Signed Bond Graph for multiple faults diagnosis

(1)

Signed Bond Graph for multiple faults diagnosis

Nizar Chatti

^a,n

, Belkacem Ould-Bouamama

^b

, Anne-Lise Gehin

^b

, Rochdi Merzouki

^b

aISTIA, LARIS EA 7315, Angers university, 62 Avenue Notre Dame du Lac, 49000 Angers, France

bLAGIS, CNRS-UMR 8219, Polytech'Lille, Avenue Paul Langevin, 59655 Villeneuve d'Ascq, France

a r t i c l e i n f o

Article history:

Received 12 June 2013 Received in revised form 13 June 2014

Accepted 21 July 2014

Keywords:

Diagnosis Bond Graph

Consistency-Based Diagnosis Fault Detection and Isolation Intelligent vehicle Multiple faults

a b s t r a c t

Different approaches have been developed to perform diagnosis and supervision on continuous systems.

On the one hand, Consistency-Based Diagnosis (CBD) as a qualitative approach has proved its convenience to diagnose multiple faults. However, it faces some problems regarding robustness in decision step and difficulties to obtain an accurate qualitative model. On the other hand, the quantitative approaches based Fault Detection and Isolation (FDI) enable to generate a set of fault indicators called residuals in order to carry out on-line diagnosis. The performances of such methods depend mainly on the behavioural model accuracy and their implementation is sometimes difficult to realise, especially when the possibility of multiple faults is taken into account. To overcome the drawbacks of such methods and to fully exploit their strengths, we give a formal description of a graphical model called Signed Bond Graph (SBG). This formalism exploits its qualitative and quantitative structural properties enabling the generation of multiple behaviour predictions (i.e. possible conflicts). Furthermore, since the SBG is constructed from the Bond Graph (BG) model, the use of this latter as a quantitative method for residuals generation allows to compare the results emanating from the qualitative reasoning based SBG in order to eliminate the possible conflicts which are inconsistent or not physically possible even though they sound logical from a qualitative point of view. The proposed approach is illustrated by a real application to a traction system of an intelligent and autonomous vehicle performed within the European project InTraDE. The result shows its good applicability and efficiency.

1. Introduction

Automated systems are vulnerable to faults. A fault in a dynamical system is a deviation of the system structure or the system parameters from the nominal situation. A fault causes a change in the character- istics of a component such that the mode of operation or performance of component is changed in an undesired way (Blanke et al., 2006). In the strict sense, the fault or fault origin is the primary cause of a dysfunction. It has to be distinguished from the effects of a fault, which are the consequences of the fault on the system's behaviour, described by modiﬁcations in the relations between the inputs and the outputs of the system. A symptom of a fault corresponds to a change of an observable quantity from normal behaviour. Diagnosis aims to detect symptoms and to isolate fault origin.

Different diagnosis approaches based on Fault Detection and Isolation (FDI) have been developed. Among the diagnosis approaches based on models, one can distinguish between qualitative and

quantitative ones. Venkatasubramanian et al. (2003a, 2003b) have reviewed quantitative and qualitative model-based approaches for fault diagnosis. Aﬁrst class of quantitative FDI approaches is based on models comparison between actual behaviour (provided by sensors) and theoretical analytical model given under a set of dynamic equations (Åström et al., 2001). Theﬁrst step of the FDI system rests on the evaluation of residuals that may reveal the inconsistency between the observed and predicted behaviours. If there is an inconsistency, the signature of the residual vector enables the isolation of faulty components. Data-driven models as multivariate analysis techniques such as Principal Component Analysis (PCA) and Partial Least Squares (PLS) are another class of approaches widely applied in process industry to perform FDI (Russell et al., 2000). These quantitative approaches essentially formulate the diagnosis problem as a pattern recognition problem by extracting statistical information from process data (Venkatasubramanian et al., 2003).

Two different communities have developed model-based approaches for diagnosis namely FDI community and DX community emanating from Artificial Intelligencefield and dealing with qualitative models (Pulido, 2004). Among the residual generation methods, BG as a multidisciplinary and graphical modelling language has proved its efficiency to generate fault indicators in Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/engappai

Engineering Applications of Arti ﬁ cial Intelligence

nCorresponding author. Tel:þ33 244688806.

E-mail addresses:nizar.chatti@polytech-lille.fr(N. Chatti), belkacem.ouldbouamama@polytech-lille.fr(B. Ould-Bouamama), anne-lise.gehin@polytech-lille.fr(A.-L. Gehin).

(2)

a systematic and generic way using speciﬁc algorithms. These algorithms are based on covering causal paths (Ould-Bouamama and Samantaray, 2008) and are implemented in dedicated software (Ould-Bouamama et al., 2006). But as many residual generation methods, BG models are limited to a single fault diagnosis.

Generally, residual generation methods dealing with multiple failed components use beyond the residuals evaluation, a sequence of measurements which may locate the failing components. Basically, when one entertains the possibility of multiple faults by considering only residual generation method, the space of potential candidates grows exponentially with the number of faults under consideration. In the presence of multiple sensors, linear combinations of residuals vector's elements can produce independent residuals when they share common variables. Never- theless, this strategy can be used only for sensor fault isolation because these residuals can not provide additional structural isolation capabilities to the diagnosis system. Among approaches in FDI community addressing the problem of multiple faults occurrence,Venkatasubramanian et al. (2003a)propose an extension of the dynamic structured residual approach with maximised sensitivity (Åström et al., 2001) which suffered from the use of scalar in the structured residuals vector. In order to overcome this limitation, authors suggest a vector-based FDI approach in multivariate dynamic systems to sensor faults diagnosis. This approach presented many advantages but it is limited to the sensor faults and can not be applied for actuators or parameter faults diagnosis.

Furthermore, FDI approaches based on residual evaluation assume the observation of only one symptom (the fault signature) to a fault origin. In other words, it is supposed that to a fault corresponds the deviation of only one measurement acquired from a sensor. Nevertheless, in some cases, a same fault can manifest several effects corresponding to different symptoms not observed on the same time windows. This is the case, for example, if two measured values are linked by an integration relation. In the speciﬁc case where sensors do not react at the same time but one after the other, it is said that we are in the case of a fault propagation.

