Optimization Based Clearance of Flight Control Laws: A Civil Aircraft Application

452 Pages·2012·7.274 MB·English

by  Anders Hansson, Andreas Varga (auth.), Andreas Varga, Anders Hansson, Guilhem Puyou (eds.)

Lecture Notes
in Control and Information Sciences 416
Editors:M.Thoma,F.Allgöwer,M.Morari
Andreas Varga, Anders Hansson,
and Guilhem Puyou (Eds.)
Optimization Based Clearance
of Flight Control Laws
A Civil Aircraft Application
ABC
SeriesAdvisoryBoard
P.Fleming,P.Kokotovic,
A.B.Kurzhanski,H.Kwakernaak,
A.Rantzer,J.N.Tsitsiklis
Editors
Dr.AndreasVarga Dr.GuilhemPuyou
DLR-Oberpfaffenhofen AIRBUS
GermanAerospaceCenter StabilityandControlDepartment
InstituteofRoboticsandMechatronics 316routedeBayonne
82234Wessling 31060ToulouseCedex03
Germany France
E-mail:[email protected] E-mail:[email protected]
Prof.AndersHansson
LinköpingsUniversitet
DepartmentofElectricalEngineering
DivisionofAutomaticControl
SE-58183Linköping
Sweden
E-mail:[email protected]
ISBN978-3-642-22626-7 e-ISBN978-3-642-22627-4
DOI10.1007/978-3-642-22627-4
LectureNotesinControlandInformationSciences ISSN0170-8643
LibraryofCongressControlNumber:2011934146
(cid:2)c 2012Springer-VerlagBerlinHeidelberg
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Foreword
As far as the laws of mathematics refer
to reality, they are not certain; and as far
as they are certain, they do not refer to
reality.
Albert Einstein
Since theearlydaysofaviation,engineerswithinventivetalentshadtoover-
cometremendouschallenges.Frombuildingvehiclesthatcanflyinacontrol-
lablemannerdesignedandflownbypioneerslikeLilienthal,Wrightbrothers,
Sperry in the beginnings, to developing todays modern comfortable and re-
liable vehicles for daily routine all-weather operations (Airbus, Boeing and
others), there were many creative efforts to improve performance (aircraft
size,endurance,speed),minimizestructuralweight,providenecessarythrust,
andguaranteesafeflightoperations.Nowadaysalltheseinventionsassurethe
high mobility of the modern human society in a global world. The aeronau-
tical challenges were drivers for many new technologies and methodologies
that are commonly used in other industries today.
In the last three decades, the requirements for the design of high-perfor-
mance flight control systems that enhance automation of flight, initiated a
number ofingenious technologicaldevelopments.Flight controllaw designis
oneoftheareaswhereaeronauticalengineersarepioneeringnewtechnologies.
During development(designandtest) offlightcontrollaws,engineersrely
onmathematicalmodels.Inevitably suchmodels cannotmimic allaspectsof
a highly complex, physical plant as a modern high-performance jet airplane
andits environment(atmosphere,airtraffic, etc.) with absolute fidelity. The
abovequotationfromEinsteindescribesthe fundamentaldifficulty that con-
trolengineersarefacing,whenstrivingtheclearanceofflightcontrollawson
basis of mathematical models.
However,engineers(namedaftertheLatinwordingenium,meaninginnate
quality,especiallymentalpower,henceacleverinvention)oftenfindsolutions
eventothemostchallengingproblems.Toprovethatanaircraftissafetofly
VI Foreword
they develop aircraftdynamics models with quantified uncertainties and ap-
ply adequate mathematical theories for their analysis. Those models can be
usedtoverifythatthecontrollawsoperateasspecifiedevenwhendeviations
from the nominal conditions occur. In addition, test engineers consider all
imaginable test conditions using experience from the past including knowl-
edge about abnormal cases, incidents and accidents. This approach extends
the uncertain parameter space with many additional test cases, with special
care to investigate the critical ones. The huge amount of possible parameter
combinations as well as the presence of model nonlinearities result in an in-
credibly high number of test cases to be checked for the clearance of flight
control laws. The objective of the EC-sponsored project COFCLUO aims
at mastering this Herculean task, by applying efficient optimization-based
search techniques to discover hidden weaknesses of flight control laws and
to determine worst-case parameter combinations to aid possible control law
redesigns. Eventually the clearance of control laws is achieved, thus guaran-
teeing the safe aircraft operation.
This book describes and demonstrates the main achievements of the
COFCLUO project. When you study it, you will find out that this endeav-
our has significantly enhanced the state-of-the-art of the clearance of flight
controllawsby providinginnovativeideas andadvancedanalysistechniques.
Thereportedprojectachievementsareaconvincingproofofaverysuccessful
European cooperation.
