Model-based multi-component adaptive prognosis for hybrid dynamical systems

Control Engineering Practice 72 (2018) 1–18

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Model-based multi-component adaptive prognosis for hybrid dynamical systems Om Prakash, Arun Kumar Samantaray *, Ranjan Bhattacharyya Systems, Dynamics and Control Laboratory, Department of Mechanical Engineering, Indian Institute of Technology (IIT), Kharagpur, 721302, India

a r t i c l e

i n f o

Keywords: Hybrid dynamical system Dynamic degradation patterns Hybrid bond graph Analytical redundancy relations Sensitivity signatures Constrained parameter estimation Adaptive prognosis

a b s t r a c t A bond graph model-based prognosis method for multiple components with unknown degradation patterns in a hybrid dynamical system is proposed. The traditional approach for remaining useful life prediction with single degradation model is inappropriate for hybrid systems where the dynamics changes according to operating mode. Therefore, multiple degradation models are suggested and these are adapted with new information of the degradation states of the monitored system. Sensitivity-based dynamic signature matrix is utilized for degradation hypothesis generation which provides the deviation directions of fewer hypothesized degradation parameters and thereby accelerates parameter and degradation trend estimation. The results are supported by experiments. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Many modern systems and processes are hybrid dynamical systems which include both continuous and discrete dynamics. Such hybrid systems show different continuous behaviour according to various operating regimes (termed as modes or discrete states) where each mode (controlled or autonomous) is triggered according to certain conditions. Improving reliability, availability, maintainability and safety (RAMS) of such modern critical systems, prompt fault detection, robust fault isolation and effective remaining useful life (RUL) prediction of degrading components have become an active and challenging research area over the last few decades. Thus, more accurate real time fault diagnosis and prognosis techniques have to be evolved according to the present demands of critical machinery performance for the predictive maintenance (PM). There are considerable literatures available in the field of diagnosis and prognosis such as in Heng, Zhang, Tan, and Mathew (2009), Jardine, Lin, and Banjevic (2006), Sikorska, Hodkiewicz, and Ma (2011) and Vachtsevanos, Lewis, Roemer, Hess, and Wu (2006). Diagnosis and prognosis methods may be broadly categorized into three types: model-based method, data-driven-based method, and the fusion of various methods, i.e. hybrid methods. This paper considers the modelbased approach for prognosis based on continuous process monitoring. In model-based method, degradation of components is closely related to the respective component’s parameter in the system’s model (Chelidze, Cusumano, & Chatterjee, 2002; Kulkarni, 2013) and this provides one of the techniques to track degradation behaviour of components

under faulty conditions. However, model-based approach requires the precise and reliable mathematical model of the real physical system. Bond Graph (BG) approach is well suitable for modelling continuous behaviour of a real system (Borutzky, 2010; Karnopp, Margolis, & Rosenberg, 2012; Mukherjee, Karmakar, & Samantaray, 2006) and its extended form called hybrid BG (HBG) is used for modelling the hybrid systems (Borutzky, 2015; Mosterman & Biswas, 1995; Roychoudhury, Daigle, Biswas, & Koutsoukos, 2011; Umarikar & Umanand, 2005; Wang, Yu, Low, & Arogeti, 2013). The BG technique has proved itself as a very useful and convenient tool not only for modelling and control analysis, but also for diagnosis and prognosis for various dynamical systems (Borutzky, 2015; Samantaray and Ould Bouamama, 2008; Wang et al., 2013). Thus, in this article, BG approach is adopted for system modelling, rule development for detection and isolation of the degrading components, parameter estimation for degradation identification and RUL prediction. Generally, various sensors are deployed in the system for the process control and the fault detection of the system and those sensors can be used for real time prognosis. However, in multi-components system, the degradation states of the components are not directly measured. Thus, to obtain the degradation states for real time prognosis, periodic online parameter estimation may be used. This can be inefficient for hybrid systems and may provide incorrect estimates for hybrid components (e.g. on–off resistors, valves, etc.) whose estimation depend on the operating mode of the system. Certain system parameters can only be

* Corresponding author.

E-mail address: [email protected] (A.K. Samantaray). https://doi.org/10.1016/j.conengprac.2017.11.003 Received 31 May 2017; Received in revised form 9 November 2017; Accepted 10 November 2017 0967-0661/© 2017 Elsevier Ltd. All rights reserved.

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estimated during transient system dynamics, i.e., they do not influence the system dynamics in its steady state. Therefore, estimation of those parameters usually requires data from fast time dynamics. However, any component’s degradation is usually a slow process where the severity level of degradation changes in the time units of hours, days, weeks, months, or even years depending on the type of the system, its dynamics and the environmental conditions. These severity change time points for different components may be different, i.e., different components degrade at different rates. A considerable amount of research can be found in the field of modelbased prognostic and health management (PHM) of continuous dynamic systems such as in Huang and Niu (2015), Jha, Dauphin-Tanguy, and Bouamama (2016), Medjaher, Gouriveau, and Zerhouni (2009) and Medjaher and Zerhouni (2009, 2013). However, these techniques may not be appropriate for hybrid system because the mode transitions or mode faults change monitorability of parameters, i.e., direct influence on measured outputs or their projected form called residuals. On the other hand, many works are available for diagnosis of hybrid systems such as in Arogeti, Wang, and Low (2010), Borutzky (2012, 2014), Ghoshal, Samanta, and Samantaray (2012), Levy, Arogeti, and Wang (2014), Levy, Arogeti, Wang, and Fivel (2015), Low, Wang, Arogeti, and Luo (2010), Low, Wang, Arogeti, and Zhang (2010), Narasimhan and Biswas (2007) and Prakash, Samantaray, and Bhattacharyya (2017) But, very few works are available for prognosis of hybrid systems. Recently, few approaches for the model-based prognosis of hybrid systems are shown in Daigle, Roychoudhury, and Bregon (2015), Luo, Pattipati, Qiao, and Chigusa (2008), Yu (2012), Yu, Wang, and Luo (2015) and Zabi, Ribot, and Chanthery (2013). In Daigle et al. (2015), Yu (2012) and Yu et al. (2015), HBG approach is used for modelling the hybrid system, but in Yu (2012), degradation models and faults are identified by using a particle swarm optimization algorithm, whereas Monte Carlo framework (particle filtering technique) is utilized in Daigle et al. (2015) and Yu et al. (2015). In Luo et al. (2008), a prognostic method is proposed to predict the RUL of a suspension system with multiple operating modes. This method considered a single crack fault in the suspension spring and RUL is predicted by mixing mode-based life predictions via time-averaged mode probabilities under different road conditions. In Zabi et al. (2013), an integrated diagnosis and prognosis approach is proposed for hybrid systems by using the model and experience-based approaches for RUL estimation with the ageing laws assumed in the form of Weibull models. In this article, only progressive parametric faults (classified as degradations in the prognosis framework) are considered. Here, there is as no difference in the treatment of additive and multiplicative faults. Sensor bias which has to be modelled as an additive fault is neglected. Any additive actuator fault and parametric fault can be equivalently represented as a multiplicative fault. In bond graph framework, multiplicative faults have better physical meaning. For example, let there be a leakage from a tank storing some fluid. One may include leakage as a negative flow source in the model (additive fault). This is physically inconsistent because the rate of leakage would then be independent of the fluid pressure in the tank. Rather, it is physically consistent to treat the leakage through a hole (or orifice) as a resistor in the model so that the leakage flow rate depends on the fluid pressure. Once this correspondence of the additive leakage fault is made to a physical parameter, it can be treated as a multiplicative fault. Since this article deals with progressive parameter deviations/faults/degradations, the case of multiplicative faults (progressive degradations) only is considered. Most of the existing model-based prognosis approaches assume the occurrence of single parameter degradation in a system for the simplification of algorithms. However for a large complex hybrid system, it is possible that more than one component may degrade and that too at different rates. So, the traditional approach for RUL prediction that uses a single degradation model is unsuitable if the subsystems or components are activated in different operating modes or and operate in different environmental conditions. Further, identification of degradation pattern for a hybrid system which operates under different modes

or environmental conditions becomes more and more difficult with the existing approaches, since the existing diagnosis/prognosis approaches are typically proposed for continuous dynamical systems under single or limited operating conditions. Internal stresses (speed, force, load, etc.) and external stresses (temperature, humidity, wind, etc.) are the two main causes for the component’s degradation in a system, so it is essential to consider how and where the components will be used and what will be their corresponding operational modes and environments. Also, degradation of a component is usually irreversible and the associated parameter value deviates monotonically in a certain direction (either increasing or decreasing parameter value). This information of parameter deviation direction can be used for improving parameter estimation algorithms through specification of appropriate constraints. Moreover, the trend of parameter deviation in a degrading component may not conform to any specific pattern and there can be sudden or unforeseen changes to the trend. Thus, an adaptive real time prognosis approach has to be evolved for multiple degrading components in a hybrid system. As stated earlier, different components degrade at different rates. Thus, an intelligent prognosis scheme should detect the time instances of severity changes of degradation of different components and such an activity can be complemented by utilizing concepts borrowed from fault diagnosis approach for detection of time points where degradation severity level changes, and fault isolation approach for identifying the component whose degradation severity has changed. For the real time prognosis of hybrid dynamical system, the following questions arise: (i) What parameters can be estimated at an instant by considering the current mode of the hybrid system? (ii) Which parameters identified in (i) need to be estimated? (iii) When to estimate the parameters identified in (ii)? (iv) How to estimate the parameters in (ii), i.e., how to improve estimation process? (v) How to accommodate changes to degradation patterns/trends of parameters in (ii)? This paper addresses these above listed issues and proposes the following innovative solutions: (i) Only those parameters in the current hybrid mode that influence the observable outputs, measurements or the so-called residuals in a projected space, i.e., monitorable parameters, can be estimated. (ii) Only those parameters which may have deviated from their last known values by some a priori fixed degree, i.e. sufficiently degraded to the next level of degradation, need to be estimated. (iii) Only when the influences of parameter deviations on the measurements or residuals have exceeded certain thresholds, the parameters influencing those measurements or residuals may be estimated. This would avoid unnecessary computational load. If any such parameter to be estimated relates to a dynamic element such as inertia, inductance and capacitance, then it can be estimated only during the transient period or mode change. This would avoid false/failed estimation of that parameter, if attempted during any steady state period. (iv) The earlier known parameter values (last estimates), known upper and lower bounds of parameter values, sensitivities of measurements or residuals to parameter changes and the hypothesized directions of deviations of the parameter values can reduce the parameter search space and improve the computational efficiency and accuracy of parameter estimation. (v) Instead of assuming any particular fixed degradation pattern, it may be generated dynamically through an interpolation function from the estimated parameter values at discrete intervals (as per (iii)) and extrapolated for RUL prediction. In case of drastic change in degradation trend, local interpolation (forgetting old trend of estimated values) may be used. For example, material 2

