- Email: [email protected]
Diesel Engine Actuator Fault Isolation Using Multiple Models Hypothesis Tests
Copyright e IFAC Fault Detection. Supervision and Safety for Technical Processes. Espoo. Finland. 1994
DIESEL ENGINE ACTUATOR FAULT ISOLATION USING MULTIPLE MODELS HYPOTHESIS TESTS S0REN B0GH NIELSEN Aalborg University. Department of Control Engineering. Fredrik Bajers Vej 7. DK 9220 Aalborg Tel: +4598158522. Fax: +4598151739. E-mail: [email protected]
~.
Denmark.
Abstract. Detection of current faults in a D.C. motor with unknown load torques is not feasible with linear methods and threshold logic. This paper suggests an FDl scheme with a bank of observers where a nonlinear observer is designed using the structure of the current fault. Multiple hypothesis testing of the observer residuals is applied to discriminate between faults and unknown input. The strategy has been applied to a simulation of a diesel engine actuator with an encouraging progress on performance compared to other results. Keywords. Failure detection; actuator failures; D.C motors; fault classification; control applications; decision theory; hypothesis testing.
task is to design a detector for a fault in the motor current ili .. that is robust to load disturbances . Linear observer design methods optimize the directional properties of the residuals to facilitate isolability (Patton & Chen (1991). Frank. P.M. (1991», but as the fault input and the load disturbance input enter at the same point, these approaches fail the isolability criteria.
1. INTRODUcnON Fault detection and isolation (FDI) in dynamic systems has been widely discussed in the literature. In the field of qualitative methods. the focus has been on linear techniques used to design fault detection observers (Frank & Wtinnenberg (1989)) or parameter estimators (Isermann (1991», where the key difficulties are fault isolability and robustness to unknown inputs. Faults are traditionally modelled as additive inputs to a dynamic model or as parameter changes. This failure representation is not always adequate as faults may appear multiplicative and their occurrence can be temporary. Additional details on the failure structure shall then be included in the FDI design.
The proposed method in this paper uses nonlinear observer design in a bank of observers. where each observer represents a different operational mode of the plant (nominal and faulty). Multiple hypothesis test is used to discriminate between current fault events and load disturbances. The analysis is carried out all the way to the point of decision making, which is often missed in FDI studies. 2. MULTIPLE MODELS HYPOTHESIS TEST
This paper presents a study in FDI on a diesel engine actuator, where linear design methods are not sufficient. Nonlinear filtering is used to detect a specific fault and isolate to unknown inputs. An introduction to the actuator is found in Blanke et al. (1994) and a detailed description is available in B~gh et al. (1993). A block diagram of the equipment is shown in Figure 1.
The classical approach to FDI using a set of filters in an observer bank has been proposed by several authors (Clark et al. (1975), Montgomery & WiIliams (1989». The paper by Frank & Wtinnenberg (1989) gives an overview of the unknown input observers for both sensor and component failure detection. Each observer in the bank is optimized for sensitivity to a specific subset of faults and is made invariant to another subset. If each observer is designed for a separate fault hypothesis. both fault detection and isolation can be accomplished using multiple hypothesis tests on the observer residuals. Surveys on statistical testing are given in Willsky (1976). Gertler (1988). Tzafestas & Watanabe (1990). and Basseville & Nikiforov (1993).
Figure 1 Block diagram of diesel engine actuator with velocity control loop. Current fault is ill.. and load disturbance is Qt
This paper uses an event detection and fault isolation scheme as illustrated in Figure 2.
A D.C. motor (with shaft velocity n.. and arm position so) is driven by a PI velocity controller with input reference nrrr An unknown load torque Q, is present and the
457
below a threshold B a current fault is concluded . The complete decision logic is listed in Table I. Table 1. Decision logic for current faults
Decision
SPRT Decision Variables
Activity
(L, ,=O)A(L,z=O)
No Decision
(L,,~)A(LI1~A)
L-......:-~~r2
Figure 2. FDl scheme for event detection and current fault isolation.
and:
(L2J~)v(L22~)
NOT«(L, , =O)A(LI1=O»v«L, ,~)A(L'2~»)
The residual from observer I is used to detect whether a current fault is present or a dominant load torque is acting. When an event has been detected (r, exceeds a threshold), the residual from the second observer is used to distinguish between the two causes . Observer I is designed from a nominal linear model of the plant, so the output will be sensitive to (i) current faults, (ii) load disturbances, and (iii) excitation of unmodelled dynamics and nonlinearities. Observer 2 is designed from a model of the plant when a current fault is active. When a current fault is not active, the residual is then expected to be large. If a current fault is active and load activity is small , the res idual is expected to be small. The output will, anyway, be sensitive to load disturbances and unmodelled dynamics.
3. OBSERVER FOR NOMINAL PLANT An observer is designed for the nominal plant shown in Figure I with current fault and load included as unknown additive inputs . The observer residual is designed to be sensitive to the unknown input and the current fault. The residual is also sensitive to effects from model uncertainties as unmodelled dynamics and nonlinearities are not included in this linear observer. The discrete time state space description of the plant x(k+l) =Ax(k) + Bu(k) + Ed(k) + F.f.(k) y(k) = Cx(k)
Fault isolation is not feasible using only one sample of the residuals. After an event has been detected, confidence is accumulated that r, is persistently large and that r2 is persistently small. This information is used to conclude about current faults. For this purpose, statistical methods are required. A two-sided CUSUM (Cumulated Sum) algorithm implemented as an SPRT (Sequential Probability Ratio Test) is used in this case as it is computational simple . The strategy of the fault isolation scheme is pictured in Figure 3.
A"
0.507 0.635
[ 5.42' \0-5
-0.358 0] [0.380] [ -0.493] -9.15' \0-2 0 ;B- 1.064 ;F - 0.635 3.93'\0-5 I
7.07.\0-5 ' 5.42·\0-5 (2)
10*---7\;"::' , .
