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Integration of prognostics at a system level: a Petri net approachManuel Chiachıo1, Juan Chiachıo2, Shankar Sankararam3, and John Andrews4

1,2,4 Resilience Engineering Research Group, Faculty of Engineering, University of Nottingham, NG7 2RD, Nottingham, [email protected]

[email protected]@nottingham.ac.uk

3 NASA Ames Research Center, Intelligent Systems Division, Moffett Field, CA, 94035-1000, [email protected]

ABSTRACT

This paper presents a mathematical framework for model-ing prognostics at a system level, by combining the prognos-tics principles with the Plausible Petri nets (PPNs) formal-ism, first developed in M. Chiachıo et al. [Proceedings of theFuture Technologies Conference, San Francisco, (2016), pp.165-172]. The main feature of the resulting framework re-sides in its efficiency to jointly consider the dynamics of dis-crete events, like maintenance actions, together with multiplesources of uncertain information about the system state likethe probability distribution of end-of-life, information fromsensors, and information coming from expert knowledge. Inaddition, the proposed methodology allows us to rigorouslymodel the flow of information through logic operations, thusmaking it useful for nonlinear control, Bayesian updating,and decision making. A degradation process of an engineer-ing sub-system is analyzed as an example of application us-ing condition-based monitoring from sensors, predicted statesfrom prognostics algorithms, along with information comingfrom expert knowledge. The numerical results reveal how theinformation from sensors and prognostics algorithms can beprocessed, transferred, stored, and integrated with discrete-event maintenance activities for nonlinear control operationsat system level.

1. INTRODUCTION

In prognostics, the integration of the predicted information ata system level encompasses two distinct research challenges.First is about predicting the change in system performanceand remaining life estimation through an adequate combina-tion of the degradation rates and states of health of individ-ual components. Second, and perhaps most important, to

Manuel Chiachıo et. al. This is an open-access article distributed under theterms of the Creative Commons Attribution 3.0 United States License, whichpermits unrestricted use, distribution, and reproduction in any medium, pro-vided the original author and source are credited.

integrate system-level nonlinearities and uncertainties withthe predicted information from prognostics. Here, system-level nonlinearities are understood as uncertain environmen-tal elements that affect the system operation irrespectivelyof the component-wise state of health or degradation, likeintervention processes (e.g. maintenance actions), resourceavailability, ad hoc synchronization of components, influenceof expert knowledge, etc. In the literature, the majority ofprognostics research to date has been focused on individualcomponents, and determining their end-of-life (EOL) and re-maining useful life (RUL), e.g.: (Chiachıo, Chiachıo, Saxena,& Goebel, 2016; Zio & Peloni, 2011; Myotyri, Pulkkinen,& Simola, 2006; Saha, Celaya, Wysocki, & Goebel, 2009;Daigle & Kulkarni, 2013). Besides, few attempts can beencountered describing system-level prognostics methodolo-gies. Generally, these attempts provide the EOL of a sys-tem based on its constituent components and how they inter-act, like in Gomez, Rodrigues, Galvo, and Yoneyama (2013),where a system-level approach was developed using fault treeanalysis from the RUL of individual components. Daigle,Bregon, and Roychoudhury (2014) provided a distributed ar-chitecture to model system-level prognostics based on theconcept of structural model decomposition whereby the so-lution of independent local prognostics subproblems were in-tegrated to obtain prognostics at system-level. Liu, Xu, Xie,and Kuo (2014) studied multi-component maintenance mod-els with economic dependence for components that degradein a continuous manner. More recently, an analytic frame-work has been provided in (Khorasgani, Biswas, & Shankarara-man, 2016) for combining the degradation rate of individualcomponents to predict the variation in system performanceover time. Nonetheless, to the authors best known, holisticmethodologies for prognostics with integration of system orsub-system level nonlinearities and uncertainties, still remainmissing in the literature.

In this paper, a novel holistic framework is proposed for mod-

1

ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 2017

eling prognostics at a system level by using Petri nets (PNs)(Petri, 1962). In particular, the newly developed PlausiblePetri nets (PPNs) (Chiachıo, Chiachıo, Prescott, & Andrews,2016) are used to integrate uncertain information about thesystem, like the probability density function (PDF) of EOLfor multiple components, information from sensors, expertknowledge, etc., with the dynamics of discrete events likesystem-level health indicators, maintenance activities, resour-ce availability, to cite but any. Consequently, the approach hasthe advantage of being able to integrate by first time informa-tion from prognostics (e.g. EOL, RUL, etc.), with decisionmaking aspects like go/no-go decisions for maintenance ac-tions. Moreover, the uncertainty is intrinsically accounted forin PPNs, since its formulation stems from combining the in-formation theory principles with the PNs technique, as willbe shown further below.

To exemplify some of the problems that can be solved by theproposed methodology, several toy examples are formulatedthrough a set of PPN architectures. Next, the framework istested using a numerical example to model decision-makingover a two-component engineering system under degradationusing prognostics information, expert knowledge, and main-tenance actions, which is just some of the challenges faced insystem-level prognostics applications using PPNs.

The remainder of the paper is organized as follows. Section 2briefly overviews basic concepts about prognostics and PNsbefore introducing the fundamentals of PPNs. The PPNs for-malism as well as their execution semantics, are succinctlydescribed in §3. A set of examples of PPN architectures forsystem-level prognostics are provided and discussed in §4.Section 5 illustrates our approach through a numerical exam-ple about two-component degrading system. Finally, Section6 gives concluding remarks.

