- Email: [email protected]
Fault Detection of a Diesel Injection System by Qualitative Modelling
Copyright e IFAC Advances in Automotive Control, Karlsruhe, Germany, 2001
FAULT DETECTION OF A DIESEL INJECTION SYSTEM BY QUALITATIVE MODELLING D. Forstner· and J. Lunze··
• Robert Bosch GmbH, Dep. FV/FLI P.D.Box 10 60 50, D-70049 Stuttgart, Germany Tel. +49711 811-6933, Fax +49 711 811-7165 email: [email protected] •• Technical University Hamburg-Harburg Institute of Control Engineering Eissendorfer Str. 40, D-21071 Hamburg, Germany Tel. +494042878-3015, Fax +494042878-2112 email: [email protected]
Abstract: The paper presents a method for onboard- diagnosis which combines a model-based diagnostic scheme with a qualitative modelling approach. It is shown how a qualitative model which represents the qualitative, discrete-event behaviour of the system can be set up in a systematic way by abstraction or by qualitative identification. Experimental results are given for the qualitative modelling and fault detection of a Common-Rail diesel injection system. Copyright @20011FAC Keywords: abstraction algorithm, fuel injection system, identification algorithm , qualitative modelling, model-based diagnosis
1. INTRODUCTION
approaches have been reported, for, instance, in (Baroni et al., 1999; Ramkumar et al., 1998; Sampath et al., 1996). The qualitative approach is motivated by the fact that due to tolerances it is often not possible to set up a quantitative model. Furthermore, the complexity of the diagnostic system can be reduced by using qualitative instead of quantitative information. As faults change the dynamical behaviour qualitatively, qualitative models capture sufficient information to perform diagnostic tasks .
Scope of the paper. In automotive on-board applications the need to monitor safety relevant functions and emission limits leads to a high demand of powerful diagnostic solutions. Among the diagnostic approaches that have been elaborated in the recent years, model-based approaches offer a systematic design process because the modelling process is separated from the general diagnostic algorithm . For example, model-based systems using differential equations have been developed that perform state observation (Frank, 1996) , parameter estimation (Isermann , 1984) , or parity relation checks (Gertler , 1998) . In artificial intelligence approaches have been presented that use qualitative representations of the system (Hamscher et al., 1992).
Figures 1 and 2 summarise the basic idea of the qualitative model-based diagnostic method described in this paper. The quantitative input and output signals of the dynamical system are transformed into a discrete-event sequence of qualitative values (figure 1) where each qualitative value represents an interval of quantitative values . If the border between two intervals is passed, an event occurs. The dynamical system together
In this paper the concerned system is represented by a qualitative, discrete-event model which has the form of a non deterministic automaton . Similar
263
2. QUALITATIVE MODELLING OF QUANTISED SYSTEMS
2.1 The quantised system
!
l
r mr r
Continuous-variable system. As shown in figure 2, the core of the system under consideration is a continuous-variable continuous-time system
r
~,------7"~----~2------72'~----~
x(t)
TIrfw'.
Fig. 1. Signal quantisation
=
f(x(t), u(t), F),
x(O)
= Xo
(1)
with state vector x E IR n and input vector u E IRm. The system behaviour depends on the fault F E :F where :F is the set of faults considered and Fo E :F symbolises the faultless system. Signal quantisation. The right quantiser in figure 2 introduces a partition of the state space IR71. into the sets Qx(z), z E N x = {I , ... , n x }. The qualitative value of the state x(t) at time t is given by the index z of the set Qx(z) to which the state belongs:
Dynamical System
[x(t)]
=z
{::::::::>
x(t) E Qx(z).
(2)
The change of the qualitative value [x(t)] from j to i is called the event eij' If the event eij occurs at time tk the new qualitative state value is given by Zk = [X(tk)] = i . Thus, an event ek represents the same information as the pair of qualitative states (Zk-l, Zk) where for ek = e;j the relations Zk-l = j, Zk = i hold. At the event time tk, the system state assumes a value of the boundary set
sequence Ex
Fig. 2. Quantised system within a model-based diagnostic scheme with the quantisers is called the "quantised system" (figure 2). In this paper, the quantised system is modelled by a nondeterministic automaton (Lunze, 1994) which will be described in Section 2.
8Qx(z, z') with Qx(z)
Aim of the paper. The paper concerns three problems:
= Qx(z) n Qx(z')
= { x 13x E Qx(z) : Ilx -xii
(4)
~t:
for an arbitrarily small
(1) The qualitative modelling problem concerns the task to set up the nondeterministic model for the system that has to be diagnosed. Solutions for this problem are given in Section 3 where two algorithms are presented that determine the qualitative model by abstraction from a continuous-variable system description, or by qualitative identification based on a set of measurements. (2) A diagnostic algorithm has to find faults based on the qualitative model and the observed discrete--event sequence. This should be performed in a recursive way to allow online applications. Section 4 yields a diagnostic algorithm that satisfies these requirements. (3) Within an application example the qualitative modelling and diagnosis is applied to a Common-Rail diesel injection system. Experimental results for the fault detection of the injection system are presented in Section 6.
(3)
€ }
It is assumed that the state trajectory is continuous. Then, only state events between adjacent qualitative states occur where two states ZA, ZB are called adjacent if
(5) holds. The quantised system is considered in the time interval [0, Th], where the continuous-variable system follows the trajectory X[O,ThJ and the quantiser generates the state event sequence
Hx is the number of state events that the system generates within a given time horizon. The same quantisation principle is applied to the input u(t) . The input space IR m is partitioned into the sets Qu(v), v E Nu = {I, ... , nu} . Thus,
264
Nu. The dynamical behaviour of the automaton
an input trajectory corresponds to a sequence of input events: Eu(l. .· Hu) == Quant( U[O ,Th]) ==
is described by the state transition relation
(VO,VI ,· .. ,VHJ = (el , ... ,eHJ.