~o
1
o
0 0.978
0]
A Luenberger observer is designed using eigenstructure assignment by applying the dual control problem. A tutorial is available in Burrows (1990). Increased robustness to noise on the velocity measurement , n .. , is achieved by using left eigenvector assignment, described in Patton & Chen (199\).
Cunent Fault
t;;"'
(l)
is derived from Blanke et al. (1994), where x = [ i2 ' n .. • sof is the state vector, u = n"" is input (output from position controller), d = [Q~ v] is unknown load torque and noise on velocity measurement, f. = tli.. is additive current fault, and y = [n .. , so). The matrices A, B. E , F., and C are found for a sampling time of tlT=lO ms:
-1.21 .\0-2 -3.79.\0--'] E .. 1.55·\0-2 -1.06 ;C[ 1.33-10-<1 -7.07 ·\0--5
L21.Ln
Current Fault Load Disturb. Continue
(L2J <.B)A(L22 <.B)
LoadDist
The observer structure is
B
Load Dist Figure 3. When an event has been detected, isolation between cu"ent fault and load disturbance is accomplished by statistical testing in the outlined scheme. Current
i(k+ 1) = (A -K. C) i(k) + Bu(k) + K. y(k)
Fault
y(k)
(3)
= Ci(k)
The desired discrete time eigenvalues and left eigenvectors of the observer are chosen to be
The variables L" or LI2 increase from zero when the mean of r, changes. Similarly, L2J or L22 increase from zero when the mean of r2 changes. If L" or LJ2 exceed a threshold A, a conclusion is drawn that the event is persistent and a decision on the cause is based on the size of the decision variable Lz, and Lzl' If these are both
458
AI = e
-20dT
Az
= e
-I06T
~
=e
- 2306T
(4)
Current fault:
leading to a gain matrix -6.24.10-.
4.40.10-5 ]
K) = -5.71·10-) -7.77.10-5 [ 7.16.10-5 1.85·10-)
[1
(10)
Aim(t) = -(imO(t)
(11)
(5)
Current saturation:
Output from the observer is a residual computed as a weighted sum of the observer estimation errors. The residual of observer 1 shall be sensitive to the un-known inputs. An appropriate weight matrix is r) =
imo(t) = K/n,Jt) - nm(t)) + iit)
(12)
AiStIl(t) =-(i';'o(t) < -imax)(i';'o(t) +imax )
(6)
5·1Q3] (y(k) - j(k))
(13)
-(i';'o(t) > imax)(i';'o(t) -imax )
4. OBSERVER FOR FAULTY PLANT The second observer is designed from a model of the plant with an active current fault. The observer residual is then small when the current fault is present, but sensitive to load disturbances. To improve the isolability between current fault events and load disturbances, the observer is robustified by including two significant nonlinear characteristics: Velocity controller integrator limit and current saturation in the power drive.
An observer is developed in continuous time by using only the linear part of the plant in the observer design. The nonlinear term is later added in the differential equations of the observer. Since the system is stable in both the saturation region and outside, it is assumed that the observer will be stable. Stability is not guaranteed. Robust nonlinear observer design for FDI has been considered in Seliger & Frank 1991. The method is based on a state transformation that is only applicable in a limited number of plant structures, and cannot be implemented in this case.
The structure of the current fault is recognized as a rectification of the signal i m (an ideal diode). This effect is integrated as a nonlinear element. The character of the nonlinear elements are described in Blanke et aI. (1994) and they can be arranged into additive terms to the existing linear model by a transformation as seen in Figure 4.
Left eigenvector assignment is applied as in the observer for the nominal plant. The observer structure is x(t) = (Ac-K"C)x(t)+Bcu(t)+K"y(t)+/(x,u,I,,) (15) jet)
fi2 Linlted Integrator
Cul'l'8t'1t Fault
The desired eigenvalues and left eigenvectors of the linear part of the observer is chosen to be
current
Saturation
Figure 4. Nonlinear elements of the actuator arranged as additive quantities to the linear model. The nonlinear continuous time state-space equation becomes x(t)
yet)
= Acx(t) = Cx(t)
+ I(x, u, I,,) + Bcu(t)
A.)
= -10
~
= -50
[I, 12 I3l =
~ = -130
r~ ~ ~] l~
(16)
0 0
leading to a gain matrix
(7)
-66.4
where I includes nonlinear terms in states x, input u, and fault I". The matrices A c and Bc and the parameters below can be found in Blanke et al. (1994). The nonlinear terms can be found to be j(x,ula ) fnm(t), O]T, where
= Cx(t)
K2 =
[
°]
2
8.92
9.7.103.8·10-)9
1.12.10-2
10.2
(17)
The residual from observer 2 is chosen to be the velocity estimation error
= [fi2(t),
r2
(8)
= [1
0] (y(k) - j(k))
(18)
The nonlinear continuous time observer eq.(15) is simulated with a 4'th order Runge-Kutta algorithm with a fixed step length of 5 msec.
Integrator limit: 5. CUSUM TEST AS AN SPRT
f 12(t)- -[(i2(t) ~ -ilim )(d'7(t) <0) + (i2(t) ~ilim)(dj2(t)> 0)] dj2(t) (9)
The statistical testing method applied in this paper is the two-sided CUSUM algorithm (Basseville & Nikiforov (1993)). A single CUSUM calculates the cumulative sum of a log-likelihood ratio. In this paper the hypothesis of a fixed change in mean is used. When both a positive
459
6. EXPERIMENTAL RESULTS
and negative change can occur (J.1-J.Io=±v), the two-sided CUSUM is used. The extended case of composite hypotheses (1J.1-J.1o I>v), can be solved with a sequential x2-test (Basseville & Nikiforov (1993».