2. BASIC CONCEPTS

2.1. Foundations of prognostics

Prognostics is concerned with predicting the future healthstate of engineering systems or components given current de-gree of wear or damage, and, based on that, estimating theremaining time beyond which the system is expected not toperform its intended function within desired specifications(Chiachıo, Chiachıo, Sankararaman, Saxena, & Goebel, 2015).In the PHM community, the aforementioned remaining timeis typically referred as RUL. The potential of prognostics inpositively contributing to safety and cost relies in its capac-ity to provide anticipated information about an anomalous orfaulty condition. This information can be used for risk re-duction in go/no-go decision, cost optimization through thescheduling of maintenance as-needed, and improved asset avail-ability.

Broadly, formal approaches for prognostics fall into three pri-

mary categories (Javed, Gouriveau, & Zerhouni, 2017; Kho-rasgani et al., 2016): (1) data driven techniques, (2) modelbased, and (3) hybrid approaches, depending on how the faultpropagation process is modeled. Irrespective of the type ofmodeling approach chosen for prognostics, two main distinctresearch problems can be devised: (i) the estimation prob-lem, which determines the current state of health of the sys-tem, and (ii) the failure prediction problem, by which theEOL and RUL can be obtained from predictions of the futurestate of the system `-steps forward in time in absence of newobservations. For the estimation problem, the component,sub-system, or system state of health or degradation is typ-ically assumed to be represented using a stochastic variablexk. This state variable evolves over time k following a spe-cific dynamic equation given in state-space form (Chiachıo etal., 2015; Zio & Peloni, 2011), as follows:

xk = fk(xk−1,uk,vk,θ) (1a)yk = hk(xk,uk,wk,θ) (1b)

where uk ∈ Rnu is an input vector, yk ∈ Rny is a measure-ment vector, and vk ∈ Rnv and wk ∈ Rnw are uncertainvariables introduced to account for the model error and mea-surement error, respectively. The functions fk and hk are pos-sibly nonlinear functions for the state transition evolution andobservation equation, respectively. This model is sequentiallyevaluated at every time step k, and produces updated informa-tion about the system state xk as long as new measurementsare available. Next, prognostics algorithms can be employedto project the state predictions into future in absence of newobservations. Then, by having defined a failure region F , theEOL can be obtained as the earliest time index k + `, ` > 1when the event [xk+` ∈ F ] occurs. It can be computed as:

EOLk = inf{k + ` ∈ N : ` > 1 ∧ I(F)(xk+`) = 1

}(2)

where I(F) is an indicator function that maps a given point inthe x-space to the Boolean domain {0, 1}, such that I(F) = 1if x ∈ F , and 0 otherwise. The RUL predicted from timek can be straightforwardly obtained from EOLk as RULk =EOLk−k. Note that due to the probabilistic nature of the statevariable given by xk, then EOL and RUL are also describedas stochastic variables.

Finally, it is important to remark that in this work, the focus isnot on predicting the EOL or RUL of a system as a probabil-ity, but on methodology that integrates EOL and RUL withinan asset management framework at a system level, as will bedescribed next.

2.2. Basis of Petri nets

Petri Nets (PNs) are bipartite directed graphs (digraph) usedfor modeling the dynamics of systems. The underlying graphof a PN consists of two kinds of nodes, transitions and places,where arcs are either from a place to transition or vice versa.

2


p1

p2

p32 t1

Figure 1. Example of Petri net of three places and one transi-tion.

A place represents a particular state of the system or activitybeing modeled (e.g. considering health management model-ing, places can be used to indicate the current state of a com-ponent or sub-system, if the performance of this componentor sub-system is currently being tested by a maintenance teamto determine its state, or if any maintenance activity is cur-rently in progress). Places are temporarily visited by tokens,the abstract moving units of a PN. The distribution of tokensover the PN at a specific time of execution is referred to asmarking, which is expressed as a vector indicative of the stateof the PN. The transitions are responsible of the dynamic be-havior of the PN, and enable the system to move from onestate to another. For example, a component wear process isone of such processes which can be reflected using a tran-sitions (Andrews, Prescott, & De Rozieres, 2014). In graphi-cal representation, places are typically expressed using circleswhile transitions are drawn as bars or boxes. Arcs are labeledwith their corresponding weights, non-negative integer valuesindicating the amount of parallel arcs (1 by default). Figure 1is provided to serve as illustrative example of a PN of threeplaces (p1, p2, p3), and one transition (t1).

From a mathematical point of view, a PN can be defined asan ordered 6-tuple N as follows (Murata, 1989):

P ,⟨P,T,E,M0,D,W

⟩(3)

where P ∈ Nnp and T ∈ Nnt denote the set of np placesand nt transitions of the PN respectively, M0 ∈ Nnp is theinitial marking, and D ∈ Rnt is the non-negative vector ofswitching delays of the transitions (0 by default). The setE ⊂ Nnp × Nnt represents the edges (also referred to asarcs), which are expressed through ordered pairs of nodesto indicate the connections between places and transitions,i.e. E ⊆ (P×T)∪ (T×P). As mentioned above, each edgehas assigned a weight (1 by default) within the set of weightsW.

