NASA/TM--2003-212111 ARL-TR-2899

U.S. ARMY

RESEARCH LABORATORY

Kalman Filtering With Inequality Constraints

for Turbofan Engine Health Estimation

Dan Simon

Cleveland State University, Cleveland, Ohio

Donald L. Simon

U.S. Army Research Laboratory, Glem_ Research Center, Cleveland, Ohio

February 2003

https://ntrs.nasa.gov/search.jsp?R=20030018910 2018-05-15T15:38:17+00:00Z
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NASA/TM--2003-21_21_11. ARL-TR-2899

U.S. ARMY

RESEARCH LABORATORY

Kalman Filtering With Inequality Constraints

for Turbofan Engine Health Estimation

Dan Simon

Cleveland State University, Cleveland, Ohio

Donald L. Simon

U.S. Army Research Laboratory, Glem_ Research Center, Cleveland, Ohio

National Aeronautics and

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February 2003
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Available electronically at http://gltrs.grcnasa.gov
Kalman Filtering with Inequality Constraints

for Turbofan Engine Health Estimation

[)an Simon

Cleveland State University

Cleveland, Ohio 44115

Donald L. Simon

U.S. Army Research Laboratory

Glenn Research Center

Cleveland, Ohio 44135

Abstract

Kalman filters are often used to estimate the state variables of a dynamic system.

However, in the application of Kalman filters some known signal information is often

either ignored or dealt with heuristically. For instance, state variable constraints

(which may be based on physical considerations) are often neglected because they

do not fit easily into the structure of the Kalman filter. This paper develops two analytic methods of incorporating state variable inequality constraints in the Kalman

filter. The first method is a general technique of using hard constraints to enforce

inequalities on the state variable estimates. The resultant filter is a combination of a

standard Kalman filter and a quadratic programming problem. The second method

uses soft constraints to estimate state variables that are known to vary slowly with

time. (Soft constraints are constraints that are required to be approximately satisfied rather than exactly satisfied.) The incorporation of state variable constraints

increases the computational effort of the filter but significantly improves its estimation accuracy. The improvement is proven theoretically and shown via simulation

*Corresponding author. This work was supported in part by a NASA/ASEE Surnrner

Faculty Fellowship.

NASA/T_2003-212111 1
results. The use of the algorithm is demonstrated on a linearized simulation of a

turbofan engine to estimate health parameters. The turbofan engine model contains 16 state variables, 12 measurements, and 8 component health parameters. It

is shown that the new algorithms provide improved performance in this example

over unconstrained Kalman filtering.

1 Introduction

For linear dynamic systems with white process and measurement noise, the Kalman

filter is known to be an optimal estimator. However, in the application of Kalman

filters there is often known model or signal information that is either ignored or

dealt with heuristically [1]. This paper presents two ways to generalize the Kalman

filter in such a way that known inequality constraints among the state variables are

satisfied by the state variable estimates.

The first method presented here for enforcing inequality constraints on the state

variable estimates uses hard constraints. It is based on a generalization of the approach presented in [2], which dealt with the incorporation of state variable equality

constraints in the Kalman filter. Inequality constraints are inherently more complicated than equality constraints, but standard quadratic programming results can be

used to solve the Kalman filter problem with inequality constraints. At each time

step of the constrained Kalman filter, we solve a quadratic progrmnming problem

to obtain the constrained state estimate. A family of constrained state estimates is

obtained, where the weighting matrix of the quadratic programming problem determines which family member forms the desired solution. It is stated in this paper,

on the basis of [2], that the constrained estimate has several important properties.

The constrained state estimate is unbiased and has a smaller error covariance than

the unconstrained estimate. We show which member of all possible constrained solutions has the smallest error covariance. We also show the one particular member

that is always (i.e., at each time step) closer to the true state than the unconstrained

estimate.

NASA/T_2003-212111 2
The second method for enforcing inequality constraints uses soft constraints

via a penalty term in the Kalman filter optimization problem. This prevents the

state estimate from changing too rapidly. It essentially smooths the unconstrained

Kalman filter estimate when the state variables are known to vary slowly with time.

It is shown that the constrained state estimate is unbiased, approaches the unconstrained estimate as time approaches infinity, and (under certain special conditions)

is equal to the running average of the unconstrained estimate.

The application considered in this paper is turbofan engine health parameter

estimation [3]. The performance of gas turbine engines deteriorates over time. This

deterioration reduces the fuel economy of the engine. Airlines periodically collect

engine data in order to evaluate the health of the engine and its components. The

health evaluation is then used to determine maintenance schedules. Reliable health

evaluations are used to anticipate future maintenance needs. This offers the benefits

of improved safety and reduced operating costs. The money-saving potential of such

health evaluations is substantial, but only if the evaluations are reliable. The data

used to perform health evaluations are typically collected during flight and later

transferred to ground-based computers for post-flight analysis. Data are collected

each flight at the same engine operating points and corrected to account for variability in ambient conditions. Typically, data are collected for a period of about

3 seconds at a rate of about 10 or 20 Hz. Various algorithms have been proposed

to estimate engine health parameters, such as weighted least squares [4], expert

systems [5], Kalman filters [6], neural networks [6], and genetic algorithms [7].

This paper applies constrained Kalman filtering to estimate engine component

efficiencies and flow capacities, which are referred to as health parameters. We can

use our knowledge of the physics of the turbofan engine in order to obtain a dynmnic

model [8, 9]. The health parameters that we try to estimate can be modelled as

slowly varying biases. The state vector of the dynamic model is augmented to include

the health parameters, which are then estimated with a Kalnmn filter [10]. The

model formulation in this paper is similar to previous NASA work [11]. However, [11]

was limited to a 3-state dynamic model and 2 health parameters, whereas this

NASA/T_2003-212111 3
presentworkincludesamorecomplete16-statemodeland8healthparameters.In

addition,wehavesomea priori knowledge of the engine's health parameters: we

know that they never improve. Engine health always degrades over time, and we can

incorporate this information into state constraints to improve our health parameter

estimation. (This is assuming that no maintenance or engine overhaul is performed.)

This is similar to the probabilistic approach to turbofan prognostics proposed in [12].

