by Wen Fan - University of Toronto T-Space · Fault Tolerant Control of a Flexible Aircraft by Wen...

Fault Tolerant Control of a Flexible Aircraft

by

Wen Fan

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Institute for Aerospace StudiesUniversity of Toronto

c© Copyright 2018 by Wen Fan

Abstract

Fault Tolerant Control of a Flexible Aircraft

Wen Fan

Doctor of Philosophy

Institute for Aerospace Studies

University of Toronto

2018

Fault tolerant control (FTC) designs that deal with control surface faults while handling

undesirable aeroelastic problems for flexible aircraft are presented. The goals of control de-

sign are to achieve flutter suppression or perform structural load alleviation while recovering

nominal flight and maintaining acceptable performance in the event of control surface faults.

The contributions of this thesis are two fold. First, the influence of control surface faults

on not only the rigid-body motions but also the aeroelastic modes has been analyzed and

mitigated through FTC design. Second, fault handling ability has been incorporated into

an integrated flight control design that accounts for three different aeroelasticity require-

ments for flexible aircraft: flutter suppression, gust load alleviation (GLA) and maneuver

load alleviation (MLA). A mathematical model based on the Lagrange’s equations for quasi-

coordinates that can describe the rigid-body motions of the aircraft and elastic deformations

of the flexible wings as well as the coupling between them is developed. The flexible wing

structure is modeled as a cantilever beam based on the Euler-Bernoulli beam elements under

unsteady aerodynamic loads. For the flutter suppression problem, a linear parameter-varying

FTC controller is developed to address stuck faults, actuator saturation and aeroelastic in-

stability caused by parameter variations. The controller, in which the gain schedules with

airspeed, can eliminate the effects of faults on the rigid-body motion while suppressing the

unwanted vibrations of the wing and guaranteeing no closed-loop performance degradation

caused by actuator saturation. For the GLA problem, a mixed H2/H∞ FTC controller that

can simultaneously achieve rigid-body motion stabilization, load alleviation on flexible wing

structures, and on-line accommodation of loss of control effectiveness fault is developed.

ii

For the MLA problem, a model predictive control formulation with reference adjustment is

presented to handle stuck and loss of control effectiveness faults. For each fault case, the

controller can automatically adjust the tracking reference to guarantee an admissible track-

ing trajectory. During the maneuver, the structural load is constrained within given upper

and lower bounds. The effectiveness of the three FTC designs are all demonstrated on an

aircraft model with high aspect-ratio wings in numerical simulations.

iii

Acknowledgements

First and foremost, I would like to express my deepest gratitude to my supervisors, Prof.

Hugh Liu and Prof. Raymond Kwong, who have supported me throughout my thesis and

PhD studies, for their continuous guidance, helpful advice and kind encouragement. I sin-

cerely appreciate those weekly meetings we had to discuss my research progress, the freedom

they gave me to explore research ideas, the precious time they spent to revise my work in

detail and each piece of advice they gave to help me grow in both research and life.

Secondly, I would like to thank the other two members of my PhD committee Prof. Craig

Steeves and Prof. Chris Damaren who always took their time to critically review my work

and examine my research progress. Their insightful questions and constructive suggestions

during my DEC meetings guided me on the right track in my PhD studies.

Thirdly, I appreciate the friendly environment built by all members and alumni of the

Flight System and Control Lab.

Fourthly, I would like to thank the nice and helpful staff members of UTIAS. Without

their collaborative efforts, UTIAS would not be like a big warm family.

Last but not least, I want to express my deepest thanks to my loving and caring mother,

who has always been supporting me in any decisions I made, including pursuing my PhD.

Through my upbringing, she had patiently guided me to keep a positive attitude towards

difficulties and challenges. It was from her that I learned how to be a self-reliant and diligent

person.

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Contents

Ackowledgements iv

Contents v

List of Tables viii

List of Figures ix

Nomenclature xiii

Acronyms and Abbreviations xviii

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Active Flutter Suppression . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Gust Load Alleviation (GLA) . . . . . . . . . . . . . . . . . . . . . . 7

1.2.3 Maneuver Load Alleviation (MLA) . . . . . . . . . . . . . . . . . . . 7

1.2.4 Fault Tolerant Control (FTC) . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Flexible Aircraft Model 13

2.1 Equations of Motion for Flexible Aircraft . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Structural Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Aerodynamic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 State-space Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

v

3 FTC Design for Flutter Suppression Problem 28

3.1 Influence of Stuck Faults on Flexible Aircraft . . . . . . . . . . . . . . . . . . 29

3.1.1 Linearized Model and Nominal Controller . . . . . . . . . . . . . . . 30

3.1.2 Fault Injection and Post-Fault Responses . . . . . . . . . . . . . . . . 33

3.2 FTC Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.1 Linear Parameter-Varying (LPV) Representation . . . . . . . . . . . 51

3.2.2 Faulty System Description . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.3 Design Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 LPV FTC Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.1 Controller Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.2 Set-invariance Conditions for Tolerating Stuck Faults . . . . . . . . . 54

3.3.3 LMI Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4.1 Fault Scenario 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4.2 Fault Scenario 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 FTC Design for Gust Load Alleviation Problem 70

4.1 Faulty Flexible Aircraft Model with Gust . . . . . . . . . . . . . . . . . . . . 71

4.2 Fault Tolerant GLA Control Design . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.1 Fault Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.2 Fault Tolerant Mixed H2/H∞ Controller Design . . . . . . . . . . . . 74


4.3.1 Gust Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3.2 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 FTC Design for Maneuver Load Alleviation Problem 89

5.1 MLA Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2 Feasible Reference-Tracking Model Predictive Control (MPC) . . . . . . . . 92

5.2.1 Admissible Invariant Set for Tracking . . . . . . . . . . . . . . . . . 92

5.2.2 MPC Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.3 Fault Tolerant MPC for MLA . . . . . . . . . . . . . . . . . . . . . . . . . . 97


5.4.1 Nominal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4.2 Fault Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4.3 Fault Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

vi

5.4.4 Fault Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Conclusions and Future Work 110

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Copyright Permission 113

Bibliography 114

vii

List of Tables

3.1 Properties of the HALE aircraft . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Comparison of Free Vibration Mode Natural Frequencies (rad/s) . . . . . . . 32

3.3 Comparison of Aeroelastic Results . . . . . . . . . . . . . . . . . . . . . . . . 32

viii

List of Figures

2.1 Schematic Drawing of Flexible Aircraft Frames: Inertial Frame XY Z with

Origin O, Fuselage Frame xfyfzf with Origin Of , Wing Frame xwywzw with

Origin Ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 HALE Aircraft Model Geometry Top View (not to scale) . . . . . . . . . . . 29

3.2 Root Locus of the Open-loop Flexible Aircraft System as Airspeed Varies from

25 m/s to 35 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Closed-loop and Open-loop Responses of Wing Deformations at Speed = 32

m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Closed-loop and Open-loop Responses of Rigid-body States at Speed = 32 m/s 34

3.5 Closed-loop and Open-loop Responses of Wing Deformations at Speed = 34

m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6 Closed-loop and Open-loop Responses of Rigid-body States at Speed = 34 m/s 35

3.7 Post-fault Responses of Wing Deformations at Speed = 32 m/s for δa1 = 1 ,

3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.8 Post-fault Responses of Rigid-body States at Speed = 32 m/s for δa1 = 1 ,

3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.9 Control Surface Deflections at Speed = 32 m/s for δa1 = 1 , 3 and 5 . . . 37


3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37


3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38



3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39


3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


ix


3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41


3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42


3.19 Post-fault Responses of Wing Deformations at Speed = 32 m/s for δa1,2 = 1

, 3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.20 Post-fault Responses of Rigid-body States at Speed = 32 m/s for δa1,2 = 1 ,

3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.21 Control Surface Deflections at Speed = 32 m/s for δa1,2 = 1 , 3 and 5 . . . 44

3.22 Post-fault Responses of Wing Deformations at Speed = 34 m/s for δa1,2 = 1

, 3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.23 Post-fault Responses of Rigid-body States at Speed = 34 m/s for δa1,2 = 1 ,

3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.24 Control Surface Deflections at Speed = 34 m/s for δa1,2 = 1 , 3 and 5 . . . 46

3.25 Post-fault Responses of Wing Deformations at Speed = 32 m/s for δe = 1 ,

3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.26 Post-fault Responses of Rigid-body States at Speed = 32 m/s for δe = 1 , 3

and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.27 Control Surface Deflections at Speed = 32 m/s for δe = 1 , 3 and 5 . . . . 48

3.28 Post-fault Responses of Wing Deformations at Speed = 34 m/s for δe = 1 ,

3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.29 Post-fault Responses of Rigid-body States at Speed = 34 m/s for δe = 1 , 3

and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.30 Control Surface Deflections at Speed = 34 m/s for δe = 1 , 3 and 5 . . . . 49

3.31 Closed-loop Responses of Wing-tip Bending and Torsion with the FTC and

Nominal Controllers at 32 m/s under Fault Scenario 1 . . . . . . . . . . . . . 60

3.32 Closed-Loop Responses of Pitch Angle and Rate with the FTC and Nominal

Controllers at 32 m/s under Fault Scenario 1 . . . . . . . . . . . . . . . . . . 60

3.33 Control Surface Deflections of the FTC and Nominal Controllers at 32 m/s

under Fault Scenario 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.34 Closed-loop Responses of Wing-tip Bending and Torsion with the FTC and

Nominal Controllers at 33.5 m/s under Fault Scenario 1 . . . . . . . . . . . . 62


Controllers at 33.5 m/s under Fault Scenario 1 . . . . . . . . . . . . . . . . . 62

x

3.36 Control Surface Deflections of the FTC and Nominal Controllers at 33.5 m/s


3.37 Closed-Loop Responses of Wing-Tip Bending and Torsion with the FTC and

Nominal Controllers at 32 m/s under Fault Scenario 2 . . . . . . . . . . . . . 64


Controllers at 32 m/s Under Fault Scenario 2 . . . . . . . . . . . . . . . . . 64

3.39 Control Surface Deflections of the FTC and Nominal Controllers at 32 m/s


3.40 Closed-Loop Responses of Wing-Tip Bending and Torsion with the FTC and

Nominal Controllers at 33.5 m/s under Fault Scenario 2 . . . . . . . . . . . . 66


Controllers at 33.5 m/s under Fault Scenario 2 . . . . . . . . . . . . . . . . . 66

3.42 Control Surface Deflections of the FTC and Nominal Controllers at 33.5 m/s


3.43 Closed-Loop Responses of Wing-Tip Bending and Torsion under Fault Sce-

nario 2 with Switched on FTC Controller and Nominal Controller at 33.5

m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.44 Closed-Loop Responses of Pitch Angle and Rate under Fault Scenario 2 with

Switched on FTC Controller and Nominal Controller at 33.5 m/s . . . . . . 68

3.45 Control Surface Deflections of Switched on FTC Controller and Nominal Con-

troller at 33.5 m/s under Fault Scenario 2 . . . . . . . . . . . . . . . . . . . 68

4.1 Structure diagram of fault tolerant GLA Design . . . . . . . . . . . . . . . . 73

4.2 HALE Aircraft Model Geometry Top View(not to scale) . . . . . . . . . . . 77

4.3 A Discrete Gust Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 A Dryden Gust Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.5 Altitude Responses in Nominal Case with “1- cosine” Gust Excitation . . . . 80

4.6 Wing Root Bending Moment Responses in Nominal Case with “1- cosine”

Gust Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.7 Control Surface Deflections in Nominal Case with “1- cosine” Gust Excitation 81

4.8 Altitude Responses in Nominal Case with Dryden Gust Excitation . . . . . . 82

4.9 Wing Root Bending Moment Responses in Nominal Case with Dryden Gust

Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.10 Control Surface Deflections in Nominal Case with Dryden Gust Excitation . 83

4.11 Estimates of Effectiveness Factors with “1-cosine” Gust Excitation . . . . . . 84

4.12 Altitude Responses in Faulty Case with “1- cosine” Gust Excitation . . . . . 84

xi

4.13 Wing Root Bending Moment Responses in Faulty Case with “1- cosine” Gust

Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.14 Control Surface Deflections in Faulty Case with “1- cosine” Gust Excitation 85

4.15 Estimates of Effectiveness Factors with Dryden Gust Excitation . . . . . . . 86

4.16 Altitude Responses in Faulty Case with Dryden Gust Excitation . . . . . . . 86

4.17 Wing Root Bending Moment Responses in Faulty Case with Dryden Gust

Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.18 Control Surface Deflections in Faulty Case with Dryden Gust Excitation . . 87

5.1 Reference Command, Artificial Reference, Tracking Output Responses with

and without MLA in Nominal Case . . . . . . . . . . . . . . . . . . . . . . . 100

5.2 Wing Root Bending Moment Responses with and without MLA in Nominal

Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3 Control Surface Deflections with and without MLA in Nominal Case . . . . . 101

5.4 Rigid-body State Responses with and without MLA in Nominal Case . . . . 102

5.5 Reference Command, Artificial Reference and Tracking Output in Fault Case 1103

5.6 Wing Root Bending Moment Response in Fault Case 1 . . . . . . . . . . . . 103

5.7 Control Surface Deflections in Fault Case 1 . . . . . . . . . . . . . . . . . . . 104

5.8 Rigid-body State Responses in Fault Case 1 . . . . . . . . . . . . . . . . . . 104









xii

Nomenclature

Flexible Aircraft Model

Equations of Motion

C damping matrix of the wing

Cf transform matrix between the fuselage frame and the inertial frame

Cw transformation matrix from the fuselage frame to wing frame

dmf , dmw a mass element on the fuselage, a mass element on the wing

E stiffness matrix of the aircraft model

Eij block of aircraft stiffness matrix, i, j = 1, 2, 3

Ef matrix relating Euler velocity vector and fuselage frame velocity vector

F generalized resultant forces, [N]

F Rayleigh’s dissipation of the wing structure

fe generalized forces that act on the aircraft wing

g vector of gravitational acceleration [m/s2]

H damping matrix of the aircraft model

Hij block of aircraft damping matrix, i, j = 1, 2, 3

K stiffness matrix of the wing

L Lagrangian for the whole aircraft

M generalized resultant forces and moments [N·m]

M mass matrix of the aircraft

Mij block of aircraft mass matrix, i, j = 1, 2, 3

m,mf ,mw mass of the whole aircraft, mass of the fuselage, mass of the wing [kg]

Nu,Nψ shape function matrices for bending and torsion

O,Of , Ow origins of the inertial frame, fuselage frame and wing frame

qw modal coordinates vector

Rf position vector of origin of fuselage frame relative to inertial frame [m]

Rf absolute position of a mass element on the fuselage frame [m]

Rw position vector of a mass element on the wing relative to Ow [m]

xiii

rf position of a mass element on the fuselage relative to Of [m]

rfw position vector from Of to Ow [m]

rw rigid distribution vector of a mass element on the wing relative to Ow [m]

T kinetic energy for the whole aircraft

Tf , Tw kinetic energy for the fuselage, kinetic energy for the wing

uw bending displacement vector of a mass element on the wing [m]

U strain energy

Ug gravitational energy

Ui strain energy of element i

V aircraft system velocity vector

Vf translational velocity vectors of fuselage frame [m/s]

Vf absolute velocity of a point on the fuselage [m/s]

Vw absolute velocity of a point on the wing [m/s]

V potential energy for the whole aircraft

vw bending velocity vector of a mass element on the wing [m/s]

XY Z the inertial frame

xfyfzf fuselage frame

xwywzw wing frame

αw torsional velocity vector of a mass element on the wing [rad/s]

θf vector of Euler angles, rad

θ pitch angle [rad]

φ roll angle, rad

ψw torsional displacement vector of a mass element on the wing [rad]

ψ yaw angle [rad]

ωf angular velocity vector of fuselage frame [rad/s]

Structure Modeling

Ci damping matrix of element i

Ciu element damping matrix for bending

Ciψ element damping matrix for torsion

cui, cψi bending and torsion damping functions for element i

d vector of all the nodal degrees of freedom

di nodal displacement vector for element i

EIi element bending rigidity [N ·m2]

Fi Rayleigh’s dissipation of element i

GJi element torsional rigidity [N ·m2]

Ki stiffness matrix of element i

xiv

Kiu element stiffness matrix for bending

Kiψ element stiffness matrix for torsion

Nele total element number

ui bending displacement of a finite element node i [m]

βi bending slope of a finite element node i

ζ structural damping factor

Λu,Λψ lowest natural frequencies for bending and torsion [rad/s]

Φ shape function matrix for all element nodes

Φu bending shape function matrix for all element nodes

Φψ torsional shape function matrix for all element nodes

Φi shape function matrix for element i

Φi element shape functions, i = 1, · · · , 6Φq mode shape matrix

ψi torsional displacement of a finite element node i [rad]

Aerodynamic Modeling

Ai coefficient matrix in induced-flow state equation, i = 1, 2, 3, 4

a dimensionless parameter for the elastic axis location

Bi coefficient matrix in induced-flow state equation, i = 1, 2, 3

b half chord length [m]

bi coefficient for average induced-flow velocity calculation, i = 1, · · · , NCfa transformation matrix from local aerodynamic frame to fuselage frame

Fac,Mac,fac generalized loads from the aerodynamics and controls

F distac ,M dist

ac distributed aerodynamic loads under fuselage frame

L0,M0 unsteady lift and moment of a 2D airfoil about aerodynamic center

Lδ,M δ aerodynamic lift and moment from control surface

Lac,Mac total lift and moment of an airfoil about the aerodynamic center

Lea,Mea total lift and moment of an airfoil about the elastic axis

N number of induced-flow states

y, z velocity vector components along and perpendicular to the chord [m/s]

δu control surface deflection

θw torsional angle of a 2D airfoil [rad]

λ Induced-flow state vector

λ0 induced-flow velocity [m/s]

λi induced-flow state, i = 1, · · · , N [m/s]

ρ air density [kg/m3]

xv

Control Systems

A system matrix

Aa augmented system matrix

Ad discrete-time system matrix

ai reciprocal of time constant of the i-th actuator, i = 1, · · ·mB control distribution matrix

Ba augmented distribution matrix of healthy control surfaces

Bd discrete-time control distribution matrix

B1 distribution matrix of healthy control surfaces

B2 distribution matrix of stuck control surfaces

C output matrix

D diagonal matrix associated with upper bound of stuck fault values

Ea augmented distribution matrix of stuck control surfaces

Fs set of numbers of stuck control surfaces

G distribution matrix of gust disturbance

Hc auxiliary control gain

hci ith row of auxiliary control gain, i = 1, · · ·m1

K control gain

Kc augmented control gain

KI , Kx integral control gain, proportional control gain

m total number of control surfaces

m1,m2 number of healthy control surfaces, number of stuck control surfaces

n dimension of system matrix

N control horizon

O∞ maximal invariant for tracking

Oε∞ approximation of O∞

P terminal weighting matrix

Pc Lyapunov matrix

Q state weighting matrix

R input weighting matrix

T weighting matirx for artificial reference

U future control input sequence

U airspeed [m/s]

U,U lower and upper bounds of airspeed range, m/s

Ur airspeed range, m/s

u control inputs

xvi

uc control input vector of healthy control surfaces

us stuck fault value vector

ui actual deflection of the i-th control surface

uci control input command of the i-th control surface

ui saturation limit of the ith control surface, i = 1, · · ·mVc Lyapunov function

Vi Lyapunov function for the adaptive estimator

w gust disturbance

X predicted state sequence

x state vector

xr rigid-body state vector

xe vector of elastic and aerodynamic states

y output vector

yc constrained output vector

yt tracking output vector

Zu, Zy input constraint set, output constraint set

zrigid rigid-body performance output

zflexible flexible performance output

α index for minimization of the invariant set

γ index for convex optimization

γ1, γ2 indices for mixed-norm convex optimization

δa1, δa2, δe deflections of Flap 1, Flap2, elevator, deg

δi upper bound of stuck fault value, i = 1, · · · ,m2

εi lower bound of effectiveness for the i-th control surface, i = 1, · · · ,mµ weight parameter for mixed-norm optimization

ρ effectiveness factor matrix

ρi effectiveness factor of the i-th control surface, i = 1, · · · ,mρi estimate of the effectiveness factor ρi

ξ(t) augmented state vector

Ωc invariant set of augmented system

Ωd ellipsoidal set of stuck fault values

xvii

Acronyms and Abbreviations

AFS Active Flutter Suppression

BACT Benchmark Active Control Technology

DOFs degrees of freedom

ERAST Environmental Research Aircraft and Sensor Technology

FDD Fault Detection and Diagnosis

FTC Fault Tolerant Control

FTFC Fault Tolerant Flight Control

GLA Gust Load Alleviation

GPC Generalized Predictive Control

HALE High-Altitude Long-Endurance

HAPS High-Altitude Pseudo-Satellite

ISR Intelligence, Surveillance and Reconnaissance

LCOs Limit Cycle Oscillations

LFT Linear Fractional Transformation

LMI Linear Matrix Inequality

LPV Linear Parameter-Varying

LQG Linear Quadratic Guassian

LQR Linear Quadratic Regulator

MLA Maneuver Load Alleviation

MMST Multiple Models, Switching and Tuning

MPC Model-Predictive Control

NATA Nonlinear Aeroelastic Test Apparatus

OCC Output Covariance Constraint

OIST Output Input Saturation Transform

PAAW Performance Adaptive Aeroelastic Wing

PDC Parallel Distributed Compensation

PI Proportional and Integral

xviii

PSD Power Spectral Density

QP Quadratic Programming

RNNs Recurrent Neural Networks

TS Takagi-Sugeno

UAVs Unmanned Aerial Vehicles

VCCTEF Variable Camber Continuous Trailing Edge Flap

2D two-dimensional

3D three-dimensional

xix

Chapter 1

Introduction

1.1 Background

The concept of energy-efficient aircraft designs has attracted considerable attention in air-

craft industries. It is crucial for the sustained growth of commercial aviation, from both

environmental and economic points of view. Reducing the airframe weight by using light-

weight composite materials in modern aircraft designs is a major means to improve energy

efficiency. Commercial transport aircraft manufacturers have begun to employ a high pro-

portion of light-weight composites in their new aircraft: the Boeing 787 Dreamliner uses

50% composites [1], the A350 XWB airframe is made out of 53% composites [2] and the

Bombardier CSeries aircraft contains 46% advanced composite materials [3]. The substan-

tial weight reduction results in less fuel consumption. For example, Boeing claims that its

787 comsumes 20% less fuel than the similar-sized aircraft 767 due to the use of composites

[1].

Another design trend in transport aircraft to improve fuel efficiency is the increasing wing

aspect ratio. For example, the Boeing 787 Dreamliner has a higher aspect ratio than the old

generation, which is 11 compared to 10 of the Boeing 777-300ER wing and 8 of the Boeing

747-400 wing. The Fixed Wing project of NASA’s Fundamental Aeronautics program has

also listed the high aspect ratio wing as one of the advanced concept studies for commercial

subsonic transport aircraft entering service in the 2030-2035 period. The high aspect ratio

wing work aims to address the challenge of reducing fuel burn by increasing wing aspect

ratio, which leads to the decreasing of lift-induced drag.