At a certain state k, the PN dynamics can be described throughan algebraic equation defined by:

Mk+1 = Mk +ATuk (4)

where uk is the firing vector at k, a nt-dimensional vector

of Boolean values whose elements are obtained according tothe firing rule. A ∈ Nnt×np is the incidence matrix of thegraph, whose elements are the result of subtracting the for-ward (A+) and backward (A−) incidence matrices respec-tively, as follows:

A = A+ −A− (5)

where

A+ =

a+11 a+12 ··· a+1np

a+12 a+22 ··· a+2np

.... . .

a+nt1a+nt2

··· a+ntnp

A− =

a−11 a−12 ··· a−1np

a−12 a−22 ··· a−2np

.... . .

a−nt1a−nt2

··· a−ntnp

(6)

The element a+ij from Eq. (6) represents the weight of the arcfrom transition ti to output place pj , whereas a−ij is the weightof the arc to transition ti from input place pj , i = 1, . . . , nt,j = 1, . . . , np. If transition ti is activated at state k, thenui,k ∈ uk is modified according to the firing rule, which canbe expressed as follows:

ui,k =

{1, if M(j) > a−ij ∀pj ∈ •Pti0, otherwise

(7)

where M(j) ∈ N is the marking for place pj , and •Pti de-notes the set of places that belong to the preset of transi-tion ti, i = 1, . . . , nt. For example in the PN from Fig. 1,•Pt1 = {p1, p2}.

Note that by means of PNs and their marking, the behaviorof complex engineering systems can be described in termsof discrete system states and their changes over time. Thefollowing rules summarize the algebra of PNs as explainedabove:

1. Transitions always consume from all the input arcs atthe same time and produce from all out-coming arcs thesame amount;

2. Transition ti is enabled if every input place pj from itspreset •Pti is marked with at least a−ij tokens;

3. An enabled transition ti will fire once the delay time τi ∈D has passed;

4. After firing, transition ti removes a−ij tokens from pj , andadds a+ij tokens to each j-th output place of ti.

Similarly to other formalisms for system modeling like Baye-sian networks, Artificial Neural Networks, etc., PNs sufferfrom the well-known “state explosion” problem, which is par-ticularly common in real-life engineering applications, wherethe number of states increases exponentially with the size ofthe system. Furthermore, it should be noted that PNs arenot well-suited to deal with uncertainty since their output are

3


p(N )1

p(S)1

p(N )2

a−11

a′11−

a′11+

a+12

t1

Preset of t1 Postset of t1

Figure 2. Illustration of a sample PPN with two numericalplaces (p(N )

1 , p(N )2 ), one symbolic place (p(S)1 ), and one tran-

sition (t1).

based on sequences of Boolean operations. Thus, over the lastdecades, researchers have enhanced the original PNs to han-dle (1) uncertainty (Konar & Mandal, 1996; Looney, 1988;Cao & Sanderson, 1993; Bugarin & Barro, 1994; Zhou &Zain, 2016; Cardoso, Valette, & Dubois, 1999), and (2) con-tinuous variables to efficiently reproduce the hybrid aspectof systems without the need of employing complex graphs(David, 1997; Antsaklis, 2000; Julvez, Di Cairano, Bempo-rad, & Mahulea, 2014; Vazquez & Silva, 2015; Silva, 2016).Plausible Petri Nets (PPNs) (Chiachıo, Chiachıo, Prescott,& Andrews, 2016) can be classified as one of these variantsof the PNs developed to better reproduce the nature of engi-neering systems, hence giving partial response to the referreddrawbacks, as will be shown next.

3. PLAUSIBLE PETRI NETS (PPNS)

Plausible Petri nets (PPNs) are a class of PNs recently de-veloped by the authors, which are based on a combinationof discrete and continuous numerical processes whose val-ues may be uncertain (plausible). Two interacting subnetsform the PPN graph: 1) a symbolic subnet, where the to-kens are objects in the sense of integer moving units, as inclassical PNs (Petri, 1962), 2) a numerical subnet, wheretokens are states of information 1. The resulting frameworkis a hybrid variant of PNs where the sets of nodes {P,T}are partitioned into two disjoint subsets, namely numericaland symbolic, which are denoted using superscripts (N ) and(S), respectively. In particular, the set of places P are par-titioned into subset P(N ) ∈ Nnp and P(S) ∈ Nn

′p , such

that P(N ) ∪ P(S) = P, and P(N ) ∩ P(S) = ∅. Super-scripts np, n′p represent the number of numerical and sym-bolic places, respectively. Analogously, transitions T are par-titioned into numerical transitions T(N ) ∈ Nnt and symbolictransitions T(S) ∈ Nn′t , where T(N ) ∪ T(S) = T, and1A state of information can be described by a set of numerical values about astate variable, along with a mapping over them that assigns each numericalvalue with its relative plausibility (Tarantola & Valette, 1982; Rus, Chi-achıo, & Chiachıo, 2016).

T(N ) ∩ T(S) 6= ∅. In this case, nt, n′t denote the numberof numerical and symbolic transitions, respectively. Observethat those transitions that belong to T(N ) ∩T(S) are referredto as mixed transitions (Chiachıo, Chiachıo, Prescott, & An-drews, 2016).

In PPNs, the referred states of information about a systemstate variable xk ∈ X are denoted by fp(xk) and f t(xk)for numerical places and transitions, respectively. In prac-tical terms, these states of information can be understood asprobability density functions (PDFs) except for a normalizingconstant. Thus, the marking Mk of a PPN at a certain time kconsists in a combined vector Mk =

(M

(N )k ,M

(S)k

), where

M(N )k and M

(S)k are column vectors of normalized PDFs and

integer values, respectively. Moreover, as in classical PNs,there exist arc weights for the symbolic places, denoted bya′+ij , a

′−ij ∈ W(S) ⊂ N, whereby the incidence matrix A(S)

can be obtained by Eq. (5). The arc weights for the numeri-cal places are denoted by a+ij , a

−ij ∈W(N ) ⊂ R+, such that

A(N ) =[a+ij]−[a−ij], and i = 1, . . . , nt, j = 1, . . . , np,

where nt, np represent the amount of numerical transitionsand numerical places of the PPN, respectively. Note that thearc weights from the symbolic subnet, e.g. (a′11

−) are dif-ferentiated from the numerical ones using an accent (′). Ingraphical representation, numerical nodes are represented us-ing double line, whereas single line is used for the rest. APPN model is shown in Fig. 4 for illustration purposes. Thedashed rectangles shown in Fig. 4 are to highlight the presetand postset of t1.