Berlin, January 12, 2011 Robert Luckner
Preface
ThisbookaddressestheClearanceOf FlightControlLawsUsingOptimisa-
tion(COFCLUO)andsummarizesthemainachievementsoftheECfounded
6th Framework Programme COFCLUO project. It is well known that be-
fore an aircraft can be tested in flight, it has to go through a rigorous certi-
fication and qualification process to prove to the authorities that the flight
control system is safe and reliable. Currently significant time and money is
spent by the aeronautical industry on this task. An important part of the
certification and qualification process is the clearance of flight control laws
(CFCL).The overallobjectiveofthe COFCLUOProjectwastodevelopand
apply optimisation techniques to CFCL in order to improve efficiency and
reliabilityofthecertificationandqualificationprocess.Theapplicationofan
optimisation-basedapproachreliesonclearancecriteriaderivedfromthecer-
tification and qualification requirements. To evaluate these criteria different
types of models of the aircraft are employed, which usually both serve for
clearance as well as for controllaw design purposes.The developmentof dif-
ferentmodelsandofsuitable clearancecriteriawerethereforealsoobjectives
of the project. Because of wider applicability, the optimisation-based CFCL
will open up the possibility to design innovative aircraft that today are out
oftheapplicationfieldofclassicalclearancetools.Optimisation-basedCFCL
will not only increase safety but it will also simplify the whole certification
and qualification process, thus reduce costs. The speedup achieved by us-
ing the new optimisation-based approach also supports rapid modelling and
prototyping and reduce ”time to market”. The COFCLUO project success-
fullycontributedtotheachievementofatop-levelobjectivetomeetsociety’s
needs for a more efficient, safer and environmentally friendly air transport
byprovidingnewtechniquesandtoolsforsignificantlyimprovedtechnologies
for CFCL.
May 2011 Andreas Varga
Anders Hansson
Guilhem Puyou
Contents
Part I: Clearance of Civil Aircraft
1 Introduction............................................. 3
Anders Hansson, Andreas Varga
1.1 Background ......................................... 3
1.2 The COFCLUO Project............................... 7
1.3 Outline of the Book .................................. 8
References ................................................ 9
2 Clearance Benchmark for a Civil Aircraft................ 11
Guilhem Puyou, Yannick Losser
2.1 Introduction......................................... 11
2.1.1 Nonlinear Benchmark ......................... 12
2.1.2 Integral Benchmark ........................... 12
2.2 Description of Flight Control Laws ..................... 13
2.2.1 Flight Control Laws Philosophy................. 14
2.2.2 Longitudinal Axis............................. 15
2.2.3 Lateral Axis.................................. 16
2.3 The Nonlinear Benchmark Model....................... 17
2.3.1 Flight Envelopes.............................. 17
2.3.2 Pilot Inputs and Pilot Model ................... 18
2.3.3 Actuators and Sensors......................... 19
2.3.4 Flight Mechanics.............................. 19
2.3.5 Control Laws................................. 22
2.4 Clearance Criteria for the Nonlinear Benchmark.......... 23
2.4.1 Un-piloted Aircraft Stability.................... 23
2.4.2 Manoeuvrability Requirements for the
Longitudinal Axis............................. 25
2.4.3 Flight Domain Protection...................... 26
2.5 The Integral Benchmark Model ........................ 27
2.5.1 Flexible Aircraft Model........................ 28
X Contents
2.5.2 Mass Configurations and Flight Points........... 29
2.6 Clearance Criteria for the Integral Benchmark ........... 30
2.6.1 Aeroelastic Stability........................... 30
2.6.2 Stability Margins ............................. 31
2.6.3 Comfort with Respect to Turbulence ............ 31
2.7 Current AIRBUS Practices ............................ 32
2.7.1 Validation Methods ........................... 32
2.7.2 Validation Means ............................. 34
2.7.3 Validation Coverage........................... 35
2.8 AIRBUS Expectations ................................ 36
References ................................................ 36
Part II: Generation of Linear Uncertain Models
3 Generation of LPV Models and LFRs for a Nonlinear
Aircraft Model .......................................... 39
Simon Hecker, Harald Pfifer
3.1 Introduction......................................... 39
3.2 Basic Procedure for the Generation of LPV Models....... 41
3.2.1 Specification of Known Relations................ 42
3.2.2 Element-Wise Significance Check ............... 42
3.2.3 Multivariable Polynomial Fitting................ 43
3.2.4 Full Rank Basis Reduction ..................... 45
3.3 Optimisation of the Linear Parameter Varying Model ..... 46
3.3.1 Optimisation with ν-Gap Metric Constraint ...... 47
3.3.2 Optimisation of the Polynomial Coefficients ...... 47
3.4 Application to the COFLCUO Nonlinear Aircraft Model .. 48
3.4.1 The Aircraft Model ........................... 48
3.4.2 Trimming and Linearisation .................... 49
3.4.3 Generation of an LFR for the Actuator Model
and the Sensor Model ......................... 51
3.4.4 Generation of LPV Models and LFRs for the
Flight Dynamics Model........................ 52
3.4.5 Validation of the LPV Models and LFRs of the
