ON-LINE TRANSIENT ENGINE DIAGNOSTICS IN A KALMAN FILTERING FRAMEWORK. P. Dewallef. ASMA Department Chemin des chevreuils 1

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1 Proceedings of GT25 25 ASME TURBO EXPO Reno-Tahoe, Nevada, USA, June 6-9, 25 GT ON-LINE TRANSIENT ENGINE DIAGNOSTICS IN A KALMAN FILTERING FRAMEWORK S. Borguet University of Liège ASMA Department Chemin des chevreuils 1 4 Liège, Belgium s.borguet@ulg.ac.be P. Dewallef University of Liège ASMA Department Chemin des chevreuils 1 4 Liège, Belgium p.dewallef@ulg.ac.be O. Léonard University of Liège ASMA Department Chemin des chevreuils 1 4 Liège, Belgium o.leonard@ulg.ac.be ABSTRACT A common assumption made in the performance assessment of a turbine engine for aircraft propulsion consists in restricting the data processing to steady-state data. This especially holds for on-board performance monitoring of a commercial aircraft which spends up to 9% of the time in cruise flight where such conditions are satisfied. The present contribution is intended to investigate the ability of a diagnosis method to process unsteady data rather than steady-state data. The aim of this unsteady approach is to strongly reduce the time and the efforts spent to obtain a reliable diagnosis. In order to assess the improvements in terms of diagnosis efficiency and engine operability, the resulting diagnostic method is tested for different degradations that can be expected on commercial turbofans. The results are also compared to those obtained from cruise flight steady-state data in order to balance the two approaches. NOMENCLATURE â estimation of an unknown variable a A8IMP nozzle exit area ( : ) ε k the measurement noise hpc high pressure compressor hpt high pressure turbine k discrete time index lpc low pressure compressor lpt low pressure turbine n total number of data samples ν k the process noise p i total pressure at station i SEi efficiency degradation of the component whose entry is located at section i ( : 1.) SWiR flow capacity of the component whose entry is located at section i ( : 1.) Ti total temperature at station i u k actual command parameters v k actual but unknown external disturbances w k actual but unknown health parameters actual but unknown state variables observed measurements INTRODUCTION The steadily rising importance of performance monitoring in the maintenance process of a modern turbine engine for aircraft propulsion has been promoted, these last years, by the advent of condition-based maintenance [1] for which the maintenance schedule is continuously updated based on the recording and the analysis of turbine engine performances over a long period of time. Basically, such a trend analysis is intended to provide a means of detecting significant changes in the performance parameters which result from changes in the mechanical conditions of the engine. In the application detailed further in the document, the health parameters constitute a set of 11 multiplying factors representing the degradations of the main engine 1 Copyright c 25 by ASME

2 components (fan, lpc, hpc, hpt, lpt, nozzle) in terms of efficiency and flow capacity. The approach considered herein is referred to as gas path analysis (GPA) and uses several measurements taken along the gas path of the engine to assess those health parameters. This method, more thoroughly described in [2], has been successfully used by many authors in the turbine engine community. However, when used in on-board performance monitoring, the health parameter estimation must cope with a negative redundancy since only 7 to 8 gas path measurements are available which is less than the 11 health parameters to estimate and results in a non-unique solution. Increasing the number of samples could make the redundancy positive but if they are related to the same operating point (GPA single-point method) they do not represent a set of independent observations and therefore the estimation remains unstable due to the sensitivity of the estimation method to the spurious correlations found in the measurement noise. Many contributions such as [3 7] have addressed this instability issue in different ways. In particular, a so-called softconstrained Kalman filter (SCKF) has been developed and tested by the authors in [6] on steady-state cruise flight data. This approach, based on a regularisation scheme, improved the diagnosis in terms of stability and fault isolation. Moreover, a combination of the SCKF to a Bayesian Belief Network (BBN) has been tested in [7] which decreases the number of undetected faults but still, some faults related to the low pressure turbine remain difficult to isolate