Other than commercial transport aircraft, the high-altitude long-endurance (HALE) un-

manned aircraft, which also feature light-weight airframe and high aspect ratio wings, have

been developed and received a lot of attentions in past decades. These air-borne vehicles, as

indicated by the name, are capable of flying at altitude as high as 15 to 30 km and stay aloft

1

Chapter 1. Introduction 2

for considerable periods of time without recourse to landing. They can have wide-range appli-

cations in both military and civilian areas, including airborne Intelligence, Surveillance, and

Reconnaissance (ISR), telecommunications relay, environmental data monitoring, weather

observing and atmospheric research. For example, NASA’s Environmental Research Aircraft

and Sensor Technology (ERAST) program was a multi-year effort to develop cost-effective,

slow-flying unmanned aerial vehicles (UAVs) for performing upper atmospheric science mis-

sions at altitudes above 60,000 feet (18,288 m) [4]. Before its termination in 2003, ALTUS

II, and a series of solar- and fuel cell system-powered UAVs such as Pathfinder, Centurion,

and Helios aircraft were developed. Such aircraft, according to NASA, could be used to col-

lect, identify, and monitor environmental data to assess global climate change and assist in

weather monitoring and forecasting. They also could serve as airborne telecommunications

platforms, performing functions similar to communications satellites at a fraction of the cost

of lofting a satellite into space.

Aiming at filling a capability gap between satellites and UAVs, Airbus Defence and Space

has been developing the Zephyr aircraft as a High-Altitude Pseudo-Satellite (HAPS). The

series of lightweight solar-powered UAVs, which was originally designed and built by QinetiQ,

is famous for breaking the world records for longest endurance flight. The Zephyr 7, which

flew for 14 days between 9 July to 23 July 2010, holds the official long-endurance record for

unrefueled aircraft of 336 hours, 22 minutes and 8 seconds.

Another effort worth mentioning in the development of the HALE aircraft is the Vulture

program. Starting in 2008 and terminated in 2012, the Defence Advanced Research Projects

Agency (DARPA) had funded Aurora Flight Sciences, Boeing and Lockheed Martin for the

development of Phase 1 and given Boeing an 89 USD million contract for developing the

SolarEagle aircraft in Phase 2. The SolarEagle, which was proposed to have a wingspan of

393.7 feet (120 m), was intended to remain airborne and stay on station uninterrupted for

five years. In addition to the SolarEagle, Boeing also developed a liquid hydrogen-fueled

HALE demonstrator aircraft Phantom Eye. It had a wingspan of 150 ft (46 m) and could

maintain its altitude for up to four days while carrying a 450-pound (204-kg) payload at

65,000 ft (19,800 m). Interests on HALE UAVs also come from companies that are outside

the traditional aerospace industries. Both Facebook and Google, seek to develop solar-

powered HALE UAVs as atmospheric satellites that are capable of staying aloft for months

to provide Internet services in remote parts of the world.

Due to the mission requirements, the design of HALE aircraft are more likely to have

large high aspect ratio wings, both for purposes of staying aloft and for accommodating solar

cells and slender fuselages. For example, Facebook’s Aquila has a wingspan comparable

to a Boeing 737 but weighs only one-third as much as a car. For either HALE aircraft


or modern commercial transport aircraft, the attributes of light-weight airframe and high

aspect ratio wings result in a flexible aircraft configuration that is prone to large wing

deformations. Owing to this structural flexibility, these aircraft face two main challenges:

first is that the stronger interactions between rigid-body dynamics, structural dynamics and

aerodynamics can no longer be ignored; second is the increased tendency to suffer aeroelastic

problems, which may lead to undesirable vibration, flying performance deterioration, and

even catastrophic structural failure.

NASA’s Helios mishap in June 2003 is a lesson learned in the developing history of flex-

ible aircraft. During a flight test, one of the turbulence encounters caused the aircraft to

deform into a persistent high dihedral configuration, which led to an unstable divergent

pitch oscillation. With airspeed diverging rapidly from the nominal flight speed, the air-

craft eventually broke apart [5]. According to the mishap investigation report [5], there

were complex interactions among the flexible structure, unsteady aerodynamics, flight con-

trol system, propulsion system, the environmental conditions, and flight dynamics on the

stability and control characteristics of the aircraft. Conventional analysis techniques could

not handle these interactions properly. As pointed out in one of the recommendations from

Helios mishap investigation report, control systems are included in the development of more

advanced multidisciplinary time-domain analysis methods appropriate to highly flexible air-

craft [6]. Alongside with aerodynamics and structural dynamics, control system design is

also an important aspect of complex flexible aircraft design. Active control, in contrast to

traditional preventive techniques which add weight penalty on the structure design, can pro-

vide a promising alternative to mitigate negative aeroelastic effects on the flexible aircraft

structures, including flutter suppression and structural load alleviation. It can also improve

aircraft performance and even bring more benefits such as further weight savings. For ex-

ample, in a survey of applications of active control technology for lighter-weight aircraft,

it is pointed out that the gust response on operational aircraft is effectively mitigated and

empty weight reduction is achieved by using active controls [7]. It is also reported from the

results of NASA/Rockwell Active Flexible Wing program that further weight savings of at

least 15% of takeoff gross weight can be achieved by additionally using flutter suppression,

gust load alleviation, and/or maneuver load alleviation controls [8].

As important as the control system is in maintaining the normal operation of flexible

aircraft, any failure or damage associated with components in the control feedback loop

could also bring potential risks to the aircraft. Control surfaces, which execute control com-

mands from flight control computers, are subject to anomalies. From an analysis conducted

by NASA, documenting the causes of loss-of-control accidents and incidents in commercial

flights during 1988 to 2003, faults associated with control surfaces caused 144 incidents and 8


accidents [9]. This corresponds to the fourth highest frequency in all thirteen counted failure

and damage subtypes. Hydraulic systems failures, actuator failures or structural damages

due to excessive loads can cause control surfaces to get stuck at a certain trim, become

more difficult to position, or even become completely ineffective. For flexible aircraft that

rely on active controls to deal with undesirable aeroelastic impacts, it is critical to take into

account the control surface effectiveness. What is more, due to the strong coupling between

rigid-body and flexible modes, faults in control surfaces not only affect the normal operation

and maneuver of aircraft but also may induce or accelerate negative aeroelastic impacts on

aircraft structures. To improve the reliable operation of flexible aircraft, a fault tolerant

flight control system that can suppress unwanted aeroelastic phenomena such as flutter,

alleviate extra structural loads caused by either gust disturbances or maneuver flight, and

maintain overall closed-loop system stability and acceptable performance in the event of

control surface faults is highly desirable.

1.2 Literature Review

The main topics studied in this thesis are active flutter suppression (AFS), gust load al-

leviation (GLA), maneuver load alleviation (MLA) and fault tolerant control (FTC). The

first three topics have been covered in the literature for flexible aircraft but mostly in the

fault-free case while the fourth topic is addressed mainly for rigid-body ones. In this section,

literature review of a number of existing works related to the above topics is given.

1.2.1 Active Flutter Suppression

Flutter is an aeroelastic instability phenomenon caused by the coupling between aerody-

namic, structural dynamics and inertial forces as the speed of aircraft increases, which can

lead to catastrophic structural failure of aircraft wings. By making use of the control sur-

faces, AFS becomes a promising technique to ensure the safety of aircraft and expand the

flutter boundary, hence the flight envelope.

Control designs have been undertaken in AFS for aeroelastic systems, which are typically

two-dimensional wing section models, for many years. A survey paper by Mukhopadhyay

in 2003 [10] gives a brief historical account of the development of aeroelastic analysis and

control. A wide variety of control methodologies have been applied in flutter suppression

for The Benchmark Active Control Technology (BACT) project. Those results have been

presented in a special section on three issues of Journal of Guidance, Control, and Dynamics

[11, 12, 13]. Another survey paper by Librescu and Marzocca [14] reviews the advances


in control of aeroelasitc system before the year 2005. The development of AFS for a wing

section or a wing model is now briefly reviewed.

Adaptive output feedback control has been used to deal with wing section systems that

either have unknown system parameters or structural nonlinearities and with only output

measurements available for feedback control design. For a typical aeroelastic wing section

with only pitch angle or plunge-displacement as the output feedback, a model reference vari-

able structure adaptive control is designed in [15], a backstepping adaptive output controller

is developed in [16] and feedback linearization is used in [17]. In [18, 19], adaptive output

feedback control is proposed to suppress flutter, limit cycle oscillations (LCOs) and reduce

the vibrational level in the subcritical flight speed range for a nonlinear 2-D wing-flap sys-

tem. Two control surfaces (leading- and trailing-edge control surface) have been used to

improve the control performance of aeroelastic vibration suppression[20, 21, 22, 23, 24]. Be-

sides all the above efforts on 2-D wing section model, flutter suppression of a 3-D aeroelastic

wind-tunnel model via an adaptive output feedback control scheme has also been studied

[25].

It is desirable to have a flutter suppression controller that can work in a large operating

range, as aeroelastic responses are dependent on parameters such as airspeed and dynamic

pressure which can be time-varying during operation. Conventionally, a time-invariant con-

troller is designed at a fixed operating point to cover an anticipated range of parameter

variation. Another method is to interpolate between controllers designed at several fixed

points. However, the time-varying nature of the parameters make it difficult to guarantee

stability and performance using these methods [26]. To deal with the situation, two control

methods, i.e. linear parameter-varying (LPV) control and µ-method have been used in AFS

design.

LPV controllers can automatically gain-schedule with the varying parameters and offer

performance guarantees. In [27], an LPV controller based on H∞ performance is designed

for AFS of the BACT wing section with varying Mach numbers and dynamic pressures.

An LPV controller self-schedules with airspeed is synthesized for the nonlinear aeroelastic

test apparatus (NATA) to suppress LCOs over a range of airspeeds [28]. Reference [29]

also presents an LPV controller design to suppress flutter over a range of airspeeds for a

high-fidelity reduced-order wing model.

The µ-method solves the problem by constructing a model with structured uncertainties

that capture the parameter variations around a nominal model and designing a robust control

law for the nominal model. A linear fractional transformation (LFT) with parameterization

around perturbation in dynamic pressure for robust aeroservoelastic stability analysis is

derived in [30]. A polynomial function of airspeed is formulated in place of the dynamic


pressure to compute the robust flutter speeds [31]. A µ-controller for flutter suppression

based on the state-space LFT model incorporating airspeed and air-density variations as

structured parametric uncertainties is design in [32]. In [33], a reduced-order modelling

scheme is presented and a robust flutter suppression controller is designed for a multiple-

actuated wing with airspeed and air density variations.

Despite the extensive aforementioned studies of AFS on a 2-D wing section model or

a 3-D wing model, the relatively recent development of integrated flight and flutter sup-

pression control, which takes care of both rigid-body motion and wing deformation, is of

more interest and importance for flexible aircraft. In [34], by using the output covariance

constraint (OCC) algorithm, a controller is developed for aircraft with flexible wings. The

control design specifications include not only rigid body dynamics but also constraints on the

vibrational behavior of the wings so that both handling qualities and wing fluttering motion

suppression are achieved. A multi-objective flight control framework is envisioned in [35] to

address multiple control objectives for an aircraft with Performance Adaptive Aeroelastic

Wing (PAAW) technology. A simulation study of an optimal control designed for three ob-

jectives: flight path angle control, flutter suppression and drag minimization is also presented

in this work. In [36], an aeroelastic aircraft model with Variable Camber Continuous Trailing

Edge Flap (VCCTEF) system to suppress the fluttering motion of the wing is studied. A

linear matrix inequality (LMI) based optimal control law is designed which minimizes the

vibration motions of the wing subject to the VCCTEF actuation constraints characterized by

OCC. In [37], an H∞ controller is designed to increase the structural damping and suppress

flutter on a small flexible unmanned aircraft.

To cope with the variations in aeroelastic dynamics under varying flight conditions, LPV

control has been applied to suppress flutter across the flight envelope for aeroservoelastic

aircraft in [38, 39]. Another idea to develop a parameter-varying flight and flutter suppression

controller for flexible aircraft is to use a Takagi-Sugeno (TS) fuzzy-model, which combines

a set of local linear aeroelastic models at different flight conditions, to approximate the

parameter-varying flight dynamics and design controllers based on the Parallel Distributed

Compensation (PDC) technique [40, 41]. In [42], an AFS strategy based on recurrent neural

networks (RNNs) is used to to move flutter instabilities outside the flight envelope of an

unconventional three-surface transport aircraft. The design objective is to adaptively tune

the controller according to current operating conditions, which can be achieved by taking

advantages of two RNNs: one for real-time identification of the aircraft, the other for control

based on the identified model.


1.2.2 Gust Load Alleviation (GLA)

Diminished structural rigidity makes flexible aircraft more sensitive to gust encounters. To

reduce the structural deformations and improve the fatigue life of aircraft structure and ride

quality, attenuating gust loads on aircraft is necessary.

Several methods have been used to design control systems that are capable of both

rigid-body motion control and gust load alleviation for flexible aircraft. A linear quadratic

Guassian (LQG) based gust load alleviation controller with an integral pitch angle tracking

control is designed by Dillsaver et al. [43] for the flexible X-HALE aircraft. The designed

control system can reduce the wing curvatures and track a pitch angle command in the

presence of a gust disturbance. A model-predictive controller with prediction enhancement

is proposed by Haghighat et al. [44] for both stabilization and gust load alleviation of a

full flexible aircraft model, and compared with traditional model-predictive control (MPC)

and linear quadratic regulator (LQR). MPC is also employed by Giesseler et al. [45] and

Simpson et al. [46] to alleviate gust loads for flexible aircraft. An H∞ controller is designed

by Cook et al. [47] for a very flexible aircraft, which has been shown to have effective load

alleviation in root bending moments and be able to stabilize the unstable phugoid mode.

Since the trade-off between high modeling fidelity and low system dimension is a signif-

icant challenge in the control system design for flexible aircraft, some researchers develop

their work based on experimental models using system identification techniques. Impulse

response method and the generalized predictive control (GPC) method are applied in [48] to

develop aeroservoelastic analytical models of a SensorCraft wind tunnel model and an GLA

control law is designed based on the GPC method. An experimental model-based feedback

control framework is developed in [49] and demonstrated on the S4T wind-tunnel models to

perform flutter suppression and GLA.

Besides the related work on GLA mentioned above, there are other control methods that

have been used to alleviate gust loads, such as adaptive feedforward control [50], neuro-fuzzy

thoery based control [51, 52], optimal control allocation [53], etc.

1.2.3 Maneuver Load Alleviation (MLA)

Due to the coupling between rigid body dynamics and aeroelastic modes, a maneuver of

flexible aircraft can excite dynamic responses involving the aeroelastic modes and introduce

extra structural loads on the flexible wing structure. Therefore, a unified flight and load

controller is needed to guarantee that structural loads during a selected maneuver are allevi-

ated to conform to the structural load limitations while keeping the nominal flight behavior


unmodified.

Structural load alleviation objectives have been included in [54] in conjunction with the

flutter suppression and drag minimization to extend the previously proposed multi-objective

flight control framework for high aspect ratio flexible aircraft [35]. A simulation study of

multi-objective optimal control design with MLA is presented, in which the wing root bending

moment is reduced to stay within the load limits during a pull-up maneuver but the down

side is that the tracking of pitch rate command is poor. In [55] an adaptive controller based

on two RNNs is applied to perform MLA under different flight conditions on the aeroelastic

model of a fighter aircraft. The controller can alleviate the wing root bending moment

without loss of maneuvering performance over a certain range of Mach numbers.

The objective of guaranteeing that the structural loads do not violate load upper and

lower limits can be characterized as constraints on an output (e.g. wing root bending mo-

ment) of the flexible aircraft model. MPC, which is capable of handling system state con-

straints, is utilized in [56] to alleviate the bending moment at the external wing during

a sudden and strong roll maneuver for a flexible transport aircraft. In [57], on top of a

nominal H∞ flight controller that achieves load tracking performance, an output saturation

mechanism based on Output Input Saturation Transform (OIST) technique [58] is added,

which shapes the wing root bending moment response to remain within the limitations for

a longitudinal maneuver of a flexible aircraft.

1.2.4 Fault Tolerant Control (FTC)

FTC is a control design methodology that aims at tolerating potential malfunctions in system

components while maintaining acceptable performance and stability properties of the system.

With the capability of handling and recovering from faulty situations, it can be an effective

way to improve aircraft reliability and ensure flight safety. A large amount of research work

and increasing attention have been devoted to fault tolerant flight control (FTFC) over the

last two decades in both the control theory and aerospace communities, most of which have

been reviewed and summarized in several survey papers and books [59, 60, 61, 62].

In general, FTC can be classified into two main categories: passive and active. In passive

methods, the controller is fixed and predesigned to be robust against certain types of faults

that can be modelled as uncertainty regions around a nominal model. Active methods are

based on an on-line redesign of the control law or selection of predesigned controllers. A fault

detection and diagnosis (FDD) scheme is usually required to provide the fault information for

active FTC. With no requirement of FDD, a passive FTC controller is simple to implement

and do not need any reconfiguration mechanism since it can work in both nominal and faulty


situations. Examples of such passive FTFC designs can be found in [63, 64, 65], where a

reliable robust controller solved from LMI optimization for a certain performance objective

is developed against a set of actuator outage and control surface impairment faults.

Owing to the need to handle a number of fault scenarios with one controller, passive

FTC designs are generally rather conservative. On the other hand, active FTC designs

can achieve better control performance and are more flexible in dealing with a wider class

of faults. Several control methods have been applied to active FTFC designs: multiple

models method [66, 67, 68, 69, 70], which can deal with different fault scenarios by switching

controllers from a bank of predesigned ones corresponding to different faulty models; control

allocation [71, 72, 73], which redistributes the control signal to remaining healthy actuators

without redesigning the control law; adaptive control, which accommodates changes in the

system structures and parameters and has been used to cope with control surface failures

[74, 75, 76] and even structural damages [77, 78, 79]; MPC, which is well-known for its

ability to handle constraints and changing model dynamics systematically and has been

used to handle jammed actuators without the need to explicitly model the failure [80, 81];

and others including sliding mode control [82], and LPV techniques[83, 84, 85].

But there is still a critical issue in active FTC designs, which is the limited time window

allowed for FDD between fault occurrence and when the faulty system becomes irrecoverable

even with a reconfigured controller or loses stability. Both passive and active designs have

their distinct merits and disadvantages. This inspires the development of hybrid FTC designs

[86], which exploit and combine the advantages of the two methods. The idea is to use a

passive controller to slow down the deterioration of the faulty system after fault detection and

switch in the active controller after complete fault diagnosis to improve the performance. For

example, in [87], a hybrid FTC design is proposed to counteract loss of control effectiveness

in hydraulically-driven control surfaces for an aircraft model.

Control surfaces are important components in flight control system that exert physical

deflections from control commands on aircraft. Faults associated with them are mostly

discussed in FTFC literature. The two most frequently seen fault types that could happen

in aircraft control surfaces are stuck fault (also named as jamming or lock-in-place fault) and

loss-of-effectiveness fault. Other faults types such hard-over and float can be seen as special

cases of the two.

Stuck fault means that the faulty control surface is stuck at a position and no longer

responds to control commands. It can be regarded as an additional constant disturbance

imposed onto the system while the system loses this faulty control channel. In [88], the

effect of a stuck fault on the closed-loop system is characterized by the induced L∞ norm.

Then a fault-tolerant flight tracking controller against stuck-actuator faults is developed


based on minimization of the induced L∞ norm index. A more common treatment for

the stuck fault is to regard it as a matched disturbance that can be fully counteracted by

resorting to redundancy of the remaining actuators [89, 90]. But this method requires that

the control distribution matrix of the stuck control channel must lie in the range of the

control distribution matrix of the remaining control channels.

Loss of effectiveness means that the faulty control surface cannot execute the control

command 100% effectively as in the healthy case. To describe the severity of such a fault, a

factor of effectiveness is used to approximate the percentage of control command that can be

exerted on the aircraft. LMI-based approaches can be used to cope with loss of effectiveness.

In passive designs, the controller is usually synthesized by satisfying the conditions for sta-

bilization and closed-loop performance (e.g. H∞, H2) optimization in the worst case, which

can guarantee the accommodation for any loss of effectiveness that is less severe [63, 87, 91].

In active designs, the controller can be synthesized for the corresponding fault scenario with

effectiveness factors obtained from FDD [87]. To reduce the conservativeness, LPV method

can be incorporated by using the estimation of the control effectiveness factor as a scheduling

parameter [92].

All the aforementioned FTFCs are designed for rigid-body aircraft models, aiming at

the recovery or maintenance of the rigid-body motions of aircraft. The relevant work that

considers flexible aircraft and fault influence on flexible modes is quite limited. Worth men-

tioning is that Boskovic et al. [93] have included the aeroelastic coupling and augment

pitch-and-plunge states in the system equation. A multiple models, switching and tuning

(MMST) strategy is proposed to accommodate small wing damage and the controllers are

able to suppress pitch-and-plunge states.

Through the above literature review of the four topics, we can see that:

• From the perspective of flexible aircraft control, the influence from control surface faults

on aeroelastic modes and FTC designs have not been well analyzed and developed.

• From the perspective of FTFC, aircraft with high structural flexibility have rarely been

studied.

As discussed in Section 1.1, a fault tolerant flight control system that addresses issues raised

in the above discussion is highly desirable for flexible aircraft.


1.3 Thesis Contributions

This thesis studies fault tolerant control (FTC) strategies dealing with control surface faults

while handling undesirable aeroelastic problems for flexible aircraft. The FTC designs will

aim at both mitigation of the negative aeroelastic responses induced by control surface faults

through the coupling between rigid-body and flexible modes and recovery of the aircraft

operation and maneuver from the adverse conditions. The main goal of the work is to develop

an FTFC framework for flexible aircraft that can achieve flutter suppression or perform

structural load alleviation, recover nominal flight and maintain acceptable performance in

the event of control surface faults.

The contributions of this work are in two main directions. First, fault handling ability

has been incorporated into flutter suppression or structural load alleviation control design

for flexible aircraft by developing FTC designs for each of the following three problems

that are particularly important for flexible aircraft: flutter suppression, GLA and MLA.

These aeroelasticity requirements can still be well addressed regardless of control surface

anomalies. Second, different from a conventional FTFC design, influence of control surface

faults on aeroelastic modes has been analyzed and taken into account. The FTC designs

described in the thesis can minimize the effects of faults on not only rigid-body motion but

also aeroelastic modes.

All the FTC designs are based on a state-space form of a flexible aircraft model that

captures the coupling of rigid-body and flexible modes. By adopting the Lagrange’s equa-

tions for quasi-coordinates developed by Meirovitch [94], the mathematical model for flexible

aircraft can describe the rigid-body motions of the aircraft and the relatively small elastic

deformations of the flexible wings as well as the coupling between them.