As the sequences of state and input events are to be investigated together, the joined sequence of qualitative input and state events is built which equals a quantisation of the vector s(t) == (U(t)T,X(t)T)T. The qualitative value [s(t)] of this vector is symbolised by the 2-vector q which represents the qualitative input v = [u(t)] and the qualitative state Z = [x(t)J . The joined event sequence
The automaton can step from the current state Zk to the successor state Zk+l provided that the input Vk is present if (11)
holds. For a given initial qualitative state 20, fault F, and time horizon H , the set of possible state event sequences can be generated recursively by multiple evaluations of the automaton's transition relation:
(7)
(~;) )
== {E(l. . . H)
describes the behaviour of the quantised system. Hence , the quantised system has a discrete-event behaviour.
The crucial task when using the presented modelbased diagnostic method is to set up the transition relation R of the automaton. Four methods are available to solve this task: • Knowledge acquisition: The transition relation is set up manually by an expert. This method is only feasible for small systems with only few qualitative inputs and states (small numbers nu , nz) ' • Abstraction: If a quantitative system description (1) is given , the transition relation can be determined by an abstraction procedure given in the subsequent section 3.3 that analyses the quantitative system dynamics. • Identification: A set of measurements that covers the possible behaviours for each concerned fault can be used to set up the transition relation of the automaton. A qualitative identification algorithm is presented in section 3.2. • Model composition: If automata for linked components are available they can be composed. The partitioning of the linked signals need to be equal in the considered component models.
X[O,Th ]
u(t) E
Qu(Vk)
for
tk ~
t<
E Q",(Zo) , tHI
with event times tl ... tH of E , to
=0
R , k=O ... H-1} .
3.1 Modelling task
E == Quant (U[O ,Th]) , Xo
1
3. DETERMINATION OF THE AUTOMATON'S TRANSITION RELATION
X[O ,ThJ :
:i;(t) == f(x(t) , u(t),F) ,
(VZHH))
For each fault F E :F a nondeterministic automaton N(F) has to be set up with a transition relation R(F) that depends on the fault F. The sets Nz and Nu are the same for all models. The model behaviour BM (20 , F, H) for a fault F is defined by applying R(F) within equation (12).
(8)
= { E(l . .. H) 1 3 U[O ,Th]'
= ((:~) -v
(ZHl,Zk , Vk) E
Nondeterminism of the quantised system. As only the qualitative values are measured , the event sequence E generated by the quantised system cannot be uniquely predicted (Lunze et al., 1999) . The reason for this is given by the fact that for a given initial qualitative state Zo the initial state Xo of the system (1) is not exactly known but merely restricted to the set Q",(Zo) , cf. eqn. (2) . Depending on x(O) E Q",(Zo) and the fault F the system may produce one sequence of the set
BS(Zo , F,H)
(12)
BM(Zo , H)
}.
2.2 Representation of quantised systems by means of automata Instead of the quantitative description (1) and the quantisers (2) a compact discrete-event representation of the quantised system by a nondeterministic automaton (9) will be used , with the qualitative state set Nz , the initial model state 20 and the qualitative input set
265
abstraction method. It is assumed that the boundary set 6Qx(z, z') defined by equation (3) can be described by a continuous, differentiable function. For each boundary point x B E 6Qx (z , z') the boundary direction vector a(xB , z, z') denotes the vector with length one that is orthogonal to the boundary and that is directed from the partition Q., (z) to the partition Qx (z'). In the case of partitions separated by linear hyperplanes, a(xB, z , z') is constant for all XB E 6Qx(z, z'). The inner product
In all cases, the transition relation has to be selected such that the relation
holds . Then, the model is called complete. Relation (13) says that the model generates all possible event sequence that the quantised system may produce. This is important because only if the relation (13) is satisfied by the model for each initial qualitative state zo, fault F, and time horizon H, a fault F can be excluded for the quantised system, if it is not consistent with the model behaviour (Forstner and Lunze, 1999b) .
D(XB, u,F)
= (f(XB, u,F), a(xB))
(14)
is a measure that can be used to define possible transitions:
3.2 Qualitative Identification '..
The subsequent identification algorithm determines all events represented by the given measurements and adds a transition for each event to the model transition relation R(F i ) of the present fault Fi.
T(z , z,v , F)
Given: Set of quantitative input- state-trajectories Bmeas = {Sto,Thl J ... S~ThM J} where the fault Fl ... FM was present. Partitions Qx, Qu.
D(XB , U, F) > 0
(15)
o else.
=
Initialise the transition relations R(F) := 0 for all F E :F.
=1...M:
Quantise the ith measurement
={
I if 3 XB E 6Qx(z, z'), u E Qu(v) :
If all input and state partition boundaries are parallel to the signal axes, 8Qx(z , z') and Qu(v) can be represented by intervals. Equations (14) and (15) can then be solved by interval arithmetics (cf. (Preisig et al. , 1997)) : If the interval solution of D(6Q.,(z , z'), Qu(v), F) has an element greater than zero, T(z', z, v, F) 1 holds, where the boundary direction vector a is given by the normalised vector from the centers of the partitions Qx(z) and Qx(z') . Based on these results the following abstraction algorithm can be stated:
Algorithm 1. Identification algorithm
For i
,
Algorithm 2. Abstraction algorithm
S!O ,Th; J E Bmeas:
E (I ... H i ) = Quant(slo,Th ,J) ' For 1 = 1 ... Hi - 1: Define (Z/+l , Z/,Vt) E R(Fi) . end 1 end i
Given:
Result: transition relations R(F) of the automata N(F) .
