- Email: [email protected]
Robust fault detection for a jet engine using robust l1 Estimation1
IFAC [;UC>
Copyright 0 IFAC Fault Detection, Supervision and Safety of Technical Processes, Washington, D.C., USA, 2003
Publications www.elsevier.comllocatelifac
ROBUST FAULT DETECTION FOR A JET ENGINE USING ROBUST i1 ESTIMATION 1 Tramone Curry· Emmanuel G. Collins·
• Department 01 Mechanical Engineering Florida A&M Univer"itl/ - Florida State Universitl/ Tallahauee, FL 3£310
Abstract: This paper considers the application of robust i1 estimation to robust fault detection for a jet engine. It reviews the design of robust i1 estimators based on multiplier theory and the resulting fixed threshold approach to fault detection (FD). Although this methodology has been established, this paper investigates its application to a realistic system. As the jet engine is a highly nonlinear system, a linear representation of the system incorporates uncertainty to account for modeling errors. Due to the modeling errors and some unmeasurable disturbances, it is difficult to distinguish between the effects of an actual fault and those caused by uncertainty and disturbances. Of the various types of faults that may occur, it is typical for a jet engine to experience a drift in the sensor reading, where the error between the actual and sensed values increases over time. It is the aim of a robust FD system to be sensitive to faults, such as this, while remaining insensitive to uncertainty and disturbances. In addition to uncertainty in the system and output matrices A and C, respectively, this paper considers uncertainty in the input matrix B, whereas current theory does not apply to this uncertainty. Using a discrete-time, linear uncertain model of a jet engine and implementing an LQG controller, the closed-loop system is formulated. A fault is represented by simulating sensor drift in the system. The results of FD using robust il estimation with a fixed threshold are demonstrated for the closed-loop system with very positive results. Copyright © 2003 [FAC Keywords: Fault Detection, Robust Estimation, Uncertainty
and type of the fault after detection. In modern systems such as aircraft and spacecraft, there is an increasing demand for reliability and safety. For example, a jet engine is very critical for an aircraft and if faults occur, the consequences can be extremely serious (Chen and Patton, 1992). As with all dynamic systems, a jet engine can be subjected to actuator and/or sensor faults. In the realm of sensor faults, it is typical for the jet engine to experience a drift in the sensor reading, where the error between the actual and sensed values increases over time. It is the aim of a robust
1. INTRODUCTION
A fault can be defined as an unexpected change in a system, such as a component malfunction (Chen et al., 1996) that tends to degrade or cause undesirable behavior in the overall system performance. The purpose of fault detection is to determine the time of occurrence of a fault, whereas fault isolation seeks to also determine the location I This work wa6 supported in part by NASA under grant NAG3-2193 and in pan by AFOSR under grant F49620-
01-0550.
161
FD system to be sensitive to faults, such as this, while remaining insensitive to uncertainty and disturbances. Hence, over the last two decades fault detection (FD) and fault detection and isolation (FDI) have attracted considerable research interest (Chen and Patton, 1996; Chen et al., 1996; Collins and Song, 2001; Dexter and Benouarets, 1997; Faitakis and Kantor, 1996; Frank and Ding, 1997). However, there remains a need for more effective methodologies for FD and FDI. This paper exclusively focuses on FD, as applied to a jet engine. Fault detection schemes can be classified into two categories, physical redundancy and analytical redundancy. The latter methods are usually modelor estimator-based approaches (Frank, 1990) in which the key component is the computation of the residual or estimation error, which is the difference between the actual and estimated plant output. A number of investigators have studied the use of these estimator-based methodologies, which consist of two-stages: (1) re6idual generation and (2) decision making (Chen and Patton, 1992). The residuals &re used to accentuate faults in the system. In principle, these residual values should be zero in fault free operation and nonzero in the presence of a fault. However, due to factors such as modeling uncertainty and exogenous disturbances, i.e., plant disturbances and measurement noise, the residuals are almost always nonzero, even in fault free operation. One way of accounting for this condition, is to use a threshold to determine the occurrence of a fault. Comparison of some function (e.g., a norm) of the residual to the threshold is used as the indicator that separates the faulty system from the fault free case. The selection of the threshold can be dependent on some measure of the magnitude of the exogenous system input. As the jet engine is a highly nonlinear system, a linear representation of the system incorporates uncertainty to account for modeling errors. Due to the modeling errors and unmeasurable disturbances, it is difficult to distinguish between the effects of an actual fa.ult and those caused by uncertainty and disturbance. This ambiguity is the main source of false alarms and gives rise to the need for robustness in the FD system (Chen et al., 1996). It is the aim of a robust FD system to be sensitive to faults while remaining insensitive to uncertainty and disturbances. In order to reduce false alarms rates and increase the accuracy in FD the residuals themselves must be robust, which requires robust estimation.
