Of Highly Augmented Aircraft - University of Leicester · Of Highly Augmented Aircraft ... 1.2...

[i-A nalysis Tools For T he Flight C learance O f H ighly A ugm ented Aircraft

T hesis subm itted for th e degree of

D octor o f Philosophy

at the U niversity o f Leicester

by

Ridwan K ureem un B E ng(H ons) M Sc

D epartm ent o f Engineering

U niversity o f Leicester

UK

D ecem ber 18, 2002

UMI Number: U485987

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A bstract

This study describes the development and application of new analysis techniques using the

structured singular value /x for the clearance of flight control laws for highly augmented

aircraft. M otivated by the limitations of the classical m ethods currently used by industry,

these new techniques have been developed to provide a more rigorous and efficient analysis

of worst-case aircraft stability and performance characteristics. The classical approaches

currently used in industry for the clearance of flight control laws are explained and eval

uated. New /x-tools are developed to directly address specific clearance criteria currently

used by industry. Different approaches to represent uncertain linear and nonlinear systems

as Linear Fractional Transform ation (LFT)-based uncertainty models are described: a fast

numerical approach th a t can easily generate LFT-based uncertainty models for nonlinear

aircraft models at the expense of some conservatism, a more complex symbolic approach

th a t requires detailed information about the uncertain param eters in the aircraft dynamic

equations, and a physical modelling approach which can generate LFT-based uncertainty

models in a straightforward manner assuming the availability of the aircraft model in a block

diagram representation. Two new algorithms for com puting tight bounds on real fi are intro

duced. Both m ethods are shown to be capable of generating good lower bounds on /x, even

for high-order uncertainty models. The application of the new ^x-analysis tools developed

in this study to the flight control law clearance process is illustrated for a detailed fighter

aircraft model called the HIRM +, for a VSTOL fighter aircraft model called the HWEM

and for a civil transport aircraft model. Comparisons between the newly developed analysis

techniques and the classical approaches dem onstrate th a t ^x-analysis tools can significantly

improve both the reliability and efficiency of the flight control law clearance process.

A cknow ledgem ents

I am indebted to my supervisor, Dr Declan Bates, for his invaluable guidance throughout

this research. Thanks also to my co-supervisor, Dr Michael Pont, and internal examiner,

Prof Ian Postlethwaite.

I am grateful to Mr Glenn D ’Mello from Qinetiq for his discussions on the HWEM project.

My thanks also go out to all of the Control and Instrum entation research group.

Members of the GARTEUR group are also gratefully acknowledged.

Finally, I would like to express my warmest thanks to my wife and to my parents for their

continued encouragement and support.

C ontents

A bstract i

A cknow ledgem ents ii

N om enclature xvii

1 T he current industrial flight control law clearance process for highly aug

m ented aircraft 1

1.1 In tro d u c tio n .......................................................................................................................... 1

1.2 Description of a typical aircraft analysis m o d e l ..................................................... 3

1.2.1 Non linear m o d e l .................................................................................................. 3

1.2.2 Linear m o d e l ......................................................................................................... 4

1.2.3 Inclusion of uncertainties in the m o d e l .......................................................... 4

1.3 Stability margin c r i t e r io n .............................................................................................. 5

1.3.1 Single loop analysis ............................................................................................ 5

1.3.2 M ulti loop a n a ly s is ................................................................................................ 6

1.4 Unstable eigenvalue criterion ....................................................................................... 7

1.5 Average phase rate and absolute am plitude c r i t e r io n ........................................... 7

1.6 Clonk c r i t e r io n .................................................................................................................. 9

1.7 Largest exceedance of angle of attack and normal load factor limits criterion . 11

1 .8 C o n c lu s io n s ......................................................................................................................... 1 2

2 N ew ^/-analysis tools for the clearance o f flight control laws 14

2.1 Introduction to the structured singular v a l u e ......................................................... 14

2 . 2 C om putation of / i ............................................................................................................... 18

2.2.1 Com putation of complex and mixed f i .......................................................... 19

2.2.2 Com putation of real fi using standard m e th o d s ......................................... 19

2.2.3 Com putation of tight bounds on real /i for high order systems using

H sen s itiv itie s ............................................................................................................. 2 0

2.2.4 C om putation of tight bounds on real /i for high order systems using

an optim isation a p p r o a c h ......................................................................................2 2

2.2.5 Com putation of [i w ithout a frequency g r id d in g ............................................. 23

2.3 LFT-based uncertainty modelling for aerospace s y s t e m s ..........................................24

2.3.1 LFT-based uncertainty modelling for linear systems: a physical mod

elling a p p ro a c h .......................................................................................................... 24

i i i

2.3.2 LFT-based uncertainty modelling for nonlinear systems: a physical

modelling approach ............................................................................................ 26

2.3.3 LFT-based uncertainty modelling for nonlinear systems: an uncer

tain ty bands a p p ro a c h 29

2.3.4 LFT-based uncertainty modelling for nonlinear systems: a symbolic

approach .................................................................................................................... 32

2.4 ii tools for the stability margin c r i t e r io n ......................................................................... 33

2.4.1 Single loop elliptical Nichols plane exclusion regions a s a / i problem . 33

2.4.2 M ulti loop elliptical Nichols plane exclusion regions as a / i problem . . 43

2.4.3 (i as a Nichols plane stability m a rg in ................................................................... 43

2.4.4 Single loop trapezoidal Nichols plane exclusion regions a s a / i problem 46

2.4.5 M ulti loop trapezoidal Nichols plane exclusion regions as a /i problem 47

2.4.6 C om putation of /i for elliptical and trapezoidal Nichols plane exclusion

reg io n s...........................................................................................................................47

2.5 /i tools for the unstable eigenvalue c r i t e r io n .................................................................. 48

2.6 C o n c lu s io n s ........................................................................................................................... 50

3 C learance o f a civil transport aircraft flight control law 52

3.1 The civil transport aircraft m o d e l.................................................................................. 52

3.2 Design of a control law for a civil transport aircraft model using constrained

output feedback ................................................................................................................... 53

3.2.1 The Tioo loopshaping design m e th o d ...................................................................53

3.2.2 The constrained static Tioo loopshaping m e th o d .......................................... 56

3.3 LFT-based uncertainty modelling of a civil transport aircraft using a physical

modelling a p p ro ac h ............................................................................................................. 58

3.4 Results for the stability margin c r i te r io n .........................................................................60

3.4.1 Single loop analysis ..................................................................................................61

3.5 Results for the unstable eigenvalue c r i te r io n .................................................................. 61

3.6 C o n c lu s io n s ............................................................................................................................... 64

4 C learance o f th e H IR M + R ID E flight control law 65

4.1 The HIRM-I- aircraft m o d e l .................................................................................................65

4.2 The RIDE control l a w ...........................................................................................................6 6

4.3 The H IRM + flight en v e lo p e .................................................................................................6 6

4.4 LFT-based uncertainty modelling of the H IRM + using the uncertainty bands

ap p ro ach ..................................................................................................................................... 6 8

i v

4.5 Analysis cycle for the clearance process ....................................................................... 69

4.6 Analysis of the longitudinal H IRM + dynamics ..........................................................71

4.6.1 Results for the unaccelerated 7 t r i m ................................................................. 71

4.6.2 Results for the pullup-pushover-a t r i m ..............................................................8 6

4.6.3 Results for the pullup-pushover-n2 t r i m .......................................................... 97

4.6.4 Summary of analysis results for the pullup-pushover-n2 t r im .................... 108

4.7 Analysis of the lateral HIRM-f d y nam ics ......................................................................109

4.7.1 Results for the unaccelerated 7 t r i m ............................................................... 110

4.7.2 Results for the pullup-pushover-a t r i m ............................................................118

4.7.3 Results for the pullup-pushover-n2 t r i m ........................................................ 130

4.8 Comparison of results with the classical a p p ro a c h .................................................... 147

4.9 C o n c lu s io n s ........................................................................................................................... 152

C learance o f th e H W EM CL002 flight control law 153

5.1 The HWEM aircraft m o d e l ............................................................................................. 153

5.2 The CL002 control la w ....................................................................................................... 154

5.3 The HWEM flight e n v e lo p e ............................................................................................. 154

5.4 LFT-based uncertainty modelling of the HWEM using a symbolic approach . 156

5.4.1 G athering the non-linear equations of m o t io n ................................................ 156

5.4.2 In te rp o la tio n s ............................................................................................................. 158

5.4.3 Assumptions and app rox im ations........................................................................158

5.4.4 Symbolic linearisation of the H W E M ................................................................. 159

5.4.5 Validation of the symbolic m o d e l ........................................................................160

5.4.6 Difficulties with the symbolic LFT-based uncertainty modelling tech

nique 161

5.5 LFT-based uncertainty modelling of the HWEM using a physical modelling

ap p ro ach ................................................................................................................................... 163

5.6 Validation of the LFT-based uncertainty m o d e l ........................................................164

5.7 Analysis of the longitudinal HWEM d y n am ics ...........................................................172

5.7.1 Stability over the flight e n v e lo p e ........................................................................172

5.7.2 Results for the stability margin criterion - single loop an a ly sis ................... 173

5.7.3 Results for the unstable eigenvalue c r ite r io n ................................................... 198

5.8 Comparison between the classical and fi m e th o d s .................................................... 198

5.8.1 Single loop analysis of the stability margin c r i t e r i o n .................................. 198

5.8.2 Comparison of com putation times .................................................................... 206

5.8.3 M ulti loop analysis of the stability margin c rite r io n ......................................206

5.9 Comparison between the elliptical Nichols exclusion region test and the trape

zoidal Nichols exclusion region t e s t ................................................................................ 207

5.9.1 Single loop analysis ..............................................................................................207

5.9.2 M ulti loop a n a ly s is ................................................................................................. 209

5.10 C o n c lu sio n s ............................................................................................................................209

6 Conclusions 211

6.1 C o n c lu sio n s ............................................................................................................................211

6.2 Future w o rk ............................................................................................................................213

v i

List o f Figures

1 Longitudinal aircraft variab les.........................................................................................xvii

2 Directional aircraft variab les............................................................................................ xvii

3 Lateral aircraft v a ria b le s .................................................................................................... xviii

1.1 Unstable eigenvalue p l o t ................................................................................................. 5

1.2 Nichols stability margin boundaries (single loop analysis) ................................. 6

1.3 Nichols stability margin boundaries (multi loop an a ly sis).................................... 6

1.4 Boundaries for the unstable eigenvalue re q u ire m en t.............................................. 8

1.5 Average phase ra te criterion b o u n d a r ie s ................................................................... 9

1.6 The Clonk m a n o e u v re .................................................................................................... 10

1.7 Pilot commands for testing largest exceedance of a and n z l i m i t s ................... 12

2.1 Interconnection structure of a general uncertain closed-loop s y s te m ................ 15

2.2 Upper LFT uncertainty d esc rip tio n ............................................................................. 15

2.3 Robust performance requirement as fictitious uncertainty b lo c k ....................... 17

2.4 Introduction of a scalar weight in the interconnection s t r u c t u r e ........................... 21

2.5 Introducing a scalar weight in the interconnection s t r u c t u r e ..................................21

2.6 Block diagram representation of the linear spring-mass-damper system . . . 25

2.7 Block diagram representation of the linear spring-mass-damper system with

uncertainty in tro d u c e d ..........................................................................................................26

2.8 LFT uncertainty model for spring-mass-damper system ........................................ 26

2.9 P art of nominal HWEM VSTOL m o d e l ................................................................... 27

2.10 P art of HWEM VSTOL model with uncertainties in t r o d u c e d .......................... 28

2.11 Upper LFT uncertainty d esc rip tio n ................................................................................. 28

2.12 Interconnection structure of uncertain closed-loop s y s t e m ................................. 30

2.13 S tandard block diagram for ^ -a n a ly s is ...........................................................................30

2.14 M ultivariable feedback control s y s t e m ...........................................................................34

2.15 Nichols plane exclusion r e g io n s ........................................................................................34

2.16 Elliptical Nichols plane exclusion regions ....................................................................35

2.17 General interpretation of Nyquist plane exclusion regions ..................................... 36

2.18 Corresponding circular Nyquist plane exclusion regions .................................... 36

2.19 Single-loop feedback control system with uncertainties - test for exclusion

region B ..................................................................................................................................... 38

2.20 S tandard MA form for robustness analysis using / i ...................................................38

v i i

2.21 S tructured singular value plot for closed-loop system .............................................40

2.22 Nichols plots for nominal and worst-case s y s te m s ....................................................... 40

2.23 Responses to pilot step demands on a and 6 - Controller 1 and Controller 2 42

2.24 Multi-loop system with uncertainties - test for loop 1, exclusion region B . . 42

2.25 Standard MA form for robustness analysis using f i ....................................................43

2.26 P lot of ii bounds for closed-loop system: loopl (-), loop2 ( - . - ) ............................... 44

2.27 Nichols plots for worst-case systems: loop 1 and loop 2 44

2.28 S tructured singular value plot for closed-loop system - Loopl (-), Loop2 (-.-),

Controller 2 45

2.29 Nichols plots for worst-case systems - Loop 1 and Loop 2, Controller 2 . . . 45

2.30 LFT representation of trapezoidal Nichols exclusion region using first order

Pade a p p ro x im a tio n ............................................................................................................. 47

2.31 Trapezoidal Nichols exclusion regions ............................................................................48

2.32 Possible real-fi tests in the s - p l a n e .................................................................................. 49

3.1 Im plem entation structure of the Hoc loopshaping controller K ( s ) .........................53

3.2 Normalised coprime factor uncertainty d e s c r ip t io n ............................................... 55

3.3 Block diagram representation of (3 to $ term of equation (3.1)...... .......................59

3.4 Block diagram representation of j3 to {3 term of equation (3.1), with uncertain

Cy0 p a r a m e t e r ................................................................................................................... 59

3.5 LFT uncertainty model for lateral axis analysis e x a m p le ..........................................60

3.6 Mixed fi bounds, Nichols exclusion region test ...........................................................61

3.7 Nominal and worst case Nichols plot (aileron loop cut) ..........................................62

3.8 Mixed fi bounds, Nichols exclusion region test ...........................................................62

3.9 Nominal and worst case Nichols plot (rudder loop cut) ..........................................63

3.10 fi bounds for worst-case e ig e n v a lu e .................................................................................. 63

3.11 W orst case eigenvalues .......................................................................................................64

4.1 Flight envelope of the H IRM + ......................................................................................... 6 6

4.2 Interconnection structure of uncertain closed-loop s y s t e m .................................. 70

4.3 S tandard block diagram for ^ -a n a ly s is ............................................................................70

4.4 Regions and points in flight envelope for a n a ly s i s ................................................... 72

4.5 Real fi upper and lower bounds for point FC1 ....................................................... 72



4.8 Real fi upper and lower bounds for region F C 4 ....................................................... 74

v i i i

4.9 Real /z upper and lower bounds for flight conditions FC5 to FC 8 ................... 74

4.10 Cleared portion of the flight envelope - stability .................................................. 75

4.11 Mixed /z bounds, Nichols exclusion region test, flight condition FC3 ..................76

4.12 Nominal and worst case Nichols plot, flight condition FC3 .............................. 76

4.13 Mixed fx bounds, Nichols exclusion region test, flight condition FC4 ..................77

4.14 Nominal and worst case Nichols plot, flight condition FC4 .............................. 77

4.15 Mixed /z bounds, Nichols exclusion region in n e r e llip se test, FE region FC5

to FC8 ................................................................................................................................. 78

4.16 Nominal and worst case Nichols plot, FE region FC5 to FC 8 .............................. 78

4.17 Mixed /z bounds, Nichols exclusion region o u te r e llip se test, FE region FC5

to FC 8 ................................................................................................................................. 79

4.18 Comparison of portions of FE cleared for stability and stability margin criteria 79

4.19 /z test for worst case unstable eigenvalue, flight condition F C 1 ............................... 81

4.20 fx test for worst case unstable eigenvalue, flight condition FC 2 ............................... 82

4.21 fi test for worst case unstable eigenvalue, flight condition F C 3 ............................... 82

4.22 W orst case unstable eigenvalue for flight condition FC3 .................................... 83

4.23 M igration of the short-period mode eigenvalues for flight condition FC3 . . . 83

4.24 M igration of the phugoid mode eigenvalues for flight condition F C 3 .................... 84

4.25 Close-up of Figure 4 .2 4 .........................................................................................................84

4.26 Summary of results for the worst-case stability margins due to symmetrical

taileron loop cut (Unacclerated 7 trim ) ........................................................................ 85

4.27 Summary of results for the worst-case eigenvalues (Unacclerated 7 trim) . . 85

4.28 Real z upper and lower bounds for flight condition FC1 (33° < a < 35°) . . 87

4.29 Real /z upper and lower bounds for flight condition FC2 (33° < a < 35°) . . 87

4.30 Real /z upper and lower bounds for flight condition FC3 (33° < a < 35°) . . 8 8

4.31 Real /x upper and lower bounds for flight condition FC4 (26° < a < 28°) . . 8 8

4.32 Real /x upper and lower bounds for flight condition FC5 (33° < a < 35°) . . 89

4.33 Real /x upper and lower bounds for flight condition FC 6 (17° < a < 19°) . . 89

4.34 Real /z upper and lower bounds for flight condition FC7 (7° < a < 9°) . . . . 90

4.35 Real /z upper and lower bounds for flight condition FC 8 (33° < a < 35°) . . 90

4.36 Mixed fx bounds, Nichols exclusion region test, F C l ............................................... 92

4.37 Nominal and worst case Nichols plot, F C l ................................................................ 92

4.38 Mixed (x bounds, Nichols exclusion region test, FC3 ........................................... 93

4.39 Nominal and worst case Nichols plot, FC3 ................................................................ 93

4.40 Mixed fx bounds, Nichols exclusion region test, FC7 ............................................... 94

i x

4.41 Nominal and worst case Nichols plot, FC7 ...................................................................94

4.42 M igration of the short period eigenvalues for F C l (Longitudinal axis, Pullup-

pushover a t r i m ) .................................................................................................................... 95


taileron loop cut (Pullup-pushover-a trim ) ................................................................. 96

4.44 Summary of results for the worst-case eigenvalues (Pullup-pushover a trim ) . 96

4.45 Real /x upper and lower bounds for flight condition F C l (1 < n z < 2) . . . . 98

4.46 Real fi upper and lower bounds for flight condition FC2 (1 < n z < 2) . . . . 98

4.47 Real ^ upper and lower bounds for flight condition FC3 (1 < n z < 2) . . . . 99

4.48 Real fi upper and lower bounds for flight condition FC4 (2 < n z < 3) . . . . 99

4.49 Real /x upper and lower bounds for flight condition FC5 (2 < n z < 3) . . . . 100

4.50 Real /x upper and lower bounds for flight condition FC 6 (5 < n z < 6) . . . . 100

4.51 Real fi upper and lower bounds for flight condition FC7 ( 6 < n z < 7) . . . . 101

4.52 Real fi upper and lower bounds for flight condition FC 8 (2 < n z < 3) . . . . 101

4.53 Mixed fi bounds, Nichols exclusion region test, FC4 .................................... 103

4.54 Nominal and worst case Nichols plot, FC4 .............................................................. 103

4.55 Mixed fi bounds, Nichols exclusion region test, FC5 .................................... 104

4.56 Nominal and worst case Nichols plot, FC5 .............................................................. 104

4.57 Mixed fi bounds, Nichols exclusion region test, FC 6 .................................... 105

4.58 Nominal and worst case Nichols plot, FC 6 .............................................................. 105

4.59 Mixed /x bounds, Nichols exclusion region test, FC7 .................................... 106

4.60 Nominal and worst case Nichols plot, FC7 ..............................................................106

4.61 Mixed n bounds, Nichols exclusion region test, FC 8 .................................... 107

4.62 Nominal and worst case Nichols plot, FC 8 .............................................................. 107

4.63 M igration of the short period mode eigenvalues for FC 8 (Longitudinal axis,

Pullup-pushover-n2 trim ) ................................................................................................. 108

4.64 M igration of the phugoid mode eigenvalues for FC 8 (Longitudinal axis, Pullup-

pushover-n2 t r im ) .................................................................................................................. 109


taileron loop cut (Pullup-pushover-n2 t r i m ) ............................................................... 109

4.66 Summary of results for the worst-case eigenvalues (Pullup-pushover-n2 trim ) 110

4.67 Real /x upper and lower bounds for point F C l ............................................. I l l

4.68 Real ix upper and lower bounds for point FC2 ............................................. I l l

4.69 Real fi upper and lower bounds for region F C 4 ............................................. 112

4.70 Real n upper and lower bounds for region FC5 to FC 8 ............................ 1 1 2

x

4.71 Real p, upper and lower bounds for point FC3 .........................................................113

4.72 Mixed ^ bounds, Nichols exclusion region test, FE point F C l ..........................115

4.73 Nominal and worst case Nichols plot, FE point F C l (Differential taileron loop

cut) ..........................................................................................................................................115

4.74 Mixed n bounds, Nichols exclusion region test, FE point F C 2 ..........................116

4.75 Nominal and best worst case Nichols plot, FE point FC2 (Differential taileron

loop cut) ............................................................................................................................... 116

4.76 Mixed fi bounds, Nichols exclusion region test, FE point F C 3 ..........................117

4.77 Nominal and worst case Nichols plot, FE point FC3 (Rudder loop cut) . . . 117

4.78 M igration of the dutch roll eigenvalues for F C l (Lateral axis, Unaccelerated

7 trim ) ...................................................................................................................................119

4.79 Summary of results for the worst-case stability margins due to differential

taileron loop cut (Unaccelerated 7 t r i m ) .................................................................. 119

4.80 Summary of results for the worst-case stability margins due to rudder loop

cut (Unaccelerated 7 trim) ..............................................................................................1 2 0

4.81 Summary of results for the worst-case eigenvalues (Unaccelerated 7 trim ) . . 120

4.82 Real fi upper and lower bounds for flight condition F C l .........................................122

4.83 Real ji upper and lower bounds for flight condition F C 2 .........................................122

4.84 Real fi upper and lower bounds for flight condition F C 3 .........................................123

4.85 Real n upper and lower bounds for flight condition F C 4 ......................................... 123

4.86 Real fi upper and lower bounds for flight condition F C 5 .........................................124

4.87 Real (jl upper and lower bounds for flight condition FC 6 .........................................124

4.88 Real /1 upper and lower bounds for flight condition F C 7 .........................................125

4.89 Real // upper and lower bounds for flight condition FC 8 .........................................125

4.90 Mixed fi bounds, Nichols exclusion region test, FC5 .............................................. 127

4.91 Nominal and worst case Nichols plot, FC5 (Differential taileron loop cut) . . 127

4.92 Mixed /u bounds, Nichols exclusion region test, FC7 .............................................. 128

4.93 Nominal and worst case Nichols plot, FC7 (Differential taileron loop cut) . . 128

4.94 Mixed fi bounds, Nichols exclusion region test, FC5 .............................................. 129

4.95 Nominal and worst case Nichols plot, FC5 (Rudder loop c u t ) ..........................129

4.96 M igration of the dutch roll eigenvalues for FC3 (pullup-pushover-a trim) . . 131

4.97 Summary of results for the worst-case stability margins due to differential

taileron loop cut (pullup-pushover-a trim ) ............................................................... 131

4.98 Summary of results for the worst-case stability margins due to rudder loop

cut (pullup-pushover-a trim ) ...........................................................................................132

x i

4.99 Summary of results for the worst-case eigenvalues (pullup-pushover-a trim ) . 132

4.100Real // upper and lower bounds for flight condition F C l .................................. 134

4.101Real fi upper and lower bounds for flight condition F C 2 .................................. 134

4.102Real // upper and lower bounds for flight condition F C 3 .................................. 135

4.103Real fi upper and lower bounds for flight condition F C 4 .................................. 135

4.104Real // upper and lower bounds for flight condition F C 5 .................................. 136

4.105Real // upper and lower bounds for flight condition FC 6 .................................. 136

4.106Real /i upper and lower bounds for flight condition F C 7 .................................. 137

4.107Real // upper and lower bounds for flight condition FC 8 .................................. 137

4.108Mixed // bounds, Nichols exclusion region test, F C l .............................................. 139

4.109Nominal and worst case Nichols plot, F C l (Differential taileron loop cut) . . 139

4.110Mixed // bounds, Nichols exclusion region test, FC2 .............................................. 140

4.111Nominal and worst, case Nichols plot, FC2 (Differential taileron loop cut) . . 140

4.112Mixed fi bounds, Nichols exclusion region test, FC3 .............................................. 141

4.113Nominal and worst case Nichols plot, FC3 (Differential taileron loop cut) . . 141



4.116Mixed // bounds, Nichols exclusion region test, FC5 .............................................. 143


4.118Mixed // bounds, Nichols exclusion region test, FC 6 .............................................. 144

4.119Nominal and worst case Nichols plot, FC 6 (Differential taileron loop cut) . . 144



4.122Mixed fi bounds, Nichols exclusion region test, FC 8 .............................................. 146


4.124Migration of the dutch roll mode eigenvalues for FC4 (Lateral axis, Pullup-

pushover-n2 trim ) .............................................................................................................. 148

4.125Summary of results for the worst-case stability margins due to differential

taileron loop cut (pullup-pushover-n2 t r i m ) ............................................................... 148

4.126Summary of results for the worst-case stability margins due to rudder loop

cut (pullup-pushover-n2 trim ) .......................................................................................149

4.127Summary of results for the worst-case eigenvalues (pullup-pushover-n2 trim ) 149

4.128Summary of results for the worst-case stability margins due to symmetrical

taileron loop cut using //-analysis a p p r o a c h ............................................................... 151

x i i

4.129Summary of results for the worst-case stability margins due to symmetrical

taileron loop cut using classical approach ...................................................................151

5.1 Pitch rate response to a step dem and on tailplane .................................................161

5.2 Pitch rate response to a step dem and on nozzle .................................................... 162

5.3 Pitch a ttitude response to a step dem and on t a i l p l a n e ..........................................162

5.4 Responses to a 1° step demand on ta i lp la n e .............................................................. 165

5.5 Responses to a 5° step demand on ta i lp la n e .............................................................. 166

5.6 Responses to a 1 ° step demand on nozzle ................................................................. 167

5.7 Responses to a 5° step demand on nozzle ................................................................. 168

5.8 Responses to a 1 % step demand on t h r o t t l e .............................................................. 169

5.9 Responses to a 10% step demand on th ro ttle...............................................................170

5.10 Perturbed responses to a 1° step dem and on tailplane ............................................ 171

5.11 P itch rate responses to a step demand on pitch r a t e .................................................172

5.12 Real fi bounds - FC l ....................................................................................................... 173

5.13 Real fi bounds - FC2.... ....................................................................................................... 174

5.14 Real n bounds - FC3............................................................................................................ 174

5.15 Real fi bounds - FC4.... ....................................................................................................... 175

5.16 Real fi bounds - FC5 ....................................................................................................... 175

5.17 Real fi bounds - FC 6 ....................................................................................................... 176

5.18 Real [i bounds - FC7 ....................................................................................................... 176

5.19 Mixed /i bounds for the Nichols exclusion test - F C l ........................177

5.20 Nominal and worst case Nichols plot - F C l ..............................................................177

5.21 Mixed fi bounds for the Nichols exclusion test - F C 2 ........................178

5.22 Nominal and worst case Nichols plot - F C 2 ..............................................................178


5.24 Nominal and worst case Nichols plot - FC3 ..............................................................179

5.25 Mixed /i bounds for the Nichols exclusion test - F C 4 ........................180

5.26 Nominal and worst case Nichols plot - F C 4 ..............................................................180


5.28 Nominal and worst case Nichols plot - FC5 ..............................................................181

5.29 Mixed // bounds for the Nichols exclusion test - FC 6 ........................182

5.30 Nominal and worst case Nichols plot - FC 6 ................................................... 182

5.31 Mixed /i bounds for the Nichols exclusion test - F C 7 ........................183

5.32 Nominal and worst case Nichols plot - F C 7 ................................................... 183

5.33 Mixed fi bounds for the Nichols exclusion test - F C l ........................184

x i i i

5.34 Nominal and worst case Nichols plot - F C l ............................................................... 184

5.35 Mixed n bounds for the Nichols exclusion test - F C 2 ............................................... 185

5.36 Nominal and worst case Nichols plot - FC2 ............................................................... 185

5.37 Mixed n bounds for the Nichols exclusion test - F C 3 ............................................... 186


5.39 Mixed fi bounds for the Nichols exclusion test - F C 4 ............................................... 187

5.40 Nominal and worst case Nichols plot - F C 4 ............................................................... 187

5.41 Mixed [l bounds for the Nichols exclusion test - F C 5 ............................................... 188


5.43 Mixed [i bounds for the Nichols exclusion test - FC 6 ............................................... 189

5.44 Nominal and worst case Nichols plot - FC 6 ............................................................... 189

5.45 Mixed ^ bounds for the Nichols exclusion test - F C 7 ...............................................190


5.47 Mixed p bounds for the Nichols exclusion test - F C l ...............................................191

5.48 Nominal and worst case Nichols plot - F C l ............................................................... 191

5.49 Mixed p bounds for the Nichols exclusion test - F C 2 ...............................................192


5.51 Mixed /i bounds for the Nichols exclusion test - F C 3 ...............................................193




5.55 Mixed fi bounds for the Nichols exclusion test - F C 5 ...............................................195


5.57 Mixed p bounds for the Nichols exclusion test - FC 6 ...............................................196

5.58 Nominal and worst case Nichols plot - FC 6 ............................................................... 196



5.61 W orst case eigenvalues for F C l, ‘o’ fi approach, ‘*’ classical approach . . . . 199

5.62 Worst case eigenvalues for FC2, ‘o’ fi approach, classical approach . . . . 199

5.63 Worst case eigenvalues for FC3, ‘o’ fi approach, ‘*’ classical approach . . . . 200

5.64 Worst case eigenvalues for FC4, ‘o’ fi approach, classical approach . . . . 200

5.65 Worst case eigenvalues for FC5, ‘o’ /j, approach, ‘*’ classical approach . . . . 201

5.66 W orst case eigenvalues for FC 6 , ‘o ’ n approach, ‘*’ classical approach . . . . 201

5.67 Worst case eigenvalues for FC7, ‘o’ p, approach, ‘*’ classical approach . . . . 202

5.68 Stability margin criterion for FC4: fi worst case (-*-) and classical (-) . . . . 203

x i v

5.69 Close-up of Figure 5.68 203

5.70 Stability margin criterion for FC5: fi worst case (-*-) and classical (-) . . . . 204


5.72 Stability margin criterion for FC 6 : fi worst case (-*-) and classical (-) . . . . 205


5.74 Com putational effort of n and classical techniques ................................................. 206

5.75 T hrottle loop cut: fi worst case (-*-) and classical (-)....... ....................................... 207

5.76 Mixed fi bounds for the elliptical Nichols exclusion region test ........................... 208

5.77 Real ^ bounds for the trapezoidal Nichols exclusion region t e s t ..........................208

5.78 Worst case Nichols plots for tailplane loop cut ........................................................ 209

x v

List o f Tables

4.1 Test points in the flight e n v e lo p e ......................................................................................67

4.2 Longitudinal uncertain param eters of the HIRM + ....................................................67

4.3 Most relevant lateral uncertain param eters of the H I R M + ....................................... 6 8

4.4 Reduction factors for simultaneous aerodynamic u n c e r ta in tie s ............................6 8

4.5 Worst-case stability margins for the symmetrical taileron (Unaccelerated 7

t r i m ) ............................................................................................................................................80

4.6 Unstable eigenvalues (Unaccelerated 7 t r i m ) .................................................................. 81

4.7 Allowable range of a for the longitudinal pullup-pushover-a t r i m .................... 8 6

4.8 Worst-case stability margins (Pullup-pushover-a t r i m ) .............................................91

4.9 W orst-case unstable eigenvalues (Pullup-pushover a t r i m ) ...................................... 95

4.10 Allowable range of load factor for longitudinal pullup-pushover n z trim . . . 97

4.11 W orst-case stability margins (Pullup-pushover-n2 trim ) ........................................102

4.12 Worst-case unstable eigenvalues (Pullup-pushover n z trim ) ........................... 108

4.13 W orst-case stability margins for the differential taileron (Unacclerated 7 trim ) 114

4.14 W orst-case stability margins for the rudder (Unacclerated 7 t r im ) ..................... 114

4.15 Worst-case unstable eigenvalues (Unaccelerated 7 trim ) ........................................118

4.16 Allowable range of a for the lateral pullup-pushover-a t r i m .................................1 2 1

4.17 W orst-case stability margins due to the differential taileron (Pullup-pushover-

a trim ) ...................................................................................................................................126

4.18 W orst-case stability margins due to the rudder (Pullup-pushover-a trim ) . . 126

4.19 Worst-case unstable eigenvalues (Pullup)-pushover-a t r i m ) .................................... 130

4.20 Allowable range of load factor for the pullup)-pushover-n2 trim (lateral axis) 133

4.21 Worst-case stability margins due to the differential taileron (Pullup-pushover-

n z trim ) ................................................................................................................................138

4.22 Worst-case stability margins due to the rudder (Pullup-pushover-n2 trim) . . 138

4.23 Worst-case unstable eigenvalues (Pullup-pushover-n2 trim ) .................................147

4.24 Timing results ..................................................................................................................... 150

5.1 Flight envelope for the H W E M .......................................................................................155

5.2 Most relevant longitudinal u n c e r ta in t ie s ...................................................................... 155

5.3 Most relevant lateral u n c e r ta in t ie s ................................................................................ 155

5.4 /x se n s itiv itie s .........................................................................................................................202

x v i

Nom enclature

X

VT

VH

VDZ

Fig. 1: Longitudinal aircraft variables

x v l

Fig. 2: Directional aircraft variables

FCS Flight Control System

FCL Flight Control Laws

FE Flight Envelope

GARTEUR Group for Aeronautical Research and Technology in Europe

HIRM + High Incidence Research Model (augmented model)

x v i i

Fig. 3: Lateral aircraft variables

HWEM Harrier Wide Envelope Model

KIAS Knots Indicated Air Speed

LFT Linear Fractional Transformation

LMI Linear M atrix Inequality

LTI Linear Time Invariant

MAC Mean Aerodynamic Chord

RCV Reaction Control Valve

RIDE Robust Inverse Dynamic Estim ation

STOVL Short Take Off and Vertical Landing

VAAC Vectored-thrust Aircraft Advanced flight Control

V D Vertical speed

V H Horizontal speed

V L Lateral speed

V T Velocity along flight path

X Total x body axis force

Y Total y body axis force

Z Total z body axis force

M Mach number

m Mass

I Inertia

X cg X-position of the centre of gravity

Ycg Y-position of the centre of gravity

Z cg Z-position of the centre of gravity

U Body-axis X velocity

x v i i i

V

w

Vt a s

Q

P

9

r

a

Ixx

lyy

I z z

q

sc

n z

Vet z e

Q-nx

G-ny

Q-nz

h

F s l v l l

F slvl2

suction

TUT

m S T S

c mSTD

a mscs

a W6CD

C r

crc m a

a TTlabr

c.mo

a m o t n n

C l wbod

Body-axis Y velocity

Body-axis Z velocity

True airspeed

Pitch rate

Roll rate

Acceleration due to gravity

Yaw rate

Linear acceleration

Moment of inertia about x-axis

Moment of inertia about y-axis

Moment of inertia about z-axis

Dynamic pressure

W ing reference area

Mean aerodynamic chord

Normal load factor

E arth axes position

Body axis x-acceleration

Body axis y-acceleration

Body axis z-acceleration

A ltitude

T hrust of engine 1 (left engine)

T hrust of engine 2 (right engine)

Nose suction

Pitch moment coefficient due to tailplane deflection at zero incidence

Pitch moment coefficient due to symmetrical taiplane

Pitch moment coefficient due to differential tailplane

Pitch moment coefficient due to symmetrical canard

Pitch moment coefficient due to differential canard

Pitch moment coefficient due to pitch rate

Pitch moment coefficient due to incidence rate

Pitch moment coefficient due to incidence

Airbrake pitch moment coefficient

Pitch moment coefficient due to basic aircraft at zero incidence

Pitch moment coefficient due to tailplane at zero incidence

W ing/Body lift coefficient

x i x

0 ,„„ Lift coefficient due to tailplane deflection

C , Lift coefficient due to pitch rate

Cl STD Rolling moment coefficient due to differential tailplane deflection

ClsCD Rolling moment coefficient due to differential canard deflection

Ci6R Rolling moment coefficient due to rudder deflection

CiP Rolling moment coefficient due to roll rate

Cir Rolling moment coefficient due to yaw rate

Cl, Rolling moment coefficient due to sideslip

Cn0 Yawing moment coefficient due to sideslip

Cnr Yawing moment coefficient due to yaw rate

Cnp Yawing moment coefficient due to roll rate

CnsTD Yawing moment coefficient due to differential tailplane deflection

CnscD Yawing moment coefficient due to differential canard deflection

Cn6R Yawing moment coefficient due to rudder deflection

C'd\i)bod W ing/body drag coefficient

Cdflap Flap drag coefficient

C(laSr Airbrake drag coefficient

Cdporf Under-fuselage pod drag coefficient

Xaro X-component of aerodynamic force (body axis)

Zaro Z-component of aerodynamic force (body axis)

ro X-component of moment (body axis)

X saro X-component of aerodynamic force (aerodynamic axis)

Zsaro Z-component of aerodynamic force (aerodynamic axis)

Xjrisaro X-component of moment (aerodynamic axis)

Xsint X-component of interference force

Zsuit Z-component of interference force

t X-component of interference pitching moment

Xthrst X-component of propulsive force

^thrst X-component of propulsive force

$CD Differential canard deflection

$CS Symmetrical canard deflection

St d Differential tailplane deflection

5t s Symmetrical tailplane deflection

$R Rudder deflection

Structured singular value

XX

a Angle of attack

a Angle of sideslip

A Uncertainty set

7 Flight path angle

e Pitch attitude

<t> Bank angle

V Tailplane deflection

Oj Nozzle angle

n Angular velocity

p Stability margin

ip Heading angle

x x i

C hapter 1

T he current industrial flight control law clearance process

for highly augm ented aircraft

1.1 In tro d u ctio n

Modern high performance aircraft are often designed to be naturally unstable for perfor

mance reasons and, therefore, can only be flown by means of a controller which provides

artificial stability. As the safety of the aircraft is thus dependent on the controller, it must

be proven to the clearance authorities th a t the controller functions correctly throughout the

specified flight envelope in all normal and various failure conditions, and in the presence of

all possible param eter variations.

