Detectability and Isolability Conditions in Presence of Measurement and Parameter Uncertainties Using Bond Braph Approach

8th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes (SAFEPROCESS) August 29-31, 2012. Mexico City, Mexico

Detectability and Isolability Conditions in Presence of Measurement and Parameter Uncertainties Using Bond Braph Approach Y. TOUATI ∗ R. MERZOUKI ∗ B. OULD BOUAMAMA ∗ R. LOUREIRO ∗ ∗

Polytech’lille - University of Lille1, (e-mail: [email protected]).

Abstract: In this paper, a procedure of fault modeling, estimation and isolation is developed using the the analytical redundancy relation in order to study the diagnosability conditions of faults. The ARR generation is performed by using the bond graph model of the system in presence of parameter and measurement uncertainties. The procedure of fault modeling and estimation is developed by using the same procedure of uncertain modeling and the bi-causality notion. The transfer function between the residual and the fault is generated systematically from the graph. A co-simulation to a suspended quarter of an electrical vehicle is given to verify the work. Keywords: Diagnosis, Fault detection and isolation, Fault estimation, Bond graph, uncertainties, sensitivity. 1. INTRODUCTION Fault detection and isolation has been the subject of many researches because of its crucial role for systems, human and environment safety. The objective is to detect the faults with a minimum delay and fault values. Many diagnosis methods has been developed using the qualitative and quantitative approaches. The latter is more used such as the observer (Chen, 1999), the parity space(Staroswiecki, 1999) and the graphical approaches (Djeziri, 2006). These methods is mainly based on the generation of analytical redundancy relations (ARRs) which are close to zero in normal operating and up to a certain value named threshold in faulty situation. The threshold is calculated using the known information about measurement and parameter uncertainties, or it can be deduced experimentally after the evaluation of the residuals. In order to derive mathematically the thresholds, an approach based on parity space has been developed in Han (2002). In the latter, the state equations have been used in discrete time in order to represent the uncertainties to detect sensor and actuator faults. The measurement and parameter uncertainties are represented by bounded variables in order to generate the residual envelop. A survey of residual generation and evaluation using generally observer approach can be found in Frank (1997). Most of published papers in these last years try to eliminate the effect of the parameter or measurement uncertainties on the residuals, which can cause the non-detection of certain faults, such as the filtering approach. For example, in Casavola (2008), the robust FDI problem is treated using deconvolution filters (H∞ and H− ) with a quasi-convex ⋆ Sponsor and financial support acknowledgment goes here. Paper titles should be written in uppercase and lowercase letters, not all uppercase.

978-3-902823-09-0/12/$20.00 © 2012 IFAC

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linear time invariant formulation. In Jinchao (2009), the uncertainties are modeled using the formulation of the robust fault detection filter design, as H∞ model-matching problem, the problem is optimized using the Linear Matrix Inequality (LMI) technique. This paper deals with fault detection and isolation using bond graph approach. The latter consist of using the graphical model for structural analysis, ARRs and threshold generation. The causal proprieties of the graph is used to study the observability and the monitorability of the system Sueur (1991) . Moreover, the structural proprieties are used to generate the ARRs by a systematic way using the causal paths to eliminate the unknown variables. The FDI using bond graph for qualitative and quantitative approaches is well presented in Samantaray (2008). In Djeziri (2006), a robust diagnosis method with respect to parameter uncertainties has been developed using the bond graph approach in the linear fractional transformation (LFT) form. The method has been applied to a real dynamic system in order to detect the effect of the backlash phenomenon on the dynamic of a mechatronic system. In this work, an algorithm to fault estimation is proposed based on the bond graph modeling in linear functional transformation (LFT) and the bi-causality notion. The Algorithm consists in the generation of the transfer function between the residual and the fault automatically from the graph. The latter is used after to calculate the minimum detectable and isolable fault or the condition of fault detection and isolation. This work is organized as follows: in section 2, the measurement and parameter uncertainties modeling is presented. In the section 3, The fault modeling and the generation of