completely decouple the effects of the disturbance inputs from those caused by faults (Chen and Patton, 1996; Chen et al., 1996; Frank and Ding, 1997; Gertler, 1997; lsermann, 1997). Another approach to robust FD models-the uncertainty as complex uncertainty with bounded magnitude (Frank and Ding, 1994) and develops a solution to the robust detection problem using the small gain theorem. For real parametric uncertainties a less conservative approach is one that is based on multiplier theory (essentially mixed structure singular value theory) (Collins and Song, 1998, 2OOOa,b, 2001). In order to address time-domain point-wise-in-time performance, II theory, which captures worst-case peak amplitude response due to bounded persistent loo disturbances, is clearly an appropriate performance criterion (Collins and Song, 2001). This paper will use robust 11 estimation based on multiplier theory (Collins and Song, 2001) to generate the residuals for fault detection. These robust estimators minimize an upper bound (over the entire uncertainty set) of the II norm of the residual due to bounded amplitude persistent loo disturbances. Just as with the previous methods, this robust residual generation technique may be used in conjunction with fixed thresholds (Collins and Song, 2000a,b, 2001). Using a discrete-time, linear uncertain model of an advanced afterburning turbofan engine and implementing a low authority LQG controller, these estimation and FD techniques will be applied to the closed-loop system. This paper is organized as follows. Section 2 presents the general descriptions of the uncertain system and controller, and the formulation of the closed-loop uncertain system to which estimation will be applied. Section 3 reviews robust II estimation and its application to robust fault detection. Section 4 presents and discusses simulated results for the jet engine while Section 5 gives concluding remarks. 2. FORMULATION OF CLOSED-LOOP UNCEIU'AIN SYSTEM A jet engine can be expressed as a discrete-time, linear uncertain dynamic system
= (A" + dA,,)x,,(k) + (B" + dB,,)u,,(k) + Doo,IWoo(k), k E Z+ (1) 1I,,(k) = (C" + dC,,)x,,(k) + Doo,2Woo(k),
x,,(k + 1)
(2) Up E 'R tJp
where xI' E 'Rn,. is the state vector, is the control input, 11" E 'R'" denotes the plant measurements, Woo E'RtJ .., denotes an loo disturbance signal satisfying IIwoo Oll oo ,2 ::; 1, and the uncertainties dA", dB" and dC" satisfy
One of the more common robust residual genera.tion approaches models the uncertainties as extra disturbance input terms and utilizes the technique of disturbance decoupling, i.e., it attempts to
162
~Ap E UA •
E 'Rnpxnp : ~Ap
!: {~Ap
=
-HA,.FA ,.GA., FA. E FA.},(3 ) n ABp E US,. £ {ABp E 'R ,.xt1,. : ~Bp = -HS,.F s.GS,., FS,. E FS,.},(4 )
ACp E Ue,. £
{~Cp
E 'Rp,.xn,. : ~Cp
=
-He,.Fe ,.Ge,., Fe. E Fe.}, (5) where
FA,. £ {FA,. E V
r
:
(6)
Fs,. £ {Fs,. E ~ : Ms,.,1 $ Fs ,. $ Ms,. ,2},
(7) (8) r
r
with MA, ... MA,.., E V , Ms, ... Ms." E V , Mc, .• , Mc•.• E V', MA,. ., - MA •.• ~ 0, Ms •.• Ms •.• ~ 0, and Mc,.., - Mc,. .• ~ O. Note that the system in (1) and (2) has uncerta inty in the input matrix Bp. This is significant in that current mixed structur e singular value (MSSV) theory does not allow this uncertainty. Assume that the dynamic system in (1) and (2) is controlled by a linear controller,
xe(k + 1) up(k)
= Aexe(k) + Bel/p(k), = Cexe(k) + Del/p(k).
3. REVIEW OF ROBUST i1 ESTIMATION & FAULT DETEC TION Given the closed-loop system in (11)-(12 ), it is desired to design a predictive filter with the form xe(k + Ilk) Aex.(kl k -1) + W[y(k) (19) - Cx.(klk - 1)]
=
MA,.,1 $ FA,. $ MA,.,2},
Fe,. £ {Fe. E V' : Me.,1 $ Fe. $ Me,.,2},
matrices. Hence, the robust estimati on and fault detection results of (CoUina and Song, 2001) may be applied.
(9) (10)
Then, the closed-loop system is described by,
to estimat e the state vector xl" where A. E 2npxp• are the paramet ers 'R2n,.x2n,. and W E 'R to be determined. The estimati on error is defined as (20) e(k) £ x(k) - x.(klk - 1), which using (11), (12), and (19) can be shown to obey the evolution equation e(k) = (A. - WC)e(k ) + (A + AA - W ~C (21) - A.)x(k) + (D1 - W D2)Woo (k). Now define the error output z E 'R 9, as z(k) £ Eooe(k). Then augmenting (11) with (21) yields i(k + 1) = (A + ~A)i(k) + D1Woo(k), (22) (23) z(k) E!i(k),
=
where
x(k + 1) = (A + ~A)x(k) + D 1w oo (k), (11) y(k) = (C + AC)x(k ) + D2Woo(k), (12) where
A _ [AI' + BpDeCp BpCe] (k) - [Xp(k)] Ae' BeC" - xe(k)' x D D1 = [B ;,1 ], C = [Cl' 0] D2 = Doo,2. e
Furthermore,
~A
=
[:m] ,A = [A ~A. A.
Dl
=
[Dl
and
~C
satisfy
E 'R 2n,.x2np : AA
Eoo] .
(24)
= (26)
F;.
= [ FA0
0] '
Fe
GA
0 ] HA Ho= [ HA WHe '
0]
(27)
Go = [ -Gc 0 '
and
(16)
Ml M2
$ Fe $ Me,2}, (17)
= diag(MA,lJMe,I) , = diag(MA,2,Mc,2) .
(28)
The robust il estima tion proble m is to find the estimato r paramet ers A. and W such that the combined system (19), (22)-(23) is asymptotically stable over the uncertai nty sets UA and Ue, and the i1 performance function
where
MA,l = diag(MA,. .• , Mc,. .• , Ms,. .• ), MA ,2 = diag(MA •.• ,Me.... Ms •.,), Me,2 = Mc,..,. Me.1 = Mc•."
