Bond Graph Model Based and Fuzzy Logic For Robust FDI of Mechatronic Systems

Bond Graph Model Based and Fuzzy Logic For Robust FDI of Mechatronic Systems

N. Chatti^∗ A-L. Gehin^∗ R. Merzouki^∗ B. Ould Bouamama^∗ Y. Touati^∗

∗Polytech-Lille, LAGIS, CNRS-FRE 3303, Avenue Paul Langevin, 59655 Villeneuve D’Ascq, France

E-mail: nizar.chatti@polytech-lille.fr, anne-lise.gehin@polytech-lille.fr

Abstract: Fault diagnosis is crucial for ensuring the safe operation of complex engineering systems and avoiding to execute an unsafe behaviour. This paper deals with robust decision making (RDM) for fault detection of an electromechanical system by combining the advantages of Bond Graph (BG) modelling and Fuzzy logic reasoning. The proposed fault diagnosis method is implemented in two stages. In the first stage, the residuals are deduced from the BG model allowing to build a Fault Signature Matrix (FSM) according to the sensitivity of residuals to different parameters. In the second stage, the result of FSM and the robust residual thresholds are used by the fuzzy reasoning mechanism in order to evaluate a degree of detectability for each set of components. Finally, in order to make robust decision according to the detected fault component, an analysis is done between the output variables of the fuzzy system and components having the same signature in the FSM. The performance of the proposed fault diagnosis methodology is demonstrated through experimental data of an omnidirectional robot.

Keywords:Bond graphs, Fault diagnosis, Fuzzy logic, Fault detection, Robust decision making, Mobile robotics.

1. INTRODUCTION

Nowadays, the growing demand for safety and reliability of modern engineering systems motivates the development of new fault diagnosis algorithms for the decision support system. It is worth noting, that a wide variety of concepts, methods and tools have been developed to address deci- sions challenges that confront a large degree of uncertainty for evaluating residuals including both the fault detection and the isolation capabilities. Fault detection and isolation (FDI) procedures consist of comparison between the real and the reference process behaviors. In the literature, dif- ferent approaches for the FDI have been developed based on quantitative and qualitative models. A good amount of these is given in Chatti et al. (2011). Among robust diag- nosis quantitative methods, Djeziri et al. (2007) deals with the generation of fault indicators and residual thresholds in the presence of parameter uncertainties by using a bond graph representation.

Models allowing to produce fault indicators are very im- portant in the design of a safe system. Nevertheless, to de- termine the exact mathematical relationships between the physical values characteristic of the system is not always easy, especially if the system makes appear phenomena of different natures as in an electromechanical system. Bond Graph is a multidisciplinary and unified graphical mod- elling language which has proved its adequacy to represent energy exchanges in mixed systems Ould-Bouamama and

This work was supported by European funds in the framework of InTraDE (Intelligent Transport for Dynamic Environment), from Interreg IVB NWE program.

Samantaray (2008). Bond Graphs have first been used as modelling tool, and their causal and structural prop- erties (observability, controllability, monitorability) have been subsequently used to generate fault indicators in a systematic and generic way Merzouki et al. (2009), Boon- low et al. (2010). Moreover, Bond Graph modelling has also been used in the past for different FDI approaches Ould-Bouamama et al. (2005).

In Bouallegue et al. (2010), residual expressions are gen- erated from Bond Graph model in derivative causality and detection is based on residual fixed thresholds around a working point and adaptive thresholds that vary ac- cording to the set point. One of the drawbacks of this classical method is the uncertainty and instability of the decision especially with regard to time when the residual values are near the thresholds. In order to remedy to this situation, Henry and Zolghari (2010) proposed a linear fractional transformation (LFT) uncertain model with fil- tering methods for residuals generation and evaluation. A robust FDI with respect to parameter uncertainties using Bond Graph modeling approach in the LFT configuration is also proposed in Djeziri et al. (2007).

Nevertheless, even if some applications of these approaches exist, fault detection and isolation procedure is not trivial.

