | Our first result can be stated as follows:

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Proposition 1 A solution for (15) is given by   −1    Diag exp −(λ i )T Ai Πi−1 Πi = Z i λ i

v x

for

i = 1, . . . , |M| (16) where Π1 is the initial value for probabilities measurements, λ i is the solution of   Ci (Πi−1 ) exp −(λ i )T Ai = 0 (17) and      (18) Z i λ i = Πi−1 , exp −(λ i )T Ai |U | . Proof. In the literature the solution of (15) (see for instance [12]) is obtained by using the technical of Lagrange’s multipliers. More precisely, we seek a solution of the form    −1  Πi = Z i (λ ) Diag exp λ T Ai Πi−1 (19) where λ is a solution of    T (20) ∇|M| ln Z i (λ ) = − ai Straightforward computations show that equation (20) is equivalent to (17). Therefore, (16) is a solution for (15) which finishes the proof of the proposition.

m l1

m1

l2

m2

Road h1

h2

Fig. 2. Half-car suspension

4.2 Result on prognosis

This process is assumed to evolve under three types of operating modes depending on three road kind, i.e severe ”red”, fair ”bleu” or good ”green” (see [17] and [18]). The goal of this example is to analyze the suspension availability for a mission.

The main result of this work gives a sufficient condition for testing the success of a mission (4) under a confident rate in sense of probabilities and is given as follows.

First, we identify the different damage modes (environments) and their degradation curves (damage behaviors). The Fig. 3 shows 3 damage abacus wich depend on the road severity.

Proposition 2 Let ε ≥ σX R (t|M| , d0 ) a small real. We define θε D as follows : 2 (t , d ) + α 2 (t σX 0 T |M| |M| − t0 ) D θε = . ε2   Let Pˆi = Πi (i = 1, . . . , |M|) where Πi i=1,...,|M| is defined by (16). If µXt T (d0 ) + ε ∈ Yn (21) |M|

then the mission M(4) will succeed with a confidence level greater than 1 − θε .

Proof. This result is an application of Chebyshev’s inequality. Indeed, we have, for all t ∈ [t0 ,t|M| ] 2 (d )    σX 0 R   t Pˆ µXt R (d0 ) − Xt R  < ε ≥ 1 − 2 ε

then 2 2    σX T (d0 ) + α (t|M| − t0 )   t|M| R ≥ 1− Pˆ  T (d0 ) − Xt  < ε  µXt|M| ε2 or   ˆP Xt R < µ T (d0 ) + ε ≥ 1 − θε Xt

|M|

By using (21) we deduce that   Pˆ Xt R ∈ Yn ≥ 1 − θε

which clearly implies the statement of the Proposition 2.

Fig. 3. Abacus representing damage in 9 environments An abacus is composed by three curves where each curve describes the damage caused on the suspension when it is used under a specific environment(road severity+speed). Furthermore, 9 environments are distinguish and regrouped in following set U = {u1 , . . . , u9 }. The damage behaviors set, F , is composed by polynomials functions fui = αi t βi (for i = 1, . . . , 9). Each function from F is considered to approximate a curve, given in the Fig. 3, by using the following equation  ln(yia ) − ln(yib )   βi = i=1,. . . , 9 ln(tai ) − ln(tbi ) (22)   i i αi = exp(ln(yb ) − βi ln(tb ))

Here (yia ,tai ) and (yib ,tbi ) are two different point from the approximated curve. The numerical values of the parameters βi and αi are given in Table 5.1.

5. NUMERICAL EXAMPLE 5.1 Process under consideration

5.2 Mission modeling

To illustrate the relevance of the proposed methodology we consider a half of bus suspension (see Fig.2).

