Dan Simon

Cleveland State University, Cleveland, Ohio

Donald L. Simon

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

Kalman Filter Constraint Tuning for Turbofan

Engine Health Estimation

NASA/TM—2005-213962

November 2005

ARL–MR–621

U.S. ARMY

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Dan Simon

Cleveland State University, Cleveland, Ohio

Donald L. Simon

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

Kalman Filter Constraint Tuning for Turbofan

Engine Health Estimation

NASA/TM—2005-213962

November 2005

National Aeronautics and

Space Administration

Glenn Research Center

ARL–MR–621

U.S. ARMY

RESEARCH LABORATORY
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7121 Standard Drive

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Available electronically at http://gltrs.grc.nasa.gov
Abstract

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

However, in the application of Kalman  ̄lters 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  ̄t easily into the structure of the Kalman  ̄lter. Recently published work has

shown a new method for incorporating state variable inequality constraints in the

Kalman  ̄lter. The resultant  ̄lter is a combination of a standard Kalman  ̄lter and

a quadratic programming problem. The incorporation of state variable constraints

has been shown to generally improve the  ̄lter's estimation accuracy. However, the

incorporation of inequality constraints poses some risk to the estimation accuracy.

After all, the Kalman  ̄lter is theoretically optimal, so the incorporation of heuristic

constraints may degrade the optimality of the  ̄lter. This paper proposes a way

NASA/TM—2005-213962 1

Kalman Filter Constraint Tuning for

Turbofan Engine Health Estimation

Dan Simon 

Cleveland State University 

Cleveland, Ohio 44115 

Donald L. Simon 

U.S. Army Research Laboratory 

Glenn Research Center 

Cleveland, Ohio 44135 
to tune the  ̄lter constraints so that the state estimates follow the unconstrained

(theoretically optimal)  ̄lter when the con ̄dence in the unconstrained  ̄lter is high.

When con ̄dence in the unconstrained  ̄lter is not so high, then we use our heuristic

knowledge to constrain the state estimates. The con ̄dence measure is based on

the agreement of measurement residuals with their theoretical values. If some measurement residuals are low, and those residuals are highly sensitive to a given state,

then we are con ̄dent that the unconstrained estimate of that state is correct. Otherwise, we incorporate our heur stic knowledge as state constraints. The algorithm

is demonstrated on a linearized simulation of a turbofan engine to estimate engine

health.

1 Introduction

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

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

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

dealt with heuristically [1]. Previous work by the authors [2, 3] resulted in a new

method for incorporating state variable inequality constraints in the Kalman  ̄lter.

This method is based on a generalization of [4], which dealt with the incorporation

of state variable equality constraints in the Kalman  ̄lter. Constraints are enforced

by projecting out-of-bound state estimates onto the contraint surface. Inequality

constraints are inherently more complicated than equality constraints, but standard

quadratic programming techniques can be used to solve the Kalman  ̄lter problem

with inequality constraints. At each time step of the constrained Kalman  ̄lter, we

solve a quadratic programming problem to obtain the constrained state estimate.

It was shown earlier [2, 3] that the constrained estimate has several important

properties. For example, the constrained state estimate is unbiased and has a smaller

error covariance than the unconstrained estimate. Also, the constrained estimate

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

NASA/TM—2005-213962 2

i
estimate. The incorporation of state variable constraints was shown to improve the

 ̄lter's estimation accuracy for turbofan health estimation.

However, these properties of the constrained  ̄lter hold true only if the state

constraints that are enforced are correct. In practice, state constraints are often

based on heuristic knowledge { that is, the constraints are more correctly viewed as

\soft" constraints. The use of inequality constraints therefore poses some risk to the

estimation accuracy. The Kalman  ̄lter is theoretically optimal, so the incorporation

of heuristic constraints is a modi ̄cation to the optimal  ̄lter. We want to be able

to incorporate our heuristic knowledge into the  ̄lter, but we do not have absolute

con ̄dence in our heuristic knowledge.

The constrained  ̄lter is theoretically superior to the unconstrained  ̄lter, but

only if the constraints are accurate. The incorporation of constraints is not always

exact, and some judgment must be used in their de ̄nition. This paper proposes a

way to tune the constraints so that the state estimate is equal to the unconstrained

(theoretically optimal) estimate when the con ̄dence in the unconstrained estimate

is high. When con ̄dence in the unconstrained  ̄lter is not so high, we use our

heuristic knowledge to constrain the state estimates. The con ̄dence measure is

based on the agreement of measurement residuals with their theoretical values. If

some measurement residuals are low, and the measurements corresponding to those

residuals are highly sensitive to a given state, then we are con ̄dent that the unconstrained estimate of that state is correct. Otherwise, we incorporate our heurstic

knowledge as state constraints.

The application considered in this paper is aircraft turbofan engine health parameter estimation [5]. Health parameters represent engine component e±ciencies

and °ow capacities. The performance of a gas turbine engine 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 o®ers the

bene ̄ts of improved safety and reduced operating costs. The money-saving potential of such health evaluations is substantial, but only if the evaluations are reliable.

NASA/TM—2005-213962 3
The data used to perform health evaluations are typically collected during °ight and

later transferred to ground-based computers for post-°ight analysis. Data are collected each °ight at the same engine operating points and corrected to account for

variability in ambient conditions. Various algorithms have been proposed to monitor engine health, such as weighted least squares [6], expert systems [7], Kalman

 ̄lters [8], neural networks [8], and genetic algorithms [9].

This paper applies constrained Kalman  ̄ltering, along with constraint tuning on

the basis of measurement residuals, to estimate engine health parameters. We use

heuristic knowledge of the health parameter dynamics to constrain their estimate.