[0
_0WC] ,
(25) . - HoFAGo , FA. E F . d, }, M $ F;. $ M1 : 2 ~r+2' E {FA £ FA where
=
E=
~A E Uj ~ {~A E 'R4npx4n p : ~A
(14) - HAFAGA, FA E FA}, p x2n ,. : AC = AC E Ue £ {AC E 'R • (15) - HeFeG e, Fe E Fe}, r MA,2}, $ FA $ ,1 MA : +. V E {FA £ :FA :Fe ~ {Fe E ~. : Me,1
~WDJ '
Furtherm ore, AA satisfies
(13)
00 ,2
~A e UA !: {AA
x(k)
(18)
.7(A., W)
Notice that in the closed-loop system (11)-(12) all of the uncerta inty appears in the AA and AC
163
=
sup IIH"tu~lh
~AEUJ
(29)
is minimized, where H zlII _ is the transfer function from the loo disturbance Woo to the performance variable z. To make this problem mathematically tractable multiplier theory (Collins and Song, 1998, 2001) is used to develop a differentiable upper bound on j(Ae, W) ~ .1(Ae, W) and a continuation algorithm is used to solve for Ae and W which optimize this upper bound (CoUinB and Song, 2001).
Jell :! ([tr(EooQooE;;')fl~
+ 2umax (D2) . [tr(EooQooE;;')fl* + U!.,.(D2)} .
IIwlI~00.2).[No.Nl' (37) Robust fault detection can be accomplished by comparing IIrll(oo.2).[No.N] with Jeh. A fault occurs if IIrll(00.2).[No.N] > JUIJ i.e., IIrll(oo.2) .[No.N]
> Jeh :} a fault occurred. (38)
Now consider the uncertain discrete-time system
x(k + 1) = (A
+ ~A)x(k) + DIWoo(k) + Rtf(k), 4. JET ENGINE SIMULATION RESULTS
(30)
y(k)
= Cx(k) + D2Woo(k) + R2/(k),
(31)
where x, y, and Woo are previously discussed, and I E 'R-n , is the fault vector. Rtf(k) represents actuator and component faults while R2/(k) denotes the sensor faults. The fault distribution matrices RI and R2 are assumed to be known. It should be mentioned that the determination of the fault vector I and the fault distribution matrices Rt and R2 are normally determined on a case-by-case basis by inspection of the state-space model and the characteristics oC the particular process. A more general design procedure can be perCormed by employing component lault analysis techniques (Blanke et al., 1997) which guarantees that a complete set of fault effects is used.
The robust fault detection problem is to generate a robust residual signal r( k) that satisfies
=
IIr(k)IIp ~ JtI. if I(k) 0, Ilr(k)llp > Jth if I(k) '" 0,
=y -
CXe(k) =Ce(k) + D2W(k)
!lp(k)
0
(39)
where the sampling period is 0.01. The state vector is xp:! [zp, zP2 zp,
f,
(41)
and the states are described as:
x PI :! High Pressure Spool Speed (RPM) Low Pressure Spool Speed (RPM) Temperature (Degrees C). The control input vector is
Up:! [uPI "P2 up,]T,
(42)
where the inputs are defined as
up, ~ Main Burner Fuel Flow (Kg/HR) P2 ~ Exhaust Nozzle Throat Area (M2) u" !: Bypass Duct Area (M 2 ).
=
U
The output vector is
IIrll~00.2) .[No,Nl ~ {[tr(EooQooE~)ql~ + 2um.,. .
YP :! [YPI
(D'J)[tr(EocQooE;;')fl*
IIp~
y"
f'
(43)
where the outputs are defined as
+ u!ax(D2)}lIwlI~00.2) .[No.Nl' (35)
YPI
!: Actual High Pressure Spool Speed (RPM)
YJ12 :@: Actual Low Pressure Spool Speed (RPM)
where Qoo is a positive-definite matrix satisfying
y" !: Actual High Pressure Compressor Inlet Temperature (Degrees C).
+ ~1 Voo , a -
~
(40)
+ R2/(k).
WC)Qoo(A - WC)T
k
= (Cp + ~Cp)zp(k) + D oo .2 w(k) + v(k)
x" ~ High Pressure Compressor Inlet
As derived in (Collins and Song, 2001), if !(k) residual can be represented by the norm inequality
= a(A -
= (Ap + ~Ap)zp(k) + (Bp + ~Bp)up(k) + Doo.IW(k),
X P2 ~
(34)
o the
Qoo
xp(k + 1)
(32) (33)
where 11 • lip denotes the p norm oC a Lebesgue signal and Jth is the threshold value. IT the estimator (19) is applied to the system described by (30) and (31) the residual generated is
r(k)
An example is presented in this section to illustrate robust II estimator design using the PopovTsypkin multiplier and the application of the robust It estimator to robust Cault detection oC dynamic systems. The model used was supplied by NASA Glenn Research Center and is given as
(36)
The variable w denotes a vector of disturbance signals and v is sensor noise.
with Voo :! (Dl - WD 2 )(D 1 - WD 2)T and a > 1. The residual norm is defined as IIr(· )1100.2 !: ess sUPt>ollr(t)1I2, where [No, NJ corresponds to a certain tUne interval. The threshold can be chosen as
The uncertainty matrices, ~Ap, ~Bp and ~Cp, are representative oC some engine degradation over time. Thus, it is assumed that a newly constructed
164
nonzero values of their upper or lower bounds. For example OA" whose lower and upper bounds are 0 and 0.0011, respectively, was assigned its upper bound 0.0011. Whereas, OA., whose lower and upper bounds are -0.0008 ana 0, respectively, was assigned ' its lower bound -0.0008. Random white noise signals with zero mean were added as both the disturbance inputs and sensor noise. The variances of the disturbance inputs, W1, W2 and W3, were 0.03, 0.08 and 0.2, respectively, while the variances of the sensor noises, VI, V2 and va, were 0.004, 0.003 and 0.002, respectively.
engine can be modeled with the nominal matrices A p, Bp and Gp and with use, the parameters of the degraded engine are encompassed in the uncertainty. The system parameter matrices are 0.8938
Ap
Bp
= [ -0.0001 o
=
0.0034 0.0020] 0.8940 0.0014 , -0.0001 0.89960
0.0059 0.1119 -0.0160] 0.0042 0.0720 0.0083 , [ 0.0001 -0.0003 0
1.0000 0.0077 0.0044] -0.0003 1.0000 0.0031 , [ 0.0299 -0.0208 0.0010 = diag{O.l, 0.1, 0.01}, D p ,2 = 0.1 X I sx3 •
Gp = D p ,1
This paper only considered the occurrence of sensor faults within the system. As previously discussed, a typical sensor fault in the jet engine is a drift in the sensor reading. Thus, a. slow drifting (or ramping) sensor fault was added to the YP2 sensor reading at the time instant t = 40 sec. Specifically, the error between the actual value and the value of the faulty sensor's reading increased over time. Due to the disturbance and uncertainty, the finite-horizon infinity norm (35) of the residual with N - No 60 (corresponding to a time interval of 0.6 sec.), was nonzero even in the absence of faults.