3.1. Execution semantics

As stated in the last section, the marking at time k of PPNsconsists of both types of information given by M

(N )k for the

numerical places, and M(S)k for the case of symbolic places.

The marking evolution of the symbolic subnet correspondsto the state equation of a PN (Murata, 1989) (recall Eq. [4]).However, the evolution of M(N )

k relies on an ad hoc informa-tion flow dynamics based on two basic operations (Chiachıo,Chiachıo, Prescott, & Andrews, 2016): the conjunction anddisjunction of states of information (Tarantola & Valette, 1982;Rus et al., 2016). Both markings evolve in a synchronizedmanner, as will be described in next section. In these op-erations, the first principles from Boolean logic, in particu-lar the logic operators AND (∧) and OR (∨), are invoked toallow the continuos information from the numerical subnetto be exchanged into the PPN. More specifically, they en-able the combination and aggregation of states of informa-tion across the numerical subnet of a PPN. To avoid repeatingliterature but conferring a sufficient conceptual framework,the conjunction and disjunction of states of information havebeen briefly explained and illustrated in Fig. 3. The interestedreader is referred to (Tarantola & Valette, 1982) for further

4


fa(x)

fb(x)

f(x) = (fa ∧ fb) (x) = 1αfa(x)fb(x)

µ(x)

α : constant

x

fa(x)

fb(x)

f(x) = (fa ∨ fb) (x)= 1

β

(fa(x) + fb(x)

), β : constant

x

Figure 3. Illustration of the conjunction (left) and disjunction (right) of two arbitrary states of information, namely fa(x) andfb(x).

details.

From this standpoint, the dynamics of PPNs is formulated un-der the adoption of the following rules (Chiachıo, Chiachıo,Prescott, & Andrews, 2016):

1. An input arc from place p(N )j to transition ti ∈ T(N )

conveys a state of information given by a−ij(fpj∧f ti

)(xk),

which remains in p(N )j after transition ti has fired;

2. Transition ti ∈ T(N ) produces to an output arc a stateof information given by a+ij

(f ti ∧ f•Pti

)(xk), where

f•Pti (xk) denotes the resulting density from the dis-

junction of the states of information of the preset of ti. Asstated in Fig. (3), the normalized version of f

•Pti (xk)can be obtained as:

f•Pti (xk) =

1

β

(fp1 + fp2 + · · ·+ fpm

)(xk) (8)

where β is a constant, and p1, . . . , pm ∈ •Pti ⊂ P(N );

3. After firing numerical transition ti, the state of informa-tion resulting in place p(N )

j from the postset of ti, is thedisjunction of the state of information fpj (xk) (the pre-vious state of information), and a+ij

(f ti∧f•Pti

)(xk) (the

information produced after firing transition ti). In math-ematical terms:

fpj (xk+1) =(fpj ∨ a+ij

(f ti ∧ f

•Pti

))(xk) (9)

The rules given above are sufficient to explain the informa-tion flow dynamics of PPNs. Notwithstanding, observe thatthey are mostly based on conjunction of states of informationwhich requires the evaluation of normalizing constants in-volving an intractable integral. Also, note that there are situa-tions where the conjunction is conducted using density func-tions which are not completely known, perhaps because theyare defined trough samples. Particle methods (Arumlampalam,Maskell, Gordon, & Clapp, 2002; Doucet, De Freitas, & Gor-don, 2001) can be used in these cases to circumvent the eval-

uation of the normalizing constant with a feasible computa-tional cost. In particle methods, a set ofN samples

{x(n)

}Nn=1

with associated weights{ω(n)

}Nn=1

are used to obtain an ap-proximation for the required density function [e.g.

(fa ∧

fb)(x)], as follows:

(fa ∧ fb

)(x) ≈

N∑n=1

ω(n)δ(x− x(n)

)(10)

where δ is the Dirac delta and x(n) ∼(fa∧fb

)(x). The parti-

cle weight ω(n) represents the likelihood value of x(n), and isrepresentative of the plausibility of x(n) when it is distributedaccording to

(fa ∧ fb

)(x). It can be evaluated for the case of

X being a linear space as follows (Tarantola & Mosegaard,2007):

ω(n) =fa(x

(n))fb(x(n))∑N

n=1 fa(x(n))fb(x(n))

(11)

A pseudocode implementation to obtain a particle approxi-mation from the conjunction

(fa ∧ fb

)(x), is provided in the

Appendix as Algorithm 1.

