using linear fractional transformations for clearance of flight control laws

Using Linear FractionalTransformations for Clearance Of Flight

Control Laws

Master’s thesisperformed in Automatic Control

byJorgen Hansson

Reg nr: LiTH-ISY-EX-3420-2003

1st October 2003

Using Linear FractionalTransformations for Clearance Of Flight

Control Laws

Master’s thesis

performed in Automatic Control,Dept. of Electrical Engineering

at Linkopings universitet

Performed for SAAB ABby Jorgen Hansson

Reg nr: LiTH-ISY-EX-3420-2003

Supervisor: M.Sc. Fredrik KarlssonSAAB AB

Lic. Karin Stahl GunnarssonSAAB AB

Lic. Ragnar WallinISY, Linkopings Univeristet

Examiner: Doc. Anders HanssonISY, Linkopings Universitet

Linkoping, 1st October 2003

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¤URL for elektronisk version

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SammanfattningAbstract

NyckelordKeywords

Flight Control Systems are often designed in linearization points overa flight envelope and it must be proven to clearance authorities thatthe system works for different parameter variations and failures all overthis envelope.

In this thesis µ-analysis is tried as a complement for linear analysis inthe frequency plane. Using this method stability can be guaranteed forall static parameter combinations modelled and linear criteria such asphase and gain margins and most unstable eigenvalue can be includedin the analysis. A way of including bounds on the parameter variationsusing parameter dependent Lyapunov functions is also tried.

To perform µ-analysis the system must be described as a Linear Frac-tional Transformation (LFT). This is a way of reformulating a param-eter dependent system description as an interconnection of a nominallinear time invariant system and a structured parameter block.

A linear and a rational approximation of the system are used to makeLFTs. These methods are compared. Four algorithms for calculation ofthe upper and lower bounds of µ are evaluated. The methods are triedon VEGAS, a SAAB research aircraft model.

µ-analysis works quite well for linear clearance. The rational approx-imation LFT gives best results and can be cleared for the criteria men-tioned above. A combination of the algorithms is used for best results.When the Lyapunov based method is used the size of the problemgrows quite fast and, due to numerical problems, stability can only beguaranteed for a reduced model.

Division of Automatic Control,Dept. of Electrical Engineering581 83 Linkoping

1st October 2003

—

LITH-ISY-EX-3420-2003

—

http://www.ep.liu.se/exjobb/isy/2003/3420/

Using Linear Fractional Transformations for Clearance Of Flight Con-trol Laws

Klarering av Styrlagar for Flygplan med hjalp av Linjara RationellaTransformationer

Jorgen Hansson

××

Linear Fractional Transformation (LFT), Structured Singular Value(SSV), Linear Matrix Inequality (LMI), µ-analysis, Lyapunov function,Flight Clearance, Stability Margin

Abstract

Flight Control Systems are often designed in linearization points overa flight envelope and it must be proven to clearance authorities thatthe system works for different parameter variations and failures all overthis envelope.

In this thesis µ-analysis is tried as a complement for linear analysisin the frequency plane. Using this method stability can be guaranteedfor all static parameter combinations modelled and linear criteria suchas phase and gain margins and most unstable eigenvalue can be in-cluded in the analysis. A way of including bounds on the parametervariations using parameter dependent Lyapunov functions is also tried.

To perform µ-analysis the system must be described as a LinearFractional Transformation (LFT). This is a way of reformulating a pa-rameter dependent system description as an interconnection of a nom-inal linear time invariant system and a structured parameter block.

A linear and a rational approximation of the system are used tomake LFTs. These methods are compared. Four algorithms for calcu-lation of the upper and lower bounds of µ are evaluated. The methodsare tried on VEGAS, a SAAB research aircraft model.

µ-analysis works quite well for linear clearance. The rational ap-proximation LFT gives best results and can be cleared for the criteriamentioned above. A combination of the algorithms is used for best re-sults. When the Lyapunov based method is used the size of the problemgrows quite fast and, due to numerical problems, stability can only beguaranteed for a reduced model.

Keywords: Linear Fractional Transformation (LFT), Structured Sin-gular Value (SSV), Linear Matrix Inequality (LMI), µ-analysis,Lyapunov function, Flight Clearance, Stability Margin

v

Preface

This master’s thesis is the final step between me and my Master ofScience Degree in Applied Physics and Electrical Engineering. It wasperformed at SAAB AB, Division of Gripen Aeronautical Engineering,Section of Flying/Handling Qualities, GDFF, in corporation with theDivision of Automatic Control at Linkopings Universitet. This workwas done during the spring semester and summer of 2003. This thesisis part of the FMV FoT 25 research project.

Acknowledgments

I would like to thank Dr.Martin Hayes and Ph.D student Peter Iordanovat the University of Limerick for helping me out with the optimizationprocedure for lower bounds on µ. I would also like to thank ProfessorAnders Helmersson for big help with calculation of the frozen analysisand the FFM. A big thank you to my supervisor and examiner at theuniversity Ragnar Wallin and Docent Anders Hansson for rewardingdiscussions and hints. Thanks also to my supervisors at SAAB, FredrikKarlsson and Karin Stahl Gunnarsson for taking time for me in theirbusy working days. Thank you to all the people at section GDFF atSAAB for making my thesis time a pleasant one. My classmates PetterFrykman proofread the report and Anders Ekman opposed on the the-sis. Finally I thank my girlfriend Malena and my family for putting outwith my wining and giving me support in general during my studies.

vii

Contents

Abstract v

Preface and Acknowledgments vii

I Introduction 1

1 Introduction 3

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Relation to previous work . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

II Theory 7

2 Flight Clearance 9

2.1 Gain Scheduled Controllers . . . . . . . . . . . . . . . . . . . . . 9

2.2 Parameter dependence . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Clearance Process . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Basic Flight Mechanics 13

3.1 Coordinate frames . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Euler angles. . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Forces, moments and velocities . . . . . . . . . . . . . . . . . . 14

3.3 Moment of inertia . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Static Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4.1 Centre of gravity . . . . . . . . . . . . . . . . . . . . . . 16

ix

3.5 Nonlinear Equations of Motion . . . . . . . . . . . . . . . . . 173.6 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.6.1 Phugoid and short period mode . . . . . . . . . . . . . 18

4 Linear Fractional Transformation 214.1 Upper and Lower LFTs . . . . . . . . . . . . . . . . . . . . . . 21

4.1.1 State-space systems . . . . . . . . . . . . . . . . . . . . 224.2 The Star Product. . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 LFT generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3.1 Linear Parameter Varying Systems . . . . . . . . . . . 254.3.2 Trends & Bands approach . . . . . . . . . . . . . . . . 264.3.3 Approximation with rational functions. . . . . . . . . . 27

4.4 Preparation of approximation data . . . . . . . . . . . . . . . . 284.4.1 Normalizing states . . . . . . . . . . . . . . . . . . . . . 284.4.2 Normalizing parameter data . . . . . . . . . . . . . . . 28

4.5 Model reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 284.5.1 Tree decomposition . . . . . . . . . . . . . . . . . . . . 284.5.2 n-D decomposition. . . . . . . . . . . . . . . . . . . . . 294.5.3 µ-sensitivities . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Structured Singular Values 315.1 Structured Singular Values . . . . . . . . . . . . . . . . . . . . 31

5.1.1 Small gain theorem . . . . . . . . . . . . . . . . . . . . 315.1.2 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.1.3 Structured Singular Value. . . . . . . . . . . . . . . . . 335.1.4 Upper and lower bounds . . . . . . . . . . . . . . . . . 34

5.2 µ-sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 Modelling of Clearance Criteria . . . . . . . . . . . . . . . . . . 35

5.3.1 Uncertainty modelling . . . . . . . . . . . . . . . . . . . 355.4 Phase and Gain margins . . . . . . . . . . . . . . . . . . . . . . 37

5.4.1 Nichols exclusion regions . . . . . . . . . . . . . . . . . 375.4.2 Stability margin and worst case parameter combination 39

5.5 Most unstable eigenvalue . . . . . . . . . . . . . . . . . . . . . 405.5.1 Specifications. . . . . . . . . . . . . . . . . . . . . . . . 40

6 µ-calculations 416.1 Upper µ-calculations . . . . . . . . . . . . . . . . . . . . . . . . 41

6.1.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.1.2 µ-Analysis and Synthesis Toolbox . . . . . . . . . . . . 426.1.3 Finite Frequency Method . . . . . . . . . . . . . . . . . 42

6.2 Lower µ-calculations . . . . . . . . . . . . . . . . . . . . . . . . 446.2.1 mu command augmented with imaginary part . . . . . 446.2.2 Basic Optimization Algorithm . . . . . . . . . . . . . . 456.2.3 Pole Placement Approach. . . . . . . . . . . . . . . . . 45

x

7 Bounds on Parameter Rates 477.1 Lyapunov Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.1.1 Lyapunov functions . . . . . . . . . . . . . . . . . . . . 487.1.2 Quadratic Stability. . . . . . . . . . . . . . . . . . . . . 497.1.3 Parameter dependent Lyapunov functions . . . . . . . 49

7.2 Implicit system approach . . . . . . . . . . . . . . . . . . . . . 507.2.1 Linear Matrix Inequalities . . . . . . . . . . . . . . . . . 507.2.2 Augmentation of LFT . . . . . . . . . . . . . . . . . . . 517.2.3 Connection to Lyapunov . . . . . . . . . . . . . . . . . 547.2.4 Alternative formulation . . . . . . . . . . . . . . . . . . 55

7.3 Difference from µ-analysis . . . . . . . . . . . . . . . . . . . . . 56

8 The VEGAS Model and the GAM 598.1 VEGAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608.2 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

III Results 63

9 µ-calculation comparison 659.1 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

9.2.1 Workflow for µ-calculation . . . . . . . . . . . . . . . . 68

10 LFT-generation and validation 6910.1 Generation of LFTs . . . . . . . . . . . . . . . . . . . . . . . . 6910.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7210.3 µ-sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

10.3.1 Trends & Bands . . . . . . . . . . . . . . . . . . . . . 7310.3.2 Rational LFT . . . . . . . . . . . . . . . . . . . . . . . 74

10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

11 Clearance of VEGAS 7711.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7711.2 Phase and gain margins . . . . . . . . . . . . . . . . . . . . . 78

11.2.1 Stability margins . . . . . . . . . . . . . . . . . . . . . 7811.2.2 Worst case parameter combination . . . . . . . . . . . 79

11.3 Most unstable poles . . . . . . . . . . . . . . . . . . . . . . . 8111.4 Bounds on Parameter Rates . . . . . . . . . . . . . . . . . . . 8211.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

12 Conclusions and future work 8312.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8312.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

12.2.1 µ-analysis . . . . . . . . . . . . . . . . . . . . . . . . . 85

xi

12.2.2 Parameter dependent Lyapunov functions . . . . . . . 8512.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

References 87

Notation 89

A LFT data 91

B Matlab scripts manual 93B.1 µ-analysis and Synthesis Toolbox . . . . . . . . . . . . . . . . 93B.2 Onera LFR-toolbox . . . . . . . . . . . . . . . . . . . . . . . . 94B.3 BOA and PPA . . . . . . . . . . . . . . . . . . . . . . . . . . . 95B.4 Written Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

xii

List of Figures

2.1 Flight envelope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Centre of gravity diagram.. . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Inertial and aircraft fixed coordinate frames. . . . . . . . . . . . . 143.2 Angles and rates for roll (φ, p), pitch (θ, q), yaw (ψ, r) . . . . . . 143.3 Forces, moments and velocities. . . . . . . . . . . . . . . . . . . . 153.4 Static stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5 Centre of gravity and moment. . . . . . . . . . . . . . . . . . . . . 173.6 Motion of short period mode. . . . . . . . . . . . . . . . . . . . . . 183.7 Motion of phugiod mode. . . . . . . . . . . . . . . . . . . . . . . . 19

4.1 Linear mapping.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Feedback. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 State-space system as LFT. . . . . . . . . . . . . . . . . . . . . . . 234.4 State-space system. . . . . . . . . . . . . . . . . . . . . . . . . . . 244.5 The star product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.6 1-dimensional Trends & Bands approximation. . . . . . . . . . . . 264.7 Conservative approximation.. . . . . . . . . . . . . . . . . . . . . . 274.8 1-dimensional rational approximation. . . . . . . . . . . . . . . . . 27

5.1 Interconnected system.. . . . . . . . . . . . . . . . . . . . . . . . . 315.2 System with ∆-feedback. . . . . . . . . . . . . . . . . . . . . . . . 335.3 s-plane interpretation of SSV.. . . . . . . . . . . . . . . . . . . . . 345.4 Uncertainty in frequency plane. . . . . . . . . . . . . . . . . . . . . 365.5 Complex uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . 365.6 Gain and phase margins on Nyquist plot. . . . . . . . . . . . . . . 375.7 Exclusion regions in the Nichols and Nyquist plane. . . . . . . . . 385.8 Adding exclusion region. . . . . . . . . . . . . . . . . . . . . . . . . 395.9 Stability margin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.10 Eigenvalue specification. . . . . . . . . . . . . . . . . . . . . . . . 40

6.1 Adding scaling matrices. . . . . . . . . . . . . . . . . . . . . . . . . 426.2 System augmented with frequency. . . . . . . . . . . . . . . . . . . 43

xiii

6.3 LFT for dividing frequency. . . . . . . . . . . . . . . . . . . . . . . 446.4 Imaginary augmentation of real LFT description. . . . . . . . . . . 456.5 Offset into RHP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.1 Stability regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7.2 Allowed ∆ and ∆ intervals. . . . . . . . . . . . . . . . . . . . . . . 577.3 Models of envelope. . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8.1 VEGAS in SYSTEMBUILD. . . . . . . . . . . . . . . . . . . . . . . . . 598.2 Control surfaces on VEGAS. . . . . . . . . . . . . . . . . . . . . . 608.3 LFT controller used for analysis. . . . . . . . . . . . . . . . . . . . 61

9.1 Comparison of µ-calculations. . . . . . . . . . . . . . . . . . . . . . 669.2 Upper µ with different grids. . . . . . . . . . . . . . . . . . . . . . 669.3 Peak zoomed, all algorithms used. . . . . . . . . . . . . . . . . . . 66

10.1 CMαas function of Mach number. . . . . . . . . . . . . . . . . . 69

10.2 SIMULINK interconnection. . . . . . . . . . . . . . . . . . . . . . . 7110.3 Envelope covered by model. . . . . . . . . . . . . . . . . . . . . . 7210.4 Comparison of LFT:s and linearization point in nominal case. . . 7210.5 Comparison of LFTs with adjusted compensation parameters. . 7310.6 Trends & Bands LFT results. . . . . . . . . . . . . . . . . . . . . 7410.7 Rational LFT results. . . . . . . . . . . . . . . . . . . . . . . . . . 75

11.1 Stability of Vegas.. . . . . . . . . . . . . . . . . . . . . . . . . . . 7711.2 Upper µ for exclusion regions . . . . . . . . . . . . . . . . . . . . 7811.3 Nichols plot of nominal and worst case system. . . . . . . . . . . 7911.4 Comparison of signals, nominal and worst case Mach 0.2. . . . . 8011.5 Migration of short period poles. . . . . . . . . . . . . . . . . . . . 8011.6 µ for eigenvalue specifications. . . . . . . . . . . . . . . . . . . . 8111.7 Worst case poles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

12.1 Difference in parameters. . . . . . . . . . . . . . . . . . . . . . . . 84

xiv

Part I

Introduction

1

Chapter 1

Introduction

1.1 Background

The clearance of a flight control system is a very important and time-consuming process. Due to hard regulations on aircraft systems forsafety reasons it must be proven that the systems will work. Since theaircraft dynamics and the aerodynamics are very nonlinear and thecontrol system is linear and often designed in discrete points, verifi-cation of the control system is very important. To consider variationsin aircraft dynamics and aerodynamic coefficients, a few variations inthe parameters are investigated and then simulation of the nonlinearsystem is performed to make sure that the system will work and re-main stable for all possible variations. Since the number of parametersthat can vary in a complete aircraft system is quite large, this is a verytime consuming process and although simulations are used to find allstrange and normal flight cases a full certainty of having covered allcases can never be given. The methods tried in this thesis covers allstatic parameter variations modeled and can be used for linear analysisto make sure the system is stable and fulfils requirements on stabilityand eigenvalues. The worst case parameter combinations found fromthis analysis can then be used to find out which configurations andflight cases it is important to look at in the nonlinear simulations.