It should be emphasized that in this paper we are confining the problem to the

estimation of engine health parameters in the presence of degradation only. There

are specific engine cases that can result in abrupt shifts in filter estimates, possibly

even indicating an apparent improvement in some engine components. An actual

engine performance monitoring system would need to include additional logic to

detect and isolate such faults.

Section 2 presents a discussion of the standard discrete time Kalman filter. Some

important properties of the Kalman filter that will be used later in this paper are

also reviewed. Section 3 generalizes the results of [2] to hard inequality constraints.

This inequality-constrained Kalman filter has several attractive theoretical properties, including state variable estimates that are unbiased, an estimation error variance smaller than the unconstrained filter, and a time-domain estimation error that

is always smaller than the unconstrained estimation error. Section 4 extends the

standard Kalman filter in a different way for those cases where it is known that the

state variables change slowly with time. This constraint is enforced by finding a new

state estimate that is "close" to the unconstrained estimate in some sense, but that

is slowly time varying. It is shown that this new estimate is unbiased, approaches

the unconstrained estimate as time goes to infinity, and (under certain conditions)

is equal to the running average of the unconstrained estimate.

Section 5 discusses the problem of turbofan health parameter estimation, along

with the dynamic model that we used in our simulation experiments. Although the

health parameters are not state variables of the model, it is shown how the dynamic

model can be augmented in such a way that a Kalman filter can estimate the health

parameters [10, 11]. We then show how this problem can be expressed in such

a way to be compatible with the constraints discussed in the preceding sections.

NASA/T_2003-212111 4
Section6 presentssomesimulationresultsbasedon a turbofanmodellinearized

arounda knownoperatingpoint. Weshowthat the Kalmanfilter canestimate

healthparameterswith anaverageerrorof lessthan 0.2%,and the constrained

Kalmanfiltersperformbetter than the unconstrainedfilter. Section7 presents

someconcludingremarksandsuggestionsforfurtherwork.

2 Kalman Filtering

This section reviews standard (unconstrained) state estimation via the Kalman filter

and some important properties of the filter that will be used later in this paper. The

results and notation are taken from [13]. Consider the discrete linear time-invariant

system given by

:Ck+l = A:ck +Buk +wk (1)

Yk = C:ck+ek

where k is the time index, x is the state vector, u is the known control input, y

is the measurement, and {wk} and {ek} are noise input sequences. The problem

is to find an estimate xk+l of xk+l given the measurements {Yo, Yl,"" ,Yk}. We

will use the symbol Yk to denote the column vector that contains the measurements

{Y0, Yl,"', Yk}. We assume that the following standard conditions are satisifed.

E[xo] = xo (2)

E[_] = E[_] = o (3)

E[(xo - xo)(xo- xo)_] = ro (4)

E[_] = R__,_ (6)

E[_] = E[:_] = E[:_] = 0 (7)

where E[.] is the expectation operator, x is the expected value of x, and 5k_,_ is the

Kronecker delta function (5k_,_ = 1 if k = m, 0 otherwise). Q and R are positive

semidefinite covariance matrices. The Kalman filter equations are given by

Kk = AEkcT(cEkc T + _)-1 (S)

NASA/T_2003-212111 5
Xkd-1 = Agk + Buk + Kk(yk - Cgk) (9)

2k+l : (AEk - I(kCEk)A r + Q (]0)

where the filter is initialized with _?o= xo, and Eo given above. It can be shown [13]

that the Kalman filter has several attractive properties. For instance, if Xo, {wk},

and {ek} are jointly gaussian, the Kalman filter estimate _?k+l is the conditional

mean of Xk+l given the measurements Yk; i.e., _?k+l = E[Xk+llYk]. Even if xo,

{wk}, and {ek} are not jointly gaussian, the Kalman filter estimate is the best affine

estimator given the measurements Yk; i.e., of all estimates of Xk+l that are of the

form FYk + g (where F is a fixed matrix and g is a fixed vector), the Kalman filter

estimate is the one that minimizes the variance of the estimation error. It can be

shown [13, pp. 92 ff.] that the Kalman filter estimate (i.e., the minimum variance

estimate) can be given by

Xk-[-1 : Xk-[-1 _ Xk-[-1 _- ExyE]_(Yk - Yk) (11)

where xkd_1 is the mean of xkd-1, Y_xy is the variance matrix of xkd- 1 and Yk, Y_yy

is the covariance matrix of Yk, and Xk+l is the conditional mean of Xk+l given

the measurements Yk. In addition, from [13, p. 93] we know that the Kalman

filter estimate _?k+l and Yk are jointly gaussian, in which case _?k+l is conditionally

gaussian given Yk. The conditional probability density function of Xk+l given Yk is

p(xly) = exp[-(x - x)Ty_-l(x -- x)/2]

(2_r),,_/2IEI1/2 (12)

where n is the dimension of x mad

-1

E = Exx - ExyEyy Eyx (13)

The Kalman filter estimate is that value of x that maxinfizes the conditional probability density function P(xlY), and E is the covariance of the Kalman filter estimation error.

NASA/T_2003-212111 6
3 Kalman Filtering with Hard Inequality Constraints

This section extends the well known results of the previous section to cases where

there are known linear inequality constraints among the state components. Also,

several important properties of the constrained filter are discussed. Consider the

dynamic system of (1) where we are given the additional constraint

Dxk _<dk (14)

where D is a known s × n constant matrix, s is the number of constraints, n is the

number of state variables, and s < n. It is assumed in this paper that D is full

rank, i.e., that D has rank s. This is an easily satisfied assumption. If D is not full

rank that means we have redundant state constraints. In that case we can simply

remove linearly dependent rows from D (i.e., remove redundant state constraints)

until D is full rank. Three different approaches to the constrained state estimation

problem are given in this section. The time index k is omitted in the remainder of

this section for ease of notation.