This task is a very lengthy and expensive process, particularly for high performance aircraft,

where many different combinations of flight param eters (e.g. large variations in mass, in

ertia, centre of gravity positions, highly non-linear aerodynamics, aerodynamic tolerances,

air da ta system tolerances, structural modes, failure cases) must be investigated so tha t

guarantees about worst-case stability and performance can be made.

The complex aircraft models used for clearance purposes describe the actual aircraft dynam

ics, but only within given uncertainty bounds. One reason for this is the limited accuracy of

the aerodynamic d a ta set determined from theoretical calculations and wind tunnel tests.

These param eters can even differ between two aircraft of the same type, due to production

tolerances.

The purpose of the clearance process is to dem onstrate th a t a set of selected criteria express

ing stability and handling requirements is fulfilled. Typically, criteria covering both linear

and non-linear stability, as well as various handling and performance requirements are used

for the purpose of clearance. The clearance criteria can be grouped into four classes:

1. Linear stability criteria

2. Aircraft handling/P ilot Induced Oscillation (PIO) criteria

3. Non-linear stability criteria

4. Non-linear response criteria

In this chapter, a brief description of the following five criteria will be given, which cover

all of the above classes:

1 . Stability margin criterion (class 1 )

1

2. Unstable real eigenvalues criterion (class 1 )

3. Average phase rate and absolute am plitude criterion (class 2)

4. Clonk criterion (class 3)

5. Largest exceedance of angle of attack and normal load factor limits criterion (class 4).

To perform the clearance, for each point of the flight envelope, for all possible configurations

and for all combinations of param eter variations and uncertainties, violations of the clearance

criteria and the worst-case result for each criterion have to be found. Based on the clearance

results, flight restrictions are imposed where necessary.

Any analysis technique used in the clearance process must be able to identify where the

aircraft is safe to fly in the flight envelope for a given control system and set of uncertainties.

In particular, answers to the following questions are required:

• Are there any violations of the stability margin in the required flight envelope? W hat

is the strongest violation and which uncertainty combination has caused it?

• Are there any unstable eigenvalues which are outside the requirement limits? For

which uncertainty combination do the unstable eigenvalues have the largest real part?

• Is there any handling criterion which indicates handling worse than Level 2? W here

and for which uncertainty combination does this occur?

• Is there any exceedance in the positive or negative limits of the angle of attack, angle

of sideslip, load factor or roll rate? Which uncertainty combination yields the worst

case and for which manoeuvre?

• Which uncertainty combination gives the largest deviation from nominal manoeuvre

response?

Faced with limited time and resources, the current flight clearance process employs a grid-

ding approach, whereby the uncertain param eter space is gridded over intervals, and the

various clearance criteria are checked for all possible combinations of the grid points. Tradi

tionally, only the extrem e points of the uncertain param eters are considered. Obviously, the

effort required for the clearance assessment increases exponentially with the number of grid

points. Moreover, it is possible th a t the worst-case com bination of uncertain param eters

does not lie on their extreme points, and thus the risk of missing a worst-case closed loop

behaviour potentially exists. In this thesis, new analysis techniques, based on the struc

tured singular value and /i-analysis, are developed to address these limitations. These new

2

tools are designed to provide increased autom ation and reliability of the flight control law

clearance problem with reduced effort and cost.

The thesis is organised as follows. This chapter describes the current industrial clearance

process for flight control laws of high performance aircraft. Some linear and nonlinear clear

ance criteria are discussed. Chapter 2 introduces some new tools based on the structured

singular value fi th a t can be used to improve the flight control law clearance process. The

problem of generating LFT-based uncertainty models is also described. In Chapter 3, a

physical approach to LFT-based uncertainty modelling is illustrated for a civil transport

aircraft model. The application of the new //-analysis tools developed is also presented.

C hapter 4 describes an uncertainty bands approach to LFT-based uncertainty modelling for

a detailed fighter aircraft model called the HIRM +. The clearance of a flight control law

for a high performance short take off and vertical landing aircraft model called the HWEM

is investigated in C hapter 5. This chapter considers two approaches to LFT-based uncer

tainty modelling - a symbolic approach and a physical modelling approach. Finally, some

conclusions and suggestions for future work are presented in C hapter 6 .

The results contained in this thesis have been published in [1, 2, 3, 4, 5, 6 , 7, 8 , 9, 10, 11].

1.2 D e sc r ip tio n o f a ty p ica l aircraft an a lysis m o d el

The process of validating the flight control laws of an aircraft consists of a number of

steps. The generation of a representative analysis model is the basis of this clearance

task. This section also gives a brief description of certain linear and non-linear clearance

criteria/requirem ents currently used in industry [12, 13].

1.2.1 N on linear m odel

For the purpose of flight control system analysis, a full size non-linear model of the aircraft

is necessary. This is in contrast to design problems, where only a reduced-order linear model

is often used. Typically, the required model must include the non-linear aerodynamic da ta

set, the non-linear equations of motion of the aircraft with configuration dependent data

as well as the non-linear control laws and hardware models (e.g. actuators, sensors, filters,

hydraulics, hinge moments, engine, computing and da ta processing). In addition, models

for atmospheric turbulence or gusts are included.

The non-linear model is used to assess the stability, handling and flying qualities of the

aircraft through simulations. These simulations are carried out in both realtime (manned)

and non-realtime. Moreover, rig tests with real hardware in the loop (i.e. flight control com

puters with control law software and redundancy management, actuators and hydraulics)

3

complement these tests and are used to check the transient behaviour of the aircraft in the

event of system failures.

1.2.2 Linear m odel

A small perturbation model is derived from the non-linear model and the dynamics are

separated into longitudinal and lateral motions. This is done by trimming and linearising

the non-linear model for a large number of grid points over the whole flight envelope. The

grid points are dependent on Mach number (M ), angle of attack (a) and altitude or dynamic

pressure. This gridding must be narrow enough to cover fast changes in the non-linear

aerodynamics as M or a is varied. For instance, in the transonic area, it may be necessary

to reduce steps in M to A M = 0.05 and steps in a to A a = 1° whereas in the subsonic

area, steps of A M = 0.2 and A a = 2° can be sufficient.

In the linear model, the hardware (e.g. pilot inceptors, actuators, sensors, delay times) is

represented by transfer functions which approxim ate the non-linear model in the frequency

range of interest (up to 7 Hz for linear stability and handling investigations). As the linear

model with the command path included can easily reach a very high order, numerically

stable algorithms are required. Moreover, the inclusion of structural modes in the model

will further increase the order of the system.

1.2.3 Inclusion o f uncertainties in the m odel

Before stability and handling quality evaluations are performed, im portant stability deriva

tives like Cma , Cnp, Ci0 and control power derivatives like CmSts, CiStd, CnSr of the unaug

mented aircraft are plotted as a function of A/, a and control surface positions to identify

possible problems such as loss of control effectiveness.

For naturally unstable aircraft, it is useful to plot the unstable (positive) real roots as a

function of M and a. Figure 1.1 shows a typical plot of the unstable pole of the short period

mode versus a for a fixed M . It can be observed th a t there is a sudden instability peak

from a value of +1 to +3.5 and back to +0.7 for a A a of only 2°. In this region, stability

problems are very likely to arise. Under these circumstances, if the control laws are to be

scheduled with a measurement, errors may occur in the air da ta system which can lead to

the wrong choice of gains.

In order to prove th a t the aircraft is safe for flight, the clearance assessment must be per

formed not only for the nominal model but also for all possible deviations from the nominal.

Thus, the model has to be extended to take into account uncertainties arising in the flight

param eters. Typical uncertainties include variations in the nominal values of the stability

4

4

3

2

g■3®DC

1

0

•10 2 4 6 8 10 12 14

o(° )

Fig. 1.1: Unstable eigenvalue plot

and control derivatives, changes in mass and centre of gravity positions. A more detailed

list of these uncertainties for the highly augmented aircraft considered in this study will be

given in Chapters 4 and 5.

1.3 S ta b ility m arg in criter ion

A basic requirement of all clearance work is to prove th a t the aircraft is stable over the

entire flight envelope with sufficient margin against instability for all known uncertainties

(worst-case combinations). The process consists of calculating linear stability margins for

the open-loop frequency response in pitch, roll and yaw. These frequency responses are

obtained by breaking the loop at the input of each actuator or of each sensor and are

then plotted in Nichols diagrams where the required phase and gain margins are shown as

exclusion regions which must not be violated by the plot.

1.3.1 Single loop analysis

In single loop analysis, the open-loop frequency response is obtained by breaking the loop

at the input of each actuator or of each sensor, one a t a time, while leaving the other loops

closed. For the nominal case, these Nichols plots should not violate the outer exclusion

region shown in Figure 1.2, which corresponds to a minimum gain margin of ±6 dB and

a minimum phase margin of 35°. W hen uncertainties are taken into account, a boundary

corresponding to ±4.5 dB is used, as shown by the inner exclusion region in Figure 1.2.

5

Outer exclusion region

co32,c(3e>

-2

Inner exclusion region

- 8 1— -200 -140-190 -160 -150 -130-180 -170

P hase ( 0 )

Fig. 1.2: Nichols stability margin boundaries (single loop analysis)

1.3.2 M ulti loop analysis

In multi loop analysis, all the loops of the actuators or sensors are cut simultaneously.

mTO

C<8O

-2

-3

-190 -180 -160 -1 4 0 -130-200 -170 -150Phase ( 0 )

Fig. 1.3: Nichols stability margin boundaries (m ulti loop analysis)

The closed-loop system should then be able to w ithstand the application of simultaneous

and independent gain and phase offsets, as defined by the region shown in Figure 1.3, at

the actuators and sensors without becoming unstable.

6

To test for this criterion, a perturbation m atrix K j is inserted at (for example) the

input of each actuator. W ith K set to 1, T is then varied simultaneously in each loop until

the eigenvalues of the closed-loop system go unstable. The phase margin is calculated as

<t>PM — 2 * ta n ~ lujT (1-1)

where u> is the frequency of the generated undam ped oscillation. The next step consists

of increasing/decreasing K by 1.5 dB (corresponds to right corner points of the Nichols

exclusion region shown in Figure 1.3) and varying T again for the new fixed gain until the

eigenvalues become unstable. By setting T = 0 and varying K , the upper and lower gain

margins can be obtained (corresponding to the left corner points of the Nichols diagram).

These steps can be repeated for a number of points around the required Nichols exclusion

region. To keep the com putational effort reasonable, this test is usually restricted to only a

few points of the exclusion regions (e.g. the four corners of the exclusion region). In addition,

for each loop, only the same values of gain and phase variations are checked in practice, to

avoid testing over too large a number of different combinations. Hence, it can be argued

th a t this procedure can in practice lead to optimistic results.

1.4 U n sta b le e ig en v a lu e criter ion

In addition to the stability margin criterion, the eigenvalues of the closed-loop system must

be calculated in order to identify possible unstable (i.e. those with positive real part) eigen

values which do not show up in the Nichols plots. It is required to identify the flight cases

where unstable eigenvalues occur and for what tolerance combination these eigenvalues have

the largest real part. This test aims to determine the most severe cases of divergent modes

in the closed-loop system and to allow an assessment of their acceptability in term s of their

influence on aircraft handling. A typical boundary on the real part of the eigenvalues is

shown in Figure 1.4. Note tha t, for real eigenvalues, the definition of worst case as the max

imum real part among the positive eigenvalues is straightforward. For complex eigenvalues,

different definitions of worst cases can be chosen, such as the m agnitude of the complex

eigenvalue. Usually, the maximum positive real part is selected as this quantity can be

directly linked to existing handling qualities requirement on the minimum time to double

am plitude of unstable modes.

1.5 A v era g e p h a se ra te and a b so lu te a m p litu d e cr iter io n

In addition to the stability requirements described in the previous sections, the aircraft

m ust fulfil good handling requirements. This means th a t the clearance assessment must

7

0.5

0.4

0.3

0.2

0.1

0

- 0.1

- 0.2

-0 .3

-0 .4

-0 .5

-0 .05 0 0.05 0.1 0.15

Fig. 1.4: Boundaries for the unstable eigenvalue requirement

prove th a t the pilot can control the aircraft precisely and easily to accomplish the mission.

Three levels of flying qualities have been defined by the M ilitary Specification F-8785C, [14],

as follows:

1. Level 1: Handling clearly adequate for the mission flight phase.

2. Level 2: Handling adequate but some increase in pilot workload and /or degradation

in mission effectiveness exists.

3. Level 3: The aircraft can be controlled but pilot workload is excessive and /or mission

effectiveness is inadequate.

The average phase rate and absolute margin criterion is a linear handling qualities criterion

which aims at showing th a t there are no pilot induced oscillation (PIO) tendencies in pitch

and roll. The m agnitude of the average phase rate (APR) is defined as:

A P R = ~ = (' + 18° 0) (1.2)Jc JC

where f c is the frequency in Hz at -180° and 4>2/c is the phase angle in degrees at double

f c. The following requirements should be fulfilled for a given set of uncertainties:

1. The average phase rate values of the frequency response of p itch /bank a ttitude to

stick force should dem onstrate at least Level 2 handling and should therefore not lie

outside the Level 2 boundaries of the phase rate criterion (see Figure 1.5).

8

2. This second part of the requirement applies to the pitch axis only. The absolute

am plitude of the frequency response of pitch attitude to stick force (°/lbs) a t -180°

should be less than -16 dB.

200

Level 3

150

level 2

O. 100

0.2 0.4 1.2 1.60.6Frequency, fc[Hz], at -180 degrees

0.8 1.4phase

Fig. 1.5: Average phase rate criterion boundaries

1.6 C lonk criter io n

This criterion uses combined pitch/roll manoeuvres to identify the flight conditions where

non-linearities cause stability problems such as angle of attack or angle of sideslip departure.

The aim is to maximise the response of a particular param eter to combined pitch and roll

commands by finding the ‘worst case’ in term s of the responses. This is known to be governed

by the rate of application of each command and the relative phasing of the two command

inputs.

The Clonk criterion defines a combined pitch and roll manoeuvre, aimed at testing the

angle of attack margin from departure. The Clonk manoeuvre is defined by the sequence of

commands described below and in Figure 1.6.

The inputs to the manoeuvre are pilot stick commands resembling square waves, i.e., the

stick travels from an end limit in one direction to the opposite end limit in the other direction.

The pilot inputs for this manoeuvre are defined in term s of the physical displacement of the

stick (e.g. millimeters). Figure 1.6 shows the generic time histories of the relevant variables

involved in the definition of the Clonk manoeuvre which can be summarised as:

9

'SE A

SA

t im e

tim e

CXc x ,■m argin

t im e

- T - c x ,m argin

Fig. 1.6: The Clonk manoeuvre

1. S tart simulation.

2. Clonk 1: Move the longitudinal stick aft (defined by 5se , the solid line in the upper

plot of Figure 1.6) at 1000 m m /s until the backward limit is reached and roll stick

left (defined by 5s a , the dashed line in the upper plot of Figure 1.6) at 800 m m /s

until the maximum left command is reached. While maintaining the maximum pitch

command, the roll command is released at 200 m m /s (after the left end limit has been

reached) until the neutral roll stick position is reached.

3. Clonk 2: Apply maximum longitudinal stick forward at 1000 m m /s and maximum roll

stick right at 800 m m /s at time t2. After the right end limit has been reached, the

roll command is released at 200 m m /s.

4. Clonk 3: W hen 6 changes sign from positive to negative, apply maximum longitudinal

stick aft and roll stick left (t3), at a rate of 1000 m m /s and 800 m m /s respectively.

Release roll command at 200 m m /s after the end limit has been reached as in Clonk 1.

5. Clonk 4: W hen 6 changes sign from negative to positive, apply maximum longitudinal

stick forward and roll stick right (t4), at a rate of 1000 m m /s and 800 m m /s respec

tively. The roll command is then released at 200 m m /s after the end limit has been

reached.

6. Clonk 5: Repeat Clonk 3.

1 0

7. Stop excitation (by setting the stick to the neutral position in pitch and roll) if one of

the following cases occurs:

(a) after five cycles of the Clonk manoeuvres (i.e. at time 113);

(b) when the aircraft exits its flight envelope;

(c) when the angle of attack margin (otm a rg in ) is less than zero, where a m a rgin is

given by

margin (t) — min {(o:u p p e r Q:n(^))) (Ckn(^) Q lo w e r)} (1-3)

where a upper and a /^ e r are the upper and lower angle of attack limits respectively

and Qn is the actual value of angle of attack (in the presence of uncertainties).

The time values for t l and t2 are usually chosen as 2 seconds and 3 seconds respectively.

The other tim e constants t3 to t7 (and so on) are defined by the zero crossing condition

of 0 (change in sign of 6) and are further characterised by the condition tha t 0 reaches a

maximum before decreasing again. At t4 and t6, 6 reaches a minimum.

The angle of attack margin relative to the angle of attack limits is recorded. This requirement

is satisfied if the minimum ocmargin is positive. Mathematically, it is required to compute,

for each flight condition, the following parameter:

^m a rg in ,m in = ^ m a r g in { i ) ( L 4 )

1 .7 L argest ex ce e d a n c e o f an g le o f a tta ck and norm al load factor lim its

criter ion

This criterion aims at identifying all flight cases where in the pull-up manoeuvres defined in

Figure 1.7 the positive angle of attack or load factor limits are exceeded. The test is carried

out for both the nominal case and the uncertain case.

The pilot commands are (see Figure 1.7):

1. A rapid full stick pull, i.e., a 1000 m m /s stick rate on the longitudinal stick th a t brings

the stick from the initial position to the maximum am plitude in the aft direction.

2. A pull in 3 seconds, i.e., a ramp command tha t brings the stick from the initial position

to full aft longitudinal stick in 3 seconds.

For both commands, a nominal trajectory must first be generated for each flight condition

and the two performance variables a and n z . The nominal trajectory is the response to

the pilot commands shown in Figure 1.7 when all uncertainties are set to zero. For added

uncertainties, the deviation from the nominal trajectory should stay below some limit. The

1 1

Full stick rapid pull140

120 Ef 100

Slope a +1000 mm/s

20

Full stick pull in 3 s140

3 40

Time (s)

Fig. 1.7: Pilot commands for testing largest exceedance o f a and n z lim its

overshoot of the a / n z limits should be less than l° /0 A g with respect to the reference and

must not lead to departure. A m athematical interpretation of the criterion is given below.

Define the uncertainty set n . Note th a t IIo=0 (uncertainties set to zero) represents the

nominal case.

Let a (t,II) be the angle of attack response to the pilot command. If a up is the upper limit

of the allowable angle of attack, then the difference between the the angle of attack and its

upper limit can be defined as

a D(t ,U) = a( t ,U) - a up (1.5)

Then, the maximum value of o p is given by

a ex(n ) = m axa£i(f, II) (1.6)

The criterion thus consists of solving the following optimisation problem:

&ex,wc ~ rnax o ex (II) (1.7)

The criterion is satisfied if, at a given flight condition, a eXjWC is negative.

The above criterion also applies to the normal load factor response, n z .

1.8 C on clu sion s

A summary of the current approaches used by industry for checking the stability and per

formance characteristics of highly augmented aircraft was presented in this chapter. These

1 2

traditional methods are based on a gridding technique whereby combinations of the extreme

values of the uncertain param eters are considered one at a time. This process has to be

carried out for all possible param eter variations, and soon becomes inefficient, as the number

of points in the grid increases exponentially with the number of uncertain param eters. An

other drawback of this approach is tha t there is no guarantee th a t a worst-case closed loop

behaviour has not been missed, since only a few scattered points are checked in a continuous

region of param eter space.

These deficiencies have motivated the development of new analysis tools for the clearance of

flight control laws, with the aim of providing increased autom ation, reduced effort and cost

and more reliable results. W ith these aims in mind, Action Group FM-AG11 of the Group

for Aeronautical Research and Technology in Europe (GARTEUR) was established in 1999

to perform an assessment of the usefulness of new analysis techniques for the clearance of

flight control laws. Five analysis techniques were investigated within this action group over

a three year period: /i analysis (University of Leicester, UK), i'-gap analysis (University of

Cambridge, UK), polynomial-based analysis (CIRA, Italy), bifurcation analysis (University

of Bristol, UK) and optimisation-based worst-case search (DLR, Germany). Full results of

the study performed by the FM-AG11 Action Group can be found in [15].

In this thesis, the development and application of the structured singular value /i as an

analysis tool for the flight control law clearance process is described, /i analysis is a fre

quency domain technique, and is based on Tioo robust control and linear systems theory.

Therefore, it is not surprising th a t the criteria which are most easily addressed with this

tool are robust stability, avoidance of Nichols plane exclusion regions and identification of

worst case eigenvalues. In this thesis, we therefore focus on the analysis of the linear criteria

defined in Sections 1.3 and 1.4. The application of /i-analysis tools to non-linear criteria

such as the Clonk criterion and the largest exceedance of the angle of attack and load factor

limits criteria constitutes an area of future research.

13

Chapter 2

N ew /i-analysis tools for th e clearance o f flight control

laws

In both the analysis and design of robust control systems, the concept of the structured

singular value // is of fundamental importance, // analysis allows for detailed modelling of

the conditions under which the considered control system operates satisfactorily, both in

the sense of stability and performance. The calculation of // for such systems provides a

robustness margin, indicating whether the behaviour of the controlled system in the presence

of model uncertainty is likely to be satisfactory or not.

In the flight control law clearance environment, deficiencies of the classical methods have

motivated the development of new analysis tools, such as those based on //, th a t can provide

more efficiency and increased reliability. In this chapter, we show how the linear stability

clearance criteria described in Chapter 1 can be addressed using newly developed //-tools and

present new methods for computing tight bounds on // in the case of purely real param etric

uncertainties [3, 4, 16, 7].

2.1 In tro d u ctio n to th e s tru ctu red singu lar value

In order to use techniques associated with //-analysis, it is necessary to formulate the un

certain closed-loop system into a standard canonical representation which is referred to in

the literature as a Linear Fractional Transformation (LFT) [17]. Consider, in particular, a

linear time invariant (LTI) closed-loop system which is subject to some unstructured and /or

structured type of norm-bounded uncertainty, and arranged in the form shown in Figure 2.1,

where P and K denote the plant and controller respectively. W ith respect to this figure,

unstructured uncertainty means th a t the uncertainty m atrix A is fully populated, while

structured uncertainty means th a t it has some (typically diagonal or block diagonal) struc

ture. In the context of a flight control clearance problem, unstructured uncertainty could

correspond, for example, to unmodelled high frequency aircraft dynamics, while structured

uncertainty is used to represent particular aircraft param eters such as stability derivatives,

inertias, etc., which are subject to change or known only to w ithin a certain tolerance.

In effect extra inputs and outputs are introduced so th a t the system uncertainty can be

considered as part of an ‘external’ feedback loop. Partitioning M compatibly with the A

m atrix, the relationship between the input and ou tpu t signals of the closed-loop system

14

shown in Figure 2.2 is then given by the upper LFT:

y — J~u(A/, A )r = (A^22 ~t~ A/21A ( / — Afu A) 1 M \2 )r (2 .1)

Fig. 2.1: Interconnection structure o f a general uncertain closed-loop system

w

rM

Fig. 2.2: Upper L F T uncertainty description

Now, assume tha t the nominal (A = 0) closed-loop system in Figure 2.2 is asymptotically

stable and tha t A is a complex unstructured uncertainty block. Then by the Small Gain

Theorem (SGT), [18], we have the following result: The closed-loop system in Figure 2.2 is

stable if and only if

<t(A(jus)) < (2 .2 )o ( M n (ju;))

The above result defines a test for the stability (and thus a robustness measure) for a closed-

loop system subject to unstructured uncertainty in term s of the maximum singular value of

the m atrix M \\.

For aerospace systems it is often the case th a t uncertainty can be related to variations in

specific aircraft param eters, such as centre of gravity, inertias, stability derivatives etc [16].

15

In such cases, it is possible to generate models of uncertainty which have a particular struc

ture, and thus reduce the level of conservatism in the robustness analysis. The generation of

structured LFT-based uncertainty models which accurately capture the effect of uncertainty

on the original non-linear aircraft model is considered in detail in Chapter 5.

The generation of a structured LFT-based uncertainty model means th a t we have been able

to place all of the uncertainty affecting the system into an uncertainty m atrix A which has

a diagonal or block diagonal structure, i.e.,

A(jcu) = diag(Ai( ju; ) , , A p(jtv)), a(Ai(ju>)) < k (2.3)

where k defines an upper bound on the size of the maximum singular value of any uncertainty

block A{. Now again assume tha t the nominal closed-loop system is stable, and consider

the question: W hat is the maximum value of k for which the closed-loop system will remain

stable? We can still apply the SGT to the above problem, but the result will be conservative,

since the structure of the m atrix A will not be taken into account. The SGT will in effect

assume th a t all of the elements of the m atrix A are allowed to be non-zero, when we

know th a t most of the elements are in fact zero. Thus the SGT will consider a larger

set of uncertainty than is in fact possible, and the resulting robustness measure will be

conservative, i.e. pessimistic.

In order to get a non-conservative solution, Doyle [19, 20] introduced the structured singular

value p:

^ ( ^ i ) m in(k s.t. d e t(I — M \\A) = 0) ^

The above result defines a test for stability (robustness measure) of a closed-loop system

subject to structured uncertainty in term s of the maximum structured singular value of the

m atrix M \\. Note th a t the structured singular value robustness measure can be derived

directly from the multivariable Nyquist stability theorem, [18],:

M IM O N yquist Stability Theorem: For the system of Figure 2.2, let PQi denote the

number o f unstable poles in the open-loop transfer function matrix M \\ A. Then the closed-

loop system is stable i f and only i f the Nyquist plot of det(I — M n A ( s ) ) (i) does not pass

through the origin, and (ii) makes PQi anti-clockwise encirclements o f the origin.

Note th a t for the problem at hand we are looking for zero encirclements of the origin, since

Pol = 0. This arises because M , the nominal closed-loop system, is assumed to be stable,

and the uncertainty m atrix A is also constrained to be stable.

Singular value performance requirements can be combined with stability robustness analysis

in the p framework to measure the robust performance properties of the system. Consider

a modification to the standard LFT uncertainty structure of Figure 2.2, which is illustrated

in Figure 2.3. Assume, for example, th a t the signal y is the error between the reference

16

w

‘P + 1

lp+1

w

Fig. 2.3: R obust performance requirement as fictitious uncertainty block

demands and corresponding (closed-loop) measured outputs. One way of defining a per

formance specification for these variables is to require th a t the maximum singular value

of the frequency response m atrix from r to y lies below some weighting function

For example, for zero steady-state tracking error we would require cf(M2 2 (jw)) < < 1 at

low frequencies, w'hich could be specified by making the m agnitude of W(juj) very small at

those frequencies. To prove good robust performance characteristics we need to check if the

above nominal performance specifications are satisfied for all variations in the uncertainties

A i to Ap. In a ^-analysis framework this can be accomplished as follows: We assume, as

usual, th a t the model uncertainty blocks A i to Ap have been normalised to 1, so th a t the

uncertainty m atrix A is given by

A(juj) = diag(A\ ( ju j ) , , A p( j u ) ) , a ( A i ( j u j ) ) < l

Assume further th a t the performance weight W(juj) has been absorbed into M , so th a t we

require cf(M2 2 ( ju)) < 1 , Vw. The overall transfer m atrix M{ju>) can therefore be given by

z M u

1

w

y 1 M 22 r

Thus we have tha t

Robust stability o p a ( M u ) < 1 Vw

and

Nominal performance <t(M2 2) < 1 V u>

The robust performance requirement is th a t the gain from r to y must be less than 1 for

any uncertainty in the set A, thus

Robust performance <=» a(Fu( M , A )) = g(M .22 + M 21 A(7 — M \ \ A ) ~ l M 1 2 ) < 1

17

where FU( M , A ) is the upper LFT of M and A. We can replace this robust performance

requirement with a robust stability requirement using the SGT, as follows. The key idea is

to notice th a t having the closed-loop gain from r to y to be less than one at all frequencies

implies th a t we can feed back y to r through a fictitious uncertainty block Ap+i having

gain <r(Ap+i) < 1, w ithout de-stabilising the system. This is because the loop gain of this

‘fictitious’ loop will be less than one, and the system is nominally stable, thereby meeting

the SGT conditions. In turn, if a(Fu(M, A)) > 1 a t any frequency for some value of A,

then the fictitious loop could de-stabilise the system, since the phase of A p+i is completely

uncertain. Thus robust stability of the fictitious p + 1 block system is equivalent to robust

performance of the original p block system. To test for robust performance, simply define

the new uncertainty set

& p { j v ) = d iag(Ai ( jw) , , Ap+iO'u;)), a(Ai(ju;)) < 1

and then check th a t

P A p ( M ) < 1 V l j

as illustrated in Figure 2.3. Note that the above test is a necessary and sufficient condition

for robust performance, and is thus completely non-conservative as long as the actual model

uncertainty is well described by the block structure of A.

The above discussion on performance robustness has considered performance from a fre

quency domain viewpoint only. Although some specifications on closed-loop performance

can naturally be w ritten in the frequency domain (e.g. handling/flying quality specifications

given in term s of low order equivalent system transfer functions), other performance speci

fications for aerospace systems are given in the time domain, e.g. maximum allowable rise

times or overshoot for pilot demands. Worst-case performance in term s of p only has an ex

act time domain meaning in term s of sinusoids, and thus care must be taken in interpreting

^-analysis results for general time-domain problems.

