Gas-Path Health Estimation for an Aircraft Engine Based on a ... - MDPI

energies Article

Gas-Path Health Estimation for an Aircraft Engine Based on a Sliding Mode Observer Xiaodong Chang, Jinquan Huang *, Feng Lu and Haobo Sun College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Road, Nanjing 210016, China; [email protected] (X.C.); [email protected] (F.L.); [email protected] (H.S.) * Correspondence: [email protected]; Tel.: +86-139-5179-6358 Academic Editor: Chang Sik Lee Received: 7 June 2016; Accepted: 21 July 2016; Published: 29 July 2016

Abstract: Aircraft engine gas-path health monitoring (GPHM) plays a critical role in engine health management (EHM). Among model-based approaches, the Kalman filter (KF) has been widely employed in GPHM. The main shortcoming of KF-based scheme lies in the lack of robustness against uncertainties. To enhance robustness, this paper describes a new GPHM architecture using a sliding mode observer (SMO). The convergence of the error system in uncertainty-existing circumstances is demonstrated, and the proposed method is developed to estimate components’ performance degradations regardless of modeling uncertainties. Simulations using a nonlinear model of a turbofan engine are presented, in which health monitoring problems are handled by the KF and the SMO, respectively. Results indicate the proposed approach possesses better diagnostic performance compared to the KF-based scheme, and the SMO shows its strong robustness and great potential to be applied to GPHM. Keywords: aircraft engines; health estimation; modeling uncertainties; sliding mode observer (SMO); linear matrix inequalities (LMIs)

1. Introduction Modern aircraft gas turbine engines are extremely complex systems that are subjected to extreme conditions. Over repeated flight cycles, the lifespan of engine components will be affected, and malfunctions may occur [1]. An aircraft engine will exhibit gradual degradation in the gas path over its operating life [2]. This is usually caused by the fouling or erosion that happens to compressors or turbines as parts wear from regular use. Generally these deteriorations are reflected by variations of components’ efficiencies and flow capacities, which are the so-called “health parameters”. Health parameters are not directly measureable, but their degradations cause changes in other observable parameters, such as temperature, pressure and rotational speed. The health degradation estimation plays a critical role in engine gas path health management system, which provides an evaluation of components’ performance based on the level of degradation of efficiencies and flow capacities, and helps operators determine a regular maintenance schedule and arrange replacement of components when the performance reaches unacceptable levels. Several approaches have been introduced to estimate the health condition of gas-path components, which can be classified into two types: model-based methods (such as observers and filters) and data-based methods (such as fuzzy logic, neural networks and genetic algorithms). Compared to data-based approaches, model-based approaches utilize all model information available, and offer better estimation accuracy [3]. A well-developed model-based strategy for health monitoring systems is the Kalman filter (KF) strategy. Luppold et al. [4] proposed an algorithm based on a KF concept to estimate in-flight engine performance variations. Simon et al. [5] applied the constrained KF approach, along with constraint tuning on the basis of measurement residuals, to estimate engine Energies 2016, 9, 598; doi:10.3390/en9080598

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Energies 2016, 9, 598

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health parameters. Since piecewise-linear models are linearized from a component-level model (CLM), which is a complex nonlinear representation of the gas turbine engine, modeling errors that decrease the accuracy of the health estimation are inevitable. In view of this, Brotherton presented an approach that fused a physical model called self-tuning on-board real-time model (STORM) with an empirical neural net model to provide a unique hybrid model (eSTORM) base on a KF, aiming to compensate modeling errors and provide engine diagnostics [6]. Volponi et al. [7] further developed eSTORM, by providing a self-tuning technique to the engine model as the engine evolved over the course of its life, to ensure accurate performance tracking. Sliding mode observer (SMO) techniques have gained lots of attention in recent years. One of the reasons is due to their robustness against modeling uncertainties. As any model is unable to perfectly represent the system, robust state estimation is a vital property for the accuracy of state monitoring systems. Edwards et al. [8] proposed the architecture for fault detection and isolation (FDI) via sliding mode techniques. Ng at al. [9] presented a disturbance decoupled fault reconstruction approach focused on actuator faults. Tan et al. [10] reported the application of SMOs in sensor fault reconstruction. Alwi et al. [11] described a sliding mode fault reconstruction scheme based on a linear parameter varying model for an aircraft benchmark problem, to deal with aircraft sensor faults considering uncertainties and disturbances. Rahme et al. [12] proposed an adaptive SMO approach to diagnose sensor faults of gas turbines, where the degradations of health parameters are treated as constant uncertainties. So far uses of SMO in fault detection have mainly been in handling actuator/sensor fault cases, and to the authors’ knowledge, research has barely been done on the application of the sliding mode technique to engine health performance monitoring systems. In this paper, an SMO-based health monitoring scheme for an aircraft engine is proposed. With health parameters treated as state variables, a robust SMO is developed to estimate the degradations of health parameters, despite the existence of modeling uncertainties. The convergence of the error system in uncertainty-existing circumstances is demonstrated. Once sliding motion is achieved, the system is independent of the uncertainties due to the so-called equivalent output error injection term. Simulations show that in comparison with the KF scheme, the proposed SMO has better diagnostic performance in engine health degradation estimation. 2. Theory and Modeling 2.1. Modeling and Preliminaries Since the proposed SMO is a kind of model-based algorithm, a linear state variable model (SVM) is required, which is a representative of the engine dynamic nonlinear model in a small range around the steady-state operating point. The linear model of an aircraft engine is as follows: .

xptq “ Axptq ` Buptq ` Lhptq yptq “ Cxptq ` Duptq ` Mhptq

(1)

where x is the state vector, u is the input, y is the measurable output, and h is the health parameter vector. A, B, C, D, L and M are corresponding coefficient matrices. All variables in the linear model are normalized. The linear engine model is extracted from the CLM, which is a nonlinear dynamical representative of a real engine. The commonly used linearization methods for establishing engine SVM include the bias derivative method [13] and fitting method [14–16]. Compared to the bias derivative method, the fitting method is more feasible in low-order systems, and it possesses a higher modeling precision [14]. In this paper an improved least square fitting method is adopted to construct the engine SVM in Equation (1). To solve the initial value problem in fitting method, the initial values of coefficient matrices are obtained by the bias derivative method to create an initial linear model. Then the coefficient matrices are optimized by fitting the simulation data generated by the initial linear model and dynamic simulation data from the CLM [15]. The modeling method was verified in [15] and showed a satisfactory accuracy to be employed in model-based monitoring systems.

ed as:  0

 Tc  (t ) (t )   )

T T T N   c    T   N c    T   N c    (2)  (4)        c c  Ca   Ca   C a  (3)  3 of 14 

(4) 

(4) 

(4) 

f C null‐space of C e transformation, the measurement matrix    spans the null‐space of C a. With the transformation, the measurement matrix  a. With the transformation, the measurement matrix  a. With the transformation, the measurement matrix    Equation  (3),  the  information  of  health  d as:  techniques, health parameters is required to  Energies 2016, 9, 598 3 of 15 lth degradation via an observer.  mented system is created:  1 1 1 )   I pC a Tc  0C c  I pC a Tc  0C c I pCa Tc  0 I p  (5)  (5)  (5)  (5)  bserver designing process. The coordinate  (3)  B notes the identity matrix. In the new coordinates, Equation (3) can be described  In order to estimate health parameters via observer techniques, health parameters is required to entity matrix. In the new coordinates, Equation (3) can be described   coordinates, Equation (3) can be described  . In the new coordinates, Equation (3) can be described   0  u (t )   Energies 2016, 9, 598    (3),  the be Equation  information  health  3 of 14  considered of  as state variables, thus the following augmented system is created: 3 of 14  (2)  th degradation via an observer.  (4)  « . ff « ff « ff « ff Du(t ) u (tA)c xc (t ) xc B (In order to estimate health parameters via observer techniques, health parameters is required to  tc)u(tA ) c xc (t )xc (Bt )c u(tA ) c xc (t )  Bc u(t ) chniques, health parameters is required to  xptq bserver designing process. The coordinate  A L xptq B         (6)  (6) “ (6)  (6)  . ` uptq u (C t )c xc (be considered as state variables, thus the following augmented system is created:  t ) y D (t )c u(tC) c xc (t ) y (Dt )c u(tC) c xc (t )  Dc u(t ) ented system is created:  0 0 hptq 0 hptq  transformation, the measurement matrix  nted as:  « ff (2) ” ı xptq x (t )   A  L BBcc = T c = D where    x (t )   B  ∈ −1c is in the form of  d D A is in the form of   = T c c A B a a . A T , and D c , B c  = T c  = D c B a a . A , and D c  is in the form of  c  = D   where  a . A c  is in the form of    ∈ ∈ where  where  ∈ “ u0(t) u  (t )  h (t )    yptq   C  M   u(t )hptq ` Duptq (4) (3)     0 0   h(t )   0      u(t  ) (5)  (2)  (2)   x (t ) 