For the flutter suppression problem, an LPV-based FTC controller is developed to address

stuck faults as well as actuator saturation. The influence of stuck control surface faults on

a flexible aircraft model is analyzed, and the analysis shows that unwanted vibrations of the

wing and uncontrolled rigid-body motions can be caused through the interaction between

rigid-body and flexible modes. The proposed FTC controller considers two practical concerns

for flexible aircraft: actuator saturation and aeroelastic instability caused by parameter

variations. It is able to achieve the following goals: minimizing the effects of stuck control

surface on rigid-body motion and aeroelastic modes of the wing; allowing safe operation

in a range of airspeed variation under faulty scenarios by using LPV control design; and

guaranteeing no closed-loop performance degradation caused by actuator saturation.

For GLA problem, a mixed H2/H∞ FTC controller is developed for a flexible aircraft

subject to gust disturbances and loss of control effectiveness faults in control surfaces. The


control design uses an LPV approach that incorporates adaptively estimated control ef-

fectiveness factors to improve fault tolerance. The designed controller can simultaneously

achieve rigid-body motion stabilization, gust load alleviation on flexible wing structures and

on-line fault accommodation to loss of control effectiveness fault.

For MLA problem, a fault-tolerant MPC formulation with reference adjustment is pre-

sented for a flexible aircraft to handle stuck and loss of control effectiveness faults. Upper

and lower bounds are set for the structural loads during a maneuver such that the load

alleviation objective becomes an output constraint. For stuck and loss of effectiveness fault

cases, the MPC design can steer the system to track any admissible reference with respect to

each fault case and keep the structural loads staying within the given bounds. If a reference

command is not admissible, it will be adjusted to an admissible command as close to the

given one as possible.

The organization of this thesis is given below:

• Chapter 2 presents a flexible aircraft model that captures the coupling of rigid-body

and flexible modes.

• Chapter 3 presents an FTC design for flutter suppression of a flexible aircraft with

stuck control surface faults.

• Chapter 4 presents a fault-tolerant GLA control design for a flexible aircraft with loss

of control effectiveness.

• Chapter 5 presents a fault-tolerant MLA control design for a flexible aircraft with stuck

control surface and loss of control effectiveness faults.

• Chapter 6 presents concluding remarks and directions for future work.

Chapter 2


The modeling of flexible aircraft requires multidisciplinary integration of flight dynamics,

aerodynamics and structural dynamics. A mathematical model that can describe the rigid-

body motions of the aircraft and the elastic deformations of the flexible components as well

as the coupling between them is important for the analysis and control of flexible aircraft.

2.1 Equations of Motion for Flexible Aircraft

Equations of motion in terms of mean axes [95] and body-fixed axes [94, 96, 97, 98] have been

developed for flexible aircraft in the literature. The former method has several assumptions

that are difficult to meet and is less suitable to model aircraft with large elastic deformations.

In this thesis, the quasi-coordinates method proposed by Meirovitch [94] is adopted to derive

the nonlinear equations of motion for flexible aircraft. We assume that the fuselage is rigid

and the wing is flexible, which is considered as a cantilever beam undergoing bending and

torsion.

In order to describe the motion of the flexible aircraft, three frames are defined as shown

in Figure 2.1 : the inertial frame XY Z with the origin at O, the fuselage frame xfyfzf

attached to the undeformed fuselage with the origin at Of , the wing frame xwywzw attached

to the undeformed wing with the origin at Ow. Let Rf denote the position vector from

O to Of , θf = [φ θ ψ]T denote the vector of Euler angles between xfyfzf and XY Z, uw

denote the bending displacement of each point on the wing with respect to xwywzw and ψw

denote the torsional displacement. Then the motion of the aircraft can be described by six

rigid-body degrees of freedom (DOFs) of the fuselage frame: three translations Rf and three

rotations θf , and elastic deformations of the flexible wing relative to the wing frame: uw

and ψw.

According to the generic quasi-coordinates Lagrangian equations of motion given in

13

Chapter 2. Flexible Aircraft Model 14

Figure 2.1: Schematic Drawing of Flexible Aircraft Frames: Inertial Frame XY Z with OriginO, Fuselage Frame xfyfzf with Origin Of , Wing Frame xwywzw with Origin Ow

Meirovitch and Tuzcu [96], six second-order ordinary differential equations are presented

to describe the rigid-body motions and a set of partial differential equations is presented to

govern the elastic deformations of each flexible component. Since the flexible aircraft system

includes both ordinary and partial differential equations, it does not generally have a closed-

form solution. So the partial differential equations usually need to be approximated by a set

of ordinary differential equations. For this purpose, the elastic bending displacement and

torsional displacement of each point on the wing are discretized spatially in terms of mode

shapes:

uw(rw, t) = Nu(rw)qw(t)

ψw(rw, t) = Nψ(rw)qw(t),(2.1)

where Nu(rw) and Nψ(rw) are shape function matrices, qw(t) ∈ Rnq is modal coordinates

vector representing the elastic modes. The choice of shape functions will be specified later

in Section 2.2 and the shape function matrices will be determined by Equation (2.35).

Then the Lagrangian equations of motion in quasi-coordinates for the flexible aircraft are


given by

d

dt

(∂L

∂Vf

)+ ωf

∂L

∂Vf− Cf

∂L

∂Rf

= F

d

dt

(∂L

∂ωf

)+ Vf

∂L

∂Vf+ ωf

∂L

∂ωf−(ETf

)−1 ∂L

∂θf= M

d

dt

(∂L

∂qw

)− ∂T

∂qw+∂F∂qw

+∂V∂qw

= fe

(2.2)

where the first and second equations represent the rigid-body translational and rotational

motions respectively and the third equation describes the bending and torsional motions

of the flexible wing in terms of elastic modes. The variables appearing in Equation (2.2)

are defined as follows: L is the Lagrangian; T and V are the kinetic energy and potential

energy for the whole aircraft respectively; F represents the Rayleigh’s dissipation of the

wing structure; Vf and ωf are the translational and angular velocity vectors of the fuselage

frame respectively; F and M represent the generalized resultant forces and moments while

fe represents the generalized forces that act on the aircraft wing; (·) is the crossproduct

operator representing a skew-symmetric matrix for a vector ω = [ω1 ω2 ω3]T in this form:

ω =

0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

; (2.3)

Cf denotes transform matrix between xfyfzf and XY Z with 3-2-1 rotational sequence, which

is expressed by

Cf =

1 0 0

0 cosφ sinφ

0 − sinφ cosφ

cos θ 0 − sin θ

0 1 0

sin θ 0 cos θ

cosψ sinψ 0

− sinψ cosψ 0

0 0 1

=

cosψ cos θ sinψ cos θ −sinθ

cosψ sin θ sinφ− sinψ cosφ sinψ sin θ sinφ+ cosψ cosφ cos θ sinφ

cosψ sin θ cosφ+ sinψ sinφ sinψ sin θ cosφ− cosψ sinφ cos θ cosφ

; (2.4)

and Ef is the matrix relating Euler velocity vector and velocity vector under the fuselage


frame given by

Ef =

1 0 − sin θ

0 cosφ cos θ sinφ

0 − sinφ cos θ cosφ

, (2.5)

for which we assume that Euler angles will operate inside appropriate ranges so that the

rotational singularities will not appear and the inverse matrix E−1f exists.

2.1.1 Kinetic Energy

The absolute position of a mass element dmf on the fuselage is given by

Rf = Rf + rf , (2.6)

where rf is the nominal position of the mass element relative to the origin of xfyfzf . The

velocity of a point on the fuselage can be expressed as

Vf = Vf + rTf ωf , (2.7)

where rf is the skew symmetric matrix corresponding to rf .

Then the kinetic energy of the fuselage is

Tf =1

2

∫V Tf Vfdmf

=1

2mfV

Tf Vf +

1

2

∫ωTf rfVfdmf +

1

2

∫V Tf rfωfdmf +

1

2

∫ωTf rf r

Tf ωfdmf ,

(2.8)

where mf is the mass of the fuselage.

The position of a mass element dmw on the wing relative to the origin of xwywzw is given

by

Rw = rw + uw, (2.9)

where rw is rigid distribution vector of the mass element, uw is the elastic displacement of

the mass element as shown in Figure 2.1. The relative velocity with respect to xwywzw is

then given by

Vw = vw + Cwωf (rw + uw) + αw(rw + uw), (2.10)

where Cw is the coordinate transformation matrix from xfyfzf to xwywzw, vw is the defor-

mational velocity and αw is the torsional velocity of the mass element at rw. The absolute


velocity of a point on the wing will then have this expression

Vw = CwVf (rfw) + Vw

∼= CwVf + [CwrTfw + (rw + uw)TCw]ωf + rT

wαw + vw,(2.11)

where rfw is the vector from the origin of xfyfzf to the origin of xwywzw and the small

velocity term caused by wing bending and torsional deformations uTwαw is neglected.

Then the kinetic energy of the wing is

Tw =1

2

∫V Tw Vwdmw

=1

2mwV

Tf Vf +

∫V Tf

[rTfw + CT

w (rw + uw)TCw]︸︷︷︸

X1

ωfdmw

+

∫V Tf C

Tw (rT

wαw + vw)dmw

+1

2

∫ωTf [Cwr

Tfw + (rw + uw)TCw]T[Cwr

Tfw + (rw + uw)TCw]︸︷︷︸

J

ωfdmw

+

∫ωTf

[Cwr

Tfw + (rw + uw)TCw

]T︸︷︷︸X2

(rTwαw + vw)dmw

+1

2

∫qTw(NT

uNu +NTψ rwr

TwNψ)qwdmw,

(2.12)

where mw is the mass of the wing.

The kinetic energy for the whole aircraft is

T = Tf + Tw =1

2V TMV (2.13)

where V =[V Tf ωT

f qTw

]Tis the velocity vector and M is the mass matrix of the aircraft

with the following submatrices:

M11 = mI3×3, M12 =∫rTf dmf +

∫X1dmw,

M13 =∫CTw (rT

wNψ +Nu)dmw, M22 =∫rf r

Tf dmf +

∫Jdmw,

M21 = MT12, M23 =

∫X2(rT

wNψ +Nu)dmw,

M31 = MT13, M32 = MT

23, M33 =∫

(NTuNu +NT

ψ rwrTwNψ)dmw .

(2.14)


2.1.2 Potential Energy

The potential energy is due to the gravitational energy and the strain energy:

V = Ug + U (2.15)

The gravitational energy Ug is given by

Ug = −mRTf g− qTwMT

13Cfg (2.16)

where g is the vector of gravitational acceleration.

And the strain energy U is given by

U =1

2qTwKqw (2.17)

where K is the stiffness matrix of the wing, which will be specified later in Section 2.2.

2.1.3 Equations of Motion

The equations of motion will be obtained by inserting the kinetic energy and potential energy

into Lagrange’s equations (2.2). First, the partial derivatives of the Lagrangian are derived

as follows:

∂L

∂Vf=

∂T

∂Vf= M11Vf +M12ωf +M13qw, (2.18)

∂L

∂Rf

=∂V∂Rf

= −mg, (2.19)

∂L

∂ωf=

∂T

∂ωf= M21Vf +M22ωf +M23qw, (2.20)

∂L

∂ωf= 0, (2.21)

∂L

∂qw=

∂T

∂qw= M31Vf +M32ωf +M33qw. (2.22)

(2.23)


The partial derivative of kinetic energy with respect to qw is

∂T

∂qw=∂T

∂Vw

∂V Tw

∂uw

∂uw∂qw

=

∫NuCwωf

T

Vwdmw

=

∫NuCwωf

T

[CwVf + CwrTfwωf + rT

wCwωf + CwωfNuqw + (rTwNψ +Nu)qw]dmw

(2.24)

The partial derivative of potential energy with respect to qw is

∂V∂qw

=∂Ug∂qw

+Kqw (2.25)

where the first part ∂Ug∂qw

= −MT13Cfg reflects the influence of elastic deformations on the

gravitational energy. Instead of directly keeping it in the potential energy derivative, we

choose to handle the gravity as a distributed force acting on the wing structure. Then the

equivalent contribution of the gravity to the elastic deformation equation will appear in the

generalized forces fe term on the right-hand side.

The Rayleigh’s dissipation term is expressed by

F = qTwCqw (2.26)

where C is the damping matrix of the wing structure and will be specified later in Section

2.2. And the partial derivative of Rayleigh’s dissipation with respect to qw is

∂F∂qw

= Cqw (2.27)

Inserting the above kinetic and potential energy partial derivatives and their time deriva-

tives as well as the damping partial derivative into Equations (2.2), the dynamical equations

for a flexible aircraft can be re-arranged as

M

Vf

ωf

qw

+H

V f

ωf

qw

+ E

Rf

θf

qw

=

F

M

fe

(2.28)

where M = [Mij] is the mass matrix, H = [Hij] is the damping matrix and E = [Eij] is


the stiffness matrix of the aircraft model. And the submatrices of H and E are defined as

follows:

H11 = ωfM11,

H12 = ωfM12,

H13 = ωfM13 +∫CTw CwωfNudmw,

H21 = M21 + VfM11 + ωfM21,

H22 = M22 + VfM12 + ωfM22,

H23 = M23 + VfM13 + ωfM23,

H31 = −∫NT

u CwωfT

Cwdmw,

H32 = M32 +∫NT

u CwωfT

CwrTfwdmw +

∫NT

u CwωfT

rTwCwdmw ,

H33 = C +∫NT

u CwωfT

(Nu + rTwNψ)dmw,

(2.29)

E11 = O3×3, E12 = O3×3 , E13 = O3×nq ,

E21 = O3×3, E22 = O3×3, E23 = O3×nq ,

E31 = Onq×3, E32 = Onq×3, E33 = K −∫NT

u CwωfT

CwωfNudmw.

(2.30)

The terms of generalized forces F ,M and fe appearing in Equation (2.28) can be obtained by

means of virtual work once the actual distributed forces acting on the aircraft are calculated.

The actual distributed forces are assumed to come from the aerodynamic forces, gravity and

controls.

Remark 2.1: Observing the terms in submatrices of mass matrix M given in Equa-

tions(2.14), it can be seen that the rigid-body and flexible motions are not only coupled

through the off-diagonal submatrices between rigid and flexible variables M13,M23,M31 and

M32, but also the submatrices M12 and M21, which are dependent on the elastic deformations

of the wing uw. The two submatrices form the inertia matrix of the aircraft, so as the wing

deforms, the inertia matrix of the flexible aircraft will be changing.


The kinematic equations for the aircraft are given by

Rf = CTf Vf ,

θf = E−1f ωf .

(2.31)

2.2 Structural Modeling

In this thesis, the flexible wing structure is modeled as a cantilever beam undergoing coupled

bending and torsion, based on the Euler-Bernoulli beam theory. The structural model is

formulated using the finite element method so that the elastic deformations of the wing can

be easily represented in a state-space form.

Assume that the beam is divided into Nele elements. For an element i, it consists of

two end nodes i and i + 1 with coordinates yi and yi+1 respectively. For a node i, it has

three DOFs: ui, βi, ψi in which ui is the bending displacement in zw direction, βi = ∂ui∂y

is

the bending slope and ψi is the torsional displacement about the yw axis. Element shape

functions are chosen as [99]

Φ1 = 2ξ3 − 3ξ2 + 1,Φ2 = (ξ3 − 2ξ2 + ξ)li

Φ3 = 3ξ2 − 2ξ3,Φ4 = (ξ3 − ξ2)li

Φ5 = 1− ξ,Φ6 = ξ

(2.32)

where ξ = y−yili

, li = yi+1−yi, y represents the yw-axis location of a point between nodes i and

i+ 1 and satisfies yi ≤ y ≤ yi+1. The first four cubic shape functions are known as Hermite

polynomials and are used to interpolate the bending displacements between the nodal values.

And the fifth and the sixth shape functions are used to linearly interpolate the torsional

displacements between the nodal values. Then the bending and torsional displacements of a

point between nodes i and i+ 1 can be approximated as

u(y, t)

ψ(y, t)

=

Φ1(y) Φ2(y) 0 Φ3(y) Φ4(y) 0

0 0 Φ5(y) 0 0 Φ6(y)

ui(t)

βi(t)

ψi(t)

ui+1(t)

βi+1(t)

ψi+1(t)

= Φidi (2.33)


where Φi is the element shape function matrix and di denotes the nodal displacement vector

for element i.

Let d denote the vector of all the nodal DOFs, then the bending and torsional displace-

ments of a point on the wing structure can be expressed byu(y, t)

ψ(y, t)

= Φd (2.34)

where Φ =[ΦTu ΦT

ψ

]Tcan be obtained by assembling the element shape function matrix Φi.

Using modal displacement method to approximate the elastic displacement, the vector of

nodal DOFs can also be expressed as a linear combination of mode shape vectors: d = Φqqw,

where Φq is the mode shape matrix formed by the eigenvectors of the structural system and

qw is the modal coordinates vector. Then the elastic displacement and angular displacement

vectors of a point on the wing are spatially discretized as:

uw(rw, t) =[

0 0 (ΦuΦq)T]Tqw(t) = Nu(rw)qw(t),

ψw(rw, t) =[

0 (ΦψΦq)T 0

]Tqw(t) = Nψ(rw)qw(t).

(2.35)

where Nu(rw) and Nψ(rw) are the shape function matrices used in Equation (2.1) for spatial

discretization.

The strain energy of element i can be written in terms of bending and torsional elastic

displacements:

Ui =1

2

∫ li

0

[GJi(∂ψ

∂y)2 + EIi(

∂2u

∂y2)2]dy (2.36)

where GJi is the element torsional rigidity and EIi is the element bending rigidity.

Introducing the approximation from Equations (2.1) into the strain energy Equation

(2.36) and integrating over the length li will get

Ui =1

2qTwK

iqw, (2.37)

where

Ki =

∫ li

0

GJi∂(ΦψΦq)

T

∂y

∂(ΦψΦq)

∂y+ EIi

∂2(ΦuΦq)T

∂y2

∂2(ΦuΦq)

∂y2dy

= Kiψ +Ki

u

(2.38)

is the element stiffness matrix, Kiψ represents the part of element stiffness matrix for torsion


and Kiu represents the part for bending. Then the stiffness matrix K for the wing structure

can be obtained by assembling all element stiffness matrices.

The Rayleigh’s dissipation function of element i can be written in terms of bending and

torsional velocities:

Fi =1

2

∫ li

0

[cψiGJi(∂ψ

∂y)2 + cuiEIi(

∂2u

∂y2)2]dy (2.39)

where cψi and cui are torsion and bending damping functions for element i. Similarly, we

can express it as a function of qw:

Fi =1

2qTwC

iqw, (2.40)

where

Ci =

∫ li

0

cψiGJi∂(ΦψΦq)

T

∂y

∂(ΦψΦq)

∂y+ cuiEIi

∂2(ΦuΦq)T

∂y2

∂2(ΦuΦq)

∂y2dy

= Ciψ + Ci

u

(2.41)

is the element damping matrix, Ciψ represents the part of element damping matrix for torsion

and Ciu represents the part for bending. We can see that if cψi and cui are constant, Ci

ψ is

proportional to the stiffness matrix Kiψ and Ci

u is proportional to the stiffness matrix Kiu.

The damping functions can be determined by [100]

cψi = ζ/Λ0.5ψ

cui = ζ/Λ0.5u

(2.42)

where ζ is a structural damping factor, Λψ is the lowest natural frequency for torsion, Λu is

the lowest natural frequency for bending.

Then the stiffness matrix C for the wing structure can be obtained by assembling all

element damping matrices.

2.3 Aerodynamic Modeling

In this thesis, aerodynamic forces and moments are assumed to be contributed from the

flexible wing. Then the finite-state, induced-flow theory of Peters et al. [101, 102] is adopted

for the computation of two-dimensional unsteady aerodynamic loads acting on the wing.

The theory is valid for calculating aerodynamic loads of a two-dimensional (2D) airfoil in

inviscid and incompressible flow. It is a well-used theory in aeroelastic system modeling,


which can represent the unsteady aerodynamic loads in time domain and also allow them

to be easily merged into the equations of motion in a state-space form. Since it is a 2D

theory, strip theory is then used to integrate 2D aerodynamic loads over the wing span to

compute the three-dimensional (3D) aerodynamic loads. The idea about this method is to

first discretize the wing into a finite number of strips so that 2D aerodynamic theory can be

used to compute aerodynamic forces and moments for each strip, and then integrate over the

length of wing to get the 3D forces and moments. It allows the use of the same rigid-body

and elastic variables that describe the aircraft motions to express the aerodynamic forces.

The unsteady lift and moment of a 2D thin airfoil about the aerodynamic center are

given as

L0 =πρb2(−z + yθw − baθw

)+ 2πρy2b

(− zy

+ θw + b(1

2− a)

θwy− λ0

y

)

M0 =− πρb3

(−1

2z + yθw + b

(1

8− a

2

)θw

) (2.43)

where ρ is the air density, b is the half chord length of airfoil, ba is the distance of the

midchord in front of the elastic axis, y and z are velocity vector components along the chord

and perpendicular to the chord respectively, θw is the torsional angle and λ0 represents the

average induced-flow velocity of N induced-flow states λ = [λ1, λ2, . . . , λN ]T:

λ0 =1

2

N∑n=1

bnλn, (2.44)

in which the induced-flow states vector λ is governed by a set of N first-order ordinary

differential equations in the following form

λ = A1λ+ A2z + A3θw + A4θw, (2.45)

where the calculations for matrices A1, A2, A3 and A4 can be found in [101]. The coefficients

bn are calculated based on the following formulabn = (−1)n−1 (N+n−1)!(N−n−1)!

1(n!)2 , 1 < n < N

bN = (−1)n−1.(2.46)

For the choice of N , 4 to 8 induced-flow states can give adequate approximation for λ0 [102].

The local velocity components of the airfoil can be derived from the aircraft rigid-body

DOFs and wing nodal displacements. The term − zy

represents a part of the effective angle of


attack that is contributed from the aircraft’s angle of attack, pitch and roll rates and plunge

motion of the airfoil. And the plunge displacement and torsional angle can be related to the

wing nodal bending and torsional displacements. Then Equation (2.45) can be rewritten in

terms of the velocity vector V =[V Tf ωT

f qTw

]Tas

λ = B1λ+B2V +B3V (2.47)

The above governing equation for induced-flow states allows itself to be easily merged into

the equations of motion for flexible aircraft in state-space representation.

Control forces acting on the aircraft are from the aerodynamic forces generated by the

deflection of control surfaces. Denoting the deflection of a trailing-edge control surface as

δu, the 2D aerodynamic lift and moment are given by

Lδ = 2πρb(c1y2δu + c2yδu + c3δu)

M δ = 2πρb2(c4y2δu + c5yδu + c6δu),

(2.48)

where the coefficient c1 through c6 are based on the geometry of the control surface whose

detailed description is given in [101]. Since the deflection rate and double derivative terms

δu and δu are much less dominant than the deflection δu in practice, they can be neglected

and only contributions from δu to lift and moment are considered.

Then the total lift and moment of an airfoil with a trailing-edge deflection about the

aerodynamic center are

Lac = L0 + Lδ

Mac = M0 +M δ,(2.49)

which can be transferred to the wing elastic axis

Lea = Lac

Mea = (1

2+ a)bLac +Mac.