To achieve multiple-fault diagnosis, DX community proposes the Consistency-Based Diagnosis (CBD). Indeed, through the notion of Possible Conflicts, a set of candidate faults can be provided for a given scenario based on qualitative models which allow to express the cause-effect relationships between process variables. Nevertheless, these qualitative models are difficult to obtain since they are derived from human expertise or operator knowledge of the process. They can be eventually generated from equations that define the behaviour of the system but the qualitative influence is evaluated by computing the partial derivative for every equation. This operation needs deep and objective expertise or simple model such as Signed Directed Graph (SDG) (Maurya et al., 2004).

Temporal Causal Graph (TCG), proposed by Mosterman and Biswas (1999) is the extension of SDG in the sense that they formally capture causal and temporal relations among system variables in a common framework. TCG is easily computed from BG models, by deducing cause-effect relations from the power variables of the BG. TCG is used for diagnosis inManders et al.

(2000)andBiswas et al. (2003). The developed algorithm, uses a combined qualitative and quantitative approach for fault detection, isolation, and identiﬁcation. For this task, an observer (designed as a Extended Kalman Filter) is used to generate a residual. The magnitude and slope of these signals are transformed into qualitative symbols that are expressed as increases (þ) and decreases () from the nominal operating mode. Possible faults are then derived by backward propagation in the TCG structure.

During the last decade, different research groups proposed a common framework for sharing and comparing techniques from both communities. This effort has turned into what is known as BRIDGE community which attempts to propose new approaches

integrating the best from both communities and to deal with quantitative and qualitative models (Biswas et al., 2009; Cordier et al., 2004). But in cited papers, only comparisons are given using two separate methods: quantitative analytical redundancy methods based on analytical model and possible conﬂict diagnosis method where the model is based on a set of ﬁrst order logic formulas.

The Signed Bond Graph (SBG) presented in this paper, extends the fault detection procedure based on residual generation and improves diagnosis results in the case of multiple faults. This model has previously been introduced in Chatti et al. (2013a, 2013b) for health monitoring of fuel cells. It is now formally described to generate qualitative and quantitative diagnosis on the basis of a single representation. Indeed, the SBG is built from the BG and exploits a key element of this last one, the causality property that allows to maintain causal relations between system variables. This enables mapping back the changes of measured variables to possible sources of change and to determine a set of possible fault origins by backward propagating the effects of measurements. These causal paths of fault propagation are not directly represented on the BG. They are however required to correctly interpret the residual signatures directly obtained from the BG and to correctly characterise the fault candidates, speciﬁ- cally in a multiple fault context. Indeed, the inconvenient of residual analysis in a multiple fault context is to generate false alarms due to the possible compensation of faults. The idea by introducing the SBG model is to improve the Fault Detection and Isolation procedure based on residual analysis by adding a qualitative reasoning allowing the association to a residual signature a restricted set of behaviour predictions (i.e. possible conﬂicts).

Finally, a global supervision module-based SBG is proposed. This module uses both qualitative and quantitative reasoning for FDI.

The proposed approach is illustrated by a real application namely a traction system of an intelligent and autonomous vehicle designed within the European project InTraDE (InTraDE).

The rest of the paper is organised as follows.Section 2presents a brief overview of qualitative diagnosis approaches related to the FDI community as well as the DX community.Section 3presents the SBG model and its features for fault diagnosis. Section 4 presents the developed diagnosis procedure based on integration of behavioural and causal properties of BG and qualitative aspect of SBG. InSection 5, the proposed FDI methodology is applied to an intelligent and autonomous vehicle. Section 6 concludes the paper by highlighting the strengths of the proposed approach.

2. Qualitative diagnosis approaches

Different qualitative models have been already successfully used in the framework of model-based diagnosis. These models enable to check the consistency of observations over time with the theoretical system behaviour. This is why, building a robust and accurate qualitative model is a crucial task for model-based diagnosis. This section is devoted to present two existing qualitative approaches in order to point out their advantages and drawbacks. This analysis elucidates the main contribution of our developed model namely the SBG which gather both quantitative and qualitative features for single and multiple fault diagnosis. In order to highlight the main contribution of this paper with regards to a set of existing qualitative approaches, we focused on the SDG and the CBD approaches emanating from DX community.

2.1. SDG for fault diagnosis

The SDG is a qualitative model-based approach which uses a causal model representing the cause-effect relationships between

(3)

system variables. In recent years, this approach has been widely applied in the chemical industry and it proved its efﬁciency for diagnosis of single and multiple faults (Tateno et al., 2006).Iri et al.

(1979)proposed to represent the structure of the system to be monitored by a signed digraph and to apply graph theory to identify possible causes of process disturbances and faults. How- ever, even though the strategy is well adapted for dynamic systems, the development of SDG is not an easy task. Roughly speaking, the main methods used to model a system by a SDG are either based on process knowledge as described by piping and instrumentation diagrams ðP&IDÞ, or differential equations (DE), algebraic equations (AE) and differential algebraic equations (DAE) describing the system behaviour. Let us consider a SDGðN;AÞ composed of nodesN¼NX[NEand directed signed arcsAwhere NX is the set of system variables (state variables and output variables) andNEis the set of exogenous variables (independent variables, including disturbances…). Each node can take a value of (‘0’), (‘þ’) and (‘’), representing that the corresponding variable is at its normal steady state value, above or below the nominal steady state value, respectively. During the dynamic process changes, the variable values related to sensor measurement are evolving. This is why, each variable is assumed to be at its normal steady state value until a residual exceeds its threshold. At that moment, we consider the time window wherein the residuals react and we affect the sign to the variable by comparing its mean value during the same time window with its nominal value. The qualitative value is given by comparing the measured value with the average value within a speciﬁc sliding window of observation.

Let us consider the example of DE system described by:

dNX

dt ¼FðxiANX;ejANEÞ ð1Þ

whereFis a function vector.

To build the SDG, for every differential equation in(1), directed arcs are drawn from all the variables on the right-hand side to the system variable on the left-hand side in that equation. An arc is drawn fromejtoxiif the expression fordxi=dt,f_iðxkANX;ejANE) is a function of xi. The sign of the corresponding arc sign(xi,ej) is given by∂f_i=∂ej. The arcs represent the direct influence between variables and each arc is assigned with the sign‘þ’if it represents a positive influence and the sign ‘’if it represents a negative influence.

As shown inFig. 1, the directed arc fromBtoCis a negative arc, which means that nodesB and Cchange in different directions ð∂C=∂B!0Þ. The second arc fromCtoBis a positive arc, which means that nodes B and C change in the same direction (∂B=∂Cg0Þ.

Deﬁnition 1. A valid node is a node for which the measured or estimated value is not equal to the predicted value, this predicted value being determined by the normal behaviour model.