1.2 Problem formulation

In the GARTEUR AG(11) research project some advanced techniquesfor the clearance of flight control laws have been developed and tried(Fielding et al. 2002). SAAB AB, who participated in this project, areinterested in using some of these methods as a complement to parts

3

4 Chapter 1. Introduction

of the current clearance and design process. The object of this thesiswas to try some of these methods, based on a system described as aLinear Fractional Transformation. Most interesting for SAAB at thispoint was µ-analysis. The thesis had the following goals.

• Describe the steps necessary to perform the analysis.

• Try the methods on VEGAS, a SAAB research aircraft model.

• A number of envelope points should be tried and uncertaintiesshould be included in the model.

• Information about properties between the envelope points shouldbe found if possible.

• A few clearance criteria should be included e.g. stability marginand eigenvalues. The parameter combination giving the worst caseshould be found if possible.

• If time is left, try some other method also based on Linear Frac-tional Transformations.

1.3 Method

This is what was done in this thesis.

µ-analysis

The main goal was to try µ-analysis. Most of the theory was obtainedfrom Fielding et al. (2002) were this already had been done for flightclearance purposes. Other scientific articles related to this area werealso studied.

• To perform µ-analysis the system must be described as a LinearFractional Transformation (LFT). Two numerical ways of gener-ating such LFT descriptions, one linear approximation and one ra-tional approximation, were tried and compared. These were usedon data from a nonlinear simulation model VEGAS with Machnumber, centre of gravity and pitch moment coefficient depend-ing on angle of attack as varying parameters. These parameterswere chosen after consultation with the supervisors at SAAB. Acontroller which already was a LFT was used to stabilize the air-craft.

• To calculate the structured singular value, µ, reliable algorithmswere necessary. Four different algorithms for calculating upperand lower bounds were tried and compared.

1.4. Relation to previous work 5

• The stability margin and worst case unstable poles criteria wereincluded in the analysis.

Parameter dependent Lyapunov functions

This method differs from µ-analysis but it also demands a LFT descrip-tion of the system. Since it includes variations on the parameter ratesin the analysis it was interesting for SAAB. Therefore this analysismethod was applied to one of the LFT descriptions of VEGAS.

1.4 Relation to previous work

Here the differences between the scientific articles used for this thesisand the achieved results are presented. In Fielding et al. (2002) severalways of generating LFTs are suggested. None of them utilizes rationalapproximations which are used in this thesis. A rational function cangenerally approximate a surface with a lower order function comparedwith a polynomial. In Mannchen et al. (2002) the Trends & Bands ap-proach is suggested for generation of LFTs. The µ-analysis results arecompared with the results from a LFT generated by a numerical min-max approach and a LFT based on symbolical expressions of a physicalmodel. In this thesis the Trends & Bands method is compared witha LFT based on rational approximations. In Iordanov et al. (2003) acomparison of existing µ-calculation algorithms is done but the FiniteFrequency Method is not included but utilized to improve the results.Here the Finite Frequency Method is evaluated together with threeother algorithms. All of the µ-analysis clearance criteria in this thesishave earlier been tried on Trends & Bands LFTs based on the HIRM+aircraft model with the RIDE controller in Fielding et al. (2002).

1.5 Tools

The software tools used in this thesis are

MATRIXX& SYSTEMBUILD where the nonlinear aircraft simulationmodel VEGAS is implemented. This environment was used forlinearization and data collection from the nonlinear model.

MATLAB & SIMULINK was used for all other calculations and simu-lations. The toolboxes mainly used were

LFR Toolbox used to make LFT descriptions.

Optimization Toolbox used for min-max optimization and uti-lized by some of the µ-calculation algorithms.

6 Chapter 1. Introduction

Control System Toolbox for handling of state-space descrip-tions systems and Nyquist and Nichols plots.

µ-analysis and Synthesis Toolbox where the mu command isimplemented.

1.6 Thesis Outline

The thesis is organized as follows:

Chapter 2 gives a short background and introduction to the flightclearance process.

Chapter 3 provides some basic concepts from flight mechanics neces-sary to understand the model later developed.

Chapter 4 presents the main theory for Linear Fractional Transfor-mations and how they can be generated and reduced.

Chapter 5 introduces the Structured Singular Value (SSV) and presentshow flight clearance criteria such as phase and gain margins andmost unstable eigenvalue can be included in the analysis.

Chapter 6 gives an overview of the algorithms used to calculate theSSV.

Chapter 7 contains a way to include bounds on the rates of parametervariations in the analysis.

Chapter 8 introduces the VEGAS model and the controller used foranalysis.

Chapters 9–11 present the results. The µ-algorithms are compared,LFTs of VEGAS with uncertainties included are generated andcompared. Clearance of VEGAS with controller is performed.

Chapter 12 summarizes the thesis, draws conclusions and gives sug-gestions of future work.

Part II

Theory

7

Chapter 2

Flight Clearance

Modern fighter aircrafts are often aerodynamically unstable to achievebetter performance. Thus they can’t be flown without a flight controlsystem. To get the aircraft certified to fly, the flight control systemmust be approved by the clearance authority by proving it to be fullyfunctional throughout the flight envelope in presence of different failuresand parameter variations.

2.1 Gain Scheduled Controllers

Since the aircraft dynamics are very dependent on e.g. Mach number,angle of attack and altitude, gain scheduling is often used to controlthe system. The system is linearized over a flight envelope dependingon those three parameters and a control law is designed in each discretepoint. The overall control system then interpolates the different controllaws based on the current parameter state.

α

alt

M

Figure 2.1. Flight envelope.

9

10 Chapter 2. Flight Clearance

Using this design method a few points must be considered:

• The control laws are designed for linearized systems but will theywork when applied to the real nonlinear system?

• The control laws are designed in discrete points of the envelopeand fulfil the specifications in these points but what happens be-tween these points?

• The dynamics also depends on various additional parameters suchas e.g. aerodynamic coefficients, mass and centre of gravity. Willthe system work when these parameters vary?

To make sure that the control system works for all possible parametervariations, some of the possible combinations are tried out, often thedesign points and the limits of the variations. Then a lot of nonlinearsimulation is performed to verify the functionality over the whole en-velope and the values which have not explicitly been tested. This is avery time consuming task since the envelope is big and there are a lotof possible parameter combinations.

2.2 Parameter dependence

There are many parameters affecting an aircraft and its Flight ControlSystem (FCS) but generally they can be divided into four groups:

Configuration dependent variabilities such as centre of gravity (c.g.)and inertia. They are depending on e.g. the amount of fuel andthe current stores.

Aerodynamic uncertainties depending on e.g. the current aircraftprofile, the wind and so on.

Hardware dependent variabilities such as changes in the sensor oractuator dynamics.

Air data system dependent tolerances such as measurement er-rors in Mach number or angle of attack.

Often at least the probable interval variations of the parameters areknown, e.g. a specification on longitudinal c.g. position can be seen infigure 2.2.

2.3 Clearance Process

Clearance of an aircraft and its control system is a tedious task withmany steps. One example of the industrial clearance process steps couldbe (Fielding et al. 2002):

2.3. Clearance Process 11

A/C

mass

[kg]

Longitudinal c.g. position [%m.a.g]

Figure 2.2. Centre of gravity diagram.

Generation of analysis model involves making a nonlinear simula-tion model with parameter uncertainties. This model is used tomake linear (small perturbed, trimmed) models.

Familiarization with aircraft and controller to get a picture ofthe unaugumented aircraft dynamics. Plots of aerodynamic sta-bility and control derivatives are studied.

Trend studies on the effect of uncertainties to see how differentparameters affect handling and stability.

Linear stability analysis where e.g. open-loop stability margins andeigenvalues are calculated for a narrow grid of the envelope andfor different uncertainties.

Linear handling analysis where evaluations of different time andfrequency domain criteria are studied and worst cases are found.

Nonlinear analysis by off-line and manned simulation to eval-uate the flying characteristics with and without uncertainties andfind handling and control problems.

Clearance report where the manoeuvre and flight envelope limita-tions based on the earlier analysis are visualized and derived.

Improvement of clearance based on special investigations on re-duced stability margins.

Utilizing µ-analysis could facilitate the linear analysis part of the clear-ance process. Using the methods described in this thesis open loopstability margins and eigenvalues can be analyzed in and between thepoints in the envelope for a linear approximation of the system. Thisapproximation can be nonlinear in parameters. The worst case staticparameter combination can be found and utilized to perform nonlinearanalysis where needed.

Chapter 3

Basic Flight Mechanics

This chapter contains basic flight mechanics necessary for understand-ing the model and the analysis results. The longitudinal linearized equa-tions of motion are derived from the nonlinear and some properties ofthe solutions to these linear equations are discussed. The short periodapproximation of the linearized equations is later used for construc-tion of an LFT description. For a more thorough survey on aircraftdynamics e.g. Etkin (1972) is recommended.

3.1 Coordinate frames

When describing an aircraft three coordinate frames are usually used(figure 3.1).Two fixed to the aircraft, one with the x axis fixed to theaircraft for describing the motion, SA, and one with the x axis fixed tothe velocity vector for describing the aerodynamic forces acting on theplane SW . The third system is an inertial system fixed to the earth’ssurface describing the aircrafts movement relative the earth SI . Thesame vector can be described in two frames and

v = xixi + yiyi + zizi == xbxb + ybyb + zbzb

if one coordinate frame rotates with angular velocity ω relative anotherframe then the theorem of Coriolis (3.1) can be used to transform themotion from one frame to the other.

d

dt

∣∣∣∣i

v =d

dt

∣∣∣∣b

v + ω × v (3.1)

13

14 Chapter 3. Basic Flight Mechanics

xA

yA

zA

xI

yI

zI

SA

SI

Figure 3.1. Inertial and aircraft fixed coordinate frames.

3.1.1 Euler angles

The angles used for describing SA relatively SI are called Euler angles,ΦA, and defined in figure 3.2 together with respectively rate, ωA.

ΦA =

φθψ

ωA = ΦA =

pqr

φ

p

qα θ rψ

Figure 3.2. Angles and rates for roll (φ, p), pitch (θ, q), yaw (ψ, r)and angle of attack α.

3.2 Forces, moments and velocities

The directions of the forces, FA and moments, TA (3.2) acting on theplane described in the body fixed frame are defined in figure 3.3 where

3.3. Moment of inertia 15

directions of the velocities vA also can be seen.

FA =

XYZ

TA =

LMN

vA =

uvw

(3.2)

X, u

Y, v

Z,w

M

N

L

Figure 3.3. Forces, moments and velocities.

The longitudinal forces, FA,l, and moments, TA,l, depend on theaerodynamic coefficients Ci in (3.3). q is the aerodynamic pressure andS is the wing area.

FA,l =XZ

=qSCX

qSCZ

Horisontal ForceV ertical Force

TA,l = M = qSCM Pitchingmoment(3.3)

The aerodynamic coefficients are dimensionless and dependent on a lotof parameters e.g. velocity, altitude and the aircrafts profile. They cannot be described analytically but are estimated empirically from windtunnel tests and flight tests.

3.3 Moment of inertia

An aircraft can usually be assumed to be symmetric in the xz-plane.This gives the inertia tensor in (3.4).

I =

Ix 0 −Ixz

0 Iy 0−Ixz 0 Iz

(3.4)

3.4 Static Stability

When an aircraft is subject to a disturbance in α an aerodynamic mo-ment starts acting on it. If this moment brings it back, the system is


said to be statically stable. Using the sign conventions in figure 3.4a positive alpha disturbance should produce a negative pitch momentand vice versa if the system is statically stable. Modern fighter aircraftswithout a controller are often longitudinal unstable for subsonic veloc-ities. This is the case for the Swedish aircraft Gripen and the VEGASmodel used in this thesis. When the aircraft is in force and momentbalance it is said to be trimmed. The static stability is dependent one.g. the pitch moment coefficient and the centre of gravity.

M < 0

∆α > 0

Figure 3.4. Static stability

Pitch moment coefficient

The pitch moment coefficient CM describes how the pitch moment act-ing on the aircraft due to aerodynamics changes in the envelope. Itconsists of several components (3.5).

CM = CM,base + CMα. . . (3.5)

where CMαis the pitch moment coefficient depending on angle of at-

tack.

3.4.1 Centre of gravity

The centre of gravity also affects the static stability. Simply describedthe longitudinal moment depends on the moments caused by the wingsand by the tail as in figure 3.5. For a disturbance in α the tail momentshould force the plane back. This is the case when CMα

< 0. If thecentre of gravity is moved more aft the wing moment gets larger and

3.5. Nonlinear Equations of Motion 17

CMαincreases making the plane more unstable. The aerodynamic cen-

tre (a.c.), is a point on the aircraft where the moments do not dependon the angle of attack. Both the centre of gravity and the aerodynamiccentre move around due to the dynamics.

xacxcglt

Figure 3.5. Centre of gravity and moment.

3.5 Nonlinear Equations of Motion

The longitudinal nonlinear equations are derived from regular rigidbody dynamics. FT is the thrust force from the engine.

X + FT +mg sin θ = m(u+ qw − rv)Z +mg cosφ cos θ = m(w + pv − qu)

M = Iy q + rp(Ix − Iz) + Ixz(p2 − r2)θ = q cosφ− r sinφ

Table 3.1. Nonlinear longitudinal equations of rigid body motion.

3.6 Linearization

When linearizing the nonlinear longitudinal equations of motion all lat-eral components such as r, p, v disappear. For small disturbances theaircrafts lateral movement is not affected by the longitudinal. Assum-ing trimmed flight condition i.e. force and moment balance and flyingstraight forward with constant velocity gives u = u0, w = 0 and v = 0.