The FDI scheme presented above has been applied to a simulation of the diesel engine actuator benchmark. The simulation software is quite complex and reproduces the behaviour of the real equipment in close detail. The test shows the ability of discriminating between current faults and external load torque disturbances and also illustrates the area of problems.
The window length of the CUSUM is of fixed size. If the algorithm is implemented as an SPRT, the window length is arbitrary and depends on the information in past samples. The SPRT is used in this paper to reach a decision with a desired confidence interval.
A four seconds sequence is plotted in appendix A. The current fault (no negative currents) is present from 0.4 sec to 2.3 sec. The current fault causes integrator windup in the velocity controller, i m • This demonstrates the need for including integrator limits and current saturation in the observer for the faulty plant. The effect of the fault is a positive velocity error, nm-n"J'
The hypotheses in this paper are based on mean changes of the residuals as listed in Table 1 for the two test cases.
Table 2. Hypotheses for the two tests. Test
Hypothesis
Rationale
SPRT 1
Ho: J.11=0
No Event
HI: IJ.lJ/=v J
Event Persistent
Ho: J.12=0
Current Fault Active
H,: 1J.12I=v2
Load Disturbance
SPRT 2
The design parameters have been experimentally tuned to the values in Table 3.
Table 3. Design parameters for the FDI scheme.
Four log-likelihood ratios are required in this case: A positive mean change and a negative mean change for each residual r, and r2 • Let p(r; IH) p(rlk-n),rlkn+1), ...,1j(k)IH), then
=
L II
=
p(rJ· ' J.1=+v I) _
ln~
L I2
=
p(rt IJ.1=-v I) _
ln~
p(rl·1 J.1=O)
p(rJ· ' J.1=O)
l.n
p(r; IJ.1=-V2) = lnl-_ _
The probability densities are computed assuming residual j to be a Gaussian random sequence with variance In this case, the recursive formula for the decision variables for a positive mean change are for the two tests (;=1,2)
a;.
Lj/(k-l)-
2~: (v 2ri k»J j -
(20)
where a lower limit of zero is realized with the supremum function. Similar for a negative mean change, Lp(k) = sup
(0,
Lp(k-l) +
2~: (-v 2ri k»J j-
Observer 1 Threshold
T
17
SPRT 1 Mean
v,
20
SPRT 1 Variance
Value
er;
22
SPRT 1 Threshold
A
30
SPRT 2 Mean
v2
10
SPRT 2 Variance
~
6
SPRT 2 Threshold
B
15
The residuals and decision variables for the FDI scheme are plotted in appendix B. The residual from the observer for the nominal plant, rI' is seen to increase whenever a current fault is active and/or a load disturbance is present. Each time r, exceeds the threshold T, the SPRT calculation is activated, producing the decision variables. L JJ or LJ2 rapidly exceed the upper threshold A when r, remains large, demanding for a decision. The decision is based on r2 , which is seen to be small when the current fault is enabled. During active load torque r2 grows, enabling separation of the two causes. The decision variables, ~, and ~2' remain below the threshold, B, as long as only the current fault is active, but one of them exceeds the threshold when the load torque is also present. This yields the decisions, depicted in the bottom figure together with indicators for periods when current fault is active and load torque is active.
are the four decision variables. They are tested against the thresholds in Table 1.
(0,
Symbol
(19)
p(r; IJ.1=O)
Lj/(k) = sup
Parameter
(21)
The presence of a current fault is detected only about 50 msec. after the effect shows up the first time at 0.5 sec. Subsequently, a train of failure alarms indicates successful current fault detection. During periods with load disturbances, the current fault alarm rate decreases and events are classified as load changes. This shows the inevitable obstacle that current faults cannot be detected during high load excitation. During the period with no faults and high load activity, no false alarms are fired. This demonstrates the required event isolation.
The decision variables are calculated at sample k for the last n samples. A minor disadvantage of the isolation approach in this configuration is that a decision is forced to be taken when L JJ or LJ2 exceeds the threshold A. This does not guarantee that enough samples has been included in the computation of L2J and L22 for the isolation between current faults and load disturbances.
460
The quantity of missed alarms is a key issue in FDI. An example is present at 1.2 sec., that is caused by too low a sensitivity of observer I. The current fault induces only a small error, which is not manifested in the residual r l • The missed alarm rate can be reduced by increasing the sensitivity to fault events, but with a penalty on the false alarm rate.
Fault detection and isolation with nonlinear observers and statistical testing results in a relatively high computational load. The complexity is limited in this case, by calling the CUSUM algorithms only when an event has been detected by the linear observer and isolation is required. Ideally, all significant nonlinearities and higher order terms of the model should be included in both observers for increased robustness and isolability, but the computational complexity will be unrealistic for realtime applications.
A few current fault decisions are triggered in the periods after the fault is no longer active. This is caused by the subsequent settling time of the observers, and cannot be avoided. It is not considered to be a major problem, as the vital feature is to avoid missed alarms.
8. ACKNOWLEDGEMENTS This project is supported by the Danish Research Council, STVF grant number 16-4979.
The applicability of the FDI scheme in a wider range of operating modes of the actuator has been exercised on a sequence with large velocity reference changes (data is available in Nielsen et al. (1993». Excitation of current saturation and unmodelled dynamics cause triggering of the FDI scheme, but only a single false alarm is present. The current fault is perfectly recognized.
9. REFERENCES Basseville, M., Nikiforov , I. V. (1993) . Detection of Abrupt Changes: Theory and Application. Prentice Hall. ISBN 0-13-126780-9. Blanke, M., Nielsen, S.B., J~rgensen, R.B ., Patton, R.J. (1994). Fault Detection and Accommodation in Diesel Engine Actuator - a Benchmark, Preprints of SA FEPROCESS '94.
