Research ArticleActive Fault Tolerant Control Based on Bond Graph Approach
Manel Allous1 and Nadia Zanzouri1,2
1 Department of Electrical Engineering, LACS Laboratory, National Engineering School of Tunis,UniversiteΜ de Tunis El Manar, P.O. Box 37, Le Belvedere, 1002 Tunis, Tunisia
2 Preparatory Engineering Institute of Tunis 2, UniversiteΜ de Tunis, Rue Jawaher Lel Nahrou-Monfleury, 1089 Tunis, Tunisia
Correspondence should be addressed to Manel Allous; [email protected]
Received 30 April 2014; Revised 21 September 2014; Accepted 21 September 2014; Published 10 November 2014
Academic Editor: Gorazd Stumberger
Copyright Β© 2014 M. Allous and N. Zanzouri. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
This paper proposes a structural fault recoverability analysis using the bond graph (BG) approach. Indeed, this tool enablesstructural analysis for diagnosis and fault tolerant control (FTC). For the FTC, we propose an approach based on the inverse controlusing the inverse BG. The fault tolerant control method is also compared with another approach. Finally, simulation results arepresented to show the performance of the proposed approach.
1. Introduction
Due to the growing complexity of the dynamical systems,there is an increasing demand for safe operation, faultdiagnosis (FDI) (fault detection and isolation), and faulttolerant control (FTC) (strategies for control redesign). Dif-ferent approaches have been developed for the designingand the implementation of FDI and FTC procedures [1].These techniques are based on the knowledge of the systemmodel (model-based methods) [2, 3] or its structure (data-based methods) [4, 5]. FTC is categorized into two differenttechniques: passive FTC (PFTC) [6, 7] and active FTC(AFTC) [8, 9]. In PFTC, controllers are fixed and designedto be robust against a class of presumed faults. The AFTCapproach reacts to system component failures actively byreconfiguring control actions and acceptable performance ofthe entire system can be maintained.
This paper is focused on the design of a novel AFTC thatintegrates a reliable and robust fault diagnosis scheme withthe design of a controller reconfiguration system. The FDIand FTC are fully integrated in dynamic systems design inseveral fields of engineering, such as robotic and automotivesystems. Nevertheless, it must have tool that enables couplingthe diagnosis results with fault tolerant control conditions.Therefore, the BG enables integrating both structural diag-nosis results with control analysis. A BG model allows
knowledge of a large amount of structural, functional, andbehavioral information.This information enables computingappropriate control actions that compensate the faults.
The BG has proven to be a powerful tool not only forgenerating the direct model of a system but also for obtainingits inverse model. In [10], the authors have proposed aninverse control strategy based on BG model.
The innovative interest of the present paper is to combinethe inverse control strategy and observer designs to generatethe FDI and FTC algorithms from the BG model. Theproposed approach takes into account the parameter uncer-tainties and considers the fault recoverability with respect tofault compensation, without complex calculations.
In the first part of the paper, we propose a methodologybased on BG model for fault detection and fault tolerantcontrol. In the second part, we have developed a methodwhich compensates the faults in the absence of complexcalculations. Finally, an illustrative example is developedand simulation results show the advantage of the proposedapproach.
2. FDI and FTC Approaches Based onBond Graph
The bond graph approach is proposed by [11] and thendeveloped by [12, 13]. This tool allows the multidomain
Hindawi Publishing CorporationAdvances in Electrical EngineeringVolume 2014, Article ID 216153, 8 pageshttp://dx.doi.org/10.1155/2014/216153
2 Advances in Electrical Engineering
PA < 0 PB < 0
A B
Figure 1: Power and orientation symbol on BG.
Table 1: BG elements.π
β
π
π ππ
π
β
π
; ππ
π
β
π
π=ππ/ππ‘
β
π
πΌ
π1
β
π1
Gyπ2
β
π2
π
β
π=ππ/ππ‘
πΆ
π1
β
π1
TFπ2
β
π2
systems (mechanical, electrical, thermal, etc.) to be describedwith the same components. Its causal structure was initiallyexploited to determine structural conditions of controllabil-ity, observability [14], and diagnosability [15, 16].