(44)
The uncertainty matrices ~Ap = -HA.FA.GA., ~Bp = -HB.FB.GB" ~Gp = -He. Fe, Gc" where
FA. FB. Fc.
= diag{bAp bA., bA" bA., bAs, bAo} = diag{bBp bB., bB.} = diag{bel> bO., oo.}, (45)
=
with
o ~ OA, ~ 0.0011, -0.0008 ~ bA. ~ 0, o ~ OA. ~ 0.0002, -0.0001 ~ OA. ~ 0, o ~ OA6 ~ 0.0005, 0 ~ OAe ~ 0.0001, o ~ OB, ~ 0.0008, 0 ~ OB. ~ 0.0115, -0.0005 ~ OB. ~ 0, -0.0015
~
00.
~
-0.0001 ~ 00, ~ 0, 0, 0
~
bo.
~
0.0004.(46)
The matrices HA" GA., HB., GB., Ho. and Go. are chosen so that the uncertain parameters OA, through OAe correspond to parameter fluctuations in the first two rows of matrix Ap , OB, through OB. and oc, through 00. to the first row of Bp and Gp, respectively. Fig. 1. FD Using Robust L1 Estimation for a "Slow" Drift Fault
It is desired to design a predictive filter of the form
(19) for which the estimation error is given by (20). For this particular example, the error output is defined as z(k) = Eooe(k) where Eoo = Gp.
It can be seen from Figure 1 that both the nominali} estimator and the robust i1 estimator can
By applying the continuation algorithm described in (Collins and Song, 2001) to the closed-loop control system given by (11) and (12), the nominal and robust L1 estimator gains were generated. For the nominal estimator the state matrix A. is set equal to A, but for the robust estimator an optimal state matrix is selected.
detect this fault since both residuals surpassed their threshold values. But it should be noted that the residual generated by the nominali 1 estimator tends to give false alarms since it surpassed its threshold value at two time intervals before t 40 sec even though there is no fault in the system in the time interval [0,40 sec). This false alarm resulted from the fact that the nominal estimator cannot distinguish the fault effect from the effects caused by system uncertainties. It is also noted in both estimators that the fault was detected sometime after it was initially added to the system. However, due to the nature of the simulated fault it is initially very small. When a
=
In order to illustra.te the application of the robust L1 estimator to robust fault detection, FD of the closed-loop system in (22) and (23) subject to plant uncertainties and disturbances was performed. In order to show the extent of robustness, uncertainty for all system matrices was considered. The uncertain parameters are assigned the
165
.. Fig. 2. FD Using Robust 11 Estimation for a. "Fast" Drift Fault "harder" fault (one with a faster time constant) was simulated, the detection was faster and more closely reflected the initial occurrence of the fault. This can be seen in Figure 2 where the fault introduced at t 25sec. has a time constant twice as large as the one simulated in Figure 1. Fault detection was performed with a variety of system uncertainties, each yielding very similar results to those described in Figures 1 and 2.
=
5. CONCLUSIONS This paper considered the application of robust i1 estimation for uncertain, linear discrete-time systems to the robust fault detection of dynamic systems. The multiplier-based approach of (Collins and Song, 2001) was used to design robust 11 estimators and the resulting fixed threshold approach was further investigated. By considering a discrete, linear model of a jet engine with real parametric uncertainties, an LQG controller was implemented to fonn a closed-loop system. In addition to uncertainty in the system and output matrices A and C, respectively, this paper considered uncertainty in the input matrix B . Using this closed-loop system and introducing a drifting sensor fault, it was shown that the robust fault detection methodology based on fixed thresholds was capable of significantly reducing the false alarm rate. The application of this methodology to a realistic system proved to be feasible and very effective. Coupled with the closed-loop and uncertainty considerations, the application to ajet engine constitutes the main contributions of this paper.
REFERENCES M. Blanke, R. Izadi-Zamanabadi, S.A. Bogh, and C.P. Lunau. Fault tolerant control systems: a holistic view. Control Engineering Proctia, 5 (1):693-702, 1997.
