NASA Contractor Report 178083

The Evaluation of the

OSGLR Algorithm

for Restructurable Control

W. Bonnice, E. Wagner, S. Hall, P. Motyka

THE CHARLES STARK DRAPER LABORATORY, INC.

555 Technology Square

Cambridge, Massachusetts 02139

CONTRACT NAS1-17556

May 1986

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23665
NASA Contractor Report 178083

The Evaluation of the

OSGLR Algorithm

for Restructurable Control

W. Bonnice, E. Wagner, S. Hall, P. Motyka

THE CHARLES STARK DRAPER LABORATORY, INC.

555 Technology Square

Cambridge, Massachusetts 02139

CONTRACT NAS1-17556

May 1986

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23665
TABLE OF CONTENTS

Section

INTRODUCTION .............................................

BACKGROUND ...............................................

2.1 Introduction ........................................

2.2 C-130 Simulation Description ........................

2.3 Description of FDI Algorithms Evaluated .............

2.3.1 The Detection Filter .........................

2.3.2 The Modified Detection Filter ................

2.3.3 The Generalized Likelihood Ratio (GLR)

Test for Dynamic Systems .....................

2.3.4 Orthogonal Series Generalized Likelihood

Ratio (OSGLR) Test .........................

2.4 Evaluation and Comparison of the FDI Algorithms .....

2.4.1 Failure Modes ................................

2.4.2 Type of Failure ..............................

2.4.3 Magnitude of Failure .........................

2.4.4 False Alarm Performance ......................

2.4.5 Detection Time ...............................

2.4.6 Computational Burden .........................

2.4.7 Robustness ...................................

2.4.8 Maturity .....................................

2.4.9 Summary and Overall Comparison ...............

2.5 System Monitoring Strategy ..........................

2.5.1 Local Actuator FDI ...........................

2.5.2 Analytic FDI .................................

2.5.3 Control Surface Position Sensors .............

2.5.4 System Monitoring Strategy Selected ..........

2.6 Conclusions .........................................

A SURVEY OF FDI TECHNIQUES APPLICABLE TO AIRCRAFT

CONTROL SURFACE FAILURE DETECTION AND ISOLATION ..........

3.1 Introduction ........................................

3.2 Failure Sensitive Filters ...........................

3.3 Innovations-based Systems ...........................

Page

5

5

9

9

12

15

21

26

28

28

29

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33

33

34

34

37

37

38

39

40

42

42

42

43

iii
TABLEOFCONTENTS(Cont.)

Section Page

3.4 Multiple Hypothesis Filtering ....................... 44

3.5 Generalized Bayesian Tests .......................... 45

3.6 Conclusions ......................................... 45

ACCOMMODATION OF MODELING ERRORS FOR THE OSGLR

ALGORITHM ................................................ 47

4.1 Dynamic Thresholds .................................. 48

4.2 Dynamic Decision Regions ............................ 52

4.3 Model Error Estimation .............................. 53

4.4 Robust Residual Generation .......................... 53

4.5 Summary and Conclusions ............................. 67

DETAILED EVALUATION OF THE OSGLR ALGORITHM USING

A SINGLE CRUISE LINEAR MODEL ............................. 72

5.1 Introduction ........................................ 72

5.2 Sensor Errors and Dynamics .......................... 73

5.3 Off-Nominal, No-Failure Flight Conditions ........... 75

5.3.1 Maneuvering Test Case ........................ 75

5.3.2 Turning Test Case ............................ 78

5.3.3 Maximum Rate-of-Climb Test Case .............. 81

5.3.4 Accelerating Test Cases ...................... 81

5.3.5 Decelerating and Descending Test Cases ....... 84

5.3.6 Non-Zero Flap Deflections .................... 84

5.3.7 Off-Nominal Turbulence and Steady Winds ...... 89

5.4 Failures ............................................ 92

5.4.1 Elevator Failure Test Case ................... 92

5.4.2 Off-Nominal Elevator Failure Test Cases ...... 95

5.4.3 Right Aileron Failure Test Case .............. 99

5.4.4 Rudder Failure Test Case ..................... 99

5.4.5 Right Flap Failure Test Case ................. 103

5.4.6 Largest Failure Decision Functions as a

Function of Time and Failure Magnitude ..... 106

5.5 Summary and Conclusion .............................. 113

SCHEDULING ............................................... 120

6.1 Introduction ........................................ 120

6.2 Scheduling of Zero Flap Deflection Linear Models .... 120

6.3 Nonzero Flap Deflections ............................ 127

iv
TABLE OF CONTENTS (Cont.)

Section

7

Appendix

A

B

6.4 Failure Detection and Scheduling ....................

6.5 Scheduling of Age-Weighted Filter and OSGLR

Steady-State Matrices .............................

6.6 Summary and Conclusions .............................

SUMMARY AND CONCLUSIONS ..................................

REFERENCES ...............................................

Page

129

135

139

141

145

SUMMARY OF TEST CASES .................................... 148

SCHEDULING IMPLEMENTATION DETAILS ........................ 156

B.1 Introduction ........................................ 156

B.2 Linear Model Formulation ............................ 156

B.3 Steady-State OSGLR Equations ........................ 157

B.4 Scheduling .......................................... 158

V
LIST OF ILLUSTRATIONS

Figure

2.1

2.2

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

Detection filter block diagram ...........................

Block diagram of the modified detection filter ...........

Sensitivity measure for the OSGLR algorithm ..............

State estimation errors and standard deviations

for a nonlinear, no-failure, nominal cruise

flight condition test case ...............................

Decision functions for the nonlinear, no-failure,

nominal cruise flight condition test case ................

OSGLR decision functions for a -0.01745 tad (-I ° )

elevator bias failure which occurred at 10 seconds

at the nominal cruise flight condition of 77.2 m/s

(150 knots) and 304.8 m (1000 ft) ........................

OSGLR decision functions for a no-failure case at an

off-nominal cruise flight condition of 102.9 m/s

(200 knots) and 1524 m (5000 ft) .........................

OSGLR decision functions for a -0.01745 tad (-1 ° )

elevator bias failure at 10 seconds at the nominal

cruise condition of 77.2 m/s and 304.8 m using a

Kalman filter with process noise added ...................

OSGLR decision functions for a no-failure case at an

off-nominal cruise flight condition of 102.9 m/s

and 1524.0 m using a Kalman filter with process

noise added ..............................................

Decision functions produced by an age-weighted OSGLR

algorithm: no failures, off-nominal cruise flight

condition of 102.9 m/s and 1524.0 m, age-weighting

time constant of 15 s ....................................

Page

I0

16

49

54

56

58

59

60

61

65

vi
LIST OF ILLUSTRATIONS (Cont.)

Figure

4.9

4.10

4.11

4.12

5.3

5.4

5.5

5.6

5.7

Decision functions produced by an age-weighted OSGLR

algorithm: -0.01745 rad (-1 °) elevator bias failure

at 10 s, nominal flight condition of 77.2 m/s and

304.8 m, age-weighting time constant of 15 s .............

Estimated rudder basis function coefficients for a

0.001745 rad/s (0.1 ° deg/s) rudder ramp failure

starting at t=0 produced by an age-weighted OSGLR

algorithm with age-weighted time constant of 3

seconds ..................................................

Decision functions produced by an age-weighted OSGLR

algorithm: no failures, off-nominal cruise flight

condition of 102.9 m/s and 1524 m, age-weighting

time constant of 3 s .....................................

Decision functions produced by an age-weighted OSGLR

algorithm: -0.01745 rad (-1 °) elevator bias failure

at 10 s, nominal cruise flight condition of 77.2 m/s

and 304.8 m, age-weighting time constant of 3 s ..........

Decision functions for the maneuvering test case .........

Decision functions for the maneuvering test case with

thrust as an input .......................................

Decision functions with the aircraft turning right

for 3.32 rad (190 deg) and then left for 3.14 rad

(180 deg) ................................................

Decision functions for the maximum rate-of-climb test

case .....................................................

Decision functions for the acceleration from 51.44 m/s

(100 knots) to 102.9 m/s (200 knots) test case ...........

Decision functions for the decelerating at constant

altitude from 102.9 m/s to 51 m/s test case ..............

Decision functions for the descending from 1524 m

(5000 ft) test case ......................................

Page

66

68

69

70

77

79

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82

83

85

86

vii
LIST OF ILLUSTRATIONS(Cont.)

Figure

5.8

5.9

5.10

5.11

5.12

5.13

5.14

5.15

5.16

5.17

5.18

5.19

Decision functions for the 50% flap deflection test

case .....................................................

Decision functions for the 75% flap deflection test

case .....................................................

Decision functions for the maneuvering test case

with light turbulence ....................................

Decision functions for the aircraft flying at the

nominal cruise flight condition in thunderstorm

turbulence ...............................................

Decision Functions for a 10.3 m/s (20 knots)

crosswind test case ......................................

Decision functions for a -0.0873 red (-5 deg)

elevator bias failure ....................................

Isolation decision functions for a -0.0873 red

(-5 deg) elevator bias failure ...........................

Decision functions for a -0.0873 red (-5 deg)

elevator bias failure with the aircraft flying

at an off-nominal cruise airspeed of 64.3 m/s

(125 knots) at the time of failure .......................

Decision functions for a -0.0873 red (-5 deg)

elevator bias failure with the aircraft flying

at an off-nominal cruise airspeed of 90 m/s

(175 knots) at the time of failure .......................

Decision functions for a 0.0873 red (5 deg) right

aileron bias failure .....................................

Isolation decision functions for a 0.0873 red

(5 deg) right aileron bias failure .......................

Decision functions for a -0.0873 red (-5 deg)

rudder bias failure ......................................

Page

87

88

9O

91

93

94

96

97

98

1 O0

101

102

viii
LIST OF ILLUSTRATIONS (Cont.)

Figure

5.20

5.21

5.22

5.23

5.24

5.25

5 •26

5.27

5.28

6.1

6.2

Page

Decision functions for a 5% right flap bias

failure .................................................. 104

Isolation decision functions for a 5% right

flap bias failure ........................................ 105

The largest decision function values produced

by elevator bias failures ................................ 107

The largest decision function values produced

by right aileron bias failures ........................... 108

The largest decision function values produced

by right flap bias failures .............................. 109

The largest decision function values produced by

rudder bias failures ..................................... 110

The smallest of the isolation decision functions

DFli(i # 1) produced by elevator bias failures ........... 114

The smallest of the isolation decision functions

DF2i(i # 2) produced by right aileron bias

failures ................................................. 115

The smallest of the isolation decision functions

DF5i (i # 5) produced by right flap bias failures ........ 116

Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for the acceleration

from 51.44 m/s test case: 51.44, 77.2, 102.9 m/s

linear models, 77.2 m/s age-weighted filter and

OSGLR steady-state matrices .............................. 122

Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for an acceleration

from 51.44 m/s test case: 46.3, 51.44, 77.2,

102.9 m/s linear models, 77.2 m/s age-weighted

filter and OSGLR steady-state matrices ................... 124

ix
LIST OF ILLUSTRATIONS(Cont.)

Figure

6.3

6.4

6.5

6.6

6.7

6.8

6.9

Page

Decision functions for a 51.44 m/s, no-failure,

cruise test case using the 51.44 m/s linear

model and associated steady-state Kalman filter

and OSGLR matrices ....................................... 125

Decision functions for a 77.2 m/s cruise no-failure

test case using the 77.2 m/s linear model and

associated steady-state Kalman filter and OSGLR

matrices .. .... ......... ...... .................. .... ...... 126

Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for a 50% flap deflection

test case: 46.3, 51.44, 77.2, 102.9 m/s linear models,

77.2 m/s age-weighted filter and OSGLR steady-state

matrices ...... ...... .......... ........................... 128

Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for a 50% flap deflection

test case: 38.6, 51.44, 77.2 m/s with 50% flap linear

models, 77.2 m/s With 50% flap age-weighted filter

and OSGLR steady-state matrices .......................... 130

Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for a 75% flap deflection

test case: 38.6, 51.44, 77.2 m/s with 50% flap linear

models, 77.2 m/s With 50% flap age-weighted filter

and OSGLR steady-state matrices .......................... 131

Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for a 100% flap deflection

test case: 100% flap deflection from the start of the

test case, 38.6, 51.44, 77.2 m/s with 100% flap linear

models, 77.2 m/s With 100% flap age-weighted filter

and OSGLR steady-state matrices .......................... 132

Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for a -0.0873 rad

(-5 deg) elevator bias failure at 90.0 m/s (175 knots)

test case: 46.4, 51.44, 77.2, 102.9 m/s linear models,

77.2 m/s age-weighted filter and OSGLR steady-state

matrices ................................................. 133
LIST OFILLUSTRATIONS(Cont.)

Figure

6.10

6.11

6.12

Page

Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for a -0.0873 tad (-5 deg)

elevator bias failure at 64.3 m/s (125 knots) test

case: 46.4, 51.44, 77.2, 102.9 m/s linear models,

77.2 m/s age-weighted filter and OSGLR steady-state

matrices .................................. .... ... ..... ... 134

Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for a -0.0873 tad (-5 deg)

elevator bias failure at 64.3 m/s (125 knots) test

case: 46.3, 51.44, 77.2, 102.9 m/s linear models,

steady-state age-weighted filter and OSGLR matrices

also scheduled ........................................... 136

Decision functions for a -0.0873 rad (-5 deg)

elevator bias failure at 175 knots test case

produced by an age-weighted, linearly scheduled

OSGLR algorithm: linear interpolation scheduling

approach, 46.3, 51.44, 77.2, 102.9 m/s linear models,

steady-state age-weighted filter and OSGLR matrices

also scheduled ........................................... 138

xi
LIST OF TABLES

Table Page

2.1 Inputs, Outputs, and States of the C-130 Aircraft ..... 7

2.2 Standard Deviation of Sensor Noise .................... 8

2.3 Comparison of FDI Algorithms .......................... 35

5.1 C-130 control surface definition ...................... 73

5.2 Sensor errors ......................................... 74

5.3 Maneuvering test case ................................. 76

5.4 Symbols for time after onset of bias failure .......... 111

A.1 Summary of Test Cases ................................. 149

xii
LIST OFACRONYMS

CSDL

FDI

GLR

LLR

LR

NASA

OSGLR

STOL

The Charles Stark Draper Laboratory

Failure Detection and Isolation

Generalized Likelihood Ratio

Log Likelihood Ratio

Likelihood Ratio

National Aeronautics and SpaceAdministration

Orthogonal Series Generalized Likelihood Ratio

Short Take Off or Landing

xiii
LIST OFSYMBOLS

ai

a

A

Aa

A¢

h°

--i

b(k)

B

c

Ci

C

d.

--l

d(k)

unknown coefficients of series expansion of failure mode shape

for OSGLR algorithm

column vector which augments A-II matrix; vector of ai's

linear system state matrix, dimension n xn

state transition matrix of the unknown coefficient vector a in

the discrete time OSGLR algorithm

state matrix that defines the OSGLR basis functions _i in

discrete time

ith column of B matrix

known vector denoting type of failure and its effect on the

system

linear system state input matrix of dimension n xm

t

P _; projection of residuals onto the normalized C(b. - Kd.)

~ ~ --i --i

vectors and --INd and _2N; also column vector which augments C

components of c which correspond to d. and have been filtered with the same eigenvalue as the detection filter

ith component of the C vector

linear system output matrix, dimension r xn

ith column of the D matrix

known vector denoting type of failure and its effect on the

measurement

linearly independent normalized columns of D

linear system control input matrix for the output equation,

dimension r xm

xiv
LIST OF SYMBOLS(Cont.)

DFl

DE, .

13

DF(k)

e(k)

E( )

f

f(k)

f(k,8)

F(k)

g

g(k)

_(k,e)

G(k)

h(k)

H 0

H I

I

k

k.

K

£(kf,e,9)

£(kf)

detection decision function for control surface i, i = 1, 2,

..., 6

isolation decision function between control surfaces i and j,

i,j = I, 2, ..., 6

OSGLR decision function at time k; GLR decision function at k

estimation error

mean or expected value

vector of nonlinear functions of the states and controls

mode shape of failure

influence of failure mode on the state estimation error at

increment k, failure occuring at time e

influence of vector of coefficients a on the estimation error

vector of nonlinear functions of the states and controls

known bias vector of dimension n

signature of failure at increment k, failure occuring at time

e

influence of vector of coefficients a on the residuals

known bias vector of dimensional r

no failure hypothesis

failure hypothesis

identity matrix

kth discrete time step

ith column of the K matrix

detection filter, Kalman filter, or age-weighted filter gain

matrix

log likelihood ratio at final iteration kf for failure of

magnitude v occurring at time 8

log likelihood ratio at time kf

xv
LIST OF SYMBOLS (Cont.)

M(k)

n(t)

n(k,@)

N(a,b)

P

p(x/y)

P

P(k)

q!

Q(k)

r

r

rf

R

R(k)

s

S

covariance of Kalman or age-weighted filter residual at time

step k

unexpected control input

failure mode shape

normal distribution with mean a and variance b

number of basis functions of OSGLR algorithm

conditional probability of x given that y has occurred

the columns of P are the normalized C(b. - Kd.) dlN

and _1 N vectors

covariance of Kalman or age-weighted filter estimation error

_- _', difference between the system state vector and the

detection filter state vector

state vector of the secondary filter used in the modified

detection filter

state driving noise covariance

number of measurements

Z - Z', residual vector

components of r which correspond to d. and have been fil-

-- --I

tered with the same eigenvalue as the detection filter

matrix operator which, when applied to _, nulls out the components corresponding to the projections of the residual onto

the _IN and _2N vectors

measurement noise covariance

Laplace transform variable

variance or matrix operator which, when applied to c, forms a

two-dimensional vector containing the components corresponding

to the projections of the residual onto the diN and d2N

vectors.

xvi
LIST OF SYMBOLS(Cont.)

S(kf, 8)

t

T,T2

u

ut

u

--c

u

--nominal

v(k)

w(k)

x

x I

Y_

y_'

Y--nominal

z

l(k)

Au'

Ax

Ax'

A_y

8

X

discrete-time information matrix at increment kf, failure

occurring at time e

time

threshold

nonlinear or linear system control input vector, dimension m

detection filter control input vector

control system commands to the actuators

nominal control surface deflections

zero mean Gaussian white noise vector with covariance matrix

R(k)

zero mean Gaussian white noise vector with covariance matrix

Q(k)

nonlinear or linear system state vector, dimension n

detection filter state vector, dimension n

nonlinear or linear system output vector, dimension r

detection filter output vector, dimension r

nominal output vector, dimension r

z-transform variable or exponential decay term

Kalman or age-weighted residual at time step k

linear input vector, dimension m

linear input vector to the detection filter, dimension m

linear state vector, dimension n

linear detection filter state vector, dimension n

linear output vector, dimension r

linear detection filter output vector, dimension r

time of failure

detection filter eigenvalue

xvii
LIST OFSYMBOLS(Cont.)

A(kf, 8,9)

_i(k)

l(k )

_(k)

xCk )

likelihood ratio at final iteration kf for failure of magnitude _ occuring at time 8

magnitude of failure

ith basis function of OSGLR algorithm at increment k

vector of basis functions

linear system discrete time state transition matrix,

dimension nxn

information vector at time k

Superscript

-I

t

+

inverse

pseudo inverse

quantity prior to filter update

quantity after filter update

Subscript

a

d

f

N

0

refers to OSGLR basis function coefficients

discrete

final

vector has been normalized

initial or nominal

refers to OSGLR basis functions

xviii
SECTIONI

INTRODUCTION

Recently, there has been much interest in the problem of restructuring the control law of aircraft following the failure of control surfaces or actuators (References 1,2,3). This interest is motivated in

part by two incidents involving commercial aircraft. In the case of

Delta flight 1080 on April 12, 1977, the left elevator became stuck in

the 19 ° up position at takeoff (Reference 4). The pilot was able to compensate for the failure, in part by manipulating thrust to control the

pitch axis. However, the pilot of the DC10 that crashed in Chicago

(Reference 5) was unable to recover from the left engine breaking loose

and the resulting retraction of the left wing's outboard leading edge

slats. Simulations indicate that the aircraft could have been flown if

impending stall conditions had been recognized and the proper corrective

action taken. Restructuring the control system on-line to counteract the

effect of these failures may be one solution to such problem situations.

Although the need for restructurable control has been demonstrated for

state-of-the-art aircraft systems, it can be expected to be most applicable to future aircraft where redundant control surfaces will very likely

be extensively employed.