stness of SMO, Equation (6) is added with modeling uncertainties as  O, Equation (6) is added with modeling uncertainties as  (6) is added with modeling uncertainties as  oit the robustness of SMO, Equation (6) is added with modeling uncertainties as  (t ) t ) presented as: u y (t )   C M   (2) Du(be For analysisof  convenience Equation  can transformation, the measurement matrix  In  Equation  (3),  the  information  health  oordinates, Equation (3) can be described   h(t )  ealth degradation via an observer.  . x a ptq “ A a x a ptq ` Ba uptq ed as:  For analysis convenience Equation (2) can be presented as:    Q ( t ξ ) x (  t , ( B x t ) , u u  ( ) t A )  x Q ( t ξ ) ( x  t , ( B x t , ) u  ) ( t A )  x Q ( t ξ ) (  t , B x , u ( ) t )  Q ξ ( t , x , u ) e observer designing process. The coordinate  (3) c  c c c   (t ) c c   c c c c c   c c c (7)    (7)  yptq “ C(7)  (7)  a x a ptq ` Da uptq  y (Dt )c u(tC) c xc (t ) y (Dt )c u(tC) c xc (t )  Dc u(t ) (5)  ((tt)  (6)  x a (t )  Aa xa (t )  Ba u(t ) t)     (3)  (3)  oordinates, Equation (3) can be described  t) y(t )  Ca xa (pt )ˆnDa u(t ) nˆn nˆm pˆm ribution  tainty    represents  the  uncertainty  distribution  matrix.  the  where uncertainty  distribution  matrix.  , ,Aa P∈ ⌷ distribution   matrix.  , ,denotes  ,Ba P∈ ⌷  matrix.  , denotes  , ∈ ⌷   , denotes  , ∈ ⌷   denotes  , Ca P , Da P . In Equation (3), the information of health (4)   in the form of    where  ∈ wn but bounded:  ded:   are unknown but bounded:    Equation  ∈parameters ⌷information  ,  ∈are ⌷ of  , xa (t), ∈ ⌷thus,  it’s possible ∈⌷ . to In estimate Equation  (3),  the  information  of observer. health  where (3),  the  all health  in health degradation via an t) parameters are all in x alth degradation via an observer.  a (t), thus it’s possible to estimate health degradation via an observer.  A change of coordinates is introduced to simplify the observer designing process. The coordinate   (8)  ξ ( t , x , u ) || ||β ξ  ( t , x , u ) || ||β ξ ( t , x , u ) || β (6)  (8)  (8)  (8)  he transformation, the measurement matrix  ) 6) is added with modeling uncertainties as  observer designing process. The coordinate  A change of coordinates is introduced to simplify the observer designing process. The coordinate  transformation is associated with the invertible matrix: real scalar. The notation  he notation  ents the Euclidean norm for vectors and the  represents the Euclidean norm for vectors and the  ||  ||   transformation is associated with the invertible matrix:  ||  ||   represents the Euclidean norm for vectors and the  ||  ||   represents the Euclidean norm for vectors and the  ff « T in the form of    where  ∈ N rm for matrices.  ces.  c I p    (5)  (4)  N T  Tc “ c ξ (t , x , u)   Ca (7)  Tc   c    (4)  (4)  w coordinates, Equation (3) can be described   Ca  ng Architecture  Gas‐Path Health Monitoring Architecture  server‐Based Gas‐Path Health Monitoring Architecture  lth Monitoring Architecture  ) is added with modeling uncertainties as  nˆpn´ pq

bution  matrix. ∈where ⌷   denotes  Pspans the null‐space of C spans the null-space of Ca . With the transformation, the measurement matrix has where  e transformation, the measurement matrix  ⌷ , , Nc  ∈ a. With the transformation, the measurement matrix  o fulfill robust degrading estimation based on Equation  ion, a SMO is designed to fulfill robust degrading estimation based on Equation  s designed to fulfill robust degrading estimation based on Equation  st degrading estimation based on Equation  u ( t ) the form: as output estimation error. Consider a SMO of the form:  mation error. Consider a SMO of the form:    as output estimation error. Consider a SMO of the form:  ” ı ξcr. Consider a SMO of the form:  (t , x , uhas the form:  )   ´1 (7) (6)  u(t ) C “ C T “ (5) 0 I c a p c  1 (8)  ˆ cl(ety)( (5)  (xˆ tc)B(tc)uG(ntν)A(ctx)G (tc)G e)y xˆt )cB u(ntν)A(tc )xˆGc l(et y) (t )Bc u G(ntν)(t )Gl e y ((5)  t )  Gn ν(t )Cc  C aTc  0 I p            (9)  (9)  (9)  (9)  pˆ p nts the Euclidean norm for vectors and the  ˆ ˆ (tc)uwhere (tc)u(tC ) matrix.  (tC ) , xˆ ,(t )Ic∈ yˆD t )   denotes  ution  is in the form of  ∈ identity matrix. In the new coordinates, Equation (3) can be described as: PDc⌷u  (where  denotes the c xc (t ) yD coordinates, Equation (3) can be described  ∈ ⌷ c c  denotes the identity matrix. In the new coordinates, Equation (3) can be described  where  . of  e ∈∈estimates  ⌷⌷ as:  ,  , are  .  of  the  ∈∈⌷⌷linear  ,   , are  .  gain  the  ∈∈⌷⌷ matrix  linear    , are  gain  and  the  ∈ ⌷matrix  the  linear    are  gain  and  the the  matrix  linear  gain  and  the  matrix  and  the  xc ptq “ Ac xc ptq ` Bc uptq  (6) is added with modeling uncertainties as  (8)  (6) ely. v(t)∈ ix, respectively. v(t)∈ ⌷ ⌷ s:  is defined as:    is defined as:    is defined as:  (t ) x c (t )  Ac xcyptq (t )  “ Bc uC(t )x ptq ` D uptq  Architecture  c  c c   (6)  (6)  t) y (t )  C c xc (t )  Dc u(t ) ts the Euclidean norm for vectors and the  « ff t degrading estimation based on Equation  Qc ξ (t , x , u) A A c11 c12   ´ 1 (7)   Consider a SMO of the form:  where xc (t) = cA T caT xac−1 (t), , Bca. A = Tc is in the form of  form of ∈ c Aa Tc c = D c Ba , and Dc = Da . Ac is in the s in the form of  cxa(t), A c = T   where  , BA c = T ∈c =cBTa, and D   where  where xc(t) = T Ac21 Ac22 (t ) Gn⌷ ν(t ) p n ´ p qˆp n ´ p q Architecture  ribution    matrix. . where, A, c11 ∈ P ⌷   denotes  (9)  . 6) is added with modeling uncertainties as  In order to exploit the robustness of SMO, Equation (6) is added with modeling uncertainties as   degrading estimation based on Equation  In order to exploit the robustness of SMO, Equation (6) is added with modeling uncertainties follows:  Consider a SMO of the form:  as follows: ∈ ⌷   are  the  linear  gain  matrix  and  the    β (8)  . ξ ( t , x , u ) x c (t )  Acxxcc ptq (t ) “BcA uc(tx)cptq Qc` ξ (tB , xc ,uptq u) ` Qc ξpt, x, uq c  t ) G ν(t )   (7)   (7)  (7)  n sents the Euclidean norm for vectors and the  yptq “ C x ptq ` D uptq   y ( t )  C x ( t )  D u ( t ) c c c (9)  c c c nˆ h

h

where  bution  matrix.  where ∈ ⌷ , , Q crepresents  ∈ the  uncertainty  distribution  matrix.    denotes  P ⌷   denotes  represents the uncertainty distribution matrix. ξ pt,, x,, uq ∈ P ⌷ denotes uncertainties, ⌷   are  the  linear  gain  matrix  and  the  uncertainties, which are unknown but bounded:  ng Architecture  which are unknown but bounded:

