Re-entry flight clearance

Re-entry Flight Clearance

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 12 september 2006 om

15:00 uur

door

Sinar Juliana

Prof. dr. ir. J.A. Mulder

Toegevoegd promotor: Dr. Q.P. Chu

Samenstelling promotiecommissie:

Rector Magnificus voorzitter

Prof. dr. ir. J.A. Mulder Technische Universiteit Delft, promotor Dr. Q.P. Chu Technische Universiteit Delft,

toegevoegd promotor Prof. I. Postlethwaite, Ph.D University of Leicester, UK

Prof. dr. ir. M.H. van Emden University of Victoria, BC, Canada Prof. dr. ir. P.M.J. van den Hof Technische Universiteit Delft Prof. ir. B.A.C. Ambrosius Technische Universiteit Delft Prof. ir. K.F. Wakker Technische Universiteit Delft

Prof. dr. ir. M.J.L. van Tooren Technische Universiteit Delft, reservelid

Copyright c
2006 by S. Juliana

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without the prior permission of the author.

Cover: the DART re-entry vehicle with the earth in the background (earth image source: NASA; DART image source: TU Delft). isbn-10: 90-9021013-X

isbn-13: 978-90-9021013-1

Summary

Human exploration of outer space began more than half a century ago. Since then, many space missions were launched and many more will be launched in the future. Among these space missions, a significant number included a re-entry into the earth atmosphere for the recovery of the human crew or samples from the moon, planets or from outer space. The safety of these re-entry flights is of crucial importance, in particular if they serve to bring human crew back to earth. The safety of re-entry flight is the subject of the present thesis.

Re-entry flight is one of the most challenging parts in the spaceflight missions mentioned above. Prior to actual flight, the vehicle has to be certified to fly. This certification process is known as re-entry flight clearance. Within this process, vehicle mathematical models have to be developed as realistic as possible for evaluation purposes. However, for complex re-entry flights, vehicle models cannot be formed without errors or uncertainties. Therefore, model evaluations have to be performed by including all possible uncertainties. This process is actually called re-entry flight model clearance. However, in this thesis we simply call this process as re-entry flight clearance.

The objective of this research is to identify and evaluate promising mathemat-ical techniques for re-entry flight clearance. Two vehicle models were developed and used as case studies for the evaluation of the clearance techniques. The first case study was based on DART (Delft Aerospace Re-entry Test Demonstrator), a ballistic re-entry vehicle. An attitustabilising flight control system was de-signed for this vehicle using a combination of Non-linear Dynamic Inversion (NDI) and Proportional Integral Derivative (PID) control laws. The second case study was base on SPHYNX (Subscale Precursor Hypersonic X), a lifting-body re-entry vehicle. This model was equipped with an attitude-tracking flight control system, which was designed using a gain-scheduled Linear Quadratic Regulator (LQR) control law.

factory and had provided some robustness against uncertainties, the maximum level of the introduced uncertainties for which the system still perform as re-quired, needs to be investigated. Simulations of the closed-loop systems, in which model errors were present, were performed for both DART and SPHYNX sys-tems. In the simulations for SPHYNX, errors were introduced into the location of the centre of gravity relative to the moment reference point. For DART, the errors were introduced into the magnitude of the aerodynamic moments. The simulation results showed that SPHYNX and DART control systems could not fulfil the required performance, and eventually became unstable in the cases that relatively large uncertainties were introduced. These results have shown that it is essential to have a controller that is robust against these models uncertainties, or to apply clearance techniques to analyse the robustness of the closed-loop system such that the worst-case of model uncertainties can be found. The latter has been performed in this thesis.

The clearance process for SPHYNX was performed to evaluate the model in its nominal flight trajectory, while for DART, in its flight envelope region. The stability of the system in the presence of model uncertainties was chosen as the clearance criterion. To compare the capabilities of different mathematical clearance techniques, DART and SPHYNX were modeled as both linear and non-linear systems. Two mathematical methods were investigated and developed in this research. These are µ analysis for linear models and interval analysis for both linear and non-linear models. For the clearance using µ analysis, the models with uncertain parameters were represented in Linear Fractional Representation (LFR) forms. As the uncertainties are defined explicitly, they can be ’pulled-out’ from the nominal system, and modeled as a feedback loop. By doing this, the effect of the uncertainties on the model can be analysed using µ analysis. For the clearance using interval analysis, the use of the model set with uncertainties was more straightforward, since interval analysis can directly be applied to analyse a set of models which are defined within certain intervals. Moreover, interval analysis can be applied to both linear and non-linear models, due to its capacity of guaranteed global optimum searching.

The stability of the system was evaluated using two mathematical criteria: worst-case eigenvalue analysis (linear) and the Lyapunov analysis (non-linear). µ analysis was applied to evaluate the worst-case eigenvalues of linear systems, while interval analysis was applied to evaluate the worst-case eigenvalues of linear systems and Lyapunov functions of non-linear systems. The suitability of the two mathematical techniques for re-entry flight model clearance was evaluated based on the results of the clearance processes. Non-linear simulations were also performed to verify the clearance results generated by the two techniques.

iii

Notations

List of Symbols

D drag force, N F external force, N FA aerodynamic force, N FT thrust force, N I identity matrix In inertia tensor J cost function K controller

KD derivative control gain KI integral control gain KP proportional control gain L lift force, N

M Mach number

Mb external moment, N m MA aerodynamic moment, N m

MT thrust moment, N m

R distance to earth center, m S side force, N V velocity, m s−1 V Lyapunov function e control error g gravitational acceleration, m s−2 h altitude, m m mass, kg

p roll rate, rad s−1 q pitch rate, rad s−1

¯

q dynamic pressure, N m−2 r yaw rate, rad s−1

r distance vector, m

u input variable, input vector x state variable, state vector y output variable, output vector ∆ uncertainty matrix, with elements δ Ωe earth rotational velocity, rad s−1 α angle of attack, rad

β sideslip angle, rad χ heading angle, rad δ uncertain parameter δ latitude, rad

δa aileron deflection, rad δe elevator deflection, rad λ eigenvalue

γ flight path angle, rad µ structured singular value

ν input defined by non-linear dynamic inversion ρ air density, kgm−3

σ aerodynamic roll angle, rad ¯

σ maximum singular value τ longitude, rad

vii

Abbreviations

CFD Computational Fluid Dynamics

DART Delft Aerospace Re-entry Test Demonstrator

DOF Degree of Freedom

ESA European Space Agency

FADS Flush Air Data Sensor

GESARED General Simulator for Atmospheric Re-entry Dynamics GNC Guidance, Navigation, and Control

GPS Global Positioning System INS Inertial Navigation System LFR Linear Fractional Representation LQR Linear Quadratic Regulator

LTI Linear Time Invariant

NASA National Aeronautics and Space Administration NDI Non-linear Dynamic Inversion