The bond graph is based on the graphical representationof the energy exchange within the system to be modeled.Table 1 represents the BG elements: resistor (π ), compliance(πΆ), and inertia (πΌ) are passive elements. Effort source (π
π)
and flow source (ππ) are the active elements.
Figure 1 indicates the power direction in the system.There are only two types of junctions: the 1 and the 0
junctions (Figure 2). 1 junction has equality of flows and theefforts sum up to zero. 0 junctions have equality of efforts andthe flows sum up to zero (Figure 2).
2.1. Luenberger Observer Based on BG. Bond graph ap-proaches for observers design were developed in someworks,such as Luenberger observers [17, 18], reduced order Luen-berger observers [19], proportional integral observers [20],and nonlinear observers applied to electrical transformers[21].
The objective of this work is to design a Luenbergerobserver by bond graph for fault detection and isolation.
2.2. BG Modeling Bicausality Concept for System Inversion.The concept of βbicausalβ introduced by Gawthrop [22]enlarges the possibilities of computation models that can bederived from a bond graph. The bicausal bond graph modelis seen as half-strokes each associated with an effort and aflow variable that can be imposed independently at each endof the bond. Causal half-strokes indicate the fixed or knownvariables of the bond and so determine the right-hand side ofthe assignments form [23] (Figure 3).
The bicausality is used to get systemsβ inversion byimposing the output variable without modifying the energystructure. System inversion is an interesting analysis to knowan input considering a given output. Therefore, in the nextsection, we use the bicausality property of the bond graphs.
Some conditions (structural invertibility) are proposed topresent the bond graph-based procedure for system inversion[20].
Proposition 1. A linear system modeled by bond graph isinvertible if there is at least one causal path between the inputvariable and the output variable of the system.
e1
e3e4
f1f2
f3
f401
e2
e1
e3e4
f1f2
f3
f4e2
Figure 2: BG junctions.
System 2 System 1 System 2
e1 e2
f1 f2
e1 e2
f1 f2
System 1
e1 = e2f1 = f2
e2 = e1f2 = f1
} }Figure 3: Bicausal bond graph concept.
2.3. Control Law Design Directly on Inversion Bond Graph.The control strategy proposed by [24] computes the desiredinputs based on the system objectives. Also, in [10], theauthors have proposed an inverse control strategy from theBG model with parameter uncertainties estimated directlyfrom the inverse BG-LFT (linear fractional transformation).The system inversion concept gives the basis to computeappropriate control actions that compensate the faults.Figure 4 shows the control design based on bond graphdeveloped by [10].
In [10], the BG-LFTwas used to estimate the faulty power.Then, to validate these structural results, a local adaptivecompensation based on the inverse control strategy using theinverse BG-LFT was proposed. This strategy computes thedesired inputs based on the system objectives and on theundesired power caused by the fault.
Limitations of this approach are as follows.
(i) The fault estimation is necessary for the controldesign. The inverted model of bond graph uses theestimate fault to compensate it.
(ii) The fault estimationwith a BG-LFT causes FTC delay.
3. Proposed Approach
The principal of our proposed approach AFTC system ispresented in Figure 5. There are basically two parts.
(i) The first part concerns the diagnosis by Luenbergerobserver using BG approach; in this part, the faultestimation and fault isolability are not necessary forsystem recoverability; just the residual is injected tothe control loop.
(ii) The second part shows the control part determined byinverse BG for nominal system.
The following symbols have been used.
I.BG.N.S: Inverse BG for nominal system.π¦ref: Desired value.π¦sys: Measured output of system.π¦obs: Estimated output of observer.
Advances in Electrical Engineering 3
Cn, In Rn
JunctionsDf, De
Cn, In Rn
Junctions
Online
Inverse BGmodel
System
fΜ
BG-LFT
Offline
Se
Sf0, 1, TF, GY
0, 1, TF, GY
Figure 4: Control design on bond graph developed by [10].