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where p denotes momentum, the time integration represents the past history and hence the force F is the cause and the velocity of the body 𝑣 is the effect. This equation form representation where only one effect appears on the left hand side and one or more causes appear on the right hand side is called a computational or mathematical causality. The equation represents a model which may be logical, statistical, empirical, or derived from deep knowledge, i.e. the physics of the problem. In this work, equation models are based on physics. Thus, in the cited example, physical and computational causalities have the same meaning. The concept of causality in the Aristotelian philosophy assumes that the causes always lie in the past and it is good enough for building forward simulation models of physical systems. On the other hand, diagnosis task is different where the causes that lie in the past have to be inferred from the observed effects in the present. For the considered example where the force F acts on the point mass m, the equations in inverse causal form can be written as 𝐹 (𝑡) = 𝑚 d𝑣(𝑡) ≈ d𝑡 𝐹 (𝑡 − 𝛥𝑡) = 𝑚 𝑣(𝑡)−𝑣(𝑡−𝛥𝑡) where 𝛥𝑡 is a small time window corresponding 𝛥𝑡 to the measurement sampling interval. The velocity may be directly measured or an observable variable that can be computed from the other known measurements and states. Consequently, the velocity becomes the cause and the force becomes the effect as per the computational causality whereas the normal connotation of force as the cause and velocity as the effect remains valid as per the physical or Aristotelian philosophical causality. In a BG model, causality decides the computational order of power variables (e, f ) based on cause and effect relation for mathematical equation generation. It defines whether the effort in a power bond is computed from the flow, or vice versa. The causality is represented by a vertical stroke at one end of a power bond. In the causalled power bond, the effort (e) variable is directed towards the causal stroke end, while the flow (f ) variable is directed towards another end, i.e. opposite to causal stroke end. This concept is also shown with the dotted arrow for each element of BG in Table 1. Generally, preferred integral causalities are assigned to the all energy storage elements (C and I) for dynamical behaviour study where initial values of states have to be specified. On the other hand, preferred derivative causalities are assigned to C and I elements for diagnosis because values of initial states are unknown. A technique, called sequential causality assignment procedure (SCAP) is used to assign these causalities to a BG model and the causality rules for the elements, sources, junctions and sensors mentioned in Table 1 are strictly followed. A casual bond graph reveals the computational structure and is equivalent to a block diagram. However, unlike block diagram with fixed computation structure, a bond graph can be given new causality after any modification to it. The differential equations of state are generated from a bond graph model with storage elements in integral causality and those can be solved (simulated) using various types of integration methods.

wear due to friction has three phases: a fast rate of initial wear which gradually saturates and follows an exponential trend, then a slow and almost linear wear rate for considerable duration, and finally an end of life sudden wear at a very fast rate. The initial estimate of RUL can be done using exponential trend, then the next estimate of RUL can be done with linear trend assumption while discarding the initial data points and the third phase occurs after end of life. In this article, a heated capacitor used in the experiments shows sudden change in degradation trend and the proposed degradation model substitution is applied there. In summary, paper proposes an integrated approach for real time prognosis of multiple degrading components in a hybrid dynamical system using HBG as a common framework. A constrained parameter estimation technique with dynamically updated constraints supported by the information of parameter drift direction is proposed to speed up the degradation pattern identification and RUL prediction. Moreover, a sensitivity bond graph (SBG) approach is utilized to provide the gradient information of the objective function used in the parameter estimation. The proposed method accommodates the influences of different operating modes and is adapted with new information of degradation states continuous monitoring of the system. Experiments on an electrical/electronic system with accelerated ageing of components are used to demonstrate the proposed approach and its usefulness for hybrid system PHM. This paper is organized into five sections. After the introduction section, Section 2 introduces the BG model-based fault detection and isolation (FDI) methodology and the basic terms used in FDI which will be used in the proposed continuous prognosis framework. Section 3 presents the adaptive degradation model identification and RUL prediction in case of components having dynamic degradation patterns. Section 4 deals with the application of the proposed method on an electrical/electronic hybrid system, where results from the experiments are presented and discussed. Finally, Section 5 concludes the work and draws perspectives for future research. 2. Bond graph model-based fault/degradation diagnosis methodology Bond graph (BG) is a graphical modelling approach used for the multi-physics dynamic systems (Borutzky, 2010; Karnopp et al., 2012; Mukherjee et al., 2006). A BG model is an interconnected graph of different generic elements as nodes and the edges between the nodes are called bonds. This model mainly consists of half arrow power bonds with causal strokes, two active (Se and Sf), three passive (I, C and R), four power conservation junctions (0, 1, TF and GY) and two sensors/detectors (De and Df) as generalized elements. The generic elements of BG with their causalities, causal equations, block diagrams and the set of associated rules are presented in Table 1. The concept is to represent the power/energy exchange between the lumped parameter representations of the components and to use the reference power directions as a unified coordinate system across different energy domains. The orientation of a half arrow in a power bond represents direction of power or energy flow, and is associated with the two generalized power variables, i.e. effort (e) and flow (f ) variables. The product of these power variables gives power in the corresponding bond. These power variables represent different variables in different domains. For example, voltage and electrical current in electrical domain correspond to effort and flow power variables, respectively. Causality connects one process (cause) to another process (effect) where the cause is fully or partially responsible for the effect. The concept of causality can be explained from physical or philosophical viewpoints. In both, the causes have to lie in the past and the effect lies in the future. In a lumped parameter modelling framework, the velocity v of a point mass m at time t under application of a force F is mathe𝑡 𝑡 matically written as 𝑣 (𝑡) = 𝑚1 𝑝 (𝑡) = 𝑚1 ∫−∞ 𝐹 (𝜏) 𝑑𝜏 = 𝑚1 ∫0 𝐹 (𝜏) 𝑑𝜏 + 𝑝 (0)

2.1. Residual and fault signature matrix for robust FDI In model-based diagnosis, a residual is an indicator of the deviation of the behaviour of a system from its expected/normal behaviour (Gertler, 1993). Various approaches can be used for the generation of residual which acts as a fault indicator in model-based diagnosis (Blanke, Kinnaert, Lunze, & Staroswiecki, 2003) and those may be classified as observer-based (Patton & Chen, 1997), parity relationbased (Frank, 1990), parameter estimation/identification-based (Isermann, 1984; Simani, Fantuzzi, & Patton, 2003) and analytical redundancy relations (ARRs) based (Borutzky, 2015; Chow & Willsky, 1984; Staroswiecki & Comtet-Varga, 2001) methods. The ARRs are usually manifestations of different conservation relations represented in the form of constraints or balance equations. For hydraulic system, these can be energy (Bernoulli equation), mass (continuity equation) and momentum (Navier–Stokes equation) conservation relations. For electrical circuits, these can be Kirchhoff’s current and voltage laws, loop 3

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Table 1 Generic bond graph elements with possible causal form.

laws, element constitutive laws, and energy/power balance equations, etc. Bond graph being a graphical representation of the physics of dynamical systems offers a systematic approach to generate ARRs from the BG model. For simpler systems, ARRs can be written in closed symbolic form by eliminating the unknown power variables of the BG model while retaining the measurable system variables (Ould Bouamama, Samantaray, Staroswiecki, & Dauphin-Tanguy, 2003). Bond graph causality provides the algorithm to eliminate the unknown variables from the model.

The diagnostic bond graph (DBG) model as proposed in Samantaray, Medjaher, Bouamama, Staroswiecki, and Dauphin-Tanguy (2006) can be used for ARR generation from the DBG model in symbolic form. A DBG model of a system is obtained by changing the effort (or flow) sensors De (or Df) of BG model into modulated effort (or flow) sources MSe (or MSf), respectively, which are equivalent to changing the causality of different sensors. Such changes imply that for the diagnosis problem, measurements are known variables. The substitution of sensors by sources changes the causalities of junctions and elements in the bond 4

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Fig. 1. (a) Electric circuit example, (b) BG model, (c) DBG model, and (d) fault signature matrix of electric circuit.

graph model (compare the causalities of sensors and sources in Table 1). Also, all the storage elements are assigned with derivative causalities so that unknown initial conditions are eliminated. Usually, for each sensor one ARR is generated, thus the number of ARRs is the same as the numbers of sensors in the plant. For instance, the BG model of an electric circuit (Fig. 1(a)) is shown in Fig. 1(b), where Se: 𝑉s , R: 𝑅1 , R: 𝑅2 , C: 1/𝐶1 and C: 1/C2 model the voltage source, resistances, 𝑅1 and 𝑅2 , and capacitances 𝐶1 and 𝐶2 , respectively. The current (flow) sensor is modelled by Df: I and the two voltage (effort) sensors are modelled by De: 𝑉1 and De: 𝑉2 . The 1 junctions in the model indicate common flow (current) or series connection and the 0 junctions indicate common effort (potential difference) or parallel connection. Note that flow detector Df always appears at 1 junction and effort detector De always appears at 0 junction. The causalities assigned in Fig. 1(b) have integral causality in storage (here C) elements and the sensors have usual causality shown in Table 1. The model in Fig. 1(b) has no causal conflict and can be used to simulate the behaviour of the system for given inputs, parameter values and initial conditions (initial charges in the two capacitors). The DBG model of the system is shown in Fig. 1(c), where the flow (current) sensor Df: I and effort (voltage) sensors De: 𝑉1 and De: 𝑉2 of BG model (Fig. 1(b)) are replaced by Msf: I and MSe: 𝑉1 , MSe: 𝑉2 , respectively. In Fig. 1(c), one imaginary effort detector (De1∗ ) and two imaginary flow detectors (Df1∗ and Df2∗ ) are introduced at new outputs. For a nominal plant with nominal parameters, these new outputs should be zero (Samantaray et al., 2006) and hence the equations for these outputs give ARRs. Accordingly, one of the ARRs is derived from De∗1 = 𝑒4 = 0 after eliminating the unknown variables by using the causality assignment. An ARR(U, Y, 𝜽) is a constraint written in terms of input vector U ∈ (Se, Sf), known measurements imposed on DBG Y ∈ (MSe,.. MSf,..) in place of (De, Df) in original BG, and a parameter vector 𝜽 = [𝜃1 , 𝜃2 , … , 𝜃𝑗 , … , 𝜃𝑝 ]𝑇 comprising p number of known nominal parameters. For the causality assignment shown in Fig. 1(c), at junction 11 , 𝑒4 = −𝑒1 + 𝑒2 + 𝑒3 , where 𝑒1 = 𝑉S , 𝑒2 = 𝑓2 𝑅1 = 𝑓4 𝑅1 = 𝑓5 𝑅1 = 𝐼𝑅1 and 𝑒3 = 𝑒8 = 𝑒9 = 𝑉1 . Thus, after putting all known effort values in De∗1 = 0, it provides the ARR1 as 𝑉S − 𝐼𝑅1 − 𝑉1 = 0 which is an expression containing only the known (measured) variables 𝑉S , 𝐼 and 𝑉1 and parameter 𝑅1 . Disregarding measurement faults, the constraint (or consistency check) expressed by ARR1 is satisfied as long as the system operates without any deviation in the value of 𝑅1 . The evaluation of an ARR is( called a residual. Here, ( expression ) ) the first residual 𝑟1 (𝑡) = Eval ARR1 = Eval 𝑉S (𝑡) − 𝐼 (𝑡) 𝑅1 − 𝑉1 (𝑡) at time t. If the value of 𝑅1 has changed and the measurements have changed due to that then 𝑟1 ≠ 0 with new measurements and old/nominal value of 𝑅1 and that indicates a degradation of 𝑅1 . The influence of parameter deviations on specific residuals is represented in a fault signature matrix (FSM) which will be described later. The FSM for the electrical system in Fig. 1(a) is shown in Fig. 1(d) where only one residual 𝑟1 (corresponding to ARR1 )

would deviate with deviation of 𝑅1 value and such a residual with one to one map is called a structured residual. In fact, to accommodate sensor noise and uncertainties, the residual consistency check is performed by specifying a threshold allowable deviation of residual, i.e. residual 𝑟1 will be assumed to be consistent if ||𝑟1 (𝑡)|| ≤ 𝜀1 (𝑡) where 𝜀1 is a small threshold that will be discussed subsequently. In the prognosis problem, attempt is made to find the instantaneous true value of 𝑅1 and the trend of its deviation which would keep ||𝑟1 || ≤ 𝜀1 at all times, and then postulate the time after which the true value of 𝑅1 would violate a threshold limit of acceptable performance. Likewise, the DBG in Fig. 1(c) shows Df ∗1 = 𝑓8 = 0 and Df ∗2 = 𝑓13 = 0, which provide the ARR2 and ARR3 after elimination of all unknown variables. Thus the ARRs of the electric circuit are expressed as De∗1 ∶ 𝐴𝑅𝑅1 = 𝑉S − 𝐼𝑅1 − 𝑉1 = 0, (𝑉 − 𝑉2 ) d Df ∗1 ∶ 𝐴𝑅𝑅2 = 𝐼 − 𝐶1 𝑉1 − 1 , d𝑡 𝑅2 (𝑉 − 𝑉2 ) d Df ∗2 ∶ 𝐴𝑅𝑅3 = 1 − 𝐶2 𝑉 2 . 𝑅2 d𝑡

and (1)