10.3182/20120829-3-MX-2028.00155

SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico

transfer function between the residuals and the faults are developed. Finally, a co-simulation to a suspended quarter of an electrical vehicle is done to verify the procedure. 2. DIAGNOSIS BY BOND GRAPH IN PRESENCE OF UNCERTAINTIES 2.1 Parameter uncertainties Robust diagnosis to parameters uncertainties using bond graph approach is based essentially on the decoupling of the uncertain part from the nominal part of the ARR, using directly the bond graph model of the system. This approach has been developed in Djeziri (2007) using the bond graph model in linear fractional transformation named BG-LFT representation Kam (2005). This approach consist in replacing the bond graph elements by BG-LFT elements in order to obtain decoupled residuals. The nominal part of the ARR is used to calculate the fault indicator while the uncertain part is used to calculate the threshold in real time. 2.2 Measurement and input uncertainties As shown in Y. TOUATI (2011), the measurement uncertainty modeling on the bond graph model can be done as follows: • The measurement uncertainty of an effort detector can be modeled by a virtual source of effort M Se∗ : ζSSe (Fig.1-a).

• Model the measurement and parameter uncertainties directly on the model. • Write the ARRs of the model using the equations of energy conservation, and use the causal paths to eliminate the unknown variables. • Write the ARRs of detectors redundancy. • For all ARRs derived from the equations of energy conservation, the threshold is obtained by adding the maximal absolute values of the different parts of the ARRs containing the measurement errors. 3. RESIDUAL/FAULT TRANSFER FUNCTION GENERATION The parameter, input and output faults can be represented in the bond graph using the LFT form as the same as we represent the uncertainties. The sensitivity function of a fault can be easily generated from the bond graph model using the bi-causality notion. The idea is to use the causal path from the detector (which is connected to the observed junction) to the concerned faults. The observed junction is the junction which has at least one detector connected to it. As Fig. 2 shows, the Fault Fm can be calculated from the unknown variables Z1 and wF 1 . These unknown variables can be calculated using the causality and bicausality notion. The sensitivity function can be deduced from the bond graph by following the causal path from the detector to the source that represents the fault. The function of the sensitivity is equal to the gain of this causal path. Proof: Let us consider the linear system represented in Fig. 2. Two residual can be generated from the model

SSe = SSer + ζSSe Where SSer is the real effort and ζSSe is the measurement error. • The measurement uncertainty of a flow detector can be modeled by a virtual source of flow M Sf ∗ : ζSSf (Fig.1-b).

MSf : Fl

C : C1

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0

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SSf = SSfr + ζSSf Where SSfr is the real flow and ζSSf is the measurement error. The input uncertainties are modeled by a source of effort if the input is a source of effort and by a source of flow if the input is a source of flow.

MSe* : ] SSe

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Fig. 2. Bond graph model in derivative causality with dualized detectors.

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Fig. 1. Measurement uncertainty modeling. 2.3 Thresholds generation in presence of uncertainties

SSe1 : P1

• Put the model in preferred derivative causality if possible. 959

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Fig. 3. Faulty bond graph model in derivative cauality.

SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico

in derivative causality 1: 1 dP1 − (P1 − N P2 ) − fSSe1 = 0 dt R1 dP2 1 N (P1 − N P2 ) − C2 − fSSe2 = 0 (1) ARR2 : R1 dt Where fSSe2 and fSSe2 is equal theoretically to zero. So we can write from the second ARR the following equation 2: 1 dP2 fSSe2 = N (P1 − N P2 ) − C2 = 0. (2) R1 dt ARR1 : F l − C1

When we apply the bi-causality to the detector in faulty bond graph, the residual is also equal to the effort of the detector because we represent the fault on the model. The relation between the residual and the source that represent the fault can be deduced by following the causal path by starting from the effort of the sensors in bi-causality to the corresponding parameter fault. And in this case, the sensitivity function between the residual and the fault in the system 2 is equal to the following function: FR = −