(8)   || ξ ( t , x , u ) ||||ξpt, β   x, uq|| ă β (8)  (8)  ust degrading estimation based on Equation  where β is a known real scalar. The || ¨ || represents the Euclidean norm for vectors and the or. Consider a SMO of the form:  where β is a known real scalar. The notation  nts the Euclidean norm for vectors and the  || notation ||   represents the Euclidean norm for vectors and the  induced spectral norm for matrices. induced spectral norm for matrices.  e y (t )  G n ν (t )   (9)  2.2. Sliding Mode Observer-Based Gas-Path Health Monitoring Architecture g Architecture  2.2. Sliding Mode Observer‐Based Gas‐Path Health Monitoring Architecture  thismatrix  sub-section, a SMO is designed to fulfill robust degrading estimation based on Equation (7). ∈ ⌷   are  the  linear In gain  and  the  t degrading estimation based on Equation  In this sub‐section, a SMO is designed to fulfill robust degrading estimation based on Equation  Define ey ptq “ yˆ ptq ´ y ptq as output estimation error. Consider a SMO of the form: s:  (7). Define  . Consider a SMO of the form:    as output estimation error. Consider a SMO of the form: 

(t ) Gn ν(t )

∈⌷  

.

 

“ Ac(xˆt c) ptq (t )c uptq (t )Gl ey ptq ` Gn νptq xˆ c (t )  Axˆc cxˆptq  Gl` ey B  Gn ν´ c (t )  Bc u   (9)  ˆxˆ (t )“ cx (ˆtc)ptq ` Dc uptq yˆ (t )  C yptq  DCu c

c

(9) 

c

where    are  the  linear  , )  are  gain  the matrix  estimates  and of  the  , .  ∈ ⌷ ,  ∈⌷ nonlinear gain matrix, respectively. v(t)∈ ⌷   is defined as: 

  are  the  linear  gain  matrix  and  the 

(9)

   (4)  (3)    tB) (ta)u(tc = D x (c−1 Aa cxBaL(at, and D y, B )xD )  B a. Ac is in the form of  (t ) c = T C T   where  ∈ ( ) ( )  t  t u u .  entity matrix. In the new coordinates, Equation (3) can be described           h (t0)   0 0   h(t )   0    ⌷   (3), (3)  der to exploit the robustness of SMO, Equation (6) is added with modeling uncertainties as  t the robustness of SMO, Equation (6) is added with modeling uncertainties as  ,  ∈ .  In  Equation  the  information  of  health  ransformation, the measurement matrix  (2)  (2) 

 x (t )  sible to estimate health degradation via an observer.  (t )(t )C M   Du  Du(t ) obustness of SMO, Equation (6) is added with modeling uncertainties as  x c (t )h(t )Ac xc (t )  Bc u(t )   health  ced to simplify the observer designing process. The coordinate  ation  (3),  the  information  (6)  A x (t2016, )x (Btof  )9,u(598 tA) cxQ ξ)(t ,Bxc,u()t )  Qc ξ (t , x , u) c (ct(5)  y (t ) x cC(tc)xEnergies c (t )c cDc u(ct ) c     nvertible matrix:  egradation via an observer.  on (2) can be presented as:  ted as:  yc(tx)c  t )Q c(ξC t()tc,xxc ,(ut )) Dc u(t ) x c (t )  A (t )Cc xBcc(ut )(ty) (D cu

(7) 

(7) 

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  ver designing process. The coordinate  (7)  T c = Da. Ac is in the form of  ordinates, Equation (3) can be described  , B   where  (t )Ba uD(tc)u(t ) n∈ ˆp nˆ p u (t )c = T x a (tcy)B(ta), and D AaxCNac(cxt )c epresents  ∈ ⌷ Tc   represents  the   uncertainty  uncertainty  distribution  distribution  matrix.  matrix.  , , ∈ ⌷  , ,denotes  ,Gn P ∈ ⌷   denotes   where the  pˆxc , y) ˆ are the estimates of pxc , yq.(4)  Gl P are the linear gain matrix and the (3)  (3)  (t ) y(t )  Ca x aC(ta)  Da u(t ) p re unknown but bounded:  ies, which are unknown but bounded:  nts  the  uncertainty  distribution  matrix.  , , v(t)P ∈ ⌷ is  denotes  nonlinear gain matrix, respectively. defined as: stness of SMO, Equation (6) is added with modeling uncertainties as  nown but bounded:  ⌷ n   Equation  ,  a. With the transformation, the measurement matrix  ∈ ⌷(3),  the  .  In information  (3),  health  of  health  (6)  pace of C ||Equation  ξ ( t , x , u ) || ||(4)  βof  ξ (the  t3 of 14  , x , information  u ) || β   $ (8)  (8)  P0 ey ptq & ssible to estimate health degradation via an observer.  alth degradation via an observer.  ´ρ e ptq ‰ 0 y || ξ ( t , x , u ) || β   ptq|| ||P0 ey(8)  al scalar. The notation   a known real scalar. The notation  νptq “ (10) x c (t )  A1c xc (t )  Bc u(t )||Q||c ξ  (represents the Euclidean norm for vectors and the  t , x , u) ||  ||   represents the Euclidean norm for vectors and the  uced to simplify the observer designing process. The coordinate  observer designing process. The coordinate  chniques, health parameters is required to  sformation, the measurement matrix  % n the form of    where  ∈         C C T 0 I 0 otherwise (7)  (5)  ar. The notation  m for matrices.  pectral norm for matrices.  yc (t ) a Ccc xc (t) ||Dcp||u( t )represents the Euclidean norm for vectors and the  invertible matrix:  nted system is created:  atrices.  pˆ p matrix. In the new coordinates, Equation (3) can be described  Bthe   N cT where ver‐Based Gas‐Path Health Monitoring Architecture   uncertainty  distribution  matrix.  , , P0 ∈ (5)  is added with modeling uncertainties as  g Mode Observer‐Based Gas‐Path Health Monitoring Architecture  ρ is a positive real scalar. P ⌷   denotes  is a symmetric positive matrix that will be defined later. (4)  (4)     0  u(t ) Tc   wn but bounded:  sed Gas‐Path Health Monitoring Architecture    C a Gn is designed to be in the structure:   « ff (2)  n, a SMO is designed to fulfill robust degrading estimation based on Equation  s sub‐section, a SMO is designed to fulfill robust degrading estimation based on Equation  inates, Equation (3) can be described  x c (t )  Ac xc (t )  Bc u(t ) ´E   || ξ ( t , x , u ) ||  β   (8)  ( t , x , u ) (t ) ue transformation, the measurement matrix  e    as output estimation error. Consider a SMO of the form:    as output estimation error. Consider a SMO of the form:  (6) Gn “ space of C a. With the transformation, the measurement matrix  MO is designed to fulfill robust degrading estimation based on Equation  (11) y (t )   C c xc (t )  Dc u(t ) (7)  Ip   as output estimation error. Consider a SMO of the form:  The notation    represents the Euclidean norm for vectors and the  ˆ ˆ c B (tc)u(tA ) c xˆGc l(et )y (t )Bc uG(ntν) (t )Gl e y (t )  Gn ν(t ) xˆ c ||(t ) || A  c xc (t ) x     where    ∈ (6)  ed as:  pn´ pqˆ p (9)  (9)   1 Bp a, and D c = D a . A c is in the form of       ices.    ution  matrix.  , , ∈ ⌷  C C T 0 I (5)  ˆc(xtˆ)c (t )Cwhere D e(tyC)(tc)xˆ cdenotes  ν(D t )c u(5)  represents the design freedom, and no special structure is imposed on Gl . cy (tG (G (t ) xˆ cpc(ut()t)yˆEP t )n  xˆc c (t ) a A l cB c)u   (9)  (t ) e ptq “ xˆ c ptq ´ xc ptq as state estimation error. The following error system is obtained from ) yˆ (t )  Cc xˆ c (t )  Dc u(tDefine (3)   , are  the   matrix. In the new coordinates, Equation (3) can be described  coordinates, Equation (3) can be described  estimates  ,)  )  are  the  of estimates  , .  of  ∈ ⌷ , , .  ∈∈ ∈⌷⌷ ∈ ⌷linear    are  gain  the matrix  linear  gain  and  the  matrix  and  the  Gas‐Path Health Monitoring Architecture  e form of    where  of SMO, Equation (6) is added with modeling uncertainties as  Equations (7) and (9): (8)  .  respectively. v(t)∈ gain matrix, respectively. v(t)∈ ⌷ ⌷   is defined as:    is defined as:  tes  of  , .  ∈ ,  ∈ ⌷   are  the  linear  matrix  and ` the  eptq “gain  pAc ´ Gl Cc qeptq Gn νptq ´ Qc ξpt, x, uq (12)  is designed to fulfill robust degrading estimation based on Equation  Equation  (3),  the  information  of  health  ctively. v(t)∈ ⌷   is defined as:  s the Euclidean norm for vectors and the   u ( t ) x ( t )  A x ( t )  B u ( t ) added with modeling uncertainties as  c c c c as output estimation error. Consider a SMO of the form:   A c xc (t )  Bc u(t )  where Qc ξ (t , x  ,euy)ptq is replaced lth degradation via an observer.  ptq “ Cc e ptq. Due to the structure of Cc , ey ptq is directly (6) considering that e(6)    u(t ) y (t )  C c xc (t )  Dc u(t ) (7) y bserver designing process. The coordinate  Cxˆc x(tc)( t) A D u ( t ) extracted from the last p components of e(t), which means the last p components of e(t) are c ˆ c c xc (t )  Bc u(t )  Gl e y (t )  Gn ν (t ) « ff ey (t), (9)  ´ ¯T   , u) n´ p A p11 A p12 ˆa (, and D Architecture  (t ) distribution  t )  Cc xˆ cc = D t ) e “ ematrix.  Tuncertainty  s in the form of  cBy aD . A c( is in the form of   Twhere  T (7) where , ∈, e1 ∈   .denotes  cu i.e., P  ⌷where  Define∈Ap = Ac ´ Gl Cc , in the form of where 1 , ey A p21 A p22 bounded:  degrading estimation based on Equation    of  , .  ∈ ⌷ ,  ⌷pqˆp nare  (4)  linear  gain  matrix  and  the  p∈n´ ´ pq the  n  matrix.  s of SMO, Equation (6) is added with modeling uncertainties as  (6) is added with modeling uncertainties as  P ⌷   denotes . Then the error system Consider a SMO of the form:  || ξ ( t , x⌷, u  )is defined as:  ||, βA, p11 ∈ ely. v(t)∈ (8)  can be partitioned as:

.  transformation, the measurement matrix  Gn ν(t||)  ||   represents the Euclidean norm for vectors and the  ation  e1 ptq “ A p11 e1 ptq ` A p12 ey ptq ´ Eνptq ´ Qc1 ξpt, x, uq   (9)  (8)  Q (tc)ξ(t ,Axc ,xuc )(t )  Bc u(t )  Qc ξ (t , x , u) (13) .     ` A p22 ey ptq ` νptq ´ Qc2 ξpt, x, uq (7) ey ptq “ A p21 e1 ptq(7)  )  C c xc (t )  Dc u(t ) he Euclidean norm for vectors and the    (5)  ⌷   are  the  linear  gain  matrix pnand  the  ´ pqˆ h pˆ h th Health Monitoring Architecture  ibution  uncertainty  matrix.  distribution  , , ∈ matrix.  ⌷   denotes  , Q , c2 ∈ where Qc1 P and P ⌷   .denotes  oordinates, Equation (3) can be described  t bounded:  Define a sliding surface: gned to fulfill robust degrading estimation based on Equation  hitecture  S “ te | Cc e “ 0u (14) ut estimation error. Consider a SMO of the form:    || ξ ( t , x , u ) || β   (8)  (8)  (rading estimation based on Equation  t)   (6)  the stability of the observer, it is required to prove that the error system can ents the Euclidean norm for vectors and the   Gl e y (t ) In Gnorder ν(t ) to guarantee otation  tsider a SMO of the form:  )Ac xˆ c (t ) ||B c||u( t )represents the Euclidean norm for vectors and the    (9)  and remained on it by appropriate design. In view of this, be driven to the sliding surface in finite time

Cc xˆ c (t )  Dc u(t ) G in the form of  n ν (t )

the following conclusion   where  ∈ is presented. .    ∈ ⌷ ,  ∈ ⌷   are  the (9)  linear  gain  matrix  and  the  ath Health Monitoring Architecture  g Architecture  Theorem 1: Considering the error in Equation (12), with Gn in the structure of Equation (11), assume )∈ ⌷   is defined as:  that matrix  there exists matrix Gl and a Lyapunov matrix P of the form: 6) is added with modeling uncertainties as  igned to fulfill robust degrading estimation based on Equation  st degrading estimation based on Equation    are  the  linear  gain  and  athe  « ff r. Consider a SMO of the form:  put estimation error. Consider a SMO of the form:  P1 P1 E P“ ą0 (15) xˆ, u)t )(t) B u(t )  G e (t )  G ν(t ) cξ ((tt)A,G    c ˚ P0 ` ET P1 E y c xc n(ν l y n (7)    (9)  (9) 

 Cc xˆ c (t )  Dc u(t )

pn´ pqˆpn´ pq

bution  matrix.  where , , P1 ∈ ⌷   denotes , that satisfies: ∈, ⌷ .    are  ∈ ⌷ the , linear  ∈ ⌷gain   Pmatrix  are  the and  linear  the  gain  matrix  and  the  (t)∈ ⌷   is defined as:  s:  PA p ` A Tp P ă 0 (8) 

(16)

and ρ is chosen to satisfy: nts the Euclidean norm for vectors and the  ρ ą 2||Ac21 ||p1 ` ||E||qµ1 β{µ0 ` ||Qc2 ||β ` η0

 Architecture 

(17)

´ ¯ where µ0 “ ´λmax PA p ` ATp P , µ1 “ ||PQc || and η0 is a small positive real scalar, then the error t degrading estimation based on Equation  Equation (12) is driven to the sliding surface in finite time.  Consider a SMO of the form: 

(t ) Gn ν(t )

∈⌷  

 

(9) 

  are  the  linear  gain  matrix  and  the 

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Proof: The following Lyapunov function is introduced: V1 ptq “ e T ptqPeptq

(18)

V 1 ptq “ e T ptqpPA p ` A Tp Pqeptq ` 2e T ptqPGn νptq ´ 2e T PQc ξpt, x, uq

(19)

The derivative of V 1 (t) is given by: .

Note that PGn “ CcT P0 , and with inequality in Equation (16) and definitions of µ0 and µ1 , V 1 ptq becomes: ˇˇ ˇˇ ˇˇ ˇˇ . ˇˇ ˇˇ ˇˇ ˇˇ V 1 ptq ď ´µ0 ˇˇeptqˇˇ2 ` 2eyT ptqP0 νptq ` 2ˇˇeptqˇˇµ1 β ˇˇ ˇˇ ˇˇ ˇˇ ˇˇ ˇˇ (20) “ˇˇeptqˇˇp´µ0 ˇˇeptqˇˇ`2µ1 βq ´ 2ρˇˇP0 ey ptqˇˇ .