Xu∆u+Xw∆w −mg = m∆uZu∆u+ Zw∆w + Zδe

∆δe + Zδc∆δc +mg = m∆w −mu0∆q

Mu∆u+Mw∆w +Mq∆q +Mδe∆δe +Mδc

∆δe = Iy∆q∆θ = ∆q

(3.6)


where Xi = ∂X∂i

∣∣i=i0

, Zi = . . . and so on. Dividing (3.6) with m in forceequations, Iy in the moment equation and rearranging to a state-spacedescription gives

∆u∆w∆q∆θ

=

Xu Xw 0 0Zu Zw u0 0Mu Mw Mq 00 0 1 0

∆u∆w∆q∆θ

+ (3.7)

+

0 0Zδe

Zδc

MδeMδc

0 0

[∆δe∆δc

]

3.6.1 Phugoid and short period mode

When solving the linearized longitudinal equations of motion typicallytwo complex conjugated solutions are achieved. These are the linearsystem modes. One corresponds to a fast heavily damped oscillationand the other to a slow lightly damped oscillation. By making someapproximations these modes can be separated. When designing a lon-gitudinal controller these modes must be suppressed, especially the fastshort period mode.

Short period mode

The short period mode is fast and heavily damped, typical values can bee.g. a frequency of ωsp = 3 rad/s and a damping coefficient of ξsp = 0.4.This mode has the properties that

• ∆u is small compared to ∆w.

• ∆w and ∆θ are in phase.

• ∆α ≈ ∆θ

which describes a pitch oscillation as in figure 3.6 This motion is ap-

Figure 3.6. Motion of short period mode.

proximated by (3.8)[∆w∆q

]=

[Zw u0

Mw Mq

] [∆w∆q

]+

[Zδe

Zδc

MδeMδc

] [∆δe∆δc

](3.8)

3.6. Linearization 19

Utilizing that ∆α ≈ ∆wu0

for small disturbances a state transformationas in section 4.4.1 can be done on (3.8) changing ∆w to ∆α (3.9)

[∆α∆q

]=

[Zα 1Mα Mq

] [∆α∆q

]+

[Zδe

Zδc

MδeMδc

] [∆δe∆δc

](3.9)

Phugoid mode

The phugoid mode is lightly damped and slow, e.g. frequency ωp = 0.1rad/s and damping ξp = 0.04.

• ∆w is small compared to ∆u and ∆w ≈ 0 so the velocity vectorand x vector are the same, i.e. the aircrafts nose is always in thedirection of motion.

• ∆u is approximately 90 before ∆θ.

which give the motion in figure 3.7.

Figure 3.7. Motion of phugiod mode.

Chapter 4

Linear Fractional Transfor-mation

In this chapter the Linear Fractional Transformation referred to asLFT is introduced. The big advantage of LFTs is that linear time-varying systems that depend rationally on parameters can be describedas an interconnection of a nominal linear time invariant system and astructured parameter block. Analysis of e.g. stability can then be doneusing these LFTs. The interested can read more in e.g. Zhou et al.(1996).

4.1 Upper and Lower LFTs

Consider a linear mapping or system (figure 4.1) where M can be com-plex M : C 7→ C

w1

w2

z1

z2

M11 M12

M21 M22

Figure 4.1. Linear mapping.

z1 = M11w1 +M12w2

z2 = M21w1 +M22w2 (4.1)

Now put another block called ∆ ∈ C in feedback on the upper ports ofthe system (figure 4.2)

21

22 Chapter 4. Linear Fractional Transformation

w2 z2

∆

M11 M12

M21 M22

Figure 4.2. Feedback.

This gives the following transfer function from w2 → z2

z1 = M11w1 +M12w2

z2 = M21w1 +M22w2

w1 = ∆z1z1 = M11∆z1 +M12w2 ⇔ z1 = (I −M11∆)−1M12w2

z2 = M21∆z1 +M22w2 ⇔ z2 = (M21∆(I −M11∆)−1M12 +M22)w2

This is called an upper linear fractional transformation. If the feedbackis put on the two lower ports instead a similar transfer function can bederived.

Definition 4.1 The upper and lower linear fractional transformationsof M and ∆ are defined by

Fl(M,∆l) , M11 +M12∆l(I −M22∆l)−1M21 (4.2)Fu(M,∆u) , M22 +M21∆u(I −M11∆u)−1M12 (4.3)

(I −M11∆u) and (I −M22∆l) must be invertible for respective LFTto exist. M11 in (4.2) or M22 in (4.3) can be considered the nominalsystem and the rest describes how the system vary with the parameters∆.

4.1.1 State-space systems

A regular state-space system can be described using LFTs by consid-ering the integration 1

s as an artificial feedback (figure 4.3).

z = Aw +Bu (4.4)y = Cw +Du (4.5)

w =1sz (4.6)

4.1. Upper and Lower LFTs 23

u y

1s

A BC D

Figure 4.3. State-space system as LFT.

Using this approach a linear state-space system that depends nonlin-early on parameters can be described as an interconnection of a lineartime invariant system and a parameter block.

Example 4.1Consider a scalar state-space system depending nonlinearly on δ

x =δ2 + 11 + δ

x+ bu = x+ δ

(δ − 11 + δ

)x+ bu

y = cx+ du

Taking w1 = δ(

δ−11+δ

)x = δw2 gives

z1 = w2

z2 = δx− x− w2

x = x+ w1 + bu

w1 = δz1

w2 = δz2

Finally let δx = w3

x = 1 1 0 0 bz1= 0 0 1 0 0z2=−1 0 −1 1 0z3= 1 0 0 0 0y = c 0 0 0 d

which is an upper LFT as in figure 4.4.


1s

δI3x3

M

u y

Figure 4.4. State-space system.

4.2 The Star Product

If two linear mappings such as (4.1) are interconnected as in figure 4.5the star product is obtained. It is a generalization of LFTs.

y1

z1

z2

y2

u1

w1

w2

u2

Q11 Q12

Q21 Q22

M11 M12

M21 M22

Figure 4.5. The star product.

4.3. LFT generation 25

y1 = Q11u1 +Q12w1

z1 = Q21u1 +Q22w1

z2 = M11w2 +M12u2

y2 = M21w2 +M22u2

w1 = z2 ⇔ z1 = (I −Q22M11)−1Q21u1 + (I −Q22M11)−1Q22M12u2

w2 = z1 ⇔ z2 = (I −M11Q22)−1M11Q21u1 + (I −M11Q22)−1M12u2

y1 = (Q11 +Q12(I −M11Q22)−1M11Q21)u1 +Q12(I −M11Q22)−1M12u2

y2 = M21(I −Q22M11)−1Q21u1 + (Q11 +M21(I −Q22M11)−1Q22M12)u2

Definition 4.2 The star product Q ?M ∈ C is defined as

Q ?M =[Q11 Q12

Q21 Q22

]?

[M11 M12

M21 M22

]

=[ Fl(Q,M11) Q12(I −M11Q22)−1M12

M21(I −Q22M11)−1Q21 Fu(M,Q22)

](4.7)

(I −M11Q22) and (I −Q22M11) must be invertible. The star productnotation can be used for upper and lower LFTs which are special casesof the star product.

Fl(M,∆l) = M ?∆l (4.8)Fu(M,∆u) = ∆u ? M (4.9)

4.3 LFT generation

There are many ways to make LFT descriptions of nonlinear systems,but generally they can be divided into two approaches. Making ananalytical LFT description from the nonlinear symbolic equations orapproximating functions from numerical data. In this thesis the latterapproach is chosen since it normally can be done with less effort and theemphasis of this thesis is on the analysis methods. Two ways of makingnumerical approximations are tried. Both ways adapt functions to statespace data.

4.3.1 Linear Parameter Varying Systems

State space data can be obtained by e.g. finding stationary points fora nonlinear aircraft model and then linearize the model around thesepoints for different flight cases and parameter combinations. A systemdescribed by such functions is called a Linear Parameter Varying Sys-tem (LPV) since it is linear in states but the state coefficients varies


with the parameters as in (4.10). Note that one parameter could betime. [

x1

x2

]=

[A11(δ) A12(δ)A21(δ) A22(δ)

] [x1

x2

](4.10)

This LPV can always be reformulated as an LFT if Aii(δ) is rationalor polynomial by doing steps similar to example 4.1.

4.3.2 Trends & Bands approach

The Trends & Bands approach is a linear approximation suggested byMannchen et al. (2002) and is used quite successfully in the GARTEURproject. The idea is to approximate the trend of the data with linearparameters and adding the largest deviation, if it is significant, as anadditional parameter making a band around the function. By doing thisthe physical meaning of the function is preserved in the trend param-eters and the whole physical system is covered by the approximation.The one parameter case is seen in figure 4.6. The aij element in the

δerr

aij

aij = f(δk)

aij0 + aijkδk

δk

Figure 4.6. 1-dimensional Trends & Bands approximation.

system matrix, A, depending on a parameter aij = f(δk) would get thefollowing approximation

aij = aij0 + aijkδk + δerr (4.11)

If the element depends on two parameters the approximation is a planeand when the element is depending on more than two parameters theapproximation is a multidimensional linear regression plane. To get thecoefficients e.g. min-max optimization can be used. In min-max opti-mization the coefficients are chosen such that the maximal deviation isminimized. The advantage of this method is the simplicity and sinceall matrix elements have the same form in the approximation it canbe automatically generated. However, this is a linear approximation,so if the true function has a narrow peak, as seen in figure 4.7, the

4.3. LFT generation 27

approximation will cover a lot of systems besides the real and intro-duce conservatism. This way of making LFTs will later be referred toas T&B.

δerr

aij

δk

Figure 4.7. Conservative approximation.

4.3.3 Approximation with rational functions

To get a less conservative LFT description, rational functions can beadapted to data from linearization points (4.12). Rational functionsare preferred before polynomials since more advanced surfaces can beapproximated with lower order functions.

aij =b0 + b1δk + b1δ

2k + . . .

1 + a0δk + . . .+ δerr (4.12)

Using this approach the model will be more accurate compared to T&Bsince the approximated surface will follow the original function betteras shown in figure 4.8.

δk

aij

δerr

Figure 4.8. 1-dimensional rational approximation.

This will make the compensation parameters, wich represents the er-ror, smaller and reduce the conservatism. More effort is needed to findthe functions but if some physical knowledge of the system exists thiseffort can be reduced. The number of parameters in the ∆- block can


be quite large if a high order rational function is needed for a goodapproximation.

4.4 Preparation of approximation data

The data used to approximate functions can be normalized in severalways to achieve better numerical properties.

4.4.1 Normalizing states

If a state is transformed with a constant matrix T such that Tx = xthen T x = ˙x since T is constant and

T−1 ˙x = AT−1x+Bu⇔ ˙x = TAT−1x+ TBu (4.13)y = CT−1x+Du (4.14)

The two systems[A B

C D

]=

[TAT−1 TBCT−1 D

](4.15)

will have the same input-output properties. This can be used e.g. toscale the state-space matrices for better numerical properties or tochange some of the states.

4.4.2 Normalizing parameter data

If the parameter data used in the estimation is normalized to the in-terval [−1, 1] the size of estimated coefficients can be compared. A co-efficient which is small compared to the other is probably insignificantand can be removed without making the approximation error larger.This is useful in the rational function approach.

4.5 Model reduction

The LFT model should have as low order as possible. Low order reducesnumerical problems, time for calculation and makes it easier for theoptimization algorithms. The methods below are used in this thesis forclever realization and model reduction.

4.5.1 Tree decomposition

A LFT will have the same number of parameters in the ∆-block asthe number of parameters in the LPV if the reformulation is done in

4.5. Model reduction 29

the simplest possible way, without any simplifications. This is called theorder of the LFT. This order can be reduced for a polynomial LPV, suchas the T&B, if the LFT is built using tree decomposition (Magni 2001).The idea is to factorize and sum decompose the polynomial matrix inturns, until no more transformations can be done. This is implementedin the Onera Toolbox, (Magni 2001), command symtreed.

Example 4.2Consider a polynomial matrix of order six,

S(∆) =[δ1δ2 + δ3δ1δ2δ3

]

by first applying a sum decomposition[δ1δ2 + δ3δ1δ2δ3

]=

[δ30

]+

[δ1δ2δ1δ2δ3

]

then a factorization[δ30

]+

[δ1δ2δ1δ2δ3

]=

[δ30

]+

[1δ3

]δ1δ2

the order has been reduced from 6 to 4.

There are of course several ways of doing this by factorizing other pa-rameters and doing the decompositions and factorizations in anotherorder. The best way is not known in advance so a number of combina-tions must be tried and the one giving the lowest order is used.

4.5.2 n-D decomposition

n-D decomposition is implemented in the Onera Toolbox commandminlfr. By doing transformations and step by step eliminating thenonobservable and noncontrollable states in the transformed LFT theorder can be reduced (Magni 2001).

4.5.3 µ-sensitivities

By using µ-sensitivities as described in section 5.2, parameters withsmall influence on the µ-value are found. These can be removed withoutmaking the model considerably more optimistic in the sense that it willproduce only a slightly lower µ-value.

Chapter 5

Structured Singular Values

5.1 Structured Singular Values

The Structured Singular Value (SSV) , µ, was first suggested by Doylein 1982 as a way of analyzing systems with uncertainties. The SSVis applied to systems described by Linear Fractional Transformationsand is the inverse of the largest parameter combination, in a 2-normsense, which will make the system unstable. Using this analysis, calledµ-analysis, stability can be verified and parameter combinations whichgive the worst performance can be found.

5.1.1 Small gain theorem

A conservative yet useful theorem is the small gain theorem. The the-orem is sufficient but not necessary, meaning that if the theorem doesnot grant stability the system can still be stable.

+

+

y1

y2 e2

r1

r2

S1

S2

Figure 5.1. Interconnected system.

31

32 Chapter 5. Structured Singular Values

Theorem 5.1 (Small gain theorem) Consider two stable systems in afeedback interconnection as in figure 5.1. The closed loop system is sta-ble if the product of the gains is smaller than 1.

‖S2‖‖S1‖ < 1 (5.1)

This can be interpreted as when the loop is circulated the signal shouldnot be amplified as it internally would go towards infinity. Thereforethe total gain should always be smaller than 1. The limit, i.e. when thetotal gain is 1, can algebraically be verified by (5.2)

det(I − S2S1) = 0 (5.2)

The theorem is conservative in at least two senses.

• The sign of the feedback doesn’t affect the theorem.

• If the norms are taken for a whole frequency range they couldtake their maximum value at different frequencies.

5.1.2 Norms

To get a value on the gain of a system with multiple inputs and outputsvector and matrix norms can be used. The vector p-norm is

‖u‖p = p

√√√√ m∑i=1

|ui|p (5.3)

with two important special cases p = 2,∞

‖u‖2 = 2

√√√√ m∑i=1

|ui|2 = u∗u (5.4)

‖u‖∞ = ∞

√√√√ m∑i=1

|ui|∞ = maxi

(|ui|) (5.5)

(5.6)

the matrix norm induced by the vector norm is

‖A‖p = supx6=0

‖Ax‖p

‖x‖p(5.7)

with p = 2,∞‖A‖2 =

√λmax(A∗A) = σ(A) (5.8)

‖A‖∞ = max1≤i≤m

n∑j=1

|aij | (row sum) (5.9)

5.1. Structured Singular Values 33

The 2-norm is also called the maximal singular value and used as ameasurement of the largest gain of a matrix. If A is square this is theabsolute value of the largest eigenvalue.