I.BG.N.S
Process
Observer
Error
Residual
I.BG.N.S
I.BG.N.S Residualur
β+
++
+
+
β
uΜ
yref
yobs
unom
uFTC ys ys
Figure 5: AFTC strategy based on bond graph.
Computing the Control (π’πΉππΆ). Various methods have beenproposed to recover as close as possible the system perfor-mance according to the considered fault representation.
Some extensions of the classical pseudoinverse method(PIM) have been proposed to guarantee both the perfor-mance and the stability of the faulty system. The authors in[25, 26] have synthesized a suitable feedback controlπΎfeedback.In [27, 28], the authors have proposed to compute a reconfig-urable forward gain πΎforward controller in order to eliminatethe steady-state tracking error in faulty case. Therefore, thecontrol signal applied to the system is represented in
π’
FTC= πΎforwardπ¦ref β πΎfeedbackπ₯. (1)
A novel technique to adjust the command equation (1) isproposed by [29] in
π’
FTC= πΎforwardπ¦ref + οΏ½ΜοΏ½. (2)
οΏ½ΜοΏ½ given by
οΏ½ΜοΏ½ = πΎfeedback (π₯ β π₯π
) . (3)
β
Inverse BGfor nominal
systemProcess
Observer
f
ErrorResidual
Residual
β+
+
+
+
+
yref
yobs
ysysuFTC
Figure 6: New control based on bond graph.
According to the control law in (1) and (2), we propose a newcontrol which uses the residual signal provided by the FDIbased observer and error. So, the control law is expressed by
π’
FTC= πΎforwardπ¦ref + πΎforward(π¦ref β π¦sysβββββββββββββββββ
error
)
+ πΎforward(π¦sys β π¦obsβββββββββββββββββresidual
) .
(4)
Or πΎforward is a gain of inverse BG for nominal system.In Figure 5, the compensation term π’
π(residual control)
is useful to compensate the fault. Also οΏ½ΜοΏ½ (error control) isadded to the nominal control (π’nom); this term (οΏ½ΜοΏ½) improvesthe compensation of the fault effect. So this additive controlresultsβ role is to reproduce the control signal (π’FTC: con-trolled input resulting from (4) for compensating for theeffect of the fault every moment that the fault is detected).
To simplify the calculus of the control input representedin Figure 5, we propose to replace the three inverse BGmodels by a single inverse BG (Figure 6).
4 Advances in Electrical Engineering
Table 2: Parameter values of the DC motor.
π = 1Ξ© Rotor resistanceπΏ = 5mH Rotor inductanceπ = 10
β4Nm/rdβ sβ1 Coefficient of viscousπ½ = 0.001Kgm2 Moment of inertiaππ= 0.2Nm/A Coefficient of the torque
π’nom = 220V Motor voltage
1 1
3
1
2
4
6
7
5
8kn
I : JI : L
R : bR : R
usys
GyDf : Wsys
Figure 7: Bond graph model of DC motor.
So, the new control law is expressed by
π’
FTC= πΎforward (π¦ref + error + residual) . (5)
4. Illustrative Example
An example of a DC motor is used to illustrate our new FTCtechnique. The BG model of the system is given in Figure 7,and the state-space equations are presented in (6);
[
οΏ½ΜοΏ½1
οΏ½ΜοΏ½2
] =
[
[
[
β
π
π½
ππ
π½
β
ππ
πΏ
β
π
πΏ
]
]
]
[
π₯1
π₯2
] + [
0
1
πΏ
] π’,
π¦ = [
1 0
0 1
] [
π₯1
π₯2
] + [
0.1
0
] π,
(6)
with π₯ = (π₯1, π₯2) = (π€, π) being the state vector, π¦ the
measured output variable, π’ the control input variable, andπ the disturbance input variable.
The parameters of the DCmotor are presented in Table 2.
Closed Loop System. Figure 8 shows that the AFTC strategyintegrates the FDI module with an inverse BG for nominalsystem and the bond graphmodel is controllable and observ-able [14].
From the controlled input and output signals, the FDImodule provides the residual which is injected to the controlloop, in order to compensate the effect of fault.