ď||eptq||p´µ0 ||eptq||`2µ1 βq which shows the magnitude of e(t) decreases when ||e(t)|| > 2µ1 β/µ0 . This implies that in finite time ||e(t)|| will be bounded with respect to the set: Ωε “ te : ||e|| ă 2µ1 β{µ0 ` εu

(21)

where ε is an arbitrary small positive scalar. In order to prove a sliding motion can be induced on the sliding surface, it’s convenient to introduce another change of coordinates to Equation (12) associated with nonsingular matrix: « TE “

In´ p 0

E Ip

ff

0 Ip

ff

(22)

Applying the change of coordinates T E yields: « A“

A11 A21

A12 A22

«

ff ” , C“

ı 0

Ip

, Gn “

« , Gl “

G l1 G l2

ff

« , Q“

Q1 Q2

ff (23)

where A11 “ Ac11 ` EAc21 , A21 “ Ac21 , Q1 “ Qc1 ` EQc2 and Q2 “ Qc2 . The error system in Equation (12) with the change of coordinates e “ TE e can be rewritten and partitioned as: .

e1 ptq “ A11 e1 ptq ` pA12 ´ G l1 qey ptq ´ Q1 ξpt, x, uq . ey ptq “ A21 e1 ptq ` pA22 ´ G l2 qey ptq ` νptq ´ Q2 ξpt, x, uq

(24)

Consider another Lyapunov function candidate V2 ptq “ eyT ptq P0 ey ptq where P0 has been defined in Equation (15). The time derivative of V 2 (t) along the trajectories of the error system is given by: .

V 2 ptq

.T

.

“ ey ptqP0 ey ptq ` eyT ptqP0 ey ptq T

“ eyT ptqppA22 ´ G l2 q P0 ` P0 pA22 ´ G l2 qqey ptq `2eyT ptqP0 pA21 e1 ptq ` νptq ´ Q2 ξpt, x, uqq Note that:

« T pTE´1 q PTE´1

and:

« TE A p TE´1 “

“

A11 A21

P1 0

0 P0

(25)

ff

A12 ´ G l1 A22 ´ G l2

(26)

ff (27)

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Combining Equations (16), (26) and (27) yields: « T pTE´1 q pPA p

`

A Tp PqTE´1

T

“

ff

P1 A11 ` A11 P1

N1

N2

P0 pA22 ´ G l2 q ` pA22 ´ G l2 q P0

T

ă0

(28)

where N 1 and N 2 are elements that are not necessary in the following analysis. Since Equation (28) is symmetric, it is obvious that: T

P0 pA22 ´ G l2 q ` pA22 ´ G l2 q P0 ă 0

3 of 14 

therefore from Equations 3 of 14  (25) and (29) it follows that: chniques, health parameters is required to  . nted system is created:  V 2 ptq ď 2eyT ptqP0 pA21 e1 ptq ` νptq ´ Q2 ξpt, x, uqq hniques, health parameters is required to  B ted system is created:  Since e1 ptq “ e1 ptq ` Eey ptq, considering Equation (21) yields:  0  u (t ) B     (2)   u (t )  | |e u (t )   1 ptq|| ă ||e1 ptq|| ` ||E||||ey ptq|| ă p1 ` ||E||qp2µ1 β{µ0 ` εq (2) 

t)

(29)

(30)

(31)

then we combine Equations (30) and (31) to get:

ed as: 

.

d as:  (t )  

V 2 ptq (3) 

))

ˇˇ ˇˇ ˇˇ ˇˇ ˇˇ ˇˇ ˇˇ ˇˇ ď 2ˇˇP0 ey ptqˇˇp2ˇˇ A21 ˇˇp1`ˇˇEˇˇqµ1 β{µ0 `ˇˇQ2 ˇˇβq P e ptq

(32) ´2ρpP0 ey ptqqT P0 ey ptq   (3)  ˇˇ ˇˇ ˇˇ ˇˇ ˇˇ ˇ|| ˇ 0 y ˇ||ˇ ˇˇ   Equation  (3),  the  information  of  health  ď ´2ˇˇP0 ey ptqˇˇpρ ´ 2ˇˇ A21 ˇˇp1`ˇˇEˇˇqµ1 β{µ0 ´ˇˇQ2 ˇˇβq lth degradation via an observer.  Equation  (3),  the  information  of  health  Note that A “ Ac21 and Q2 “ Qc2 . Then, substituting Equation (17) into Equation (32) yields: bserver designing process. The coordinate  h degradation via an observer.  21 . server designing process. The coordinate  V 2 ptq ă 0 (33)

(4)  which means the reachability condition is satisfied, and a sliding motion on S is attained in finite time. Thus Theorem 1 is (4)  proved. .  transformation, the measurement matrix  Once the sliding mode surface is reached, which indicates ey “ ey “ 0. Note that A p11 “ Ac11 and ransformation, the measurement matrix  A p21 “ Ac21 , the error system in Equation (13) can be simplified as:    (5)  . e1 ptq “ Ac11 e1 ptq ´ Eveq ptq ´ Qc1 ξpt, x, uq   (34) (5)  oordinates, Equation (3) can be described  0 “ Ac21 e1 ptq ` veq ptq ´ Qc2 ξpt, x, uq ordinates, Equation (3) can be described  where veq (t) is the so-called equivalent output error injection term, which represent the average (t )   behavior of the discontinuous term v(t) and can be obtained by a low pass filter from v(t). Eliminating (6)  )t ) veq (t) from Equation (34) yields a reduced order system:   (6) 

 in the form of 

  where 

∈

.

e1 ptq “ pAc11 ` EAc21 qe1 ptq ´ pQc1 ` EQc2 qξpt, x, uq

n the form of    where  ∈ Assume that Qc is in the structure: 6) is added with modeling uncertainties as 

 is added with modeling uncertainties as  ξ (t , x , u)

 

(t , x , u)

« Qc “

(7) 

(35)

ff ´EX X

(36)

  (7)  pˆ h bution  matrix.  where , , X∈ P ⌷   denotes  is an arbitrary real matrix. The structure of Qc indicates Qc satisfies Qc = Gn X for pˆ h ution  matrix.  some , , X∈ P ⌷   .denotes  Combining Equations (35) and (36) yields: (8) 

(8)  nts the Euclidean norm for vectors and the 

s the Euclidean norm for vectors and the 

 Architecture 

Architecture  t degrading estimation based on Equation   Consider a SMO of the form:  degrading estimation based on Equation 

.

e1 ptq “ pAc11 ` EAc21 qe1 ptq

(37)

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This is an ideal sliding motion which is entirely independent of ξ pt, x, uq. The remaining dynamics e1 (t) is governed by (Ap11 + EAp21 ). From Equation (28) it can be shown that P1 (Ac11 + EAc21 ) + (Ac11 + EAc21 )T P1 < 0 and P1 > 0, thus the matrix (Ac11 + EAc21 ) is stable, which means the reduced order motion can be maintained on the sliding surface. Notice that for the existence of an ideal sliding motion, the match condition Equation (36) is not required, and a large enough ρ is sufficient to induce a sliding motion, like Equation (17). The match condition is only required for the reduced sliding motion Equation (35) to be independent of ξ pt, x, uq. The design philosophy is to synthesize P and Gl so that inequality Equation (16) can be achieved. 3 of 14  The design method adopted is to use linear matrix inequalities (LMIs) [17–19]. Here P and Gl will be chosen so that the inequality: hniques, health parameters is required to  ted system is created:  PA ` A T P ă ´PW P ´ PG V G T P (38) p

B u (t ) 0   

t)

l

l

is satisfied, where the design weighting matrices W and V are diagonal positive definite, and P has (2)  the structure in Equation (15). Let Z = PGl and substituting Ac ´ Gl Cc for Ap , the inequality can be written as: T PAc ` AcT P ` pZ T ´ V ´1 Cc q V pZ T ´ V ´1 Cc q ´ CcT V ´1 Cc ` PW P ă 0 (39)

d as: 

)

p

Choosing:

Z T “ V ´1 Cc (40) (3)  to eliminate the third term in Equation (39) yields: Equation  (3),  the  information  of  health  PA p ` A Tp P ´ CcT V ´1 Cc ` PW P ă 0 (41) th degradation via an observer.  server designing process. The coordinate  Thus the inequality Equation (16) is replaced by Equation (41), which is required to be satisfied. From Equation (40) Gl can be directly calculated as:

 