PD Proportional and Derivative PI Proportional and Integral

PID Proportional, Integral, and Derivative

RCS Reaction Control System

SIGI Space Integrated GPS and INS SIVIA Set Inverter via Interval Analysis

Contents

Summary

i

Notations

v

1 Introduction

1

1.1 Historical Notes . . . 1

1.1.1 Development of Re-entry Activities . . . 2

1.1.2 Spaceflight Accidents . . . 4

1.2 The Challenges of Re-entry . . . 5

1.3 Problem Statement and Objectives of the Research . . . 6

1.4 Research Scope . . . 7

1.5 Contribution of the Thesis . . . 8

1.6 Thesis Outline . . . 9

2 Flight Clearance

11 2.1 Clearance in Re-entry Flight Perspective . . . 11

2.2 Clearance Approaches . . . 12

2.3 Mathematical Clearance Methods . . . 14

2.3.1 Grid Point Method . . . 15

2.3.2 Advanced Methods . . . 16

2.4 Method Selection . . . 18

3 µ Analysis for Re-entry Flight Clearance

21 3.1 Structured Singular Value . . . 22

3.1.1 Singular Value Decomposition . . . 22

3.1.2 Structured Singular Value . . . 23

3.2 The Algorithms . . . 28

3.2.1 Lower Bound . . . 28

3.2.2 Upper Bound . . . 31

3.2.3 Example of µ Computation . . . . 34

3.3 Robust Stability Analysis . . . 36

3.3.1 Linear System with Uncertain Parameters . . . 36

3.3.2 Numerical Result . . . 37

3.4 Multiple Connected Sets of Instability . . . 39

3.5 Summary . . . 41

4 Interval Analysis for Re-entry Flight Clearance

43 4.1 Interval Analysis . . . 44

4.1.1 Arithmetic Operations . . . 44

4.1.2 Set Operations . . . 45

4.1.3 Inclusion . . . 46

4.2 Interval Analysis Algorithm . . . 47

4.2.1 Subpavings and Set Inversion . . . 47

4.2.2 The Set Inverter via Interval Analysis Algorithm . . . 47

4.3 Stability of a Linear System with Eigenvalue Analysis . . . 49

4.3.1 Multiple Connected Sets of Instability . . . 49

4.3.2 Routh-Hurwitz Stability Criterion . . . 52

4.3.3 Numerical Example . . . 54

4.3.4 Comparison with µ Analysis . . . 56

4.4 Combination of µ and Interval Analysis . . . . 56

4.4.1 System Description and Analysis Objective . . . 56

4.4.2 The Analysis Result . . . 57

4.5 Evaluation of the Lyapunov Function for Stability . . . 59

4.5.1 Definition . . . 59

4.5.2 Linear System Example . . . 60

4.5.3 Non-linear System with Uncertain Parameters . . . 62

4.5.4 Numerical Example . . . 64

4.6 Summary . . . 66

5 Re-entry Flight Control

67 5.1 Flight Mechanics . . . 67

5.1.1 Translational Motion . . . 68

5.1.2 Rotational Motion . . . 70

5.2 Re-entry Flight Control Overview . . . 73

5.3 Delft Aerospace Re-entry Test Demonstrator . . . 74

5.3.1 Control objectives . . . 75

5.3.2 Mathematical Model . . . 76

CONTENTS xi

5.3.4 Control Implementation and Simulation . . . 85

5.4 SPHYNX . . . 94

5.4.1 Control Objectives . . . 95

5.4.2 Mathematical Model . . . 95

5.4.3 Controller Design . . . 96

5.4.4 Control Implementation and Simulation . . . 99

6 Modeling Parametric Uncertain Systems

103 6.1 Linear Fractional Representation . . . 104

6.1.1 Examples . . . 106

6.1.2 Approaches for LFR Modeling . . . 113

6.2 NDI for Parametric Uncertain Systems . . . 113

6.2.1 NDI for Re-entry Vehicle with Uncertainties . . . 118

6.2.2 NDI with Aerodynamic Parameter Errors . . . 120

6.3 NDI Control System in LFR . . . 121

6.3.1 LFR Generation . . . 122

6.3.2 LFR Validation . . . 126

7 SPHYNX Re-entry Flight Clearance

129 7.1 System Model with Parametric Uncertainties . . . 130

7.2 Linear Analysis . . . 132

7.2.1 µ Analysis Result . . . 135

7.2.2 Interval Analysis Result . . . 137

7.3 Non-linear Analysis . . . 140

7.4 Non-linear Simulations . . . 144

8 DART Re-entry Flight Clearance

155 8.1 System Model . . . 155

8.2 Linear Analysis . . . 158

8.2.1 Nominal System Stability . . . 158

8.2.2 System with Aerodynamic Parameter Uncertainty . . . 161

8.2.3 µ Analysis Result . . . 163

8.2.4 Interval Analysis Result . . . 164

8.3 Non-linear Analysis . . . 171

8.3.1 Nominal System Stability . . . 172

8.3.2 System with Aerodynamic Parameter Uncertainty . . . 174

8.4 Non-linear Simulations . . . 183

9 Conclusions and Recommendation

189 9.1 Conclusions . . . 189

9.1.1 Re-entry Flight Control System Designs . . . 189

9.1.3 Systematic Methods for Re-entry Flight Clearance . . . 191

9.1.4 Computational Effort . . . 192

9.2 Recommendation . . . 193

A Norms

195 A.1 Vector Norms . . . 195

A.2 Induced Matrix Norms . . . 196

A.3 Signal Norms . . . 197

A.4 System Norms . . . 198

B Lie Derivatives and Lie Brackets

199 B.1 Lie Derivatives . . . 199

B.2 Lie Brackets . . . 200

C Approximation for DART Aerodynamic Moments

201

References

203

Samenvatting

211

Acknowledgements

215

Chapter

1

Introduction

This chapter presents the motivation behind the research that gave birth to this thesis. Originally, this study was initiated as part of the project performed at Delft University of Technology, aiming to design, build and fly a reusable vehicle into space. The vehicle flight missions were to obtain aerothermodynamic data, i.e. selected fundamental data concerning the physics of re-entry flight, and to test the design of a hot metallic load-carrying vehicle external structure in real re-entry flights (TU Delft [2003a]) .

For the success of the mission, flight safety is considered as one of the most important aspects. The safety criteria should be defined and then used to eval-uate the designed vehicle in its flight trajectory. This evaluation of these safety criteria ’clears’ sets of operational flight conditions where the vehicle can safely be operated. This process is called ’flight clearance’. The present research was dedicated to develop and apply systematic mathematical methods for the clear-ance of re-entry flight models with non-linear system characteristics. Although in this research the methods were applied to specific re-entry vehicle cases, they can also be used for any non-linear mechanical system.

1.1

Historical Notes

The launch of Sputnik on October 4, 1957 by the Soviet Union marked the begin-ning of the space era (NASA [2005a]; Siddiqi [2003b]). This event was followed by many more spaceflights to date. Although initially space missions were dominated only by the Soviet Union and the United States, culminating in the race to the moon (Siddiqi [2003a]), later on many more countries also showed their interests in space exploration. European countries began a consortium for space exploration in 1964, which is called European Space Agency (ESA) (Graham [1995]). In 2005, it had in total 17 countries as members (ESA [2005a]). Its most recent success was

the Mars Express project, which resulted in a succesful deployment of an orbiter to explore the planet Mars in June 2003 (ESA [2005b]). The vehicle is still work-ing on mappwork-ing the red planet and sendwork-ing the data to the earth. China began its space program in 1956, marked by the establishment of the ’Fifth Academy’, the research institute of space technology (Harvey [2004]). The country launched its first human into earth orbit on October 15 2003 and became the third country in the world to send human into space (Harvey [2004]).