3.1.1. Firing rule

In PPNs, any transition ti ∈ T is fired at time k if the delaytime has passed and:

1. Every symbolic place from the preset of ti has enoughtokens according to their input arc weight, as in classicalPNs;

2. Each of the conjunction of states of information betweenf ti and fpjk is possible, where p(N )

j belongs to the presetof ti;

3. Conditions (a) and (b) are both satisfied when ti is amixed transition, i.e. ti ∈ (T(S) ∩T(N )).

Note from Condition (b) that a conjunction, e.g.(fpjk ∧f

tik

)(xk),

is possible if(fpjk ∧ f

tik

)(xk) 6= ∅ (Tarantola & Valette,

1982). Note also that when any of the states of information

5


p(N )1

p(S)1

Workingcondition

p(N )2

p(N )3

Buffer

p(S)2

Maintenanceneeded

a−12

a −11

a′11−

a′12+

a+13

t1

(a)

p(N )1

p(S)1

Workingcondition

p(N )2

p(N )3

Buffer

p(S)2

Maintenanceneeded

a−11

a′11−

a′12+

a +13

t1

a−12

a ′12 − a

+23

t2

(b)

Figure 4. Plausible Petri nets of the examples given in §4.1.

involved in a conjunction is the homogenous density (alsoreferred to as “non-informative density”) µ(xk) of the statespace of consideration X , then the conjunction is always pos-sible (Tarantola & Valette, 1982; Chiachıo, Chiachıo, Prescott,& Andrews, 2016), thus Condition (b) is automatically ful-filled. This argument is important in terms of using PPNs inpractical applications, as will be demonstrated in next section.

4. SYSTEM LEVEL PHM BY PPNS

In this section, a set of PPN sample architectures are providedto illustrate how our PPNs can be used for decision mak-ing at a system level in applications where information fromprognostics along with other sources of information (like ex-pert knowledge, sensors, etc.) may coexist, as usual in prac-tice. The examples have been kept as simple as possible sincethey are mostly intended to serve as guideline to built morecomplex PPNs. Moreover, the examples provide represen-tative architectures whereby to conceptualize the proposedPNN methodology in a prognostics and health managementcontext.

4.1. Integrative decision making for multi-component prog-nostics

A couple of PPN architectures are exemplified here for mod-eling decision making aspects in presence of multiple infor-mation about the EOL from different components of an en-gineering system. Figure 4 shows two PPNs of three numeri-cal {p(N )

1 , p(N )2 , p

(N )3 } and two symbolic {p(S)1 , p

(S)2 } places,

along with one mixed transition t1 (two transitions for Fig. 4b).This example assumes that numerical places p(N )

1 and p(N )2

enclose uncertain information about the EOL of two com-ponents or sub-systems, which is expressed through a PDFdenoted by fEOL1 and fEOL2 , respectively. Transition t1 isdefined based on condition (Chiachıo, Chiachıo, Prescott, &

Andrews, 2016) using an indicator function for the state spacethat assigns a value of 1 when the expected value of EOLreaches a specific threshold denoted by ε ∈ R, and 0 other-wise, as indicated in Table 1. Provided that place p(S)1 hasone token (assumed that a′11 = 1 in this example), then, ac-cording to the firing rule given in §3.1.1, transition t1 is acti-vated once the expected EOL from both components or sub-systems has reached the threshold value ε (not necessarily atthe same time), whereupon the system turns to a “mainte-nance needed” state. The resulting information is collectedin place p(N )

3 which acts as a buffer of information. Thisbuffer can be used for diagnostics purposes since it collects aweighted distribution of plausible EOL values from the two-component system, conditioned to E(EOLk) > ε. As statedby the execution semantics of PPNs (refer to §3.1), the result-ing PDF in p(N )

3 can be described as:

fp3 =a+13

a−11 + a−12

(a−11f

EOL1 + a−12fEOL2

)(12)

Note from the last equation that when a−11 = a−12 = a+13 = 1,

then fp3 = 12

(fEOL1 + fEOL2

), an averaged sum of both

PDFs of EOL.In Fig. 4b, an analogous PPN architecture is illustrated for thetwo-component system. However in this case, the graph rep-resents a two-component system acting in series ( t2 ≡ t1),thus it turns to the “maintenance needed” state once the ex-pected value of EOL from any of the components reachesthe threshold value ε. Note that, both PPN architectures canbe straightforwardly extended to the case of multiple com-ponents or sub-systems provided that the PDF of EOL fromthose components are known. Finally observe that throughthese simple graph architectures, go/no-go decisions can bemade using uncertain information from multiple componentsacting in series, parallel, or a combination from them.

6


p(N )1

p(S)1

p(N )2

Expertinformation

p(S)3

p(N )3

a−11

a′11−

a+13

a′13+

t1

a−12

a+

23a′23−

t2

(a)

p(S)1

Workingcondition

p(N )1

p(S)5

Componentbeing

repared

p(N )2

Buffer

p(S)3

Maintenanceneeded

a−11

a′11−

a+12

a′13+

a′15+

t1p(S)4

Availableengineers

a′23−

a′24−

a′25+

t2

(b)

Figure 5. Illustration of the exemplified PPN architecture explained in §4.2 (left) and §4.3 (right).

4.2. Combining expert knowledge and prognostics mea-sures

In this section, an example of PPN is provided which includesinformation about the PDF of EOL of a component or sub-system, along with information coming from expert knowl-edge. Figure 5a illustrates the PPN consisting of three numer-ical places, two symbolic places, and two transitions. Let usnow assume that the numerical place p(N )

1 comprises the PDFof EOL of a component given by fEOL, whereas fp3 = ∅, ini-tially. Next, the proposed PPN architecture also encompassesinformation from one expert in place p(N )

2 , which is repre-sented by a PDF given by fExpert (e.g. fExpert can be a uni-form PDF of EOL representing an interval of possible valuesof EOL). As in the last example, transition t1 is defined basedon condition, such that it is fired if the uncertainty2 of thePDF fEOL reaches or exceeds a specific threshold value. If t1is fired, then place p(S)2 receives a token that enables the infor-

2For example by calculating the differential entropy of the PDF of EOL(Chiachıo, Beck, Chiachıo, & Rus, 2014; Chiachıo, Chiachıo, Prescott, &Andrews, 2016)

mation from the expert to be transferred to place p(N )3 . Note

that through this exemplified PPN, a decision making processcan be modeled about the adoption of information from anexpert if the uncertainty about the EOL is higher than a cer-tain value ξ. For example, this can be as a consequence of afaulty sensor, or a perturbed prognostic estimation. The re-sulting information in p(N )

3 includes the PDF fEOL comingfrom p

(N )1 and that from the expert, as follows:

fp3 =a+13

a−11 + a−12

(a−11f

EOL1 + a−12fExpert

)(13)

If required, higher relevance can be conferred to the expert in-formation by increasing the weight a−12 with respect to a−11. Fi-nally, observe that the PDF fp3 can be further used in an op-erational context (e.g., for diagnostics), as was described inlast section.