7. CONCLUSIONS Current fault isolation on the diesel engine actuator using multiple hypothesis test on the residuals from an observer bank shows promising results. A standard linear observer of the nominal plant is used to detect events, ie. either current faults or load disturbances . Isolation between the two events is carried out using a new approach where the structure of the current fault is included in the design of a second observer. The classical approach with faults modelled as unknown additive inputs is inadequate in this case. The new approach is often avoided primarily because the detailed knowledge is generally unavailable and furthermore because it requires nonlinear observer design that is difficult to assure stability . The stability question has not been in focus in this paper.
B~gh,
S., Blanke, M., J~rgensen, R.B. (1993). Nonlinear Characterization and Simulation of Industrial Position Control Loop. Aalborg University Report R93-4017. Feb. 1993. Burrows, S.P. (1990). Robust Control Design Techniques Using Eigenstructure Assignment. D.Phil Thesis, University of York, sept. 1990. Clark, R.N ., Fosth, D.C ., Walton, V.M. (1975). Detecting Instrument Malfunctions in Control Systems. IEEE Trans. Aerospace Electronic Systems . VoU!. No.4. July 1975. pp.465-73.
The performance of the FDI scheme is very satisfactory with respect to failure hit rate , false alarm rate, and detection time. The detection time is about 50 msec . corresponding to only 5 samples . Only a few faults are missed because they have a very short duration and a negligible effect. The FDI method is not able to detect faults when load excitation is high, a drawback that is expected by the design.
Frank, P .M . (1991) . Enhancement of Robustness in Observer-Based Fault Detection. Preprints of SA FEPROCESS '91, Vol. I, Sept. 1991, pp. 275-87. Frank, P.M., Wiinnenberg, 1. (1989). Robust Fault Diagnosis Using Unknown Input Observer Schemes. In: Fault Diagnosis in Dynamic Systems Theory and Application (Ed . Patton, R.1 ., Frank, P., Clark, R.) , Prentice Hall , 1989, pp. 47-98.
The paper indicates the necessity of employing statistical testing in FDI. Using only threshold logic on observer residuals is not sufficient and even the simple CUSUM (Cumulative Sum) algorithm implemented as an SPRT (Sequential Probability Ratio Test) improves the decision confidence considerably . In this paper a two-sided CUSUM test is applied to detect mean changes in the observer residuals.
Gertler, 1.1 . (1988) . Survey of Model-Based Failure Detection and Isolation in Complex Plants. IEEE Control Systems Magazine. Dec. 1988, pp. 3-11. Isermann, R. (1991) . Fault Diagnosis of Machines via Parameter Estimation and Knowledge Processing . Preprints of SAFEPROCESS '91, Vol. 1, Sept. 1991, pp. 121 -33
The FDI approach has been analyzed using a simulation of laboratory equipment, but further tuning is anticipated when the algorithm is tested on longer sequences from the real equipment.
Nielsen, S.B., Patton, R.J ., Blanke, M., J~rgensen , R.B. (1993) . Industrial Actuator Benchmark Test . Aalborg University Report R93 -4020, ISSN 0908-1208 . May 1993.
461
Montgomery, R.C., Williams, I .P. (1989) . Analytical Redundancy Management for Systems with Appreciable Structural Dynamics. In : Fault Diagnosis in Dynamic Systems Theory and Application (Ed. Patton, R.J., Frank, P. , Clark, R.), Prentice Hall, 1989, pp. 361-86.
Appendix B. Fault Detection Results Obsefver 'I residual and Threstml
Patton, R.J ., Chen, I. (1991). Robust Fault Detection Using Eigenstructure Assignment: A Tutorial Consideration and Some New Results. IEEE Proc. of the 30th Con! on Detection and Control, pp.2242-7. Seliger, R., Frank, P.M. (1991). Robust Component Fault Detection and Isolation in Nonlinear Dynamic Systems Using Nonlinear Unknown Input Observers. Preprints of SAFEPROCESS '91, Vo!. I, Sept. 1991, pp. 313-8.
Observer 12 Residual 20
Tzafestas, S., Watanabe, K. (1990). Modem Approaches to System/Sensor Fault Detection and Diagnosis. Journal A, vol 31, no. 4, pp. 42-56. Willsky, A.S . (1976) . A Survey of Design Methods for Failure Detection in Dynamic Systems. Automatica Vo!. 12.,1976, pp. 601-11.
~0·L---~0.-5--~--~1~.5--~2~--~2.5----i----3~ . 5--~
Tme[s] SPRT Decision variables from observer'l Residual 50
Appendix A. Test Sequence Plot Load 0isIUIbance a..I
40
A
30
1
,
f
10
O~~--------~
'11
,
CtmeIt taul OFF
u
~,
, ,I' ~
11
I,
1.5
~
2 Tme[s]
n
n:'','I \"
",
'I
0.5
t,
1,.\
~U2
" ,,~
I I
,
11
20
E ~
1\
Ul
,
2.5
~
3.5
3
~ Ctmn taut ON -500
0
0.5
1.5
2 Trne Is]
2.5
SPRT Decision variables from observer.2 Residual
3.5
50
, ~ ,~~ I
1111~ll
40
AcIuaIor Ctmn Lm and 11th Fat,j! j'_m
111~ll
L_21
III
30 20
\
j
10
" L 22
A 0.5 0.5
1.5
2 Trnels]
2.5
'0/' 1.5
,I
,',~',' ,W,' ,,,",,
Threshold B
2 Tme[l]
,I,ll,' ,Ill,' ,"
~ 2.5
1. 3.5
P..-e Indicatora (CIuhed) IIId Dec:iIiona (solid)
3.5 Load
~.
i
0
~3
1 0
0.5
1.5
2 Tmels]
2.5
3.5
The observer residuals r, and r2 are used to calculate the decision variables LII , L/2' Lz" and L22 . Applying threshold logic on the decision variables, the decisions shown in the bottom graph are obtained.