WFR = ZR

1 R1

−N r2 (P1 − N P2 )

(3)

In general the relation between the residual and the fault can be deduced using the following rule 4: Parameter faults: P WFe − G (ri → WF ) Fe = − = · ri . (4) Ze Ze Where ri is the residual i, G (ri → WF ) is the gain of the causal path that link the inactive variable of the detector to the source which represent the effect of the fault. Ze is the input of the FBG-LFT element. e is the faulty element. Actuator fault: X WF = G (ri → WF ) ·ri Sensor fault:

c1 c2 .. . cm

r1 s1,1 s2,1 .. . sm,1

r2 s1,2 s2,2 .. . sm,2

··· ··· ··· .. .. .. ... ···

rn s1,n s2,n .. . sm,n

a1

a2

···

an

Ib g1 g2 .. . gm

Ibc b1 b2 .. . bm

Db e1 e2 .. . em

Dbc f1 f2 .. . fm

Table 1. Robust Fault Signature Matrix. Let us define S m×n as a matrix of boolean values si,j , where:  1 if the expresion of rj contains ci . si,j 0 otherwise. Let us define G (column Ib in the RFSM) as an isolability vector, where:  1 if the signature {si,1 ,si,2 , ..., si,m }is unique, gi = 0 otherwise. Let us define the transfer function between the fault and the residual as Fr = ΓF |r . If the signature of the fault is unique, then the fault is isolable if its effect on the residuals which are sensible to this fault is bigger then all the associated thresholds values. The minimum detectable fault values are represented the B (Ibc column)vector called the isolability condition vector (Table 1), where: −1 −1 bi = 2max(si,1 Γ−1 F |ri (a1 ), si,2 ΓF |ri (a2 ), ..., si,n ΓF |ri (an ))

In the robust fault signature matrix, we can define E (Db column in the RFSM) as a detectability vector and F (Dbc column in the RFSM) as detectability condition vector that represents the minimum detectable value of the fault, where: −1 −1 fi = 2min(si,1 Γ−1 F |ri (a1 ) , si,2 ΓF |ri (a2 ) , ..., si,n ΓF |ri (an )) ei = si,1 ∨ si,2 ∨ ... ∨ si,n . 5. CASE STUDY: INTELLIGENT AUTONOMOUS VEHICLE

∗ G ri → FS,i   ri In this section, we validate the presented approach of P  n ∗ 1 → F∗ − · · · − G F → F FS,i 1−G S,i S,i S,i robust fault diagnosis on a quarter part of Intelligent and Autonomous Vehicle (Fig. 4). This part describes The fault of the sensor must be represented by a mod- the dynamics of electromechanical system used to drive ulated source on all bonds which are connected to the the decentralized controlled wheel. For this case, two observed junction, and only one source can be in bi- types of uncertainties are considered: measurement and ∗ causality, so we define FS,i as the latter. The others are parameter uncertainties. The vehicle itself called RobuCar n defined by Fs,i where n = 1, 2, .., n. is test bench vehicle, available at LAGIS Laboratory LAGIS (2011). It is a vehicle with four electromechanical decentralized controlled wheels and two controlled steering 4. DETECTABILITY AND ISOLABILITY systems Merzouki (2009). Three flows measurements are CONDITIONS available, for current, and angular velocities of the rotor The detectability and the isolability is studied using the and the wheel. Fault signature Matrix The fault isolation can be done using the fault signature matrix (FSM), that can be 5.1 Bond graph modeling directly deduced either from the analytical redundancy relations or from the bond graph model directly. Noting First, the bond graph model of the studied system is that the ARRs can be deduced directly from the graphical given in preferred integral causality (Fig. 5), where three model. In this section, we propose a robust FSM shown dynamic parts are distinguished: the electrical part repin Table 1, where the columns are the residuals (rj , j = resents the dynamic of electrical power in the actuator, 1, 2, ...n), and the rows are the parameters that represent where the mechanical part describes the dynamic of the the components (ci , i = 1, 2, ...m). rotor. Finally the load part shows the wheel side dynamic ∗ FS,i =