5.1.3 Structured Singular Value

Consider a LFT description as in figure 5.2 with a LTI system M(jω)interconnected with a structured parameter block ∆, which is nominallystable for ∆ = 0.

y u

∆

M11(s) M12(s)M21(s) M22(s)

Figure 5.2. System with ∆-feedback.

Then according to the small gain theorem, this system is stable if

σ(M11(s))σ(∆) ≤ 1 ⇔ σ(∆) ≤ 1σ(M11(s))

(5.10)

This is a stability test for an unstructured ∆ but it is conservative inthis case since it doesn’t utilize the structure of ∆ which is diagonal orblock diagonal.

∆(s) = diag∆1(s), . . . ,∆N (s)σ(∆i(s)) ≤ k (5.11)

Instead the largest singular value of ∆ i.e. the largest k that won’t turnthe system unstable is desired. This is defined as

Definition 5.1 (Structured Singular Value) Given a matrix M ∈ Cand an uncertainty structure ∆ ∈ C, the structured singular value µ is

µ∆(M) =(

min∆∈∆

σ(∆) : det(I −M∆) = 0)−1

(5.12)

and if no ∆ fulfils det(I −M∆) = 0 then µ∆ = 0


Real axisIm

agin

ary

axis

Figure 5.3. s-plane interpretation of SSV.

The system must be nominally stable i.e. all nominal eigenvalues mustbe in the left half plane (LHP) for µ-analysis to be performed. Usuallyµ is calculated along the imaginary axis and an s-plane interpretation ofthe structured singular value can be seen in figure 5.3. A system that isnominally stable has all it’s eigenvalues in the LHP. As the parameterschange, some of the eigenvalues move towards the imaginary axis andwill eventually reach the axis. When they reach the axis the gain of thesystem is one (5.13).

‖∆(s)‖2‖M11(s)‖2 = 1 (5.13)

If the smallest parameter combination making the poles reach the axisis larger than the allowed parameter values, the system is stable forall allowed parameter variations. The possible parameter (∆) intervalsare usually normalized so that ‖∆‖∞ ≤ 1 implies that the system isstable for µ∆ ≤ 1. Since µ is the inverse of the largest singular value ofthe smallest ∆ making the system unstable, it can also be seen as thelargest allowed gain of M11. If ∆ is diagonal then σ(∆) is the largestabsolute value of ∆.

5.1.4 Upper and lower bounds

Practically the exact µ-value is hard to find. Therefore upper and lowerbounds are calculated. Finding the upper bound can be formulated as aconvex optimization problem such that the true maximum for a scaledsystem can be found. The scaling makes the bound a little conservativeand the bound is not the true µ-value. The lower bound problem isnonconvex why a local minimum are often found. If the global minimumis found it is the true µ-value.

5.2. µ-sensitivities 35

5.2 µ-sensitivities

To find out which parameters of the LFT that have the most or leastinfluence on the µ-value, µ-sensitivities can be used (Braatz & Morari1991). This can be interesting for model reduction where the parametersthat has the least influence on µ can be removed without making themodel more optimistic. It can also be used to find the most importantparameters to realize what part of the model to improve to make itmore accurate.

Definition 5.2 (µ-sensitivities)

∂µi ≡[

lim∆αi

→0+

µ(M(αi)) − µ(M(αi − ∆αi))∆αi

]αi=1

(5.14)

α = diagα1Ir1 , . . . , αnIrn (5.15)

Where ri is the uncertainty block dimension.

M(α) =[

αM11√αM12√

αM21 M22

](5.16)

It can be interpreted as the change in the maximum µ-value whena parameter is reduced a little bit. If the change is significant, theparameter has a large influence on the µ-value and the system.

5.3 Modelling of Clearance Criteria

Clearance criteria given in the frequency domain can be modelled andincluded in the analysis. This makes µ-analysis quite useful for thiskind of clearance. However, in the flight industry many criteria aregiven in the time domain, which makes it harder to include them in theµ-analysis framework.

5.3.1 Uncertainty modelling

For most control designs and control analysis, a mathematical modelof the real plant is used. This model will always differ from the realplant and will not behave exactly as the plant. An example can be seenin figure 5.4. These differences will be referred to as uncertainties andmust be modelled to make the model more accurate.


|G(jω)| nominalreal plant

ω

Figure 5.4. Uncertainty in frequency plane.

Real Uncertainties

To model a parameter that varies in an interval, real uncertainties areused. This models a parameter which is not known, but the probableinterval is known or a parameter which is known to vary within aninterval.

c ∈ [0.8, 1.6]c ∈ 1.2 + (0.4)δ : δ ∈ R, |δ| ≤ 1

Complex uncertainties

Complex uncertainties are used to model e.g. uncertain dynamics. Acomplex uncertainty is a region in the s-plane as in figure 5.5

c ∈ 1.2 + (0.4)δ : δ ∈ C, |δ| ≤ 1

0.8 1.6

0.4

Imag

inar

yax

is

Real axis

Figure 5.5. Complex uncertainty.

5.4. Phase and Gain margins 37

5.4 Phase and Gain margins

Linear stability criteria such as gain and phase margins can be in-cluded in the LFT-description and µ-analysis, making it easy to verifyspecifications on them. The Nyquist criterion says that if the Nyquistcurve does not enclose -1 in the Nyquist plane, the system is stable.The distance and angle to this point, as seen in figure 5.6, is the gainrespectively phase margin.

-1

1Gm

ϕm

Figure 5.6. Gain and phase margins on Nyquist plot.

5.4.1 Nichols exclusion regions

Based on the Nyquist criterion, Nichols exclusion regions can be used toverify gain and phase margins. The critical point i.e. -1 in the Nyquistplane or −180 in the Nichols plane is surrounded with a region. Themargins are fulfilled if the Nichols curve of the system doesn’t passthrough this region. This region is not exact but based on experience.It can quite easily be approximated and included in the LFT descriptionmaking it possible to use µ-analysis (Bates et al. 2001, Fielding et al.2002). The simplest approach is to approximate the region with anellipse as seen in figure 5.7 and describe this as a multiplicative uncer-tainty in the frequency region (5.19).

|L(jω)|2dB

G2m

+(∠L(jω) + 180)2

P 2m

= 1 (5.17)

This ellipse is described by (5.17) where ∠ is the argument of L(jω). Itcan be mapped to a circle with centre −a and radius r in the Nyquist


−220 −210 −200 −190 −180 −170 −160 −150 −140−8

−6

−4

−2

0

2

4

6

8

Open Loop Phase [deg]

Ope

n Lo

op G

ain

[dB

]

(a) Nichols plane

−2.5 −2 −1.5 −1 −0.5 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Re L(jω)

Im L

(jω)

(b) Nyquist plane

Figure 5.7. Exclusion regions in the Nichols and Nyquist plane.

plane using (5.18)

Gm = 20 log10(a+ r)

Pm = cos−1(a2 − r2 + 1

2a) (5.18)

P (s) = P1(s)(−a+ ∆) (5.19)

where P1(s) is the nominal model and ||∆||∞ < r, ∆ ∈ C . This canalso be written as

P (s) = aP1(s)(1 +WN∆) (5.20)

where ||∆||∞ < 1 and WN = − ra . This is added to all loops simultane-

ously or one loop at a time to check the stability. A multivariable systemcan be unstable for simultaneous perturbations although it passes theone loop at a time test. The loops can be broken on the sensor side orthe actuator side. A block diagram description is seen in figure 5.8.Constants for the two usual margins shown in figure 5.7 are found intable 5.1.

Gain Phase a Wn

Nominal 6dB 36.87 1.25 0.6Perturbed 4.5dB 28.44 1.14 0.47

Table 5.1. Gain and phase margins.

5.4. Phase and Gain margins 39

WN ∆N

a P1

K

Figure 5.8. Adding exclusion region.

5.4.2 Stability margin and worst case parameter com-bination

The stability margin is used as a tool for comparing stability. Theinner Nichols exclusion region is defined to have stability margin oneand a stability margin less than one is hence considered insufficient.How much this region can be scaled before it touches the system is thestability margin ρ as in figure 5.9. The region is scaled in the analysis byscaling r which is the same as scaling WN in (5.20). To find the worst

ρ = 1ρ = 1.75

Figure 5.9. Stability margin.

case parameter combination the region must be scaled and µ calculatediteratively until a peak value of one is reached. The scale factor on WN

when the µ value is one is the stability margin. Another way suggestedby Kureemun et al. (2001) is to define ρ by (5.21).

ρ =1µ

(5.21)

When the region just touches the systems Nichols curve a µ-value ofone should be achieved. A µ smaller than one corresponds to a largerregion and so on.


5.5 Most unstable eigenvalue

Usually µ is calculated by doing a frequency grid i.e. calculating µ onthe imaginary axis in the s-plane as described in section 5.1.3. But thegrid can be done over some other line in the s-plane if the nominalpoles all are inside the region surrounded by the frequency grid. Theusual calculating algorithms will work. This line in the s-plane can beutilized for including specifications on the most unstable eigenvalues inthe analysis.

5.5.1 Specifications

A specification on unstable eigenvalues can be seen in (5.22). It can betranslated into five lines in the s-plane as in figure 5.10.

<λ = 0 , for ω ∈ Ω1 = |ω| ≥ 0.15rad/s<λ ≤ ln 2

20 , for ω ∈ Ω2 = 0 ≤ |ω| ≤ 0.15rad/s<λ ≤ ln 2

7 , for ω ∈ Ω3 = 0(5.22)

By calculating µ on a grid consisting of those lines it can be verified if

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Re

Im

section 1section 2

section 3

section 4section 5

µ ≤ 1µ ≥ 1

Figure 5.10. Eigenvalue specification.

any of the poles pass the lines (Mannchen et al. 2001). To find out thevalue of the most unstable pole the lines must be moved iteratively untilµ becomes one. The point of the line corresponding to the peak value isthe most unstable eigenvalue. By doing this with a lower µ-algorithmthe parameter combination that generates this eigenvalue can be found.The worst case poles can also be found by using the Pole PlacementApproach described in section 6.2.3 and moving a line into the RHPuntil a µ-value of one is found.

Chapter 6

µ-calculations

There exist several approaches to calculate upper and lower bounds onµ. Tight bounds are wanted to draw the right conclusions about thesystem since if 1 is between the bounds not much can be said. In thisthesis a few calculating algorithms are tried out and compared. Com-puting µ when the problem consists of real repeated uncertainties is amuch more difficult task then for mixed or strictly imaginary uncertain-ties. In a physical model, the uncertainties are often real and repeatedso such problems need to be solved. There are basically two types ofalgorithms.

Grid based algorithms which calculate a µ-value on each predefinedpoint of a frequency axis grid.

Peak finding algorithms where frequency is included in the algo-rithm and a µ peak value and the corresponding frequency arefound.

6.1 Upper µ-calculations

All the upper bound algorithms recast the problem as a convex op-timization problem. They will always find the maximal value but theresult can differ a bit from the true value since the system is scaled.

6.1.1 Scaling

To find an upper bound on µ the system is usually scaled. How torecast the problem when the parameters are imaginary is shown. Ifthe parameters are real or both real and imaginary other scalings areintroduced. The problem is scaled with a matrix D as in figure 6.1. This

41

42 Chapter 6. µ-calculations

D should be positive definite, Hermitian i.e. D∗ = D, and commutewith ∆ so that D∆ = ∆D.

∆∆ DD D−1D−1

MMuu yy

Figure 6.1. Adding scaling matrices.

The new largest singular value is σ(D∆D−1). A search for both a Dmatrix and the upper µ value should be done (6.1).

infDσ(D∆D−1) (6.1)

This is a convex problem so a global solution can always be found if itexists. Even if an upper bound for the original problem is impossibleto find, the new scaled problem may be solved if the scaling matrix Dis found.

6.1.2 µ-Analysis and Synthesis Toolbox

The command mu, (Balas et al. 1993), solves the upper bound overa grid of frequency points. Static µ is calculated in each point. Thisalgorithm can handle large problems with up to 100 uncertainties. Itgives conservative bounds on the real µ and since the frequency axisis calculated on a grid it can miss narrow points. The bound found isµ∆(M(jω)) = β > 0 if

σ

((I +G2

l )− 1

4 (DlMD−1

r

β− jGm)(I +G2

r)− 1

4

)≤ 1 (6.2)

and there exists scaling matrices Gl, Gm, Gr,Dl and Dr.

6.1.3 Finite Frequency Method

In the finite frequency method, (Helmersson 1995), frequency is addedas an additional uncertainty to the LFT description. By doing this, acomplete frequency interval is covered in one computation. The compu-tation can be done with any upper bound algorithm e.g. mu describedabove. When large frequency intervals are added the resulting upperµ-value can be much larger than the real value. This conservatism can

6.1. Upper µ-calculations 43

be reduced by using a branch and bound scheme. The intervals are di-vided into smaller pieces and the one with the largest value is chosenfor the next computation. By doing this the peak value can be found.Even though the value can be somewhat conservative depending on thesize of the interval, narrow peaks will not be missed.

Frequency mapping

The interval δ ∈ [−1, 1] can be mapped to 1s = −j 1

ω using a bilineartransformation

1s

=1jω

= −j 1 − δ

1 + δ= −j

(1 − 2δ

1 + δ

)(6.3)

which can be written as an LFT

Nω =[ −I √

2Ij√

2I −jI]

(6.4)

The interconnection in figure 6.2 gives the whole frequency range

Nω

M

Figure 6.2. System augmented with frequency.

ω ∈ [0,∞]. To divide it into smaller intervals, an interconnection witha third LFT is necessary. To get the intervals [ωl, ωm],[ωm, ωr] e.g. (6.5)and (6.6) can be used

NR =[

0 Iδl−δm

2 I δl+δm

2 I

], NL =

[0 I

δm−δl

2 I δm+δl

2 I

](6.5)

where the δ:s are a mapping from frequency (6.6).

δi =ωi − 1ωi + 1

(6.6)

Finally everything is interconnected using the star product accordingto figure 6.3. The µ-value is calculated using e.g. mu in MATLAB’s


NINT

M

Nω

Figure 6.3. LFT for dividing frequency.

µ-Analysis and Synthesis toolbox with a constant matrix. NINT is theinterval NR or NL. The best results are achieved if the algorithm getsa starting peak frequency. The peak upper bound is then calculated intwo intervals around this frequency. The interval yielding the largestµ-value is chosen for the next calculation. This new interval is dividedin two and so on. As starting point e.g. the peak frequency from thepole placement approach or the mu upper calculation can be used.

6.2 Lower µ-calculations

The lower µ problem is a nonconvex problem so local minima are oftenfound. If a global minimum is found it is the true µ-value.

6.2.1 mu command augmented with imaginary part

The mu command totally fails to converge when calculating the reallower bound. By augmenting the system with a small imaginary partthe algorithm gets better convergence, (Balas et al. 1993). Doing thischanges the structure of the model and it is not the original problemthat is solved. The expansion of the system can be seen in figure 6.4.