166
J. Chen and R. J. Patton. A robust study of model-based fault detection for jet engine systems. In lEE Proceeding8-Control Theory and Applications, pages 871-876, July 1992. J. Chen and R. J. Patton. Optitnal filtering and robust fault diagnosis of stochastic systems with unknown disturbances. In lEE Procet!ding8Control Theory and Applicationl, volume 143, pages 31-36, January 1996. J. Chen, R. J. Patton, and H. Zhang. Design of unknown input observers and robust fault detection filters. International Journal of Control, 63(1):8&-105, 1996. E. G. Collins and T. Song. Robust H, estimation using the popov-tsypkin multiplier. In Procet!ding8 0/ IEEE Con/erenCL. on Decision and Contro~ pages 1189-1194, December 1998. Tampa-Florida. E. G. CoUins and T. Song. Multiplier-based robust Hoo estimation with applications to robust fault detection. Journal of GuidanCL., Control and Dt/namics, 23(5):857-864, 2000a. E. G. Collins and T. Song. Robust H, estimation with application to robust fault detection. Journal 0/ Guidance, Control and Dvnamics, 23 (6):1067-1071, 2000b. E. G. CoIlins and T. Song. Robust 11 estimation using the popov-tsypkin multiplier with applications to robust fault detection. International Journal of Control, 74(3):303-313, 2001. A. L. Dexter and M. Benouarets. Model-based fault diagnosis using fuzzy matching. IEEE 7hJnsactions on Systems, Man, Cybernetics, 27 (5):673-682, September 1997. Y. E. Faitakis and J . C. Kantor. Residual generation and fault detection for discrete-time systems using an loo technique. International Journal of Control, 64(1):155-174, 1996. P. M. Frank. Fault diagnosis in dynamic systems using analytical and knowledge -based redundancy - a survey and some new results. Automatica, 26(3):459-474, 1990. P. M. Frank and X. Ding. Frequency domain approach to optimally robust residual generation and evaluation for model-based diagnosis. Automatica, 30(5):789-804, 1994. P. M. Frank and X. Ding. Survey of robust residual generation and evaluation methods in observer-based fault detection systems. Journal of Process Contro~ 7(6):403-424, 1997. J . J . Gertler. Fault detection and isolation using parity relations. Control Engineering Practice, 5:653-661, 1997. R. Isermann. Supervision, fault-detection and fault-diagnosis methods - an introduction. Control Engineering Prodice, 5(5):639-652, 1997.
Copyright 0 IFAC Fault Detection, Supervision and Safety of Technical Processes, Washington, D.C., USA, 2003
Publications www.elsevier.comllocatelifac
ROBUST FAULT DETECTION FOR A JET ENGINE USING ROBUST i1 ESTIMATION 1 Tramone Curry· Emmanuel G. Collins·
• Department 01 Mechanical Engineering Florida A&M Univer"itl/ - Florida State Universitl/ Tallahauee, FL 3£310
Abstract: This paper considers the application of robust i1 estimation to robust fault detection for a jet engine. It reviews the design of robust i1 estimators based on multiplier theory and the resulting fixed threshold approach to fault detection (FD). Although this methodology has been established, this paper investigates its application to a realistic system. As the jet engine is a highly nonlinear system, a linear representation of the system incorporates uncertainty to account for modeling errors. Due to the modeling errors and some unmeasurable disturbances, it is difficult to distinguish between the effects of an actual fault and those caused by uncertainty and disturbances. Of the various types of faults that may occur, it is typical for a jet engine to experience a drift in the sensor reading, where the error between the actual and sensed values increases over time. It is the aim of a robust FD system to be sensitive to faults, such as this, while remaining insensitive to uncertainty and disturbances. In addition to uncertainty in the system and output matrices A and C, respectively, this paper considers uncertainty in the input matrix B, whereas current theory does not apply to this uncertainty. Using a discrete-time, linear uncertain model of a jet engine and implementing an LQG controller, the closed-loop system is formulated. A fault is represented by simulating sensor drift in the system. The results of FD using robust il estimation with a fixed threshold are demonstrated for the closed-loop system with very positive results. Copyright © 2003 [FAC Keywords: Fault Detection, Robust Estimation, Uncertainty
and type of the fault after detection. In modern systems such as aircraft and spacecraft, there is an increasing demand for reliability and safety. For example, a jet engine is very critical for an aircraft and if faults occur, the consequences can be extremely serious (Chen and Patton, 1992). As with all dynamic systems, a jet engine can be subjected to actuator and/or sensor faults. In the realm of sensor faults, it is typical for the jet engine to experience a drift in the sensor reading, where the error between the actual and sensed values increases over time. It is the aim of a robust
1. INTRODUCTION
A fault can be defined as an unexpected change in a system, such as a component malfunction (Chen et al., 1996) that tends to degrade or cause undesirable behavior in the overall system performance. The purpose of fault detection is to determine the time of occurrence of a fault, whereas fault isolation seeks to also determine the location I This work wa6 supported in part by NASA under grant NAG3-2193 and in pan by AFOSR under grant F49620-
01-0550.
161
FD system to be sensitive to faults, such as this, while remaining insensitive to uncertainty and disturbances. Hence, over the last two decades fault detection (FD) and fault detection and isolation (FDI) have attracted considerable research interest (Chen and Patton, 1996; Chen et al., 1996; Collins and Song, 2001; Dexter and Benouarets, 1997; Faitakis and Kantor, 1996; Frank and Ding, 1997). However, there remains a need for more effective methodologies for FD and FDI. This paper exclusively focuses on FD, as applied to a jet engine. Fault detection schemes can be classified into two categories, physical redundancy and analytical redundancy. The latter methods are usually modelor estimator-based approaches (Frank, 1990) in which the key component is the computation of the residual or estimation error, which is the difference between the actual and estimated plant output. A number of investigators have studied the use of these estimator-based methodologies, which consist of two-stages: (1) re6idual generation and (2) decision making (Chen and Patton, 1992). The residuals &re used to accentuate faults in the system. In principle, these residual values should be zero in fault free operation and nonzero in the presence of a fault. However, due to factors such as modeling uncertainty and exogenous disturbances, i.e., plant disturbances and measurement noise, the residuals are almost always nonzero, even in fault free operation. One way of accounting for this condition, is to use a threshold to determine the occurrence of a fault. Comparison of some function (e.g., a norm) of the residual to the threshold is used as the indicator that separates the faulty system from the fault free case. The selection of the threshold can be dependent on some measure of the magnitude of the exogenous system input. As the jet engine is a highly nonlinear system, a linear representation of the system incorporates uncertainty to account for modeling errors. Due to the modeling errors and unmeasurable disturbances, it is difficult to distinguish between the effects of an actual fa.ult and those caused by uncertainty and disturbance. This ambiguity is the main source of false alarms and gives rise to the need for robustness in the FD system (Chen et al., 1996). It is the aim of a robust FD system to be sensitive to faults while remaining insensitive to uncertainty and disturbances. In order to reduce false alarms rates and increase the accuracy in FD the residuals themselves must be robust, which requires robust estimation.
