design of a robust controller for the vega tvc using the

UPC-ETSEIB ESTEC TEC-ECN

MASTER’S THESIS

Design of a robust controller for the VEGA TVCusing the µ− synthesis technique

Author: GADEA DIAZ Enrique

ESA Technical Advisor: BENNANI Samir

Noordwijk, October 10, 2011

MASTER’S THESIS

Design of a robust controller for the VEGA TVC

using the µ− synthesis technique

Author: GADEA DIAZ Enrique

ESA Technical Advisor: BENNANI SamirNoordwijk, October 10, 2011

3

Acknowledgments

This Master’s Thesis project would not have been possible without the support of manypeople. The author wishes to express his gratitude to his supervisor, Dr. Samir Bennaniwho was abundantly helpful and offered invaluable assistance, support and guidance andto all the GNC section team especially to Dr. Guillermo Ortega for his support and help.The author also wishes to express his gratitude to all the stagiaires and trainees that madehis stay in the Netherlands so incredible.

Special thanks to Marine Parahy and to all his friends from the ETSEIB and SU-PAERO that brought an inestimable support during these years.

The author wishes to express his love and gratitude to his beloved family membersEduardo, Josefina and Javier; for their understanding and endless love, through the dura-tion of his studies.

5

Abstract

This study has been developed in the European Space Research and Technology Centre(ESTEC) in the TEC-ECN section.

The purpose of this study is to design a robust controller for the thrust vector controlof the VEGA launcher using the µ−synthesis design. To reach this objective a first studyof the equations of motions together with the impact and effect of a classical controllerwill be done.

The studied model will include the rigid body motions together with three bendingmodes. To get the desired performances it will be necessary to design three different con-trollers. One for the rigid body behaviour, another for filtering the bending modes anda last one to reduce the angle of attack. In practice the filter for the bending modes isdesigned by knowing the shape that it should have, without taking the requirements intoaccount. Using a µ−synthesis design for the filter definition the requirements introducedby the weighting functions are satisfied in the better way.

The main part of the study will be carried out for the pitch axis. However, at theend of this study a model taking the coupling between pitch and yaw into account will bepresented simultaneously with a controller being able to stabilise overall

7

Contents

1 Introduction 171.1 Main Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2 Study Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.1 The Vega Launcher . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3.2 System elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Scientific context 322.1 Technical specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.1.1 General requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 322.1.2 Problems of launchers control . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Control laws used in launch vehicles . . . . . . . . . . . . . . . . . . . . . . 37

3 Classical controller 403.1 Launch Vehicle Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.1 Impact of Kp and Kd . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1.2 Finding the value of Kp and Kd . . . . . . . . . . . . . . . . . . . . 433.1.3 System Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Complete Rigid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.1 α feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2.2 z feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.2.3 z feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.4 z + z feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Flexible Launch Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . 643.3.1 Bending modes impact . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Robust Control 684.1 Uncertainty Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1.1 Kp and Kd as a function of a6 . . . . . . . . . . . . . . . . . . . . . 684.1.2 Using the PD controller defined by slides . . . . . . . . . . . . . . . 68

4.2 Building a new synthesis model . . . . . . . . . . . . . . . . . . . . . . . . . 704.2.1 Weighting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Robustness Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.1 Robust Stability And Performance . . . . . . . . . . . . . . . . . . . 78

9

4.3.2 Worst Case Performance and Skew µ . . . . . . . . . . . . . . . . . 794.4 Rigid Body controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.5 Bending Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.6 Drift control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.7 One controller for all the Pay Loads . . . . . . . . . . . . . . . . . . . . . . 904.8 Time simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5 Multi-axis controller 935.1 Problems due to the coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 935.2 Building the system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.3 Designing a new controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4 Bending modes and drift control . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4.1 Bending modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.5 Limitations of this approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

A Complete rigid body dynamic 110A.1 Rigid body dynamic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110A.2 External Forces and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 112

A.2.1 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112A.2.2 Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

A.3 Building the synthesis model . . . . . . . . . . . . . . . . . . . . . . . . . . 116

B Tuning a controller 118B.1 Introducing time domain constraint . . . . . . . . . . . . . . . . . . . . . . . 118B.2 Introducing frequency domain constraint . . . . . . . . . . . . . . . . . . . . 120

C Getting the controller in one step 122

D Time Simulation 124D.1 Interpolation of Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 124D.2 Time Simulation for the classical controller . . . . . . . . . . . . . . . . . . 126

D.2.1 Time simulation with α as feedback . . . . . . . . . . . . . . . . . . 126D.2.2 Time simulation with z as feedback . . . . . . . . . . . . . . . . . . 129D.2.3 Time simulation with z as feedback . . . . . . . . . . . . . . . . . . 131D.2.4 Time simulation with z + z as feedback . . . . . . . . . . . . . . . . 133

D.3 Time Simulation for the robust controller . . . . . . . . . . . . . . . . . . . 135D.3.1 Time simulation for the rigid body controller . . . . . . . . . . . . . 135D.3.2 Time simulation including the bending modes . . . . . . . . . . . . . 136D.3.3 Time simulation for the drift control . . . . . . . . . . . . . . . . . . 137

E Robustness 138E.1 Robustness of the classical controller . . . . . . . . . . . . . . . . . . . . . . 138E.2 Robustness of the µ− synthesis controller . . . . . . . . . . . . . . . . . . . 140

10

List of Figures

1.1 Study Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2 Structure of the Vega Launcher . . . . . . . . . . . . . . . . . . . . . . . . . 211.3 Vega Mission Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.4 Launch vehicle forces and torques in flight . . . . . . . . . . . . . . . . . . . 241.5 Evolution of k1 and a6 as a function of time . . . . . . . . . . . . . . . . . . 271.6 Flexible LV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.1 Model with uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2 Maximum roll rate allowed . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3 Low frequency gain margin as a function of the roll rate . . . . . . . . . . . 362.4 Synthesis model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5 Model with a first stabilisation (H) . . . . . . . . . . . . . . . . . . . . . . . 39

3.1 Roots evolution depending on Kp value . . . . . . . . . . . . . . . . . . . . 413.2 Model with Kp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 Model with Kd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Roots evolution depending on Kd value . . . . . . . . . . . . . . . . . . . . 423.5 Model with Kp and Kd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.6 Bode and Step response for GK

1+GK . . . . . . . . . . . . . . . . . . . . . . . . 44

3.7 Black plot for GK(S) Bode and Step response for GK1+GK . . . . . . . . . . . 45

3.8 Evolution of k1 and a6 as a function of time . . . . . . . . . . . . . . . . . . 453.9 Evolution of W and ξ as a function of time . . . . . . . . . . . . . . . . . . 463.10 Evolution of Phase Margin and Roots as a function of time . . . . . . . . . 473.11 Frequency evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.12 Evolution of Kp and Kd as a function of time . . . . . . . . . . . . . . . . . 483.13 Delay Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.14 Rigid body model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.15 Black plot for GK(S) and Bode and Step response for GK

1+GK . . . . . . . . 52

3.16 Black plot for GK(S) and Bode and Step response for GK1+GK . . . . . . . . 53

3.17 Theta response for a Step in αw . . . . . . . . . . . . . . . . . . . . . . . . . 543.18 Step response with αw = 0.03 and θref = 0 as input . . . . . . . . . . . . . . 553.19 Step response with αw = 0.03 and θref = 0.08 as input . . . . . . . . . . . . 553.20 Root locus depending on parameter Kα . . . . . . . . . . . . . . . . . . . . . 563.21 Alpha response for a step with different gain feedback . . . . . . . . . . . . . 573.22 Alpha response for a step with different gain feedback . . . . . . . . . . . . . 58

11

LIST OF FIGURES

3.23 Root Locus plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.24 θ and drift response for a θref = 0.08 . . . . . . . . . . . . . . . . . . . . . . 603.25 Theta response for a Step in αw . . . . . . . . . . . . . . . . . . . . . . . . . 603.26 Root Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.27 Theta response for different Kz values . . . . . . . . . . . . . . . . . . . . . 623.28 Theta response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.29 Step Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.30 Black plot of G(s) and GK(s) . . . . . . . . . . . . . . . . . . . . . . . . . . 663.31 Black plot for GK(S) and Bode and Step response for GK

1+GK . . . . . . . . 663.32 Step Response and close loop Bode plot . . . . . . . . . . . . . . . . . . . . 67

4.1 Evolution and boundaries of Kp and Kd on time . . . . . . . . . . . . . . . 694.2 Evolution and boundaries of A6,W and k1 as a function of time . . . . . . 694.3 Root Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.4 Nyquist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.5 Root Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.6 W-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.7 W-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.8 LFT Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.9 N∆-structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.10 M∆- structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.11 Rigid body controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.12 3D plot of the controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.13 S,T,KS,and Alpha Bode plots . . . . . . . . . . . . . . . . . . . . . . . . . . 824.14 Margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.15 Shapes of Wflex filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.16 Rigid body with bending modes system . . . . . . . . . . . . . . . . . . . . . 854.17 Shape of a filter designed using µ− synthesis . . . . . . . . . . . . . . . . . 864.18 3D plot of the notch filter and the complete controller . . . . . . . . . . . . 864.19 S,T,KS,and Alpha Bode plots . . . . . . . . . . . . . . . . . . . . . . . . . . 874.20 Synthesis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.21 Weighting function for the wind input . . . . . . . . . . . . . . . . . . . . . 894.22 Comparison between both rigid body controllers . . . . . . . . . . . . . . . . 894.23 Shape of the Kdrift controller . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.1 Pitch response for different roll rates (p) . . . . . . . . . . . . . . . . . . . 945.2 Yaw response for different roll rates (p) . . . . . . . . . . . . . . . . . . . . 945.3 Multi-axes rigid body model . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4 Multi-axis rigid body model . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.5 Controller shape for different roll rates . . . . . . . . . . . . . . . . . . . . 975.6 Roll rate profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.7 S,T,KS,and Alpha Bode plots . . . . . . . . . . . . . . . . . . . . . . . . . . 985.8 S and T Bode plot from θref . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.9 KS, Alpha and Beta bode plot from θref . . . . . . . . . . . . . . . . . . . . 995.10 S and T Bode plot from ψref . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.11 S,T,KS,and Alpha bode from ψref . . . . . . . . . . . . . . . . . . . . . . . . 100