For example, we know that health parameters 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.) It should be emphasized that in this paper we

are con ̄ning the problem to the estimation of engine health parameters in the

presence of degradation only. There are speci ̄c engine faults that can result in

abrupt shifts in  ̄lter 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.

This paper is organized as follows. Section 2 presents a review of the constrained

Kalman  ̄lter, along with a proposed method for how the residuals can be used for

constraint tuning. Section 2 also shows how a matrix quantifying the sensitivity of

measurements to state variables can be obtained, and how the entries of that matrix

can be used to quantify our con ̄dence in the accuracy of the unconstrained Kalman

 ̄lter estimates. Our con ̄dence can then be used to decide whether or not to enforce

heuristic constraints on the state variable estimates. Section 3 discusses the problem

of turbofan health parameter estimation, along with the dynamic model that we use

in our simulation experiments. Although the health parameters are not state variables of the model, the linearized dynamic model is augmented in such a way that a

Kalman  ̄lter can estimate the health parameters following a previously published

approach [10, 11]. We then show how this problem can be expressed in a way that is

compatible with the constraints discussed in Section 2. Section 4 discusses the apNASA/TM—2005-213962 4
plication of the sensitivity analysis and Kalman  ̄lter constraint tuning technique to

the turbofan engine health parameter estimation problem. Section 5 presents some

simulation results based on a turbofan model linearized around a known operating

point. We show that the constrained Kalman  ̄lter can estimate health parameters

better than the unconstrained  ̄lter, and the addition of constraint tuning further

improves estimation accuracy. Section 6 presents some concluding remarks and

suggestions for further work.

2 Kalman Filtering with Constraint Tuning

In this section we  ̄rst summarize the standard Kalman  ̄lter equations. We then

review constrained state estimation via the Kalman  ̄lter, and propose a method

for residual-based constraint tuning.

Consider the discrete linear time-invariant system given by

x(k + 1) = Ax(k) + Bu(k) + w(k) (1)

y(k) = Cx(k) + e(k)

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

measurement, and fw(k)g and fe(k)g are noise input sequences. The problem is to

 ̄nd an estimate ^x(k + 1) of x(k + 1) given the measurements fy(0); y(1); ¢¢¢ ; y(k)g.

We will use the symbol Y (k) to denote the column vector that contains the measurements fy(0); y(1); ¢¢¢ ; y(k)g. We assume that the following standard conditions

are satisifed.

E[x(0)] = 1x(0) (2)

E[w(k)] = E[e(k)] = 0

E[(x(0) ¡ x1(0))(x(0) ¡ x1(0))T ] = §(0)

E[w(k)wT (m)] = Q±km

E[e(k)eT (m)] = R±km

E[w(k)eT (m)] = 0

NASA/TM—2005-213962 5
where E[¢] is the expectation operator, 1x is the expected value of x, and ±km is the

Kronecker delta function (±km = 1 if k = m, ±km = 0 otherwise). Q and R are

positive semide ̄nite covariance matrices. The Kalman  ̄lter equations are given by

K(k) = A§(k)CT (C§(k)CT + R)

¡1 (3)

x^(k + 1) = Ax^(k) + Bu(k) + K(k)(y(k) ¡ Cx^(k))

§(k + 1) = (A§(k) ¡ K(k)C§(k))AT + Q

where the  ̄lter is initialized with ^x(0) = 1x(0), and §(0) given above. It can be

shown [12] that the Kalman  ̄lter has several attractive properties. For instance, if

x(0), fw(k)g, and fe(k)g are jointly Gaussian, the Kalman  ̄lter estimate ^x(k + 1)

is the conditional mean of x(k + 1) given the measurements Y (k), i.e., ^x(k + 1) =

E[x(k + 1)jY (k)]. Even if x(0), fw(k)g, and fe(k)g are not jointly Gaussian, the

Kalman  ̄lter estimate is the best linear estimator given the measurements Y (k),

i.e., of all estimates of x(k + 1) that are of the form F Y (k) + g (where F is a  ̄xed

matrix and g is a  ̄xed vector), the Kalman  ̄lter estimate is the one that minimizes

the variance of the estimation error. Also, the Kalman  ̄lter estimate ^x(k) is that

value of 3 that maximizes the conditional probability density function P(3jY (k)).

Finally, §(k) is the covariance of the Kalman  ̄lter estimation error at time k.

2.1 Constrained Kalman Filtering

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

D(k)x(k) · d(k) (4)

where D(k) is a known s£n 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(k) is full rank, i.e.,

that D(k) has rank s. This is an easily satis ̄ed assumption. If D(k) is not full rank

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

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

D(k) is full rank. The time index k is omitted in the remainder of this section for

ease of notation.

NASA/TM—2005-213962 6
The problem of  ̄nding a constrained estimate for the state of the system (1)

can be posed in three di®erent ways [2, 3]. Regardless of how we pose the problem,

we want to make sure that our constrained estimate ~x satis ̄es the constraint (4).

That is,

Dx~ · d (5)

The solution to the constrained estimation problem turns out to be the solution to

min

x~ (~x ¡ x^)

TW(~x ¡ x^) such that Dx~ · d (6)

where ^x is the unconstrained (standard) Kalman  ̄lter estimate, and W is a symmetric positive de ̄nite weighting matrix. Note that if the unconstrained estimate

satis ̄es the constraint, then the solution of the above equation is simply ~x = ^x.

That is, if the standard Kalman  ̄lter estimate satis ̄es the constraints, then the

constrained estimate is equal to the unconstrained estimate.