2 .2 C o m p u ta tio n o f p

The com putation of p is a non polynomial time problem, i.e. the com putational burden

of the algorithms th a t compute the exact value of p is an exponential function of the

number of uncertainties. It is consequently impossible to compute the exact value of p

for large dimensional problems associated with real industrial systems. A usual solution in

this case is to com pute upper and lower bounds on p. If these are sufficiently tight, then

little information is lost. Note th a t to fully exploit the power of the structured singular

value theory, tight upper and lower bounds on p are required. The upper bound provides a

18

sufficient condition for stability/perform ance in the presence of a specified level of structured

uncertainty. The lower bound provides a sufficient condition for instability, and also returns

a worst-case A, i.e. a worst-case combination of uncertain param eters for the problem.

The degree of difficulty involved in computing good bounds on /i depends on (a) the order

of the A matrix, and (b) whether A is complex, real or mixed, as discussed below.

2.2.1 C om putation o f com plex and m ixed ( i

W hen all the blocks in A are complex, tight bounds on / j , may be computed relatively easily.

Polynomial time algorithms are available to compute upper and lower bounds on complex (i,

[17]. Both bounds converge to exact n for low order problems and extensive com putational

experience, [21], has shown th a t the bounds remain quite tight even for high order problems.

For mixed real and complex uncertainty, polynomial time algorithms are also available for

calculating both upper and lower bounds on /r. The upper bound algorithms use linear

m atrix inequality based optimisation, [22, 23], while the lower bound is generated via power

algorithms, [24, 25]. The upper bound is generally quite tight, but the quality of the lower

bound depends heavily on the amount of complex versus real uncertainty present in A. If

A is dom inated by real uncertainty then the lower bound algorithm may fail to converge,

especially for high order problems.

2.2.2 C om putation o f real f i using standard m ethods

For purely real /i problems, examples appear in the literature which show tha t fi can even

be a discontinuous function of the problem data, [26, 27]. For real /i problems with a

physical engineering motivation, however, it is shown in [26] th a t discontinuity problems

do not arise, and convergent upper, [28], and lower, [29, 30], bound algorithms for /i exist.

Unfortunately both of these algorithms are exponential time, and thus in practice this limits

the size of the A m atrix to about 11. A fix for the upper bound problem is to apply the

polynomial time mixed fi upper bound algorithms - this generally gives good results, even

for high order systems. However, the lower bound for real /i obtained from the mixed /a

algorithm is generally poor and the algorithm often fails to converge, particularly for high

dimensional problems. This represents a particular problem for the application of fi to flight

control law analysis problems, since the number of uncertain param eters which needs to be

considered can often result in high order real uncertainty matrices, and the com putation of

a tight lower bound is essential in order to identify the ‘worst-case’ combination of uncertain

param eters. One convenient engineering ‘fix’ for the problem is to add small amounts of

‘artificial’ complex uncertainty to each real A* in order to improve the lower bound derived

19

from the mixed p algorithm, [17]. While this approach can work reasonably well, it has some

disadvantages. One is th a t the introduction of ‘enough’ complex uncertainty to generate

a tight lower bound can significantly increase the associated upper bound, thus making

the analysis results more conservative and more difficult to interpret. Secondly, it has been

shown tha t the real part of the worst-case structured A determined using existing p software

is particularly sensitive to the addition of uncertainty in this manner, [27]. This can result

in the com putation of a combination of uncertain real param eters th a t may be very different

from the real worst-case combination.

In [31, 32], a state-space approach for computing the peak values of the lower bound on

real p is proposed. The approach basically consists of extracting the real part from a

destabilising mixed uncertainty m atrix and increasing it until one of the closed-loop poles

migrates through the imaginary axis. While this approach is polynomial time and can thus

be applied to high dimensional problems, it only returns the peak value of the lower bound

over a specified frequency range, and cannot be used to generate a tight lower bound at each

point of a frequency grid. Information on the ‘shape’ of the p lower bound as a function

of frequency can provide insight into the type of uncertainty which is causing the problem,

e.g. narrow peaks on the p plot due to aircraft structural modes.

The uncertain aircraft param eters are real in nature and therefore it is necessary to use

real p algorithms in the analysis. In this thesis, two new m ethods have been developed

for generating tight bounds on real p. These tools are described in detail in the following

sections.

2.2.3 C om putation o f tight bounds on real p for high order system s using p

sensitivities

In this section, we describe a method which seeks to selectively reduce the size of the high-

order real A uncertainty m atrix until the use of exponential time lower bound algorithms

become feasible [3, 4]. The method uses the recently introduced p-sensitivities, [33]. To

understand the concept of p-sensitivities, consider an uncertainty matrix:

A i 0A = (2.6)

0 A 2

where A i and A 2 are themselves structured uncertainty blocks. W ithout loss of generality,

we will consider the sensitivity of the structured singular value p a (M ) with respect to the

first model perturbation A i. Let the complex m atrix M be partitioned compatibly with

2 0

equation (2.6) as:

M =M u M\2

M 21 M 22

(2.7)

so th a t M u and A/2 2 correspond to A i and A 2 respectively. The uncertainty A i is then

weighted by a scalar a as shown in Figure 2.4 such th a t the model perturbation A becomes:

A =—

1 > 0

1

a i 0

1O<0t

0 a 2 1

O1

— “1

C4 <10

1

(2 .8)

Fig. 2.4: Introduction o f a scalar weight in the interconnection structure

a i 0The weighting m atrix

0 IFigure 2.5 so th a t A/ becomes:

in Figure 2.4 is transferred from A to M as shown in

M( a ) =

1P O

I

M u M\2 a M n otM12=

0 I 1 s: to to to 1 A /2 1 A /2 2

(2.9)

Fig. 2.5: Introducing a scalar weight in the interconnection structure

2 1

The n sensitivity with respect to A i is then defined as:

n(M{a) ) — — Aa) )A a

a = 1

The /i-sensitivities allow a comparison to be made between the different elements Ai of the

A matrix, in order to determine which elements are contributing most (and least) to the

maximum value of f i . The matrix A can thus be reduced to a size for which the exponential

tim e lower-bound algorithms of [29] can be applied, by discarding those elements A* with the

lowest //-sensitivities. The approach is validated by comparing the resulting lower bounds

w ith the upper bounds calculated via mixed f i algorithms - if no ‘significant’ A*s have been

eliminated, these bounds should then be close. Note th a t this approach evaluates the relative

im portance of each A* from a closed-loop point of view, i.e. w ith respect to its effect on the

maximum value of /1 . This is in contrast to the approach often adopted in the literature,

whereby those A^s with smallest relative magnitude are considered least im portant (and

thus discarded). The application of the proposed technique in [1] shows clearly tha t there

is no correlation between relative size of the A*s and their effect on f i . An illustration of

the proposed approach for computing tight bounds on real fi is also given in [3, 10]. This

technique will be used in Chapter 4.

2.2.4 C om putation o f tight bounds on real f i for high order system s using an

optim isation approach

The second method developed in this study casts the problem of computing a lower bound

for fi as an optimisation-based search for the worst-case (i.e. smallest) real de-stabilising

uncertain A m atrix [3, 4]. Denote the vector of A * diagonal entries of A by x. Thus, if

a g npxpx = [A], . . . , a p]t e n p (2.io)

For real scalar uncertainty, this search can be formulated as an equivalent non-convex opti

misation problem:

m in / ( r ) = ^nin^d(A) where A = diag ([#]) (2-11)

with a non-linear constraint cT (x):

|det (I - A M u (juj)\ < e (2.12)

e in the above constraint is a user defined param eter which can be used to trade-off com puta

tion time versus tightness of the resulting lower bound. Commercially available optimisation

[59, 58], has been adopted to implement this algorithm. A feature of the approach is tha t

2 2

a minimum destabilising A of appropriate structure is computed after each iteration of the

line search algorithm. Exit criteria can easily be chosen for a particular problem so th a t at

each frequency point along the jia;-axis a good estim ate of the worst case destabilising A

will be computed.

This offers a significant improvement over existing [i lower bound algorithms [17], where con

vergence often does not occur with strictly real param eter uncertainty. The destabilising A

is used as the starting vector at the next frequency step for another iteration of the optimisa

tion if necessary. As the search for a worst case destabilising A is non-convex, local minima

can occur. This problem is addressed, in part, using a detection procedure which forces a

restart of the optim isation with a different starting ‘seed’ vector if the lower bound on fi at

a particular frequency is significantly different, (by more than a user specified percentage),

from the lower bound result obtained at the preceding frequency. A user-defined maximum

is placed on the number of restarts tha t are allowed. Whenever a restart is necessary, and

at the very outset of the optimisation, the ‘seed’ vector is extracted from a Singular Value

Decomposition (SVD) on M. This technique will be used in Chapters 3, 4 and 5.

2.2.5 C om putation o f w ithout a frequency gridding

In general, the structured singular value /x(u;) is computed at each point of a frequency grid.

In the case of purely complex or mixed real-complex perturbations, is a continuous

function of lj. However, purely real uncertainties can cause discontinuities in the value

of /i w ith respect to w. In addition, in the case of flexible systems, it is possible to miss

the critical frequency at which narrow and high peaks occur if the grid is not fine enough.

Invariably, refining the grid means increasing the com putational effort involved in computing

the n bounds. One solution to this problem is to recast the standard frequency dependent

/i analysis problem as an augmented skewed -fi problem, where the frequency u> is treated

as an additional uncertainty [32]. In this approach it is then possible to compute n as a

continuous function of frequency over some interval [a;,a;]. The main drawback with this

approach is th a t the size of the augmented skewed-^ problem can be much larger than the

size of the original problem. In fact, the uncertain frequency lu appears as a repeated real

scalar A u In whose size increases with the order (n) of the original system.

In this thesis, to minimise the computational cost, the skewed-fi problem was cast as an

optim isation procedure around the critical frequency. Using standard // algorithms, the

critical frequency is computed using a coarse frequency grid. Optim isation is then performed

around this critical frequency to compute /i.

23

2.3 L F T -b ased u n certa in ty m o d ellin g for aerosp ace sy stem s

In order to apply ^-analysis tools, the original uncertain system must be represented as

an LFT-based uncertainty model, and in fact it can be argued th a t the major difficulty in

applying ^-analysis to real-world applications lies in this initial modelling step. In recent

years, the subject of LFT-based uncertainty modelling has received much attention and

has been found to be a very deep problem - see [32] for an overview. For aircraft analysis

problems, obtaining a unique linear model based on an LFT description of the uncertain

ties, which covers all flight conditions and all possible param eter variations, is a difficult

modelling task. Different techniques for generating LFT-based uncertainty models have

their advantages and disadvantages and their application is geared towards the nature of

the problem being considered. For instance, in the flight clearance process, the usefulness

of the chosen analysis technique is assessed, based on characteristics such as

• the conservatism of the method

• the ability to identify the worst-case param eter combination

• the com putation time and effort required

• the complexity involved in the implementation

LFT-based uncertainty models can be derived for both linear and non-linear systems. For

linear systems, LFT-based uncertainty models can be derived in a straightforward manner

from a given set of linearised equations of motions, provided th a t the uncertain param eters

are explicitly defined in the equations of motion. This approach to LFT-based uncertainty

modelling will be illustrated for a simple linear spring-mass-damper system in Section 2.3.1.

Three different approaches to LFT-based uncertainty modelling for nonlinear systems are

presented in Sections 2.3.2, 2.3.3 and 2.3.4.

2.3.1 LFT-based uncertainty m odelling for linear system s: a physical m odelling

Consider a typical linear spring-mass-damper system whose input Uf is an external force

acting on the system and whose output yd is the displacement of the mass. Let the states

of this second order system be x\ and X2 • Then, the following equations hold

approach

X2 = X \ X2 H Ufm m m

X\ = x 2

Vd = x i

k b 1(2.13)

(2.14)

(2.15)

24

where k is the spring constant and b is the friction constant.

Now, assume th a t uncertainties are introduced in the values of the spring constant (k ) and

damping constant (b) so tha t these quantities are allowed to vary, for instance, by ±10%

of their nominal values. Thus, the spring constant can be rew ritten as k = k°( 1 + W1A 1),

where k° is the nominal value of the spring constant, W\ is a weight on the uncertainty

(W\ = 0.1 in this case) and A i, which varies in the interval [-1,1], represents the normalized

uncertainty in the spring constant. Similarly, the damping constant can be rewritten as

b — 6°(1 + W2 A 2 ). The approach proposed is to represent the uncertain equation (2.15)

in a block diagram form as shown in Figure 2.6. Now, the uncertainties in the spring

and damping constants can be represented physically in block diagram form as shown in

Figure 2.7. In Figure 2.7, extra ‘fictitious’ inputs w\ and u?2 , and outputs z\ and Z2 , have

been added at the point in the system where the uncertainties A i and A 2 occur. Using

standard block diagram manipulation software such as the linm od function in MATLAB,

it is then straightforward to compute the transfer m atrix M w ith inputs u = [wi,W2 ,Uf]

and outputs y = [zi, 2 2 , Vd]- The resulting LFT-based uncertainty model for the system

is shown in Figure 2.8. This method is simple and intuitive and moreover, the approach

provides an exact description of joint param etric dependencies in the model. It can thus be

used to non-conservatively model the effect of the uncertainties on the closed loop system.

As a result, the exact worst-case set of uncertain param eters can be computed using p.

The main problem with this approach is th a t for complex high order systems, the task of

building the block diagram model can become quite tedious. Another disadvantage of this

method is th a t it tends to produce A matrices with many repeated parameters, since each

instance of the same uncertainty is modelled individually. However, by clever manipulation

of the block diagrams, this problem can be eased.

This technique will be further illustrated for a civil transport aircraft model in Chapter 3.

m

Fig. 2.6: Block diagram representation o f the linear spring-mass-damper system

25

XI = yd

< >m

W i

w 2

m

Fig. 2.7: Block diagram representation o f the linear spring-mass-damper system with un

certainty introduced

W-2

Fig. 2.8: L F T uncertainty model for spring-mass-damper system

2.3.2 LFT-based uncertainty m odelling for nonlinear system s: a physical m od

elling approach

In this approach, the uncertainties are directly introduced into the physical non-linear model

in the form of multiplicative (or additive) uncertainties. For an illustration, consider part

of the nominal non-linear model of the HWEM VSTOL aircraft [2] shown in Figure 2.9.

Assuming th a t the pitch moment coefficient due to tailplane deflection, Cmta i l , is subject

to uncertainty of ±20%, we can write C Ajtail = C%iTAIL (1 + U-CAItail A i), where C%Ita il

is the nominal value of Cmt a i l , U-Cmta i l is a weight on the uncertainty (IJ-Cmta i l = 0.2

in this case) and A i varies in the interval [-1,1]. Note th a t any variation in a param eter

of the form x min < x < x max can be w ritten as above in term s of a nominal value and a

weighted uncertainty in the interval [-1,1]. Now, the uncertainty in the aircraft dynamics

due to Cmta i l can be represented physically in block diagram form as shown in Figure 2.10.

In Figure 2.10, we have added extra ‘fictitious’ inputs and outputs w\ and z\ at the point

in the system where the uncertainty A i occurs. This step is then repeated for each A *

26

Blend CONORxmadi

DMPTABCMWBAR

dtail XLTA1

CMTAILCREF XLTRAT

CMQCMAERO

CONQGCMQ CMWDOT

y ~ x

0.328 ■GCMSW

Fig. 2.9: Part o f nominal H W E M VSTO L model

representing the uncertainty in the other uncertain param eters. Now using standard block

diagram m anipulation software (e.g. the function linm od in MATLAB), the resulting non

linear model can be linearised to calculate the transfer m atrix of the system M with inputs

u = [wi, ..., wn, uc] and outputs y = [2:1 , ..., zn , ym], where uc are the controlled inputs and

ym are the measured outputs. The LFT-based uncertainty model for the system, shown in

Figure 2.11, is then given by the relation

Vm = Fu( M ( s), A )uc (2.16)

where A = diag(A i, ..., A n)

Clearly, the approach outlined above is simple and intuitive and provides an exact description

of joint param etric dependencies in the model. Thus, it can be used to non-conservatively

model the effect of the param etric uncertainties on the closed-loop system. As a result,

the exact worst case set of uncertain param eters can be computed. If each uncertainty

is introduced in only one location in the SIMULINK block diagram, the resulting LFT-

based uncertainty model will also be of minimal order and thus the occurrence of repeated

uncertain param eters can be avoided. In addition, this physical modelling approach allows

additional uncertainties in the physical param eters (such as product of the uncertainties) to

be easily implemented in the model.

However, this method has some limitations. In particular, detailed information about the

27

Blend CONQPxmach

DMPTAB

CMWBAR

U.CM TM . ^

| * ^ i tini outldtail XLTAIL

CMTAIL_nomCREF XLTRAT

CMQCMAERO

CONQGCMQ CMWDOT

GCMSW0.328

Fig. 2.10: Part o f H W EM VSTO L model with uncertainties introduced

out in

w

Dm

Fig. 2.11: Upper L F T uncertainty description

model and the uncertainties is required. Hence, its application is restricted to those models

th a t can be implemented in a SIMULINK block diagram type representation. Another

drawback is th a t the dependence of linearisations on the uncertain param eters is ignored, and

thus it is not clear how the LFT-based uncertainty model can represent variations in flight

param eters such as the angle of attack or Mach number. It is therefore not straightforward

to generate LFT-based uncertainty descriptions th a t are valid over particular regions of the

flight envelope using this approach. This physical modelling technique will be used for the

LFT-based uncertainty modelling of the HWEM aircraft model in Chapter 5.

28

2 .3 .3 L F T -based u n certa in ty m od ellin g for non linear system s: an u n certa in ty

bands approach

Here we describe an alternative method for LFT-based uncertainty modelling which can be

used when precise information about the way in which the uncertain aircraft param eters

enter the aircraft dynamic equations is not known. This may be due to lack of knowledge

about the precise effect of these param eters and /o r difficulties with the software used to

construct the aircraft model and its various subsystems. Obviously, in such a situation,

the physical modelling approach described above cannot be used at all. Instead, a simple

uncertainty bands approach can be adopted, [1], which easily and quickly generates LFT-

based uncertainty models at the expense of a certain amount of conservatism.

This technique uses the differences in the linearised trim s th a t arise from the nonlinear

simulation model of the aircraft evaluated over all combinations of the extreme points of

the uncertain param eters. These models form a so-called multi-model state description

Sx(t) = AiSx( t ) + B i5u(t) (2-17)

Sy(t) = CiSx(t) + Di6u{t).

For each varying element of each state-space m atrix we can now calculate its minimum

(e.g. a™m), maximum (e.g. a™jax), and ‘nominal’ ( e.g. (a™ax -I- a™in) j 2) values. We can

thus replace the multi-model system (2.17) by an affine param eter dependent representation

of the form

A B _ ^ 0 B 0n A

+ X ^ A i

1........o

*

C D. C °

D o _ i = i 0 0+ ]C Ai

1 = 71,4 + 1

0 B i

0 0+

E A-1 = 7 1 0 + 1

0 0n o

+ ^ 2 A i

0 0

0 l = 7 l c + l 0 D i

( 2 . 1 8 )

(Ao, B o, Co, Do) thus corresponds to the new ‘nominal’ system. Then, for each varying entry

in the m atrices A, we have a A* and an Ai in the above expression, where Ai is an uncertain

real scalar param eter which varies between 1 and -1, and A{ is equal to ((a™ax — a™in)/2).

Each of the matrices associated with each Ai has rank one and can be factored using the

singular value decomposition into row and column vectors:

A i 0 E i r -i= G i H i

0 0 . F '

(2.19)

29

If we now define the linear system P w ith ex tra inputs and outputs via the equations

where

which

X

V

Z\ =

1.....

..o s 1

A q B q E \ .... E n

Co Do F\ .... Fn

Gi H i 0 .... 0

Gn H n 0 .... 0

X

u

Wi

. Wn .

(2 .20)

n = n D, we can form the closed-loop interconnection structure shown in Figure 2.12,

can then be converted to the standard form shown in Figure 2.13. The uncertainty

w.

W\

Fig. 2.12: Interconnection structure o f uncertain closed-loop system

A i 0

0 A

z 1 y

Fig. 2.13: Standard block diagram for p-analysis

30

bands approach described above is fast and easy to implement. Also, quantities such as, for

example, Mach number and altitude can be easily incorporated in the vector of unknown

param eters to generate LFT-based uncertainty models which are valid over particular regions

of the flight envelope. Given a good trimming routine, this entire process can be fully

autom ated. Moreover, a knowledge of the explicit relationship between each uncertain model

param eter and each element of the system state-space matrices is not required, thus greatly

simplifying the task of LFT-based uncertainty modelling. The resulting parametric model

covers all possible linearisations arising from the uncertain non-linear model. However, this

representation of physical uncertainties may be conservative since possible joint param eter

dependencies among the state space elements in the model will be ignored. It also means

th a t while we can identify the worst-case state-space system for a given criterion, we cannot

trace this back to identify the corresponding worst-case uncertainty in term s of the actual

uncertain flight param eters.

W ith these drawbacks in mind, the use of the uncertainty bands approach for generating

LFT-based uncertainty models is proposed as a tool for use in the initial analysis stage of

the flight control law clearance process. For the flight cases which cannot be cleared by this

approach, it may be necessary to resort to more ‘exact’ symbolic LFT-based uncertainty

modelling techniques in order to determine whether the problem lies with any conservatism

introduced in the LFT-based uncertainty modelling or with the control law itself.

An alternative approach, referred to as the Trends and Bands approach, which reduces the

conservatism introduced in the uncertainty bands method was investigated in [34] and is

briefly summarised here. The method seeks to find trends established by the dependency

of the state space elements on the uncertain param eters by making use of compensation

param eters. Each compensation param eter acts independently on one state space m atrix

element. In general, these trends are modelled by a multi-dimensional regression plane,

which describes the linear variation, and a band structure tha t is limited by planes parallel

to the regression plane, above and below, to include nonlinear deviations. The actual value

of the uncertain state-space element is then assumed to lie somewhere within this band

structure. Compared with the uncertainty bands approach, the sizes of the bands are now

reduced, thus decreasing conservatism. Moreover, the trends and bands method returns the

worst-case in term s of the actual physical uncertain param eters. Further details about this

method can be found in [34].

The uncertainty bands m ethod will be used to generate LFT-based uncertain models for a

fighter aircraft called the HIRM + in Chapter 4.

31

2 .3 .4 L F T -based uncerta in ty m od ellin g for nonlinear system s: a sym bolic ap

proach

This method focuses on the generation of symbolic LFT-based uncertainty models. Ini

tially, this approach requires the formulation of the non-linear dynamic equations of the

aircraft model. Symbolic linearisation is then performed to obtain a linearised LFT-based

uncertainty model. The resulting linearised equations should explicitly contain the uncer

ta in param eters in a rational form. For a complex aircraft model, extensive m athematical

manipulations will be involved and it is sometimes necessary to make simplifications to the

symbolic equations (e.g. by using approximations). The advantage of this method is th a t

the exact worst-case set of uncertainties is obtained in term s of the actual uncertain flight

param eters. However, the modelling effort involved can be substantial, particularly if a large

set of uncertainties is to be investigated.

To illustrate this m ethod, consider the nonlinear HWEM model. The method necessitates

the derivation of the dynamic HWEM equations. The equations of motion for the longitu

dinal axis can be expressed as

Consider, for instance, the equation for the longitudinal acceleration, u, shown in equation

(2.21). The moments, aerodynamic forces and propulsive forces in equation (2.21) can be

further expanded as

To show how these equations relate to the uncertain param eters of the HWEM, equations

(2.25) to (2.29) are substituted in equation (2.21) to give

aro q x W — g x sin{6 )

(Xmeng + Xmaro)

(2 .21 )

(2 .22)

(2.23)

9 (2.24)

( X s a r o “t- - ^ C m t ) X C O s ( c k ) ( Z s a r o T Z s i n t ) X SZTl^Ck)

~ (pdwbod + Cdflap + Cdabr + Cdpod) x q

- {Ctwbod + Cttail + Ctq) X q

- W F x R G x (0.35 x q + U)

a r o

s a r o

s a r o

(2.25)

(2.26)

(2.27)

(2.28)

(2.29)

Xthrst W F x RG x (0.35 x q + U) ^ (Cdwbod + Cdflap + Cdabr + Cdpod) x q x cos(a) ^m m m

Xgint x cos (a) { iwbod T Citaii T Cfq) x q x sin cx') ZSint x 5Z7i(q:) q x W — g x sin(0) (2.30)m m m

32

It can be observed th a t the uncertain param eters (m, Cd and Ci coefficients) now appear

explicitly in equation (2.30). Symbolic linearisation can now be performed using commer

cial software such as M APLE or MATLAB. Note th a t the uncertain elements still appear

symbolically in the results. For instance, element(2,3) of the A m atrix was found to be

172.3 - (16.4 x ^ ) _ 8 6 . 7 (2.31)m

where m and X cg are the uncertain parameters. Once these equations have been converted

into this form, it is easy to represent them physically in SIMULINK block diagrams. This

method will be further discussed and illustrated for the HWEM analysis in Chapter 5.

2 .4 // to o ls for th e sta b ility m argin criter ion

In this section we describe two ways in which the structured singular value robustness

measure can be used to address the stability margin clearance criterion defined in Chapter 1 .

As we shall show, fi has a straightforward interpretation in term s of classical gain/phase

margin and Nichols exclusion region robustness specifications. In fact, /i-analysis provides

the capability to efficiently test for the avoidance of exclusion regions in the Nichols plane,

for all combinations of param etric uncertainty, w ithout resorting to the gridding approach

traditionally used by industry.

The first approach follows th a t in [4, 35, 11] and involves the use of elliptical exclusion

regions. In the second approach [36], trapezoidal exclusion regions are used to address this

criterion.

2.4.1 Single loop elliptical N ichols plane exclusion regions as a f i problem

The standard requirement in this criterion for single-loop analysis can be stated as follows:

R obustness Specification 1

For the multivariable feedback control system shown in Figure 2.14, the open loop Nichols

plot of the frequency response between each actuator demand u and the corresponding error

signal e, obtained by breaking the loop at the point shown in Figure 2.14, while leaving the

other loops closed, should avoid the region A shown in Figure 2.15 for A = 0, and and avoid

the region B shown in Figure 2.15 for all A 6 A m - This property should hold for each loop

of the system.

In the framework of //-analysis, it turns out to be convenient to work with alternative Nichols

plane exclusion regions of the form shown in Figure 2.16. The first step is to convert the

Nichols exclusion regions of the type shown in Figure 2.15 to ellipses. These elliptical regions

33

break single loop for open-loop analysis here

Plant

SensorsController

Actuators

Fig. 2.14: Multivariable feedback control system

Open-Loop Gain (dB)

Exclusion region A

4.5

Exclusion region B

° 1.5 Open-Loop

Phase(°)

-180' -145'

Fig. 2.15: Nichols plane exclusion regions

34

are centered around the critical point (-180,0) and satisfy the equation

I L U ^ ) IdB + + 180)2 = 1 (2.32)G 2 P 2m

where L(juj) is the open-loop frequency response, Gm is the desired gain margin and Pm

is the desired phase margin. A key point is th a t for certain values of Gm and Pm, the

corresponding exclusion regions in the Nyquist plane are circles with centre (a) and radius

(r) given by the following simultaneous equations:

cos

201og10(u + r) = Gr

2 ~ r 2 + 1 , = P 2 a 1 m

(2.33)

(2.34)

To see this, consider the set-up shown in Figure 2.16. In order to ensure tha t the gain margin

is symmetric in the Nichols plane (i.e. ± G m), we must have the corresponding circle in the

Nyquist plane to pass through points A and B (see Figure 2.17), where the coordinates of A

and B are given as (10 2cT, 0) and (1 0 ~2or, 0) respectively. Points A and B define the centre

(a) and radius (r) of the circle in the Nyquist plane, and hence the corresponding value of

the phase margin Pm .

Nichols Plot

Exclusion region A

Exclusion region BmT5c<3O)Q.8ic

O

-4

-6

-8

-220 -200 -180Open-loop phase (degrees)

-160 -140 -120

Fig. 2.16: Elliptical Nichols plane exclusion regions

Thus for example, any feedback system whose open-loop frequency response avoids the

regions A and B in Figure 2.16 provides gain and phase margins of ±.6 d B / ± 36.87° and

±4.5d B / ± 28.44° respectively (note th a t these values are very close to those required under

the classical exclusion regions defined in C hapter 1). For these particular choices of gain

35

Imaginary

Circle with centre -a and radius r

Unit circle

-a+r

Real

cos {(a - r +l)/(2a)}

Fig. 2.17: General interpretation o f Nyquist plane exclusion regions

and phase margins the corresponding exclusion regions in the Nyquist plane are circles

with (centre,radius) given by (-1.25,0.75) for region A, and (-1.14,0.54) for region B - see

Figure 2.18. Note tha t, as shown in [35], only specific values of phase and gain margins can

be chosen to map the ellipses in the Nichols plane into circles in the Nyquist plane (hence

the smaller phase margin of 28.44° for exclusion region B in Figure 2.16). Now, as shown

Nyquist Plot

Exclusion region A0.5

Exclusion region B

E

-0 .5

-3 -2 .5 -2 -1 .5 ■1 -0 .5 0Re L(jw)

Fig. 2.18: Corresponding circular N yquist plane exclusion regions

in [35], another way to interpret the requirement for avoidance of a circle with centre a and

radius r in the Nyquist plane by the open-loop frequency response L(juj), is to consider a

plant subject to multiplicative disc uncertainty of (centre,radius) given by (C,R) at each

36

frequency where

C = (2.35)az — rz

R = (2.36)a* — r 1

Thus, for example, we can interpret the avoidance of the circle B in the Nyquist plane by

the open-loop frequency response L(juj) by considering a plant subject to multiplicative disc

uncertainty of (centre,radius) given by (+1.14,0.54) at each frequency. It is then easy to

see th a t avoidance of the (-1,0) critical point in the Nyquist plane by L(jto) for all possible

plants in this set is exactly equivalent to avoidance of the exclusion region B by L(juj) for

the original plant. The set of possible plants can be represented as

P (s) = Pi(a)(1.14 + A n ) (2.37)

where P\ is the original plant, Ajy is complex and || Ajv ||oo< 0.54. This is of course the

same as writing

P(s) = 1 .14Pi(s)(l + W N A n ) (2.38)

with Wpj — 0.47 and || Ajv ||oo< 1- In this waY we can represent the Nichols exclusion

region as a ‘fictitious’ multiplicative input uncertainty for the scaled nominal plant. It

now remains for us to represent the actual uncertainty present in the original model of the

plant. This uncertainty can be structured (i.e. resulting from uncertainty in actual aircraft

param eters) or unstructured (i.e. resulting from unmodelled dynamics in the aircraft model).

For simplicity, here we consider an unstructured multiplicative uncertainty on the original

plant P\ of the form

P i(s ) = Po(s)(l + W m (s)A m ), || A a/ !!«,< 1 (2.39)

We thus end up with the system shown in Figure 2.19. We can now do some block diagram

m anipulations to ‘pull-out-the-deltas’ from Figure 2.19, and thus convert this system into

the standard M A form for robustness analysis under the n framework, as in Figure 2.20.

Now consider the following robustness specification:

R obustness Specification 2:

For the control system in Figure 2.20, we require < 1 , V o ;

This specification is exactly the same as saying th a t the open-loop frequency response of

every plant in the set of uncertain plants P\(s) lies outside the Nichols plane exclusion region

B of Figure 2.16 - it is thus equivalent to the second half of R obustness Specification

1, for the single-loop case. The first half of R obustness Specification 1 can be re-cast,

again in the single-loop case, in term s of an even simpler test on the H°°-norm of the

37

1.14

Fig. 2.19: Single-loop feedback control system with uncertainties - test for exclusion region

B

Fig. 2.20: Standard MA form for robustness analysis using p

complementary sensitivity function T(s) - see [35] for details.

We can illustrate the ideas presented above via a robustness analysis of a simple single-loop

flight control system. Consider a linearised short period model for the longitudinal axis of

an aircraft, G(s), [35], which in state-space form is given by:

x = A x -t- B u ; y = Cx;

with sta te vector x, control input u and controlled output y given by

x = [a,q, 8 e], u = Se, y — ex.

where a is angle of attack (deg), q is pitch rate (deg/s) and Se is elevator deflection (deg).

Numerical values for the state-space matrices at a given flight condition are as follows:

A =-0.9381 0.9576 -0.3765

,B =1.6630 -0.8120 10.8791

C = 1 0

We include a simple model for the dynamics of the actuators and other hardware elements:

0.000697s2 - 0.0397s + 1H w(s) =

0.000867s2 + 0.0591s + 1

38

to give a nominal plant model Po(s) = H w(s)G(s). Finally we let this nominal model be

subject to multiplicative input uncertainty to take account of modelling errors and neglected

high frequency dynamics, to give:

Pi = Po(s)(l + W M (s )A M)

where2 0 s + 2000 .( . .I

A/ = ' s + 1 0 0 0 ’ I | A a ; I 1 - 1

A feedback controller was designed for this system using the technique of H°° Loop-Shaping,

[38]. We can then use R o b u s tn e s s S p ec ifica tio n 2 above to check the robustness of this

controller to variations in the plant dynamics due to model uncertainty. The resulting fi-

plot was calculated using standard software routines, [17], and is shown in Figure 2.21 -

note tha t the upper and lower bounds are equal for this problem. The maximum value

of fi is equal to 1 .0 1 , indicating tha t the worst case plant in the set P i(s) will just cause

the open-loop frequency response to enter the Nichols exclusion region B . This result is

verified in Figure 2.22, which shows the open-loop frequency response for the nominal plant

Pq{s) and for the worst-case plant in the set P\{s). Finally, in the case where a particular

controller satisfies R o b u s tn e s s S p ec ifica tio n 2 we may be interested in the answers to

the following questions: (a) by how much can the plant uncertainty be increased before the

worst case plant causes L(juj) to enter a specified exclusion region, and (b) by how much can

the exclusion region be increased before a specified level of plant uncertainty causes L(jcu)

to enter the exclusion region? Both of these questions correspond to so-called skewed-y

calculations [32], and thus can be answered exactly in our robustness analysis framework.

