Integrated Diagnostics and Prognostics for the Electrical Power System of a Planetary Rover

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1 Integrated Diagnostics and Prognostics or the Electrical Power System o a Planetary Rover Matthew Daigle, Indranil Roychoudhury, and Anial regon 3 NASA Ames Research Center, Moett Field, Caliornia, 94035, USA matthewjdaigle@nasagov SGT Inc, NASA Ames Research Center, Moett Field, Caliornia, 94035, USA indranilroychoudhury@nasagov 3 Department o Computer Science, University o Valladolid, Valladolid, Spain anial@inoruvaes ASTRACT For electric vehicles, technology or monitoring, diagnosis, and prognosis o the electrical power system (EPS) ecomes essential or sae and eicient operation To this end, we develop a general system-level integrated diagnosis and prognosis ramework, which detects, isolates, and identiies EPS aults, and predicts when the EPS will ail to deliver suicient power The approach takes advantage o recent work in structural model decomposition in order to distriute the gloal diagnosis and prognosis prolems into local suprolems that can e solved in parallel, thus enaling implementation on distriuted computational platorms The ramework is applied to the EPS o a planetary rover tested, and is demonstrated using data rom ield experiments INTRODUCTION For electric vehicles, technology or monitoring, diagnosis, and prognosis o the electrical power system (EPS) is critical In order to ensure saety, algorithms are needed that are ale to predict the end-o-discharge (EOD) o the atteries powering the vehicle The EOD time depends oth on the current state o the atteries, including state-o-charge (SOC), and the uture power requirements o the atteries The uture power requirements or the atteries depend oth on the power required or uture vehicle maneuvers and on any ault present in the system, which may cause increases in power demands Thereore, oth diagnosis (determining the current system state and aults) and prognosis (predicting the EOD o the system) are required Matthew Daigle et al This is an open-access article distriuted under the terms o the Creative Commons Attriution 30 United States License, which permits unrestricted use, distriution, and reproduction in any medium, provided the original author and source are credited A large ody o research exists or oth model-ased diagnosis (Gertler, 998; lanke et al, 006) and prognosis methods (Luo et al, 008; Saha & Goeel, 009; Orchard & Vachtsevanos, 009), however, most o the approaches in the literature ocus in either solely the diagnosis or the prognosis task A ew works have proposed the integration o oth tasks within a common ramework (Patrick et al, 007; Orchard & Vachtsevanos, 009; Roychoudhury & Daigle, 0; Zai et al, 03), however, unlike our approach, these approaches perorm the diagnosis and prognosis tasks in a centralized way, thus suering rom scalaility issues due to the large numer o states and parameters in real-world systems Moreover, most solutions do not approach the system-level prolem To the est o our knowledge, there is no approach in the literature which comines, in a distriuted ashion, the system-level diagnosis and prognosis tasks In previous work, we have developed an integrated modelased diagnosis and prognosis ramework (Roychoudhury & Daigle, 0) The main contriution o this work was a uniied modeling ramework In an extension o this work, we used structural model decomposition to develop a distriuted integrated diagnosis and prognosis ramework (regon, Daigle, & Roychoudhury, 0), ased on other work in distriuted diagnosis (regon et al, 04) and distriuted prognosis (Daigle, regon, & Roychoudhury, 0, 04) Through structural model decomposition, a gloal model is transormed into a set o local sumodels For model-ased diagnosis and prognosis, this results in the gloal diagnosis and prognosis prolems eing transormed into local diagnosis and prognosis suprolems These suprolems can e solved independently y assigning them to dierent processing units, thus enaling a scalale and computationally eicient distriuted diagnosis and prognosis solution

2 ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 04 In this paper, we apply these rameworks and ideas to the EPS o a planetary rover tested at NASA Ames Research Center (alaan et al, 03) The applied architecture constitutes a new ramework or integrated system-level diagnosis and prognosis For the rover, we are interested in a system-level prediction, that is, when the EPS can no longer supply suicient power to the loads The rover is powered y several atteries, and this condition is a unction o the state o all the atteries Hence, component-level prognostics algorithms cannot e used, and a system-level prognosis ramework is required (Daigle, regon, & Roychoudhury, 0) We utilize recent work in structural model decomposition (Roychoudhury, Daigle, regon, & Pulido, 03) to achieve a distriuted implementation o the ramework We demonstrate the complete approach using real experimental data rom the rover operating in the ield The paper is organized as ollows Section ormulates the system-level diagnosis and prognostics prolems Section 3 descries the ackground on structural model decomposition, distriuted diagnosis, and distriuted diagnosis Section 4 presents the rover EPS case study Sections 5 and 6 present the system-level diagnosis and prognostics solutions, respectively, or the rover EPS Section 7 presents the results or dierent scenarios Finally, Section 8 concludes the paper PROLEM FORMULATION In this section, we ormulate the integrated system-level diagnosis and prognosis prolem Ultimately, the goal is to predict when some event occurs in the system, such as the rover running out o power In order to make such a prediction, we need to know the state o the system, including any aults that are present, thereore, diagnosis is needed in order to perorm prognosis We irst ormulate the system-level diagnosis prolem, ollowed y the system-level prognosis prolem System-Level Diagnosis The prolem o system-level diagnosis consists o three parts: (i) detecting whether a ault is present, (ii) isolating the correct ault, and (iii) identiying the aulty system state In each o these parts, dierent models may e used We assume that a model M can e succinctly represented in the ollowing general ormulation: x(k + ) = (k, x(k), θ(k), u(k), v(k)), () y(k) = h(k, x(k), θ(k), u(k), n(k)), () where k is the discrete time variale, x(k) R nx is the state vector, θ(k) R n θ is the unknown parameter