because, in effect, there is not enough information in the available data to completely describe the fault. Of course, this non-separability problem could be solved if more sensors were made available but, since it would result in significant additional costs, this solution is not considered by engine manufacturers. A more reasonable alternative consists in gathering data related to different operating points in order to increase the information contained in the data samples (see [8] for an example application). For sake of simplicity and low computational effort, GPA multi-point methods are often restricted to the processing of steady-state data. Nevertheless, such a procedure overlooks the way data are made available to the user. Indeed, for a commercial turbofan, waiting for steady-state conditions to be achieved (typically 5 to 1 minutes) is an important constraint since those conditions are rarely met outside the cruise flight. This limits the number of operating points available for the diagnosis and, as a consequence, limits the applicability of those methods in a practical application. The purpose of this contribution is to investigate the ability of a diagnosis method to cope with unsteady data. The advantage of processing unsteady data is to extend the range of operating points available to the user which may enable a better diagnosis. This contribution is organised as follows : a statement of the diagnosis problem in unsteady conditions is given first and is followed by a short introduction of the unsteady health parameter estimation procedure. Afterwards, the developed estimation method is tested on a specific application intended to highlight the most significant aspects of the transient diagnosis method in the specific framework of aircraft turbine engine diagnosis. Finally, some conclusions and suggestions for future research are presented. PROBLEM STATEMENT The GPA approach to diagnosis is depicted in figure 1 where the engine to monitor is stimulated by the command parameters (e.g. the fuel flow) and by the external disturbances (aircraft movements, wind, turbulence, flight Mach number, altitude, ambient temperature, humidity,... ). As a response, it generates observable signals which are also a function of both its health condition and the transient effects taking place inside the engine (heat transfers, shaft inertia and fluid transport delays). external disturbances v k u k command parameters v k estimated external disturbances Figure 1. engine to monitor engine performance simulation model estimated state variables w k estimated health parameters observed measurements estimated measurements Structure of the identification procedure - residuals The scope of the engine performance model is to generate some measurement estimates ŷ k in order to build a residual r k whose purpose is to represent the distance between the estimated health parameters and the actual health condition of the engine. In the case of unsteady turbine engine simulation, the engine performance model can be expressed in the state-space form : = F (u k,v k,w, 1 ) + ν k (1) = G(u k,v k,w, ) + ε k (2) where u k are the command variables set by the user, v k are the external disturbances which have the character of inputs except that they are not controlled by the operator 1, w are the aforemen- 1 External disturbances are also called exogenous inputs in system theory 2 Copyright c 25 by ASME r k

3 tioned health parameters and are the state variables which are defined as the minimal set of data sufficient to uniquely describe the unforced dynamical behaviour of the system. The engine performance model represented by equation (2) is supported by the nonlinear discrete equation (1) which models the transient effects of the engine. In other words it means that fault indicators can be predicted on the sole basis of the current inputs (u k and v k ), the state variables and the health parameters w. ε k represents the measurement noise coming from sensor inaccuracies. ν k stands for the process noise which represents the model inaccuracies. Both measurement noise and process noise are assumed generated by a white and Gaussian random process. The state-space formulation became an increasingly dominant approach after Kalman s work on prediction and linear quadratic control [9]. Yet, the residuals r k will be a faithful image of the health condition provided that the model is unbiased and also that it is stimulated by the right inputs. Since the command variables are set by the user, they are assumed to be known without inaccuracies and the model is directly fed by the desired command parameters (see figure 1). Conversely, all