A feasible and practical restructurable control system requires

the correct and timely detection and isolation of the control surface

failure so that the proper corrective action can be taken. The Charles

Stark Draper Laboratory, Inc. (CSDL) has investigated and evaluated practical FDI techniques for the NASA Langley Research Center under contract

NASI-17556 entitled, "Evaluation of Failure Detection and Identification

-I-
Techniques for Application in Aircraft Restructurable Control Systems."

Reference 6 and this report document the results of this contractual

effort.

The initial phase of the CSDL study, Reference 6, compared three

failure detection and isolation techniques for detecting and isolating

control surface or actuator failures: the detection filter, the generalized likelihood ratio (GLR) test, and the orthogonal series generalized

likelihood ratio (OSGLR) test. A modification to the basic detection

filter was also considered. The algorithms were evaluated by testing

their ability to detect and isolate control surface failures in a nonlinear simulation of a C-130 aircraft. The testing was in general limited to a single cruise operating condition. All the algorithms were capable of detecting failures. However isolating aileron and flap failures

was difficult since these surfaces produce similar effects on the aircraft. Since the algorithms had difficulty isolating wing surface failures, the system monitoring strategy for implementing the FDI algorithms

would augment the FDI algorithm with position measurements of these surfaces.

The detection filter (References 7,8,9) has the form of an observer with the feedback gain matrix chosen so that each type of failure produces a uniquely defined residual. With direct input-output coupling resulting from the use of acceleration measurements, the residuals produced

by control surface failures may only be constrained to a known plane.

The modification to the basic detection filter uses secondary filtering

of the residuals to produce unidirectional control surface failure signatures with acceleration measurements.

The GLR and OSGLR algorithms (References 10 and 11 respectively)

are sequential hypothesis tests. The failure hypothesis for the GLR algorithm assumes an additive failure of a known behavior or, equivalently,

failure mode with unknown onset time and unknown magnitude. The failure

mode assumed for this study was a step. Based on the residuals from the

Kalman filter designed for the no-failure system, the maximum likelihood

-2- i
estimate of the failure magnitude is determined for each possible failure

onset time. Using this estimate of the failure magnitude, the likelihood

ratio is maximized over the possible onset times to produce the generalized likelihood ratio test statistic.

Instead of assuming a fixed failure mode, the OSGLR algorithm expresses the failure mode as a truncated series of orthonormal basis functions. Based on the residuals of a no-failure Kalman filter, the coefficients of the series expansion are estimated using maximum likelihood estimation and a generalized likelihood ratio is formed using these estimates. The rationale for adopting this approach is that most failures

should be represented fairly well using a truncated orthogonal series expansion and this algorithm should be more robust to failure mode uncertainty than the conventional GLR test. Unlike the GLR algorithm, the onset time of the failure does not need to be estimated and therefore the

algorithm is computationally more efficient.

The OSGLR algorithm provided the best detection and isolation performance. The disadvantage of the OSGLR algorithm is its robustness to

modeling errors was worse than the other algorithms. The GLR algorithm

performed almost as well as the 0SGLR algorithm. However, the computational burden of the GLR algorithm is very heavy and it is not robust to

failure mode uncertainty. The OSGLR and GLR algorithms were less sensitive to severe turbulence than the basic and modified detection filter

algorithms because the turbulence model was incorporated into the nofailure Kalman filter. The sensitivity of the detection filter algorithms to turbulence degraded their detection performance relative to the

OSGLR and GLR algorithms. The advantage of the detection filter algorithms is their relatively low computational processing. Based on this

comparison, the OSGLR algorithm was selected for further study as the

most promising of the algorithms.

This report focuses on an in-depth evaluation of the OSGLR algorithm. This evaluation considers a variety of steady winds and wind turbulence levels and a number of flight conditions or flight regimes repre-

-3-
sentative of the range of environments in which a transport aircraft must

operate. Detection and false alarm performance issues are also addressed. Sensor dynamics and errors are included in the simulation.

In addition, to ensure all promising practical FDI techniques have

been evaluated, the literature is searched for FDI techniques that might

be applicable to control surface and actuator failure detection and

isolation in addition to those already studied. Any promising techniques

are to be included in the detailed evaluation described above.

This report is organized as follows. Section 2 reviews the initial phase of this study and summarizes the results. Section 3 surveys

the literature for other applicable FDI algorithms. Since the major

drawback to the OSGLR algorithm is robustness to modeling errors, Section

4 examines possible methods of reducing the effects of modeling errors on

the algorithm. The testing at a number of off-nominal flight and wind

conditions of the algorithm designed using a single cruise linear model

is described in Section 5. Also a number of failure cases are described

in order to determine the detection performance of the algorithm.

Section 6 examines scheduling as a means of extending the algorithm to a

larger flight envelope. The report is summarized and the major conclusions presented in Section 7. Appendix A contains a table summarizing

the test cases presented in this report for easy reference and comparison. Appendix B contains additional implementation details of the scheduling approach.

-4-
SECTION2

BACKGROUND

2.1 Introduction

The previous work done on this contract (Reference 6) analyzed and

compared the feasibility of using three FDI algorithms to detect and isolate aircraft control surface and actuator failures. The three algorithms were the detection filter, the generalized likelihood ratio (GLR)

test, and the orthogonal series generalized likelihood ratio (OSGLR)

test. A modification to the detection filter was also developed and

tested. In addition, a system monitoring strategy was developed. The

present effort further investigates the most promising of the FDI

algorithms evaluated - the OSGLR algorithm. In order to provide the

necessary background for this effort and to motivate our selection of the

OSGLR algorithm as the most promising, this section reviews the prelimary

evaluation of the FDI algorithms and system monitoring strategy described

in Reference 6. The nonlinear C-130 aircraft simulation used in the

evaluation is described in the next section. A description of the algorithms evaluated, a comparison of the algorithms, and the selection of

the system monitoring strategy follows.

2.2 C-130 Simulation Description

The FDI algorithms were evaluated by testing their ability to detect and isolate control surface failures that have occurred in a

simulation of a Lockheed C-130 aircraft. The C-130 aircraft is a military, medium- to long-range transport propelled by four turboprop engines

-5-
located on a high wing. The particular version of the C-130 aircraft

used for this program has short takeoff and landing (STOL) capability

provided by trailing edge double-slotted flaps.

The simulation of the C-130 aircraft uses the standard six

degree-of-freedom aircraft nonlinear equations of motion. The aerodynamic forces and moments are described by one-, two-, or three-dimensional look-up tables. These look-up tables are functions of angle of

attack, sideslip angle, thrust, and the controlsurface deflections.

Each of the four engines are assumed to provide the same thrust.

Actuator dynamics have been included; however, sensor dynamics were not

included.

The surfaces available for control of the aircraft are the

ailerons, flaps, rudder and elevator. The simulation allows for independent motion of the left and right ailerons and the left and right flaps.

Since aileron and flap failures are similar in their effect on the

dynamics of the vehicle, detecting and isolating aileron and flap

failures provided an adequate test for the algorithms that were

evaluated.

The eleven measurements available for detecting and isolating

failures are those typically available onboard aircraft. These measurements are listed in Table 2.1, along with the six control inputs described above and the ten states that describe the aircraft dynamics. The

measurements are generated in the simulation by superimposing zero-mean

Gaussian distributed noise on the output variables. The noise statistics

used for this study are shown in Table 2.2.

Wind turbulence is also incorporated in the simulation. The turbulence velocity along each body axis is modeled by passing white noise

through shaping filters to produce signals with desired one-dimensional

power spectral densities. The Dryden form of the spectra, defined in

Reference 12, is modeled. The turbulence intensities used in the preliminary evaluation are presented later in this section. The turbulence

scale lengths were the clear air values defined in Section 3.7.3.2 of

Reference 12.

-6-
Table 2.1. Inputs, Outputs, and States of the C-130 Aircraft

Inputs

Elevator

Right aileron

Left aileron

Right flap

Left flap

Rudder

Outputs

Airspeed

Acceleration at the cg along the y body axis

Acceleration at the cg along the z body axis

Angular velocity about the x body axis

Angular velocity about the y body axis

Angular velocity about the z body axis

Roll

Pitch

Yaw

Altitude rate

Altitude

States

Airspeed

Sideslip angle

Angle of attack

Angular velocity about the x body axis

Angular velocity about the y body axis

Angular velocity about the z body axis

Roll

Pitch

Yaw

Altitude

--7--
Table 2.2. Standard Deviation of Sensor Noise

SENSOR

Airspeed

Accelerometers

Roll Rate Gyro

Pitch and Yaw Rate Gyros

STANDARD DEVIATION

3.35 m/s

•3 m/s 2

°0024 rad/s

•0007 rad/s

11 ft/s

•98 ft/s 2

o1375 deg/s

•04 deg/s

Attitude Gyros

Altitude Rate

Altitude

•01 radians

•08 m/s

3.05 m

•573 degrees

.25 ft/s

10 ft
2.3 Description of FDI Algorithms Evaluated

The three FDI algorithms considered, the detection filter, the generalized likelihood ratio test, and the orthogonal series generalized

likelihood ratio test, are described in this section. A modification to

the detection filter was also developed and tested. The modification

will be referred to as the modified detection filter and will be discussed as a separate algorithm.

2.3.1 The Detection Filter 1

The detection filter (References 7,8,9) is a linear, time-invariant

observer of the system (see Figure 2oi)o Therefore, the nonlinear system

excluding the actuator dynamics must be linearized about a nominal operating condition for use in the observer. The model can be expressed as

x(t) = Ax(t) + Bu(t) (2.1)

y(t) = Cx(t) + Du(t) (2.2)

where x(t), u(t), and y(t) are the deviations in the states, controls,

and outputs respectively from their nominal values. The superscript

tilde in Figure 2.1 denotes the actual nonlinear qualities. Detection

filter theory cannot account for the effects of disturbances, measurement noise, or mismodeling. Their effects on the performance of the

detection filter will be determined via simulation.

The detection filter state and residual satisfy

x'(t) = Ax'(t) + Bu'(t) + Kr(t)

r(t) = y(t) - y'(t)

(2.3)

(2.4)

1 This description was taken from Reference 13.

-9-
0

"1

-10-
where

_y' (t) = Cx'(t) + Du'(t) (2.5)

Here, the prime denotes the detection filter state, output, and expected (no-failure) control input. In the absence of failures, the detection filter state estimation error satisfies the differential equation

_(t) = (A - _C)q(t) (2.6)

and the residual in terms of the state estimation error is given by

r(t) = Cq(t) (2.7)

-- m

In the presence of a failure of the ith actuator or control surface,

the filter error and residual dynamics are given by

4(t) = (A - KC)q(t) + (_i - K_i)n(t) (2.8)

r(t) = Cq(t) + d.n(t) (2.9)

where b. and d. are the ith columns of the B and D matrices, and n(t) is

-I -i

a function of time which represents the failure mode (i.e., the behavior

of the failed surface relative to the expected behavior). Detection filter theory allows n(t) to be any arbitrary function of time.

Since the gain matrix K must be chosen such that the filter is

stable, the detection filter will track the system when there are no

failures. However, when a failure occurs, the actuator models or the

linear model incorporated into the filter no longer accurately model the

system, and a nonzero residual results. The objective of detection filter design is to constrain the residual produced by a particular failure

to a single direction or plane. A failure is detected, then, when one or

-ii-
more of the residual projections along a known failure direction or in a

known failure plane are sufficiently large. Isolating a failure maybe

possible if one of the residual projections is significantly greater than

the rest.

Prior to this work, the development of the detection filter assumed

no direct coupling between the inputs and the outputs (i.e., no D

matrix). Without input-output coupling, present design procedures maybe

able to determine K such that the residual is unidirectional in response

to a control surface or actuator failure. If such a gain matrix exists,

the residual will be constrained to the direction cAkb. where k is the

-l

smallest positive integer for which cAkb. is nonzero. Since n(t) does

-l

not affect the failure direction, the failure direction is independent of

the failure mode. This is an important advantage of the detection

filter.

Using acceleration measurements to detect actuator and control surface failures, however, results in direct input-to-output coupling. In

this case, the residual produced by such failures may be constrained to a

plane spanned by the vectors d. and C(b. - Kd.) if the part of the

-i -l -l

residual produced by the (b. - Kd.)n(t) term can be restricted to a

-i -I

single direction. The difficulty in determining a gain matrix K to do

this is that present design procedures assume the direction of the additive term to the filter state error equation is known. However, the

direction of the (b. - Kd.)n(t) term is not known until the gain matrix

-I -I

is determined. Fortunately, knowledge of this direction is not necessary

when the system is fully measured (i.e., the rank of C is equal to the

number of states). The system used in the preliminary evaluation was

fully measured.

2.3.2 The Modified Detection Filter 2

Secondary filtering of the detection filter residual was found to

restore the unidirectional failure signature property even when there is

direct coupling between the inputs and the measurements. Assuming that

2 This description was taken from Reference 13.

-12-
the detection filter gain matrix, K, is calculated to satisfy the

relationship K = A - AI, the transfer function between the residual and

the unexpected input from the ith actuator is

C(b. - Kd.)

r(s) = [ -i -i S - A + d.]_l n(s) (2.10)

If the contribution d.n(s) could be filtered with the same time

-i

constant as in the detection filter, the failure signature would be

unidirectional, lying along C(b. - Kd.) + d..

-I -i -I

The secondary filtering scheme, then, has several elements. First,

the components of the residual along the event vectors d. are separated

-I

from the residual. Next they are filtered using the detection filter

time constant. Finally, these filtered components are then added to

the other components, forming a new residual which is used for failure

detection and isolation.

In order for the initial separation of the components along d. to be

-I

possible, all event vectors d. must be mutually independent, and each

-1

must be independent of the hyperplane formed by the C(b. - Kd.) vectors.

-i -I

To obtain the components of the residual in the directions d., first

-I

write the residual

r(t) Cl(t)

c2(t)

c8(t)

(2.11)

= Pc(t) (2.12)

Here, C(b. - Kd.) has been .... n_ mallzed• for each _v_ the six actuators

-I -I

considered in the preliminary evaluation, giving C(b_i - Kd_i)N, and _IN

-13-
and

_2N are the two distinct directions among the columns of the D matrix

resulting from the two acceleration measurements. The vector c(t) is

obtained using

c(t) = pTr(t) (2,13)

m

where P% is the Moore-Penrose pseudo inverse of P.

The magnitudes c 7 and c 8 are passed through a secondary two-state

filter:

4'(t)= Iq,(t)+ I,cTCt)q (2.14)

8(t)J

Substituting these filtered components for the unfiltered ones in c(t)

leads to

cf(t)

m

= cI (t)

c6(t)

!

ql (t)

q_(t)

(2.15)

Transforming this vector of components back into the original residual

space results in

_f(t) = P_f(t) (2,16)

-14-
Actuator failures maybe detected and (simultaneously) isolated through

projection of rf(t) onto each of the six signature vectors

C(b. - Kd.) + d..

-l -l -l

In using the transformation P, _(t) has been assumed to lie, in this

case, in the eight-dimensional space spanned by columns of P. Applying

transformations P% and P leads to suppression of noise in the residual

that is in directions orthogonal to the range of P. Because noise in

these directions only interferes with failure detection and isolation,

this suppression could be very beneficial. However, the transformation

is model-dependent and therefore will be sensitive to modeling errors.

The modified detection filter is shown in Figure 2.2

2.3.3 The Generalized Likelihood Ratio (GLR) Test for

Dynamic Systems 3

The GLR test (Reference 10) assumes a linear, time-varying system.

The discrete-time case will be considered here. In the normal or unfailed mode of operation, the state equation and measurement equation

are given by

HO: _(k+1) = _(k)_(k) + B(k)_(k) + _(k) (2.17)

z(k) = C(k)_(k) + D(k)_(k) + z(k) (2.18)

where _(k) is an n-dimensional state vector, _(k) is an m-dimensional

measurement vector, _(k) is the state dynamics matrix, and C(k) is the

measurement matrix. B(k) and D(k) are the input matrices for state and

output equations with _(k) being the control input, w(k) and v(k) are

zero-mean, independent, Gaussian white processes with intensities Q(k)

and R(k), respectively. For the analysis that follows, the effects of

the control input may be omitted, due to the linearity of the equations.

3 This description of the GLR test was partially taken from Reference 6.

-15-
Av

+

x

El

Ioo

(J

I,.I 0

4a

*r-I

4.4 'I 0 '°

-,-I

4J0

4-I

¢}

-,-I

4-I

-_I

"0

4=1_

00

¢N

-f.t

-16-_
The GLR failure hypothesis is that

H 1: x(k+1) = #(k)x(k) + w(k) + b(k)n(k, 8) 9

y(k) = C(k)x(k) + v(k) + d(k)n(k, 8)9

(2.19)

(2.20)

where _(k) and _(k) are known vectors that depend on the type of

the failure. For example, if an actuator failure is modeled, b(k)

will be the column of the matrix B(k) corresponding to that acuator,

and _(k) will be the corresponding column of D(k). n(k, 8) is the

mode shape, or simply mode, of the failure, which occurs at time e.

Generally, we have that

n(k, e) = o, k < 8 (2.21)

For example, if a bias failure is assumed, then

0 k < @

n(k, e) = (2.22)

1 k> e

Finally, v is the magnitude of the failure.

One of the difficulties with the GLR test is that the failure mode

is assumed to be known in advance. While specifying the mode of the

failure will most likely improve the detection performance of a

failure for that specific mode shape, the detection performance of

other failure modes probably will be degraded. Adequately enumerating

all the possible failure modes is a practical consideration which must

be dealt with.

Suppose the data _(k) are observed over the observation interval

k 0 ! k _ kf (2.23)

-17-
For a given time of failure, 8, and magnitude of failure, 9, the

Likelihood Ratio (LR) is given by

P(Y(k0), _Y(k0+1), ..., y(kf)IH I, 8,9)

A(kf,e, _) = (2.24)

p(y(k O),y(kO+1), ..., y(kf)IHo)

Because the _(k) are not independent from time step to time step, the

evaluation of the conditional probabilities is difficult. To evaluate

the LR, a Kalman filter is implemented, based on the normal mode (H 0)

system. The filter equations are

x-(k+1) = _(k)x+(k) (2.25)

__+(k) = _-(k) + K(k)!(k) (2.26)

where K(k) is the Kalman gain matrix, and _(k) is the residual, given by

l(k) = z(k) - C(k)x-(k) (2.27)

The Kalman gain matrix is given by

K(k) = P- (k)cT(k)M -I (k) (2.28)

where P-(k) is the covariance of the estimation error

^

e-(k) = x(k) - x-(k) (2.29)

and M(k) is the covariance of !(k), given by

M(k) = C(k)P-(k)CT(k) + R(k) (2.30)

-18-
The covariance is propagated by

P-(k+1)

P+(k)

= #(k)P+(k)#T(k) + Q(k)

= [I - K(k)C(k)]P-(k)

(2.31)

(2.32)

The LR may then be written in terms of the residual sequence_(k) rather

than the measurementsequence. Becausethe residual sequence is (conditionally) a white Gaussian sequence, the LR is easier to determine in

terms of !(k) than in terms of z(k).

Due to the linearity of the state equation and the filter equations, the residual may be expressed under each hypothesis as

H0: _(k) = Y_o(k) (2.33)

HI: !(k) = Y0(k) + _(k,@)9 (2.34)

where __o(k) is a zero-mean, white Gaussian sequence with covariance

M(k). g(k, 8) is the failure signature of a failure occurring at time @.

g(k, 8) is given by

g(k, 8) = C(k)f(k, 8) + d(k)n(k, 8) (2.35)

where f(k, 8) is the influence of the failure mode n(k, @) on the state

estimation error, f(k, 8) may be generated recursively by

f(k+1,@) #(k)[I- K(k)C(k)]f(k,@)+ [b(k)- _(k)K(k)d(k)]n(k, 8)

(2.36)

with the initial condition

f(k 0, 8) = O (2.37)

-19-
It can be seen therefore that

_(k,8) = _, k _ 8 (2.38)

Because the residual sequence is (conditionally) Gaussian and

white, the Log Likelihood Ratio (LLR) ratio has a particularly simple

form:

where

£(kf, 8, _)

I

x(kf, 8) - _ 92S(kf, 8)

×(kf,e) = _f g_Ck,e). -_(k)!ck)

k= 0

(2.39)

(2.40)

SCkf, e) : _ g_(k,e). -_(k)g_(k)

k=O

(2.41)

Now, the generalized likelihood ratio is given by

£(kf) = max £(kf, 8, _)

_,$

Performing the maximization over _ first, we have that

(2.42)

^ x(kf, 8)

v(kf, e) =

S(kf, e)

Hence, the GLR test statistic is given by

1 X2 (kf, 8)

£(kf ) = max

2 s(kf,e)

(2.43)

(2.44)
As a matter of convenience, the GLR decision function is defined by

2

X (kf, 8)

DF(kf) = 2 £(kf) = max ^ (2.45)

S(kf, 8)

A failure is detected when the decision function exceeds the detection

threshold.