1.1.1

Development of Re-entry Activities

In the beginning of the space age, space vehicles functioned as satellites. They were launched from the earth into any desired orbit, from where they transferred information through telemeters to an earth station. After the life-period of a vehicle was finished, it would be de-orbited and burned in the earth atmosphere. The first living creature orbiting the earth, a dog called Laika, was incinerated together with the vehicle Sputnik 2 that carried it after being put to sleep (Harford [1997]). This destructive technique to end space missions is still practiced to date for unmanned earth satellites.

The need to bring collected materials, experimental samples, and crews safely back to earth stimulates research activities in the fields of re-entry flights. The term re-entry defines the part of a flight in which a spacecraft re-enters the at-mosphere of the earth. This definition is not to be confused with a related term ’entry’, which refers to the part of a flight on which a spaceflight enters the at-mosphere of other celestial bodies (such as the entry of Jupiter in 1995 (NASA [2003]) or the recent Mars entry (NASA [2005b]; Wilson [2004])).

The first safely recovered vehicle from space was Discoverer 13, the reconnais-sance satellite of the United States in the Corona program (Heppenheimer [1998]), on August 10, 1960. Later in the same month, some living creatures were sent into space by the Soviet Union in the spacecraft Korabl Sputnik 2, which re-entered the earth safely after one day travel in space. There were two dogs, four mice, a rat, flies and some plants on board (Sobel [1965]). These first two were followed by many more re-entry flights.

1.1 HISTORICAL NOTES 3

Figure 1.1: Top: Apollo; Mid: Gemini; Bottom: Mercury (NASA [1987])

Figure 1.2: Left: Soyuz (NASA [1975]); Right: ARD (ESA – D.Ducros [1998])

To generate more lift for spacecraft manoeuvrability improvement, winged re-entry vehicles were developed, e.g. Buran and the Space Shuttle. This type of vehicle is illustrated in Figure 1.3, which shows the take-off and landing of the Space Shuttles Atlantis. Until the present time, Space Shuttles are the only reusable space vehicles.

Figure 1.3: The Space Shuttle: launch (NASA [1991]) and landing (NASA [1985])

Figure 1.4: X-38: re-entry concept (NASA [1997]) and free flight test (NASA [1999])

1.1.2

Spaceflight Accidents

Besides abundant successes, spaceflight activities also experienced catastrophic failures. From all spaceflights with human crews, there have been four accidents to date resulting in 18 fatalities of astronauts. Three accidents happened during the re-entry phase and one during launch.

The first re-entry failure happened in Soyuz 1: its crew was killed as the parachute system in the capsule did not open properly. Another accident hap-pened to Soyuz 11 descent module with its three crew members. A valve was accidentally opened, leading to the leakage of air into space when the spacecraft was de-orbited from the Salyut space station. The depressurisation of the cabin and the lack of oxygen caused death to the crews as they re-entered the earth atmosphere.

1.2 THE CHALLENGES OF RE-ENTRY 5

from the fleet. The first accident happened to the space shuttle Challenger, which failed during launch due to a fault that caused its external fuel tank to explode. The second happened to the space shuttle Columbia, which was burnt during its re-entry. A piece of insulation foam had broken away from the external fuel tank, hit the underside and damaged the heat shield of one of its wings. According to the post-accident analysis, it was the result of an incident happened during launch.

1.2

The Challenges of Re-entry

Re-entry flight is challenging for engineers and scientists due to its complex char-acteristics that differ from space flight or atmospheric flight. In the outer-space flight, a vehicle travels in an empty outer-space with hypersonic speed; meanwhile, in the atmospheric flight, a vehicle travels in the atmospheric environment with lower speed. The re-entry flight is a complex combination of these two flight char-acteristics. Through the re-entry paths, the vehicle travels from the emptiness of space to the dense atmosphere of the earth, where the vehicle, moving in hy-personic speed, encounters high temperatures as a result of the friction with the atmospheric particles. As the vehicle travels through the different atmospheric layers, it experiences different effects from the surrounding environment. In the ionospheric region of the atmosphere, for example, the vehicle experiences radio communication blackout, i.e. the phase in which signals from and to the vehicle cannot penetrate ionized gas particles surrounding them. Moreover, as it gets closer to the ground, the high velocity has to be reduced dramatically and a soft landing has to be conducted to keep the vehicle and their contents intact. All these re-entry aspects have to be considered in order to ensure the safety of the flight.

Trajectory tracking, flight stability and landing accuracy are the main issues for the success of a entry flight mission. Thorough understanding of the re-entry flight characteristics is required to deal with these issues. For atmospheric flight, the study of the (aerodynamic) flight characteristics can be performed by means of wind-tunnel tests and experiments. For re-entry flight, however, the available facilities of wind-tunnels to date are not yet adequate to capture all phenomena experienced during re-entry; hence, the study about the re-entry flight characteristics, especially the aerothermodynamics characteristics during hypersonic flight, cannot be obtained sufficiently through wind-tunnel tests.

Other means to understand the re-entry flight characteristics is by perform-ing direct flight tests. In this case, the actual data are gathered through direct measurement in real flight condition. The operational cost of the re-entry flight test is very high. Therefore, a re-entry flight test is normally considered as the final test, which will be performed only after extensive on-ground tests have been passed successfully.

direct flights make computational analysis and simulation inevitable tools in the study of the re-entry flight characteristics. In computational analysis and simula-tion, the vehicle and the surrounding environment are represented by mathemat-ical models. The study of the re-entry flight characteristics is done by performing mathematical analysis and computer simulation based on the these models.

The main challenge in computaional analysis and simulation comes from the inevitably limited actual data of the re-entry flight characteristics. The synthesis of an adequate mathematical model is rather complicated due to the complex nature of the flight environment and the unique characteristics of each re-entry vehicle. Some allowances should be given to the model to anticipate parameter uncertainties. Another challenge is to identify and include in the model cru-cial non-linearity. Reliable mathematical analysis methods are needed to deal with both the parameter uncertainties and the non-linearity of the model. These methods should be able to give accurate evaluation result within reasonable com-putation time.

1.3

Problem Statement and Objectives of the Research

As mentioned in the beginning of this chapter, the research is part of a project to design a reusable atmospheric re-entry flight vehicle. The vehicle is called DART, which stands for Delft Aerospace Re-entry Test Demonstrator. The scenario of the DART re-entry mission is as follows: the vehicle is carried as a payload in a rocket and launched from the earth. After separation from its carrier in the outermost layer of the atmosphere, the vehicle has to perform the assigned aerothermodynamic experiments during its re-entry flight, and returns back to the earth with the collected data.

The success of the mission requires the vehicle to be launched and return safely. One of the most challenging part of the mission is the re-entry phase. Before the actual flight takes place, the vehicle has to be certified as ’safe to fly’ during re-entry. The designers have to evaluate the vehicle in its flight domain, with different kinds of flight conditions and parameters, based on predefined safety criteria. The process of evaluation and certification in this design stage is called the flight model clearance.

The present research was aimed to find suitable mathematical methods for general re-entry flight model clearance. The methods suitability was evaluated by using DART vehicle model as a case study. The mathematical model of DART, including its control system, and its flight trajectory should be first developed, since they were not yet available. Parameter uncertainty should be taken into ac-count in the modeling to accomodate the unpredictable parameter variation. The mathematical methods were then evaluated by applying them for the clearance process of the flight model. System stability in the presence of model uncertainties was chosen as the clearance criterion in this process.

