Mayank Shekhar

Mayank S Jha
email: jha.mayank.jha@gmail.com

G Dauphin-Tanguy

B Ould-Bouamama

Particle Filter based Integrated Health Monitoring in Bond Graph Framework

Keywords: Prognostics, Bond Graph, Degradation Model, Particle Filters, Remaining Useful Life, Robust Fault Detection, Bayesian estimation, Monte Carlo

This chapter presents a holistic method to addresses the issue of health monitoring of system parameters in Bond Graph (BG). The advantages of BGs are integrated with Bayesian estimation techniques for efficient diagnostics and prognostics of faults. In particular, BG in Linear fractional transformations (LFT) are used for modelling the global uncertain system and sequential Monte Carlo method based Particle filters (PF) are used for estimation of state of health (SOH) and subsequent prediction of the remaining useful life (RUL). In this work, the method is described with respect to a single system parameter which is chosen as prognostic candidate. The prognostic candidate undergoes progressive degradation and its degradation model is assumed to be known a priori. The system operates in control feedback loop. The detection of degradation initiation is achieved using BG LFT based robust fault detection technique. The latter forms an efficient diagnostic module. PFs are exploited for efficient Bayesian inference of SOH of the prognostic candidate. Moreover, prognostics is achieved by assessment of RUL in probabilistic domain. The issue of prognostics is formulated as joint stateparameter estimation problem, a hybrid prognostic approach, wherein the fault model is constructed by considering the statistical degradation model of the prognostic candidate. The observation equation is constructed from nominal part of the BG-LFT derived Analytical Redundancy Relations (ARR). Various uncertainties which arise because of noise on ARR based measurements, degradation process, environmental conditions etc. are effectively managed by PF. This allows the production of effective predictions of the RUL of the prognostic candidate with suitable confidence bounds. The method is applied over a mechatronic system in real time and performance is assessed using suitable metrics.

Introduction

Besides the abrupt faults that have been considered in the previous chapters, incipient system faults and degradations of the system parameters pose significant hurdles in efficient maintenance of the system. For example, fatigue enabled wear in turbine blades, incipient leakage in valves of process engineering systems, friction induced jamming of rod in aircraft actuators etc. pose great threat to system reliability and safety. Such problems are efficiently resolved when addressed under the realm of so-called condition based maintenance (CBM) and prognostics and health management (PHM) [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF]. The latter represent a predictive maintenance philosophy that has emerged only recently on contrary to the traditional strategies based upon preventive and corrective maintenance.

The main feature of CBM is the consideration of the -actual‖ condition of system component for designing maintenance actions rather than on an elapsed time or running hours' basis. Thus, CBM primarily depends upon current assessment of system health, or state and involves real time data monitoring and processing. The two basic aspects of CBM are diagnostics and prognostics. As seen in the previous chapters, Diagnostics involves detection of fault and thereby, identification and quantification of the root cause of a problem. Prognostics involves prediction of the future health of the equipment either before or after a problem occurred [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF][START_REF] Sikorska | Prognostic modelling options for remaining useful life estimation by industry[END_REF]. As stated in [START_REF]Condition Monitoring And Diagnostics Of Machines -Prognostics -Part 1: General Guidelines[END_REF], prognostics is -estimation of time to failure and risk for one or more existing and future failure modes‖.

The Remaining Useful Life (RUL) becomes a reliable estimate of the time to failure; it denotes how long system can function safely/reliably and within the prescribed limits of system functionalities. Thus, assessment of RUL involves predictions in future. In this context, the major motivation remains in providing sufficient lead-time between detection of a fault (diagnostic step) and occurrence of the system/component failure so that pro-active maintenance actions can be strategized in advance [START_REF] Vachtsevanos | Intelligent Fault Diagnosis and Prognosis for Engineering Systems[END_REF].

RUL prediction is not a trivial task as it involves future predictions which not only require precise information of current health, but also remain sensitive to various types of uncertainties to a large degree. These uncertainties involve stochastic evolution of incipient degradations, failure modes, varying operational conditions, measurement noise etc. In face of all such uncertainties, the prognostic procedure must be able to accurately assess the rapidity of system degradation till failure and novel events that may significantly influence the assumed/learnt degradation trend. Due to inherent stochastic phenomena and uncertainty involved, evaluation of confidence on RUL predictions is given a significant weightage. In fact, several business decisions are based upon confidence limits associated with RUL predictions rather than the specific value of RUL itself [START_REF] Saxena | Metrics for offline evaluation of prognostic performance[END_REF]. In essence, determination of accurate and precise RUL estimate forms the core objective of any prognostics procedure.

On the other hand, the term PHM describes the systems that implement a CBM philosophy [START_REF] Vachtsevanos | Intelligent Fault Diagnosis and Prognosis for Engineering Systems[END_REF]. However, in the context of PHM, prognostics gains a wider meaning encompassing the tasks of fault detection, fault-identification, current health assessments, performance monitoring and RUL predictions [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF]. Thus, diagnostics and prognostics form building blocks of any CBM enabled PHM architecture. When these two essential tasks are achieved in an integrat-ed manner, such a common paradigm may be given the designation of integrated health monitoring framework [START_REF] Bregon | An integrated framework for model-based distributed diagnosis and prognosis[END_REF]7].

In BG framework, diagnostics and prognostics task can be achieved in an integrated way by exploiting the properties of Analytical Redundancy Relations (ARRs) and their numerical evaluations or residuals. In this context, due to deterministic nature of ARRs, most of the existing works have neglected the inherent randomness in damage progression [START_REF] Medjaher | Hybrid prognostic method applied to mechatronic systems[END_REF][START_REF] Medjaher | Residual-based failure prognostic in dynamic systems[END_REF][START_REF] Djeziri | Data driven and model based fault prognosis applied to a mechatronic system[END_REF][START_REF] Djeziri | Fault prognosis based on fault reconstruction: Application to a mechatronic system[END_REF], which in turn, has led to RUL predictions that do not incorporate associated uncertainties and inherent stochasticity.

This chapter details ARR based integrated health monitoring methodology where the benefits of BG in Linear Fractional Transformations (BG-LFTs) have been integrated with advantages of Bayesian inference techniques to obtain accurate and precise estimate of parametric health in probabilistic domain. The inherent randomness in degradation progression is effectively managed by using sequential Monte Carlo based Particle Filters (PF) for estimation of state of a system parameter and subsequent RUL prediction in probabilistic domain.

After this Introduction, Sect. 7.2 details various approaches of prognostics, BG-LFT method and non-linear Bayesian inference technique using PFs. Sect. 7.3 discusses Degradation models (DM). The method of prognostics is described in the next section. Sect. 7.4 and 7.5 discuss the integrated health monitoring strategy and evaluation metrics, respectively. Sect. 7.6 details the application of methodology on a mechatronic system in real system. Sect. 7.7 draws conclusions.