(2.50)

The above aerodynamic loads can be further transformed to the fuselage frame

F distac = Cfa

0

0

Lea

, M distac = Cfa

Mea

0

0

(2.51)


where Cfa is the transformation matrix from the local aerodynamic frame to the fuselage

frame. By using virtual work and integrating the distributed aerodynamic loads obtained

from Equations (2.51) and (2.50) over the wing span, the generalized loads Fac,Mac and

fac from the aerodynamics and controls, which are the parts of F ,M and fe in Equations

(2.28), can be obtained.

2.4 State-space Representation

Define the state vector

xT =

[RTf θT

f qTw V T

f ωTf qT

w λT

]T

. (2.52)

By re-arranging Equations (2.28) and (2.47), the nonlinear equations of motion for the

flexible aircraft can be written in state-space formxr(t)xe(t)

︸︷︷︸

x(t)

= f(xr(t),xe(t), U)

xr(t)xe(t)

︸︷︷︸

x(t)

+B(U)u(t) (2.53)

where xr =

[RTf θTf V T

f ωTf

]Tdenotes the rigid-body states, xe contains the remaining

states associated with the elastic states and the aerodynamic states and u denotes control

inputs, which are the deflections of control surfaces. It should be noted that the aeroelastic

characteristics of flexible aircraft which has been included in the nonlinear function f(x) are

highly dependent on the airspeed denoted by U .

2.5 Summary

In this chapter, the equations of motion for a flexible aircraft are presented. The developed

model is based on the quasi-coordinate Lagrange’s equations and describes the rigid-body

dynamics, elastic deformations of the flexible wing and the coupling between them. The

flexible wing structure is modeled as a Euler-Bernoulli beam undergoing coupled bending

and torsion and finite element method is used to represent the elastic deformations. The

finite-state, induced-flow theory of Peters et al. [101, 102] is used to capture the unsteady

aerodynamics. The deflections of trailing-edge control surfaces are considered as the control


inputs to the flexible aircraft model. The state-space representation of the model is given

and will be used to develop FTC designs in the following chapters.

Chapter 3

FTC Design for Flutter Suppression

Problem

Traditional aircraft with an essentially rigid body may only have very limited aeroelastic

effects that are neglected or separated from most flight control designs. For flexible aircraft,

one challenge arising in the modeling and control design is that the significant aeroelastic

modes must be taken into account. Among negative aeroelastic effects, an oscillatory insta-

bility phenomenon called flutter has to be dealt with, otherwise it could cause vibrations,

flying performance deterioration, stability degradation and even catastrophic structural fail-

ures. As introduced in Section 1.2.1, active control, in contrast to traditional preventive

techniques which add weight penalty on the structure design, provides a promising way to

suppress flutter for aeroelastic aricraft. Integrated control designs have been developed for

flexible aircraft by several researchers that are capable of simultaneous rigid-body motion

control and wing flutter suppression [35, 36, 40]. These control designs can effectively miti-

gate the negative aeroelastic responses and ensure the normal operation and safety of flexible

aircraft when all the control surfaces are healthy.

The functioning status of control surfaces affects whether the flight control commands

can be correctly executed, but healthy conditions cannot be always guaranteed as control

surfaces are also subject to anomalies. Faults associated with control surfaces, according to

an analysis conducted by NASA [9], correspond to the fourth highest frequency among all

counted failure and damage subtypes that have caused loss-of-control accidents and incidents

in commercial flights during 1988 to 2003. The malfunctions of control surfaces include being

stuck at a certain trim, difficult to position or even completely ineffective.

For flexible aircraft with strong coupling between rigid-body and flexible modes, control

surface faults not only affect the normal operation and maneuver of aircraft but also may

induce or accelerate negative aeroelastic impacts on aircraft structures. Therefore, before

28

Chapter 3. FTC Design for Flutter Suppression Problem 29

jumping into the FTC designs, attention must be paid to analyzing the post-fault responses of

flexible aircraft to a certain type of faults. If undesirable aeroelastic responses of the flexible

wing structure become serious consequences of control surface faults, then an integrated FTC

strategy which accounts for both rigid-body performance and aeroelastic stability should be

custom designed for flexible aircraft. However, as most FTFC designs only aim at the

recovery or maintenance of the rigid-body motions of aircraft [60, 61, 62], neither such

post-fault analysis nor integrated FTC strategies have been well addressed in the existing

literature, to the best of the author’s knowledge.

To fill the need in post-fault analysis and follow-up FTC designs for flexible aircraft, in

this chapter the two aspects are addressed. First, the influence of stuck control surface faults

on a flexible aircraft is investigated through a post-fault response analysis, which shows the

impacts of faults on both the rigid-body and aeroelastic modes. Second, an FTC design is

developed to cope with stuck faults for the flexible aircraft.

3.1 Influence of Stuck Faults on Flexible Aircraft

5m

10

m16m

1m

0.5m

Elevator

Flap 2 Flap 1 Flap 1 Flap 2

Figure 3.1: HALE Aircraft Model Geometry Top View (not to scale)

In this section, in order to investigate the influence of control surface faults on flexible

aircraft, stuck faults will be injected to control surfaces of a flexible aircraft model with a

nominal baseline controller to analyze the post-fault responses. A model which is loosely

based on the HALE aircraft proposed in several papers by Patil et al. [103] is considered.

Figure 3.1 shows the geometry of the aircraft. The aircraft has a rigid fuselage with a 50 kg


payload, a flexible high aspect-ratio wing with two flaps implemented from 75% chord to

the trailing edge on each side, and a rigid tail with a 50% chord elevator. Properties of this

aircraft are given in Table 3.1. The aircraft is assumed to cruise at the altitude of 20 km.

Table 3.1: Properties of the HALE aircraft

Properties Wing Tail

Half Span 16 m 2.5 m

Chord 1 m 0.5 m

Mass per Unit Length 0.75 kg/m 0.08 kg/m

Elastic Axis 0.5 chord NA

Center of Gravity 0.5 chord 0.5 chord

Bending Rigidity 2× 104 N ·m2

Torsional Rigidity 1× 104 N ·m2

Moment of Inertia about Elastic Axis 0.1 kg·m 0.01 kg·m

3.1.1 Linearized Model and Nominal Controller

A linearized model is needed for the flutter analysis and linear control design. The lineariza-

tion is done around the trimmed value of the aircraft states and wing deformations for a

given flight condition. If we consider a steady level flight, the linearized state-space model

is given by

δx = Aδx+Bδu (3.1)

where δx and δu denote the deviations of the states and control input from their trimmed

conditions. The following simulation study will be focused on the interaction between lon-

gitudinal dynamics and aeroelasticity during steady-level flight. The bilaterally symmetric

flaps are assumed to be deflected equally. For the rest of the chapter, Flap 1 will be used to

indicate the two symmetric flaps closer to the wing root and Flap 2 to indicate the two sym-

metric flaps closer to the wing tip. And the system states are composed of the pitch angle θ,

the pitch rate q, the horizontal velocity u, the vertical velocity w, the modal coordinates qw,

their derivatives qw and aerodynamic states. Four elastic modes are considered: the first,

second and third bending modes and the first torsion mode. As airspeed varies from 25 m/s

to 35 m/s, their damping ratios and frequencies, reflected through the real and imaginary


parts of the eigenvalues, are shown in Figure 3.2. From a linear flutter analysis, the flutter

speed is obtained as 32.6 m/s with the flutter frequency of 22.1 rad/s. The first torsional

mode crosses the imaginary axis and becomes unstable at the flutter speed. The responses

of flexible wing show unstable oscillatory behavior when the airspeed exceeds this speed.

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 20

5

10

15

20

25

30

35

40

Real part of eigenvalue (damping ratio)

Imagin

ary

part

of

eig

envalu

e (

frequency)

3rd Bending

1st Torsion

2nd Bending

1st Bending

32.6 m/s

25 m/s

35 m/s

Figure 3.2: Root Locus of the Open-loop Flexible Aircraft System as Airspeed Varies from25 m/s to 35 m/s

Remark 3.1: The above modal characteristics are determined upon using 8 finite el-

ements for each wing. The natural frequencies of the first four free vibration modes using

different numbers of finite elements are listed in Table 3.2 and compared to the results given

in Patil et al. [103]. Frequencies for the first two bending modes are quite accurate when

only using 8 elements. Although increasing the finite element number can increase the result

accuracy for the first torsion and third bending modes, the improvement is not substantial.

The aeroelastic results for using 8 elements are also compared to the ones from Patil et al.

[103] in Table 3.3. It can be seen that our model with 8 elements shows consistency with

the results of Patil et al. [103] in describing the elastic modes as well as the aeroelasticity of

the wing. Based on this analysis and consideration for the model size, 8 elements are used.

Remark 3.2: The first four elastic modes are used in the flexible aircraft model. The


Table 3.2: Comparison of Free Vibration Mode Natural Frequencies (rad/s)

8 elements 12 elements 16 elements 24 elements Patil et al. [103]

First Bending 2.243 2.243 2.243 2.243 2.243

Second Bending 14.056 14.056 14.056 14.054 14.056

First Torsion 31.095 31.069 31.058 31.050 31.046

Third Bending 39.379 39.363 39.358 39.352 39.356

Table 3.3: Comparison of Aeroelastic Results

Flutter speed (m/s) Flutter frequency (rad/s) Divergence speed (m/s)

8 elements 32.60 22.10 37.16

Patil et al. [103] 32.51 22.37 37.15

reason why the low-frequency modes should be kept is that they are most likely coupled with

flight dynamic modes. For this bending-torsion coupled aeroelastic wing, the torsion mode

is the first one that goes unstable, which makes the flutter occur at a frequency around 22

rad/s. The third bending mode, which is the fourth mode, has a natural frequency very

close to the first torsion mode whereas the the fifth mode with 77 rad/s natural frequency is

further away from it. Therefore, we have used the first three low-frequency bending modes

and the first torsion modes.

Remark 3.3: The spillover problem is not considered in this work as the controller

bandwidth is unlikely to overlap with the frequency range of higher modes (from 77 rad/s

to 1027 rad/s) that have not been included in the model. We also did simulations to apply

controllers designed using the model with four elastic modes on a model with eight modes

at various airspeeds. Comparing to simulation results of applying same controllers to the

four-mode model, the closed-loop responses looked almost the same. No destabilization of

the higher modes was observed.

A baseline LQR controller is designed to keep the trimmed steady level flight as well

as suppress aeroelastic modes of the wing for the flexible aircraft flying at a fixed airspeed.

The Q and R matrices are chosen as identity matrices for simplicity, as varying their values

does not lead to significant performance improvement in the simulation experiments. The


LQR controller is then applied to the nonlinear aircraft model at the chosen airspeed. The

resulting closed-loop responses at 32 m/s and 34 m/s are given in Figures 3.3 and 3.4 and

Figures 3.5 and 3.6 respectively, comparing to the corresponding open-loop responses of the

nonlinear aircraft model.

At 32 m/s, which is below the flutter speed (referred to as pre-flutter for simplicity), the

system has stable open-loop responses but small-magnitude oscillations can be observed in

the wing-tip rotation. The nominal controller suppresses the oscillations and stabilizes the

pitch motion as well.

0 5 10 15-7.83

-7.82

-7.81

-7.8

-7.79

tip d

ispla

cem

ent (m

)

Closed-loop Open-loop

0 5 10 15

time (s)

4.6

4.7

4.8

4.9

5

tip r

ota

tion (

deg)

0 2 4

4.634

4.636

4.638

Figure 3.3: Closed-loop and Open-loop Responses of Wing Deformations at Speed = 32 m/s

At 34 m/s, which is beyond the flutter speed (referred to as post-flutter for simplicity), the

open-loop responses show that the wing undergoes unstable vibrational motion. The pitch

motion gets affected due to the coupling with flexible modes and shows unstable oscillatory

behaviors as well. The nominal controller does its job in suppressing the unstable vibrational

motion of the wing and stabilizing the pitch motion. The nominal closed-loop responses are

presented here also for the later comparison with post-fault responses.

3.1.2 Fault Injection and Post-Fault Responses

To observe the influence of control surface faults on the flexible aircraft, faults are injected

to the control surfaces Flap 1, Flap 2 and elevator, the deflections of which are denoted by

δa1, δa2 and δe respectively. Since the study is focused on the influence of control surfaces


0 5 10 151.2

1.4

1.6

θ (

deg) Closed-loop Open-loop

0 5 10 150.7

0.75

w (

m/s

)

0 5 10 15

time (s)

-0.5

0

0.5

q (

deg/s

)

Figure 3.4: Closed-loop and Open-loop Responses of Rigid-body States at Speed = 32 m/s

0 5 10 15-8.2

-8.18

-8.16

-8.14

-8.12

tip d

ispla

cem

ent (m

)

Closed-loop Open-loop

0 5 10 15

time (s)

4

5

6

tip r

ota

tion (

deg)

Figure 3.5: Closed-loop and Open-loop Responses of Wing Deformations at Speed = 34 m/s


0 5 10 15

0.7

0.8

0.9

θ (

deg) Closed-loop Open-loop

0 5 10 150.4

0.45

0.5w

(m

/s)

0 5 10 15

time (s)

-1

0

1

q (

deg/s

)

Figure 3.6: Closed-loop and Open-loop Responses of Rigid-body States at Speed = 34 m/s

faults, to avoid confusion in the analysis of post-fault responses, disturbances like wind gusts

or noise effects are not included in the simulations.

Fault Scenario: Flap 1 Stuck

The first fault scenario is when Flap 1 gets stuck at a certain angle while Flap 2 and the

elevator work normally. The fault is injected at t = 5 s for both pre-flutter and post-flutter

systems.

In Figures 3.7 and 3.8, the responses of elastic deformations and rigid-body states at

the speed of 32 m/s with Flap 1 stuck at 1 , 3 and 5 are shown respectively. And the

deflections of control surface are shown in Figure 3.9. At the speed of 34 m/s, the responses

of elastic deformations and rigid-body states with Flap 1 stuck at 1 , 3 and 5 are shown

in Figures 3.10 and 3.11 respectively. The deflections of control surface are shown in Figure

3.12.

For both speeds, the system states settle down after the occurrence of stuck fault in Flap

1. The stuck flap causes both the rigid-body states and elastic deformations to oscillate

lightly from the nominal trimmed conditions. The post-fault steady-state values are close to

but deviate from the nominal steady-state values. The larger the stuck angle is, the larger

the deviations are. This is because when Flap 1 gets stuck at a certain angle, the flexible

wing finally gets trimmed to a different shape with the other two control surfaces designed


0 5 10 15-7.84

-7.82

-7.8

-7.78

tip d

ispla

cem

ent (m

)

δa1

=1°

δa1

=3°

δa1

=5°

0 5 10 15

time (s)

2

3

4

5

tip r

ota

tion (

deg)

Figure 3.7: Post-fault Responses of Wing Deformations at Speed = 32 m/s for δa1 = 1 , 3

and 5

0 5 10 15

1.2

1.4

1.6

θ (

deg) δ

a1=1

°δ

a1=3

°δ

a1=5

°

0 5 10 150.6

0.7

0.8

w (

m/s

)

0 5 10 15

time (s)

-1

-0.5

0

0.5

q (

deg/s

)

Figure 3.8: Post-fault Responses of Rigid-body States at Speed = 32 m/s for δa1 = 1 , 3

and 5


0 5 10 15

0

5

10

δa1 (

deg) δ

a1=1

°δ

a1=3

°δ

a1=5

°

0 5 10 15-2

0

2

δa2 (

deg)

0 5 10 15

time (s)

-10

0

10

δe (

deg)

Figure 3.9: Control Surface Deflections at Speed = 32 m/s for δa1 = 1 , 3 and 5

0 5 10 15-8.2

-8.15

-8.1

-8.05

tip d

ispla

cem

ent (m

)

δa1

=1°

δa1

=3°

δa1

=5°

0 5 10 15

time (s)

2

3

4

5

tip r

ota

tion (

deg)

Figure 3.10: Post-fault Responses of Wing Deformations at Speed = 34 m/s for δa1 = 1 ,3 and 5


0 5 10 15

0.7

0.8

0.9

θ (

deg) δ

a1=1

°δ

a1=3

°δ

a1=5

°

0 5 10 15

0.4

0.6

w (

m/s

)

0 5 10 15

time (s)

-0.5

0

0.5

q (

deg/s

)


and 5

0 5 10 15

0

5

10

δa1 (

deg) δ

a1=1

°δ

a1=3

°δ

a1=5

°

0 5 10 15-2

0

2

δa2 (

deg)

0 5 10 15

time (s)

-10

0

10

δe (

deg)



to go back to their trimmed values, which are zero in this case.

Fault Scenario: Flap 2 Stuck

The second fault scenario is when Flap 2 gets stuck at a certain angle while Flap 1 and the

elevator work normally. The fault is injected at t = 5 s for both pre-flutter and post-flutter

systems.

In Figures 3.13 and 3.14, the responses of elastic deformations and rigid-body states at the

speed of 32 m/s with Flap 2 stuck at 1 , 3 and 5 are shown respectively. The deflections

of control surface are shown in Figure 3.15. Figures 3.16 and 3.17 show the responses of

elastic deformations and rigid-body states at the speed of 34 m/s with Flap 2 stuck at 1 ,

3 and 5 respectively. The deflections of control surface are shown in Figure 3.18.

0 5 10 15-9.5

-9

-8.5

-8

-7.5

tip d

ispla

cem

ent (m

)

δa2

=1°

δa2

=3°

δa2

=5°

0 5 10 15

time (s)

2

4

6

8

tip r

ota

tion (

deg)


Similar to the Flap 1 stuck cases, for both speeds, the elastic and rigid-body states get

settled down after the occurrence of stuck fault in Flap 2 with deviations of the steady-state

values from the nominal trimmed conditions. It can also be seen that the post-fault steady-

state deviations are larger than the Flap 1 stuck cases for a same stuck angle, which implies

that the stuck fault of the outer flaps is more detrimental than of the inner ones.

If we look at the Flap 1 deflection with the δa2 = 5 case in Figure 3.15, the steady-state

value of deflection angle is nearly -20. For the elevator deflections shown in Figure 3.18, the

peak value of the δa2 = 5 case almost reaches -20. Based on the observations of post-fault


responses, the larger the stuck angle is, the larger control surface deflections would be. It

can be inferred that for a larger stuck angle, more control efforts from the remaining healthy

control surfaces are required, which may exceed their saturation limits. For stuck faults

occurring in either Flap 1 or Flap 2, although their major influence on the flexible aircraft

model is to make the attitude and wing deformations deviate from the nominal values, a

large stuck angle does have the potential of causing actuator saturation in the healthy control

surfaces, which could lead to more severe consequences such as nonlinear oscillations and

instability.

0 5 10 151

1.5

2

θ (

deg) δ

a2=1

°δ

a2=3

°δ

a2=5

°

0 5 10 150.6

0.8

1

w (

m/s

)

0 5 10 15

time (s)

-1

0

1

2

3

q (

deg/s

)


and 5


0 5 10 15-20

0

20

δa1 (

deg)

δa2

=1°

δa2

=3°

δa2

=5°

0 5 10 15

0

5

10

δa2 (

deg)

0 5 10 15

time (s)

-10

0

10

δe (

deg)


0 5 10 15-10

-9

-8

-7

tip d

ispla

cem

ent (m

)

δa2

=1°

δa2

=3°

δa2

=5°

0 5 10 15

time (s)

1

2

3

4

5

tip r

ota

tion (

deg)



0 5 10 150

2

θ (

deg) δ

a2=1

°δ

a2=3

°δ

a2=5

°

0 5 10 150.4

0.6

0.8

w (

m/s

)

0 5 10 15

time (s)

-2

0

2

4

q (

deg/s

)


and 5

0 5 10 15-2

0

2

δa1 (

deg)

δa2

=1°

δa2

=3°

δa2

=5°

0 5 10 15

0

5

10

δa2 (

deg)

0 5 10 15

time (s)

-20

-10

0

δe (

deg)


Fault Scenario: Flap 1 and Flap 2 Stuck

The third fault scenario is when Flap 1 and Flap 2 get stuck at a certain angle while the

elevator works normally. The fault is injected at 5 seconds for both pre-flutter and post-

flutter systems.


Figures 3.19 and 3.20 show the responses of elastic deformations and rigid-body states

at the speed of 32 m/s with Flap 1 and Flap 2 stuck at 1 , 3 and 5 respectively. The

deflections of control surfaces are shown in Figure 3.21. The system starts to show decaying

oscillatory responses after the two flaps get stuck in wing-tip rotation, rigid-body states and

elevator response. Wing elastic deformations and rigid-body states deviate away from the

nominal trimmed values.

0 5 10 15-10

-9

-8

-7

tip d

ispla

cem

ent (m

)

δa1,2

=1°

δa1,2

=3°

δa1,2

=5°

0 5 10 15

time (s)

-2

0

2

4

6

tip r

ota

tion (

deg)

Figure 3.19: Post-fault Responses of Wing Deformations at Speed = 32 m/s for δa1,2 = 1 ,3 and 5

In Figures 3.22 and 3.23, the responses of elastic deformations and rigid-body states at

the speed of 34 m/s with Flap 1 and Flap 2 stuck at 1 , 3 and 5 are shown respectively.

At this speed, stuck faults of the two flaps cause divergent oscillations of the system. The

oscillatory elastic modes cannot be suppressed due to the loss of control of both flaps. It

should be noted that saturation constraints are not set for control surfaces in this simulation

study. The unconstrained response of elevator is enough to show the failure of the system

when both Flap 1 and Flap 2 are stuck.

Fault Scenario: Elevator Stuck

The fourth fault scenario is when the elevator gets stuck at a certain angle while Flap 1 and

Flap 2 work normally. The fault is injected at 5 seconds for both pre-flutter and post-flutter

systems.