completely decouple the effects of the disturbance inputs from those caused by faults (Chen and Patton, 1996; Chen et al., 1996; Frank and Ding, 1997; Gertler, 1997; lsermann, 1997). Another approach to robust FD models-the uncertainty as complex uncertainty with bounded magnitude (Frank and Ding, 1994) and develops a solution to the robust detection problem using the small gain theorem. For real parametric uncertainties a less conservative approach is one that is based on multiplier theory (essentially mixed structure singular value theory) (Collins and Song, 1998, 2OOOa,b, 2001). In order to address time-domain point-wise-in-time performance, II theory, which captures worst-case peak amplitude response due to bounded persistent loo disturbances, is clearly an appropriate performance criterion (Collins and Song, 2001). This paper will use robust 11 estimation based on multiplier theory (Collins and Song, 2001) to generate the residuals for fault detection. These robust estimators minimize an upper bound (over the entire uncertainty set) of the II norm of the residual due to bounded amplitude persistent loo disturbances. Just as with the previous methods, this robust residual generation technique may be used in conjunction with fixed thresholds (Collins and Song, 2000a,b, 2001). Using a discrete-time, linear uncertain model of an advanced afterburning turbofan engine and implementing a low authority LQG controller, these estimation and FD techniques will be applied to the closed-loop system. This paper is organized as follows. Section 2 presents the general descriptions of the uncertain system and controller, and the formulation of the closed-loop uncertain system to which estimation will be applied. Section 3 reviews robust II estimation and its application to robust fault detection. Section 4 presents and discusses simulated results for the jet engine while Section 5 gives concluding remarks. 2. FORMULATION OF CLOSED-LOOP UNCEIU'AIN SYSTEM A jet engine can be expressed as a discrete-time, linear uncertain dynamic system
= (A" + dA,,)x,,(k) + (B" + dB,,)u,,(k) + Doo,IWoo(k), k E Z+ (1) 1I,,(k) = (C" + dC,,)x,,(k) + Doo,2Woo(k),
x,,(k + 1)
(2) Up E 'R tJp
where xI' E 'Rn,. is the state vector, is the control input, 11" E 'R'" denotes the plant measurements, Woo E'RtJ .., denotes an loo disturbance signal satisfying IIwoo Oll oo ,2 ::; 1, and the uncertainties dA", dB" and dC" satisfy
One of the more common robust residual genera.tion approaches models the uncertainties as extra disturbance input terms and utilizes the technique of disturbance decoupling, i.e., it attempts to
162
~Ap E UA •
E 'Rnpxnp : ~Ap
!: {~Ap
=
-HA,.FA ,.GA., FA. E FA.},(3 ) n ABp E US,. £ {ABp E 'R ,.xt1,. : ~Bp = -HS,.F s.GS,., FS,. E FS,.},(4 )
ACp E Ue,. £
{~Cp
E 'Rp,.xn,. : ~Cp
=
-He,.Fe ,.Ge,., Fe. E Fe.}, (5) where
FA,. £ {FA,. E V
r
:
(6)
Fs,. £ {Fs,. E ~ : Ms,.,1 $ Fs ,. $ Ms,. ,2},
(7) (8) r
r
with MA, ... MA,.., E V , Ms, ... Ms." E V , Mc, .• , Mc•.• E V', MA,. ., - MA •.• ~ 0, Ms •.• Ms •.• ~ 0, and Mc,.., - Mc,. .• ~ O. Note that the system in (1) and (2) has uncerta inty in the input matrix Bp. This is significant in that current mixed structur e singular value (MSSV) theory does not allow this uncertainty. Assume that the dynamic system in (1) and (2) is controlled by a linear controller,
xe(k + 1) up(k)
= Aexe(k) + Bel/p(k), = Cexe(k) + Del/p(k).
3. REVIEW OF ROBUST i1 ESTIMATION & FAULT DETEC TION Given the closed-loop system in (11)-(12 ), it is desired to design a predictive filter with the form xe(k + Ilk) Aex.(kl k -1) + W[y(k) (19) - Cx.(klk - 1)]
=
MA,.,1 $ FA,. $ MA,.,2},
Fe,. £ {Fe. E V' : Me.,1 $ Fe. $ Me,.,2},
matrices. Hence, the robust estimati on and fault detection results of (CoUina and Song, 2001) may be applied.
(9) (10)
Then, the closed-loop system is described by,
to estimat e the state vector xl" where A. E 2npxp• are the paramet ers 'R2n,.x2n,. and W E 'R to be determined. The estimati on error is defined as (20) e(k) £ x(k) - x.(klk - 1), which using (11), (12), and (19) can be shown to obey the evolution equation e(k) = (A. - WC)e(k ) + (A + AA - W ~C (21) - A.)x(k) + (D1 - W D2)Woo (k). Now define the error output z E 'R 9, as z(k) £ Eooe(k). Then augmenting (11) with (21) yields i(k + 1) = (A + ~A)i(k) + D1Woo(k), (22) (23) z(k) E!i(k),
=
where
x(k + 1) = (A + ~A)x(k) + D 1w oo (k), (11) y(k) = (C + AC)x(k ) + D2Woo(k), (12) where
A _ [AI' + BpDeCp BpCe] (k) - [Xp(k)] Ae' BeC" - xe(k)' x D D1 = [B ;,1 ], C = [Cl' 0] D2 = Doo,2. e
Furthermore,
~A
=
[:m] ,A = [A ~A. A.
Dl
=
[Dl
and
~C
satisfy
E 'R 2n,.x2np : AA
Eoo] .
(24)
= (26)
F;.