Deﬁnition 2. For a given arcaijbetween two nodesni,njsuch asni

is the source andnjis the destination node, the arcaijis consistent ifsignðaijÞ ¼signðniÞnsignðnjÞ. The sign of the arc is the product of its nodes' signs.

Deﬁnition 3. A consistent pathp1xbetween two nodesn1,nxis made of consistent arcsaijsuch as8ia1 and8jax,ni¼nj1and

nj¼niþ1. In other words, a consistent path is formed of a sequel of consistent arcs all oriented in the same sense.

In the normal operating mode, all nodes of a SDG take the value of zero. When a fault appears, the faulty nodeﬁrst changes sign and becomes a valid node. The fault effects are then propagated along directed arcs throughout the whole SDG. Hence, one or several system variables will deviate from the normal steady state and the corresponding values will change from‘0’to‘þ’or from’0’ to‘’. The fault occurrence can be deduced from the deviation of the process variables caused by the combination of all the symptoms caused by this fault. The different traversed elements allowing toﬁnd the origin of the fault, correspond to its pattern.

For example, if fault node A (inFig. 1) is (‘þ’), then a possible pattern of the fault is½AðþÞ;BðþÞ;CðÞ;DðÞ;EðþÞ. If the sign of the arc between nodesAandBis‘þ’and if the sign of the fault nodeAis‘þ’then the sign of the nodeBmust be‘þ’to have a consistent arc between the nodes A and B. With the same reasoning, the sign of the node C, respectively D must be‘’, respectively‘’to have a consistent path between nodesAandD.

The SDG diagnosis aims to identify the component responsible for the fault from the valid nodes and the consistent paths of the graph.

2.2. Consistency based diagnosis (CBD)

Reiter (1987)proposed a logical theory of diagnosis using First- Order Logic principles. A diagnosis speciﬁes whether each component of a system is abnormal or not, according to the observations of the system behaviour and the system description. This approach is known as the Consistency Based Diagnosis. The main purpose of this approach is to generate the set of minimal fault candidates that is consistent with the available measurements (called observations). The following deﬁnitions of the CBD are used:

Deﬁnition 4. A system is a triple (SD, COMPS, OBS) where:

1.SD, the system description, is a set ofﬁrst-order logic formulas.

2.COMPS, the system components, is aﬁnite set of constants (the number of components is deﬁned).

3.OBS, the observations of a system, is aﬁnite set ofﬁrst-order sentences.

Let AB(c) be a predicate meaning that the component c is abnormal. AB(c) means that c is nonfaulty. The fault detection problem can be formalised by

SD[OBS[ f:ABðcÞjcACOMPSg ð2Þ

is inconsistent. In other words, the observationOBSconﬂicts with what the system should do if all its components were behaving correctly. The localisation step aims to explain the system fault.

Reiter (1987)deﬁnes the notion of diagnosis as follows.

Deﬁnition 5. A diagnosis for (SD, COMPS, OBS) is a minimal set

Δ

^COMPS^{such that:}

SD[OBS[ fABðcÞjcA

Δ

g [ f:ABðcÞjcACOMPS

Δ

^g ^ð3Þ

is consistent.

With this deﬁnition,

Δ

corresponds to the set of possible faulty components consistent with observations.

Much more details about the CBD can be found in Pulido (2004). Furthermore, Cordier et al. (2004) propose a formal framework in order to compare the FDI analytical redundancy approach and the DX consistency-based model. They point out

A B C D

E +

+ - +

+

Fig. 1.Simple SDG description.

(4)

some equivalences between the two approaches under sufﬁcient conditions.

2.3. Discussion

The main idea behind the SDG method is to use a graphical model to analyse in an exhaustive manner the fault propagation by showing the pathways of causality. This enables the definition of a set of fault candidates. However, the SDG approach presents some limits. Thefirst one is that only variables (systems variables and exogenous variables) are taken into account. There is no representation of hardware components which are at the origin of the fault. A direct consequence of this lack of knowledge is that it is impossible to extract structural and behavioural information about the physical system and its dynamic. Moreover, there is virtually no real system where all the state variables are measured. Another difficulty is that even if SDG can be built from mathematical (Maurya et al., 2003a, 2003b), precise equations are usually unavailable. SDG is then, in most cases built on the basis of process knowledge as described byP&ID. This is then a complex and experience dependent task, and the resulting SDG has to be validated by process data before being used for analysis or diagnosis (Yang et al., 2012).

The CBD presents also some limits when it is applied to the diagnosis of dynamic systems because of its lack of robustness. In fact, the generated solutions are based on deﬁned thresholds that are not robust with regard to disturbances, modelling errors, and noise.

The proposed SBG as the SDG enables the qualitative reasoning based on the notion of Possible Conﬂicts generation since they allow the expression of the cause-effect relationships between process variables. However, the two forms are radically different and the advantage of the SBG approach compared to the SDG, is that it deals with structural and behavioural information allowing to give much relevant dynamic description of the system's components. Indeed, the SBG (by modelling power exchanges between system components with respect of energy conservation and power continuity) overcomes the lack of component description of the SDG approach and offers the structural and behavioural information required to describe dynamic systems. Moreover, BG approach proved its reliability for dealing with uncertainties and generating robust thresholds (Djeziri et al., 2007;

Touati et al., 2012). The formal model of the SBG is presented in the next section.

3. SBG: mathematical formalism and systematic generation

3.1. BG model

A SBG is a graphical model directly built from the BG model by applying adequate transformations. A BG combines vertices and power bonds. Power bonds are oriented to represent the direction of the power transfer between vertices and are associated with two variables: the effort (above the bond) and theﬂow (below the bond). The vertices are divided into three sets:

Active elements: These elements are associated with energy sources allowing to supply power to the process. They are four types: source of effortSe, source ofﬂowSf, modulated source of effortMSe, modulated source ofﬂowMSf.

Passive elements: These elements allow the transformation of the received power into a dissipated power (R-element), and stored under kinetic (I-element) or potential energy (C-element).

Conservative multiport elements: These elements allow the reproduction of the constraint architecture of the whole system to be modelled. They includeTransFormers,junction structures andGYrators. Junction structures are used in order to connect several elements of the BG model (R,CandI) by a 0-junction

when the effort variable is the same and theﬂows are different and by a 1-junction when the ﬂow is the common variable.