3.1 The Maximum Probability Method

In this section we derive the constrained Kalman filtering problem by using a maximum probability method. From [13, pp. 93 ff.] we know that the Kalman filter

estimate is that value of x that maximizes the conditional probability density function P(xlY), which is given in (12). The constrained Kalman filter can be derived

by finding an estimate J: such that the conditional probability P(5:IY ) is maximized

and _ satisfies the constraint (14). Maximizing P(Y_IY) is the same as maximizing

its natural logarithm. So the problem we want to solve can be given by

maxlnP(Y_lY ) _ nfin(Y_- x)TE-I(Y_-- x) (15)

09

such that DY_ < d

NASA/T_2003-212111 7
Usingthe factthat theunconstrainedstateestimate2 = x (the conditional mean

of x), we rewrite the above equation as

I_n(2TE-12 -- 22TE-12) such that D2 _< d (16)

Note that this problem statement depends on the conditional gaussian nature of 2,

which in turn depends on the gaussian nature of x0, {w_}, and {e_} in (1).

3.2 The Mean Square Method

In this section we derive the constrained Kalman filtering problem by using a mean

square minimization method. We seek to minimize the conditional mean square

error subject to the state constraints.

n_nE(][x - 2][2[Y) such that D2 < d (17)

where ][. ][ denotes the vector two-norm. If we assume that :r and Y are jointly

gaussian, the mean square error can be written as

Z(llx- 21121Y) = f(x- 2)r(x-- 2)P(xlY)dx (18)

Noting that the Kalman filter estimate is the conditional mean of x, i.e.,

2 = f xP(xlY )& (20)

we formulate the first order conditions necessary for a minimum as

_n(2% - 22%) such that D2 _<_ (21)

X

Again, this problem statement depends on the conditional gaussian nature of 2,

which in turn depends on the gaussian nature of x0, {w_}, and {e_} in (1).

3.3 The Projection Method

In this section we derive the constrained Kalman filtering problem by directly projecting the unconstrained state estimate 2 onto the constraint surface. That is, we

NASA/T_2003-212111 8
solvetheproblem

i,On(_ -- _)rw(_ -- _) such that D_ _<_ (22)

X

where W is any symmetric positive definite weighting matrix. This problem can be

rewritten as

i_n(_W_ - 2_W_) such that D_ _< d (23)

The constrained estimation problems derived by the maximum probability method (16)

and the mean square method (21) can be obtained fl'om this equation by setting

W = E -1 and W = I respectively. Note that this derivation of the constrained

estimation problem does not depend on the conditional gaussian nature of _; i.e.,

z0, {wk}, and {ek} in (1) are not assumed to be gaussian.

3.4 The Solution of the Constrained State Estimation

Problem

The problem defined by (23) is known as a quadratic programming problem [14, 15].

There are many algorithms for solving quadratic programming problems, almost all

of which fall in the category know1 as active set methods. An active set method

uses the fact that it is only those constraints that are active at the solution of the

problem that are significant in the optimality conditions. Assume that t of the s

inequality constraints are active at the solution of (23), and denote by/) and cl the t

rows of D and t elements of d corresponding to the active constraints. If the correct

set of active constraints was known a priori then the solution of (23) would also be

a solution of the equality-constrained problem

nfin(_TW_ - 2_TW_) such that/)Y: = d (24)

This shows that the inequality constrained problem defined by (23) is equivalent to

the equality-constrained problem defined by (24). The equality-constrained problem

was discussed in [2], and so those results can be used to investigate the properties

of the inequality-constrained problem.

NASA/T_2003-212111 9
3.5 Properties of the Constrained State Estimate

In this section we examine some of the statistical properties of the constrained

Kalman filter. We use _? to denote the state estimate of the unconstrained Kalman

filter, and _ to denote the state estimate of the constrained Kalman filter as given

by (23), recalling that (16) and (21) are special cases of (23).

Theorem 1 The solution Yc of the constrained state estimation problem given by (23)

is an unbiased state estimator for the system (1) for any symmetric positive definite

weighting matrix W. That is,

E(_) = E(z) (25)

Theorem 2 The solution _ of the constrained state estimation problem given by (23)

with W = E -1, where E is the covariance of the unconstrained estimate given in (10)

and (13), has an error covariance that is less than or equal to that of the unconstrained state estimate. That is,

Cov(x- _) __Cov(x - _) (26)

At first this seems counterintuitive, since the standard Kalman filter is by definition

the minimum variance filter. However, we have changed the problem by introducing

state variable constraints. Therefore, the standard Kalman filter is no longer the

minimum variance filter, and we can do better with the constrained Kalman filter.

Theorem 3 Among all the constrained Kalman filters resulting from the solution

of (23), the filter that uses W = E -1 has the smallest estimation error covariance.

That is,

Cov(_-l) __ Cov(_w) for' all W (27)

Theorem 4 The solution 2 of the constrained state estimation problem given by (23)

with W = I satisfies the inequality

IIx_- _11-<IIx_- _11 for all k (28)

whe_'e I1 II is the vector' two-no_'_ and _ is the unconstrained Kalman filter estimate.

NASA/T_2003-212111 10
Theorem 5 The error of the solution Yc of the constrained state estimation problem

given by (23) with W = I is smaller than the unconstrained estimation error in the

sense that

_T'[Cov(_)] _< _T'[Cov(_)] (29)

where Tr[.] indicates the trace of a rnatriz, and Coy(.) indicates the covariance rnatriz

of a random vector.

The above theorems all follow from the equivalence of (23) and (24), and the

proofs presented in [2]. We note that if any of the s constraints are active at the

solution of (23), then strict inequalities hold in the statements of Theorems 2 5. The

only time that equalities hold in the theorems is if there are no active constraints at

the solution of (23); that is, if the unconstrained Kalman filter satisfies the inequality

constraints.

4 Kalman Filtering with Soft Inequality Constraints

In this section we are interested in obtaining a Kalman filter-based state estimate

for state variables which we know a priori vary slowly with time. Since we are

concerned with using the Kalman filter as a parameter estimator, we will assume for

this problem that the A matrix in (1) is the identity matrix and the B matrix is zero.