(4) 

Gl “ P´1 CcT V ´1

(42)

ransformation, the measurement matrix  From the structure of Gl it can be shown that the “magnitude” of Gl is directly determined by the ”magnitude” of P´1 , and the larger Gl is, the more difficult it can be realized in hardware. The target considered here is to minimize trace(P´1 ) subject to P satisfying Equation (41). In fact the   (5)  motivation here is to seek optimal value of linear gain Gl by the standard linear quadratic Gausssian ordinates, Equation (3) can be described  (LQG) optimal observer design method, as described in [20]. The associated optimal cost is given by trace(P´1 ). The method can be summarized as:

)

minimize trace(S) (6) with regard to P1 , P0 , E and S subject to: « ff P1 P1 E P“ ą0 n the form of    where  ∈ ˚ P0 ` E T P1 E « ff PAc ` AcT P ´ Cc V ´1 Cc P ă0  is added with modeling uncertainties as  ˚ ´W ´1 « ff ´P I n (t , x , u) ă0   (7)  ˚ ´S

 

‚

nˆn

(43)

(44)

(45)

, , S∈ where P ⌷   denotes  is symmetric positive definite. This represents a convex optimization problem, which can be handled by standard LMI software. Then Gn and v(t) can be both obtained with E and P0 . A new architecture (8)  for estimating engine performance deterioration is described based on the SMO designed above. A two-spool turbofan engine is considered in this paper, of which the schematic ts the Euclidean norm for vectors and the  model is shown in Figure 1. The airflow is supplied by a single inlet. Airflow passes through the fan and separates into two streams: one passes through engine core path, and the other passes through

ution  matrix. 

Architecture 

degrading estimation based on Equation  Consider a SMO of the form: 





where  ∈ ⌷   is  symmetric  positive  definite.  This  represents  a  convex  optimization  problem,  which can be handled by standard LMI software. Then Gn and v(t) can be both obtained with E and P0.  A new architecture for estimating engine performance deterioration is described based on the  SMO  designed  Energies 2016, 9, 598 above.  A  two‐spool  turbofan  engine  is  considered  in  this  paper,  of  which  8 ofthe  15 schematic  model  is  shown  in  Figure  1.  The  airflow  is  supplied  by  a  single  inlet.  Airflow  passes  through the fan and separates into two streams: one passes through engine core path, and the other  the bypass duct. Fuel is injected in the combustor and burned to produce the hot gas to drive the passes through the bypass duct. Fuel is injected in the combustor and burned to produce the hot gas  turbines. The fan and low pressure compressor (LPC) are driven by low pressure turbine (LPT), while to drive the turbines. The fan and low pressure compressor (LPC) are driven by low pressure turbine  high pressure compressor (HPC) is driven by high pressure turbine (HPT). The airflow leaves through (LPT), while high pressure compressor (HPC) is driven by high pressure turbine (HPT). The airflow  the nozzle, which has a variable cross section area A8 . The notations used in this paper and their leaves through the nozzle, which has a variable cross section area A 8. The notations used in this paper  descriptions are shown in Notations Section. and their descriptions are shown in Notations Section. 

  Figure 1. Schematic model of a two‐spool turbofan engine.  Figure 1. Schematic model of a two-spool turbofan engine.

As engine parts wear from regular operation, the lifespan of components will be reduced. The  As engine parts wear from regular operation, the lifespan of components will be reduced. aging of components is reflected as changes in internal flow capacities and component efficiencies,  The aging of components is reflected as changes in internal flow capacities and component efficiencies, and  components’  sudden  damages  also  result  in  the  change  of  component  performance  and  and components’ sudden damages also result in the change of component performance and characteristics. Thus these capacities and efficiencies of components are chosen as “health parameters”  characteristics. Thus these capacities and efficiencies of components are chosen as “health parameters” to reflect component health performance. The degradations of health parameters are described as:  to reflect component health performance. The degradations of health parameters are described as: h hi  i  1 i  1, 2,..,8   (46)  hihi , r ∆hi “ ´ 1 i “ 1, 2, .., 8 (46) hi,r where hi,r denotes the nominal value of hi. A normalized health parameter varies between 0 and 1,  where hi,r denotes the nominal value of hi . A normalized health parameter varies between 0 and 1, with 1 representing a healthy component and 0 a “fully deteriorated” one. The maximum level of  with 1 representing a healthy component and a “fully deteriorated” one. The maximum level of deterioration  indicates  an  engine  overhaul  is 0necessary.  The  health  parameters  are  employed  to  deterioration indicates an engine overhaul is necessary. The health parameters are employed to handle with both slowly aging and sudden fault. The purpose of a gas-path health monitoring (GPHM) system is to estimate these health parameters for performance monitoring and fault detection. Mechanical system dynamics due to rotating inertias constitute the most important contribution to engine transient behavior [21]. Thus rotating dynamics are the most important dynamics to be considered. In view of this, x is chosen as col(NL ,NH ). Newton’s law for rotating masses is applied to each shaft as: . N L “ f 1 pNL , NH , u, v, hq . (47) N H “ f 2 pNL , NH , u, v, hq where f 1 and f 2 are the net torques delivered by LPT and HPT. u contains the control input components, which is given by col(W f ,A8 ), and v denotes the external parameters (flight condition). Because of the intricate geometry of the engine components and the complexity of airflow, algebraic expressions for f 1 and f 2 are unavailable. A practical modeling approach is the piecewise linear SVM. Employing perturbation techniques at each equilibrium condition, a set of linear relationships which depict the interactions of the engine states with inputs and outputs are established, as described in Equation (1). Compared to the nonlinear CLM, SVM takes a smaller computation burden and is more feasible in real-time applications. Most of sensors used to conduct GPHM are installed on the engine for control purposes, and some of these sensors are employed to enable GPHM system. In this paper the available measurements are defined by the standard suite of sensors found in the tactical turbofan engine control system. The output vector is chosen as col(NL , NH , T25 , P25 , T3 , P3 , EGT). Given this sensor suite constraint,

interactions of the engine states with inputs and outputs are established, as described in Equation (1).  Compared to the nonlinear CLM, SVM takes a smaller computation burden and is more feasible in  real‐time applications.  Most of sensors used to conduct GPHM are installed on the engine for control purposes, and  Energies 9, 598 sensors  are  employed  to  enable  GPHM  system.  In  this  paper  the  available  9 of 15 some  2016, of  these  measurements  are  defined  by  the  standard  suite  of  sensors  found  in  the  tactical  turbofan  engine  control system. The output vector is chosen as col(N L, NH, T25, P25, T3, P3, EGT). Given this sensor suite  observability studies indicate seven health parameters could be discerned properly. Table 1 lists constraint, observability studies indicate seven health parameters could be discerned properly. Table  the statistical data of engine deterioration level for certain flight cycles. Compared to other health 1  lists  the statistical  of  engine  certain  flight  cycles. Compared  to  other  parameters, the flow data  capacity of LPTdeterioration  deteriorates level  muchfor  less, so it should then be ignored. Thus the health parameters, the flow capacity of LPT deteriorates much less, so it should then be ignored. Thus  health parameters monitored are given by col(h1 , h2 , h3 , h4 , h5 , h6 , h7 ). the health parameters monitored are given by col(h1, h2, h3, h4, h5, h6, h7).  Table 1. The variation of health parameters with different flight cycles (%). Table 1. The variation of health parameters with different flight cycles (%).  Cycle Number

Cycle Number  3000  3000 4500 4500  6000 6000 

h

h11  ´1.46 −1.46  ´2.04 −2.04  ´2.61 −2.61 

h2

h2  ´2.08 −2.08  ´3.04 −3.04  ´4.00 −4.00 

h3

h

h

h

4 5 h3 h4 h5 6 ´2.94 ´3.91 ´2.63 1.76 −2.94  −3.91  −2.63  ´6.17 ´8.99 ´3.22 2.17 −6.17  ´14.06−8.99 ´3.81 −3.22  ´9.40 2.57 −9.40  −14.06  −3.81 

h

h

h6  7 ´0.53 1.76  ´0.81 2.17  ´1.11 2.57 

h7 8 0.26 −0.53  0.34 −0.81  0.42 −1.11 

h8 0.26  0.34  0.42 

Since health parameters  parameters are  areaugmented  augmentedas asstate  statevariables,  variables, described in Equation (2), Since  health  as  as described  in  Equation  (2),  the  the estimation of component degradation can be easily obtained by the described SMO. The proposed estimation of component degradation can be easily obtained by the described SMO. The proposed  GPHM architecture is shown in Figure 2. GPHM architecture is shown in Figure 2. 