Background and Techniques

This section discusses different techniques of prognostics. Moreover, BG-LFT technique of modelling uncertain systems and associated fault detection technique is discussed briefly. The latter is employed for detection of degradation initiation for the integrated health monitoring purposes. Additionally, non-linear Bayesian filtering using Particle Filters (PF) is described as it plays a significant role in the prognostics method presented in this chapter.

Approaches of Prognostics

Last decade has witnessed an extensive surge in development of various prognostics techniques and its application in diverse technical domains. Due to the inherent versatility, approaches of prognostics have been attempted to be classified in different ways [START_REF]Condition Monitoring And Diagnostics Of Machines -Prognostics -Part 1: General Guidelines[END_REF][START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF][START_REF] Lee | Intelligent prognostics tools and e-maintenance[END_REF][START_REF] Vachtsevanos | Intelligent Fault Diagnosis and Prognosis for Engineering Systems[END_REF][START_REF] Liao | Expert system methodologies and applications-a decade review from 1995 to 2004[END_REF] etc. Here, the authors have preferred to adapt the classification presented in [START_REF] Sikorska | Prognostic modelling options for remaining useful life estimation by industry[END_REF]. The modified classification groups are presented and discussed in brief.

Probabilistic life-usage models: These approaches depend upon the statistical information collected to assess the historical failure rate of the components and develop life-usage models [START_REF] Blischke | Reliability: modeling, prediction, and optimization[END_REF][15][START_REF] Rausand | System reliability theory: models, statistical methods, and applications[END_REF]. Various functions can be applied to model sta-tistical failure data such as exponential, normal, lognormal, Weibull functions etc.

[17]. Moreover, the RUL is described as a probability density function (PDF) [START_REF] Vachtsevanos | Intelligent Fault Diagnosis and Prognosis for Engineering Systems[END_REF][START_REF] Sikorska | Prognostic modelling options for remaining useful life estimation by industry[END_REF]. Accurate assessment of RUL demands huge sets of failure database and extensive testing.

Data Driven Prognostics: The data associated with system functionality, degradation patterns etc. are exploited using machine learning techniques to extract system signals, and features which can be used to obtain behavior of damage progression, health index etc. Broadly, two major strategies can be identified as discussed below.

Degradation Trend extrapolation and time series predictions: In broad terms, the signals that indicate the state of the system are mapped as function of time and extrapolated in future using various techniques until a pre-fixed failure threshold is reached/crossed [START_REF] Engel | the real issues involved with predicting life remaining[END_REF]. Mainly time series forecasting techniques are borrowed for this purpose such as: linear/non-linear regression techniques, auto-regressive models [START_REF] Wu | Prognostics of machine health condition using an improved arima-based prediction method[END_REF], exponential smoothing techniques [START_REF] Byington | Prognostic enhancements to diagnostic systems for improved condition-based maintenance [military aircraft[END_REF], autoregressive moving average (ARMA), Autoregressive integrated moving average (ARIMA) [START_REF] Box | Time series analysis: forecasting and control[END_REF]. The ARMA models and associated variants prove efficient for short-term predictions. Due to noise and inefficient uncertainty management, they prove less reliable for long term predictions.

Learning damage progression: The degradation trends, failure patterns etc. are learnt for training mathematical models. The latter in turn is used to model the relationship between damage progression and RUL. Employment of artificial neural networks (ANNs) and their numerous variants fall under this category. Feed-forward ANNs are extensively employed to estimate the current degradation index (state) by using system features (extracted signals, feature pattern etc.) as inputs. Then, one step-ahead prediction is generated by using previous state of degradation values (degradation index). The next iteration uses this prediction to produce long term predictions [22]. Major drawback in this context is that the efficiency of predictions remain limited in face of variable degradation trends, novel failure modes etc. As such, accurate RUL predictions are not obtained on individual component unit to unit basis, but rather over large sets of component population. A comprehensive updated review of data-driven techniques can be found in [START_REF] An | Practical options for selecting data-driven or physics-based prognostics algorithms with reviews[END_REF][START_REF] Tsui | Prognostics and Health Management: A Review on Data Driven Approaches[END_REF].

Model Based Prognostics: Under this category, physics-of-failure models or degradation models (DM) are typically used to assess the damage progression and state of health (SOH). These DMs are derived from the first principles of physics. As such, they possess the capability of attaining maximum accuracy and versatility (scope of adaptation under varying degradation trend). There is a clear understanding of the underlying degradation process. There exists vast literature such as: fatigue models for modelling initiation and propagation of cracks in structural components [START_REF] Zio | Particle filtering prognostic estimation of the remaining useful life of nonlinear components[END_REF], electrolytic overstress ageing [START_REF] Celaya | A model-based prognostics methodology for electrolytic capacitors based on electrical overstress accelerated aging[END_REF], Arrhenius equation for prediction of resistance drift [START_REF] Kuehl | Using the Arrhenius equation to predict drift in thin film resistors[END_REF], physics inspired power model [START_REF] Maricau | A methodology for measuring transistor ageing effects towards accurate reliability simulation[END_REF] or log-linear model for degradation of current drain [START_REF] Lu | Statistical inference of a time-to-failure distribution derived from linear degradation data[END_REF], physics-inspired exponential degradation model for aluminum electrolytic capacitors [START_REF] Kulkarni | Physics based electrolytic capacitor degradation models for prognostic studies under thermal overstress[END_REF] etc.

Given the behavioral model of damage progression, the current SOH is popularly obtained in probabilistic domain with the help of Bayesian estimation techniques. Based upon the current SOH estimate, prediction of RUL is done. Such a probabilistic framework involving recursive Bayesian techniques efficiently addresses the main issues related to SOH under variable degradation; efficient management of uncertainty, environmental noise, future loading conditions, associated confidence limits for RUL predictions [START_REF] Daigle | Model-based prognostics under limited sensing[END_REF][START_REF] Daigle | A Model-Based Prognostics Approach Applied to Pneumatic Valves[END_REF][START_REF] Daigle | Model-based prognostics with concurrent damage progression processes[END_REF][START_REF] Roychoudhury | An integrated model-based diagnostic and prognostic framework[END_REF]. Filter for estimation and prediction process is chosen based upon the modelling hypothesis and desired performances [START_REF] Daigle | A comparison of filter-based approaches for model-based prognostics[END_REF]. Well-known Kalman filter, an optimal estimator for linear systems has been used for prognostics in [START_REF] Celaya | A model-based prognostics methodology for electrolytic capacitors based on electrical overstress accelerated aging[END_REF]. Extended Kalman filter (EKF) or unscented Kalman filter may also be used for joint state-parameter estimation as presented in [START_REF] Plett | Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 3. State and parameter estimation[END_REF] and [START_REF] Daigle | Distributed prognostics based on structural model decomposition[END_REF] respectively. However, they remain restricted to additive Gaussian noise. Additionally, EKF being sub-optimal diverges quickly if the initial estimate of state is different from the reality by big measure or the model considered for estimation is not correct [START_REF] Saha | Comparison of prognostic algorithms for estimating remaining useful life of batteries[END_REF].