The extension of the above results to the multivariable case is quite straightforward. Ro

bustness (in terms of avoidance of a prescribed Nichols exclusion region) can be measured

(a) cutting one loop at a time, with all the other loops closed, and all loops subject (simulta

neously) to an LFT-based uncertainty model representing the uncertain aircraft param eters,

or (b) cutting all loops simultaneously, with all loops subject (simultaneously) to an LFT-

based uncertainty model representing the uncertain aircraft param eters. We illustrate the

approach for the former case via a two-input two-output flight control system design for the

HIM AT aircraft model, [17, 39]. The HIM AT model represents the dynamics of a scaled,

remotely piloted version of an advanced fighter aircraft, and has been widely used in the

robust control literature as a benchmark for evaluation of controller synthesis and analysis

techniques. Assuming effective decoupling between the aircraft’s longitudinal and lateral

dynamics, a linearised model Pq f°r the longitudinal rigid body dynamics is given by:

x = A x + P u ; y = C x + D u (2.40)

39

1.4

1.2

0.8

0.6

0.4

0.2

10'o) (rod/s)

Fig. 2.21: Structured singular value plot for closed-loop system

Nichols Plot

L(jw) for worst case plant in P1

Exclusion region A

Exclusion region B

-5

L(jw) for nominal plant Po

-1 0

-1 5-200 -190-2 1 0 -180 -170 -120-160 -150 -140 -130

O pen-loop phase (degrees)

Fig. 2.22: Nichols plots for nominal and worst-case systems

40

with state vector x , control inputs u and controlled outputs y given by

x = [8 V ,a ,q ,6 ], u = [ 6 e, 6 c], y = [a,0] (2.41)

where SV is forward speed, 6 e is elevon deflection, Sc is canard deflection, and the other

variables have their usual meanings. Numerical values for the state-space matrices at a

given flight condition can be found in [17, 39]. Potential differences between the nominal

model Pq and the actual behaviour of the real aircraft P\ due to uncertainty in the actuator

dynamics, aircraft stability derivatives, etc, are represented by a simple diagonal, frequency

dependent uncertainty model at the plant input, so th a t

P i = P o (s)(h x 2 + W M (s) A m ) (2.42)

where

WhI = s + 10000 X ^2x2’ ^ M ^°° “ 1 (2.43)

For the above model, a controller was designed to independently control a and 6 in order

to provide vertical translation, pitch pointing and direct lift manoeuvring capabilities. Two

separate controllers, Controller 1 and Controller 2 , were designed for this problem, via the

m ethod of 7f°° Loop-Shaping, [38]. Step responses for pilot demands on a and 6 are given

in Figure 2.23, and show good tracking and decoupling properties for both controllers. Con

sider now the robustness specification th a t the open-loop frequency response of loop 1 avoids

Nichols exclusion region R, with loop 2 closed, and both loops subject to the multiplicative

plant uncertainty defined above. The block diagram corresponding to this test is shown in

Figure 2.24, with W n = 0.47 and || A n ||oo< 1- Converting this system into the standard

M A form for /Li-analysis, Figure 2.25, we have the following:

R obustness Specification 3:

For the system in Figure 2.25, we require th a t < 1 , Vo;

This robustness specification, repeated for each loop of the system, is thus equivalent to the

second half of R obustness Specification 1 , for the multivariable case. The correspond

ing /i and Nichols plots for the controller are shown in Figures 2.26 and 2.27. Although

Controller 1 was found to achieve good levels of robustness using standard fi-analysis and

7i°° Loop-Shaping robustness margin measures, we can see from Figures 2.26 and 2.27 tha t

Loop 1 fails the Nichols exclusion region test for the worst case plant in P\. The H°°

Loop-Shaping weighting functions used to synthesise this controller were thus adjusted to

decrease the roll-off rate for Loop 1 around crossover. The resulting controller, Controller

2 , was then found to avoid exclusion region B , for both loops, and for all plants in Pi - see

41

Controller 1 - a lpha d em an d Controller 1 - th e ta d em an d

co»(00.5 0.5

Controller 2 - alpha dem and Controller 2 - theta dem and

0.5

time (s)

0.5

time (s)

Fig. 2.23: Responses to pilot step demands on a and 0 - Controller 1 and Controller 2

O

o

A/a

1.14

Fig. 2.24: Multi-loop system with uncertainties - test for loop 1, exclusion region B

Figures 2.28 and 2.29. Note tha t for these ‘toy’ examples, both the plant uncertainty and

the ex tra uncertainty introduced by the Nichols region test, are purely complex. In the case

of a plant subject to purely real param etric uncertainty, the introduction of the complex

uncertainty associated with the Nichols region test means th a t algorithms for computing

bounds on mixed p will then be required. Finally, also note th a t the problems of computing

the maximum allowable level of plant uncertainty or size of exclusion region before insta

bility occurs can be cast as skewed-p calculations [17], and thus can be answered exactly in

42

■ M l

Fig. 2.25: Standard M A form for robustness analysis using n

the structured singular value robustness analysis framework.

2.4.2 M ulti loop elliptical Nichols plane exclusion regions a s a / i problem

For multi loop analysis, a weighted uncertainty associated with the Nichols exclusion region

of Figure 2.16 is introduced at the input of each actuator and /o r sensor in each loop of

the system. The weight on this uncertainty is initially selected according to an arbitrary

choice of gain and phase margins, fi is then computed for the closed loop system and if its

value is not equal to 1 , the exclusion regions are shrunk or expanded accordingly using this

weighting factor. In this way, the weight of the uncertainty is varied until /x = 1 . At this

point, the limit of stability has been reached and the worst-case phase and gain margins can

be obtained.

This m ethod allows simultaneous variations of the uncertainty in all loops and thus every

possible combination of the phase/gain offset to be considered. In contrast, the classical

m ethod explained in Section 1.3.2 (as used in practice) assumes the same phase and gain

margin variations in each loop and also checks for only a few points on the edges of the

specified exclusion region. Hence, optimistic results are likely to occur with the classical

approach. This approach to evaluating the multiloop Nichols criterion will be further illus

tra ted in Chapter 5.

2.4.3 /x as a Nichols plane stability margin

To address the stability margin criterion, a new technique, based on the idea presented in

the last section, is used to provide an easy alternative to the classical methods described in

Section 1.3. It was shown, in Section 2.4, th a t a one-to-one mapping can be made between

a particular Nichols plane robustness test and a certain bound on fi. This means th a t each

value of /x will correspond to a certain phase margin and gain margin. Thus, we propose

to use the value of /x as a measure of the amount of non-compliance or distance to non-

compliance with a specified Nichols plane G ain/Phase margin requirement. This approach

43

1.5

Loopl exclusion region

- exclusion region B test

Lopp2 exdusion: region A test

0.5

© (rad/s)

Fig. 2.26: Plot o f p bounds for closed-loop system: loopl (-), loop2 (-.-)

Nichols Plot

20Loop 1

Exclusion region Am2.

& Exclusion region Ba.8JC

I -10

Loop 2

-2 0

-3 0

-200 -180 -160 -140 -120 -100Open-loop phase (degrees)

Fig. 2.27: Nichols plots for worst-case systems: loop 1 and loop 2

44

1.4

Loop 1 - exclusion region A test

test

Loop 2 - exclusion0.8

0.6

0.4

0.2

(o (rad/s)

Fig. 2.28: Structured singular value plot for closed-loop system - Loopl (-), Loop2

Controller 2

Nichols Plot

Exclusion region A

Exclusion region B

-1 0

Loop 2Loop 1

-1 5

-2 0

-2 5

-180 -120 -100-200 -160O pen-loop phase (degrees)

-140-220(degrees)

Fig. 2.29: Nichols plots for worst-case system s - Loop 1 and Loop 2, Controller 2

45

eliminates the need for scaling the Nichols exclusion region since the stability margin is

already normalised ( /1 is equal to one on the boundary of the exclusion region). We shall

therefore define the stability margin, p, as the reciprocal of p (p — 1 /p ) such tha t robust

stability is achieved if p > 1 .

2.4.4 Single loop trapezoidal Nichols plane exclusion regions as a p problem

This method, developed in [36], models the Nichols exclusion regions of Figure 2.15 using a

Pade approximation. The variations in the phase and gain can be represented by equations

(2.44) and (2.45) respectively. The phase offset is given by

, I tymax ~ (frmin \ r , / (fimax T 0 min \ a a \4 = { ------- 2------- ) 62 + { --------2--------) (2'44)

The gain offset a (in dB) is represented as

a = S i(t — 77162) (2.45)

where <5i and 6 2 are normalised real uncertainties, and t and m characterise the top limit

line of the exclusion region. For instance, the exclusion region B in Figure 2.15 for the single

loop analysis requires th a t t = 3 and m = 1.5.

To cast this problem into a p framework, it is necessary to convert these equations to the

polar form re~ ie, where the negative sign denotes phase lag. This gives

re~3e — ecSi(t-mS2)-j('nS2+'r2)

— e- i l 2 ec&\{t-m52) - 3 'y\52 (2.46)

where c = (ln l0 ) / 2 0 , 7 1 = and 7 2 =

To generate the LFT-based uncertainty description, a first order Pade approximation is then

used:

e " Ts = 1 - i r k ( 2 -4 7 )1 "T" 2

where, as shown in [36, 37], T s can be replaced by cS\(t — 1x16 2) — jli& 2 -

This first order approxim ation is adequate for phase margins of up to 90°. The resulting

LFT-based uncertainty model for this first order Pade approxim ation is shown in Figure 2.30.

The uncertainty block, A margins, is made up of two real scalars, £1 and 6 2 , the latter being

repeated twice.r 61 0

A margins (2.48)0 S2I2

As shown in Figure 2.31, this method matches the exclusion region used by the classical

approach (see Figure 1.2) very well. The m ethod uses real uncertainties only, as opposed

46

t

m

0.5

Fig. 2.30: L F T representation o f trapezoidal Nichols exclusion region using first order Pade

approximation

to the m ethod described in Section 2.4.1, wherein a complex uncertainty associated with

the elliptical Nichols exclusion region is introduced. As shown in [36], these trapezoidal

exclusion regions are less conservative than the elliptical exclusion regions. To investigate

this m atter further, we will compare the two methods in the analysis of the HWEM CL002

law presented in C hapter 5.

2.4.5 M ulti loop trapezoidal Nichols plane exclusion regions as a p problem

The same procedure is carried out as in the previous section to approximate the exclusion

region using the Pade approximation. For the multiloop case, the gain and phase margin

requirements are 3 dB and 30° respectively and therefore, the param eters m and t from

equation (2.45) have to be chosen as m = 1 and t = 2. The multiloop criterion is then

checked by scaling the exclusion region by applying a scaling factor to m, t and 71 until

p = 1 . At this point, the gain and phase margins can be computed by back-substituting

these values in equations (2.44) and (2.45). This idea will be further illustrated in Chapter 5.

2.4.6 C om putation o f p for elliptical and trapezoidal Nichols plane exclusion

regions

As m entioned in Section 2.4.1, the uncertainty associated with the elliptical Nichols exclusion

is complex and it is therefore necessary to use mixed-p algorithms to compute bounds on

p. The quality of the resulting bounds will depend on the amount of complex versus real

uncertainty. In the case of the trapezoidal Nichols exclusion region, purely real uncertainties

47

are introduced by the Pade approximation and thus this problem necessitates the use of real-

fi algorithms for generating bounds on p.

Phase ( 0 )

Fig. 2.31: Trapezoidal Nichols exclusion regions

2 .5 fi to o ls fo r t h e u n s ta b le e ig e n v a lu e c r i t e r io n

In this section we describe how the structured singular value robustness measure can be

used to address the unstable eigenvalue clearance criterion defined in Chapter 1 . Prom the

standard block diagram for /i-analysis, stability of the closed loop system is equivalent to

stability of the quantity ( I — M \\ A )-1 . By testing the stability of ( / — M \\ A )-1 as the A*

elements vary, we can find the worst case, or smallest, set of simultaneous changes in Ai

which drive the system unstable. From m atrix theory,

(I - M n A )_l = adj(7 - M n A )/d e t( / - M n A)

Thus, for a given set of model perturbations A , and a given complex number so that is not

an open loop pole of M n ( s ) or A (s), so is a closed loop pole if and only if

d e t(7 - M n ( s 0) A ( s 0 )) = 0

Suppose we want to find the smallest set of Ai elements which places a pole at so:

km = : min {A; 6 [0 , oo] such tha t det(J — M n(so)A(so)) = 0 }

48

where A = diag(A i....A p) and cr(Af(so)) < k V i

Then

^ a ( M i i ) = 1 /km

Most published work on //-analysis has assumed th a t // must be computed on a frequency

sweep along the s = juo axis. However, computing // away from the imaginary axis can

provide a lot of useful information about changes in the closed-loop performance as well.

Some possible tests are shown in Figure 2.32. The unstable eigenvalue criterion is checked

Im(s)constantdampingtest

stability test

worst-case unstable eigenvalue test

Re(s)real-axis eigenvalue test

worst-case stable eigenvalue test

Fig. 2.32: Possible real-fi tests in the s-plane

by shifting the imaginary axis into the left half plane until an uncertainty combination is

found which places a closed loop pole on the axis. By sweeping so along a line of constant

dam ping, such as £ = 0 . 4 , one may find the smallest perturbation which reduces dam p

ing below this level. Since km is typically discontinuous as so moves from the real axis to

neighbouring complex points, it is also useful to check stability along the real axis. Another

useful way to present the data is to compute km on a grid in the s-plane around a nom

inal closed-loop pole, and then make a contour map of km . This shows directly how the

closed-loop poles migrate in the s-plane as a result of the uncertainty, i.e., it corresponds to

a m ulti-param eter root locus - see [40] for more on this approach.

Another possibility is to study the behaviour of some specific eigenvalues (e.g., those asso

ciated with the longitudinal and lateral modes of the aircraft) as the aircraft control law

is subjected to increasing percentages of the worst-case uncertainty. In this approach, we

first of all plot the eigenvalues of the nominal closed-loop system, (A =0). We then use fi

49

to calculate the worst-case A matrix, and plot the associated closed-loop eigenvalues. The

nature of the movement of the closed-loop eigenvalues from the nominal to the worst case

can then be shown by plotting the eigenvalue positions for different percentages of the worst

case A, and ‘joining the dots’ to make a root-locus type plot. Plots of this type can provide

much useful information about the relative movement of the different eigenvalues with re

spect to increasing uncertainty. In addition, for the case of systems which are de-stabilised

by the worst-case A, this method can help to understand which pole goes unstable first

(an additional requirement for the unstable eigenvalue criterion) and also at what level of

uncertainty this occurs.

2 .6 C on clu sion s

In this chapter, we have shown how certain classical flight control law clearance criteria

can be recast into a /z framework. The resulting new analysis tools allow a more rigor

ous analysis of the control law to be performed. Indeed, clearance criteria can be checked,

and worst cases found, for all possible combinations of the values of the uncertain aircraft

param eters. This provides a stronger guarantee th a t a criterion is not violated than tha t

provided by traditional gridding approaches, which generally only check th a t the criterion is

not violated for all combinations of the extreme values of the uncertain parameters. Whole

portions of the flight envelope can be cleared by including flight param eters such as Mach

number or a ltitude as additional uncertain param eters in the LFT-based uncertainty model,

thus removing the need to grid the flight envelope itself in the analysis. Clearance criteria

can also be checked for continuous intervals of, for example, angle of attack and load factor.

Com putation times for classical gridding approaches increase exponentially with the number

of uncertain parameters. Com putation times for fi bounds are generally polynomial func

tions of the number of uncertainties. Thus, for clearance problems involving large number

of uncertain param eters, /z-analysis tools can offer significant com putational savings.

Since /z-analysis results are given as a function of frequency, they also convey more informa

tion about the nature of the worst-case uncertainty and how it affects the aircraft dynamics

than is generally provided by classical methods.

Different approaches to LFT-based uncertainty modelling have also been presented for both

linear and nonlinear systems. These LFT-based uncertainty modelling techniques each have

their own advantages and disadvantages, and their suitability for a particular application

depends on the nature of the model being considered. The uncertainty bands approach is

fast and easy but does not provide the worst case uncertainty combination in terms of the

actual uncertain parameters. In addition, conservatism can be introduced in the results.

50

The physical modelling approach can be relatively straightforward to implement given a

detailed model in a SIMULINK block type representation, and the exact worst case uncer

tain ty combination is returned. However, this approach does not easily allow the generation

of LFT-based uncertainty models which cover whole regions of the flight envelope. The

symbolic approach is much more complex to implement and verify, but can return the exact

worst case uncertainty combination, even over a whole region of the flight envelope.

The new /i-analysis tools presented in this chapter are used in the clearance of the flight

control laws for three different aircraft models, in Chapters 3, 4 and 5 of this thesis.

51

C hapter 3

C learance o f a civil transport aircraft flight control law

This chapter considers the problem of generating LFT-based uncertainty models for linear

the aircraft wherein the uncertain param eters are explicitly defined. A physical modelling

approach is then used to generate the LFT-based uncertainty models. The main advantage

However, detailed information about the way in which the uncertainties affect the aircraft

dynamics is required. Moreover, there is the risk of the occurrence of repeated uncertainties

which can be problem atic in the com putation of bounds on real /i. Two linear stability

clearance criteria, namely the stability margin criterion and unstable eigenvalue criterion,

are investigated. The proposed approach is illustrated for the clearance of a control law

designed using a constrained Hoo loopshaping approach for a civil transport aircraft model

3.1 T h e c iv il tra n sp o r t a ircraft m od el

The problem considered consists of the analysis of a flight control system for a civil transport

aircraft. Only the lateral axis is considered and the rigid model (taken from [32, 41]) is

characterized by four states x = r, 0 ]T , four outputs y = [ny,p, r, 4>] and two control

inputs u = [<5p, <5r]. The linearized lateral equations of motion, at a trim value (<*o,0o) °f

the angle of attack and of the pitch angle, are given as:

aircraft models. This approach assumes the availability of the linear equations of motion of

of this method is th a t the worst-case is obtained in term s of the actual uncertain parameters.

[5, 6],

qSbCy„4- sinao)p + ( qSbCYr

2 mVr— cosao)r + j-cj) + ^ - - Sp 6p + - - - - -- Sr (3.1)V m m

, qSbCL„ o , qSb2 CLp _ , qSb2CLr _ , qSbCL&p , qSbCLsr T) — 1 ■ 1 f j 4* T) --f" ■ ■ T “f“ vJ) ~j“* 0 VF 40.07m 80.14V'Tm 80.14VVm 40.07m 40.07m

(3.2)

qSbCnil Q ( qSb2 CUp _ , qSb2Cn„ _ , qSbCns„ ^f — --------------- f j 4* " 7) -----------------------V ---------------- OT

99.92m 199.84Vrm 199.84VTm 99.92m(3.3)

4> = p + tanOor (3.4)

The acceleration at the centre of gravity is given by

n yqSbCYp qSbCYr (3.5)

52

A second order actuator is added at the aileron control input Sp:

N i - 1 .7 7 s + 399 ~D{ ~ s 2 + 48.2s + 399

(3.6)

and a third order actuator is added at the rudder control input 8 r:

N 2 _ 2 .6 s2 - 1185s + 27350D 2 ~ s3 + 77.7s2 + 3331s + 27350 '

3 .2 D esig n o f a con tro l law for a c iv il tra n sp o rt aircraft m od el u sin g

con stra in ed o u tp u t feedback

The development in this section follows th a t in [6 ].

3.2.1 The H o c loopshaping design m ethod

The Hoo loopshaping controller design method, introduced in [38], has been used to suc

cessfully design and implement controllers for a wide variety of aerospace applications

[42, 43, 44, 45, 46, 47]. The keys to the success of the method are the transparent, largely

classical nature of the design process and also the excellent robustness properties of the

resulting controllers. A comprehensive tutorial on the Hoc loopshaping method is given in

[48]. In this section, only a summary of the key features of the procedure is given. The

Hoo loopshaping design m ethod is essentially a two stage process. First, the open-loop

plant is augmented by (generally diagonal) weighting matrices to give a desired shape to

the singular values of the open-loop frequency response. Then, the resulting shaped plant is

robustly stabilised with respect to coprime factor uncertainty using Hoo optimisation. An

im plem entation structure for Hoo loopshaping controllers is shown in Figure 3.1.

W 2K o o (s)

Fig. 3.1: Implementation structure o f the Hoo loopshaping controller K {s )

53

W ith reference to this figure, the weighting m atrix W i(s) is chosen to add integral action

and ensure reasonable roll-off rates for the open-loop singular values around the desired

crossover frequencies. The scalar weighting m atrix k is then used to adjust control actuation

requirements with respect to the the various actuator rate and magnitude limits. Note tha t

in this configuration the plant G is assumed to be scaled so as to be approximately normalised

w ith respect to maximum allowable input signals. The scalar diagonal m atrix W 2 is used

to prioritise certain controlled variables over others.

The second stage of the Hoo loopshaping design method involves the use of H 00 optimisation

to compute a controller block Koq which robustly stabilises the shaped plant against a

particular type of uncertainty description, based on stable perturbations to each of the

factors in a coprime factorisation of the plant. For a plant G w ith a normalised left coprime

factorisation [18],

G = M ~ l N (3.8)

an uncertain plant model Gp can then be w ritten as

Gp = (M + A m ) - 1(W + A at) (3.9)

where A m and A n are stable unknown transfer function matrices which represent the un

certainty in the nominal plant model G - see Figure 3.2. The objective of robust stabilisation

is to stabilise the class of perturbed plants defined by

Gp = (M + A m ) l (N + Ayv) : IHA^Aa/IHoo < e (3.10)

Now, as shown in [38], the largest possible class of such systems, i.e, the maximum value of

e, €m,ax, is given by:

K7 min — inf (.I - G K )~ 1M ~ 1

K I< 1

(3.11)

where 7 min is the stability margin. Note tha t 7 m*n is the Hoo norm from 0 to in

Figure 3.2.

The solution of equation (3.11) is particularly attractive in th a t the optimal 7 can be found

without recourse to 7 -iteration which is normally required to solve Hoo control problems.

Given a minimal realisation [A, B ,C , D] of a controllable and observable plant, the solutions

X and Z of the two Riccatti equations

v42Z + Z A l - Z C *R ~ l C Z + B S ~ l B* = 0

A tX + X A , - X B S ~ l B * X + C *R ~ 1C = 0

(3.12)

(3.13)

54

oo

- 1

Fig. 3.2: Normalised coprime factor uncertainty description

where

A z — A — B S ~ l D*C

R = I + DD*

S = I + D*D

give the optim al 7

= {1 - ||[iV M ]||^}-^ = (1 + p {X Z ))± (3.14)

where ||.||// denotes the Hankel norm and p denotes the spectral radius. The central con

troller K qq which guarantees tha t

K(I - G K ) ~ l M ~ l

I-

< 7

for a specified 7 > 7 m*n, is then given by

Koo =A k B k

c k o k _

with

A k = A + B F + ■y2 (L , )~ 1 Z C '(C + D F)

B k = y i L T ' z c r

Ck = B - X

D k = - z r

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

where

F = - S ~ l (D*C + B * X )

L = ( 1 - 7 2) I + X Z

55

(3.21)

(3.22)

For 7 < 4 (i.e. allowable coprime factor uncertainty of > 25%), it can be shown theoretically,

[38], th a t the controller K qo(s) does not significantly change the shapes of the open-loop

singular values. Thus, robust stability is achieved w ithout significant degradation in per

formance characteristics. If 7 is greater than 4, this indicates th a t the chosen loop shapes

are incompatible w ith robust stability, and further adjustm ent of the weighting functions is

then required.

T he final step of the design procedure is to add the constant prefilter K oo(0 )W 2 in order

to ensure zero steady sta te tracking error, assuming integral action in W \. Note tha t the

Koc controller block calculated from (3.16) is an unstructured compensator of order equal

to th a t of the shaped plant. In the next section, we show how to compute a static feedback

m atrix K ^ which optimises the stability margin 7 subject to constraints on the controller

structu re and order.

3.2.2 T he constrained static H o o loopshaping m ethod

As described in the previous subsection, the Hoo loopshaping design procedure is a two step

process - first, the singular values of the open-loop plant are shaped using weighting func

tions to give desirable performance and robustness properties, then a feedback compensator

Koo is com puted to optimise the robustness margin given in (3.11).

Since the designer is free to choose the weighting functions to reflect the various constraints

on the structu re of the overall controller, we only need to change the second step of the pro

cess, i.e., a constrained static controller Koos must be computed to optimise the robustness

margin 7 . As analytical solutions to this problem are not currently available, we propose

to use numerical optim isation techniques to choose the gain m atrix K ^ which provides

the minimum value of 7 subject to the various constraints on the elements of Koos• An

advantage of this approach is th a t by formulating the controller synthesis problem as a

non-linear constrained optim isation problem, extra constraints on the closed-loop system

can easily be incorporated directly into the design; for instance, to satisfy stability margin

specifications, we can require th a t all the closed-loop eigenvalues to lie to the left of -0.5

in the complex plane. S tandard software from, for example, the MATLAB Optimisation

Toolbox, can be used to formulate and sub-optimally solve the resulting controller synthesis

problem as follows.

Let [A ,B ,C , D] be the shaped open-loop plant, Kinfs be the constrained static Hoo con

troller and G be the transfer function m atrix (3.11) from </> to [u, y\ as shown in Figure 3.2.

The following MATLAB code can then be used to compute Kinfs:

56

% Set constraints

> > Kinfs = [x(l) 0 x(3) x(4); 0 x(2) 0 0];

> > AC = A + B*Kinfs*C;

> > g(l) = max(real(eig(AC))) + 0.5;

% Formulate optimisation problem

> > f = normhinf(G);

> > function [f,g] = FUN(x);

% Set bounds on the controller gains

> > vlb = -30*[1 1 1 1];

> > vub = -l*vlb;

% Initial ‘guess’ for controller parameters

> > xO = [19 0.2 -5.3 9.5];

% Compute Kinfs

> > x = constr(‘FU N ’,x0,0,vlb,ulb);

Note th a t the optim isation problem formulated above is non-convex, and thus the solution

may be a local optim um . Also, the iterative algorithm requires an initial stabilising ‘guess’

for the controller gain m atrix. One way to find an initial stabilising gain is proposed in [60]:

firstly, a stabilising state-variable feedback gain K is computed for the shaped plant and

then to determ ine a stabilising ou tpu t feedback gain, those elements of the state feedback

m atrix th a t do not correspond to the measured outputs are weighted in a linear quadratic

(LQ) cost function. Those elements will then be forced to zero, and an initial stabilising

ou tpu t feedback m atrix may be com puted.

A nother approach for com puting this initial gain is to generate an initial dynamic compen

sator Koc using standard Hoo optim isation, and then to apply model reduction techniques,

such as the Hankel norm approxim ation [49], to reduce the order of the designed controller to

zero. This gain m atrix can then be used as the initial ‘guess’ in the optimisation algorithm.

We note in passing th a t while order reduction techniques can be useful in finding an initial

stabilising sta tic controller [6 ], they will generally not yield an optimal static Koos- This

was clearly shown in the com parative study of reduced order Hoo loopshaping controllers

versus equivalent order fixed-structure PID controllers calculated using optimisation meth

ods in [42].

Proceeding according to the m ethod described above, a constrained static output feedback

control law is synthesised using Hoo loopshaping techniques to provide the desired closed-

loop stability.

57

The weighting m atrices W \ and W 2 (see Figure 3.1) were chosen as:

m ( s) (3.23)

1 0 0 0

0 1 0 0w 2 (3.24)

0 0 1 0

0 0 0 1

W \ was chosen to ensure good disturbance rejection at low frequencies and a good roll-off

ra te around crossover. W 2 is generally used to reflect the relative importance of the outputs

to be controlled and was chosen as the identity m atrix for this example.

The control input used is an ou tpu t feedback of the type

where A;i = -3 .995 , k2 = -2 .9348, k3 - 5.8936, k4 = -2.0510

3 .3 L F T -b a sed u n ce r ta in ty m o d e llin g o f a c iv il tra n sp o rt aircraft u sin g a

p h y sica l m o d e llin g a p p roach

The development in this section follows th a t in [5]. Uncertainties are introduced in the mass

of the aircraft and in the 14 coefficients which characterize the aerodynamic model, namely

the stability derivatives CYp, CYp, Cyr , CYsp, Cy6r, CLf3, CLp, CLr, CLgp, CLsr, CNf3, CNp,

C/vr and C]s/6r • Each of the uncertain param eters is allowed to vary within ±10% of its

nominal value. For instance, the mass can be rew ritten as m = m°{ 1 + IT iA i) where m°

is the nominal value of the mass, W \ is a weight on the uncertainties (10%) and A i, which

varies in the interval [-1,1], represents the normalized param etric uncertainty in mass.

Since the symbolic linearised dynamic equations (3.1) to (3.5) of the civil transport aircraft

model already contain the uncertain param eters, it is possible to generate an LFT-based

uncertainty model using the procedure explained in Section 2.3.1.

The approach proposed is to represent the uncertain equations of motion in a block diagram

form. Consider for example the first term of equation (3.1) relating (3 to (3:

u = —K y (3.25)

with

K =ki 0 0 0

0 k2 k3 k4

O ther term s in (3.1)

m

Fig. 3.3: B lock diagram representation o f (3 to $ term o f equation (3.1)

This p art of equation (3.1) can be represented in block diagram form as shown in Figure 3.3.

Assuming th a t the Cy0 param eter is subject to uncertainty of ±10%, we can write C y a =

C y 0 { \ + W jA i) where C y g is the nominal value of C y0 , W \ is a weight on the uncertainty

(W \ = 0.1 in th is case) and A i varies in the interval [-1,1]. Note th a t any variation in

a param eter of the form x min < x < Xmax can be w ritten as above in terms of a nominal

value and a weighted uncertainty in the interval [-1,1]. Now, the uncertainty in the aircraft

dynam ics due to C y 0 can be represented physically in block diagram form as shown in

Figure 3.4. In Figure 3.4 we have added ex tra ‘fictitious’ inputs and outputs w\ and z\ at

W\

O ther term s in (3.1)

Fig. 3.4: B lock diagram representation o f (3 to $ term o f equation (3.1), with uncertain C y0

param eter

the point in the system where the uncertainty A i occurs. This step is then repeated for each

Ai representing the uncertainty in the other stability derivatives and the mass. Now, using

s tandard block diagram m anipulation software (e.g. the function linm od in MATLAB) it is

then easy to calculate the transfer m atrix of the system M w ith inputs u = [w\, ....wn , £p, <5r]

and ou tpu ts y = [z\, ....zn ,n y ,p, r, (/>]. The LFT uncertainty model for the system, shown in

59

Figure 3.5, is then given by the relation

n y

P

r

<t>

= Fu(M (s), A)5 p

<5r

where A = diag(A i, A 2 , A 2 8 ) contains for this example 28 real parametric uncertainties,

14 of which are repeated scalars.

w 1 , ...W28M(s)

r

Fig. 3.5: L F T uncertainty m odel for lateral axis analysis example

Clearly, the approach outlined above is simple and intuitive. The approach allows an exact

description of joint param etric dependencies in the model and thus, it can be used to non-

conservatively model the effect of the param etric uncertainties on the closed loop system. As

a result, the exact worst-case set of uncertain param eters can be computed. It is obvious,

however, th a t the task of building the block diagram model can be tedious for complex

high order systems. A nother drawback of this approach is th a t it tends to produce A

m atrices with many repeated param eters, since each instance of the same uncertainty is

modelled individually. Thus in the above example the A* associated with the uncertain

mass term is repeated 14 times. A lthough this problem can sometimes be eased by clever

arrangem ents of the block diagram representation (by factoring out m in the equations of

m otion for exam ple), it again becomes difficult to avoid for complex systems. Note also

th a t the aircraft equations are not affine with respect to the uncertainties in mass and theqSCy«

stability derivatives since products such as — appear. Thus the method in Section 2.3.3

cannot be used to generate an LFT-based uncertainty model for this system.

3 .4 R e su lts for th e s ta b ility m argin cr iter ion

In th is section, we present results for the stability margin criterion with respect to the

avoidance of the Nichols exclusion region defined in Section 2.4.

60

3 .4 .1 S in g le loop an alysis

The test consists of plo tting the open loop Nichols response by breaking the loop at the

input of each of the aileron or rudder actuators. Here, we test for the avoidance of the

exclusion region providing gain and phase margins of ±4.5 dB and ±28.44° respectively.

T he resulting n bounds and the corresponding Nichols plot for the aileron loop cut (with the

rudder loop closed) are shown in Figures 3.6 and 3.7. It can be observed tha t the Nichols

exclusion region is not violated.

W ith the rudder ac tuato r loop cut, it can be further noted th a t the test is also passed as

shown by the /i and Nichols plots in Figures 3.8 and 3.9.

0.9

0.8

0.7

0.6

0.4

0.3

0.2

10'to (rad/s)

Fig. 3.6: M ixed fi bounds, Nichols exclusion region test

3 .5 R e su lts for th e u n sta b le e igen va lu e criter ion

Using the m ethod explained in Section 2.5, n was com puted along lines parallel to the

im aginary axis into the LH-plane until its value reaches 1. At this point, the magnitude of

the real p art of the worst-case eigenvalue is obtained. The fi plot is shown in Figure 3.10.

Figure 3.11 shows a plot of the worst-case eigenvalues for the closed loop system. It can be

noted th a t all the eigenvalues lie w ithin the boundaries defined in Section 1.4, thus satisfying

the unstable eigenvalue criterion.