vector, u(k) R nu is the input vector, v(k) R nv is the process noise vector, is the state equation, y(k) R ny is the output vector, n(k) R nn is the measurement noise vector, and h is the output equation We will descrie in Section 3 an equivalent structural representation o a model M that will e used or structural model decomposition In the model-ased paradigm, we assume that in the nominal (ault-ree) case, the system ehaves according to some model M n, and, given the inputs u(k), produces measured outputs y(k) The prolem o ault detection is to determine when model-predicted (nominal) outputs ŷ n (k) are dierent rom the measured outputs y(k) in a statistically signiicant manner The dierence y(k) ŷ n (k) is called a residual; a (statistically signiicant) nonzero residual indicates a ault Faults are generally represented as changes in the model (ie, in parameter values and/or model structure) So, in general, each ault F, where F is the complete set o potential aults, is represented as a new model, M Given that a ault is present, the prolem o ault isolation is to determine which model M now represents the system The prolem o ault identiication is to determine the ault parameter estimate or the isolated ault, p(θ (k) y(k 0 :k)), where y(k 0 :k) denotes all measurements oserved rom the initial time k 0 to the current time k System-Level Prognosis Rather than eing ocused on individual components, systemlevel prognostics is ocused on the system as a whole, and on predictions or the system As such, it is a more general ormulation o the prognostics prolem System-level prognostics was previously deined in (Daigle, regon, & Roychoudhury, 0) Here, we generalize the prolem ormulation ased on (Daigle & Kulkarni, 04) and explicitly integrate it with the diagnosis prolem Speciically, predictions must e made or a given ault hypothesis, which consists o a ault model M and joint state-parameter estimate p(x (k), θ (k) y(k 0 :k)) Fault identiication computes an estimate o θ (k), and the initial step o prognostics is to compute the ull joint-state parameter estimate or the new aulty model System-level prognostics is concerned with predicting the occurrence o some system-level event E that is deined with respect to the states, parameters, and inputs o the system We deine the event as the earliest instant that some event threshold unction T E : R nx R n θ R n u, where {0, } changes rom the value 0 to That is, the time o the event k E at some time o prediction k P given some ault is deined as k E (k P ) in{k N: k k P T E (x (k), θ (k), u(k)) = } (3) old typeace denotes vectors, and n a denotes the length o a vector a

3 ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 04 The time remaining until that event, k E, is deined as k E (k P ) k E (k P ) k P (4) The prognostics prolem is inherently uncertain, due to the random nature o the system evolution (represented with v(k)), and unknown uture inputs (u(k) or k > k P ) Thereore, k E and k E are random variales, and we must compute the proaility distriution p(k E (k P ) y(k 0 :k P )) (Daigle, Saxena, & Goeel, 0; Sankararaman, Daigle, Saxena, & Goeel, 03; Sankararaman, Daigle, & Goeel, 04) 3 ACKGROUND For a large system, oth the diagnosis and prognosis prolems are correspondingly large A centralized approach does not scale well, can e computationally expensive, and prone to single points o ailure Thereore, we propose to decompose the gloal diagnosis and prognosis prolems into independent local suprolems In this work, we uild on the ideas rom structural model decomposition (lanke et al, 006; Pulido & Alonso-González, 004) to compute local independent suprolems, which may e solved in parallel, thus providing scalaility and eiciency We adopt here the structural model decomposition ramework descried in (Roychoudhury et al, 03) This approach allows us to make guarantees o the minimality o the derived sumodels and allows to generate dierent sumodels or each one o the diagnosis and prognosis tasks In the ollowing, we review the main details and reer the reader to (Roychoudhury et al, 03) or additional explanation We deine a model as ollows: Deinition (Model) A model M is a tuple M = (V, C), where V is a set o variales, and C is a set o constraints among variales in V V consists o ive disjoint sets, namely, the set o state variales, X; the set o parameters, Θ; the set o inputs, U; the set o outputs, Y ; and the set o auxiliary variales, A Each constraint c = (ε c, V c ), such that c C, consists o an equation ε c involving variales V c V Input variales, U, are known, and the set o output variales, Y, correspond to the (measured) sensor signals Parameters, Θ, include explicit model parameters that are used in the model constraints Auxiliary variales, A, are additional variales that are algeraically related to the state and parameter variales, and are used to reduce the structural complexity o the equations The notion o a causal assignment is used to speciy the computational causality or a constraint c, y deining which v V c is the dependent variale in equation ε c Deinition (Causal Assignment) A causal assignment α to a constraint c = (ε c, V c ) is a tuple α = (c, vc out ), where vc out V c is assigned as the dependent variale in ε c We write a causal assignment o a constraint using its equation in a causal orm, with := to explicitly denote the causal (ie, computational) direction Deinition 3 (Valid Causal Assignments) We say that a set o causal assignments A, or a model M is valid i For all v U Θ, A does not contain any α such that α = (c, v) For all v Y, A does not contain any α = (c, vc out ) where v V c {vc out } For all v V U Θ, A contains exactly one α = (c, v) The deinition o valid causal assignments states that (i) input or parameter variales cannot e the dependent variales in the causal assignment, (ii) a measured variale cannot e used as an independent variale in any constraint, and (iii) every variale, which is not input or parameter, is computed y only one (causal) constraint ased on this, a causal model is a model extended with a valid set o causal assignments Deinition 4 (Causal Model) Given a model M = (V, C), a causal model or M is a tuple M = (V, C, A), where A is a set o valid causal assignments 3 Structural Model Decomposition To decompose a model into sumodels, we need to reak internal variale dependencies We do this y selecting certain variales as local inputs Given the set o potential local inputs (in general, selected rom V ), and the set o variales to e computed y the sumodel (selected rom V U Θ), we create rom a causal model M a causal sumodel M i, in which a suset o the variales in V are computed using a suset o the constraints in C In this way, each sumodel computes independently rom all other sumodels A causal sumodel can e deined as ollows Deinition 5 (Causal Sumodel) A causal sumodel M i o a causal model M = (V, C, A) is a tuple M i = (V i, C i, A i ), where V i V, C i C, and A i A When using measurements (rom Y ) as local inputs, the causality o these constraints must e reversed, and so, in general, A i is not a suset o A The procedure or generating a sumodel rom a causal model is given as Algorithm (GenerateSumodel) in (Roychoudhury et al, 03) Given a causal model M, a set o variales that are considered as local inputs, U, and a set o variales to e computed, V, the GenerateSumodel algorithm derives a causal sumodel M i that computes V using U The algorithm works y starting at the variales in V, and propagating ackwards through the causal dependencies Propagation along a dependency chain stops once a variale in U is reached, or once a constraint is reached in which the causality can e reversed so that a variale in U can ecome a local input We reer 3

4 ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 04 System-level ault detection System-level diagnosis System-level ault identiication u(k) System y(k) u (k) y (k) u (k) y (k) u m (k) y m (k) M n M n M m n ˆx n(k), ˆ n(k), ŷn(k) ˆx n(k), ˆ n(k), ŷn(k) ˆx m n (k), ˆ m n (k), ŷn m (k) Centralized ault isolation F(k) (k) (k) j(k) M M M j p( (k) y(k 0:k)) p( (k) y(k 0:k)) p( j (k) y(k 0:k)) Figure System-level diagnosis architecture the reader to (Roychoudhury et al, 03) or the algorithm and additional details 3 Structural Model Decomposition or System-Level Diagnosis In this work, we use model decomposition to simpliy the ault detection and ault identiication prolems (regon, iswas, & Pulido, 0; regon, Daigle, & Roychoudhury, 0) For ault detection, we compute a set o residuals ased on the sensors, and so derive a set o minimal local sumodels to compute the nominal values o these sensors, ie, one sumodel or each y Y In the sumodel computing the output y, we use the other sensors Y {y} as local inputs, thus allowing decomposition So, given the nominal model M n, or each output y Y, we create a sumodel with V = {y} and U = {U (Y {y})} Fault identiication requires estimating a set o parameters associated with aults Here, we also add Y as local inputs Given a ault model M, we create a sumodel with V = θ, where θ denotes the set o ault parameters, and U = U Y 3 Structural Model Decomposition or System-Level Prognosis Prediction requires determining k E or a given ault hypothesis, which is computed ased on T E, which, in turn, is a unction o the system states, parameters, and inputs Oten, the system-level, gloal threshold T E can e expressed as the logical or o other local thresholds, ie, T E = T E T E T E n or n conditions With each local threshold T E i we can associate a local event E i and compute times k E i, such that k E can now also e deined as min(k E, k E,, k E n ) This leads to a natural decomposition where each k E i is computed independently, and allows us to decompose the prediction prolem So, to create the prediction sumodels, we use the GenerateSumodel algorithm in (Roychoudhury et al, 03) with U set to {U P } and V set to {k E i } or each local threshold T E i, where U P V is the set o variales that can e predicted a priori The decomposition that can e achieved depends also on the selected U P I no variales exist that can e predicted a priori outside o U, then the GenerateSumodel algorithm may not result in any decomposition and it will suice to simply use the gloal model The initial state needed or prediction can e generated rom a set o local estimators The gloal prediction model is decomposed into local state estimators or the needed states, in the same way as in estimation or diagnosis 3 Integrated System-level Diagnosis and Prognostics Architecture Figs and illustrate the architecture or our system-level diagnosis and prognosis rameworks, respectively Regarding system-level diagnosis (Fig ), at each discrete time step, k, the system takes as input u(k) and produces outputs y(k) These are split into local inputs u i (k) and local outputs y i (k) or each one o the m system-level ault detection sumodels, M i n Within each sumodel M i n, nominal tracking is perormed, computing estimates o nominal states, ˆx i n(k), parameters, ˆθi n(k), and the measurements, ŷn(k) i The ault isolator perorms detection irst y comparing the estimated measurement values against the oserved values, to determine statistically signiicant deviations or the residual, r i (k) = y i (k) ŷ i (k) Deviations in the residuals are then transormed to qualitative symols used y the centralized ault isolation lock to generate a set o isolated ault candidates, F(k) For each one o the isolated ault candidates, i (k), local models or ault identiication, M θi, are used to compute local parameter estimates p(θ i (k) y(k 0 :k)) These 4

5 ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 04 p( (k) y(k 0:k)) p( (k) y(k 0:k)) p( j (k) y(k 0:k)) System-level estimation M est p(x (k), (k) y(k 0:k)) System-level prediction M est M pred M j System-level prognosis est p(x (k), (k) y(k 0:k)) p(x j (k), j (k) y(k 0:k)) M M j p(k E (k P ) y(k 0:k P )) p(k E (k P ) y(k 0:k P )) pred p(k E j (k P ) y(k 0:k P )) pred p(k E(k P ) y(k 0:k P )) Figure System-level prognosis architecture System-level prognosis detail or M est M pred p( e (k) ye (k 0:k)) M e p(x e (k), e (k) ye (k 0:k)) M p p p(k E (kp ) yp (k 0:k P )) p( (k) y(k 0:k)) p( e (k) ye (k 0:k)) p( ke (k) yke (k 0:k)) M e M ke p(x e p(x ke (k), e (k) ye (k 0:k)) (k), ke (k) yke (k 0:k)) p(x (k), (k) y(k 0:k)) M p M lp p p(k E (kp ) yp (k 0:k P )) p(k E (k P ) y(k 0:k P )) lp p(k E (kp ) ylp (k 0:k P )) Figure 3 Detail o the system-level prognosis architecture local parameter estimates are then used as input to systemlevel prognosis (Fig ) The system-level prognosis lock o the architecture is divided into two phases: system-level estimation and system-level prediction Parameter estimates rom the local ault identiication locks, together with the inputs and outputs o the system, are used as input or the local estimation locks, M i est, to compute state-parameter estimates p(x i (k), θ i (k) y(k 0 :k)) Finally, the local state-parameter estimates are used as input to the systemlevel prediction locks, M i pred, to compute