With this in mind, we can use the results of the previous section, especially (22), to

formulate a Kalman filter-based estimate as follows

min(2k - 3ck)Tw(jck - d:k) such that 2{i} varies slowly (30)

zk

where, as before, W is a constant symmetric positive definite weighting matrix. This

is a type of regularization; that is, some additional structure is incorporated into

the Kalman filter estimate [16, 17, 18]. The above problem can be formulated as

II_lin[(xk -- ffgk)Tw(ffgk -- ffgk) q- (ffgk -- ffgk-1)Tvk(ffgk -- xk-1)] (31)

zk

NASA/T_2003-212111 11
whereVk is a (possibly time-varying) symmetric positive definite weighting matrix

that balances the desire for a close approximation to 5: and smooth estimate 5:. The

solution to the above problem is

5:0 = E[x0] (32)

5:k = (w + Vk)-l(w5:k + Vkh:k-1)

Since W and Vk are both positive definite, we know that (W + Vk) -1 exists.

Theorem 6 Assume (as stated above) that A = I and B = 0 in (1). Then the

solution 5: of the constrained state estimation problem given by (32) is an unbiased

state estimator for the system (1) for any symmetric positive definite weighting

matrices W and Vk. That is,

E(5:) = E(x) (33)

Proof: The theorem can be proven by induction. Since A = I and B = 0 we know

that E[x d = x0 for all k. We therefore know from (32) that 5:0 = x0. From (32)

with k = 0 we see that E[5:1] = x0. We repeat this process to show that E[5:k] =

E[5:k] = x0 for all k.

QED

Theorem 7 Assume (as stated above) that A = I and B = 0 in (1). Further

assume that wk = 0 in (1) (since we are trying to estimate constant parameters).

Then the constrained state estimate 5: approaches the unconstrained estimate 5: in

the limit as time goes to infinity. That is,

lira 5:k = lira 5:k (34)

k--_ k--_

Proof." We see fi'om (8) (10) that, under the conditions stated here, /(k --_ 0 as

k _ _. Therefore 5:k approaches a constant value as k _ _. From (32) we see

that, in steady state

5: = (w + vk)-l(ws: + vks:)

= [i- (w+ _)-lvd-l(w+ vk)-lws:

= g + w-lvk)(w + _)-1w5:

(35)

NASA/T_2003-212111 12
wherethe lastequalityfollowsfrom the matrix inversionlemma.Premultiplying

bothsidesof theaboveequationby W we obtain WE = W_?, so if W is invertible

(which it is, since we are assuming in this section that W is positive definite), we

obtain 2 = _? (in steady state). Note that the theorem is true even if Vk does not

approach a steady state value as k --_ oc.

QED

Theorem 8 If Vk = (k - 1)W in (32) then _k is the runnin 9 average of _k.

Proof." The running average of _?k is defined as

] k

= # Z (36)

i 1

which implies that

1

Xk+l -- k -_- I (Xk+I -_- kXk)

Now if Vk = (k - 1)W then (32) shows that

(37)

ff:k+l : [(/_ _- ])W]-I(w:Ek+I _-/_Wff:k) (38)

1

-- k -_- ] (2k+l -_- k_2k)

which is exactly the running average shown in (37).

QED

5 Turbofan Engine Health Monitoring

Figure 1 shows a schematic representation of a turbofan engine. A single inlet

supplies airflow to the fan. Air leaving the fan separates into two streams: one

stream passes through the engine core, and the other stream passes through the

annular bypass duct. The fan is driven by the low pressure turbine. The air passing

through the engine core moves through the compressor, which is driven by the high

pressure turbine. Fuel is injected in the main combustor and burned to produce

hot gas for driving the turbines. The two air streams combine in the augmentor

NASA/T_2003-212111 13
duct,whereadditionalfuelisaddedto furtherincreasetheairtemperature.Theair

leavesthe augmentorthroughthenozzle,whichhasa variablecrosssectionarea.

Variousturbofansimulationpackageshavebeenproposedovertheyears[19,20,

21].Thismodelisbasedonagasturbineenginesimulationsoftwarepackagecalled

DIGTEM (DigitalTurbofanEngineModel)[8,22].DIGTEMis writtenin Fortran

andincludes16statevariables.It usesa backwarddifferenceintegrationscheme

becausetheturbofanmodelcontainstimeconstantsthat differbyupto fourorders

of magnitude.

The nonlinearequationsusedin DIGTEM canbefoundin [8,9]. Thetimeinvariantequationscanbesummarizedasfollows.

5: = f(22,u,p) @Wl(t) (39)

V = .V(22,_,V)+_(t)

22 is the 16-element state vector, u is the 6-element control vector, p is the 8-element

vector of health parameters, and y is the 12-element vector of measurements. The

noise term wl(t) represents inaccuracies in the model, and e(t) represents measurement noise. The elements in these vectors are summarized in Tables 14, along with

their values at the nominal operating point (220, u0,/90, Y0) considered in this paper.

Table 4 also shows typical signal-to-noise ratios for the measurements, based on

NASA experience and previously published data [23]. Sensor dynamics are assumed

to be high enough bandwidth that they can be ignored in the dynamic equations [23].

Equation (39) can be linearized about the nominal operating point by using the first

order approximation of the Taylor series expansion

f(22, u,p) _-. f(220,uo,Po) + (40)

0f(.) (22 _ 220) + Of(.) (u - uo) + Of(.)_

022 07£ " _VP--PO) +Wl(t)

.q(22,,_,p)_ .q(220,,_o,po) + 0g¢.)(22_ 220)"+ og¢.)(,__ ,_o)"+ og(.)_ _(t) 022 " Ou _ _p- po) +

Therefore, a linear small signal system model can be defined for small excursions

fl'om the nominal operating point.

NASA/T_2003-212111 14
Wenotethat

&b= :b- :_0 = AI&r + B6u + A2@ + wl(t) (41)

6y = y - Yo = C16x + D6u + C28p + e(t)

of

A1 -

c_x

A1 (i, j)

Ax(j)

(42)

Similar equations hold for the A2, C1, and C2 matrices. We obtained numerical

approximations to the A1, A2, C1, and C2 matrices by varying x and p fl'om their

nominal values (one element at a time) and recording the new :b and y vectors in

DIGTEM.

Turbofan engine health monitoring is typically a two-step process [3]. In the

first step, engine data is collected each flight at the same engine operating points

and corrected to account for variability in ambient conditions. Data are typically

collected for a period of about 3 seconds per flight at a rate of about 10 or 20 Hz. In

the second step, the data are transferred to ground-based computers for post-flight

analysis to determine engine health.