Depending on the particular optimality criterion that is employed, W can take

on several di®erent values [2, 3]. If a mean square error criterion is used then

W = I. If a maximum probability criterion is used then W = §¡1. If a projection

method is used then W is an arbitrary positive de ̄nite matrix. The optimality of

the constrained estimate does not depend on the conditional Gaussian nature of ^x,

i.e., x(0), fw(k)g, and fe(k)g in (1) are not assumed to be Gaussian.

The problem de ̄ned by (6) is known as a quadratic programming problem [13,

14]. There are many algorithms for solving quadratic programming problems, almost

all of which fall in the category known 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 signi ̄cant in the optimality conditions. Assume that t of the s

inequality constraints are active at the solution of (6), and denote by D^ and ^d 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 (6) would also be

a solution of the equality constrained problem

min

x~ (~xTWx~ ¡ 2^xTWx~) such that D^x~ = ^

d (7)

This shows that the inequality constrained problem is equivalent to an equality

constrained problem. The constrained estimate ~x has several attractive properties.

NASA/TM—2005-213962 7
1. The solution ~x of the constrained state estimation problem given by (6) is an

unbiased state estimator for the system (1) for any symmetric positive de ̄nite

weighting matrix W.

2. The solution ~x of the constrained state estimation problem given by (6) with

W = §¡1, where § is the covariance of the unconstrained estimate given

in (3), has an error covariance that is less than or equal to that of the unconstrained state estimate.

3. Among all the constrained Kalman  ̄lters resulting from the solution of (6),

the  ̄lter that uses W = §¡1 has the smallest estimation error covariance.

4. The solution ~x of the constrained state estimation problem given by (6) with

W = I satis ̄es the inequality

kx(k) ¡ x~(k)k·kx(k) ¡ x^(k)k for all k (8)

where k¢k is the vector two-norm.

The above properties all follow from the proofs presented in [4] and the equivalence

of (6) and (7).

2.2 Constraint Tuning

Many times the constraints of (4) are more heuristic than exact. We have some

con ̄dence in the constraints, but we also have some con ̄dence in the unconstrained

Kalman  ̄lter estimates. We therefore need to somehow moderate our enforcement

of the constraints.

In this subsection we analyze the sensitivity of the measurements to the states.

We then propose using this information to decide if an unconstrained state variable

estimate is reliable. We examine residuals that correspond to measurements that

are highly sensitive to a given state. If those residuals are small, then we have a

high con ̄dence in the estimate of that state, and we relax the constraints. However,

if those residuals are large, then we have a low con ̄dence in the estimate of that

state, and we enforce constraints.

NASA/TM—2005-213962 8
Recall our system equations from (1).

x(k + 1) = Ax(k) + Bu(k) + w(k) (9)

y(k) = Cx(k) + e(k)

We see that C can be interpreted as the sensitivity matrix of the measurements to

the states. The element Cij gives the sensitivity of the ith measurement to the jth

state. In practice we should normalize C by dividing each row by the corresponding

measurement value. This gives a normalized sensitivity matrix ¢ as follows.

¢ =

2

6

4

1=y1 ¢¢¢ 0

.

.

. ... .

.

.

0 ¢¢¢ 1=yq

3

7

5

C (10)

where q is the number of measurements. During the execution of the Kalman  ̄lter,

the measurement residuals are given by

o(k) = y(k) ¡ Cx^(k) (11)

The theoretical mean and covariance of the residuals are given as [12, 18]

E[o(k)] = 0 (12)

S(k) = E[o(k)oT (k)]

= C§(k)CT + R

Therefore, if the measurement residuals satisfy their theoretical statistical properties, we can have con ̄dence that the state estimates are reliable.

Residual based constraint tuning proceeds as follows. We generate a list of the

measurements that are most sensitive to each state. This can be obtained by sorting

each column of the sensitivity matrix ¢ in descending order. Use the notation Mji

to denote the measurement number that has the jth largest sensitivity to the ith

state. That is, Mji is the row number in the ith column of ¢ that has the jth largest

magnitude. As an example, suppose that we have a system with three states and

three measurements, and the ¢ matrix ends up being equal to

¢ =

2

6

4

234

415

178

3

7

5 (13)

NASA/TM—2005-213962 9
The M matrix is then given as

M =

2

6

4

233

112

321

3

7

5 (14)

The  ̄rst column of M is h

213 iT. This is because in the  ̄rst column of

¢ in (13), we see that ¢21 has the largest magnitude, ¢11 has the second largest

magnitude, and ¢31 has the third largest magnitude. This means that measurement

2 is the measurement that is most sensitive to the  ̄rst state, measurement 1 is the

second most sensitive, and measurement 3 is the third most sensitive. The same

reasoning can also be applied to the other states.

Now we take the  ̄rst 1 rows of the M matrix, where 1 is a user de ̄ned threshold.

This tells us the 1 measurements that are most sensitive to each state. In the above

example, if we choose 1 = 2, then we will see that measurements 2 and 1 are most

sensitive to the  ̄rst state, measurements 3 and 1 are the most sensitive to the second

state, and measurements 3 and 2 are the most sensitive to the third state.

Looking at the  ̄rst 1 rows of the  ̄rst column of M, we see that if residuals 2

and 1 are small, then we can have a high con ̄dence in our unconstrained estimate

of the  ̄rst state. From the second column of M, we see that if residuals 3 and 1

are small, then we can have a high con ̄dence in our unconstrained estimate of the

second state. From the third column of M, we see that if residuals 3 and 2 are small,

then we can have a high con ̄dence in our unconstrained estimate of the third state.