Set in Monte-Carlo framework, PFs form a suitable filter choice in this context [39,[START_REF] Doucet | An introduction to sequential Monte Carlo methods[END_REF]. PF can be applied to non-linear systems corrupted with non-Gaussian noises, for which optimal solutions may be unavailable or intractable. Comprehensive comparison of filters for prognostic purposes are found in [START_REF] An | Practical options for selecting data-driven or physics-based prognostics algorithms with reviews[END_REF], [START_REF] Daigle | A comparison of filter-based approaches for model-based prognostics[END_REF][START_REF] Saha | Comparison of prognostic algorithms for estimating remaining useful life of batteries[END_REF]. Recently, PFs have been extensively for prognostic purposes [41]. Significant works include prediction of end of life (EOL) in lithium-ion batteries [42], battery health monitoring [START_REF] Saha | Prognostics methods for battery health monitoring using a Bayesian framework[END_REF], prediction of battery grid corrosion [START_REF] Abbas | An intelligent diagnostic/prognostic framework for automotive electrical systems[END_REF], estimation and prediction of crack growth [START_REF] Cadini | Monte Carlo-based filtering for fatigue crack growth estimation[END_REF], fuel cell prognostics [START_REF] Jouin | Prognostics of PEM fuel cell in a particle filtering framework[END_REF], application to damage prognostics in pneumatic valve [START_REF] Daigle | A model-based prognostics approach applied to pneumatic valves[END_REF][START_REF] Daigle | Model-based prognostics under limited sensing[END_REF], estimation-prediction of wear as concurrent damage problem in centrifugal pumps with a variance control algorithm [START_REF] Daigle | Model-based prognostics with concurrent damage progression processes[END_REF], employment in distributed prognosis [START_REF] Roychoudhury | An integrated model-based diagnostic and prognostic framework[END_REF], uncertainty management for prognostics [START_REF] Baraldi | Investigation of uncertainty treatment capability of model-based and data-driven prognostic methods using simulated data[END_REF]. Particle filters attract considerable attention [START_REF] An | Prognostics 101: A tutorial for particle filterbased prognostics algorithm using Matlab[END_REF], owing to the ever growing efforts being made for betterment in performances and computational efficiency, such as the use of correction loops [START_REF] Orchard | Advances in uncertainty representation and management for particle filtering applied to prognostics[END_REF], fixed-lag filters [START_REF] Daigle | Model-based prognostics with fixed-lag particle filters[END_REF], kernel smoothing method [START_REF] Hu | A particle filtering and kernel smoothing-based approach for new design component prognostics[END_REF] etc.

The major issue in this type of approach is the accurate and reliable modelling of underlying degradation progression. Often, such accurate degradation models are not available.

Hybrid Prognostics: The problem of non-availability of highly accurate degradation models is alleviated by fusing the advantages of model based and datadriven techniques. This way, there is significant amelioration in the overall prognostic approach [START_REF] Jha | Particle filter based hybrid prognostics for health monitoring of uncertain systems in bond graph framework[END_REF][START_REF] Jouin | Prognostics of PEM fuel cell in a particle filtering framework[END_REF]. The basic philosophy remains in capturing the damage progression using DMs that can be: (i) based upon physics of failure, first principles of behavioral physics (ii) derived using machine learning techniques, (iii) obtained statistically by finding a mathematical model that best fits a given set of degradation data such as: linear model () () Dt as an index representing the degradation (change, percentage change etc.) and a and b as the model parameters. In this context, significant works are: obtaining capacitance loss DM using non-linear least square regression [START_REF] Celaya | A model-based prognostics methodology for electrolytic capacitors based on electrical overstress accelerated aging[END_REF], relevance vector machine regression performed over ageing tests data [START_REF] Saha | Comparison of prognostic algorithms for estimating remaining useful life of batteries[END_REF], DM approximated by a linear part and logarithmic/exponential part [START_REF] Jouin | Prognostics of PEM fuel cell in a particle filtering framework[END_REF] and residual based statistical DM [START_REF] Jha | Particle filter based hybrid prognostics for health monitoring of uncertain systems in bond graph framework[END_REF]. Once the DM has been obtained with acceptable accuracy, recursive Bayesian techniques as discussed previously can be employed to estimate SOH and obtain subsequent RUL predictions. This way, benefits of Bayesian estimators are integrated with data-driven approaches to learn the DM as the current information arrives sequentially.

D

Prognostics in BG framework

Almost all of the existing attempts in BG framework for prognostics have been ARR based and deterministic in nature. Moreover, DMs are considered deterministic so that the SOH and subsequent RUL predictions are obtained deterministically [START_REF] Medjaher | Hybrid prognostic method applied to mechatronic systems[END_REF][START_REF] Medjaher | Residual-based failure prognostic in dynamic systems[END_REF][START_REF] Djeziri | Data driven and model based fault prognosis applied to a mechatronic system[END_REF][START_REF] Djeziri | Fault prognosis based on fault reconstruction: Application to a mechatronic system[END_REF][START_REF] Yu | Prognosis of hybrid systems with multiple incipient faults: augmented global analytical redundancy relations approach[END_REF]7]. Being restricted in deterministic domain, the randomness associated with variable damage progression, novel events, noises etc. are simply ignored that does not lead to an efficient management of the latter in prognostication process and render RUL predictions without confidence limits. Recently, [START_REF] Jha | Particle filter based hybrid prognostics for health monitoring of uncertain systems in bond graph framework[END_REF] proposed a methodology of hybrid prognostics where the benefits of Bayesian filtering techniques and BG enabled ARRs are integrated for efficient prognostics in probabilistic domain. In fact, this chapter is inspired by the work detailed in [START_REF] Jha | Particle filter based hybrid prognostics for health monitoring of uncertain systems in bond graph framework[END_REF].

Bond Graph in Linear Fractional Transformations

BG-LFT is an efficient and systematic way of representing parametric uncertainty over nominal models. An uncertainty on a parameter value θ can be introduced under either an additive form or a multiplicative one, as shown in (7.1) and (7.2) respectively.

n θ = θ ±Δθ; θ 0  n θθ n Δθ θ = θ ( 1 ± δ ); δ = θ (7.1) (7.2)
where θ  and θ δ are respectively, the absolute and relative deviations around the nominal parametric value θ n . When the element characteristic law is written in terms of 1 θ , (7.2) becomes:

1/θ 1/θ nn 1 1 -Δθ = .( 1+δ
); δ = θ θ θ +Δθ (7.3)

Representation on BG

The representation technique is illustrated briefly by taking a pedagogical example of R-element in resistance causality. The characteristic law corresponding to R-element in the linear case (see Fig. 7.1) is given as, .

RR e R f  (7.4)

In case of uncertainty on R, (7.4) becomes

(1 ). . . .