The test sequence includes excitation of the load disturbance and a repetitive and temporary current fault. The bottom graph shows the effect on motor shaft velocity.
462
DIESEL ENGINE ACTUATOR FAULT ISOLATION USING MULTIPLE MODELS HYPOTHESIS TESTS S0REN B0GH NIELSEN Aalborg University. Department of Control Engineering. Fredrik Bajers Vej 7. DK 9220 Aalborg Tel: +4598158522. Fax: +4598151739. E-mail: [email protected]
~.
Denmark.
Abstract. Detection of current faults in a D.C. motor with unknown load torques is not feasible with linear methods and threshold logic. This paper suggests an FDl scheme with a bank of observers where a nonlinear observer is designed using the structure of the current fault. Multiple hypothesis testing of the observer residuals is applied to discriminate between faults and unknown input. The strategy has been applied to a simulation of a diesel engine actuator with an encouraging progress on performance compared to other results. Keywords. Failure detection; actuator failures; D.C motors; fault classification; control applications; decision theory; hypothesis testing.
task is to design a detector for a fault in the motor current ili .. that is robust to load disturbances . Linear observer design methods optimize the directional properties of the residuals to facilitate isolability (Patton & Chen (1991). Frank. P.M. (1991», but as the fault input and the load disturbance input enter at the same point, these approaches fail the isolability criteria.
1. INTRODUcnON Fault detection and isolation (FDI) in dynamic systems has been widely discussed in the literature. In the field of qualitative methods. the focus has been on linear techniques used to design fault detection observers (Frank & Wtinnenberg (1989)) or parameter estimators (Isermann (1991», where the key difficulties are fault isolability and robustness to unknown inputs. Faults are traditionally modelled as additive inputs to a dynamic model or as parameter changes. This failure representation is not always adequate as faults may appear multiplicative and their occurrence can be temporary. Additional details on the failure structure shall then be included in the FDI design.
The proposed method in this paper uses nonlinear observer design in a bank of observers. where each observer represents a different operational mode of the plant (nominal and faulty). Multiple hypothesis test is used to discriminate between current fault events and load disturbances. The analysis is carried out all the way to the point of decision making, which is often missed in FDI studies. 2. MULTIPLE MODELS HYPOTHESIS TEST
This paper presents a study in FDI on a diesel engine actuator, where linear design methods are not sufficient. Nonlinear filtering is used to detect a specific fault and isolate to unknown inputs. An introduction to the actuator is found in Blanke et al. (1994) and a detailed description is available in B~gh et al. (1993). A block diagram of the equipment is shown in Figure 1.
The classical approach to FDI using a set of filters in an observer bank has been proposed by several authors (Clark et al. (1975), Montgomery & WiIliams (1989». The paper by Frank & Wtinnenberg (1989) gives an overview of the unknown input observers for both sensor and component failure detection. Each observer in the bank is optimized for sensitivity to a specific subset of faults and is made invariant to another subset. If each observer is designed for a separate fault hypothesis. both fault detection and isolation can be accomplished using multiple hypothesis tests on the observer residuals. Surveys on statistical testing are given in Willsky (1976). Gertler (1988). Tzafestas & Watanabe (1990). and Basseville & Nikiforov (1993).
Figure 1 Block diagram of diesel engine actuator with velocity control loop. Current fault is ill.. and load disturbance is Qt
This paper uses an event detection and fault isolation scheme as illustrated in Figure 2.
A D.C. motor (with shaft velocity n.. and arm position so) is driven by a PI velocity controller with input reference nrrr An unknown load torque Q, is present and the
457
below a threshold B a current fault is concluded . The complete decision logic is listed in Table I. Table 1. Decision logic for current faults
Decision
SPRT Decision Variables
Activity
(L, ,=O)A(L,z=O)
No Decision
(L,,~)A(LI1~A)
L-......:-~~r2
Figure 2. FDl scheme for event detection and current fault isolation.
and:
(L2J~)v(L22~)
NOT«(L, , =O)A(LI1=O»v«L, ,~)A(L'2~»)
The residual from observer I is used to detect whether a current fault is present or a dominant load torque is acting. When an event has been detected (r, exceeds a threshold), the residual from the second observer is used to distinguish between the two causes . Observer I is designed from a nominal linear model of the plant, so the output will be sensitive to (i) current faults, (ii) load disturbances, and (iii) excitation of unmodelled dynamics and nonlinearities. Observer 2 is designed from a model of the plant when a current fault is active. When a current fault is not active, the residual is then expected to be large. If a current fault is active and load activity is small , the res idual is expected to be small. The output will, anyway, be sensitive to load disturbances and unmodelled dynamics.
3. OBSERVER FOR NOMINAL PLANT An observer is designed for the nominal plant shown in Figure I with current fault and load included as unknown additive inputs . The observer residual is designed to be sensitive to the unknown input and the current fault. The residual is also sensitive to effects from model uncertainties as unmodelled dynamics and nonlinearities are not included in this linear observer. The discrete time state space description of the plant x(k+l) =Ax(k) + Bu(k) + Ed(k) + F.f.(k) y(k) = Cx(k)
Fault isolation is not feasible using only one sample of the residuals. After an event has been detected, confidence is accumulated that r, is persistently large and that r2 is persistently small. This information is used to conclude about current faults. For this purpose, statistical methods are required. A two-sided CUSUM (Cumulated Sum) algorithm implemented as an SPRT (Sequential Probability Ratio Test) is used in this case as it is computational simple . The strategy of the fault isolation scheme is pictured in Figure 3.
A"
0.507 0.635
[ 5.42' \0-5
-0.358 0] [0.380] [ -0.493] -9.15' \0-2 0 ;B- 1.064 ;F - 0.635 3.93'\0-5 I
7.07.\0-5 ' 5.42·\0-5 (2)
10*---7\;"::' , .