1.4 RESEARCH SCOPE 7

Precursor Hypersonic X) re-entry vehicle, was also employed for a case study. In principle, the methods evaluation process for SPHYNX is the same as that for DART. The main differences between the two are the type of control system and the uncertainty parameters employed in the model. Unlike DART, the flight trajectory for SPHYNX was taken from available data.

In general, the objective of the research can be formulated as follows: to identify and evaluate promising mathematical techniques for re-entry flight model clearance. These techniques should be able to account for the non-linearity and uncertainties in the dynamical models of re-entry vehicles.

To meet the research goal, the following tasks need to be performed:

1. Development or modification of available modelling methods for dynamic systems with uncertainties. Application of these methods to derive represen-tative models of re-entry vehicles, which are easy to generate and applicable to the clearance problems.

2. Development or modification of systematic mathematical methods for re-entry flight model clearance applications.

3. Analysis of the re-entry vehicle flight models using the developed clearance methods to determine the cleared condition.

4. Simulations of the re-entry flights to validate the clearance results.

5. Evaluation of the final results in terms of the ability of the methods to account for the non-linearities and uncertainties in the dynamical models of re-entry vehicles.

1.4

Research Scope

The research was focused on the identification and evaluation of mathematical clearance techniques for re-entry flight applications. The scope of the identifica-tion and evaluaidentifica-tion process was defined with regards to the vehicle mathematical models, the re-entry flight trajectory, and the development and application of the mathematical clearance methods.

the models. In this research, the DART model was assumed to contain uncer-tainties in the magnitude of the aerodynamic moments; meanwhile, the SPHYNX model was assumed to contain uncertainties in the center-of-gravity location.

Both DART and SPHYNX were designed to start the re-entry phase with a hypersonic speed from the altitude of higher than 100 km. During the re-entry, the vehicles would experience rapid changes in speed and altitude. At Mach number 2 and at altitude close enough to the ground, parachute-landing systems would be deployed. The scope of the clearance process in this research was limited to the phase of flight between the beginning of the re-entry and before the deployment of the parachute-landing systems.

During the re-entry flight, the vehicle has to fly along a predefined trajectory. For some reasons, however, the flight may deviate from the nominal trajectory. In this case, the vehicle needs to fulfil the safety criteria within some boundaries around the nominal trajectory, known as the flight envelope. These boundaries were not distinctly defined. They can be shrunk or enlarged according to the result of the clearance. The clerance process for DART was performed to clear the model not only in its nominal flight trajectory, but also within the flight envelope region. For SPHYNX, however, the clearance process was only performed in the nominal flight trajectory.

The stability of the system in the presence of model uncertainties was chosen as the clearance criterion. Two types of stability criteria to be analysed were worst-case eigenvalue (linear) and the Lyapunov stability (non-linear) criterion. Two mathematical methods were selected to be investigated and developed for the re-entry flight model clearance applications, which are µ-analysis and interval analysis. The worst-case eigenvalue criterion was used in the application of both µ-analysis and linear interval µ-analysis. The Lyapunov stability criterion was used in non-linear interval analysis application. Both µ- and interval analysis were applied to the DART and SPHYNX re-entry flight model clearance. The suitability of the methods in dealing with non-linear characteristics of the re-entry flight was determined based on the clearance results. Non-linear simulations were performed for both vehicle flight models to verify the clearance results.

1.5

Contribution of the Thesis

This thesis gives several contributions to both applied mathematics and aerospace research field in certain manners. They are summarized into three points as follow: 1. Application of non-linear mathematical clearance for re-entry flight models. To date, the applications of mathematical clearance methods are limited only to atmospheric flight models. This thesis utilises non-linear mathe-matical clearance method, for the first time, for re-entry flight model appli-cations.

non-1.6 THESIS OUTLINE 9

linear system models with physical parameter uncertainties. NDI needs a perfect non-linear model to perform complete linearisation of the systems by using the model inversion as a feedback to the system. A perfect model, however, does not exist, since all models may contain errors that come from computation and/or measurement data. To accomodate these errors, they can be incorporated in the model as uncertainties. In this thesis, an effort was made to quantify the error bounds of physically meaningful parameters as uncertainties in the non-linear system models. After applying NDI, the errors were still present in the closed-loop system; hence, the expression could be used to analyse the closed-loop system for stability. This approach might also be useful for for designing a control system in the outer-loop, which has robustness against the uncertainties.

3. Development of interval analysis for non-linear mathematical re-entry flight clearance. Literature (e.g. Fielding et al. [2002]) have shown that interval analysis can be used to perform linear mathematical model clearance. This thesis shows that, beside the linear mathematical model clearance, interval analysis can also be used to perform non-linear mathematical model clear-ance. It has been shown in this thesis that as long as the clearance criteria can be expressed as algebraic function, interval analysis can solve the clear-ance problem. In this case, the Lyapunov stability of non-linear systems with uncertainties was used as the non-linear clearance criterion.

1.6

Thesis Outline

Figure 1.5 shows the organisation of the thesis. Chapter 2 contains the background of the research activity. A survey on the state-of-the-art of the mathematical clearance methods in general is presented. An example of the so-called grid point method to perform a mathematical clearance is given to show the need of a more reliable method. The selection of the analysis methods to be used in this research is presented at the last section of this chapter.

Chapter 3 and 4 present the description of the selected tools for the re-entry flight model clearance. Chapter 3 presents the µ analysis method, while Chap-ter 4 presents the inChap-terval analysis method. Each chapChap-ter gives an overview of the method, accompanied by small cases of clearance process to exemplify the performance of the algorithms. These chapters are by no means extensive mono-graphs of the methods; readers who are interested in more rigorous explanation are advised to consult the references given in the text.

Chapter 5 presents a survey on the flight mechanics and flight control tech-niques for re-entry vehicles. Two re-entry vehicle models are presented in this chapter: a ballistic (DART) and a lifting-body (SPHYNX) vehicle model. For these models, two types of flight control systems were designed based on a linear and a non-linear control system design approach. For the SPHYNX model, a Lin-ear Quadratic Regulator (LQR) control law was applied. For the DART model, a Non-linear Dynamic Inversion (NDI) technique, combined with a Proportional Integral Derivative (PID) control law, were applied. The nominal stability and performance of the closed-loop systems were analysed and verified with non-linear simulations. The two re-entry vehicle closed-loop system models were used as the case studies for the re-entry flight model clearance application in Chapter 7 and 8.

In the clearance process by using µ analysis, a system model has to be ex-pressed in a so-called Linear Fractional Representation (LFR) form (Magni et al. [2002]). Chapter 6 presents an overview of dynamic system modelling in the LFR form. The dynamic systems to be modelled were assumed to contain uncertainties in the parameters. To illustrate the modelling process, an NDI control law was derived for the DART model, which contains uncertainties in the aerodynamic parameters. Subsequently, the model was expressed in an LFR form.

Chapter 7 and 8 present the implementation of the clearance tools. Chapter 7 presents the SPHYNX model clearance results, whereas Chapter8 presents the DART model clearance results. The two clearance methods presented in Chapter 3 and 4 were employed to analyse the models based on chosen stability criteria. The results showed the accuracy of the methods, as well as their reliability and computational tractability.