external disturbances are not measurable and are, most of the time, restricted to the inlet total temperature and pressure together with the ambient pressure which are sufficient to estimate the static pressure and temperature as well as the flight velocity ( v k in figure 1). The remaining disturbances are left unknown and their effects on the fault indicators are assumed negligible. Hence, if u k and v k are known, the residuals r k are a function of both w and which requires that two tasks must be simultaneously carried out during an unsteady engine performance monitoring, namely 1. the estimation of the engine state variables associated to the dynamic behaviour of the engine, and 2. the identification of the engine health parameters w reflecting the degradation of the engine. This explains why, for sake of simplicity, most of the multipoint diagnosis methods assume steady-state conditions. Doing so, the state variables are discarded from the engine performance model and the residuals can be computed on the sole basis of the health parameters w. Thus stated, the estimation problem comes out to determine that value of ŵ which minimises a given objective function. An efficient approach consists in assuming a Gaussian measurement noise within a maximum a posteriori framework 2 which leads to the following objective function : J = (w w c ) T D 1 (w w c ) + n k=1 r T k R r r k (3) 2 The maximum a posteriori approach is also named ridge regression, regularisation or Bayesian identification. where k is a discrete time index, n the total number of data samples and R r the covariance matrix of the measurement noise (which is usually strictly diagonal). The first term in relation (3) restricts the identified health parameters ŵ in the neighbourhood of some prior values w c. The range of the neighbourhood is controlled by the diagonal covariance matrix D. The use of such a regularisation term implicitly assumes that the process generating the data obeys certain smoothness constraints. In the present contribution, a method is proposed to simultaneously estimate the state variable and identify the health condition of the engine. The procedure followed to introduce this unsteady dual estimation is to split up the complete estimation process in two steps. The problem of the health parameter estimation is first addressed assuming that the state variables are known which enables the health parameters to be estimated regardless of the steady or unsteady nature of the data through relations (3) and (2). Secondly, the state variable estimation is considered assuming that the health parameters are known. Finally, the dual estimation problem is introduced where both the state variables and the health parameters must be simultaneously estimated from the observed noisy signals. ON-LINE HEALTH PARAMETER ESTIMATION In a multi-point framework, relation (3) can be solved either by batch, where data are gathered and processed all at once, or through a sequential approach which doesn t involve the batch processing of the full block of data but only a simple update of the parameters each time new data are available. Generally speaking, the advantage of the latter approach resides in the quick and simple update formula provided by those algorithms. Moreover, sequential estimation is very well suited for real time applications since diagnosis results are available on-line without the need to store any database. The sequential minimisation of relation (3) based on a socalled soft-constrained Kalman filter has been the subject of a previous publication by the authors [6] to which the interested reader is referred to for more details. A more thorough description of such parameter estimation technique can also be found in [1]. However, for ease of understanding, the Kalman filter estimation procedure is shortly introduced by means of the block diagram depicted in figure 2. unit delay Figure 2. w k w k-1 u k v k G (u k,v k,w k-1, ) K yk - Block diagram of the Kalman filter for the on-line estimation of the health parameters. r k - 3 Copyright c 25 by ASME