The goal of our turbofan engine health monitoring problem is to obtain an

accurate estimate of @, which varies slowly with time. We therefore assume that

@ is constant between measurement times. We also assume that the control input

is perfectly known, so 6u = 0. This gives us the following equivalent discrete time

system [24, pp. 90 ff.].

&ck+l = AldS:ck + A2d@k + wlk (43)

5Yk = ClSzk + C2@k + ek

where A_d = exp(A_T) and A2d = A_ _(A_d -- I)A2 (assuming that A_ is invertible,

which it is in our problem). We next augment the state vector with the health

parameter vector [11] to obtain the system equation

NASA/T_2003-212111 15
6pk+l z laid0 A2d] I 6pk w2k

(44)

where w2k is a small noise term (uncorrelated with wlk) that represents model

uncertainty and allows the Kalman filter to estimate time-varying health parameter

variations. The discrete time small signal model can be written as

t_Pk+ 1 (_Pk -I- Wk (45)

6pk + ek

where the definitions of A and C are apparent from a comparison of the two preceding equations. Now we can use a Kalman filter to estimate 6xk and 6pk. Actually,

we are only interested in estimating 6pk (the health parameter deviations), but the

Kalman filter gives us the bonus of also estimating 6xk (the exem'sions of the original

turbofan state variables).

It is known that health parameters do not improve over time. That is, 6p(1),

6p(2), 6p(3), 6p(4), 6p(6), and 6p(8) are always less than or equal to zero and

always decrease with time. Similarly, 6p(5) and 6p(7) are always greater than or

equal to zero and always increase with time. In addition, it is known that the health

parameters vary slowly with time. As an example, since 6p(1) is the constrained

estimate of 6p(1), we can enforce the following constraints on 6p(1).

&(1) _< 0 (46)

tSpk+l(1 ) < _k(1) q-T +

_k+l(1) _ _k(1)--"/1

where 71+ and 71 are nonnegative factors chosen by the user that allows the state

estimate to vary only within prescribed limits. Typically we choose 71 > 71+ so that

the state estimate can change more in the negative direction than in the positive

NASA/T_2003-212111 16
direction.Thisisin keepingwith our a priori knowledge that this particular state

variable never increases with time. Ideally we would have 7 + = 0 since @(1) never

increases. However, since the state variable estimate varies around the true value of

the state variable, we choose 71+ > 0. This allows some time-varying increase in the

state variable estimate to compensate for a state variable estimate that is smaller

than the true state variable value.

These constraints are linear and can therefore easily be incorporated into the

form required in the constrained filtering problem statement (14). Note that this

does not take into account the possibility of abrupt changes in health parameters

due to discrete damage events. That possibility must be addressed by some other

means (e.g., residual checking [3]) in conjuction with the methods presented in this

paper.

6 Simulation Results

We simulated the methods discussed in this paper using MATLAB. We simulated

a steady state 3 second burst of engine data measured at 10 Hz during each flight.

Each of these routine services was performed at the single operating point shown

in Tables 1 4. The signal-to-noise ratios were determined on the basis of NASA

experience and previously published data [23] and are shown in Table 4. We used a

one-sigma process noise in the Kalman filter equal to 1% of the nominal state values

to allow the filter to be responsive to changes in the state variables. We set the one

sigma process noise for each component of the health parameter portion of the state

derivative equation to 0.01% of the nominal parameter value. This was obtained by

tuning. It was small enough to give reasonably smooth estimates, and large enough

to allow the filter to track slowly time-varying parmneters. For the filter with hard

constraints, we chose the 7 variables in (46) such that the maximum allowable rate

of change in (Sp was a linear 9% per 500 flights in the direction of expected change,

and 3% per 500 flights in the opposite direction. The true health parameter values

never change in a direction opposite to the expected change. However, we allow

the state estimate to change in the opposite direction to allow the Kalman filter

NASA/T_2003-212111 17
to compensateforthefactthat thestateestimatemightbeeithertoo largeor too

small.Wesettheweightingmatrix W in (23) and (31) equal to E -1 in accordance

with Theorem 3. We found by experimenting that setting the weighting matrix Vk

in (31) equal to 120W resulted in good performance for the Kalman filter with soft

constraints.

The first test case we simulated was a linear degradation of the first health

parameter (fan airflow) over 500 flights, while the other seven health parameters

remained constant. Figure 2 shows the Kalman filters' performances in this case. We

ran eight simulations like this. In each simulation, one of the eight health parameters

degraded linearly by a factor of 3% during the course of the simulation, while the

other seven health parameters remained constant. The 3% degradation over 500

flights is in line with turbofan performance data collected by NASA and reported in

the literature [25]. Each of the eight cases exhibit performance similar to Figure 2.

Table 5 shows the performance of the filters averaged over all eight simulations.

All of the filters estimate the health parameters to within less than 0.2% of their

nominal values. It can be seen that (on average) the filter with soft constraints

offers an 11% improvement over the unconstrained filter, and the filter with hard

constraints offers a 22% improvement over the unconstrained filter. These numbers

should not be interpreted as having any statistical signficance (due to our limited

sample size of eight cases) but they do show the improvement that is possible with

constrained Kalman filters. Table 5 also shows that a couple of health parameters

(fan airflow and LPT airflow) were actually estimated better with the unconstrained

filter than with the constrained filter. We therefore see that the constrained filter

does not guarantee better estimation in every individual sample run, but it does

guarantee better performance statistically.

The next scenario we considered was the case where all eight health parameters degrade at the same time. We simulated a degradation over 500 flights of

-1% for fan airflow, -2% for fan efficiency, -3% for compressor airflow, -2% for

compressor efficiency, +3% for high pressure turbine airflow, -2% for high pressure

turbine enthalpy change, +2% for low pressure turbine airflow, and -1% for low

pressure turbine enthalpy change. This is summarized in Table 6. Figure 3 shows

NASA/T_2003-212111 18
the performanceof theKalmanfiltersin this case.Table7showsthe performance

of the filtersaveragedover16simulationslike this (eachsimulationbeingsubject

to adifferentrandomnoisehistory).It canbeseenthat (onaverage)thefilter with

softconstraintsoffersa9%improvementovertheunconstrainedfilter_andthefilter

with hardconstraintsoffersa 38%improvementovertheunconstrainedfilter. As

mentionedabove_thesenmnbersshouldnot beinterpretedashavinganystatistical signficance(dueto ourlinfitedsamplesizeof 16cases)but theydo showthe

improvementthat ispossiblewith constrainedKalmanfilters.