For all F E F: For all v E Nu: For all z E N x : For all z' E N x with z', z being adjacent: Determine T(z' , z,v,F) by equation (15) . IfT(z',z,v,F) = 1 define (z',z,v) E R(F) . end Define (z , z , v) E R( F) representing input events. end end end
i
Differential equation x(t) = f(x(t) , u(t) , F) . Partitions Qx, Qu'
Initialise the transition relations R(F) := 0 for all F E :F.
The relations R(F) set up by this algorithm represents all events that have been observed during the past experiments. However, it can not be guaranteed that all events are considered that the system may generate. That is, the relation (13) may be violated.
3.3 Abstraction from a differential equation If according to the differential equation (1) the boundary between two adjacent states can be passed for a given fault and qualitative value the concerned transition has to be a member of the transition relation R(F) of the nondeterministic automaton. This is applied by the subsequent
Result: transition relations R(F) of the automata N(F) . In (Forstner and Lunze , 2000) it is proved for a more general version of this abstraction algorithm
266
Algorithm 3. Diagnostic algorithm
that the obtained models are complete. That is , the algorithm guarantees that relation (13) is satisfied.
Given:
4. DIAGNOSTIC ALGORlTHM
1. Initialisation: H = 0, Fo = :F. 2. Wait for the next event, set H := H + I , measure VH and ZH · 3. Test (ZH ' ZH-l , vH-d E R(F) for all FE :F. 4. Determine F H by eqn . (17). 5. If H < H proceed with step 2.
It is assumed that an unknown fault F E F has occurred at time t ~ 0 and is present until the diagnostic algorithm is stopped. E(I . . . H) denotes the sequence of input events and state events of the quantised system for a given time horizon H. The main idea of consistency-based diagnosis is to compare the observed sequence with the behaviour of the model. The diagnostic task is to answer the question whether the automaton can generate the event sequence E(I ... H) , i.e. whether the relation E(l. .. H) E BM(2'o , F,H) holds. The diagnostic result is denoted by F H as follows : FH
= {F I E(l. . . H)
E BM(Zo , F, H)}.
Result:
E
= 1,2, . . . ,H.
(1) Specification: The set of faults to be diagnosed has to be specified. Additionally, available signals and the ressource restrictions concerning calculation time and memory have to be given. To verify the diagnostic system a set of test cases should be specified , too. (2) Partition selection: For the input and output signals that will be used to solve the diagnostic task, partitions have to be selected. This is a critical point in the development process because the partition design determines the model complexity and the quality of the diagnostic result . (3) Qualitative modelling: The automata for the normal behaviour and the considered faults have to be set up by one of the methods described in section 3. (4) Test: The performance of the diagnostic algorithm based on the given automata has to be evaluated by the test cases specified in the first step. The most important properties are the fault detection rate, the fault identification rate, and the fault alarm rate . (5) Implementation: Due to the model-based approach , the diagnostic algorithm may already have been implemented in a preceding application. Then, only the model has to be implemented.
(16)
(17)
1\
(ZH , ZH - l , VH-d
for H
To apply the presented qualitative modelling and diagnostic approach the following tasks have to be performed:
To determine the diagnostic result F recursively, it has to be tested whether changes in the input or output are consistent with the transition relation R of the automaton. The diagnosis starts with no information about the occurrence of a fault . Therefore, it is assumed that all faults FE F may have occurred. Furthermore, it is assumed that the initial qualitative state 2'0 is known . The diagnostic algorithm determines F H recursively for given FH-l as follows :
= {F I FE FH-l
FH
5. DEVELOPMENT PROCESS
The diagnostic problem is solved for increasing time horizon H. Then, F E F H for time horizon H = 0, 1,2, .. . says that the observed behaviour until the H - th event is consistent with the automaton and F ~ F H means that the automaton cannot produce the event sequence E(I . .. H) and, hence, the fault F cannot have occurred in the quantised system.
FH
model N, initial qualitative state 2'0 , initial qualitative input vo , time horizon H.
R(F)} .
The first part of eqn. (17) concerns the case that fault F could not be excluded by using the measurement data with time horizon H - 1. Then it is tested whether the newly observed qualitative state ZH+l satisfies the state transition relation R(F) with the qualitative input VH and qualitative state ZH . The fault F is not a solution of the diagnostic problem with time horizon H if it has been already excluded due to measurements obtained until time H -1.
In practice, this procedure has to be used in an iterative way. If the test results don't satisfy the specifications, the partition should be varied, i.e. the developement continues at step (2). As the qualitative modelling is done automatically by the abstraction or identification algorithm the steps (2) to (4) can be performed many times for different partitions which allows to iteratively optimise the qualitative models.
The value of F H can be determined for one observed input-state pair after another . Thus , the following diagnostic algorithm can be used online.
267
Qualitative modelling of the fuel injection system. Each of the three input signals have been partitioned into five intervals which yields 125 different qualitative input values (nu = 125) . The state signal, i.e. the rail pressure, is partitioned into six intervals (nx = 6) . Only a model for the faultless case Fo has been set up , i.e. :F = { Fo } as fault detection has to be performed which requires no fault models. The identification method was used here because a large set of measurements was available from the demonstrator car. The relation R(Fo) represents about 1000 transitions . The measurements used to identifiy the relation R(Fo) cover a broad range of operation conditions. Thus, relation (13) is satisfied with high probability, and the model can be used for diagnosis.