4 At the beginning of the estimation process, k = 1 and ŵ k 1 is initialised by the prior value and ŵ = w c. Based on this prior value, a measurement estimate ŷk is produced by the engine performance model through relation (2) and is compared to the measurement sample to build the residual r k. The previous health parameter estimate ŵ k 1 is then updated based on this residual r k through the Kalman gain K. The Kalman gain appears as the key quantity of the method. The procedure is repeated for each new data sample. When k reaches n, the Kalman filter gives the best possible estimate in the case of a linear model, and is equivalent to a batch method where all the data samples from k = 1 to n are processed all at once. For nonlinear models like relation (2), the generic Kalman filter can be extended to nonlinear system models through either the Extended Kalman filter or the Unscented Kalman filter described in [11]. In the application detailed further, the Unscented Kalman filter is chosen since it achieves second order accuracy for any non-linearity compared to the Extended Kalman filter which only achieves first order accuracy. In terms of computational effort, the Unscented Kalman filter requires 2p + 1 calculations of relation (2) per iteration, where p is the number of health parameters. The Extended Kalman Filter scheme, where the Jacobian of the system is evaluated by means of central differences, also requires 2p + 1 model resolutions per iteration so that the UKF is preferred. However, if the Jacobian is evaluated by means of forward (or backward) differences or if an analytical relation is available and provided that the non-linearities are not too stiff, then the EKF may be more attractive due to its lower computational burden. ON-LINE STATE VARIABLE ESTIMATION The purpose of the Unscented Kalman filter is to address the problem of trying to estimate the state variables of a discrete time controlled process that is governed by a nonlinear stochastic difference equation. Therefore, if the health parameters ŵ k are assumed to be known, the on-line estimation of the state variables described by relation (1) is a straightforward application of the Unscented Kalman filter [11]. The estimation procedure is described in figure 3. The update procedure is similar to the one depicted in figure 2 except that, conversely to the health parameters, the dynamic behaviour of the state variables is driven by equation (1). The previous state estimate 1 is used together with the state transition equation defined by relation (1) to predict a prior estimate of the state variables x k. Based on this prior estimate, the measurement prediction equation defined by relation (2) produces the measurement estimates ŷk which allows the estimation of the residuals r k. The correction part consists in updating the prior estimate x k based on the residuals through the Kalman gain K. Each iteration requires 2q + 1 calculations of relations (1) and (2), where q is the number of state variables. -1 Figure 3. u k v k F (u k,v k,w k,-1 ) unit delay of the state variables. w k - G (u k,v k,w k, -) - r - k K Block diagram of the Kalman filter for the sequential estimation ON-LINE TRANSIENT ESTIMATION Several approaches have been proposed in the literature [1] to solve the on-line dual estimation of the state variables and the health parameters from the same sequence of measurements. In [12], the authors have proposed a methodology characterised by two Kalman filters running concurrently. This approach, called marginal estimation is summarised in figure 4 where the health parameter estimation Kalman filter and the state variable estimation Kalman filter stand for the estimation procedures detailed respectively in figures 2 and 3. unit delay w k-1-1 unit delay Figure 4. health parameter estimation state variable estimation u k v k dual estimation Kalman filter w k Block diagram of the marginal estimation procedure for the dual health parameter and state variable assessment. The previous estimates 1 and ŵ k 1 are used to generate an estimation of ŵ k which is, in turn, used to correct the state variables. The state variable estimates are expected to improve as the estimated health parameters ŵ k converge toward their true values. In addition to a good convergence, this formulation results in a rather modular formulation. The computational effort involved by the marginal estimation procedure is 2(p + q + 1) calculations of relations (1) and (2). APPLICATION OF THE METHOD The application used as a test case is a large bypass ratio mixed-flow turbofan. The engine layout is described in figure 5 where the health parameter location and the station numbering 4 Copyright c 25 by ASME