A closed-form solution to the above maximization does not exist

in general. Therefore, in order to implement the GLR, the statistic

£(kf,8) must be computed for all possible times of failure, 8. As a

result, a bank of matched filters that grows linearly with time is

required. To avoid this unlimited growth in the amount of computation,

the assumed time of failure may be restricted, say, to be in the range

kf - N < @ _ kf. Even so, the amount of computation required to

implement the GLR can be quite large, especially if N is large. This

problem arises because the unknown parameter 8 enters the equations in a

nonlinear fashion. Hence, a nonlinear estimation structure is required

to estimate 8. This problem is typical of FDI tests in which the onset

time of the failure appears as an unknown parameter of the failure hypothesis.

2.3.4 Orthogonal Series Generalized Likelihod Ratio (OSGLR) Test 4

The OSGLR test (Reference 11) is similar in many ways, to the GLR

test discussed previously. The discrete-time version will be presented

here. A linear, time-varying system is assumed. The unfailed system is

the same as the unfailed system assumed for the GLR test (Eqs. 2.17 and

2.18). However, the OSGLR failure hypothesis is assumed to have the form

HI: _(k+1) = _(k)_(k) + w(k) + b(k)f(k) (2.46)

y(k) = C(k)_(k) + z(k) + d(k)f(k) (2.47)

4 This description of the OSGLR test was partially taken from

Reference 14.

-21-
where b(k) and d(k) are known vectors that depend on the type of the -- I

failure, and f(k) is the mode shape, or simply mode, of the failure.

Because the failure mode is generally not known a priori, it would be

desirable to allow the failure mode f(k) to be completely arbitrary. The

approach that will be taken here is to represent the mode shape f(k) by a

truncated series expansion. The motivation is that if the basis functions of the expansion are chosen properly, it should be possible to approximate any well-behaved failure mode. In this sense, the 0SGLR test

is similar to the detection filter which assumes the failure mode to be

unknown.

The data is assumed to be observed over the interval [ko,kf]. A

constraint is imposed on the form of the series expansion, namely, that

it has the form

P

f(k) = _ ai(k f) _i(kf-k) (2.48)

i=I

where p is the number of basis functions used in the series expansion.

That is, the series expansion is expressed in terms of basis functions

which are defined relative to the end of the observation interval, kf.

Eq. 2.48 may be expressed more conveniently in vector form as

f(k) = _T(kf-k)_(kf) (2.49)

Eqs. 2.46, 2.47, and 2.49 specify the failure hypothesis H I .

Note that the left side of Eq. 2.49 is a function only of k,

whereas the right side is a function of both k and kf. This apparent

contradiction is resolved by choosing the basis functions and

coefficients such that the right side of Eq. 2.49 is invariant with

respect to kf. This requires that _(kf) satisfy the difference

equation

a(k f+1 ) = Aa_a (k f )

(2.50)
where A is a constant matrix. Furthermore, the vector of basis functions

a

_(i) must satisfy the difference equation

i(i+1 ) = A_(i) (2.51)

where A_is a constant matrix and i is a dummy time index defined

by

A

i = kf-k (2.52)

That is, i defines a time scale that runs backwards from the end

of the observation interval, kf.

related by

The matrices Aa and A_ are

-T

Aa = AS (2.53)

The measurement process y(k) over the interval [k0,kf] is used

to form the likelihood ratio for the hypotheses H 0 and H I . Because

the measurement process is correlated in time, it is difficult to determine the likelihood ratio in terms of y(k). Instead, the data are

filtered using a Kalman filter based on H 0 to produce the residual

process, !(k). Because the residual process is white (when no failure

has occurred), it is more easily used to obtain the likelihood ratio.

The Kalman filter discrete-time equations are given in the GLR test

description (Eqs. 2.25 - 2.32).

Due to the linearity of the state and measurement equations

and of the Kalman filter, the OSGLR hypotheses may be written as

H0: _(k) = Y_0 (k) (2.54)

Hi: l(k) = _(k) + G(k)a(k) (2.55)
where _o(k) is a zero-mean, Gaussian white processes with intensity

M(k) o G(k) represents the influence of the coefficient vector a(k)

on the residual. The residual influence matrix G(k) is given by

G(k) = C(k)F(k) + d(k)_T(0)

where F(k) is the matrix that represents the influence of a(k) on

the estimation error. F(k) satisfies the difference equation

_(k) = {A(k)[_ - KCk)C(k)]F(k)

+

(2.56)

(2.57)

with initial condition

F(k 0) = 0

(2.58)

The information in the residual process may be reduced to a

sufficient statistic _(k) that contains all the information in the

residuals about the two hypotheses (Reference 15). x(k) satifies the

difference equation

x(k+l) = A_(k) + GT(k+l)M-l(k+1)l(k+1)

(2.59)

with initial condition

x(k 0) = GT(k0)M-I(k0) l(k0 )

Now, _(k) is a Gaussian random vector, because l(k) is a Gaussian

random process. Under H0, the mean of _(k) is the zero vector.

The covariance of _(k), S(k), satisfies the difference equation

S(k+1) = A S(k)A_ + GT(k+I)M-I(k+I)G(k+I)

(2.60)

(2.61)

-24-
with initial condition

S(k 0) = GT(k0)M-1 (k0)G(k 0) (2.62)

Under H I , _(k) has the same covariance, but its mean is given by

 [xCk)] = s(k) Ck) (2.63)

Therefore, the OSGLR hypotheses may be rewritten as

HO: x(kf) ~ N(_OO,S(kf)) (2.64)

H I : x(kf) ~ N(S(kf)a(kf),S(kf)) (2.65)

Finally, the log likelihood ratio for these hypotheses is given by

£(kf,_(kf)) = _aT(kf)_(kf) _ _1 a T(kf)S(kf)a(kf) (2.66)

Because the vector of coefficients a(kf) is unknown, an appropriate test statistic to use is the generalized likelihood ratio.

this case, the decision function DF(k) is defined by

DF(kf) = max^ 2£(kf,_(kf))

_(kf)

= _T(kf)S-1 (kf)_(kf)

In

(2.67)

A failure is detected whenever the decision functions exceeds a

threshold. That is, the detection test is given by

declare a failure

> 2

DF(k) T

<

continue testing

(2.68)

-25-
where T 2 is the detection threshold. The decision function is

written with k as the time argument, rather than kf, to emphasize

the fact that the test is to be implemented as a sequential test, although it is derived as a fixed-data test.

Note that the OSGLR test has a relatively simple structure. The

differential equations for _(k) and S(k) are simply the Kalman filter

equations for estimating a(k) in information form. The state equation

for _(k) is given by Eq. 2.50, and the measurement equation is given by

Eq. 2.55. The only other equation required to implement the OSGLR test

is the decision function calculation of Eq. 2.68. Hence, the OSGLR test

is no more difficult to implement than a single Kalman filter of order p.

In many respects, the OSGLR test is similar to the GLR test. The

GLR test also employs a Kalman filter to estimate the unknown failure

magnitude, u. Because _ is a constant, and the Kalman filter is implemented in information form, the resulting estimator is simply a matched

filter. However, a bank of filters is required, because 9 must be estimated for each possible time of failure, 8. Thus, the GLR test may be

computationally burdensome, especially if the number of possible failure

times (i.e., the width of the "data window") is large.

2.4 Evaluation and Comparison of the FDI Algorithms

The evaluation and qualitative comparison of the four FDI algorithms described in Section 2.3 in detecting and isolating control surface failures is presented here (Reference 6). The evaluation was conducted as follows. Each algorithm was designed and tested at a single

flight condition of 1000 ft. altitude and 150 knot airspeed. An additional flight condition was tested to determine the effects of modeling

errors. Only the time-invariant, steady state implementation of the GLR

and OSGLR algorithms was tested. The assumed failure mode of each GLR

failure hypothesis was a step function of unknown magnitude occurring at

time @. For each OSGLR failure hypothesis, six basis functions were used

-26-
in the truncated series expansion. The basis functions used were the

discrete-time equivalent of the Laguerre function with a 3 s time constant. The GLRand OSGLRalgorithms were tested in turbulence of intensity 1.98 m/s (6.5 ft/s) while the detection filter and the modified detection filter were tested mostly in turbulence of intensity 0.3 m/s (I

ft/s) although turbulence intensity of 1.98 m/s (6.5 ft/s) was also

briefly considered. The Kalman filter, used by both the GLR and OSGLR

algorithms, incorporated the turbulence model. Bias failures were introduced into the nonlinear C-130 simulation to test the two detection filter algorithms• To test the GLR and OSGLR algorithms, bias failures were

introduced into a linear simulation of the C-130 aircraft• In addition,

the nonlinear C-130 simulation was used to determine the effects of nonlinearities on the GLR and OSGLR algorithms and to test their ability to

detect ramp and stuck failures. Thresholds were not selected for the two

detection filter algorithms and a general detection and isolation logic

was not developed. In spite of these limitations, we believe that a sufficiently accurate picture of the capabilities of each algorithm was obtained to allow the algorithms to be qualitatively compared.

In comparing the four algorithms, the following issues were considered:

• Failure modes (bias, stuck, ramp, etc.) that can be detected

• Type of failure (rudder, elevator, etc.) that can be detected

and isolated

• Magnitude (or degree) of failures that can be detected

• False alarm performance

• Detection time (time delay between failure and detection)

• Computational burden

• Robustness

• Maturity

Each issue will now be individually addressed.
2.4.1 Failure Modes

Failure modes describe the behavior of the failed surface. Some

of the important failure modes for restructurable control applications

are stuck, hardover, and bias failures and the loss of part or all of a

surface. The ability of the algorithms to detect these and other failure

modes will be considered here.

The GLR test uses models of failure modes to detect failures. It

is therefore, most capable of detecting the modes that are modeled. However, each mode modeled requires a separate bank of filters which makes

modeling even a small number of possible modes computationally costly.

With only the bias failure mode modeled, the GLR algorithm was able to

detect bias and ramp failures. The GLR algorithm had difficulty deciding

if a stuck failure existed with only a 2 second data window; but had no

difficulty in doing so with a 5 second data window.

In the OSGLR algorithm, the failure modes are represented by a

truncated series expansion rather than a fixed function. The series

expansion chosen and the number of terms used determine how well a particular failure mode can be represented, using the first six terms in the

expansion, the OSGLR test was able to detect bias, ramp, and stuck failures (the only failure modes tested). These same six terms should be

adequate to detect most other failure modes.

One advantage of both the unmodified and the modified detection

filter is that all failure modes should be detectable as the residual

direction is independent of the failure mode, depending only on the surface which failed.

2.4.2 Type of Failure

All four algorithms were able to detect elevator, rudder, right

and left aileron, and right and left flap failures. The left and right

elevators were assumed to move together as a unit. Elevator and rudder

failures could be isolated by all four algorithms. However, isolating

-28- i
wing surface (aileron and flap) failures was difficult for all of the

algorithms. Based on the few test cases simulated, the modified detection filter could isolate a moderate wing surface failure (0.0349 rad

(2 °) aileron bias, 10% flap bias) to one of two possible surfaces in

minor turbulence. The detection filter algorithm could eliminate one

wing surface from consideration at most. It was demonstrated that false

isolation could result with the GLR algorithm. Isolation to a specific

wing surface is possible with the OSGLR algorithm with a significant delay of perhaps 10 seconds or more.

2.4.3 Magnitude of Failure

The magnitude of the failures that can be detected depends on the

sensor noise, disturbances, and modeling errors. The GLR and OSGLR

algorithms were able to detect moderate (0.01745 tad or I° elevator,

rudder and aileron) bias failures in the presence of noise and severe

turbulence (_ = 1.98 m/s or 6.5 ft/s).

The detection filter and the modified detection filter were able

to detect moderate (-0.0349 tad or -2 ° elevator, aileron, and 10% flap)

failures in minor turbulence (0 = 0.3 m/s or I ft/s). However,

detecting moderate failures in severe turbulence was more difficult.

While hardover failures were not tested, they should be easily detected

even in severe turbulence.

Modeling errors also degraded the ability of all the algorithms to

detect small failures. Detection of hard failures, though, should still

be possible.

2.4.4 False Alarm Performance

False alarm rates can only be determined via simulation because

analytic estimates are not available. (The analytic estimates of false

alarm rates for the OSGLR a_uL_t,hm __A__ cannot account for the effect of

modeling errors.) However, determining even large false alarm rates

using simulation is difficult because of the limited number of conditions

-29-
that can be tested and the large computational burden. Therefore, the

false alarm performance of each of the algorithms will be qualitatively

assessed by its sensitivity to noise, turbulence, and modeling errors.

Algorithms that are sensitive to these effects can be expected to have

larger false alarm rates than algorithms that are less sensitive.

The OSGLR and GLR algorithms were least sensitive to noise and

turbulence. This is true for two reasons: the system model incorporated

into these algorithms included a turbulence model, and a Kalman filter is

used to provide an estimate of the state. The unmodified detection filter was sensitive to turbulence while the modified detection filter was

very sensitive to turbulence.

Since each algorithm incorporated a linear model of the aircraft,

all four of the algorithms were very sensitive to modeling errors. The

OSGLR algorithm appeared to be the most sensitive to modeling errors.

2.4.5 Detection Time

The failure detection times depend on the magnitude of the failure, the sensor noise, the disturbances present, and the thresholds

selected. Selecting thresholds for the GLR and OSGLR algorithms higher

than the largest no-failure decision function levels at the nominal

cruise flight condition (nonlinear simulation) would result in detection

time for 0.01745 rad (I °) elevator and right aileron bias failures in

severe turbuleance on the order of a second. Using the same thresholds

for detecting rudder failures as for other control surface failures,

would result in longer detection times. The rudder decision functions

are smaller than those of the other control surface decision functions

for the same magnitude failure. However, the effects of mismodeling

affect the rudder decision functions less than the other decision functions. Therefore, smaller thresholds for the rudder decision function

could be used, allowing the rudder detection times to be comparable to

the other control surface detection times.

-30-
Detection times for both the modified and unmodified detection

filters also depend on the filter eigenvalue chosen and the time constant

of the low-pass filter, if any, required to suppress noise in the residual. The detection times were estimated based on the time delay between

failure onset and a clear indication that a failure has occurred. For

the unmodified detection filter, approximately two seconds would be

required to detect 0.0349 rad (2 deg) elevator or right aileron failure

in minor turbulence with the residual being low-pass filtered with a time

constant of one second. A 0.0349 rad (2 deg) rudder failure would take a

second longer. Detection times for the modified detection filter would

be on the sameorder as for the unmodified detection filter.

While the detection times were comparable for all the algorithms,

the GLRand OSGLRalgorithms were tested in turbulence of intensity 1.98

m/s (6.5 ft/s) as opposed to 0.3 m/s (I ft/s) turbulence intensity for

the detection filter algorithms.

2.4.6 Computational Burden

The computational burden of all of the algorithms has not been

quantitatively determined. Yet, some approximate comparisons will be

made here. Each of the algorithms consists of a filter of the system to

generate a residual, some type of residual processing, and a test for

failures. The filter portion of each algorithm is computationally equivalent. Therefore, the relative computational burden can be determined by

examining the computations required to process the residual and to test

for the failure.

The least additional computations are required by the unmodified

detection filter. The residual processing would likely consist of several banks of low-pass filters to give the algorithm the ability to quickly

detect hard failures and still detect soft failures. To test for a failure, the residual must be projected onto the failure signature plane segments for each control surface.

-31-
The modified detection filter requires slightly more computational

processing than the detectiton filter. The additional computations

result from the secondary filtering of the residual which restores the

property of a unidirectional residual in response to a control surface

failure.

The relative computational burden of the OSGLRalgorithm is primarily determined by the numberof actuators or control surfaces and by

the numberof terms in the series used to represent the possible failure

modes. The residual is used to drive an additional filter for each control surface to produce an information vector whose dimension is the

number of terms used in the series expansion to represent the failure

mode. The test for a failure in each actuator or control surface is the

information vector weighted by an information matrix. For the timeinvariant case considered in the preliminary evaluation, the information

matrix is a constant matrix. For a large number of both actuators and

terms used in the series, the computational burden would be very heavy.

However, keeping only the first six terms of the series expansion was

found to be adequate in the preliminary evaluation. With the six control

surfaces considered in the preliminary evaluation, the computational

burden of the OSGLR algorithm is at least 50% greater than the unmodified

detection filter.

The relative computational burden for the GLR algorithm is mainly

a function of the number of actuators, the number of failure modes modeled, and the data window chosen. Each actuator requires N correlation

receivers for each failure mode modeled where N is the length of the data

window divided by the time step. For a realistic data window of 2

seconds, a 0.02 second time step, six actuators, and only modeling the

bias failure mode, the GLR algorithm would be computationally very burdensome. A comparison of CPU times for this window suggests that the

GLR algorithm would require approximately 18 times more computation than

the OSGLR test.

-32-
2.4.7 Robustness

As each of the algorithms considered here relies upon a linear

model of the system to detect and isolate failures, these algorithms will

be sensitive to modeling errors. Modeling errors caused large GLR and

OSGLR decision functions. However, The 0SGLR test was shown to be more

sensitive to modeling errors than the GLR test. An off-nominal cruise

flight condition produced biases in the unmodified and modified detection

filter residual projections. With these biases, detecting small wing

failures at the nominal flight condition would be difficult, if not impossible, unless dynamic thresholds or some other compensating approach

is used.

Another source of modeling errors for the GLR and OSGLR algorithms

are the failure mode models incorporated into each of the algorithms.

The OSGLR algorithm is likely to be robust to actuator failure mode

modeling errors as the model is sufficiently general to represent most

modes adequately. However, the GLR algorithm required specific models of

failure modes such as bias failures. As only bias failures were modeled

in the preliminary evaluation, the GLR algorithm is likely to be less robust to other failure modes than the other three algorithms.

2.4.8 Maturity

The GLR algorithm is mature in both theory and in application.

The OSGLR algorithm is almost as mature in theory as the GLR algorithm.

However, as the OSGLR test is a recently developed FDI algorithm, there

is very little experience in applying it.

Detection filter theory is mature for restructurable controls application to linear, time-invariant systems with no input-to-output coupling (i.e. no acceleration measurements). However, no theory exists for

applying the detection filter to time-varying systems. In addition, for

systems with input-to-output coupling, systematic methods of using the

extra degrees of freedom in the gain matrix calculation (which result

-33-
from having more measurementsthan states) and scaling to improve detec

tion filter performance are needed. Finally, there is limited experience

in applying the detection filter with only a couple of applications hav

ing been reported.

The modified detection filter, developed for this application,

needs additional investigation to be considered mature in both theory and

application. The problems of time-varying systems, improving performance

through scaling, and the extra degrees of freedom in the gain matrix calculation mentioned above for the detection filter also apply to the modified detection filter.

2.4.9 Summary and Overall Comparison

The eight issues addressed in this memorandum are summarized in

Table 2.3. The GLR and OSGLR algorithms performed the best, especially

in severe turbulence. However, the computational burden of the GLR algorithm is heavy and its ability to isolate wing surface failure modes is

uncertain. The most significant advantage of the detection filter algorithms is their relatively low computational processing requirements. If

the sensitivity of the detection filter algorithms to turbulence could be

reduced, their performance might be comparable to the GLR and OSGLR algorithms. The most significant disadvantage of the OSGLR algorithm is its

sensitivity to modeling errors. Still, the OSGLR algorithm is the most

promising of these four algorithms evaluated.

2.5 System Monitoring Strategy

A second program task was to develop a system monitoring strategy

for the detection and isolation of aircraft actuator and control surface

failures. Before discussing the issues, the advantages and limitations

of three applicable FDI techniques will be considered: local actuator

FDI, analytic FDI algorithms such as those considered in this program,

and the use of surface position measurements.

-34-
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2.5.1 Local Actuator FDI

Direct duplication of actuation has been the practice for most

military and large commercial aircraft for quite some time. Whole actuation channels may be duplicated several times. There have been several

methods devised for dealing with failures of elements in these channels.

The subsystems and channels must be substantially identical in order to

give the same control inputs and to enhance the performance of the

system. Some differences are inevitable because of tolerances, and these

must be taken into account so that disengagement of a channel will occur

only under genuine failure conditions. The performance of the subsystems

and channels is continuously adjusted, in a process called equalization

(Reference 16). As a part of equalization, inter-channel differences are

minimized through feedback. If a difference is too great, then the failed channel is disengaged or bypassed. Frangible elements (shear pins,

for example) have also been used, allowing a jammed actuator to be broken

by the others. Thus, a large degree of actuator FDI (and reconfiguration) already takes place on a local level. However, there are failures

such as jams, actuator bias, inappropriate overall actuation gain, or

control surface damage which the local FDI scheme will be unable to detect.