Figures 3.25 and 3.26 show the responses of elastic deformations and rigid-body states


0 5 10 151

1.5

2

θ (

deg) δ

a1,2=1

°δ

a1,2=3

°δ

a1,2=5

°

0 5 10 150.6

0.8

1

w (

m/s

)

0 5 10 15

time (s)

-2

0

2

q (

deg/s

)

Figure 3.20: Post-fault Responses of Rigid-body States at Speed = 32 m/s for δa1,2 = 1 ,3 and 5

0 5 10 15

0

5

10

δa1 (

deg) δ

a1,2=1

°δ

a1,2=3

°δ

a1,2=5

°

0 5 10 15

0

5

10

δa2 (

deg)

0 5 10 15

time (s)

-10

0

10

δe (

deg)

Figure 3.21: Control Surface Deflections at Speed = 32 m/s for δa1,2 = 1 , 3 and 5


0 5 10 15-10

-9.5

-9

-8.5

-8

-7.5

tip d

ispla

cem

ent (m

)

δa1,2

=1°

δa1,2

=3°

δa1,2

=5°

0 5 10 15

time (s)

-10

0

10

20

tip r

ota

tion (

deg)

Figure 3.22: Post-fault Responses of Wing Deformations at Speed = 34 m/s for δa1,2 = 1 ,3 and 5

0 5 10 150

2

4

θ (

deg) δ

a1,2=1

°δ

a1,2=3

°δ

a1,2=5

°

0 5 10 150

0.5

1

w (

m/s

)

0 5 10 15

time (s)

-10

0

10

q (

deg/s

)

Figure 3.23: Post-fault Responses of Rigid-body States at Speed = 34 m/s for δa1,2 = 1 ,3 and 5


0 5 10 15

0

5

10

δa1 (

deg) δ

a1,2=1

°δ

a1,2=3

°δ

a1,2=5

°

0 5 10 15

0

5

10

δa2 (

deg)

0 5 10 15

time (s)

-50

0

50

δe (

deg)

Figure 3.24: Control Surface Deflections at Speed = 34 m/s for δa1,2 = 1 , 3 and 5

at the speed of 32 m/s with the elevator stuck at 1 , 3 and 5 respectively. The deflections

of control surface are shown in Figure 3.27. Both the wing elastic deformations and rigid-

body states are affected by stuck fault of the elevator, showing responses of slow decaying

oscillations. The two flaps on the wing cannot settle down the system states within the given

simulation time length. The amplitude and frequency of oscillations become larger as the

stuck angle increases. The stuck faults of elevator not only affect the attitude of aircraft but

also the elastic deformations of the wing, resulting in unsettled vibrations.

Figures 3.28 and 3.29 show the first 10-second responses of elastic deformations and rigid-

body states at the speed of 34 m/s with the elevator stuck at 1 , 3 and 5 respectively. The

deflections of control surface are shown in Figure 3.30. At this speed, the system becomes

unstable after the elevator gets stuck.

Comparing with the flap stuck fault scenarios, it can be inferred that the fault of elevator

and flaps influence the system through rigid-body states and elastic deformations respec-

tively. Due to the coupling between the rigid-body and flexible modes, the fault of either

the elevator or flaps eventually influences the whole dynamics of flexible aircraft. For cases

shown in Figures 3.28 – 3.30 and 3.22 – 3.24, the unstable responses of rigid-body modes

and flexible modes accelerate each other to uncontrolled situations.

To summarize, the influence of different control surface faults on a flexible aircraft model

is investigated. Due to the strong coupling between rigid-body and flexible modes, control

surface faults will have impacts on the flexible aircraft system through either rigid-body

modes or elastic modes, and eventually influence the whole dynamics. From the above post-


0 5 10 15 20 25 30-8.5

-8

-7.5

tip d

ispla

cem

ent (m

)

δe=1

°δ

e=3

°δ

e=5

°

0 5 10 15 20 25 30

time (s)

0

5

10

15

tip r

ota

tion (

deg)

Figure 3.25: Post-fault Responses of Wing Deformations at Speed = 32 m/s for δe = 1 , 3

and 5

0 5 10 15 20 25 30-5

0

5

θ (

deg) δ

e=1

°δ

e=3

°δ

e=5

°

0 5 10 15 20 25 30-1

0

1

w (

m/s

)

0 5 10 15 20 25 30

time (s)

-2

0

2

q (

deg/s

)

Figure 3.26: Post-fault Responses of Rigid-body States at Speed = 32 m/s for δe = 1 , 3

and 5


0 5 10 15 20 25 30

-20

0

20

δa1 (

deg) δ

e=1

°δ

e=3

°δ

e=5

°

0 5 10 15 20 25 30-1

0

1

δa2 (

deg)

0 5 10 15 20 25 30

time (s)

0

5

10

δe (

deg)

Figure 3.27: Control Surface Deflections at Speed = 32 m/s for δe = 1 , 3 and 5

0 1 2 3 4 5 6 7 8 9 10-8.3

-8.2

-8.1

tip d

ispla

cem

ent (m

)

δe=1

°δ

e=3

°δ

e=5

°

0 1 2 3 4 5 6 7 8 9 10

time (s)

2

3

4

5

6

tip r

ota

tion (

deg)

Figure 3.28: Post-fault Responses of Wing Deformations at Speed = 34 m/s for δe = 1 , 3

and 5


0 1 2 3 4 5 6 7 8 9 10-10

0

10

θ (

deg) δ

e=1

°δ

e=3

°δ

e=5

°

0 1 2 3 4 5 6 7 8 9 100.2

0.4

0.6w

(m

/s)

0 1 2 3 4 5 6 7 8 9 10

time (s)

-4

-2

0

q (

deg/s

)

Figure 3.29: Post-fault Responses of Rigid-body States at Speed = 34 m/s for δe = 1 , 3

and 5

0 1 2 3 4 5 6 7 8 9 10

0

5

10

δa1 (

deg)

δe=1

°δ

e=3

°δ

e=5

°

0 1 2 3 4 5 6 7 8 9 10-1

0

1

δa2 (

deg)

0 1 2 3 4 5 6 7 8 9 10

time (s)

0

5

10

δe (

deg)

Figure 3.30: Control Surface Deflections at Speed = 34 m/s for δe = 1 , 3 and 5

fault analysis of four different fault scenarios at pre-flutter and post-flutter speeds, stuck

faults of different control surfaces have been shown to cause unwanted vibrations of the wing

and uncontrolled rigid-body motions through the interaction between rigid-body modes and


elastic modes. In the rest of this chapter, an FTC design is developed to handle this type of

fault for flexible aircraft.

3.2 FTC Problem Formulation

When studying FTC designs for flexible aircraft, it is important to take actuator saturation

into consideration. One obvious reason is that the magnitude of control surface deflections is

physically constrained. This fact has been addressed in some control designs for flexible air-

craft [41, 36, 40]. It is also well-known that the presence of actuator saturation can degrade

the closed-loop system performance, leading to nonlinear oscillations and even catastrophic

consequences. Aircraft mishaps have been reported to be related to the detrimental effects

of actuator saturation [104, 105]. What is more important, the negative interplay between

actuator saturation and faults cannot be overlooked since the two phenomena actually can

happen at the same time. As explained in [106], occurrence of actuator faults can result in

undesired transients or large steady-state errors. Without the consideration of saturation,

FTC design will likely demand large control input to compensate for the undesired perfor-

mance, which may saturate the actuators. As the actuators become saturated, the errors

further increase and in turn require larger control input, so on and so forth.

Another aspect that needs to be considered in FTC designs for flexible aircraft is the

varations of aeroelastic dynamics at varying flight conditions. The aeroelastic responses of

the flexible wing are highly dependent on parameters such as airspeed which are time-varying

in operation. It is well-known that flutter onset will be expected once the airspeed increases

to the so-called flutter speed. This means that even a simple manuever like increasing the

flight speed may lead to unstable wing vibrations if the flight controller is only designed

for a fixed lower speed. Thus, for flexible aircraft, it is especially desirable to have a flight

controller that can work for a large operating range, not only for the purpose of extending

flight envelope, but also for preventing the potential excitation of aeroelastic modes during

maneuvers. Linear parameter-varying (LPV) control is a well-suited method to handle this

situation by scheduling control gains according to varying parameters while guaranteeing

system stability and performance along the time-varying parameter trajectory. It has been

applied to suppress flutter across the flight envelope for aeroservoelastic aircraft in [39]. In

the FTC designs for flexible aircraft, it is similarly desirable that the FTC controller retains

the attribute of scheduling with varying parameters.

Motivated by the above background and the two practical concerns for flexible aircraft

FTC design, a gain-scheduling FTC controller is developed for a flexible aircraft to address

flutter suppression, stuck faults as well as actuator saturation. The proposed FTC controller


is aimed at achieving the following goals: minimizing the effects of stuck control surface on

rigid-body motion and aeroelastic modes of the wing; allowing safe operation in a range of

airspeed variation under faulty circumstances; and no closed-loop performance degradation

caused by actuator saturation. Methodologically, the polytopic method by Hu and Lin [107]

is applied to represent the saturation nonlinearity, and LPV control design is used to develop

a gain-scheduling controller. The aforementioned design goals can be unified to give a set

of LPV set-invariance conditions, which can be formulated under the framework of Linear

Matrix Inequalities (LMIs).

3.2.1 Linear Parameter-Varying (LPV) Representation

For each airspeed U , the aircraft is first trimmed in a given steady flight condition, then

linearization is applied to Equation (2.53) at the corresponding trimmed states. By doing

this, an LPV representation for flexible aircraft model can be described by

δx(t) = A(U(t))δx(t) +B(U(t))δu(t)

y(t) = Cδx(t)(3.2)

where δx ∈ Rn and δu ∈ Rm denote the deviations of the states and control input from their

trimmed conditions, y ∈ Rp denotes the output, airspeed U is the varying parameter and

assumed to be measurable. For convenience, the δ before x and u will be omitted without

causing any confusion in what follows.

3.2.2 Faulty System Description

With stuck control surface faults, the LPV flexible aircraft model subject to the actuator

saturation can be written as

x(t) = A(U)x(t) +B1(U)sat(uc(t)) +B2(U)us

y(t) = Cx(t)(3.3)

where uc ∈ Rm1 is the control input associated with the healthy control surfaces, us ∈Rm2 is a constant vector formed by the values at which faulty control surfaces get stuck,

B1 and B2 are the distribution matrices with m1 + m2 = m. The constrained deflec-

tions of the remaining control surfaces are described by the saturation function sat(uc) =

[sat(u1), sat(u2) · · · sat(um1)]T with sat(ui) = sign(ui)min|ui|, ui in which ui > 0 is the

deflection limit of the i-th control surface.

The occurrence of stuck faults in the system means that additional constant and persistent


disturbances are imposed onto the system while certain control channels are lost. For flexible

aircraft, these kinds of fault can drive the aircraft motions away from their desired values,

and may even cause unwanted vibrations of the flexible wings [108]. To recover the desired

aircraft motions and prevent negative aeroelastic responses, the design freedoms provided

by the remaining control surfaces should be utilized to counteract the influence of stuck

control surfaces on the flexible aircraft dynamics. Thus, the control problem studied in this

thesis is how to design the remaining control inputs subject to deflection limits to achieve

the aforementioned goals for the LPV system (3.3).

To carry out control design, the following assumptions need to be made.

Assumption 3.1: For system(3.3), the pair (A(U), B1(U)) is stabilizable.

Assumption 3.2: The stuck fault values are bounded, i.e., there exists a positive con-

stant δi such that us ∈ us : |usi | ≤ δi.Assumption 3.3: There are no transmission zeros at the origin for system (3.3).

Remark 3.4: Assumption 3.1 is a basic requirement for using the remaining control

surfaces to design a controller for system (3.3). Assuming the boundedness of stuck faults

in Assumption 3.2 is natural and reasonable. Assumption 3.3 guarantees the use of integral

control to eliminate the fault influence on the output y. In most existing FTC literature

dealing with stuck faults, an assumption about the linear dependence of columns in B1

and B2 is made, which is usually given as the requirement of “rank(B) = rank(B1) ” (Refs.

[109, 110]) or “B1F1 = B2, where F1 is a constant matrix with appropriate dimensions ” (Ref.

[111]). This means that the constant inputs from the stuck channels can be compensated

by a linear combination of the remaining control channel inputs. But in this work, such an

assumption is not made as the current configuration of the flexible aircraft model does not

satisfy this assumption. Thus the effects of stuck faults on the full system states cannot be

completely eliminated through any control design. However, a modified goal may still be

achieved, which is eliminating the effects on some of the states that designers are particularly

interested in, namely the output y. Provided Assumption 3.3 holds, this goal is achievable

by using integral control, which will be presented in the Section 3.3.

3.2.3 Design Objectives

The design objectives are given as follows. Assume that the stuck faults belong to the fault

set S = us : |usi | ≤ δi, i ∈ Fs where Fs is a known set of the numbers of control surfaces

that get stuck and δi is a known positive constant. Let Ur = U |U ≤ U ≤ U denote the

operating airspeed range. The design objective is to find a controller uc to ensure that for

any U ∈ Ur


(i) the states of the system (3.3) are bounded and converge asymptotically to values which

are as close to the origin as possible;

(ii) the output y is stabilized at the origin.

If we define the rigid-body motion states as the output y, the above design objectives

mean that for flexible aircraft subject to a prescribed control surface stuck scenario, an

LPV FTC controller will be designed to guarantee that the rigid-body motions of aircraft

are stabilized to the nominal positions and aeroelastic vibrations of the wings settle down

with wing deformations as small as possible. For different prescribed fault scenarios, a set

of such LPV FTC controllers can be designed and prepared off-line. Once one of these

prescribed fault scenarios happens, with good fault detection and isolation information, the

corresponding predesigned controller can be switched in. In other words, this paper aims at

designing a set of these preventive FTC controllers to handle possible stuck fault scenarios

for flexible aircraft.

3.3 LPV FTC Design

3.3.1 Controller Structure

As discussed in Remark 3.2, the effects of stuck faults on the full system states cannot be

completely compensated by any control law due to the linear independence of the remaining

control channels and the stuck control channels. So a more modest goal, stated as design

objective (ii), is pursued to ensure the stabilization of output y will not be affected. For

simplicity, this paper assumes that all the states are available for feedback design. Then to

achieve design objective (ii), a proportional and integral (PI) controller structure is proposed,

which can guarantee zero steady-state error of output in the presence of stuck faults:

uc(t) = Kx(U)x(t) +KI(U)

∫ t

0

y(τ)dτ (3.4)

Denote the integral of output by ζ(t) =∫ t

0y(τ)dτ and define a new state vector ξ(t) =[

xT (t) ζT (t)

]T. An augmented system can be formulated as

ξ(t) = Aa(U)ξ(t) +Ba(U)sat(uc(t)) + Ea(U)us (3.5)


where Aa =

A(U) 0n×p

C 0p×p

, Ba =

B1(U)

0p×m1

, and Ea =

B2(U)

0p×m2

.

Recall from Assumption 3.2 that boundedness of stuck fault values is assumed. Let us

be inside the following predefined ellipsoidal set: Ωd(D) = us : uTs (t)Dus ≤ 1, where

D = diag1/δ2i ∈ Rm2×m2 .

Then the controller structure given in Equation (3.4) can be interpreted as a state-

feedback controller uc(t) = Kc(U)ξ(t) where Kc(U) = [Kx(U) KI(U)] for the augmented

system (3.5) subject to stuck faults us ∈ Ωd(D) .

With Assumptions 3.1 and 3.3, the stabilizability of the pair Aa(U), Ba(U) is ensured.

This means that, if there is no saturation nonlinearity in system (3.5), we can find the control

gain Kc(U) that achieves the second design objective by making Aa(U)+Ba(U)Kc(U) stable.

The next question is how can both design objectives (i) and (ii) be achieved with saturation

nonlinearity present.

3.3.2 Set-invariance Conditions for Tolerating Stuck Faults

With the controller structure given by uc(t) = Kc(U)ξ, the first design objective can be

interpreted in a language of set invariance as follows.

Given a set X∞ ⊂ Rn+p, design Kc(U) such that the closed-loop system of (3.5) has

an invariant set Ωc(Pc(U)) ⊂ αX∞ with α minimized, i.e., all state trajectories start from

Ωc(Pc(U)) will remain inside it, where Pc(U) ∈ R(n+p)×(n+p) is a positive definite matrix and

Ωc(Pc) is an ellipsoid set is defined by Ωc(Pc) = ξ ∈ Rn+p| ξTPcξ ≤ 1.If such a Kc(U) is found, then all system states will be bounded in an invariant set

Ωc(Pc(U)) with values as close to the origin as possible since the size of the set is minimized.

The problem becomes how to achieve set invariance for an LPV system subject to actuator

saturation and to minimize the size of the invariant set.

Control of linear time invariant systems subject to actuator saturation has been studied

in the control community for decades. A popular way of solving this problem is to model the

saturation effect in a way such that it can be dealt with under a linear framework(see e.g.

Refs.[107, 112] and references therein). Results have also been extended to LPV systems

[113, 114], which however are often assumed to be polytopic. In this thesis, the polytopic

representation of saturation nonlinearity proposed by Hu and Lin [107] is adopted. By using

this method, the set-invariance conditions for an LPV system can be then formulated in an

LMI framework so that the control problem will be readily solved by convex optimization

tools. Since the flexible aircraft model does not satisfy as a polytopic LPV system, a gain-

scheduled controller based on a parameter-dependent Lyapunov function will be synthesized


by using the basis function method proposed by Wu et al. [115].

Lemma 3.1 [107] Let G be the set of m1×m1 diagonal matrices whose diagonal elements

are either 1 or 0, and uc, vc ∈ Rm1 . Suppose that |vi| ≤ ui for all i ∈ [1,m1], where vi is the

ith element of vc and ui is the saturation limit of the ith element of uc. Then

sat(uc) ∈ coGiuc + Givc : i ∈ [1, 2m1 ] (3.6)

where co denotes the convex hull, Gi ∈ G and Gi = I −Gi.

By Lemma 3.1, if an auxiliary control law is defined also in a state-feedback form vc(t) =

Hc(U)ξ(t) and |hciξ| ≤ ui, i = 1, 2, . . . ,m1 where hci is the ith row of the matrix Hc, then

the saturation term sat(Kcξ) can be expressed by a convex combination of a group of linear

feedbacks

sat(Kcξ) =2m1∑i=1

ηi(GiKc + GiHc)ξ (3.7)

where 0 ≤ ηi ≤ 1 and∑2m1

i=1 ηi = 1.

Let L (Hc) = ξ ∈ Rn+p : |hciξ| ≤ ui, i = 1, 2, . . . ,m1, and sym(A ) denote A T + A

for any square matrix A . The set-invariance conditions are given in the following theorem.

Theorem 3.1 For a given ellipsoid set Ωc(Pc(U)), and any U inside a prescribed param-

eter range Ur, if there exist parameter dependent matrices Pc(U) > 0, Kc(U) and Hc(U)

such that the following matrix inequalities are satisfiedsym(Pc(U)[Aa(U) +Ba(U)(GiKc(U) + GiHc(U))]) + Pc(U) + β1Pc(U) Pc(U)Ea(U)

ETa (U)Pc(U) −β2D

< 0

(3.8)

for i = 1, 2, . . . , 2m1 , where β1 > 0, β2 > 0, β2 − β1 ≤ 0, and Ωc(Pc) ⊂ L (Hc), then under

control law (3.4), Ωc(Pc(U)) is an invariant set for system (3.5), i.e., the state ξ(t) starting

in Ωc(Pc) remains in the set Ωc(Pc).

Proof. For any ξ ∈ Ωc(Pc), if Ωc(Pc) ⊂ L (Hc), then the conditions in Lemma 1 are

satisfied, and the saturation term in system (3.5) can be represented by Equation (3.7).

Define a parameter dependent Lyapunov function Vc = ξTPc(U)ξ. Then the derivative


of Vc is given by

Vc =ξT[Aa +Ba

2m1∑i=1

ηi(GiKc + GiHc)]TPc

+ Pc[Aa +Ba

2m1∑i=1

ηi(GiKc + GiHc)] + Pcξ + 2ξTPcEaus

=ξT (PcAacl + ATaclPc + Pc)ξ + 2ξTPcEaus

(3.9)

where Aacl = [Aa +Ba

∑2m1

i=1 ηi(GiKc + GiHc)].

To prove the set invariance, i.e., all ξ start in Ωc(Pc) always remain inside it, we require

for system (3.5) subject to stuck fault bounded by uTsDus ≤ 1 that Vc < 0 in the set

ξ : ξTPcξ ≥ 1. By S-procedure [116], this condition can be satisfied if there exist scalars

β1 > 0 and β2 > 0 such that

Vc + β1(ξTPcξ − 1) + β2(1− uTsDus)

=ξT (PcAacl + ATaclPc + Pc)ξ + 2ξTPcEaus + β1(ξTPcξ − 1) + β2(1− uTsDus) < 0(3.10)

If inequality (3.8) holds for all i = 1, 2, . . . , 2m1 and β2 − β1 ≤ 0, then we can conclude

that

[ξT uTs

]PcAacl + ATaclPc + Pc + β1Pc PcEa

ETa Pc −β2D

ξus

+ β2 − β1 < 0 (3.11)

which is equivalent to inequality (3.10).

Thus, the conditions given in Theorem 1 are sufficient to prove the invariance of the set

Ωc(Pc(U)).

Remark 3.5: The set-invariance conditions ensure all states in ξ(t) =

[xT (t) ζT (t)

]Tare bounded. Since LMI (14) holds, Aacl(U) is stable, thus it is not difficult to show x and

ζ will settle down to constant steady-state values. Furthermore, by setting ζ = 0 in system

(3.5), we can see that the steady-state value for output y = Cx is zero. This means that

the second design objective has been achieved.

Based on Theorem 3.1, the following optimization problem of minimizing the size of the


invariant set Ωc(Pc(U)) is formulated:

minPc(U)>0,Kc(U),Hc(U),β1≥β2>0

α

s.t. (a)|hciξ| ≤ ui, ∀ξ ∈ Ωc(Pc(U), i = 1, 2, . . . ,m1

(b)(3.8) holds, ∀i = 1, 2, . . . , 2m1

(c)Ωc(Pc(U)) ⊂ αX∞

(3.12)

where the reference set X∞ will be specified in the following subsection where the above

constraints are converted into LMIs.

If the optimization problem (3.12) is feasible, then the control gain Kc(U) is guaranteed to

achieve the design objectives for the LPV aircraft model (3.3) subject to actuator saturation

and stuck faults. In what follows, we will show that this problem can be solved as an LMI

optimization problem.

3.3.3 LMI Formulation

Let Pc(U) = P−1c (U), Kc(U) = Kc(U)Pc, Hc(U) = Hc(U)Pc, then Condition (a) in (3.12)

holds if hciP−1c (U)hTci ≤ u2

i , which by Schur’s complement is converted into u2i hci(U)

hTci(U) Pc(U)

≥ 0 (3.13)

where hci = hciPc denotes the i-th row of Hc(U), for i = 1, 2, . . . ,m1.

By using the congruence transformation diagP−1c (U), I, Condition (b) is equivalent to

sym([Aa(U)Pc(U) +Ba(U)(GiKc(U) + GiHc(U)])− ˙Pc(U) + β1Pc(U) Ea(U)

ETa (U) −β2D

< 0 (3.14)

for i = 1, 2, . . . , 2m1 .

For Condition (c), the reference set X∞ which is used to characterize the size of Ωc(Pc(U))

can also be chosen as an ellipsoid

X∞ = ξ(t) : ξT (t)Qξ(t) ≤ 1, Q = QT > 0 (3.15)


Then Condition (c) is equivalent to

Pc(U) ≤ γQ−1 (3.16)

where γ = α2.

Then the optimization problem (3.12) can be formulated as a convex optimization prob-

lem with LMI constraints

minPc(U)>0,Kc(U),Hc(U),β1≥β2>0

γ

s.t. (a)LMI(3.13) holds, ∀ i = 1, 2, . . . ,m1

(b)LMI(3.14) holds, ∀i = 1, 2, . . . , 2m1

(c)LMI(3.16) holds

(3.17)

Theorem 3.2 For an LPV flexible aircraft model (3.3) subject to actuator saturation

and prescribed stuck faults, the control gain Kc(U) = [Kx(U) KI(U)] for a controller uc

defined in (3.4) that achieves the two design objectives given in Section 3.2.3 can be solved

by

Kc(U) = Kc(U)P−1c (U) (3.18)

where Pc(U) > 0, Kc(U) are the solutions of the LMI optimization problem (3.17).