Enrique Gadea - Robust TVC Control 12

LIST OF FIGURES

5.12 Step response with and without filter . . . . . . . . . . . . . . . . . . . . . . 1015.13 Multi-axis model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.14 S,T and KS from θref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.15 S and T bode plot from θref . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.16 KS bode plot from ψref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

A.1 Thrust Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.2 Aerodynamic Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

B.1 Comparison of the step response before and after using the PD optimization tool 119

C.1 Rigid body and bending modes filter . . . . . . . . . . . . . . . . . . . . . . . 122C.2 Time simulation with drift control . . . . . . . . . . . . . . . . . . . . . . . 123

D.1 Theta response with 5s of transition duration and 0.1s . . . . . . . . . . . . 124D.2 3D shape of two different controllers . . . . . . . . . . . . . . . . . . . . . . 125D.3 Time simulation for the different controllers . . . . . . . . . . . . . . . . . . 125D.4 Time simulation for θref = 1 and αw = 0 . . . . . . . . . . . . . . . . . . . 126D.5 Time simulation for θref = 0 and αw = 1 . . . . . . . . . . . . . . . . . . . 127D.6 Time simulation for θref = 0.08 and αw = 0.03 . . . . . . . . . . . . . . . . 127D.7 Time simulation for θref = 1 and αw = 0 . . . . . . . . . . . . . . . . . . . 128D.8 Time simulation for θref = 0 and αw = 1 . . . . . . . . . . . . . . . . . . . 128D.9 Time simulation for θref = 0.08 and αw = 0.03 . . . . . . . . . . . . . . . . 129D.10 Time simulation for θref = 1 and αw = 0 . . . . . . . . . . . . . . . . . . . 129D.11 Time simulation for θref = 1 and αw = 0 . . . . . . . . . . . . . . . . . . . 130D.12 Time simulation for θref = 0 and αw = 1 . . . . . . . . . . . . . . . . . . . 130D.13 Time simulation for θref = 0.08 and αw = 0.03 . . . . . . . . . . . . . . . . 131D.14 Time simulation for θref = 1 and αw = 0 . . . . . . . . . . . . . . . . . . . 131D.15 Time simulation for θref = 1 and αw = 0 . . . . . . . . . . . . . . . . . . . 132D.16 Time simulation for θref = 0 and αw = 1 . . . . . . . . . . . . . . . . . . . 132D.17 Time simulation for θref = 0 and αw = 1 . . . . . . . . . . . . . . . . . . . 133D.18 Time simulation for θref = 0.08 and αw = 0 . . . . . . . . . . . . . . . . . . 133D.19 Time simulation for θref = 0.08 and αw = 0 . . . . . . . . . . . . . . . . . . 134D.20 Time simulation for θref = 0 and αw = 0.03 . . . . . . . . . . . . . . . . . . 134D.21 Time simulation for θref = 0.08 and αw = 0.03 . . . . . . . . . . . . . . . . 135D.22 Time simulation for the rigid body . . . . . . . . . . . . . . . . . . . . . . . 136D.23 Time simulation for the bending modes . . . . . . . . . . . . . . . . . . . . . 137D.24 Time simulation for the drift control . . . . . . . . . . . . . . . . . . . . . . 137


LIST OF FIGURES

Acronyms

AV UM Altitude and Vernier Upper ModuleCOG Center Of GravityCOP Center of PressureEOM Equations of MotionGM Gain MarginDM Delay MarginPM Phase MarginIMU Inertial Measurement UnitLTI Linear Time InvariantLTV Linear Time VaryingLPV Linear Parameter VaryingLV Launch VehicleSISO Single Input Single OutputMIMO Multi Input Multi OutputPID Proportional Integral DerivativeRHP Right Half PlaneRS Robust StabilityRP Robust PerformanceTV C Thrust Vector Control

List of Symbols

A Aerodynamic center of pressure

a1 = Lα+T−Dm

a2 = LαmU

a3 = Tm

a4 = mEℓECm

a5 = mEℓEC ·lCG+JEJy

a6 = CzαℓGA

Jy

C Nozzle pivot pointCA Coefficient of axial aerodynamic forceCLα Lift Coefficient derivativeCNα Coefficient of normal aerodynamic force slope w.r.t. the incidenceCnα Local coefficient of z-aerodynamic force slope w.r.t. the incidenceD qSCxF Aerodynamic forceFA Aeroelastic forceG Center of mass of the entire launcherN Center of mass of the nozzleJL Launcher inertia without nozzle w.r.t. GJE Nozzle inertia w.r.t. C


LIST OF FIGURES

k1 = TℓCGJy

ℓ Curvilinear coordinate from the head and positive in backward directionL Vehicle lengthℓOA Distance between O and AℓOC Distance between O and CℓOP Distance between O and the payload center of mass PℓCG Distance between C and GℓGA Distance between G and AℓGS Distance between G and SℓGU Distance between G and UℓEC Distance between N and CℓIMU Distance between vehicle head and the Inertial Measurement Unit (IMU)Lα = qSCNα

m Total mass| Mc Control torque about C for the nozzle rotationmE Nozzle massmL Launcher mass without nozzleO Origin of the body frameq = 1

2ρU2

qh The generalized coordinateQh The generalized forcesS Reference surfaceT ThrustU Longitudinal SpeedUr Air relative longitudinal speedw Wind disturbanceWind Wind vector[G;xt; yt, zt] Trajectory reference frame[G;xb; yb; zb] Body fixed reference frame[O;x, y, z] Launch Vehicle (LV) geometric reference frame

Matrices

Af Flexible state matrixAr Rigid state matrixBf Flexible control matrixBr Rigid control matrixCf Flexible output matrixCr Rigid output matrixDf Flexible feed-through matrixDr Rigid feed-through matrix


Chapter 1

Introduction

World of controllers design is very huge, complex and hard to understand. Eachmachine with a minimum level of mechanical complexity and some autonomy needs acontroller to guarantee a good behaviour and stability. Nowadays different controllertechniques design exist, including a lot of variations but it is true that depending on themachine or the system that needs to be controlled one or another procedure can be chosen.However, when control theory started, the way to avoid the open-loop problems was intro-ducing feedbacks to control some outputs. The main objective was to stabilise the systemand to guarantee some performances but without looking things like robustness. Systemswere not very complex and perhaps the need to look for robustness was not yet born.Results were good and with some classical controllers like a PID (Proportional IntegralDerivative) 90% of problems were solved and still do.

In the late 70’s early 80’s robust control was born and a new set of techniques with it.Since then, complexity of mechanisms never stop growing and consequently an evolutionin the requirements appeared. Nowadays, systems have to satisfy some performances, bestable and allow uncertainties in some parameters. For not very well-known, complex orcritical systems (like for example a launch vehicle) this is very important.Current Launch Vehicles (LV) typically use classical controllers but since mass, thrust andaerodynamic properties of the LV are varying all the time use of Gain Scheduling is neces-sary. This procedure (using a classical controller) is cost expensive and has no robustnessat all. As it will be proved, a classical controller is designed in a linearised point of timedomain and do not support any uncertainties in the design. Even more, a variation inthe nominal design value stability cannot be guarantee. This is why robust concept is soimportant and necessary in systems like a LV where there are not only some known timevarying parameters but there are also some unknown time varying parameters like windspeed for example.In addition LV are flexible structures with their own bending modes that can induce toinstability. To avoid this problems, bending modes are filtered. Nevertheless, this filterdesign is usually done by a previous experience without taking into account system per-formances.

17

CHAPTER 1. INTRODUCTION

1.1 Main Objectives

The main purpose of this study is to present the different steps to design a robustcontroller for the TVC using the µ − synthesis technique and to study the advantagesobtained comparing to a classical one. This will be done for one axis first then for multi-axis, taking the coupling between pitch and yaw into account. A robust controller includesa controller for the rigid body and a filter for the bending modes and a drift control ifnecessary. As a tool to better achieve the objective, some controllers will be designed usingthe data from the Vega Launcher. These controllers will be the base to show the maindifferences between the robust and the classical control, but in any case they represent aboundary of what robust control can achieve. For sure some of them can be improved byinvesting more time and resources. The main constraint, as it will be explained further,is the non-convexity of the problem and some times the computational constraint.

Making a robust controller using the µ− synthesis algorithm present the main draw-back of the non-convexity of the algorithm. That means there is no way to guarantee anoptimal result. The second problem will be the definition of the weighting functions, usedfor the synthesis model of the robust controller. To define this functions it is importantto first, have an idea of the desired closed-loop shape of the system and second, to adjustthe gains of all them to achieved the desired level of performances and robustness. It isa trial and error process where the experience of the designer and his know-how has animportant role.

There is also an important computational constraint. When defining the synthesismodel, the number of states defined increases very fast (rigid body, bending modes, weight-ing functions, controllers,etc.) producing a lot of limitations and forcing to reduce the orderof the elements thus losing in precision and in realism. Even doing so, there were somepoints where it was impossible to find a way to get any result so and interpolation of theresults from the nearest points was necessary.

This problem was not only present for the design of the controller but it also wasan important constraint for the time simulations too where the controllers were alwaysreduced.

1.2 Study Logic

The study logic of this document is shown in figure 1.1

1. The methodology of this study is focused in the progressive increase of the modelcomplexity together with the controller complexity. Starting from a very simpletwo states rigid body model with a PD controller and evolving until reaching amulti-axes controller considering the rigid body dynamic with roll coupling, bendingmodes and the TVC actuator. This procedure, starting from the most simple modeland evolving into a complex one, will allow to introduce the basic notions of launchvehicles and to increase the complexity of the system gradually making the impact of



Figure 1.1: Study Logic

each new element introduced together with the interaction between all the elementseasier to understand. This kind of approach allows to manage a complex problemlike a multi-axis controller design in a more familiar and easy way.