Note that ARR1 is represents a Kirchhoff’s voltage law (KVL), whereas ARR2 and ARR3 represent Kirchhoff’s current laws (KCLs) for the subparts of the electrical circuit. The decoupling of these sub-parts happens according to the causality assignment. In normal operation, ideally, residuals should be zero. However, due to modelling and process uncertainties, evaluated residuals show small non-zero values. In order to account for these uncertainties in a robust diagnosis/prognosis, the system is modelled in DBG-LFT (linear fractional transformation) form (Djeziri, Merzouki, Ould Bouamama, & Dauphin-Tanguy, 2007; Merzouki, Samantaray, Pathak, & Ould Bouamama, 2012; Touati, Merzouki, & Bouamama, 2012). For uncertain dynamic systems, nominal and uncertain parts of the ARRs are separated from the outputs of the DBGLFT model and the uncertain parts are used to specify the residual thresholds, called adaptive thresholds. The adaptive thresholds should envelope the residuals evaluated by using the nominal ARRs during normal operation of the system or the system which has not degraded sufficiently. Generally, the parameter uncertainty of any parameter value 𝜃𝑗 ∈ (I, C, R, TF, GY) can be represented either in a multiplicative form or an additive form as follow: 𝜃𝑗 = 𝜃𝑗n (1 ± 𝛿𝜃𝑗 )

(2)

or 𝜃𝑗 = 𝜃𝑗n ± 𝛥𝜃𝑗 , where 𝛿𝜃𝑗 = (𝛥𝜃𝑗 ∕𝜃𝑗n ) and 𝛥𝜃𝑗 are the relative and the absolute deviations of nominal parameter value 𝜃𝑗n . In DBG-LFT form model, the nominal value of parameter 𝜃𝑗n is decoupled from its uncertain part ±𝛥𝜃𝑗 . The uncertain part is treated as a disturbance either in the form of additional flow or effort that depends 5

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Fig. 2. (a)–(b) Modelling parameter uncertainty, and (c)–(d) modelling measurement uncertainty in DBG-LFT.

on the type of BG element and its causality in the model. In DBG-LFT model, parameter 𝜃𝑗 ∈ (C) and 𝜃𝑗 ∈ (I), in differential causality, are modelled in the form as shown in Fig. 2(a) and Fig. 2(b), respectively, where J indicates a junction (0 or 1). Likewise, 𝜃𝑗 ∈ (R) is modelled in either form as shown in Fig. 2(a) or Fig. 2(b). Thus, the additional disturbance flow (or effort) due to uncertain part is brought to the junction 0 (or 1) by the uncertainty ∓𝛿𝜃∗ as shown in Fig. 2(a) (or Fig. 2(b)). This is achieved by introducing the virtual flow (or effort) ′ sensor Df ′ ∶ 𝑧𝜃∗ (or De ∶ 𝑧𝜃∗ ) and virtual modulated source of flow ′ ′ (or effort) input MSf ∶ ±𝑤𝜃∗ or (MSe ∶ ±𝑤𝜃∗ ) into the uncertain ∗ BG model. Note that subscript 𝜃 depends on the constitutive law of respective element. For instance, let us consider the linear R-element in the conductive causality (causal form in Fig. 2(a)) with associated power variables 𝑒R (effort) and 𝑓R (flow). If the true parameter value of a resistor R-element is not known exactly then it can be expressed as 𝑅n ± 𝛥𝑅 = 𝑅n (1 ± 𝛿R ) where 𝑅n denotes nominal parameter value and ±𝛥𝑅 = ±𝛿R 𝑅n is the uncertain part of R. The constitutive law of linear R-element modelled in conductive causality is given as 𝑓R =

) 𝑒 1 1 ( 𝑒 = 1 ∓ 𝛿1∕R 𝑒R = R ∓ 𝑤1∕R = 𝑓Rn ∓ 𝑤1∕R 𝑅n ± 𝛥𝑅 R 𝑅n 𝑅n

small 𝜆𝑆𝑖 part may be neglected. A coherence vector (C) whose standard form is 𝐶 = [𝑐1 (𝑡), 𝑐2 (𝑡), … , 𝑐n (𝑡)], where 𝑐𝑖 (𝑡) ∈ {0, 1, }(𝑖 = 1, 2, … , 𝑛) is used to generate the alarms during on-line supervision. The element 𝑐𝑖 (t ) ( ) of coherence vector (C) depends on the decision procedure, 𝛩 𝑟𝑖 (t) , and is obtained as { 0, if − 𝜀𝑖 (𝑡) ≤ 𝑟𝑖 (𝑡) ≤ 𝜀𝑖 (𝑡), ( ) 𝑐𝑖 (𝑡) = 𝛩 𝑟𝑖 (t) = (6) 1, otherwise. During normal operation of the system, all elements of coherence vector (C) show zero values; otherwise non-zero value of any element of C indicates the abnormal behaviour of the system and it generates the alarm. In the present case, abnormal behaviour implies sufficient deviation of at least one parameter value from the corresponding nominal value. After an alarm is raised, the next step in the diagnosis is the isolation of the parameters whose values have deviated sufficiently (more than the uncertainty values), i.e. the components which have degraded beyond specified limits. For isolation of degraded component, the standard fault/degradation signature matrix (FSM) is used which includes the fault/degradation symptoms of different components of a system and it can be generated systematically by testing the sensitivity of each ARR with respect to each component’s parameter. In FSM, columns represent the set of residuals and rows represent the set of components or parameters. Each row contains signature for the respective parameter in the form of a binary number (1 or 0) according to sensitivity of each ARR. If ith ARR is sensitive to jth parameter deviation then the entry in ith column and jth row of FSM is 1, which is otherwise 0. A degradation/fault of a component is monitorable if at least any one of the residuals is sensitive to it whereas it is isolatable only when the row vector corresponding to that component/parameter is different from the row vectors corresponding to all other components/parameters. So, monitorability index (𝑀b ) and isolatability index (𝐼bs ) are also included in the FSM to represent detectability and isolatability of the degradations/faults. These indices are shown as binary numbers 1 and 0, respectively, for TRUE and FALSE. The coherence vector (C) is continuously evaluated at small time intervals during the system monitoring and if it is nonzero at any time then there is some fault/degradation of a component and that particular component can be isolated by uniquely matching the coherence vector with rows of the FSM. For the detection and isolation of any degradation/fault, the corresponding monitorability index and isolatability index values must be 1. For instance, FSM of electric circuit example (Fig. 1(a)) are obtained by analysing the ARRs in (1) and is shown in Fig. 1(d) in the tabular form with isolatability indices 𝐼bs for of single fault/degradation case and 𝐼bm for multiple faults/degradations case. For isolation of multiple faults/degradations, FSM should be in diagonal or structured form. However, most often unstructured form of FSM is obtained due to the limitation of sensors placement in the plant. For example, from the FSM in Fig. 1(d), fault/degradation in 𝑅1 belongs to structured part of the FSM and is always isolatable (both in single and multiple faults/degradations cases). However, other parameters (𝐶1 , 𝑅2 and 𝐶2 ) belong to unstructured part of FSM and are not isolatable in multiple faults/degradations cases. Thus, for isolation of actual faults/degradations in such case of unstructured FSM, parameter estimation (Isermann, 1984; Samantaray & Ghoshal, 2007; Samantaray,

(3)

where 𝑓Rn is nominal flow and ∓𝛿1∕R (𝑒R ∕𝑅n ) = ∓ 𝑤1∕R is the additional contribution of flow due to uncertain part of the parameter R and may be treated as a disturbance. Note that 𝛿1∕R is the uncertainty in estimating the value of 1/R. Likewise, other BG-elements (TF-element and GY-element) with uncertainties in the parameter values can be modelled by using BG-LFT form model. Also, the error in the measurement of effort (or flow) 𝛥𝑀𝑆𝑒 (or 𝛥𝑀𝑆𝑓 ) may be detached from its nominal effort (or flow) part 𝑀𝑆𝑒n (or 𝑀𝑆𝑓n ) and can be expressed as 𝑀𝑆𝑒 = 𝑀𝑆𝑒n ± 𝛥𝑀𝑆𝑒 𝑀𝑆𝑓 = 𝑀𝑆𝑓n ± 𝛥𝑀𝑆𝑓

(4)

These measurement errors 𝛥𝑀𝑆𝑒 and 𝛥𝑀𝑆𝑓 can be modelled by the ′ ′ virtual sources MSe (shown in Fig. 2(c)) and MSf (shown in Fig. 2(d)), respectively, in DBG-LFT form model at the respective junctions. The DBG-LFT form model separates the nominal and uncertain parts of ARRs for robust fault diagnosis of an uncertain system. Without the loss of generality, the ARR(U, Y, 𝜽) of an uncertain system may be expressed as 𝐴𝑅𝑅n𝑖 (𝐔, 𝐘, 𝜽) ± (𝜆𝑖 + 𝜆𝑆𝑖 ) = 0,

(5)

where 𝐴𝑅𝑅n𝑖 , 𝜆𝑖 and 𝜆𝑆𝑖 represent the ith nominal ARR that gives residual (𝑟𝑖 ) (𝑖 = 1, 2, … , 𝑛; n is number of residuals), the uncertain part due to parameter uncertainties and the static uncertain part needed to account for measurement noise, respectively. Also, U ∈ (Se, Sf) is the known input vector, Y ∈ (MSen , MSfn ) is the nominal measurement vector, 𝜽 = [𝜃1 , 𝜃2 , … , 𝜃𝑗 , … , 𝜃𝑝 ]𝑇 is the nominal parameter vector comprising p parameters, and 𝜆𝑖 ∈ 𝑤𝜃𝑗 , 𝜆𝑆𝑖 ∈ (𝛥𝑀𝑆𝑒 , 𝛥𝑀𝑆𝑓 ). Online evaluation of each nominal part, 𝐴𝑅𝑅n𝑖 , and uncertain part, (𝜆𝑖 + 𝜆𝑆𝑖 ), using U, Y, and 𝜽 along with the different specified uncertainties bounds provides residual (𝑟𝑖 ) and adaptive thresholds (𝜀 = ±|(𝜆𝑖 + 𝜆𝑆𝑖 )|), respectively. Note that as the absolute values of different uncertain parts contribution is added in the adaptive threshold, the 6

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where 𝑎𝑘 is the 𝑘th row discrete parameter of the MCSSM, 𝑎𝑘 ∈ {0, 1}, 𝑘 ∈ {1, 2, … , 𝑚}, and m is the number of discrete parameters in the system. Since discrete parameters can take up only two values, there is no slow degradation associated with them and hence they are kept outside the purview of prognosis. The nature of faults in a system can be classified as abrupt, intermittent and progressive. Abrupt fault is due to a sudden irreversible change in a parameter value. Intermittent fault occurs in irregular time intervals and is often unpredictable. Progressive fault is a gradual drift in the parameter value due to usage and natural ageing of the components. The deviations of residuals corresponding to such faults are shown in Fig. 3. Abrupt fault generally requires immediate plant reconfiguration, if possible, or shutdown. Intermittent fault is a forewarning of a major fault and requires immediate intervention or maintenance. However, progressive fault is a slow process and is more common. This type of slow parameter drift is generally tolerable up to some extent as long as the output performance is satisfactory. This is the domain for prognosis where one can estimate the limit of parameter deviation for acceptable output performance and schedule predictive maintenance intervention accordingly.