5 is also supplied. The engine performance model has been developed in the frame of the OBIDICOTE 3 project and is detailed in [13]. 1 inlet Figure SW12R SE12 fan lpc 13 the turbofan layout. SW2R SE2 hpc SW26R SE26 combustor SW41R SE41 hpt lpt SW49R SE49 A8IMP nozzle Health parameter location and section numbering related to The dynamic model is available in the state-space form specified by relations (1) and (2). The dynamic effects taken into account by this model are restricted to the shaft inertia and the heat soakage in the hpc, the hpt and the combustor which result in the set of 7 state variables listed in table 1. on a PowerMac G5 desktop computer, the nonlinear dual estimation procedure may simply become unachievable due to the relatively low computational power of a conventional on-board engine controller whose computational power is close to a Pentium at 9MHz. For this reason, it is chosen to restrict the present application to ground testing where the required computational power is more likely to be available. In such a situation, two ambient variables (ambient pressure and temperature) together with the command variable (the fuel flow) are sufficient to define the engine operating point. Sea-Level Static for a standard day (ISA-SLS) conditions are assumed in this application. The evolution of the fuel flow with time is depicted in figure 6. It consists of two successive slow acceleration-deceleration manoeuvres between idle and max-continuous regimes followed by a slam between idle and take-off regimes. The simulation was run for 75 seconds with data collected at a sampling rate of 5 Hz. which is a typical sampling rate for dynamic characterisation in current test benches. fuel flow WFE [kg/s] time [s] Figure 6. Evolution of the fuel flow with time used in the test cases. Label N l p N hp Table 1. T M3b T M3c T M4b T M42b T M42c Description low pressure spool rotational speed high pressure spool rotational speed high pressure compressor blade temperature high pressure compressor casing temperature combustion chamber casing temperature high pressure turbine blade temperature high pressure turbine casing temperature Available state variables modelling the transient effects according to the original OBIDICOTE nomenclature. The available model is capable of simulating the behaviour of the engine in the whole flight envelope. However, the application of the marginal estimation procedure depicted in figure 4 results, in this case, in 2(p + q + 1) = 38 calculations of the nonlinear system model. If real-time performance is easily achieved 3 A Brite/Euram project for On-Board Identification, Diagnosis and Control of Turbofan Engine To assess the efficiency of the methodology, several fault cases, taken from [14] and summarised in table 2, are considered. They are intended to represent degradations that can be expected on modern turbofans for all individual components (fan, lpc, hpc, hpt, lpt, nozzle). Considered faults involve single as well as multiple health parameters. In the context of this application, faults are present from the beginning of the test (step at time t= s) and are not expected to vary during the test. Due to the high costs involved with testing on a real engine, simulated data was used for this study. Gaussian noise, whose magnitude is specified in table 3, is added to each generated measurement. Three measurement sets (detailed in table 3) are considered in the test cases. Measurement set A is related to a usual set of sensors available on-board. Measurement set B is based on set A with two optional sensors which are generally proposed by engine manufacturers. Finally, measurement set C is composed of measurement set A completed by the total air mass flow rate ) and the thrust (FGN) measurements which are available at test bed. Initial estimates for the health parameters are imposed at their since the engine is assumed to undergo its first maintenance session and, accordingly, no other a priori in- (W1 air 5 Copyright c 25 by ASME

6 a -1% on SW12R -.5% on SE12 FAN, LPC -.7% on SW2R -.4% on SE2 b -1% on SE12 c -1% on SW26R -.7 % on SE26 HPC d -1% on SE26 e -1% on SW26R f +1% on SW42R HPT g -1% on SW42R -1 % on SE42 h -1% on SE42 i -1% on SE49 LPT j -1% on SW49R -.4 % on SE49 k -1% on SW49R l +1% on SW49R -.6 % on SE49 m +1% on A8IMP Nozzle n -1% on A8IMP Table 2. Considered fault cases. RESULTS ON MEASUREMENT SET A To assess the accuracy of the identification procedure, a bias is calculated according to the following relation: w k = ŵ k w k wk nom (4) where w k and wk nom are respectively the actual and the nominal values of the health parameters. The vector-valued w k from equation (4) represents the relative distance between the estimated values of each parameter and the actual ones at time index k. Then a figure of merit, called asymptotic efficiency further in the text, is obtained by averaging the biases over the last two seconds of simulation (i.e. over 1 samples). In order to summarise the efficiency of the method in terms of fault isolation, the maximal value of the figure of merit is reported in table 4 for each test-case. For example, the value reported in table 4 for