2. First it is necessary to understand what is a Launch Vehicle and what kind of be-haviour has, so a first study of the equations of motion together with the requirements(to know what kind of result can be expected) is necessary. From this study onemay define the basic synthesis model to start working with, together with a list ofspecifications to be satisfied by the system.

3. With this first model a first classical and basic controller is implemented in orderto see how the system reacts and what are the different response and behavioursfor different inputs and perturbations. The effect of the classical controller (a PDcontroller) is studied and it will be used as a way to reach more complex and re-alistic models. It will allow to have some orders of magnitude of the possibilitiesintroduced by a controller in the system, i.e error and drift minimization, angle ofattack reduction, etc. On the other hand, all the results obtained with the classicalcontroller will be used as a reference to the ones obtained by the robust controller.

4. Taking the same equations used in the classical controller and the desired per-formances, a robust controller is designed. This controller is designed using the



µ − synthesis technique and the results are compared with the ones obtained withthe classical controller to show the advantages of a robust controller in front of theclassical controller. One critical point while designing the robust controller is thedefinition of the weighting functions. This will be managed thanks to the knowledgeof the system behaviour, acquired during the study and implementation of a classicalcontroller. With the weighting functions defined, a first controller is designed and infunction of the performances achieved, the weighting functions will be modified. It isa trial and error process but knowing the effect produced in the system when modi-fying the weighting functions, a good solution can be found quickly. The robustnessresults of each controller are compiled in annex E.

5. Once the one axis controller is done a new study will be performed introducingthe basis for the multi-axes situation. First of all, the rigid body dynamics will bereviewed because when introducing the multi-axes there are some couplings effects toadd to the equations. After modifying the rigid body dynamics and using the resultsand the knowledge from the classical and the robust controller, a new controllerwill be designed in a robust way using the µ − synthesis technique and taking theroll-coupling into account.

6. For the classical controller and for the robust controller in one axis a rigid bodycontrol is done first, then the bending modes are taken into account and a driftcontrol is done at the end. For the drift control design, the bending modes are nottaken into account.

7. For each controller designed a complete time simulation is done. All the plots canbe found in annex D

1.3 System Description

1.3.1 The Vega Launcher

While this study can be used for any launch vehicle or rocket, on first focus for imple-mentation is the Vega Launcher.

The Vega Launcher was developed within a European Program organised under theaegis of the European Space Agency. The launcher’s prime contractor is ELV S.p.a, a jointcompany of Fiat Avio and the Italian Space Agency (ASI) but it also has the support ofBelgium, the Netherlands, Spain, Sweden, Switzerland and France. Although there is agrowing tendency for satellites to become larger, there is still a need for a small launcher toplace 300 to 2000 kg satellites, economically, into the polar and low-Earth orbits used formany scientific and Earth observation missions. Europe’s answer to these needs is Vega,named after the second brightest star in the northern hemisphere. Vega makes access tospace easier, quicker and cheaper.

Costs are being kept to a minimum by using advanced low-cost technologies and byintroducing an optimised synergy with existing production facilities used for Ariane launch-ers. Vega has been designed as a single body launcher with three solid propulsion stages



and an additional liquid propulsion upper module used for attitude and orbit control, andsatellite release. Unlike most small launchers, Vega will be able to place multiple payloadsinto orbit. Development of the Vega launcher started in 1998. The first launch is plannedfor December 2011 from Europe’s Spaceport in French Guiana where the Ariane-1 launchfacilities have been adapted for its use.

History

The origins of the Vega program go back to the early 90th, when some studies wereperformed to investigate the possibility of complementing the Ariane family with a smalllaunch vehicle using Ariane solid booster technology. Vega began as a national Italianconcept. BPD Difesa y Spazio in 1988 proposed a vehicle to the Italian Space Agency toreplace the retired US Scout launcher by a new one based on the Zefiro motor developedfrom the company’s Ariane expertise.After about ten years of definition and consolidationactivities, the Italian Space Agency and Italian industry proposed Vega as a Europeanproject based on their know-how in solid propulsion taken from development and pro-duction Ariane 4 solid strap-on boosters (PAP) and components of the Ariane 5 solidstrap-on boosters (EAP). In April 1998, ESA’s Counsil approved a Resolution authorizingpre-development activity. As a result, the present configuration was chosen with first stagethat could serve also as an improved Ariane-5 strap-on. The Vega program was approvedby ESA Ariane Programme Board on 27-28 November 2000, and the project officiallystarted on 15 December 2000 when seven countries subscribed to the Declaration.

Figure 1.2: Structure of the Vega Launcher

The Vega Launcher has four stages : three with solid rocket motors and one liquidpropellant stage. The term stage is used to refer to a complete element of a launchvehicle. By complete element is meant propellant tanks, one or more engines and electrical,mechanical and fluid equipment.

Typical mission profiles

A typical mission profile consists of the following three phases:

1. Phase I: Ascent of the first three stages of the LV into the low elliptic trajectory(sub-orbital profile);

2. Phase II: Payload and upper stage transfer to the initial parking orbit by first AVUMburn, orbital passive flight and orbital manoeuvres of the AVUM stage for payloaddelivery to final orbit;



3. Phase III: AVUM deorbitation or orbit disposal manoeuvres.

Figure 1.3: Vega Mission Profile

Phase I: Ascent of the first three stages The flight profile is optimized for eachmission. It is based on the following flight events:

• 1st stage flight with initial vertical ascent, programmed pitch manoeuvre and a zero-incidence flight;

• 2nd stage zero-incidence flight;

• 3rd stage flight, fairing separation and injection into sub-orbital trajectory.

The typical Vega three-stage ascent profiles and associated sequence of events are shownin figure 1.3

Phase II: AVUM flight profile After third stage separation at the sub-orbital tra-jectory the multiple AVUM 1 burns are used to transfer the payload to a wide variety ofintermediate or final orbits, providing the required plane changes and orbit raising. Up to5 burns can be provided by the AVUM to reach the final orbit or to deliver the payloadto different orbits. Additionally, at the first burn, AVUM can provide the compensationof up to 3σ errors accumulated during the first three stage flight.

1The AVUM is the 4th multifunctional stage of the Vega launch vehicle that is designed to pilot thefirst three stage flight, to finalise orbit injection, to increase injection accuracy and to provide orbitalmanoeuvres and payload separation.



PHASE III - AVUM deorbitation or orbit disposal manoeuvre After spacecraftseparation and following the time delay needed to provide a safe distance between theAVUM and the spacecraft, the AVUM typically conducts a deorbitation or orbit disposalmanoeuvre. This manoeuvre is carried out by an additional burn of the AVUM mainengine.

1.3.2 System elements

In this study the first stage is the only one that will be studied, specifically the TVC.Because all the attention is put in the TVC control, the other stages are not importantso they will not be included at any point. To make possible the study it is importantto know the EOM of the launcher including the rigid body dynamics and the bendingmodes. These are the two main elements that are going to manage the behaviour of thelauncher during its flight. However, to increase the proximity between the designed modeland reality, the TVC actuator is also included

Rigid Body

The rigid LV dynamics are not the only motion that will appear in the final EOM, butare also interesting on their own since they can be used to make preliminary considerationsas regards to controller design.

One can start the derivation of the EOM by considering the forces and torques thatact on the LV during its atmospheric flight. These are shown in figure 1.4 in which the LVis depicted as ascending along a predetermined trajectory. The reference frames, angles,forces, torques, etc. appearing in figure 1.4 are the following:

α = arctan( zb−Wxb ) angle of attack w.r.t. the x-body axis

U velocity vectorW wind vectorU air relative velocityFA aerodynamic force applied at the aerodynamic point AT thrustFE engine nozzle inertia forceME engine nozzle inertia torquemg weight

Gravity One force present all over the flight is gravity. The force of gravity will bepresent in axes x and z. The expression of this force resolved along the body axes is

Fxg = −mg sin θFzg = mg cos θ

(1.1)

Thrust The thrust due to the rocket engines is one of the major forces acting on the LVduring its flight. Swivelling of the thrust vector is also the primary means by which theLV attitude is controlled.



Figure 1.4: Launch vehicle forces and torques in flight

The forces and torque generated are given by:

FTx = T (t) cos(δ)

FTz = −T (t) sin(δ)MT = −T (t)ℓCG sin(δ)

(1.2)

where ℓCG is the distance between the nozzle swivel point C and the center of gravity G.Equation A.15 emphasize that the thrust is a function of time.

Aerodynamic Forces and Torques The interaction between the LV and the atmo-sphere in which it flies generates aerodynamic forces and torques. This interaction is onlysignificant during the early stages of flight and typically has a destabilizing effect on theLV dynamics.

The forces and torques due to aerodynamic loads will be derived using quasi-steady-state aerodynamic theory. The aerodynamic components of interest for plane (pitch)motion are given by (Axial and Normal):

FAx = −q(t)SCAFAz = −q(t)SCNMAy = q(t)SCNℓGA(t)

(1.3)

S reference surface



CA coefficient of x-aerodynamic forceCN coefficient of z-aerodynamic force

Where it was assumed that FAx and FAz act in the negative xb and zb directions respec-tively as shown in figure 1.4. It is important to emphasize that the dynamic pressure qand the distance ℓGA between the aerodynamic center and the COG both depend stronglyon time.