6. FAULT DETECTION OF A FUEL INJECTION SYSTEM The Common Rail fuel injection system. The qualitative model-based diagnosis has been applied to a demonstrator car with Common Rail fuel injection system (Bauer et al., 1996; Cascio et al. , 1999) . The demonstrator car allows to trace the data of the electronic control unit (ECU) and to process the captured data by an external PC (figure 3). Several faults can be stimulated electronically within the demonstrator car which allows to test and demonstrate the diagnostic system.
Dia~nostic
results. The diagnostic algorithm described in section 4 has been implemented as a Matlab function. It checks whether an observed sequence of input and state events is consistent with the given automaton (figure 2). A measurement example and the associated diagnostic result is shown in figure 5. The first three plots concern the input signals. In the fourth plot the rail pressure is given. The horizontal, dashed lines in this plot denote the borders between the partitions. The last plot of figure 5 depicts the event times of input and state events by vertical bars and the diagnostic result by a grey bar that denotes the time horizon for which the observed events are consistent with the model. The end of the grey bar represents the occurrence of an inconsistency which signals the occurrence of a fault .
Fig. 3. Signal acquisition
Figure 5 shows a case where an intermittent fault occurred at t = 9.2s . At tl = 10.ls the normal condition was restored. The presence of the fault is marked by a box on the time axis of figure 5. The fault was realised by an intermittent shortcircuit of the electrical pump that delivers the high pressure pump. The model-based diagnostic system detects the fault at tD = 9.6s shortly after the fault occurred. The fault is detected when the pressure passes the border of p = 400bar. The pressure is still within a pressure range that the system may assume under normal condition. A simple maximum and minimum threshold monitoring could not detect the occurrence of the fault at this time. The model-based diagnosis yields better results because the observed state event would not take place in the faultless case for the given values of the three input signals. Additionally, the first part of figure 5 where the normal operation is given shows that the diagnostic algorithm classifies this measurement correctly to be faultless although the signal values change dynamically.
Fig. 4. Common Rail fuel injection system The fuel injection system is depicted in figure 4. Fuel is delivered from the tank (1) into the common rail (8) by a high pressure pump (6) leading to fuel pressures up to 1300 bar. Injectors (9) which are opened and closed by magnetic valves inject the fuel into the cylinders of the diesel engine. The electronic control unit (ECU, ll) captures various sensor signals, for instance the rail pressure and the engine speed. It determines the fuel that has to be injected and the needed pressure value. It controls the pressure by an pressure regulator valve and defines the injection begin and duration . Four signals are relevant for the diagnostic task: The rail pressure is the state of the system, the injected fuel amount , the engine speed and the pressure control signal represent input signals.
268
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Fig. 5. Diagnosis for a measurement trajectory which includes an intermittent fault International Journal of Control. submitted for publication. Forstner, D. and J . Lunze (2000) . Discrete-event abstraction of quantised systems with asynchronous input and state events. Automation of mixed processes - hybrid dynamic systems. Dortmund, Germany. (accepted for presentation) Frank, P.M. (1996) . Analytical and qualitative model- based fault diagnosis - a survey and some new results. European Journal of Control 2, 6- 28. Gertler , J .J. (1998). Fault Detection and Diagnosis in Engineering Systems. Marcel Dekker. Hamscher , W ., J. de Kleer and L. Console (Eds.) (1992) . Readings in Model-Based Diagnosis. Morgan-Kaufmann. San Mateo. Isermann, R. (1984). Process fault detection based on modeling and estimation methods - a survey. automatica 20(4) , 387-404. Lunze, J. (1994) . Qualitative modelling of linear dynamical systems with quantized state measurements . automatica 30, 417-431. Lunze, J ., B. Nixdorf and J . Schroder (1999) . On the non determinism of discrete-event representations of continuous-variable systems. automatica 35(3) , 395-406. Preisig, H. , M. Pijpers. and M. Weiss (1997). A discrete modelling procedure for continuous processes based on state-discretisation. Proc. 2nd Mathmod. Vienna. 395- 406. Ramkumar, K.B ., P. Philips, H.A . Preisig, W.K Ho and KW . Lim (1998). Structured fault- detection and diagnosis using finitestate automaton . Proc. 24th Annual Con/. of the IEEE Industrial Electronics Society. Aachen, Germany 1667-1672. Sampath, M., R. Sengupta, S. Lafortune, K Sinnamohideen and D. C. Teneketzis (1996) . Failure diagnosis using discrete-event models . IEEE Trans. Control Systems Technology 4(2) , 105-124.
7. CONCLUSION A model- based diagnostic method has been presented that uses a non deterministic automaton to represent the dynamical behaviour of the given system. The paper focused on the question how the non deterministic automaton can be found for a given system. Two solutions and related algorithms have been presented to solve this task, first the abstraction from a quantitative system description, second the qualitative identification from measurements. The paper showed that this method can be applied to the fault detection of a Common Rail fuel injection system. Faults could be detected shortly after their occurrence. The fuel injection system owns typical characteristics of automotive applications (availability of various sensor signals, powerful central processor unit with real time computing capabilities, fast process dynamics). Consequently, the method is equally applicable to other automotive systems.
8. REFERENCES Baroni, P., G. Lamperti, P. Pogliano and M. Zanella (1999). Diagnosis of large active systems. Artificial Intelligence 110, 135- 183. Bauer, H. , A. Cypra, A. Beer and H. Bauer (Eds.) (1996). Automotive handbook. 4th ed .. Robert Bosch GmbH. Stuttgart. (distributed by SAE Society of Automotive Engineers). Cascio, F. , L. Console, M. Guagliumi , M. Osella, A. Panatti, S. Sottano and D. Theseider Dupre (1999) . Generating on-board diagnostics of dynamic automotive systems based on qualitative models. Proc. 13th International Workshop on Qualitative Reasoning (QR99). Loch Awe , Scotland. pp. 27- 35. Forstner, D. and J. Lunze (1999b). Discrete-event models of quantised systems for diagnosis .