R n R R n R R n R Rn Runc e R f R f R f e e        (7.5)
Constitutive equation (7.5) can be represented as uncertain R-element as shown in Fig. 7.1(b), wherein a modulated source MSe is introduced. The latter is associated with auxiliary input w R and a virtual effort sensor associated with auxiliary output z R. It must be noted that negative (-) sign appears in the BG-LFT representation (see Fig. Similarly, parametric uncertainty on the other passive elements can be represented. The technique remains similar for various other BG elements.

BG-LFT based robust fault detection

Fault diagnosis in BG-LFT framework is mainly dependent upon ARR generation [START_REF] Djeziri | Robust fault diagnosis by using bond graph approach[END_REF]. ARRs are constraint relationships involving only known variables. In the context of BG modeling, an :

) 0 ARR f t t t t  θ SSe SSf ( ( ), ( ), ( ), ( )
Se Sf , , where θ is vector of system parameters.

Generation of Uncertain ARRs:

The generation of robust analytical redundancy relations from an observable bond graph model is explained by the following steps:

1 st Step: Preferred derivative causality is assigned to the nominal model and detectors De (Df) are dualized to SSe (SSf) ; wherever possible. The BG-LFT model is constructed. 2 nd Step: The candidate ARRs are generated from -1‖ or -0‖ junction, where power conservation equation dictates that sum of efforts or flows, respectively, is equal to zero, as:

 for 0-junction:

, .0 i i n i i s f Sf s w       (7.6)
 for 1-junction:

 , .0 i i n i i s e Se s w       (7.7)
with s being the sign rendered to the bond due to energy convention, i w is the uncertain effort (flow) brought by the multiplicative parametric uncertainty δ i  associated with i th system parameter θ i , at 1(0) junction.

rd

Step: The unknown effort or flow variables are eliminated using covering causal paths from unknown variables to known (measured) variables (dualized detectors), to obtain the ARRs which are sensitive to known variables as,

  , , , , , , , , , , n n n
n n n i R Se Sf SSe SSf R C I TF GY RS w     (7.8)
where subscript n represents the nominal value of the corresponding BG element.

Generation of Adaptive Thresholds:

The ARR derived in (7.8) consists of two perfectly separable parts due to the properties of the BG-LFT model: a nominal part noted r shown in (7.9) and an uncertain part noted i bw   shown in (7.10).

 

, , , , , , , , ,

n n n n n n r Se Sf SSe SSf R C I TF GY RS  (7.9)   , , , , , , , , , , , , , , , i i
n n n n n n R I C TF GY RS bw w Se Sf SSe SSf R C I TF GY RS          (7.10)
The uncertain part generates the adaptive threshold over the nominal part. From (7.8), (7.9) and (7.10), following may be obtained:

0 i rb r b w       (7.11)
The thresholds are formed in form of envelop as:

a r a    (7.12) where || i aw   (7.13)
The use of absolute values to generate the thresholds of normal operation ensures the robustness of this algorithm to false alarms. BG-LFT technique is well developed and detailed in literature. Readers are referred to [START_REF] Djeziri | LFT Bond Graph Model-Based Robust Fault Detection and Isolation[END_REF][START_REF] Djeziri | Robust fault diagnosis by using bond graph approach[END_REF] for details.

7.2.4

Non-Linear Bayesian Inference using Particle Filters Consider a dynamic system whose state at time step k t is represented by the vector k

x . The evolution of the system state is described by a state-space model, p x is assumed to be known. Given

11 ( , ) k k k k   xx fv   , k k k k  yw hx (7.
k k k k k k p     x x x x fv   1 , ( | ) k k k k k k p   y x y x hw (7.16) (7. 1 1 ( , ) ( | )
1: 1 1 1: 1 1 1: 1 1 1 1: 1 1 ( | ) ( | , ) ( | ) ( | ) ( | ) k k k k k k k k k k k k p p p p p d              x y x x y x y x x x y x (7.18)
Here,

1 ( | ) kk p  xx is obtained from (7.16
), where the system is assumed to follow 1 st order Markov dynamics. Update: Bayes rule is used to update the prior as the new measurement k y ar- rives, to obtain the posterior distribution of k x as,

1 : 1 1: 1 : 1 ( | ) ( | ) ( | ) ( | ) k k k k kk kk pp p p    x y y x xy yy (7.19)
with the normalizing constant being,

1: 1 1: 1 ( | ) ( | ) ( | ) k k k k k k k p p p d   
w1 i k i   (7.22)
The posterior PDF is approximated as, 0:

1 :

0: 0: 1 ( | ) w . ( ) N ii k k k k k i p     x y x x (7.23)
where  denotes the Dirac delta function. This gives discrete weighted approxi- mation to the true posterior state distribution x is generated from the proposal distribution () qx , then the weighted approximation of the density () px is given as,

( )

N ii i p x x x     (7.24)
where normalized weight can be obtained as, () w (x )

i i i px q  (7.25)
For a set of samples t  , in a sequential manner. To this end, a constraint on importance density is placed so that it is factorable as, 0:

1:

0: 1 1: 0: 1 1: 1 ( | ) ( | , ) ( , ) k k k k k k k q q q     x y x x y x y (7.27)
Then, the new state

0: i k x 0: 1 1: ( | , ) k k k q  x
x y can be appended with existing samples

0: 1 i k  x 0: 1 1: 1 ( | ) kk q  xy
to obtain new sets of samples

0: i k x 0: 1: ( | ) kk q
xy . This is fol- lowed by update of particle weights. The posterior state PDF is expressed as,

1 0: 1 : 0: 1 0 : 1 1 : 1 ( | ) ( | ) ( | ) ( | ) ( , ) k k k k k k k k kk pp pp p     y x x x x y x y yy (7.28)
Then, using (7.26), (7.27) and (7.28), particles are updated recursively as, 0: 1: 0: 1: In SIS PF, the importance density is set equal to a priori PDF of state i.e.

0: 1 0 : 1 1 0: 1 1 : 0: 1 1 : 1 1 1 0: 1 1 : ( | ) w ( | ) ( | ) ( | ) ( | ) ( | , ) ( , ) ( | ) ( | ) w ( | , ) i kk k kk k k k k k k k k k k k i k k k k k k k k p q p p p qq pp q            
0 : 0 : 1 1 1 ( | ) ( | ) ( | ) k k k k k k k q p f     x x x x x x
. This translates to the fact that new particles can be generated from the previous set of particle by simulating the state transition function

1 ( | ) k k k f  xx
. Moreover, assumption of Markov dynamics implies that

0: 1 1: 1 ( | , ) ( | , ) i i i i k k k k k k
qq   x x y x x y . This renders the whole procedure suitable for online implementation as only the filtered estimate x can be neglected. Weight update step (7.29) can be modified as,

1 1 0: 1 1 : 1 ( | ) ( | ) ww ( | , ) w ( | )
i i i ii k k k k kk ii k k k ii k k k pp q p       y x x x x x y yx (7.30)
Then, the posterior filtered PDF