Basically, when the residual is weakly sensitive to the fault or to a value very near from the thresholds boundaries, the fault alarms still exist. To overcome these limitations, we propose in this paper a robust decision making approach (RDMA). It represents a particular set of methods and tools designed to support decision making under condi- tions of deep uncertainty. This RDMA is based on Fuzzy

logic methodology and aims to facilitate the identifica- tion of vulnerabilities by identifying the residuals’ change through time. The proposed RDMA is implemented in two stages. In the first stage, the residuals are deduced from the Bond Graph model and then a Fault Signature Matrix (FSM) is built according to the sensitivity of residuals to different parameters. In the second stage, the result of FSM and the robust chosen thresholds are used by the fuzzy reasoning mechanism to evaluate a degree of detectability for each set of parameters having the same signature in the FSM.

The proposed paper will be organized as follows. In a second part, the problem statement is given. In the third section, a short introduction recalls the principles of Bond Graph (BG) methodology. The section 4 deals with the generation of Analytical Redundancy Relations (ARRs) from BG. Section 5 is devoted for the proposed Fuzzy logic methodology. In section 6, we illustrate the proposed approach to a traction system in omni directional mobile robot named Robotino and finally section 7 concludes the paper by highlighting the strengths of the proposed approach.

2. PROBLEM STATEMENT

The contribution in this paper consists in an enhancement of the availability of the information for a robust decision making by generating a set of fuzzy rules. This is impor- tant, in situations, where the thresholds of residuals cannot be accurately determinate due to the presence of perturba- tions or uncertainty of thresholds fault detection especially with regard to time when the residual values approach the thresholds values. In a first step, the dynamic model of the physical system is done using Bond Graph (BG) in order to represent all the system parameters, uncertainties and energy variables. Then, the generation of Analytical Redundancy Relations (ARRs) is done for the uncertain system by decoupling the nominal part from the uncertain part. The residual represents the nominal part of the ARR, while their adaptive thresholds are calculated from the uncertain part of the ARRs. After that, residual sensitivity analysis is done, by using the ARR uncertain part, in order to calculate the fault detectability indexes. Finally, using the robust thresholds, the membership functions of each residual is defined by using the Fuzzy Reasoning Mechanism (FRM) in order to evaluate more accurately the degree of detectability from the output variables of the Fuzzy system, which are the faults that affect different parameters. As a result, the procedure allowing to make an accurate decision in presence of fault is improved. An overview of the approach is given in Fig. 1.

3. BOND GRAPH METHODOLOGY

The Bond Graph Model (BGM) is a graphical description of dynamic behaviour of physical systems. This means that systems from different domains (electrical, mechanical, hydraulic, thermodynamic ) are described in the same way.

Indeed, a BGM consists of subsystems linked together by a set of half arrows representing power bonds and direction of power. Each process is then described by a pair of variables (effort e and flow f) and their product is the power Thoma (1975), Borutzky et al. (1999). Furthermore,

Bond Graph modelling is a powerful tool for modelling engineering systems, especially when different physical domains are involved. A Bond-graph model is an oriented linear graphG(E, B, J) where E∪J are the vertices and B are the edges.

• E = e is the set of the multi-port elements repre- senting fundamental energetic processes. It groups the following elements:

· Seand Sf represents an effort source or aflow source, associated with energy sources from func- tional point of view.

· I represents the inductive element, for example an inductance for an electrical system or mass for a mechanic system, associated with a dynamic phenomenon storage of kinetic energy.

· C represents the capacitive element, such as the capacity for an electrical system or rigidity for a mechanical system, associated with a function of energy storage (e.g., spring).

· Rrepresents the resistive element, such asresis- tance for electrical system or the viscous friction for a mechanical system, associated with elements that dissipate energy (e.g., shock).

· T F transformer and GY gyrator are used to express the transformation of energy from one domain to another while respecting the conser- vation laws of energy, which are associated with transformation functions.

· DeandDfrepresent respectively an effort detec- tor and a flow sensor, which are associated with measurement functions.

• B={b} is the set of the orientedbonds representing the power energy transmission. Each of which is defined by bij = {ni, nj} where ni ∈ E∪J is the origin node andnj ∈E∪J is the destination node.