We consider the case of a bus performing 5 round trips/day. One round trip is described by M = ((Ei , Pi ) , [ti−1 ,ti ])i=1,...,10 with :

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Radouane Ouladsine et al. / IFAC-PapersOnLine 48-21 (2015) 1327–1332

αi 2.90 × 10−36 5.68 × 10−29 1.23 × 10−28 4.94 × 10−33 1.34 × 10−31 4.02 × 10−28 1.49 × 10−27 2.42 × 10−21 2.24 × 10−29

Environment u1 u2 u3 u4 u5 u6 u7 u8 u9

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βi 4.77 3.78 3.76 4.44 4.26 3.78 3.73 2.87 4.02

Table 1. The numerical values of fui parameters • Way to destination (go) : t0 = 0, t1 = 7min, t2 = 22min, t3 = 32min, t4 = 38min, t5 = 45min. • Way return (back) : t5 = 45min, t6 = 52min, t7 = 58min, t8 = 68min, t9 = 83min and t1 0 = 90min. The available partial information is described in the Appendix A. 5.3 Simulation results We consider the problem of upstream prediction. By considering that Pˆ0 is the equiprobability and by using the result of Proposition 2, we get the following estimates for the probability distributions. u1 u2 u3 u4 u5 u6 u7 u8 u9

Fig. 4. Evolution of ε in function of θε is only partially known. The developed methodology consists on completing the lack of the expert information, by using the Maximum of relative entropy principal, then, predcting the system availability during a mission. Note that, because of the complex phenomena function which describes damage behaviour is supposed to be stochastic. Consequently, the estimated degradation is given through a confidence interval and the mission success is probabilistic. Besides, we prove that the prediction precision depends directly of the provided information quality. To illustrate the relevance of the methodology the prediction of a bus suspension availability, for a long user profile, is treated.

Pˆ1 0.42 0.42 0.1 0.01 0.01 0.01 0.01 0.01 0.01

REFERENCES [1]

Pˆ2 0.05 0.05 0.02 0.21 0.21 0.43 0.01 0.01 0.01 Pˆ3 0.02 0.02 0.01 0.03 0.03 0.08 0.55 0.13 0.13

[2]

Pˆ4 0.01 0.01 0.01 0.01 0.01 0.02 0.41 0.41 0.11 Pˆ5 0.13 0.35 0.31 0.01 0.01 0.02 0.08 0.08 0.01

(23)

Pˆ6 0.13 0.35 0.31 0.01 0.01 0.02 0.08 0.08 0.01

[3]

Pˆ7 0.01 0.04 0.04 0.01 0.01 0.01 0.39 0.39 0.1 Pˆ9 0.01 0.01 0.01 0.01 0.01 0.01 0.62 0.2 0.12 Pˆ9 0.01 0.01 0.01 0.21 0.21 0.41 0.06 0.05 0.03

[4]

Pˆ10 0.13 0.35 0.31 0.01 0.01 0.02 0.08 0.08 0.01 The simulation results are performed for a global mission M1 composed of 200 days. Note that during each day the system is used like in M. The estimated values of the degradation 2 mean and dispersion are µX T (d0 ) = 0.033 and σX (d0 ) = T 0.0184 × 10−6 ,

t| M1 |

respectively.

t| M1 |

[5]

[6]

The confidence interval, given by ε, and the probability of the mission success given by 1 − θε are linked by the equation (21) and illustrated graphically in the Fig.4 . For example, the interval which allows to assure 99% of mission success is given by ε = 0.036.

[7] [8]

6. CONCLUSION This paper is a contribution of the prognostic based on expert’s knowledge, and we deal with the case when this knowledge

[9]

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G. Vachtsevanos, L. F.L., M. Roemer, A. Hess, and W. B., Intelligent Fault Diagnosis and Prognosis for Engineering Systems. Hoboken, NJ: John Wiley and Sons, 2006. C. S. Byington, P. W. Kalgren, R. Johns, and R. J. Beers, Prognosis enhancements to diagnostic system for improved condition based maintenance, in IEEE Systems Readiness Technology Conference, California, USA, pp. 320-329, Sep. 2003. M. J. Roemer, C. S. Byington, G. J. Kacprzynski, and G. Vachtsevanos, An overview of selected technologies with reference to integrated phm architecture, First International Forum on Integrated System Health Engineering and Management in Aerospace Napa Valley, California, USA, October 2005. C. S. Byington, M. J. Roemer, and T. R. Galie, Prognosis enhancements to diagnostic system for improved condition based maintenance, IEEE Aerospace Conference, vol.6, pp. 2815−2824, 2002. M. Lebold and M. Thurston, Open standards for condition-based maintenance and prognostic systems, 5th Annual Maintenance and Reliability Conference, Gatlinburg, USA, 2001. D. Gucik, R. Outbib and M. Ouladsine, Estimation of damage Behaviour for model-based-prognostic, Safe Process, Barcelona, Spain, 2009. D. Chelidze, Multimode damage tracking and failure prognosis in electromechanical system, in SPIE Conference, vol. 4733, 2002, pp. 1-12, 2002. F. L. Greitzer and R. A. Pawlowski, Embedded prognostics health monitoring,in International Instrumentation Symposium, May 2002. T. Brotherton, G. Jahns, J. Jacobs, and D. Wroblewski, Prognosis of faults in gas turbine engines, in IEEE Aerospace Conference, vol. 6, pp. 163-171, 2000.