269
FAULT DETECTION OF A DIESEL INJECTION SYSTEM BY QUALITATIVE MODELLING D. Forstner· and J. Lunze··
• Robert Bosch GmbH, Dep. FV/FLI P.D.Box 10 60 50, D-70049 Stuttgart, Germany Tel. +49711 811-6933, Fax +49 711 811-7165 email: [email protected] •• Technical University Hamburg-Harburg Institute of Control Engineering Eissendorfer Str. 40, D-21071 Hamburg, Germany Tel. +494042878-3015, Fax +494042878-2112 email: [email protected]
Abstract: The paper presents a method for onboard- diagnosis which combines a model-based diagnostic scheme with a qualitative modelling approach. It is shown how a qualitative model which represents the qualitative, discrete-event behaviour of the system can be set up in a systematic way by abstraction or by qualitative identification. Experimental results are given for the qualitative modelling and fault detection of a Common-Rail diesel injection system. Copyright @20011FAC Keywords: abstraction algorithm, fuel injection system, identification algorithm , qualitative modelling, model-based diagnosis
1. INTRODUCTION
approaches have been reported, for, instance, in (Baroni et al., 1999; Ramkumar et al., 1998; Sampath et al., 1996). The qualitative approach is motivated by the fact that due to tolerances it is often not possible to set up a quantitative model. Furthermore, the complexity of the diagnostic system can be reduced by using qualitative instead of quantitative information. As faults change the dynamical behaviour qualitatively, qualitative models capture sufficient information to perform diagnostic tasks .
Scope of the paper. In automotive on-board applications the need to monitor safety relevant functions and emission limits leads to a high demand of powerful diagnostic solutions. Among the diagnostic approaches that have been elaborated in the recent years, model-based approaches offer a systematic design process because the modelling process is separated from the general diagnostic algorithm . For example, model-based systems using differential equations have been developed that perform state observation (Frank, 1996) , parameter estimation (Isermann , 1984) , or parity relation checks (Gertler , 1998) . In artificial intelligence approaches have been presented that use qualitative representations of the system (Hamscher et al., 1992).
Figures 1 and 2 summarise the basic idea of the qualitative model-based diagnostic method described in this paper. The quantitative input and output signals of the dynamical system are transformed into a discrete-event sequence of qualitative values (figure 1) where each qualitative value represents an interval of quantitative values . If the border between two intervals is passed, an event occurs. The dynamical system together
In this paper the concerned system is represented by a qualitative, discrete-event model which has the form of a non deterministic automaton . Similar
263
2. QUALITATIVE MODELLING OF QUANTISED SYSTEMS
2.1 The quantised system
!
l
r mr r
Continuous-variable system. As shown in figure 2, the core of the system under consideration is a continuous-variable continuous-time system
r
~,------7"~----~2------72'~----~
x(t)
TIrfw'.
Fig. 1. Signal quantisation
=
f(x(t), u(t), F),
x(O)
= Xo
(1)
with state vector x E IR n and input vector u E IRm. The system behaviour depends on the fault F E :F where :F is the set of faults considered and Fo E :F symbolises the faultless system. Signal quantisation. The right quantiser in figure 2 introduces a partition of the state space IR71. into the sets Qx(z), z E N x = {I , ... , n x }. The qualitative value of the state x(t) at time t is given by the index z of the set Qx(z) to which the state belongs:
Dynamical System
[x(t)]
=z
{::::::::>
x(t) E Qx(z).
(2)
The change of the qualitative value [x(t)] from j to i is called the event eij' If the event eij occurs at time tk the new qualitative state value is given by Zk = [X(tk)] = i . Thus, an event ek represents the same information as the pair of qualitative states (Zk-l, Zk) where for ek = e;j the relations Zk-l = j, Zk = i hold. At the event time tk, the system state assumes a value of the boundary set
sequence Ex
Fig. 2. Quantised system within a model-based diagnostic scheme with the quantisers is called the "quantised system" (figure 2). In this paper, the quantised system is modelled by a nondeterministic automaton (Lunze, 1994) which will be described in Section 2.
8Qx(z, z') with Qx(z)
Aim of the paper. The paper concerns three problems:
= Qx(z) n Qx(z')
= { x 13x E Qx(z) : Ilx -xii
(4)
~t:
for an arbitrarily small
(1) The qualitative modelling problem concerns the task to set up the nondeterministic model for the system that has to be diagnosed. Solutions for this problem are given in Section 3 where two algorithms are presented that determine the qualitative model by abstraction from a continuous-variable system description, or by qualitative identification based on a set of measurements. (2) A diagnostic algorithm has to find faults based on the qualitative model and the observed discrete--event sequence. This should be performed in a recursive way to allow online applications. Section 4 yields a diagnostic algorithm that satisfies these requirements. (3) Within an application example the qualitative modelling and diagnosis is applied to a Common-Rail diesel injection system. Experimental results for the fault detection of the injection system are presented in Section 6.
(3)
€ }
It is assumed that the state trajectory is continuous. Then, only state events between adjacent qualitative states occur where two states ZA, ZB are called adjacent if
(5) holds. The quantised system is considered in the time interval [0, Th], where the continuous-variable system follows the trajectory X[O,ThJ and the quantiser generates the state event sequence
Hx is the number of state events that the system generates within a given time horizon. The same quantisation principle is applied to the input u(t) . The input space IR m is partitioned into the sets Qu(v), v E Nu = {I, ... , nu} . Thus,
264
Nu. The dynamical behaviour of the automaton
an input trajectory corresponds to a sequence of input events: Eu(l. .· Hu) == Quant( U[O ,Th]) ==
is described by the state transition relation
(VO,VI ,· .. ,VHJ = (el , ... ,eHJ.