Basically, a 0-junctionJ0 is a multiport element which means that the effort over all connected bonds is the same, and that theﬂows sum is zero, considering the sign. The 1-junctionJ1is also a multiport element but it means, unlike theJ0junction, that theﬂow through all connected bonds is the same, and that the efforts sum is zero, considering the sign.

TransFormer noted (TF) and GYrator (GY) are used to represent energy transformation from one domain to another while respecting the conservation laws of energy. Sensors are represented by effort (De) andﬂow (Df) detectors.

In addition to physical and mathematical levels, BG represents algorithmic level of modelling by causality concept. This causality represented by a causal stroke position in the power bond, determines the order according to which the unknown variables (effort orflow) can be calculated from the known ones. By convention, the cross stroke is placed near (respectively far from) the element for which the effort (respectively flow) is known. The example shown in Fig. 2a means that the system A imposes effort (orflow) to system B.Fig. 2b gives the corresponding block diagrams. The causality for basic BG elements can be summarised as follows:

The sources impose always one causality, imposed effort by effort sources and imposedﬂow byﬂow sources.

For the elements C and I, the choice of the integral causality:

eðtÞ ¼1 C

Z t 0

fðtÞ dt ð4Þ

fðtÞ ¼1 IZ t

0

eðtÞ dt ð5Þ

is preferable when BG is used for simulation purpose. To avoid the problem of initial unknown condition, the derivative causality:

fðtÞ ¼CdeðtÞ

dt ð6Þ

eðtÞ ¼IdfðtÞ

dt ð7Þ

is required when BG is used for fault indicators generation.

Only one effort is imposed on a 0-junction. Only oneﬂow is imposed on a 1-junction.

3.2. SBG deﬁnition and properties

The SBG is an extension of the BG and rests on the following deﬁnition and properties.

Deﬁnition 6. A Signed Bond Graph G(X,A) is a directed labelled and signed graph where:

X¼ fxi;IAIxgis a set of nodes corresponding to BG elements.

More precisely:X¼XS[XCo[XTR[XD[XCewith:

Fig. 2.BG principle and the notion of causality.

(5)

○ XS¼ fxs=xsASe[Sf [MSe[MSfg, the set of the active elements which supply energy to the system.

○ XCo¼ fxCo=xCoAR[C[Ig, the set of the passive elements as deﬁned inSection 3.

○ XTr¼ fxTr=xTrATF[GYg, the set of the conservative elements used to express the energy transformation from one domain to another.

○ XD¼ fxD=xDADe[Dfg, the set of detectors.

○ XCe¼ fxCe=xCeAJ₀[J₁g, the set of the junction elements.

A¼ faij; i;jAIxg is a set of directed labelled and signed arcs.

Each of which is deﬁned byaij¼ ðxi;lij;sij;xjÞwhere

○ xiAXis the origin node.

○ xjAXis the destination node.

○ lijAfe;f;em;f_mg is a label used to express the variable representing the power exchanged between the origin and the destination nodes of the arc. This power can be either an unmeasured effort (e), an unmeasuredﬂow (f), a measured effort (em) or a measuredﬂow (fm).

○ sAfþ;;0;∅g is a sign used to express the sense of the energy variation between the origin node and the destination node. Signs þ, , 0 and ∅are respectively used to represent a power supply, a power consumption, a power conservation and no power exchange (when signals emanating from sensors are considered, for instance).

The following properties hold.

Property 1.

8xSASe[MSe;(aAA:a¼ ðxS;e;þ;xCeÞ where xCeAJ₁ ð8Þ 8xSASf [MSf;(aAA:a¼ ðxS;f;þ;xCeÞ where xCeAJ₀ ð9Þ In other words, each source element is connected to a junction element and provided a power to this central element.

Property 2. Let e(a)denote the effort associated with the arc a,and f (a)denotes theﬂow associated with the arc a,let subscript i for arc ai

denotes an incoming arc,let j for arc ajdenote an outgoing arc,

8xCeAJ₀,

(one and only one aiAA:ai¼ ðx;ei;s;xCeÞ ð10Þ eðaiÞ ¼eðajÞ 8ajAA:aj¼ ðxCe;ej;s;xÞ ð11Þ

_8x_Ce_A_J₁_,

(one and only one aiAA:ai¼ ðx;f_i;s;xCeÞ ð12Þ fðaiÞ ¼fðajÞ 8ajAA:aj¼ ðxCe;f_j;s;xÞ ð13Þ

The effort(respectively theflow),on a junction J0(respectively J1),is imposed through a unique incoming arc. Other arcs are outgoing arcs transporting effort(flow)to other elements. The effort(respectively theflow)variables on the outgoing arcs of a junction J0(respectively J1) is equal to the effort (respectively the flow) variable of the incoming arc. The sum of theflow(respectively the effort)variables of the outgoing arcs of a junction J0(J1)is equal to the sum of theflow (effort)variables of the incoming arcs.

Property 3.

_8x_Tr_A_X_Tr_,

( one and only one a_i;a_iAA:a_i¼ ðxCe;e_i;0;xTrÞ ð14Þ ( one and only one aj;ajAA:aj¼ ðxTr;ej;0;xCeÞ ð15Þ ( one and only one al;alAA:al¼ ðxCe;f_l;0;xTrÞ ð16Þ

(one and only one a_h;a_hAA:a_h¼ ðxTr;f_h;0;xCeÞ ð17Þ where subscripts i, l for arcs denote incoming arcs and subscripts j, h denote outgoing arcs.

TF and GY elements act as propagator of power while respecting power conservation law. They have only one effort input,only one effort output,only oneﬂow input and only oneﬂow output. They are always connected to junction elements.

Property 4.

₈_x_Co_A_X_Co_,

(one and only oneai;aiAA and one and only one aj;ajAA:

ai¼ ðxCe;ei;;xCoÞ3aj¼ ðxCo;f_j;;xCeÞ ð18Þ or

ai¼ ðxCe;fi;;xCoÞ3aj¼ ðxCo;ej;;xCeÞ: ð19Þ

Passive elements are connected to junction elements. If the incoming arc imposes effort,the outgoing arc provides theﬂow and vice versa.

The energy between the two nodes varies in opposite direction.

Preferential causality may exist for C and I elements. They are expressed as:

8xCoAC;ai¼ ðxCo;ei;;xCeÞ3aj¼ ðxCe;f_j;;xCoÞ ð20Þ 8xCoAI;ai¼ ðxCo;f_i;;xCeÞ3aj¼ ðxCe;ej;;xCoÞ ð21Þ Unless the system requires otherwise,an element C imposes the effort while an element I provides aﬂow.