Ghoshal, Chakraborty, & Mukherjee, 2005; Simani et al., 2003) is required. For model-based diagnosis and prognosis of hybrid system, the model has to consider the hybrid dynamics under various operation modes (Lunze, Nixdorf, & Richter, 2001). For hybrid systems, the BG modelling approach is extended by using the switched junction concept, called the hybrid bond graph (HBG) modelling (Mosterman & Biswas, 1995; Roychoudhury et al., 2011). Since, hybrid system has different dynamics in different mode, its model structure changes due to mode change and sometimes, causality reassignment is required. To avoid causality reassignment, sequential causality assignment for hybrid systems (SCAPH) was proposed in Low, Wang, Arogeti, and Luo (2010) and Low, Wang, Arogeti, and Zhang (2010). There are 2𝑚 numbers of different continuous dynamics possible for m numbers of discrete state variables 𝑎𝑘 ∈ (0, 1), 𝑘 = 1, 2, … , 𝑚. Using SCAPH, causality of each controlled/switched junction is assigned in such a way that a single model is valid for every mode with the assigned causalities. For hybrid system, DHBG-LFT model is used, in place of DBG-LFT model, for the generation of global ARRs (GARRs) which are obtained by replacing sensors of the HBG model by sources and then applying preferred differential causalities to I and C elements as discussed earlier. Consequently, GARRs are obtained in global forms in terms of measurable system variables (U, Y), nominal parameters (𝜽) and mode vector of discrete state variables (MD). The GARR of an uncertain hybrid system may be written as 𝐺𝐴𝑅𝑅n𝑖 (𝐔, 𝐘, 𝜽, 𝐌𝐃) ± (𝜆𝑖 + 𝜆𝑆𝑖 ) = 0.

3. Prognosis framework for hybrid systems Prognosis predicts the time to failure (TTF) of a degrading component or subsystem, also called remaining useful life (RUL), by assessing the current degradation state of a component or subsystem, its past degradation trend or profile given by the continuous monitoring module with the known future operating conditions of the component or subsystem (Jardine et al., 2006). RUL is the time left before a component or subsystem, which is assigned for a particular job or a function, reaches its end of life (EOL) and it may be expressed as

(7)

In hybrid system, residuals are sensitive to both parametric deviations and discrete mode changes. Thus, the extended form of FSM for parametric deviations/faults and discrete mode faults, respectively, termed as global fault/degradation signature matrix (GFSM) and mode change signature matrix (MCSM), are used for fault/degradation isolation. The GFSM and MCSM are further extended to global fault/degradation sensitivity signature matrix (GFSSM) and mode change sensitivity signature matrix (MCSSM) as proposed in Levy et al. (2015) where the FSM and coherence vector’s elements are signed, i.e., 𝑐𝑖 (t ) ∈ {0, +1, −1}. The signed FSM and coherence vector add more depth of information and are useful to predict the direction of deviation of the associated parameter. GFSSM is dynamic in nature as every element of this matrix is updated with the evolution of time according to the sign operation of residual sensitivity with respect to corresponding parameter deviation. This matrix also has an ability to provide the fault/degradations direction, i.e. increasing (𝜃𝑗 ↑) or decreasing (𝜃𝑗 ↓) trend for the corresponding degrading parameter. Its elements are obtained as { −sign(𝜕𝑟𝑖 ∕𝜕𝜃𝑗 ), if 𝑟𝑖 is sensitive to increasing 𝜃𝑗 ↑, 𝐺𝐹 𝑆𝑆𝑀𝑗𝑖↑ = 0, otherwise, (8) { sign(𝜕𝑟 ∕𝜕𝜃 ), if 𝑟𝑖 is sensitive to decreasing 𝜃𝑗 ↓, 𝑖 𝑗 ↓ 𝐺𝐹 𝑆𝑆𝑀𝑗𝑖 = 0, otherwise,

RUL = 𝑡𝑓 𝑙 − 𝑡0 | 𝑡𝑓 𝑙 > 𝑡0 , 𝐷(𝑡)

(10)

where 𝑡𝑓 𝑙 signifies the random variable of TTF, 𝑡0 signifies the current age and D (t ) signifies the past operation profile up to the current time. 3.1. Proposed adaptive prognosis for multiple degradations A complete schematic description of the proposed adaptive prognosis for hybrid system is presented in Fig. 4. It uses concepts borrowed from model-based diagnosis for prognosis task by using BG approach. Moreover, the RUL prediction scheme using parameter estimation technique for progressive fault is also shown in Fig. 5. Fault/degradation Detection, Isolation and Estimation of Degrading Parameters In a hybrid system, the residuals (𝑟n𝑖 ) are sensitive to both discrete fault and parametric fault/degradation. Some or all of the residuals which are sensitive to a specific parameter or mode fault in a system cross either lower or upper threshold if any kind of the inconsistency arises. If any one of the nominal mode (say 𝑎𝑘 ↓) of MD is inconsistent or any one of the nominal parameter (say 𝜃𝑗 ↓ decreasing trend) of 𝜃 deviates (more than the uncertainty value), then some or all of the elements of coherence vector (C) show non-zero values depending upon the violation of residuals thresholds, i.e. +1 for violation of upper threshold, −1 for violation of lower threshold and 0 if there is no any violation of threshold. Then obtained coherence vector (C) is matched with MCSSM and GFSSM and the faults/degradations which match with the obtained coherence vector (C) are considered in the hypothesized fault list (HFL) that includes both mode faults and parametric faults along with the direction of faults (increasing or decreasing values in parameters). It may be possible that there are more than one mode fault and parametric fault present in the initial HFL, since fault signatures of some of the mode faults and parametric faults cannot be uniquely matched in MCSSM and GFSSM, respectively. After fault/degradation hypothesis generation, it is initially assumed that the threshold violation

where 𝑟𝑖 is the 𝑖th column residual, 𝑖 ∈ {1, 2, … , 𝑛}, n is the number of residuals, 𝜃𝑗 is the 𝑗th row component’s parameter in the GFSSM, 𝑗 ∈ {1, 2, … , 𝑝}, p is the number of components. For parameters which can have both increasing and decreasing trend, two rows are used in GFSSM. However, for parameters which have increasing or decreasing trend only (such as tank leakage), one row is used in GFSSM. MCSSM is also dynamic in nature like GFSSM and it has ability to distinguish between increasing (𝑎𝑘 ↑, from 0 to 1) or decreasing (𝑎𝑘 ↓, from 1 to 0) trend of discrete mode faults. Its elements are obtained as { −sign(𝜕𝑟𝑖 ∕𝜕𝑎𝑘 ), if 𝑟𝑖 is sensitive to increasing 𝑎𝑘 ↑, ↑ 𝑀𝐶𝑆𝑆𝑀𝑘𝑖 = 0, otherwise (9) { sign(𝜕𝑟 ∕𝜕𝑎 ), if 𝑟𝑖 is sensitive to decreasing 𝑎𝑘 ↓, 𝑖 𝑘 ↓ 𝑀𝐶𝑆𝑆𝑀𝑘𝑖 = 0, otherwise 7

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Fig. 3. Residual responses in (a) abrupt fault (b) Intermittent fault and (c) Progressive fault (degradation).

Fig. 4. Schematic description of proposed adaptive prognosis method.

Fig. 5. Triggering time points of parameter estimation and RUL prediction scheme for progressive fault/degradation.

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where W ∈ R𝑛×𝑛 is a positive semi-definite weighing function and may ( ) ( ) ( ) be assumed as identity matrix, 𝐞 𝑡𝑗 = 𝐲 𝑡𝑗 , 𝜽 − 𝐲̂ 𝑡𝑗 is error vector, ( ) ( ) 𝐲 𝑡𝑗 , 𝜽 is output vector from a simulation model and 𝐲̂ 𝑡𝑗 is actual measurement vector at time instant 𝑡𝑗 , k is the current sample time, 𝑞 ≥ 0 is the number of collection of past sampled data during monitoring, and 𝜽L and 𝜽U are known vectors of lower and upper bounds on the parameter vector 𝜽. However, if one uses simulation model output and tries to match the responses (so-called model matching) then the initial conditions also need to be estimated. To avoid this problem, Samantaray and Ghoshal (2007) proposed minimization of residual errors where ( ) ( ) 𝐞 𝑡𝑗 , 𝜽 = 𝑟 𝑡𝑗 , 𝜽 is the residual vector obtained by evaluating the GARRs, which have been derived from DHBG-LFT model with dynamic elements (I and C) in differential causality (initial conditions appear in the equations when dynamic elements are in integral causality). The objective of the optimization now turns out to be the process of finding 𝜽 that minimizes the residuals so that they are bounded by the residual thresholds. Furthermore, there is no need to consider all the residuals in the objective function; only those residuals which have deviated due to the unknown parameter variation need to be used to estimate only the parameters appearing in deviated residuals and not appearing in any other residuals. The relative magnitude of residual thresholds may be used to define the weights. This approach gives good and quick parameter estimates (Samantaray & Ghoshal, 2007) and hence, it is followed here. However, note that this approach needs sufficient sensors to decouple parametric influences on the residuals. On the other hand, parameter estimation by using measurement error attempts simultaneous estimation of all parameters and initial condition and hence the uncertainty over estimated parameter values becomes large and the estimation may not succeed at all times. The constrained parameter estimation technique gives the degradation of 𝜃𝑗 at 𝑘th instant of time 𝑡𝑘 corresponding to mode Z = 𝑧(𝑖) (See Fig. 5). Note that discrete mode fault and large variation in a parameter magnitude are treated as abrupt fault and hence require immediate plant maintenance. That is why those are kept outside the purview of prognosis discussed in this article. It is assumed that the first estimate of 𝜃𝑗 is denoted by 𝜃𝑗f (𝑡𝑘 , 𝑧(𝑖) ) at 𝑘th instant of time at the mode 𝑧(𝑖) . After getting the first estimate of degrading parameter 𝜃𝑗 and assuming monotonic decreasing trend of 𝜃𝑗 , initial known bound of 𝜃𝑗 ∈ [𝜃𝑗L , 𝜃𝑗U ] is also updated to 𝜃𝑗 ∈ [𝜃𝑗L , 𝜃𝑗f (𝑡𝑘 , 𝑧(𝑖) )] for subsequent parameter estimation steps. The bounds are likewise updated for increasing trend of parameter value. If the nominal part of each 𝐺𝐴𝑅𝑅n𝑖 is evaluated again with the first estimate of the fault/degradation, 𝜃𝑗f (𝑡𝑘 , 𝑧(𝑖) ) and adaptive thresholds are also updated then the evaluated residuals again lie within the updated thresholds (Borutzky, 2015; Prakash & Samantaray, 2017). So, the original parameter vector 𝜽 is updated by swapping the nominal 𝜃𝑗 by the first estimate of the fault/degradation, 𝜃𝑗f (𝑡𝑘 , 𝑧(𝑖) ), of the real system. Thus, the updated parameter vector 𝜽 = [𝜃1 , 𝜃2 , … , 𝜃𝑗f (𝑡𝑘 , 𝑧(𝑖) ), … , 𝜃𝑃 ]𝑇 at 𝑘th instant (supposed to be the new nominal parameter vector) is used to update DHBG-LFT form model. Consequently, the same residuals set sensitive to the variation of parameter 𝜃𝑗 (t, 𝑧(𝑖) ) would cross the updated thresholds again with certain time delay. Again, new estimate of degrading parameter (𝜃𝑗 ) with updated bound 𝜃𝑗 ∈ [𝜃𝑗L , 𝜃𝑗f (𝑡𝑘 , 𝑧(𝑖) )] is generated when the same set of updated residuals cross the updated thresholds. This was, constrained parameter estimation technique quickly converges as the search region is dynamically reduced. The second new estimate of degradation and this estimate is denoted by 𝜃𝑗f (𝑡𝑘 + 𝛥, 𝑧(𝑖) ) at (𝑡𝑘 + 𝛥)th time instant at the operating mode 𝑧(𝑖) . For precise identification of degradation profile, adequate number of estimated data points of degrading parameter (𝜃𝑗 ) are required corresponding to any operating mode 𝑧(𝑖) . Likewise, more estimated parameter values are obtained from the continuously monitored system.