The improvedperformanceof the constrainedfilterscomeswith a price_and

that priceis computationaleffort. Thefilter with soft constraintsrequiresonly

slightly (14%)morecomputationaleffortthan the unconstrainedfilter_but the

filter with hard constaintsrequiresaboutfour timesthe computationaleffortof

the unconstrainedfilter. Thisis becauseof the additionalquadraticprogramnfing

problemthat isrequiredfor hardconstraints.However_computationaleffortis not

acriticalissuefortheparticularapplicationof turbofanhealthestimationsincethe

filteringis performedonground-basedcomputersaftereachflight.

7 Conclusion and Discussion

We have presented two methods for incorporating linear state inequality constraints

in a Kalman filter. The first method incorporated hard constraints into the Kalman

filter to maintain the state variable estimates within a user-defined envelope. The

second method incorporated soft constraints into the Kalman filter to ensure that

the state variable estimates vary slowly with time. The simulation results demonstrate the effectiveness of these methods, particularly for turbofan engine health

estimation.

If the system whose state variables are being estimated has known state variable

constraints, then those constraints can be incorporated into the Kalman filter as

shown in this paper. However, in practice, the constraints enforced in the filter

might be more relaxed than the true constraints. This allows the filter to correct

state variable estimates in a direction that the true state variables might never

NASA/T_2003-212111 19
change.Thisis a departurefrom strict adherenceto theory,but in practicethis

improvesthe performanceof the filter. This is animplementationissuethat is

conceptuallysimilarto tuningastandardKalmanfilter.

It wasseenin Theorem2 that the filter with hardconstraintshasa smaller

estimationerrorcovariancethantheunconstrainedKalmanfilter.At firstthisseems

counterintuitive,sincethe standardKalmanfilter is by definitionthe minimum

variancefilter. However,wehavechangedtheproblembyintroducingstatevariable

constraints.Therefore,the standardKalmanfilter is not the minimumvariance

filter fortheturbofanenginehealthestimationproblem,andwecandobetterwith

theconstrainedKalmanfilter.

Wesawthat the filter with hardconstraintsrequireda muchlargercomputational effortthanthe standardKalmanfilter. This is dueto the additionof the

quadraticprogrmnmingproblemthat mustbesolvedin the constrainedKalman

filter. Theengineermustthereforeperforma tradeoffbetweencomputationalelfort andestimationaccuracy.For realtimeapplicationsthe improvedestimation

accuracymaynot beworththeincreasein computationaleffort.

It wasseenin Figures2 and 3 that althoughthe constrainedfilters improve

the estimationaccuracy,thegeneraltrendof thestatevariableestimatesdoesnot

changewith theintroductionof stateconstraints.Thisis becausethe constrained

filtersarebasedontheunconstrainedKalmanfilter. Theconstrainedfilter estimates

thereforehavethesameshapeasthe unconstrainedestimatesuntil theconstraints

areviolated,at whichpoint the statevariableestimatesareprojectedontothe

edgeof the constraintboundary.Theconstrainedfilterspresentedin this paper

arenot qualitativelydifferentthanthe standardKalmanfilter; theyarerathera

quantitativeimprovementin thestandardKalmanfilter.

Notethat theKalnmnfilterworkswellonlyif theassumedsystemmodelmatches

reality fairly closely.Themethodpresentedin this paper,by itself,will not work

well if therearelargesensorbiasesor hardfaultsdueto severecomponentfailures.

A mission-critical implementationofaKalmanfiltershouldalwaysincludesomesort

of residualcheckto verifythevalidityof theKalmanfilter results,particularlyfor

theapplicationofturbofanenginehealthestimationconsideredinthispaper[3,26].

NASA/T_2003-212111 20
Althoughwehaveconsideredonlylinearstateconstraints,it isnotconceptually

difficultto extendthis paperto nonlinearconstraints.If thestateconstraintsare

nonlineartheycanbelinearizedasdiscussedin [2].

Furtherworkalongthelinesof thisresearchcouldfocusoncombiningourwork

with [27]in orderto guaranteeconvergencein thepresenceof nonlinearconstraints.

Othereffortscouldexploretheincorporationofstateconstraintsforoptimalsmoothing,ortheuseofstateconstraintsin H_ filtering [28]. Further work could also focus

on integrating the nonlinear simulation logic in DIGTEM [8, 22] with the Kalman

filter to obtain more complete results. This would also allow us to more easily

test the Kalman filter at various operating points without translating data from

DIGTEM to MATLAB.

References

[1] D. Massicotte, R. Morawski, and A. Barwicz, Incorporation of a positivity

constraint into a Kalman-filter-based algorithm for correction of spectrometric

data, IEEE Transactions on Instrumentation and Measurement 44(1) pp. 2-7,

February 1995.

[2] D. Simon and T. Chia, Kalman filtering with state equality constraints, IEEE

Transactions on Aerospace and Electronic Systems, 39(1) pp. 128-136, January

2002.

[3] D. Doel, TEMPER A gas-path analysis tool for commercial jet engines, ASME

Journal of Engineering for Gas Turbines and Power (116) pp. 82-89, Jan. 1994.

[4] D. Doel, An assessment of weighted-least-squares-based gas path analysis,

ASME Journal of Engineering for Gas Turbines and Power (116) pp. 366-373,

April 1994.

[5] H. DePold and F. Gass, The application of expert systems and neural networks

to gas turbine prognostics and diagnostics, ASME Journal of Engineering for

Gas Turbines and Power (121) pp. 607-612, Oct. 1999.

NASA/T_2003-212111 21
[6]A. Volponi,H. DePold,andR. Ganguli,TheuseofKalmanfilter andneural

networkmethodologiesin gasturbineperformancediagnostics:acomparative

study,ProceedingsofASMETurboExpo2000,pp.1-9,May2000.