61

Gain

(dB)

Aileron loop cut3 0

— Worst C ase— Nominal

20

-1 0

-2 0

-3 0

-4 0

-5 0

-2 0 0 -180 -1 6 0 -80-140 -120 -100P hase (°)

Fig. 3.7: Nom inal and worst case Nichols plot (aileron loop cut)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

o) (rad/s)

Fig. 3.8: Mixed p bounds, Nichols exclusion region test

62

Gain

(dB

)

R u d d er Loop Cut20

Nominal Worst C ase

-10

-20

-3 0

—40—220 -2 0 0 -180 -1 6 0 -140 -120 -100 -80

P hase (°)

Fig. 3.9: Nom inal and worst case Nichols plot (rudder loop cut)

0.9

0.8

0.7

0 6

0.4

0.3

0.2

0.1

cd (rad/s)

Fig. 3.10: /i bounds for worst-case eigenvalue

04 I I--------- 1--------- 1--------- 1--------- 1--------- ---------

0.3 - .................................................................................... X -

0 .2 - -

0.1 - • -

Jl 0 - X ■ X ------------------

-0.1 - -

-0 .2 - -

- 0 . 3 - ......................................................................... X -

-0 41----------'--------- '--------- '--------- 1--------- 1--------- '--------- ----------0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0.1 0 0.1

Re

Fig. 3.11: Worst case eigenvalues

3 .6 C o n c lu s io n s

A simple physical modelling approach for generating LFT-based uncertainty descriptions

was presented for a linearised civil transport aircraft model. A control law for this aircraft

was synthesised using the m ethod of Tioo loopshaping in combination with optimisation tech

niques. The LFT-based uncertainty modelling approach presented is simple to implement

and also generates the worst-case uncertainty set in term s of the actual uncertain parame

ters. However, it requires detailed inform ation about the effect of the uncertain param eters

in the model. Moreover, it is necessary to obtain the equations of motion in a rational form,

with the uncertainties explicitly defined in them. Another drawback of this method is that

repeated param eters can occur and th is may lead to conservatism in the upper bound of /i.

j ___________ i___________ i___________i___________ L..7 -0 .6 -0 .5 -0 .4 -0.3

Re- 0.2 - 0.1 0 0 .

Fig. 3.11: Worst case eigenvalues

64

C hapter 4

C learance o f th e H IR M + R ID E flight control law

In th is chapter, we consider a numerical approach (based on uncertainty bands) to LFT-

based uncertainty modelling for a detailed nonlinear fighter aircraft model called the HIRM+.

This m ethod requires little inform ation about the way the uncertain parameters enter the

model and can quickly generate LFT-based uncertainty models. Although it introduces some

conservatism, this LFT-m odelling technique can be easily implemented and fully autom ated

[1, 7, 8 ]. The stability m argin criterion and unstable eigenvalue criterion are investigated

and the clearance results com pared w ith those produced using the classical approach [50].

4.1 T h e H I R M + a ircraft m o d el

The HIRM -f model has been developed from the HIRM, a m athem atical model of a generic

fighter aircraft originally developed by QinetiQ. The HIRM is based on aerodynamic data

obtained from wind tunnel tests and flight testing of an un-powered, scaled drop model.

The model was set up to investigate flights a t high angles of attack (-50° to +120°) and over

a wide sideslip range (-50° to -(-50°), bu t does not include compressibility effects resulting

from high subsonic speeds. The origin of the model explains the unconventional configura

tion w ith bo th canard and tailplane, plus an elongated nose.

The aircraft is basically stable. However, there are combinations of angle of attack and con

trol surface deflections which cause the aircraft to become unstable longitudinally and/or

laterally. Engine, ac tuato r and sensor dynamics models have been added to create a rep

resentative, nonlinear sim ulation model of a twin-engined, modern fighter aircraft. The

model building was done using the object-oriented equation-based modelling environment

DYMOLA [51]. In building the H IR M +, the emphasis has been laid on the realistic mod

elling of param etric uncertainties. A dditional param eters have been defined to allow for

uncertainty in mass, inertial data, position of the centre of gravity, aerodynamic control

power derivatives, stability derivatives, and some coefficients in the actuator and engine dy

namics. In spite of these changes, the nominal models of H IRM + and HIRM are essentially

the same [52]. The states, inputs and outputs of the flight dynamics are:

x \Vt a s a 0 p q r (l> 6 4> x e ye ze}' (4.1)

u [&rs $ td $cs $cd $r suction Fsivn Fsivi2 ugust vgust wgust}' (4.2)

y = \p q r 6 <f> if) anx any anz Vt a s M h a (3]' (4.3)

65

4 .2 T h e R ID E co n tro l law

R obust Inverse Dynamic Estim ation (RIDE) has been used to synthesise the control law.

T he RIDE control law was developed for the HIRM model by QinetiQ. This design method

is based around a multivariable proportional plus integral structure with an additional

dynam ic inverse input th a t inverts the model w ith respect to the outputs. The control law

has an inverse dynam ics loop acting to decouple the outputs, a proportional feedback loop

to provide stability and an integral loop to compensate for errors in the estimate of the

inverse dynamics. In addition, a feedforward component can also be added to shape the

transient response.

The dem ands to the control law are pitch rate, roll rate and sideslip. A speed controller is

also implemented. A full description of the design process and implementation of RIDE is

provided in [51].

4 .3 T h e H I R M + fligh t en v e lo p e

The flight envelope of the H IRM + is shown in Figure 4.1. Eight flight conditions are defined

x 1 o4 Flight Envelope4.5

FC8FC3

3.5

FC5

FC2

FC6

FC4

FC7FC10.5

0.6 0.7 0.8 0.90.3 0.4 0.50.2Mach

Fig. 4.1: Flight envelope o f the H IRM +

in Table 4.1 and shown in Figure 4.1. The flight cases considered are all at equilibrium

conditions, in straight and level flight (for given 7 , M and h), and in pull-up manoeuvres

(for given a , M and h), characterised by different values of a and n z . The chosen range of

66

variation of a has been chosen as [-15°,+35°] and a typical step used in the analysis is A a

= 2°. The uncertain param eters of the H IRM + are shown in Tables 4.2 and 4.3. For the

F lig h t c o n d it io n FC1 FC2 FC3 FC4 FC5 FC 6 FC7 FC8

M a c h n u m b e r 0.2 0.3 0.5 0.5 0.6 0.7 0 .8 0 .8

H e ig h t ( f t) 5000 25000 40000 15000 30000 20000 5000 40000

Table 4.1:: Test points in the flight envelope

purpose of this analysis, bo th the longitudinal and lateral uncertain parameters are rated

into two categories: Category 1 is the most relevant for clearance and is therefore m andatory

in the analysis task while Category 2 is generally considered the less im portant.

P a ra m e te r V ariable namei [Min,Max] values U nits Category

X-centre-of-gravity offset

from nominal centre of gravity Xcgunc [-0.15,+0.15] m 1

Uncertainty in the moment of

inertia about the y-axis lyyunc [-0.05,+0.05] 1

Uncertainty in pitching moment

due to incidence (-'maUnc [-0 .1 ,+0 .1] 1/rad 1


derivative due to symmetrical

tailplane deflection Cm&TSUixc [-0.04,+0.04] 1 /rad 1


due to pitch rate CrriqUnc [-0 .1 ,+0 .1] 1

Uncertainty level of aircraft mass TTlCLSSiJnc [-0 .2 ,+0 .2 ] - 2

Z-centre-of-gravity offset

from nominal centre of gravity Zcgunc [-0.04,+0.04] m 2

Uncertainty level of Ixz 7X Zy tic [-0 .2 ,+ 0 .2 ] - 2


derivative due to symmetrical

canard deflection Cmscsunc [-0 .0 2 ,+0 .0 2 ] 1 /rad 2

Table 4.2: Longitudinal uncertain parameters o f the HIRM +

W hen several aerodynam ic uncertainties are simultaneously used in the analysis, reduction

factors m ust be applied on their absolute values. The values of reduction factors for different

num ber of aerodynam ic uncertainties are given in Table 4.4. Due to load factor limitations

and control surface deflection limits, it is not possible to trim all flight conditions of Table 4.1

67

Parameter Variable name [Min,Max] values Units

Y-centre-of-gravity offset

from nominal centre of gravity y1 cgunc [-0.10,+0.10] m


inertia about the x-axis IxxVnc [-0.2,+0.2]


inertia about the z-axis IzZUnc [-0.08,+0.08]

Uncertainty in rolling moment

due to sideslip ClpUnc [-0.04,+0.04] 1/rad

Uncertainty in yawing moment

due to sideslip Cflpunc [-0.04,+0.04] 1/rad

Uncertainty in yawing moment

due to yaw rate C n runc [-0.05,0.05]

Table 4.3: M ost relevant lateral uncertain parameters o f the HIRM +

N um ber o f aerodynam ic uncertainties 2 3 4 > 5

R eduction factor 0.62 0.46 0.37 0.31

Table 4.4: Reduction factors for simultaneous aerodynamic uncertainties

for all angles of a ttack between -15° and +35°. This is already true for the nominal model,

for which all the uncertain param eters are set to zero. The number of non-trimmable points

in the flight envelope increases w ith the number of uncertainties used. This fact has been

accounted for during the assessment.

4 .4 L F T -b a sed u n ce r ta in ty m o d e llin g o f th e H IR M + u sing th e un cer

ta in ty b a n d s ap p roach

LFT-based uncertainty models are derived using the uncertainty bands method described in

Section 2.3.3 since direct relationships between the HIRM + uncertain param eters and the

m odel’s non-linear or linear equations are not readily available. We consider the uncertainty

associated w ith all of the specified uncertain model param eters at various flight conditions

over the envelope. In this approach we also include Mach number and altitude as uncer

ta in param eters in order to assess robustness properties over regions of the flight envelope

(note th a t th is reduces/rem oves the need to grid the flight envelope). For the longitudinal

H IR M + dynamics, linearising at all combinations of extreme points of the resulting vector

of uncertain param eters (including Mach number and altitude) gives 211 linear models plus

68

the nominal. Now for each varying element of each state-space m atrix we calculate its min

imum, maximum and ‘nom inal’ (centre) values. Thus the uncertain state-space system can

be w ritten as:

A B _ Ao Bo TLA

+" Ai 0 T I B

+ E *0 A

C D. C° Do _ 1=1 0 0 i=nA+l 0 0

nc

+ Sii=n,B + l

0 0 ri£)

+ ]C Si0 0

0 i= n c+1 0 A _(4.4)

Each of the m atrices associated with each Si has rank one and can be factored using the

singular value decom position into row and column vectors:

(4.5)

If we now define the linear system P w ith ex tra inputs and outputs via the equations

Ai 0 E i= G i H i

0 0 Fi

X

y

Z\=

1 —.. o s 1

Ao Bo E \ .... E n

Co Do F\ .... Fn

G i H i 0 .... 0

X

u

W\

. Wn .Gn H n 0 .... 0

where n = tid, we can form the closed loop interconnection structure shown in Figure 4.2,

which can then be converted to the standard form shown in Figure 4.3.

4 .5 A n a ly s is c y c le for th e c lea ra n ce p rocess

As sta ted in the previous section, the m ethod for LFT-based uncertainty modelling used in

this chapter is fast and easy, but conservative. As a result, the corresponding fi tests provide

only sufficient conditions for the clearance criterion in question, and the corresponding worst

case uncertainty is given in term s of a particular state-space system, not in term s of the

original uncertain param eters. In order to provide necessary and sufficient conditions, and

in order to be able to identify the corresponding worst case set of uncertain parameters, a

more complex LFT-based uncertainty modelling approach is required. Clearly, the question

as to w hether this ex tra modelling effort is justified can only clearly be answered by first

69

w,

Wi

Fig. 4.2: Interconnection structure o f uncertain closed-loop system

w-

W1

Fig. 4.3: Standard block diagram for p-analysis

applying the fast bu t conservative approach and analysing the results. If for example, we

can positively clear a control law over the whole envelope with the fast approach, little

will be gained by deriving a more exact LFT-based uncertainty model. Also, if only a

few flight cases cannot be cleared using the conservative approach, it may be sufficient to

examine these cases in detail using traditional approaches and /o r time domain simulations.

If, on the other hand, large numbers of flight conditions throughout the flight envelope

cannot be cleared, there is no option bu t to resort to a more complex approach, in order

to determ ine w hether the problem lies with the conservatism introduced in the LFT-based

uncertainty modelling or w ith the control law itself. We thus propose the following cycle

for the application of fi analysis methods to the H IRM + clearance problem:

1. G enerate a fast but conservative LFT-based uncertainty model for the HIRM + using

autom ated tools.

2. Check selected clearance criteria over a sample set of flight conditions.

70

(a) If results m ostly negative: S tart again with more exact approach to LFT-based

uncertainty modelling.

(b) If results m ostly positive: Continue for all regions of the flight envelope.

3. Examine any failure cases using traditional methods and /or simulations.

4 .6 A n alysis o f th e longitud inal H IR M -f dynam ics

In this section, we detail results from the analysis of the longitudinal dynamics of the

HIRM-h RIDE control law, using the n tools and methods described in Chapter 2. The three

flight cases (straight level flight and pull-up manoeuvres) for the longitudinal dynamics are

considered. It should be noted th a t all of the bounds generated were computed using the

procedure described in Section 2.2. The upper bounds were calculated using the mixed fi

upper bound in [17]. The lower bounds were computed using the exponential time algorithm

of [29], after the order of the A m atrix was reduced by eliminating those A*s with smallest

^-sensitivities. The procedure is seen to work well, yielding tight real fi bounds for a very

high order system.

4 .6 .1 R e s u lts fo r th e u n a c c e le ra te d 7 t r im

In the following we present results for the unaccelerated 7 trim over the flight envelope

proposed in [13]. Tests for the various criteria are performed for longitudinal dynamics only,

w ith all specified uncertainties applied simultaneously.

4.6 .1.1 Stability over the flight envelope

Before attem pting to identify over which regions of the flight envelope the RIDE control

law satisfies the stability margin criterion, we first of all identify the regions over which the

aircraft remains stable. The region th a t satisfies the stability margin criterion will then

be a subset of this region. We present results for three points in the flight envelope, FC1,

FC2 and FC3, see Figure 4.4, for which our /i tests fail to guarantee robust stability, see

Figures 4.5, 4.6 and 4.7, and two regions of the flight envelope (regions A and B shown in

shaded) covering FC4 to FC 8 , for which robust stability is guaranteed. By refining these two

regions (region A includes FC4 and region B encloses FC5, FC 6 , FC7 and FC 8 ), the portion

of the flight envelope over which robust stability is guaranteed is found to be as shown in

Figure 4.10 (shaded region). Finally, note th a t although we cannot guarantee stability over

the full flight envelope, the worst case fi values are not much greater than 1 anywhere. Given

the conservatism introduced by the method adopted for LFT-based uncertainty modelling,

71

these results indicate th a t the control law preserves stability over most if not all of the flight

envelope.

Flight Envelopex 10

FC3

FC6

FC2

FC8

FC1..

0.4 0.6 0.7 0.90.1 0£ 0.3

Fig. 4.4: Regions and points in flight envelope for analysis

2.4

2.2

0.8

0.6

0.4

0.2

w (rad/s)

Fig. 4.5: Real p upper and lower bounds for point FC1

72

0.8

0.6

0.4

02

10' 10'

Fig. 4.6: Real gi upper and lower bounds for point FC2

0.8

0 6

0.4

0.2

w (rad/s)


73

0 .7

0.6

0.5

0.4

0.3

0.2

0.1

w (rad/s)

Fig. 4.8: Real p upper and lower bounds for region FC4

0.8

0.7

0.6

0.5

2 0.4

0.3

0.2

0.1

w (rad/s)

Fig. 4.9: Real p upper and lower bounds for flight conditions FC5 to FC8

74

x io4 Flight Envelope4 .5

FC3 PI: (0,23,0)

P2: (0.3,15000) P3: (0.37,25000) P4: (0.53,40000)

:C8

3.5

FC5

FC2 P3

FC6

0.5

0.1 0.2 0.3 0.4 0.5 0.80.7 0.9Mach

Fig. 4.10: Cleared portion o f the flight envelope - stability

4.6 .1.2 W orst-case stability margin over the flight envelope

In this section we present clearance results for the stability margin criterion, with respect to

violation of Nichols plot exclusion regions as defined in Section 2.4.1. We focus on the two

regions of the flight envelope, covering flight conditions FC4 to FC 8 , for which our initial p

tests guaranteed stability. For these regions we test for avoidance of the elliptical exclusion

regions providing Gain and Phase M argins of ±4.5d B / ± 28.44° respectively, as described

in C hapter 2 . We also consider flight condition FC3 for which the p robust stability test

failed initially. Only single-loop analysis is performed and the open-loop Nichols plots of the

frequency response are obtained by breaking the loop of the symmetrical taileron actuator,

leaving all the other loops closed. The resulting p bounds and corresponding worst case

Nichols plots are shown in Figures 4.11 and 4.12 for flight condition FC3, in Figures 4.13

and 4.14 for region FC4, and in Figures 4.15, 4.16 and 4.17 for flight conditions FC5 to

FC 8 . Note the degradation in the quality of the lower bound for p - this is because the

in troduction of th e complex uncertainty A associated with the Nichols region test means

th a t we can no longer use the real p lower bound algorithm of [29], but must instead use

the mixed p lower bound algorithm of [17].

75

a -0 .8

0.6

0.4

0.2

cd (rad/s)

Fig. 4.11: M ixed fi bounds, Nichols exclusion region test, Sight condition FC3

W orst-caseNominal

40

20

-10

-2 0 — -2 2 0 -160 -120-1 8 0 -140 -100-200

Fig. 4.12: Nominal and worst case Nichols plot, Sight condition FC3

76

0 .8

0.7

0.6

0.5

5 0.4

0.3

0.2

0.1

10'1w (rad/s)

Fig. 4.13: M ixed p bounds, Nichols exclusion region test, flight condition FC4

Effect of symmetrical taileron loop cut50

40

30

20

10

0

10

-20

-30

-40

-50-240 -2 3 0 -220 -210 -200 ■190 ■180 -170 -160 -150 -140 -130 -120 -110 -100 -90

Fig. 4.14: Nominal and worst case Nichols plot, Eight condition FC4

77

0 .9

0.8

0.7

0.6

0.532

0.4

0.3

0.2

0.1

10'w (rad/s)

Fig. 4.15: M ixed \i bounds, Nichols exclusion region in n e r e llip se test, FE region FC5 to

FC8

Effect of symmetrical taileron loop cut50

40

30

20

10

0

10

-20

-30

-40

-50-240 -230 -220 -210 -200 -190 -180 -170 -160 -150 -140 -130 -120 -110 -100 -90

Fig. 4.16: Nominal and worst case Nichols plot, FE region FC5 to FC8

78

0.832

0.6

0.4

0.2

10~110‘w (rad/s)

Fig. 4.17: M ixed f± bounds, Nichols exclusion region o u te r e llip se test, FE region FC5 to

FC8

Based on the definition in Section 2.4.3, the worst-case stability margins for the symmetrical

taileron were com puted for the different flight conditions and are shown in Table 4.5. A

comparison of the different regions of the flight envelope cleared for the basic stability

criteria, and the stability margin criteria is shown in Figure 4.18.

Flight Envelope4.5

(0.23,0)(03,15000)(0.37,25000)(0.53,40000)

4 -M9

3.5

(0.25,0)(0.26,5000)(0.28,10000)(0 31,15000)(0 34,20000) (0.38,25000) (Q.43.3QQQQ) . (0.49,35000) (0.54,40000)

M7

S - 2.5 M6

WI4

M3

M2.0.5

0.7 0.8 0.90.4 0.5 0.60.2 0.30.1Mach

Fig. 4.18: Comparison of portions of FE cleared for stability and stability margin criteria

79

F lig h t c o n d itio n P

F C 1 0.91

FC2 0.93

FC3 0.98

FC4 1.42

FC5 1.43

FC 6 1.46

FC7 1.57

FC 8 1.37

Table 4.5: Worst-case stab ility margins for the symmetrical taileron (Unaccelerated 7 trim)

4.6 .1.3 W orst-case unstable eigenvalues over the flight envelope

In this section we present results of fi tests for worst-case (in terms of the magnitude of the

real part) unstable eigenvalue for the three flight conditions FC1, FC2 and FC3, and the

two FE regions enclosing flight conditions FC4 to FC 8 shown in Figure 4.4. The tests are

performed according to the procedure described in Section 2.5. The values of the real parts

of the worst-case unstable eigenvalues for flight conditions FC1, FC2 and FC3 are found to

be 0.435, 0.460 and 0.008 respectively. The corresponding fi tests are shown in Figures 4.19,

4.20 and 4.21.

As an illustrative example, the worst-case eigenvalues of the flight condition FC3 are shown

in Figure 4.22 w ith a superim posed plot of the boundaries for the unstable eigenvalue re

quirem ent proposed in [13]. Table 4.6 summarises the unstable eigenvalues for the eight

flight conditions considered separately.

Before applying the multivariable root-locus method described in Section 2.5, the eigenvalues

in the nom inal system corresponding to the short-period and phugoid modes are identified.

The position of these eigenvalues for different percentages (10%, 20%,..., 100%) of the worst

case uncertain ty A are then identified. Figure 4.23 shows the root-locus plot of the two

eigenvalues of the short-period mode for the flight condition FC3. It can be observed how

the pole m igrates tow ards the RH-plane as the uncertainty level increases. The movement

of the eigenvalues associated with the phugoid mode is illustrated in Figure 4.24. It can be

observed th a t the first 1 0 % of the worst-case uncertainty causes the pole to migrate a rela

tively long distance towards the imaginary axis, with subsequent increase in the uncertainty

having a negligible im pact on the position of this pole (see zoomed Figure 4.25). Note tha t

80

Flight condition U nstable eigenvalue

FC1 4.35 x 10' 1

FC2 4.60 x 10- 1

FC3 8 .0 0 x 1 0 '3

FC4 1 .2 0 x 1 0 '3

FC5 1.43 x 10' 3

FC 6 1 .2 1 x 1 0 '3

FC7 1.90 x 10' 3

FC 8 1.81 x IQ"3

Table 4.6: Unstable eigenvalues (Unaccelerated 7 trim)

for this flight condition bo th oscillatory modes satisfy the stability criteria (see [13]) for the

worst-case eigenvalue position.

0.8

2 0 6

0 4

0.2

10-’w (rad/s)

Fig. 4.19: fr test for worst case unstable eigenvalue, flight condition FC1

4.6 .1.4 Sum m ary of analysis results for the unaccelerated 7 trim

Figures 4.26 and 4.27 summarise the results for the worst-case stability margins (symmetrical

taileron loop cut) and worst-case eigenvalues respectively. The values of the worst-case

stability margins, shown in Figure 4.26, show th a t flight conditions FC1, FC2 and FC3 fail

81

0.8

2 0.6

0.4

0.2

w (rad/s)

Fig. 4.20: [i test for worst case unstable eigenvalue, flight condition FC2

0.8

32

0.6

0.4

0.2

w (rad/s)

Fig. 4.21: fi test for worst case unstable eigenvalue, Right condition FC3

82

W o rs t-C a se D elta P o le -Z e ro

r“ ------- , , ---------1 r "

-

________I . ...J 1

Fig. 4.22: W orst case unstable eigenvalue for flight condition FC3

0.7

0.4

Worst case

E

- 0.2 0100%-0 .5

- 0.8

-1 .4 20%

-1 .7- 0.2- 0.8 -O.i -0- 1.2

Re

Fig. 4.23: M igration o f the short-period mode eigenvalues for flight condition FC3

83

0.1

0.08

0.06

0.04

0.02

E

- 0.02

-0 .04

-0 .06

-0 .08

- 0.1- 9 -8 - 7 - 5 -3 -2-6

Re -3x 10'

Fig. 4.24: M igration o f the phugoid m ode eigenvalues for Bight condition FC3

x 1 0 '3

10%20%

30%40% 50%

100%

60% 90%0.570%

80%

E

- o ' 80% 70%

-0 .5 90%60%

100%

30%

20%

10%-1 .5

- 2 0 2-1 0 -8 -6 - 4-12-1 4-1 6x 10'S

Fig. 4.25: Close-up o f Figure 4.24

84

HIRM+ 1.2, RIDE 1.0, (i-analysis x 1 o4 12 -D ec-2000 , c1 _symm, Long Cat 1 +2

4 .5

FC81.37FC3

0.98

3.5

FC5

k FC2 0.93

FC61.46

FC41.42

FC7 ' T.57

FC10.5 0.91

0.2 0.5Mach

0.80.1 0.3 0.4 0.6 0.7 0.9

Fig. 4.26: Sum m ary o f results for the worst-case stability margins due to symmetrical

taileron loop cut (Unacclerated 7 trim )

HIRM+ 1.2 Ride 1.0, ^-analysis 1 2 -D ec-2000, c3, Long Cat 1 +2

4.5

A FC8 1.81e-3FC3

0.008

3.5

FC51.43e-3

“ FC2 0.460

FC61 .2 ie -3

FC41.20e-3

FC1 A k FC7..1.92e-3

0.50.435

0.80.5Mach

0.7 0.90.2 0.3 0.4 0.60.1

Fig. 4.27: Summary of results for the worst-case eigenvalues (Unacclerated 7 trim)

85

th is criterion. In Figure 4.27, the worst-case eigenvalue criterion fails for flight conditions

F C 1 and FC 2 as the real parts of these two eigenvalues lie outside the boundaries specified

in [13].

4.6 .2 R esu lts for th e pullup-pushover-c* trim

In this trim condition, the angle of attack is considered as an uncertain param eter which

varies between -15 and +35 degrees and thus enters the model as an additional uncertainty.

T he model is linearised for each flight condition using this range of a. Owing to the fact that

the trim routine fails for certain flight conditions, it was decided to restrict the analysis to

the eight discrete points of the flight envelope specified in [13] rather than trying to consider

regions of the flight envelope as was done in the previous section for the unaccelerated 7

trim .

4.6.2 .1 Stability over the flight envelope

In linearising the model for this trim condition, it must be ensured th a t the load factor

lim its (-3g < n z < 7g) are not violated. However, there exist certain values of a which

cause the load factor to go beyond these limits for some flight conditions. Table 4.7 below

summarises the allowable range of a for each of the eight flight conditions.

Flight condition a (°)

FC1 -15 to +35

FC2 -15 to +35

FC3 -15 to +35

FC4 - 1 2 to +28

FC5 -15 to +35

FC 6 - 6 to + 2 1

FC7 -1 to +9

FC 8 -13 to +35

Table 4.7: Allowable range o f a for the longitudinal pullup-pushover-a trim

Linearisation using a large a step interval is found to introduce excessive conservatism which

gradually decreases as the a interval is reduced. Therefore, a sensible step to be used in the

analysis has been found to be A q = 2°. Of the eight flight conditions, two were found to

be unstable (FC1 and FC2) as shown by the p bounds generated in Figures 4.28 and 4.29,

while the rem aining six (FC3 to FC 8 ) ensure robust stability (see Figures 4.30 to 4.35).

86

0.8

0.6

0.4

0.2

w (rad/s)

Fig. 4.28: Real p upper and lower bounds for Right condition FC1 (33° < a < 33°)

0.8

0.6

0.4

0.2

w (rad/s)

Fig. 4.29: Real fi upper and lower bounds for Right condition FC2 (33° < a < 35°)

87

0.8

0.7

0.6

0.5

J 0.4

0.3

0 2 -

0.1

10'w (rad/s)

Fig. 4.30: Real fx upper and lower bounds for flight condition FC3 (33° < a < 35°)

0.35

0.3

0.25

0.2

0.15

0.1

0.05

w (rad/s)

Fig. 4.31: Real p upper and lower bounds for flight condition FC4 (26° < ot < 28°)

88

0.3 5

0.3

0.25

0.232

0.15

0.1

0.05

10'w (rad/s)

Fig. 4.32: Real p upper and lower bounds for Bight condition FC5 (33° < a < 35°)

0.3

0.25

0.2

J 0.15

0.1

0.05

10'w (rad/s)

Fig. 4.33: Real p upper and lower bounds for flight condition FC6 (17° < a < 19°)

89

0.2 5

0.2

0.15

32

0.1

0.05

w (rad/s)


0.25

0.2

0.15

0.1

0.05

0

w (rad/s)


90

4.6 .2 .2 W orst-case stability margin over the flight envelope

T he clearance results of the Nichols exclusion test for the elliptical regions providing gain

and phase margins of ± 4 .5d B / ±28.44° respectively are presented in Figures 4.37, 4.39 and

4.41 for F C l, FC3 and FC7. The corresponding p bounds are shown in Figures 4.36, 4.38

and 4.40. Table 4.8 summarises the worst-case stability margins due to the symmetrical

taileron for the eight flight conditions. It can be observed tha t the worst-case stability


F C l 0.653

FC 2 0.694

FC3 0.859

FC4 0.847

FC5 0.751

FC 6 0.734

FC7 1.531

FC 8 0.896

Table 4.8: Worst-case stability margins (Pullup-pushover-a trim)

m argins are all less th an 1 except for flight condition FC7. In fact, for these seven flight

conditions, the stability m argin criterion fails at higher values of angle of attack (a > 28°).

4.6.2.3 W orst-case eigenvalues over the flight envelope

To com pute the worst-case unstable eigenvalues, the p tests described in Chapter 2 are

applied and the results shown in Table 4.9. It can be observed tha t only flight conditions

FC2 and FC7 satisfy this criterion. The maximum real part of the worst-case eigenvalues

occurred for FC4 and this was found to be 1.153. In order to study the migration of the short

period modes of the aircraft for different amounts of uncertainty at FC l, the multivariable

root locus technique is applied to the system. The two eigenvalues of the short-period mode

are seen to move close to the imaginary axis and also become lightly damped as shown in

Figure 4.42. For the worst-case system, the short period satisfies the requirements laid for

th is analysis.

91

0.8

0.6

0.4

0.2

ci) (rad/s)

Fig. 4.36: M ixed g bounds, Nichols exclusion region test, FCl

— W orst-case Nominal

30

20

-10

-1 0 0-200 -1 4 0 -120-180 -160-2 2 0

Fig. 4.37: Nominal and worst case Nichols plot, FCl

92

0.8

0.6

0.4

0.2

10'ca (rad/s)

Fig. 4.38: M ixed p bounds, Nichols exclusion region test, FC3

40

30

20

10

0

•10

-20

-30

-40 •130 •120 -110 -100•150 -140-200 -190 -180 -170 •160-210-220-230-240

Fig. 4.39: Nominal and worst case Nichols plot, FC3

93

0 .4 5

0.4

0 35

0.3

0.25

0.2

0.15

0.1

0.05

w (rad/s)

Fig. 4.40: M ixed fi bounds, Nichols exclusion region test, FC7

50

40 Worst case

30

20

10

0

10

•20

■30

■40

-50-240 -230 -220 -210 -200 -190 -180 -170 -160 -150 -140 -130 -120 -110 -100 -90


94

Flight condition W orst-case unstable eigenvalue

F C l 0 .1 2 1

FC2 0.095

FC3 0.135

FC4 1.153

FC5 0.831

F C 6 0.117

FC7 0.063

F C 8 0.327

Table 4.9: Worst-case unstable eigenvalues (Pullup-pushover a trim)

0.5

Worst . .

-0 .5

10% 20% 30% 40%

-1 .5 -0.1 -0 .05 0.05-0 .25 -0 .2 -0 .15-0 .3-0 .35-0 .4

Fig. 4.42: M igration o f the short period eigenvalues for F C l (Longitudinal axis, Pullup-

pushover a trim )

95

4.6 .2 .4 Summary of analysis results for the pullup-pushover-a trim

Figures 4.43 and 4.44 sum m arise the results for the worst-case stability margins (symmetrical

taileron loop cut) and worst-case eigenvalues respectively. In both analysis, four aerody

namic param eters were used. The figures show the ranges of a for which the flight envelope

is not cleared. For instance, the stability margin criterion fails for FC4 (see Figure 4.43)

if the a is in the range 28° < a < 35°. The same interpretation applies to Figure 4.44 to

identify the violation of the unstable eigenvalue criterion.

HIRM+ 1.2, RIDE 1.0, ^-analysis12-D8C-2000, c1_symm, Long Cat 1 +2

4.5

FC3 [27,35] ; [Ua4,Ua4]

3.5

► FC2 [-15,-12][27,35] '■ [Ua4,Ua4] [Ua4,Ua4]

FC6 [-15,-4124,35] [Ua4,Ua4][Ua4,Ua4]

1.5FC4 [28,35] [Ua4,Ua4]

k FC1 [-15,—8][28,35] [Ua4,Ua4][Ua4,Ua4]

................FC7 4 L: [-15,-3][9,35]

[Ua4,Ua4][Ua4,Ua4]

0.5

0.6 0.70.2 0.4 0.5Mach

0.8 0.90.1 0.3


taileron loop cut (Pullup-pushover-a trim)

HIRM+ 1.2 Ride 1.0, ^-analysis . 0* 12-Dec-2000, c3, Long Cat 1 +2

4.5

lFC8 [-15,-13] [33,35] [Ua4,Ua4] [Ua4,Ua4]

FC3 [30,35] [Ua4,Ua4]

3.5

FC5: [29,35] [Ua4,Ua4]

FC6 [—15,—5][22,35] [Ua4,Ua4][Ua4,Ua4]

FC4 [-15,-11][23,35] : [Ua4,Ua4][Ua4,Ua4]

0.5 .................FC7[—15,—2][10,35]

[Ua4,Ua4][Ua4,Ua4]

0.7 0.90.4 0.6 0.80.5Mach

0.2 0.3

Fig. 4.44: Sum m ary of results for the worst-case eigenvalues (Pullup-pushover a trim)

96

4 .6 .3 R esu lts for th e p u llu p -p ush over-n 2 trim

In this trim condition, the load factor is an additional variability th a t enters the model. The

acceptable range for the variation of the load factor is between -3g and 7g.