Chapter

2

Flight Clearance

Clearance can be defined as a process to certify a model ability to fulfil certain criteria within certain parameter boundaries. In aerospace applications, flight clearance is performed to evaluate a flying vehicle according to predefined criteria in a designated flight region. The predefined criteria could be criteria for stabil-ity, performance and handling qualities. For transportation media as aerospace vehicles, this evaluation is very important, since it concerns the safety of the ve-hicles, crews, and environment. It is for this reason the flight clearance should be performed early in the design phase of the flight vehicle.

This chapter describes the background of the re-entry flight clearance, the typical clearance approaches, and the state-of-the-art of mathematical clearance. At the end of this chapter, the selection of the mathematical clearance methods in this research is presented.

2.1

Clearance in Re-entry Flight Perspective

Frequent transportations with space vehicles will be feasible in the future if the operational cost can be reduced significantly (Bertin and Cummings [2003]). The payload of these transportations will include human passengers and goods, as in the present atmospheric flights. This means that requirements on the space-craft capability, reliability and safety will be much higher than in the present spaceflights: the spacecraft will have to be certified in a fashion similar to airlin-ers (Bertin and Johnson [1997]).

Re-entry flight is inseparable from the present and future spaceflight missions. During the re-entry flight, a vehicle has to follow a predefined trajectory towards a designated landing area (Figure 2.1). Trajectory tracking, flight stability, and landing precision are important aspects in the re-entry flight. The predefined trajectory has to be tracked with a sufficient margin of errors. Stability margin

has to be fulfilled against the variation and uncertainty in flight parameters such as aerothermodynamic parameters, mass, and inertia. The accuracy of landing is important as well to ensure the vehicle and its contents reaching the target ground intact. All these aspects need to be treated in systematic approaches in order to analyse and evaluate the stability and performance of the re-entry vehicle.

Figure 2.1: A re-entry flight trajectory

To ensure the flight safety, the re-entry flight model has to be evaluated prior to actual flight. This is known as the entry flight model clearance, or re-entry flight clearance in short. One example of the flight model clearance is the evaluation of the stability and performance of the re-entry vehicle model in its flight envelope. As illustrated in Figure 2.2, the re-entry flight envelope is defined along the nominal flight trajectory. The flight safety is checked by evaluating the regions in the flight envelope using some predefined criteria, e.g. the flight stability under the influence of aerodynamic moment uncertainties. The region, in which the criteria are fulfilled, is then ’cleared’. It means that, under the defined flight condition, the vehicle can fly safely in this region. The right-hand side figure in Figure 2.2 shows a blow-up view of the envelope with flyable and unflyable regions as a result of the clearance. The flyable areas are shown by the shaded rectangles in the flight envelope.

2.2

Clearance Approaches

The usual flight clearance practice is performed by combining computational and experimental approaches. These include mathematical analyses, computer simu-lations, and flight tests.

2.2 CLEARANCE APPROACHES 13

Figure 2.2: Illustration of re-entry flight envelope clearance

clearance approach is also called mathematical clearance. The use of mathemati-cal analyses for clearance process has been shown to be applicable to atmospheric flight application (see e.g. Fielding et al. [2002]). As it is proven later on in Chap-ter 7 and 8, some of those analyses can also be used for flight model clearance in re-entry flight applications. More elaborate discussion about the mathematical clearance methods is given in the next section.

Another flight clearance approach is computer simulations based on the Monte Carlo analysis. In this approach, a set of system models are run in a simulation environment. To obtain adequate result accuracy, the simulations have to in-clude various combinations of parameters considered to be critical for the vehicle stability and performance; therefore, a large number of simulations are required. Some examples of clearance using Monte Carlo simulations are the assessments of trajectory dispersion for several re-entry vehicles given by Desai et al. [1997a]; Desai and Cheatwood [2001]; Desai and Knocke [2004]; Desai et al. [1997b]. The purpose of the simulations was to estimate the downrange and crossrange of the landing area. Figure 2.3 shows the result of the Monte Carlo simulations for one vehicle, GENESIS (Desai and Cheatwood [2001]). The statistical results of the landing dispersions were obtained after 3000 numerical simulations involving vari-ations in parameters. The densest area in the figure shows the probable landing area predicted by the simulations.

Figure 2.3: Monte Carlo simulations for landing dispersion of GENESIS (Desai and

Cheatwood [2001])

[2004]).

The third approach, flight test, can be considered as the most accurate ap-proach to perform flight clearance, since the vehicle is evaluated in the actual flight conditions. A large number of trials are required to cover the variations in flight parameters. This approach is not common for re-entry flight clearance, since the cost involved in the tests is very high.

Until recently, flight tests for the re-entry flight clearance were performed only for partial flight region, not the entire re-entry trajectory. One example of such flight tests is presented by Winchenbach et al. [2002]. The objective of these flight tests was to evaluate the dynamic stability of re-entry vehicles in the atmospheric region. For this evaluation, 83 flight tests were performed on six scaled models, which were designed based on the Huygens, Mars Micro Probe, Stardust, and Genesis re-entry vehicles (Winchenbach et al. [2002]). The tests were only conducted from Mach numbers 0.7 to 3.5; hence, they did not cover the hypersonic flight regime.

2.3

Mathematical Clearance Methods

2.3 MATHEMATICAL CLEARANCE METHODS 15

2.3.1

Grid Point Method

The flight clearance process using grid point method is usually performed in sev-eral steps (Korte [2002]). First, system models, possibly non-linear, with or with-out control systems, are generated. The engineers get themselves familiarised with these models and perform trend study on the effects of uncertain parameters to the stability, performance and handling qualities of the models. In the next step, the models are linearised. Stability analysis is performed by calculating the stability margin on a set of grid points in the flight region. Finally, non-linear simulations may be performed to verify the clearance results.

The application of grid point method for clearance process is illustrated by the following example. Take a system with two uncertain parameters, p1and p2.

The system can be represented by a linear model,

˙x = Ax, (2.1)

with the system matrix

A = ⎡ ⎣ −2 − p1− p2 −2 − p1− p2 −2 − σ 2_{− 6p} 1− 6p2− 2p1p2 1 0 0 0 1 0 ⎤ ⎦ , (2.2) where p1 and p2 are uncertain parameters and σ is a constant. The uncertain

parameters p1and p2may vary within the two-dimensional space of p1× p2, with

p1∈ [−3, 7] , and p2∈ [−3, 7] . (2.3)

The constant is set as σ = 0.5.

The characteristic equation of the model is (Jaulin et al. [2001])

λ3+ (p1+ p2+ 2)λ2+ (p1+ p2+ 2)λ + 2p1p2+ 6p1+ 6p2+ 2 + 0.25 = 0. (2.4)

The system stability, which can be determined by the eigenvalues of the model, is defined as the clearance criterion. The eigenvalues are the roots of Equa-tion (2.4), which depend on the parameters p1and p2. By evaluating this criterion

for a set of grid points in the parameter space, stable and unstable conditions on all the points can be found.

Figure 2.4 shows the clearance result. The parameter space is covered by 121 equally spaced grid points, on which eigenvalues of the system are evaluated. The parameter space contains multiple connected sets of instability, i.e. the points where the eigenvalues have positive real parts. These are represented by the cross marks in the figure. The circle marks represent the stable points, i.e. the points where the eigenvalues have negative real parts.