Particle Degeneracy and Resampling

During the propagation steps, the approximation density is adjusted through re-weighting of the particles. Previous steps lead to an inevitable situation where due to increase in weight variance, the importance weights become increasingly skewed. After few iterations, all but one particle have negligible weights (particle degeneracy) [START_REF] Doucet | A tutorial on particle filtering and smoothing: Fifteen years later[END_REF]. To avoid the latter, a new swarm of particles are resampled from the approximate posterior distribution obtained previously in the update stage, constructed upon the weighted particles [START_REF] Li | Fight sample degeneracy and impoverishment in particle filters: A review of intelligent approaches[END_REF]. The probability for a particle to be sampled remains proportional to its weight. This way, particles with smaller weights (signifying less contribution to estimation process) are discarded and particles with large weights are used for resampling. To resolve this issue, the standard SIS is accompanied by a resampling step (referred to as Sampling-Importance resampling (SIR) PF [39]. The different ways of resampling can be referred in [START_REF] Douc | Comparison of resampling schemes for particle filtering[END_REF]. In this work, SIR PF is employed for estimation of SOH and RUL predictions. In general, the particles are forced in the region of high likelihood by multiplying high weighted particles and abandoning low weighted particles. In other words, resampling step involves elimination of those particles that have small weights so that focus shifts on the particles with large weight.

Degradation Models

DMs capture the underlying degradation of a given component/subsystem with time, environmental and operational conditions etc. DMs can be obtained based upon physics of degradation or statistical approaches [START_REF] Gebraeel | Prognostic degradation models for computing and updating residual life distributions in a time-varying environment[END_REF] and [START_REF] Guo | Practical Approaches for Reliability Evaluation Using Degradation Data[END_REF]. Given a prognostic candidate (system parameter) θ d , the associated DM can be expressed as,

Obtaining Degradation Model in BG framework

In BG framework, the DM of a system parameter θ d  θ ,  θ can be obtained from the time evolution profile of the respective ARR to which it is sensitive, assuming that the rest of the system parameters sensitive to the same ARR, do not undergo any kind of progressive fault or degradation [START_REF] Medjaher | Hybrid prognostic method applied to mechatronic systems[END_REF], [START_REF] Borutzky | Failure Prognosis for Hybrid Systems Based on ARR Residuals[END_REF]. Here, consider the point valued part of the d th I-ARR, () ) is a function of system measurements inputs (known variables), signal derivative(s) etc., it is always corrupted with noise. It should be noted that residual based DM should be obtained prior to prognostics. This routine can be performed offline i.e. prior to the phase when system's health monitoring is of interest.

d rt such that with \ θ ( ) d t   θθ , 0, ( ) 0 d t r t  ,   1 () θ ( ), , ( ), ( ),

Methodology of Hybrid Prognostics

In this section, the methodology for prognostics is described. Following assumptions are made:

 Only system parameters are considered uncertain. Sensors are considered non-faulty;  A single system parameter (prognostics candidate) is assumed to be under progressive degradation. In fact, it is assumed that single mode of degradation affects the system parameter.  The system parameter (prognostics candidate) that undergoes degradation is assumed to be known a priori. The issue of isolation or isolability of the prognostic (faulty) candidate is assumed resolved. Let θ ( ) d t  θ be such prognostic candidate.  Degradation model (DM) of θ ( ) d t  θ is assumed to be known a priori.  For an ARR derived, only one system parameter sensitive to it (known a priori) varies with time.  Noise associated with measurements (residuals) is assumed normally distributed Gaussian in nature. Objectives are:  Reliable estimation of prognostic candidate's SOH and state of hidden degradation parameters that accelerate or vary the degradation progression.  Reliable prediction of the RUL of the prognostic candidate.

Robust detection of Degradation Initiation

The problem of detecting the degradation beginning is treated as robust fault detection problem. The BG -LFT enabled fault detection method presented in Sect. 7.2.3 is exploited in form of an efficient diagnostic module. To this end, following steps are taken.

Step 1: Preferred derivative causality is assigned to nominal model and sensors are dualized.

Step 2: BG-LFT model of the nominal system is obtained.

Step 3: ARR sensitive to θ d is derived. Let the ARR be R(t) and the associated residual (numerical evaluation of ARR) be () d rt.

Step 4: Robust thresholds are derived as explained in Sect. 7.2.3. Degradation initiation is detected when the residual goes out of the BG-LFT thresholds. The corresponding pseudo algorithm is given in Table 7-1.

Fault Model Construction

This section describes the fault model constructed for estimating the state of the prognostic candidate which denotes the state of health of the parameter.

State Equation

The parameter under degradation θ ( )

d

ARR based Observation equation

The nominal residual used for detection of degradation initiation can be further exploited used for SOH estimation if the corresponding ARR expression is altered to obtain the observation equation. To this end, following theorem is enunciated.

Theorem: Under the single degradation hypothesis, assuming that the nominal part ()

d
    : ( ) , SSe( ), ( ), ( ), ( ) A θ
( ) 0 d T d n ARR r t t t t t t      θφ SSf Se Sf (7.38) or,         ( ) , ( ), ( ), , , A( θ )

State of Health Estimation In discrete time step

, the fault model can be described as,

11 ( , ) d d d d k k k k   x x f x v   d d d d k k k k yw  hx (7.43) (7.44)
The initial state PDF

1 1 1 (θ , | ) d d d k k k py    γ is
N i d d i d i d d j d j k k k k k k k j p y p y    γγ (7.

47)

Resampling: There exist many types of resampling techniques [START_REF] Douc | Comparison of resampling schemes for particle filtering[END_REF]. In this work, systematic resampling is preferred owing to its simplicity in implementation, O(N) computational time and modular nature. The resampling method is well detailed in literature and thus, not described here.

The prediction, update and resample procedures form a single iteration step; they are applied at each time step k.

Fig. 7.2 Illustration of estimation process in Particle Filters

The pseudo algorithm is provided in Table 7-2.

Inputs:   ,, 1 1 1 1
(θ , ), w

N d i d i i k k k i     γ , d k y Output:   ,, 1 (θ , ), w N d i d i i k k k i γ for i=1 to N do , di k γ ,, 1 ( | ) d i d i kk p  γγ , θ di k , , , 11
(θ | θ , ) d i d i d i k k k p  γ i k w ,,
(| θ , )

d d i d i k k k py γ end for 1 N i k i Ww    for i=1 to N do / ii kk w w W  end for     , , , , 11 (θ ,
), w RESAMPLE (θ , ), w