= [ FA0
0] '
Fe
GA
0 ] HA Ho= [ HA WHe '
0]
(27)
Go = [ -Gc 0 '
and
(16)
Ml M2
$ Fe $ Me,2}, (17)
= diag(MA,lJMe,I) , = diag(MA,2,Mc,2) .
(28)
The robust il estima tion proble m is to find the estimato r paramet ers A. and W such that the combined system (19), (22)-(23) is asymptotically stable over the uncertai nty sets UA and Ue, and the i1 performance function
where
MA,l = diag(MA,. .• , Mc,. .• , Ms,. .• ), MA ,2 = diag(MA •.• ,Me.... Ms •.,), Me,2 = Mc,..,. Me.1 = Mc•."
[0
_0WC] ,
(25) . - HoFAGo , FA. E F . d, }, M $ F;. $ M1 : 2 ~r+2' E {FA £ FA where
=
E=
~A E Uj ~ {~A E 'R4npx4n p : ~A
(14) - HAFAGA, FA E FA}, p x2n ,. : AC = AC E Ue £ {AC E 'R • (15) - HeFeG e, Fe E Fe}, r MA,2}, $ FA $ ,1 MA : +. V E {FA £ :FA :Fe ~ {Fe E ~. : Me,1
~WDJ '
Furtherm ore, AA satisfies
(13)
00 ,2
~A e UA !: {AA
x(k)
(18)
.7(A., W)
Notice that in the closed-loop system (11)-(12) all of the uncerta inty appears in the AA and AC
163
=
sup IIH"tu~lh
~AEUJ
(29)
is minimized, where H zlII _ is the transfer function from the loo disturbance Woo to the performance variable z. To make this problem mathematically tractable multiplier theory (Collins and Song, 1998, 2001) is used to develop a differentiable upper bound on j(Ae, W) ~ .1(Ae, W) and a continuation algorithm is used to solve for Ae and W which optimize this upper bound (CoUinB and Song, 2001).
Jell :! ([tr(EooQooE;;')fl~
+ 2umax (D2) . [tr(EooQooE;;')fl* + U!.,.(D2)} .
IIwlI~00.2).[No.Nl' (37) Robust fault detection can be accomplished by comparing IIrll(oo.2).[No.N] with Jeh. A fault occurs if IIrll(00.2).[No.N] > JUIJ i.e., IIrll(oo.2) .[No.N]
> Jeh :} a fault occurred. (38)
Now consider the uncertain discrete-time system
x(k + 1) = (A
+ ~A)x(k) + DIWoo(k) + Rtf(k), 4. JET ENGINE SIMULATION RESULTS
(30)
y(k)
= Cx(k) + D2Woo(k) + R2/(k),
(31)
where x, y, and Woo are previously discussed, and I E 'R-n , is the fault vector. Rtf(k) represents actuator and component faults while R2/(k) denotes the sensor faults. The fault distribution matrices RI and R2 are assumed to be known. It should be mentioned that the determination of the fault vector I and the fault distribution matrices Rt and R2 are normally determined on a case-by-case basis by inspection of the state-space model and the characteristics oC the particular process. A more general design procedure can be perCormed by employing component lault analysis techniques (Blanke et al., 1997) which guarantees that a complete set of fault effects is used.
The robust fault detection problem is to generate a robust residual signal r( k) that satisfies
=
IIr(k)IIp ~ JtI. if I(k) 0, Ilr(k)llp > Jth if I(k) '" 0,
=y -
CXe(k) =Ce(k) + D2W(k)
!lp(k)
0
(39)
where the sampling period is 0.01. The state vector is xp:! [zp, zP2 zp,
f,
(41)
and the states are described as:
x PI :! High Pressure Spool Speed (RPM) Low Pressure Spool Speed (RPM) Temperature (Degrees C). The control input vector is
Up:! [uPI "P2 up,]T,
(42)
where the inputs are defined as
up, ~ Main Burner Fuel Flow (Kg/HR) P2 ~ Exhaust Nozzle Throat Area (M2) u" !: Bypass Duct Area (M 2 ).
=
U
The output vector is
IIrll~00.2) .[No,Nl ~ {[tr(EooQooE~)ql~ + 2um.,. .
YP :! [YPI
(D'J)[tr(EocQooE;;')fl*
IIp~
y"
f'
(43)
where the outputs are defined as
+ u!ax(D2)}lIwlI~00.2) .[No.Nl' (35)
YPI
!: Actual High Pressure Spool Speed (RPM)
YJ12 :@: Actual Low Pressure Spool Speed (RPM)
where Qoo is a positive-definite matrix satisfying
y" !: Actual High Pressure Compressor Inlet Temperature (Degrees C).
+ ~1 Voo , a -
~
(40)
+ R2/(k).
WC)Qoo(A - WC)T
k
= (Cp + ~Cp)zp(k) + D oo .2 w(k) + v(k)
x" ~ High Pressure Compressor Inlet
As derived in (Collins and Song, 2001), if !(k) residual can be represented by the norm inequality
= a(A -
= (Ap + ~Ap)zp(k) + (Bp + ~Bp)up(k) + Doo.IW(k),
X P2 ~
(34)
o the
Qoo
xp(k + 1)
(32) (33)
where 11 • lip denotes the p norm oC a Lebesgue signal and Jth is the threshold value. IT the estimator (19) is applied to the system described by (30) and (31) the residual generated is
r(k)
An example is presented in this section to illustrate robust II estimator design using the PopovTsypkin multiplier and the application of the robust It estimator to robust Cault detection oC dynamic systems. The model used was supplied by NASA Glenn Research Center and is given as
(36)
The variable w denotes a vector of disturbance signals and v is sensor noise.