−3 −2 −1 0 1 2 3 4 5 6 7 −3 −2 −1 0 1 2 3 4 5 6 7 p 1 p2 isolated unstable point

Figure 2.4: Grid point analysis result for the system in Equation (2.1) – (2.4): the

parameter space with× ≡ unstable points, ◦ ≡ stable points

As it is surrounded by stable points, one could ask how far the unstable area extends around this point. To find the answer, one has to define finer grids in the surrounding of the unstable point and then evaluate each point in the new grids. The grid point clearance practice in general takes long computation time and the result is not highly reliable. The flight region is usually divided into fine grids to achieve higher result accuracy. Finer grids means higher computational load; meanwhile, there is still no guarantee that clearance can be fully obtained as the grids do not cover the flight region continuously. Moreover, when linearised models are used, the linearisation errors may reduce the reliability of the results.

2.3.2

Advanced Methods

Fielding et al. [2002] described recent development in the clearance of flight control laws done by the Group of Aeronautical Research and Technology in Europe (GARTEUR). The work was applied to a military aircraft with flight control system, the HIRM+RIDE benchmark. The research was aimed to provide fast, efficient, and numerically reliable analysis tools for the clearance process.

According to Fielding et al. [2002], there are several approaches that can be used for mathematical clearance process. They are as follows:

• µ analysis

2.3 MATHEMATICAL CLEARANCE METHODS 17

is more reliable than the traditional grid approach and gives a stronger guarantee of the clearance results. The computational load of this method is generally a polynomial function of the number of uncertainties (Bates and Postlethwaite [2002]). Although it may offer computation time reduction compared to the traditional approach, the computation of bounds for µ sometimes can be difficult. This may happen especially in the case of real µ, which represents parametric uncertainties (Bates and Postlethwaite [2002]). • bifurcation and continuation method

This method utilises non-linear differential equations to describe dynamic system models. It can be used to find possible stable and unstable equilibria in the model (Lowenberg [2002]). The method does not guarantee that the so-called ’worst cases’ of stability or performance will always be found, but it can be used to identify the critical area in the flight envelope for closer investigation (Karlsson and Fielding [2002]). It offers significant reduction in computational effort, as compared to the classical grid point approach. • interval analysis

Verde and Corraro used this technique to evaluate linear stability criteria of uncertain linear systems. They called this method the polynomial-based clearance method, since the criteria are defined as polynomial functions of uncertain parameters. The result achieved using this method is more reli-able compared to the grid point approach (Verde and Corraro [2002b]). The computational load of this method is lower than the grid point approach for the same result accuracy. The use of this method is simple and straightfor-ward, and the results can be interpreted easily. It can be used to investigate the whole region of uncertain parameters in the flight envelope as a con-tinuous system (Karlsson and Fielding [2002]). This method is applicable for both linear and non-linear criteria, as long as they can be expressed in algebraic functions.

• ν-gap analysis

Similar to µ analysis, the uncertain linear model is expressed in an LFR form. Steele and Vinnicombe [2002] claimed that the method is less con-servative than µ analysis, since it facilitates the use of a simple algorithm, instead of the lower bound for µ, to find the worst-case of parameter combi-nations. The computation time of this method is lower than the grid point technique for the same result accuracy. However, it can only be applied to linear systems with uncertainties.

• optimization based analysis

Its computation time is comparable to the classical grid-point approach, but will improve as the number of uncertainties increases. However, the method reliability depends on the optimisation method chosen in the analysis. If no global optimisation algorithm available, the worst cases can be missed. In this case, the classical approach has to be employed as a complementary to this method (Karlsson and Fielding [2002]).

Two examples of the method application have been reported by Gunnarsson et al. [2004] and Goto et al. [2004]. Clearance of flight control law by using LFR and µ analysis was given in the paper of Gunnarsson et al. [2004], where linear stability requirements were applied to the flight envelope of an Uninhabited Aerial Vehicle (UAV). Goto et al. [2004] conducted a non-linear dynamic analysis by using a continuation method for HOPE-X re-entry vehicle. This analysis was performed to find equilibrium points of the vehicle, which were needed to design an automatic flight control system. It should be noted, however, that all the clearance methods mentioned above are used so far for atmospheric flight applications.

2.4

Method Selection

In general, the characteristics of computational clearance methods can be distin-guished according to the type of numerical analysis to be performed, which are discrete and continuous, and the types of system to be handled (or criteria to be used), which are linear and non-linear. Continuous analysis is more accurate than discrete analysis since it covers the entire clearance area. The case example of the grid-point method application in Section 2.3.1 has shown that worst cases can be easily missed due to the gridding system in discrete analysis. The accuracy of discrete analysis can be improved by making finer grids; as a consequence, how-ever, the computation time will increase. Despite these disadvantages, discrete analysis might be preferable in certain applications due to its simplicity.

Clearance process using linear method is simpler than non-linear method. For this reason, sometimes a non-linear system is modelled as a linear system by trimming and linearisation. Hence, the clearance process can be performed by using linear method. In this way, considerable time and effort can be saved. The result accuracy, however, might be lower than when using non-linear method.

The characteristics of the computational clearance methods discussed in pre-vious sections can be summarised in the matrix shown in Table 2.1.

2.4 METHOD SELECTION 19

L NL D C

grid point analysis × ×

Monte Carlo analysis × ×

µ analysis × ×

ν-gap analysis × ×

bifurcation and continuation method × ×

interval analysis × × × ×

optimization-based analysis × × × ×

L = linear; NL = non-linear; D = discrete; C = continuous Table 2.1: Categorization of the mathematical clearance methods

method is considered low since it cannot guarantee the finding of worst cases (Karlsson and Fielding [2002]). Interval analysis offers possibilities to solve any clearance criteria, linear or non-linear, which can be expressed in algebraic func-tion. The optimization based analysis translates clearance criteria into optimiza-tion problem. If no global optimizaoptimiza-tion algorithm available, worst-cases can be missed (Karlsson and Fielding [2002]). Comparing the main property of the three methods, it is obvious that interval analysis is more suitable for re-entry flight clearance application than the other two. Therefore, interval analysis was chosen to be used in this research.

Chapter

3

µ Analysis for Re-entry Flight

Clearance

As explained in Chapter 2, one major problem in flight clearance activities is that it involves massive computation with a large number of parameters. The main cause of this difficulty is that the classical clearance methods involve grid point evaluation within the parameter space. These grid points, which are defined by various relevant system properties and configurations, as well as possible model uncertainties and different environmental conditions, amount to a large model set. Moreover, the accuracy of the analysis also depends on the density of the grid points.

Systematic approaches to flight clearance are needed to improve the clearance results. These are the accurate yet efficient methods which can avoid the grid points and at the same time give more accurate clearance result. In this chapter, µ analysis is presented, to serve as one of the clearance methods investigated in this present thesis. The structured singular value µ was first introduced in Doyle [1982], and at the same time also independently published as multivariable sta-bility margin in Safonov [1982]. µ analysis and synthesis have been extensively used in the field of robust control analysis and synthesis. The methods employ the structured singular value µ, which can be used as a measure of stability and performance robustness for systems with uncertainties or perturbations in their dynamics and parameters.

An overview of the structured singular value and how it can be used for sta-bility robustness analysis are given in the next section. Particularly, the real structured singular value is emphasised, since we are interested in systems with parametric uncertainties. It is followed by the description of the employed algo-rithms to find the upper- and lower bounds for µ. The later sections of the chapter give examples on robust stability analysis for systems with parametric

ties. The results serve as a benchmark to the ones obtained by a more innovative approach, the interval analysis method, which will be discussed in Chapter 4.