Two variables are associated with a bond bi: the instantaneousenergye_iand the instantaneousflow of energyfi. The power is then obtained by the relation:

p=e×f

• J ={j} is the set of the junctions representing the energy conservation laws. Each of which is defined by j ={B_I, BO} where BI ⊂B is the set of the input bonds coming to the junction and BO ⊂ B is the set of the output bonds going from the junction. Two types of junctions are distinguished:

· junction 0: at this node ei =ej ∀bi, bj ∈BI ∪ BO and

n

i=1

fi =

m

j=1

fo where bi ∈BI, bo ∈BO

andn is the number of element of BI, m is the number of element ofBO,

· junction 1: at this node fi =fj ∀b_i, bj ∈BI ∪ BO and

n

i=1

ei =

m

j=1

eo where bi ∈ BI, bo ∈BO

andn is the number of element of BI, m is the number of element ofBO.

From the functional point of view, a junction is associated with energy conservation functions. This feature is not available if a candidate fault indicator generated from this junction is not consistent with the actual condition.

Physical System

Generation of

ARRs Robust Threshold

Detection

Membership Functions Fault Detection and Isolation

Bond Graph Model

Determination of residuals and thresholds

Fuzzy Reasoning Mechanism

Robust Decision Making

Fig. 1. Overview of the proposed approach.

4. ALGORITHM OF ARRS AND THRESHOLDS GENERATION

4.1 ARRs generation

FDI procedures considered in this paper are based on residuals deduced from the BGM and rely on the exis- tence of analytical relations. This model based residuals provides many interesting features comparing to observer based residuals because it corresponds to relations between system variables which can be explained from the physical model so it is simple to understand. Also, these relations can be easily displayed graphically using Bond Graphs and they can be defined under a symbolic form using symbolic computation software. The structure of any physical sys- tem can be represented by a bond graph model. For the BGM, constraints and variables are the two partitions of the bipartite graph that are directly deducted from the graphical model. The set of the known variables is:

K = (De, Df, Se, Sf, M Se, M Sf, θm) where De, Df are the effort and flow detectors corresponding to the sensors, S_e, S_f, M S_e, M S_f, are the effort sources, flow sources, modulated effort sources, modulated flow sources corre- sponding to the actuators andθmis the set of the process parameters. A bond graph model with correct causal- ity means that the corresponding system of equations is solvable and then the set of unknown variables can be calculated. The over-determined sub-systems correspond to the observable sub-graphs in the bond graph model of the system. The residuals are essentially combinations of observable systems and thus they can be built using the graph analysis techniques. These combinations of observed process variables are obtained through identification of sub-graphs of the bond graph model in which all the outer vertices are either actuators (inputs) or detectors (outputs). Strong decoupling with respect to unknown inputs occurs naturally since only sub-graphs with known outer vertices give rise to residuals Staroswiecki (2000).

The outer vertices of each such sub-graph form a constraint whose structure is given by the edges. This constraint cor- responds to an analytical redundancy, which when avail- able in equation form (i.e. symbolically resolvable) leads to an ARR (residual computation form). Then each residual can be rewritten asr−f(De, Df, Se, Sf, M Se, M Sf, θm) = 0. These ARR deducted from the BGM allows to detect and isolate fault that affect system components and then to evaluate the potential the system has to continue to achieve its objectives from the services provided by the non-faulty components.

4.2 Residuals Thresholds Generation

The following rules are used to generate the thresholds:

• Get a BG model and put it in preferred derivative causality.

• Model the measurement uncertainties directly on this model.

• Use the equations of energy conservation to write the ARRs, and use the causal path to eliminate the unknown variables.

• Write the ARRs of detectors redundancy.

• For all ARRs derived from the equations of energy conservation, add the maximal absolute values of the different parts of the ARRs containing the measure- ment errors to obtain the threshold.

• For the ARRs generated from the redundant de- tectors, the thresholds are equal to the sum of the maximum measurement errors of the two redundant detectors.

• Repeat the procedure until all the junctions are considered and all ARRs with distinctly separate signatures are obtained.