6.2. Lower µ-calculations 45

∆R

∆C

MM

αα

∆R + α2∆C

Figure 6.4. Imaginary augmentation of real LFT description.

The analysis should be done on

Maug =[Iα

]M

[Iα

]′(6.7)

∆aug = diag(∆R,∆C) (6.8)

6.2.2 Basic Optimization Algorithm

The problem is formulated as a constrained nonlinear optimizationproblem, (Hayes et al. 2002). The frequency axis is divided into a gridand the problem is solved in each point. The µ-value from the previ-ous frequency point can be used as a starting value for the next point.Since the problem is nonconvex the algorithm is restarted with differ-ent initial values to find the true or at least as good µ as possible.The nonlinear optimization formulation is to find min σ(∆) such thatdet(I −M∆) ≤ ε. The ε is introduced to make a numerical limit onhow close to zero it should be. This is calculated using e.g. Matlab’sOptimization Toolbox.

6.2.3 Pole Placement Approach

Instead of dividing the frequency axis into a grid and finding the largestsingular value in each point, the system is scaled until one of the closedloop poles migrates through the imaginary axis and into the right halfplane, (Iordanov et al. 2003). A lower LFT description of a system(6.9).

x = Ax+Bw

y = Cx+Dw (6.9)u = ∆w


has the transfer function (6.10)

x = (A+B∆(I −D∆)−1C)x = A∆x (6.10)

The system A∆ (6.10) becomes unstable when any pole passes the imag-inary axis. This takes place when the largest real part of any eigenvalueis zero (6.11). <A is the real part of A and =A is the imaginarypart.

λmax(A∆) = maxi

<λi(A∆) = 0 (6.11)

Instead of using definition 5.1 for µ (6.12) is used.

µ(M(s)) =(min σ(∆) : max

i<λi(A+B∆(I −D∆)−1C) = 0

)−1

(6.12)This is the smallest σ(∆) which makes any pole cross the imaginaryaxis. The frequency is an implicit function of ∆ (6.13).

ωp = |=λmax(A∆)| (6.13)

Since the µ-value is not calculated on a frequency grid narrow pointswill not be missed and a value closer to the true value will be found. Thecalculations can be done with e.g. MATLAB’s Optimization Toolbox.A few different starting seeds can be used to avoid finding local minima.The simplest choice is to start from a random ∆.

Offset imaginary axis

This algorithm can be used for any vertical line in the s-plane, as shownin figure 6.5, if all the nominal poles are left of this line. The conditionfor this is (6.14) where α is the value on the real axis where the line ismoved.

λmax(A∆) = maxi

<λi(A∆) = α (6.14)

Real axis

Imag

inar

y ax

is

0 α

Figure 6.5. Offset into RHP.

Chapter 7

Bounds on Parameter Rates

In the earlier chapters analysis methods for static parameter variationswere described but parameters usually change over time. In this chaptera way to guarantee stability with a bound on the parameter rate is de-scribed. The implicit system approach and the steps to formulate theoptimization problem are taken from Iwasaki & Shibata (2001). Thetheory is only explained briefly. The main idea is to augment the LFTdescription with derivatives of the parameters. Conditions on this LFTdescription are formulated as Linear Matrix Inequalities (LMI) whichcan be solved quite efficiently. If solutions to these conditions are foundstability can be guaranteed for certain limits on the parameters. Solv-ing this new formulation can be shown to be the same as solving aparameter dependent Lyapunov function which depends on the origi-nal LFTs parameters ∆. Finally a way of starting from the parameterdependent Lyapunov function (Helmersson 1999) is shortly described.

7.1 Lyapunov Theory

Since the methods in this chapter are based on quadratic Lyaponovfunctions a short review of this follows. First a graphical definition ofsome stability concepts in the state space. If the initial states x∗(0) arewithin the δ region (figure 7.1) then the solutions will move no furtheraway than ε from x(t) if the system is stable. The solutions will movetowards x(t) if asymptotically stable. If globally asymptotically stablethe solutions will move towards x(t) regardless of the starting point,i.e. δ is infinite.

47

48 Chapter 7. Bounds on Parameter Rates

t

x(t)

x(0)

x*(0)

x*(t)

x(t)

1

2

δ

δ ε

ε

Figure 7.1. Stability regions.

7.1.1 Lyapunov functions

By looking at how the distance from an equilibrium point evolves overtime conclusions about the systems stability can be made. If the dis-tance decreases over time the system is asymptotically stable i.e. thesolutions moves towards the equilibrium point.

Definition 7.1 Let x0 be an equilibrium point for the system x = f(x).A function that satisfies

V (x0) = 0 (7.1)V (x) > 0, ∀x 6= x0 (7.2)

d

dtV (x) ≤ 0 (7.3)

is a Lyapunov function.

Simplified a Lyapunov function can be seen as a function which mea-sures the distance to the equilibrium point. This function should de-crease along the solutions if the system is stable. Equation (7.1)–(7.2)says that the equilibrium point V (x0) must be a global minimum. Equa-tion (7.3) says that V (x) which is the distance to the origin should de-crease with time. There is no general way of finding V (x), but if it canbe found the system is stable. Often a global Lyapunov function cannot be found, but the conditions can be used to prove local stability.This means that the Lyapunov function found is only valid in a limitedregion around the stationary point.

7.1. Lyapunov Theory 49

7.1.2 Quadratic Stability

A quadratic matrix P = PT is positive definite if all eigenvalues arelarger than zero. Using P in a quadratic Lyapunov function for a linearsystem A would give (Glad & Ljung 1997)

x = Ax(t) (7.4)V (x) = x(t)TPx(t) (7.5)

d

dtV (x(t)) = xTPx(t) + xT (t)P x(t) = (7.6)

= xT (t)(ATP + PA)x(t) (7.7)

ifd

dtV (x(t)) = −xT (t)Qx(t)

where Q is positive definite then (7.8) is a Lyapunov equation. AT isthe transpose of A.

ATP + PA = −Q (7.8)

If A is globally asymptotically stable i.e. all eigenvalues are in the LHP apositive definite P can always be found for any positive Q. This resultcan also be used to draw conclusions for a nonlinear system whichcan be approximated with a linear system with an additional nonlinearpart (7.9). This additional part should be small close to the equilibriumpoint.

x = Ax+ h(x) (7.9)

For this case usually only a local Lyapunov function can be found.

7.1.3 Parameter dependent Lyapunov functions

If the nonlinear system is described by an LFT a parameter dependentLyapunov function can be used. Section 7.2 shows how this can becalculated.

x = A∆x(t) (7.10)d

dtV (x(t)) = x(t)P∆x(t) + xT (t)P∆x(t) + xT (t)P∆x(t) =(7.11)

= xT (t)(AT∆P∆ + P∆A∆ + P∆)x(t) (7.12)

The stability region will be larger when the derivative is included inthe Lyapunov function. The stability region is the ε in figure 7.1. If anonlinear system is described by a true LPV and a nonlinear part

g(x,∆) = A∆truex+ htrue(x,∆)


this true LPV can be approximated by

A∆true = A∆ +A∆err

where A∆ e.g. could be a LFT approximation. Then the nonlinear sys-tem is

g(x,∆) = A∆x+A∆err + htrue(x,∆)︸︷︷︸herr(x,∆)

The whole error herr(x,∆) can be included in the Lyapunov function,(Hansson et al. 2003). The two ways of making LFTs suggested in chap-ter 4 includes the approximation error in the LFT. Including the errorin the Lyapunov function instead reduces the size of the LFT and willreduce the computational complexity.

7.2 Implicit system approach

One way of including bounds on parameter rates is to simply augmentthe LFT with derivatives of ∆. When this is done the µ-calculationalgorithms described in earlier chapters cannot be used. Instead con-ditions on the system are formulated as Linear Matrix Inequalities.These conditions are convex and can be solved with efficient optimiza-tion routines such as interior point methods. The theory on how toreformulate the problem is only briefly discussed in this chapter andtaken from Iwasaki & Shibata (2001), where more details can be found.They also show that solving the problem this way is the same as findinga parameter dependent Lyapunov function.

7.2.1 Linear Matrix Inequalities

A Linear Matrix Inequality (LMI) is an affine matrix valued function,(Helmersson 2003),

F (w) = F0 +m∑

i=1

wiFi > 0 (7.13)

where wi and Fi = FTi are real. This function defines a convex set

and can therefore be solved quite efficiently. The notation F (w) > 0means that F is positive definite i.e all eigenvalues are positive. LMIsare usually not formulated as (7.13) but they can always be rewrittenon this form.

Example 7.1The Lyapunov inequality

P = PT > 0ATP + PA < 0

7.2. Implicit system approach 51

is an LMI since it is linear in P and can be rewritten on the standardform (7.13). If P is parameterized as

P =[p1 p2

p2 p3

]= p1

[1 00 0

]+ p2

[0 11 0

]+ p3

[0 00 1

]=

= p1P1 + p2P2 + p3P3

then the whole inequality can be written as a big LMI condition.

F (p) = p1

[P1 00 −ATP1 − P1A

]+ p2

[P2 00 −ATP2 − P2A

]+

+ p3

[P3 00 −ATP3 − P3A

]> 0

7.2.2 Augmentation of LFT

In Iwasaki & Shibata (2001) a way of including bounds on the param-eter rates and an efficient way of solving the problem with LMIs is sug-gested. The LFT description of the system is augmented with deriva-tives of ∆ in a clever way so an implicit description of the system isobtained. Consider the standard time varying LFT

x(t) = Ax(t) +Bw(t)z(t) = Cx(t) +Dw(t)w(t) = ∆(t)z(t) (7.14)

To include the derivatives, the system is augmented by differentiatingz and w

z = CAx+ CBw +Dw

w = ∆z + ∆z

introducing ϕ = ∆z this is put together to an extended LFT

xwzzz

∆z∆z∆z

=

A B 0 00 0 I 0C D 0 0CA CB D 0C D 0 00 I 0 00 0 I −I0 0 0 I

︸︷︷︸24 A B

C D

35

xwwϕ

(7.15)


with the augmented uncertainty block ∇ = diag(∆,∆, ∆). The feed-

back block should be[I∇

]= K. Now this must be proven to be stable

for all allowed variations in ∇. This is algebraically the same as veri-fying that (7.16) has full column rank. Full column rank implies thatthe matrix never is singular i.e. has no eigenvalues on the imaginaryaxis. The problem is that there are an infinite number of constraintssince the frequency axis is continuous and the same goes for ∇ whichis continuous within the allowed intervals of variation.

[C(jωI − A)−1B + D, K] (7.16)

This can be transferred to another condition by introducing an IntegralQuadratic Constraint (IQC) Θ. G∗ is the complex conjugate transposeof G.

Lemma 7.1 The following statements are equivalent:i) [Gω K] has full column rangii) there exist a matrix Θ such that

G∗ωΘωGω < 0 K∗ΘωK > 0

where K∗ΘωK puts a constraint on the parameters. If Θ is assumed tobe constant the frequency variable can be eliminated using a version ofKalman-Yakubovich-Popov lemma on G∗

ωΘωGω.

Lemma 7.2 Given matrixes A, B and Θ then the following are equiv-alenti) for all real ω (7.17) holds

[(jωI −A)−1B

I

]∗Θ

[(jωI −A)−1B

I

]< 0 (7.17)

ii) there exists a P = P ∗ such that

[PA+ATP PB

BTP 0

]+ Θ < 0

This lemma states the equivalence between a frequency condition and aLMI. Now by adding C and D the lemma can be used on the augmentedLFT.

[C(jωI − A)−1B + D]∗Θ[C(jωI − A)−1B + D]∗ =[(jωI − A)−1B

I

]∗[C D]∗Θ[C D]

[(jωI −A)−1B

I

]< 0

is by 7.2 equivalent to[PA+ ATP PB

BTP 0

]+ [C D]T Θ[C D] < 0


which can be reformulated as (7.18) A B

C DI 0

∗ 0 0 P

0 Θ 0P 0 0

A B

C DI 0

< 0 (7.18)

where P and Θ are real and symmetric. But one problem is left, thereis still an infinite number of ∇ within the allowed intervals.

Multiconvexity

The allowed ∇ can be described as a multiconvex set, if the conditionsare a number of LMIs. If the set is multiconvex it is sufficient to checkonly the vertices of ∇ and still be able to guarantee that this will holdfor every allowed parameter combination.[

I∇i

]∗Θ

[I∇i

]= 0

[I∇i

]∗ [Θ11 Θ12

Θ21 Θ22

] [I∇i

]> 0 (7.19)

An additional condition that Θ22 < 0 must be added to make (7.19)convex. If the matrixes are evaluated this term is quadratic in ∇ as in(7.20).

Θ11 + Θ12∇i + ∇∗i Θ21 + ∇∗

i Θ22∇i. (7.20)Alternatively, conditions that the blocks in the diagonal of Θ22 whichcorresponds to each ∆ should be negative definite could be added. Thiscondition is less conservative.

Example 7.2This shows how the conditions on Θ for the vertices are implemented.The conditions for a ∇ = diag(∆1,∆2) where ‖∆i‖∞ ≤ 1 is[

I∇

]∗Θ

[I∇

]> 0

becomes four conditions.∆1 = ∆2 = 1

1 00 11 00 1

∗

Θ

1 00 11 00 1

∆1 = 1,∆2 = −1

1 00 1

1 00 −1

∗

Θ

1 00 1

1 00 −1


...

and so on. This is convex if Θ22 < 0Since ∇ is quadratic in Θ22 the alternative condition would be

∇Θ22∇ =[

∆1 00 ∆2

] [Θ221 Θ222

Θ223 Θ224

] [∆1 00 ∆2

](7.21)

and instead of Θ22 < 0 the conditions Θ221 < 0 and Θ224 < 0 shouldbe added.

Utilizing everything described above the optimization problem to besolved is summarized in (7.22)–(7.24). If a P and a Θ are found whichfulfil this then the augmented system (7.15) is stable for all allowedvariations in the parameters and the rate of parameters.

A B

C DI 0

∗ 0 0 P

0 Θ 0P 0 0

A B

C DI 0

< 0 (7.22)

[I∇i

]∗Θ

[I∇i

]≥ 0 (7.23)

Θ22 < 0 (7.24)

∇i are all the vertices of ∇. Θ and P are real symmetric matrices. Thesize of this problem grows quite fast as a ∆ ∈ Rn×n gives a ∇ ∈ R3n×3n

which results in a Θ ∈ R6n×6n. The size of (7.23) grows exponentiallywith the number of parameters. This makes this formulation limitedwhen it comes to how many parameters the LFT can have. Eventu-ally numerical problems will occur due to the size of the problem.Iwasaki & Shibata (2001) also suggests another way which can be moreconservative but only grows polynomially with the number of parame-ters.

7.2.3 Connection to Lyapunov

The formulation above can be shown to be equivalent to as solving theparameter dependent Lyapunov equation (7.25). A∆ and P∆ depend


on the matrices of the original LFT (7.14) as

0 > P∆ +AT∆P∆ + P∆A∆ (7.25)

A∆ = A+B∆(I −D∆)−1C

N∆ = (I − ∆D)−1∆CN∆ = (I − ∆D)−1∆(I −D∆)−1

P∆ =[

IN∆

]T

P0

[IN∆

]

P∆ =[

0N∆

]T

P0

[IN∆

]+

[IN∆

]T

P0

[0N∆

]

where P0 is a constant, real, symmetric matrix.