predictions, p(k E i (k P ) y(k 0 :k P )), at given prediction time k P Predictions or each ault hypothesis are comined into the gloal prediction p(k E (k P ) y(k 0 :k P )) Fig 3 shows the detail o the system-level estimation and prediction locks or ault, namely M est and M pred The system-level estimation task is decomposed using local estimation sumodels, M e to M ke As shown in the igure, susets o the the local parameter estimates p(θ (k) y(k 0 :k)), the system inputs, u(k), and the system outputs, y(k), are used as input or each one o the local stateparameter estimation sumodels (this, o course, is similar to the estimation prolem using the nominal model in the diagnosis part) The output o all the local sumodels is then comined to compute the local state-parameter estimate or ault, p(x (k), θ (k) y(k 0 :k)) The system-level prediction prolem is also decomposed using local prediction sumodels The state estimate or the ault is split into local estimates or the prediction sumodels, which then each compute a local ke i value; these are then merged into the system-level prediction k E or the ault 4 ROVER EPS MODELING We are interested in integrated diagnosis and prognosis o the EPS o the rover Thus, our system under consideration consists o the atteries, the attery current sensor, and the voltage sensors The rover motors, which produce the electrical loads experienced y the EPS, are considered outside o our system under consideration, and so the loads the motors demand are viewed as inputs to the EPS 5

6 ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 04 i L R p i p V i p R p V i L V i L V : V 4 V V V 4 V V V k E Figure 4 Rover EPS schematic The circuit schematic or the rover EPS is shown as Fig 4 There are 4 lithium-ion cells in total, with two parallel ranches o cells in series In parallel is a parasitic load, modeled as a resistance, R p, that may appear as a ault The attery current,, is split into the current going to the load, i L, and the current going to the parasitic load (i present), i p The total voltage provided y the EPS to the load is denoted as V The cell model computes the voltage as a unction o time given the current drawn rom the cell, and is descried in detail in (Daigle & Kulkarni, 03) For completeness, the model is summarized in the appendix, and we reer the reader to (Daigle & Kulkarni, 03) or additional explanation We assume that all cells start ully charged, so the voltage over each parallel ranch is the same, and the current is split evenly ( /) As the cells discharge, the total voltages must stay alanced, since the two sets o cells are in parallel, and thereore the current into each ranch remains / The causal graph corresponding to the EPS model is shown in Fig 5 The oxes in the igure indicate the attery cell models (or revity, the internal variales are not shown) Also indicated are the sensor models A measured value y (the superscript indicates the measured value o a physical variale y) is equal to the physical variale y plus a ias, indicated with the superscript The iases, when present, produce a constant oset to the true value Here, it also makes clear that we use the measured value o the load current, i L, as an input to the system, which we assume is aultless The causal graph also indicates the computation o the time k E (in the ollowing, and in the igures, we drop the suscript, as these sumodels are not speciic to a given ault) For the rover, E corresponds to any o the atteries reaching end-o-discharge (EOD), which is what must e predicted EOD is deined y a voltage threshold V EOD, where T E is deined y V < V EOD or V < V EOD,, V 4 < V EOD When any cell voltage is less than V EOD, EOD is reached or that attery and T E evaluates to The rover cannot e used eyond that point, as it will damage any atteries whose voltage is elow the cuto voltage Figure 5 Causal graph or rover EPS model R p i p V i L V i L V : V 4 V V V 4 Figure 6 Causal graph or gloal nominal model 5 ROVER EPS DIAGNOSIS V V V 4 As descried in Section, or diagnosis, models are used or the three phases o the diagnosis process: (i) ault detection, consisting o state estimation and residual generation, (ii) ault isolation, and (iii) ault identiication We descrie the models used or each in the ollowing susections 5 Fault Detection Recall that in order to detect aults, we produce residuals, or which we need to compute model-predicted values o the outputs We denote a residual using r y, where y is the variale name or the sensor output The causal graph or the gloal model or residual generation is shown as Fig 6 It is generated y calling GenerateSumodel with U = {i L }, and V = {V, V,, V4, i } For residual generation, only the nominal model is needed, ecause the aim is only to detect when the nominal model is no longer valid, due to the appearance o a ault In Fig 6, the nominal parts o the model are colored lack, and the ault-related parts in red Since the aults are ree rom the nominal version o the model, only the lack portion is needed or residual generation We retain the red parts in the igures to indicate that the measured values will e causally eected y the aults As descried in Section 3, we can decompose the residual generation prolem, y creating local models or each sen- 6

7 ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 04 R p i L i L i p V V : V 4 Figure 7 Causal graph or local i residual generator V i V i Figure 8 Causal graph or local V i V V i residual generator sor to compute predicted values The causal graph or the local model or i is shown in Fig 7, and is generated y calling GenerateSumodel on the gloal model with U = {i L, V, V,, V4} and V = {i } The predicted value o i, in the nominal case, is simply equal to the measured load current, i L For each residual generator that has states we use the unscented Kalman ilter (UKF) or estimation (Julier & Uhlmann, 004) The causal graph or the local model or Vi (i [, 4]) is shown in Fig 8, and is generated y calling GenerateSumodel on the gloal model with U = {i L, i, V, V,, V4} {Vi } and V = {Vi } The voltage or each cell is computed independently, using i as an input (this is divided y to e used as input to the cell model) 5 Fault Isolation Fault isolation is perormed y analysis o the residual signals Due to the decomposition used in the residual generation step, each ault maniests in only a suset o the complete residual set As is clear in Fig 7, r i will deviate (in a statistically signiicant way rom zero) due only to the R p ault and the i ault As is clear in