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[10] A. Eleuteri, R. Tagliaferri, L. Milano, S. De Placido and M. De Laurentiis, A novel neural network-based survival analysis model, Neural Networks, vol.16(5-6), p.855-864, 2003. [11] S. Martorell, A. Sanchez, and V. Serradell, Age-dependent reliability model considering effects of maintenance and working conditions, Reliability Engineering and System Safety, vol. 64, pp. 19−31, 1999. [12] Jaynes, E.T. Information Theory and Statistical Mechanics, Physical review, vol.106(4), pp.620-630, 1957. [13] S. Kullback and R. A. Leibler, On information and sufficiency, Annals of Mathemathical Statistics, vol. 22, pp. 79−86, 1951. [14] F. Peysson, M. Ouladsine, R. Outbib, J-B. Leger O. Myx, C. Allemand, A Generic Prognostic Methodology Using Damage Trajectory Models, IEEE Transactions on Reliability, vol. 58, Issue 2, pp. 277−285, 2009. [15] R. Ouladsine and R. Outbib, A probabilistic approach for prognosis of complex systems. Australian Control Conference, Melbourne, Australia, 2011. [16] R. Ouladsine and R. Outbib, Prognosis of Complex Systems by Analysis Damage Trajectory. IEEE International Conference on Control and Automation, Santiago, Chile, 2011 [17] D. E. Adams, Nonlinear damage models for diagnosis and prognosis in structural dynamic systems, in SPIE Conference, vol. 4733, pp. 180-191, 2002. [18] D. Hrovat, Survey of advanced suspension developments and related optimal control applications, Automatica, vol.33, no.10, pp.1781−1817, 1997. [19] R. Ouladsine and R. Outbib, Pronostic a` Base D’exp´erience: Une Approche Probabiliste. IEEE Conf´erence Internationale Francophone d´Automatique, Grenoble, France, 2012.

5 Task 10 E5 = {{u2 , u3 }, {u1 }} with P5 ({u2 , u3 }) = and 6 1 P5 ({u1 }) = . 5

Appendix A. PARTIAL INFORMATION 7 and P1 ({u3 }) = 8 1 Task 2 E2 = {{u4 , u5 }, {u6 }} with P2 ({u4 , u5 }) = and P2 ({u6 }) = 2 1 Task 3 E3 = {{u8 , u9 }, {u7 }} with P3 ({u8 , u9 }) = and P3 ({u7 }) = 3 9 Task 4 E4 = {{u7 , u8 }, {u9 }} with P4 ({u7 , u8 }) = and 10 1 P4 ({u9 }) = . 9 5 Task 5 E5 = {{u2 , u3 }, {u1 }} with P5 ({u2 , u3 }) = and P5 ({u1 }) = 6 7 Task 6 E1 = {{u1 , u2 }, {u3 }} with P1 ({u1 , u2 }) = and P1 ({u3 }) = 8 1 Task 7 E2 = {{u4 , u5 }, {u6 }} with P2 ({u4 , u5 }) = and P2 ({u6 }) = 2 1 Task 8 E3 = {{u8 , u9 }, {u7 }} with P3 ({u8 , u9 }) = and P3 ({u7 }) = 3 9 and Task 9 E4 = {{u7 , u8 }, {u9 }} with P4 ({u7 , u8 }) = 10 1 P4 ({u9 }) = . 9

Task 1 E1 = {{u1 , u2 }, {u3 }} with P1 ({u1 , u2 }) =

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1 . 8 1 . 2 2 . 3

1 . 5 1 . 8 1 . 2 2 . 3