As the sequences of state and input events are to be investigated together, the joined sequence of qualitative input and state events is built which equals a quantisation of the vector s(t) == (U(t)T,X(t)T)T. The qualitative value [s(t)] of this vector is symbolised by the 2-vector q which represents the qualitative input v = [u(t)] and the qualitative state Z = [x(t)J . The joined event sequence
The automaton can step from the current state Zk to the successor state Zk+l provided that the input Vk is present if (11)
holds. For a given initial qualitative state 20, fault F, and time horizon H , the set of possible state event sequences can be generated recursively by multiple evaluations of the automaton's transition relation:
(7)
(~;) )
== {E(l. . . H)
describes the behaviour of the quantised system. Hence , the quantised system has a discrete-event behaviour.
The crucial task when using the presented modelbased diagnostic method is to set up the transition relation R of the automaton. Four methods are available to solve this task: • Knowledge acquisition: The transition relation is set up manually by an expert. This method is only feasible for small systems with only few qualitative inputs and states (small numbers nu , nz) ' • Abstraction: If a quantitative system description (1) is given , the transition relation can be determined by an abstraction procedure given in the subsequent section 3.3 that analyses the quantitative system dynamics. • Identification: A set of measurements that covers the possible behaviours for each concerned fault can be used to set up the transition relation of the automaton. A qualitative identification algorithm is presented in section 3.2. • Model composition: If automata for linked components are available they can be composed. The partitioning of the linked signals need to be equal in the considered component models.
X[O,Th ]
u(t) E
Qu(Vk)
for
tk ~
t<
E Q",(Zo) , tHI
with event times tl ... tH of E , to
=0
R , k=O ... H-1} .
3.1 Modelling task
E == Quant (U[O ,Th]) , Xo
1
3. DETERMINATION OF THE AUTOMATON'S TRANSITION RELATION
X[O ,ThJ :
:i;(t) == f(x(t) , u(t),F) ,
(VZHH))
For each fault F E :F a nondeterministic automaton N(F) has to be set up with a transition relation R(F) that depends on the fault F. The sets Nz and Nu are the same for all models. The model behaviour BM (20 , F, H) for a fault F is defined by applying R(F) within equation (12).
(8)
= { E(l . .. H) 1 3 U[O ,Th]'
= ((:~) -v
(ZHl,Zk , Vk) E
Nondeterminism of the quantised system. As only the qualitative values are measured , the event sequence E generated by the quantised system cannot be uniquely predicted (Lunze et al., 1999) . The reason for this is given by the fact that for a given initial qualitative state Zo the initial state Xo of the system (1) is not exactly known but merely restricted to the set Q",(Zo) , cf. eqn. (2) . Depending on x(O) E Q",(Zo) and the fault F the system may produce one sequence of the set
BS(Zo , F,H)
(12)
BM(Zo , H)
}.
2.2 Representation of quantised systems by means of automata Instead of the quantitative description (1) and the quantisers (2) a compact discrete-event representation of the quantised system by a nondeterministic automaton (9) will be used , with the qualitative state set Nz , the initial model state 20 and the qualitative input set
265
abstraction method. It is assumed that the boundary set 6Qx(z, z') defined by equation (3) can be described by a continuous, differentiable function. For each boundary point x B E 6Qx (z , z') the boundary direction vector a(xB , z, z') denotes the vector with length one that is orthogonal to the boundary and that is directed from the partition Q., (z) to the partition Qx (z'). In the case of partitions separated by linear hyperplanes, a(xB, z , z') is constant for all XB E 6Qx(z, z'). The inner product
In all cases, the transition relation has to be selected such that the relation
holds . Then, the model is called complete. Relation (13) says that the model generates all possible event sequence that the quantised system may produce. This is important because only if the relation (13) is satisfied by the model for each initial qualitative state zo, fault F, and time horizon H, a fault F can be excluded for the quantised system, if it is not consistent with the model behaviour (Forstner and Lunze, 1999b) .
D(XB, u,F)
= (f(XB, u,F), a(xB))
(14)
is a measure that can be used to define possible transitions:
3.2 Qualitative Identification '..
The subsequent identification algorithm determines all events represented by the given measurements and adds a transition for each event to the model transition relation R(F i ) of the present fault Fi.
T(z , z,v , F)
Given: Set of quantitative input- state-trajectories Bmeas = {Sto,Thl J ... S~ThM J} where the fault Fl ... FM was present. Partitions Qx, Qu.
D(XB , U, F) > 0
(15)
o else.
=
Initialise the transition relations R(F) := 0 for all F E :F.
=1...M:
Quantise the ith measurement
={
I if 3 XB E 6Qx(z, z'), u E Qu(v) :
If all input and state partition boundaries are parallel to the signal axes, 8Qx(z , z') and Qu(v) can be represented by intervals. Equations (14) and (15) can then be solved by interval arithmetics (cf. (Preisig et al. , 1997)) : If the interval solution of D(6Q.,(z , z'), Qu(v), F) has an element greater than zero, T(z', z, v, F) 1 holds, where the boundary direction vector a is given by the normalised vector from the centers of the partitions Qx(z) and Qx(z') . Based on these results the following abstraction algorithm can be stated:
Algorithm 1. Identification algorithm
For i
,
Algorithm 2. Abstraction algorithm
S!O ,Th; J E Bmeas:
E (I ... H i ) = Quant(slo,Th ,J) ' For 1 = 1 ... Hi - 1: Define (Z/+l , Z/,Vt) E R(Fi) . end 1 end i
Given:
Result: transition relations R(F) of the automata N(F) .