To reflect the dependence on U of the variables in the LMI optimization problem, we

parametrize them using basis functions. Based on the suggestion in [115] and noting the

model’s parameter dependency in U and U2, three basis functions 1, U, U2 are chosen. To

solve (3.17) for the given airspeed range Ur, the three U -dependent variables are first ex-

pressed using the basis functions as:

Pc = P0 + P1U + P2U2 (3.19)

Kc = K0 + K1U + K2U2 (3.20)

Hc = H0 + H1U + H2U2 (3.21)

Assume the variation rate of airspeed is bounded. Choose N0 grid points over the airspeed

range Ur. For each chosen point of U , substitute the above expressions into (3.17) to give

one set of LMIs governing the variables Pi, Ki, Hi, i = 0, 1, 2. Solving the N0 sets of LMIs

at all chosen grid points simultaneously determines the variables Pi, Ki, Hi, i = 0, 1, 2. Then


the control gain Kc(U) is obtained as a function of U :

Kc(U) = (K0 + K1U + K2U2)(P0 + P1U + P2U

2)−1 (3.22)

3.4 Numerical Simulations

In this section, the designed controller is applied to the flexible aircraft model used in Section

3.1. The simulation study will be focused on the longitudinal attitude stabilization and

wing vibration suppression of the flexible aircraft during steady-level flight. To prevent the

attitude stabilization from being influenced by the stuck control surface faults, the pitch

angle is chosen as the output.

The baseline LQR controller designed for each fixed speed in Section 3.1.1 is still used as

the nominal controller, which is able to stabilize the longitudinal attitude as well as suppress

aeroelastic modes of the wing. The open-loop and closed-loop responses of the system at 32

m/s and 34 m/s have been shown in Figures 3.3 – 3.4 and Figures 3.5 – 3.6 respectively.

The objective of the FTC controller is to stabilize the pitch angle and effectively suppress

the vibrational motion of the flexible wing subject to stuck faults and actuator saturation

for a given airspeed range. In what follows, two stuck control surface fault scenarios are

simulated to illustrate that the corresponding FTC controllers designed in this chapter are

able to achieve such an objective. In the simulations, the deflections of Flap 1, Flap 2 and

elevator, denoted by δa1, δa2 and δe respectively, are set to be limited by ±20 deg. All the

state responses shown are the differences with respect to the corresponding nominal trimmed

conditions. The airspeed range is from a pre-flutter speed of 31 m/s to a post-flutter speed

of 34 m/s. Grid points for LPV controller synthesis over this interval are chosen as 31, 32,

33, and 34 m/s.

3.4.1 Fault Scenario 1

The first fault scenario is when a stuck fault no greater than 5 deg happens in Flap 2. We

inject a stuck fault of 5 deg into Flap 2 at t = 0 s to see how a gain-scheduling FTC controller

designed for this fault scenario can handle it. Closed-loop responses of the flexible aircraft

with the FTC controller at 32 m/s are shown in Figures 3.31 and 3.32, with comparison to

the responses with a nominal LQR controller designed at this airspeed. The control surface

deflections of the two controllers are compared in Figure 3.33.

As what can be seen from the three figures, the nominal controller cannot stabilize the

pitch angle to zero due to the stuck fault in Flap 2. What is worse is that Flap 1 saturates


0 5 10 15 20 25 30 35 40 45 50−2

−1.5

−1

−0.5

0

0.5

tip d

ispla

cem

ent

(m)

FTC nominal

0 5 10 15 20 25 30 35 40 45 50−4

−2

0

2

4

6

tip r

ota

tion (

deg)

time (s)

20.5 21 21.5 223.2

3.25

3.3

Figure 3.31: Closed-loop Responses of Wing-tip Bending and Torsion with the FTC andNominal Controllers at 32 m/s under Fault Scenario 1

0 5 10 15 20 25 30 35 40 45 50−0.5

0

0.5

1

1.5

θ (

deg)

FTC nominal

0 5 10 15 20 25 30 35 40 45 50−2

0

2

4

6

q (

deg/s

)

time (s)

20.5 21 21.5 22−0.05

0

0.05

Figure 3.32: Closed-Loop Responses of Pitch Angle and Rate with the FTC and NominalControllers at 32 m/s under Fault Scenario 1


0 5 10 15 20 25 30 35 40 45 50

−20

−10

0

10

δa1 (

deg)

FTC nominal

0 5 10 15 20 25 30 35 40 45 504

5

6δ

a2 (

deg)

0 5 10 15 20 25 30 35 40 45 50

−20

−10

0

10

δe (

deg)

time (s)

20.5 21 21.5 220.04

0.05

0.06

Figure 3.33: Control Surface Deflections of the FTC and Nominal Controllers at 32 m/sunder Fault Scenario 1

at -20 deg after around 3 seconds. The presence of actuator saturation causes the small-

magnitude high-frequency oscillations in the system states, which are shown in the small

magnified box in the responses of wing-tip rotation, pitch rate and the elevator deflection. On

the other hand, our designed FTC controller stabilizes the pitch angle to zero and suppresses

the vibrations which can happen in the wing with no remaining control surfaces saturated.

The steady-state deviations of wing-tip bending and rotation from zero (see Figure 3.31)

are the effects of the stuck fault on system states that cannot be fully eliminated through

control. But as required by our design objectives, the deviations are bounded and have been

minimized in the controller design procedure.

The system with the FTC controller is also simulated at 33.5 m/s, a speed in between

of two grid points, to show the effectiveness of the gain scheduling design. The closed-

loop responses are compared with the responses with a nominal LQR contoller design at

the airspeed, as shown in Figures 3.34 and 3.35 The control surface deflections of the two

controllers are compared in Figure 3.36. In this case, the nominal controller still cannot

stabilize the pitch angle to zero and has a large elevator deflection. The FTC controller also

works for this speed and shows better transient performance over the nominal one.


0 5 10 15 20 25 30 35 40 45 50−2

−1.5

−1

−0.5

0

0.5

tip d

ispla

cem

ent

(m)

FTC nominal

0 5 10 15 20 25 30 35 40 45 50−4

−2

0

2

4

6

tip r

ota

tion (

deg)

time (s)

Figure 3.34: Closed-loop Responses of Wing-tip Bending and Torsion with the FTC andNominal Controllers at 33.5 m/s under Fault Scenario 1

0 5 10 15 20 25 30 35 40 45 50−0.5

0

0.5

1

1.5

θ (

deg)

FTC nominal

0 5 10 15 20 25 30 35 40 45 50−2

0

2

4

6

q (

deg/s

)

time (s)

Figure 3.35: Closed-Loop Responses of Pitch Angle and Rate with the FTC and NominalControllers at 33.5 m/s under Fault Scenario 1


0 5 10 15 20 25 30 35 40 45 50

−20

−10

0

10

δa1 (

deg) FTC nominal

0 5 10 15 20 25 30 35 40 45 504

5

6δ

a2 (

deg)

0 5 10 15 20 25 30 35 40 45 50

−20

−10

0

10

δe (

deg)

time (s)

0 0.5 1 1.5−2

0

2

Figure 3.36: Control Surface Deflections of the FTC and Nominal Controllers at 33.5 m/sunder Fault Scenario 1

3.4.2 Fault Scenario 2

The second fault scenario is when a stuck fault no greater than 3 deg happens in the elevator.

We first inject a stuck fault of 3 deg into the elevator at t = 0 s to see how a gain-scheduling

FTC controller designed for this fault scenario can handle it.

Closed-loop responses of the flexible aircraft with the FTC controller at 32 m/s are

shown in Figures 3.37 and 3.38, with comparison to the responses with a nominal LQR

controller designed at this airspeed. The control surface deflections of the two controllers

are compared in Figure 3.39. In this case, the nominal controller fails to settle down the

system states, which show slow decaying oscillations in responses, for the given time length

of 50 s. The FTC controller, on the other hand, achieves the design objectives in pitch

motion stabilization and wing vibration suppression without any remaining control surface

saturated for this speed.


0 5 10 15 20 25 30 35 40 45 50−1.5

−1

−0.5

0

0.5

tip d

ispla

cem

ent

(m)

FTC nominal

0 5 10 15 20 25 30 35 40 45 50−2

0

2

4

6

tip r

ota

tion (

deg)

time (s)

Figure 3.37: Closed-Loop Responses of Wing-Tip Bending and Torsion with the FTC andNominal Controllers at 32 m/s under Fault Scenario 2

0 5 10 15 20 25 30 35 40 45 50−4

−2

0

2

4

θ (

deg)

FTC nominal

0 5 10 15 20 25 30 35 40 45 50−2

−1

0

1

2

q (

deg/s

)

time (s)

Figure 3.38: Closed-Loop Responses of Pitch Angle and Rate with the FTC and NominalControllers at 32 m/s Under Fault Scenario 2

Similar to what has been done for the first fault scenario, the system with the FTC

controller is also simulated at 33.5 m/s to show the effectiveness of gain scheduling design.


0 5 10 15 20 25 30 35 40 45 50−20

−10

0

10

δa1 (

deg)

FTC nominal

0 5 10 15 20 25 30 35 40 45 50

0

2

4

δa2 (

deg)

0 5 10 15 20 25 30 35 40 45 502

3

4

δe (

deg)

time (s)

Figure 3.39: Control Surface Deflections of the FTC and Nominal Controllers at 32 m/sunder Fault Scenario 2

In Figures 3.40 and 3.41, the closed-loop responses are compared with the responses with

a nominal LQR contoller design at this airspeed. The control surface deflections of the two

controllers are shown in Figure 3.42. The closed-loop responses with the nominal controller

clearly show how a control surface fault can affect the rigid motion and cause unexpected

vibrations of the flexible wing through the coupling of rigid-body and flexible modes. In this

case, the stuck fault in elevator causes the flexible aircraft states to become uncontrolled by

the nominal controller. As expected, the FTC controller still works for this speed and shows

great superiority to the non-FTC and non-gain-scheduling nominal controller.

For this fault scenario, we also simulate how the system performs with FTC controller

switching in after fault occurrence. The closed-loop responses and control surface deflections

at 33.5 m/s are shown in Figs. 3.43–3.45 with comparison to the responses without FTC

controller switching in. The nominal LQR controller is used to stabilize the aircraft pitch

motion and wing vibrations before the elevator gets stuck at t = 0.5 s. After the switch on

of FTC controller, the pitch angle is effectively stabilized and the wing motions settle down

smoothly.


0 5 10 15 20 25 30 35 40 45 50−1.5

−1

−0.5

0

0.5

tip d

ispla

cem

ent

(m)

FTC nominal

0 5 10 15 20 25 30 35 40 45 50−2

0

2

4

6

tip r

ota

tion (

deg)

time (s)

Figure 3.40: Closed-Loop Responses of Wing-Tip Bending and Torsion with the FTC andNominal Controllers at 33.5 m/s under Fault Scenario 2

0 5 10 15 20 25 30 35 40 45 50−4

−2

0

2

4

θ (

deg)

FTC nominal

0 5 10 15 20 25 30 35 40 45 50−2

−1

0

1

2

q (

deg/s

)

time (s)

Figure 3.41: Closed-Loop Responses of Pitch Angle and Rate with the FTC and NominalControllers at 33.5 m/s under Fault Scenario 2


0 5 10 15 20 25 30 35 40 45 50−20

−10

0

10

δa1 (

deg)

FTC nominal

0 5 10 15 20 25 30 35 40 45 50

0

2

4

δa2 (

deg)

0 5 10 15 20 25 30 35 40 45 502

3

4

δe (

deg)

time (s)

Figure 3.42: Control Surface Deflections of the FTC and Nominal Controllers at 33.5 m/sunder Fault Scenario 2

0 5 10 15 20 25 30 35 40 45 50−0.3

−0.2

−0.1

0

0.1

tip d

ispla

cem

ent

(m)

FTC nominal

0 5 10 15 20 25 30 35 40 45 50

−2

0

2

tip r

ota

tion (

deg)

time (s)

Figure 3.43: Closed-Loop Responses of Wing-Tip Bending and Torsion under Fault Scenario2 with Switched on FTC Controller and Nominal Controller at 33.5 m/s


0 5 10 15 20 25 30 35 40 45 50−3

−2

−1

0

1

2

θ (

deg)

FTC nominal

0 5 10 15 20 25 30 35 40 45 50−3

−2

−1

0

1

2

q (

deg/s

)

time (s)

Figure 3.44: Closed-Loop Responses of Pitch Angle and Rate under Fault Scenario 2 withSwitched on FTC Controller and Nominal Controller at 33.5 m/s

0 10 20 30 40 50

−10

0

10

δa1 (

deg)

0 10 20 30 40 50−6

−4

−2

0

0 10 20 30 40 50

−2

0

2

4

δa2 (

deg)

0 10 20 30 40 50

−0.4

−0.2

0

0 2 4 6 8 101

1.5

2

δe (

deg)

time (s)0 2 4 6 8 10

1

1.5

2

time (s)

FTC nominal

Figure 3.45: Control Surface Deflections of Switched on FTC Controller and Nominal Con-troller at 33.5 m/s under Fault Scenario 2


3.5 Summary

In this chapter, the influence of different control surface faults on a flexible aircraft model

is first investigated and then a fault tolerant control (FTC) design that can gain-schedule

with the airspeed is developed for a flexible aircraft with actuator saturation and stuck con-

trol surface faults. The FTC design features a linear parameter-varying (LPV) proportional

and integral (PI) controller structure with control gains determined based on set-invariant

conditions. For flexible aircraft flying at any airspeed in a given airspeed range, the FTC

controller is able to eliminate the effects of stuck control surface faults on the system output

while minimizing the effects on other system states and guaranteeing no closed-loop per-

formance degradation caused by actuator saturation. In simulation case studies, the FTC

controllers based on the proposed design for given stuck fault scenarios are shown to be

effective in eliminating the influence of stuck faults on the longitudinal rigid-body motion

and suppressing potential vibrational motions of the flexible wing, as well as gain-scheduling

with airspeed without saturating the remaining control surfaces. The design developed in

this chapter provides a solution to the design of FTC controllers that handle prescribed stuck

fault scenarios for flexible aircraft.

Chapter 4

FTC Design for Gust Load

Alleviation Problem

With greater structural flexibility, flexible aircraft wings are more prone to large deforma-

tions. Besides the increasing tendency to undesirable aeroelastic effects due to the strong

interaction between rigid-body dynamics and flexible modes, the wing structures are more

sensitive to gust encounters. The sudden changes in aerodynamic forces caused by atmo-

spheric gusts can excite dynamic responses involving both the rigid-body and flexible modes,

which may reduce the ride quality, introduce extra structural loads and shorten the structural

fatigue life. This means that not only the rigid-body motions can be affected, unwanted wing

vibrations may also be induced. Therefore, for flexible aircraft, it is necessary to develop

an integrated flight control that performs structural load alleviation in the event of gusts in

addition to rigid-body motion control. Several methods, such as LQG [43], MPC [44] and

H∞ [47] have been used to design such control systems as reviewed in Section 1.2.2. These

designs are able to effectively alleviate gust loads while achieving rigid-body motion control.

On the other hand, faults associated with control surfaces, which execute control com-

mands in physical actions, can also influence the flexible aircraft in both rigid-body motions

and aeroelastic responses of the wings. In Haghighat et al. [44], where gust load alleviation

controllers are designed for a flexible aircraft, it has been shown that the loss of control

surface effectiveness can adversely affect the closed-loop performance of a traditional MPC

controller and an LQR controller, resulting in rigid-body tracking performance degradation

as well as high frequency oscillatory responses of stresses at both wing tip and roots. Since

control surface faults can give rise to undesirable aeroelastic responses of the wing structure

and make the integrated GLA flight controller less effective, FTC designs which aim at the

recovery or maintenance of both rigid-body motions and GLA performance are needed for

flexible aircraft.

70

Chapter 4. FTC Design for Gust Load Alleviation Problem 71

In this chapter, a mixed H2/H∞ FTC controller is developed for a flexible aircraft subject

to gust disturbances and loss of control effectiveness fault in control surfaces. The control

design uses a gain-scheduling approach that incorporates adaptively estimated control ef-

fectiveness factors to improve fault tolerance. The designed controller can simultaneously

achieve rigid-body motion stabilization, gust load alleviation on flexible wing structures and

on-line accommodation to loss of control effectiveness fault.

4.1 Faulty Flexible Aircraft Model with Gust

Extending the nonlinear equations of motion given in Equation (2.53), the state-space equa-

tions of the flexible aircraft subject to the gust disturbance are described byxr(t)xe(t)

︸︷︷︸

x(t)

= f(xr(t),xe(t))

xr(t)xe(t)

︸︷︷︸

x(t)

+Bu(t) +Gw(t) (4.1)

where xr =

[RTf θTf V T

f ωTf

]Tdenotes the rigid-body states, xe contains the remaining

states associated with the elastic states and the aerodynamic states, u =

[u1 u2 · · · um

]Tdenotes control surface deflection vector, w denotes the gust disturbance and G is its distri-

bution matrix.

Applying linearization to Equation (2.53) in a chosen steady-level flight condition, a

linear representation for flexible aircraft model under the gust disturbance is

δx(t) = Aδx(t) +Bδu(t) +Gw(t) (4.2)

where δx ∈ Rn and δu ∈ Rm denote the deviations of the states and control input from

their trimmed conditions respectively. For convenience, the δ before x and u will be omitted

without causing any confusion in what follows.

The actuator dynamics for control surfaces are assumed to be modeled by a first-order

transfer functionuiuci

=ai

s+ ai(4.3)

which can also be expressed by

ui = −aiui + aiuci (4.4)


where ui is the actual deflection of the i-th control surface, uci is its control input command

and ai > 0 is the reciprocal of time constant of the i-th actuator for i = 1, 2, · · ·m.

Then the loss of control effectiveness fault can be modeled by

ui = −aiui + aiρiuci (4.5)

where ρi ∈ [εi, 1] denotes the effectiveness factor of the i-th control surface: 1 means the

control surface is healthy, εi > 0 denotes the lower bound of its remaining effectiveness.

Assumption 4.1: Assume that the actuator dynamics are much faster than the aircraft

dynamics, i.e., for i = 1, 2, · · ·m, the parameter of actuator dynamics ai > 0 is far larger

than the aircraft damping.

With Assumption 4.1, we can let ui = 0 in the loss of control effectiveness fault model

(4.5) and take ui = ρiuci into Equation (4.2). The faulty linearized flexible aircraft model is

then expressed by

x(t) = Ax(t) +B(ρ)uc(t) +Gw(t) (4.6)

where uc =

[uc1 uc2 · · · ucm

]Tis the control input command vector, B(ρ) = Bρ and

ρ = diagρ1, · · · , ρm. We can see the linear dependence of the linearized faulty model (4.6)

on each diagonal element of ρ.

With the faulty system description (4.6), the fault tolerant GLA control problem is how

to achieve the regulation of rigid-body states xr and attenuate the structural loads caused

by the gust w on the flexible wing, with control surfaces corrupted by loss of effectiveness

fault. This problem will be addressed in the following section.

4.2 Fault Tolerant GLA Control Design

The fault tolerant GLA control design consists of two parts: a fault estimation mechanism

and a mixed H2/H∞ FTC controller. As shown in Figure 4.1, the fault estimation mechanism

is designed to adaptively estimate the control effectiveness factors ρ for all control surfaces.

The on-line estimated fault information will be incorporated by the fault tolerant mixed

H2/H∞ controller, which is designed to gain-schedule with the corresponding effectiveness

factors to simultaneously achieve rigid-body motion stabilization, gust load alleviation and

fault accommodation.


Actuator Model


Fault Estimation

Mixed Control Gain

Figure 4.1: Structure diagram of fault tolerant GLA Design

4.2.1 Fault Estimation

An adaptive observer is designed for each control surface to estimate its effectiveness factor,

given in the following form

˙ui = −aiui + aiρiuci − li(ui − ui) (4.7)

where li is a positive scalar and ρi is the estimate of the effectiveness factor ρi for the i-th

control surface, i = 1, · · · ,m.

The adaptive law for ρi is given by

˙ρi = Proj[εi,1]Λi

=

0, if ρi = 1 and Λi ≥ 0

or ρi = εi and Λi ≤ 0

Λi, otherwise

(4.8)

where Λi = −kiaiuiuci, ui = ui − ui and ki > 0 is the adaptive gain that can be tuned by

designers.

The error equation of (4.5) and (4.7) is

˙ui = (−ai − li)ui + aiρiuci (4.9)

where ρi = ρi − ρi.


Choose the following Lyapunov function

Vi =1

2u2i +

1

2kiρ2i (4.10)

Then the derivative of Vi with respect to time is

Vi = ui ˙ui +1

kiρi ˙ρi

= (−ai − li)u2i + aiρiuiuci +

1

kiρi ˙ρi

(4.11)

with the adaptive law (4.8),

Vi =

(−ai − li)u2

i + aiρiuiuci, if ρi = 1 and Λi ≥ 0

or ρi = εi and Λi ≤ 0

(−ai − li)u2i , otherwise

=

(−ai − li)u2

i − Λiki

(1− ρi), if ρi = 1 and Λi ≥ 0

(−ai − li)u2i − Λi

ki(εi − ρi), if ρi = εi and Λi ≤ 0

(−ai − li)u2i , otherwise

≤ (−ai − li)u2i ≤ 0.

(4.12)

Since Vi is a decreasing function of time, ui and ρi are bounded. It is also reasonable to

assume the boundedness of the control command uci and its rate uci. Then it is a standard

technique by using Barbalat’s Lemma to show that limt→∞ ui(t) = 0. From (4.9) we can

see that limt→∞ ˙ui(t) = limt→∞ aiρi(t)uci(t) = 0. If we assume limt→∞ uci(t) 6= 0, we can

conclude that limt→∞ ρi(t) = 0.

4.2.2 Fault Tolerant Mixed H2/H∞ Controller Design

As mentioned in Section 4.1, the fault tolerant GLA controller needs to achieve the following

three objectives: rigid-body motion regulation, gust load alleviation and fault accommoda-

tion. For the first two objectives, they can be integrated by solving a multiobjective control

problem with mixed H2/H∞ specifications. Then a gain-scheduling technique will be in-

corporated in the synthesis of the mixed H2/H∞ controller to accommodate loss of control

effectiveness fault.


For the purpose of designing a controller that integrates both rigid-body motion reg-

ulation and gust load alleviation, two performance outputs of the flexible aircraft system

are defined: the first performance output is denoted by zrigid, which is related to the rigid-

body motion states that we want to regulate and control surface deflections; the second

performance output is denoted by zflexible, which represents the structural loads on the flex-

ible wing. Then two norms will be used to characterize the performance requirements in

regulating rigid-body states and alleviating gust loads respectively.