Fig 8, r V i will deviate due to V i and i Note that the relation i = +i holds, so when i is used as a local input, the causal relation is modiied so that ecomes the dependent variale, and the causal constraint is := i i That is, the true value o is equal to the measured value minus the ias For residual generation, the ias is not included, so y using the measured value, i as a local input, when a ias is present the wrong (ie, iased) current will e ed to the cell model and used to compute V i, thus causing a deviation in the corresponding residual The eects o the aults on the residuals are shown in Tale Faults are indicated oth y the model parameter and the direction o its change, eg, Rp denotes a decrease in the parasitic resistance Fault eects on residuals are represented as qualitative ault signatures (Mosterman & iswas, 999) and relative residual orderings (Daigle, Koutsoukos, & iswas, 007) Fault signatures express the qualitative change in a signal as the result o a ault In general, they can e used to represent changes in magnitude, slope, and higher-order derivatives o a signal, ut here, we represent changes in magnitude only, as this is suicient to otain unique diagnoses For example, the parasitic load ault causes an increase in r i An ordering etween a residual r and r or ault, denoted as r r, indicates that the ault will cause an oservale deviation in r eore r For example, a ias in the V sensor will produce a deviation in r V eore every other residual (since the ault aets no other residuals) oth signatures and orderings can e derived rom the model automatically (Daigle, 008) oth signatures and orderings are reasoned over in an eventased ramework to perorm ault isolation (Daigle, Koutsoukos, & iswas, 009) When a residual deviation is irst detected, the ault isolation algorithm checks or the aults that could have produced that deviation As more residuals deviate, the algorithm checks or consistency with the current sequence o deviations, retaining only aults that can produce the oserved sequence according to the predicted signatures and orderings In addition, we can also eliminate candidates as inconsistent when no deviation is oserved in a residual y using timeouts (this is equivalent to oserving a 0 signature) (Daigle, Roychoudhury, & regon, 03) For each residual we set a time limit under which we expect a residual deviation to occur ater a ault I we detect a ault and that residual has not deviated y that time, we oserve a 0 signature and reason with that inormation Including this inormation, we can distinguish qualitatively etween all aults, and thereore otain unique diagnoses ased on the qualitative signatures and orderings alone 3 53 Fault Identiication The ault identiication sumodels are generated rom the gloal model shown in Fig 5, with the aulty parts included In the call to GenerateSumodel, U is set to the set o measured variales, and V is set to the ault parameter that is to e estimated In the nominal model, when the parasitic load is asent, this is equivalent to an ininite resistance in parallel Thus, the appearance o the parasitic load is denoted as a decrease in the parasitic resistance 3 Without using the 0 signatures or isolation, i a a voltage sensor ias occurred, we would have to wait ininitely long to ensure there were no urther deviations and rule out i as a possiility 7

8 ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 04 Tale Fault Signatures and Residual Orderings Fault r V r V r V 4 r i Residual Orderings Rp r i, r i,, r i 4 V r V,, r V 4, r V r i V r V,, r V 4, r V r i V r V,, r V 4, r V r i V r V,, r V 4, r V r i V r V 4,, r V 4, r V 4 r i V r V 4,, r V 4, r V 4 r i i i i L R p i p V i L i L V V i L V : V 4 V : V 4 Figure Causal graph or local i estimation Figure 9 Causal graph or local R p estimation R p i p V V i V i Figure 0 Causal graph or local V i V i estimation For the parasitic load ault, the causal graph or the local estimation model is shown in Fig 9 The parasitic resistance R p is computed using i P and V, where i p is computed ased on the dierence etween the measured load and attery currents, and V is computed ased on the measured voltages The causal graph or the local model or the voltage sensor ias estimation is shown in Fig 0 The voltage ias is computed ased on the measured voltage and the model-predicted voltage, computed using the measured attery current The causal graph or the local model or the current sensor ias estimation is shown in Fig i is computed as the dierence etween the measured attery and load currents 6 ROVER EPS PROGNOSIS As descried in Section, prognosis requires a prediction model, an initial state estimate, and uture trajectories o the inputs, U kp and the process noise, V kp The prediction model must e ale to compute the event threshold T E, given the local inputs or prediction i L V P L V : V 4 Figure Causal graph or system-level prediction The causal graph or the gloal model or prediction is shown in Fig We need only to compute T E, so none o the sensor outputs are included Note also that or prediction, we use as an input the load power, P L, instead o the load current This is ecause it is much easier in practice to predict load power a priori With a given speed command to the rover motors, power is constant, ut current will increase as the attery cells discharge and V decreases It is important also to note that the prediction prolem cannot e decomposed in general Given, we can compute each V i independently, and evaluate V i < V EOD Since E occurs when any one o the cells drops elow the cuto voltage, we can compute EOD or each cell and take the minimum to determine when E will occur (since E occurs when the irst cell reaches EOD) However, depends on i P and i L, oth k E 8