[7]T. Kobayashiand D.L. Simon, A hybrid neural network-geneticalgorithm techniquefor aircraft engine performancediagnostics,37th

AIAA/ASME/SAE/ASEEJointPropulsionConference, July2001.

[8]C. Daniele,S.Krosel,J.Szuch,andE.Westerkamp, Digitalcomputerprogram

for generatingdynamicturbofanenginemodels(DIGTEM),NASATechnical

Memorandum83446, September 1983.

[9] J. Szuch, S. Krosel, and W. Bruton, Automated procedure for developing hybrid

computer simulations of turbofan engines, NASA Technical Paper 1851, August

1982.

[10] B. Friedland, Treatment of bias in recursive filtering, IEEE Transactions on

Automatic Control AC14(4) pp. 359-367, Aug. 1969.

[11] H. Lambert, A simulation study of tubofan engine deterioration estimation

using Kalman filtering techniques, NASA Technical Memorandum 104233, June

1991.

[12] M. Roemer and G. Kacprzynski, Advanced diagnostics and prognostics for turbine engine risk assessment, IEEE Aerospace Conference, pp. 345-353, March

2000.

[13] B. Anderson and J. Moore, Optimal Filtering (Prentice Hall, Englewood Cliffs,

New Jersey, 1979).

[14] R. Fletcher, Practical Methods of Optimization Volume 2: Constrained Optimization (John Wiley & Sons, New York, 1981).

[15] P. Gill, W. Murray, and M. Wright, Practical Optimization (Academic Press,

New York, 1981).

NASA/T_2003-212111 22
[16]T. Kailath,A. Sayed,andB. Hassibi,LinearEstimation(PrenticeHall,Upper

SaddleRiver,NewJersey,2000).

[17]A. Sayed,A frameworkforstate-spaceestimationwithuncertainmodels,IEEE

TransactionsonAutomaticControl46(7),pp.998-1013,July2001.

[18]J. Tse,J.Bentsman,andN. Miller,Minimaxlongrangeparameterestimation,

IEEEConferenceonDecisionandControl,LakeBuenaVista,Florida,pp.277-

282,December1994.

[19]I. Ismail,andF. Bhinder,Simulationof aircraftgasturbineengines,ASME

JournalofEngineeringfor GasTurbinesandPower(113)1pp. 95-99,1991.

[20]Y. Najjar,Comparisonofmodellingandsimulationresultsfor singleandtwinshaftgasturbineengines,InternationalJournalofPowerandEnergySystems

(18)1pp.29-33,1998.

[21]Z. Xie, M. Su,andS.Weng,Extensibleobjectmodelfor gasturbineengine

simulation,AppliedThermalEngineering21(1),pp.111-118,Jan.2001.

[22]C. DanieleandP.McLaughlin,Thereal-timeperformanceof a parallel,nonlinearsimulationtechniqueappliedto aturbofanengine,in ModelingandSimulationonMicrocomputers:1984(R.Swartz,Ed.)Societyfor ComputerSimulation,pp.167-171,1984.

[23]W. Merrill, Identificationof multivariablehigh-performanceturbofanengine

dynamicsfromclosed-loopdata,AIAA Journalof Guidance,Control,andDynamics(7)6pp. 677-683,Nov.1984.

[24]C. Chen,LinearSystemTheoryandDesign(OxfordUniversityPress,New

York,1999).

[25]O.Sasahara,JT9Dengine/moduleperformancedeteriorationresultsfromback

to backtesting,InternationalSymposiumonAir BreathingEngines,pp. 528-

535,1985.

NASA/T_2003-212111 23
[26]A. Gelb,AppliedOptimalEstimation(MIT Press,Cambridge,Massachusetts,

1974).

[27]J. De Geeter,H. VanBrussel,and J. DeSchutter,A smoothlyconstrained

Kalmanfilter,IEEETransactionsonPatternAnalysisandMachineIntelligence

19(10)pp. 1171-1177, October1997.

[28]D. SimonandH. E1-Sherief,Hybrid Kalman/ Minimax Filtering in PhaseLocked Loops, Control Engineering Practice 4(5) pp. 615-623, October 1996.

NASA/T_2003-212111 24
State Nominal Value

Low Pressure Turbine Rotor Speed

High Pressure Turbine Rotor Speed

Compressor Mass Flow

Combustor Inlet Temperature

Combustor Mass Flow

High Pressure Turbine Inlet Temperature

High Pressure Turbine Mass Flow

Low Pressure Turbine Inlet Temperature

Low Pressure Turbine Mass Flow

Augmentor Inlet Temperature

6140 RPM

9395 RPM

0.457 kg/s

965 K

0.264 kg/s

1593 K

1.48 kg/s

1129 K

1.79 kg/s

790 K

Augmentor Mass Flow

Nozzle Inlet Temperature

Duct Fluid Momentum

Augmentor Fluid Momentum

Duct Mass Flow

Duct Temperature

1.46 kg/s

790 K

53.6 kg/s 2

103 kg/s 2

4.52 kg/s

571 K

Table 1: Turbofan states.

Control Nominal Value

Combustor Fuel Flow

Augmentor Fuel Flow

Nozzle Throat Area

Nozzle Exit Area

Fan Vane Angle

Compressor Van Angle

0.37 kg/s

0 kg/s

430 cm 2

492 cm 2

25 deg

20 deg

Table 2: Turbofan controls.

NASA/T_2003-212111 25
HealthParameter NominalValue

FanAirflow

FanEfficiency

CompressorAirflow

CompressorEfficiency

High PressureTurbineAirflow

High PressureTurbineEnthalpy Change

Low PressureTurbineAirflow

Low PressureTurbineEnthalpyChange

102 kg/s

0.82

48.7 kg/s

0.83

41.0 kg/s

101 J/kg

48.3 kg/s

27.1 J/kg

Table 3: Turbofan health parameters.