Table 1. Description of the transitions of the PPN shown in Figs. 4a, 4b, 5a, and 5b.

Reference Transition Type Condition State of information

Fig. 4t1 Mixed C1 =

{xk ∈ X : EfEOL1 (xk) 6 ε

}f t1 ∼ IC1(xk)

t2 (Fig.4b) Mixed C2 ={xk ∈ X : EfEOL2 (xk) 6 ε

}f t2 ∼ IC2(xk)

Fig. 5at1 Mixed C1 =

{xk ∈ X : EfEOL(xk) 6 ε

}f t1 ∼ IC1(xk)

t2 Mixed None (homogeneus) f t2 ∼ µ(xk)

Fig. 5bt1 Mixed C1 =

{xk ∈ X : EfEOL(xk) 6 ε

}f t1 ∼ IC1(xk)

t2 Symbolic τ (delay) Not applicable

7


4.3. Integration of prognostics and resource availability

This example illustrates the case of a decision making pro-cess, like activation of a maintenance action, taken after thePDF of EOL has reached a certain value ε as in §4.1, exceptthat in this case it occurs contingent upon availability of en-gineers. Fig. 5b shows the idealized system using a PPN oftwo numerical places, four symbolic places, one mixed tran-sition, and one symbolic transition. Numerical place p(N )

1 isassumed to represent the PDF fEOL of a component. As inthe example shown in Fig. 4a, t1 is fired once the expecta-tion of fEOL has reached the threshold ε provided that p(S)1

has enough amount of tokens according to a′11−, which for

the sake of illustration, is assumed to be a′11 = 1. Next, letus also assume that symbolic transition t2 represents the acti-vation of the maintenance activity (an activation delay τ canbe assigned to t2 (Andrews et al., 2014), although this infor-mation is irrelevant here). Observe that when t1 is fired, thent2 will be activated if the number of available engineers ishigher or equal to a′24

−. In such case, p(S)5 is marked, thenthe system turns to “component being repaired” state.

5. NUMERICAL EXAMPLE

The PPN framework presented above is exemplified here us-ing a numerical example to illustrate some of the advantagesof using PPNs for integration of prognostics at a system level.In this example, a self-managed two-component system ismodeled through a PPN which comprises uncertain informa-tion about the EOL of two degrading components acting inseries, along with expert knowledge about the whole system’sEOL. Figure 6 illustrates the idealized system model througha PPN of four numerical places, four symbolic places, and

five transitions. Noisy measurements about the components’state of degradation are assumed to be available, whereby anestimation of the EOL can be obtained, provided that thereexists a specified degradation threshold, and an appropriateprognostics algorithm to make predictions (see for example(Chiachıo, Chiachıo, Shankararaman, & Andrews, 2017)).Let us denote by EOL(j)

k ∈ X ⊂ R+ a stochastic variablecorresponding to the EOL of the j-th component, j = 1, 2,which evolves over time k ∈ N following a dynamic equa-tion EOL(j)

k = hk(xEOLj

k−1 , θj) given by:

EOL(j)k = e−θjkEOL(j)

k−1 + vk (14)

where θj is an uncertain decay parameter whose values aremodeled as a Gaussian centered at 0,006 and 0,008 for j =1, 2, respectively, and 100% of coefficient of variation in bothcases. The term vk represents a measurement error whichis assumed to be modeled as a zero-mean Gaussian densityfunction with standard deviation given by σv = 5. In thePPN, the mixed transitions ti, i = {1, 2, 4} are defined bycondition, thus their states of information can be expressedusing Diract Delta density functions (Tarantola, 2005), i.e.,f ti(EOLk) ∼ ICi(EOLk). Henceforth, their activation isprescribed for the stochastic variable EOL(j)

k on fulfilling thecondition EOL(j)

k ∈ Ci, such that:

C1 ={

EOLk ∈ X : Efp1 (EOLk) 6 ε}

(15a)

C2 ={

EOLk ∈ X : Efp2 (EOLk) 6 ε}

(15b)

C3 ={

EOLk ∈ X : H(EOLk) > ξ}

(15c)

where ε = 20 and ξ = 5. In (15c), H denotes the differ-ential entropy of EOLk, that can be obtained by evaluating

p(N )1 p

(S)1

p(S)2

p(N )2

p(N )3

t1

p(S)4

p(S)3

t2

fEOL1

k

fEOL2

k

Workingcondition Fail

p(N )4

Expertinformation

Entropythreshold

Warning

µ

t3 t4

τ

t5

Figure 6. Illustration of the PPN from the numerical example given in Section 5.

8


0 5 10 15 20 25 30k

0

20

40

60

80

100

120

140EOLk

Threshold (ε2 = 20)

(a)

0 5 10 15 20 25 30k

0

20

40

60

80

100

120

140

EOLk

Threshold (ε2 = 20)

(b)

Figure 7. Plots of the evolution over time of the states of information about the stochastic variable EOL(j)k for components

j = 1, 2 (panels [a] and [b], respectively) from the Plausible Petri net of the example given in §5. The expectation of the statesof information is represented using gray circles. The dashed lines represent the 5th-95th probability band.