2.5.2 Analytic FDI

Analytic FDI schemes, such as those considered in this program,

look for control surface failure signatures in the whole-system

dynamics. That they detect failures at the system level (ultimately the

most important one) is what makes these schemes potentially of great

value. They can be considered most useful in the context of aircraft

actuator FDI in identifying failures that the low-level FDI schemes have

missed. Such failures might include inaccuracy or breakdown of the local

schemes and actual physical damage to the control surface itself. However, analytic FDI schemes have been shown to have difficulty in fully

isolating failures among control surfaces that are functionally redun-

-37-
dant. Thus, isolating a wing surface failure to a specific flap or aileron was difficult. The quantitative measuresdeveloped to determine the

maximumdiscrimination of such failures showedthe inherent difficulty.

The value of actually isolating a failure of a surface that is

functionally redundant might be questioned to somedegree. After a surface fails, it is required that sufficient capability remain to end the

flight in an acceptable way. If there is sufficient capability and if

this can be appropriately determined then complete failure isolation

might be considered optional. However, the ability to quickly and accurately determine the functional capability of the aircraft while in

flight does not exist at the present time. Furthermore, there are reasons why isolation to a specific surface would be preferred. It is likely that a failure could be more quickly and appropriately compensatedif

it were fully isolated. Moreover, full isolation might allow for more

flexibility and confidence in continuing a flight after a failure has occurred. Therefore, full isolation capability is taken as the simpler and

more desirable option.

2.5.3 Control Surface Position Sensors

Position sensors mounted on or near the control surface itself

could be considered to provide information on a level just below that of

the whole system. Information from these transducers can be expected to

be reasonably easy to obtain and use. Position transducers of the synchro, potentiometer, and linear variable differential (LVDT) types have

been extensively used and are simple (Reference 17). For these position

transducers to be an independent source of information on any type of

failure, however, they should not be part of the actual flight control

loop. (Position sensors are not presently used by aircraft control systems.)

Flight control systems in which control loops are closed using

aircraft dynamic information will automatically compensate for some actuator failures, such as small surface bias errors. To identify larger

-38-
biases, comparing actuator position expected (using a reference model)

with that actually measuredby position transducers might suffice to isolate a failed surface. Alternately, in a separate actuator positioning

flight control modein benign flight conditions, commandingthe surface

to moveto someabsolute position or to movea certain fixed amount could

suffice to detect and isolate biases or incorrect gains. If the surface

is jammed, FDI using outputs from position transducers is also possible,

using tests similar to those described for biases.

Employing an FDI schemebased on surface position measurements

involves additional hardware and perhaps the design of separate flight or

test modes. In addition, FDI of the surface position sensors may also be

required. It should be stressed, too, that position transducers would

still be of only limited use in identifying certain types of failures,

such as actual control surface damage.

2.5.4 System Monitoring Strategy Selected

An analytic whole-system FDI scheme is preferable as it can detect

failures that may not be detectable by either the local FDI scheme or by

position sensors. However, since analytic algorithms have some

difficulty isolating failures, they could be augmented with direct surface position sensors to provide information for complete failure isolation, eliminating the uncertainties inherent in fully identifying the

failure of a functionally redundant surface. Only a limited nunmber of

transducers might then be required, and limited use made of their information. Use of position sensors to augment analytic FDI schemes also implies that each scheme could serve as a limited check on the other, providing a more reliable FDI system. While information from the local actuator FDI scheme may be of limited use also in this regard, this information will not be considered.

The system monitoring strategy for the C-130 aircraft considered

in this evaluation (assuming that the elevators move together) would be a

-39-
combination of the best analytic FDI algorithm (the OSGLRtest) with

position sensors on each of the wing surfaces.

2.6 Conclusions

The major conclusion which may be drawn from the results of this

study is that algorithmic failure detection and isolation may be feasible

for restructurable control applications. This conclusion must be qualified by the results obtained during this preliminary evaluation which

have been basically limited to a single operating condition and to the

investigation of a small subset of the potential failures. In particular, failure detection at the nominal flight condition does not appear to

be a problem. Each of the algorithms was able to detect small elevator,

rudder, aileron and flap failures for the C-130 aircraft in turbulence.

The isolation of control surface failures was not a problem for elevator

and rudder failures. However, difficulties arose in the isolation of

wing surface, i.e., aileron and flap, failures. These failures have a

similar effect on the responses of the aircraft system. This difficulty

highlights the potential need to augment the analytic FDI algorithms with

the direct measurement of failures using, for example, position sensors

on some of the control surfaces.

The OSGLR algorithm performed best of those evaluated. The general representation of the failure mode implies that most failure modes

should be detectable. All failure types investigated during this study

were detected and isolated, although there was a long time delay associated with isolating the wing surface failures. Bias failures on the order of a degree of surface deflection were detected in approximately a

second for a system without modeling errors. The computation and burden

associated with this algorithm is moderate relative to the others, and

the theory associated with it is mature. However, robustness properties

of the OSGLR test are worse than those of the other algorithms. In addition, the OSGLR algorithm has not been previously applied to any system.
The GLRalgorithm also performed well in the C-130 application.

The major drawbacks associated with it are its heavy computational burden, its lack of robustness to failure mode uncertainty, and its poor

wing surface isolation performance. The most significant advantage of

the detection filter algorithms is their relatively low computational

processing requirements. If the sensitivity of these algorithms to turbulence could be reduced, their performance might be comparable to the

GLR and OSGLR algorithms. None of the algorithm was robust to model uncertainty.

In our judgement, the OSGLR algorithm is the most promising of the

algorithms evaluated and, therefore, will be investigated further. The

poor robustness to modeling errors of the OSGLR algorithm can, we

believe, be improved; reducing the sensitivity to these errors of the

OSGLR algorithm is the subject of Section 4. Many of the ideas presented

there are applicable to other FDI algorithms as well.

-41-
SECTION3

A SURVEY OF FDI TECHNIQUES APPLICABLE TO AIRCRAFT

CONTROL SURFACE FAILURE DETECTION AND ISOLATION

3.1 Introduction

In order to determine if there are practical FDI alternatives to

the three algorithms already evaluated, a survey of FDI techniques that

might be applicable to detecting and isolating aircraft control surface

and actuator failures is presented. This survey is based largely on

Reference 18. A discussion of several general categories of FDI algorithms will follow, with attention to those algorithms that might be

usable in the aircraft application. The conclusions section summarizes

the results of this search.

3.2 Failure Sensitive Filters

The detection filter can be considered one of the class of failure

sensitive filters used for FDI. This class also includes limited memory

filters and filters augmented with the parameters of hypothesized failures included as state variables. Limited memory filters are effectively

used only for failure detection, because they are not failure specific.

By keeping the residual gain high, it is hoped that a failure can be

expected to result in a sudden and pronounced change in the filter

behavior. Filter state augmentation does help in this regard, but at the

expense of potentially quite large dimensionality. The augmented filter

will not be able to match the detection filter's ability to respond to

any failure manifested in a known direction, and the augmentation serves

to decrease performance of the filter when there are no failures.

-42- i
3.3 Innovations-based Systems

The GLR and OSGLR algorithms belongs to the group of FDI schemes

known as innovations-based systems. These monitor the innovations of the

(typically) Kalman filter designed for the unfailed system. This group

also includes simple residual magnitude monitoring for failure detection. The GLR algorithm attempts to detect and isolate additive failures

using knowledge of the different effects such failures have on the

(linear) system innovations. Because the exact form of the failure is

generally unknown, some estimate is needed in order to have an accurate

indication of failure. This failure estimation is inherent in any GLR

scheme. Matched correlation detectors, in general each assuming a

particular direction, mode, and time of failure, use as inputs the

residuals of the normal mode filter and estimate the unknown associated

with the failure. These estimates are then used in composite tests for

the presence of particular failures.

The OSGLR algorithm is similar to the GLR algorithm in many ways.

However, the OSGLR algorithm represents the failure mode by a truncated

series expansion. A matched filter for each control surface is used to

estimate the coefficients of the truncated series expansion. The order

of the matched filter is the number of terms in the truncated series expansion. Also, no explicit estimation of the time of failure is required

by the OSGLR approach.

In aircraft application, in which the direction of the failure is

known, effective use of the traditional GLR algorithm would probably involve hypothesizing one or more different failure modes, e.g., bias or

ramp. Assuming only certain limited modes can give the GLR algorithm

some robustness with respect to other, different modes of failures. The

OSGLR algorithm, however, avoids the complexity of specifying different

modes, since inherently it allows for any.

-43-
The traditional GLR algorithm explicitly estimates time of failure onset, resulting in a time-linear growth in the number of detectors.

Limiting the possible times of failure onsets to the recent past limits

this growth. However, this "windowing" may degrade detection and isolation performance, especially if the window is chosen such that the GLR

algorithm is computationally efficient. The windowing approximation need

not be made in the OSGLR algorithm, however. Although the OSGLR

algorithm cannot explicitly estimate time of failure, no new matched

filters need be created.

The FDI algorithms evaluated so far have been used with a linearized model of the C-130. FDI performance suffers when the operating condition is far from nominal. Caglayan and Lancraft have applied the GLR

ideas to systems for which the no-failure filter is an EKF (Reference

19). The nonlinear filter formulation gives a more complex but flight

path-independent scheme, eliminating the need for gain scheduling. In

the C-130 application, much of the model information is available only

through tables. This makes the prospect of EKF filtering unlikely without extensive changes in the modeling representation. This may merit

some consideration, however.

3.4 Multiple Hypothesis Filtering

Multiple hypothesis filtering schemes have also been proposed for

FDI. In the exact application of this concept, a "bank" of full-order

linear filters is constructed, with each filter based on a different

hypothesis concerning system behavior. The innovations from the filters

are then used to compute the probability that each system model is the

correct one. This type of scheme has all of the drawbacks of the traditional GLR algorithm, in that the number of filters grows with time.

Windowing of data is also possible here. Unlike the GLR algorithm, state

estimation for the failed system is immediately available with the

multiple hypothesis filter schemes since there is a filter valid for each

failure hypothesis. However, state estimation for the failed system with

-44-
the innovation-based systems is possible by approximately compensating

for the effects of an additive failure on the no-failure filter. If a

complete set of failure types and modescan be delineated, multiple

hypothesis filtering schemescan be expected to give the best possible

FDI performance. Their complexity, however, renders them impractical.

3.5 Generalized Bayesian Tests

Generalized Bayesian tests (Reference 20), as another class of sequential algorithms, seek to provide optimal failure detection and isolation through the assignment of costs for making certain decisions at certain times. The optimal FDI strategy minimizes the expected overall

cost. This class of algorithms can be considered the most general at the

present time. Potential application is to any system for which a priori

information concerning possible failures is available. (This is implied

in the term Bayesian.) Moreover, unlike most schemes, probabilistic information about the overall FDI scheme is a byproduct of the optimiztion.

The optimal FDI rule is generally not computable, but the structure of the approach can be expected to be useful in deriving a practical

suboptimal FDI algorithm. Reference 20 considers several possible

approximations.

3.6 Conclusions

The detection filter can readily be considered the best of the

failure-sensitive filters and the OSGLR algorithm the best of the innovations-based schemes for the linear time-invariant system with additive

failures. They have been implemented, and it is doubtful that more complex schemes would significantly enhance practical FDI performance. It is

likely that simpler schemes would not be able to provide timely and accurate full failure identification. It should be noted that generalized

Bayesian schemes are relatively new and undeveloped; a good characterization of their potential suitability will come only as a result of future

-45-
research. The attractive decision-cost optimality comes, however, with

solution of a difficult problem.

Based on this survey and the results of the initial evaluation

(described in Section 2), the OSGLR algorithm, in our opinion, has the

most potential of the algorithms that are mature for the Restructurable

Controls Application. Therefore, only the OSGLR algorithm will be investigated further.

-46-
SECTION4

ACCOMMODATION OFMODELINGERRORS FOR THE OSGLR ALGORITHM

Effective analytic fault detection and isolation schemes require

accurate system models. All the FDI algorithms investigated in the

preliminary evaluation, especially the OSGLR algorithm, were found to be

sensitive to modeling errors. Certain modeling errors are introduced in

linearizing aircraft nonlinear dynamics and in using a model during

periods of off-nominal flight conditions. These errors will, at least to

some extent, be attributed to control surface failures, and, consequently, lead to larger decision functions than if there were no such errors.

In fact, with significant mismodeling, the decision functions are

unbounded in their growth. Therefore, it is not always possible to avoid

false alarms associated with modeling errors by judicious selection of

the detection thresholds.

There are several ways of better accommodating modeling errors in

analytic FDI algorithms. These might be considered as the following:

• Dynamic thresholding - choosing decision thresholds based on

estimates of the system state or, perhaps, the disturbances

present (e.g., turbulence).

• Dynamic decision regions - determining the statistics of the

FDI residual, and choosing the threshold given those

statistics.

• Actual model error estimation, and use of those estimates in

offsetting residual errors.

• Robust design of the actual residual generator, leading to a

sub-optimal scheme over the expected range of model errors.

-47-
While someof the above approaches have been used to reduce modeling

errors in a number of FDI applications, no applications of FDI techniques

in which the errors are propagated through a filter have been discovered. The F-8 study (References 21 and 22), for example, uses some fairly

sophisticated modeling error accommodation, but the FDI is for sensors

whose output can be more or less directly compared.

This section will consider each of the four approaches in

attempting to reduce the sensitivity of the OSGLR algorithm to

midmodeling.

4.1 Dynamic Thresholds

Detection thresholds can be tied to estimates of the state or of

system disturbances or expressed as increasing functions of time during a

maneuver, perhaps, e.g., T = (a + bt) 2. While it is difficult to see

how dynamic thresholds can be systematically designed for filter

residual-based FDI schemes (like the detection filter and OSGLR), one

approach was investigated.

One measure of the sensitivity of the OSGLR algorithm to

mismodeling is

s-lx

rain (4.1)

f_-I

where _ is the information vector, S is the nominal information matrix

used to calculate the decision function, and S is the actual information

matrix. A graphical interpretation of this measure is shown in Figure

4.1. This minimization both determines the direction (in _ space) in

which the nominal implementation is sensitive to a particular mismodeling

T -1

and the largest a ellipse of the actual decision function _ S _ that is

contained in the I _ ellipse of the nominal decision function _rs-1l.--

-48- r

i

J
X SPACE ELLIPSE FOR xTs-I_xt

a a ELLIPSE FOR _xTs-1

WHERE

xTs -1 X

a = min

x-'r_-lK

"_ lo

II

/

ELLIPSE FOR xTs-Ix

Figure 4.1 Sensitivity measure for the OSGLR algorithm

-49-
If determining the actual information matrix S is possible, one

approach for compensating for the mismodeling effects might be to

increase the threshold, T 2, of the nominal decision function such that

the T _ ellipse would contain or mostly contain a TI_ ellipse for

2 is the threshold that yielded adequate performance with no fS-1_. T 1

mismodeling.

For mismodeling that results in changes in the linear model, the

actual information vector S can be determined by combining the Kalman

filter and OSGLR equations as implemented with the actual linear model

(including the effects of mismodeling) into a single linear system.

x(k+1 )

x- (k+l)

X1 (k+1) ,;

-- --i

= #*K*C

G*TM*-I C

0

_*(I-K*C*)

_G*TM*-Ic *

0

u

x(k)

_-(k)

XI (k)

+

m

B

B*+#*K*AD u(k) +

m

_(k)

#*k*v(k)

G*TM*-Iv(k)

(4.2)

where x(k) and x-(k) are the linear system state and the Kalman filter

state estimate respectively. _1(k) is the information vector time

shifted one time step:

Xl(k) = x(k-1 )

-50- i
The superscript (*) indicates that the matrix is either a nominal linear

model matrix or steady-state Kalmanfilter and OSGLRmatrices based on

the nominal linear model. The actual linear model matrices are not

superscripted and

AD _ D - D* (4.4)

The process and measurementnoise are represented by w(k) and v(k).

Since the information matrix is simply the covariance of _, the actual

information matrix S is the lower right-hand block of the augmentedstate

covariance matrix:

E[x(k)xT(k)] E[x(k)xT(k)] ...

• .. E[x(k)xT(k)] .o.

ego .go S

(4.5)

One difficulty with this approach is that the actual information

matrix is a function of the covariance of the inputs unless modeling

errors do not affect the B and D matrices. Instead of trying to appropriately describe the covariance of the inputs, the control system could

be included in the augmented linear system (Eq. 4.2). However, the

actual information matrix then becomes a function of the covariance of

the commands to the control system. One approximation might be to assume

the covariance of the commands to be zero. A much better approximation

would be to assume the pilot is a feedback system.

Assuming the covariance of the inputs is somehow described, the

actual information vector can be determined by propagating forward in

time the augmented state covariance matrix or by solving a Lyapunov

-51-
equation for the steady-state covariance matrix. In the latter case, the

actual linear model must be stable unless the control system is

included. Then the sensitivity of the OSGLRalgorithm to this change in

the linear model could be determined and the threshold compensated. This

analysis alone is not adequate in selecting thresholds as nonlinearities

not represented in the linear models are not taken into account. Also,

mismodeling may cause the information vector to becomebiased, or worse,

to grow without bound.

This sensitivity analysis was not performed in this effort as

describing the control system of the C-130 aircraft was prohibitive.

Without a systematic method to design dynamic thresholds, a trial-anderror procedure would have to be used.

4.2 Dynamic Decision Regions

Filter sensitivity results can also be used to implement dynamic

decision regions for the detection test used in the OSGLR algorithm. The

form of the detection test is the information vector weighted by the

information matrix and compared to a threshold.

ZS-Ix <> T (4.6)

x

The information matrix reflects the expected uncertainty associated with

the basis function coefficient estimates. The certainty predicted in the

filter generally does not take into account modeling errors, however. In

order to reduce false alarms, one might like S to reflect the uncertainties associated with modeling errors even in the absence of failures.

The basic idea then is to use the actual information matrix as

opposed to the nominal information matrix in calculating the decision

function --_Ts-I_. The actual information matrix could be calculated as

outlined in the previous section, instead of continuously updating the

information matrix, it would be more efficient to bound S in an

-52-
approximate manner, Even better would be to approximate S as a function

of operating parameters, states, or inputs. As discussed in the previous

section, certain nonlinearities and biases or unbounded growth in the

information vector caused by mismodeling cannot be dealt with using this

approach.

4.3 Model Error Estimation

A third option in accommodating modeling errors in FDI involves

estimating these errors directly. This could be, of course, a highly

effective option, but it would probably be difficult to account for much

of the error and would thus be computationally prohibitive. Moreover,

this extensive estimation could probably be foregone without bad

consequences if one of the other accommodation options were employed.

4.4 Robust Residual Generation

The OSGLR algorithm nominally depends on residuals produced by a

Kalman filter to detect and isolate failures. Either improving the

robustness of the Kalman filter or using a more robust filter than the

Kalman filter will reduce the effects of modeling errors on the OSGLR

algorithm. The need for reducing the sensitivity of the Kalman filter to

modeling errors is apparent even at the nominal cruise flight condition.

Figure 4.2 shows the predicted state covariance and the actual

aircraft state estimation errors for a segment of nominal, no-failure

flight, using a single linear model in the Kalman filter. Even here, a

case for which the modeling errors are expected to be small, the estimation errors are not well bounded by the covariance. Figure 4.3 shows the

corresponding OSGLR decision functions, which are large. 1 Practical

improvements might involve adding process noise to the Kalman filter (in

I Appendix A contains a summary of test cases presented in this report

for easy reference and comparison.

-53-
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.O i °

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no-failure, nominal cruise flight

condition test case.

-56-

i
addition to the turbulence process noise already present), or perhaps

using so-called limited memory filtering in the OSGLR. Either technique

curtails growth of the actual estimation error covariance and increases

predicted covariance. Use of either technique could also be expected to

desensitize the decision functions to errors due to both linearization of

the nonlinear dynamics and off-nominal maneuvering of the aircraft.