−

P

P




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SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico

R : Rm

R : Re

MSe : wRe

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Fig. 4. A quarter of an electrical vehicle R : Rm

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Fig. 5. Bond graph model of the electromechanical system in preferred integral causality. where the interaction with the ground is considered. The bond graph model in preferred derivative causality of this electromechanical system is illustrated in Fig. 6. The system is composed of two inertia elements (Jm and Js ), two 1 ) and resistance elements (fm and fs ), a capacitor (C = K a velocity reducer (T F ). The system is equipped by three sensors (Df : i, Df : θ˙e and Df : θ˙s ) for measuring the current, the angular velocity of motor axis and load respectively. After representing the parameter and measurement C:

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Fig. 6. Bond graph model of the electromechanical system in preferred derivative causality. uncertainties on the system model (Fig. 6), we obtain the uncertain model of Fig. 7. From the latter, three ARRs can be obtained: di ARR1 : U − Le − Re i − Ke θ˙e + a1 = 0, dt ARR2 : Ke i − fm θ˙e − Jm θ¨e − K(θe − N θs ) + a2 = 0, ARR3 : KN (θe − N θs ) − fs θ˙s − Js θ¨s + φ(F x) + a3 = 0. Where a1 and a2 represent the uncertain part of ARR1 and ARR2 respectively: a1 = WRe + WRe|i + WLe + WLe|i + Ke ζθm a2 = Wfm + Wfm |θm + WJm + WJm |θm + Ke ζi ˜ θ + KN ˜ Wθ + WK (θm − N θs ). +KW s m a3 = Wfs + Wfs |θs + WJs + WJs |θs ˜ 2 Wθ + WK N (θm − N θs ) + ζφ(F x) . ˜ Wθ + KN +KN s m

MSe : wJs

s

1

I : Js

Fig. 7. The model of the electromechanical system with measurement and parameter uncertainties. Where WRe , WLe , Wfm , WJm , Wfs , WJs and WK are the fictitious inputs of parameter uncertainties: di WRe = −δRe Re .i ; WRe = −δLe Le . dt Wfm = −δfm fm θ˙e ; WJm = −δJm Jm θ¨e Wfs = −δfs fs θ˙s ; ; WJs = −δJs Js θ¨s WK = −δK K Wfm |θ˙m , WJm |θ˙m , Wfs |θ˙s , WJs |θ˙s , Wθm and WθS represent the virtual inputs of measurement uncertainties: ˜ e ζi ; R ˜ e = Re (1 + ζR ) WRe |i = R e d(ζ ) ˜ i ˜ ; Le = Le (1 + ζLe ) WLe |i = Le dt d(ζθm ) Wfm |θ˙m = f˜m = f˜m ζθ˙m ; f˜m = fm (1 + δfm ) dt d(ζθ˙m ) WJm |θ˙m = J˜m = f˜m ζθ¨m ; J˜m = Jm (1 + δJm ) dt d(ζθs ) = f˜m ζθ˙s ; f˜s = fs (1 + δfs ) Wfs |θ˙s = f˜s dt d(ζθ˙s ) = f˜s ζθ¨s ; J˜s = Js (1 + δJs ) WJs |θ˙s = J˜s dt W θm = ζ θm . W θs = ζ θs . ˜ = K(1 + δK ) K where ζi , ζθe and ζθs are respectively the measurement errors on the current (i) and the angular position (θe and θs ). The gear constant N is supposed well known without uncertainty. The measurement error ζSn is bounded by an additive measurement uncertainty δSn (max (ζSn ) = δSn ).

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ζφ(F x) ≤ ∆φ(F x) . where, δ0 , δ1 and δ2 are the dry, the stiction, and the viscous friction, respectively. β is the stiction coefficient. These values are dependent on the conditions of the road that the vehicle operates on. rw is the radius of the wheel.