~o
1
o
0 0.978
0]
A Luenberger observer is designed using eigenstructure assignment by applying the dual control problem. A tutorial is available in Burrows (1990). Increased robustness to noise on the velocity measurement , n .. , is achieved by using left eigenvector assignment, described in Patton & Chen (199\).
Cunent Fault
t;;"'
(l)
is derived from Blanke et al. (1994), where x = [ i2 ' n .. • sof is the state vector, u = n"" is input (output from position controller), d = [Q~ v] is unknown load torque and noise on velocity measurement, f. = tli.. is additive current fault, and y = [n .. , so). The matrices A, B. E , F., and C are found for a sampling time of tlT=lO ms:
-1.21 .\0-2 -3.79.\0--'] E .. 1.55·\0-2 -1.06 ;C[ 1.33-10-<1 -7.07 ·\0--5
L21.Ln
Current Fault Load Disturb. Continue
(L2J <.B)A(L22 <.B)
LoadDist
The observer structure is
B
Load Dist Figure 3. When an event has been detected, isolation between cu"ent fault and load disturbance is accomplished by statistical testing in the outlined scheme. Current
i(k+ 1) = (A -K. C) i(k) + Bu(k) + K. y(k)
Fault
y(k)
(3)
= Ci(k)
The desired discrete time eigenvalues and left eigenvectors of the observer are chosen to be
The variables L" or LI2 increase from zero when the mean of r, changes. Similarly, L2J or L22 increase from zero when the mean of r2 changes. If L" or LJ2 exceed a threshold A, a conclusion is drawn that the event is persistent and a decision on the cause is based on the size of the decision variable Lz, and Lzl' If these are both
458
AI = e
-20dT
Az
= e
-I06T
~
=e
- 2306T
(4)
Current fault:
leading to a gain matrix -6.24.10-.
4.40.10-5 ]
K) = -5.71·10-) -7.77.10-5 [ 7.16.10-5 1.85·10-)
[1
(10)
Aim(t) = -(imO(t)
(11)
(5)
Current saturation:
Output from the observer is a residual computed as a weighted sum of the observer estimation errors. The residual of observer 1 shall be sensitive to the un-known inputs. An appropriate weight matrix is r) =
imo(t) = K/n,Jt) - nm(t)) + iit)
(12)
AiStIl(t) =-(i';'o(t) < -imax)(i';'o(t) +imax )
(6)
5·1Q3] (y(k) - j(k))
(13)
-(i';'o(t) > imax)(i';'o(t) -imax )
4. OBSERVER FOR FAULTY PLANT The second observer is designed from a model of the plant with an active current fault. The observer residual is then small when the current fault is present, but sensitive to load disturbances. To improve the isolability between current fault events and load disturbances, the observer is robustified by including two significant nonlinear characteristics: Velocity controller integrator limit and current saturation in the power drive.
An observer is developed in continuous time by using only the linear part of the plant in the observer design. The nonlinear term is later added in the differential equations of the observer. Since the system is stable in both the saturation region and outside, it is assumed that the observer will be stable. Stability is not guaranteed. Robust nonlinear observer design for FDI has been considered in Seliger & Frank 1991. The method is based on a state transformation that is only applicable in a limited number of plant structures, and cannot be implemented in this case.
The structure of the current fault is recognized as a rectification of the signal i m (an ideal diode). This effect is integrated as a nonlinear element. The character of the nonlinear elements are described in Blanke et aI. (1994) and they can be arranged into additive terms to the existing linear model by a transformation as seen in Figure 4.
Left eigenvector assignment is applied as in the observer for the nominal plant. The observer structure is x(t) = (Ac-K"C)x(t)+Bcu(t)+K"y(t)+/(x,u,I,,) (15) jet)
fi2 Linlted Integrator
Cul'l'8t'1t Fault
The desired eigenvalues and left eigenvectors of the linear part of the observer is chosen to be
current
Saturation
Figure 4. Nonlinear elements of the actuator arranged as additive quantities to the linear model. The nonlinear continuous time state-space equation becomes x(t)
yet)
= Acx(t) = Cx(t)
+ I(x, u, I,,) + Bcu(t)
A.)
= -10
~
= -50
[I, 12 I3l =
~ = -130
r~ ~ ~] l~
(16)
0 0
leading to a gain matrix
(7)
-66.4
where I includes nonlinear terms in states x, input u, and fault I". The matrices A c and Bc and the parameters below can be found in Blanke et al. (1994). The nonlinear terms can be found to be j(x,ula ) fnm(t), O]T, where
= Cx(t)
K2 =
[
°]
2
8.92
9.7.103.8·10-)9
1.12.10-2
10.2
(17)
The residual from observer 2 is chosen to be the velocity estimation error
= [fi2(t),
r2
(8)
= [1
0] (y(k) - j(k))
(18)
The nonlinear continuous time observer eq.(15) is simulated with a 4'th order Runge-Kutta algorithm with a fixed step length of 5 msec.
Integrator limit: 5. CUSUM TEST AS AN SPRT
f 12(t)- -[(i2(t) ~ -ilim )(d'7(t) <0) + (i2(t) ~ilim)(dj2(t)> 0)] dj2(t) (9)
The statistical testing method applied in this paper is the two-sided CUSUM algorithm (Basseville & Nikiforov (1993)). A single CUSUM calculates the cumulative sum of a log-likelihood ratio. In this paper the hypothesis of a fixed change in mean is used. When both a positive
459
6. EXPERIMENTAL RESULTS
and negative change can occur (J.1-J.Io=±v), the two-sided CUSUM is used. The extended case of composite hypotheses (1J.1-J.1o I>v), can be solved with a sequential x2-test (Basseville & Nikiforov (1993».
The FDI scheme presented above has been applied to a simulation of the diesel engine actuator benchmark. The simulation software is quite complex and reproduces the behaviour of the real equipment in close detail. The test shows the ability of discriminating between current faults and external load torque disturbances and also illustrates the area of problems.