There is no method of systematically selecting process noise to

reduce the effects of mismodeling on the Kalman filter. The designer

must resort to a trial-and-error procedure of selecting the process noise

and testing the resulting Kalman filter. The other difficulty with this

approach is that the choice of process noise may reduce the effect of a

failure on the Kalman filter residuals, degrading the ability of the

OSGLR algorithm to detect and isolate that failure. In spite of these

difficulties, this approach was briefly investigated. A set of state

covariance weightings were determined such that 0.01745 rad (I ° ) elevator

and right aileron bias failures, 0.0349 rad (2 ° ) rudder bias failures,

and 5% right flap bias failures still could be detected and isolated at

the nominal cruise condition 77.2 m/s (150 knots) at 304.8 m (1000 ft)

while greatly reducing the no-failure decision function at the offnominal cruise condition 102.9 m/s (200 knots) at 1524.0 m (5000 ft).

(The process noise added was in addition to the turbulence process noise

already driving the system.) The decision functions for the 0.01745 rad

(I ° ) elevator bias failure and the off-nominal no-failure test cases

using the nominal (no process noise) Kalman filter and OSGLR algorithm

are shown in Figures 4.4 and 4.5, respectively. The decision functions

for the same two test cases using the Kalman filter and OSGLR algorithm

designed using process noise are shown in Figures 4.6 and 4.7. Scheduling the threshold in terms of distance from the nominal point might allow

for quick detection of failures close to the nominal cruise condition

while reducing the effects of mismodeling.

Limited memory filtering is to be preferred as it can be implemented in such a way that it is general, simple, and quite effective. The

-57-
o

O_

o

._

•_ C),

u

!

10.00 20.00 30.00 I&O.O0

Time (s)

50.00

Figure 4.4 OSGLR decision functions for a -0.01745 rad

(-i °) elevator bias failure which occurred

at i0 seconds at the nominal cruise flight

condition of 77.2 m/s (150 knots) and

304.8 m (i000 ft)
o

! !

O0 I0.00 20.00 30.00 qo.o0

Time (s)

Figure 4.5 OSGLR decision functions for a no-failure case

at an off-nominal cruise flight condition of

102.9 m/s (200 knots) and 1524 m (5000 ft)
Time (s)

Figure 4.6 OSGLRdecision functions for a -0.01745 rad

(-i °) elevator bias failure at i0 seconds at

the nominal cruise condition of 77.2 m/s and

304.8 m using a Kalman filter with process

noise added

-60-
O

O.

o

,-4

x

v O

O0 ! ! ! !

20.00 WO.O0 60.00 80.00 lO0.O0

Time (s)

Figure 4.7 OSGLR decision functions for a no-failure

case at an off-nominal cruise flight condition of 102.9 m/s and 1524.0 m using a

Kalman filter with process noise added

-61-
basic idea of limited memoryfiltering is the more or less gradual elimination of old data, as it is no longer thought to be meaningful. Oneapproximate method that can be used in the current application is the ageweighted filter (Reference 23). In this filter, discarding old measurement data is accomplished by weighting them according to when they occurred. In other words, the covariance of the measurementnoise is increased for past measurements. For assumedconstant measurementnoise

covariance R, one mannerof accomplishing this is to set

_-m = (emAt/T)R' m = 0,1 ,2... (4.7)

where R' is the new noise covariance and k denotes the current time in

the filtering. At is the measurement (sampling) interval, and T is a

time constant associated with the length of time the model is considered

valid.

A recursive filter can be constructed under these assumptions.

The only difference between the resulting algorithm and the standard

Kalman filter is the appearance of the age-weighting factor s = e At/T in

the state covariance update equation:

Pk (-) = SQk-lP'k-1 (+) _-I + Qk-1 (4.8)

Otherwise the age-weighted filter equations are the same as the Kalman filter equations (Eqs. 2.25-2.30 and Eq. 2.32). Compared with the

original Kalman filter, the factor s, being greater than 1, leads to

larger state and measurement covariances and, consequently, a larger

filter update gain. The result is a faster tracking filter. Only the

steady-state implementation of the age-weighted filter is tested in

this report.

If the measurement noise covariance matrix is actually constant

and the age-weighting is being used to produce a faster tracking filter
|

as in this report, the Pk is not the state error covariance matrix

^

Xk-Xk)T ] . The state error covariance matrix can be calculated

using results for a filter of the form of the Kalman filter but with an

arbitrary gain matrix. The measurement error covariance matrix is calculated by using the correct state error covariance matrix. For convenience, since the actual covariance matrices are not of interest, the covariance matrices including the effects of age-weighting will be referenced

to the age-weighted filter state error and residual covariance matrices.

The (normal) OSGLR FDI scheme involves the use of a no-failure

Kalman filter and what are essentially matched Kalman filters for the

various control surface failure hypotheses driven by the residuals of the

no-failure filter. (Recall that for each control surface there is basically an estimator for the coefficients of the failure signature function.) Implementing an age-weighting filter for only the first, nofailure filter has been found to result in little change in the size of

the decision functions. Age-weighting should thus be implemented in the

matched filters as well. These filters are implemented in "information

form", a form equivalent to the more common form of the Kalman filter.

The change in the standard filter equations can be mapped into the

information form equations. This results in changes in the propagation

equations for the information vector and information matrix. In the

matched filters,

X' (k+1) = sl A_'(k) + GT(k+I)M -1(k+1)_(k+1) (4.9)

S' (k+1) - sl A S' (k)A T$ + GT(k+I)M -1(k+1)G(k+1) (4.10)
Here,

A is the single-step transition matrix for the coefficients,

a, being estimated;

G is the residual influence matrix for the system;

y is the residual of the no-failure filter; and

M is the residual covariance of the incoming measurements

including the effect of age-weighting.

As with the age-weighted filter, only the steady-state implementation of

matched filters with age-weighting will be tested in this report.

The effect of age-weighting on the OSGLR algorithm can be seen

from Figures 4.8 and 4.9. The decision functions for the off-nominal,

no-failure test case and the -0.01745 rad (-I ° deg) elevator bias failure

(occurring at 10 s) test case with age-weighting time constant of 15 s

are shown in Figures 4.8 and 4.9, respectively. The largest value of the

decision functions for the off-nominal test case is reduced by ageweighting from 1,100,000 (at 50 s and increasing) to 45,000 for this

age-weighting time constant of 15 s. The major benefit of age-weighting

is to limit the decision functions caused by modeling errors. Otherwise,

false alarms would be certain to occur. However, age-weighting also reduces and limits the decision functions caused by failures. The largest

value of the decision functions for the -0.01745 rad (-I deg) elevator

bias failure test case was reduced by age-weighting from 400,000 (at 50 s

and increasing) to 40,000 (and increasing at 50 s although more slowly)

for the time constant of 15 s.

The effect of age weighting ultimately depends on the age-weighting time constant chosen. As the time constant is decreased, past information will be discarded sooner and the size of the decision functions

caused by both modeling errors and failures will decrease. However, in

order to allow the state estimates to converge, the age-weighting time

constant should be larger than the time constant associated with the

smallest closed-loop filter eigenvalue. With no age-weighting, the

-64-

:
O

U9

00

I I I

20,00 gO.O0 60.00 80.00

Time (s)

I00.00

Figure 4.8 Decision functions produced by an age-weighted

OSGLR algorithm

• no failures

• off-nominal cruise flight condition of

102.9 m/s and 1524.0 m

• age-weighting time constant of 15 s

-65-
00

!

I0.00 20.00 30.00 I_0.O0

Time (s)

50.00

Figure 4.9

Decision functions produced by an age-weighted

OSGLR algorithm

• -0.01745 tad (-i °) elevator bias failure

at I0 s

• nominal flight condition of 77.2 m/s

and 304.8 m

• age-weighting time constant of 15 s

-66-

J
largest filter time constant is 1.4 s. Small age-weighting time

constants mayalso cause difficulty in the convergence of the higher

order basis function coefficient estimates. Figure 4.10 shows the

estimated rudder basis function coefficients for a 0.001745 rad/s

(0.1°/s) rudder ramp failure starting at the beginning of the test case.

The age-weighting time constant is 3 s. The first four coefficients

converge reasonably well, whereas the last two, although they remain

fairly well behaved, have more difficulty. Still, the convergence of the

basis function coefficients wasadequate for this case.

The 3 s age-weighting time constant was investigated in addition

to the 15 s time constant already considered. The decision functions for

the off-nominal, no-failure test case and the -0.01745 rad (-1.0 deg)

elevator bias failure test case with age-weighting time constant of 3 s

are shown in Figures 4.11 and 4.12. The 3 s time constant reduces the

largest value of the decision functions for the off-nominal test case

from 45,000 for the 15 s time constant to 1600. Likewise, the maximum

magnitude of the decision functions for the -I deg elevator bias failure

test case was reduced from 45,000 (and increasing) to 3500. Based on

these two test cases, the 3 s time constant produces the greatest

reduction in the decision function caused by mismodeling while causing

the least reduction in the decision functions resulting from a failure.

But an age-weighted time constant this small will make detecting failures

of limited time duration difficult as the OSGLR algorithm will forget

that a failure had occurred. These failures may not be significant

though. Therefore, in the interest of keeping the decision functions as

insensitive to modeling errors as possible, the 3 s age-weighting time

constant will be used from henceforth.

4.5 Summary and Conclusions

Several options for accommodating modeling errors in failure detection and isolation have been discussed. These included dynamic

thresholds, dynamic decision regions, model error estimation, and robust

-67-
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-68-
O

O

C)

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%'oo 2b.oo _b.oo 6b.oo 8b.oo

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Figure 4.11 Decision functions produced by an age-weighted

OSGLR algorithm

• no failures

• off-nominal cruise flight condition of

102.9 m/s and 1524 m

• age-weighting time constant of 3 s

-69- ;

J
O

O

=r

O

oo. x

v (D

o

4J

%•00 I0.00 30.00 WO.O0 50.00

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Figure 4.12 Decision functions produced by an age-weighted

OSGLR algorithm

• -0.01745 tad (-i °) elevator bias failure

at i0 s

• nominal cruise flight condition of 77.2 m/s

and 304.8 m

• age-weighting time constant of 3 s

-70-
residual generation. Using dynamic thresholds is a simple approach to

reducing false alarms caused by modeling errors. However, since modeling

errors will, in general, cause the decision functions to grow in an unlimited manner, this approach would not give satisfactory performance

alone. Also, there is no convenient systematic approach for selecting

the dynamic thresholds. The dynamic decision region approach alters the

information matrix used to weight the information vector in calculating

the decision function to better reflect the uncertainties associated

with modeling errors. Similar to the dynamic threshold approach, this

approach alone could not reduce the effects of mismodeling satisfactorily. In addition, the procedure suggested to design these dynamic decision regions mayas in the case of the C-130 aircraft, require significant effort. Model error estimation probably would be the most difficult

and computationally burdensome of the four approaches. Two ideas were

presented to improve the robustness of the filter used to generate residuals for the OSGLR algorithm. The first was to add process noise in

certain directions (e.g., specific states). While adding process noise

worked well for the few cases tested, the selection of the noise is an

iterative process and may only be valid for that particular linear system

model. The second idea, age-weighted filter, was found to be simple,

general, and very effective. Age-weighted filtering allows the consideration of only the more recent information for estimating aircraft state

and the basis function coefficients. Consequently, modeling errors due

to linearization or aircraft maneuvering do not cause the filters to diverge.

Age-weighted filtering, because of its simplicity and effectiveness, will be used to reduce the effects of modeling errors. The ageweighting time constant will be 3 s.

-71-
SECTION5

DETAILED EVALUATION OF THE OSGLR ALGORITHM

USING A SINGLE CRUISE LINEAR MODEL

5.1 Introduction

In the preliminary evaluation described in Section 2, the

steady-state implementation of the OSGLR algorithm was tested at a nominal and an off-nominal cruise flight condition. Furthermore, only a few

failure cases at the nominal cruise flight condition were tested. This

section further examines the flight regime in the neighborhood of the

nominal cruise flight condition 77.2 m/s (150 knots) at 304.8 m (1000

ft). The effects of sensor errors, off-nominal flight conditions including maneuvers, different turbulence intensity levels, steady winds, and

nonzero flap deflections are considered. 1 Also, a number of bias failures are tested. Based on these results, the magnitude of bias failures

that can be detected and the detection times (for this nominal flight

condition) can be determined for a particular threshold.

In this and the following section, DF i is the decision function

whose failure hypothesis is that the ith control surface has failed. The

control surfaces are defined in Table 5.1

A summary of the test cases is presented in Appendix A for easy reference and comparison of the test cases presented in this section and

in sections 4 and 6.
Table 5-I. C-130 control surface definition

Input Control Surface

I

2

3

4

5

6

elevator

right aileron

left aileron

rudder

right flap

left flap

5.2 Sensor Errors and Dynamics

Misalignment, scale factor error, and quantization were added to

the acceleration and angular rate measurements. Biases were added to the

acceleration, angular rate, and attitude measurements. Sensor models

including first-order lag filter dynamics were included for all the

measurements except for the altitude rate.

The altitude rate measurement was eliminated because a sensor

model was not readily available. The performance of the OSGLR algorithm

was not affected in detecting and isolating elevator, right aileron, and

right flap failures. However, eliminating the altitude rate measurement

reduced the ability of the algorithm to isolate a 0.01745 rad (I deg)

rudder bias failure. Larger rudder failures, such as a 0.0349 rad (2

deg) bias failure, still could be easily isolated. Therefore, the

elimination of the altitude measurement caused only a minor degradation

in performance.

The OSGLR algorithm was tested with the sensor noise, errors, and

dynamics shown in Table 5.2; the effect on the performance was too small

to be evident. All the results presented in this report were produced

with these effects of sensor noise, errors, and dynamics included.

-73-
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-74-
5.3 Off-Nominal, No-Failure Flight Conditions

The steady-state implementation of the OSGLR algorithm based on

the 77.2 m/s (150 knot) linear model was tested for a variety of

maneuvering and other off-nominal flight conditions to determine their

effects on the performance of the algorithm. This testing served to

determine the range of flight conditions for which the 77.2 m/s (150

knot) linear model is valid for the purposes of FDI. Light and

thunderstorm turbulence and steady wind conditions were also tested.

5.3.1 Maneuvering Test Case

The first test case considered was a combination of cruise flight

conditions and turning, acceleration, and climb maneuvers as described in

Table 5.3. This test case will be referred to as the maneuvering test

case. The aircraft begins cruising at 77.2 m/s (150 knots), then turns

at 0°0524 rad/s (3 deg/s), returns to level flight, accelerates with full

thrust, climbs at first with nominal thrust level and later full thrust,

and finally turns slowly while climbing at full thrust. The decision

functions for this test case are shown in Figure 5.1. The decision functions increase slightly as the aircraft rolls to begin the turn and again

as the aircraft rolls to level flight. As full thrust is applied and the

aircraft accelerates at an approximately constant altitude, the decision

functions increase significantly. The decision functions decrease and

become small again with the thrust level returning to the nominal value

and the velocity decreasing with the aircraft climbing. In order to

maintain the climb, the thrust level is increased and the decision functions increase significantly once again.

As thrust level, or alternatively thrust, is an actual input to

the system, incorporating thrust level or thrust into the linear model as

an input might reduce the decision functions at off-nominal thrust

levels. Thrust was found to be more effective than thrust level. Thrust

is actually a nonlinear function of both thrust level and velocity. With

thrust level as the input, some of the velocity dependence of thrust

-75-
Table 5.3 Maneuvering test case

Time(s)

0-20

20-25

20-55

50-55

55-60

60-100

100-1 25

105-165

1 25-I 65

145-165

Maneuver

Nominal cruise condition 77.2 m/s (150 knots) at

304.8 m (1000 ft), no flaps, 55% thrust level

Roll to 0.4363 rad (25 degrees)

Turn at approximately 0.0524 rad/s (3 deg/s)

Roll back to level flight

Nominal cruise condition

Thrust level increased to 100% causing the velocity to increase from 77.2 m/s (150 knots) to

102.9 m/s (200 knots)

Thrust level decreased to nominal value

Climb at an average rate of 6.7 m/s (22 ft/s)

Thrust level increased to 100%

Roll and turn slightly in addition to the climb
O

O

O

DF 5

DF 2

DF 3

DF 6

DF 1

DF 4

30.00 60.00 90.00 120.00

Time (s)

150.00 80.00

Figure 5.1 Decision functions for the maneuvering

test case

-77-
could be accounted for in the linear model. Still, less mismodeling will

result with thrust as the input. (Thrust was added to the linear model

simply by adding another column to the B and D matrices.)

The decision functions for the maneuvering test case with thrust

incorporated as an input are shown in Figure 5.2. The decision functions

during the periods of off-nominal thrust levels are significantly decreased. The largest decision functions are now during the acceleration

at constant altitude portion of the test case rather than the climb with

100% thrust portion. The decision functions during the acceleration at

constant altitude portion increase as the velocity and therefore dynamic

pressure increases. As the thrust level begins to decrease to the nominal value at 100 s, the decision functions also decrease. The increased

thrust level during the climb portion of the trajectory does not cause

any difficulty. The probable reason for this is that the dynamic pressure is close to the nominal dynamic pressure. Based on this test, the

largest no-failure decision functions occur when both the dynamic pressure and the thrust are significantly off-nominal.

All further results use thrust as an input.

5.3.2 Turning Test Case

A longer turning maneuver, where the aircraft rolls and begins a

0.0524 rad/s (3 deg/s) right turn at 5 s, was also considered. At 65 s,

the aircraft returns to level flight having turned 190 deg. The aircraft

then turns left starting at 75 s, returning to level flight at 130 s.

The decision functions, shown in Figure 5.3, are small with the two

largest spikes occuring when the aircraft first rolls to turn right and

when the aircraft returns to level flight after the left turn. These

spikes appear to correspond to off-nominal aileron activity required to

roll the aircraft.

-78-
O

O

6

O

:P

O

O

_0

o

x (D v

_6

0

0_o

gg

DF I

DF 2

30. O0 60. O0 90. O0 120. O0 150. O0

Time (s)

180.00

Figure 5.2 Decision functions for the maneuvering

test case with thrust as an input

-79- i
O

O

o_.

A DF

0 _IIDF6

it01

C:b'.oo 30.oo 60.oo 90.oo 120.oo 150.oo 180.oo

Time (s)

Figure 5.3 Decision functions with the aircraft turning

right for 3.32 rad (190 deg) and then left

for 3.14 rad (180 deg)

-80-
5.3.3 Maximum Rate-of-Climb Test Case

In the maneuvering test case, the climbing rate of the aircraft is

significant but less than the maximum possible. In order to determine

the largest decision functions that could be produced by a sustained

climbing maneuver, the algorithm was tested with the aircraft climbing at

an average rate of 10 m/s (33 ft/s) which is approximately the maximum

rate of climb. At the beginning of the test case, the aircraft is flying

at the nominal cruise condition. Five seconds into the flight, the

thrust level is increased to 100% with the climb initiated at 15s. The

decision functions (Figure 5.4) are larger than in the climbing portion

maneuvering test case but only 800 larger than the largest decision

function produced by the maneuvering test case.

5.3.4 Accelerating Test Cases

A number of accelerating cases were tested with the most severe

case being the acceleration from 51.44 m/s (100 knots) to I02.9 m/s (200

knots) test case. The decision functions for this test case are shown in

Figure 5.5. The largest decision functions are at the beginning of the

test case. Initially, the airspeed of the aircraft is 51.44 m/s (100

knots) but the airspeed drops a few seconds later to 33.44 m/s (65

knots). This airspeed drop is caused partly by a large turbulence level

at the beginning of the simulation run and also because the control

surfaces are being moved to control the aircraft. The result of this

airspeed drop is that the actual dynamic pressure a few seconds after the

beginning of the test case is only 20% of the nominal dynamic pressure of

the 77.2 m/s (150 knot) linear model. The linear model is clearly not

valid in the neighborhood of 51.44 m/s (100 knots). The thrust level was

increased to 100% starting at 10 s, causing the airspeed to increase. As

the actual dynamic pressure approaches the nominal dynamic pressure,

decision functions decrease. At 41 s, the airspeed is approximately 77.2

m/s (150 knots) which is the nominal airspeed of the linear model and the

decision functions are very small in this region. They increase again as

-81- i
_) ,,

U_

Time (s)

Figure 5.4 Decision functions for the maximum

rate-of-climb test case

-82-
O

O

6

CD

o

v •

0

DF I

20. O0 LIO.O0 60. O0 80. O0 100. O0

Time (s)

Figure 5.5 Decision functions for the acceleration from

51.44 m/s (i00 knots) to 102.9 m/s (200 knots)

test case

-83-
the velocity increases to 102.9 m/s (200 knots). Based on this test

case, there appears to be more mismodeling at low dynamic pressures than

at high dynamic pressures.