3.1

Structured Singular Value

This section summarises the notion of structured singular value µ with its appli-cation to robust stability analysis. The textbook of Skogestad and Postlethwaite [2003] forms a theoretical background of this chapter.

3.1.1

Singular Value Decomposition

For any real or complex matrix M of size l× m, its singular values can be defined as positive square roots of the k = min(l, m) largest eigenvalues of both matrices

ˆ

M = MH_{M and ˜M = M M}H_{, where M}H _{is the conjugate transpose of M . In} mathematical notation, a singular value σi(M ) can be written as

σi(M ) =  λi  ˆ M  =  λi  ˜ M  , (3.1) where λi  ˆ M 

is the ith eigenvalue of ˆM , which is equal to the ith eigenvalue of ˜

M .

Any matrix M can be decomposed into

M = U ΣVH, (3.2)

where U and V are unitary matrices of size l× l and m × m, respectively. A unitary matrix A has its conjugate transpose equal to its inverse, i.e.

AH = A−1. (3.3)

The eigenvalues of a unitary matrix have their absolute values equal to one, and the singular values of a unitary matrix are equal to one. Σ is an l× m matrix contains a diagonal matrix Σ1, which has the singular values σi in its diagonal,

whereas the other elements of Σ are zero. Equation (3.2) is called the singular value decomposition of M . Σ1 can be written as

Σ1= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ σ1 0 . . . 0 0 σ2 ... .. . . .. 0 0 . . . 0 σk ⎤ ⎥ ⎥ ⎥ ⎥ ⎦, k = min(l, m); (3.4) with the singular values arranged in descending order,

¯

3.1 STRUCTURED SINGULAR VALUE 23

¯

σ is called the maximum singular value of M , and σ its minimum singular value. The matrix Σ can be written as

Σ = Σ1 0  , if l > m, (3.6) or Σ =  Σ1 0 , if l < m, (3.7) or Σ = Σ1, when l = m. (3.8)

A physical interpretation of singular value decomposition is as follows. Take a multivariable system M (s) with m inputs and l outputs. The frequency response of M (s) in a fixed frequency ω can be denoted as M (jω). M (jω), or M in short, is, therefore, a constant l× m complex matrix. We can define the singular value decomposition of M (jω) as in Equation 3.2, where

• U is an l × l unitary matrix of output singular vector ui, • V is an m × m unitary matrix of input singular vector vi.

The column vectors of U , denoted as ui, represent the output direction of M , whereas the column vectors of V , denoted as vi, represent the input direction of M . These input and output directions are related through the singular values σi, i.e.

M vi = σiui. (3.9)

The singular value σi also represents the gain of the matrix M . Since ui and vihave unit length (ui2=vi2= 1), we can write

σi(M ) =Mvi2=Mv

i2

vi2

, (3.10)

which defines the gain of M in its ith direction (Skogestad and Postlethwaite [2003]).

3.1.2

Structured Singular Value

The structured singular value µ can be used to determine the stability of closed-loop systems with uncertainty in the frequency domain. In particular, a system with a transfer function matrix M (s) and a structured uncertainty ∆(s) is con-sidered, as shown in Figure 3.1. ∆(s), or ∆ in short, is assumed to be a matrix with diagonal blocks.

The definition of the structured singular value is given as a function which provides a generalization of the maximum singular value σ, and the spectral radius ρ. The structured singular value, denoted as µ, is defined mathematically in the following (Skogestad and Postlethwaite [2003]):

µ(M ) = 1

min∆{¯σ(∆)| det(I − M∆) = 0, for structured ∆}

Figure 3.1: M∆ loop for stability analysis

The definition in Equation (3.11) involves varying ¯σ(∆). However, we can also scale ∆ such that ¯σ(∆)≤ 1. This is performed by scaling ∆ by a factor k, and looking for the smallest k which makes the matrix I − kM∆ singular, and µ is the reciprocal of this value, i.e. µ = 1

k. Therefore, the definition of µ can be given as follows (Skogestad and Postlethwaite [2003]):

Let M be a given complex matrix and ∆ = diag {∆i} be a set of complex matrices, which are normalised by a factor k such that σ(∆) ≤ 1. The ∆ has a block-diagonal structure, i.e.

∆ = diag {∆i} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ∆1

0

. ._. ∆i

0

. .. ⎤ ⎥ ⎥ ⎥ ⎥ ⎦, (3.12)

where ∆irepresents a specific source of uncertainty. Figure 3.1 shows the M− ∆ structure for a system M (s) with uncertainty ∆(s) in a closed-loop. The real non-negative function µ(M ), called the structured singular value, is defined as

µ(M ) = 1

min{k| det(I − kM∆) = 0 for structured ∆, σ(∆) ≤ 1}. (3.13) When there is no uncertainty ∆(s) which can cause det(I − M∆) = 0, then the structured singular value is defined as µ(M ) = 0.

The structured singular value robustness measures can be derived from the multivariable Nyquist stability criterion:

For the system in Figure 3.1, if there is no unstable open-loop poles in the open-loop system M ∆, the closed-loop system is stable if and only if the Nyquist plot of det(I− M∆) does not pass through the origin.

3.1 STRUCTURED SINGULAR VALUE 25

(I− M∆) singular. On the other hand, a higher value of µ implies that a smaller perturbation ∆ will cause (I− M∆) to be singular.

The example in the following (Skogestad and Postlethwaite [2003]) illustrates that µ depends on the structure of the uncertainty ∆. Take a matrix M and its singular value decomposition,

M = 2 2 −1 −1  = U ΣVH = −0.8944 0.4472 0.4472 0.8944  3.1623 0 0 0  −0.7071 −0.7071 −0.7071 0.7071 H . (3.14) The perturbation ∆ = 1 σ1 v1uH1 = 1 3.1623 −0.7071 −0.7071   −0.8944 0.4472 = 0.2 −0.1 0.2 −0.1  , (3.15) with ¯σ(∆) = 1/¯σ(M ) = 1/3.1623 = 0.316 makes det(I− M∆) = 0. Therefore, µ(M ) = 3.1623 when ∆ is a full matrix.

The smallest diagonal perturbation ∆ which makes det(I− M∆) = 0 is ∆ = 1 3 1 0 0 −1  , (3.16)

with ¯σ = 0.3333. Therefore, µ(M ) = 3 when ∆ is a diagonal matrix, which is smaller than µ(M ) for a full perturbation matrix ∆.

3.1.3

Stability Robustness

We will demonstrate that µ can be used to determine the stability robustness of a system with structured uncertainty in frequency domain, as shown in Figure 3.1. In the figure, ∆ is a set of norm-bounded (see Appendix A) perturbations, which are ’pulled out’ from the nominal system into a block diagonal matrix, i.e. Equa-tion (3.12). Each perturbaEqua-tion is assumed to be stable and normalised, with the largest singular value equal to one,

σ(∆i(jω))≤ 1 ∀ω. (3.17)

An important property demonstrated in Skogestad and Postlethwaite [2003], is that the maximum singular value of a block diagonal matrix is equal to the largest of the maximum singular values of the individual blocks, i.e.

Therefore, for ∆ = diag{∆i}, it follows

σ(∆i(jω))≤ 1 ∀ω, ∀i ⇔ ¯σ(∆) = ∆_∞≤ 1. (3.19) Figure 3.2 shows an uncertain system with M ∆-structure, whose transfer func-tion from w to z can be represented in the form of a ’Linear Fracfunc-tional Represen-tation’ (LFR) (see Chapter 6, Section 6.1 for the derivation of this representation).