Notice that a second approach could also be taken to determining our con ̄dence

in the state estimates. For example, instead of seeing which residuals are most

sensitive to the  ̄rst state, we could see which states have the most e®ect on the

 ̄rst residual. Then, for example, if residual 1 was small we could say that we

have a high con ̄dence in our unconstrained estimate of the second and third states

(note that the largest entries in the  ̄rst row of ¢ in (13) are the entries in the

second and third columns). The question of which of these two approaches to take

remains an open issue. In this paper we took the  ̄rst approach, which consists of

checking which residuals were the most sensitive to each state, one state at a time.

This seems to be a more natural method (from an algorithmic point of view) since

NASA/TM—2005-213962 10
we can accomplish constraint tuning one state at a time. The constraint tuning

algorithm can be summarized as follows.

1. We are given the following system with n states, q measurements, and s

constraints.

x(k + 1) = Ax(k) + Bu(k) + w(k) (15)

y(k) = Cx(k) + e(k)

D(k)x(k) · d(k)

We initialize the Kalman  ̄lter quantities ^x(0), ~x(0), and §(0).

2. At each time step k = 0; 1; ¢¢¢, perform the following.

(a) Run the unconstrained and constrained Kalman  ̄lters as follows.

K(k) = A§(k)CT (C§(k)CT + R)

¡1 (16)

x^(k + 1) = Ax^(k) + Bu(k) + K(k)(y(k) ¡ Cx^(k))

§(k + 1) = (A§(k) ¡ K(k)C§(k))AT + Q

min

x~(k+1) [~x(k + 1) ¡ x^(k + 1)]T W(k + 1) [~x(k + 1) ¡ x^(k + 1)]

such that D(k + 1)~x(k + 1) · d(k + 1)

where W(k) is our weighting matrix (see Section 2.1). This gives us an

unconstrained estimate ^x(k + 1) and a constrained estimate ~x(k + 1).

(b) Compute the theoretical residual covariance S(k + 1) from (12).

(c) For i = 1; ¢¢¢ ; n, perform the following.

i. Find the rows with the 1 largest magnitudes in the ith column of

the ¢ matrix. Label these row numbers Mji (j = 1; ¢¢¢ ; 1).

ii. Examine the 1 residuals that correspond to measurement numbers

Mji (j = 1; ¢¢¢ ; 1). If all 1 of these residuals have been smaller than

®Srr(k + 1) (where r = Mji) for · consecutive time steps, then use

x^i(k + 1) as the estimate of the ith state. Otherwise, use ~xi(k + 1)

as the estimate of the ith state

NASA/TM—2005-213962 11
In practice, the decision of how many residuals to use for each state variable (the

value of 1), what relative threshold values to use for those residuals (the value of

®), and how long a residual must remain \small" before we trust the unconstrained

estimate (the value of ·) are open questions. Nevertheless, the theory presented

in this section gives a general approach for deciding when to relax constraints and

when to enforce constraints.

3 Turbofan Engine Health Monitoring

Figure 1 shows a schematic representation of a turbofan engine [15]. A single inlet

supplies air°ow 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

duct, where additional fuel is added to further increase the air temperature. The air

leaves the augmentor through the nozzle, which has a variable cross section area.

Figure 1: Schematic representation of a turbofan engine.

The simulation used in this paper is a gas turbine engine simulation software

NASA/TM—2005-213962 12
package called MAPSS (Modular Aero Propulsion System Simulation) [15]. MAPSS

is written using Matlab Simulink. The MAPSS engine model is based on a low

frequency, transient, performance model of a high-pressure ratio, dual-spool, lowbypass, military-type, variable cycle, turbofan engine with a digital controller. The

controller update rate is 50 Hz, and the component level model balances the mass /

energy equations of the system at a rate of 2500 Hz. The three state variables used

in MAPSS are low-pressure rotor speed (XNL), high-pressure rotor speed (XNH),

and the average hot section metal temperature (TMPC) (measured from aft of the

combustor to the high pressure turbine).

The discretized time invariant equations that model the turbofan engine can be

summarized as follows.

x(k + 1) = f[x(k); u(k); p(k)] + wx(k) (17)

p(k + 1) = p(k) + wp(k)

y(k) = g[x(k); u(k); p(k)] + e(k)

where k is the time index, x is the 3-element state vector, u is the 3-element control

vector, p is the 10-element health parameter vector, and y is the 11-element measurement vector. Note that the noise terms and health parameter degradation are

not modeled in MAPSS but have been added to the model for the problem studied in

this paper. The health parameters change slowly over time. Between measurement

times their deviations can be approximated by the zero mean noise wp(k) (although

in our study the health parameters only changed once per °ight). The noise term

wx(k) represents inaccuracies in the system model, and e(k) represents measurement

noise. A Kalman  ̄lter can be used with (17) to estimate the state vector x and the

health parameter vector p.

The states, controls, health parameters, and measurements are summarized in

Tables 1{4, along with their values at the nominal operating point considered in

this paper, which is a power lever angle of 21o at sea level static conditions (zero

altitude and zero mach). Table 4 also shows typical signal-to-noise ratios for the

measurements, based on NASA experience and previously published data [16]. Sensor dynamics are assumed to be high enough bandwidth that they can be ignored

NASA/TM—2005-213962 13
in the dynamic equations. In Tables 1{4, LPT is used for Low Pressure Turbine,

HPT is used for High Pressure Turbine, LPC is used for Low Pressure Compressor,

and HPC is used for High Pressure Compressor.

State Nominal Value

LPT Rotor Speed 7264 RPM

HPT Rotor Speed 12152 RPM

Average Hot Section Metal Temperature 1533 oR

Table 1: MAPSS turbofan model states and nominal values.