occurs due to a mode fault and any inconsistency for the mode fault is first checked before checking for the parametric fault/degradation. The mode fault identification procedure can be consulted in Prakash and Samantaray (2017) and Wang et al. (2013). In this article, specifically, progressive parametric faults/ degradations will be dealt with. If the inconsistencies in the residuals are not mode fault then it is suspected that those are due to parametric faults/degradations and the mode faults are removed from the initial HFL. The next step is to isolate the parametric faults/deviations present in the refined HFL and provide the estimates of these parameters to the prognosis module for degradation model identification and RUL prediction. Note that some parametric fault (𝜃𝑗 ) may be directly isolatable under certain operating modes if the obtained coherence vector (C) has a unique match in GFSSM, i.e., the fault corresponds to the structured part of the GFSSM; otherwise, multiple suspected parametric faults (𝜃𝐅 ) will be present in the refined HFL. The parameter estimation is used to isolate the true faults/ degradations from the list of multiple suspected/hypothesized parametric faults/deviations (𝜽𝐅 ) present in the HFL (Low, Wang, Arogeti, & Luo, 2009; Samantaray & Ghoshal, 2007; Samantaray et al., 2005) and for the estimation of the fault/degradation magnitudes of the actual degrading parameter at different time instances (see Fig. 5). For obtaining the estimates of degrading parameter at different operating modes (Z), a window (𝛥𝑡𝑒 ) of sampled data is used. The window length is an important parameter. If a certain HFL element is active in some specific mode then the window must cover operation in that specific mode. Likewise, for dynamic elements (I and C elements), the window must contain transient dynamics, such as a mode change induced transient. Fig. 5 illustrates how the concepts from diagnosis are utilized for prognosis. When a parameter drifts progressively, the residual sensitive to it also drifts and goes out of the threshold envelope after sufficient degradation, i.e., more than the uncertainty in parameter value considered in defining the adaptive thresholds (DHBG-LFT model). Then using a window of data, the drift in the parameter value is estimated and the estimated parameter value is treated as the new nominal parameter value. Once that is done, the residual reverts back to its nominal value (near zero) and continues to drift with further parameter drift. When the next threshold violation occurs, another estimate of the drifted parameter value is obtained and again the estimated value is treated as the new nominal parameter value. This way, a set of estimated parameter values (trend) is obtained at different time instants. Depending upon the number of such estimates, an interpolation curve is generated and that curve is extrapolated to obtain the RUL. For a parameter with known direction of deviation obtained from GFSSM, constrained parameter estimation technique with dynamically updated parameter bound is proposed here to speed up the degradation pattern identification and RUL prediction. The bounds of the suspected parametric faults/degradations, 𝜽𝐅 ∈ [𝜽𝐅𝐋 , 𝜽𝐅𝐔 ], are created by using deviation direction information from GFSSM and the earlier known values (nominal/last estimated values) of the parameters (𝜽𝐅 ) and the possible maximum deviation information, which is derived from deep knowledge of the system or the so-called technological specifications. To further speed up the parameter estimation, an SBG (Gawthrop, 2000; Samantaray & Ghoshal, 2007) approach is utilized to provide the gradient information of the objective function in the parameter estimation. The proposed constrained parameter estimation technique is based on the gradient projection method with known bounds on the parameters in which least squares of residuals are minimized by using Gauss–Newton optimization algorithm. The gradient projection method is most efficient, particularly, when constraints include only bounds on the parameters (Nocedal & Wright, 2006). The minimization of the objective function is formulated as: min 𝐽 (𝜃) = 𝜃

𝑘 1 ∑ T 𝐞 (𝑡𝑗 ).𝐖.𝐞(𝑡𝑗 ) 2 𝑗=𝑘−𝑞

Degradation Model Identification and RUL Prediction After successive parameter estimation, a set of estimated data points of parameter 𝜃𝑗 up to current sample time 𝑡𝑘 at mode Z = 𝑧(𝑖) is obtained

(11)

subject to ∶ 𝜽L ≤ 𝜽 ≤ 𝜽U , 9

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( ( )) as 𝐷𝜃𝑗 (𝑡𝑘 ) = 𝑡𝑘 , 𝜃𝑗f 𝑡𝑘 , 𝑧(𝑖) , 𝑘 = 0, 1, … , 𝑘, during continuous monitoring. Then, these data are used for degradation model identification (𝑖) 𝑀𝜃𝑧 at the respective mode for RUL prediction. However, it is assumed 𝑗 that the degradation patterns of the components or subsystems are unknown to us. But, the degradation model is identified from a bank of general degradation models according to the best fit to the estimated data of the parameter. The general form of degradation model is written as (𝑖)

𝑀𝜃𝑧

𝑗

𝑡

∶ 𝜃𝑗 (𝑡, 𝑧(𝑖) ) = 𝜇(𝑧(𝑖) )

∫ 𝑡0

𝜒(𝑡, 𝜃𝑗 (𝑧(𝑖) ))𝑑𝑡 + 𝜃𝑗 (𝑡0 , 𝑧(𝑖) ),

(12)

where 𝜃𝑗 (𝑡0 , 𝑧(𝑖) ) is the magnitude of 𝜃𝑗 at an instant 𝑡0 , 𝜇(𝑧(𝑖) ) is modedependent rate of degradation and 𝜒(𝑡, 𝜃𝑗 (𝑧(𝑖) )) represents the linear or nonlinear function of time t at mode Z = 𝑧(𝑖) . For example, if 𝜒(𝑡, 𝜃𝑗 (𝑧(𝑖) )) is any constant value then (12) represents linear degradation model. If 𝜒(𝑡, 𝜃𝑗 (𝑧(𝑖) )) = 𝜃𝑗 (𝑡, 𝑧(𝑖) ) then (12) represents exponential degradation model and if 𝜒(𝑡, 𝜃𝑗 (𝑧(𝑖) )) = 2𝑃1 𝑡 + 𝑃2 , where 𝑃1 and 𝑃2 are constants, and 𝜇(𝑧(𝑖) ) = 1, then (12) represents polynomial second order degradation model. Likewise, different nonlinear degradation models can be obtained by using generalized degradation model as presented in (12). Note that linear and exponential degradation models are the most commonly used degradation models in engineering industries for the RUL prediction (Randall, 2011). Generally, component’s degradation has a tendency to accelerate in the latter stage of ageing, particularly when approaching to the EOL. In this regard, it is suggested that exponential model and the second order polynomial model should be used. It is also suggested that the polynomial models of higher orders more than the second order model should be avoided in curve fitting unless there is some known physical reason or past experience of such type of degradation of the component or subsystem. The RUL prediction with higher order polynomial models may provide good interpolation, but bad extrapolation. Among the various degradation models, the model which has the best fitting to the data set of parameter estimates obtained at corresponding operating mode, 𝑧(𝑖) , can be selected as a best degradation model for RUL prediction. Thus, when there is less number of parameter estimates, it is proposed that a linear degradation model be used for initial prediction of RUL. This initially predicted RUL provides the some indication to the maintenance technicians for planning and scheduling the maintenance activities or the other tasks. Further, the linear degradation model may be adapted with modified model like exponential or higher order polynomial model, according to best fit equation, when more number of parameter estimates is found during continuous monitoring. Thus, the selection of degradation model which depends on the information of estimated data points up to current time 𝑡𝑘 at mode Z = 𝑧(𝑖) is represented as { 𝜃𝑗 (𝑡, 𝑧(𝑖) ) = MLIN , if 𝑘 < 𝑘𝑆 𝑧(𝑖) (13) 𝑀𝜃 ∶ 𝑗 𝜃𝑗 (𝑡, 𝑧(𝑖) ) = 𝜁1 .MLIN + 𝜁2 .MPL2 + 𝜁3 .MEXP , if 𝑘 ≥ 𝑘𝑆

Fig. 6. (a) Various working modes (b) Mode-dependent RUL prediction of the degrading parameter (𝜃j ).

𝜃𝑗𝑓 𝑙 then the component is declared as EOL component at 𝑡𝑓 𝑙 . Thus, the 𝑡𝑓 𝑙 (TTF) and RUL are defined as 𝑡𝑓 𝑙 = inf {𝑡 ∈ R ∶ 𝜃𝑗 (𝑡, 𝑍) ≥ 𝜃𝑗𝑓 𝑙 |𝜃𝑗 (𝑡0 , 𝑍) < 𝜃𝑗𝑓 𝑙 }

(14)

RUL (𝑡, 𝑍) = 𝑡𝑓 𝑙 − 𝑡0

(15)

For instance, RUL prediction of 𝑗th component (𝜃𝑗 ) with the transition of known working modes (Z) of a hybrid system is presented in Fig. 6, where time instances of transition of operating mode are represented by 𝑡(𝑖−1) , (𝑖 = 1, 2, …) and the transition of operating mode to a new mode after 𝑡(𝑖−1) is represented by 𝑧(𝑖) ∈ 𝒁. The identified (𝑖) multiple degradation models 𝑀𝜃𝑧 are utilized for RUL prediction. Note 𝑗 that here multiple means different uncorrelated degradation models for different components as well as different uncorrelated degradation models of a given component under different modes. Finally, the cumulative degradation profile according to different known future operating modes is expected/predicted to reach the failure threshold 𝜃𝑗𝑓 𝑙 at time 𝑡𝑓 𝑙 and the RUL is computed. Generally, this failure threshold (𝜃𝑗𝑓 𝑙 ) is considered based on a certain percentage of the nominal parameter value. Note that RUL is predicted for every degrading component and the component having the least predicted RUL requires more attention by the maintenance technicians. 4. Experimental study on an electrical hybrid system In this article, an electrical hybrid system, shown in Fig. 7, is considered as an application example for experimental studies. It is chosen for easy and low cost experimental implementation. This electrical hybrid system is equivalent to a complex hydraulic hybrid system described in Prakash & Samantaray (2017).

where k is number of estimated data points up to current time, 𝑘𝑆 is sufficient number of estimated data points decided by the user for precise degradation model identification, MLIN = 𝜇(𝑧(𝑖) )(𝑡 − 𝑡0 ) + 𝜃𝑗 (𝑡0 , 𝑧(𝑖) ), MPL2 = 𝑃1 (𝑡 − 𝑡0 )2 + 𝑃2 (𝑡 − 𝑡0 ) + 𝜃𝑗 (𝑡0 , 𝑧(𝑖) ) and MEXP = 𝜃𝑗 (𝑡0 , 𝑧(𝑖) ) exp{𝜇(𝑧(𝑖) )(𝑡 − 𝑡0 )}. Thus, if 𝑘 ≥ 𝑘𝑆 , a particular degradation model (DM) is selected for data fitting according to the value of degradation model vector DMV = [𝜁1 , 𝜁2 , 𝜁3 ]. If DMV = [1, 0, 0] then linear model MLIN is selected, if DMV = [0, 1, 0] then second order polynomial model MPL2 is selected and if DMV = [0, 0, 1] then exponential degradation model MEXP is selected and the model which has least root mean square error (RMSE) to the data fit is selected as a best degradation model at mode Z = 𝑧(𝑖) . Likewise, other equation models can be plugged into the bank of degradation models in (13). RUL is predicted for degrading parameter {𝜃𝑗 (𝑡, 𝑍), 𝑡 > 0} by (𝑖) extrapolating the data using the identified model 𝑀𝜃𝑧 with the future 𝑗 operating modes (Z) of the system known up to the current time. Thus, when the extrapolated trend of 𝜃𝑗 reaches a well set failure threshold

4.1. Electrical hybrid system description The system includes capacitors (𝐶1 and 𝐶2 ), resistors (𝑅in , 𝑅1 , 𝑅2 , 𝑅d1 , and 𝑅d2 ), diodes (𝐷1 and 𝐷2 ) and controlled voltage source (𝑉in ). The voltage source (𝑉in ) is controlled by PI-controller to maintain a set voltage (𝑉set = 5V) across the capacitor 𝐶1 . The considered hybrid system includes both supervisory controlled mode (𝑎R1 ) and autonomous modes (𝑎d1 and 𝑎d2 , respectively, which depend on the state of capacitors 𝐶1 and 𝐶2 ). Dynamics of resistor 𝑅1 is mode dependent (𝑎R1 ) which is operated at different states by using a switch Sw1 , i.e. either on or off state, according to the controller command given to the switch Sw1 . Diodes 𝐷1 and 𝐷2 operate as switches corresponding to autonomous modes 𝑎d1 and 𝑎d2 of the system which allow the current to flow through 10

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Fig. 7. Schematic diagram of the electrical system. Fig. 9. DHBG-LFT model of electric hybrid system derived from HBG in Fig. 8.