In order to achieve good temporal performance in rigid-body motion regulation, the

following quadratic cost function is chosen for minimization

J =

∫ ∞0

(xTrz(t)Qxrz(t) + uT

c (t)Ruc(t)) dt = ‖zrigid‖22 (4.13)

where xrz(t) = Crzx(t) denotes the rigid-body states to be regulated, Q is the positive

definite weighting matrix for xrz(t) and R is the weighting matrix for uc(t).

The performance output zrigid(t) can thus be expressed by

zrigid(t) = Cz1x(t) +Dz1uc(t) (4.14)

where Cz1 = [(CTrzQCrz)

12 0]T and Dz1 = [0 R

12 ]T. The cost function (4.13) can be

minimized through minimizing the H2 norm ‖Tzrigidw‖2, where Tzrigidw is the transfer function

from w to zrigid. The rigid-body motion regulation becomes a classical H2 control problem.

For the second performance output zflexible, we choose the wing root bending moment as

an indicator of the structural loads that need to be alleviated on the flexible wing, which is

defined as

zflexible(t) = Cz2x(t) (4.15)

Since the gust disturbance is a finite-energy signal, the influence from gust w(t) to the

output zflexible(t) can be characterized by the H∞ norm ‖Tzflexiblew‖∞, where Tzflexiblew is the

transfer function from w to zflexible. The H∞ norm represents the maximum gain between

the L2 norms of zflexible and w. Then gust load alleviation can be done through minimizing

‖Tzflexiblew‖∞.

Re-arrange Equations (4.6), (4.14) and (4.15), we have

x(t) = Ax(t) +B(ρ)uc(t) +Gw(t)

zrigid(t) = Cz1x(t) +Dz1uc(t)

zflexible(t) = Cz2x(t)

(4.16)


To satisfy both the specifications for rigid-body motion regulation and structural load al-

leviation, a mixedH2/H∞ controller can be designed for system (4.16) to minimize µ‖Tzrigidw‖2+

‖Tzflexiblew‖∞, where µ is a weight parameter. The controller synthesis can be cast into the

following convex optimization problem

minY >0,M(ρ)

γ1 + µγ2

s.t. (a)

(AY +BM(ρ)) + (AY +BM(ρ))T G (Cz2Y )T

∗ −γ1I 0

∗ ∗ −γ1I

< 0

(b)

Y (Cz1Y +Dz1M(ρ))T

∗ W

> 0

(c)Trace(W ) < γ2

(4.17)

where ∗ represents the symmetric structure of the matrices in linear matrix inequalities

(LMIs).

In (4.17), LMI (a) and Y > 0 guarantee that ‖Tzflexiblew‖∞ ≤ γ1. And LMIs (a)-(c) are

sufficient to have ‖Tzrigidw‖2 ≤ γ2. The state-feedback controller gain K can then be obtained

as

K(ρ) = M(ρ)Y −1 (4.18)

Note that system (4.16) is dependent on the effectiveness factors ρ = diagρ1, · · · , ρm,it would be inconvenient to solve the optimization problem (4.17) each time we obtain an

on-line estimate of ρ. And the stability and performance cannot be guaranteed by doing

this when ρ changes in time. To synthesize a controller that meets the mixed H2/H∞

specifications and incorporates the effectiveness factors at the same time, the technique used

in LPV control design is adopted here.

First we predefine several fault modes P1, · · · ,PN , which correspond to the possible fault

scenarios that can happen in different combinations of control surfaces. The fault modes are

defined by Pl = ρ : εj ≤ ρj(t) ≤ 1, j ∈ Fl where Fl = n1l , · · · , n

mll is a prescribed set of

the location number corresponds to each faulty control surface, and ml is the total number

of faulty control surfaces, l = 1, · · · , N .

To design a controller for faults that belong to a given fault mode Pl, we express the


controller gain K in the following structure by using the basis function method [115]

K = K0 +

ml∑i=1

ρnilKi (4.19)

where the basis functions are chosen as 1, ρn1l, · · · , ρnmll .

Accordingly the matrix M which is solved for in the optimization problem (3.12) is

expressed by

M = M0 +

ml∑i=1

ρnilMi (4.20)

Then the LMIs given in (4.17) can be solved simultaneously at chosen grid points over

[εnil , 1], i = 1, · · · ,ml to get a set of solution for Y and M0,M1, · · · ,Mml . The control gain

matrices are obtained as K0 = M0Y−1, Ki = MiY

−1. The design method is analogous to

the design of gain scheduling controllers for linear parameter-varying systems.

Once we get the control gain matrices for a given fault mode, a fault tolerant mixed

H2/H∞ controller that “gain schedules” with the on-line estimates of effectiveness factors is

given by

u(t) = (K0 +

ml∑i=1

ρnil(t)Ki)x(t) (4.21)


In this section, the designed controller is applied to the flexible aircraft model that is loosely

based on the one from [103]. Properties of this aircraft have been given in Table 3.1 and

more descriptions can be found in Section 3.1.

10m

16m

0.5m

Flap 2 Flap 1 Flap 2Flap 1

Elevator

Figure 4.2: HALE Aircraft Model Geometry Top View(not to scale)


The simulation study will be focused on the longitudinal altitude control and gust load

alleviation for the flexible aircraft in steady-level flight. Therefore, the rigid-body states

consist of the pitch angle θ, the pitch rate q, the horizontal velocity u, the vertical velocity

w and the altitude h. Control surfaces are placed as shown in Figure 4.2. The bilaterally

symmetric flaps are assumed to be deflected equally. Flap 1 indicates the two symmetric

inner flaps and Flap 2 indicates the two outer ones. The deflections of Flap 1, Flap 2 and

the elevator are denoted by δa1, δa2 and δe respectively.

4.3.1 Gust Profile

There are two common methods for modeling the gust disturbance: discrete method, where

the gust is usually modeled using a “1- cosine” profile; and the continuous method, where

the gust is modeled as a stationary Gaussian stochastic process with a known power spectral

density (PSD). In this simulation study, a discrete profile and a continuous gust profile will

be used to evaluate the effectiveness of our designed controller.

The “1- cosine” discrete gust model is given in the following form:

wg =wg2

(1− cos2πxgLg

) (4.22)

where wg is the vertical gust velocity, wg is the gust amplitude, Lg is the gust length and

0 ≤ xg ≤ Lg is the aircraft position in the spatial description of the gust relative to a fixed

origin.

For continuous gust modeling, the Dryden model is used with the PSD for vertical gusts

given by [117]

Φdw(ω) = σ2 L

πV(

1 + 3L2ω2

V 2

(1 + L2ω2

V 2 )2) (4.23)

where ω denotes the angular frequency, the σ is the turbulence intensity, L is the turbulence

scale length, and V is the aircraft speed. In order to generate a time history of the continuous

gust, a coloring filter is used. By passing a white noise through the filter, an output signal

that has the desired PSD can be obtained. For a Dryden vertical gust disturbance, the

transfer function of the filter is given by [117]

Hdw(s) = σ

√L

πV

1 +√

3LVs

(1 + LVs)2

(4.24)

Both gust profiles are created for the flexible aircraft flying at V = 25 m/s at the altitude

of 20 km. And only vertical gust velocity is used.


For discrete gust, a “1- cosine” gust profile is used in the simulation as shown in Figure

4.3.

0 2 4 6 8 10time (s)

-0.5

0

0.5

1

1.5

2

2.5

wg (

m/s

)

Figure 4.3: A Discrete Gust Profile

For continuous gust, a Dryden gust profile is generated from the Aerospace Blockset of

Simulink. The turbulence intensity is chosen as severe with exceedance probability of 10−5.

The time history of the gust profile is shown in Figure 4.4, which lasts from t = 0 s to t = 10

s.

0 5 10 15 20 25time (s)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

wg (

m/s

)

Figure 4.4: A Dryden Gust Profile


4.3.2 Simulation Examples

The objective of the designed controller is to maintain the altitude of the aircraft and effec-

tively alleviate the structural loads on the flexible wing in the event of gust disturbances.

We will test the effectiveness of our designed controller for a given fault mode: Flap 1 and

Flap 2 work normally or lose their control effectiveness and the elevator works normally,

described by ε1 ≤ ρ1 ≤ 1, ε2 ≤ ρ2 ≤ 1 and ρ3 = 1, where ε1 and ε2 are set to be 0.3.

First, we show the results of how the designed controller works in nominal case, i.e. all the

control surfaces are working normally. With the “1- cosine” gust excitation, the responses of

altitude deviation and extra wing root bending moment caused by the gust with the designed

controller are shown in Figure 4.5 and Figure 4.6 respectively, comparing with the open-loop

response and the responses with an LQR controller that regulates the altitude. The control

surface deflections of the designed controller and the LQR controller are shown in Figure

4.7.

0 2 4 6 8 10time (s)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

altit

ude

(m)

Designed controllerOpen-loopLQR

Figure 4.5: Altitude Responses in Nominal Case with “1- cosine” Gust Excitation

With the Dryden gust excitation, the responses of altitude deviation and extra wing root

bending moment caused by the gust with the designed controller are shown in Figure 4.8

and Figure 4.9 respectively, comparing with the open-loop response and the responses with

an LQR controller that regulates the altitude. Control surface deflections of the designed

controller and the LQR controller are shown in Figure 4.10.


0 2 4 6 8 10time (s)

-250

-200

-150

-100

-50

0

50

100

150

200

root

ben

ding

mom

ent (

Nm

)


Figure 4.6: Wing Root Bending Moment Responses in Nominal Case with “1- cosine” GustExcitation

0 2 4 6 8 10

-30-20-10

010

/a1

(de

g)

Designed controllerLQR

0 2 4 6 8 10

-10

0

10

/a2

(de

g)

0 2 4 6 8 10time (s)

-10

0

10

/e (

deg)

Figure 4.7: Control Surface Deflections in Nominal Case with “1- cosine” Gust Excitation


0 5 10 15 20 25time (s)

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

altit

ude

(m)


Figure 4.8: Altitude Responses in Nominal Case with Dryden Gust Excitation

0 5 10 15 20 25time (s)

-150

-100

-50

0

50

100

150

200

root

ben

ding

mom

ent (

Nm

)


Figure 4.9: Wing Root Bending Moment Responses in Nominal Case with Dryden GustExcitation


0 5 10 15 20 25-20

0

20

40

/a

1

(de

g) Designed controllerLQR

0 5 10 15 20 25-20

0

20/

a2

(de

g)

0 5 10 15 20 25time (s)

-20

0

20

/e (

deg)

Figure 4.10: Control Surface Deflections in Nominal Case with Dryden Gust Excitation

From the results shown in these figures, for both types of gusts, the designed controller

can effectively regulate the altitude and reduce the extra wing root bending moment caused

by the gust disturbance. In contrast to our designed controller, although the LQR controller

effectively regulates the altitude, neglecting the need of gust load alleviation results in com-

parable wing root bending moment level to the open-loop case, which is not desirable for

flexible aircraft.

Second, we show the results of how the designed controller accommodates a loss of ef-

fectiveness fault that belongs to the given fault mode. The following fault is injected to the

system:

ρ1(t) = ρ2(t) = 0.5, t ≥ 0 s (4.25)

For “1- cosine” gust excitation case, the estimates of two effectiveness factors are shown

in Figure 4.11. It can be seen that our fault estimation scheme is able to estimate the

effectiveness factors accurately and in a timely manner. The responses of altitude deviation

and extra wing root bending moment caused by the gust with the designed controller are

shown in Figure 4.12 and Figure 4.13 respectively, comparing with the responses with a

fixed-gain H2/H∞ controller designed for the nominal case. Control surface deflections of

the designed controller and the fixed-gain controller are shown in Figure 4.14.

For Dryden gust excitation case, the estimates of two effectiveness factors are shown


0 2 4 6 8 10

0.5

0.6

0.7

0.8

0.9;1

0 2 4 6 8 10time (s)

0.5

0.6

0.7

0.8

0.9;2

Figure 4.11: Estimates of Effectiveness Factors with “1-cosine” Gust Excitation

0 2 4 6 8 10time (s)

-0.05

0

0.05

0.1

0.15

0.2

altit

ude

(m)

Designed controllerFixed-gain controller

Figure 4.12: Altitude Responses in Faulty Case with “1- cosine” Gust Excitation


0 2 4 6 8 10time (s)

-60

-50

-40

-30

-20

-10

0

10

20

root

ben

ding

mom

ent (

Nm

)


Figure 4.13: Wing Root Bending Moment Responses in Faulty Case with “1- cosine” GustExcitation

0 1 2 3 4 5 6 7 8 9 10−20

0

20

40

δ a1 (

deg)

0 1 2 3 4 5 6 7 8 9 10

−10

0

10

δ a2 (

deg)

0 1 2 3 4 5 6 7 8 9 10

−10

0

10

20

δ e (de

g)

time (s)

Designed controllerFixed−gain controller

Figure 4.14: Control Surface Deflections in Faulty Case with “1- cosine” Gust Excitation


0 2 4 6 8 100.4

0.5

0.6

0.7

0.8

0.9

11;1

0 2 4 6 8 10time (s)

0.4

0.5

0.6

0.7

0.8

0.9

11;2

Figure 4.15: Estimates of Effectiveness Factors with Dryden Gust Excitation

0 5 10 15 20 25time (s)

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

altit

ude

(m)


Figure 4.16: Altitude Responses in Faulty Case with Dryden Gust Excitation


0 5 10 15 20 25time (s)

-50

-40

-30

-20

-10

0

10

20

30

40

50

root

ben

ding

mom

ent (

Nm

)


Figure 4.17: Wing Root Bending Moment Responses in Faulty Case with Dryden GustExcitation

0 5 10 15 20 25

0

20

40

/a

1

(de

g) Designed controllerFixed-gain controller

0 5 10 15 20 25-10

0

10

/a

2

(de

g)

0 5 10 15 20 25time (s)

-20

-10

0

10

/e (

deg)

Figure 4.18: Control Surface Deflections in Faulty Case with Dryden Gust Excitation


in Figure 4.15. Similar to the “1-cosine” gust case, our fault estimation scheme is able

to estimate the effectiveness factors accurately and in a timely manner. The responses

of altitude deviation and extra wing root bending moment caused by the gust with the

designed controller are shown in Figure 4.16 and Figure 4.17 respectively, comparing with

the responses with a fixed-gain H2/H∞ controller designed for the nominal case. Control

surface deflections of the designed controller and the fixed-gain controller are shown in Figure

4.18.

For both types of gusts, it can be seen from above figures that the designed controller can

still effectively regulate the altitude and reduce the wing root bending moment in the faulty

case. It also shows better performance than the fixed-gain controller in the transient response

for regulating altitude and bending moment reduction when both of them are applied in the

same faulty case.

4.4 Summary

In this chapter, a gain-scheduled mixed H2/H∞ controller is designed for a flexible aircraft

subject to loss of control effectiveness fault and gust disturbance. The flexible aircraft model

that describes the aircraft rigid-body motions, the elastic deformations of the flexible wing

and the coupling between rigid-body and flexible modes is presented. The control design

features an adaptive fault estimator and a gain-scheduled mixed H2/H∞ controller which

is able to regulate the rigid-body motion of the aircraft, alleviate gust loads on the flexible

wing as well as accommodate the loss of effectiveness faults that occur in control surfaces.

In the simulation case studies, the designed controller demonstrates its effectiveness in alti-

tude regulation, wing root bending moment reduction and fault accommodation under both

discrete and continuous gusts. Future work will consider incorporating actuator saturation

constraints in the controller design and expanding the results to multiple flight conditions.

Chapter 5

FTC Design for Maneuver Load

Alleviation Problem

In addition to wind gusts, aircraft maneuvers can also excite dynamic responses involving

the aeroelastic modes and cause extra structural loads on the flexible wing structures. Like

the requirement of performing GLA in the event of gusts, it is also important to reduce the

structural loads during a maneuver. For flexible aircraft, a unified flight and load controller

that alleviates the maneuver loads while keeping the flight behavior unmodified is desired.

MLA has been taken into account as a part of multi-objective optimal flight control design

for high aspect ratio flexible aircraft in [35]. The wing root bending moment is reduced to

conform to the required structural load limit during a pull-up maneuver, but the tracking of

pitch rate command is sacrificed. In [55], a design based on two recurrent neural networks

(RNNs) is able to alleviate the wing root bending moment for an aeroelastic fighter model

without loss of the maneuvering performance. However, choices of the neural networks’

initial weights affect the results and require a lot of trial and error tests. Instead of directly

minimizing a cost function that has a term associated with the wing root bending moment,

another idea to handle MLA problem is to characterize the conformity to structural load

limit, i.e. the structural loads during a maneuver do not violate certain upper and lower

bounds, as an output constraint. In [57], an output saturation mechanism based on output

input saturation transform (OIST) technique [58] is applied to shape the wing root bending

moment response and keep it within an load limit interval for a longitudinal maneuver of a

flexible aircraft. Although it achieves both tracking and load alleviation targets, the OIST

method itself has limitations that the closed-loop stability can only be proved for single-

input single-output minimum phase systems [118]. By regarding the bending moment at the

external wing as a constrained output, MPC has been utilized in [56] to alleviate the loads

during a sudden and strong roll maneuver for a flexible transport aircraft. As a method

89

Chapter 5. FTC Design for Maneuver Load Alleviation Problem 90

that is well-known for its capability of handling input and output constraints, MPC shows

promise in handling MLA design.

While various MLA designs have been developed for flexible aircraft, control surface

faults and fault tolerant MLA controllers have not been considered. There are at least

two reasons why we need to develop FTC designs for MLA problem. First, control surface

faults, which can affect both rigid-body motions and aeroelastic responses of the wings,

would degrade the performance of tracking and load alleviation. Second, when control

surface faults occur, tracking the original reference command may become infeasible for the

faulty aircraft system subject to the input constraint and limitations of structural loads. If

an unachievable reference is given to the system, the state trajectories will evolve to violate

these constraints, causing actuator saturation, or excessive structural loads or both and

even catastrophic instability of the aircraft. Thus, for the MLA problem of flexible aircraft,

it is necessary to take into consideration the influence of faults in both tracking and load

reduction performance as well as the feasibility of a given reference command. FTC designs

should be developed to recover the performance of load alleviation, and achieve tracking of

an admissible reference command such that the safety of post-fault flexible aircraft can be

guaranteed.

In this chapter, a fault-tolerant MPC formulation with reference adjustment is presented

for the MLA problem of a flexible aircraft. Upper and lower bounds are set for the struc-

tural loads during a maneuver such that the load alleviation objective becomes an output

constraint. In the cases of stuck fault and loss of effectiveness fault, the MPC design can

steer the system to track any admissible reference with respect to each fault case and keep

the structural loads within the given bounds. If a reference command is not admissible, it

will be adjusted to an admissible command as close to the original one as possible.

5.1 MLA Problem Description

Recall the linearized state-space equations of the flexible aircraft given in Equation (3.1).

By discretizing it with a sampling time Ts, a discrete-time model is obtained as

x(k + 1) = Adx(k) +Bdu(k) (5.1)

where x(k) ∈ Rn is the state vector of the discrete-time model that corresponds to the

continuous time state variable x(t) from (3.1) for t = kTs , u(k) ∈ Rm is the corresponding

control input vector for the discrete-time model, and (Ad, Bd) is stabilizable.

As mentioned earlier, the MLA control design can be treated as a tracking control prob-


lem with output constraints. To be more specific, the goal of MLA can be achieved by making

some flight dynamics outputs track given reference trajectories while keeping a certain struc-

tural load output within an interval of amplitude constraints. Therefore, we separate the

system outputs into two categories: one is associated with the tracking flight dynamic states

and the other represents constrained outputs of the aircraft.

yt(k) = Ctx(k)

yc(k) = Ccx(k)(5.2)

where yt(k) ∈ Rp, p ≤ m, denotes the tracking output vector, and yc(k) ∈ R2 denotes the

constrained output vector, which is chosen to be the wing root bending moment Mx = Crbmx

and the forward airspeed u = Cux in the following MLA design.

To achieve the load reduction, upper and lower bounds denoted by Mxmax and Mxmin

respectively are imposed on Mx. As for u, since the variation of airspeed may change

the aeroelastic dynamics behavior, we want to keep the maneuver of aircraft around the

trimmed airspeed. Thus, the forward airspeed u, which is the dominant airspeed part, is

also included in the constrained output and confined by |u − utrim| ≤ δu, δu > 0. Let

yc,max = [Mxmax utrim + δu] and yc,min = [Mxminutrim − δu], then the constraints on the

output yc can be described by

yc ∈ Zy = yc ∈ R2 : Aycyc ≤ byc (5.3)

where Ayc = diag([I2 − I2]) and byc = [yc,max − yc,min]T.

Similarly, assume the upper and lower bounds for control input vector u are umax ∈ Rm

and umin ∈ Rm respectively, then the input constraints can be described by

u ∈ Zu = u ∈ Rm : Auu ≤ bu (5.4)

where Au = diag([Im − Im]) and bu = [uTmax − uT

min]T.

To simplify the notation, we can put the two constraints together and rewrite them in a

unified linear inequality form with respect to z = [xT uT]T:

z ∈ Z = z ∈ Rn+m : Azz ≤ bz (5.5)

The objective of a nominal MLA design is to find a control law u(k) for the system model

(5.1) such that for a given reference command yr, the tracking output yt(k) is steered to yr

as close as possible while satisfying the constraints (5.5).


5.2 Feasible Reference-Tracking Model Predictive Con-

trol (MPC)

In this section, a formulation of MPC is presented for the nominal flexible aircraft system to

track any admissible reference command in an admissible way and automatically steer the

system to the closest admissible steady state if the given command is not admissible. Based

on the approach proposed by Limon et al.[119], a parameter vector that characterizes an

artificial steady state and input is added as a decision variable of the quadratic programming

(QP) problem for MPC. By doing this, an artificial reference that ensures the feasibility of

MPC is calculated. Given an admissible reference command, the system can asymptotically

track it and the artificial reference will be the same as the desired one. If an inadmissible

one is given, the system state will be driven to track the artificial reference that is closest to

the desired one by the MPC controller.

5.2.1 Admissible Invariant Set for Tracking

Assume that the output yt can asymptotically track a reference rss, then the steady state

xss and input uss with respect to the given reference must satisfy the following equation:

Ad − In Bd 0n×1

Ct 0p×m −Ip

︸︷︷︸

Ass

xss

uss

rss

=

0n×1

0p×1

(5.6)

A non-trivia solution for Equation (5.6) can be parametrized by

xss = Mη1η (5.7)

uss = Mη2η (5.8)

rss = Nηη (5.9)

where η ∈ Rnη is a basis vector for the nullspace of Ass, which can be used to characterize

any solution with the minimal dimensions, and Mη1, Mη2 and Nη are appropriate matrices.

The steady state and input should satisfy the constraints (5.5):[MT

η1 MTη2

]T

η = Mηη ∈ Z (5.10)


In the next step, we will find the maximal admissible invariant set for tracking, O∞,

which is the set of all possible initial states and η such that the closed-loop system with the

initial states can be admissibly steered to the steady states xss and inputs uss characterized

by η via the following control law:

u = K(x− xss) + uss = Kx+ Lη (5.11)

where K is a stabilizing control gain such that Ad +BdK is Hurwitz and L = [−K Im]Mη.