For all F E F: For all v E Nu: For all z E N x : For all z' E N x with z', z being adjacent: Determine T(z' , z,v,F) by equation (15) . IfT(z',z,v,F) = 1 define (z',z,v) E R(F) . end Define (z , z , v) E R( F) representing input events. end end end
i
Differential equation x(t) = f(x(t) , u(t) , F) . Partitions Qx, Qu'
Initialise the transition relations R(F) := 0 for all F E :F.
The relations R(F) set up by this algorithm represents all events that have been observed during the past experiments. However, it can not be guaranteed that all events are considered that the system may generate. That is, the relation (13) may be violated.
3.3 Abstraction from a differential equation If according to the differential equation (1) the boundary between two adjacent states can be passed for a given fault and qualitative value the concerned transition has to be a member of the transition relation R(F) of the nondeterministic automaton. This is applied by the subsequent
Result: transition relations R(F) of the automata N(F) . In (Forstner and Lunze , 2000) it is proved for a more general version of this abstraction algorithm
266
Algorithm 3. Diagnostic algorithm
that the obtained models are complete. That is , the algorithm guarantees that relation (13) is satisfied.
Given:
4. DIAGNOSTIC ALGORlTHM
1. Initialisation: H = 0, Fo = :F. 2. Wait for the next event, set H := H + I , measure VH and ZH · 3. Test (ZH ' ZH-l , vH-d E R(F) for all FE :F. 4. Determine F H by eqn . (17). 5. If H < H proceed with step 2.
It is assumed that an unknown fault F E F has occurred at time t ~ 0 and is present until the diagnostic algorithm is stopped. E(I . . . H) denotes the sequence of input events and state events of the quantised system for a given time horizon H. The main idea of consistency-based diagnosis is to compare the observed sequence with the behaviour of the model. The diagnostic task is to answer the question whether the automaton can generate the event sequence E(I ... H) , i.e. whether the relation E(l. .. H) E BM(2'o , F,H) holds. The diagnostic result is denoted by F H as follows : FH
= {F I E(l. . . H)
E BM(Zo , F, H)}.
Result:
E
= 1,2, . . . ,H.
(1) Specification: The set of faults to be diagnosed has to be specified. Additionally, available signals and the ressource restrictions concerning calculation time and memory have to be given. To verify the diagnostic system a set of test cases should be specified , too. (2) Partition selection: For the input and output signals that will be used to solve the diagnostic task, partitions have to be selected. This is a critical point in the development process because the partition design determines the model complexity and the quality of the diagnostic result . (3) Qualitative modelling: The automata for the normal behaviour and the considered faults have to be set up by one of the methods described in section 3. (4) Test: The performance of the diagnostic algorithm based on the given automata has to be evaluated by the test cases specified in the first step. The most important properties are the fault detection rate, the fault identification rate, and the fault alarm rate . (5) Implementation: Due to the model-based approach , the diagnostic algorithm may already have been implemented in a preceding application. Then, only the model has to be implemented.
(16)
(17)
1\
(ZH , ZH - l , VH-d
for H
To apply the presented qualitative modelling and diagnostic approach the following tasks have to be performed:
To determine the diagnostic result F recursively, it has to be tested whether changes in the input or output are consistent with the transition relation R of the automaton. The diagnosis starts with no information about the occurrence of a fault . Therefore, it is assumed that all faults FE F may have occurred. Furthermore, it is assumed that the initial qualitative state 2'0 is known . The diagnostic algorithm determines F H recursively for given FH-l as follows :
= {F I FE FH-l
FH
5. DEVELOPMENT PROCESS
The diagnostic problem is solved for increasing time horizon H. Then, F E F H for time horizon H = 0, 1,2, .. . says that the observed behaviour until the H - th event is consistent with the automaton and F ~ F H means that the automaton cannot produce the event sequence E(I . .. H) and, hence, the fault F cannot have occurred in the quantised system.
FH
model N, initial qualitative state 2'0 , initial qualitative input vo , time horizon H.
R(F)} .
The first part of eqn. (17) concerns the case that fault F could not be excluded by using the measurement data with time horizon H - 1. Then it is tested whether the newly observed qualitative state ZH+l satisfies the state transition relation R(F) with the qualitative input VH and qualitative state ZH . The fault F is not a solution of the diagnostic problem with time horizon H if it has been already excluded due to measurements obtained until time H -1.
In practice, this procedure has to be used in an iterative way. If the test results don't satisfy the specifications, the partition should be varied, i.e. the developement continues at step (2). As the qualitative modelling is done automatically by the abstraction or identification algorithm the steps (2) to (4) can be performed many times for different partitions which allows to iteratively optimise the qualitative models.
The value of F H can be determined for one observed input-state pair after another . Thus , the following diagnostic algorithm can be used online.
267
Qualitative modelling of the fuel injection system. Each of the three input signals have been partitioned into five intervals which yields 125 different qualitative input values (nu = 125) . The state signal, i.e. the rail pressure, is partitioned into six intervals (nx = 6) . Only a model for the faultless case Fo has been set up , i.e. :F = { Fo } as fault detection has to be performed which requires no fault models. The identification method was used here because a large set of measurements was available from the demonstrator car. The relation R(Fo) represents about 1000 transitions . The measurements used to identifiy the relation R(Fo) cover a broad range of operation conditions. Thus, relation (13) is satisfied with high probability, and the model can be used for diagnosis.
6. FAULT DETECTION OF A FUEL INJECTION SYSTEM The Common Rail fuel injection system. The qualitative model-based diagnosis has been applied to a demonstrator car with Common Rail fuel injection system (Bauer et al., 1996; Cascio et al. , 1999) . The demonstrator car allows to trace the data of the electronic control unit (ECU) and to process the captured data by an external PC (figure 3). Several faults can be stimulated electronically within the demonstrator car which allows to test and demonstrate the diagnostic system.