4.6.3.1 Stability over the flight envelope

W hilst linearising the model, it m ust be ensured th a t the limits of the angle of attack

( — 15° < a < 35°) are not exceeded. However, there exist certain values of the load factor

which cause this condition to be violated. As a result, the load factor range must be reduced

for these particu lar flight conditions. Table 4.10 lists the allowable range of the load factor

which satisfies the above condition for a.

A step of A n 2 = 1 g is used as it was found th a t increasing the step size tends to introduce

excessive conservatism in the LFT-based uncertainty model. The p bounds generated in

Figures 4.45 to 4.52 indicate th a t three flight conditions (FC l to FC3) fail to satisfy the

robust stability test.

F lig h t c o n d itio n n z (g )

F C l -3 to 2

FC2 -3 to 2

FC3 -3 to 2

FC4 -3 to 3

FC5 -3 to 3

FC 6 -3 to 6

FC7 -3 to 7

FC 8 -3 to 3

Table 4.10: Allowable range o f load factor for longitudinal pullup-pushover n z trim

97

2.5

32

0.5

10'w (rad/s)

Fig. 4.45: Real p upper and lower bounds for flight condition FC l (1 < n z < 2)

0.5

w (rad/s)

Fig. 4.46: Real p upper and lower bounds for flight condition FC2 (1 < nz < 2)

98

0.835

0.6

0.4

0.2

10'10'w (rad/s)

Fig. 4.47: Real p upper and lower bounds for flight condition FC3 (1 < n z < 2)

0.7

0.6

0.5

0.4

350.3

0.2

0.1

10w (rad/s)


99

0 .2 5

0.2

0.15

35

0.1

0.05

10' 10w (rad/s)


0.8

0.7

0.6

0.5

2 0.4

0.3

0.2

0.1

10'w (rad/s)


100

0.35

0.3

0.25

0.2

0.15

0.1

0.05

w (rad/s)


0.7

0.6

0.5

0.4

32

0.3

0.2

0.1

10w (rad/s)


101

4.6 .3.2 W orst-case stability margin over the flight envelope

For all stable flight conditions the stability margin requirement is satisfied as shown in

Figures 4.54, 4.56, 4.58, 4.60 and 4.62. The resulting p bounds are illustrated in Figures 4.53,

4.55, 4.57, 4.59 and 4.61. Note the poor quality of the fi lower bounds in some cases - as a

result it is not possible to guarantee th a t the ‘real’ worst cases have been found. However,

since the p upper bounds are always less than 1 , the stability margin criterion is guaranteed

to be satisfied for all cases. A possible way of tightening the gap between the upper and

lower bounds is to cast this mixed-p problem as an optimisation problem using a similar

type of algorithm to th a t explained in Chapter 2. This, however, constitutes an area of

future work.

Table 4.11 shows the worst-case stability margins due to the symmetrical taileron for the

eight flight conditions.


F C 1 1.25

FC2 1.29

FC3 1.28

FC4 1.31

FC5 1.27

FC 6 1.30

FC7 1.43

FC 8 1.41

Table 4.11: Worst-case stab ility margins (Pullup-pushover-nz trim)

4.6 .3.3 W orst-case eigenvalues over the flight envelope

By com puting p away from the jtc-axis, the worst-case unstable eigenvalues can be computed

and are shown in Table 4.12. It can be observed th a t the requirement for the worst-case

unstable eigenvalues specified in [13] is violated for FC1, FC2 and FC3. Applying the

m ultivariable root locus technique yields the plots shown in Figures 4.63 (short period

mode) and 4.64 (phugoid mode) for flight condition FC 8 . The two worst-case eigenvalues

for each mode are found to be stable.

102

0 .8

0.7

0.6

0.5

5 0.4

0.3

0.2

0.1

10'w (rad/s)

Fig. 4.53: M ixed p, bounds, Nichols exclusion region test, FC4

50

40

30

20

10

0

• 10

-20

•30

-40

-50 100•160 -140 •120•180-200-220


103

0 .4 5

0.4

0.35

0.3

0.2532

0.2

0.15

0.1

0.05

10'w (rad/s)


50

40

30

20

10

0

10

-20

-30

-40

-50 -120■140 -100•160-180-200-220


104

0.9

0.8

0.7

0.6

2 0.5

0.4

It0.3

0.2

0.1

10'10'w (rad/s)


50

40

30

20

10

0

10

-20

-30

-40

-50■120 -100■160 -140-180-200-220


105

0 .4

0.35

0.3

0.25

5 0.2

0.15

0.1

0.05 -j

10' 10'w (rad/s)

Fig. 4.59: M ixed /i bounds, Nichols exclusion region test, FC7

50

40

30

20

10

0

•10

-20

-30

-40

-50•100•120■160 •140-180-200-220


106

0 .7

0.6

0.5

0.4

0.3

0.2

w (rad/s)

Fig. 4.61: M ixed fi bounds, Nichols exclusion region test, FC8

50

40

30

20

10

0

10

-20

-30

-40

-50•160 •140 •120 -100-180-200•220


107


F C l 0.195

FC2 0.198

FC3 0.175

FC4 0.054

FC5 0.063

FC 6 0.008

FC7 0.009

F C 8 0.047

Table 4.12: Worst-case unstable eigenvalues (Pullup-pushover n z trim)

30%59%

0.5

Wor;t

-0 .5

50%30%Nominal 20%

-1 .5

-2 — -0 .9 -0 .2-0 .3 - 0.1-0 .5 -0 .4-0 .7 - 0.6- 0.8

Fig. 4.63: M igration o f the short period m ode eigenvalues for FC8 (Longitudinal axis,

Pullup-pushover-nz trim )

4.6.4 Sum m ary o f analysis results for the pullup-pushover-nz trim

Figures 4.65 and 4.66 sum m arise the results for the worst-case stability margins (symmetrical

taileron loop cut) and worst-case eigenvalues respectively. Four aerodynamic parameters

were used and in bo th analysis, only flight condition FC7 was cleared for the whole range

of load factor specified.

108

0 .0 4

0% 10%0.03

Nominal

Worst0.02

0.01

- 0.01

- 0.02 Worst

100%-0 .0 3 50%10%

-0 .0 4 — -4 .6 -4 .4 -4 .2 -4 -3 .8 -3 .6 -3.4

x 10~3

Fig. 4.64: M igration o f the phugoid mode eigenvalues for FC8 (Longitudinal axis, Pullup-

pushover-nz trim )

HIRM+ 1.2, RIDE 1.0, n-analysis12 -D ec-2000 , c1_symm, Long Cat 1+2

4.5

3.5

£ "2 .5

FC6[7] [Ua4] :

FC1 {2,7] [Ua4,Ua4]

■4 k. f c 70.5

0.3 0.4 0.5Mach

0.6 0.7 0.80.2


taileron loop cut (Pullup-pushover-nz trim)

4 .7 A n a ly s is o f th e la tera l H I R M + d yn am ics

This section presents results obtained from the analysis of the lateral dynamics of the

HIRM-f- R ID E control laws. Six Class 1 uncertain param eters (Ycg, Ixx, I zz, Q 0 , Cn/3 and

CUr) are taken into account and a reduction factor of 0.46 is applied since three aerody-

109

HIRM+ 1.2 R ide 1.0, ^ -a n a ly s isx 10* 12-D ec-2000, c3, Long Cat 1+2

4.5

■< ^FC8 [3,7] [Ua4,Ua4]FC3 [3,7]

[Ua4,Ua4]3.5

FC6 [7] [Ua4]

FC4 [3,7] [Ua4,Ua4]

0.5

0.2 0.3 0.4 0.5Mach

0.6 0.7 0.8 0.9

Fig. 4.66: Sum m ary o f results for the worst-case eigenvalues (Pullup-pushover-nz trim)

namic uncertainties are used. For the unaccelerated 7 trim, the flight envelope is partitioned

as shown in Figure 4.4. Due to trim m ing problems, the analysis of the last two flight cases

(pullup m anoeuvers) has been restricted to only the eight discrete flight conditions men

tioned in [13]. T he analysis can however be readily extended to cover portions of the flight

envelope (as in the unaccelerated 7 trim ) where Mach number and altitude are treated as

additional variabilities.

All the stability criteria m entioned in C hapter 2 are addressed. Single loop analysis is

performed for the Nichols boundary criterion which is evaluated for both the differential

tailplane and rudder.

4 .7 .1 R e s u l ts fo r th e u n a c c e le r a te d 7 t r im

This section deals w ith the straight and level flight case over the flight envelope shown in

Figure 4.4.

4.7.1.1 S tability over the flight envelope

For this analysis bo th M ach num ber and altitude are varied and thus portions of the flight

envelope can be considered rather than limiting the analysis to discrete points. The p

bounds from Figures 4.67 to 4.71 show th a t robust stability is guaranteed for all the flight

regions.

110

0 .3 5

0.3

0.25

0.2

32

0.15

0.1

0.05

10'w (rad/s)

Fig. 4.67: Real /i upper and lower bounds for point FC1

0.35

0.3

0.25

0.2

■D20.15

0.1

0.05

10'w (rad/s)


111

08

0.7

0.6

0.5

5 0.4

0.3

0.2

0.1

10‘1 10 w (rad/s)

10'

Fig. 4.69: Real p upper and lower bounds for region FC4

0.9

0.8

0.7

0.6

J 0.5

0.4

0.3

0.2

0.1

10 w (rad/s)

10'

Fig. 4.70: Real (i upper and lower bounds for region FC5 to FC8

112

0 .3 5

0.3

0 .25

0.2

32

0 .15

0.1

0 .05

w (rad/s)


4.7.1.2 W orst-case stab ility margin over the flight envelope

Single-loop analysis is performed and the effects of both differential taileron and rudder loop

cuts are investigated. The Nichols exclusion test is conducted at flight conditions FC1, FC2

and FC3.

Figures 4.73 and 4.75 show the effect of a differential taileron loop cut and the corresponding

p bounds are found in Figures 4.72 and 4.74 respectively.

Results of the Nichols robustness criterion evaluated for the rudder loop cut are illustrated

in Figures 4.77 and 4.76 for FC3. For either loop cut, all the flight conditions avoid the

specified Nichols exclusion region.

Tables 4.13 and 4.14 give an indication of the worst-case stability margins, p, for the dif

ferential taileron and rudder respectively. It is seen th a t none of the flight conditions has

violated the stab ility m argin criteria (p > 1 ).

113

F lig h t c o n d itio n p

F C 1 1.48

FC2 1.50

FC3 1.59

FC4 1.78

FC5 1.71

FC 6 1.85

FC7 1.61

FC 8 1 .6 8

Table 4.13: Worst-case s tab ility margins for the differential taileron (Unacclerated 7 trim)


FC1 1.39

FC2 1.40

FC3 1.42

FC4 1.53

FC5 1.48

FC 6 1.60

FC7 1.70

FC 8 1.50

Table 4.14: Worst-case stability margins for the rudder (Unacclerated 7 trim)

114

0 .3 5

0 3

0.25

0.2

0.15

0.1

0.05

10 10'w (rad/s)

Fig. 4.72: M ixed p bounds, Nichols exclusion region test, FE point FC1

50

40

30

20

10

0

10

-20

-30

-40

-50•120 •110•170 •160 •150 -130-180 •140 -100 -90•190-200-220 -210

Fig. 4.73: N om inal and worst case Nichols plot, FE point FC1 (Differential taileron loop

cut)

115

0 .4

0.35 / I

0.3

0.25

§ 0.2

0.15

0.1/\

0.05

10' 10'w (rad/s)

Fig. 4.74: M ixed (i bounds, Nichols exclusion region test, FE point FC2

50

• Nominal40

30

20

10

0

10

•20

-30

-40

-50 -170 -160 -140 •130•150 •120 •110 ■100-200 •190 ■180 -90-210-220

Fig. 4.75: N om inal and best worst case Nichols plot, FE point FC2 (Differential taileron

loop cut)

116

0 .4

0.35

0.3

0.25

5 0.2

0.15 - 11

0.1

0.05 I t

10' 10'w (rad/s)

Fig. 4.76: M ixed p bounds, Nichols exclusion region test, FE point FC3

30

20

10

0

10

-20

-30

-40

•50

-60

-70 -110 -100 -90•190 -180 -170 -160 -150 ■140 -130 •120-230 -220 -210 -200-240

Fig. 4.77: Nominal and worst case Nichols plot, FE point FC3 (Rudder loop cut)

117

4.7.1.3 Worst-case eigenvalues over the flight envelope

Results of the fj, tests carried out to identify the worst-case unstable eigenvalues for the

eight flight conditions are presented here. I t can be observed, from Table 4.15, tha t all

the eight worst-case eigenvalues are located in the LH-plane, implying tha t this clearance

criteria is fulfilled. In the la teral mode, we will focus on the dutch roll mode of the aircraft

F light condition W orst-case unstable eigenvalue

F C 1 9.57 x 10' 3

FC2 1 .0 1 x 1 0 - 14

FC3 6 .6 6 x 1 0 ‘ 14

FC4 3.19 x 10' 12

FC5 1.05 x 10- 12

FC 6 1.62 x lO' 12

FC7 1.80 x lO' 13

FC 8 1 .1 1 x 1 0 '12

Table 4.15: Worst-case unstable eigenvalues (Unaccelerated 7 trim)

and thus, the appropria te eigenvalues have to be identified. The eigenvalues corresponding

to the dutch roll is a com plex-conjugate pair. On implementing the multivariable root locus

technique, the m igration of these two eigenvalues for the flight condition FC 1 is plotted in

Figure 4.78. It can be seen th a t bo th eigenvalues generally move rightwards towards the

jic-axis.

4.7.1.4 Sum m ary of analysis results for the unaccelerated 7 trim

Figures 4.79, 4.80 and 4.81 sum m arise the results for the worst-case stability margins (dif

ferential ta ileron and rudder loop cuts) and the worst-case eigenvalues.

4.7.2 R esu lts for th e pullup-pushover-a trim

Here, the model is linearised w ith an additional uncertainty, a, which is varied between the

limits of —15° and +35°. Trimming problems confined the analysis to a gridding approach

where only th e eight discrete points shown in Figure 4.4 are considered.

118

0 .4

0.350%

Worst

0.2Nominal

-0.1

-0 .2

70% . 80% 90% . .100%-0 .3

-0 .4-0 .3 -0 .25 -0 .2 -0 .15 -0.1 -0 .05 0 0.05

Fig. 4.78: M igration o f the dutch roll eigenvalues for FC1 (Lateral axis, Unaccelerated 7

trim )

HIRM+ 1.2, RIDE 1.0, |i-analysis 12-D ec-2000 , c1_diff, Lat Cat 1

4.5

FC81.68FC3

3.5

FC5

c 2.5 t FC2 1.50

1.85

FC41.78

0.5

0.4 0.5Mach

0.6 0.70.3 0.8 0.90.2

Fig. 4.79: Sum m ary of results for the worst-case stability margins due to differential taileron

loop cut (Unaccelerated 7 trim)

119

HIRM+ 1.2, RIDE 1.0, n-analysis x io 4 12 -D ec-2000 , c1_rudd, Lat Cat 1

4 .5

FC81.50FC3

1.42

3.5

FG5

k FC2 1.40

FC61.60

FC41.53

FC10.5

0.80.2 0.3 0.4 0.5Mach

0.7 0.90.1 0.6

Fig. 4.80: Sum m ary o f results for the worst-case stability margins due to rudder loop cut

(Unaccelerated 7 trim )

HIRM+ 1.2 Ride 1.0, n-analysis 12-D ec-2000 , c3, Lat Cat 1

4.5

FC8 1.106-12FC3

6.66e-14

3.5

FC51.056-12

c 2.5 FC21.00e-1

FC6 1.60e-12

FC43.1-9e-12

FC1 k.F C 7 ...:1 .80e-130.5 9 .57e-14

0.7 0.8 0.90.4 0.5Mach

0.60.30.20.1

Fig. 4.81: Sum m ary of results for the worst-case eigenvalues (Unaccelerated 7 trim)

120

4.7.2 .1 Stability over the flight envelope: results

Linearisation of the model was done with an a step of 2 ° whilst ensuring tha t the load

factor limits were not exceeded. However, this condition is violated for some values of a for

some flight conditions as detailed in the Table 4.16. Once again, it is found th a t increasing

the step range of a introduces conservatism. For instance, using a step size of A q = 2°

ra ther than the full range (A q = 50°) can reduce the conservatism up to 60% for FC 6 . The

F lig h t c o n d itio n « n

F C 1 -15 to +35

FC2 -15 to +35

FC3 -15 to +35

FC4 - 1 2 to +28

FC5 -15 to +35

FC 6 - 6 to + 2 1

FC7 -1 to +9

FC 8 -13 to +35

Table 4.16: Allowable range o f a for the lateral pullup-pushover-a trim

resulting p bounds generated in Figures 4.82 to 4.89 indicates robust stability for all the

eight flight conditions.

4.7.2.2 W orst-case stability m argin over the flight envelope

The worst case stability m argin criterion with respect to the violation of the Nichols plot

exclusion regions is now considered for all the eight flight conditions. For the differential

taileron loop cut, some illustrative Nichols plots for flight conditions FC5 and FC7 and their

corresponding p bounds are shown in Figures 4.90 to 4.93. Results for the rudder loop cut

are shown in Figures 4.94 and 4.95 for FC5.

It can be observed th a t FC5 fails the stability margin criterion for both the differential

taileron and rudder loop cuts. Referring to Tables 4.17 and 4.18, it can be further deduced

from the values of p th a t, for bo th loop cuts, this criterion is also failed for all the other

flight conditions except FC7.

1 2 1

0 .7

0.6

0.5

0.4

0.3

0.2

0.1

10'w (rad/s)

Fig. 4.82: Real p upper and lower bounds for flight condition FC1

0.7

0.6

0.5

0.4

350.3

0.2

0.1

w (rad/s)

Fig. 4.83: Real fi upper and lower bounds for flight condition FC2

1 2 2

0 .7

0.6

0.5

0.4

32

0.3

0.2

0.1

10' 10' 10'w (rad/s)


0.35

0.3

0.25

0 235

0.15

0.1

0.05

10'10'10'w (rad/s)

Fig. 4.85: Real p upper and lower bounds for Right condition FC4

123

0 .3 5

0.3

0.25

0.232

0.15

0.1

0.05

10'w (rad/s)

Fig. 4.86: Real p upper and lower bounds for Bight condition FC5

0.35

0.3

0.25

0.2

0.15

0.1

0.05

10 w (rad/s)

10'


124

0 .3 5

0.3

0.25

0.2

0.15

0.1

0.05

w (rad/s)

Fig. 4.88: Real p upper and lower bounds for Eight condition FC7

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

10'10'w (rad/s)


125


FC 1 0 .1 1 0

FC2 0.152

FC3 0.213

FC4 0.385

FC5 0.381

FC 6 0.314

FC7 3.471

FC 8 0.498

Table 4.17: Worst-case stab ility margins due to the differential taileron (Pullup-pushover-a

trim )


FC1 0.018

FC2 0 .0 1 2

FC3 0.009

FC4 0.345

FC5 0.285

FC 6 0.753

FC7 3.172

FC 8 0.579

Table 4.18: Worst-case stab ility margins due to the rudder (Pullup-pushover-a trim)

126

3 .2

2.8

2.4

3 .1 .6

0.8

0.4

o) (rad/s)

Fig. 4.90: M ixed \i bounds, Nichols exclusion region test, FC5

40

• Nominal - W orst-case

30

20

10

0

■10

•20

-30

-40•150 •130 -120 •110 •100-170 •160 •140•180-190-200-210-220

Fig. 4.91: Nominal and worst case Nichols plot, FC5 (Differential taileron loop cut)

127

0 .5

0.45

0.4

0.35

0.3

§ 0.25

0.2

0.15 /I0.1

0.05

10 w (rad/s)



128

0.8

0.6

0.4

0.2

10'co (rad/s)

Fig. 4.94: M ixed [i bounds, Nichols exclusion region test, FC5

40

30

20

10

0

10

-20

-30

-40 ■140 ■120 ■100■160 •150 -130 ■110-180 •170-190-200-210-220

Fig. 4.95: Nominal and worst case Nichols plot, FC5 (Rudder loop cut)

129

4.7.2.3 Worst-case eigenvalue over the flight envelope

T he application of the p, tests to identify the worst-case unstable eigenvalues yields the

results in Table 4.19. F light conditions FC1, FC3, FC4, FC5, FC 6 and FC 8 fail the unstable

F light cond ition W orst-case unstable eigenvalue

F C 1 0.114

F C 2 0.033

FC3 0.123

FC4 1.025

FC5 0.629

FC 6 0.099

FC7 0.052

FC 8 0.298

Table 4.19: Worst-case unstable eigenvalues (Pullup-pushover-a trim)

eigenvalue criterion as they lie outside the boundaries specified in [13].

To investigate the effect of different levels of uncertainty on the pair of eigenvalues associated

to the dutch roll mode, the m ultivariable root locus technique is applied to the model at

flight condition FC3 as an illustrative example. Figure 4.96 shows the migration of both

eigenvalues tow ards the j w -axis as the proportion of worst-case uncertainty is increased.

4 .7 .2.4 Sum m ary of analysis results for the pullup-pushover-a trim

Figures 4.97, 4.98 and 4.99 sum m arise the results for the worst-case stability margins (dif

ferential taileron and rudder loop cuts) and the worst-case eigenvalues. Three aerodynamic

param eters were used in all the analysis.

4.7.3 R esu lts for th e pullup-pushover-n2 trim

In this trim option, an additional uncertainty, the load factor, which is varied between -3g

and 7g, is to be considered in our model. Results for the different stability criteria are

presented.

130

0 .3 5

0.380% 90%

0.2530%

0.2

0.15

o%0.05

-0.05

-0.1

-0.15

-0 .210% 20%

-0.2550%

-0.3

-0.35 — -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

Fig. 4.96: M igration o f the dutch roll eigenvalues for FC3 (pullup-pushover-a trim)

HIRM+ 1.2, RIDE 1.0, n-analysis 12-Dec-2000, c1_diff, Lat Cat 1

4.5

-AFC8 [27,35] [Ua3,Ua3]FC3 [26,35]

[Ua3,Ua3]

3.5

FC5 [28,35] [Ua3,Ua3]

FC2 [-15,-13X32,35] [Ua3,Ua3] [Ua3,Ua3]

FQ6 [-15,-8][28,35] [Ua3,Ua3][Ua3,Ua3]

1.5FC4 [27,35] : [Ua3,Ua3]

A FC1 [-15,-8X33,35] [Ua3, Ua3][Ua3,Ua3]

FC7 ^ [-15,-2][12,35]

[Ua3,Ua3][Ua3,Ua3]

0.5

0.4 0.5Macti

0.6 0.80.3 0.7 0.90.20.1


loop cut (pullup-pushover-a trim)

131

HIRM+ 1.2, RIDE 1.0, n -an a ly s is12-D ec-2000, c1_rudd, Lat Cat 1

4.5

FC3 [26,35] [Ua3,Ua3]

3.5

F05 [27,35 [Ua3,Ua3]

FC2 [-1 5 ,-1 3][29;i35] [Ua3,Ua3][Ua3,Ua3]

FC6 [-15,-8][28,35] [Ua3,Ua3][Ua3,Ua3]

FC4 [28,35] : [Ua3,Ua3] ;

l FCI {-i5,-9][30,35} [Ua3, Ua3][Ua3, Ua3]

0.5 ................. FC7: [-15,-2][13,35]

[Ua$,Ua3][Ua3,Ua3]

0.1 0.2 0.3 0.4 0.5Mach

0.6 0.8 0.90.7


(pullup-pushover-a trim )

HIRM+ 1.2 Ride 1.0, (j-analysis12 -D ec-2000 , c3, Lat Cat 1

4.5

■< LFC8 H 5][3 3 ,3 5 ]- [Ua3][Ua3,Ua3]

3.5

FC6 [-15,-8][24,35] [Ua3,Ua3][Ua3,Ua3]

FC4 [-15 ,-14][26,35] [Ua3,Ua3][Ua3,Ua3]

l FC1 [-1S,-8][33,35] [Ua3,Ua3][Ua3,Ua3]

0.5; [-15,-1][13,35]

[Ua3,Ua3][Ua3,Ua3]

0.7 0.8 0.90.4 0.5Mach

0.60.30.2

Fig. 4.99: Sum m ary o f results for the worst-case eigenvalues (pullup-pushover-a trim)

132

4.7.3.1 Stability over the flight envelope

Here, the angle of attack , which is a free variable, must stay within the limits —15° < a < +35°

whilst performing the linearisation. For some values of n z, this condition is violated and

thus, the allowable range for the load factor was reduced as shown in Table 4.20. A step

F lig h t c o n d itio n n 2 (9 )

FC I -3 to 2

FC2 -3 to 2

FC3 -3 to 2

FC4 -3 to 5

FC5 -3 to 3

FC 6 -3 to 7

FC7 -3 to 7

FC 8 -3 to 4

Table 4.20: Allowable range o f load factor for the pullup-pushover-nz trim (lateral axis)

of A n z = lg is used since larger step sizes introduce excessive conservatism in the resulting

LFT-based uncertain ty model. For all the eight flight conditions, the criterion for robust

stability is satisfied (p < 1) as shown in Figures 4.100 to 4.107.

4.7.3.2 W orst-case stability m argin over the flight envelope

Figures 4.108 to 4.123 show the Nichols plots and the resulting p bounds due to the differ

ential taileron loop cut, for the elliptical exclusion region test. It can be seen that, for all

flight cases, the exclusion region is avoided for both the nominal case and the worst case.

The worst-case stab ility m argins were also computed (see Tables 4.21 and 4.22) and were

all found to be greater th an 1 in accordance with the above results.

133

0 .3 5

0.3

0.25

0.235

0.15

0.1

0.05

10'w (rad/s)

Fig. 4.100: Real p upper and lower bounds for flight condition FCI

0.35

0.3

0.25

0.2

0.15

0.1

0.05

10'w (rad/s)


134

0 .3 5

0.3

0.25

0.2

0.15

0.1

0.05

010'

w (rad/s)


0.35

0.3

0.25

0.2

0.15

0.1

0.05

10'10'w (rad/s)


135

0 .3 5

0.3

0.25

0.232

0.15

0.1

0.05

10'w (rad/s)


0.35

0.3

0.25

02

0.15

0.1

0.05

10'w (rad/s)


136

0 .3 5

0.3

0.25

0.2

320.15

0.1

0.05

10'10' 10'w (rad/s)


0.35

0.3

0.25

0.2

0.15

0.1

0.05

w (rad/s)


137


FC I 1.49

FC2 1.51

FC3 1.54

FC4 1.76

FC5 1.53

FC 6 1.59

FC7 1.67

FC 8 1 .6 6

Table 4.21: Worst-case s tab ility margins due to the differential taileron (Pullup-pushover-nz

trim )


FC I 1.35

FC2 1.36

FC3 1.40

FC4 1.53

FC5 1.47

FC 6 1.63

FC7 1.69

FC 8 1.48

Table 4.22: Worst-case s tab ility margins due to the rudder (Pullup-pushover-nz trim)

138

0 .7

0.6

0.5

0.4

0.3

0.2

0.1

w (rad/s)

Fig. 4.108: M ixed p bounds, Nichols exclusion region test, FCI

50

40

30

20

10

0

10

-20

-30

-40

-50 •140 -110 -100 -90•160 •150 -130 •120•170-190 -180-200-210-220

Fig. 4.109: Nominal and worst case Nichols plot, FCI (Differential taileron loop cut)

139

0 .5

0.45

0.4

0.35

0.3

§ 0.25

0.2

0.15

0.1

0.05

/ l /

w (rad/s)

Fig. 4.110: M ixed (i bounds, Nichols exclusion region test, FC2

50

• Nominal40

30

20

10

0

10

-20

•30

•40

•50 -140 •130 •110 -100 -90•160 •150 •120•170180•190-200-210-220


140

0 .7

0.6

0.5

0.4

0.3

0.2

0.1

w (rad/s)

Fig. 4.112: M ixed /i bounds, Nichols exclusion region test, FC3

NominaJ Worst case


141

&

0 .5

0.45

0.4

0.35

0.3

5 0.25

0.2

0.15

0.1

0.05

10'10'w (rad/s)


-180


142

0 .5

0.45

0.4

0.35

0.3

§ 0.25

0.2

0.15

0.1

0.05

w (rad/s)


50

40 Worst case

30

20

10

0

10

■20

■30

■40

-50-100•160 -150 -120 •110•170 -140 -130 -90-180•190-210 -200-220


143

0 .5

0.45

0.4

0.35

0.3

5 0.25

0.2

0.15

0.1

< I:0.05

1010'w (rad/s)

Fig. 4.118: M ixed ji bounds, Nichols exclusion region test, FC6

50

40

30

20

10

0

• 10

■20

■30

-40

-50■110 -100■160 -150 •130 •120 -90■170 -140•180•190-210 ■200■220


144

0 .4 5

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

10'w (rad/s)

Fig. 4.120: M ixed (i bounds, Nichols exclusion region test, FC7

50

Nominal40

30

20

10

0

10

■20

-30

-40

-50 -100■120 -90•170 •160 -150 •140 •130 ■110•180•190-220 -210 -200


145

0 .5

0.45

0.4

0.35

0.3

5 0.25

0.2

0.15

0.1

0.05

w (rad/s)


-3 0


146

4.7.3.3 Worst-case eigenvalue over the flight envelope

All the unstable eigenvalues shown in Table 4.23 were found to lie within the acceptable

boundaries as defined in Section 1 .4 .


F C I 0.013

F C 2 0.009

FC3 0.008

FC4 0.003

FC5 0.004

FC 6 0.007

FC7 0 .0 0 1

FC 8 0.008

Table 4.23: Worst-case unstable eigenvalues (Pullup-pushover-nz trim)

The m ultivariable root locus technique, applied to flight condition FC4, describes the move

ment of the two eigenvalues of the dutch roll mode for increasing levels of the worst-case

uncertainty. 1 0 0 % uncertain ty brings bo th eigenvalues very close to the jw -axis as shown

in Figure 4.124.

4.7.3.4 Sum m ary of analysis results for the pullup-pushover-nz trim

Figures 4.125, 4.126 and 4.127 sum m arise the results for the worst-case stability margins

(differential taileron and rudder loop cuts) and the worst-case eigenvalues.

4 .8 C o m p a r iso n o f r e su lts w ith th e classica l approach

We com pare results obtained in our analysis with those produced using the classical approach

of [50]. The com parison is mainly focused on the m ajor agreement and/or discrepancies

between the various analysis results for the pullup-pushover-a trim.

Analysis of the longitudinal axis, in which the effect of the symmetrical taileron loop was

investigated, was perform ed using the full set of uncertain param eters. A general observation

is th a t m ost of the results agree largely between our m ethod (see Figure 4.128) and that of

the classical approach (see Figure 4.129). The discrepancies can be explained by the different

approaches used to com pute the worst-case stability margins. The classical approach used

147

0 .7

0.6

0.5 Nominal50%

0.4

0.3

02

-0.1

-0 .2

-0 .3

-0 .4

50%-0 .5

20%-0.6

-0 .7 L -0.6 -0 .5 -0 .4 -0 .3 -0 .2 -0.1 0 0.1

Fig. 4.124: M igration o f the dutch roll mode eigenvalues for FC4 (Lateral axis, Pullup-

pushover-nz trim )

HIRM+ 1.2, RIDE 1.0, ^-analysis12-D ec-2000, c1_diff, Lat Cat 1

4.5

3.5

FC5 [4,7] [Ua3,Ua3]

k FC2 [2,7] [Ua3,Ua3]

FC6 [7] [Ua3]

i FC1 [2,7] [Ua3,Ua3]

■* FC70.5

0.5Mach

0.3 0.4 0.6 0.90.2 0.7 0.80.1


loop cut (pullup-pushover-nz trim)

148

HIRM+ 1.2, RIDE 1.0, n-analysis x 104 12-D ec-2000, c1 _rudd, Lat Cat 1

4 .5

FC3 [3,7] [Ua3,Ua3]

3.5

FC5 [4,7] [Ua3,Ua3]

: FC6 [7] : [Ua3]

FC4 [4,7] [Ua3,Ua3] ;

FC1 [2,7] [Ua3,Ua3]

U FC70.5

0.3 0.4 0.6 0.90.2 0.5Mach

0.80.7


(pullup-pushover-nz trim )

HIRM+ 1.2 Ride 1.0, n-analysis12 -D ec-2000 , c3, Lat Cat 1

4.5

FC3 [3,7] [Ua3,Ua3]

[Ua3,Ua3]

3.5

FC5:[4,7][Ua3,Ua3]

g 2.5 ‘ FC2 [2,7] [Ua3,Ua3]

FC6 [7] [Ua3]

FC4 [4,7] [Ua3,Ua3]

i L FG70.5

0.90.4 0.6 0.80.3 0.5Mach

0.70.20.1

Fig. 4.127: Sum m ary of results for the worst-case eigenvalues (pullup-pushover-nz trim)

149

the s tandard trapezoidal Nichols exclusion regions while performing the single-loop analysis

while we used the more stringent elliptical Nichols exclusion region test in our approach.

This conservatism is clearly reflected in the clearance results where the ranges of angle of

a ttack for which the control law is cleared are smaller than those of the baseline method. For

instance, flight condition FC3 was cleared for —15° < a < 26° while the result of the classical

approach for the same flight condition gave a bigger clearance range of —15° < a < 29°.

Similar explanation holds for the other flight conditions.

T he analysis of the la teral axis was performed using the most relevant set of uncertain

param eters for bo th the differential taileron and rudder loops. Most results agree largely.

Again conservatism due to (a) the elliptical Nichols exclusion regions, and (b) the LFT-

based uncertain ty approach, has reduced the cleared range of angle of attack for all the

flight conditions.