1/2 ln [(2πe)var(EOLk)] as a measure quantifying the uncer-tainty of EOLk. The initial marking of the numerical placesis given by fp10 ∼ N (100, 5), fp20 ∼ N (100, 5), fp30 = ∅,and fp4 = U [18, 22]. The latter represents expert knowledgeabout EOL, which is given by a uniform PDF defined overthe interval [18, 22].

Initially at k = 0, the system starts in the “working condi-tion” state represented by one token at p(S)1 , thus M

(S)0 =

(1, 0, 0, 0)T . Once any of the expected values of the predicted

EOL of components 1 and 2 (represented in place p(N )1 and

p(N )2 , respectively) has reached the threshold value ε, then

transitions t1, t2 (not necessarily both nor simultaneously)produce one token to p(S)2 which enables transition t3 to befired, whereupon the expert information about system’s EOLenters into play and is transferred to p(N )

3 . Next, the systemturns to a “warning” state, and a decision is made conditionedupon: 1) the quality of the information given by fp3k , and 2)the total time spent by the system under no failure states. Thedifferential entropy (DE) is used in this example as a qual-ity indicator of the information in place p(N )

3 , so that thetransition t4 is activated if the DE of fp3k is higher than thethreshold ξ = 5. In such case, the system turns to “inspec-tion needed” state, otherwise the system remains in “warn-ing” mode. In this example, the time spent by the system inperforming transition t5 (which corresponds to an activationtime given by τ ) is assumed to represent a scheduled periodicmaintenance activity such that if k > τ , then the system di-

rectly turns to the “maintenance needed” condition irrespec-tive of the component’s degradation state nor their predictedEOL.

For the numerical evaluation of the PPN in Fig. 6, the ex-ecution semantics rules (recall §3.1) along with Eq. (4), areapplied in confluence with the firing rule for the system stateevolution described through the marking Mk. The algorithmfor particle approximation of conjunction of states of infor-mation described in the Appendix, has been applied usingN = 1000. Note that the disjunction of states of information

Table 2. Summary of the discrete events taking place whenrunning the PPN shown in Fig. 6. The second and third col-umn show the symbolic marking and firing vector, respec-tively.

Time M(S)k uk Event

k = 0 (1 0 0 0) (0 0 0 0 0) –k = 1 (1 0 0 0) (0 0 0 0 0) –

.... . .

k = 16 (1 0 0 0) (1 0 0 0 0) t1 firedk = 17 (0 1 0 0) (0 0 0 0 0) t3 enabledk = 18 (0 1 0 0) (0 0 1 0 0) t3 firedk = 19 (0 0 1 0) (0 0 0 0 0) t4 enabledk = 20 (0 0 1 0) (0 0 0 1 0) t4 firedk = 21 (0 0 0 1) (0 0 0 0 0) –

9


0 5 10 15 20 25 30 35 40EOL

(3)k

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

Figure 8. Kernel density estimates of the state of informationabout EOL in place p(N )

3 for k = 17 and k > 18 (darker graycolor).

can be straightforwardly evaluated using samples by just join-ing the samples from the component-wise density functions,and affecting their particle weights using an appropriate nor-malizing constant so as to obtain a bone fide density. Figure 7shows the results for the two-components’ EOL (from placesp(N )1 and p(N )

2 ) for time indexes k = 0 to k = 30. Notefrom Fig. 7 that component 1 first reaches the threshold, i.e.,the expectation of EOL(1)

k first reaches the value ε = 20 atk = 16. Next, transition t3 is enabled and the informationcoming from the expert is aggregated to p(N )

3 at k = 18. Fig-ure 8 shows the resulting PDF in p(N )

3 before and after theinformation from the expert was incorporated. Observe theinfluence of the expert knowledge in terms of gain of infor-mation, which makes the distribution of plausible EOL val-ues being more concentrated around the recommended valuesfrom the expert, namely [18,22]. Notwithstanding, the DE ofthe resulting state of information given by fp3 at k > 18 ishigher than ξ = 5, then the system finally turns to the ”main-tenance needed” state. A summary of the results for the sym-bolic subnet of the PPN model is provided in Table 2.

This example illustrates that uncertain information about theEOL of different components along with information fromexperts, can be integrated into a system level model for prog-nostics and decision making. The numerical results confirmthat system non linearities, apart from that attributable to thestochastic variable EOLk, can be taken into account. Exam-ples of the referred non linearities are: ad hoc synchroniesbetween components, user-defined time to failure, resourcesavailability, etc., which makes our PPNs useful for prognos-tics and health management at a system level.

6. CONCLUSIONS

This paper presented a novel prognostics methodology to in-tegrate information from prognostics with decision makingaspects at a system level using Plausible Petri nets. The ap-proach has the advantage of addressing the prognostics prob-lem as a unified system-level approach where multiple sourcesof uncertain information can be integrated with discrete-events,conferring high versatility to better reproduce real-world prob-lems of prognostics without the need of using extremely com-plex nets, as usual when adopting other formalisms. Furtherresearch is needed to investigate suitable PPN architectures tobetter integrate the influence of intervention activities withinthe predicted state of health at a system level.

ACKNOWLEDGMENT

This work was supported by the Lloyd’s Register Founda-tion, a charitable foundation in the UK helping to protectthe life and property by supporting engineering-related ed-ucation, public engagement, and the application of research.