The objective is to synthesize a controller so that thestructure of the closed loop system is as close as possibleto that of the desired reference model under the normaloperation or in the presence of fault.
(i) Computing the Residual (r) in Normal Operating. By causalpath, we deduce the structural equations from the BG ofFigure 8. We compute the residual π
1with the following
equations:
(a) structural equations for system model:
π1π
= π2π
= π3π
= π4π
,
π1π
= π2π
+ π3π
+ π4π
;
π4π
= πππ
5π
,
π5π
= πππ
4π
;
π5π
= π6π
= π7π
= π8π
,
π5π
= π6π
+ π7π
+ π8π
;
π5π
=
π4π
ππ
,
π5π
=
π1π
β π2π
β π3π
ππ
;
(7)
(b) structural equations for Luenberger observer:
π1ob= π2ob= π3ob= π4ob,
π1ob= π2ob+ π3ob+ π4ob;
π4ob= πππ
5ob,
π5ob= πππ
4ob;
π5ob= π6ob= π7ob= π8ob,
π5ob= π6ob+ π7ob+ π8ob;
π5ob=
π4ob
ππ
,
π5ob=
π1obβ π2obβ π3ob
ππ
.
(8)
From junctions equations (7) and (8), we generate theresidual π
1:
π1= π8π
β π8ob= π6π
β π6ob= π5π
β π5ob=
π4π
ππ
+
π4ob
ππ
,
π5π
=
1
ππ
[ππ β πΏ
ππ2π
ππ‘
β π π2π
] ,
π5ob=
1
ππ
[πob β πΏππ2obs
ππ‘
β π π2obβ π1πΎ11] ,
π1=
1
ππ
[ππ β πΏ
ππ2π
ππ‘
β π π2π
β πob + πΏππ2ob
ππ‘
+ π π2ob] .
(9)
Advances in Electrical Engineering 5
MSeMSe MSe MSe
1 1 1 1
1 1
09
10
MSf
MSf
MSf
Process
Observer
Inverse BG fornominal system
+ β
fΜ8
1c3c
2c
4c
R : b R : R
5c
6c
7c8c
I : J I : L
kn
1s
3s
2s
4s 5s
6s
7s
8s
ys
r1r2+
β
R : bR : R
9c
I : JI : L
R : R R : b
I : L I : J+
kn
β
kn 8s
yref
Wobs
Gy uFTC Gy
Gy
9ob 10ob 11ob 12ob
3ob 7ob
1ob 2ob 4ob 5ob 6ob
8ob
K22 K21 K12 K11
Figure 8: FTC based on BG model.
Or
ππ = πobs,
π2π
=
π5π
ππ
=
1
ππ
[π½
ππ6π
ππ‘
+ ππ6π
] ,
π2ob=
π5ob
ππ
=
1
ππ
[π½
ππ6ob
ππ‘
+ ππ6π
β πΎ11π1] ,
π1=
1
ππ
[
[
ππ β πΏ
π (1/ππ) [π½(ππ
6π
/ππ‘) + ππ6π
]
2
ππ‘
β π
1
ππ
[π½
ππ6π
ππ‘
+ ππ6π
]βπob
+ πΏ
π (1/ππ) [π½ (ππ
6ob/ππ‘) + ππ
6π
β πΎ11π1]
ππ‘
π
Γ
1
ππ
[π½
ππ6ob
ππ‘
+ ππ6π
β πΎ11π1] β π1πΎ12
]
]
.
(10)
The residual π1of the system is realized as
π1=
βπ½πΏ
π
2
π
π
2
π1
ππ‘
β
ππ1
ππ‘
[
π π½ + πΏπ + πΎ11πΏ
π
2
π
]
β π1[
π π + πΎ11π
π
2
π
+
πΎ12
ππ
] ,
[1 +
π π + πΎ11π
π
2
π
+
πΎ12
ππ
] π1+ [
π π½ + πΏπ + πΎ11πΏ
π
2
π
]
ππ1
ππ‘
+
π½πΏ
π
2
π
π
2
π1
ππ‘
= 0.