Inspection of ARRs with respect to the known variables and component parameters they link leads to a so-called fault signature matrix (FSM). Looking at ARRs, an occur- rence matrix can be set up with one row for each known variable or component parameter and one column for each residual. A known variable or component parameter present in an ARR is indicated by1, its absence by 0. That is, the resulting matrix shows which components contribute to which residuals.

5. FUZZY LOGIC METHODOLOGY

The essence of fault diagnosis based on Fuzzy Logic is to accumulate the operating experience to organize the control structure to improve control performance automat- ically Zhang et al. (1999). Especially, it appears very useful when the system parameters are uncertain or imprecise or when the system is too complex to be analysed by mathe- matical methods. Nevertheless, because this methodology does not use structural information but the expert’s expe- rience, its several steps in design will be difficult to gen- eralize. In contrast, BGM contains the accurate structural information required to generalize design steps for robust decision making based detection using Fuzzy logic. This is why, we proposed in this paper to combine BGM and Fuzzy Logic in order to detect more accurately the fault when it appears. The following paragraph describes the Fuzzy logic methodology.

A fuzzy logic model (see. Fig.2) with its fundamen- tal input-output relationship consists of four components namely, the fuzzifier, the inference engine, the defuzzifier, and a fuzzy rule base Zadeh (1965).

Fuzzifier The Inference Engine Defuzzifier Crisp

Values

Crisp Values

Residuals DecisionMaking

Membership Functions based on Robust Thresholds

IF-THEN rules Fuzzy variables

Fig. 2. Fuzzy Logic Model.

5.1 Fuzzification

For each input and output variable selected, we define three membership functions (MF). A qualitative category is also defined for each one of them based on the thresholds values for the inputs (residuals) according to the residual sensitivity to certain faults and a degree of detectability for the outputs (faults having the same signature in the FSM). The shape of these functions can be diverse but in our work, only triangles and trapezoids are used.

Indeed, a residual will be coherent with the model of the system if it is null or inferior to a chosen threshold and it represents a fault indicator that reflects the faulty situation of the monitored system. The residual can be also positive or negative. This is why, we associate for example for each value taken by the residual a membership function corresponding to “Negative”, “Zero” or “Positive” value and the point definitions of each MF is based on the robust thresholds and experimentations in both normal and faulty situations.

5.2 Rule base

A supervisory expert system is built using the rule based structure IF -THEN representing the correlation between facts and states or actions Graham et al. (1988). In this paper, a rule derives operating knowledge from given resid- uals deduced from the BGM and is generated from the expert knowledge and in our case from the FSM logic and experimentation which have been done by introduc- ing some different faults and by analyzing the effect on each residual. A fact is a description of the relationship between an input variable and its output variable. The rule based structure is made using the Fuzzy Logic tool- box for MATLAB. The Fuzzy outputs are distinguished in three levels (Insensitive, Uncertain, and Sensitive) in order to be used by the diagnosis. The Fuzzy Rule Base algorithm embedded in MATLAB Fuzzy Logic toolbox has the following steps: the inputs are transformed into memberships of fuzzy sets by fuzzifying functions. This information is given to the inference engine. After that, membership values are transformed into required output variables by a defuzzification step.

5.3 Defuzzification

The MFs of the output have always the same shape and configuration in our risk model: the risk of any problem has the same ranks for the MFs of the output:

“Insensitive”, “Uncertain” and “Sensitive” and always without overlapping for the certain values which are equal to 1. In order to obtain a percentage of detectability of each set of faults, the output is defuzzyfied. The equations of the straight lines of each MF of the output are calculated.

The calculations for each of the MFs are done by using the formalism described by Mamdani. In fact, Mamdani FIS is the most known or used in developing fuzzy models.

The output of the system is generally defuzzified resulting fuzzy sets which are combined using aggregation operators from the consequent of each rule of the input as used by Akter et al. (2005). Basically, the comparison of the outputs allows to identify the higher value corresponding to the fault signature detected in the system.