Property 5.

8xDADe;(aAA:a¼ ðxCe;em;∅;xDÞ ð22Þ 8xDADf;(aAA:a¼ ðxCe;f_m;∅;xDÞ ð23Þ Each detector is connected to a junction element. The detector is the destination node of the arc. There is no energy exchange between the junction element and the detector since only the effort or theﬂow is measured.

Property 6.A junction xCeis said to be observable if:

(aAA:a¼ ðxCe;em;∅;xDÞ ð24Þ

3.3. Construction of the SBG

The SBG can be automatically built from the BG model, by an appropriate algorithm, according to the following steps:

1. Build the BG as described inOuld-Bouamama et al. (2006).

2. Keep all the BG elements and remove all the bonds.

3. UseProperties 1 to 5to connect nodes by signed and labelled arcs.

4. Mark measured nodesxD by a square and other nodes by a circle.

The algorithm to automatically perform step 3 is given in the appendix.

Example 1. Let us consider the electrical circuit given in (Fig. 3a).

In this circuit, the voltages (effort variables) over the elements R, L, C are different and the current (ﬂow variable) through these elements is the same. The corresponding BG is given in integral causality in (Fig. 3b). The 1-junction expresses the current conservation and indicates that the voltages (efforts) sum is equal to zero and the 0-junction expresses the voltage conservation and indicates that the current (ﬂows) sum is equal to zero. The SBG in

(6)

(Fig. 3c) is directly deduced from the BG model according to the following steps.

1. Only BG elements are kept and considered as SBG nodes.

2. Property 1 is used to connect nodeSeto nodeCe1.

3. Property 4 is used, with preferential causality, to connect node L1to nodeCe1and nodeCe1to nodeCe0.

4. Property 2 is used, to connect nodeR1to nodeCe1, nodeCe1to nodeCe₀and nodeR₂to nodeCe₀.

4. SBG for qualitative diagnosis

Considerobssubset of global observationsOBS. For any obser- vationobsOBSof the system in a given moment, a SBG model can determine qualitatively whether each measured node has deviated from its normal state, as well as the direction of deviation, according to a set of threshold values (depending on measurement uncertainties and noises). By the use of reasoning in the SBG model, the observationobspropagates through consistent paths. The nodes belonging to this path are elements of

Δ

COMPS, the set of possible conﬂicts. The propagation is stopped either when a consistency is noted or when a measured node is reached. Let just introduce three deﬁnitions before to give a complete example.

Deﬁnition 7. A pattern of a SBG model is a function

Γ

: flig-fþ;0;g that associates to each label li a speciﬁed sign according to the observed value of the observation node xD_i. Hence,

Γ

^ðlⁱ^Þis the sign of the arc labelli,iANⁿ, i.e.

Γ

^ðliÞ ¼0 ifjlilinj!

ε

^li ð25Þ

Γ

^ðlⁱ^{Þ ¼ þ}^if^lⁱ^lⁱⁿ^Z

ε

^lⁱ ^ð26Þ

Γ

^ðlⁱ^{Þ ¼}^if^lⁱⁿ^lⁱ^Z

ε

^lⁱ ^ð27Þ

where

ε

^lⁱis the threshold andlinis the nominal value.

The SBG helps to determine the set of conﬂict and hence the set of fault candidates.

Definition 8. A possible conflict is a set of elementsxiAXsuch that if one of them does not operate correctly, it creates an inconsistency. Then, when a measured value is different from the value predicted by the model, a discrepancy is noticed and becomes the source of a conflict.

Deﬁnition 9. A discrepancy is a symptom related to a deviation of a measured variable.

In a fault free situation (seeFig. 4(a)), all nodes of the SBG take the value of zero. The set of Possible Conﬂicts of a system can be deduced from the SBG model by propagating back from every discrepancy node along the directed edges of the SBG until an inconsistency is reached (see fault propagation scheme in Fig. 4 (b)). To explain how they are obtained, let us consider the assumption

Γ

^ðfm₁Þ ¼‘þ’ corresponding to the observation node related to the detectorDf1. The propagation paths are given by the top of theFig. 4(b). First, the propagation starts by the arc which imposes the ﬂow (because Ce1¼J₁) corresponding to the only incoming arc for the nodeCe1. In other words, iffm1is too high it is becausef2is too high. Since the behaviour (normal or not) of the elementL1is unknown,e2can evolve into two directions:e₂^þ,e₂. The two assumptions have to be considered. First case, ife2is too high, it is becausee3ore4is too low. Ife3is too low, thenf3must be too low. This assumption refutes the observation

Γ

^ðfm₁Þ ¼‘þ’. The possible conﬂictPC1¼ ðR1;L1Þis then deduced. Second case, if e2is too low, it is becausee3ore4is too high. The assumptione3

too high does not contradict with the observation

Γ

^ðfm₁Þ ¼‘þ’. The possible conﬂict set is thenfL1g. The assumptione₄too high is consistent with the observation

Γ

^ðe^m1Þ ¼‘þ’. Two assumptions have then to be taken for the componentC1since its behaviour is unknown. An increase of f₅ is explained by a decreasing of e₆ which refutes the observation

Γ

^ðe^m1Þ ¼‘þ’. The possible conﬂict PC2¼ ðL1;C1;R2Þis then deduced.

Hence, on-line diagnosis will establish discrepancies between expected system behaviour and observations. Discrepancies imply the presence of faults, thus triggering the fault isolation and identification processes that are able to determine the cause of the fault, i.e. the element or the set of elements which deviate from the normal operating mode. However, this qualitative reasoning lacks validation of the possible conflicts. This reasoning is based on the SBG which is emanating from the BG model. This is why, we propose to use the structural and behavioural features of the BG (as a quantitative model) in order to generate fault indicators corresponding to the residuals based on the deviation between the measurements and the model-based predicted behaviour. The evaluation of these fault indicators allows to keep or to fire the determined possible conflicts. This strategy is possible because all the results come from one common model namely the BG. The next section describes the procedure of fault indicators generation.

5. SBG model based for quantitative FDI design

Within the BG methodology, the Analytical Redundancy Rela- tion (ARR) generation procedure rests on the BG causal features and uses the theory of unknown variables elimination by propagating the causation graphically from one modelling element to the other (from unknown variables to known ones) by using the junction's structural constraints.