the test case a states that the relative gap between identified and actual values for each health parameter is less (or equal to, for the parameter SW49R) than.6% at the end of the simulation. The faults on the fan and on the lpc are consequently accurately diagnosed. Table 3. Label uncertainty set A set B set C W1 air ±5kg/s T13 ±2K p 13 ±1Pa p 26 ±5Pa T3 ±2K p 3 ±5Pa N l p ±6rpm N hp ±12rpm p 49 ±5Pa T6 ±2K FGN ±5N Total Measurement set definition. Uncertainties are assumed to be three times the standard deviation. steady-state unsteady-state a.3% on SW49R.6 % on SW49R b.9% on SE12.2 % on SW26R c.7% on SW26R.5 % on A8IMP d.3% on SE12.11 % on A8IMP e.3% on SW26R.8 % on SW12R f.2% on SE42.1 % on SW12R g.17% on SW49R.5 % on SW12R h.24% on SW49R.9 % on SW12R i.19% on SW49R.1 % on A8IMP j.82% on SW49R -.4 % on SE2 k.53% on SW49R -.5 % on SW12R l.41% on SW49R -.1 % on SW12R m.6% on SE41.3 % on SE2 n.6% on SW2R.2 % on SW49R Table 4. Asymptotic efficiency of the identification for each fault cases with measurement set A. formation is available about its condition. The initialisation of the state variables is easier since some of them are measurable, the remaining, unmeasurable ones, being initialised with steadystate s. The standard deviations (i.e. the square root of the diagonal terms of the health parameter covariance matrix assessed by the Kalman filter procedure) related to the results summarized in 6 Copyright c 25 by ASME

7 table 4 are about.1%. Consequently, a diagnosis characterised by a bias below.25% is considered reliable and is indicated in table 4 by a checkmark. Moreover, results previously obtained in [6] in the case of steady-state cruise flight data are also added in order to enable some comparison between the two approaches. In the vast majority of the test cases, the unsteady health parameter estimation achieves accuracies which remain within.1%. It represents a major improvement in comparison with the results obtained from steady-state data. This especially holds for the fault cases j, k and l related to the low pressure turbine. The most outstanding conclusion resides in the resolution of the fault case j which has been noticed as especially difficult since no measurement is available between the two turbines which makes it difficult to distinguish a fault on the lpt from a fault on the hpt [7]. The better diagnosis results brought by the unsteady identification method can be explained by the more important amount of information provided by an unsteady data sequence compared to the one provided by steady-state cruise flight data. In order to have a deeper understanding about the efficiency of the unsteady identification method, it is interesting to take a closer look at fault case j (-1% on SW49R and -.4% on SE49). Results obtained from steady-state data are recalled in figure 7 where the fault on SW49R is under-estimated and the fault on SE49 is not detected. Moreover, a false alarm on SE41 spoils the fault isolation. This poor diagnosis is not due to the level of noise in the measurements since the solution does not improve as more data are gathered. The problem comes from the fact that, in steady-state, the effects of SW49R and SE41 on the measurements are exactly the same. Therefore, from the point of view of an external observer, those parameters cannot be distinguished Figure 7. SW41R SE41 SW49R SE49 A8IMP time (s) Steady-state health parameter identification on fault case j extracted from [6] (measurement set A). Dotted lines show actual parameter values The health parameters identified from transient operation are sketched in figure 8. It can be seen that, at the end of the test (at time t=75 s), the fault on the lpt is clearly separated from the others which hints at the more important amount of information contained in an unsteady data sequence. Consequently, the observability of SW49R and SE41 is enhanced and a reliable localisation of the fault is allowed. The effect of the measurement noise is exhibited in the beginning of the test where not enough data are gathered to filter the measurement noise. As a consequence, all the faults which are not separable within the measurement noise are confused and the fault is spread over several parameters. The measurement noise filtering achieved by the Kalman filter is shown, in figure 8, by the concentration of the fault on the minimum subset of parameters. This allows a clear localisation as well as an accurate assessment of the engine fault Figure 8. SW12R SE12 SW2R SE2 SW26R SE26 SW41R SE41 SW49R SE49 A8IMP time [s] Unsteady health parameter identification on fault case j (measurement set A). Dotted lines show actual