Control Nominal Value

Main Burner Fuel Flow 2454 lbm / hr

Variable Nozzle Area 343 in2

Rear Bypass Door Variable Area 154 in2

Table 2: MAPSS turbofan model controls and nominal values.

NASA/TM—2005-213962 14
Health Parameter Normalized Value

Fan air°ow 1

Fan e±ciency 1

Booster tip air°ow 1

Booster tip e±ciency¤ 1

Booster hub air°ow 1

Booster hub e±ciency 1

High pressure turbine air°ow 1

High pressure turbine e±ciency 1

Low pressure turbine air°ow 1

Low pressure turbine e±ciency 1

Table 3: MAPSS turbofan model health parameters and nominal values.

(¤) The fourth health parameter is not yet implemented in MAPSS.

Measurement Nominal Value SNR

LPT exit pressure 19.33 psia 100

LPT exit temperature 1394 oR 100

Percent low pressure spool rotor speed 63.47% 150

HPC inlet temperature 580.8 oR 100

HPC exit temperature 965.1 oR 200

Bypass duct pressure 20.66 psia 100

Fan exit pressure 17.78 psia 200

Booster inlet pressure 20.19 psia 200

HPC exit pressure 85.06 psia 100

Core rotor speed 12152 RPM 150

LPT blade temperature 1179 oR 70

Table 4: MAPSS turbofan model measurements, nominal values, and signalto-noise ratios. SNR is de ̄ned here as the nominal measurement value divided

by one standard deviation of the measurement noise.

NASA/TM—2005-213962 15
Constraints can be incorporated in the state estimator by using heuristic knowledge of the behavior of the health parameters. For example, it is known that health

parameters do not improve over time.

pm(k) · pm(k ¡ 1); m 2 [1 ¡ 6; 8; 10] (18)

pm(k)  ̧ pm(k ¡ 1); m 2 [7; 9]

In addition, it is known that the health parameters vary slowly with time. For

example, since ~p1(k) is the constrained estimate of p1(k), we can enforce the following

constraints on ~p1(k).

p~1(k) · p1(0) (19)

p~1(k) · p~1(k ¡ 1) + °+

1

p~1(k)  ̧ p~1(k ¡ 1) ¡ °¡

1

where °+

1 and °¡1 are nonnegative factors chosen by the user. These factors allow

the health parameter estimate to vary only within prescribed limits from one time

step to the next. Typically we choose °¡

1 > °+1 so that the parameter estimate

can change more in the negative direction than in the positive direction. This is

in keeping with our a priori knowledge that this particular health parameter never

increases with time. Ideally we would have °+

1 = 0 since p1(k) never increases with

time. However, since the health parameter estimate varies around the true value of

the health parameter, we choose °+

1 > 0. This allows some time-varying increase

in the health paramter estimate to compensate for a previous estimate that was

smaller than the true value.

Constraints (19) are linear and can therefore easily be incorporated into the form

D(k)~x(k) · d(k) as required in the constrained  ̄ltering problem statement (4). Note

that the constrained  ̄ltering approach presented here 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 [5])

in conjuction with the methods presented in this paper.

NASA/TM—2005-213962 16
4 Turbofan Engine Health Parameter Sensitivity Analysis

In this section we apply the constrained Kalman  ̄ltering constraint tuning procedure introduced in Section 2.2 to the turbofan engine health parameter estimation

problem. This includes analyzing the sensitivity of the measurements to the health

parameter values as was done in reference [17]. As discussed in Section 2.2, we then

use this information to decide if an unconstrained health parameter estimate is reliable. If measurement residuals that are highly sensitive to a given health parameter

are near zero, then we have a high con ̄dence in the estimate of that health parameter, and we relax the constraints. However, if the measurement residuals are large,

then we have a low con ̄dence in the estimate of that health parameter, and we

enforce constraints that correspond to our heuristic knowledge of health parameter

behavior.

Suppose we linearize and augment (17) to obtain the system

"

x(k + 1)

p(k + 1) #

=

"

A1 A2

0 I

# " x(k)

p(k)

#

+

"

B

0

# h

u(k)

i

+

"

wx(k)

wp(k)

#

(20)

y(k) = h

C1 C2

i "

x(k)

p(k)

#

+ e(k)

= C

"

x(k)

p(k)

#

+ e(k)

If we followed the approach given in Section 2.2 we would use C2 as the sensitivity of

the measurements to the health parameters. However, if the system is operating in

steady state so that x(k + 1) = x(k) and p(k + 1) = p(k), then the coupling between

x(k) and p(k) can exploited to obtain more complete sensitivity information. In this

case (20) can be solved for y(k) as

y(k)=[C1(I ¡ A1)

¡1A2 + C2]p(k) + C1(I ¡ A1)¡1wx(k) + (21)

C1(I ¡ A1)

¡1Bu(k) + e(k)

= ¢p(k) + C1(I ¡ A1)

¡1(Bu(k) + wx(k)) + e(k)

where ¢, de ̄ned by the above equation, is the sensitivity matrix of the measurements to the health parameters. The element ¢ij gives the sensitivity of the ith

NASA/TM—2005-213962 17
measurement to the jth health parameter. In practice we normalize ¢ by dividing

each row by the corresponding nominal measurement value given in Table 4.

The di®erence between the sensitivity matrix obtained using the approach of

Section 2.2 (C2), and the sensitivity matrix obtained here (¢), is analogous to the

di®erence between a partial derivative and a total derivative. ¢ is a more accurate

measure of the sensitivity (assuming that the system is in steady state).