5.3.5 Decelerating and Descending Test Cases

To investigate decelerating flight, the aircraft was started at an

airspeed of 102.9 m/s (200 knots) with a thrust level of 30%. (The

thrust level required to maintain 102.9 m/s airspeed at 304.8 m (1000 ft)

is 80%.) The altitude of 304.8 m (1000 ft) was approximately maintained. The airspeed at the end of the simulation run is approximately 51

m/s (100 knots). The decision functions, shown in Figure 5.6, are small

until the airspeed drops below 61 m/s (118.5 knots). As in the maneuvering test case, the 150 knot linear model is not adequate for airspeeds in

the neighborhood of 51 m/s (100 knots).

Decision functions for a descending test case are shown in Figure

5.7. The aircraft has an initial airspeed of 103 m/s at an altitude of

1524 m (5000 ft) with a thrust level of 20%. The aircraft descends to an

altitude of 762 m (2500 ft) at the end of the test case. The final airspeed is approximately 77 m/s (150 knots). The decision functions are

small with the larger magnitudes occurring in the first 35 s where the

dynamic pressure is the most off-nominal.

5.3.6 Non-Zero Flap Deflections

In addition to maneuvers, the OSGLR algorithm was tested with the

flaps extended 50, 75, and 100%. In each of the test cases, the flaps

were extended starting at 5 s, and reached 50% to 100% in 5 to 15 s. The

decision functions for the 50% test case shown in Figure 5.8 are small.

However, the decision functions for the 75% test case, shown in Figure

5.9 are large. The decision functions for the 100% test case are approximately four times larger than those of the 75% case. One explanation

for the large decision functions is that large deflections of the flaps

-84-
O

o

O

DF 1

DF 5

• 00 15.00 30.00 L&5.00

Time (s)

DF 4

60.00 75.00

Figure 5.6 Decision functions for the decelerating at

constant altitude from 102.9 m/s to 51 m/s

test case

-85-
O

=r

DF 5

DF 2

DF 3

DF 6

00

DF 4

!

20. O0 l&O.O0 60. O0 80. O0

Time (s)

I00.00

Figure 5.7 Decision functions for the descending

from 1524 m (5000 ft) test case

-86-
O

O

O

6

DF 1

DF 2

DF 3

DF 5

DF 4

I

!O. O0 20. O0 30. O0 L&O.O0

Time (s)

50.00

Figure 5.8 Decision functions for the 50%

flap deflection test case
O

O.

O0 l O. O0 20. O0 30. O0

Time (s)

DF 1

DF 5

DF 2

DF 3

DF 6

DF 4

qO.O0 50.00

Figure 5.9 Decision functions for the 75%

flap deflection test case

-88-
causes significant changes in the dynamics. Also, large flap deflections

cause a significant decrease in the airspeed and therefore dynamic

pressure.

5.3.7 Off-Nominal Turbulence and Steady Winds

The nominal turbulence used to test the OSGLR algorithm is the

Dryden model with clear air scales and intensity of 1.98 m/s (6.5 ft/s)

(Reference 12). As the turbulence model is incorporated into the linear

model of the system, the nominal turbulence does not cause significant

no-failure decision functions. To determine the effect of off-nominal

turbulence, light turbulence of intensity 0.3 m/s (I ft/s) and thunderstorm turbulence of intensity 6.4 m/s (21 ft/s) were tested. The decision functions for the maneuvering test case with light turbulence, shown

in Figure 5.10, are slightly smaller than with the nominal turbulence

intensity. The decision functions for the aircraft cruising at the

nominal flight condition in thunderstorm turbulence are shown in Figure

5.11. The validity of this test case is questionable for the first 20 s

as the turbulence causes the angle of attack to exceed the largest angle

of attack for which the aerodynamic forces are modeled. (The simulation

needs a pilot to control the aircraft adequately in thunderstorm turbulence.) In general, the decision functions due to thunderstorm turbulence are significant. One approach to reduce the effect of thunderstorm

turbulence might be to develop the age-weighted filter gain and the OSGLR

influence and information matrices for the thunderstorm case and schedule

these matrices as a function of turbulence level. A simpler approach

would be to use a higher nominal turbulence level. This approach should

be adequate if detection and isolation at lower turbulence levels is not

significantly degraded.

The effect of steady winds on the performance of the algorithm was

determined by testing a number of wind conditions. The steady winds

tested were headwinds, crosswinds, and a direction in between. These

horizontal winds were all 10.3 m/s (20 knots). A downwind of 2.57 m/s

sg- i
O

O

=P

O

O0

DF 5

DF 2

DF 1

DF 3

DF 6

DF 4

30. O0 60. O0 90. O0 120. O0

Time (s)

150.00 180.00

Figure 5.10 Decision functions for the maneuvering test

case with light turbulence

-90-
OO

o

o

DF1

DF6

DF

DF2

DF5

DF4

.00 20. O0 I&O.O0 60. O0 80. O0 100. O0

Time (s)

B

Figure 5.11 Decision functions for the aircraft flying

at the nominal cruise flight condition in

thunderstorm turbulence

-91-
(5 knots) was also tested. The crosswind test case produced the largest

decision functions (Figure 5.12). The large decision functions in the

first ten seconds of the test case are caused by the aircraft responding

to the crosswind. In response to the crosswind, the thrust was increased

momentarily and the ailerons and rudder were used to control the aircraft. After the initial transient, steady winds do not result in

increased decision functions.

5.4 Failures

Elevator, right aileron, rudder, and right flap bias failures are

considered in this section. With two exceptions, all the failures occur

with the aircraft flying at the nominal cruise flight condition of 77.2

m/s (150 knots) at 304.8 m (1000 ft). The exceptions are elevator bias

failures with the aircraft flying at two off-nominal airspeeds, 64.3 m/s

(125 knots) and 90.0 m/s (175 knots). All the failures occurred 10 s

into the simulation run.

A representative failure test case for each of the surfaces tested

as well as the two off-nominal flight condition test cases will be

presented. In lieu of the individual failure cases, results for the

nominal flight condition failure test cases are then summarized, for each

surface tested, in terms of the maximum decision function as a function

of the time from the occurrence of the failure and the failure magnitude. The ability to isolate a failure of each of the surfaces tested is

similarly examined.

5.4.1 Elevator Failure Test Case

The decision functions for the -0.0873 tad (-5 deg) elevator bias

failure with the aircraft at the nominal cruise flight condition are

shown in Figure 5.13. The growth of the decision functions is limited by

the age-weighting, as in the off-nominal, no-failure cases. The failure

-92-
O

DF 5

DF 2

DF

3

DF 6

I

!O. O0 20. O0 30. O0 i&O.O0 50. O0

Time (s)

Figure 5.12 Decision functions for a 10.3 m/s (20 knots)

crosswind test case

-93- J
CD

CD

CD

O

CD

4.

O3

o

v

O

"_ .

CD

CD

C3

C3

_'.00

DF I

DF 5

DF 2

DF 3

DF 6

DF 4

I0. O0 20.00 30.00 40. O0 50. O0

Time (s)

Fi_Jre 5.13 Decision functions for a -0.0873 rad

(-5 deg) elevator bias failure

-94-
is easily detected if the detection threshold is less than 60,000. The

failure can be detected in approximately 2 s if the threshold is less

than 20,000. Four of the isolation decision functions, DFij , are shown

in Figure 5.14 where DFij is defined to be

DF. = DF. - DF.

z3 i 3

Isolation decision function DF14 , not shown in Figure 5.14, would have

been even larger than those shown. Isolating this failure is

straightforward.

5.4.2 Off-Nominal Elevator Failure Test Cases

The decision functions for the same failure with the aircraft

flying at an airspeed of 64.3 m/s (125 knots) at the time of the failure

are shown in Figure 5.15. The decision functions rise more slowly than

for the same magnitude failure occurring with the aircraft flying at the

nominal cruise flight condition. The effect of the failure is less

because the dynamic pressure for this case is less than the nominal

flight condition. Still, a detection threshold that could detect a

failure of the same magnitude at the nominal airspeed could detect this

failure. At approximately 35 s, the decision functions peak and decrease

afterwards. This decrease is caused by a decrease in the airspeed. In

Section 5.3, the OSGLR algorithm based on the 77.2 m/s (150 knot) linear

model produced large no-failure decision functions at low speeds in the

neighborhood of 100 knots. Apparently, the 77.2 m/s (150 knot) linear

model does not model the aircraft well at these low speeds. Also, as

just mentioned, a decrease in dynamic pressure will reduce the effect of

the failure.

The decision functions for the -0.0873 rad (-5 deg) elevator bias

failure with the aircraft flying with an airspeed of 90.0 m/s (175 knots)

are shown in Figure 5.16. The decision functions rise more quickly than

-95-
CD

CD

CD

DFI 3

12

DFI5

! | %'.00 to.oo 2b.oo 30.00 ,o.oo

Time (s)

50.00

Figure 5.14 Isolation decision functions for a -0.0873 rad

(-5 deg) elevator bias failure

-96-
DF 1

DF 5

DF 2

DF 3

DF 6

DF 4

i i i

10.00 20.00 30.00 I&O.O0

Time (s)

50.00

Figure 5.15 Decision functions for a -0.0873 rad (-5 deg)

elevator bias failure with the aircraft flying

at an off-nominal cruise airspeed of 64.3 m/s

(125 knots) at the time of failure

-97-
O

O

O.

e9

o

v

0

-_1

4_

o

o

-,4

0

DF 1

DF 5

DF 2

DF 3

DF 6

DF 4

°0_00 !0.00 20.00 30.00 LtO. O0

Time (s)

50.00

Figure 5.16 Decision functions for a -0.0873 rad (-5 deg)

elevator bias failure with the aircraft flying

at an off-nominal cruise airspeed of 90 m/s

(175 knots) at the time of failure

-98-
the nominal flight condition -0.0873 rad (-5 deg) elevator bias failure

test case. Again, this is to be expected because of the higher dynamic

pressure. The steady-state magnitude of the elevator decision function

is approximately the same as for the nominal flight condition failure

test case,

5.4.3 Right Aileron Failure Test Case

The decision functions for a 0.0873 rad (5 deg) right aileron bias

failure at the nominal cruise flight condition test case are shown in

Figure 5.17. The wing surface decision functions (DF2, DF3, DF5, DF6)

become large quickly. As with the -0.0873 rad (-5 deg) elevator bias

failure at the nominal flight condition test case, a detection threshold

of 20,000 would allow detection of a failure within 2 s. Isolating the

failure to the right aileron surface is more difficult. The isolation

decision functions DF23 , DF25 , and DF26 are shown in Figure 5.18. If an

isolation threshold of 100 was realistic, this failure could be correctly

isolated. However, testing of right aileron failures in a variety of

flight conditions would be required to determine if this isolation

threshold is realistic.

5.4.4 Rudder Failure Test Case

The decision functions for a -0.0873 rad (-5 deg) rudder bias

failure are shown in Figure 5.19. For the same magnitude failure, the

decision functions produced by a rudder failure are much smaller than

produced by the other surfaces. Apparently, the rudder has less effect

on the motion of the aircraft than the other surfaces. If the same

detection threshold was used for each of the decision functions, this

rudder failure could not be detected. Based on the no-failure results

presented in Section 5.3, the smallest detection threshold that could be

possibly selected would be 5000. This assumes the aircraft is flying

near the nominal flight condition, the aircraft is not flying in thunderstorm turbulence, the flaps are not deflected and the maneuvers presented

-99-
O

O

6

O

CD

O

6

CO_

.--4

o O

O

-,-4

o

C

0

-4 C

m GO

.4

o

CD

CD

_.00

DF 2

DF 3

DF 5

DF 6

DF 1

DF 4

I I !

10.00 20.00 30.00 qO.O0

Time (s)

50.00

Figure 5.17 Decision functions for a 0.0873 rad (5 deg)

right aileron bias failure

-i00-
O

(D

o

O

DF26

DF25

DF23

CD

O

I I i • 00 I0.00 20.00 30.00 _I0. O0

Time (s)

50.00

Figure 5.18 Isolation decision functions for a 0.0873 rad

(5 deg) right aileron bias failure
CD

CD

o

CD

CD

CD

a3

O

moo ×

v

.r4

O

CD

CD

"5o_"

.H

u

O

_" / DF5

DF 6

I :;: '

1. O0 i | ! 10.00 20.00 30.00 _0.00 50.00

Time (s)

Figure 5.19 Decision functions for a -0.0873 rad

(-5 deg) rudder bias failure

-102-
were the most severe normally attempted by commercial aircraft. However,

in the no-failure test cases presented in Section 5.3, the rudder decision function is always small. The rudder decision function, excluding

the times the airspeed was in the neighborhood of 51.44 m/s (100 knots),

is less than 200 except for the large flap deflection and the thunderstorm turbulence test cases. With some compensation for thunderstorm

turbulence and large flap deflections, a detection threshold of 400 for

only the rudder decision function might be possible. If so, this rudder

failure could be detected in approximately 3 s.

Isolation of this rudder failure is easy since the other decision

functions are all less than 100. However, in general, isolation would

have to be based simply on the magnitude of the rudder decision function

since the decision functions of the other surfaces may be larger due to

modeling errors. One approach would be to simultaneously detect and isolate a rudder failure when the rudder decision function reaches the rudder detection threshold. Since failures of other control surfaces also

produce large rudder decision functions, a rudder failure should not be

declared if another failure has already been detected.

5.4.5 Right Flap Failure Test Case

The decision functions for a 5% right flap failure test case are

shown in Figure 5.20. For a detection threshold of 20,000, this failure

would be detectable in 12 s. The failure would not be detectable for

higher thresholds. As with the right aileron test case, isolating this

failure to the correct wing surface may not be possible. Isolation decision functions DF52 , DF53 , and DF56 are shown in Figure 5.21. If an

isolation threshold of a 100 was realistic, this failure could be isolated at the same time the failure was detected. Again, failure test cases

at off-nominal flight conditions and wind conditions are required to determine realistic isolation thresholds.

-103-

i
O

O

0

DF 5

DF 2

DF

DF 1

DF 4

I I

I0.00 20.00 30.00

Time (s)

50. O0

Figure 5.20 Decision functions for a 5%

right flap bias failure

-104-
CD

CD

pDF53

DF52

i I i

10.00 20.00 :30.00 U,0.00

Time (s)

50.00

Figure 5.21 Isolation decision functions for a 5%

right flap bias failure

-105-
5.4.6 Largest Failure Decision Functions as a Function

of Time and Failure Magnitude

The purpose of testing the OSGLR algorithm with a number of failure cases is to determine, for a given detection threshold, the size of

the failures that can be detected and the time required for detection. A

number of bias failures with the aircraft flying at the nominal flight

condition were used to test the algorithm. The results were then plotted

for each surface to readily answer the two questions raised. The largest

values of the decision functions as a function of the size of the bias

failure are plotted for each surface tested in Figures 5.22 - 5.25. Each

figure contains five curves where each curve is for a particular time

after the failure onset. For example, one curve describes the largest

value of the decision functions I s after the failure. The symbols and

their corresponding times after the failure onset are defined in Table

5.4. The symbols correspond to actual test results. Considering the

decision functions for only the first 15 s following a bias failure is

sufficient because the decision functions, in most cases, grow slowly if

at all after this. Since only a few bias failure cases for each surface

were tested, the curves are approximate.

In addition, the elevator and rudder plots only considers negative

bias failures, and the right aileron plot only considers positive bias

failures. Yet, one might expect the plots to be approximately the same

for failures with opposite signs but the same magnitude.

First consider elevator, right aileron, and right flap failures.

Excluding the 75% and full flap deflection and the acceleration from

51.44 m/s (100 knots) test cases, the largest value of the decision functions for the off-nominal flight conditions have been less than 10,000.

Therefore, selecting a threshold of 20,000 might be reasonable at least

for some flight regimes. Based on Figure 5.22, -0.0524 tad (-3 deg)

elevator bias failures should be detectable with a 20,000 threshold.

Similarly, 0.0436 rad (2.5 deg) right aileron and 5% right flap bias

failures should be detectable. These bias failures would take 10 to 15 s

-106-
O

X

o

00 -2.00 -q. O0 -6.00 -8.00 -I0.00

(-0.0349) (-0.0698) (-0.1047) (-0.1396) (-0.1745)

Size of elevator bias failure

degrees (radians)

Figure 5.22 The largest decision function values

produced by elevator bias failures

-107-
O

O

6

o

o

O

O

6.

O0

Figure 5.23

I I I

• 2. O0 q. O0 6. O0 8. O0

(0.0345) (0.0698) (0.1047) (0.1396)

Size of right aileron bias failure

degrees (radians)

The largest decision function values

produced by right aileron bias failures

-108-

10.00

(o.o1745
0

0

I I | I I

00_00 5.00 !0.00 15.00 20.00 25.00

Size of right flap bias failure (percent)

Figure 5.24 The largest decision function values produced

by right flap bias failures

30.00

-109- !
o

Figure

L._S -I0.00 ' -IS.O0 ' -20.00 '

(-0. 0873) (-0.1745) (-0.2618) (-0.3491)

Size of rudder bias failure

degrees (radians)

5.25 The largest decision function values

produced by rudder bias failures

-2S. O0

(-0.4363)

-ii0-
Table 5.4 Symbols for time after onset of bias failure

Symbol

Q

+

X

Time After Onset of Bias Failure

1 sec

2 sec

5 sec

10 sec

15 sec

-iii-
to detect. However, -.1222 rad (-7 deg) elevator, 0.1571 rad (9 deg)

right aileron, and 21%right flap bias failures, should be detectable in

approximately 1 s. Ignoring thunderstorm turbulence, a threshold of

10,000 would be reasonable. With this threshold, a -0.0873 tad (-5 deg)

elevator bias failure, a 0.1134 rad (6.5 deg) right aileron bias failure,

and a 15%right flap bias failure could be detected in Is.

In the rudder bias failure test case discussion, a separate detection threshold for the rudder decision function of 400 was motivated.

This threshold maybe reasonable for someoperating envelope about the

nominal flight condition that excludes thunderstorm turbulence and large

flap deflections. With this threshold, a -0.07 tad (-4 deg) bias failure

should be detected in approximately 15 s and a -0.1396 rad (-8 deg) bias

failure in approximately 1 s.

As detection thresholds will be selected to minimize false alarms,

the size of the decision functions produced by off-nominal, no-failure

flight conditions will determine the thresholds. Since these plots approximately define the detection performance or a given detection threshold, they help determine the off-nominal flight conditions for which the

decision functions need to be reduced to achieve acceptable performance.

While these figures are only for bias failures, they still might

be of use to estimate the detectability of other failures whose failure

modes can be reasonably approximated by a bias failure. For example, a

failure that ramps at 0.01745 rad/s (I deg/s) to a bias failure of 0.0873

rad (5 deg) will produce larger decision functions than a 0.0873 rad (5

deg) failure that occurs 5 s after the ramp-to-a-bias failure.

It should be emphasizedthat Figures 5.22 - 5.25 are only valid in

a neighborhood of the nominal airspeed of 77.2 m/s (150 knots). At lower

airspeeds, the decision functions are likely to be smaller for the same

magnitude failure as less aerodynamic forces are produced for the same

deflection. Similarly, the decision functions are likely to be larger

for the samemagnitude failure at higher airspeeds.

-112-
Similar plots of the smallest failure isolation decision functions

as a function of the time after failure and the failure magnitude can be

used to determine the isolation performance. In Figure 5.26, the minimum

of the isolation decision functions DF12 , DF13 , DF14 , DF15 , and DF16 are

plotted for two elevator bias magnitudes of -0.01745 rad (-I deg) and

-0.0873 tad (-5 deg). Approximate curves for the minimum of these isolation decision functions for five times after the failure onset are also

shown in the figure. No effort was made to define what might be reasonable isolation thresholds. Yet, even small failures on the order of

0.045 tad (2.58 deg) should be easily and quickly isolated.

The minimum of the isolation decision functions DF21 , DF23 , DF24 ,

DF25 , and DF26 are plotted for the right aileron bias failure test cases

in Figure 5.27. Even for a small isolation threshold of 100, only 0.07

rad (4 deg) and larger bias failures that can be isolated. The OSGLR

algorithm might be able to quickly isolate large right aileron failures

if an isolation threshold of 500 was reasonable.

The minimum of the isolation decision functions for the right flap

bias failure test case DF51 , DF52 , DF53 , DF54 , and DF56 are plotted in

Figure 5.28. While larger than the minimum of the isolation decision

functions for the right aileron failures, an isolation threshold of less

than 400 would be required to isolate 10% and smaller bias failures.