parameter values Figure 9 represents the evolution of the absolute error on the estimated state variables. It shows the rapid convergence of the spool speeds toward their actual values. The convergence of the metal temperatures is slower since no direct measurement is available which makes them loosely correlated with the available gas path measurements. This is also a consequence of the marginal estimation filter for which the state estimation improves as the health parameters converge toward their actual values. In order to make a comparison with the convergence speed obtained on fault case j, figure 1 shows the identification of an hpc fault represented by fault case c (-1% on SW26R and -.7% on SE26). In this case, the asymptotic solution is reached after 1 seconds which is better than for fault case j. This is an effect of the observability of the parameters which is more favourable in test case c than in test case j for which more data are needed to filter the measurement noise. EFFECTS OF ADDITIONAL SENSORS The purpose of this section is to study the influence of the choice of the measurement set on the identification efficiency. 7 Copyright c 25 by ASME

8 Spool speed [RPM] Temperatures [K] N LP N HP TM3B TM3C TM4B TM42B TM42C time [s] Figure 9. (measurement set A) % deviation from Absolute error in estimated state variables on fault case j SW12R SE12 SW2R SE2 SW26R SE Figure 1. time [s] Unsteady health parameter identification on fault case c (measurement set A). Dotted lines show actual parameter values Table 5 reports the figure of merit defined above for the three aforementioned measurement sets. Values in table 5 must be interpreted in the same way as those in table 4. For sake of brevity, the health parameter corresponding to the the tabulated value doesn t appear in the table. The increased number of sensors available in measurement sets B and C provides the dual filter with a more detailed information about the engine faults. As a consequence, the diagnosis tool performs slightly better than for measurement set A. Surprisingly, the availability of the thrust and the air mass flow rate measurements does not seem to improve the diagnosis in comparison with the two additional pressure measurements (p 26, p 49 ). More studies are necessary to completely understand this effect. Nonetheless, based on the results from table 5, measurement set B should be preferred to measurement set C since it is composed of sensors located inside the engine (i.e. in the gas path), which makes it usable at test bench, as well as in flight. The behaviour of the transient dual filter with respect to time is depicted on figure 11 for fault case j and measurement set B. Even if the fault is first spread on SW12R, the addition of the two sensors enhances the convergence of the identification since the scatter of the parameters associated to the lpc and the hpc is reduced. Table 5. set A set B set C a.6 %.6 %.3 % b.2 %.3 %.3 % c.5 %.2 %.6 % d.11 %.6 %.7 % e.8 %.5 %.2 % f.1 %.3 %.5 % g.5 %.4 %.4 % h.9 %.2 %.6 % i.1 %.6 %.8 % j.4 %.6 %.7 % k.5 %.2 %.3 % l.12 %.9 %.5 % m.3 %.3 %.3 % n.2 %.2 %.1 % Asymptotic efficiency of the unsteady identification for each fault case and for each measurement set (tabulated values are the maximum values of the biases assessed with relation (4) Figure 11. SW12R SE12 SW2R SE2 SW26R SE26 SW41R SE41 SW49R SE49 A8IMP time [s] Unsteady health parameter identification on fault case j (measurement set B). Dotted lines show actual parameter values DISCUSSION The quality of the proposed results is also a consequence of the perfect matching between the model used to generate the data (simulating the real engine) and the model supporting the dual filter - the only difference being the artificial measurement noise. It is mandatory to investigate the behaviour and the robustness of the dual filter when the model dynamics differs from that of the real system (e.g. no heat transfers taken into account in the dual filter). This issue is of primary importance for a practical application in which modelling errors will always be present. 8 Copyright c 25 by ASME