The window length of the CUSUM is of fixed size. If the algorithm is implemented as an SPRT, the window length is arbitrary and depends on the information in past samples. The SPRT is used in this paper to reach a decision with a desired confidence interval.
A four seconds sequence is plotted in appendix A. The current fault (no negative currents) is present from 0.4 sec to 2.3 sec. The current fault causes integrator windup in the velocity controller, i m • This demonstrates the need for including integrator limits and current saturation in the observer for the faulty plant. The effect of the fault is a positive velocity error, nm-n"J'
The hypotheses in this paper are based on mean changes of the residuals as listed in Table 1 for the two test cases.
Table 2. Hypotheses for the two tests. Test
Hypothesis
Rationale
SPRT 1
Ho: J.11=0
No Event
HI: IJ.lJ/=v J
Event Persistent
Ho: J.12=0
Current Fault Active
H,: 1J.12I=v2
Load Disturbance
SPRT 2
The design parameters have been experimentally tuned to the values in Table 3.
Table 3. Design parameters for the FDI scheme.
Four log-likelihood ratios are required in this case: A positive mean change and a negative mean change for each residual r, and r2 • Let p(r; IH) p(rlk-n),rlkn+1), ...,1j(k)IH), then
=
L II
=
p(rJ· ' J.1=+v I) _
ln~
L I2
=
p(rt IJ.1=-v I) _
ln~
p(rl·1 J.1=O)
p(rJ· ' J.1=O)
l.n
p(r; IJ.1=-V2) = lnl-_ _
The probability densities are computed assuming residual j to be a Gaussian random sequence with variance In this case, the recursive formula for the decision variables for a positive mean change are for the two tests (;=1,2)
a;.
Lj/(k-l)-
2~: (v 2ri k»J j -
(20)
where a lower limit of zero is realized with the supremum function. Similar for a negative mean change, Lp(k) = sup
(0,
Lp(k-l) +
2~: (-v 2ri k»J j-
Observer 1 Threshold
T
17
SPRT 1 Mean
v,
20
SPRT 1 Variance
Value
er;
22
SPRT 1 Threshold
A
30
SPRT 2 Mean
v2
10
SPRT 2 Variance
~
6
SPRT 2 Threshold
B
15
The residuals and decision variables for the FDI scheme are plotted in appendix B. The residual from the observer for the nominal plant, rI' is seen to increase whenever a current fault is active and/or a load disturbance is present. Each time r, exceeds the threshold T, the SPRT calculation is activated, producing the decision variables. L JJ or LJ2 rapidly exceed the upper threshold A when r, remains large, demanding for a decision. The decision is based on r2 , which is seen to be small when the current fault is enabled. During active load torque r2 grows, enabling separation of the two causes. The decision variables, ~, and ~2' remain below the threshold, B, as long as only the current fault is active, but one of them exceeds the threshold when the load torque is also present. This yields the decisions, depicted in the bottom figure together with indicators for periods when current fault is active and load torque is active.
are the four decision variables. They are tested against the thresholds in Table 1.
(0,
Symbol
(19)
p(r; IJ.1=O)
Lj/(k) = sup
Parameter
(21)
The presence of a current fault is detected only about 50 msec. after the effect shows up the first time at 0.5 sec. Subsequently, a train of failure alarms indicates successful current fault detection. During periods with load disturbances, the current fault alarm rate decreases and events are classified as load changes. This shows the inevitable obstacle that current faults cannot be detected during high load excitation. During the period with no faults and high load activity, no false alarms are fired. This demonstrates the required event isolation.
The decision variables are calculated at sample k for the last n samples. A minor disadvantage of the isolation approach in this configuration is that a decision is forced to be taken when L JJ or LJ2 exceeds the threshold A. This does not guarantee that enough samples has been included in the computation of L2J and L22 for the isolation between current faults and load disturbances.
460
The quantity of missed alarms is a key issue in FDI. An example is present at 1.2 sec., that is caused by too low a sensitivity of observer I. The current fault induces only a small error, which is not manifested in the residual r l • The missed alarm rate can be reduced by increasing the sensitivity to fault events, but with a penalty on the false alarm rate.
Fault detection and isolation with nonlinear observers and statistical testing results in a relatively high computational load. The complexity is limited in this case, by calling the CUSUM algorithms only when an event has been detected by the linear observer and isolation is required. Ideally, all significant nonlinearities and higher order terms of the model should be included in both observers for increased robustness and isolability, but the computational complexity will be unrealistic for realtime applications.
A few current fault decisions are triggered in the periods after the fault is no longer active. This is caused by the subsequent settling time of the observers, and cannot be avoided. It is not considered to be a major problem, as the vital feature is to avoid missed alarms.
8. ACKNOWLEDGEMENTS This project is supported by the Danish Research Council, STVF grant number 16-4979.
The applicability of the FDI scheme in a wider range of operating modes of the actuator has been exercised on a sequence with large velocity reference changes (data is available in Nielsen et al. (1993». Excitation of current saturation and unmodelled dynamics cause triggering of the FDI scheme, but only a single false alarm is present. The current fault is perfectly recognized.
9. REFERENCES Basseville, M., Nikiforov , I. V. (1993) . Detection of Abrupt Changes: Theory and Application. Prentice Hall. ISBN 0-13-126780-9. Blanke, M., Nielsen, S.B., J~rgensen, R.B ., Patton, R.J. (1994). Fault Detection and Accommodation in Diesel Engine Actuator - a Benchmark, Preprints of SA FEPROCESS '94.