relations GARR3 , GARR4 and GARR5 , respectively, as d (𝑉 (𝑡)) d𝑡 s1 (𝑉s1 (𝑡) − 𝑉s2 (𝑡)) ⋅ sign(𝑉s1 (𝑡) − 𝑉s2 (𝑡))

GARR3 ∶ 𝐼in − 𝐶1 ⋅ − 𝑎R1 ⋅

| | | | d 1 𝜆3 = ||𝛿C1 ⋅ 𝐶1 ⋅ (𝑉s1 (𝑡))|| + ||𝑎R1 ⋅ 𝛿1∕R1 ⋅ (𝑉 (𝑡) − 𝑉s2 (𝑡))|| d𝑡 𝑅1 s1 | | | | | | 1 | | ⋅ (𝑉s1 (𝑡) − 𝑉set1 (𝑡))| , + |𝑎d1 ⋅ 𝛿1∕Rd1 ⋅ 𝑅d1 | | | | 𝜆4 = ||𝑎R1 .𝛿1∕R1 ⋅ 𝑅1 (𝑉s1 (𝑡) − 𝑉s2 (𝑡))|| 1 | | | | + ||𝑎d1 ⋅ 𝛿1∕Rd1 ⋅ 𝑅1 ⋅ (𝑉s1 (𝑡) − 𝑉set1 (𝑡))|| d1 | | | | + |𝛿C2 .𝐶2 . d𝑡d (𝑉s2 (𝑡))| | | | | | + ||𝑎R2 .𝛿1∕R2 ⋅ 𝑅1 (𝑉s2 (𝑡))|| + ||𝑎d2 ⋅ 𝛿1∕Rd2 ⋅ 2 | | |

(16)

(17)

where 𝑉max is the maximum input voltage, 𝑈PI is the output of PIcontroller, 𝑉set is a voltage set point, 𝐾P is proportional gain and 𝐾I is the integral gain of PI-controller, respectively.

| | 𝜆5 = |𝛿Rin ⋅ 𝑅in ⋅ 𝐼in (𝑡)| . | |

(23)

(24) 1 𝑅d2

| ⋅ (𝑉s2 (𝑡) − 𝑉set2 (𝑡))|| , | (25)

Using (8)–(9) on (20)–(22), the GFSSM and MCSSM for the electrical system are presented in Tables 2 and 3, respectively. The present article focuses on parametric degradations linked with unstructured residuals (𝑟3 and 𝑟4 ) of GFSSM for which isolation and degradation pattern identification are difficult. Note that the residuals 𝑟1 , 𝑟2 and 𝑟5 belonging to the structured part of the GFSSM and corresponding to the voltage source actuation (𝑉in ), PI-controller and resistor 𝑅in have not been considered because those can be estimated in a straight-forward way. It is further assumed in this study that the sensors are faultless, but can have small measurement uncertainty like bias and noise. If required, the ARRs for the detection of sensor faults can be easily generated and

4.2. Generation of ARRs/GARRs The DHBG-LFT model of the electrical system is derived from the HBG and is shown in Fig. 9. The ARRs for the voltage source actuation and the controller (assumed to have no uncertainty) are simply obtained from comparisons of output and input relationships as ARR1 ∶ 𝑉in − 𝛷V (𝑈PI ) = 0, ( ) ARR2 ∶ 𝑈PI − 𝛷PI 𝑉s1 (𝑡) = 0.

(21)

GARR5 ∶ 𝑉in (𝑡) − 𝑉s1 (𝑡) − 𝐼in (𝑡) ⋅ 𝑅in ± 𝜆5 = 0, (22) { { 0, if 𝑉 (𝑡) ≤ 𝑉 0, if 𝑉 (𝑡) ≤ 𝑉 where 𝑎d1 = 1, if 𝑉s1 (𝑡) > 𝑉set1 and 𝑎d2 = 1, if 𝑉s2 (𝑡) > 𝑉set2 . s1 set1 s2 set2 Also, (20) to (22) contain uncertain parts 𝜆𝑖 of GARR𝑖 , (𝑖 = 3, 4 and 5). The effects of various parameter uncertainties on a GARR are un-correlated with the possibility of cancelling out each other. Hence, absolute values of the individual effects are considered for adaptive thresholds (small 𝜆𝑆𝑖 part are neglected) and obtained as

resistors 𝑅d1 and 𝑅d2 at the prescribed threshold voltages 𝑉set1 and 𝑉set2 , respectively. 𝐼d1 is the current source whose value is same as the current that flows (unmeasured current) through the resistor 𝑅d1 . Sensors for 𝑉in , 𝑉s1 , 𝑉s2 and 𝐼in are also installed in the system which measure the input controlled voltage, voltages in capacitors 𝐶1 , 𝐶2 and current flow through resistor 𝑅in , respectively. The ground voltage (𝑉G ) is assumed to be reference voltage. The HBG model of the system is shown in Fig. 8 where 1-junctions with subscript 𝑎R1 , 𝑎d1 and 𝑎d2 are switched junctions related with discrete modes. Voltage source saturation characteristic (𝛷V ) and PI-controller output law (𝛷PI ) are, respectively, given as

𝑈PI = 𝐾P (𝑉set − 𝑉s1 (𝑡)) + 𝐾I (𝑉set1 − 𝑉s1 (𝑡)) d𝑡, ∫ ( ) = 𝛷PI 𝑉s1 (𝑡) ,

(20)

1 ⋅ (𝑉s1 (𝑡) − 𝑉set1 (𝑡)) ± 𝜆3 = 0, − 𝑎d1 ⋅ 𝑅d1 1 GARR4 ∶ 𝑎R1 ⋅ (𝑉 (𝑡) − 𝑉s2 (𝑡)) ⋅ sign(𝑉s1 (𝑡) − 𝑉s2 (𝑡)) 𝑅1 s1 1 d + 𝑎d1 ⋅ ⋅ (𝑉s1 (𝑡) − 𝑉set1 (𝑡)) − 𝐶2 ⋅ (𝑉s2 (𝑡)) 𝑅d1 d𝑡 1 1 − ⋅ (𝑉s2 (𝑡)) − 𝑎d2 ⋅ ⋅ (𝑉s2 (𝑡) − 𝑉set2 (𝑡)) ± 𝜆4 = 0, 𝑅2 𝑅d2

Fig. 8. HBG model of the electrical system.

⎧𝑈PI , if 0 ≤ 𝑈PI ≤ 𝑉max , ( ) ⎪ 𝑉in = ⎨0, = 𝛷V 𝑈PI , if 𝑈PI ≤ 0, ⎪ ⎩𝑉max , if 𝑈PI > 𝑉max ,

1 𝑅1

(18) (19)

Two virtual flow sensors (Df∗ ) and one virtual effort sensor (De∗ ) used in the DHBG-LFT model (Fig. 9) provide the three constraint 11

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Control Engineering Practice 72 (2018) 1–18 Table 2 GFSSM for the electrical system. Parameter

ARR1 (𝑟1 )

ARR2 (𝑟2 )

GARR3 (𝑟3 )

GARR4 (𝑟4 )

GARR5 (𝑟5 )

𝑅1 ↓ 𝑅1 ↑ 𝑅2 ↓ 𝑅2 ↑ 𝑅d1 ↓ 𝑅d1 ↑ 𝑅d2 ↓ 𝑅d2 ↑ 𝐶1 ↓ 𝐶1 ↑ 𝐶2 ↓ 𝐶2 ↑ 𝑅in ↓ 𝑅in ↑ 𝑉in ↓ 𝑉in ↑ 𝑈PI ↓ 𝑈PI ↑

0 0 0 0 0 0 0 0 0 0 0 0 0 0 +1 −1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +1 −1

𝑎R1 sign(𝑉s1 (𝑡) − 𝑉s2 (𝑡)) −𝑎R1 sign(𝑉s1 (𝑡) −𝑉s2 (𝑡)) 0 0 𝑎d1 −𝑎d1 0 0 −sign(V̇ s1 ) sign(V̇ s1 ) 0 0 0 0 0 0 0 0

−𝑎R1 sign(𝑉s1 (𝑡) − 𝑉s2 (𝑡)) 𝑎R1 sign(𝑉s1 (𝑡) − 𝑉s2 (𝑡)) sign(𝑉s2 (𝑡)) −sign(𝑉s2 (𝑡)) −𝑎d1 𝑎d1 𝑎d2 −𝑎d2 0 0 −sign(V̇ s2 ) sign(V̇ s2 ) 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 −sign(𝐼in (𝑡)) sign(𝐼in (𝑡)) 0 0 0 0

Table 3 MCSSM for the electrical system. Parameter

ARR1 (𝑟1 )

ARR2 (𝑟2 )

GARR3 (𝑟3 )

GARR4 (𝑟4 )

GARR5 (𝑟5 )

𝑎R1 ↑ 𝑎R1 ↓

0 0

0 0

sign(𝑉1 (𝑡) −𝑉2 (𝑡)) − sign(𝑉1 (𝑡) −𝑉2 (𝑡))

− sign(𝑉1 (𝑡) −𝑉2 (𝑡)) sign(𝑉1 (𝑡) −𝑉2 (𝑡))

0 0

Table 4 Nominal parameters of the electrical system.

applied in a hybrid system by using a technique of hardware redundancy as proposed in Medjaher, Samantaray, Bouamama, and Staroswiecki (2006) and Ould Bouamama, Medjaher, Samantaray, and Staroswiecki (2006). ARRs for redundant sensors are statistical, algebraic or empirical relations between measurements from the different sensors which check the consistency of the sensor measurements from complimentary sources and there is no difference in their fault isolation/identification technique in a hybrid system as compared to a continuous system. Thus, sensor faults have been kept outside the scope of this article. 4.3. Experimental setup An experimental test-bed is developed for the considered electrical system (refer Fig. 7) and is presented in Fig. 10(a). The different circuit’s components including the arrangement to introduce the degradations are marked in Fig. 10(b) by numerals. The PI-controller is constructed by using a combination of different operational amplifiers (op-amps), resistors and capacitor components. A controlled relay switch, Sw1 , is used to set the resistor 𝑅1 at on and off state according to output command (𝑎R1 ) given by a microcontroller (Arduino Uno). The nominal model parameters are shown in Table 4 and it is assumed that the uncertainties in the values of resistances and capacitances are ±2%. In this article, evaluations of residuals and adaptive thresholds are performed by a program developed in MATLAB which evaluates the nominal GARRs presented in (20) to (21), and the uncertain parts presented in (23) to (24), respectively. These, respectively, provide the numerical values of residuals 𝑟3 and 𝑟4 and adaptive thresholds 𝜀3 = ±𝜆3 and 𝜀4 = ±𝜆4 . Input and output measurements, nominal values of parameters with their uncertainties values and mode information are the inputs to the MATLAB program. It was noticed from the experiments that measurement 𝑉s1 is higher than measurement 𝑉s2 at all times, so the GFSSM and MCSSM as presented in Tables 2 and 3 can be always presented as shown in Tables 5 and 6 after removing the unnecessary parts related to structured residuals. Note that the absolute value of sensitivity fault/degradation signature presented in Table 5 (GFSSM) provides the standard GFSM in terms of binary signatures without deviation sign. It may, however, be noted that the there is no need of simplifying the GFSSM presented in Table 2; it is done so in this paper with the sole purpose of simplifying the explanations of the results presented hereafter.