with Voo :! (Dl - WD 2 )(D 1 - WD 2)T and a > 1. The residual norm is defined as IIr(· )1100.2 !: ess sUPt>ollr(t)1I2, where [No, NJ corresponds to a certain tUne interval. The threshold can be chosen as
The uncertainty matrices, ~Ap, ~Bp and ~Cp, are representative oC some engine degradation over time. Thus, it is assumed that a newly constructed
164
nonzero values of their upper or lower bounds. For example OA" whose lower and upper bounds are 0 and 0.0011, respectively, was assigned its upper bound 0.0011. Whereas, OA., whose lower and upper bounds are -0.0008 ana 0, respectively, was assigned ' its lower bound -0.0008. Random white noise signals with zero mean were added as both the disturbance inputs and sensor noise. The variances of the disturbance inputs, W1, W2 and W3, were 0.03, 0.08 and 0.2, respectively, while the variances of the sensor noises, VI, V2 and va, were 0.004, 0.003 and 0.002, respectively.
engine can be modeled with the nominal matrices A p, Bp and Gp and with use, the parameters of the degraded engine are encompassed in the uncertainty. The system parameter matrices are 0.8938
Ap
Bp
= [ -0.0001 o
=
0.0034 0.0020] 0.8940 0.0014 , -0.0001 0.89960
0.0059 0.1119 -0.0160] 0.0042 0.0720 0.0083 , [ 0.0001 -0.0003 0
1.0000 0.0077 0.0044] -0.0003 1.0000 0.0031 , [ 0.0299 -0.0208 0.0010 = diag{O.l, 0.1, 0.01}, D p ,2 = 0.1 X I sx3 •
Gp = D p ,1
This paper only considered the occurrence of sensor faults within the system. As previously discussed, a typical sensor fault in the jet engine is a drift in the sensor reading. Thus, a. slow drifting (or ramping) sensor fault was added to the YP2 sensor reading at the time instant t = 40 sec. Specifically, the error between the actual value and the value of the faulty sensor's reading increased over time. Due to the disturbance and uncertainty, the finite-horizon infinity norm (35) of the residual with N - No 60 (corresponding to a time interval of 0.6 sec.), was nonzero even in the absence of faults.
(44)
The uncertainty matrices ~Ap = -HA.FA.GA., ~Bp = -HB.FB.GB" ~Gp = -He. Fe, Gc" where
FA. FB. Fc.
= diag{bAp bA., bA" bA., bAs, bAo} = diag{bBp bB., bB.} = diag{bel> bO., oo.}, (45)
=
with
o ~ OA, ~ 0.0011, -0.0008 ~ bA. ~ 0, o ~ OA. ~ 0.0002, -0.0001 ~ OA. ~ 0, o ~ OA6 ~ 0.0005, 0 ~ OAe ~ 0.0001, o ~ OB, ~ 0.0008, 0 ~ OB. ~ 0.0115, -0.0005 ~ OB. ~ 0, -0.0015
~
00.
~
-0.0001 ~ 00, ~ 0, 0, 0
~
bo.
~
0.0004.(46)
The matrices HA" GA., HB., GB., Ho. and Go. are chosen so that the uncertain parameters OA, through OAe correspond to parameter fluctuations in the first two rows of matrix Ap , OB, through OB. and oc, through 00. to the first row of Bp and Gp, respectively. Fig. 1. FD Using Robust L1 Estimation for a "Slow" Drift Fault
It is desired to design a predictive filter of the form
(19) for which the estimation error is given by (20). For this particular example, the error output is defined as z(k) = Eooe(k) where Eoo = Gp.
It can be seen from Figure 1 that both the nominali} estimator and the robust i1 estimator can
By applying the continuation algorithm described in (Collins and Song, 2001) to the closed-loop control system given by (11) and (12), the nominal and robust L1 estimator gains were generated. For the nominal estimator the state matrix A. is set equal to A, but for the robust estimator an optimal state matrix is selected.
detect this fault since both residuals surpassed their threshold values. But it should be noted that the residual generated by the nominali 1 estimator tends to give false alarms since it surpassed its threshold value at two time intervals before t 40 sec even though there is no fault in the system in the time interval [0,40 sec). This false alarm resulted from the fact that the nominal estimator cannot distinguish the fault effect from the effects caused by system uncertainties. It is also noted in both estimators that the fault was detected sometime after it was initially added to the system. However, due to the nature of the simulated fault it is initially very small. When a
=
In order to illustra.te the application of the robust L1 estimator to robust fault detection, FD of the closed-loop system in (22) and (23) subject to plant uncertainties and disturbances was performed. In order to show the extent of robustness, uncertainty for all system matrices was considered. The uncertain parameters are assigned the
165
.. Fig. 2. FD Using Robust 11 Estimation for a. "Fast" Drift Fault "harder" fault (one with a faster time constant) was simulated, the detection was faster and more closely reflected the initial occurrence of the fault. This can be seen in Figure 2 where the fault introduced at t 25sec. has a time constant twice as large as the one simulated in Figure 1. Fault detection was performed with a variety of system uncertainties, each yielding very similar results to those described in Figures 1 and 2.
=
5. CONCLUSIONS This paper considered the application of robust i1 estimation for uncertain, linear discrete-time systems to the robust fault detection of dynamic systems. The multiplier-based approach of (Collins and Song, 2001) was used to design robust 11 estimators and the resulting fixed threshold approach was further investigated. By considering a discrete, linear model of a jet engine with real parametric uncertainties, an LQG controller was implemented to fonn a closed-loop system. In addition to uncertainty in the system and output matrices A and C, respectively, this paper considered uncertainty in the input matrix B . Using this closed-loop system and introducing a drifting sensor fault, it was shown that the robust fault detection methodology based on fixed thresholds was capable of significantly reducing the false alarm rate. The application of this methodology to a realistic system proved to be feasible and very effective. Coupled with the closed-loop and uncertainty considerations, the application to ajet engine constitutes the main contributions of this paper.
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