Indeed, the relations between system variables can be easily displayed graphically using BG and they can be deﬁned under a symbolic format using symbolic computation software (Ould- Bouamama et al., 2006). Moreover, graph analysis techniques allow the identiﬁcation in the BG of observable sub-graphs in which all the outer vertices are either actuators (inputs) or detectors (outputs). These outer vertices are linked by a constraint whose structure is given by the edges of the corresponding subgraph. This constraint, when it is available in equation form (i.e. symbolically resolvable), leads to an ARR (residual computation form). Then each residual can be rewritten as:

rfðDe;D_f;Se;S_f;MSe;MS_f;

θ

^m^{Þ ¼}⁰ ^ð28Þ

wherefDe;Df;Se;Sf;MSe;MSf;

θ

^m^gis a set of known variables.

Fig. 3.Electrical system (a) its BG model in integral causality (b) and its SBG (c).

(7)

According to the deduced ARR, a fault signature matrix which crosses ARR in rows and faultsF in columns is built in order to evaluate the possibilities the system has to detect and isolate faults. Boolean matrix element aijequals 1 if the ith residual is affected by thejth fault. The on-line residuals evaluation leads to the formulation of a binary coherence vector:

C¼ ðc1c2⋯cnÞ ð29Þ

whose elements, ciði¼1;…;nÞ, are determined from a decision procedure

φ

, which generates the alarm conditions.

A simple decision procedure can be used for instance:

C¼

φ

^ðr¹^r²⋯rnÞ ð30Þ

whereby each residual, ri is tested against a threshold

ε

^, ﬁxed according to parameter uncertainties, sensor noises and so on. In the simple decision procedure,

ε

can be taken to be equal to twice the standard deviation

σ

of the residual in a normal operating mode:ð

ε

^¼72

σ

Þ. In a single fault case, a pair of component faults are isolable if their signatures are different. In a multiple fault case, the obtained signatures can lead to a insufﬁcient result which can not determine the set of faulty components.

The algorithm of ARR generation can be found inDjeziri et al.

(2007). This procedure is implemented in a software developed by one of the authors as a toolbox in Symbols2000 (Ould-Bouamama et al., 2006).

Let us now return to the example given inFig. 3a. To avoid the problem of the initial conditions, which are unknown in a real system, and to be able to generate directly the ARR, the BG model has to be put in derivative causality and theﬂow detector (corresponding to the current sensor) is transformed into a signal sourceSSf¼f_m~₁ modulated by the measured value and the effort detector (corresponding to the voltage sensor) is transformed into a signal sourceSSe¼em~₁ (see Fig. 5). These imposed signals are the starting point for the elimination of unknown variables. The ﬁrst ARR candidate is generated from conservative law energy modelled by the 1-junction:

e1e2e3e4¼0 ð31Þ

The unknown variablese1,e2,e3ande4are eliminated using covering causal paths from unknown to known ones. This leads to an oriented graph well known in bipartite graph theory. For instance e3 is eliminated using the path:

e3-R:R1-SSf:fm₁

The details of this algorithm are provided inOuld-Bouamama et al.

(2012). Thus theﬁrst ARR is uL1

df_m₁

dt R1f_m₁em1 ð32Þ

whereuis the imposed voltage,fm₁is the measured current. This ARR is sensitive to the faults that affect the sourceu, the sensors which Fig. 4.(a) Possible conﬂicts based inconsistency generated from SBG of the electrical system and (b) fault propagation within the SBG.

(8)

measuref_m₁andem1and the componentsR1,L1. The second candidate ARR deduced from the 0-junction can be written as:

f_m₁em1

R2

C1

dem1

dt ð33Þ

This ARR is sensitive to the faults that affect the sensorsf_m₁andem1

and the componentsR2andC1.

6. SBG for global supervision

Two steps can be distinguished in the global diagnosis procedure we propose. Theﬁrst one is its design determined off-line and the second one is its on-line exploitation.

6.1. The design of the diagnosis module

The design of the diagnosis module rests on four steps which are illustrated inFig. 6 and summarised as follows. First, the BG

model is built based on the physical system architecture given by itsP&ID. Second, the SBG is directly derived from the BG model as explained in the previous sections. The BG model is also used for formal ARR generation from which (based on technical specifications) a Fault Signature Matrix (FSM) is deduced. Third, the FSM is used for monitorability analysis (which fault which may affect the system can be detected and localised). Finally, a sensor placement can then proposed to satisfy the fixed specifications using a software FDIpad (Ould-Bouamama et al., 2006).

6.2. The on-line exploitation

Once the diagnosis module is correctly designed, it can be exploited on-line to detect and isolate faults whenever possible. This diagnosis procedure is illustrated inFig. 7and can be summarised as follows.

First, ARR are evaluated using the measured outputs and the control inputs which represented the known variables of the system. From the values of these ARR, a coherence vector is obtained. If its value is different from (0, 0 …0), it is then compared to the fault signatures regrouped in the FSM. This leads to a list of potential faults and allows the extraction of a subset of fault candidates. Using the measured values, the consistency of the possible conflicts generated off-line from the SBG, is checked and abnormal situations are identified. In the final step, the consistency between the results obtained according to the ARR evaluation and the possible conflicts generation is tested. If there is consistency, some faults may be isolated, otherwise a partial result is given.

7. Application: the electrical vehicle robutainer

7.1. RobuTainer hardware description

This work is carried out within the European project InTraDE (Intelligent Transportation for Dynamic Environment) (InTraDE).

This project contributes to improving the trafﬁc management and space optimisation inside conﬁned space by developing a clean and safe, intelligent transportation system called Robutainer (see Fig. 8).

Robutainer is an over-actuated electric vehicle that can be manu- ally or automatically piloted inside conﬁned areas. It encompasses four Fig. 6. Ofﬂine design step.

Fig. 5.BG model in derivative causality.

Fig. 7.Online exploitation.

(9)

actuated wheels. The rear and front wheels can be independently steered by four steering motors. The traction part is deﬁned by 4 DC traction motors, delivering a relatively important mass torque with a decentralised input control. This vehicle example is structured as a graph of four independent quarters of vehicle and hence it presents four redundant parts composed of an electromechanical system namely the DC motor, the steering motor and two wheels (traction wheel and brake wheel) in interaction with the road. Moreover, in each electromechanical system, three sensors are embedded in order to provide current (i_mj), angular velocity of the rotorð

θ

_ejÞ, and angular

velocity of the wheelð

θ

_^sjÞ, and one inertial sensor enabling to acquire longitudinal, lateral, and vertical accelerations, as well as yaw velocity.