During the execution of the Kalman  ̄lter, the measurement residuals are given

by

o(k) = y(k) ¡ [C1x^(k) + C2p^(k)] (22)

The theoretical mean and variance of the residuals are given in (12). Therefore, if

the measurement residuals satisfy their theoretical statistical properties, we can have

con ̄dence that the state and health parameter estimates are reliable. We generate

a list of the measurements that are most sensitive to each health parameter. This

is obtained by sorting each column of the sensitivity matrix ¢ in descending order.

In the case of MAPSS at the operating point used in this paper, the normalized

sensitivity matrix is given as

¢ =

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

0:01 0:06 0 :12 0:00 0 :27 0:39 0 :06 0 :27 0:14 0:02

0 :43 0 :33 0:09 0:00 0:16 0:17 0:04 0:15 0 :45 0 :19

0 :14 0 :21 0:10 0:00 0:09 0:07 0:01 0:02 0 :35 0:11

0:05 0 :25 0:11 0:00 0:12 0:01 0:02 0:02 0 :41 0 :14

0:04 0:03 0:02 0:00 0:03 0:01 0:01 0:02 0:11 0:04

0:01 0:07 0:09 0:00 0:01 0:16 0 :21 0:18 0:03 0:04

0:00 0:19 0:04 0:00 0:07 0:08 0:03 0:08 0:28 0 :11

0 :08 0:10 0 :23 0:00 0 :17 0 :64 1 :22 0 :52 0:13 0:09

0:03 0:10 0:04 0:00 0:10 0 :44 0:05 0:18 0:00 0:05

0:06 0:12 0 :13 0:00 0 :16 0 :64 0:05 0 :43 0:02 0:10

0:01 0:12 0:03 0:00 0:07 0:17 0:03 0:14 0:10 0:07

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

(23)

The three largest sensitivities in each column are italicized. (The fourth health

parameter is not yet implemented in MAPSS, so the fourth column of ¢ is zero.)

We see that the measurements that are most sensitive to the  ̄rst health parameter

are measurement numbers 2, 3, and 8; the measurements that are most sensitive to

the second health parameter are measurement numbers 2, 3, and 4; and so on. This

tells us that if residuals 2, 3, and 8 are small, then we can have a high con ̄dence

NASA/TM—2005-213962 18
in our unconstrained estimate of health parameter 1; if residuals 2, 3, and 4 are

small, then we can have a high con ̄dence in our unconstrained estimate of health

parameter 2; and so on. In practice, the decision of how many residuals to use for

each health parameter, and what threshold values to use for those residuals, is an

open question. Nevertheless, the theory presented in this section gives a general

approach for deciding when to relax constraints and when to enforce constraints.

5 Simulation Results

We simulated the methods discussed in this paper using Matlab. We measured a

steady state 3 second burst of open-loop engine data at 10 Hz during each °ight.

These routine data collections were performed over 100 °ights at the single operating

point shown in Tables 1, 2, and 4. The engine's health parameters were initialized

to the values shown in Table 3 and then deteriorated a small amount once each

°ight (i.e., once every 30 time steps). The signal-to-noise ratios were determined

on the basis of NASA experience and previously published data [16] and are shown

in Table 4. In the Kalman  ̄lter we used a one-sigma state process noise equal to

0.005% of the nominal state values to allow the  ̄lter to be responsive to changes in

the state variables. We also set the one sigma process noise for each component of

the health parameter to 0.01% of the nominal parameter value. These values were

obtained by tuning. They were small enough to give reasonably smooth estimates,

and large enough to allow the  ̄lter to track slowly time-varying parameters. In the

enforcement of constraints we chose the ° variables in (19) such that the maximum

allowable change in ~p was a linear-plus-exponential function of time that reached a

maximum of 9% after 500 °ights in the direction of expected change, and 3% after

500 °ights in the opposite direction. The true health parameter values never change

in a direction opposite to the expected change. However, we allow the estimate

to change in the opposite direction to allow the Kalman  ̄lter to compensate for

the fact that the previous estimate might be either too large or too small. The

constraint boundaries are illustrated in Figure 2. In our simulations the true health

parameters changed once per °ight. However, the constraints that we imposed on

NASA/TM—2005-213962 19
our estimates varied with each time step.

Figure 2: The constraints are determined by allowing the state estimate to

change a maximum of 3.0 times the expected magnitude in the expected direction of health parameter change, and 1.0 times the expected magnitude in

the opposite direction.

We set the weighting matrix W in (6) equal to the identity matrix. Although

§¡1 is the optimal value of W in terms of the error covariance, we found from

experience that setting W = I results in only a small loss of performance, but it

generates a signi ̄cant savings in computational e®ort. This is because we avoid

inverting the 13 £ 13 covariance matrix at each time step.

NASA/TM—2005-213962 20
We simulated a linear-plus-exponential degradation of the 10 health parameters

over 100 °ights. The initial health parameter estimation errors were assumed to be

zero. The simulated health parameter degradations were representative of turbofan

performance data reported in the literature [19]. Figure 3 shows a typical plot of the

true deviation of health parameters 1 and 5, along with the deviations estimated

by the unconstrained Kalman  ̄lter. We can see from the plot that shortly after

°ight 83 the unconstrained estimates are quite good. In this case we would not

want to enforce constraints on the health parameter estimates at this particular

time (although we may want to enforce constraints again later). But how can we

know that the unconstrained health parameter estimates are good?

Figure 3: Unconstrained Kalman  ̄lter estimates of health parameters 1 and

5. The estimates are noisy, but between °ight 83 and 83 the estimates are

quite accurate.

NASA/TM—2005-213962 21
If we look at (23) to  ̄nd the three measurements that are most sensitive to health

parameters 1 and 5, we come up with measurements 1, 2, 3, 8, and 10. A closeup of

the normalized residuals of these measurements, shown in Figure 4, indicates that

they are indeed small between °ight 83 and 84. This indicates that we can have

a high con ̄dence in the unconstrained estimates and relax our constraints at that

moment.