7. CONCLUSIONS Current fault isolation on the diesel engine actuator using multiple hypothesis test on the residuals from an observer bank shows promising results. A standard linear observer of the nominal plant is used to detect events, ie. either current faults or load disturbances . Isolation between the two events is carried out using a new approach where the structure of the current fault is included in the design of a second observer. The classical approach with faults modelled as unknown additive inputs is inadequate in this case. The new approach is often avoided primarily because the detailed knowledge is generally unavailable and furthermore because it requires nonlinear observer design that is difficult to assure stability . The stability question has not been in focus in this paper.
B~gh,
S., Blanke, M., J~rgensen, R.B. (1993). Nonlinear Characterization and Simulation of Industrial Position Control Loop. Aalborg University Report R93-4017. Feb. 1993. Burrows, S.P. (1990). Robust Control Design Techniques Using Eigenstructure Assignment. D.Phil Thesis, University of York, sept. 1990. Clark, R.N ., Fosth, D.C ., Walton, V.M. (1975). Detecting Instrument Malfunctions in Control Systems. IEEE Trans. Aerospace Electronic Systems . VoU!. No.4. July 1975. pp.465-73.
The performance of the FDI scheme is very satisfactory with respect to failure hit rate , false alarm rate, and detection time. The detection time is about 50 msec . corresponding to only 5 samples . Only a few faults are missed because they have a very short duration and a negligible effect. The FDI method is not able to detect faults when load excitation is high, a drawback that is expected by the design.
Frank, P .M . (1991) . Enhancement of Robustness in Observer-Based Fault Detection. Preprints of SA FEPROCESS '91, Vol. I, Sept. 1991, pp. 275-87. Frank, P.M., Wiinnenberg, 1. (1989). Robust Fault Diagnosis Using Unknown Input Observer Schemes. In: Fault Diagnosis in Dynamic Systems Theory and Application (Ed . Patton, R.1 ., Frank, P., Clark, R.) , Prentice Hall , 1989, pp. 47-98.
The paper indicates the necessity of employing statistical testing in FDI. Using only threshold logic on observer residuals is not sufficient and even the simple CUSUM (Cumulative Sum) algorithm implemented as an SPRT (Sequential Probability Ratio Test) improves the decision confidence considerably . In this paper a two-sided CUSUM test is applied to detect mean changes in the observer residuals.
Gertler, 1.1 . (1988) . Survey of Model-Based Failure Detection and Isolation in Complex Plants. IEEE Control Systems Magazine. Dec. 1988, pp. 3-11. Isermann, R. (1991) . Fault Diagnosis of Machines via Parameter Estimation and Knowledge Processing . Preprints of SAFEPROCESS '91, Vol. 1, Sept. 1991, pp. 121 -33
The FDI approach has been analyzed using a simulation of laboratory equipment, but further tuning is anticipated when the algorithm is tested on longer sequences from the real equipment.
Nielsen, S.B., Patton, R.J ., Blanke, M., J~rgensen , R.B. (1993) . Industrial Actuator Benchmark Test . Aalborg University Report R93 -4020, ISSN 0908-1208 . May 1993.
461
Montgomery, R.C., Williams, I .P. (1989) . Analytical Redundancy Management for Systems with Appreciable Structural Dynamics. In : Fault Diagnosis in Dynamic Systems Theory and Application (Ed. Patton, R.J., Frank, P. , Clark, R.), Prentice Hall, 1989, pp. 361-86.
Appendix B. Fault Detection Results Obsefver 'I residual and Threstml
Patton, R.J ., Chen, I. (1991). Robust Fault Detection Using Eigenstructure Assignment: A Tutorial Consideration and Some New Results. IEEE Proc. of the 30th Con! on Detection and Control, pp.2242-7. Seliger, R., Frank, P.M. (1991). Robust Component Fault Detection and Isolation in Nonlinear Dynamic Systems Using Nonlinear Unknown Input Observers. Preprints of SAFEPROCESS '91, Vo!. I, Sept. 1991, pp. 313-8.
Observer 12 Residual 20
Tzafestas, S., Watanabe, K. (1990). Modem Approaches to System/Sensor Fault Detection and Diagnosis. Journal A, vol 31, no. 4, pp. 42-56. Willsky, A.S . (1976) . A Survey of Design Methods for Failure Detection in Dynamic Systems. Automatica Vo!. 12.,1976, pp. 601-11.
~0·L---~0.-5--~--~1~.5--~2~--~2.5----i----3~ . 5--~
Tme[s] SPRT Decision variables from observer'l Residual 50
Appendix A. Test Sequence Plot Load 0isIUIbance a..I
40
A
30
1
,
f
10
O~~--------~
'11
,
CtmeIt taul OFF
u
~,
, ,I' ~
11
I,
1.5
~
2 Tme[s]
n
n:'','I \"
",
'I
0.5
t,
1,.\
~U2
" ,,~
I I
,
11
20
E ~
1\
Ul
,
2.5
~
3.5
3
~ Ctmn taut ON -500
0
0.5
1.5
2 Trne Is]
2.5
SPRT Decision variables from observer.2 Residual
3.5
50
, ~ ,~~ I
1111~ll
40
AcIuaIor Ctmn Lm and 11th Fat,j! j'_m
111~ll
L_21
III
30 20
\
j
10
" L 22
A 0.5 0.5
1.5
2 Trnels]
2.5
'0/' 1.5
,I
,',~',' ,W,' ,,,",,
Threshold B
2 Tme[l]
,I,ll,' ,Ill,' ,"
~ 2.5
1. 3.5
P..-e Indicatora (CIuhed) IIId Dec:iIiona (solid)
3.5 Load
~.
i
0
~3
1 0
0.5
1.5
2 Tmels]
2.5
3.5
The observer residuals r, and r2 are used to calculate the decision variables LII , L/2' Lz" and L22 . Applying threshold logic on the decision variables, the decisions shown in the bottom graph are obtained.
The test sequence includes excitation of the load disturbance and a repetitive and temporary current fault. The bottom graph shows the effect on motor shaft velocity.
462