  Figure 2. Schematic representation of gas‐path health monitoring (GPHM).  Figure 2. Schematic representation of gas-path health monitoring (GPHM).

Due to the SVM of the turbofan engine being time‐invariant, the gain matrix of the SMO can be  Due to the SVM of the turbofan engine being time-invariant, the gain matrix of the SMO can computed  off‐line.  Once  the  SMO  is  well  designed,  it  enables  the  GPHM  system  to  monitor  the  be computed off-line. Once the SMO is well designed, it enables the GPHM system to monitor components’  performance  in  real‐time  for  in‐flight  health  tracking.  The  SMO  possesses  a  strong  the components’ performance in real-time for in-flight health tracking. The SMO possesses a strong robustness against a class of uncertainties, which is precious to the estimation of aero-engine health parameter problem, considering the inevitable modeling error from linearizing process of an aircraft engine. 3. Results and Discussion In this paper, a CLM introduced in [3], which captures the nonlinear behavior of a twin-spool turbofan engine with highly fidelity is considered, working as a simulation platform for a real engine. The CLM consists of a set of individual components, each of which requires a number of inputs and generates one or more variables. The steady state simulation of CLM is based on mass flow balance and power balance equations, while the transient simulation, initialed by steady state calculation, follows mass flow balance and rotor dynamics equations. The Newton-Raphson method is employed to solve the nonlinear expressions both in steady state case and transient dynamics. Iterative solution of nonlinear equations in each step stops once the iteration number reaches 10, or the iteration error is less than 0.01. The CLM is written using C language and packaged with dynamic link library (DLL) for use in the Matlab environment [22,23]. The health parameters are modeled and health degrading injection is available in the CLM. Sensor dynamics are ignored in the simulations, with the assumption that they have high enough bandwidth. The model has been validated versus the testing data. The engine is simulated at the reference flight condition and at a nominal cruise power setting,

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and the effectiveness of the proposed health estimation approach will be demonstrated by using the simulated data extracted from the CLM. The positive scalar ρ is a critical parameter to be determined. In fact, ρ provides a degree of freedom in terms of the trade-off between robustness and chattering. The greater the value of ρ, a stronger robustness is present to ensure sliding mode occurrence, but a higher chattering problem follows. Assume that β = 1.0, from Equation (17) it can be calculated that ρ is required to satisfy ρ > 0.0092. In this paper ρ is set to 0.01. W and V are user-defined matrices and specify W = I9 and V = L7 . Considering aero-engines are prone to noises, the nominal white Gaussian noises are introduced to the measurements. The standard deviation of noises is given by σnoise = 0.0015, which is determined by practical experience. 3.1. Single Health Parameter Degradation Case In this case (Case 1) a ´5% abrupt shift is imposed on h5 from its normal value, to simulate a typical fault scenario in engine real operations. When engines are operated near the stall limit, the steady axisymmetric flow pattern in LPC becomes unstable. This instability may cause the phenomenon of surge and rotating stall in LPC, which are generally reflected by h5 degradation. In order to testify the robustness of SMO, the modeling uncertainty term Qc ξ pt, x, uq is introduced here. In the simulation specify Qc = Gn to satisfy matching condition. Assume that the modeling mismatches are in system matrix Ac , then ξ pt, x, uq is given by: » — — — — — ξ“— — — — –

0.13 0.30 0.10 0.02 0.11 0.02 0.01

0.04 0.02 0.04 0.02 0.05 0.03 0.14

0.03 0.01 0.01 0.03 0.06 0.03 0.12

0.11 0.05 0.10 0.09 0.01 0.14 0.01

0.09 0.04 0.02 0.04 0.07 0.09 0.02

0.06 0.05 0.02 0.04 0.16 0.04 0.06

0.18 0.10 0.07 0.02 0.10 0.08 0.12

0.03 0.07 0.03 0.03 0.03 0.03 0.06

0.05 0.02 0.01 0.02 0.07 0.09 0.04

fi ffi ffi ffi ffi ffi ffi xc ffi ffi ffi fl

(48)

The health parameter estimation results are shown in Figure 3a. The output errors between the engine and the observer are shown in Figure 3b, which indicates the sliding mode is quickly achieved soon after the malfunction occurs. The degraded heath parameter is estimated with acceptable errors as shown in Figure 3c. The maximum root-mean square error (RMSE) between the real values and the estimated ones is shown in Table 2. The diagnosing time is defined as the time when estimated values catch up with real values after fault injection. The diagnosing time is shown in Table 3. Table 2. The maximum root-mean square error (RMSE) (%). Case 1 Methods SMO KF

Case 2

Matched Qc

Unmatched Qc

Matched Qc

0.45 0.87

0.50 1.45

0.34 0.67

Table 3. The diagnosis time (s). Case 1 Methods SMO KF

Case 2

Matched Qc

Unmatched Qc

Matched Qc

0.67 3.71

1.21 4.55

0.61 3.52

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  3

 

(a) 

(b)

(c) Figure  3. 3.  Single  parameter  degradation degradation  (matched (matched  uncertainties): uncertainties):  (a)  health  Figure Single health  health parameter (a) estimation  estimation of  of health parameters by sliding mode observer (SMO); (b) errors between    and    by SMO; and (c) degraded  parameters by sliding mode observer (SMO); (b) errors between y and yˆ by SMO; and (c) degraded parameter estimation by KF and SMO.  parameter estimation by KF and SMO. Table 2. The maximum root‐mean square error (RMSE) (%). 

In order to show the advantage of the developed SMO in robustness, the same degrading Case 1 Case 2  problem is introduced to the nominal KF based monitoring system with the same model uncertainties. Methods  Matched Q c  Unmatched Q c  Matched Q c  The results are also shown in Figure 3c, Tables 2 and 3. It can be seen that the estimation of h5 SMO  0.45  0.50  0.34  is inaccurate due to the uncertainties, which emphasizes the advantage of the SMO in robustness KF  0.87  1.45  0.67  against uncertainties. As mentioned above, the matching condition for Qc is not necessary for the existence of an ideal Table 3. The diagnosis time (s).  sliding motion, which a large enough ρ is sufficient to induce. The match condition is only required Case Case 2  for the reduced sliding motion to be independent of ξ1 pt, x, uq. Here an unmatched Qc is introduced to Methods  Matched Q c   Unmatched Q c   Matched Q c  the system in the same degrading problem mentioned above. Qc is specified to be: SMO  KF 

0.67  3.71 

«

ff 1.21 

4.55 

0.61  3.52 

0 (49) Ip In  order  to  show  the  advantage  of  the  developed  SMO  in  robustness,  the  same  degrading  problem  to that the innominal  based  monitoring  problem system  with  the  same  uq introduced  and ξ pt, x,is  is identical to EquationKF  (47). The degradation is handled by themodel  SMO uncertainties. The results are also shown in Figure 3c, Tables 3 and 4. It can be seen that the estimation  framework and the KF framework, respectively. Figure 4 depicts the health estimation, in the case of hunmatched 5 is inaccurate due to the uncertainties, which emphasizes the advantage of the SMO in robustness  of uncertainties, via the SMO and the KF approaches. As can be seen from Figure 4a,   against uncertainties.  the degraded health parameter can be found and tracked, with an accuracy of about 95% by the  is not necessary for the existence of an ideal  SMO,As mentioned above, the matching condition for Q and other health parameters drift slightly from ctheir nominal values after the degradation sliding motion, which a large enough ρ is sufficient to induce. The match condition is only required  begins. This indicates that though Qc does not satisfy Equation (36), the sliding motion is still for the reduced sliding motion to be independent of  , uncertainties. . Here an unmatched Q c is introduced  attained. The estimation results, however, are polluted, by In comparison, the KF to the system in the same degrading problem mentioned above. Q c is specified to be:  framework performs a poorer quality in handling the same degrading problem, as shown in Figure 4b. Qc “