Longitudinal and lateral velocities (namely u_ and v) are directly_ estimated from the acceleration measurements.

7.2. RobuTainer modelling

We considered in this paper only the dynamics of the chassis and the electromechanical system composed of the DC motor and the traction wheel. The corresponding BG model is given inFig. 9.

7.3. Fault diagnosis and results 7.3.1. Quantitative reasoning based BG

We focused on the BG including the DC Motor and the wheel to explain the procedure of ARR generation for fault diagnosis. In fact, the architecture of the vehicle presents some redundancy because each DC motor as well as each wheel has the same presentation within the BG model. The BG Model of thejth electromechanical system (including the DC motor as well as the wheel) is elaborated in derivative causality (seeFig. 10). The electric power is provided by the electrical part of a DC motor which is equivalent to an input voltage sourceU_0jin serial with a resistanceR_ejand an inductance Fig. 8. Overview of Robutainer.

Df

Fig. 9.BG model of the half vehicle.

(10)

Lej. The electrical current is measured by the sensorDf :imj. The gyrator elementGYdescribes the power transformation from the electrical part of the DC motor to its mechanical part which is characterised by its rotor inertiaJejand its viscous friction para- meterfej. The angular velocity is measured by the sensorDf: _

θ

^ej^.

The mechanical gear which links the mechanical and wheel parts is represented by a transformer element TF. The wheel is characterised by its inertia Jsj, its viscous friction parameter fsj, the wheel angular velocity sensorDf : _

θ

^sj, a lateral forceFljand the constant wheel radiusR. To obtain ARR, the BG model has to be put in preferred derivative causality to avoid the problem of unknown initial conditions. This is done by converting eachflow or effort detector into a signal source modulated by the measured value. It is worth noting that all the dynamic elements are linked by causal paths to at least one detector and all dynamic elementsI andCadmit a derivative causality on the BG model in preferred derivative causality. The system is therefore observable. Never- theless, after dualisation of the detectors, a conflict of causality appears on the system located before the transformerTFwhen we move to a derivative causality, this part of the system is under- determined. However, the part after the transformerTFis over- determined because no causal conflict appears in dualising the detector. As the initial conditions are known because the real system is equipped with detectors, the ARR can be generated even if the causal elementCis kept in integral causality. The resulting BG model of thej^thelectromechanical system is given inFig. 10.

The parameter values of the RobuTainer are given inTable 1. From the BG model, the constitutive relations of the junctions, the gyrator and the transformers, the following ARR linking only known variables and parameters, can be found:

ARR1j:U0jRejimjLejd

dtðimjÞkej

θ

_^ej^¼⁰ ^ð34Þ

ARR2j:J_ejd

dtð

θ

_^ej^Þþfej

θ

_^ej^þK^ej^ð

θ

^ej^N^j

θ

^sj^Þⁱ^mj^k^ej^¼⁰ ^ð35Þ

ARR3j:J_sjd

dtð

θ

_^sj^Þþ^fsj

θ

_^sj^N^j^K^ej^ð

θ

^ej^N^j

θ

^sj^ÞR^Flj¼0 ð36Þ The associated residuals are:

R1j¼U0jRejimjLejd

dtðimjÞkej

θ

_ej ð37Þ R2j¼J_ejd

dtð

θ

_^ej^Þþ^fej

θ

_^ej^þK^ej^ð

θ

^ej^N^j

θ

^sj^Þi^mj^k^ej ^ð38Þ

R_3j¼J_sjd

dtð

θ

_sjÞþf_sj

θ

_sjN_jK_ejð

θ

ejN_j

θ

sjÞRF_lj ð39Þ From the BG model in Fig. 9, we can deduce three other residuals corresponding to the chassis dynamics. The associated residuals are as follows:

R4¼Fx1þFx2þFx3þFx4þm

ψ

_ ^v^_^m^u^€ ^ð40Þ R5¼Fy1þFy2þFy3þFy4m

ψ

_ ^u^_^m^v^€ ^ð41Þ R6¼ ½Fx1þF_x2þF_x3F_x4 cþ½F_y1þF_y2 a½F_y3þF_y4 bJ

ψ

€

ð42Þ The corresponding Fault Signature Matrix (FSM) is given in Table 2. The value, on the residuals columns, is equal to‘1’if the residual dynamic is affected by a faulty behaviour of the corresponding component. The vector (R1j, R2j, R3j, R4, R5, R6) is the signature of the fault. Decision making procedure is considered for residuals evaluation by taking into account speciﬁc thresholds.

When a residual exceeds the speciﬁc thresholds, a boolean value

‘1’ is affected in the FSM as explained before; otherwise, the residual evaluation remains equal to‘0’. As shown inTable 2, the only isolated faults are those corresponding toDf: _

θ

ej,Df: _

θ

sjand Jbecause each coherence vector related to these variables has only one match in the signature matrix. Hence, the fault can be detected and isolated because its signature is unique, i.e. different from the signatures of all other components. This result is explained by the sensors redundancy (i.e. motor velocity sensor, wheel velocity sensor…). However, due to the lack of unique signatures, faults affecting the other variables can not be isolated without exploiting the SBG, and this is a requirement to perform multiple faults diagnosis.

Fig. 10.BG model of thejth electromechanical system scheme (in derivative causality).

Table 1

Parameter values of the RobuTainer.

Subsystem Parameter Value

jth traction motor (electrical part) Lej 0.075H

Rej 0.32Ω

Kej 0.122 Nm/A

jth traction motor (mechanical part) Jej 0.0095 kg m²

fej 0.043 Nm s/rad

Kej 1 Nm/A

jth Gear part Nj 13

jth traction wheel fs 0.04 Nm s/rad

Js 4 kg m²

R0.38 m

Vehicle body m 2917.2 kg

a 2.3 m

b 2.3 m

c 2.3 m

h 0.486 m

Table 2

Fault signature matrix (FSM).