Since the LV is not a lifting body there are no intrinsic aerodynamic torques, so thatthe expression of the aerodynamic torque in equation A.16 has the very simple expressionof a force times a length. Furthermore, FAx is simply the aerodynamic drag which is essen-tially independent from perturbations. Instead, the CN coefficient appearing in equationA.16 is typically a function of the angle of attack α, its rate α and the pitch rate θ. Alinear dependence of CN with respect to some steady state values of the aforementionedvariables, can be obtained by classical Taylor series, where the coefficients of the seriesare the classical stability derivatives. For LV having little or no lifting surfaces the onlystability derivative of real importance is typically the one associated with α, which wedenote with CNα = ∂CN/∂α. This stability derivative is, however, a function of Mach fortransonic and supersonic speeds. Furthermore, in the case of long slender LV, CNα is afunction of position along the vehicle, CNα(x) is therefore written that way to emphasizethis fact.

In order to completely define FAz one can start by introducing the local angle of attackfor the rigid/flexible LV,

αloc(x) = α+(ℓOG − x)

Uθ − ∂ξ(x, t)

∂x− ξ(x, t)

U(1.4)

where α = arctan((zb −W )xb), ℓOG is the distance between the origin and the center ofgravity of the LV and ξ(x, t) is the displacement due to the bending modes.

Since both CNα(x) and αloc(x) depend on x the whole expression must be integrate onthe whole length L of the LV in order to obtain the desired expression of the aerodynamicforce. This gives:

FAz = −1

2ρU2S

∫ L

0CNα(x)αloc(x) dx

= −1

2ρU2S

(∫ L

0CNα(x) dxα+

1

U

∫ L

0CNα(x)(ℓOG − x) dx θ

+∑i

∫ L

0CNασi(x) dx qi(t)−

∑i

1

U

∫ L

0CNαϕi(x) dx qi(t)

) (1.5)

The last two terms in equation 1.5 represent the aeroelastic terms.



Rigid Launch Vehicle Dynamics The equations of the rigid motion, written withrespect to the the body axes [G;xb, yb, zb], are given in vector form by:

MG = Iω

FG = m(V + ω × V )(1.6)

where FG and MG are the forces and torques; m and I are mass and moment of inertiaand V and ω are the linear and angular velocity respectively. All the aforementionedquantities are considered relative to the center of gravity of the whole LV system.

By the moment only the motion of the LV in the x−z plane is considered and acceptingan axial symmetry of the LV, the equation1.6 can be specialized to:

Iyy θ =MG

m(xb + θbzb) = Fx

m(zb − θbxb) = Fz

(1.7)

where V = [xb, 0, zb]T and ω = [0, θb, 0]

T . Coupling between pitch and yaw will be intro-duced in 2.1.2

Above equations were written assuming that the variations of the mass (m) and mo-ment of inertia (Iyy) with respect to time are small compared to the dynamics of interest.

In order to obtain the final form of the EOM one must define the forces and torquesthat appear in the above equation. Due to the fact that the LV typically flies at very smallangles of attack αmax, one can assume that the normal and axial aerodynamic forces arejust the lift and drag respectively.

MG = ℓGA Lα α− T ℓCG sin δ

Fx = T cos δ −D −mg sin(θb)

Fz = −Lαα− T sin δ +mg cos(θb)

(1.8)

where Lα ≃ qSCNα, D ≃ qSCA and the angle of attack is given as α = arctan( zb−Wxb ).Combining the equations 1.7 and 1.8 together:

Iyy θb = ℓGA Lα α− T ℓCG sin δ

m(xb + θbzb) = T cos δ −D −mg sin(θb)

m(zb − θbxb) = −Lαα− T sin δ +mg cos(θb)

(1.9)

The above equations can be rewritten in a more LV ”traditional” way by introducing theaerodynamic parameter a6 = LαℓGA

Iyyand the controllability parameter k1 = TℓCG

Jyso that

the equation1.9 becomes:

θb = a6α− k1 sin δ

xb =T cos δ −D

m− g sin θb − θbzb

zb = −Lαmα− T

msin δ + g cos θb + θbxb

(1.10)



The nominal variations with time of the aerodynamic parameter a6 and the controllabilityparameter k1 are shown in 1.5.

10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3

3.5

time

A6

(a) a6 evolution

10 20 30 40 50 60 70 80 900.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

time

K1

(b) k1 evolution

Figure 1.5: Evolution of k1 and a6 as a function of time

For control synthesis purposes it is necessary to linearise the rigid body motion of theLV about a trajectory fixed reference frame and to obtain a perturbed motion about thereference trajectory, given by a time scheduled look-up table. Introducing the transforma-tion matrix from the body rates to the local-level coordinates:

[T ]LO =

[cos θ sin θ− sin θ cos θ

](1.11)

The perturbed equation of motion in body axes can be obtained as:[∆x

∆z

]= [T ]LO

[xBzB

]−

[xLzL

](1.12)

Writing the body states in term of reference variables and assuming small angles thelinear perturbed equations of motions become:

∆θ = a6∆α− k1∆δ

∆x = −(LαmαL +

T

mδL

)θ

∆z =T −D

mθ − Lα

m∆α− T

m∆δ

(1.13)

Bending Modes

A schematic of the deflected shape of the vehicle in the pitch plane is shown in 1.6.The elastic deflection at any point along the vehicle is given by

ξ (x, t) =∞∑i=1

qi (t) ϕi (x) (1.14)



Figure 1.6: Flexible LV

where x is the abscissa along the LV longitudinal axis shown in 1.6, ϕi denotes the massnormalized ith mode shape in the pitch plane. Finally, qi is the generalized coordinate ofith mode which satisfies the equation:

qi + 2ζiωiqi + ωi2qi = Qi (1.15)

where Qi is the generalized force associated with the ith mode and is given by:

Qi =

∫ L

0

(∑Fzϕi(x) +

∑Myσi(x)

)dx (1.16)

where the local rotation is

σi =∂ϕi∂x

(1.17)

Qi is thus a function of the moments and normal forces acting on the LV, and indicateshow a particular bending mode is coupled to all other dynamic modes of the system.Considering only first order effects, the ith bending mode is excited primarily by rocketengine effects:

Qi =

∫ L

0

[−(Tδ +mEℓEC δ)ϕi(x) + JE δ

∂ϕ(x)

∂x

]dx

=(−Tδ −mEℓEC δ

)ϕi(ℓOC) + JE δ

∂ϕ(ℓOC)

∂x

(1.18)

where ℓOC is the coordinate of the pivot point.

In order to have a launch vehicle autopilot controlled it is necessary to measure the LVattitude(angular displacement and/or rate) and position (acceleration). As already men-tioned, these measurements are typically obtained by an IMU, which, however, measure



not only the rigid body motion (the principal quantity interesting for trajectory following)but also the local elastic distortion at the IMU location ℓOU .

The local rotation due to the bending modes produces an output of the IMU simplygiven by :

θimu = θ − ∂ξ(x, t)

∂x

∣∣∣∣x=ℓOU

= θ −∑i

qi(t)∂ϕi(x)

∂x

∣∣∣∣x=ℓOU

= θ −∑i

qi(t)σi(ℓOU )(1.19)

As can be observed in equation 1.19 each mode gives a contribution θi to rotation outputof the IMU given by,

θi = qi(t)σi(ℓOU ) (1.20)

It is possible thus to consider the transfer function from the excitation of the modegiven by δ to the output at the IMU for the ith mode:

θiδ

=

[mEℓECϕi(ℓOC)− JE

∂ϕ(ℓOC)∂x

]s2 + Tϕi(ℓOC)

s2 + 2ζiωis+ ωi2σi(ℓOU ) (1.21)

If the engine inertia effects are neglected the above expression simplifies to

θiδ

=Tϕi(ℓOC)σi(ℓOU )

s2 + 2ζiωis+ ωi2(1.22)

TVC actuator

The dynamic performance of the TVC/Nozzle assembly can be represented as twoserial connected transfer functions: a second order model for simulating the nozzle and apure delay for the actuator response:

WTV C(s) =e−Ts

B2s2 +B1s+B0(1.23)

Nevertheless, the simulated TVC will omit the time delay.


Chapter 2

Scientific context

The main point of this study is to design a robust controller for the TVC control, afilter for the bending modes and to take the coupling between pitch and yaw into account.However, it is good to take a look to the actual scientific context about this matter notonly to take some references and ideas but also to see the possible deficiencies and howthis study can contribute.

2.1 Technical specification

2.1.1 General requirements

It is necessary to know the basic requirements expected for the closed-loop behaviourof the system. These requirements guarantee the desired behaviour and stability [2]. It istrue that there are much more needed requirements to take into account but a selectionof the most important will be enough to show what is expected to get.

• 6db Gain Margin for Low Frequency

• 20db Gain Margin for High Frequency

• 30 Phase Margin degrees

• Stability : Stability must be guarantee

• Tracking and drift minimization : This point is quite important for the mission suc-cess since a launcher not following the desired trajectory, even stable, is useless.

• Aerodynamic load minimization. It is important to avoid having excessive loads onthe launcher, and even more important is to avoid overpassing the maximum allowed.

31

CHAPTER 2. SCIENTIFIC CONTEXT

When the launcher enters in the maximum dynamic pressure it is more importantto keep the launcher structurally safe than guarantee any tracking.

2.1.2 Problems of launchers control

Control of launch vehicles has some problems that need to be studied. The main pointsare the introduction of uncertainties into the model due of the non-well known parametersor modes, the time variant parameters like mass, thrust or the aerodynamic coefficientsand the coupling between pitch and yaw due to roll.

Introducing the uncertainties

First problem appears when introducing the uncertainties in the model. One possiblesolution was presented by Dale F. Enns [10]. In his article, Dale proposed to introduce amultiplicative uncertainty for the forgiven modes or the truncated ones, two to compensatethe inaccuracy when placing the poles and the zeros of the bending modes and anotherone for the aerodynamic parameters (see figure 2.1). Of course it is necessary to introducesome weighting functions to expand properly the effect all long the frequencies.

Figure 2.1: Model with uncertainties

These uncertainties will be also used for the robustness study.