No figure for the rudder bias failure test cases is shown because

there might be circumstances where a rudder failure would be detected and

isolated even though the decision functions for the other surfaces are

larger due to an off-nominal flight condition. The detection and isolation of a rudder failure would be based simply on the rudder decision

function unless another failure has already been detected.

5.5 Summary and Conclusions

The OSGLR algorithm, based upon the linear model developed for the

nominal cruise flight condition of 77.2 m/s (150 knots) at 304.8 m

-113-
Figure 5.26

I I I I

- I. O0 -2. O0 -3. O0 -W. O0 -S. O0

(-0.0175) (-0.0349) (-0.0524) (-0.0698) (-0.0873)

Size of elevator bias failure

degrees (radians)

The smallest of the isolation decision functions

DFIi (i _ i) produced by elevator bias failures

-114-

-6. O0

(-0.1047)
CD

CD

CD

A CD

X

v

o

/

/

I

2.00

(0.0349)

Figure 5.27

I I !

4. O0 6. O0 8.00

(0.0694) (0.1047) (0.1396)

Size of right aileron bias failure

degrees radians)

!

I0.00

(0.1745)

r

12.00

(0.2094)

The smallest of the isolation decision functions

DF2i (i _ 2) produced by right aileron bias failures

-i15- i
O

6

°/

®

5. O0

! I I

10.00 15.00 20.00 2_.00 3

Figure 5.28

Size of right flap bias failure (percent)

The smallest of the isolation decision functions

_ ...... by _ flap _;=s F=_I_-_

•00

-116-
(1000 ft), was tested with sensor errors and dynamics at a number of

off-nominal flight conditions. These off-nominal flight conditions

included maneuvers, nonzero flap deflections, off-nominal turbulence

intensity levels, and steady winds. Also, a number of bias failures were

tested. Based on these tests, the following conclusion have been made.

o Sensor errors and dynamics have a negligible effect on the

performance of the OSGLR algorithm.

o Including thrust as an input to the linear model reduces the

mismodeling caused by off-nominal thrust.

o Mismodeling increases as the actual dynamic pressure diverges

from the nominal dynamic pressure.

o The 77.2 m/s (150 knot) linear model is not valid for large

(75% and greater) flap deflections. The decision functions

produced by these flap deflections were 50,000 and larger. The

same linear model also had difficulty in the neighborhood of

51.44 m/s (100 knots) and slower. In one case, the decision

functions produced in this low airspeed environment were almost

40,000. Setting thresholds to accommodate these large decision

functions would result in fair to poor FDI performance. For

example, a threshold of 60,000 could detect a -0.1571 rad

(-9 deg) elevator bias failure, a 0.1396 rad (8 deg) right

aileron bias failure, and a 20% right flap bias failure in 2

s. Yet, flap deflection greater than 75% would still produce

false alarms with this threshold.

o The cruise linear model was adequate for maneuvering flight.

Ignoring airspeeds in the neighborhood of 51.44 m/s (100 knots)

for which the linear model is not valid, the largest decision

function produced by maneuvering flight was 4500 for the maximum rate-of-climb test case. Therefore, a threshold of 10,000

in the neighborhood of 77.2 m/s (150 knots) and excluding

larger flap deflections should provide excellent false alarm

-117-
performance. With this threshold, a -0.0873 rad (-5 deg)

elevator bias failure, a 0.1134 rad (6.5 deg) right aileron

bias failure, and a 15%right flap bias failure could be

detected in Is. A -0.0611 rad (-3.5 deg) elevator bias

failure, a 0.0611 rad (3.5 deg) right aileron bias failure, and

an 8%right flap failure could be detected in 2s.

o Light turbulence of intensity 0.3 m/s (1 ft/s) and steady winds

did not significantly effect the performance of the OSGLR

algorithm. Thunderstorm turbulence of intensity of 6.4 m/s (21

ft/s) may possibly cause decision functions on the order of

6,000 to 8,000 when the aircraft is flying at the nominal

cruise flight condition. Thunderstorm turbulence combined with

maneuvers will most likely produce decision functions greater

than 10,000. Some compensation for thunderstorm turbulence may

be desirable.

o A separate detection threshold is required for the rudder decision function. A rudder failure produces a rudder decision

function several orders of magnitude smaller than the same

magnitude elevator and aileron failures. Therefore, if the

same detection threshold used for the other surface was also

used for the rudder, rudder failures would not be detectable.

Fortunately, the rudder decision function is much smaller than

the other decision functions in the off-nominal, no-failure

cases tested. A detection threshold of 400 for the rudder

decision function appears reasonable if low airspeeds in the

neighborhood of 51.44 m/s (100 knots), large flap deflections,

and thunderstorm turbulence are excluded. With this threshold,

-0.1396 tad (-8 deg) rudder bias failure can be detected in I

s, a -0.1047 rad (-6 deg) bias failure in 2 s and a -0.0648 rad

(-4 deg) in 10 to 15 s.

-118-
o Isolation of elevator failures should be straightforward. It

may be possible to isolate large aileron and flap failures with

the OSGLR algorithm. Failure test cases at off-nominal flight

conditions are required to define adequate isolation thresholds

for the ailerons and the flaps. Rudder failures should be

simultaneously detected and isolated when the rudder decision

function reaches the rudder detection threshold unless another

failure has already been detected.

The next section will examine reducing the decision function produced at off-nominal dynamic pressures and nonzero flap deflections.

-119-
SECTION6

SCHEDULING

6.1 Introduction

Airspeeds of 51.44 m/s (I00 knots) and less and flap deflections

of 75% and greater were shown in Section 5 to produce large no-failure

decision functions with an OSGLR algorithm based on the 77.2 m/s (150

knots) cruise linear model. The large decision functions are due to the

fact that the 77.2 m/s cruise linear model does not adequately characterize these off-nominal flight conditions. The scheduling of the linear

models of the aircraft, the steady-state age-weighted filter gain and

covariance matrices, and the steady-state OSGLR influence and information

matrices is considered in this section as a means of reducing these

mismodeling effects. 1 First, a simple approach of scheduling zero flap

deflection linear models only is described and tested with no-failure

cases. Scheduling for nonzero flap deflections is then examined. The

effect of scheduling only the linear models on the detection of failures

is also briefly investigated. Finally, the scheduling of the

age-weighted filter and OSGLR matrices in addition to the linear models

is discussed.

6.2 Scheduling of Zero Flap Deflection Linear Models

Mismodeling was found in Section 5 to increase as the dynamic

pressure deviated more from the nominal value. Scheduling only the

i A summary of the test cases is presented in Appendix A for easy

reference and comparison.

-120-
linear models used by the age-weighted filter as a function of dynamic

pressure was investigated as a means of reducing this mismodeling. (See

Appendix B for further details.) By reducing the effects of mismodeling

as reflected in the residual of the age-weighted filter, the no-failure

decision functions should also be reduced. A linear model is chosen for

use to minimize the ratio

.

l

where _ is the actual dynamic pressure and % is the nominal dynamic

1

pressure of the ith linear model.

Three linear models corresponding to cruise flight conditions at

304.8 m (1000 ft) for airspeeds of 51.44 m/s (100 knots), 77.2 m/s (150

knots), and 102.9 m/s (200 knots) were developed to test this simple

scheduling approach. Using these linear models, the scheduling approach

was tested on the acceleration from 51.44 m/s test case. The decision

functions for this test case with the age-weighted filter and OSGLR

steady-state matrices corresponding to the 77.2 m/s linear model are

shown in Figure 6.1. The decision functions for the same test case with

no scheduling of the linear model are shown in Figure 5.5. The decision

functions are reduced by the scheduling approach except in the neighborhood of 77.2 m/s (41 s) with the maximum decision function levels reduced

from 39,000 to 11,500. One possible reason the decision functions are

still large at the beginning of the test case is that the dynamic

pressure drops 57% below the nominal value of the 51.44 m/s linear model

in the first several seconds. (As mentioned earlier, this drop in

dynamic pressure is caused mainly by a large turbulence level and also by

the control surfaces moving to control the aircraft as the control system

starts up). Another linear model with nominal airspeed less than 51.44

-121- =
O

O

o

o

X

0

4_

u

0

D)

.r4

0

0

A DF 2

DF 5

DF 3

DF 6

_ 3]'i DF I

•O0 20 •O0 WO. O0 60. O0 80. O0 100. O0

Time (s)

Figure 6.1 Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for the acceleration

from 51.44 m/s test case

• 51.44, 77.2, 102.9 m/s linear models

• 77.2 m/s age-weighted filter and OSGLR

steady-state matrices

-122-
m/s may reduce the decision functions at the beginning of the test case

further.

Finding a steady-state cruise flight condition at low speeds about

40 m/s (77.75 knots) with no flap deflection was not possible as the

angle of attack required exceeded the validity of the simulation.

Instead, a linear model corresponding to a cruise flight condition of

46.3 m/s (90 knots) at 304.8 m (1000 ft) was developed and tested. Using

the 46.3 m/s linear model as the fourth model with the age-weighted

filter and OSGLR steady-state matrices corresponding to the 77.2 m/s

linear model produced the decision functions shown in Figure 6.2. Even

though the dynamic pressure for the 46.3 m/s model (1274.6 N/m 2 or

26.62 ibs/ft 2) was not significantly lower than the dynamic pressure

for the 51.44 m/s model (1573.8 N/m 2 or 32.87 ibs/ft2), adding the

46.3 m/s linear model was sufficient to reduce the decision functions

from 11,500 to 3,500 during the first 15 s of the test case. The largest

decision function levels are reached at approximately 20 s when the ageweighted filter switches from the 51.44 m/s linear model to the 77.2 m/s

linear model.

Another reason that the decision functions are not reduced even

more at low speeds is that the 51.44 m/s linear model does not as accurately describe the aircraft dynamics at its nominal flight condition as

well as the 77.2 m/s linear model characterizes the 77.2 m/s cruise

flight condition. The decision functions for a no-failure test case with

the aircraft flying at 51.44 m/s and 304.8 m using the 51.44 m/s linear

model and associated steady-state Kalman filter and OSGLR matrices (no

age-weighting) are shown in Figure 6.3. These decision functions are

significantly larger than the decision functions for a no-failure test

case at 77.2 m/s with the 77.2 m/s linear model and associated Kalman

filter and OSGLR steady-state matrices (no age-weighting) shown in Figure

6.4. Either the nonlinearities are much greater at slow airspeeds and

high angles of attack or the linear model needs improvement to better

describe the aircraft dynamics. Even with this large mismodeling, the

-123-
O

CD

O

o

o

DF 2

DF 5

DF 3

DF 1

DF 4

T T

20.00 LtO.O0 60.00 80.00

Time (s)

IO0.O0

Figure 6.2 Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for an acceleration

from 51.44 m/s test case

• 46.3, 51.44, 77.2, 102.9 m/s linear models

• 77.2 m/s age-weighted filter and OSGLR

steady-state matrices

-124-
O

O

6

O

DF 2

DF 3

DF 5

DF 6

I I I

lO.O0 20.00 30.00 _0.00

Time (s)

50.00

Figure 6.3 Decision functions for a 51.44 m/s, no-failure,

cruise test case using the 51.44 m/s linear

model and associated steady-state Kalman filter

and OSGLR matrices (no age-weighting)
I I I I

20.00 40.00 60.00 80.00

Time (s)

I00.00

Figure 6.4 Decision functions for a 77.2 m/s cruise

no-failure test case using the 77.2 m/s

linear model and associated steady-state

Kalman filter and OSGLR matrices (no ageweighting)

-126-
51.44 m/s model used with an age-weighting of time constant 3 s is adequate to reduce the effects of mismodeling for the acceleration from

51.44 m/s test case. Someeffort was madeto determine why the mismodeling is larger for the 51.44 m/s linear model in the limited time available but without success.

Despite the inadequacy of the 51.44 m/s linear model, scheduling

of only the linear models used by the age-weighted filter was effective

in reducing the decision functions at off-nominal dynamic pressures.

6.3 Nonzero Flap Deflections

Flap deflections of 75% and larger were shown in Section 5 to produce very large no-failure decision functions using only the 77.2 m/s

linear model. The scheduling of zero-flap linear models as discussed in

the previous subsection actually increases the no-failure decision functions. The decision functions for a 50% flap deflection test case with

the scheduling of the 46.3, 51.44, 77.2, and 102.9 m/s linear models and

the age-weighted filter and OSGLR steady-state matrices corresponding to

the 77.2 m/s model are shown in Figure 6.5. The flaps are deflected

approximately I0%/s, starting at 5 s, until the desired deflection is

achieved. Extending the flaps causes the airspeed to drop. With the

scheduling of the linear models used by the age-weighted filter as a

function of dynamic pressure, linear models more accurate at low

airspeeds than the 77.2 m/s model are used. Interestingly, as a result,

the decision functions increase. The 77.2 m/s linear model, for some

reason, is not as sensitive to the combination of the flaps extending and

the airspeed dropping as the 51.44 m/s linear model is to the flaps

extending. Similarly, scheduling with zero-flap deflection linear models

does not help the 75% and 100% flap deflection test case. Scheduling

based on flap deflection as well as dynamic pressure appears necessary.

Using linear models of 38.6, 51.44, and 77.2 m/s with a nominal

flap deflection of 50% and age-weighted filter and OSGLR steady-state

-127-
O

DF 5

DF 2

DF 3

DF 6

DF 1

DF 4

IO. O0 20. O0 30. O0 _0. O0 50. O0

Time (s)

Figure 6.5 Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for a 50% flap

deflection test case

• 46.3, 51.44, 77.2, 102.9 m/s linear models

• 77.2 m/s age-weighted filter and OSGLR

steady-state matrices

-128--
matrices corresponding to the 77.2 m/s with 50% flap linear model significantly reduced the decision functions for the 50% and 75% test cases

(Figures 6.6 and 6.7). The sudden increase in one of the decision functions in Figure 6.7 may be caused by a switch from the 77.2 m/s to the

51.44 m/s models. The decision functions for the 100% flap test case

using the 50% flap linear models, however, were larger than with the

zero-flap deflection linear models. Using linear models with 100% nominal flap deflection to reduce the decision functions for the 100% flap

test was also briefly investigated without success (Figure 6.8). More

effort is required, though, to determine why scheduling the 100% flap

linear models was not adequate for the 100% flap test case. Also, the

transition from zero to nonzero flap deflections still needs to be tested.

6.4 Failure Detection and Scheduling

The scheduling of linear models has been generally effective in

reducing the effects of mismodeling on the no-failure decision functions. Detecting failures with the linear models scheduled is briefly

considered here. Two failures at off-nominal flight conditions are presented. The first case is a -0.0873 tad (-5 deg) elevator failure at a

cruise flight condition of 90.0 m/s (175 knots). The decision functions

for this test case with the linear models scheduled and the 77.2 m/s

Kalman filter and OSGLR steady-state matrices are shown in Figure 6.9.

The failure causes a decrease in dynamic pressure and therefore a switch

from the 102.9 m/s linear model to the 77.2 m/s linear model 5 s after

the failure occurred. The change of linear models does not cause any

noticeable change in the decision functions. However, this is not true

for the second case.

The decision functions of a -0.0873 rad (-5 deg) elevator failure

at a cruise flight condition of 64.3 m/s (125 knots) is shown in Figure

6.10. For the first five seconds following the failure, the decision

functions increase. However, when the linear model used by the age-

-129-
C)

O

6

O

O

v

C_

4_

DF 3

DF 2

DF6 DF 5 /_

DF4 _/

DFI

O_

%:00 'o.oo 20. ' oo so. ' oo 40. ' oo so.oo

Time (s)

Figure 6.6 Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for a 50% flap

deflection test case

• 38.6, 51.44, 77.2 m/s with 50% flap

linear models

• 77.2 m/s with 50% flap age-weighted filter

and OSGLR steady-state matrices

-130-
O

::31'

O

_m

0

DF I

DF 2

DF 3

DF 4

DF 5

DF 6

I ! I

.00 10.00 EO. O0 30.00 qO.O0

Time (s)

50.00

Figure 6.7 iDecision functions produced by an age-weighted,

scheduled OSGLR algorithm for a 75% flap

deflection test case

• 38.6, 51.44, 77.2 m/s with 50% flap

linear models

• 77.2 m/s with 50% flap age-weighted filter

and OSGLR steady-state matrices

-131-
20.00 30.00 qO.O0 50.00

Time (s)

Figure 6.8 Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for a 100% flap

deflection test case

• 100% flap deflection from the start of

the test case

• 38.6, 51.44, 77.2 m/s with 100% flap

linear models

• 77.2 m/s with 100% flap age-weighted filter

and OSGLR steady-state matrices

-132-

{
O

C).

._ DF1

DF 3

i i l ''= i

DF 4

•00 I0.00 20.00 30.00 _0.00 50.00

Time s )

Figure 6.9 Decision functions 3roduced by an age-weighted,

scheduled OSGLRalgorithm for a -0.0873 rad

(-5 deg) elevator bias failure at 90.0 m/s

(175 knots) test case

@ 46.4, 51.44, 77.2, 102.9 m/s linear models

• 77.2 m/s age-weighted filter and OSGLR

steady-state matrices

-133-
O

o

o

DFI

DF4

I0.00 20.00 30.00 ilO.O0 50.00

Time (s)

Figure 6.10 Decision functions produced by an age-weighted,

scheduled OSGLR algorithm for a -0.0873 rad

(-5 deg) elevator bias failure at 64.3 m/s

(125 knots) test case

• 46.4, 51.44, 77.2, 102.9 m/s linear models

• 77.2 m/s age-weighted filter and OSGLR

steady-state matrices

-134-
weighted filter switches from the 77.2 m/s model to the 51.44 m/s model

because the dynamic pressure has decreased, the decision functions

decrease signficantly. The decision functions increase again when the

dynamic pressure increases and the 77.2 m/s model was used again. Two

possible reasons for this undesirable behavior are that the 51.44 m/s

linear model is inadequate or that the 77.2 m/s age-weighted filter and

OSGLR steady-state matrices are not appropriate for the 51.44 m/s linear

model. The second possibility is eliminated as a reason for the decrease

in the decision functions in the following subsection on scheduling of

the age-weighted filter and OSGLR steady-state matrices. The 51.44 m/s

linear model was shown in Section 6.2 not to model the 51.44 m/s cruise

flight condition well so the first reason is the probable explanation.

6.5 Scheduling of Age-Weighted Filter and OSGLR Steady-State Matrices

Scheduling the linear models used by the age-weighted filter while

assuming constant age-weighted filter and OSGLR steady-state matrices is

unlikely to be acceptable over an entire flight envelope. Therefore, the

scheduling of the steady-state age-weighted filter gain and residual

covariance matrices as well as the steady-state OSGLR influence and

information matrices was also investigated. (See Appendix B for more

details.) The simple scheduling approach used for the linear models was

employed (see Section 6.2).

The decision functions for the -0.0873 tad (-5 deg) elevator failure at 64.3 m/s (125 knots) is shown in Figure 6.11. There are several

discontinuities in the decision functions due to changes in the steadystate information matrix used to calculate the decision functions. In

fact, the elevator decision function jumps to 1.8 x 106 . A smaller

vertical scale was chosen to show that even with the correct 51.44 m/s

age-weighted filter and OSGLR steady-state matrices, the decision functions decay to very small values with the 51.44 m/s linear model. As

discussed in the last subsection, the most probable reason for this undesirable behavior is a poor 51.44 m/s linear model.

-135-
DF1

DF2

DF3

DF5

DF4

•O0 IO. O0 20. O0 30. O0 tlO.O0 50. O0

Time (s)

Figure 6.11 Decision functions produced by an age-weighted,

+scheduled OSGLR algorithm for a -0.0873 rad

(-5 deg) elevator bias failure at 64.3 m/s

(125 knots) test case

• 46.3, 51.44, 77.2, 102.9 m/s linear

models

• steady-state age-weighted filter and

OSGLR matrices also scheduled

-136-
The discontinuities resulting from changes in the steady-state information matrix reveal the difficulty with the simple approach of scheduling the steady-state information matrix. Using the simple scheduling,

the linear model as well as the age-weighted filter and OSGLRsteadystate matrices all change at the sametime. The difficulty is that the

steady-state information matrix is not valid for the old linear model,

age-weighted filter matrices, and the OSGLRinfluence matrix, all of

which were used to produce the information vectors up to the time of the

switch of models. In order to avoid this difficulty, one might simply

use a constant information matrix. However, in implementing this algorithm over an entire flight envelope, somechange in the information

matrix will most likely be required.