7.2.4 Alternative formulation

A more intuitive way to formulate the problem could be to start fromthe parameter dependent Lyapunov function and reformulate it to anLMI similar to (7.22). However, it will probably not be possible to uti-lize the structure of the problem as well which will result in a somewhatlarger optimization problem. The steps taken are similar to the onesdescribed by Helmersson (1999). Starting from the parameter depen-dent function (7.26) where A∆ and P∆ depends on the LFT descriptionof the system as in (7.25).

P∆ +AT∆P∆ + P∆A∆ < 0 (7.26)

(7.27)

This can be rewritten as

G(∆)∗MG(∆) < 0 (7.28)

where M contains P0 and G(∆) is an LFT description such that

(D + C∆(I − A∆)B)TM(D + C∆(I − A∆)B) =P∆ +AT

∆P∆ + P∆A∆

This can be solved using another lemma related to the Kalman-Yakobovich-Popov lemma.

Lemma 7.3 Given A ∈ Rn×k, B ∈ Rn×m,M = M∗ ∈ R(k+m)×(k+m)

and Θ ∈ R(n+k)×(n+k)such that

M +[A BI 0

]∗Θ

[A BI 0

]< 0


Then [∆(I −A∆)−1B

I

]∗M

[∆(I −A∆)−1B

I

]< 0

holds for any ∆ ∈ Rk×n such that[I∆

]∗Θ

[I∆

]≥ 0

To make sure that G(∆)∗MG(∆) is fulfilled where

G(∆) = D + C∆(I − A∆)B

it is sufficient to verify that

[C D

]∗M

[C D

]+

[A BI 0

]∗Θ

[A BI 0

]< 0

This can be reformulated into C D

A BI 0

∗ M 0 0

0 Θ11 Θ12

0 Θ∗12 Θ22

C D

A BI 0

< 0 (7.29)

under the condition that[I

∆

]∗Θ

[I

∆

]≥ 0 (7.30)

Θ22 < 0 (7.31)

which is very similar to (7.22). The size of the optimization variable Θwill depend on how clever the LFT of (7.28) is formulated but ∆ willat least be diag(∆,∆, ∆) which results in a problem of the same sizeas the implicit approach.

7.3 Difference from µ-analysis

The Lyapunov based method has two important properties which differfrom the µ-analysis earlier described.

• Since demands on the parameter rates are included in the analysisthe allowed parameter interval, for which stability for the modelcan be guaranteed, will probably decrease. A larger parameterinterval will result in a smaller allowed parameter variation rateas in figure 7.2. The allowed intervals are within the boxes.

7.3. Difference from µ-analysis 57

0

0

∆

∆

Figure 7.2. Allowed ∆ and ∆ intervals.

• The envelope can not be divided into parts and cleared indepen-dently like in µ-analysis (figure 7.3(a)). Even if the models whichare valid for different parts of the envelope pass the analysis forthe same parameter intervals, |∆| ≤ γ and |∆| ≤ ρ, there is nocertainty that the aircraft will remain stable when moving fromone part to another. This is since the derivative of the parametershas to be continuous on the boundary of the models. By doingmodels which overlap (figure 7.3(b)) this problem could possiblybe solved but that is yet to be proven.

alt

M

(a) Envelope divided

alt

M

(b) Overlapped mod-els

Figure 7.3. Models of envelope.

Chapter 8

The VEGAS Model and theGeneric Aerodata Model

The nonlinear simulation model used to make the LFT-model and forsimulations, is VEGAS (Versatile Engineering Aircraft Simulator). VE-GAS is developed in the SYSTEMBUILD environment in MATRIXX(figure8.1). The aerodynamics integrated in VEGAS come from the GenericAerodata Model developed at SAAB AB.

2-SEP-103

Continuous SuperBlockvegasus

Inputs18

Outputs23

26CODE

USER

aero

5aero

2~91427333

10CODE

USER

engine

4engine

1 PLA_PILOT

3

16CODE

USER

incidence

2incidence

10:189

6

3

3

3

19CODE

USER

rigid_body

3rigid_body

76103

9

6

-0.01se

Delay

15lowpass

6

gY = 9.80299

TURBOY = 1.5

97

rxp85

ryp85

rzp85

Y1 = 0Y2 = 0Y3 = 0

95

44CODE

USER

integrator_rgb

6integrator_rgb

7

6

7

IFPROC_USR

X_cg_USR

Z_cg_USR

Y1 = 60

Y2 = 0

Y3 = 0

8

11CODE

USER

inertia

9inertia

3

7CODE

USER

atmos

7atmos

DT_ISAY = 0

10

19:22

23

1:18

Figure 8.1. VEGAS in SYSTEMBUILD.

59

60 Chapter 8. The VEGAS Model and the GAM

8.1 VEGAS

The aircraft modelled in VEGAS is a delta winged fighter/attack air-craft which in the subsonic region is unstable in pitch. The dynamicsof the aircraft are described in GAM (Generic Aerodata Model) usingthe nonlinear equations of motion for dynamics and look-up tables foraerodynamics. The model is valid for the envelope in table 8.1. Moreinformation is available in Backstrom (1997).

Mach 0 → 2.5α −10→ 30

Altitude 0 →15000 m

Table 8.1. Envelope for GAM.

The maximum possible deflection for control surfaces can be seen intable 8.2 and the positions of the surfaces are shown in figure 8.2

Control surface DeflectionCanards δc −55→ 25

Elevons δe −25→ 30

Rudder −25→ 25

Leading edge flaps −10→ 30

Airbrakes max 55

Table 8.2: Maximum allowable deflection for VEGAS control surfaces.

canard δc

elevon δeleading edge flap

rudder

Figure 8.2. Control surfaces on VEGAS.

8.2. Controller 61

8.2 Controller

The controller used in the analysis is a research controller designed atSAAB AB (Stahl-Gunnarsson & Jacobsen 2001) using an LMI basedsynthesis technique. The model used for controller design had a flightenvelope with Mach 0.2–0.8 and an altitude of 1000 m. The controllerwill therefore perform best in this area. The model used includes onlythe short period dynamics. The design technique used gives a controllerdescribed as an LFT and that is way it was chosen for the analysisin this thesis. The LFT has Mach number as a nine times repeatedparameter. The controller controls angle of attack α and pitch rateq from the reference and actuates the elevons δe and the canards δc(figure 8.3).

refαq

δeδcController

∆

Figure 8.3. LFT controller used for analysis.

Part III

Results

63

Chapter 9

µ-calculation comparison

To realize how to perform µ-analysis in the best way and how to utilizethe algorithms fully an evaluation of the algorithms described in chapter6 was necessary. The algorithms that calculate µ on a grid of frequencypoints will probably not find the peak value if there is a narrow peakon the µ curve which corresponds to an unstable mode. The algorithmsevaluated are seen in table 9.1.

Algorithm Bound Freq. Notationmu Upper & lower Grid mu

Finite Frequency Method Upper Included FFMBasic Optimization Approach Lower Grid BOA

Pole Placement Approach Lower Implicit PPA

Table 9.1. µ-calculation algorithms.

9.1 Calculation

To evaluate the algorithms a T&B model of the complete longitudinaldynamics of VEGAS interconnected with the LFT controller describedin chapter 8 is analysed. This model will have a very narrow peakcorresponding to the uncontrolled phugoid mode and a smaller peakcorresponding to the controlled short-period mode (9.1). The two non-grid algorithms PPA and FFM generate good values close to the narrowpeak as shown in figure 9.3. For the grid based algorithms a grid of 200logarithmically spaced points was used at first. This grid was then mademore dense around the peaks which improved the results as shown infigure 9.2. The mu algorithm fails to converge when all the parametersare real and therefore the system is augmented to get a small imaginary

65

66 Chapter 9. µ-calculation comparison

part. This doesn’t give as good results as the PPA and will not be usedin later analysis.

10−2

10−1

100

101

102

0

2

4

6

8

10

12

14

ω [rad/s]

µ

mu upper boundBOA lower boundFFM upper boundPPA lower boundmu aug lower bound

Figure 9.1. Comparison of µ-calculations.

10−1

8

9

10

11

12

13

14

ω [rad/s]

µ

mu upper bound refined gridmu upper bound 200 pointsFFM upper bound

Figure 9.2. Upper µ with different grids.

10−1

11

11.5

12

12.5

13

13.5

14

14.5

15

ω [rad/s]

µ

mu upper boundBOA lower boundFFM upper boundPPA lower boundmu aug lower bound

Figure 9.3. Peak zoomed, all algorithms used.

9.2. Evaluation 67

9.2 Evaluation

mu

mu in the µ-Synthesis and Analysis toolbox in MATLAB is robust andwill work for problems with many parameters. It has a few differentapproaches for calculating the bounds implemented. The algorithm hastwo major drawbacks.

• For purely real problems the upper bound on µ can be conserva-tive and the lower bound algorithm doesn’t converge.

• Since µ is calculated on a frequency grid, narrow peaks can bemissed and in general the peak is missed. This result can be im-proved by first calculating on a sparse grid and then refining thegrid in the interesting area.

This algorithm is used to get a first estimation of interesting frequencyregions and peaks.

Finite Frequency Method

FFM uses the above algorithm but the frequency is included as anadditional uncertainty to find the peak value. The implementation inthis thesis starts the algorithm from a point close to the peak e.g. anestimation found from the mu algorithm. Then the first two intervalsare a quarter of a decade around the starting point. This will reduce theconservatism that would be introduced if large frequency intervals wereused. Doing this gives results very close to the peak on all problemstried. This algorithm is used to refine the peak value produced from agrid based algorithm such as mu.

Basic Optimization Approach

BOA solves the optimization problem described in section 6.2.2 forevery frequency point on a grid. A number of restarts are done to avoidfinding local minima. Gives quite good results but takes relatively longtime.

Pole Placement Approach

PPA gives very good results fast, but sometimes needs a few restarts tocome right. PPA can find a local minimum so if the peak value founddiffers significantly from the upper value found from mu, the calculationshould be restarted until the lower µ is somewhere near the mu result.Different starting seeds can be used for the initial value of the ∆-vector.

68 Chapter 9. µ-calculation comparison

• If the peak is clear the signtest starting seed finds the spot quitefast. If several restarts are necessary this seed should not be usedsince it is almost the same every time calculated.

• If a number of restarts are necessary the random starting seed isrecommended. This seed is randomly generated every time.

PPA is used to find a lower µ peak value and frequency. It also generatesthe parameter combination corresponding to this µ-value.

9.2.1 Workflow for µ-calculation

This is the workflow recommended for finding upper and lower boundson µ and worst case ∆

1. Calculate upper µ with the mu algorithm. This gives a good pic-ture of what the µ curve looks like. Use this to check that theother algorithms give the right results.

2. Use FFM, start with a frequency around the peak found in step1 to get the upper peak value.

3. Run PPA to find the lower µ peak and worst case ∆. If it doesn’tcome near the expected place repeat using random starting seed.Remember that the mu real upper bound can be conservative sothe frequency where the peak occurs can differ a bit. It shouldhowever still be in the neighbourhood of this value.

4. If a lower bound on the whole frequency axis is needed, run BOA.

Chapter 10

LFT-generation and valida-tion

In this chapter LFT-descriptions of the VEGAS model are generatedusing the linear Trends & Bands approach and rational approximations.These LFTs are validated against linearization points and comparedwith each other.

10.1 Generation of LFTs

Introducing Uncertainties

The nonlinear simulation model of VEGAS in Systembuild is modifiedso that parameter variations of Xcg and CMα

can be controlled. Theseparameters have great influence on the stability of an aircraft as de-scribed in section 3.4. These two parameters have great influence onthe stability. The uncertainty intervals for ±10% change in CMα

canbe seen in figure 10.1. As expected CMα

> 0 since the Mach interval issubsonic.

0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.01

0.02

0.03

0.04

0.05

0.06

CM

,α

Mach

Figure 10.1. CMαas function of Mach number.

69

70 Chapter 10. LFT-generation and validation

Since the pitch moment coefficient should be as small as possible andnegative for static stability the worst case for this model will be at Mach0.2 with a positive 10% δCMα

. A ±0.15m uncertainty in centre of gravityis also introduced which would give a worst case for δXcg

= +0.15m asdiscussed in section 3.4.1.

Modification and linearization of Simulation Model

VEGAS is trimmed and linearized around the trim points. The intervalsfor linearization are given in table 10.1. The points are linearly spacedin the intervals, giving a total of 819 linearization points.

Parameter Interval PointsM 0.2–0.8 13δXcg

-0.15–0.15m 7δCMα

-10%–10% 9

Table 10.1. Parameter intervals for linearization.

Adaptation of surfaces to linearization data

Coefficients for the LPV system in the short period approximation(10.1) for the Trends & Bands respectively the Approximation withRational Functions approach are generated by min-max optimizationwith parameter data normalized to [−1 , 1] so the coefficients can becompared. The rational functions and coefficients used are available inappendix A.

[αq

]=

[Zα(M, δXcg

, δCMα) 1

Mα(M, δXcg, δCMα

) Mq(M, δXcg, δCMα

)

] [αq

]+

+[Zδc

(M, δXcg, δCMα

) Zδe(M, δXcg

, δCMα)

Mδc(M, δXcg

, δCMα) Mδe

(M, δXcg, δCMα

)

] [δcδe

]

(10.1)

Making the LFTs

The LFTs are put together using the Onera toolbox.

T&B is polynomial so symtreed is used for clever realization andminlfr for model reduction by n-D composition. This reducesthe number of parameters from 31 to 13.

Rational function uses minlfr for model reduction. The parametersare reduced from 31 to 19. The numbers of parameters for both

10.1. Generation of LFTs 71

approximations are seen in table 10.2. The rational approxima-tion has more parameters than the T&B but is a much betterapproximation as will be shown in section 10.2.

Par. Rat.LFT T&BM 9 2δCMα

1 2δXcg

3 2δi 6 7

Sum 19 13

Table 10.2. Number of parameters for VEGAS LFTs.

The LFTs by Onera have the parameters inputs and outputs in the Amatrix. It is repartioned to a traditional state space description withthe parameters as regular inputs in the B matrix (10.2) more suitablefor MATLAB. xzy

=

A0 A1

A2 A3

B0

B1

C0 C1 D

xwu

→

A0 A1 B0

A2

C0

A3 B1

C1 D

xwu

(10.2)

Interconnection

Finally the LFTs are interconnected with the LFT-controller in SIMULINK

as in figure 10.2 to get a LFT description of the whole system. This

1

[delta,de,dc]

delta1

[de,dc]

delta

[alfa,q]

Vegas

em

r

[alfa,q]

mach

[de,dc]

Mach

Controller

1

[delta, r]

Figure 10.2. SIMULINK interconnection.

LFT will cover a line in the flight envelope like in figure 10.3. Thecontroller has Mach number as a nine times repeated parameter whichadds on to the parameters in table 10.2.