Figure 4: Normalized residuals of measurements 1, 2, 3, 8, and 10 between

°ight 83 and 84. The residuals are less than one standard deviation for several

time steps. This indicates that we can have a high con ̄dence in our estimate

of the health parameters to which those measurements are highly sensitive.

NASA/TM—2005-213962 22
Figure 5 shows what happens if we do not relax our constraints. A comparison of

Figures 3 and 5 shows that the constrained estimate is better than the unconstrained

estimate overall. At °ight 83 the unconstrained estimate is good, but the enforcement of constraints does not allow the constrained estimate to \catch up" to the

unconstrained estimate. This is because our constrained estimator does not allow

the estimates to change as quickly as the unconstrained estimator. This smoothing

e®ect is why, overall, the constrained estimates in Figure 5 are more accurate than

the unconstrained estimates in Figure 3. However, this is also why, in Figure 5, the

constrained estimate cannot catch up to the unconstrained estimate at °ight 83,

even though the unconstrained estimate is better at that point in time.

Figure 5: Constrained Kalman  ̄lter estimates of health parameters 1 and

5. The estimates are smooth and more accurate than the unconstrained estimates, but between °ight 83 and 84 the estimates are less accurate than the

unconstrained estimates (see Figure 3).

NASA/TM—2005-213962 23
Figure 6 shows what happens when we momentarily relax the constraints on

the estimates of health parameters 1 and 5. Since the highly sensitive meaurement

residuals in Figure 4 are small at °ight 83, we relax the constraints momentarily,

allowing the constrained estimate to change abruptly for one time instant. We

reset the constrained estimates to the unconstrained estimate values, and reapply

the constraints for future estimates. The overall e®ect is an improvement in the

accuracy of the constrained health parameter estimate.

Figure 6: Constrained Kalman  ̄lter estimates of health parameters 1 and 5.

The estimates are set equal to the unconstrained estimates between °ight 83

and 84 due to the small measurement residuals at this time.

In this example we chose to look at health parameters 1 and 5, and we chose

to look at the three most sensitive residuals to each health parameter, which corresponded to measurements 1, 2, 3, 8, and 10. This example is only for illustrative

purposes. In general we will not look at combinations of health parameters; we will

NASA/TM—2005-213962 24
rather look at each health parameter individually, and a certain number of residuals

that are the most sensitive to each health parameter, where the number of residuals

per health parameter is obtained by manual tuning.

We ran 20 Monte Carlo simulations like this, each simulation consisting of 100

°ights and the same health parameter degradation, but di®erent measurement noise.

Table 5 shows the performance of the  ̄lters averaged over 100 °ights and 20 simulations. The standard Kalman  ̄lter estimates the health parameters to within 7.4%

of their  ̄nal degradations. The constrained  ̄lter estimates the health parameters

to within 6.5% of their  ̄nal degradations. The constrained  ̄lter with the addition

of residual based tuning estimates the health parameters to within 6.2% of their

 ̄nal degradations. These numbers show the improvement that is possible with the

constrained Kalman  ̄lter, and with residual based tuning of the constraints.

Estimation Error (%)

Unconstrained Constrained Tuned

Health Parameter Filter Filter Filter

Fan air°ow 12.9 9.2 8.2

Fan e±ciency 6.9 6.2 5.0

Booster tip air°ow 10.9 10.6 10.0

Booster tip e±ciency¤ N/A N/A N/A

Booster hub air°ow 7.4 6.8 6.2

Booster hub e±ciency 3.8 3.1 3.0

High pressure turbine air°ow 4.3 3.3 3.2

High pressure turbine e±ciency 4.2 3.8 3.7

Low pressure turbine air°ow 3.6 3.3 3.2

Low pressure turbine e±ciency 11.3 11.2 11.0

Average 7.4 6.5 5.9

Table 5: Health parameter estimation errors (percent) of the Kalman  ̄lters,

averaged over all °ights. The estimation error is measured as j(p¡p~)=pf j, where

p is the true health parameter value, ~p is the estimated health parameter value,

and pf is the health parameter value at the end of the simulation.

(¤) The fourth health parameter is not yet implemented in MAPSS.

The improved performance of the constrained  ̄lter comes with a price, and that

price is computational e®ort. The constrained  ̄lter requires more computational

NASA/TM—2005-213962 25
e®ort than the unconstrained  ̄lter, and the incorporation of residual based tuning

requires more e®ort yet. However, computational e®ort is not a critical issue for turbofan health estimation since the  ̄ltering is performed on ground-based computers

after each °ight.

Note that the Kalman  ̄lter works well only if the assumed system model matches

reality fairly closely. The method presented in this paper, by itself, will not work

well if there are large sensor biases or hard faults due to severe component failures.

A mission-critical implementation of a Kalman  ̄lter should always include some sort

of additional residual check to verify the validity of the Kalman  ̄lter results [18],

particularly for the application of turbofan engine health estimation considered in

this paper [5].

6 Conclusion and Discussion

We have presented a residual based method for tuning the constraints of a Kalman

 ̄lter. The constrained Kalman  ̄lter uses a projection method to maintain the state

variable estimates within a user-de ̄ned envelope. However, the constraints for many

problems, including the turbofan health estimation problem investigated in this

paper, are heuristic. Therefore the engineer incurs some risk when implementing

constraints. Although the use of constraints generally improves the accuracy of

health estimation, there may be times when the constrained estimate is worse than

the unconstrained estimate. If the unconstrained Kalman  ̄lter estimate is accurate

then the incorporation of constraints can degrade the estimate. The use of residuals

can quantify our con ̄dence in the accuracy of the unconstrained estimate and tell

us whether or not constraints should be incorporated.