NN d i d i i d i d i i k k k k k k ii   γγ

RUL Prediction

The critical/failure value θ d fail of θ ( ) (θ ,| )

d
dd d d d k k l k l py  γ as [40],         0: 1 1 1 0 : 1 (θ , | ) ... θ , | θ , (θ , | ) θ , d d d d d d d k k l k l k l k l d d d d d d d d d j j j j k k k j j j k j k p y p p y d               γ γ γ γ γ (7.49)
Obtaining this integral numerically is computationally very expensive. PFs can be employed for optimal estimation of the latter under certain assumptions. [START_REF] Orchard | Advances in uncertainty representation and management for particle filtering applied to prognostics[END_REF] reviews various methods for computation of (7.49). In [START_REF] Doucet | An introduction to sequential Monte Carlo methods[END_REF], it is proposed that weights of the particles from time step k until k+ d l can be kept con- stant for d l step ahead computation. This is based on the assumption that error generated/accumulated by keeping the weights same, is negligible compared to other error sources, such as settings of process noise, measurement noise, random walk variance, model inaccuracy etc. [41]. In our context, as illustrated in Fig. 7.3, RUL predictions be achieved by projecting the current SOH estimation into future [START_REF] Daigle | Model-based prognostics under limited sensing[END_REF][START_REF] Daigle | A Model-Based Prognostics Approach Applied to Pneumatic Valves[END_REF][START_REF] Daigle | Model-based prognostics with concurrent damage progression processes[END_REF][START_REF] Jouin | Prognostics of PEM fuel cell in a particle filtering framework[END_REF]. Once the particles

 

,,

N d i d i i k k k i γ , 1 (θ , ), w



. For d l -step ahead state distribution, each of the particles is propa- gated using the state equation of the fault model. Here, for the i th particle, the corresponding weight during the , di l -step propagation is kept equal to weight w i k at time of prediction k. Then, for i th particle,

,, i d i d i k RUL k l k l  
  and the corresponding PDF is obtained as,

0: () 1 ( | ) w ( ) i i k N di k k k k RUL i p RUL y dRUL     (7.50)
The associated pseudo algorithm is provided in Table 7-3.

N d i d i i k k k i γ Variable: l Outputs:   1 ,w N ii kk i RUL  for i=1 to N do l=0 while , θθ d i d k l fail   do , 1 di k  γ , 1 (θ , ), w
d i d i kk p  γγ , 1 θ di k  , , , 1 ( | )
θ , )

d i d i d i k k k p  γ 1 ll  end while i k RUL l  end for   1 ,w N ii kk i RUL  0: ( | ) d kk p RUL y

Integrated Health Monitoring

The degradation initiation is detected by BG-LFT based robust fault detection technique, as discussed in Sect. 7.3.3. The initial value of SOH of prognostic candidate is set as:

θ ~(θ θ ,θ θ ) ; d d d d t t n l n u d U t t       (7.51)
where, d t is the time when degradation is detected as fault. The associated uncertainty interval limits   Δθ ,Δθ lu  decide the bounds of the uniform distribution.

The complete algorithm is shown in Table 7-4. Fig. 7.4 shows the schematic description of the methodology presented in this chapter.

Evaluation Metrics

In this section, evaluation metrics are provided to assess prognostic performance. For details, readers referred to [START_REF] Saxena | Metrics for offline evaluation of prognostic performance[END_REF] and [START_REF] Daigle | Model-based prognostics with concurrent damage progression processes[END_REF]. Root mean square error (RMSE) metric expresses the relative estimation accuracy as:

2 * * () RMSE Xk mean X X Mean X          (7.52)
where, for a specie X , * X denotes its corresponding true value.

k Mean denotes the mean over all values of k. This metric is useful in assessing the estimation performance.

On the other hand, assessment of RUL predictions is possible if the actual RUL or RUL ground truth is known. The terms RUL ground truth and true RUL, are used interchangeably in this chapter. A fairly good idea of true RUL can be obtained beforehand from the corresponding DM, under the assumption that degradation proceeds with uniform speed. Obviously, the hidden DPPs influence the actual speed and SOH. As such, in reality, true RUL can only be estimated with certain degree of belief. In this chapter, it is assumed that degradation progresses with uniform speed. As such, for evaluation purposes, true RUL is assessed from DM. A detailed discussion on this subject and RUL evaluation metrics can be found in [64,[START_REF] Saxena | Metrics for offline evaluation of prognostic performance[END_REF].

Alpha-Lambda( α   ) metric [5]: An accuracy cone is formed by choosing  such that [0,1]

 

, followed by generation of accuracy cone (envelope) over true RUL at time instant k, RUL , allowed for assessment of RUL predictions. Fig. 7.5 shows ground truth RUL line and  cone that enve- lopes it. The estimated RUL PDFs must have significant amount of probability mass within the  -cone, to be accepted as -true‖ predictions. Then, accuracy of RUL predictions can be efficiently assessed by relative accuracy (RA) metric. The latter is explained by first recalling the fact that RUL predictions are obtained as PDFs (see Fig. 7.3). In this work, RUL PDFs are represented using box plot representation. As shown in Fig. 7.5, the box plot representation is capable of denoting the PDF's mean, median, 5 th and 95 th percentiles of distribution data and the associated outliers. At a particular prediction instant k, the RUL prediction accuracy for θ d is evaluated by relative accuracy (RA) metric as,

* k RUL , as ** [ (1 ) , (1 )

Application on Mechatronic System in Real Time

This section describes the application of the method over a mechatronic system [START_REF] Jha | Particle filter based hybrid prognostics for health monitoring of uncertain systems in bond graph framework[END_REF] shown in Fig. 7.6. Real time implementation is achieved through 20 SIM 4C 2.1. The SOH estimation and RUL prediction algorithms are written in Matlab Function Block in Simulink. The embedded code is generated through Simulink Coder in Matlab2013a ® .

Nominal System

The functional schematic model of the mechatronic system [65], is shown in Fig. 7.6. The designation of system variables and associated values are listed in Table 7-5. The system consists of the Maxon® servo motor that provides the controlled actuation (rotation) to disks. The high stiffness transmission belt provides torque the transmission ratio of belt k to the motor disk. The motor disk is connect- ed to load disk through a flexible shaft that constitutes the drive train. The shaft is modelled as spring-damper element. The friction in the bearings of the motor disk and load disk are modelled as viscous friction. Friction arising due to belt is lumped with viscous friction coefficient at motor disk Md b . The setup is equipped with motor encoder and load encoder that measure angular position of motor shaft and load disk (2000 pulses per revolution) respectively. Angular position motor disk is obtained by dividing the motor encoder counts by belt ratio. The BG model of the nominal system in integral causality is given in Fig. 7.9. Only the monitorable part is used for analysis. The system is considered operating in feedback closed loop with Proportional-Integral (PI) controlled input voltage. The control input from PI controller (controlled variable: motor speed m  ) modulates the input voltage MSe: U PI . . C element remains in integral causality with the initial condition given by the flow at respective 0-junction, provided by encoder readings as

r t J f J b M g r k kk k k dt b k k                                (7.60) , , , , , , , ; . ; 1
                                  ) Ld belt k   (7.61)
Robust thresholds over the residual can be formed as (see Sect. 7.2.3),

1 1 1 () a r t a    (7.62)
where,

1 || i aw   (7.63)
Remark: Only one I-ARR has been derived in this work. This serves the purpose of demonstration. Following similar steps, another independent ARR(s) can be derived. Fig. 7.11 shows the residual profile under nominal conditions, wherein the residual is well within the envelope formed by thresholds. Fig. 7.12 shows the effect of adding load (or frictional toque) in a discrete way on the system. Md  is con- trolled at 30 rad/s. Addition of load leads increase in frictional torque and degradation in speed. Due to action of PI controller, the motor disk speed is maintained at set reference value of 30rad/s. However, the residual 1 () rtis sensitive to the variation in PI enabled input voltage PI U . As such, the residual captures the variation of disk speed due to load suspension. Saturation limit PI U is reached around t=65s when the total load suspended is 1.6 Kg. Thereafter, controller is unable to compensate the change in Md  . With addition of more load thereafter (t>65s), motor disk speed decreases rapidly and stops at around t=70s. For safety reasons, the disk is stopped momentarily, after which the suspended load is removed.