0 Qc      I p 

(49) 

framework and the KF framework, respectively. Figure 4 depicts the health estimation, in the case of  unmatched uncertainties, via the SMO and the KF approaches. As can be seen from Figure 4a, the  degraded health parameter can be found and tracked, with an accuracy of about 95% by the SMO,  and other health parameters drift slightly from their nominal values after the degradation begins.  This indicates that though Q c does not satisfy Equation (36), the sliding motion is still attained. The  Energies 2016, 9, 598 12 of 15 estimation  results,  however,  are  polluted  by  uncertainties.  In  comparison,  the  KF  framework  performs  a  poorer  quality  in  handling  the  same  degrading  problem,  as  shown  in  Figure  4b.  The  The degraded health parameter is estimated with an undesirable error, and other health parameters degraded health parameter is estimated with an undesirable error, and other health parameters drift  drift away after the degrading injection. The simulation confirms that the described SMO is designed away after the degrading injection. The simulation confirms that the described SMO is designed to  to be stable regardless of the form of Q c . Figure 4c,d show the output errors in the SMO and the KF be stable regardless of the form of Q c. Figure 4c,d show the output errors in the SMO and the KF  respectively. e of the SMO can converge to zero quickly after the degradation occurs, as shown in y respectively. ey of the SMO can converge to zero quickly after the degradation occurs, as shown in  Figure 4c, while e of the KF is divergent as shown in Figure 4d. The maximum RMSE values of both Figure 4c, while ey of the KF is divergent as shown in Figure 4d. The maximum RMSE values of both  frameworks are shown in Table 2 and the diagnosis time in Table 3. frameworks are shown in Table 3 and the diagnosis time in Table 4. 

(a) 

(b) 

(c) 

(d) 

Figure  4. 4.  Single Single  health health  parameter parameter  degradation degradation  (unmatched (unmatched  uncertainties): uncertainties):  (a)  health  Figure (a) estimation  estimation of  of health parameters by SMO; (b) estimation of health parameters by KF; (c) errors between    and    by SMO;  parameters by SMO; (b) estimation of health parameters by KF; (c) errors between y and yˆ by SMO; and (d) errors between    and  and (d) errors between y and yˆ by  by KF.  KF.

3.2. Multi Health Parameter Degradation Case  3.2. Multi Health Parameter Degradation Case In  this  case  (Case  2),  the  multi  health  degradation  problem  is  tested  through  the  proposed  In this case (Case 2), the multi health degradation problem is tested through the proposed scheme, scheme, with both bias and drift degradations involved. In this scenario, h5 abruptly drops to 96% of  with both bias and drift degradations involved. In this scenario, h5 abruptly drops to 96% of the the normal condition, while h1 drifts to 98% and h7 drifts to 102% in 5 s (flow capacity of turbines rises  normal condition, while h1 drifts   to 98% and h7 drifts to 102% in 5 s (flow capacity of turbines rises in in degrading cases). The matched uncertainty term  , ,   is considered here, which is the same  degrading cases). The matched uncertainty term Qc ξ pt, x, uq is considered here, which is the same as as that in the first simulation of Case 1.  that in the first simulation of Case 1. The malfunctions are estimated accurately by the proposed approach, as shown in Figure 5a.  The malfunctions are estimated accurately by the proposed approach, as shown in Figure 5a. Figure 5b shows the output errors. The same problem is introduced to the KF‐based scheme, and  Figure 5b shows the output errors. The same problem is introduced to the KF-based scheme, comparisons are shown in Figure 5c. Results indicate that the developed SMO is able to handle multi‐ and comparisons are shown in Figure 5c. Results indicate that the developed SMO is able to handle multi-fault cases with model mismatches, while in the simulation of the KF-based scheme, the faulty health parameters fail to be tracked correctly. The maximum RMSE values of both schemes are shown in Table 2 and the diagnosis time in Table 3.

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fault cases with model mismatches, while in the simulation of the KF‐based scheme, the faulty health  parameters fail to be tracked correctly. The maximum RMSE values of both schemes are shown in  Energies 2016, 9, 598 13 of 15 Table 3 and the diagnosis time in Table 4. 

(a) 

(b) 

(c)  Figure  5. 5.  Multi  parameter  degradations degradations  (matched (matched  uncertainties): uncertainties):  (a)  health  Figure Multi health  health parameter (a) estimation  estimation of  of health parameters by SMO; (b) errors between    and    by SMO; and (c) degraded parameters estimation  parameters by SMO; (b) errors between y and yˆ by SMO; and (c) degraded parameters estimation by by KF and SMO.  KF and SMO.

4. Conclusions  4. Conclusions A SMO‐based health monitoring system for an aircraft turbofan engine has been developed in  A SMO-based health monitoring system for an aircraft turbofan engine has been developed in this paper. With health parameters regarded as state variables, the estimation of degradation is in  this paper. With health parameters regarded as state variables, the estimation of degradation is in fact a fact  a  matter  state  observation.  The  convergence  the  error  system  in  uncertainty‐existing  matter of state of  observation. The convergence of the errorof  system in uncertainty-existing circumstances circumstances  is  demonstrated,  and  a  robust  SMO  has  been  described  aiming  at  estimating  the  is demonstrated, and a robust SMO has been described aiming at estimating the degradations of health degradations  of  health  parameters  in  engine  gas‐path,  where  modeling  uncertainties  are  parameters in engine gas-path, where the modeling uncertainties arethe  considered. A set of simulations considered.  A  set  of  simulations  have  been  conducted  on  the  CLM,  and  a  comparison  to  the  have been conducted on the CLM, and a comparison to the traditional KF-based health monitoring traditional  KF‐based  health  monitoring  system  has  proven  the  ascendency  of  the  proposed  SMO.  system has proven the ascendency of the proposed SMO. Results have shown a range of degradation Results  have  shown  a  range  of  degradation  cases  can  be  faithfully  detected  and  estimated  with  cases can be faithfully detected and estimated with suitable accuracy and quick diagnosis speed by the suitable accuracy and quick diagnosis speed by the described observer.  described observer. Acknowledgments: We We are are grateful grateful  financial  support  of National the  National  Nature  Science  Foundation  of  Acknowledgments: forfor  thethe  financial support of the Nature Science Foundation of China China (No. 51276087) and Jiangsu Province Nature Science Foundation (No. BK20130802).  (No. 51276087) and Jiangsu Province Nature Science Foundation (No. BK20130802). Author Contributions: Xiaodong Chang and Jinquan Huang conceived and designed the algorithm; Feng Lu and Author Contributions: Xiaodong Chang and Jinquan Huang conceived and designed the algorithm; Feng Lu  Xiaodong ChangChang  wrote wrote  the program and performed the simulations; Jinquan Huang and Huang  Haobo Sun the and  Xiaodong  the  program  and  performed  the  simulations;  Jinquan  and analyzed Haobo  Sun  data; Xiaodong Chang wrote the paper. analyzed the data; Xiaodong Chang wrote the paper.  Conflicts of Interest: The authors declare no conflict of interest. The founding sponsors had no role in the design Conflicts of Interest: The authors declare no conflict of interest. The founding sponsors had no role in the design  of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the  decision to publish the results. decision to publish the results. 

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Notations LPC HPC HPT LPT NL NH h h1 h2 h3 h4 h5 h6 h7 h8 Wf A8 P25 T25 P3 T3 EGT

Low pressure compressor High pressure compressor High pressure turbine Low pressure turbine Low pressure rotor speed High pressure rotor speed Health parameter vector LPC efficiency HPC efficiency HPT efficiency LPT efficiency LPC flow capacity HPC flow capacity HPT flow capacity LPT flow capacity Fuel flow rate Nozzle area HPC inlet pressure HPC inlet temperature Combustor inlet pressure Combustor inlet temperature Outlet temperature of 1st stage inlet guide vane of LPT

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7. 8. 9. 10. 11.

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