0.2 0.8

1000m

alt

M

Figure 10.3. Envelope covered by model.

10.2 Validation

A simulation when the pitch command is doubled is seen in figure 10.4.There are quite big differences especially in the elevon angle. This is

0 5 10 15 20 25−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Time [s]

δ e [rad

]

Lin. pointLFT rat.LFT T&B

0 5 10 15 20 25−2.5

−2

−1.5

−1

−0.5

0

0.5

Time [s]

δ c [rad

]


0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time [s]

α [r

ad]


0 5 10 15 20 25−0.5

0

0.5

1

1.5

Time [s]

q [r

ad/s

]


Figure 10.4: Comparison of LFT:s and linearization point in nominalcase.

because the compensation parameters added for differences in the ap-proximations are all set to zero. If they are adjusted, the real lineariza-

10.3. µ-sensitivities 73

tion point can be found as seen in figure 10.5. This difference from the

0 5 10 15 20 25−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Time [s]

δ e [rad

]


0 5 10 15 20 25−2.5

−2

−1.5

−1

−0.5

0

0.5

Time [s]

δ c [rad

]


0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time [s]

α [r

ad]


0 5 10 15 20 25−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time [s]

q [r

ad/s

]Lin. pointLFT rat.LFT T&B

Figure 10.5: Comparison of LFTs with adjusted compensation param-eters.

real point can result in a larger µ-value. The real point is covered by themodel but results that do not occur in the real physical plant can occurin the model making the analysis fail. As the compensation parametersget smaller the more accurate the LFT will be.

10.3 µ-sensitivities

By using µ-sensitivities as discussed in section 4.5.3, some conclusionsabout the model can be drawn.

10.3.1 Trends & Bands

By utilizing µ-sensitivities two improvements can be done on the T&Bmodel.

• Looking at the µ-sensitivities in figure 10.6(a) it seems like δCMα

and δerr,2, δerr,4, δerr,6, δerr,8 can be removed. The difference inthe µ-bound when this is done is shown in figure 10.6(b). Theupper µ-bound is almost the same although five parameters havebeen removed.


• Since µ > 1 this model fails the stability analysis. Both modelsdescribe the same system but this one is conservative comparedwith the rational function LFT. The conservatism comes fromthe bands which describes more than the true physical model asdiscussed in section 4.3.2. This conservatism can be reduced bymaking a model for a smaller interval of M . Using M ∈ [0.2, 0.5]yields a lower value on µ (figure 10.6(b)) and a lower ∂µ for machnumber (figure 10.6(a)).

0

0.2

0.4

0.6

0.8

1

∂ µ

M δC

Mα

δX

cg

δerr,1

δerr,2

δerr,3

δerr,4

δerr,5

δerr,6

δerr,7

(a) ∂µ on T&B LFT with full andreduced mach interval.

10−2

10−1

100

101

102

0

0.2

0.4

0.6

0.8

1

1.2

1.4

ω [rad/s]

µ

µ with full δµ with reduced δµ with M∈[0.2−0.5]

(b) µ on T&B LFT with full ∆, re-duced ∆ and reduced mach interval.

Figure 10.6. Trends & Bands LFT results.

10.3.2 Rational LFT

According to figure 10.7(a) the parameter with most influence is Machnumber. This is expected since in the rational estimation this is themost represented parameter. The sensitivity for CMα

is very small andremoving this parameter doesn’t reduce the µ-value significantly asshown in figure 10.7(b). This would not make the model considerablymore optimistic.

10.4. Summary 75

0

0.5

1

1.5

2

2.5

∂ µ

M δC

Mα

δX

cg

δerr,1

δerr,2

δerr,3

δerr,4

δerr,5

δerr,6

(a) ∂µ for rational LFT.

10−2

10−1

100

101

102

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

ω [rad/s]

µ

µ with CM,α

µ without CM,α

(b) µ with and without CMα .

Figure 10.7. Rational LFT results.

10.4 Summary

The LFTs generated have different degrees of conservatism dependingon how well the surfaces are approximated. The Trends & Bands LFTgives a larger µ value than the rational approximation which makes itmore conservative, but it can be automatically generated and has fewerparameters. By dividing e.g. the Mach interval into two smaller intervalsand making two LFT descriptions the result can be improved. Therational function approximation gives much better simulation resultsand a lower µ-value but has more parameters. Using µ-sensitivities, theparameters with low influence can be identified and removed from bothmodels without changing the resulting µ-value significantly.

Chapter 11

Clearance of VEGAS

To clear VEGAS and the controller, the clearance criteria from sections5.4–5.5 are applied to the rational function LFT generated in chapter10. A logarithmic frequency grid of 200 points from 10−2 − 102 is usedfor the grid based algorithms. The Lyapunov based stability criteriondescribed in chapter 7 is also used.

11.1 Stability

To verify stability, µ-analysis is performed on the LFTs. Nothing hasto be added. The results are shown in figure 11.1. Both LFTs have a

10−2

10−1

100

101

102

0

0.2

0.4

0.6

0.8

1

1.2

ω [rad/s]

µ

BOA lower boundPPA lower boundFFM upper boundµ−toolbox upper boundµ−toolbox upper on T&BPPA lower on T&BFFM upper on T&B

Figure 11.1. Stability of Vegas.

peak value around 4 rad/s corresponding to the short period mode. All

77

78 Chapter 11. Clearance of VEGAS

algorithms give a µ value below one for the rational LFT so this modelis stable for the allowed parameter variations. The bounds are quitetight as desired. The upper and lower bound occur at slightly differentfrequencies. As mentioned in section 6.1, this can happen since thescaled upper bound is not the true value. The T&B LFT has an upperµ-value over one for all frequencies and can not be verified to be stable.

11.2 Phase and gain margins

To test the stability margin criteria, the exclusion region uncertaintymust be added to the LFT as described in section 5.4. The inner ex-clusion region is first added to the canard loop and then to both loopsusing SIMULINK. This also gives one respectively two extra complex un-certainties that must be added to the ∆-block. The results are seen infigure 11.2.

10−2

10−1

100

101

102

0

0.2

0.4

0.6

0.8

1

µ−tools UB nominal systemµ−tools UB excl.reg. δ

e

µ−tools UB excl.regFFM UB nominalFFM UB excl. reg. δ

eFFM UB excl. reg

Figure 11.2: Upper µ for nominal, Nichols exclusion region in canardloop and both loops.

As expected the µ value increases first when adding the criteria to oneloop and even more when adding it to both loops but it still doesn’tcross the critical line of µ=1.

11.2.1 Stability margins

WN , in the exclusion region added to the canard loop, is increased to1.17 which results in a µ-value of 1.0182 from the PPA. This results in

11.2. Phase and gain margins 79

a stability margin of

ρ =WN,scale

WN=

1.170.47

= 2.49

11.2.2 Worst case parameter combination

The worst case parameter result from the PPA when the region wasscaled as described above is seen in table 11.1. This is the same worst

Parameter ValueM 0.2δXcg

+0.15mδCMα

10%

Table 11.1. Worst case parameter combination.

case as the one suggested in section 10.1. The slowest Mach number,Xcg

moved as far aft as possible and CMαincreased as much as possible is

making the aircraft more unstable. The Nichols curves for the nominal

−180 −135 −90 −45 0−20

−15

−10

−5

0

5

10

15

20Worst caseNominal caseExclusion regionScaled Exclusion Region

Nichols Chart

Open−Loop Phase (deg)

Ope

n−Lo

op G

ain

(dB

)

Figure 11.3. Nichols plot of nominal and worst case system.

system and worst case system calculated by scaling WN are seen infigure 11.3. The region does not touch the Nichols plot of the worstcase system. This difference will also introduce conservatism in theanalysis. However in this case the scaling difference is only 6 %.


0 5 10 15 20 25−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time [s]

δ e [rad

]

δe worst case

δe nominal

0 5 10 15 20 25−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Time [s]

δ c [rad

]

δc worst case

δc nominal

0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time [s]

α [r

ad]

α worst caseα nominal

0 5 10 15 20 25−1

−0.5

0

0.5

1

1.5

Time [s]

q [r

ad/s

]

q worst caseq nominal

Figure 11.4: Comparison of signals, nominal and worst case Mach 0.2.

This worst case is compared in simulation with the linear flight caseMach 0.2 in figure 11.4. Finally the migration of the poles correspondingto the short-period mode from 0–100% of the worst case parametercombination is shown in figure 11.5.

−3.3 −3.25 −3.2 −3.15 −3.1 −3.05 −3 −2.95 −2.9 −2.85 −2.8−4

−3

−2

−1

0

1

2

3

4

Real axis

Imag

inar

y ax

is

Figure 11.5. Migration of short period poles.

11.3. Most unstable poles 81

11.3 Most unstable poles

The µ-value of the rational function LFT is calculated along the polespecification lines described in section 5.5. The peak value is less thanone so none of the eigenvalues will pass these lines for the allowed pa-rameter intervals (figure 11.6). Note that the x-axis not is frequencybut sample of the pole specification line. To find the worst case un-

0 20 40 60 80 100 120 140 1600.5

0.55

0.6

0.65

0.7

0.75

0.8

sample

µ

section1section2section3section4

sect

ion

5

Figure 11.6. µ for eigenvalue specifications.

stable eigenvalue the T&B LFT, which is unstable for some parametercombinations, is used. When the lines are moved, as described in sec-tion 5.5.1, it is hard to get the lower algorithms to find a value. Insteadthe imaginary axis is moved right into the RHP, using the PPA, until aµ-value of one is found. The change in µ and the worst case poles whenthis is done are shown in figure 11.7.

1 1.5 2 2.5 3 3.5 40.99

0.995

1

1.005

1.01

1.015

1.02

1.025

1.03

Imaginary axis offset

µ

(a) Change in µ when offset.

−20 −15 −10 −5 0 5−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

y A

xis

Real axis

(b) Worst case poles for VEGAS.

Figure 11.7. Worst case poles.


11.4 Bounds on Parameter Rates

The Lyapunov based approach is used to find bounds on the parameterrates. First the analysis is tried on the rational LFT with Mach num-ber as only parameter but numerical problems occur and no feasiblesolution is found. Instead the T&B LFT with reduced Mach intervaland all other parameters set to zero is used. This gives a ∆-block withMach number M as an 11 times repeated real parameter, which yieldsΘ ∈ R66×66. The A,B,C and D matrices for the LFT are used inA, B, C and D to get the augumented LFT as described in section7.2.2. This results in the LMI

24 A B

C DI 0

35

∗ 24 0 0 P

0 Θ 0P 0 0

35

24 A B

C DI 0

35 < 0

»I∇i

–∗Θ

»I∇i

–≥ 0

Θ22 < 0

These conditions are implemented using the LMI solver interface YALMIP

(Lofberg 2003) and the solver SDPT3 (Tutuncu et al. 2001). A feasiblesolution is found for a normalized ‖∆‖∞ = ‖∆‖∞ ≤ 1 which corre-sponds to a Mach number variation 0.2 ≤M ≤ 0.5 and an accelerationof |M | ≤ 0.5. This is not physically possible so this model can handleall allowed accelerations. By scaling matrices for better numerical prop-erties problems with more parameters varying can probably be solved.Scaling may also increase the allowed parameter variation. Using aspecial solver for semidefinite programs originating from the Kalman-Yakobovich-Popov lemma (Wallin 2003) could also improve the results.

11.5 Summary

The VEGAS model with the LFT controller is successfully cleared for aline in the flight envelope with variations in Xcg and CMα

. The criteriaused are stability, phase and gain margins and worst case unstableeigenvalue. The parameter combination which gives the least margin isfound and it is the same as expected from a flight mechanical point ofview. The stability margin analysis introduces some conservatism, inthis case only a 6 % difference. µ-analysis seems to work well for theanalysis described in this chapter. The Lyapunov based method raninto numerical problems when applied to the full model consisting of28 parameters but a feasible solution for a reduced T&B model wasfound. So stability for a smaller line in the envelope and for a limitedacceleration can be guaranteed. This result can be improved by properscaling of the numerical data used and by using other solvers.

Chapter 12

Conclusions and future work

Here the thesis is summarized, conclusions are drawn and the areaswhere more work should be done are presented.

12.1 Summary

• To perform µ-analysis on a system with uncertainties it must beon the LFT form. An uncertain system was created by includ-ing uncertainties in two aircraft parameters of VEGAS, Xcg andCMα

. The nonlinear simulation model was trimmed and linearizedaround the trim points for variations in Mach number which madeMach the third parameter.

• Surfaces described by rational functions and linear trend andbands approximations were adapted to the data from the lin-earization using min-max optimization. These functions describeda line in the flight envelope. The functions were converted to LFTsand reduced using the Onera toolbox. This reduced the number ofparameters with roughly 1/3. Further reduction and conclusionsabout the parameters influence on the model could be drawn us-ing µ-sensitivities. In the T&B model five parameters could beremoved without changing the resulting µ-value significantly. Bydividing the α interval in two parts and making a model for eachpart the conservatism could be reduced. The rational approxima-tion could be reduced with one parameter.

• The LFTs were verified against linearization points. Though theydidn’t perform exactly the same in the nominal case the true be-haviour was there when the compensation parameters were ad-justed. The compensation parameters introduced most of the con-servatism in the analysis.

83

84 Chapter 12. Conclusions and future work

• Stability analysis was performed using µ-analysis. Reliable µ-calculation algorithms were needed and four algorithms were com-pared. The FFM gave a good peak upper value and the PPA agood lower peak together with the parameters associated withthe peak. The PPA sometimes had to be restarted when a localminimum was found. The area were the peak should occur wasestimated with the grid based upper mu command. The optimiza-tion based grid lower bound BOA also was quite successful buttook a long time to compute.

• Using these calculation tools, criteria such as phase and gain mar-gins and worst case unstable eigenvalue were included in the anal-ysis. VEGAS was cleared for a line in the flight envelope and theallowed parameter combination which gave the worst performancewas found. The combination found was the same as the one ex-pected from a flight mechanical point of view.

• Finally a Lyapunov based approach where bounds on the rateof the parameters could be included in the analysis and solvedusing LMIs was tried. This method wasn’t as intuitive as theearlier and gave numerical problems already for a small numberof parameters, scaling may, however, improve the performance.A Lyapunov function for Mach number 0.2 ≤ M ≤ 0.5 and anacceleration M ≤ 0.5 for a reduced T&B LFT was found whichguaranteed stability for these parameter variations.

12.2 Conclusions

First of all the difference between µ-analysis and the Lyapunov basedapproach should be emphasized. µ analysis finds the worst static pa-rameter combination. This is essentially the same as trying all combi-nations and finding the one that gives the worst performance or makesthe system unstable. With the Lyapunov based approach a limit on

δworst

δmax

δmin

tt

Figure 12.1. Difference in parameters.

12.2. Conclusions 85

how fast the parameters can change is included. This will be the sameas saying, for the example tried, that if the velocity is within an allowedrange and the plane has a limit on the acceleration, the system will bestable in the sense that it will stay close to the trimmed condition.The allowed parameter intervals will probably decrease compared withµ-analysis since the parameter rates are included in the analysis.