The residual based method presented here measures residuals that are highly

sensitive to given health parameters to decide whether or not constraints should

be enforced on that health parameter. In practice there are several questions that

need to be answered in the implementation of this theory. For instance, how many

residuals should be used to decide whether or not to relax the constraints? How

small should the residuals be (and for how many time steps) before constraints are

NASA/TM—2005-213962 26
relaxed? Or, using another approach, how large should the residuals be (and for

how many time steps) before constraints are enforced? For this paper, the answers

to these questions were found by manual adjustments, but further work could focus

on a more theoretically and statistically rigorous analysis of the optimal answers to

these questions.

We have seen that the constrained  ̄lter requires more computational e®ort than

the standard Kalman  ̄lter. The incorporation of constraint tuning requires yet

more computational e®ort. This is due to the addition of the quadratic programming problem that must be solved in the constrained Kalman  ̄lter, and the residual

checking logic that must be performed in the constraint tuning process. The engineer must therefore perform a tradeo® between computational e®ort and estimation

accuracy. For real time applications the improved estimation accuracy may not be

worth the increase in computational e®ort.

Although we have considered only linear state constraints, it is not conceptually

di±cult to extend this paper to nonlinear constraints. If the state constraints are

nonlinear they can be linearized as discussed in [4]. Further work could explore the

incorporation of state constraints for optimal smoothing, or the use of constraint

tuning in constrained H1  ̄ltering [20].

References

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

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

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

February 1995.

[2] D. Simon and D.L. Simon, Aircraft turbofan engine health estimation using

constrained Kalman  ̄ltering, ASME Turbo Expo 2003, Atlanta, Georgia, Paper

GT2003-38584, June 2003.

[3] D. Simon and D.L. Simon, Aircraft turbofan engine health estimation using

constrained Kalman  ̄ltering, ASME Journal of Engineering for Gas Turbines

and Power, in print.

NASA/TM—2005-213962 27
[4] D. Simon and T. Chia, Kalman  ̄ltering with state equality constraints, IEEE

Transactions on Aerospace and Electronic Systems (39), pp. 128-136, Jan. 2002.

[5] 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.

[6] 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.

[7] 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.

[8] A. Volponi, H. DePold, and R. Ganguli, The use of Kalman  ̄lter and neural

network methodologies in gas turbine performance diagnostics: a comparative

study, ASME Turbo Expo 2000, Munich, Germany, Paper 2000-GT-547, May

2000.

[9] T. Kobayashi and D.L. Simon, A hybrid neural network-genetic algorithm technique for aircraft engine performance diagnostics, 37th

AIAA/ASME/SAE/ASEE Joint Propulsion Conference, July 2001.

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

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

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

using Kalman  ̄ltering techniques, NASA TM 104233, June 1991.

[12] B. Anderson and J. Moore, Optimal Filtering (Prentice Hall, Englewood Cli®s,

New Jersey, 1979).

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

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

New York, 1981).

NASA/TM—2005-213962 28
[15] K. Parker and K. Melcher, The modular aero-propulsion systems simulation

(MAPSS) users' guide, NASA TM 2004-212968, March 2004.

[16] W. Merrill, Identi ̄cation of multivariable high-performance turbofan engine

dynamics from closed-loop data, AIAA Journal of Guidance, Control, and Dynamics (7), pp. 677-683, Nov. 1984.

[17] B. Brunell, D. Viassolo, and R. Prasanth, Model Adaptation and Nonlinear

Model Predictive Control of an Aircraft Engine, ASME Turbo Expo 2004,

Vienna, Austria, Paper GT2004-53780, June 2004.

[18] A. Gelb, Applied Optimal Estimation (MIT Press, Cambridge, Massachusetts,

1974).

[19] O. Sasahara, JT9D engine/module performance deterioration results from back

to back testing, International Symposium on Air Breathing Engines, pp. 528-

535, 1985.

[20] D. Simon, A game theory approach to constrained minimax state estimation,

IEEE Transactions on Signal Processing, in print.

NASA/TM—2005-213962 29
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ARL–MR–621

E–15278

35

Kalman Filter Constraint Tuning for Turbofan Engine Health Estimation

Dan Simon and Donald L. Simon

Aircraft engines; Systems health monitoring; Gas turbine engines; Flight safety; Kalman

filter; State constraints; Estimation; Quadratic programming; Residuals

Unclassified -Unlimited

Subject Category: 07

WBS–22–077–41–21

1L161102AF20

Dan Simon, Cleveland State University, 2121 Euclid Avenue, Cleveland, Ohio 44115–2226 (funded by NASA Grant

NAG3–2699); and Donald L. Simon, U.S. Army Research Laboratory, NASA Glenn Research Center. Responsible person,

Donald L. Simon, organization code Z, 216–433–3740.

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 are often neglected because they do not fit easily into the structure of the Kalman filter. Recently published

work has shown a new method for incorporating state variable inequality constraints in the Kalman filter, which has

been shown to generally improve the filter’s estimation accuracy. However, the incorporation of inequality constraints

poses some risk to the estimation accuracy as the Kalman filter is theoretically optimal. This paper proposes a way to

tune the filter constraints so that the state estimates follow the unconstrained (theoretically optimal) filter when the

confidence in the unconstrained filter is high. When confidence in the unconstrained filter is not so high, then we use our

heuristic knowledge to constrain the state estimates. The confidence measure is based on the agreement of measurement

residuals with their theoretical values. The algorithm is demonstrated on a linearized simulation of a turbofan engine to

estimate engine health.