Degradation model: Offline Phase

The experiments performed are non-destructive in nature. Here, load in form of sand of M Kg is suspended in a uniform manner until a prefixed limit of fail M is reached. In this context, () Mt is treated as a system parameter under degradation, the prognostic candidate.

The experiments were conducted in two distinct phases:  Offline Phase: Mass is suspended uniformly. As explained in Sect. 7.3.1, variations of () Mt are obtained from the evolution of 1 () rt. Then, statistical techniques such as (curve fitting) are used to obtain DM of () Mt .



Online health monitoring: In real time, load is added in a similar manner under similar environmental conditions as offline phase, until the prefixed failure value () Mt is reached. In real time, estimation of () Mt and associated DPPs, and subsequent RUL predictions are obtained.

Exponential Variation of Load

Load is varied uniformly in an exponential manner. Eight experiments are carried in total. Fig. 7.13 (a) shows the experimental data and Fig. 7.13 (b) shows the exponential fit over the experimental data mean. This way, an exponential DM is obtained as,

Health Monitoring: Online Phase

In the online phase, environmental conditions are kept unaltered. It is recalled that () Mt is treated as a system parameter under degradation, the prognostic candidate. Failure value is prefixed at

Fault Model

In discrete time step k, the tuple (

1 g , () Mt , 1  ) is formulated in state space as, 1, 1 1 1, 1 1, 1, 1 1, 1
. Then, observation equation can be constructed as, ,  , achieved with 1 RMSE  =7.6%. The convergence is achieved very quickly with large initial estimation spread. This is due to a high artificial noise variance set for the desirable quick convergence. It should be noted that RMSE  obtained via experiments is higher than those ob- tained via simulations, as the true speed of degradation * 1  , does not remain per- fectly constant in reality. Also, lesser number of particles are employed here used so that RUL predictions may be achieved in real time without significant data loss. With higher number of particles, greater accuracy may be achieved. ). This makes their utility virtually null. However, after t = 32 s, with significant improvement in estimation of DPP, the RUL distributions are well within accuracy cone such that more than 50% of RUL probability mass lies within accuracy cone. Ignoring the initial period of convergence, the overall prediction performance is obtained with 97.02% RA  . Fig. 7.17 RUL Predictions

k t k k M k k k k M M e v           (

State of Health Estimation

Conclusions

Prognostics is the science of assessing the end of life of a system/component and prediction of the remaining useful life of the same. Due to various kinds of uncertainties that manifest in form of parametric uncertainties, environmental conditions, sensor noises, uncertain future conditions etc., RUL prediction becomes a very challenging issue. In this work, benefits of BG-LFT modelling based robust fault detection are integrated with the advantages of Bayesian estimation method for efficient prognostics. In particular, particle filters are exploited for optimal estimations of actual state of the prognostic candidate and subsequent RUL predictions. In this chapter, a single system parameter is chosen as the prognostic candidate and RULs are obtained with respect to that parameter. As such, the work presented here paves the future path for development of efficient system level prognostics in BG framework.

The ARR based BG-LFT technique is employed for robust detection of degradation beginning. The same ARR is then exploited for prognostic purposes. Being sensitive to the control inputs, nominal residual is able to capture the parametric degradation profile even while the system outputs remain in feedback closed loop regime. This aspect renders the approach appropriate for system level health management. Approximation of noise distribution present in residuals can be difficult or impossible, due to presence of derivative or integral terms in the ARR function arguments. As such, Particle filter algorithms form the best choice in this regard as they are not restricted by non-Gaussian noises. Moreover, degradation of nonlinear nature can be efficiently estimated using Particle filters. Additionally, this method also demonstrates that fusion of BG-LFT framework and Monte Carlo framework leads to efficient management of various types of uncertainties. While parametric uncertainties are modelled and managed by using BG-LFT for efficient detection of degradation initiation; degradation process noise, measurement (residual) noise etc. are efficiently accounted for, by PF for estimation of SOH and RUL predictions.

  7.1) due to the convention of power conservation. Moreover, the symbols * De represent virtual detectors. The virtual detectors are used to repre- sent the information exchange/transfer.

Fig. 7

 7 Fig. 7.1 R-element in resistance causality. (b): uncertain R-element in resistance causality in LFT form.

  17

  perspectives of Bayesian inference, the objective remains in recursively calculation of the distribution of the state k x , given the set of observations 1:k y up to time k t , with some degree of belief. Construction of PDF as the filtered pos- terior state PDF, provides all the information about k x , inferred from the meas- urements

  theoretically, the posterior state can be estimated in a recursive way via two sequential steps: prediction and update.Prediction: Application of Chapman-Kolmogorov equation over

p

  xy is required at each step. Thus, only i k

31 )

 31 This simplified algorithm can be used for recursive estimation of state as the observations arrive sequentially. The likelihood functions of the new observations ( | ) i kk p yx, result in evaluation of weights of particles constituting the next state estimate.

  /non-linear degradation progression function (DPF) obtained from the corresponding DM. It models the way the degradation progresses in θ ( ) process noise vector and θ d n denotes nominal value of θ d .

Fig. 7. 3

 3 Fig. 7.3 Schematic illustration of RUL prediction process

  Fig. 7.4 Schematic description of the Health Monitoring Methodology

  value of  signifies the degree of uncertainty associated with * k

Fig. 7. 5

 5 Fig. 7.5 Illustration of Box plot representation and α   accuracy cone

Fig.7. 6

 6 Fig.7.6 Mechatronic torsion bar 1.0 system

Fig. 7 .

 7 Fig. 7.11 Residual 1 () rt under nominal conditions

Fig. 7 . 1 

 71 Fig. 7.13 Exponential variation of mass. (a) experimental data (b). Exponential fit over experimental data mean

Fig. 7 .

 7 [START_REF] Blischke | Reliability: modeling, prediction, and optimization[END_REF] shows the profile of residual under exponential degradation. The degradation initiation is detected when the residual goes outside the threshold envelope at around t=22s, after which prognostic module is triggered.

Fig. 7 . 2 w

 72 Fig. 7.14 Nominal residual 2, () n rt  while system is under degradation (exponential case) Estimation of SOH: The estimation of suspended load M is shown in Fig. 7.15. The estimation of SOH is performed with number of particles N=50, sample time t  =0.1s, initial random walk variance noise

Fig. 7 .