Coupling between axes

When having some roll, pitch and yaw are coupled following the expression:

∆ψ = q + p ·∆θ∆θ = r − p ·∆ψ

(2.1)

q = a6 · (∆ψ +Z

V) + (1− Ixx

Iyy) · p · r +K1 · βψ

r = a6 · (∆θ +Y

V) + (1− Ixx

Iyy) · p · q +K1 · βΘ

(2.2)

Z = −a · (∆ψ +Z

V)− p · Y − γ ·∆ψ − γT · βψ

Y = a · (∆Θ +Y

V) + p · Z + γ ·∆θ − γT · βθ

(2.3)

with:

a6 =1

2ρV 2Sref

CNα(xcp − xCoG)

Iyy> 0

a =1

2ρV 2Sref

CNαmass

(2.4)

If roll is null or very small the hypothesis of having two split axis is valid and thecontrol can be done individually for each axes but if the roll rate is important the couplingexist.

Two different cases where presented by Lazennec [12]

• feedback en (∆ψ,∆θ) for the attitude error,(q,r) for the angular rate;

• feedback en (∆ψ,∆θ) for the attitude error, (∆ψ,∆θ) for the angular rate;

Making the necessaries modifications in the equations and defining the closed-loopfrequency and damping as:

wp =√

−(A6 +K1 · kψ 2ξwp = −K1 · kψD

The global system can be written:∆ψq

∆θr

=

0 1 p 0

−w2p −2ξwp −2vpξwp (1− Ixx

Iyy)p

−p 0 0 1

2vpξwp −(1− IxxIyy

)p −w2p −2ξwp

∆ψq∆θr

(2.5)

This coupling can even destabilise the system. Taking the precedent matrix and usingthe stability condition from Routh criteria in the determinant one can see that stability



depends on the chosen feedback. Being v a parameter having 0 as value when using thefirst feedback and 1 when using the second one, from Routh conditions the only one activeis:

vp2(v − IxxIyy

)− w2p < 0

If v = 0 then stability is always guarantee, but not for v = 1 This study forgot the effectof the lateral feedback necessary to control the drift. A study done by Roux et Cruciani[5] shows that when introducing this new feedback both cases become unstable and aboundary of the roll rate accepted appears for each moment of the fly

Figure 2.2: Maximum roll rate allowed

Roux and Cruciani [5] mention the drawbacks of using a SISO approach to study thecoupling. One cannot be sure that the margins found reflect reality. Nevertheless, thesystem is not a real MIMO neither, it is a SISO sytem with a coupling increasing as rollrate increase so the SISO approach is not so bad it just needs a refinement. ∆ is now amatrix affecting the pitch angle as well as the yaw.

∆ =

(∆βψ 00 ∆βθ

)Now, a stability study can be done for different roll rates. Because there are only two

parameters to consider, stability can be plotted in plane (∆βψ,∆βθ) (figure(2.3)).

Linear Parameters Variant System

Control of launch vehicles is not an easy problem. First of all most of the parametersare not time invariant. Mass like aerodynamic coefficients or thrust are a function oftime and in addition there is an uncertainty above them making them time variant anduncertain at the same time.



Figure 2.3: Low frequency gain margin as a function of the roll rate

A lot of different studies were done about time invariant systems, but most of realsystems like the VEGA launcher have time variant parameters. To be able to use thetheorems from the linear control, systems are linearised around some equilibrium points.The most critical points are usually chosen for the linearisation because if the system isstable at that point it could be stable all the long the flight domain. The main drawbackof doing this is that it entail to have an important loss in the margins. That is whythe flight domain is linearised in different points and at each one a different control lawis designed. Problem appears when the system needs to switch from one controller tothe other one. There are two possible strategies to manage this. One option is to switchdirectly from one law to the other depending on the flight point or the different laws can beinterpolated to have only one non-linear flight law. The interpolation technique is knownas gain scheduling. At the end of the eighty’s, gain scheduling was a qualified and efficienttechnique but with a lack of theoretical justification. This gap was solved by Shammain 1988 when he defined the stability conditions for systems using gain scheduling in hisPHD thesis.

Modification of the control needs

Depending on the flight point, control needs can be very different. The main objectiveis to have a good tracking and to be able to follow a desired trajectory but it can happenthat in some situations the main objective is to reduce the angle of attack (so the loadsuffer by the launcher is reduced). This is something important because as soon as a windprofile is applied to the launch vehicle, minimizing the angle of attack and minimizing theattitude error become antagonist strategies.



2.2 Control laws used in launch vehicles

• PD + 3rd feedback

First solution found easily in the literature is a control law using a PD controllerplus a third feedback to control the drift, usually the angle of attack. It is easyto show that a PD controller can stabilise the system. The proportional term actsimmediately reducing the stationary error and thanks to the predictive phenomenaof the derivative term, the transitory error is also reduced.

With δ as the input of the system and θ as the output, the simplified model of alauncher can be written like:

θ

δ= G(s) =

−k1s2 − a6

After the introduction of the angle of attack or the drift speed as a new state andintroducing the wind as input, it is necessary to control the drift in some way. Todo that is necessary to introduce a new feedback. Different options appear andeverything will be studied with more detail further in this document. Choosing theangle of attack as feedback the control law used is:

δ = −((Kp +Kds)θ +Kαα)

With the new transfer functions one can find the necessary conditions for the driftor angle of attack minimization.

– Drift minimizationThis approach looks for the control law making (z = 0) Using the previouscontrol law z can be defined as:

zssU

=−a0

k1(Kθ +Kα − a6)(˙zssU

+ αw) (2.6)

With a0 as independent term of the closed loop determinant. If a0 = 0 thenz = 0 for any wind input

– Angle of attack minimizationMinimization of the angle of attack is necessary to reduce the load suffer bythe launcher, specially at the maximum dynamic pressure moment. While thelauncher cross this zone is more important to guarantee the structural resis-tance of the launcher than the possibles deviations on tracking. Condition tominimise the angle of attack is given by : Kθ = 0 This kind of control law isunstable but can be accepted for small time periods.



• PIDF

Use of a PD controller can be upgraded adding an integrator and a low pas filter.Introducing an integrator the stationary error following a step will be null and thefilter will dim the amplifications done in high frequency by the derivative term. ThePIDF is define like:

PIDF (s) =

(Kp +

KI

τI(s) + 1+

KD(s)

τDs+ 1

)1

τF (s) + 1

• H∞The H∞ tries to find a controller K(s) able to stabilise the system G(s) and tominimise the infinity norm of the closed loop between the inputs and the outputs.

∥Tfyw∥∞ < γ

With γ defined as a scalar Virgilio and Kamimoto [10] defined a way to introduce aH∞ controller explaining all the steps followed.

1. Creation of a synthesis modelA synthesis model like the one in figure 2.4 has to be built.

Figure 2.4: Synthesis model

2. Creation of the weighting functionsThe different weights introduced in the outputs will define the shape and theperformances of the system. It is important to know how the function should



look like and after having an idea of the shape it is necessary to make smalladjustment in a trial and error procedure.

To find for example the weighting functions necessaries for the sensitivity outputor for the complementary sensitivity one can take the simplified model with aPD controller to have an idea of the shape. However all the weighting functionswill be explained with more detail in chapter 4.

3. Solve the optimization H∞ problem

4. Modify the weighting functions until find the desired performances

• µ− synthesis

Yasuhiro Morita wrote in 2004 [8] a way to introduce the µ − synthesis in thelauncher’s world. First of all it is important to remark than rigid body poles of alauncher are unstable by default. Because of that µ− synthesis results could not beas good as expected. Morita propose to do a first stabilisation of the system and tomake use of the µ− synthesis afterwords. Calling H the necessary feedback for thefirst stabilisation and F the controller found with the µ − synthesis technique, thefinal controller will be a combination of both 2.5.

h

Figure 2.5: Model with a first stabilisation (H)


Chapter 3

Classical controller

3.1 Launch Vehicle Modelling

To create the most basic model of LV one can start taking the equation of the pitchangle from the EOM developed in section 1.3.2

θ = a6α− k1 sin(δ)

And working with small angles and accepting than α = θ the equation can also berewritten as

θ = a6θ − k1(δ) (3.1)

Doing that it is now easy to define the transfer function between θ and δ and so the firsttwo states model is defined

θ

δ= G(s) =

−k1s2 − a6

(3.2)

Poles are in ±√a6 and due to the fact that a6 is always bigger than 0, one pole has

a positive real part and the system is then unstable. However, it is known by classicalcontrol experience that a PD controller will stabilize the model. Adding the controllerinto the system, the open-loop transfer function is

K(s) = Kp +Kds

GK(s) =−k1(Kp +Kds)

s2 − a6

(3.3)

3.1.1 Impact of Kp and Kd

Knowing that a PD controller can stabilise the system it is useful to see first the impactthatKp andKd have in it. To do that and to see how the eigenvalues evolves, a first systemwith only a proportional controller first and with only a derivative one after was created.

39

CHAPTER 3. CLASSICAL CONTROLLER

• Kp

−8 −6 −4 −2 0 2 4 6 8−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(a) roots position Kp>0

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−8

−6

−4

−2

0

2

4

6

8

(b) roots position Kp<0

Figure 3.1: Roots evolution depending on Kp value

Adding a proportional controller in the loop has a direct impact into the frequency.It is easy to see that the system is going to have two pure imaginaries poles or two realones being then one stable and the other one unstable. In any case damping is not goingto be controlled. Figure 3.1 shows that roots can only be moved all long the axis whatmeans that damping cannot be controlled. The system cannot be stabilized with only aproportional controller.