Since the analysis covers a wide range of tasks (initial acquaintance of the HIRM + RIDE

model, form ulation of LFT-based uncertainty models and the application of the different

analysis techniques, ...etc), accurate tim ing of the overall process is quite difficult. However,

some approxim ate tim ings are shown in Table 4.24. Because different step sizes for the

angle of a ttack were used in our approach (A a = 2 °) as opposed to the classical approach

(A a = 1 °), an exact com parison cannot easily be made and thus, the data enclosed in

Table 4.24 should be used as a guideline only.

A n a ly s is p ro b le m (Time shown in hours) C lass ica l a p p ro a c h /i-analysis ap p ro a c h

Sym m etrical taileron loop 16.6 10.5

Differential taileron loop 6.96 6.00

R udder loop 7.5 5.0

Longitudinal worst-case eigenvalue 17.4 11.0

L ateral worst-case eigenvalue 7.2 5.0

Table 4.24: Tim ing results

150

HIRM+ 1.2, RIDE 1.0, n -an a ly s is

4 .5

FC8 [29,35] [Ua4,Ua4]

3.5

£ "2 .5 k FC2 [-15;-12][27,35] [Ua4,Ua4] [Ua4,Ua4]

FC6 [-15 ,—4][24,35] [Ua4,Ua4][Ua4,Ua4]

FC4 [28,35] : [Ua4,ua4] ;

l FG1 {-15,—8][28,35} [Ua4,Ua4][Ua4,Ua4]

0.5 ................. FC7: [-15,-3][9,35]

[Ua4,Ua4][Ua4,Ua4]

0.1 0.2 0.40.3 0.5Mach

0.6 0.7 0.8 0.9


taileron loop cut using p-analysis approach

HIRM+ 1.2, RIDE 1.0, ^-analysis12-D ec-2000, c1_symm, Long Cat 1+2

4.5

4

3.5

3

FC2 [30;35] [Ua4,Ua4]

2FC6 [-15 ,-10][30,35] [Ua4,Ua4][Ua4,Ua4]

1.5

1

FC1 [-15,—10][30,35] [Ua4,Ua4][Ua4,Ua4]

0.5 FC7: [—15,—3][13,35]

[Ua4,Ua4][Ua4,Ua4]0

0.4 0.5 0.6 0.7 0.80.2 0.3 0.9 10.10Mach

Fig. 4.129: Sum m ary of results for the worst-case stability margins due to symmetrical

taileron loop cut using classical approach

151

4 .9 C o n c lu s io n s

We make some brief observations on the results obtained in this analysis. The approach

adopted seems to be quite successful, given th a t we can clear the RIDE control law at

most of the required flight conditions for several criteria with reasonable effort. It was also

shown, through illustrative examples for some flight cases, tha t whole portions of the flight

envelope can be cleared by including Mach number and altitude as uncertain parameters,

thus removing the need to resort to traditional gridding approaches. However, due to

trim m ing problem s w ith the HIRM -f model, this approach was not extended to all the

flight cases.

As sta ted earlier, th e approach to LFT-based uncertainty modelling adopted in this analysis

is fast and easy, bu t conservative, and does not allow the com putation of the worst case in

term s of particu lar values of the uncertain param eters. Thus, we feel tha t this approach

is best suited for an initial analysis, where the main objective is to quickly eliminate those

regions of the flight envelope where the control law passes the clearance criteria, in order to

be able to concentrate on the remaining problem atic regions.

The effort required in going from the fast but conservative approach to LFT generation to

the more exact sym bolic-linearisation based approach may be quite high, especially if the

resulting requirem ents are not considered a t the aircraft modelling stage. Estimates for

the am ount of conservatism separating the two approaches, for a realistic aircraft model

example, would help to answer the question of whether the extra effort is really worthwhile.

152

C hapter 5

C learance o f th e H W E M CL002 flight control law

In this chapter, we consider another approach to the generation of LFT-based uncertainty

models for the clearance of a flight control law for a highly detailed model of a VSTOL air

craft. This m ethod requires the availability of detailed information about the uncertain pa

ram eters. However, since the model considered is already fully implemented in SIMULINK

block diagram form, the task of generating these LFT-based uncertainty models becomes

relatively straightforw ard. The m ain advantage of this approach is tha t an exact description

of the uncertainties is obtained and thus, the exact worst-case uncertain param eter com

bination can be evaluated [2, 9]. As before, two linear clearance criteria (stability margin

criterion and unstab le eigenvalue criterion) are investigated. The derived LFT-based uncer

ta in ty models are validated against bo th the original nonlinear and linear models, and the

clearance results produced are com pared with those of the classical approach in [53].

5.1 T h e H W E M a ircra ft m o d el

The HW EM is a full non-linear model of the Vectored-thrust Aircraft Advanced flight Con

trol (VAAC) H arrier, developed by QinetiQ for research on various aspects of flight control

th a t are relevant to Short Take-Off and Vertical Landing (STOVL) operations. The HWEM,

originally im plem ented as a FORTRAN program, has been converted to a pure SIMULINK

model. D a ta for the model was derived from a variety of sources, such as wind tunnel

and flight te st m easurem ents and theoretical predictions, obtained from various aircrafts

[54]. The flight control surfaces comprise an all-moving tailplane, ailerons, flaps, rudder and

airbrake. W ith the exception of the rudder, all control surfaces are hydraulically powered.

The pow er-plant is a Rolls-Royce Pegasus Mk. 103 turbofan with four separate but coupled

nozzles th a t allow the direction of the th rust to be altered. High-pressure bleed air from the

engine com pressor provides control at low speeds when the aerodynamic control surfaces

become ineffective. The bleed air is ejected through reaction control valves (RCVs) in the

wing tips (roll control), nose (pitch control) and tail boom (pitch and yaw control). Since the

RCVs are mechanically linked to the appropriate control surfaces, the need for additional

con tro ls/ac tuato rs is avoided. Bleed air to the RCVs becomes progressively available as the

nozzle angle is increased, the system being fully pressurised when nozzle angle exceeds 34°.

W hen engaged, the Flight Control System (FCS) has full authority over all surfaces, except

the airbrake, and engine control extends to the th ro ttle and nozzles. In addition, the FCS

153

can control the roll and yaw au tostab servos, which provide limited authority control of the

ailerons and yaw RCVs respectively.

The aircraft dynam ic behaviour can be categorised by three modes: a conventional take-off

and landing mode (CTO L) in which the nozzles are at 0° relative to the horizontal body

axis, a vertical take-off and landing mode (VTOL) in which the nozzles are at an angle of

90° and finally a TR A N SITIO N mode in which the nozzles are between 0° and 90°.

5.2 T h e C L 002 c o n tr o l law

The flight control law supplied w ith the HWEM is based on VAAC Control Law 002 (CL002).

It is a full three-axis (pitch, roll and yaw) manual thrust-vectoring control law designed us

ing classical m ethods. The la teral/d irectional controller has been simplified by omitting the

lateral velocity controller used to implement the Translational Rate Command Mode in the

more advanced VAAC control laws.

CL002 em ulates the s tandard H arrier’s ‘three-inceptor’ control strategy but with some ad

ditional augm entation to reduce workload.

The longitudinal stick com m ands pitch rate w ith pitch attitude hold above 60 KIAS, blend

ing to pitch a ttitu d e com m and below 50 KIAS. The pitch attitude hold function engages

when the stick is centered, provided the undercarriage is down and airspeed is below 250

KIAS. In the hover, the datum attitude, corresponding to stick-free attitude, is preset to

7.5° (approxim ately the norm al landing a ttitude of the Harrier). Thrust magnitude and di

rection are controlled conventionally, using the th ro ttle and nozzle levers respectively. Note

th a t pitch control uses tailplane, nozzle and thro ttle actuators. Additional pitch law fea

tures include the decoupling of pitch from th rust vector and flap angle changes, automatic

flap scheduling and bank com pensation to supply the pitch rate required to maintain level

flight in a banked turn .

The response type assigned to lateral stick blends from roll rate command with bank angle

hold above 130 KIAS, to bank angle command below 100 KIAS. The response type assigned

to the pedals is sideslip com m and above 40 KIAS, blending to yaw rate command/heading

hold below 30 KIAS. Further inform ation about the implementation of the CL002 control

law can be found in [54].

5 .3 T h e H W E M fligh t e n v e lo p e

The flight envelope of the HW EM for the clearance analysis task is shown in Table 5.1. All

flight conditions are defined for 1 g straight and level flight ( 7 = (f) = 0 ) at an altitude of 2 0 0

ft AMSL. T he angle of a ttack range for all flight conditions is [—4°,+16°] with increments

154

Flight condition A irspeed (knots)

F C 1 2 0 0

FC2 150

FC3 130

FC4 1 1 0

FC5 90

F C 6 60

FC7 0 .0 1

Table 5.1: Flight envelope for the H W EM

of 2 °. Five Category 1 uncertainties are specified for the longitudinal axis analysis and are

shown in Table 5.2.

P a r a m e te r V ariab le nam e [M in,M ax] value U nits

Longitudinal position of centre of gravity TJ-Xcg [-1.72,-11.7] %MAC

Pitch moment of inertia XJJyy [56887,69529] kgm2

Uncertainty on tailplane effectiveness U_CmtaiI [-20,+20] %

Uncertainty on pitching moment due to pitch rate U-Cmq [-20,+20] %

Uncertainty on pitching moment due to a U -Cma [-20,+2 0 ] %

Table 5.2: M ost relevant longitudinal uncertainties

Table 5.3 lis ts th e five C ategory 1 uncerta in ties th a t have been specified for the lateral

uncerta in ties .

P a r a m e te r V ariab le nam e [M in,M ax] value U n its

Uncertainty on sideslip vane measurement U .(3sens [-10,+10] %

Uncertainty on rolling moment due to sideslip U-Ch [-20,+20] %

Uncertainty on rolling moment due to roll rate U_Cn„ [-20,+20] %

Uncertainty on roll moment of inertia U J xx [-10,+10] %

Uncertainty on yaw moment of inertia U J zz [-10,+10] %

Table 5.3: M ost relevant lateral uncertainties

F u rth e r in fo rm ation ab o u t th e uncerta in ties can be found in [54, 55].

155

5 .4 L F T -b a s e d u n c e r ta in ty m o d e ll in g o f th e H W E M u sin g a sy m b o lic

a p p r o a c h

In th is approach, the dynam ic equations are first extracted from the SIMULINK non-linear

model. Symbolic derivation and linearisation of the HWEM non-linear dynamic equations

are then perform ed to obta in a simplified linearised symbolic LFT-based uncertainty model.

Because of the highly non-linear equations involved (such as those modelling thrust), it is al

most always necessary to make assum ptions in order to simplify the symbolic manipulations.

The use of interpolation, particularly in deriving the th rust forces and moments which are

usually complex functions of the flight param eters, may be necessary. The method consists

of a num ber of steps which are sum marised below.

5.4.1 G athering th e non-linear equations o f m otion

The equations of m otion for the rigid symmetric HWEM are derived in the classical way

[56], based on N ew ton’s second law of motion for each of the six degrees of freedom. Thus,

the force and m om ent equations can be w ritten as

T = IQ

where F = Aerodynam ic Forces + Propulsive Forces + Interference Forces

and T = A erodynam ic M oments + T hrust Moments

The to ta l force F can be further resolved into its components X , Y and Z while the total

mom ent T can be resolved into its com ponents L, M and N .

The general equations of m otion of the aircraft dynamics can be expressed as:

F = m a

aro

aro

{ X ,nen9 + X maro + p x r x (I zz Ixx) I zx x (jy (5.3)

(5.1)

(5.2)

6 = q x cosily) — r x sin(ip) (5.4)

If we consider th e longitudinal dynamics only, we have

r — p = V = 4> = 0 (5.5)

156

Equations (5.1) to (5.4) simplify to

u = ^ —en9 rn— ~ 9 x ^ “ 9 x siniQ) (5.6)

w = ( Zen?* Zar° ' ) - q x U + g x c o s ( 6 ) (5.7)

9 = {Xnieng + ^ m aro) J (5-8)1yy

6 = q (5.9)

The forces and m om ents can be further expanded and grouped as shown in the following

sets of equations.

• Longitudinal aerodynam ic forces and m om ents (body axes)

X aro = (X saro T X C O S^ q ) ( Z s a r o T Z s i n t ) X S27T.(q;) ( 5 .1 0 )

Zaro — (-^ sa r o T -^ f - s in t ) X -f- (Zsaro T ^ s in t ) X COs(qi) ( 5 .1 1 )

X r r i a r o — ( A Tnsclro T X m sirLt) X a r o X X Cg Z a r o X X Cg ( 5 .1 2 )

• Longitudinal aerodynam ic forces and m om ents (aerodynamic axes)

Xsaro = — (Cdwbod + Cdflap + Cdabr + Cdpod) X <7 (5.13)

Zsaro = — (C^bod + Clta.il + Clq) x q (5.14)

X m sar-o = {Cmtaii T Cmq + Cmu, + CmQ + Crriabr + Cmo + (7motT.m) X <JS (5.15)

• Propulsive forces

X md = - W F x R G x ( 0 . 3 5 x q y U ) (5.16)

X eng ~ Xthr3t T Am(f (5.17)

Zeng = T Zmd + Zraci (5.18)

• Som e other term s

q = i p ( c / 2 + W 2 ) S ( 5 1 9 )

= ip(C72 + W2)Sc (5 2 0 )

After some m athem atical m anipulation, it is easy to see how these equations relate to the

uncertain param eters of the HW EM . For example, consider the equation for the derivative

of longitudinal velocity, u, which is shown in equation (5.6). Substituting for X aro and X md

from equations (5.10) and (5.17) gives

.. _ X thrst + X md + {Xsaro + X sint) X COsjOL) - (Zsaro + Zsint) X S l T l j a ) _ ^ v (521)

157

From equations (5.13), (5.14) and (5.16), X saro, Z saro and X md can be substituted into (5.21) to give

^ _ X thrst _ W F x RG x (0.35 x q + U) ^ (Cdwbod + C dflap + Cdabr + Cdpod) x q x cos(a:) m m m

X sint x cos(a) (CZu,b0(1 + CZtoiI + CZJ x q x sm (a) Zsint x sin(a) TTr . //n ^--------------------- - —------------------------- —------------------------------------- — —q x W — a x sin(0) (5.22)rn m m

It can be observed th a t the uncertain param eters (m, C d and Ci coefficients) now appear

explicitly in equation (5.22). This is the form required for the equations in order to generate

LFT-based uncertain ty models using the m ethod described in Section 3.3.

5.4.2 In terpolation s

From the d a ta available in the HW EM model, it is not clear how the forces and moments

due to reaction control are modelled. Tracing these back would need a substantial amount

of effort and tim e. Even if this can be done, we will inevitably end up with a set of highly

non-linear equations th a t will be very difficult to handle. An alternative way to proceed

is to approxim ately model these forces and moments based on interpolation techniques.

Consider, for instance, the Z-com ponent of the reaction control force Z ract, which is shown

as a complex function of the control surfaces and some other parameters:

Zract = f iVi Ma, 9 , F N P , WB N O R , p , . . .etc) (5.23)

Note th a t for modelling purposes, we are interested in finding a relation between Zract and

the input rj (tailplane deflection). We can assume tha t, at a particular trimpoint, all the

other param eters in the above equation are set to their trim values (assumed fixed). Then,

by perturb ing 77 slightly from its trim m ed value, we can investigate its effect on Zract. In

fact, w hat we are actually doing is creating a look-up table to investigate the dependency

of Zract on V- By using suitable polynomials, a fit in the data can thus be obtained. Thus,

after perform ing these steps, the force and moment equations for the reaction control can

be approxim ated as first order and second order polynomials respectively as

Zract = —161 x 77 + 298 (5.24)

x mract = 2500 XT]2 - 22025 x 77 + 39070 (5.25)

5.4.3 A ssu m p tion s and approxim ations

Because L FT -based uncertain ty models cannot be generated for non-linear functions such

as sines and cosines, it is necessary to convert them into a form where they are suitable for

158

use in LFTs. A pproxim ation such as the Taylor expansion can be used. For small values of

6 , the following equations hold

62cos(0) = 1 - — (5.26)

sin{9) = 6 (5.27)

In this study, we focus on the V TO L and TRANSITION modes in which the undercarriage

is set down, which m eans th a t the airbrake is fixed at 40% of its value. Moreover, the flap

is also fixed a t 50° which causes the input AFLAP to be set to 1.

f l ap = 1 (5.28)

At height > 58 ft, th e suction has negligible effect on lift, which results in

Z S U C G = 0 (5.29)

At a fixed trim poin t, we can assume th a t the th rust will be fixed and hence Xthrust and

Xthrusti which are forces due to th ru st, can be taken as constants. In addition, the mass

flow ra te (W F ) and the nozzle opening (P T H T P ) can also be taken as constants.

5 .4 .4 S y m b o lic l in e a r is a t io n o f th e H W E M

Once these equations have been ex tracted from the non-linear model, they can be entered as

symbolic objects in M A PLE or MATLAB (using the Symbolic M ath Toolbox). In MAPLE,

these param eters can be entered directly (just like in a normal text fashion). For instance,

equation (5.4) can be symbolically displayed in MAPLE by inputting the following line of

code:

thetadot := q*cos(phi) - r*sin(phi);

The o u tp u t will be displayed as is shown in equation (5.4). In MATLAB, these param

eters have to be defined as symbolic objects using the sy m s command. The same equation

has to be inpu t as

syms thetadot q phi r

thetadot = q*cos(phi) - r*sin(phi)

Trimming of th e H W EM is next done for straight and level flight at a speed of 170 knots

and altitude of 100 ft. In this transition mode, the nozzle angle is deflected at 45.84° with

159

respect to the horizontal body axis of the aircraft. The states, inputs and outputs are chosen

as:

s = {U,W,q,e}'

u = [OjiV]'

y = [U,W,q, 0]'

At th is stage, linearisation of the nonlinear symbolic equations can be performed using the

diff com m and in M A PLE or MATLAB. The resulting linearised state-space matrices can

thus be obtained symbolically. To get the final param etric state-space matrices, the trim

values are substitu ted in these symbolic equations and evaluated using the eval command.

Note th a t the uncertain elements will still appear symbolically in the results. For instance,

element (2,3) of the A m atrix was found to be

172.3 — (16.4 x X Cq) . .A23 = ----- ^ - 86.7 (5.30)

m

where m and X cg are the uncertain param eters. Once these equations have been converted

into this form, it is triv ial to represent them physically in SIMULINK block diagrams. In

the HW EM model, the uncertainties are expressed as a percentage of their nominal values

and these param eters can be expanded as (for instance consider mass)

m = m 0 x ( 1 + W x A mass) (5.31)

where mo is the nom inal value of the mass, W is a weight (here it equals the error in mass)

and A mass is the norm alised uncertain ty th a t varies between 1 and - 1 .

5.4.5 V alidation o f th e sym bolic m odel

Prior to building the whole LFT-based uncertainty model, it is im portant to find out how

accurate th e derived model is. Obviously, we can start by considering the nominal case

where all the uncertain ties A are set to zero (that is the uncertain param eters are set at

their nom inal values). Tim e dom ain simulations are particularly suited to compare this

simplified linearised model w ith the non-linear one. Moreover, the comparison can be fur

ther extended by validating the eigenvalues obtained from the linearisation of the non-linear

model against those of the simplified linearised symbolic model.

Com parison of th e eigenvalues for the longitudinal dynamics gives the following results:

Nominal linearised model : -0.544±1.36U, 0.015±0.06U

Symbolic linearised model: -0.114i0.48Ch, -0.135, -1.725

160

It can be readily observed th a t the phugoid eigenvalues decay into real roots in the symbolic

model. W hile a sim ilar phenom enon was observed in past analysis carried out on the HWEM

at Q inetiQ and BAE System s [57], the differences in the eigenvalues shown here are quite

significant. In addition, the short-period eigenvalues in the symbolic model have become

more lightly dam ped. These results, therefore, seem to indicate some possible problems in

the symbolic model. To confirm this point, tim e domain analysis is carried out on both the

open loop non-linear m odel and the symbolic model. Figures 5.1 and 5.2 show the pitch rate

responses to dem ands in ta ilp lane and nozzle respectively. The pitch attitude response to a

dem and on tailp lane is shown in Figure 5.3. It can be deduced tha t significant discrepancies

exist between the original SIMULINK and simplified symbolic models. Possible sources for

these errors m ight be due to two factors, (i) derivation of the nonlinear symbolic equations

and (ii) nonlinear sim plifications brought in a t the s ta rt of the modelling process. The next

section sum m arises the m ain difficulties th a t were encountered in generating the symbolic

LFT-based uncertain ty model for the HWEM.

0.3

Non-linear model Simplified analytical model0.25

0.2

0.15

0.1

0.05

0

-0 .0 5

- 0.1

-0 .1 5

-0 .2 12 14 16 2010 1886420time (s)

Fig. 5.1: P itch rate response to a step demand on tailplane

5.4.6 D ifficu lties w ith th e sym bolic LFT-based uncertainty m odelling tech

nique

The m ain problem w ith th is approach is th a t the derivation of the non-linear symbolic model

can dem and trem endous effort, and requires a detailed knowledge of flight mechanics. Be

cause of the ir complexity, m any of the non-linear symbolic equations have to be further

161

N on-linear model Simplified analytical model

0.05

-0 .0 5

-0.1

-0 .1 5

time (s)

Fig. 5.2: P itch rate response to a step demand on nozzle

0.5 Non-linear model Simplified analytical model

0.4

0.3

0.2

<x>

-0.1

-0 .2

-0 .3

time (s)

Fig. 5.3: P itch a ttitu d e response to a step demand on tailplane

simplified before th e linearisation is performed. Inevitably, inappropriate assumptions may

well lead to a significant loss of accuracy which is then reflected in the final LFT-based

uncertainty m odel (results being overly optimistic or conservative). Although compensation

param eters can be used to reduce the error margins, this ‘tuning’ process can be very time-

consuming, and provides no guarantee th a t the final LFT-based uncertainty model will be

sufficiently accurate. I t is well known th a t many stability derivatives (such as the pitching

mom ent coefficients, yawing m om ent coefficients, ... etc) can be complex functions of the

162

flight param eters and it is common practice to model them by means of look-up tables.

In m any cases, finding a relationship between these elements can be non-trivial and may

require much tria l and error. Even if interpolation is used to find a polynomial fit to the

data , it is necessary to focus on only a certain range of variation of the flight parameters in

order to keep the com plexity of the resulting equations (i.e. the order of the polynomials) to

a minimum. Again, an error m argin will exist between these interpolations and the actual

behaviour of the original param eters.

The in terpolation process is trim -dependent, th a t is, the trim values of some flight param

eters may be necessary to generate the polynomial fits. Obviously, the resulting symbolic

LFT-based uncertain ty models will then only be valid in close neighbourhood of the rele

vant trim point. Incidently, th is also means th a t changing the trim point may bring drastic

changes to the behaviour of the model so th a t the whole interpolation procedure has to be

repeated again. Clearly, as the model grows in complexity, this soon becomes extremely

difficult.

The resulting L FT -based uncertain ty model may contain several repeated parameters - for

instance mass m m ay be repeated in more than one location in the differential equations.

This may cause a problem in subsequent analysis as it is well known in the /u-literature that

the use of repeated real scalars (as is the case here) may introduce conservatism in the //

upper bounds. Moreover, the actual size of the uncertainty set will be bigger which can

increase the com putational tim e or increase the gap between the // upper and lower bounds.

Bearing in m ind these difficulties, it was decided in this study th a t the generation of sym

bolic L FT-based uncertain ty models for the HWEM for a large number of flight conditions

using the symbolic approach was not feasible. An alternative approach based on physical

modelling techniques was developed instead and is presented in the next section.

5.5 L F T -b a se d u n c e r ta in ty m o d e llin g o f th e H W E M usin g a p h ysica l

m o d e llin g a p p ro a ch

An alternative approach based on physical modelling techniques can be used for generating

LFT-based uncertain ty models, where the model of the system is given in block diagram

form. For the H W EM , th is m ethod, which is described in Section 2.3.2, has proved to be

simple to im plem ent and dem ands little modelling effort. An exact representation of the

uncertain system is ob tained and thus the exact worst case set of uncertain parameters can

be com puted. Moreover, a m inim al order LFT-based uncertainty model (i.e. the A set will

contain non-repeated uncertainties) can be obtained by clever arrangement of the block di

agram representation. Each uncertain param eter defined in Tables 5.2 and 5.3 is introduced

163

in the non-linear model as an input multiplicative uncertainty according to the procedure

explained in Section 2.3.2. T he resulting model with the uncertainties introduced can then

be linearised to give the LFT-based uncertainty model.

One drawback of th is m ethod is th a t detailed information about the way in which the uncer

ta in param eters affect the aircraft dynamics is required. As the model grows in complexity

or as the num ber of uncertain param eters is increased, this method can lead to an increase

in modelling effort. However, since the HWEM is already implemented in SIMULINK block

diagram form, th e incorporation of the uncertain param eters into this model is straightfor

ward.

The second draw back w ith th is m ethod is th a t it cannot easily be used to generate LFT-

based uncertain ty models which cover whole portions of the flight envelope. This is because

the linearisations are dependent on the trim points which are themselves dependent on the

uncertain param eters.

5 .6 V a lid a t io n o f th e L F T -b a sed u n cer ta in ty m od el

To determ ine the accuracy of the derived LFT-based uncertainty model, it is vital, prior

to any analysis, to validate it against the original linearised and nonlinear model. Time

domain sim ulations are particu larly suited for this purpose although further analysis such

as eigenvalue com parison can be carried out for the validation process. Initially, the nominal

case (where all the uncertain ties are set a t their nominal values) is considered but this can

be easily extended to take into account any combination of the uncertainties within their

ranges. O pen loop sim ulations are performed based on the following cases:

1 . Case 1: N on-linear model - Set all the uncertainties to be equal to zero in the non-linear

SIM ULINK block diagram .

2. Case 2: Linearised model - Set all the uncertainties to be equal to zero in the non-linear

model and then perform linearisation.

3. Case 3: LFT-based uncertain ty model - Generate LFT-based uncertainty model from

non-linear m odel and close all the A loops with 0.

The chosen trim po in t for the validation process is at a speed of 170 knots and an altitude

of 100 ft. T he five m ost relevant longitudinal uncertainties from Table 5.2 are considered.

Results for the open-loop case are shown in Figures 5.4 to 5.9. Figure 5.10 shows the case

when a random com bination of the uncertainties was used.

It can be noted th a t an exact m atch is obtained between the linearised model and the re

sulting L FT -based uncertain ty model. Clearly this observation indicates th a t the derived

164

a (c

leg)

8— LFT model— ■ Nonlinear model

L inearised model7

6

5

4

3

2

3 52 40tim e (s)

8N onlinear model

— LFT model L inearised model6

4

2

0

•2

-45432

time (s)

0.02— LFT model

Nonlinear model Linearised model

- 0.02

S -0.04

-0.06

-0.08

- 0.10 52 3 4

time (s)

Nonlinear model — LFT model

Linearised model

time (s)

Fig. 5.4: Responses to a 1° step demand on tailplane

165

0 (d

eg)

a (d

eg)


L inearised model

-15

-2 0

-252 3 4 50

-0.1

-0.2

— LFT model Linearised model

- Nonlinear model

-0.4

-0.50 2 43 5

tim e (s) time (s)

130Nonlinear model

— LFT model Linearised model120

110</)E4^> 100

time (s)


L inearised model-1 0

-2 0

-30

-40

-50

-60

-70

time (s)

Fig. 5.5: Responses to a 5° step demand on tailplane

166

0 (d

eg)

a (d

eg)

— LFT model Linearised model Nonlinear model

53 420time (s)

N onlinear model — LFT model

L inearised model

time (s)

0.04— LFT model

Linearised model— • Nonlinear model

0.035

0.03

0.025

a 0.02

0.015

0.01

0.005

3 50 2 4time (s)

88— LFT model

Linearised model Nonlinear model87

86

85

84

833 50 2 4

time (s)

Fig. 5.6: Responses to a 1° step demand on nozzle

167

6 (d

eg)

a (d

eg)


L inearised model

2 3 50 4

0.14 — LFT model Linearised model Nonlinear model0.12

^ 0.08

0.06

0.04

0.02

0 3 52 4time (s) time (s)

30— LFT model— • N onlinear model

L inearised model25

20

15

10

55432

90

85

80

75


Linearised model

70

653 50 2 4

time (s) *'me (s )

Fig. 5.7: Responses to a 5° step demand on nozzle

168

7.55

7.54

7.53

7.51

— LFT model— • Nonlinear model

Linearised model7.49

time (s)

N onlinear model — LFT model

L inearised model7.75

7.55

7.45

time (s)

15

10

5

0— LFT model

Linearised model Nonlinear model

•50 2 3 4 5

time (s)

87.9— LFT model

Linearised model Nonlinear model87.8

87.7wE> 87.6

87.5

87.44 50 2 3

time (s)

Fig. 5.8: Responses to a 1% step demand on throttle

169

e (d

eg)

■o 7.7


L inearised model

40 2 3 5time (s)


Linearised model10.5

time (s)

16

14

12

10

8

6

4

2 — LFT model Linearised model Nonlinear model0

•20 2 3 54

time (s)

— LFT model Linearised model Nonlinear model

time (s)

Fig. 5.9: Responses to a 10% step demand on throttle

170

8

- - Nonlinear— Linear7

6

5

8 4

3

2

53 40 2t ( s )

8

— Linear- - Nonlinear

6

4

2

0

■2

■45432

t ( s )

Fig. 5.10: Perturbed responses

— Linear- - Nonlinear- 0.01

- 0.02

-0.03

-0.04

-0.05

-0.06

-0.07

-0.08

-0.0950 2 3 4

t(s )

98— Linear- - Nonlinear96

94

92

90

88

8650 3 42

t( s )

a 1° step demand on tailplane

171

LFT-based uncertain ty model is as accurate as the original linearised model. Furthermore,

it can be seen th a t the m atch between the non-linear and linearised models is good al

though, as expected, some discrepancies are observed for responses due to larger control

inputs. Hence, a good approxim ation of the non-linear model has been obtained.

The next step consists of evaluating the closed-loop characteristics of the models. Fig

ure 5.11 shows the closed-loop pitch rate responses to a step demand on pitch rate, which

again shows good m atches between the various models.

Pitch rate responseso.z

-0 .2

" f t-0 .4

i i -0 .6

-0.8 Dem and

Nonlinear— Linear

3.51 1.5 2 2.5 3 4 4.5 50.50time (s)

Fig. 5.11: P itch rate responses to a step demand on pitch rate

5 .7 A n a ly s is o f th e lo n g itu d in a l H W E M d yn am ics

We detail results from the analysis of the longitudinal dynamics of the HWEM control

law, using the fi tools and m ethods described in Chapter 2 [2]. The analysis is carried

out for the seven flight conditions defined in Table 5.1. Only the straight and level flight

case is considered, taking into account the five Category 1 pitch uncertainties. Since three

aerodynam ic uncertain ties are included simultaneously, it is necessary to apply a reduction

factor of 0.46 to the ir ranges of variation as suggested in [13].

5 .7 .1 S ta b i l i ty o v e r th e flig h t en v e lo p e

Before a ttem p ting to identify over which regions of the flight envelope the HWEM control

law satisfies the stability margin criterion, we first of all identify the regions over which the

172

aircraft rem ains stable. The region th a t satisfies the stability margin criterion will then be

a subset of th a t region. Results are presented for the seven flight conditions FC1 to FC7 in

Figures 5.12 to 5.18 and it can be observed th a t in all cases robust stability is guaranteed

< !)•

5.7.2 R esu lts for th e stab ility margin criterion - single loop analysis

In this section, we present clearance results for the stability margin criterion, with respect

to violation of Nichols plot exclusion regions as defined in Chapter 2. For the seven flight

conditions, we te st for the avoidance of the elliptical exclusion regions providing gain and

phase m argins of ±4.5dB and ±28.44° respectively. Only single loop analysis is performed

and the open-loop Nichols plots of the frequency response are obtained by breaking the loop

of one of the three actuators (tailplane or nozzle or throttle actuators), leaving the other

two loops closed. T he resulting mixed fi bounds and the worst-case Nichols plots are shown

below.

The nozzle loop is broken whilst leaving all the other actuator loops closed. The resulting

fi bounds and the worst-case Nichols plots are shown in Figures 5.19 and 5.20 for FC1, in

Figures 5 .2 1 and 5.22 for FC2, in Figures 5.23 and 5.24 for FC3, in Figures 5.25 and 5.26

for FC4, in Figures 5.27 and 5.28 for FC5, in Figures 5.29 and 5.30 for FC6 , and in Figures

5.31 and 5.32 for FC7.