9 An other aspect concerns the assumption of a Gaussian measurement noise. The validity of such a model is not always ensured and especially when the identification method must cope with sensor biases or drifts. This issue has been the subject of a previous publication by the authors [12] but some work is still needed to cope with the very low redundancies encountered onboard. CONCLUSIONS The present methodology has been tested on unsteady data sequences for the detection of an extensive set of fault cases encountered on modern turbofan engines. Due to its transient and on-line nature, the dual filter supplies the diagnosis report in a few minutes (typically 1 minutes) which would lead, in a test bench application, to valuable savings in time and fuel consumption with respect to multi-points, steady-state methods for which the diagnosis requires at least one hour. Moreover, the dual filter outperforms steady-state methods in the case of low measurement sets such those encountered on-board, by providing the user with a more reliable localisation of the fault and a more accurate assessment of its magnitude. These two features make it possible to apply the proposed methodology to ground based, on-wing engine testing. The ground based restriction arises from the computational demand of the methodology to achieve real-time performances. A possible extension of the present method consists in using this dual filter in an adaptive, model-based controller where the control loop disposes of the health condition and of the full state of the engine in real-time. However, in the proposed application, the engine was placed on a test bed and the ambient conditions were assumed to be constant. To be applicable on-board, the present methodology has to be extended to varying ambient conditions encountered during a complete flight envelope. Furthermore, in the test cases considered herein, the fault level was constant in time that is unlikely to be the case in an on-board application. Therefore, the ability of the dual filter to track a time evolving degradation should also be tested. Finally, the bottleneck of the dual filter resides in the evaluation of the engine performance model, consisting in a set of non-linear equations which prevents the present methodology from being applicable on an embarked controller. One way to speed up the dual filter is to replace the physical model by an automatic learning technique (such as neural networks), but this solution is not yet available to us. REFERENCES [1] R. Rajamani, J. Wang, and K.Y Jeong. Condition-based maintenance for aircraft engines. In ASME Turbo Expo, number GT in Controls, Diagnostics and Instrumentation, 24. [2] A. Volponi. Foundation of gas path analysis (part i and ii). In Von Karman Institute Lecture Series, number 1 in Gas Turbine Condition Monitoring and Fault Diagnosis, 23. [3] N. Aretakis, K. Mathioudakis, and A. Stamatis. Non-linear engine component fault diagnosis from a limited number of measurements using a combinatorial approach. In ASME Turbo Expo, number 331 in Controls, Diagnostics and instrumentation, 22. [4] T. Kobayashi and D.L. Simon. Application of a bank of kalman filters for aircraft engine fault diagnostics. In ASME Turbo Expo, number 3855 in Controls, Diagnostics and Instrumentation, 23. [5] D. Simon and D.L. Simon. Aircraft turbofan engine health estimation using constrained kalman filtering. In ASME Turbo Expo, number in Controls, Diagnostics and Instrumentation, 23. [6] P. Dewallef, K. Mathioudakis, and O. Léonard. On-line aircraft engine diagnostic using a soft-constrained kalman filter. In ASME Turbo Expo, number in Controls, Diagnostics and Instrumentation, 24. [7] P. Dewallef, C. Romessis, K. Mathioudakis, and O. Léonard. Combining classification techniques with kalman filters for aircraft engine diagnostics. In ASME Turbo Expo, number in Controls, Diagnostics and Instrumentation, 24. [8] T.U.J. Grönstedt. Identifiability in multi-point gas turbine parameter estimation problems. In ASME Turbo Expo, number 32 in Controls, Diagnostics and instrumentation, 22. [9] R.E. Kalman and R.S. Bucy. New results in linear filtering and prediction theory. Trans. ASME, series D, Journal of Basic Engineering, 83, [1] A.T. Nelson. Nonlinear Estimation and Modeling of Noisy Time-Series by Dual Kalman Filtering Methods. Phd. thesis, Oregon Graduate Institute of Technology, september 2. [11] E. Wan and R. van der Merwe. The unscented kalman filter. In Kalman filtering and neural networks. Wiley series on adaptive and learning systems for signal processing, communications and control, 21. [12] P. Dewallef and O. Léonard. On-line performance monitoring and engine diagnostic using robust kalman filtering techniques. In ASME Turbo Expo, number in Controls, Diagnostics and Instrumentation, 23. [13] A. Stamatis, K. Mathioudakis, J. Ruiz, and B. Curnock. Real-time engine model implementation for adaptive control and performance monitoring of large civil turbofans. In ASME Turbo Expo, number 362 in Controls, Diagnostics and Instrumentation, 21. [14] B. Curnock. Obidicote project - word package 4: Steadystate test cases. Technical Report DNS62433, Rolls-Royce, 2. 9 Copyright c 25 by ASME