9 ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 04 P i Figure 3 Causal graph or local prediction or cell i o which depend on V There are no local inputs to reak this dependency I we make a simpliying assumption, however, we can decompose the prediction prolem and thus achieve the eneits o a distriuted implementation The causal graph or this case is shown in Fig 3 In this case, we use as a local input the cell power P, where P = P L /4, thus allowing local EOD thresholds, T E i, and, hence, local events E i, to e computed independently This assumption is only valid i the cells are all approximately equal in voltage, otherwise the assumption o P = P L /4 will e violated Further, this is not valid when R p is present, as in that case P is a unction o oth P L and i p In general, the state estimates required or the prediction models must e produced y new estimators derived using structural model decomposition, or the gloal prediction model For some aults, however, the needed estimates may e availale rom the residual generators, i those residual generators were not aected y the ault In this case, new estimators do not need to e derived For the parasitic load ault, the residual generator or each Vi has the state estimates or the attery cells, and the ault identiier has the value o R p We can then reconstruct a gloal state estimate or use in prediction For a voltage sensor ault, a new local estimator (same as that used or residual generation, see Fig 8) is needed to reestimate the states or the corresponding attery cell model From the time o ault detection onwards, the corrected value o the sensor, computed y removing the estimated ias, is used to reestimate the states For the current sensor ault, the case is more complex, ecause a aulty sensor reading was used in all o the local voltage estimators Thereore, new local estimators are needed or all cells, in which the ias-corrected value must e ed as an input rom the time o ault detection onward, once the ault ias has een identiied In this work, we assume that process noise is negligile compared to the uture input uncertainty, so represent the uncertainty only in the uture input trajectories U kp (ie, the trajectory o P L ) We use the surrogate variale method to represent the uture input trajectories (Daigle & Sankararaman, 03) In this method, we represent U kp through a set o surrogate variales, such that U kp can e constructed in a deterministic way given values o the surrogate variales In this way, we can represent the proaility distriutions o the surrogate variales to indirectly represent the proaility distriution o the input trajectories For the rover, we consider V i k E an equivalent constant-loading distriution or the uture inputs That is, we assume that the uture load power, P L, will e constant with the value drawn rom some distriution In the case o the rover, the operator really only needs to know EOD predictions or est-, average-, and worst-case usage scenarios (Daigle & Kulkarni, 04) For the state estimate, we use as samples the sigma points provided y the UKF Each sample is simulated orward three times, once or each use case From this we otain est-, average-, and worst-case EOD predictions, each with some small variance (due to the state estimate variance) It is important to note that since R p is included in the prediction model, the prediction input does not change in the nominal and aulty cases I, however, R p was considered part o the load, ie, part o P L, then P L prediction would have to change in the aulty case and would e complicated, since the additional power required y R p is actually a unction o attery voltage (as shown in Fig ) This is an advantage o viewing the prediction prolem in a system-level perspective (the EPS perspective), rather than a component-level perspective (the attery cell perspective) 7 RESULTS In this section, we demonstrate the integrated system-level diagnosis and prognosis ramework on the rover case study, using real experimental ield data The task o the rover is to travel to dierent waypoints to complete some science ojective We must predict how long the rover will e ale to execute its mission eore having to return to the start point Faults must e diagnosed so that the mission can e replanned i the rover is unale to meet all o its ojectives due to the ault, and does not ecome stranded eore returning to the start point We consider irst a nominal scenario, in which the rover has enough energy to visit all waypoints and return successully to the start point Fig 4 shows the measured and estimated values o V (results are similar or the remaining voltage sensors) With V EOD = 5 V, EOD is clearly not reached Fig 5 shows tracking o the attery current sensor Although the measured value is very noisy, the residual remains within the nominal range, and no ault is detected in any o the residuals Fig 6 shows the system-level EOD predictions or the rover Each prediction consists o three points, or est-, average-, and worst-case uture loading Here, even in the worst-case scenario the predictions indicate that the rover will e ale to complete the mission We next consider a parasitic load ault o 0 Ω, appearing as an additional load on the atteries, draining additional current and causing the atteries to discharge more quickly The ault occurs at 780 s, and is detected at 80 s on the attery current residual, as shown in Fig 7 Given the increase in the attery current, the parasitic load ault and a positive ias in 9

10 ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 04 Voltage (V) Measured Estimated Current (A) Measured Estimated Figure 4 Estimation o V Figure 7 Estimation o i or a parasitic load Current (A) Measured Estimated Figure 5 Estimation o i Resistance (Ω) Figure 8 Estimation o R p or a parasitic load est Case Average Case Worst Case 0000 est Case Average Case Worst Case ke (s) ke (s) Figure 6 Predictions o k E or worst-, average-, and estcase uture usage scenarios the attery current sensor are the only possile aults (see Tale ) At 9 s, two minutes ater ault detection, we oserve a 0 symol on all the voltage sensor residuals, since they have not yet deviated Given these oservations, the only consistent candidate is the parasitic load ault The estimated parasitic resistance over time is shown in Fig 8 The estimate converges to the true value in less than 50 s, and stays very close to the true value As descried in Section 6, the prediction prolem in this case cannot e decomposed, ecause the parasitic current depends on the attery voltages, so the local input or prediction is the total motor power The systemlevel predictions are shown in Fig 9 eore the ault is diagnosed, the predictions indicate that the rover will e ale to complete its mission Ater the ault is diagnosed, the predictions relect the act that more power is eing demanded Figure 9 Predictions o k E or worst-, average-, and estcase uture usage scenarios with a parasitic load ault rom the atteries, and EOD will e reached much sooner, requiring the mission to e shortened We next consider a attery voltage sensor ault, maniesting as a constant oset (ias) o 0 V on the voltage sensor or attery The ault is injected at 600 s and detected at 634 s in the residual or the aulty sensor, as shown in Fig 0 It is immediately diagnosed, as no other ault can produce a deviation irst in the voltage sensor, according to the residual orderings In order to recover rom this ault, the estimator or the voltage is reset ack to the estimated time o the ault, and is updated up to the current time using the uniased signal, computed as the measured signal value minus the estimated ias From the current time on, the present value o the estimated ias is used to correct the measured value sent to 0

