yProceedings of GT'032003 ASME/IGTI Turbo ExpoJune 16-19, 2003, Atlanta, Georgia, USAGT2003-38584AIRCRAFT TURBOFAN ENGINE HEALTH ESTIMATIONUSING CONSTRAINED KALMAN FILTERINGDan SimonDepartment of Electrical and Computer EngineeringCleveland State UniversityCleveland, Ohio 44115Email: [email protected] L. SimonUS Army Research LaboratoryNASA Glenn Research CenterCleveland, Ohio 44138Email: [email protected] ¯lters are often used to estimate the state variablesof a dynamic system. Ho wever, in the application of Kalman¯lters some known signal information is often either ignored ordealt with heuristically. For instance, state variable constraints(which may be based on physical considerations) are often ne-glected because they do not ¯t easily into the structure of theKalman ¯lter. This paper develops an analytic method of in-corporating state variable inequality constraints in the Kalman¯lter. The resultant ¯lter is a combination of a standard Kalman¯lter and a quadratic programming problem. The incorporationof state variable constraints increases the computational e®ort ofthe ¯lter but signi¯can tly improves its estimation accuracy. Theimprovement is proven theoretically and sho wn via simulationresults obtained from application to a turbofan engine model.This model contains 16 state variables, 12 measurements, and 8component health parameters. It is shown that the new algo-rithms provide improved performance in this example over un-constrained Kalman ¯ltering.INTRODUCTIONFor linear dynamic systems with white process andmeasuremen t noise, the Kalman ¯lter is kno wn to be anoptimal estimator. However, in the application of Kalman¯lters there is often known model or signal information thatis either ignored or dealt with heuristically [1]. This paperpresents a wa y to generalize the Kalman ¯lter in such a waythat known inequality constraints among the state variablesare satis¯ed b y the state variable estimates.The method presented here for enforcing inequalityconstraints on the state variable estimates uses hard con-strain ts. It is based on a generalization of the approachpresen ted in [2], which dealt with the incorporation of statevariable equality constraints in the Kalman ¯lter. Inequal-ity constraints are inherently more complicated than equal-ity constraints, but standard quadratic programming resultscanbeusedtosolvetheKalman¯lterproblemwithin-equality constraints. At each time step of the constrainedKalman ¯lter, we solve a quadratic programming problemto obtain the constrained state estimate. A family of con-strained state estimates is obtained, where the weightingmatrix of the quadratic programming problem determineswhich family member forms the desired solution. It is statedin this paper, on the basis of [2], that the constrained es-timate has several important properties. The constrainedstate estimate is unbiased (Theorem 1 below) and has asmaller error covariance than the unconstrained estimate(Theorem 2 below). We show which member of all possi-ble constrained solutions has the smallest error covariance(Theorem 3 below). We also show the one particular mem-ber that is alw ays (i.e., at each time step) closer to the truestate than the unconstrained estimate (Theorem 4 below).Finally, we show that the variation of the constrained es-timate is smaller than the variation of the unconstrainedestimate (Theorem 5 below).The application considered in this paper is turbofan en-gine health parameter estimation [3]. The performance ofgas turbine engines deteriorates over time. This deterio-ration can a®ect the fuel economy, and impact emissions,component life consumption, and thrust response of the en-gine. Airlines periodically collect engine data in order toevaluate the health of the engine and its components. The1 Copyrightc° 2003 by ASMEhealth evaluation is then used to determine maintenanceschedules. Reliable health evaluations are used to antici-pate future maintenance needs. This o®ers the bene¯ts ofimproved safety and reduced operating costs. The money-saving potent ial of such health evaluations is substantial,but only if the evaluations are reliable. The data usedto perform health evaluations are typically collected dur-ing °ight and later transferred to ground-based computersfor post-°ight analysis. Data are collected each °ight atapproximately the same engine operating conditions andcorrected to account for variability in ambient conditionsand power setting levels. Typically, data are collected fora period of about 3 seconds at a rate of about 10 or 20Hz. Various algorithms have been proposed to estimate en-gine health parameters, such as weighted least squares [4],expert systems [5], Kalman ¯lters [6], neural networks [6],and genetic algorithms [7].This paper applies constrained Kalman ¯ltering to es-timate engine component e±ciencies and °ow capacities,which are referred to as health parameters. We can use ourknowledge of the physics of the turbofan engine in order toobtain a dynamic model [8, 9]. The health parameters thatwe try to estimate can be modelled as slowly varying biases.The state vector of the dynamic model is augmented to in-clude the health parameters, which are then estimated witha Kalman ¯lter [10]. The model formulation in this paper issimilar to previous NASA work [11]. However, [11] was lim-ited to a 3-state dynamic model and 2 health parameters,whereas this present work includes a more complete 16-statemodel and 8 health parameters. In addition, we have someaprioriknowledge of the engine's health parameters: weknow that they never improve. Engine health always de-grades over time, and we can incorporate this informationin to state constraints to improve our health parameter es-timation. (This is assuming that no maintenance or engineoverhaul is performed.) This is similar to the probabilisticapproach to turbofan prognostics proposed in [12]. The sim-ulation results that we presen t here show that the Kalman¯lter can estimate health parameter deviations with an av-erage error of less than 5%, and the constrained Kalman¯lter performs even better than the unconstrained ¯lter.It should be emphasized that in this paper we are con-¯ning the problem to the estimation of engine health param-eters in the presence of degradation only. There are speci¯cengine fault scenarios that can result in abrupt shifts in¯lter estimates, possibly even indicating an apparent im-provement in some