6. APPLICATION

The Robust decision making approach for fault detection developed in this paper is applied to a traction system in mobile robot named Robotino (see Fig.3). The Robotino is an autonomous omnidirectional three wheel drive mobile robot. For actuation, three DC motors with two sensors for each one of them (measuring the current and the angular speed) and a reducer are used. For our experimentation, we focus on the electromechanical traction system of the Robotino composed by a DC motor, sensors and a wheel as seen in the next paragraph. Furthermore, we apply different kind of faults to the Robotino in order to demonstrate the consistency of our approach.

1 - s

2 -3 s

Nm.sec.rad 0.02 R

Kg.m 10 63 J

u 16

r 43.1mV m

-1 e 47ȝNm.sec.rad R

8.13ȍ R

8.9mH L

a a

2 -6

e 7.9510 Kg.m

J u

;

;

;

;

;

Fig. 3. Traction System in Omnidirectional Mobile Robot:

Robotino

6.1 System Bond Graph Modelling

The BG Model of the system is elaborated in integral causality (see Fig.4) as well as the corresponding model in derivative causality (see Fig.5). The electric power is provided by the electrical part of a DC motor which is equivalent to an input voltage source U in serial with a resistance Ra and an inductance Le. The electrical current is measured by the sensor Df : i. The gyrator elementGY describes the power transformation from the electrical part of the DC motor to its mechanical part which is characterized by its rotor inertia Je, its viscous friction parameterRe. The mechanical gear which links the mechanical and wheel parts is represented by a transformer elementT F. The wheel is characterized by its inertiaJs, its viscous friction parameterRsand a contact effortFx.

șm

:

Df

i : Df

1

TF MSe:II(Fx) I:Js

R:Rs 1/r 1 I:Je

1 MSe:U

R:Ra I:Le

GY m

R:R_e

Fig. 4. BG Model in preferred integral causality 6.2 ARRs Generation

From the model of Fig. 5, two ARRs are established linking only the known variables and the parameters. They are the

m 2:ș SSf i

: SSf1

1

TF MSe:II(Fx) I:J_s

R:Rs 1/r 1 I:J_e

1 MSe:U

R:Ra I:L_e

GY m

R:Re

Fig. 5. BG Model in derivative causality following:

ARR1:U−Ra.i−La._dt^d(i)−m.θ˙m= 0 ARR2:m.i−Jed

dt( ˙θm)−Reθ˙m−^J_r2^s._dt^d( ˙θm)−^R_r2^s.θ˙m+¹

r.φ(Fx) = 0

The Fault Signature matrix is then constructed by stacking the coherence vectors as seen in Table 1. The matrix is extended by two additional columns. The first one with the heading M indicates whether a fault that can be detected. The second additional column with the heading

I indicates whether a fault that can be isolated.

R1 R2 M I

Le 1 0 1 0

Ra 1 0 1 0

MSe 1 0 1 0

m 1 1 1 0

SSf1 1 1 1 0 SSf2 1 1 1 0

Re 0 1 1 0

Je 0 1 1 0

Js 0 1 1 0

Rs 0 1 1 0

Fx 0 1 1 0

Table 1. Fault Signature Matrix

Using the procedure of input and output uncertainties modeling developed in Touati et al. (2011), the following thresholds can be generated directly from the graphical model using the causal paths:

a1 = max

−Raζi−Ladζi

dt −mζθ˙m

;

=Ramax (ζi) +Lamax

_dζ

i

dt

+m·max

ζθ˙m

; a2 = max

−mζi−Reζθ˙m−Je

dζθ˙m

dt −Js

r² dζθ˙m

dt −Rs

r²ζθ˙m

;

=m· max (ζi) +Remax

ζθ˙m

+Jemax

dζθ˙m

dt

+Js

r²max

dζθ˙m

dt

+Rs

r² max

ζθ˙m

.

Where ζi and ζθ˙m are the measurement errors on the current and velocity detectors respectively.

And where:

max dζθ˙m

dt

= 2·max ζθ˙m

Δt max

dζi

dt

= 2·max (ζi) Δt and Δtis the sampling time.