The transfer function of the system in open-loop is

GK(s) =−k1Kp

s2 − a6(3.4)

and in closed-loopGK

1 +GK(s) =

−k1Kp

s2 − (a6 + k1Kp)(3.5)

Figure 3.2: Model with Kp

The closed-loop poles can be easily found solving the equation

s2 − (a6 + k1Kp) = 0

s =√a6 + k1Kp

(3.6)



As it was said before, roots can only be real or complexes conjugates, depending on ifa6 + k1Kp > 0 or a6 + k1Kp < 0

• Kd

The introduction of Kd in the system will add a term in s making the damping con-trollable. The new transfer function will be:

GK(s) =−k1Kds

s2 − a6(3.7)

And in close loopGK

1 +GK(s) =

−k1Kds

s2 − k1Kds− a6(3.8)

Figure 3.3: Model with Kd

And closed-loop roots are going to evolve like in picture 3.4

−10 0 10 20 30 40 50 60−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(a) roots position Kd>0

−60 −50 −40 −30 −20 −10 0 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(b) roots position Kd<0

Figure 3.4: Roots evolution depending on Kd value

Proceeding in the same way than before it can be inferred than system’s roots areplaced in

s =k1Kd ±

√(k1Kd)2 + 4a62

(3.9)



The only way to have a stable system is having Kd < 0 and |k1Kd| > (k1Kd)2 + 4a6

• Kp and Kd

Combining now both controller, either frequency, damping or both of them can beaffected. The system is totally controllable. Thanks to Routh criterion a first approachabout Kp and Kd values can be done. Routh criterion impose that all the coefficientsmust have the same sign for having a stable system. So Kd < 0 and Kp <

−a6K1

The newtransfer function will be:

GK(s) =−k1(Kds+Kp)

s2 − a6(3.10)

And in close loopGK

1 +GK(s) =

−k1(Kds+Kp)

s2 − k1Kds− (a6 + k1Kp)(3.11)

Figure 3.5: Model with Kp and Kd

3.1.2 Finding the value of Kp and Kd

Now that the system is totally controllable, the necessary value of Kp and Kd to getthe desired performances can be found from the requirements:

1. 6db gain margin in low frequency (w → 0)

2. 30 degrees phase margin at high frequency

From the first specification

GK(w) =−k1(Kdjw +Kp)

−w2 − a6

GK(0) =−k1Kp

−a6= −2

Kp =−2a6k1

(3.12)



and from the second one

GK(w) =−k1(Kdjw +Kp)

−w2 − a6= 1|150◦ = −0.866− j0.5 (3.13)

Equating real parts−k1 ·Kp = 0.866(w2 + a6) (3.14)

Introducing the above relation of Kp

w =√1.3a6 (3.15)

Now, equating imaginary parts

−k1 ·Kd · w = 0.5(w2 + a6) (3.16)

and replacing the value of w

Kd =

√a6

−k1(3.17)

The desired open-loop transfer and closed-loop function can now be written :

GK(s) =

√a6s+ 2a6s2 − a6

GK

1 +GK(s) =

√a6s+ 2a6

s2 +√a6s− a6

(3.18)

−40

−30

−20

−10

0

10

Mag

nitu

de (

dB)

10−1

100

101

102

−135

−90

−45

0

Pha

se (

deg)

Bode G

Frequency (rad/sec)

(a) Bode plot

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

Step Response

Time (sec)

Am

plitu

de

(b) Step Response

Figure 3.6: Bode and Step response for GK1+GK



nichols plot

Open−Loop Phase (deg)

Ope

n−Lo

op G

ain

(dB

)

−360 −315 −270 −225 −180 −135 −90 −45 0−40

−30

−20

−10

0

10

20

30

40

6 dB

3 dB

1 dB

0.5 dB

0.25 dB

0 dB

−1 dB

−3 dB

−6 dB

−12 dB

−20 dB

−40 dB

(a) Nichols plot

Figure 3.7: Black plot for GK(S) Bode and Step response for GK1+GK

Time variant parameters

Parameters a6 and k1 are time variant parameters like the inertia or the thrust. Theirevolution is plotted in figure 3.8.

10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3

3.5

time

A6

(a) a6 evolution

10 20 30 40 50 60 70 80 900.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

time

K1

(b) k1 evolution

Figure 3.8: Evolution of k1 and a6 as a function of time

A first idea to solve the fact that parameter are not constant on time, rest in tryingto define a controller for one point of the flight time domain (worst case point). Doingthat, the controller will be optimized for only one of all the flight domain points but if thepoint has been chosen properly, the system could stay stable(this needs to be checked aposteriori). Nevertheless, LV performance will change from one point to the other.The transfer function is:

GK

1 +GK(s) =

−k1(Kds+Kp)

s2 −K1Kds− (a6 + k1Kp)(3.19)



So, frequency and damping are going to be respectively

w =√

−a6 − k1Kpξ =−k1Kd

2a6(3.20)

and are going to be time variant as well.

(a) Frequency evolution (b) Damping evolution

Figure 3.9: Evolution of W and ξ as a function of time

In figure 3.9 the evolution of frequency and damping in time is plotted. Dampingstarts in a very low level and starts to increase reaching the nominal and design value of0.5 at 50s and then it decrease again. With these variations specifications where a constantdamping value of 0.5 are not satisfied.

Evolution of Kp and Kd

In the analysis done before (3.1.2) Kp and Kd were written in function of parametersa6 and k1. That means that new values of Kp and Kd can be calculated for each slide oftime. Calculating again their values the performances achieved will not be modified, sodamping and phase margin will stay constant (see figure 3.10).

Damping stays constant but the frequency (being an a6 function) varies with time.The profile for the W variation is plotted in figure 3.11.

The evolution of Kp and Kd can be deduced from the equations 3.12 and 3.17 and isplotted in figure 3.12.

To end with, taking a look to the delay margin (figure 3.13) it can be inferred that thecritical point is near to the 50s (maximum dynamic pressure point).



−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Real Part

Imag

Par

t

(a) Roots evolution

10 20 30 40 50 60 70 80 9081.7868

81.7868

81.7868

81.7868

81.7868

81.7868

81.7868

time [s]

Pha

se M

argi

n [D

egre

es]

(b) Phase Margin Evolution

Figure 3.10: Evolution of Phase Margin and Roots as a function of time

10 20 30 40 50 60 70 80 90

0.8

1

1.2

1.4

1.6

1.8

2

time [s]

Fre

quen

cy

Figure 3.11: Frequency evolution

3.1.3 System Behaviour

It is possible then to guarantee stability with a fixed PD controller but loosing thedesired performances unless a dynamical controller is implemented. However, there aresome drawbacks in this technique. First there are implementation aspects. To be ableto keep the desired goal of the linear designs, the design points must be relatively closetogether. This means that a large of linear controllers must be implemented and in addi-tion a scheme to prevent inactive controllers to drift away (bump-less transfer). Secondly,the switching surfaces need to be checked for stability in a post analysis. Nevertheless,main drawback of the interpolation scheme is that stability cannot be guaranteed even ina neighbourhood of a design point.



10 20 30 40 50 60 70 80 90−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

time

Kp

(a) Kp evolution

10 20 30 40 50 60 70 80 90−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

time

Kd

(b) Kd evolution

Figure 3.12: Evolution of Kp and Kd as a function of time

10 20 30 40 50 60 70 80 900.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

time [s]

Del

ay M

argi

n [s

]

Figure 3.13: Delay Margin

Looking to figure 3.12 one can notice how Kp and Kd evolve and looking carefully tothe a6 time evolution (figure 3.8) the flight domain can be split in two slides: from 0s to50s and from 50s to 100s. Maximum value of a6 is placed more or less for t=50s (point ofmaximum dynamic pressure).

1. From t=0 to t=50sIn this slide, a6 increase its value and k1 decrease it. Taking back the condition forstability we need:

−k1Kp − a6 > 0 → Kp <−a6k1

(3.21)

So it is normal that Kp increase its value in absolute value talking. For Kd we only



need to impose a 0.5 damping

ξ =1

2→ 1

2=

−k1Kd

2w→ Kd =

w

−K1→ Kd =

√a6

−K1(3.22)

2. From t=50s to t=100sNow both parameters decrease but a6 goes faster than k1 so Kp and Kd also decrease(in absolute value)

The following table contains all the different value of Kp and Kd that will be used intime simulations for each couple of a6 and k1 value and the time slices done.

a6 k1 Kp Kd time

0.4087 5.9938 0.1364 0.1067 10-151.0817 6.1897 0.3495 0.1680 15-252.5896 5.4397 0.9501 0.2955 25-352.9521 5.9690 0.9892 0.2879 35-453.2297 7.0738 0.9132 0.2541 45-553.2119 8.4980 0.7559 0.2109 55-652.5692 10.4260 0.4928 0.1537 65-751.6461 12.2962 0.2677 0.1043 75-850.8255 15.0942 0.1094 0.0602 85-95



3.2 Complete Rigid Model

A PD controller can guarantee stability and performances in at least a basic two statesSISO model. However, this model was built assuming that there was no drift and no windso α = θ but when introducing the drift and the wind α = θ + z

V − αw where αw is thewind incidence, v is the LV velocity and z is the lateral drift speed.To create the complete rigid model is then necessary to add at least a third state. Onechoice is to add z as a new third state and αw as a new input. Nozzle inertia and bendingmodes still neglected. Due to this modification the equations of motion are rewritten as

θ = a6α− k1δ

z =−Lαm

α− T

mδ − T −D

mθ

α = θ +z

V− αw

(3.23)

The complete procedure to find these equations is in annex A

Rewriting the EOM in a state space form which is more convenient for controller designand analysis θ

θz

=

0 1 0a6 0 a6

V−a1 0 −a2

θ

θz

+

0 0a6 −k1a2 · V −a3

[αwδ

] θ

θz

=

1 0 00 1 00 0 1

θ

θz

+

0 00 00 0

[αwδ

](3.24)

With

a1 = Lα+T−Dm a2 = Lα

mv

a3 = Tm a6 = LαℓGA

Iyy

Lα = qSCNα k1 = TℓCGIyy

(3.25)

For the time being the wind will not be considered. Wind will act like a perturbation in thesystem. One of the advantages of robust control is to manage this kind of perturbationsand to keep the system stable. However, even if stability is guarantee some propertiescan suffer a big degradation like for example the angle of attack or the drift as it willbe shown. For this reason, the wind will be introduced again further in this study, whenmaking a drift control. The transfer function between θref and θIMU can be deduced fromthe space state model knowing the kind of feedback used u = (Kp +Kds)(θref − θIMU ).Hence transfer function in open-loop is:

Hol(s) =−(Kp +Kds)(a3

a6v + a2k1 + k1s)

s3 + a2s2 − a6s+ a1a6v − a2a6

(3.26)



Figure 3.14: Rigid body model

And in closed-loop

Hcl(s) =−(Kp +Kds)(a3

a6v + a2k1 + k1s)

s3 +B2s2 +B1s+B0

B2 = a2 − k1Kd

B1 = −k1Kp − a2k1Kd − a6 − a3a6vKd

B0 = a1a6v

− a2a6 − a3a6v

− a2k1Kp

(3.27)

It is important to know how the system reacts now that it has a new state. It isalso important to emphasize that for this study all the parameters are consider as timeinvariant. Numerical values are taken from the maximum dynamic pressure moment.