Another scheduling approach of linearly interpolating between the

linear models and their associated age-weighted filter and OSGLRmatrices

based upon the actual dynamic pressure was tried. The decision functions

for the -0.0873 rad (-5 deg) elevator failure at 90.0 m/s (175 knots)

test case for this scheduling is shownin Figure 6.12. The effect of

turbulence on the dynamic pressure can be seen in the elevator decision

function. The dynamic pressure could be smoothed, however. Still, this

scheduling approach is not very satisfactory either. The elevator decision function for this test case after the failure occured using the 77.2

m/s model and associated steady-state age-weighted filter and OSGLR

matrices is approximately 55,000. For the 102.9 m/s model and associated

steady-state age-weighted filter and OSGLRmatrices, the elevator decision function is approximately 130,000. For the linear scheduling of the

steady-state information matrix to have validity, one would expect the

elevator decision functions to be within the range of 55,000 to 130,000.

Instead, the elevator decision function is muchlarger. Scheduling the

steady-state information matrix probably will not be satisfactory whenever the linear model, age-weighted filter gain and covariance matrices

and OSGLRinfluence matrix are changing. Therefore, the information matrix probably needs to be time-varying.

-137-
O

OW-4

O.

A

o

u

o_

o=

O.

o

DF I

DF 2

DF 5

DF 3

DF 6

DF 4

C)

 o'.oo ,o.oo 2o.oo 3b.oo

Time (s)

50.00

r

Figure 6.12 ) Decision functions for a -0.0873 rad (-5 deg)

elevator bias failure at 175 knots test case

produced by an age-weighted, linearly scheduled

OSGLR algorithm

• linear interpolation scheduling approach

• 46.3, 51.44, 77.2, 102.9 m/s linear

models

• steady-state age-weighted filter and

OSGLR matrices also scheduled !

-138-
6.6 Summary and Conclusions

Scheduling of the linear models used by the age-weighted filter as

a function of dynamic pressure was investigated as a means of reducing

the mismodeling effects in the residuals of the filter and, therefore,

reducing the no-failure decision functions. The age-weighted filter

steady-state gain and covariance matrices and the OSGLR steady-state influence and information matrices were assumed to be constant and corresponded to one of the linear models scheduled. The scheduling approach

simply selected the linear model whose dynamic pressure was closest to

the actual dynamic pressure. Using this scheduling approach, the decision functions produced by the acceleration from 51.44 m/s test case were

reduced from 39,000 to 6,000. This reduced decision function levels are

comparable to the decision function levels produced by a maximum rateof-climb maneuver. Further reductions are possible if a more accurate

51.44 m/s linear model can be developed.

Scheduling linear models as a function of dynamic pressure actually increased the decision function produced by nonzero flap deflections

in some cases. As these no-failure decisions are large, scheduling on

flap deflection is also necessary. Using linear models with a nominal

flap deflection of 50% reduced the decision functions for the 50% and 75%

flap deflection test cases to less than 3,000. (The age-weighted filter

and OSGLR steady-state matrices corresponded to one of the 50% flap

linear models.) However, the decision functions produced by 100% flap

deflection test case were very large even with the 50% flap linear

models. Using linear models with a nominal flap deflection of 100% did

not reduce the decision function levels for the 100% flap test case

either. More effort is required to explain why scheduling 100% flap

linear models was not sufficient to reduce the decision functions

produced by the 100% flap test case.

Two off-nominal elevator failure cases were tested with the linear

models scheduled. One case showed no difficulty in detecting the failure

with the scheduling. The other case further demonstrated the need for a

-139-

i
better 51.44 m/s linear model. Further testing of failure cases is re

quired to determine the effect of scheduling only the linear models on

detecting and isolating failures.

The scheduling of the age-weighted filter steady-state gain and

residual covariance matrices and the OSGLR steady-state influence and

information matrices was also briefly investigated. Some scheduling of

these matrices will be necessary for the OSGLR algorithm to be extended

over the entire flight envelope. Two scheduling approaches were tested:

the simple approach described earlier and linear interpolation. The same

scheduling approach was used for both the linear model and the ageweighted filter and OSGLR steady-state matrices. The results from several failure cases using both scheduling approaches suggested that timevarying information matrix may be required.

The information matrix reflects the uncertainty in the information

vector. The two scheduling approaches assume the change in uncertainty

in the information vector caused by the change in the filter gain and

residual covariance matrices and the OSGLR influence matrix is instantaneous. However, there is a time lag before the effect of the changes in

the gain, covariance, and influence matrices is reflected in the information vector. Further work is needed to define the extent of the scheduling required for the age-weighted filter gain and residual covariance

matrices and the OSGLR influence matrix.

140- !
SUMMARY AND CONCLUSIONS

This report has evaluated the orthogonal series generalized likelihood ratio (OSGLR) test for detecting and isolating commercial aircraft

control surface and actuator failures. The OSGLR algorithm was chosen

based on a preliminary evaluation of three failure detection and isolation (FDI) algorithms: the detection filter, the generalized likelihood

ratio test, and the OSGLR test. The OSGLR test offered the best performance with moderate computational requirements. However, the OSGLR algorithm was also more to sensitive to modeling errors than the other two

algorithms. This preliminary evaluation is summarized in Section 2. In

addition, FDI algorithms are surveyed in Section 3 searching for additional practical techniques that might be applicable for Restructurable

Controls. The most promising algorithms appeared to be those already

considered and, therefore, the effort concentrated on the OSGLR algorithm.

For the OSGLR algorithm to be feasible, it was necessary to improve its robustness to model uncertainty. Methods of improving the robustness of the algorithm were examined with the incorporation of ageweighting into the algorithm being the most effective approach, significantly reducing the sensitivity of the algorithm. With age-weighting,

the covariance of the past measurements and residuals is increased, causing the algorithm to rely more on the new measurements than the past measurements •

In the preliminary evaluation, the algorithms were tested basically at the nominal cruise flight condition of 77.2 m/s (150 knots) at

304.8 m (1000 ft). In this more in-depth evaluation, a number of no-

-141-
failure test cases at off-nominal flight conditions including maneuvers,

nonzero flap deflections, different turbulence levels and steady winds

were used to test the OSGLR algorithm. Maneuvering flight typical of

commercial aircraft near the nominal dynamic pressure, light and nominal

turbulence levels, and steady winds did not produce significant nofailure decision functions. Large (75% and greater) flap deflections and

airspeeds of 51 m/s (100 knots) and less produced large no-failure decision functions. The 77.2 m/s (150 knots) cruise linear model was not

valid for these flight conditions.

Since detection thresholds will be chosen such that false alarms

are improbable, the no-failure decision functions produced by the offnominal flight conditions determine the realistic detection thresholds

possible in the neighborhood of the nominal flight condition. The detection performance was examined by running a number of failure test cases,

mostly at the nominal flight condition. For maneuvering flight near the

nominal dynamic pressure, small flap deflections, light and nominal turbulence levels, and steady wind, selecting a threshold to detect 5 deg

elevator, 7 deg aileron, 8 deg rudder, and 15% flap bias failures in I s

at 77.2 m/s (150 knots) appears possible. If there is no compensation

for thunderstorm turbulence, a larger detection threshold would be required resulting in slightly worse detection performance.

Finally, extending the validity of the algorithm to off-nominal

airspeeds and large flap deflection flight conditions by scheduling was

considered. The approach investigated was to schedule the linear models

used by the age-weighted filter, choosing the linear model with the

nominal dynamic pressure closest to the actual dynamic pressure. The

steady-state age-weighted filter and OSGLR matrices corresponded to one

of the linear model scheduled. Scheduling on dynamic pressure was used

because modeling errors increased with increasing off-nominal dynamic

pressure. This approach successfully reduced the no-failure decision

functions produced at low airspeeds. Scheduling on flap deflection was

also investigated with some success to reduce the no-failure decision

-142-
functions produced by large flap deflections. Since simply scheduling

the linear models over the entire flight envelope of a commercial aircraft is unlikely to be adequate, scheduling of the age-weighted filter

gain and covariance steady-state matrices and the OSGLR influence and information steady-state matrices in the same manner as the linear models

was briefly investigated. The scheduling of the steady-state information

matrix was unsatisfactory; a time-varying information matrix may be required. Scheduling of the linear model matrices, the age-weighted filter

gain and covariance steady-state matrices, and the OSGLR influence

steady-state matrix, however, appears feasible.

Depending on the requirements of restructuring the control system,

the OSGLR algorithm should be capable of detecting aircraft control surface failures for Restructurable Controls application. Moderate and

large failures can be detected quickly. Small failures require longer

detection times or cannot be detected. However, failures that are difficult to detect may not be important and could be partially compensated

for by a control system robust to small control surface failures. Isolation of some failures may be difficult if there are several surfaces

which produce similar effects on the aircraft. Even for these surfaces,

isolation may be possible if the failure is moderate or large. If restructuring the control system requires isolation for all failures, position sensors on some of the surfaces will most likely be required.

Finally, extending the algorithm over the entire operating envelope of a

commercial aircraft, while not demonstrated, appears feasible based on

the experience of extending the algorithm to a number of flight conditions.

Further investigation of the scheduling of the linear models and

the implementation of the OSGLR algorithm is required, though, before the

algorithm can be extended over the entire envelope. Some of the issues

requiring consideration are:

o the number of linear models required to cover the entire flight

envelope.

-143-
o the complexity of the implementation approach versus

performance.

• Is the steady-state implementation of the OSGLR

algorithm (except for the information matrix)

adequate or is the time-varying implementation

necessary?

• Are there significant performance benefits to a

time-varying implementation?

• If scheduling is adequate, how much scheduling of the

age-weighted filter gain and covariance matrices and

the OSGLR influence matrices is required?

o the computational and storage requirements of the different

approaches.

• Can the requirements be reduced by, for example,

fitting scheduled matrices?

These issues are important since there is not much experience in implementing multivariable systems for practical real-time use.

-144-
REFERENCES

I •

2e

3e

Montoya, R.J., W.E. Howell, W.T. Bundick, A.J. Ostroff,

R.M. Heuschen, and C.M. Belcastro, Restructurable Controls, NASA

Conference Publication 2227, NASA Langley Research Center, Hampton,

Virginia, September 21-22, 1982.

Ostroff, A.J., and R.M. Heuschen, "Investigation of Control Law

Reconfigurations to Accommodate a Control Element Failure on a

Commercial Aircraft," Proceedings of the 1984 Automatic Control

Conference, San Diego, CA, June 6-8, 1984.

Looze, D.P., S.M. Krolewski, J.L. Weiss, J.S. Eterno, and

S.W. Gully, "An Approach to Restructurable Control System Design,"

Proceedings of the 23rd Conference on Decision and Control, Las

4o

Be

6.

Vegas, NV, December, 1984.

"National Transportation Safety Board Accident Report of the

American Airlines DCI0 Crash at Chicago - O'Hare International

Airport, May 25, 1979," NTSB-AAR-79-17, December, 1979.

McMahan, J., "Flight 1080," Air Line Pilot, July, 1978.

Motyka, P., W. Bonnice, S. Hall, and E. Wagner, The Evaluation of

Failure Detection and Isolation Algorithms for Restructurable

7o

8•

9•

Control, NASA Contractor Report 177983, November 1985.

Beard, R.V., Failure Accommodation in Linear Systems Through

Self-Reorganization, Ph.D. Thesis, Dept. of Aeronautics and

Astronautics, MIT, Cambridge, MA, February, 1971.

Jones, H.L., Failure Detection in Linear Systems, Ph.D. Thesis,

Dept. of Aeronautics and Astronautics, MIT, Cambridge, MA, August,

1973.

Meserole, Jr. J.S., Detection Filters for Fault-Tolerant Control of

Turbofan Engines, Ph.D. Thesis, Dept. of Aeronautics and

Astronautics, M_, June, 1TM •

-145-
10.

11.

12.

Willsky, A.S. and H.L. Jones, "A Generalized Likelihood Ratio

Approach to the Detection and Estimation of Jumpsin Linear

Systems," IEEE Transactions on Automatic Control, Vol. AC-21, No. I,

pp. 108-112, February, 1976.

Hall, S.R., A Failure Detection Algorithm for Linear Dynamic

Systems, Sc.D. Thesis, Department of Aeronautics and Astronautics,

Massachusetts Institute of Technology, Cambridge, MA, June, 1985

(also CSDL-T-874, June, 1985).

Chalk, C.R., et el, (1969) Background Information and User Guide for

MIL-F-8785B (ASG), "Military Specification - Flying Qualities of

13.

14.

15.

16.

17.

18.

19.

20.

Piloted Airplanes," Technical Report AFFDL-TR-69-72, Air Force

Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio,

August, 1969.

Bonnice, W.F., E. Wagner, P. Motyka, and S.R. Hall, "The Application

of the Detection Filter to Aircraft Control Surface and Actuator

Failure Detection and Isolation," AIAA Paper 85-1973-CP, AIAA

Guidance, Navigation and Control Conference, Snowmass, CO., August

1985.

Hall, S.R. and B.K. Walker, "The Orthogonal Series Generalized

Likelihood Ratio Test," presented at the American Control

Conference, June 1985.

Van Trees, H.L., Detection, Estimation, and Modulation Theory, Part

I, John Wiley & Sons, New York, 1968.

Pallett, E.H.J., Automatic Flight Control, Granada, 1983, pp.

264-270+.

Weirather, L.H., "Transducers," Chapter 4 in Basic Principles of

Flight Test Instrumentation Engineering, Vol. I, AGARDograph No.

160, April 1974.

Willsky, A.S., "A Survey of Design Methods for Failure Detection in

Dynamic Systems," Automatica, 12 (1976), pp. 601-611.

Caglayan, A.K., and R.E. Lancraft, A Fault Tolerant System for an

Integrated Avionics Sensor Configuration, NASA Contractor Report

3834, September 1984.

Chow, E.Y. and A.S. Willsky, "Bayesian Design of Decision Rules for

Failure Detection," IEEE Transactions and Aerospace and Electronic

stems, _._._,._ .j_=, _. ,v. _

-146- i
21.

22.

23.

Deckert, J.C., M.N. Desai, J.J. Deyst, and A.S. Willsky, "F8-DFBW

Sensor Failure Identification Using Analytic Redundancy," IEEE

Transactions on Automatic Control, October 1977, pp. 795-803.

Desai, M.N., J.C. Deckert, and J.J. Deyst, "Dual-Sensor Failure

Identification Using Analytic Redundancy," Journal of Guidance and

Control, May-June 1979, pp 213-220.

Gelb, A., Ed., Applied Optimal Estimation, The M.I.T. Press,

Cambridge, MA, 1974.

-147-
APPENDIXA

SUMMARY OF TEST CASES

This appendix contains a table (Table A-I) summarizing the test

cases presented in this report for easy reference and comparison. Only

the steady-state implementation of the OSGLR algorithm was tested. In

addition, all the test cases included the effects of sensor noise,

errors, and dynamics.

-148-
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-149-
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-151-
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-155-
APPENDIX B

SCHEDULING IMPLEMENTATION DETAILS

B.I Introduction

This appendix describes in more detail the scheduling considered

in this report. First, the linear model formulation and the OSGLR

steady-state equations will be presented and then the scheduling of the

linear models and OSGLR steady-state algorithm described.

B.2

where

Linear Model Formulation

The continuous-time linear model can be written

_ = f(Xo,U O)_ _ + A_x_ + B_x_ (B.I)

Z = g(_o,_o ) + c__ + DA_ (B.2)

x = f(x,u) (B.3)

y = g(x,y) (B.4)

are the nonlinear state differential and output equations, x and

-o

u are the nominal state and input vectors. _x and Au are defined to be

_x _ x - x (B.5)

-- I --O

Au _ U - U (B.6)

-156- i
An equivalent discrete-time linear model can be written

Ax_(k+1) = fd(Xo,Uo) + _x_(k) + BAu(k) (B.7)

y(k) = gd(Xo,Uo ) + CAx_(k) + DA_u(k) (B.8)

where the matrix B now is the discrete state input matrix (consistent

with Eq. (2.17)). The discrete linear model may be rewritten as

x(k+1) = #x(k) + Bu(k) + g(k) (B.9)

y(k) = Cx(k) + Du(k) + h(k) (B°I0)

m -- m

by using Eqs. (B.5) and (B.6).

are given by

The constant bias vectors, g(k) and h(k),

g(k) - = -u-f_(Xo'Uo) - + x-o - #x-o - Bu-o (B°11)

h(k) = _(Xo,Uo ) - CXo - Du_o (B.12)

B.3 Steady-State OSGLR Equations

The steady-state OSGLR equations consist of the steady-state ageweighted filter equations and the OSGLR information vectors and decision

function equations.

The steady-state age-weighted filter equations are

x-(k+1) = _+(k) + Bu(k) + g(k) (B.13)

^

x+(k) = x-(k) + Ky(k) (B.14)

_7(k) = y_(k) - _y(k) (B.15)

y(k) = Cx(k) + Du(k) + h(k) (B.16)

-157-
The steady-state information vector equation is

x(k+1) = !s A_(k) + GTM -I__(k+I) (B.17)

where s is the age-weighting constant, AS is the basis function

transition matrix, G is the steady-state influence matrix, and M is the

residual covariance matrix (with the effects of age-weighting included).

The decision function is given by

T S-I DE(k) = X (k) _x(k) (B.18)

where S is the steady-state information matrix.

B.4 Scheduling

The scheduling approach given the most consideration was the

approach of scheduling the linear model used by the age-weighted filter

(Eqs. B.13 and B.16) as a function of dynamic pressure. At each time

step, the linear model whose nominal dynamic pressure was closest to the

actual dynamic pressure was chosen. While scheduling of all the #, B, C,

and D matrices and the _ and _ vectors which represent the linear model

may not be necessary in practice, this was the approach taken in this

report.

The scheduling of the steady-state age-weighted filter gain and

residual covariance matrices (K and M respectively) and the steady-state

OSGLR influence and information matrices (G and S respectively) was also

briefly examined. Scheduling of the A_matrix and the age-weighting

factor s is not necessary as they remain constant. Actually, the product

GTM -1 found in Eq. B.17 was scheduled instead of scheduling the steadystate influence and residual covariance matrices (G and M) separately®

-158-
TECHNICAL REPORT STANDARD TITLE PAGE

1. Report No.

4. Title and Subtitle

2. Government Accession No.

The Evaluation of the OSGLR Algorithm for

Restructurable Controls

3. Recipient's Catalog No.

5. Report Date

March 1986

6. Performing Organization Code

7. Author(s) 8. Performing Organization Report No.

W.F. Bonnice, E. Wagner, S.R. Hall, P. Motyka CSDL-R-1849

9. Performing Organization Name and Address 10. Work Unit No.

The Charles Stark Draper Laboratory, Inc.

555 Technology Square

Cambridge, Massachusetts 02139

2. Sponsoring Agency Nameand Address

National Aeronautics and Space Administration

Washington, D.C. 20546

11. Contract or Grant No.

NASI-17556

13. Type of Report and Period Covered

Final Report

Dec. 1983 - April 1986

14. Sponsoring Agency Code.

]5. Supplementary Notes

Langley Technical Monitor: W. Thomas Bundick

16. Abstract The detection and isolation of commercial aircraft control surface and

actuator failures using the orthogonal series generalized likelihood ratio (OSGLR)

test was evaluated. The OSGLR algorithm was chosen as the most promising algorithm based on a preliminary evaluation of three failure detection and isolation

(FDI) algorithms (the detection filter, the generalized likelihood ratio test,

and the OSGLR test) and a survey of the literature.

One difficulty of analytic FDI techniques and the OSGLR algorithm in particular is their sensitivity to modeling errors. Therefore, methods of improving

the robustness of the algorithm were examined with the incorporation of ageweighting into the algorithm being the most effective approach, significantly

reducing the sensitivity of the algorithm to modeling errors.

The steady-state implementation of the algorithm based on a single cruise

linear model was evaluated using a nonlinear simulation of a C-130 aircraft. A

number of off-nominal no-failure flight conditions including maneuvers, nonzero

flap deflections, different turbulence levels and steady winds were tested.

Based on the no-failure decision functions produced by off-nominal flight conditions, the failure detection performance at the nominal flight condition was

determined.

The extension of the algorithm to a wider flight envelope by scheduling the

linear models used by the algorithm on dynamic pressure and flap deflection was

also considered. Since simply scheduling the linear models over the entire

flight envelope is unlikely to be adequate, scheduling of the steady-state implementation of the algorithm was briefl___nvestigated.

17. Key Words Suggested by Author _. Distributio'--------_Sta-_teme---n[

Failure Detection and Isolation Unclassified - Unlimited

Restructurable Controls Subject Category - 04

Generalized Likelihood Ratio

19. Security CJassif. (of this report)_20. Unclassified

Security Classif.(ofthispage)

Unclassified

21. No. of Pages 122. Price

158