6.3 Fuzzy Logic approach

The rule based structure is made using the Fuzzy Logic toolbox for MATLAB. The Fuzzy outputs are distin- guished in three levels (Insensitive, Uncertain, and Sensi- tive) in order to be used for the diagnosis. The Fuzzy Rule

Base algorithm embedded in MATLAB Fuzzy Logic tool- box has the following steps: the inputs are transformed into memberships of fuzzy sets by fuzzifying functions. This information is given to the inference engine. After that, membership values are transformed into required output variables by a defuzzification step. The main objective is to ensure more accurately the degree of detectability of the output variables of the fuzzy system, which are the faults that can occur in different parameters. The residuals are the input variables considered in the system by taking into account the thresholds of each residual and different ranges that can be reached by these latter.

Three types of membership functions having three mem- bership functions in each namely combinations of trape- zoidal and triangular members are considered. Linguistic variables such as “Negative”, “Zero”, “Positive” and “In- sensitive”, “Uncertain”, “Sensitive” have been considered for evaluating respectively residuals and the degrees of de- tectability of different faults namelyf01which corresponds to the last five rows of Table 1,f10 which corresponds to the first three rows of Table 1 andf11 which corresponds to the fourth, fifth and sixth rows of Table 1.

Some of the fuzzy rules are activated according to the information acquired by the fault signature matrix logic.

The outputs of the activated rules are weighted by fuzzy reasoning and the degrees of detectability of different faults are calculated. The parameters defining the input and output functions are seen in Fig.6 where a1 and a2

represent the robust thresholds for each residual.

0.5

IN UN SE

0.35 0.65

0 1

df01

df₁₀

df11 0

R₁

R2

N Z P

ai1 ai2 1

IF-THEN rules based Fuzzy Logic

Fuzzification Defuzzification

Fuzzy Inference Engine

Fig. 6. Fuzzy Logic Approach and Membership Functions

6.4 Experimental Results

The SIMULINK diagram has two parts; the first one is the model of the Robotino with the inputs coming from this latter and the second one is the Fuzzy Block Controller which takes into account the residuals deduced from BG as explained before and gives the outputs namely the degrees of detectability of each residual. Then, the comparison between these different outputs allows a robust decision making in term of which signature is observed and therefore which fault is detected. Fig.7 corresponds to the residuals’ response in normal operation. Then, many fault scenarios have been tested. For instance, in Fig. 8, the response of the residuals to a fault in the current sensor which is introduced during a 9s time interval, from 5 s to 14 s is presented. The comparison of the three outputs of the Fuzzy system (see Fig.9) reveals that the signature observed is “f11” which correspond to the introduced fault.

Furthermore, whatever the values taken by the residuals, the fuzzy outputs allow to detect accurately the signature corresponding to the fault when it occurs.

0 2 4 6 8 10 12 14 -4

-2 0 2 4

Time (s) R1

0 2 4 6 8 10 12 14

-0.02 -0.01 0 0.01 0.02 0.03

Time (s) R2

R1(V)R2(N.m)

Fig. 7. Residuals in normal functioning

0 2 4 6 8 1 0 1 2 1 4

- 0 .0 2 - 0 .0 1 0 0 .0 1 0 .0 2 0 .0 3

T im e ( s )

R2

a1 a2

a1

a2

0 2 4 6 8 10 12 14

-6 -4 -2 0 2 4

Time (s)

R1(V)

Fig. 8. Residuals in faulty situation

0 2 4 6 8 1 0 1 2 1 4

0 .2 0 .3 0 .4 0 .5 0 .6 0 .7

T im e ( s ) f0 1

0 2 4 6 8 1 0 1 2 1 4

0 . 2 0 . 3 0 . 4 0 . 5 0 . 6

T im e ( s ) f 1 0

0 2 4 6 8 1 0 1 2 1 4

0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7

T im e ( s ) f1 1

Fig. 9. Fuzzy system outputs

7. CONCLUSION

A robust decision making approach for fault detection based on fuzzy evaluation of residuals has been elaborated and implemented for an electromechanical system. The main contribution of this study concerns the use of a behavioral model namely Bond Graph (BG) and the defi- nition of membership functions by using the robust thresh- olds deduced from BG in order to detect more accurately a fault upon occurrence. The results of implementation in a real system reveal the quality of the fuzzy logic in diagnosing the faults especially for the cases that the crisp logic is not able to detect accurately a fault.

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