11 ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 04 Voltage (V) Figure 0 Estimation o V or the voltage sensor ias ault Current (A) Figure ault Measured Estimated Estimation o i with a attery current sensor the estimator ecause this ault does not actually have any eect on the energy required y the rover, the predictions are the same as in the nominal condition Finally, we consider an oset ault in the attery current sensor The ault is injected at 300 s, and is detected at 344 s Detection time is slow due to the high amount o noise in the sensor The tracking o the sensor is shown in Fig, where the ias is clear visually (c Fig 5) The initial diagnosis is either the parasitic load ault, which can also cause an increase in the current, and a current sensor ault ecause a aulty current sensor value is eing used as a local input to the voltage estimators, these residuals deviate as well Tracking or V is shown in Fig ecause a larger current is used, the estimated voltage drains aster than actual, and a deviation is detected at 45 s, thus isolating the current sensor ault as the true ault Since the state estimates or the atteries will e corrupted, this will propagate to the predictions, giving incorrect results So, to recover rom the ault, once the ault is identiied, the attery estimators are reset to the time o ault detection, and the corrected measurement value, ased on the estimated ias is ed up to the current time and in the uture There is no physical eect on the energy consumption o the rover due to the ault, and thereore the predictions match those in the nominal case 8 CONCLUSIONS In this paper, we developed and implemented an approach or integrated system-level diagnosis and prognosis o the electrical power system o a planetary rover tested The algo- Voltage (V) Measured Estimated Figure Estimation o V with a attery current sensor ault rithms monitor the ehavior o the EPS and generate symols or ault isolation in a distriuted ashion Fault isolation is perormed, and or each ault hypothesis, system-level prognosis is perormed, starting with distriuted estimation o the state and ault parameters, and ollowed y distriuted prediction The distriuted nature o the architecture is ased upon the use o local sumodels that enale the decomposition o gloal diagnosis and prognosis prolems into local suprolems, applying ideas estalished in previous works The approach was demonstrated using ield data rom the rover, showing successul detection, isolation, identiication, and prediction or a set o realistic aults Future work will extend the application o the ramework to the entire rover system, not just the EPS, which will enale the diagnosis o aults in the rover motors, and incorporation o that inormation into system-level predictions We will also apply the approach to other systems, and make urther theoretical extensions o the work, eg, y including multiple aults, and hyrid systems ACKNOWLEDGEMENTS The authors acknowledge Adam Sweet, NASA Ames Research Center, and George Gorospe, SGT, Inc, NASA Ames Research Center, or otaining the experimental data used in this work M Daigle s and I Roychoudhury s unding or this work was provided y the NASA System-wide Saety and Assurance Technologies (SSAT) Project A regon s unding or this work was provided y the Spanish MICINN DPI R grant REFERENCES alaan, E, Narasimhan, S, Daigle, M, Roychoudhury, I, Sweet, A, ond, C, & Gorospe, G (03) Development o a moile root test platorm and methods or validation o prognostics-enaled decision making algorithms Intl Journal o Prognostics and Health Management, 4() lanke, M, Kinnaert, M, Lunze, J, & Staroswiecki, M (006) Diagnosis and ault-tolerant control Springer

12 ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 04 regon, A, iswas, G, & Pulido, (0) A Decomposition Method or Nonlinear Parameter Estimation in TRANSCEND IEEE Trans Syst Man Cy Part A, 4(3), regon, A, Daigle, M, & Roychoudhury, I (0, Septemer) An integrated ramework or model-ased distriuted diagnosis and prognosis In Annual conerence o the prognostics and health management society 0 (p 46-46) regon, A, Daigle, M, Roychoudhury, I, iswas, G, Koutsoukos, X, & Pulido, (04, May) An event-ased distriuted diagnosis ramework using structural model decomposition Artiicial Intelligence, 0, -35 Daigle, M (008) A qualitative event-ased approach to ault diagnosis o hyrid systems Unpulished doctoral dissertation, Vanderilt University Daigle, M, regon, A, & Roychoudhury, I (0, Septemer) A distriuted approach to system-level prognostics In Annual conerence o the prognostics and health management society 0 (p 7-8) Daigle, M, regon, A, & Roychoudhury, I (04, June) Distriuted prognostics ased on structural model decomposition IEEE Trans on Reliaility, 63(), Daigle, M, Koutsoukos, X, & iswas, G (007, April) Distriuted diagnosis in ormations o moile roots IEEE Trans on Rootics, 3(), Daigle, M, Koutsoukos, X, & iswas, G (009, July) A qualitative event-ased approach to continuous systems diagnosis IEEE Trans on Control Systems Technology, 7(4), Daigle, M, & Kulkarni, C (03, Octoer) Electrochemistry-ased attery modeling or prognostics In Annual conerence o the prognostics and health management society 03 (p 49-6) Daigle, M, & Kulkarni, C (04) A attery health monitoring ramework or planetary rovers In Proceedings o the ieee aerospace conerence Daigle, M, Roychoudhury, I, & regon, A (03, Octoer) Qualitative event-ased diagnosis with possile conlicts: Case study on the ourth intl diagnostic competition In Proc o the 4th intl workshop on principles o diagnosis (p 30-35) Daigle, M, & Sankararaman, S (03, Octoer) Advanced methods or determining prediction uncertainty in model-ased prognostics with application to planetary rovers In Annual conerence o the prognostics and health management society 03 (p 6-74) Daigle, M, Saxena, A, & Goeel, K (0, Septemer) An eicient deterministic approach to modelased prediction uncertainty estimation In Annual conerence o the prognostics and health management society (p ) Gertler, J J (998) Fault detection and diagnosis in engineering systems New York, NY: Marcel Dekker, Inc Julier, S J, & Uhlmann, J K (004, March) Unscented iltering and nonlinear estimation Proceedings o the IEEE, 9(3), 40 4 Karthikeyan, D K, Sikha, G, & White, R E (008) Thermodynamic model development or lithium intercalation electrodes Journal o Power Sources, 85(), Luo, J, Pattipati, K R, Qiao, L, & Chigusa, S (008, Septemer) Model-ased prognostic techniques applied to a suspension system IEEE Trans on Systems, Man and Cyernetics, Part A: Systems and Humans, 38(5), Mosterman, P J, & iswas, G (999) Diagnosis o continuous valued systems in transient operating regions IEEE Trans on Systems, Man, and Cyernetics, Part A: Systems and Humans, 9(6), Orchard, M E, & Vachtsevanos, G (009) A particleiltering approach or on-line ault diagnosis and ailure prognosis Trans o the Institute o Measurement and Control, 3(3/4), -46 Patrick, R, Orchard, M E, Zhang,, Koelemay, M, Kacprzynski, G, Ferri, A, & G, V (007, Septemer) An integrated approach to helicopter planetary gear ault diagnosis and ailure prognosis In Proc o the 4nd annual systems readiness technology con altimore, MD, USA Pulido,, & Alonso-González, C (004) Possile Conlicts: a compilation technique or consistency-ased diagnosis IEEE Trans on Systems, Man, and Cyernetics, Part : Cyernetics, 34(5), 9-06 Rahn, C D, & Wang, C-Y (03) attery systems engineering Wiley Roychoudhury, I, & Daigle, M (0, Octoer) An integrated model-ased diagnostic and prognostic ramework In Proc o the nd intl workshop on principles o diagnosis (p 44-5) Roychoudhury, I, Daigle, M, regon, A, & Pulido, (03, March) A structural model decomposition ramework or systems health management In Proceedings o the 03 ieee aerospace conerence Saha,, & Goeel, K (009, Septemer) Modeling Li-ion attery capacity depletion in a particle iltering ramework In Proceedings o the annual conerence o the prognostics and health management society 009 Sankararaman, S, Daigle, M, & Goeel, K (04, June) Uncertainty quantiication in remaining useul lie prediction using irst-order reliaility methods IEEE Trans on Reliaility, 63(), Sankararaman, S, Daigle, M, Saxena, A, & Goeel, K (03, March) Analytical algorithms to quantiy the uncertainty in remaining useul lie prediction In Proc o the 03 IEEE aerospace conerence Zai, S, Riot, P, & Chanthery, E (03, Octoer) Health

13 ANNUAL CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 04 monitoring and prognosis o hyrid systems In Proc o the annual conerence o the prognostics and health management society 03 (p 300-3) IOGRAPHIES Matthew Daigle received the S degree in Computer Science and Computer and Systems Engineering rom Rensselaer Polytechnic Institute, Troy, NY, in 004, and the MS and PhD degrees in Computer Science rom Vanderilt University, Nashville, TN, in 006 and 008, respectively From Septemer 004 to May 008, he was a Graduate Research Assistant with the Institute or Sotware Integrated Systems and Department o Electrical Engineering and Computer Science, Vanderilt University, Nashville, TN During the summers o 006 and 007, he was an intern with Mission Critical Technologies, Inc, at NASA Ames Research Center From June 008 to Decemer 0, he was an Associate Scientist with the University o Caliornia, Santa Cruz, at NASA Ames Research Center Since January 0, he has een with NASA Ames Research Center as a Research Computer Scientist His current research interests include physicsased modeling, model-ased diagnosis and prognosis, simulation, and hyrid systems Dr Daigle is a memer o the Prognostics and Health Management Society and the IEEE Indranil Roychoudhury received the E (Hons) degree in Electrical and Electronics Engineering rom irla Institute o Technology and Science, Pilani, Rajasthan, India in 004, and the MS and PhD degrees in Computer Science rom Vanderilt University, Nashville, Tennessee, USA, in 006 and 009, respectively Since August 009, he has een with SGT, Inc, at NASA Ames Research Center as a Computer Scientist Dr Roychoudhury is a memer o the Prognostics and Health Management Society and the IEEE His research interests include hyrid systems modeling, model-ased diagnostics and prognostics, distriuted diagnostics and prognostics, and ayesian diagnostics o complex physical systems Anial regon received his Sc, MSc, and PhD degrees in Computer Science rom the University o Valladolid, Spain, in 005, 007, and 00, respectively From Septemer 005 to June 00, he was Graduate Research Assistant with the Intelligent Systems Group at the University o Valladolid, Spain He has een visiting researcher at the Institute or Sotware Integrated Systems, Vanderilt University, Nashville, TN, USA; the Dept o Electrical Engineering, Linkoping University, Linkoping, Sweden; and the Diagnostics and Prognostics Group, NASA Ames Research Center, Mountain View, CA, USA Since Septemer 00, he has een Assistant Proessor and Research Scientist at the Department o Computer Science rom the University o Valladolid Dr regon is a memer o the Prognostics and Health Management Society and the IEEE His current research interests include model-ased reasoning or diagnosis, prognostics, health-management, and distriuted diagnosis o complex physical systems APPENDIX: ATTERY CELL MODELING The attery cell model computes the voltage as a unction o time given the current drawn rom the cell, and is descried in detail in (Daigle & Kulkarni, 03) We summarize the model here and reer the reader to (Daigle & Kulkarni, 03) or additional explanation The voltage terms o the attery are expressed as unctions o the amount o charge in the electrodes (the states o the model) Each electrode, positive (suscript p) and negative (suscript n), is split into two volumes, a surace layer (suscript s) and a ulk layer (suscript ) The dierential equations or the attery descrie how charge moves through these volumes The charge (q) variales are descried using q s,p = i app + q s,p (5) q,p = q s,p + i app i app (6) q,n = q s,n + i app i app (7) q s,n = i app + q s,n, (8) where i app is the applied electric current The term q s,i descries diusion rom the ulk to surace layer or electrode i: q s,i = D (c,i c s,i ), (9) where D is the diusion constant The c terms are lithium ion concentrations: c,i = q,i v,i (0) c s,i = q s,i v s,i, () where, or CV v in electrode i, c v,i is the concentration and v v,i is the volume We deine v i = v,i + v s,i Note now that the ollowing relations hold: q p = q s,p + q,p () q n = q s,n + q,n (3) q max = q s,p + q,p + q s,n + q,n (4) We can also express mole ractions (x) ased on the q vari- 3