Define w =

[xT ηT

]T

, we can write down the following extended dynamic model:

w(k + 1) =

A+BK BL

0 Inη

︸︷︷︸

Aw

w(k) (5.12)

We can rewrite the constraints (5.5) and (5.10) and impose them on w:

w ∈ W = w ∈ Rn+nη : F1w ∈ Z, F2w ∈ Z (5.13)

where F1 =

In 0n×nη

K L

and F2 =

[0nη×n Mη

].

The maximal admissible invariant set for tracking can be expressed by

O∞ = w ∈ Rn+nη : Aiww ∈ W,∀i ≥ 0 (5.14)

This means that if w(k) ∈ O∞, then w(k + i) will satisfy the constraint w(k + i) ∈ W for

all i > 0. This invariant set can be constructed based on the approaches given in Gilbert

and Tan [120]. However, it might not be finitely determined since Aw has eigenvalues on the

unit circle. As suggested in [120], the following set

Oε∞ = w ∈ Rn+nη : Aiww ∈ W ε,∀i ≥ 0 (5.15)

where W ε = w ∈ Rn+nη : F1w ∈ Z, F2w ∈ εZ, is a finitely determined convex polyhedron

for any ε ∈ (0, 1) and can be used to approximate O∞ by choosing ε arbitrarily close to 1.


5.2.2 MPC Design

With admissible invariant set for tracking determined, an MPC design with η as an additional

decision variable to control input sequence U over Nu horizon is presented.

For a given reference command yr, the cost function is defined as follows:

V (x(k),U,η,yr, N) =N−1∑i=1

||x(k + i)−Mη1η||2Q +N−1∑i=0

||u(k + i)−Mη2η||2R

+||x(k +N)−Mη1η||2P + ||yr −Nηη||2T

(5.16)

where the control horizon and the prediction horizon are both given by N , Q ∈ Rn×n,

R ∈ Rm×m, T ∈ Rp×p and P ∈ Rn×n are positive definite matrices. The terminal weighting

matrix P satisfies the following condition [121]:

P = Q+KTRK + (Ad +BdK)TP (Ad +BdK) (5.17)

The cost function penalizes the deviation of state and control input to the artificial steady

state and input, as well as the difference between the artificial reference and the given ref-

erence command. It represents the tracking objective of the MLA problem. With the con-

straints (5.3) and (5.4) on aircraft output and control input, the following MPC optimization

problem is proposed:

minU,η

V (x(k),U,η,yr, N)

s.t. (a) x(j + 1) = Adx(j) +Bdu(j) ∀j = k, k + 1, . . . , k +N − 1

(b) u(j) ∈ Zu ∀j = k, k + 1, . . . , k +N − 1

(c) Ccx(j) ∈ Zy ∀j = k + 1, . . . , k +N − 1

(d)

[xT(k +N),ηT

]T

∈ Oε∞

(5.18)

where constraint (a) correspond to the dynamic model of flexible aircraft, also known as the

internal model of MPC formulation; constraint (b) represents the control input constraints;

(c) includes the output constraints on structural loads and airspeed during a maneuver; (d)

is the terminal constraint set condition which ensures that the MPC controller steers the

state to the admissible invariant set for tracking with N steps.

The above MPC optimization problem can then be formulated and solved as a constrained

QP problem. Denote X as the predicted state sequence, which contains the state vectors


x(j) at all sampling points j = k, k + 1, . . . , k +N − 1, it can be expressed as a function of

the future input sequence U and the current state:

X = HU +Gx(k), (5.19)

where the current state x(k) is assumed to be available for feedback in this work, and the

matrices H and G are given by

H =

Bd

AdBd Bd

......

. . .

AN−1d Bd AN−2

d Bd · · · Bd

, G =

Ad

A2d

...

ANd

. (5.20)

With the compact formulation for future state prediction, we can rewrite the cost function

(5.16) and discard constant terms, yielding

Vqp =

[UT ηT

]H

U

η

+ aT

U

η

(5.21)

where H =

HTQH + R −HTQM1η − RM2η

−MT1ηQH − MT

2ηR NTη TNη

, aT =

[2xT(k)GTQH −2yT

r TNη

],

in which Q and R are block diagonal matrices given by

Q = diag

[Q Q · · · P

],

R = diag

[R R · · · R

],

(5.22)

and M1η =

M1η

...

M1η

, M2η =

M2η

...

M2η

.


Expand the input constraint (5.4) to the control horizon N ,

Umin =

umin

...

umin

≤ IN ·mU ≤

umax

...

umax

= Umax (5.23)

and the output constraint (5.3)

Yc,min =

yc,min

...

yc,min

≤ HcU +Gcx(k) ≤

yc,max

...

yc,max

= Yc,max (5.24)

where Hc and Gc are given by

Hc =

CcBd

CcAdBd CcBd

......

. . .

CcAN−1d Bd CcA

N−2d Bd · · · CcBd

, Gc =

CcAd

CcA2d

...

CcANd

. (5.25)

Assume the convex polyhedron Oε∞ can be expressed by the inequality A∞w ≤ b∞, and

let the state x(k +N) = hNU + gNx(k), where hN and gN can be obtained from (5.20), we

can rewrite the constraint (d) with regards to the decision variables as follows:

HN

U

η

≤ KN (5.26)

where HN = A∞

hN 0n×nη

0nη×n Inη

and KN = b∞ −

A∞gNx(k)

0nη×1

.


The constraints (b), (c) and (d) can also be expressed in a compact form as

F

U

η

≤ c, (5.27)

where the matrix F and the vector c are formed as follows,

F =

IN ·m 0N ·m×nη

−IN ·m 0N ·m×nη

Hc 02N×nη

−Hc 02N×nη

HN

, c =

Umax

Umin

Yc,max −Gcx(k)

−Yc,min +Gcx(k)

KN

. (5.28)

Using the compact formulation for the cost function and constraints, the MPC optimiza-

tion problem is re-arranged as the following QP problem

minU,η

Vqp =

[UT ηT

]H

U

η

+ aT

U

η

s.t. F

U

η

≤ c. (5.29)

If the matrix H is positive definite, the optimization problem becomes a convex QP problem,

which can be solved efficiently by specialized algorithms and has unique solution for the

future input sequence U. In the simulations described in Section 5.4, H is positive definite.

5.3 Fault Tolerant MPC for MLA

Two types of faults are considered in the FTC design. The first type is stuck fault. If the

i-th control surface get stuck at the position usi , where i = 1, 2, · · · ,m, this stuck fault can

be represented by changing the constraint on input ui to usi ≤ ui(k) ≤ usi . Then the flexible

aircraft system subject to stuck faults can be described by the dynamic model (5.1) with the


corresponding input constraints:

u ∈ Zfsu = u ∈ Rm : Auu ≤ bfs (5.30)

where fs denotes a set of stuck control surfaces, and bfs is obtained by changing the corre-

sponding upper and lower bound values in bu.

The second type is the loss of effectiveness faults. If loss of effectiveness faults occur in

the control surfaces, it can be represented by the following dynamic model:

x(k + 1) = Adx(k) +Bd,fu(k) (5.31)

where Bd,f = Bdρ, ρ = diagρ1, · · · , ρm and ρi ∈ [εi, 1] denotes the effectiveness factor of

the i-th control surface: 1 means the control surface is healthy, 0 < εi < 1 denotes the lower

bound of its remaining effectiveness.

From the above description, we can see that the two types of faults can be represented by

changing of the input constraints and internal model respectively. And these changes can be

reconfigured systematically in an MPC framework. For stuck faults, MPC can automatically

redistribute the control efforts among remaining healthy control surfaces to accommodate

them by replacing the nominal input constraints (5.4) with the faulty ones (5.30). And for

loss of effectiveness faults, they can be handled in a natural fashion by updating the internal

dynamic model provided the faulty model is known via an FDI scheme.

Then the fault tolerant MPC design can summarized as follows:

1. Given all prescribed fault conditions, for each one, compute invariant set Oε∞,f of

augmented system (5.12) with the corresponding input constraints (5.30) if it is a

stuck fault or internal model (5.31) if it is a loss of effectiveness fault.

2. Assuming that the FDI information is available, find the precomputed Oε∞,f that cor-

responds to the fault condition. In the MPC optimization (5.18), update condition (b)

for input constraints if it is a stuck fault or condition (a) for internal model if it is a

loss of effectiveness fault as well as condition (d) to replace the terminal set with Oε∞,f .



In this section, we still use the flexible aircraft model that is loosely based on the one from

[103] as a testbed for the fault tolerant MPC design. Properties of this aircraft have been

given in Table 3.1 and more descriptions can be found in Section 3.1. The placement of

control surfaces and notations for Flap 1, Flap 2 and elevator are the same as described in

Section 4.3.

In the simulations, the structural load alleviation during a longitudinal pitch tracking

maneuver is studied. The pitch angle θ is chosen as the tracking output and the wing root

bending moment Mx represents the structural load. The upper and lower bounds for Mx

are set to be 150 Nm and −150 Nm respectively. The aircraft model is linearized at 25 m/s,

and the deviation of the forward airspeed u from its trimmed value is constrained by ±2

m/s. The deflections of Flap 1, Flap 2 and elevator, denoted by δa1, δa2 and δe respectively,

are set to be limited by ±25. All the state responses shown below are the differences with

respect to the nominal trimmed conditions.

The weighting matrices for MPC have been chosen as Q = 100 ∗ CTt Ct and R = I3. The

stabilizing control gain K is the corresponding LQR gain. The terminal weighting matrix

P is determined from the Riccati equation associated with the Q, R and K. The matrix T

is chosen as 1000. And the control horizon is chosen to be N = 5. The admissible invariant

set for tracking will be computed with ε = 0.99.

5.4.1 Nominal Case

First, we show the results of how the MPC controller works in the nominal case, i.e. all

the control surfaces are working normally. The pitch angle is required to increase to 4 and

then decrease to −1. The evolutions of the artificial reference and the pitch angle with

and without MLA constraint are shown in Figure 5.1 with comparison to the given refer-

ence command. The original 4 command is not admissible and has been adjusted to the

maximum achievable angle 3.4 in the artificial reference. The −1 command is admissible,

so no adjustment is made in the artificial reference and the pitch angle is able to track it

asymptotically. The pitch response when the constraint for Mx is removed from the MPC

formulation is also simulated. It can be seen that without imposing the MLA constraint,

the original reference 4 is admissible and can be tracked. The wing root bending moment

response with and without MLA constraint are compared in Figure 5.2. The response with

MLA stays within the given load limits as desired during the pitch maneuver. The bending

moment has also been effectively reduced comparing to the one without MLA. The con-

straints on control input and the forward airspeed are fulfilled as shown in Figure 5.3 and


5.4 respectively.

0 1 2 3 4 5 6 7 8

time(s)

-1

0

1

2

3

4

5

3(d

eg)

reference commandartificial referenceactual responsewithout MLA

Figure 5.1: Reference Command, Artificial Reference, Tracking Output Responses with andwithout MLA in Nominal Case

5.4.2 Fault Case 1

The first case is when the Flap 1 get stuck at 2. The constraint on the faulty control surface

is modified to 2 ≤ δa1 ≤ 2.001. We add 0.001 to the upper bound to avoid numerical

issues when computing for the admissible invariant set. The pitch angle is required to track

the nominal maximum admissible reference command 3.4 and then go back to 0. As shown

in Figure 5.5, the reference command is no longer admissible in this fault case and has been

adjusted to 1.13 in the artificial reference. The wing root bending moment response is shown

in Figure 5.6. The MLA objective is still achieved in this faulty case as the bending moment

stays within the given load limits during the pitch maneuver. The constraints on control

input and the forward airspeed are also fulfilled as shown in Figure 5.7 and 5.8 respectively.

5.4.3 Fault Case 2

The second case is when the Flap 2 get stuck at 0 deg. So the constraint on the faulty

control surface is modified to 0 ≤ δa2 ≤ 0.001. Similar to Fault Case 1, 0.001 is added to

the upper bound to avoid numerical issues. The pitch angle is required to track the nominal


0 1 2 3 4 5 6 7 8

time(s)

-200

-100

0

100

200

300

400

500

600

root

ben

ding

mom

ent (

Nm

)

with MLAwithout MLA

Figure 5.2: Wing Root Bending Moment Responses with and without MLA in Nominal Case

0 1 2 3 4 5 6 7 8

-20

0

20

/a1

(de

g)

with MLAwithout MLA

0 1 2 3 4 5 6 7 8

-20

0

20

/a2

(de

g)

0 1 2 3 4 5 6 7 8

time(s)

-20

0

20

/e (

deg)

Figure 5.3: Control Surface Deflections with and without MLA in Nominal Case


0 1 2 3 4 5 6 7 8-2

0

2

u(m

/s)

with MLAwithout MLA

0 1 2 3 4 5 6 7 8-2

0

2w

(m/s

)

0 1 2 3 4 5 6 7 8

time(s)

-20

0

20

q(de

g)

Figure 5.4: Rigid-body State Responses with and without MLA in Nominal Case

maximum admissible reference command 3.4 and then go back to 0. As shown in Figure

5.9, the reference command is no longer admissible in this fault case and has been adjusted

to 2.2 in the artificial reference. The wing root bending moment response is shown in Figure

5.10. The MLA objective is still achieved in this faulty case as the bending moment stays

within the given load limits during the pitch maneuver. The constraints on control input

and the forward airspeed are also fulfilled as shown in Figure 5.11 and 5.12 respectively.

5.4.4 Fault Case 3

The third case is when the elevator loses half of its effectiveness. So the internal model for

MPC is modified to the faulty model (5.31) with ρ3 = 0.5 . The pitch angle is required to

track the nominal maximum admissible reference command 3.4 and then go back to 0. As

shown in Figure 5.13, the reference command is still admissible in this fault case and the

pitch angle is able to track it asymptotically. The wing root bending moment response is

shown in Figure 5.14. The MLA objective is still achieved in this faulty case as the bending

moment stays within the given load limits during the pitch maneuver. The constraints on

control input and the forward airspeed are also fulfilled as shown in Figure 5.15 and 5.16

respectively.


0 1 2 3 4 5 6 7 8

time(s)

-1

0

1

2

3

4

5

3(d

eg)

reference commandartificial referenceactual response

Figure 5.5: Reference Command, Artificial Reference and Tracking Output in Fault Case 1

0 1 2 3 4 5 6 7 8

time(s)

-150

-100

-50

0

50

100

150

root

ben

ding

mom

ent (

Nm

)

Figure 5.6: Wing Root Bending Moment Response in Fault Case 1


0 1 2 3 4 5 6 7 80

2

4

/a1

(de

g)

0 1 2 3 4 5 6 7 8

-20

0

20

/a2

(de

g)

0 1 2 3 4 5 6 7 8

time(s)

-20

0

20

/e (

deg)

Figure 5.7: Control Surface Deflections in Fault Case 1

0 1 2 3 4 5 6 7 8-2

0

2

u(m

/s)

0 1 2 3 4 5 6 7 8-2

0

2

w(m

/s)

0 1 2 3 4 5 6 7 8

time(s)

-5

0

5

q(de

g)

Figure 5.8: Rigid-body State Responses in Fault Case 1


0 1 2 3 4 5 6 7 8

time(s)

-1

0

1

2

3

4

5

3(d

eg)



0 1 2 3 4 5 6 7 8

time(s)

-150

-100

-50

0

50

100

150

root

ben

ding

mom

ent (

Nm

)



0 1 2 3 4 5 6 7 8

-20

0

20

/a1

(de

g)

0 1 2 3 4 5 6 7 8

-20

0

20

/a2

(de

g)

0 1 2 3 4 5 6 7 8

time(s)

-20

0

20

/e (

deg)


0 1 2 3 4 5 6 7 8-2

0

2

u(m

/s)

0 1 2 3 4 5 6 7 8-2

0

2

w(m

/s)

0 1 2 3 4 5 6 7 8

time(s)

-5

0

5

q(de

g)



0 1 2 3 4 5 6 7 8

time(s)

-1

0

1

2

3

4

5

3(d

eg)



0 1 2 3 4 5 6 7 8

time(s)

-150

-100

-50

0

50

100

150

root

ben

ding

mom

ent (

Nm

)



0 1 2 3 4 5 6 7 8

-20

0

20

/a1

(de

g)

0 1 2 3 4 5 6 7 8

-20

0

20

/a2

(de

g)

0 1 2 3 4 5 6 7 8

time(s)

-20

0

20

/e (

deg)


0 1 2 3 4 5 6 7 8-2

0

2

u(m

/s)

0 1 2 3 4 5 6 7 8-2

0

2

w(m

/s)

0 1 2 3 4 5 6 7 8

time(s)

-20

0

20

q(de

g)



5.5 Summary

In this chapter, a fault-tolerant MPC design with reference adjustment is presented for the

MLA problem of a flexible aircraft. By modifying input constraints for stuck fault cases

and changing the internal model for loss of effectiveness fault cases, the MPC formulation is

able to handle control surface faults in a natural fashion and steer the system to track any

admissible reference command, with automatic reference adjustment if the given one is not

admissible.

Chapter 6

Conclusions and Future Work

6.1 Conclusions

In this thesis, fault tolerant control (FTC) designs that mitigate the negative effects of control

surface faults on both rigid-body motions and aeroelastic modes while maintaining overall

closed-loop system stability and acceptable performance have been developed for flexible

aircraft to handle each of the three problems: flutter suppression, gust load alleviation

(GLA) and maneuver load alleviation (MLA).

Due to the strong coupling between rigid-body and flexible modes, faults that happen in

control surfaces can affect the normal operation and maneuver of aircraft and may induce

or accelerate negative aeroelastic responses of the flexible wings. In addition to maintaining

the stability of the system and recovering rigid-body motion performance from faulty situ-

ations, FTC designs for flexible aircraft need to account for the aeroelasticity criteria such

as flutter suppression and load alleviation simultaneously. A unified dynamics model that

treat the rigid-body degrees of freedom and the structural dynamics as a whole is required

and developed in this thesis. Based on the model, we represent the aeroelasticity criteria

together with the rigid-body motion performance as control objectives and develop control

designs to achieve them.

In Chapter 2, a state-space form of a flexible aircraft model is derived by adopting the

Lagrange’s equations for quasi-coordinates developed by Meirovitch [94]. The model is able

to describe the rigid-body motions, the relatively small elastic deformations of the flexible

wings as well as the coupling between them.

Flutter suppression problem is discussed in Chapter 3. The influence of different control

surface faults on a flexible aircraft model is investigated and FTC design that can gain-

schedule with airspeed is developed for a flexible aircraft with actuator saturation and stuck

control surface faults. The influence of stuck control surface faults is analyzed and shown

110

Chapter 6. Conclusions and Future Work 111

to cause unwanted vibrations of the wing and uncontrolled rigid-body motions through the

interaction between rigid-body and flexible modes. For a flexible aircraft model flying at

any airspeed in a given airspeed range, an LPV proportional and integral (PI) controller is

designed with control gain that schedules with airspeed. It is able to eliminate the effects

of stuck control surface faults on the rigid-body motion output while suppressing potential

vibrational motions of the flexible wing. The control design also guarantees no closed-loop

performance degradation caused by actuator saturation.

FTC design for GLA problem is presented in Chapter 4. A mixed H2/H∞ FTC controller

is developed for a flexible aircraft subject to gust disturbances and loss of control effectiveness

faults in control surfaces. The FTC design features an adaptive fault estimator and a gain-

scheduled mixed H2/H∞ controller that is able to regulate the rigid-body motion of the

aircraft, reduce wing root bending moment as well as accommodate the loss of effectiveness

faults under both discrete and continuous gusts.

In Chapter 5, a fault-tolerant model predictive control (MPC) formulation with reference

adjustment is developed to handle stuck and loss of control effectiveness faults for MLA

problem. Under the MPC formulation, the structural load alleviation objective is achieved

by treating the wing root bending moment as a constrained output. Different control surface

fault cases can be systematically handled by modifying input constraints for stuck fault cases

and changing internal model for loss of effectiveness fault cases. By adding a parameter vector

that characterizes an artificial steady state and input as a decision variable, the feasibility of

MPC optimization is ensured. The system can asymptotically track any admissible reference

and if the desired one is not admissible, track an artificial reference which is close to the

desired one while respecting input and output constraints.

In summary, FTC designs have been developed for flexible aircraft. By taking into

account the influence of control surface faults on aeroelastic modes in the FTC designs,

fault handling ability has been incorporated into flutter suppression and structural load

alleviation controls for flexible aircraft and the effects of faults on both rigid-body motion

and aeroelastic modes have been mitigated.

6.2 Future Work

We list several topics that can be directions for future research:

• The structural model for the flexible wing is based on Bernoulli beam theory. Although

for the purpose of this work it is satisfactory to capture the aeroelastic interactions,

a more detailed nonlinear structural model will be more accurate in describing large

flexible deformations and should be incorporated in future studies.

Chapter 6. Conclusions and Future Work 112

• The availability of full state information has been assumed for all FTC designs in

this work. In practice, the aerodynamic states cannot be directly measured. And

for measurable states, their sensor signals may be corrupted by noises or affected by

aeroelastic modes. Therefore, extending the current results of state feedback control

to observer-based output feedback control is an important direction for future study.

• Integrating FDI into the current FTC designs will make the fault handling capability

of flexible aircraft more complete. After fault occurrence, it is critical to know the

maximal time window allowed for FDI before the effective FTC controller is applied

such that the closed-loop system maintains stability. Attenuation of the transients

caused by controller reconfiguration to avoid extra wing vibrations is also an interesting

aspect that can be taken into account under an integrated FDI and FTC framework.

• In the presented work, we assume that the aircraft will not fly beyond the flutter speed

or perform a maneuver in the event of strong turbulences, and do not consider distur-

bances in FTC designs for flutter suppression and MLA problems. How to increase the

robustness of FTC for the two problems will make the designs more realistic.

Copyright Permission

Partial contents of Chapter 2 and main contents of Chapter 3 and Chapter 4 have been

published in the following conference and journal papers:

• Wen Fan, Hugh H. Liu, and Raymond Kwong. “The Influence of Control Surface

Faults on Flexible Aircraft”, AIAA Guidance, Navigation, and Control Conference,

AIAA Paper 2016-0082, 2016. doi: 10.2514/6.2016-0082. [108]

• Wen Fan, Hugh H. Liu, and Raymond Kwong. “Gust Load Alleviation Control for a

Flexible Aircraft with Loss of Control Effectiveness”, AIAA Guidance, Navigation, and

Control Conference, AIAA Paper 2017-1721, 2017. doi: 10.2514/6.2017-1721. [122]

• Wen Fan, Hugh H. Liu, and Raymond Kwong. “Gain-Scheduling Control of Flexible

Aircraft with Actuator Saturation and Stuck Faults”, Journal of Guidance, Control,

and Dynamics, Vol.40, No.3(2017), pp.510-520. doi: 10.2514/1.G002222. [123]

These portions of this thesis are reprinted by permission of the copyright owner: American

Institute of Aeronautics and Astronautics, Inc..

113

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