HWEM 5.4, WEMSIM005, ^-analysisM ar-2002, Basic Stability Test, Long Cat 1

0.14

0.12

0.08

0.06

0.04

0.02

o> (rad/s)

Fig. 5.12: Real /i bounds - FC1

173

HWEM 5.4 , W EM SIM 005, ji-an a ly s isMar-2002, Basic Stability Test, Long Cat 10.16

0.14

0.12

0.1

0.06

0.04

0.02

o> (rad/s)

Fig. 5.13: Real fi bounds - FC2

HWEM 5.4, WEMSIM005, ^ a n a ly s isM ar-2002, Basic Stability Test, Long Cat 1

0.5

0.45

0.4

0.35

0.3

0.2

0.15

0.1

0.05

10'co (rad/s)

Fig. 5.14: Real /i bounds - FC3

174

HWEM 5.4, W EM SIM 005, ^ -a n a ly s isM ar-2002. B^sic ^tabilitv Test. Long Ca( 10.5

0.45

0.4

0.35

0.3

=<-0.25

0.2

0.15

0.1

0.05

10'to (rad/s)

Fig. 5.15: Real n bounds - FC4

HWEM 5.4, WEMSIM005, ^ a n a ly s is M ar-2002, Basic Stability Test, Long Cat 1

0.5

0.45

0.4

0 3 5

0.3

0.2

0.15

0.1

0.05

to (rad/s)

Fig. 5.16: Real n bounds - FC5

175

HWEM 5.4 , W EM SIM 005, ^ -a n a ly s isM ar-2002, Basic Satbility Test, Long Cat 10.5

0.45

0.4

0.35

0.3

0.2

0.15

0.1

0.05

10'at (rad/s)


HWEM 5.4, WEMSIM005, n-analysisM ar-2002, Basic Stability Test, Long Cat 1

0.25

0.2

0.15

0.05

10'10'<o (rad/s)


176

Gain

(dB)

HWEM 5.4, W EM SIM 005, ^ -a n a ly s isM ar-2002, Criterion 1A - Nozzle Loop Cut, Long Cat 10.18

0.16

0.14

0.12

0.08

0.06

0.04

0.02

10'co (rad/s)

Fig. 5.19: M ixed g, bounds for the Nichols exclusion test - FC1

HWEM 5.4, WEMSIM005, ^-analysisM ar-2002, Criterion 1A - Nozzle Loop Cut, Long Cat 1

30

20

10

-10

-20

-3 0

-4 0

-5 0

-6 0 — -2 2 0 -120 -100 -8 0-160 -140-1 8 0-200

Nominal Worst Case

P hase ( 0 )

Fig. 5.20: Nominal and worst case Nichols plot - FC1

177

Gain

(dB)

HWEM 5.4, W EM SIM 005, n -an a ly s isM ar-2002, Criterion 1A - Nozzle Loop Cut, Long Cat 10.2

0.18

0.16

0.14

0.12

0.08

0.06

0.04

0.02

o) (rad/s)

Fig. 5.21: M ixed fi bounds for the Nichols exclusion test - FC2

HWEM 5.4, WEMSIM005, ^-analysisM ar-2002, Criterion 1A - Nozzle Loop Cut, Long Cat 1

30

20

-10

-20

-3 0

-4 0

-5 0

-6 0

-7 0 ■— -220 -160 -140 -120 -100 -8 0-1 8 0-200

— Worst Case Nominal

Phase (°)


178

Gain

(dB)

HWEM 5.4, W EM SIM005, n -an a ly s isM ar-2002, Criterion 1A - Nozzle Loop Cut, Long Cat 10.6

0.5

0.4

0.2 -

0.1

<b (rad/s)


HWEM 5.4, WEMSIM005, n-analysisM ar-2002. Criterion 1A - Nozzle Loop Cut. Long Cat 130

20

-10

-20

-3 0

-4 0

-5 0

-100 -8 0-1 8 0 -160 -140 -120-200

Worst Case Nominal

P h a s e ( 0 )


179

Gain

(dB)

HWEM 5.4 , W EM SIM005, ^ a n a ly s isM ar-2002, Criterion 1A - Nozzle Loop Cutj Long Cat 10.6

0.5

0.4

0.2

co (rad/s)

Fig. 5.25: M ixed p bounds for the Nichols exclusion test - FC4

HWEM 5.4, WEMSIM005, ^-analysis M ar-2002, Criterion 1A - Nozzle Loop Cut, Long Cat 1

30

20

10

-10

-20

-3 0

-4 0

-5 0

-6 0 — -220 -100 -80-140 -120-160-1 8 0-200

Worst C ase Nominal

P hase ( 0 )


180

Gain

(dB)

HWEM 5.4, W EM SIM005, n -an a ly s isM p -2 0 0 ^ . Criterioq 1A - [vjozzle Lpop Cul,. L.onq £ ? t 10.7

0.6

0.5

0.4

0.3

0.2

0.1

co (rad/s)


HWEM 5.4, WEMSIM005, n-analysisM ar-2002, Criterion 1A - Nozzle Loop Cut, Long Cat 1

30

20

-10

-20

-3 0

-4 0

-5 0

-6 0-220 -160 -120 -100-140 -80-1 8 0-200

Worst C ase Nominal

P hase ( 0 )


181

Gain

(dB)

HWEM 5.4, W EM SIM005, ^ -a n a ly s isM ar-2002, Criterion 1A - Nozzle Loop Cut. Long Cat 10.7

0.6

0.5

0.4

3.

0.3

0.2

10‘(D (rad/s)


HWEM 5.4, WEMSIM005, n-analysisM ar-2002. Criterion 1A - Nozzle Loop Cut. Loncj Cat

30

Nominal Worst C ase20

-10

-20

-3 0

-4 0

-5 0

-100 -80-160 -120-140-1 8 0-200Phase ( 0 )


182

Gain

(dB)

HWEM 5.4, W EM SIM005, n -an a ly s isM ar-2002. Criterion 1A - Nozzle Loop Cut. Long Cat 10.35

0.3

0.25

0.2

0.15

0.05

cd (rad/s)


HWEM 5.4, WEMSIM005, n-analysisM ar-2002, Criterion 1A - Nozzle Loop Cut, Long Cat 1

30

20

-10

-20

-3 0

-4 0

—50—220 -100 -8 0-160 -120-180 -140-200

Worst Case Nominal

Phase ( 0 )


183

Similarly, the tailplane loop is broken whilst leaving all the other actuator loops closed. The

resulting p bounds and the worst-case Nichols plots are shown in Figures 5.33 and 5.34 for

F C 1 , in Figures 5.35 and 5.36 for FC2, in Figures 5.37 and 5.38 for FC3, in Figures 5.39

and 5.40 for FC4, in Figures 5.41 and 5.42 for FC5, in Figures 5.43 and 5.44 for FC6 , and

in Figures 5.45 and 5.46 for FC7.

HWEM 5.4, WEMSIM005, ji-analysisMar-2002, Criterion 1A - Tailplana Loop Cut, Long Cat 10.18

0.16

0.14

0.12

a.

0.08

0.06

0.04

0.02

a> (rad/s)


HWEM 5.4, WEMSIM005, n-analysis Mar-2002, Criterion 1A - Tailplane Loop Cut, Long Cat 1


-20

-30

-40

-50

-60 •— -220 -100 -80-140 -120-160-180-2 0 0Phase ( ° )


184

Gain

(dB

)

HWEM 5.4, W EM SIM005, p -an a ly s isM ar-2002. Criterion 1A - Tailplane Loop Cut. Long Cat 10.25

0.2

0.15

0.1

0.05

a) (rad/s)

Fig. 5.35: M ixed /i bounds for the Nichols exclusion test - FC2

HWEM 5.4, WEMSIM005, p-analysis M ar-2002, Criterion 1A - Tailplane Loop Cut, Long Cat 1

30

20

-10

-20

-3 0

-4 0

-5 0

-6 0 ■— -220 -100 -80-160 -140 -120-1 8 0-200

— Worst C ase Nominal

P hase (°


185

Gain

(dB)

HWEM 5.4, W EM SIM005, |i-a n a ly s isM ar-2002, Criterion 1A - Tailplane Loop Cut, Long Cat 10.6

0.5

0.4

a. 0.3

0.2

to (rad/s)

Fig. 5.37: M ixed /j bounds for the Nichols exclusion test - FC3

HWEM 5.4, WEMSIM005, n-analysisM ar-2002, Criterion 1A - Tailplane Loop Cut, Long Cat 1

30

20

-10

-20

-3 0

-4 0

-5 0

-6 0-220 -160 -100-140 -120 -8 0-1 8 0-200

Worst C aseNominal

Phase ( 0 )


186

Gain

(dB)

HWEM 5.4, W EM SIM005, p .-analysisM ^r-2002. Crit^ion 1A - Tailplane ^000 Cijt. LonoQipt0.6

0.5

0.4

0.2

0.1

o) (rad/s)


HWEM 5.4, WEMSIM005, ^-analysisM ar-2002, Criterion 1A - Tailplane Loop Cut, Long Cat 1

30

20

-10

-20

-3 0

-4 0

-5 0

-100 -80-160 -140 -120-1 8 0-200


Phase ( 0 )


187

Gain

(dB)

HWEM 5.4, W EM SIM005, ^ -a n a ly s isM ar-2002. Criterion 1A - Tailplane Loop Cut. Long Cat 10.6

0.5

0.4

0.2

0.1

10'co (rad/s)



20

10

-10

-20

-3 0

-4 0

-5 0

-6 0 ■— -220 -100 -80-160 -140 -120-1 8 0-200

Worst C ase Nominal___

Phase ( 0 )


188

Gain

(dB)

HWEM 5.4, W EMSIM005, (i-an aly sisM y-2002 , Cri^ericpn 1A - Tailplane Loop Cut. Long Cat0.7

0.6

0.5

0.4

0.3

0.2

0.1

100) (rad/s)

10'


HWEM 5.4, WEMSIM005, ji-analysisM ar-2002. Criterion 1A - Tailplane Loop Cut, Long Cat 1

30 Worst C ase- - Nominal

20

-10

-20

-3 0

-4 0

-5 0

-100 -80-120-140-160-180-2 0 0Phase ( 0 )


189

Gain

(dB)

HWEM 5.4, W EM SIM005, ^ -a n a ly s isM ar-2002, Criterion 1 A - Tailplane Loop Cut, Long Cat 10.35

0.3

0.25

0.2

0.15

0.1

0.05

e> (rad/s)


HWEM 5.4, WEMSIM005, ^-analysisM ar-2002. Criterion 1A - Tailplane Loop Cut. Long Cat 130

20

-10

-20

-3 0

-4 0

-5 0

-6 0 — -2 2 0 -100-160 -140 -120 -8 0-1 8 0-2 0 0

Worst C ase— - Nominal

Phase ( 0 )


190

Finally, the th ro ttle loop is broken whilst leaving all the other actuator loops closed. The

resulting fj, bounds and the worst-case Nichols plots are shown in Figures 5.47 and 5.48 for

FC1, in Figures 5.49 and 5.50 for FC 2 , in Figures 5.51 and 5.52 for FC3, in Figures 5.53

and 5.54 for FC4, in Figures 5.55 and 5.56 for FC5, in Figures 5.57 and 5.58 for FC6 , and

in Figures 5.59 and 5.60 for FC7.

HWEM 5.4, WEMSIM005, n -analysisM ar-2002, Criterion 1A - Throttle Loop Cut. Long Cat 10.25

0.2

0.15

0.05

<a (rad/s)


HWEM 5.4, WEMSIM005, ^ an a ly s isM ar-2002. Criterion 1A - Throttle Loop Cut. Long Cat 1

Worst Caso Nominal

-3 0

-4 0

-5 0

-100 -8 0-160 -140 -120-180-200P h a s e ( ° )


191

Gain

(dB)

HWEM 5.4, W EMSIM005, p -an a ly s isM ar-2002, Criterion 1A - Throttle Loop Cut. Long Cat 10.18

0.16

0.14

0.12

0.08

0.06

0.04

0.02

to (rad/s)

Fig. 5.49: M ixed ji bounds for the Nichols exclusion test - FC2

HWEM 5.4, WEMSIM005, p-analysisM ar-2002, Criterion 1A - Throttle Loop Cut, Long Cat 1

30

20

-1 0

-2 0

-3 0

-4 0

-5 0

-6 0 — -220 -180 -160 -140 -120 -100 -8 0-200

— Worst C ase Nominal___

P hase ( 0 )


192

Gain

(dB)

HWEM 5.4, W EMSIM005, ^ -a n a ly s isMar-2002, Criterion 1A - Throttle Loop Cut, Long Cat 1rTI----- !---!—' ■■■■■!------!— —: ' : ” : : I-0.6

0.5

0.4

a -0.3

0.2

0.1

10-’10'a) (rad/s)

Fig. 5.51: M ixed p, bounds for the Nichols exclusion test - FC3


30

20

-10

-2 0

-3 0

-4 0

-120 -100 -8 0-160 -140-1 8 0-2 0 0

Worst ca se Nominal

P hase ( 0 )


193

Gain

(dB)

HWEM 5.4, W EM SIM005, ^ -a n a ly s isM ar-2002, Criterion 1A - Throttle Lood Cut, Lona Cat 1 I-1"! r------!— !—: ' ‘ ' I---------- !— ^ — : : : T : i I0.6

0.5

0.4

0.2

o) (rad/s)


HWEM 5.4, WEMSIM005, n-analysisM ar-2002, Criterion 1A - Throttle Loop Cut, Long Cat 1

30

20

10

-1 0

-20

-3 0

-4 0

-5 0

-100 -80-160 -120-140-1 8 0-200

W orst C ase Nominal

P hase ( 0 )


194

Gain

(dB)

HWEM 5.4, W EM SIM005, ^ -a n a ly s isM ar-2002, Criterion 1A - Throttle Loop Cut, Long Cat 1

771-----------’------!---- I I I 77 T T ----------1— ^ ---1 ? ' T i l l—0.6

0.5

0.4

0.2

a) (rad/s)



30

20

-1 0

-2 0

-3 0

-4 0

-5 0

—60—220 -100 -80-140-160 -120-1 8 0-200

Worst Case Nominal

P hase ( 0 )


195

Gain

(dB)

HWEM 5.4, W EMSIM005, n -an a ly s isM ar-2002, Criterion 1A - Throttle Loop Cut. Long Cat 10.7

0.6

0.5

0.4

0.3

0.2

0.1

co (rad/s)


HWEM 5.4, WEMSIM005, n-analysisM ar-2002, Criterion 1A - Throttle Loop Cut, Long Cat 1

30

20

-10

-20

-3 0

-4 0

-5 0

- 6 0 '— -2 2 0 -100-160 -140 -120 -8 0-1 8 0-2 0 0

Worst Case Nominal

Phase ( 0 )


196

Gain

(dB)

HWEM 5.4, W EMSIM005, n -an a ly s isMar-2002, Criterion 1A - Throttle Loop Cut, Lona Cat 11 ! ^ I : : : : : I ! ! : : : : i0.35

0.3

0.25

0.2

0.15

0.05

10'o) (rad/s)


HWEM 5.4, WEMSIM005, p-analysisM ar-2 0 0 2 . Criterion 1A - Throttle Loop Cut. I orjg C at 130

20

-1 0

-2 0

-3 0

-4 0

-5 0

-6 0 L- -220 -80-120 -100-180 -160 -140-2 0 0

Worst C ase Nominal

Phase ( 0 )


197

5 .7 .3 R esu lts for th e u n stab le eigenvalue criterion

Here, results for the worst case eigenvalues are given in Figures 5.61 to 5.67. This criterion

has been investigated for both the fi approach and the classical method. The purpose of

this comparison is to investigate the agreement between the results generated by these two

approaches. Also shown on these figures is a superimposed plot of the acceptable boundaries

for the unstable eigenvalue requirement proposed in [13]. A general observation is that this

criterion is satisfied for all the seven flight conditions. It can be further noticed tha t almost

identical results are obtained from the /i approach and the classical technique.

5.8 C o m p a r iso n b e tw e e n th e c lassica l and \ i m eth od s

This study has revealed some attractive properties of /i for the flight clearance task and

has at the same tim e identified some pitfalls of the classical method. In what follows, a

comparison is m ade between the results obtained using the approaches.

5.8.1 Single loop analysis o f the stability margin criterion

The classical technique uses a gridding approach wherein the uncertain parameter space

is gridded over intervals. Faced with limited time, flight clearance engineers usually re

strict the gridding to only the extrem e values of the uncertainties. Obviously, the risk of

overly optim istic results potentially exists in situations where the worst-case uncertainty

combinations do not lie on the extreme points of the parameters (which would correspond

to a missed worst-case behaviour of the closed loop system). To investigate this m atter,

the stability margin criterion (single-loop analysis) was considered for the analysis of the

longitudinal dynam ics of the HWEM. Nichols diagrams were plotted for (i) the worst-case

obtained using /i and (ii) every combination of the extreme points of the uncertainties. Five

Category 1 uncertainties are considered.

It was found th a t for three flight conditions (FC4, FC5 and FC 6 ), the worst-case uncertain

ties did not lie on the extreme points of the parameters. The results are shown in Figures

5.68, 5.70 and 5.72. For clarity, these three figures have been zoomed and these zoomed

versions are shown in Figures 5.69, 5.71 and 5.73 respectively. It can be seen tha t in all

three cases, the classical approach produces optimistic results as the worst-case \i Nichols

plots are closer to the exclusion regions. For instance, the tailplane loop cut analysis at FC4

using [i (Figures 5.68 and 5.69) generated a worst-case uncertainty of Scmq = 0.9845,

= 0.9976, Scm -t = = 0-0931 and SxDxcg = 0.1438. Clearly, the last two uncertainties

are not at their extreme points. W hen the classical approach was used, the worst-case was

198

HWEM 5.4, W EM SIM005, n -an a ly s isMar-2002, Criterion 2, Long Cat 10.3

0.25

0.2

0.15

0.1

0.05

E

-0 .05

-0.1

-0 .15

-0 .2

-0 .2 5

-0 .3- 0.2 -0 .15 0.05-0.1 -0 .05 0.10

Fig. 5.61: Worst case eigenvalues for FC1, ‘o ’ p approach, classical approach

HWEM 5.4, WEMSIM005, n-analysis Mar-2002, Criterion 2, Long Cat 1

0.25

0.2

0.15

0.1

0.05

0C *-

-0 .0 5

-0.1

-0 .1 5

-0 .2

-0 .2 5

-0 .2 -0 .15 -0.1 -0 .05 0 0.05 0.1Re

Fig. 5.62: Worst case eigenvalues for FC2, 'o’ p approach, '*’ classical approach

199

HWEM 5.4, W EMSIM005, n -an a ly s isM ar-2002, Criterion 2, Long C at 1" ----- ” 1r— — ----

-0 .2 -0 .15 -0.1 -0 .05 0 0.05 0.1Re

Fig. 5.63: W orst case eigenvalues for FC3, ‘o ’ p approach, classical approach

HWEM 5.4, WEMSIM005, ^ a n a ly s is M ar-2002, Criterion 2, Long Cat 1

0.5

0.4

0.2

0.1

-0 .3

-0 .4

-0 .5

-0 .2 -0.1 -0.05Re


200

0 .5

0.4

0.2

0.1

E (X>-

-0 .4

-0 .5

HWEM 5.4, W EMSIM005, ^ -a n a ly s isM ar-2002, Criterion 2, Long C at 1

-0 .2 -0.1 -0 .05Re

0.05 0.1


HWEM 5.4, WEMSIM005, ^-analysis Mar-2002, Criterion 2, Long Cat 1

0.5

0.4

0.3

0.2

- 0.1

- 0.2 -0.05Re

0.1


201

0.4

0.2

- 0.2

- 0.6

HWEM 5.4, W EM SIM005, ^ a n a ly s isM ar-2002, Criterion 2 , Long C at 1

- 0.2 -0.15 -0.05Re

0.05 0.1

Fig. 5.67: W orst case eigenvalues for FC7, ‘o ’ p approach, classical approach

found to be 5Cmq = 1, SCrna = 1, &Cmtail = 1> &lyv = and &x cg = 1. The question then

arises as to how im portan t these last two uncertainties are, i.e, whether 8iyy and Sxcg have

a significant effect on the stability margins. This question can be easily answered using the

technique of /i-sensitivities (described in Section 2.2.3) which measures the relative impor

tance of each uncertainty in the A set.

Table 5.4 shows th a t S xcg and Sjyy are rated the second and third most im portant elements

in the set. Thus, this implies th a t changes in the values of these two parameters will affect

the stability margins. For this reason, discrepancies exist between the worst-case Nichols

plots obtained using the two approaches.

U ncertainty /^-sensitivities

8Cm°rna 0.2328

sx cg 0.1797

8i1yy 0.0287

SCmq 0.0251

Scrntail 0 .0 0 0 1

Table 5.4: fi sensitivities

202

Fig-

criterio11for FC4: i1

( *,) a: worst case (-

»d class1'cal (')

.A 36.A 38

Fig

203

HWEM 5.4, W EMSIM005, ^ -a n a ly s isM ar-2002, Criterion 1A - Tailplane Loop Cut, Long Cat 1

30

20

|x Worst Case

= 0,9864-3 0 Cmq'cma

= 0.9481 = -0 .0082 = 0.2268

-4 0

Dxcg

-5 0

-6 0 — -2 2 0 -100-160-2 0 0 -180 -140 -120 -80

P hase ( 0 )

Fig. 5.70: S tab ility margin criterion for FC5: fi worst case (-*-) and classical (-)


fi Worst C ase

GO

c.<3O

-2

-139 -138-142 -141 -140-143-145 -144Phase ( 0 )

Fig. 5.71: Close-up of Figure 5.70

204

HWEM 5.4, W EMSIM005, n -an a ly s isM ar-2002, Criterion 1A, Nozzle Loop Cut, Long Cat 130

20

H Worst Case:

-3 0 = 0.9785Cmq

*Cma= 0.9201Cmtail. .-4 0 = Q.0682 = 0.3268Dxcg

-5 0

-200 -180 -160 -140 -120 -100 -80P hase ( 0 )

Fig. 5.72: S tability margin criterion for FC6: fi worst case (-*-) and classical (-)

HWEM 5.4, WEMSIM005, n-analysisM ar-2002, Criterion 1 A, Nozzle Loop Cut, Long Cat 1

H Worst C ase

O -2

-4

-6

-144 -136 -134-148 -146 -142 -138-140-1 5 0Phase ( 0 )

Fig. 5.73: Close-up of Figure 5.72

205

5 .8 .2 C om parison o f com p u tation tim es

The com putation tim e for finding the worst case was recorded for

(i) the classical approach using 2 grid points per parameter, and

(ii) p analysis with 1 0 0 frequency points.

The results are plotted in Figure 5.74. It can be noted tha t the com putation time for

160

C lassical approach H technique140

120

100

®Ei-

60

40

20

Number of uncertainties

Fig. 5.74: Computational effort o f p and classical techniques

the classical approach increases exponentially with the number of uncertainties. For a

A size > 8 , p is seen to be less com putationally intensive than the classical technique.

Consider the case when 10 uncertainties are being investigated for the stability margin

criterion. Figure 5.75 shows a plot for the throttle loop cut. It can be observed tha t the

same worst-case behaviour of the closed loop system is being found by both p and the

gridding approach. However, from Figure 5.74, we can deduce tha t for this size of A, p is

com putationally faster than the classical gridding technique.

5 .8 .3 M u lt i lo o p a n a ly s is o f th e s ta b ili ty m a rg in c r ite r io n

M ulti loop analysis was carried out for flight condition FC5 using the methods explained

in Section 1 .3 .2 . For the p approach, the uncertainties associated with the phase/gain

offsets (elliptical Nichols exclusion regions) were independently varied until p = 1. The

corresponding gain and phase margins were found to be 13.97 dB and 37.86° respectively.

Using the classical approach, the corner points of the diamond-shaped Nichols exclusion

region were checked and the gain and phase margins were computed as 15.5 dB and 41.2°

206

Fig. 5.75: Throttle loop cut: p worst case (-*-) and classical (-)

respectively. It can be observed th a t the results obtained using the classical approach are

more optim istic than those computed from p. The explanation for these differences is two

fold. Firstly, p imposes a slightly more stringent requirement on the computation of the

phase and gain margins (since the elliptic exclusion regions used by p are slightly bigger

than the diam ond-shaped exclusion regions used by the classical approach). Secondly, every

possible com bination of the phase/gain offsets are considered in p whereas the classical

approach assumes the same phase/gain variations in each loop of the system.

5 .9 C o m p arison b e tw e en th e e llip tica l N ichols exclu sion region te st and

th e tra p ezo id a l N ich o ls exclu sion region te s t

5.9.1 Single loop analysis

This section gives the results obtained for the single loop stability margin analysis of the

tailplane loop cut using (i) the elliptical Nichols exclusion region test of Section 2.4.1

and (ii) the trapezoidal Nichols exclusion region test of Section 2.4.4, for a sample trim-

point of [170 knots, 200 ft]. The resulting mixed-p bounds are shown in Figure 5.76 for

the elliptical exclusion region test and the real-/i bounds are illustrated in Figure 5.77

for the trapezoidal exclusion region test. The different exclusion regions used for these

methods are shown in Figure 5.78. The corresponding worst-case Nichols plots are also

shown in Figure 5.78. Interestingly, different worst cases were obtained for these two

tests: the worst case param eter combination for the elliptical exclusion region test was

207

found to be [fcm, =2.0504, < 5^= 1 .4291 , SCmtM =2.0369, 6Ify= 1.0262, SxDxcg =0.3548] while

th a t of the trapezoidal exclusion region test was identified as [Scmq= 2.2716, ^cma—1.9934,

S c ^ a u =2-2119, i/„„=1.8133, «xD„„=0.6113].

0.7

0.6

0.5

0.4

0.3

0.2

0.1

10'co (rad/s)

Fig. 5.76: M ixed p bounds for the elliptical Nichols exclusion region test

0.5

0.45

0.4

0.35

0.3

0.2

0.15

0.1

0.05

co (rad/s)

Fig. 5.77: Real p bounds for the trapezoidal Nichols exclusion region test

208

Trapezoidal Elliptical

-2

- 4

-6

- 8

-175 -170 -165 -160 -155 -150 -145 -140 -135 -130P hase (°)

Fig. 5.78: Worst case Nichols plots for tailplane loop cut

5 .9 .2 M u lt i lo o p a n a ly s is

Using the the methods explained in Section 2.4, multi loop analysis was carried out using

the elliptical and trapezoidal Nichols exclusion regions. These exclusion regions were scaled

until the value of p reaches 1 and the resulting phase and gain margins were then computed.

For the elliptical Nichols exclusion region test, a phase margin of 41.2° and a gain margin

of 4.18 dB were obtained. For the trapezoidal Nichols exclusion region test, a phase margin

of 44.8° and a gain margin of 4.5 dB were obtained. Again, it can be observed that the

elliptical exclusion region is, as expected, slightly more conservative than the trapezoidal

exclusion region.

5.10 C onclusions

This chapter has detailed results of the clearance of the HWEM flight control law using p

analysis techniques. The analysis was fully investigated for the longitudinal axis dynamics

w ith the most relevant pitch uncertainties taken into account. The analysis of the full aircraft

dynamics (longitudinal plus lateral) was also considered. One flight case (the straight and

level flight) was examined at seven different trim points in the HWEM flight envelope. A

simple approach for representing the uncertain model param eters in the form of LFT-based

uncertainty models was presented and validated. This approach is seen to be particularly

attractive as it allows the worst-case set of uncertain param eters to be obtained with minimal

modelling effort.

209

Two linear stability clearance criteria were addressed in detail (worst-case stability margin

and worst-case unstable eigenvalue). All of the analysis results derived using the /i tools

were compared against the corresponding results obtained using the classical approach. For

both the single-loop and multi-loop worst-case stability margin criteria, it was shown that

unreliable (optimistic) results were generated using the classical approach. The proposed

/i-analysis tools, on the other hand, were shown to allow the computation of the true worst-

cases. In addition, for flight cases involving more than eight uncertain parameters (such

as the combined longitudinal/lateral analysis) the proposed /i-tools were shown to be more

com putationally efficient than the classical gridding approach.

210

C hapter 6

C onclusions

6 .1 C on clu sion s

In this thesis, new analysis tools have been developed for the clearance of flight control

laws for highly augmented aircraft. The classical methods currently used in industry for

checking the stability and performance characteristics of highly augmented aircraft were

explained and their disadvantages identified. The main limitation of the classical approach

is the inability to guarantee th a t the true worst case behaviour of the closed loop system

has been found, as only a few scattered points (usually the extreme values) of the uncertain

param eter space are checked. Moreover, this gridding technique can easily become time

consuming and expensive as the number of points in the grid increases exponentially with

the num ber of uncertain param eters.

The deficiencies of the classical approach have motivated the development of new analysis

tools, based on the structured singular value fi, with the aim of providing increased au

tom ation and reliability of the clearance process with reduced effort and cost. It was shown

how linear stability criteria used in the classical flight control law clearance process could

be cast into a /i framework. Two new tools for computing tight bounds on real /x were also

developed, as standard real fi algorithms were found to give poor results and often even

failed to converge. The first new method is based on the idea of using /i sensitivities to

selectively reduce the size of the uncertainty set until the use of exponential time algorithms

becomes practical. The second method casts the problem of computing a lower bound on

(i as an optimisation-based search for the worst-case real destabilising uncertainty. In this

study, both methods were seen to produce excellent results, allowing the com putation of

tight bounds on / i even for a large uncertainty set.

The application of these new fi-tools requires the uncertain aircraft models to be first con

verted into LFT-based uncertainty models. Various approaches to LFT-based uncertainty

modelling were considered, for both linear and nonlinear systems. The first method, based

on a physical modelling technique, was applied to a linearised civil transport aircraft model.

A control law was synthesised for this model using the m ethod of Tioo loopshaping in com

bination w ith optim isation techniques. It was shown th a t this approach to LFT-based

uncertainty modelling is simple to implement and also th a t the worst case uncertainty set is

obtained in term s of the actual uncertain parameters. The main drawback of this method

is th a t detailed information about the influence of the uncertain param eters on the aircraft

211

dynamics is required. Moreover, it was necessary to express the uncertainties in a rational

form, with the uncertainties explicitly defined in the equations of motion. Another disad

vantage is the likely recurrence of repeated parameters which can introduce conservatism in

the com putation of the upper bound on p.

The second m ethod for LFT-based uncertainty modelling considered in this thesis is based on

numerical techniques, wherein numerical linearisations were performed for all combinations

of the extrem e points of the uncertain parameters. The resulting LFT-based uncertainty

models contained fictitious uncertainties which represent the original uncertain parameters

only indirectly. This technique was applied to the analysis of the HIRM+ RIDE flight con

trol law. It was observed th a t this approach was quite successful, as the RIDE control law

could be cleared a t most of the required flight conditions for several clearance criteria with

modest modelling and com putational effort. It was also shown, for some flight cases, tha t

whole portions of the flight envelope can be cleared by including other flight parameters such

as Mach num ber or altitude as additional uncertainties in the vector of unknown parameters,

thus removing the need to resort to a gridding of the flight envelope. The capability to clear

the flight control system for continuous intervals of angle of attack and load factor was also

dem onstrated successfully. The approach to LFT-based uncertainty modelling adopted in

this analysis is fast and easy, but conservative, since joint param eter dependency among the

sta te space elements are ignored. Furthermore, this method does not allow the computation

of worst cases in term s of the particular values of the original uncertain parameters. Thus,

this approach is best suited for an initial analysis, where the main objective is to identify

those regions of the flight envelope where the control law passes the clearance criteria. These

regions can then be elim inated from subsequent analysis, leading to a significant saving in

tim e and effort.

A symbolic approach to LFT-based uncertainty modelling was investigated for the anal

ysis of the HWEM aircraft model. The nonlinear dynamic equations were first extracted

from the nonlinear HW EM model. Symbolic linearisation was then performed to gener

ate the LFT-based uncertainty models. The main problem with this approach is tha t the

derivation of the nonlinear symbolic model can demand tremendous effort and requires a

detailed knowledge of flight mechanics. Moreover, because of their complexity, the non

linear symbolic equations had to be further simplified before linearisation was performed.

It was also necessary to use interpolation techniques to approximate some of the stability

derivatives due to their complexity. As the interpolation process was trim-dependent, the

symbolic LFT-based uncertainty models were only valid in close neighbourhood of the rel

evant trim point and thus could only be generated at discrete points in the flight envelope.

212

Obviously, these simplifications provided no guarantee th a t the final LFT-based uncertainty

model would be sufficiently accurate. In fact, it was shown that validation of the LFT-based

uncertainty model against the original nonlinear model results gave very poor results. An

other disadvantage with this approach is tha t the resulting LFT-based uncertainty model

contained several repeated param eters which could have introduced further conservatism in

the com putation of the // upper bounds.

The last approach investigated in this thesis is based on physical modelling techniques and

is well suited for nonlinear systems which are given in block diagram form. The method

consists of introducing the uncertain parameters directly into the nonlinear block diagram

model as multiplicative (or additive) uncertainties. For the HWEM, this method proved to

be simple to implement and demanded little modelling effort. An exact representation of the

uncertain system was obtained and thus the exact worst case set of uncertain parameters

could be com puted. In addition, by clever arrangement of the block diagram representation,

it was possible to generate a minimal order LFT-based uncertainty model. One drawback

of this m ethod is th a t detailed information about the way in which the uncertain parame

ters affect the aircraft dynamics was required. As the model grows in complexity or as the

num ber of uncertain param eters is increased, the required modelling effort is also increased.

A nother drawback is th a t this approach ignores the dependence of the linearised model on

the uncertain param eters, thus restricting the analysis to discrete points in the flight enve

lope.

As a basis for comparison, all of the analysis results derived using the // tools were checked

against the corresponding results obtained using the classical approach for the HWEM air

craft model. It was shown th a t, for the stability margin criterion, different worst cases

were often identified by the two approaches. Indeed, it was clearly seen tha t unreliable

(optimistic) results could often be generated using the classical approach. The proposed

//-analysis tools, on the other hand, were in general seen to allow the com putation of true

worst cases. It was shown th a t the computation times for the classical approach increases

exponentially w ith the number of uncertain parameters whereas com putation times for //

bounds are generally polynomial functions of the number of uncertain parameters. Thus,

the //-tools developed in this study were also shown to be computationally faster than the

classical approach when more than eight uncertainties were considered simultaneously.

6 .2 F u tu re w ork

In general, //-analysis techniques are only well developed for uncertain multivariable linear

systems. This fact limits their application at present to linear, frequency domain clearance

213

criteria. Further development of the basic theory is required in order to produce /i-tools

which can be used reliably to address nonlinear and/or time-domain clearance criteria. In

corporation of nonlinear effects like actuator rate saturation, for example, into a /i framework

is possible via describing function analysis.

The extension of the real /i lower bound optimisation algorithm to the mixed /i problem,

which arises naturally from the Nichols exclusion test, is another direction for future re

search.

Finally, this thesis has confirmed tha t the generation of exact LFT-based uncertainty mod

els for nonlinear systems which can capture variations in trim-dependent parameters (e.g.

angle of attack) is a deep and difficult problem. Further research in this area is essential,

in order to allow ‘one-shot’ clearance of flight control laws over whole regions of the flight

envelope using /x-analysis techniques.

214

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