 7 Fig. 7.15 Estimation of SOH of prognostic candidate

1

 1

Fig. 7 . 1 

 71 Fig. 7.16 Estimation of DPP Fig. 7.17 shows the RUL prediction with 0.2   .The RUL distributions obtained until t = 32s, are not good predictions and suffer with large variance spread due to a large corresponding spread in 1  (see Fig. 7.16). This makes their utility

  obtaining recursive Bayesian inferences via Monte Carlo simulations. Basic philosophy rests in representing the posterior state PDF by a set of random samples or -particles‖ where each of the particles has an associated weight based upon which, the state estimates are computed[START_REF] Doucet | A tutorial on particle filtering and smoothing: Fifteen years later[END_REF]. Sequential importance sampling (SIS) PF is one of the most popular PFs in which posterior state PDF

											(7.20)
									y y	x y	y x x
	This step incorporates the latest measurement into a priori state PDF
	p	(	1: kk | xy	1 	)	to estimate the posterior state PDF	(	|	1:	)
											p	(	| xy	)
											0: kk 1:
	by a set of N number of weighted particles [39],
											 (	x	0: ii ), w kk		N	(7.21)
											1 i
	where	{ x	0: i	, k iN  1,...	}	is the set of particles representing the state value with cor-
	responding associated importance weights as {w	, k iN  1,... i	}	. Moreover,
	0: xx { kj jk , 0,...., 	}	is the set of all states up to time k. It should be noted that
	these weights are the approximations of the relative posterior probabilities of the
	particles and are normalized such that,

kk p xy

. The exact Bayesian solution obtained from recurrence relations (7.18) and (7.

[START_REF] Wu | Prognostics of machine health condition using an improved arima-based prediction method[END_REF]

), form the basis of optimal Bayesian inference. This procedure remains tractable and produces best results for ideal systems such as linear Gaussian state space models. For the latter, it leads to the formation of classical Kalman filter. In general, optimal and closed form solutions for non-linear systems with non-Gaussian noises, cannot be analytically determined. For non-linear state space models with additive Gaussian noises, sub-optimal Extended Kalman filter (EKF) has been developed. To obtain optimal solutions for non-linear systems, one resorts to Monte Carlo Methods. One such popular method is described below. Particle filter (PF) is a type of Sequential Monte Carlo method

[START_REF] Doucet | An introduction to sequential Monte Carlo methods[END_REF]

, used for

Table 7

 7

											-1 Detection of degradation
	Algorithm 1: Detection of Degradation Initiation
	Input: () d rk ,		() wt i
	Output: degradation detection
	if	r	d	(	k	)	 	i w	( t	)	and ( d r	k	)	 	i w	( t	)
	degradation detection  false
	else
		degradation detection  true
	end if

  The fault model for is constructed in state -space form by considering the parameter θ d as the state variable augmented with the DPP vector as,

												t is included as a tuple 	θ , , d d γ	g	d		to
	model the damage progression in state space form. Here,	γ	d	() t 	is the vec-
	tor of hidden parameters (DPP) that influence the speed of degradation significant-
	ly. ( d t x	)		( f x d	d	( t	),	v	d x	( t	))	(7.36)
	where,	x	d	() θ ( ), ( ) d d t t t    γ	T	is the augmented state vector, d f is state transition
	function following the Markov dynamics and d  x v	is the process noise vec-
	tor.

Table 7 -

 7 2 SIR Particle Filter for SOH estimation Algorithm

2: Estimation using SIR filter

  tained depending upon the actual SOH. Then, RUL at time k, can be obtained as,

						t must be fixed beforehand. Once the pos-
	terior PDF	(θ , | d d py γ	d	)
		k	k	0:	k
						RUL	EOL 	k	(7.48)
						kk
	Obviously, such a projection of degradation trajectory in future has to be done in
	absence of measurements. Thus, this process remains outside the domain of tradi-
	tional Bayesian filtering techniques. In practice, one of the efficient ways to
	achieve such a projection is to propagate the posterior PDF	(θ , | d d py γ	d	)	using
						k	k	0:	k
	the DM inspired state model (7.43) until the failure horizon θ d fail is reached. The
	latter may take d l time steps so that θ = θ dd fail at a time	d tl  This calls for compu-
	tation of the predicted degradation state
						0:

has been estimated at time step k, it should be projected in future in such a way that information about EOL at time step k, k EOL is ob-

  constituting the realizations of the current joint state-

	parameter estimate	(θ , | d d py γ	d	)	are obtained, each of the particles is propagat-
		k	k	0:	k
	ed into future to obtain a d l -step ahead state distribution with d l =1,… d Tk  ,
	where d T is the time until SOH remains less than failure value i.e. time until
	dd θθ d fail kl 

Table 7

 7

	-4 Integrated Health Monitoring of Prognostic candidate
	Algorithm 4: Health monitoring of 0 θ d
	while system is running do
	Detect the beginning of degradation using Algorithm 1
	if fault detection =true then
	//set initial conditions

  RUL denotes the true RUL at time k for θ d . The overall accuracy is deter-

	where	* k
	mined by	RA as shown in (7.53), where	RA	k is averaged over all the prediction
	points.
		RA	k	* kk * Median ( k RUL p RUL RUL  ) 1     	(7.53)
				RA Mean (RA ) kk p 	(7.54)

Table 7 -

 7 5 Details of system variablesFor experiments, a mechanical lever type arrangement is fabricated as shown in Fig.7.7 which introduces frictional torque

		over the motor disk by suspension
	Mech

  Following the steps described in Sect. 7.2.3, an ARR can be generated from the detectable junction 1 1 of Fig.7.10 as,

	10 f	9    13 f f	(  / ) m belt Ld k 	. Moreover, electrical torque	: MSe 	PI
	is the PI controlled input to the monitorable part of the system and is given as:
	MSe	:		PI		k	m	m i .		k	m	.	 Uk PI 	m	. 	m	  1		e	(  / ) Ra La	t		(7.58)
															Ra
	where,
												11 () R r t  	i w					(7.59)
	where,

PI

U is the PI controlled voltage input and m i is the motor stator current.

   for a suitable convergence. The magnitude of this noise should be sufficiently large for a desirable convergence of estimations and small enough for a good estimation accuracy. Usually, this noise is tuned with the help of simulations or multiple offline testing. Readers are referred to[START_REF] Jha | Particle filter based hybrid prognostics for health monitoring of uncertain systems in bond graph framework[END_REF] for a simplified variance adaption scheme proposed in this context. Observation equation is constructed from the ARR derived (Sect. 7.6.2) using the Theorem given in Sect. 7.3.4. The

															7.65)
	where	2   is an normally distributed artificial random walk noise add-1, ~(0, ) k
					1
	ed to the DPP
		1,	k
										: ARR R can be decomposed as,
										1
		1 R	:	1 r	( t	)		1, r	( nn ) ( ( ) t M t M  	).		    1, () n rt M 		0	(7.66)