Dia~nostic
results. The diagnostic algorithm described in section 4 has been implemented as a Matlab function. It checks whether an observed sequence of input and state events is consistent with the given automaton (figure 2). A measurement example and the associated diagnostic result is shown in figure 5. The first three plots concern the input signals. In the fourth plot the rail pressure is given. The horizontal, dashed lines in this plot denote the borders between the partitions. The last plot of figure 5 depicts the event times of input and state events by vertical bars and the diagnostic result by a grey bar that denotes the time horizon for which the observed events are consistent with the model. The end of the grey bar represents the occurrence of an inconsistency which signals the occurrence of a fault .
Fig. 3. Signal acquisition
Figure 5 shows a case where an intermittent fault occurred at t = 9.2s . At tl = 10.ls the normal condition was restored. The presence of the fault is marked by a box on the time axis of figure 5. The fault was realised by an intermittent shortcircuit of the electrical pump that delivers the high pressure pump. The model-based diagnostic system detects the fault at tD = 9.6s shortly after the fault occurred. The fault is detected when the pressure passes the border of p = 400bar. The pressure is still within a pressure range that the system may assume under normal condition. A simple maximum and minimum threshold monitoring could not detect the occurrence of the fault at this time. The model-based diagnosis yields better results because the observed state event would not take place in the faultless case for the given values of the three input signals. Additionally, the first part of figure 5 where the normal operation is given shows that the diagnostic algorithm classifies this measurement correctly to be faultless although the signal values change dynamically.
Fig. 4. Common Rail fuel injection system The fuel injection system is depicted in figure 4. Fuel is delivered from the tank (1) into the common rail (8) by a high pressure pump (6) leading to fuel pressures up to 1300 bar. Injectors (9) which are opened and closed by magnetic valves inject the fuel into the cylinders of the diesel engine. The electronic control unit (ECU, ll) captures various sensor signals, for instance the rail pressure and the engine speed. It determines the fuel that has to be injected and the needed pressure value. It controls the pressure by an pressure regulator valve and defines the injection begin and duration . Four signals are relevant for the diagnostic task: The rail pressure is the state of the system, the injected fuel amount , the engine speed and the pressure control signal represent input signals.
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Fig. 5. Diagnosis for a measurement trajectory which includes an intermittent fault International Journal of Control. submitted for publication. Forstner, D. and J . Lunze (2000) . Discrete-event abstraction of quantised systems with asynchronous input and state events. Automation of mixed processes - hybrid dynamic systems. Dortmund, Germany. (accepted for presentation) Frank, P.M. (1996) . Analytical and qualitative model- based fault diagnosis - a survey and some new results. European Journal of Control 2, 6- 28. Gertler , J .J. (1998). Fault Detection and Diagnosis in Engineering Systems. Marcel Dekker. Hamscher , W ., J. de Kleer and L. Console (Eds.) (1992) . Readings in Model-Based Diagnosis. Morgan-Kaufmann. San Mateo. Isermann, R. (1984). Process fault detection based on modeling and estimation methods - a survey. automatica 20(4) , 387-404. Lunze, J. (1994) . Qualitative modelling of linear dynamical systems with quantized state measurements . automatica 30, 417-431. Lunze, J ., B. Nixdorf and J . Schroder (1999) . On the non determinism of discrete-event representations of continuous-variable systems. automatica 35(3) , 395-406. Preisig, H. , M. Pijpers. and M. Weiss (1997). A discrete modelling procedure for continuous processes based on state-discretisation. Proc. 2nd Mathmod. Vienna. 395- 406. Ramkumar, K.B ., P. Philips, H.A . Preisig, W.K Ho and KW . Lim (1998). Structured fault- detection and diagnosis using finitestate automaton . Proc. 24th Annual Con/. of the IEEE Industrial Electronics Society. Aachen, Germany 1667-1672. Sampath, M., R. Sengupta, S. Lafortune, K Sinnamohideen and D. C. Teneketzis (1996) . Failure diagnosis using discrete-event models . IEEE Trans. Control Systems Technology 4(2) , 105-124.
7. CONCLUSION A model- based diagnostic method has been presented that uses a non deterministic automaton to represent the dynamical behaviour of the given system. The paper focused on the question how the non deterministic automaton can be found for a given system. Two solutions and related algorithms have been presented to solve this task, first the abstraction from a quantitative system description, second the qualitative identification from measurements. The paper showed that this method can be applied to the fault detection of a Common Rail fuel injection system. Faults could be detected shortly after their occurrence. The fuel injection system owns typical characteristics of automotive applications (availability of various sensor signals, powerful central processor unit with real time computing capabilities, fast process dynamics). Consequently, the method is equally applicable to other automotive systems.
8. REFERENCES Baroni, P., G. Lamperti, P. Pogliano and M. Zanella (1999). Diagnosis of large active systems. Artificial Intelligence 110, 135- 183. Bauer, H. , A. Cypra, A. Beer and H. Bauer (Eds.) (1996). Automotive handbook. 4th ed .. Robert Bosch GmbH. Stuttgart. (distributed by SAE Society of Automotive Engineers). Cascio, F. , L. Console, M. Guagliumi , M. Osella, A. Panatti, S. Sottano and D. Theseider Dupre (1999) . Generating on-board diagnostics of dynamic automotive systems based on qualitative models. Proc. 13th International Workshop on Qualitative Reasoning (QR99). Loch Awe , Scotland. pp. 27- 35. Forstner, D. and J. Lunze (1999b). Discrete-event models of quantised systems for diagnosis .
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