Symbol

Description

Nominal value

𝐾P 𝐾I 𝑉set 𝑉max 𝐶𝑖 𝑅𝑖 𝑅d 𝑖 𝑉set1 𝑉set2 𝑉G

Proportional gain of controller Integral gain of controller Set point of the PI-controller Maximum input voltage Capacitance value of 𝑖th capacitor (i = 1, 2) Resistance value of 𝑖th resistor 𝑅𝑖 (i = 1, 2) and 𝑅in Resistance value of 𝑖th resistor 𝑅d 𝑖 (i = 1, 2) Switch threshold voltage for 𝑅d1 Switch threshold voltage for 𝑅d2 Ground voltage

0.3066 1 5V 12 V 220 μF 3066 Ω 1000 Ω 5.1 V 2.8 V 0V

Table 5 GFSSM for the electrical system. Parameter

GARR3 (𝑟3 )

GARR4 (𝑟4 )

𝑀b

𝐼b (single fault)

𝑅1 ↓ 𝑅1 ↑ 𝑅2 ↓ 𝑅2 ↑ 𝑅d1 ↓ 𝑅d1 ↑ 𝑅d2 ↓ 𝑅d2 ↑ 𝐶1 ↓ 𝐶1 ↑ 𝐶2 ↓ 𝐶2 ↑

𝑎R1 −𝑎R1 0 0 𝑎d1 −𝑎d1 0 0 −sign(V̇ s1 ) sign(V̇ s1 ) 0 0

−𝑎R1 𝑎R1 +1 −1 −𝑎d1 𝑎d1 𝑎d2 −𝑎d2 0 0 −sign(V̇ s2 ) sign(V̇ s2 )

𝑎R1 𝑎R1 1 1 𝑎d1 𝑎d1 𝑎d2 𝑎d2 1 1 1 1

(1 − 𝑎d1 ) . 𝑎R1 (1 − 𝑎d1 ) . 𝑎R1 0 0 (1 − 𝑎R1 ) . 𝑎d1 (1 − 𝑎R1 ) . 𝑎d1 0 0 1 1 0 0

Table 6 MCSSM for the electrical system. Parameter

GARR3 (𝑟3 )

GARR4 (𝑟4 )

𝑀b

𝐼b (single fault)

𝑎R1 ↑ 𝑎R1 ↓

+1 −1

−1 +1

1 1

1 1

4.4. Injection of component degradation in the experimental model Generally, a component or subsystem does not degrade quickly under normal operation of the system/plant. Thus, in order to demonstrate 12

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Fig. 10. (a) An experimental testbed of electrical system (b) Enlarged view of breadboard showing different components with dotted box (1) PI-Controller, (2) 𝑅in , (3) 𝐶1 with nichrome coil for heating, (4) 𝐶2 , (5) Variable resistor 𝑅1 with servomotor and microcontroller, (6) Relay switch Sw1 (𝑎R1 ) with microcontroller, (7) 𝑅2 , (8) 𝐼d1 , (9) 𝐷1 , 𝑅d1 and 𝑉set1 , (10) 𝐷2 , 𝑅d2 and 𝑉set2 , (11) Buffers, (12) Relay switch with microcontroller to maintain desired temperature in thermal heating of 𝐶1 . (13) Thermocouple.

Fig. 11. (a) Accelerated ageing test bed for capacitor degradation (b) schematic flow chart with a microcontroller.

the prognosis approach, a mechanism or technique is required to inject the progressive degradation in the component. To carry out a prognosis study, degradations were injected in the capacitor 𝐶1 and resistor 𝑅1 of the experimental model.

C), a thermocouple was used and the temperature was maintained between 110 ◦ C to 130 ◦ C by using a relay switch with microcontroller (Arduino), which controlled the current through the nichrome wire. The temperature data was acquired by using a NI-USB-6211 data-acquisition (DAQ) card. Since the acquired voltage corresponding to temperature measurement was very small, it was further amplified by using an operational amplifier before sending to microcontroller program. The amplified voltage was compared with the prescribed lower and upper set voltages corresponding to desired temperatures, i.e. 110 ◦ C and 130 ◦ C, respectively. If the acquired signal value turned out to be less than the lower set value the microcontroller activated the relay switch and thus allowing the current through nichrome wire. Alternatively, if the acquired signal value was higher than the upper set value then the relay closes and stops the current flow through the nichrome wire, thus stopping it from further heating. After prolonged exposure to heat, measurements (see Fig. 12) show that capacitor degrades slowly for some duration after which it decreases linearly at a faster rate and finally reaches zero capacitance following a nonlinear trend. It is also observed that the value of the capacitance suddenly increases from its nominal value to some higher value (approximately 260 μF) as soon as heat is applied to the capacitor. This increase in capacitance value is observed for all three capacitors, which may be attributed to thermal expansion in the anode and cathode coils of the capacitor resulting in increase of effective oxide area. Time points marked in Fig. 12 distinguish two different modes of degradation. A slow initial degradation due to wear out, thermal expansion, etc. takes place until 𝑡CP1 , where subscript CP indicates an approximate change point. In the time duration between 𝑡CP1 to 𝑡CP2 , the dielectric gradually breaks down causing short-circuit. These two modes of degradation follow different trends and give different prognosis results, which will be discussed later in this article.

Technique for Capacitor (𝑪𝟏 ) Degradation Capacitor (𝐶1 ) was degraded by using accelerated ageing method under thermal overstress whose degradation was unknown beforehand. Accelerated ageing of a component is often performed under extreme environmental conditions to increase the rate of damage accumulation due to any physical or environmental phenomena in the component. In the accelerated ageing experiment, capacitor body is subjected to the temperature which is greater than the rated operating temperature (𝑇rated ) of the capacitor. During thermal overstress i.e. (𝑇applied > 𝑇rated ) the electrolyte present in the capacitor evaporates, thus reducing the effective oxide surface area, leading to decrease in capacitance value of the capacitor 𝐶1 (Kulkarni, 2013). The complete accelerated ageing test bed for capacitor degradation with schematic flow chart is shown in Fig. 11. In order to validate the proposed approach, the degradation trends of the three 220 μF capacitors (same specification and manufacturer) under thermal overstress were obtained and shown in Fig. 12. For real time measurement and data logging of capacitance value during the accelerated ageing test, a microcontroller based ‘‘capacitance meter’’ was designed. The principle behind its working is based on the measurement of time constant (T ) of a simple resistor-capacitor (RC) circuit supplied with square wave. When the capacitor is charged with constant voltage, a counter measures the time required for the capacitor to charge to 63.8% of the known input voltage which indeed is the time constant ‘T = RC’ of the circuit. Following this process the capacitance value was obtained by dividing the time constant T with known resistance R used in the RC circuit. In the accelerated ageing test, capacitor was heated by supplying the current through a nichrome wire wound over the body of the capacitor. In order to ensure the temperature of the capacitor body near the applied temperature (𝑇applied = 120 ◦

Technique for Resistor (𝑹𝟏 ) Degradation A series connection of a variable resistor and a fixed resistor (nominal value) was utilized to inject the progressive degradation in the equivalent resistor 𝑅1 at a desired time instant as per mentioned operating 13

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Fig. 12. Obtained trends of capacitor degradation using accelerated ageing test.

Fig. 13. Response of residuals and adaptive thresholds with the dynamically updated DHBG-LFT model.

mode. The progressive degradation in resistor 𝑅1 was injected in its on-state (𝑎R1 = 1) as shown below. { 𝑅1n (𝑍), if 𝑡 < 𝑡𝑓 2 𝑅1 (𝑡, 𝑧) = (26) {𝑅1n (𝑍) + 𝑘(𝑍) ⋅ 𝑡on }, if 𝑡 ≥ 𝑡𝑓 2

in Fig. 12. For the first few hours of operation, the capacitor degrades very slowly in the wear out degradation mode and the predicted RUL is very large. Therefore, the results only after the capacitor starts fast degradation, i.e., in the dielectric cracking mode that occurs after the first change point 𝑡CP1 , are shown. This change point is taken as the reference time frame. The change point identification is done heuristically by comparing the RUL estimate from a local curve fitting to the RUL estimate from a previous local curve fitting. If there is a large difference between the two predicted RULs then it is assumed that the degradation mode has changed. All the parameter estimates obtained before a change point are discarded and a new degradation model is generated thereafter. The response of residuals and adaptive thresholds evaluated with the experimentally measured data is shown in Fig. 13. Note that residuals (𝑟3 and 𝑟4 ) were filtered by using a low pass filter (with a time constant 0.2 s) in order to eliminate unwanted spikes which appear due to change in discrete modes and measurement noise. The first residual threshold violation was detected in residual 𝑟3 at 𝐶 𝑡d11 = 92.65 s in Fig. 13(a). Note that the nominal value of capacity 𝐶1 at that time was 267.2 μF. However, there is no residual violation at that time in residual 𝑟4 (see Fig. 13(b)). The coherence vector (for 𝐶 a short transient period just after 𝑡d11 = 92.65 s) from the Fig. 13 is observed as C = [−1 0]. According to C = [−1 0], use of GFSSM in Table 5 gives hypothesized faults/degradations as HFL = {𝐶1 ↑} (since, sign(V̇ s1 ) = −1 at that instant). Therefore, the degradation of resistor (𝑅1 ) is assumed to be within the uncertainty bound for 𝑅1 and its value is assumed to be its nominal value. After isolating the parametric fault/degradation 𝐶1 ↑, the constrained parameter estimation technique is triggered by creating a bound on 𝐶1 parameter value according to its increasing trend, i.e. (𝐶1 ∈ [267.2 μF, 300 ( μF]), ) which provides 𝐶 the first degradation estimate of 𝐶1 , i.e. 𝐶1f 𝑡u11 , 𝑍 = 271.52 μF at

where 𝑅1n (Z) is the nominal or last estimated resistance value of 𝑅1 at respective working mode (Z), k (Z) = 2.35 Ωs−1 at Z = z(1) = 𝑎R1 = 1, with each z(1) for 8 s and k (Z) = 0 Ωs−1 at Z = z(2) = 𝑎R1 = 0, with each 𝑡 z(2) for 3 s, 𝑡on = ∫𝑡 𝑎R1 ⋅ d𝑡, and 𝑡f2 is the time instant of degradation f2 injection. There is no degradation assumed during the off-state and for 𝑡 < 𝑡𝑓 2 due to experimental limitations mentioned later in this section, but the degradation can start from the beginning and it can be present in the off-state as well; the proposed prognosis approach can handle those situations. Degradation in resistor 𝑅1 (increased value) according to (26) was achieved by using an indigenously fabricated digital potentiometer coupled with servo motor, as shown in Fig. 10(b), by component number five. The digital potentiometer is put in series with a nominal fixed resistor (𝑅1n = 3066 Ω) which increases the value of equivalent resistor 𝑅1 only when the microcontroller gives a command to rotate the servomotor in the on state of the resistor (𝑎R1 = 1); otherwise, there is no change in equivalent resistor 𝑅1 in the off state of the resistor (𝑎R1 = 0). For controlled mode generation (𝑎R1 ), an electro-mechanical relay switch controlled by a microcontroller was used. The controlled relay switch set the resistor 𝑅1 on for 8 s and off for 3 s periodically. 4.5. Implementation and results The sampled data were collected from the experimental setup at a fixed rate of 0.002 s from the voltage sensors 𝑉s1 , 𝑉s2 and current sensor 𝐼in through a DAQ card (NI-USB-6211). The data collected from the experiment were fed into the DHBG-LFT form model of the system for the evaluation of residuals and adaptive thresholds. For this, LabVIEW-Matlab interface was used. Capacitor (𝐶1 ) and resistor (𝑅1 ) are considered to be the two degrading components. The resistor degrades linearly whereas the capacitor degrades non-linearly as shown

𝐶

time instant 𝑡u11 = 94.65 s. Note that upper bound of parameter is arbitrarily selected from past experience (see Fig. 12). Now, the DHBGLFT form model of the electrical system is updated at the time instant 𝐶 𝑡u11 by swapping the nominal value 𝐶1 = 267.2 μF by the first estimate 14

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Fig. 14. (a) Linear degradation model with less information of data points (b), Final degradation model with new information of degradation data points of parameter 𝐶1 .

Fig. 15. Estimated RUL of parameter 𝐶1 with (a) Linear degradation model with less information of data points, (b) Final identified degradation model with the new information of degradation data points of parameter 𝐶1 .