Parameter Value

a6 3.2297k1 7.0738a1 37.87a2 0.02737a3 25.54v 557Kp -0.9132Kd -0.2541

System behaviour is now different due to the new state introduced. Step performancedecrease and now the drift is also an important parameter. Bode shape is deformed dueto this new state introduction an so is the step response that is degraded but the systemstill stable.

Eigenvalues Damping Freq(rad/s)

-8.40e-1+1.54i 4.79e-1 1.75-8.40e-1-1.54i 4.79e-1 1.75-1.44e-1 1 1.44e-1



−40

−30

−20

−10

0

10

Mag

nitu

de (

dB)

10−3

10−2

10−1

100

101

102

−135

−90

−45

0

45

Pha

se (

deg)

Bode G

Frequency (rad/sec)

(a) Bode plot

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

Step Response

Time (sec)

Am

plitu

de

(b) Step Response

nichols plot


Ope

n−Lo

op G

ain

(dB

)

−360 −315 −270 −225 −180 −135 −90 −45 0−40

−30

−20

−10

0

10

20

30

40

6 dB

3 dB

1 dB

0.5 dB

0.25 dB

0 dB

−1 dB

−3 dB

−6 dB

−12 dB

−20 dB

−40 dB

(c) Nichols plot

Figure 3.15: Black plot for GK(S) and Bode and Step response for GK1+GK

Number of States Gain Margin Phase Margin Delay Margin

2 6 db 30 0.25233 5.3634db 28.3123 0.2405

The overshoot is now more than a 100% of the reference value and the final asymptoticvalue is 0,7 (for a step of 1).

Adding the TVC actuator

Adding the TVC actuator into the model will add some roll-off and will reduce thebandwidth. It is like a low-pass filter and the two poles introduced by the TVC are nat-urally stable. Of course margins will decrease a bit due to the introduction of this newelement but it is important to add this term because it will be useful to compare thisresults with the ones obtained with the robust control.

The system still stable and the new poles are:



Eigenvalues Damping Freq(rad/s)

-7.66e-1+1.65i 4.21e-1 1.82-7.66e-1-1.65i 4.21e-1 1.82-1.43e-1 1 1.43e-1-4.34e+1 1 4.34e+1-5.76e+1 1 5.76e+1

Because of the introduction of the actuator the system cross twice the line of -180ºhaving two gain margins (one in low frequency and another one in high frequency).

−300

−200

−100

0

100

Mag

nitu

de (

dB)

10−2

100

102

104

−360

−270

−180

−90

0

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

(a) Bode plot

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

Step Response

Time (sec)

Am

plitu

de

(b) Step Response

−360 −315 −270 −225 −180 −135 −90−140

−120

−100

−80

−60

−40

−20

0

20

Nichols Chart


Ope

n−Lo

op G

ain

(dB

)

(c) Nichols plot


Number of States Gain Margin LF Gain Margin HF Phase Margin Delay Margin

3 5.2641db 33.8503db 23.7623 0.2025

Adding the wind input

The introduction of a new input αw will introduce some variations and performanceswill decrease again but it is important to quantify the wind impact in each output. The



transfer function between this new input and θIMU has the same denominator (same poles)but will present a different numerator.

0 20 40 60 80 100 120−0.02

−0.018

−0.016

−0.014

−0.012

−0.01

−0.008

−0.006

−0.004

−0.002

0

Step Response

Time (sec)

Am

plitu

de

Figure 3.17: Theta response for a Step in αw

The transfer function is :

Hcl(s) =−a6s

s3 + (B2)s2 + (B1)s+B0

B2 = a2 − k1Kd


B0 = a1a6v

− a2a6 − a3a6v

− a2k1Kp

(3.28)

Because the two inputs αw and θref are going to be at the same time is also importantto see what is the effect that the wind by itself and together with θref has in the systemresponse.

0 10 20 30 40 500

100

200

300

400

500

600

(a) θIMU

0 10 20 30 40 50−0.02

−0.018

−0.016

−0.014

−0.012

−0.01

−0.008

−0.006

−0.004

−0.002

0

(b) z

Figure 3.18: Step response with αw = 0.03 and θref = 0 as input



0 10 20 30 40 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

(a) θIMU

0 10 20 30 40 50−80

−70

−60

−50

−40

−30

−20

−10

0

10

(b) z

Figure 3.19: Step response with αw = 0.03 and θref = 0.08 as input

As it has been proven, only with a PD controller the system remains stable even withthe introduction of the drift and the wind input. However, despite tracking in θ canalso be guarantee and margins are satisfied there are others requirements that cannot becontrolled only with a PD controller. This control is a requirement in order to reduce theload suffered by the launcher, so having a small angle of attack (less than 3°). To satisfyall the requirements is then necessary to introduce a new feedback, capable to control thedrift or the drift speed. It is true that θ has an impact on the angle of attack but it cannot be managed only by a θ feedback. It is not possible to keep a desired θ value whiletrying to put it at the same time near to zero to minimise the angle of attack. That iswhy it is necessary to introduce another feedback. There are some different options asnew feedback.

1.- α feedback2.- z feedback

3.- z feedback4.- z + z feedback

However, for a better understanding of the new feedback effect the PD controller willbe the same all the time, i.e it will not be tuned again.

3.2.1 α feedback

Adding a proportional term on α to the PD controller, the feedback will look like

u = (Kds+Kp) · (θref − θIMU ) +Kα · α (3.29)

If we break down alpha it is easier to understand the effect that it has in the system. Asit was said before α is composed by three terms: θ, z and αw. This feedback will allow totake into account these three parameters that directly affect the angle of attack so it willbe possible to control the angle of attack and the lateral drift in a such way.It is importantto remark that the angle of attack is not one thing that can be measured directly. It needsto be built but to do that, one has to estimate the value of the wind incidence or speedwhich is quite uncertain. In addition it is important to remark the fact that when tuningKα, θ is affected also and depending on the Kα value the system can be destabilized.



Because it exists a possibility of making the system unstable, there is a boundary on themaximum value that Kα can have.

−300 −250 −200 −150 −100 −50 0

−150

−100

−50

0

50

100

150

Root Locus Editor for Open Loop 1 (OL1)

Real Axis

Imag

Axi

s

(a) Kα > 0

−0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1

−1.5

−1

−0.5

0

0.5

1

1.5

2


Real AxisIm

ag A

xis

(b) Kα > 0 zoom

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5

0

0.5

1

1.5


Real Axis

Imag

Axi

s

(c) Kα < 0

Figure 3.20: Root locus depending on parameter Kα

Picture 3.20 shows how, while Kα varies, the system sends two poles to the right handside becoming unstable. The two complex poles that are mainly controlled are those as-sociated with the rotary motion of the vehicle about its center of gravity.



The different transfer function of θIMU , z and α with θref as input are:

θIMU

θref=

−(Kp +Kds)(a3a6v + a2k1 + k1s)

s3 +D2s2 +D1s+D0

α

θref=

−(Kp +Kds) · (a3v s2 + k1s− a1k1

v + a2k1)

s3 +D2s2 +D1s+D0

z

θref=

(Kp +Kds)(−a3s2 + a3a6 + a1k1)

s3 +D2s2 +D1s+D0

D2 = a2 − k1Kd +a3VKα

D1 = −(Kp + a2Kd)k1 − a6 −a3a6V

Kd + k1Kα

D0 =a1a6V

− a2a6 −a3a6v

Kp − a2k1Kp + (k1a2 −a1k1V

)Kα

(3.30)

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

(a) Alpha response with Kα = 0, 5

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

(b) Alpha response with Kα = −0, 3

Figure 3.21: Alpha response for a step with different gain feedback

Making Kα more negative causes an increase in the overshoot in the alpha responseas well as a reduction of the final value. However, without introducing any wind into thesystem, alpha response is very similar to theta so theta will also present an increase inthe overshoot and a reduction of the final value increasing the tracking error. There is acompromise between keeping the tracking or reducing alpha final value. A reduction inthe drift will also appear but the drift cannot be reduced without limits because drift re-ductions needs reducing Kα value but getting Kα to negative values will cause the systemto be unstable. So there is an important limitation for drift reduction.

On the other hand, having a positive value of Kα produce an increase on the drift be-cause drift rate pole is getting slower. This feedback acts more in the drift instead of actingin θ. It can be interesting if the drift level is not very high and the system can afford anincrease in the drift to get a reduction in the angle of attack and without modifying a lot θ.



0 10 20 30 40 50−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

4

(a) Alpha response with Kα = 4

0 10 20 30 40 50−3

−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

4

(b) Alpha response with Kα = 12

Figure 3.22: Alpha response for a step with different gain feedback

Wind input

Wind will have a direct impact on alpha as it can be checked in its equation. But windnot only affects alpha, it also affects θ and z so it is important to quantify the impact thatit can have on the system.

The transfer functions between α, θ, δ, z and αw can be deduced from equations 3.23and 3.29.