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BIOGRAPHIES

Manuel Chiachıo is Postdoctoral Research Fellow at the Re-silience Engineering Research Group, University of Notting-ham, UK. He holds a PhD in Structural Mechanics (interna-tional mention) awarded by the University of Granada, (Spain),and a MSc in Civil Engineering (2007), and also a MSc inStructural Engineering (2011), by the same University. Hisresearch focus on uncertainty quantification methods and al-gorithms, risk and reliability analysis, and artificial intelli-gence methods in application to a variety of engineering ar-eas, which range from structural and mechanical engineeringto bioengineering applications. During the course of his PhD

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work, Manuel worked as guest scientist at world-class univer-sities and institutions, like Hamburg University of Technol-ogy (Germany), California Institute of Technology (Caltech),and NASA Ames Research Center (USA). This research hasled to several publications in highly ranked journals. It hasbeen awarded by the National Council of Education of Spainthrough one of the prestigious FPU fellowships, by the An-dalusian Society of promotion of the Talent, by the EuropeanCouncil of Civil Engineers (ECCM) with the Silver Medalprize in the 1st European Contest of Structural Design (2008),and also by the Prognostics and Health Management Societywith a Best Paper Award in 2014. Prior to joining the Univer-sity of Granada in 2011, Manuel worked as a consultancy en-gineer for four years in top engineering companies in Spain.

Juan Chiachıo is a Research Fellow in Infrastructure AssetManagement in the Resilience Engineering Research Groupat the University of Nottingham (UK). He received his PhD inStructural Engineering (international mention) in 2014 by theUniversity of Granada, Spain. In addition, he holds a MSc inStructural Engineering (2011) and a MSc in Civil Engineer-ing (2007), both by the University of Granada. His researchis focused on translating reliability and prognostics methodsinto the life-cycle analysis of structural and infrastructuralsystems subjected in-service degradation. This research hasled to several publications in highly ranked journals, a best-paper award and nominations in major conferences, and ithas been awarded by the Spanish National Council of Educa-tion through one of the FPU annual fellowships, by the An-dalusian Society of Promotion of Talent, by the Prognosticsand Health Management Society with a Best Paper Award,and by the European Council of Civil Engineers. In addi-tion, his work has attracted the interest of world-class insti-tutions for collaborative research, like the Prognostics Centerof Excellence of NASA, the California Institute of Technol-ogy, and the Hamburg University of Technology (Germany).His current research at the University of Nottingham dealswith the development of a Bayesian prognostics frameworkfor infrastructure asset management, under EPSRC projecttitled ”Whole-life cost assessment of novel material railwaydrainage systems (EP/M023028/1)”.

Shankar Sankararaman received his B.S. degree in CivilEngineering from the Indian Institute of Technology in Ma-dras (2007) and later, obtained his Ph.D. in Civil Engineer-ing from Vanderbilt University, Nashville, Tennessee, U.S.A.in 2012. His research focuses on the various aspects of un-certainty quantification, integration, and management in dif-ferent types of aerospace, mechanical, and civil engineeringsystems. His research interests include probabilistic meth-

ods, risk and reliability analysis, Bayesian networks, systemhealth monitoring, diagnosis and prognosis, decision-makingunder uncertainty, treatment of epistemic uncertainty, and mul-tidisciplinary analysis. He is a member of the Non-Determi-nistic Approaches (NDA) technical committee at the Ameri-can Institute of Aeronautics, the Probabilistic Methods Tech-nical Committee (PMC) at the American Society of Civil En-gineers (ASCE), and the Prognostics and Health Manage-ment (PHM) Society. Currently, Shankar is a researcher atNASA Ames Research Center, Moffett Field, CA, where hedevelops algorithms for uncertainty assessment and manage-ment in the context of system health monitoring, prognostics,and decision-making. He has been recently named as virtualmember of the Resilience Engineering Research Group, Uni-versity of Nottingham (UK).

John Andrews is Head of the Resilience Engineering Re-search Group at the University of Nottingham where he holdsa Royal Academy of Engineering Research Chair in Infras-tructure Asset Management. Prior to this he worked for 20years at Loughborough University where his final post wasProfessor of Systems Risk and Reliability. The prime focusof his research has been on methods for evaluating the sys-tem resilience, unavailability, unreliability and risk. Much ofthis work has concentrated on the Fault Tree analysis tech-nique and the use of Binary Decision Diagrams (BDDs) asan efficient and accurate solution method. Recently atten-tion has turned more the degradation modeling and the ef-fects of maintenance, inspection and renewal on asset perfor-mance. In this context the modeling he has carried out hasextended the Petri net and Bayesian Network capabilities. In2005, John founded the Proceedings of the Institution of Me-chanical Engineers, Part O: Journal of Risk and Reliability ofwhich was the Editor-in-chief for 10 years. He is also a mem-ber of the Editorial Boards for 6 other international journalsin this field.

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APPENDIX

Pseudocode implementation to obtain particles from the con-junction of two arbitrary states of information fa(x) and fb(x).

Algorithm 1 Particle approximation of conjunction of statesof informationInputs: N, fa(x), fb(x) {number of particles and states of infor-

mation}Outputs:

{x(n), ω(n)

}N

n=1, where x(n) ∼ (fa ∧ fb)(x)

Begin

1: Sample{(

x(n)a , ω

(n)a

)}N

n=1from fa(x)

2: Set x(n) ← x(n)a , n = 1, . . . , N

3: Set ω(n) ← fb(x(n)

), n = 1, . . . , N {unnormalized weights}

4: Normalize weights ω(n) ← ω(n)∑Nn=1 ω(n) , n = 1, . . . , N

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integration of prognostics at a system level: a petri net ... · integration of prognostics at a...

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