( ) 𝐶 of degradation,𝐶1f 𝑡u11 , 𝑍 = 271.52 μF of the real plant, which is considered as a new nominal parameter value. Further evaluation of residuals thresholds using updated DHBG-LFT form model ( and adaptive ) 𝐶 and 𝐶1f 𝑡u11 , 𝑍 = 271.52 μF enforces the updated residuals to lie inside the updated adaptive thresholds (see Fig. 13 where this updated time 𝐶 instant is indicated as 𝑡u11 ). As the capacitor (𝐶1 ) is degraded with time due to exposure of heat, the next threshold violation occurs at 𝐶 time instant 𝑡d21 = 147.268 s, and the coherence vector (for a short 𝐶 transient period just after 𝑡d21 = 147.268 s) is observed as C = [+1 0] ̇ (since, −sign(Vs1 ) = 1 at that instant). According to C = [+1 0], use of GFSSM in Table 5 yields HFL = {𝐶1 ↓} and thus parameter 𝐶1 ↓ is again isolated as a degrading parameter. Again, parameter 𝐶1 is estimated by creating a new bound on 𝐶1 according to its decreasing trend, i.e. 𝐶1 ∈ [0 μF, 271.52 (μF], which provides the second degradation estimate ) 𝐶 𝐶 of 𝐶1 , i.e., 𝐶1f 𝑡u21 , 𝑍 = 269.9 μF at time instant 𝑡u21 = 149.268 s. Likewise, more estimated data points of degrading parameter 𝐶1 (t ) at different time instances are obtained during continuous monitoring of the system and are plotted in Fig. 14 where Fig. 14(a) shows a small range of data and Fig. 14(b) shows a longer range. The step jumps in Fig. 14(a) indicate the instances when parameter is estimated due to residual threshold violation. It is evident that the duration between two estimations is not fixed. This happens due to non-linear nature of the degradation, and the parameter and measurement uncertainties in the system. The linear degradation model is fitted with the less information of data points of parameter (𝐶1 ) estimates (see Fig. 14(a)), while the different models like exponential model and second order polynomial models, including linear model, are tried when sufficient information of data points of parameter (𝐶1 ) estimates are obtained. The second order polynomial model (see Fig. 14(b)) 𝐶

𝐶

𝑀𝐶𝑧 = 𝑃1 (𝑡a 1 )2 + 𝑃2 (𝑡a 1 ) + 𝑃3 1

for curve fitting the deviation of 𝐶1 as well as for RUL calculation, time 𝐶 instant 𝑡u11 is taken as a zero time reference. The capacitor failure threshold is assumed to be 𝐶1𝑓 𝑙 = 10 μF. Considering the obtained linear degradation model with less information of data points and finally obtained best fit degradation model, the estimated RULs for capacitor (𝐶1 ) are predicted as 18 748 s and 18 483 s (see Fig. 15). The degradation of resistor 𝑅1 is controlled by the servomotor driven potentiometer. It is a very well regulated degradation and follows a linear trend. The intention is to find whether the prognosis module is able to identify this nature of degradation using the proposed prognosis technique. The servomotor is geared and turns the potentiometer at a very slow rate. Still, the maximum allowable turns in the potentiometer was completed within 35 min, i.e., 2100 s. Due to this limitation, the servomotor was manually switched on at about 16 600 s, i.e. much after the capacitor 𝐶1 has started degradation due to dielectric cracking. Thus, there is a delay between the two degradation start points, that of 𝐶1 starts from the very beginning and that of 𝑅1 starts after a random delay. Although such a delay was intentionally introduced here due to experimental limitations, it can be justified from practical viewpoints. For example, a certain device may have a long life under normal operating conditions, but may start degrading due to some unforeseen external event. Such external events can be exposure to moisture (rain and flooding), high/low temperature, seismic activity, etc. which cannot be predicted. Note that the ability to handle degradations starting at different times no way compromises the prognosis process proposed here; it can still perform prognosis when these degradations start concurrently. The nominal value of resistor 𝑅1 is 3066 Ω. Unlike capacitor 𝐶1 which degrades continuously, it is assumed that the degradation of 𝑅1 depends on the operating mode as described in (26). There is current through that resistor in the on-state (𝑎R1 = 1) of resistor 𝑅1 and there is no current through it in its off-state (𝑎R1 = 0). One may assume two separate degradation rates for these two hybrid states. However, due to experimental limitations, the servomotor connected to the potentiometer is operated during the on-state (𝑎R1 = 1) only and

(27)

where 𝑃1 = 5.65 × 10−13 Fs−2 , 𝑃2 = −2.47 × 10−8 Fs−1 , 𝑃3 = 273.96 × 𝐶 10−6 F and 𝑡a 1 represents ageing time of 𝐶1 , is found to be the best fit model according to its goodness of fit (RMSE = 2.25 × 10−2 ) . Note that 15

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Fig. 16. (a) Linear degradation model at mode 𝑎R1 = 1 with less information of data points, (b) Final identified degradation model at mode 𝑎R1 = 1 with the new information of degradation data points of parameter 𝑅1 .

Fig. 17. Estimated RUL of parameter 𝑅1 with (a) Linear degradation model with less information of data points, (b) Final identified linear degradation model with new information of degradation data points at on state of resistor 𝑅1 and no degradation at off state.

switched off otherwise. Thus, there is a linear degradation of resistor during its on-state and no degradation during the off-state. Because of this deviation of the resistance, the first threshold vi𝑅 olation occurs at a time instant 𝑡d11 = 16811.41 s which leads to a coherence vector C = [−1, +1] (see Fig. 13). According to C = [−1 +1], use of GFSSM and MCSSM given in Tables 5 and 6, respectively, gives HFL = {𝑎R1 ↓, 𝑅1 ↑} if mode 𝑎d1 = 0, and HFL = {𝑎R1 ↓, 𝑅1 ↑, 𝑅d1 ↑} if mode 𝑎d1 = 1. Since parameter 𝐶1 is also degrading simultaneously, it should have been present in the HFL under multiple fault/degradation hypothesis (Samantaray & Ghoshal, 2007). However, the residual threshold violations due to parameter 𝐶1 degradation is asynchronous and uncorrelated to degradation of 𝑅1 and hence it is expected that an asynchronous residual threshold violation event giving C = [−1, 0] or C = [+1, 0] will eventually occur at another time and that can be used for 𝐶1 parameter estimation. Thus, the two types of residual violation events are decoupled and handled independently even when both the parameters 𝐶1 and 𝑅1 are simultaneously degrading. First, the consistency in mode 𝑎R1 is checked by evaluating all the sensitive ARRs at the suspected mode (𝑎R1 ↓) and it is observed that 𝑎R1 = 1 is the consistent mode and there is no mode fault (see Prakash & Samantaray 2017 for further details). If there happened to be a mode fault then the plant would have to be immediately repaired and prognosis of other components cannot proceed. Thus, excluding the mode fault possibility, there is either 𝑅1 ↑ or 𝑅d1 ↑ or both are the actual degraded parameters if mode 𝑎d1 = 1, otherwise 𝑅1 ↑ is the sole degraded parameter. Upon constrained parameter estimation and updating the residuals and adaptive thresholds, it is confirmed that the actual degrading component is resistor 𝑅1 ↑ and its value is found to be 𝑅 3542 Ω at 𝑡u11 = 16813.41 s. This estimated value was treated as the new nominal value and the residuals and their threshold are updated. The monitoring of the electrical system is continued for further residual threshold violations and newer estimates are obtained sequentially. The estimated resistance value of degraded resistor 𝑅1 at working mode Z = 𝑧(1) = 𝑎R1 = 1 in different time instances are plotted in Fig. 16. A linear degradation model is fitted when there as less number of parameter estimates at different times (see Fig. 16(a)) whereas different models like exponential model and higher order polynomial

models are tried when sufficient number of estimates over a long observation window are available. However, a linear degradation model (see Fig. 16(b)) (1)

𝑀𝑅𝑧

1

𝑅

= P 1 𝑡a 1 + P 2

(28) 𝑅

𝑅

𝑅

where 𝑃1 = 2.395 Ωs−1 , 𝑃2 = 3527 Ω and 𝑡a 1 = 𝑡on1 −𝑡u11 , with goodness of fit (RMSE = 67.59) was found to be best fit. Indeed, this was the actual degradation of resistor 𝑅1 injected in the experiment. The failure threshold 𝑅𝑓1 𝑙 = 6132 Ω, i.e., twice its un-degraded initial value, is considered. The initial linear degradation model with less number of data defining the parameter degradation trend predicts RUL to be 1490.96 s (see Fig. 17(a)) whereas RUL with the more data points defining the degradation trend turns out to be 1492.68 s (see Fig. 17(b)) based on considered failure threshold and known future modes of resistor 𝑅1 . Note that for RUL prediction of resistor 𝑅1 , no degradation is present in the off state of resistor 𝑅1 . In Fig. 17, time 𝑅 instant 𝑡u11 = 16813.41 s is taken as a zero time reference. 5. Conclusions This article proposes bond graph model-based prognosis of hybrid systems by synergistically combining the concepts from mode-based fault diagnosis of hybrid systems, adaptive residual thresholding for robust diagnosis, sensitivity-based constrained parameter estimation, and model-based prognosis. The proposed approach offers prognosis of simultaneous multiple parameter degradations and degradations whose trend is affected by the operating mode of a hybrid system. The key aspects of the proposed approach are (1) the use of residuals to identify the minimal set of parameters that need to be estimated at an instant, (2) identification of the instance when parameter estimation needs to be performed and the data window size and its features required for the parameter estimation, (3) the use of minimization of residuals as a parameter estimation technique with adaptive constraints on the parameter values and information of the sensitivity of residuals to parametric deviations, and (4) continuous model updating with estimated parameters in order to keep the residuals within respective adaptive residual thresholds and thereby generate the trend of parameter deviations 16

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under different operating modes. The parameter degradation trends are then used to predict the RULs of different components. The developed approach is experimentally validated with experimental application to a hybrid electrical/electronic system. It is found that the time constant (response time) of the system, active mode time of the components and degradation model identification time are the different crucial factors that govern the effectiveness of the proposed approach in real-time implementation. For systems or processes with larger time constant, the time consumed in parameter estimation is not an important issue. But, for real time implementation in processes with smaller time constant and fast transition in operating mode (like the considered electrical hybrid system in this article), the parameter estimation should converge quickly for identification of the actual degrading parameter and estimation of its degradation profile. This is accomplished by using the information of degradation directions obtained from dynamic sensitivity signature matrix to minimize the parameters appearing in an initial hypothesized degrading parameters list. The operating mode information, possible degrading parameters list and corresponding parametric degradation directions are used in SBG based constrained parameter estimation technique with dynamically updated parameter bounds. It is found that SBG based constrained parameter estimation technique with dynamically updated parameter bounds provides faster convergence of estimation process. For a hybrid system, utilization of multiple degradation models including operating modes as additional control parameter and evolution through identification of degradation model is considered for RUL prediction. Models are continually evolved with time by adapting to the new information of degradation state of the supervised system to predict the accurate RUL with bounded uncertainty value. This overcomes the drawbacks of various previously existing approaches. Furthermore, in the case of a sudden change in the trend of parameter deviation and estimated RUL, it is proposed to remove all past trend data from the model and reconstruct a new degradation model. This approach is found to be successful in predicting RUL of a heated capacitor element with non-linear degradation trend where distinct modes/mechanism of degradation cause sudden changes in degradation trend after an unknown duration of use. It is worthy to mention here that though a simple electrical hybrid system application is addressed here, the method can be applied with equal ease and effectiveness to continuous and hybrid process engineering systems (Prakash & Samantaray, 2017). It may be noted that if there are too many sensors in the model then it appears that direct parameter estimation using model matching, i.e., without using residuals, would be a more efficient method for prognosis. However, the residuals would be structured in that case leading to decoupled estimation of individual parameters and the residual direction information would be useful for setting constraints on the parameter values during the optimization process. One disadvantage of the proposed method is that with too few sensors, decoupling of the degradation effects cannot be achieved and it would not be possible to reduce the size of initial hypothesized degrading parameters list. Therefore, there is no obvious advantage of the proposed method when the system is not well-instrumented. In this article, the information of residual threshold violation has been considered in qualitative sense, i.e., +1 or −1 for upper and lower threshold deviations, respectively. It appears that if the rate of residual deviation (magnitude) and its direction (slope) can be considered in the process for generation of degradation hypothesis and parameter estimation then more accurate and reliable prognosis would be possible. This can be a subject of future research.

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