Measurement Nominal Value

Low Pressure Turbine Rotor Speed

High Pressure Turbine Rotor Speed

Duct Pressure

Duct Temperature

Compressor Inlet Pressure

Compressor Inlet Temperature

Combustor Pressure

Combustor Inlet Temperature

Low Pressure Turbine Inlet Pressure

Low Pressure Turbine Inlet Temperature

Augmentor Inlet Pressure

Augmentor Inlet Temperature

6140 RPM

9395 RPM

19.0 N/cm 2

571 K

20.5 N/cm 2

577 K

97.5 N/cm 2

965 K

26.8 N/cm 2

1130 K

17.4 N/cm 2

790 K

SNR

150

150

200

100

200

100

200

100

100

70

100

70

Table 4: Turbofan measurements.

NASA/T_2003-212111 26
EstimationError (%)

Health Unconstrained Soft Constrained Hard Constrained

Parameter Filter Filter Filter

FanAirflow

FanEfficiency

CompressorAirflow

CompressorEfficiency

HPT Airflow

HPT Enthalpy Change

LPT Airflow

LPT Enthalpy Change

0.123

0.177

0.145

0.102

0.116

0.093

0.104

0.181

0.105

0.166

0.132

0.086

0.100

0.081

0.090

0.168

0.139

0.113

0.113

0.059

0.I01

0.055

0.109

0.118

Average 0.130 0.116 0.101

Table 5: Kalman filter estimation errors. HPT = High Pressure Turbine,

and LPT = Low Pressure Turbine. The numbers shown are RMS estimation

errors (percent) averaged over eight simulations where each simulation had one

health parameter degradation while the other seven health parameters were

unchanged.

Health True Degradation

Fan Airflow

Fan Efficiency

Compressor Airflow

Compressor Efficiency

HPT Airflow

HPT Enthalpy Change

LPT Airflow

LPT Enthalpy Change

1%

2%

3%

2%

+3%

2%

+2%

1%

Table 6: Health parameter degradation amounts for test scenario.

NASA/T_2003-212111 27
EstimationError (%)

Health Unconstrained Soft Constrained Hard Constrained

Parameter Filter Filter Filter

FanAirflow

FanEfficiency

CompressorAirflow

CompressorEfficiency

HPT Airflow

HPT Enthalpy Change

LPT Airflow

LPT Enthalpy Change

0.129

0.163

0.152

0.101

0.119

0.092

0.104

0.168

0.113

0.149

0.146

0.087

0.114

0.078

0.091

0.155

0.089

0.105

0.103

0.052

0.076

0.050

0.057

0.111

Average 0.128 0.116 0.080

Table 7: Kalman filter estimation errors. HPT = High PressureTurbine,

and LPT = Low PressureTurbine. The numbersshownareRMS estimation

errors (percent)averagedover16 simulations,whereeachsimulation had a

lineardegradationof all eight healthparameters.

NASA/T_2003-212111 28
Inlet Fan

Station: 0 2

High Pressure Low Pressure

Turbine --_ _-- Turbine

_ Augmentor Nozzle

compressor combustor _

T

2.1 2.2 3 4 4.1 5 6 8 F

Figure l" Schematic representation of tuxbofan engine

NASA/T_2003-212111 29
(a) Unconstrained

Kalman filter

-2.5

0 100 200 300

flight number

400 500

(b) Kalman filter

with soft constraints

0.5

-0.5

¢0

._E -1

g

_ -1.5

"o

_ -2

-(3

-2.5

-3

100 200 300

flight number

400 500

(c) Kalman filter with

hard constraints

0.5

0

-2.5

-3 r

100

r r

200 300 400 500

flight number

Figure 2: Kalman filter estimates of health parameters. The true health parameter changes

were a -3% change in the first parameter, and zero change in the other seven parameters.

The true health parameter changes are shown as heavy lines, and the filter estimates are

shown as lighter lines.

NASA/T_2003-212111 30
(a) Unconstrained

Kalman filter

100 200 300 400 500

flight number

(b) Kalman filter

with soft constraints

O 100 200 300 400 500

flight number

(c) Kalman filter with

hard constraints

O 100 200 300 400 500

flight number

Figure 3: Kalman filter estimates of health parameters. The true health parameter changes

were various values in between-3% and +3%. The true health parameter changes are

shown as heavy lines, and the filter estimates are shown as lighter lines.

NASA/T_2003-212111 31
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4, TITLE AND SUBTITLE 5. FUNDING NUMBERS

Kalman Filtering With Inequality Constraints for Turbofan Engine

Health Estimation

6. AUTHOR(S)

[)an Simon and Donald L. Simon

7. PERFORMING ORGANiZATiON NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H_ Glenn Resem'ch Center at Lewis Field

Cleveland, Ohio 44135 - 3191

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U.S. Army Reseai'ch Laboratory ARL-TR-2899

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11. SUPPLEMENTARY NOTES

This work was supported in part by a NASA/ASEE Summer Faculty Fellowship. Dan Simon, Cleveland State

University, Cleveland, Ohio 44115; Donald L. Simon, [ J.S. Army Research Laboratory, NASA Glenn Research

Center. Responsible person, Donald L. Simon, organization code 0300, 216---433----3740.

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13. ABSTRACT (Maximum 200 words)

Kalman filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman

filters some known signal information is often either ignored or dealt with heuristically. For instance, state variable

constraints (which may be based on physical considerations) are often neglected because they do not fit easily into the

structure of the Kalman filter. This paper develops two analytic methods of incorporating state variable inequality

constraints in the Kalman filter. The first method is a general technique of using hard constraints to enfbrce inequalities

on the state variable estimates. The resultant filter is a combination of a standard Kalman filter and a quadratic programruing problem. The second method uses soft constraints to estimate state variables that are known to vary slowly with

time. (Soft constraims are constraints that are required to be approximately satisfied rather than exactly satisfied.) The

incorporation of state variable constraints increases the computational et'ti_rt of the filter but significantly improves its

estimation accuracy. The improvement is proven theoretically and shown via simulation results. The use of the algorithm

is demonstraled on a linearized simulation of a turbofan engine lo estimale health paramelers. The turbofan engine

model contains 16 state variables, 12 measurements, and 8 component health parameters. It is shown that the new

algorithms provide improved performance in this example over unconstrained Kalman filtering.

14. SUBJECT TERMS

Systems health monitoring; Aircr_t engines; Gas tm'bine engines; Kalman filter; State

constraints; Estimation; Quadratic programming

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