Flight Dynamics Model........................ 54
3.5 Conclusion .......................................... 56
References ................................................ 56
4 Generation of LFRs for a Flexible Aircraft Model ....... 59
Cl´ement Roos
4.1 Introduction......................................... 59
4.2 Problem Statement................................... 60
4.2.1 Description of the Reference Models............. 60
4.2.2 LFT Modelling Objective ...................... 61
4.2.3 Challenging Issues ............................ 62
Contents XI
4.3 Description of the Method............................. 62
4.3.1 Generation of Reduced and Consistent Models.... 63
4.3.2 PolynomialInterpolation and LFT Modelling..... 66
4.3.3 Special Case of a Coarse Grid .................. 67
4.3.4 Low Order LFR Generation Procedure .......... 68
4.4 Numerical Results.................................... 69
4.4.1 Construction of the LFR....................... 70
4.4.2 Validation on the Grid......................... 72
4.4.3 Validation on the Whole Continuous Domain ..... 72
4.4.4 Evaluation of the Low Order LFR Generation
Procedure.................................... 75
4.5 Conclusion .......................................... 76
References ................................................ 77
5 Generation of LFRs for a Nonlinear Controller and
Closed-Loop Aircraft Models ............................ 79
Carsten Do¨ll, Fabien Lescher, Cl´ement Roos
5.1 Introduction......................................... 79
5.2 Description of the Nonlinear Controller ................. 80
5.3 Generation of the Controller LFRs ..................... 84
5.3.1 LFRs for Parameter Varying Gains.............. 85
5.3.2 LFRs for Saturations and Rate Limiters ......... 87
5.3.3 LFRs for Nonlinear Input/Output Relations...... 89
5.3.4 LFRs for the Overall Nonlinear Controller ....... 91
5.4 Generation of the Closed-Loop LFR .................... 96
5.4.1 Closed-Loop LFRs for the Nonlinear Model
Performance Analysis.......................... 96
5.4.2 Closed-Loop LFRs for the Nonlinear Model
Stability Analysis ............................. 99
5.4.3 Closed-Loop LFRs for the Flexible Model
Stability and Performance Analysis.............. 100
5.5 Evaluation of the LFR Generation Process .............. 101
5.5.1 Evaluation of the LFR of the Nonlinear
Controller.................................... 101
5.5.2 Evaluation of the Closed-Loop LFR ............. 103
5.6 Conclusions ......................................... 107
References ................................................ 108
6 Identification of LPV State-Space Models Using
H2-Minimisation......................................... 111
Daniel Petersson,Johan Lo¨fberg
6.1 Introduction......................................... 111
6.2 H2-Minimisation ..................................... 113
6.2.1 Important Property of H2-Minimisation ......... 114
6.2.2 Rewriting the H2-Norm of the Error System...... 115
6.3 Method 1: General Nonlinear Optimisation .............. 116
XII Contents
6.3.1 Evaluation of the Cost Function ................ 116
6.3.2 Evaluation of the Gradient ..................... 118
6.4 Method 2: Semidefinite Programming................... 120
6.5 Regularisationof the Optimisation Problem ............. 122
6.6 Examples ........................................... 123
6.6.1 Academic Example............................ 123
6.6.2 Application Example .......................... 126
6.7 Conclusions ......................................... 127
References ................................................ 128
Part III: Analysis Techniques and Tools
7 Enhanced μ-Analysis Techniques for Clearance .......... 131
Jean-Marc Biannic, Cl´ement Roos
7.1 Introduction......................................... 131
7.2 Problem Statement and Preliminary Results ............. 132
7.2.1 Introduction to μ-Analysis ..................... 132
7.2.2 Validity of the Scaling Matrices................. 134
7.3 Computation of a Guaranteed Robustness Margin ........ 136
7.3.1 Standard Version of the Algorithm .............. 136
7.3.2 Extension to Modal Performance Analysis........ 137
7.4 Computation of a Guaranteed Stability Domain.......... 138
7.4.1 Standard Version of the Algorithm .............. 138
7.4.2 Computation of the μ-Sensitivities .............. 140
7.4.3 Other Algorithmic Variants .................... 140
7.5 Connection with Clearance of Flight Control Laws........ 141
7.5.1 Eigenvalue Criterion........................... 141
7.5.2 Stability Margin Criterion...................... 142
7.6 Conclusion .......................................... 145
References ................................................ 146
8 Worst-Case Parameter Search Based Clearance Using
Parallel Nonlinear Programming Methods ............... 149
Hans-Dieter Joos
8.1 Introduction......................................... 149
8.2 Theoretical Basis..................................... 150
8.2.1 Formulation as Global Optimisation Problem..... 150
8.2.2 Level of Confidence ........................... 151
8.2.3 Clearance Strategy............................ 151
8.2.4 Transformationof Parameter Space ............. 152
8.3 Applied Optimisation Methods......................... 153
8.4 ParallelComputation ................................. 155
8.5 Conclusions ......................................... 157
References ................................................ 158