(11)
The π2is deduced with similar method.
The inverse system enables computing appropriate con-trol actions that compensate the faults.
(ii) Computing the Control (π’πΉππΆ). The control law can bedesigned directly from the BG model:
structural equations for inverse BG model:
π1π
= π2π
= π3π
= π4π
,
π1π
= π2π
+ π3π
+ π4π
;
π4π
= πππ
5π
,
π5π
= πππ
4π
;
π5π
= π6π
= π7π
= π8π
,
π5π
= π6π
+ π7π
+ π8π
.
(12)
6 Advances in Electrical Engineering
0 5 10 15 20 25 30
Time (s)
0
5
10
15
Velocity (approach of [10])Velocity (our approach)
Desired velocity
13 13.2 13.4 13.6 13.8 14 14.2 14.4 14.6
Time (s)
8
8.5
9
9.5
10
Zoom
Velocity (approach of [10])Velocity (our approach)
Desired velocity
Figure 9: Simulation results when the fault is compensated.
The control law π’FTC of the system is shown as
π’
FTC= π8π
= π5π
+ π6π
+ π7π
,
π5π
= πππ
1π
,
π1π
= π10+
Μ
π8+ π9,
π10= π€des = π¦ref,
Μ
π8= π8π
β π8ob= π1,
π9= π€des β π8
π
= π,
π6π
=
πΏππ6π
ππ‘
, π7π
= π π
7π
= π π
6π
.
(13)
Or π1= residual and π = error.
So, the control law π’FTC is
π’
FTC= π π
6π
+
πΏππ6π
ππ‘
+ ππ(residual + π¦ref + error) . (14)
5. Simulation Results
Simulation results are carried out in the bond graph simula-tion software 20-sim [30] with parameter values described inTable 2.
Figure 9 shows the system output (velocity) evolutionwith a single fault (parameters π : π, π fault = π + πΏ, andπΏ = 0.01) introduced at the time 13 s.
We remark that the output decreases less than in the caseof control considered in [10], and then it reaches the nominalvalues quicker at instant π‘ = 13.05 s. So, the control law (FTC)is able to stabilize the system on the desired output and tocompensate the fault in the system with a very short timedelay.
0 5 10 15 20 25 300
1
2
3
4
13.2 13.4 13.6 13.8 14 14.22
2.1
2.2
2.3
2.4
2.5
Zoom
Time (s)
U FTC (approach of [10])U FTC (our approach)
U FTC (approach of [10])U FTC (our approach)
13
Time (s)
Figure 10: Control input.
Time (s)0 5 10 15 20 25 30
0
5
10
Error of velocity (approach of [10])Error of velocity (our approach)
13 13.2 13.4 13.6 13.8 14 14.2 14.4 14.6
0
0.5
1
1.5
2
Zoom
β5
Time (s)
Error of velocity (approach of [10])Error of velocity (our approach)
Figure 11: Velocity error.
From a control point of view, the reconfigurable controlmechanism requires more energy to reach the target and toguarantee system performance, as shown in Figure 10.
These results can be confirmed by the control input πFTCof Figure 10. In [10], the control input increases slowly tryingto compensate for the fault affecting the system. In ourapproach, the control input increases quickly and enablesrapid fault compensation on the controlled systemoutput andallows compensating the convergence delay.
In Figure 11, the velocity error quickly converges to zerowith the new approach.
Advances in Electrical Engineering 7
6. Conclusion
In this paper, we propose an active FTC design based onBG approach.The novel strategy combines an observer basedmodel and inverse BG model.
The proposed approach enables computing appropri-ate control actions for compensating the faults. The faultsare detected by Luenberger observer technique based onBG modeling. Fault isolation and fault estimation are notnecessary to the FTC. The comparison between the twoapproaches shows the efficiency of the proposedmethod.Theapplication of a FTC approach to induction DC motor andsimulation results illustrate the performance of the proposedFDI-FTC structure. Our future works concern the onlineimplementation of the proposed techniques on a real process.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
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