Figure 3.2: Linear Fractional Representation of system with uncertainty ∆ The transfer function from w to z for the system in Figure 3.2 is

z = LFR (M, ∆)w, (3.20) with LFR (M, ∆) =  M21∆ (I− M11∆)−1M12+ M22  . (3.21) Assume that the system is nominally stable (with ∆ = 0), and ∆ is also stable. Then the stability robustness of the system can be analysed by evaluating the singularity of the term (I− M11∆)−1. This condition is equivalent to the

stability of the M ∆ structure in Figure 3.1, where M = M11.

For the M ∆ structure in Figure 3.1, the following theorem of determinant stability condition is derived from the Nyquist Theorem, taken from Skogestad and Postlethwaite [2003], which applies to any convex set perturbations:

Assume that the nominal system M (s) and the perturbations ∆(s) are sta-ble. Consider the convex set of perturbations ∆, such that if ∆ is an allowed perturbation then so is c∆ where c is any real scalar such that|c| ≤ 1. Then the M ∆-system in Figure 3.1 is stable for all allowed perturbations, if and only if

The Nyquist plot of det(I− M∆) does not encircle the origin, ∀∆, (3.22) ⇔ det(I − M∆(jω)) = 0, ∀ω, ∀∆. (3.23) From Equation (3.23) we get the stability robustness condition which applies to both complex and real perturbation,

3.2 STRUCTURED SINGULAR VALUE 27

However, the condition in Equation (3.24) gives only a positive or negative answer to the question of stability. To obtain the factor k for which the system is robustly stable, the uncertainty ∆ is scaled by k, and the smallest k which yield worst-case stability has to be found, i.e.

det(I− kM∆) = 0. (3.25)

This number k is equal to k = _{µ(M )}1 according to Equation (3.13). This gives the definition of robust stability (Skogestad and Postlethwaite [2003]),

Take a nominal system M and a set of perturbations ∆ (real or complex) which are stable. Then the M ∆-system in Figure 3.1 is stable for all perturbations with σ(∆)≤ 1, if and only if

µ(M (jω)) < 1, ∀ω. (3.26) From this equation, it is trivial to check for stability robustness, provided that µ can be computed.

It is not possible to compute the number µ for a particular system; instead, an approximation is made by computing its upper- and lower bounds. When these bounds are tight (the upper bound approaches the lower bound), the approxima-tion is accurate. In its earlier development, only complex µ could be approximated accurately by these bounds (Packard and Doyle [1993]). The complex structured singular value is related to dynamic perturbations/uncertainties, represented by a complex uncertainty matrix ∆. In this thesis, however, the model of the system involves solely real parametric uncertainties. The µ for this problem is called the real structured singular value, since it is related to the real uncertainty matrix ∆. The real µ is hard to approximate. The problem to approximate its upper- and lower bounds is known to be NP (non-deterministic polynomial) hard (Fu [1997]). The computational time for calculating the bounds is generally an exponential function of the size of the problem. A lot of research has been conducted on the computation of these bounds (see Chang et al. [1991]; Dailey [1990]; Doell et al. [1998]; Fu and Dasgupta [2000]; Haddad et al. [1997]; Hayes et al. [2001]; Sparks and Bernstein [1995]). To obtain a good approximation on µ, its tight upper- and lower bounds have to be obtained.

For this thesis, two algorithms are employed to approximate the upper- and lower bounds for µ, respectively. The µ Analysis and Synthesis Toolbox for MATLABTM _{has capability to approximate the upper bound for real µ}

accu-rately, as shown later on in this chapter. For the approximation of the lower bound, a tool presented in Doell et al. [1998] is also available, which is also de-signed for MATLABTM_{. This algorithm is capable to estimate the lower bound}

3.2

The Algorithms

Since the value of µ is approximated by its upper- and lower bounds, one needs a tool to compute these bounds accurately. A satisfactory result is obtained as the upper and lower bounds are close to one another. In the next section, the lower bound algorithm is presented, based on the description given in Doell et al. [1998]. Sequentially, the algorithm for the µ Analysis and Synthesis Toolbox for MATLABTM_{(Balas et al. [2001]) to calculate the upper bound for µ is presented,}

based on the description given in Young et al. [1995].

3.2.1

Lower Bound

The lower bound algorithm is described in Doell et al. [1998]. This algorithm has been developed to compute the lower bound for µ, in the case of real or mixed uncertainties/perturbations. The real uncertainty problem is related to systems with parametric uncertainty, whereas the mixed uncertainty problem is related to systems in which both parametric and dynamic uncertainties exist. This algorithm has been applied to systems with parametric uncertainties (Magni and Doell [1998], Doell et al. [1998]). In this thesis, the algorithm is applied to calculate µ for a system with uncertainties in its aerodynamic parameters, as described later in this chapter.

In the following, the summary of the lower bound algorithm is presented. The technique consists of shifting the eigenvalues of the system towards the imaginary axis with a minimum perturbation, which is equivalent to the destabilising pertur-bation. The algorithm has two steps of calculation: the first step is to reach the limit of stability by searching for a perturbation with minimum Frobenius norm. The second step is to minimise the norm of this perturbation, while staying at the limit of stability.

The system with uncertainty can be represented in the following. Consider a system in LFR form as shown in Figure 3.3. The transfer function matrix M (s) can be written as

M (s) = C (sI− A)−1B + D (3.27) with A, B, C, and D the matrices in state-space representation of the system.

A class of uncertainty/perturbation ∆ acting on the system has the structure ∆ = diag (∆1, . . . , ∆i, . . . , ∆r) (3.28) in which ∆i is diagonal matrix of the possible forms:

• ∆i = δiIni with δi ∈ R (real repeated scalar block) or δi ∈ C (complex

repeated scalar block),

3.2 THE ALGORITHMS 29

Figure 3.3: M∆ structure

Step 1: Reaching the limit of stability

For the first step, consider the following lemma: Lemma 1: (Lemma 3.1 in Doell et al. [1998])

The first order approximation of an eigenvalue of A, denoted as dλ, induced by the gain variation d∆ of the perturbation matrix ∆, is calculated as

dλ = (uB + tD)d∆(Cv + Dw); (3.29) where v is the right and u is the left eigenvector of A corresponding to the eigen-value λ, respectively, and

w = ∆(I− D∆)−1Cv; t = uB∆(I− D∆)−1.

Denote λ as one of the eigenvalues of A+B∆(I−∆D)−1C. The algorithm finds d∆ that shifts λ with first order approximation to a vertical line, by a distance Ri, where

R((uB + tD)d∆(Cv + Dw)) = Ri. (3.30) This equation is a linear constraint on d∆, which will be computed such that the Frobenius norm

J1=∆ + d∆2F (3.31)

is minimised. This problem is solved at each iteration in the first step of the al-gorithm. When the criterion J1is minimum for large values of d∆, the first order

approximation of Lemma 1 is no longer valid. Therefore, there might be initial-isation problems. To avoid it, a combination of criteria J1 and J0 is minimised,

where J0is defined as

J0=d∆2F. (3.32)

Figure 3.4: Step 1 in µ lower bound computation

Step 2: Minimising the destabilising perturbation matrix norm