Parts Variables Residuals Monitorability

R1j R2j R3j R4 R5 R6 Db Ib

jth traction motor (electrical part)

Rej 1 0 0 0 0 0 1 0

Lej 1 0 0 0 0 0 1 0

U0j 1 0 0 0 0 0 1 0

kej 1 1 0 0 0 0 1 0

Df:imj 1 1 0 0 0 0 1 0

jth traction motor (mechanical part)

fej 0 1 0 0 0 0 1 0

Jej 0 1 0 0 0 0 1 0

Kj 0 1 1 0 0 0 1 0

Nj 0 1 1 0 0 0 1 0

Df: _θej 0 1 1 0 0 0 1 1

jth traction wheel R 0 0 1 1 1 1 1 0

Jsj 0 0 1 0 0 0 1 0

Df: _θsj 0 1 1 1 1 1 1 1

fsj 0 0 1 0 0 0 1 0

Chassis m 0 0 0 1 1 1 1 0

J 0 0 0 0 0 1 1 1

Df: _u 0 0 1 1 1 1 1 0

Df: _v 0 0 0 1 1 1 1 0

Df: _ψ 0 0 0 1 1 1 1 0

(11)

7.3.2. Qualitative reasoning based SBG

According to the developed model inSection 3, the corresponding SBG (see Fig. 11) is constructed directly from the BG model which is given byFig. 9. Then, the fault propagation is carried out in order to determine qualitatively the set of fault candidates according to the inconsistencies within the SBG model. A part of the fault propagation paths and the deduced possible conﬂicts is given inFig. 12.

7.3.3. Fault scenarios

Let us now take a fault scenario to illustrate the entire diagnosis procedure. The considered fault scenarios concern only components of the electromechanical system of the RobuTainer's Traction motor. The tests are performed on a vehicle dynamic simulator software (CALLAS/SCANeR Studio) and are based on real data of the RobutTainer. The software platform demonstrator CALLAS/

SCANeR studio is dedicated to testing and driving simulation applications mainly for vehicle engineering and research on improving transport safety and on implementing diagnostic algorithms before industrial design as an integrated embedded Deci- sion Support System.

In our study, a co-simulation is done using a program implemented under the Matlab/Simulink toolbox which represents the dynamics of the RobuTainer's traction motor, and another program implemented under CALLAS/SCANeR Studio platform wherein the whole dynamics of the RobuTainer are validated experimentally using real data of the vehicle.

Fault scenario. In this fault scenario, the current sensor is supposed to be nonfaulty and two faults have been introduced respectively on the inertia parameterJe1and the inductanceLe1at 100 s. The simulation results are shown inFig. 13. We notice that the value off₂, measured by the current sensor, increases and that the value off5corresponding to the rotation speed of the motor, increases as well. Hence,f_m₁takes the sign‘þ’andf_m₂takes the sign‘þ’on the SBG graph. Furthermore, according to the residuals evaluation within the samefigure (Fig. 13), we observed that the residualsR3,R4,R5andR6are insensitive to the fault. This result corresponds to the fault signature vectorV¼(1, 1, 0, 0, 0, 0). The set of possible conflicts which involves the sensitive parameters respectively to the residuals R1 and/or R2 namely the first six parameters of the FSM (seeTable 2) is presented inFig. 12. The

procedure is described inSection 4. It is worth noting that, the FSM logic concentrated on determining a single faulty component that explains all the symptoms. However, this leads to a faulty diagnosis in our scenario because from a quantitative approach point of view, the sensitivity of both R1 and R2 can only be explained by a fault which affects one common element involved in the calculation ofR₁andR₂as well. Nevertheless, the determi- nation of the propagation paths within the SBG given inFig. 11 enables the detection and isolation of suitable faults. Indeed, from Fig. 12and according to the observations (Fig. 13), the possible conflicts arePC3andPC4. However, according to the FSM (Table 2) the possible conflictPC4can befired because a fault affecting these two elements can not affect the residualsR1. Hence, the set a fault candidates corresponds toPC3 namely the two components Je 1

andLe1. We notice that the only analysis of the FSM leads to a wrong result in this case of multiple faults because of the fault's compensation. However, the fact that we rely on qualitative reasoning emanating from a common model enhanced the decision procedure. It is worth noting that even if the qualitative reasoning improves the diagnosis tasks, in some cases only partial results could be obtained, as explained inFig. 7.

8. Conclusion

Based on SBG and BG, a two level framework has been presented for incipient fault diagnosis. The basis of the framework is to integrate two different types of information namely quantitative knowledge acquired from the behavioural, causal and structural features of the BG and qualitative knowledge based on the developed SBG model. This model allows to fulﬁll fault propagation and to better understand some of the concepts related to BG. The BG is a graphical tool that make easier the generation of the ARR which have proved their adequacy to detect and isolate (when possible) single faults. By associating the ARR and the SBG, all built from the same model: the BG, we contribute to the development of a global FDI procedure which enables the detection and the isolation of single and multiple faults. The application of the integrated framework has been fulﬁlled through the electrical and autonomous vehicle namely RobuTainer designed within the framework of the European project InTraDE.

Fig. 11.SBG model of the IAV emanating from the BG model.

(12)

The fault scenarios have been tested exhaustively and prove that it is suitable for dealing with multiple fault diagnosis.

This work adds new features to the BG in order to enhance the existing fault isolation algorithms by incorporating qualitative features that take into account temporal orders of measurement deviations within a speciﬁc sliding time window, to address multiple faults, and to provide an integrated framework, we called the SBG, that extends the fault detection procedure based residuals generation. We also make necessary extensions via a set of deﬁnitions and properties allowing to check the consistency of

observations over time with the theoretical system behaviour.

Furthermore, we illustrate by the use of the SBG, how the effects of multiple faults can be formed in a consistent manner from the set of observations. Hence, we improve the efﬁciency of multiple fault isolation by using the notions of logic-based diagnosis, and by limiting the cardinality of the multiple fault hypotheses according to the quantitative diaognosis based residuals'evaluation. Using the SBG, we can reason efﬁciently about the qualitative behaviour of the system when a fault occurs with respect to nominal behaviour.

Fig. 12.Consistent paths and fault propagation.

0 100 200 300 400 500 -5

0 5

Time (s)

R3

0 100 200 300 400 500 -0.05

0 0.05

Time (s)

R4

0 100 200 300 400 500 9

10 11

Time (s)

U01

0 100 200 300 400 500 -5

0 5

Time (s)

R2

0 100 200 300 400 500 -0.05

0 0.05

Time (s)

R1

0 100 200 300 400 500 0

50 100

Time (s) MotorSpeed sensor

0 100 200 300 400 500 0

50 100

Time (s) WheelSpeed sensor

0 100 200 300 400 500 0

0.5 1

Time (s) Current sensor

0 100 200 300 400 500 -1

0 1

Time (s)

R6

0 100 200 300 400 500 -50

0 50

Time (s)

R5

Fig. 13.Measurements for the introduced faults.