12.2.1 µ-analysis

Using µ-analysis to verify linear stability criteria and find worst casestatic parameter combinations can be done without great effort. Thetheory involved is not too hard to overcome for a person with basiccontrol theory knowledge. The algorithms are quite robust and fast.However, the system must be described as an LFT and much of theconservatism introduced seems to come from the LFTs generated. Thisis e.g. seen in the comparison of the T&B and rational function ap-proximation. The LFT generation is probably where most of the effortmust be put.

+ All allowed static parameter variations are checked.

+ Clearance criteria can be included in the analysis.

+ Not too complicated theory.

- The system must be described as an LFT.

12.2.2 Parameter dependent Lyapunov functions

The Lyapunov based method is interesting since limits on the parame-ter rates can be found. The allowed parameter interval will be reducedwhen large parameter variations are allowed. However more researchhas to be performed to develop algorithms that can handle large prob-lems. The results can probably be improved by rewriting the problemin a more efficient way and scale matrices to reduce numerical prob-lems. Using solvers developed for this kind of problems would probablyalso improve the results.

+ Bounds on the parameter rates are included in the analysis.

+ If the difference from the real nonlinear system is known it canbe included in the Lyapunov function reducing the LFT size.

+ Can be formulated as a convex optimization problem and thus besolved quite efficiently.

- Quite complicated theory

86 Chapter 12. Conclusions and future work

- Becomes a large optimization problem for a small amount of pa-rameters.

- Numerical problems occur if the matrices are not properly scaled.

12.3 Future work

To proceed with this type of analysis further work has to be done. Someof the questions to be answered are:

• How is an LFT made in the best way? Numerical approximationsor analytical calculation? Should the models be done for aircraftand controller separately or with the parts interconnected? Themost desirable is of course to have separate LFTs for each partso they can be interchanged independently.

• The flight control system is digital with a sampling period of e.g.60 Hz. How should the models be translated? Should the analysisbe done in the s or z plane? The algorithms could quite easily bemodified so the unit circle is the critical line.

• How large problems can be solved using these methods? The mucommand is known to handle large problems and probably theFFM which uses the same algorithm. The optimization basedlower bounds PPA and BOA will eventually run into problems.These are not the only algorithms available though.

• Many specifications are given in the time domain. Including thesein the analysis is desirable. Research is currently taking place inthis area.

References

Backstrom, H. (1997), Report on the usage of the Generic AerodataModel, 2 edn, SAAB AB.

Balas, G., Doyle, J., Glover, K., Packard, A. & Smith, R. (1993), µ-Analysis and Synthesis Toolbox, User’s Guide, MathWorks Inc.

Bates, D. G., Kureemun, R. & Postlethwaite, I. (2001), ‘Quantifying therobustness of flight control systems using Nichols exclusion regionsand the structured singular value’, IMechE Journal of Systems andControl Engineering 215(16), 625–638.

Braatz, R. D. & Morari, M. (1991), µ-sensitvities as an aid for robustidentification, in ‘Proceedings of the American Control Conference’,pp. 231–236.

Etkin, B. (1972), Dynamics of Atmospheric Flight,John Wiley & Sons, Inc. ISBN 0-471-24620-4.

Fielding, C., Varga, A., Bennani, S. & Selier, M. (2002), Advanced Tech-niques for Clearance of Flight Control Laws, Springer. ISBN 3-540-44054-2.

Glad, T. & Ljung, L. (1997), Reglerteori, Flervariabla och olinjarametoder, Studentlitteratur. ISBN 91-44-00472-9.

Hansson, A., Helmersson, A. & Glad, T. (2003), Stability analysisof nonlinear systems using frozen stationary linearization, Techni-cal Report LiTH-ISY-R-2512, Department of Electrical Engineering,Linkopings Universitet, Sweden.

Hayes, M. J., Bates, D. G. & Postlethwaite, I. (2002), ‘New tools forcomputing tight bounds on the real structured singular value’, Jour-nal Of Guidance Control and Dynamics 24(6), 1204–1213.

Helmersson, A. (1995), A finite frequency method for µ-analysis, in‘Proceedings of the 3th European Control Conference’, Rome, Italy,pp. 171–176.

87

88 References

Helmersson, A. (1999), Parameter dependent Lyapunov functions basedon linear fractional transformation, in ‘14th World Congress ofIFAC’.

Helmersson, A. (2003), ‘Short notes on linear matrix inequalities’, In-ternet. www.control.isy.liu.se/~andersh/ecsel/lmi.ps.

Iordanov, P., Hayes, M. & Halton, M. (2003), Non-conservative real-µ-analysis and synthesis for a civil transport aircraft, Technical report,Department of Electronic & Computer Engineering, University ofLimerick, Ireland.

Iwasaki, T. & Shibata, G. (2001), ‘LPV system analysis via quadraticseparator for uncertain implicit systems’, IEEE Transactions on Au-tomatic Control 46(8), 1195–1208.

Kureemun, R., Bates, D. & Postlethwaite, I. (2001), Clearance of theHWEM control laws: A µ-analysis approach, Technical Report 119-28, GARTEUR.

Lofberg, J. (2003), YALMIP, Yet another LMI parser, Linkopings Uni-versitet. www.control.isy.liu.se/~johanl/yalmip.html.

Magni, J. (2001), Linear fractional representations with a toolbox foruse with MATLAB, Technical Report TR 240/01 DSCD, Departmentof Systems Control and Flight Dynamics, Toulouse Center, France.

Mannchen, T., Bates, D. G. & Postlethwaite, I. (2002), ‘Modelingand computing worst-case uncertainty combinations for flight con-trol system analysis’, Journal of Guidance, Control and Dynamics25(6), 1029–1039.

Mannchen, T., Klett, Y., Pertermann, C., Weinert, B. & Zobelein, T.(2001), Flight control law clearance of the hirm+ fighter aircraftmodel using µ-analysis, Technical Report 119-12-v2, GARTEUR.

Stahl-Gunnarsson, K. & Jacobsen, J.-O. (2001), Design and simula-tion of a parameter varying controller for a fighter aircraft, in ‘AIAAGuidance, Navigation and Control Conference and Exhibit’, Mon-treal, Canada.

Tutuncu, R. H., Toh, K. C. & Todd, M. J. (2001), SDPT3– a Matlabsoftware package for semidefinite-quadratic-linear programming.

Wallin, R. (2003), User’s guide to kypd-solver, Technical Report LiTH-ISY-R-2517, Department of Electrical Engineering, Linkoping Uni-versity, SE-581 83 Linkoping, Sweden.

Zhou, K., Doyle, J. & Glover, K. (1996), Robust and Optimal Control,Prentice-Hall. ISBN 0-13-456567-3.

www.control.isy.liu.se/~andersh/ecsel/lmi.ps

Notation

Symbols and acronyms used in the thesis.

Symbols

A,B,C,D state-space matricesCMα

pitch moment derivative, due to angle of attack αFl(M,∆) Lower Linear Fractional TransformationFu(M,∆) Upper Linear Fractional Transformationg Gravitational constantM Machq Aerodynamic pressurep, q, r roll rate, pitch rate, yaw rateS wing areau, v, w velocities in x, y and zX,Y,Z forces in x, y and zVx velocity in x-direction of body fixed coordinate frameVz velocity in z-direction of body fixed coordinate frameα, β angle of attack, sideslip angleδC canard angleδE elevon angleδXcg

uncertainty in center of gravity positionδCMα

uncertainty in pitch moment coefficient depending on α∆ uncertainty blockγ flight-path angleω frequencyξ damping coefficientµ the structured singular value∂µ µ-sensitivityσ largest singular valueφ, θ, ψ roll angle, pitch angle, yaw angleXcg center of gravity position along the x-axis

89

90 Notation

Acronyms

a.c. aerodynamic centreBOA Basic Optimization Approachc.g. centre of gravityGAM Generic Aerodata MaterialFFM Finite Frequency MethodGARTEUR Group for Aeronautical Research and Technology in EURopeFCS Flight Control SystemLFT Linear Fractional TransformationLHP Left Half PlaneRHP Right Half PlaneLMI Linear Matrix InequalityLTI Linear Time InvariantLPV Linear Parameter VaryingPPA Pole Placement ApproachRHP Right Half PlaneSSV Structured Singular ValueVEGAS Versatile Engineering Aircraft Simulator

Operators

A∗ Complex conjugate transposeAT Transpose∠ Argument< Real part= Imaginary partλ() Eigenvaluessup Supremuminf InfimumA > 0 A < 0 When A is a matrix this means positive respectively

negative definite.A ≤ 0 A ≥ 0 When A is a matrix this means positive respectively

negative semidefinite.

Appendix A

LFT data

Here the rational functions used for building the LFT are listed.

Zα =a0 + a1M

1 + a2M± d1

Zq = 1

Mα =a3 + a4M + a5δXcg

+ a6δCMα+ a7MδXcg

+ a8δCMαM + a9M

2

1 + a10M± d2

Mq =a11 + a12M + a13δXcg

1 + a14M± d3

Zδe=

b0 + b1M

1 + b2M± d4

Zδc= b3 + b4M

Mδe=

b5 + b6M + b7δXcg+ b8MδXcg

+ b9M2

1 + b10M + b11M2± d5

Mδc= b12 + b13M + b14M

2 ± d6

91

92 Appendix A. LFT data

a0 -1.4587949e+00 b0 -5.1239857e-01a1 -6.1027825e-01 b1 -3.5452611e-01a2 -2.3856372e-01 b2 1.7317093e-01a3 5.8077387e+00 b3 -3.4543590e-02a4 -4.6015799e-01 b4 -1.6903816e-02a5 4.2254080e+00 b5 -2.2138839e+01a6 8.0926671e-01 b6 -1.7299500e+01a7 3.5509993e+00 b7 1.4453728e+00a8 2.9162197e-01 b8 8.9064345e-01a9 -4.0198117e+00 b9 -2.1273820e+00a10 -3.7113478e-01 b10 -6.1452475e-01a11 -1.1113209e+00 b11 3.1834738e-01a12 -4.8141368e-01 b12 8.1202167e+00a13 6.2459308e-02 b13 8.1265070e+00a14 -2.7713587e-01 b14 2.0227503e+00d1 9.5685955e-02 d2 9.5458234e-01d3 4.7197762e-02 d4 2.1338030e-02d5 6.2582937e-01 d6 2.8543656e-01

Table A.1. Coefficients for rational LFT

Appendix B

Matlab scripts manual

Here the Matlab functions and the scripts written for this thesis areshortly described. All functions have the regular help command imple-mented.

B.1 µ-analysis and Synthesis Toolbox

The µ-analysis and Synthesis Toolbox comes with MATLAB. Some com-mon commands from this toolbox are described here.

sys=pck(A,B,C,D)packs a state space description into a SYSTEMmatrix of the form

A B nr of statesC D 00 0 -Inf

[A,B,C,D]=unpck(sys) Takes out the matrixes from a matrix asabove.

sys=frsp(sys,omega) where omega is a vector of frequency points.sys is a description as above. This function creates a VARYINGmatrix which is the transfer function of sys in each frequencypoint stacked on each other next to the frequencies. N is thenumber of frequency points.

Gi(s) = C(jωi ∗ I −A)−1B +D

93

94 Appendix B. Matlab scripts manual

G1(jω1) ω1

G2(jω2) ω2

......

Gi(jωi) ωN

......

GN (jωN ) 00 0 N Inf

sel(mat,inputs,outputs) selects inputs and outputs or rows andcolumns from a varying or system matrix.

minfo(sys) gives information about V ARY ING and SY STEM ma-trices.

[muVal,Dscale,sens,pert,Gscale]=mu(VAR,deltaset,options) cal-culates µ over a frequency grid.

VAR is a VARYING matrix as earlier described.

deltaset is the parameters on the form [-1 0;-2 0;1 0;2 0]where -1 is real,-2 is real and repeated, 1 is a complex and2 is complex and repeated.

pert is the lower bound pertubation.

Dscale and Gscale are scaling matrixes for upper bound andsens is the sensitivity to ‖DMD−1‖2

options where various options to the solver can be passed.

B.2 Onera LFR-toolbox

This MATLAB toolbox can be downloaded fromwww.cert.fr/dcsd/idco/perso/Magni/booksandtb.html and a com-plete documentation is found in Magni (2001).

lfrs a b c makes lfr objects of a,b,c

syms a b c makes symbolic objects of a,b,c and must be used ifsymtreed should be used later on.

LFR=symtreed(poly_mat,[symbols],[nominal values]) makes a LFRobject of a polynomial matrix using tree decomposition as de-scribed in section 4.5.1.

LFR=abcd2lfr(abcd,nr. of states) makes a LFR object of abcdwhich can be e.g. the LFR from symtreed or parameter depen-dent state space matrices [A,B;C,D] described with lfrs objects.

B.3. BOA and PPA 95

minlfr(LFR) minimizes a LFR object using n-D decomposition asdescribed in section 4.5.2.

normlfr(LFR,[min_values],[max_values]) normalizes the param-eters in a LFR object.

[A,B,C,D,E]=lfrdata(LFR) takes out the matrixes from a LFR ob-ject.

B.3 BOA and PPA

These scripts were provided from Petar Iordanov at University of Lim-erick, Ireland. A graphical interface for µ-analysis and synthesis usingthese algorithms is under construction.

[Delta, mu_peak, freq, pole, error, comp_time] == mu_pp(SYS, delta_struct, foptions)

Delta worst case delta.

mu_peak peak lower µ value.

freq frequency where lower peak occur.

pole worst case pole.

comp_time time to perform computation.

SYS system as a SYSTEM matrix as described before.

delta_struct on the form [-2 1 -1 2] where -2 is real re-peated, 1 is complex single and so on.

foptions passes options to the solver about e.g. starting seeds.

Results = boa_lb(SYS,delta_struct,options) orboa_lb(M,[],options)

SYS a SY STEM matrix as described above.

delta_struct description of ∆.

Results A cell structure with all results.

M a constant matrix.

options passes options to the solver.

96 Appendix B. Matlab scripts manual

B.4 Written Scripts

To perform analysis some scripts were written especially for this thesis.

[err,koeff,plot_cell] = modellbygge(data_array,pardata,......taljare,namnare,element) fits rational functions to lineariza-tion data.

[a,b,c,d,a_err,b_err,c_err,d_err] = modellbygge_tb(A,B,C,...D,pardata) fits linear Trends & Bands functions to linearizationdata.

[mu_ub,fu_peak] = finite_mu(SYS,deltaset,freq_init,foptions)calculates peak upper value as described in section 6.1.3

SYS a SY STEM matrix.

deltaset same as for mu.

freq_init a good guess about where the peak frequency shouldoccur.

options passes options to the solver

d_mu = mu_sens(SYS,deltaset,freq_init) calculates µ-sensitivitiesusing finite_mu. Same form on inputs as for finite_mu.

mu_aug(VAR,deltaset) calculates mu augmented with a complex partas described in section 6.2.1. Same form on inputs as for mu.

[A,B,C,D] = M2abcd(Alfr,Blfr,Clfr,Dlfr,states) repartions theLFR matrix from the Onera toolbox so parameter inputs and out-puts move from Alfr to B,C and D as described in (10.2).

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using linear fractional transformations for clearance of flight control laws

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