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× sign(u˙ − Rθ˙sj ).

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−β|(u−R ˙ θ˙sj )|

45

i(A)

The derivative of this measurement error can be bounded Sn by 2.δ ∆t. . Where ∆t represent the sampling time. The canonical form of the contact force can be calculated from the Pacejka tire model Pacejka (1991), presented in (5).

U(V)

SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico

200 100

20 24.4

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Fig. 9. The input and outputs of the system. 5.2 Co-simulation

The minimum detectable value of FSe is equal to 2a1 , and it’s shown in Fig. 10. The residuals r1 and the r2 in faulty min(FSe)(V)

The co-simulation means that we simulate separate programs under independent supports, where data information is exchanged between these programs. In our case, we co-simulate between a program implemented under Matlab/Simulink Zhan (2008) environment and CALLAS/SCANeR Studio simulator CALLAS (2011). The latter is a automotive driving simulator, dedicated to engineering and research. For the following results, the whole dynamics of RobuCar are validated from the experiments on Callas/SCANeR Studio simulator. So, we use this platform to co-simulate with the dynamic of the electromechanical system developed under Matlab/Simulink (Fig. 8). Three ARRs are derived from the model and evaluated to show the influence of uncertainties on the residuals. The obtained results are shown in Fig. 9, Figu. 12 and Fig. 13. The ARR1 , ARR2 and ARR3 are bounded

of measurement and parameters uncertainties are represented in Fig. 12. The measurement uncertainties are considered as bounded π π , − 5000 ) ≤ (ζθm , ζθs ) ≤ random variables, with (− 5000 π π ( 5000 , 5000 ), and −0.001 ≤ ζi ≤ 0.001. They correspond to the measurement uncertainty on the angular position. The residual r1 is sensitive to a fault on the actuator M Se : U . The relation between the fault and the residual is equal to the following equation: FSe = −r1 di FSe = −(U − Le − Re i − Ke · θ˙e ) dt min (FSe ) = 2a1

Matlab/Simulink

2 1

0

Measurement

QuarterofExperimentalVehicle

Fig. 8. Co-simulation between Matlab/Simulink and CALLAS/SCANeR. by (|a1 |, |a2 | and |a3 |) and (−|a1 |, −|a2 | and −|a3 |). The parameter values used in this application are: Nominal values 1.32 2.30 0.002 0.003 0.14 0.036 1 13 0.0655

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Fig. 10. The minimum detectable value of the FSe .

VirtualSimulator

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(Ω) (H) (N ms2 /rad) (N ms/rad) (dN ms2 /◦ ) (dN ms/◦ ) (dN m/◦ ) Nm/A

Uncertainty 0.01 0.02 0.02 0.026 0.08 0.1 0.07 0 0

situation with uncertainties are represented in Fig. 13. The fault is an additive fault with a value equal to 1V on the actuator M Se : U . Fig. 11 shows the input and outputs signals in case of faulty. We remark that the residuals r2 and r3 are not sensitive to the considered fault (Additive fault on the actuator M Se : U with a value equal to 1V ), except a high frequency signal variation at instance 6s, due mainly to the uncertainty of modeling. 6. CONCLUSION

The input signal U is shown in Fig. 9. The residuals r1 , r2 and r3 in normal situation in presence 962

In this work,a procedure of fault modeling and estimation is developed in order to generate systematically the transfer function between the residuals and the faults. This function is used after to study the isolability and the detectability conditions on the Robust Fault Signature Matrix in presence of uncertainties. The ARR and thresholds generation is performed by using the bond graph model of the system. The procedure of fault modeling and estimation is developed by using the GB-LFT to represent the faults form and the bi-causality notion to estimate

SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico

REFERENCES 15

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Fig. 13. r1 , r2 and r3 with uncertainties in faulty case. it. This procedure can be easily automated using the bond graph tool. An application to an intelligent electrical vehicle is done in order to verify the procedure. 963

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