α

αw=

s(s2 + k1Kds+ k1Kp

s3 +D2s2 +D1s+D0

θ

αw=

−s(k1Kα − a6)

s3 +D2s2 +D1s+D0

z

αw=

a3v s

2 + (a3Kdv )(a6 + k1

a2va3

)s+ (a3Kp

v )(a6 + k1a2va3

) + (a2 − a1v )(k1Kα − a6)

s3 +D2s2 +D1s+D0

(3.31)

Applying the final value theorem f(∞) = limx→0 f(S) one can see that the functionrejects the effect introduced by the wind in at least α and θ.

To sum up, using realistic inputs in θref and in the wind input (θref = 0.08rad andwindU = 0.03) and making a feedback in α will lead to maximum drift and angle of attack

reduction of around 14% but θ will be degraded a 17%.

Time simulations can be found in annex D

3.2.2 z feedback

Making a feedback on z will introduce a new state in the system because it is necessaryto integrate z. The main purpose of this feedback is to control the drift but manoeuvra-bility is very low. By introducing this new state the system is very close to instability.Figure 3.23 shows that Kz needs to be positive and less than a very small value.



−5 −4 −3 −2 −1 0 1 2 3 4 5

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


Real Axis

Imag

Axi

s

(a) Kz < 0

−3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4


Real Axis

Imag

Axi

s

(b) Kz > 0 zoom

Figure 3.23: Root Locus plot

The transfer functions from θref to each output are:

θIMU

θref=

−s(Kp +Kds)(a3a6v + a2k1 + k1s)

s4 +D3s3 +D2s2 +D1s+D0

α

θref=

−s(Kp +Kds) · (a3v s2 + k1s− a1k1

v + a2k1)

s4 +D3s3 +D2s2 +D1s+D0

z

θref=s(Kp +Kds)(−a3s2 + a3a6 + a1k1)

s4 +D3s3 +D2s2 +D1s+D0

D3 = a2 − k1Kd

D2 = −(Kp + a2Kd)k1 − a6 −a3a6V

Kd + a3Kz

D1 =a1a6V

− a2a6 −a3a6v

Kp − a2k1Kp

D0 = (a3a6 − k1a1)Kz

(3.32)

Margin is very small and when trying to obtain any improvement before feeling theeffects the system oscillates a lot. This feedback has a direct impact on the drift butmanoeuvrability is very very poor making this feedback not very interesting. Figure 3.24shows how the system oscillates and how drift is very reduced also. This reduction is veryimportant but without having a more damped system, a feedback which time necessaryfor stabilisation is near the 100 seconds cannot be used for a system where the time liveis about 120 seconds.

In addition zeros of the closed loop are not always stable. Their value depend on Kp,Kd, a2 and a6 value and there are some realistic combinations that can make a zero appearon the right hand side.Due to the unstable zero the system will follow the command bygoing into the opposite direction first.



0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

(a) θ response

0 20 40 60 80 100−1600

−1400

−1200

−1000

−800

−600

−400

−200

0

200

(b) drift

Figure 3.24: θ and drift response for a θref = 0.08

0 20 40 60 80 100−1

−0.5

0

0.5

1

1.5

2

Figure 3.25: Theta response for a Step in αw

3.2.3 z feedback

Having feedback directly on the drift is not a good option so the derivative is possiblya better solution. Making a feedback on z will not introduce a new state and will allow usto avoid the constant oscillatory behaviour of the z feedback. The new feedback will thenbe: u = (Kp +Kd) · (θref − θIMU ) +Kz · z

In picture 3.26 we can see how the closed loop poles evolve depending on Kz value The



−6 −5 −4 −3 −2 −1 0 1 2 3 4

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5


Real Axis

Imag

Axi

s

(a) Kz > 0

−6 −4 −2 0 2 4 6 8 10 12

−3

−2

−1

0

1

2

3


Real Axis

Imag

Axi

s

(b) Kz < 0

Figure 3.26: Root Locus

transfer functions from θref to the different outputs are:

θIMU

θref=

−(Kp +Kds)(a3a6v + a2k1 + k1s)

s3 +D2s2 +D1s+D0

α

θref=

−(Kp +Kds) · (a3v s2 + k1s− a1k1

v + a2k1)

s3 +D2s2 +D1s+D0

z

θref=

(Kp +Kds)(−a3s2 + a3a6 + a1k1)

s3 +D2s2 +D1s+D0

D2 = a2 − k1Kd + a3Kz

D1 = −(Kp + a2Kd)k1 − a6 −a3a6V

Kd

D0 =a1a6V

− a2a6 −a3a6V

Kp − a2k1Kp + (a3a6 − k1a1)Kz

(3.33)

Like with alpha feedback having a positive Kz value will only produce and increaseon drift and the system will reach instability very fast. The only option then is having anegative feedback in Kz. By doing this, drift will be reduced but it exists an upper limitbefore the system reach instability.

By adding this feedback, the impact in θIMU from θref will be huge and a very im-portant reduction from the nominal value will appear. Increasing Kz will decrease theovershoot but also the final value making the error bigger and will also introduce moreoscillations at the beginning.

This feedback presents the advantage of making and important reduction on the drift

Wind input

In spite of alpha feedback effects, now the wind effect is not completely rejected fromthe system and there is an unstable zero from the wind input to the θIMU output doing



0 10 20 30 40 50−0.5

0

0.5

1

1.5

2

(a) Kz = −0.005

0 10 20 30 40 500

0.5

1

1.5

2

2.5

(b) Kz=−0.001

Figure 3.27: Theta response for different Kz values

that the systems starts going in the opposite direction

0 10 20 30 40 50−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Figure 3.28: Theta response

The transfer functions from the wind input to the different outputs are:Using this feedback having as input a combination of θref and wind makes a reduction

of about 58% in θIMU value and 77% in drift.

3.2.4 z + z feedback

This new double feedback is a mix of the last two feedbacks seen. The feedback lawwill be:

u = (Kp +Kds) · (θref − θIMU ) +Kz · z +Kz z



The z feedback had drift reduction limited and the drift feedback, even having betterresults in drift minimization, was not feasible due to the oscillatory behaviour of thesystem. However, mixing both feedback it is possible to have a bigger drift reduction thanwith the z feedback by itself and without so many oscillations like with the z feedbackThe transfer functions between θref and θIMU , α and z are:

θIMU

θref=

−s(Kp +Kds)(a3a6v + a2k1 + k1s)

s4 +D3s3 +D2s2 +D1s+D0

α

θref=

−s(Kp +Kds) · (a3v s2 + k1s− a1k1

v + a2k1)

s4 +D3s3 +D2s2 +D1s+D0

z

θref=s(Kp +Kds)(−a3s2 + a3a6 + a1k1)

s4 +D3s3 +D2s2 +D1s+D0

D3 = a2 − k1Kd + a3Kz

D2 = −(Kp + a2Kd)k1 − a6 −a3a6V

Kd + a3Kz

D1 =a1a6V

− a2a6 −a3a6v

Kp − a2k1Kp + (a3a6 − k1a1)Kz

D0 = (a3a6 − k1a1)Kz

(3.34)

This feedback will keep the non unstable zero from the wind to θIMU One can see theeffect of adding both feedbacks keeping the numerical value of Kz used in the previoussection and looking to the modifications produced by Kz

Using the same case which includes the combinations of θref and the wind input, driftis reduced by a 99% but error between θIMU and θref is about the 71% of the referencesignal.

3.2.5 Summary

After taking a look to the different possible feedbacks the two more interesting are theangle of attack feedback because of the possibility of having some drift reduction withoutloosing too much in tracking and Z + Z because of the drift minimization it can offer.The objective of this point was not to achieve an optimal configuration with the optimizedcoefficient for the different feedbacks. The objective was to show the impact that feedbackhas in the system, how drift can be minimized and what can be expected for each differentfeedback. This will be used further when a drift control will be implemented in the robustdesign. Using the results obtained in this preliminary study one can have a first idea ofhow much drift can be reduced and what the consequences are.

As a conclusion the table below contains the numerical values of a time simulation foreach different feedback. All the plots are in annex D



VariableFeedback

Nominal Alpha Z Z + Z

K Kα = −0.3 KZ = −0.005 KZ = −0.005 KZ = −0.00304

θ 0.057 0.047 0.024 0.016Z -1740 -1500 -397 -17

Z -41 -32 -8.29 -4.64e-5α -0.047 0.039 -0.02 -0.01

3.3 Flexible Launch Vehicle Dynamics

The model of the rigid launcher is typically important for preliminary studies, but foraccurate simulations and controller analysis/design purposes, the flexibility of the launchvehicle must be considered.

The flexibility of the LV presents some control problems primarily because the sensinginstrumentation (gyros and accelerometers) pick up not only the rigid body motion but alsothe local elastic distortion. Because the vehicle is a continuous body the elastic motionis described by a system with (theoretically) an infinite number of degrees of freedom(the bending modes).In practice, either a truncation of the infinite series or a lumpedmass model is used to yield a system with a finite number of modes. The number ofmodes that are significant in a given situation depends on the bandwidth of the primarybending modes. The high frequency attenuation properties due to the inherent structuraldamping, are sufficient to dismiss the higher modes from further consideration. In thecurrent generation of LV, five elastic modes are usually sufficient to describe the significantdynamic properties of the vehicle. Often no more than three modes are needed. This isthe case for our study case.

Equations in a state space form

The state vector including N bending modes is x = [θ, θ, z, q1, q1, ..., qN , qN ]T , thus

from the 1.15θ

θz

qiqi

=

0 1 0 0 0a6 0 a6

U 0 0−a1 0 −a2 0 0

0 0 0 0 10 0 0 −ω2

i −2ζiωi

θ

θz

qiqi

+

0 0

−a6U −k1a2 −a30 00 −Tϕi(ℓC)

[wδ

]

αzimu

θimu

=

1 0 1U 0 0

0 −ℓGU 1 0 ϕi(ℓimu)1 0 0 −σi(ℓimu) 0

θ

θz

qiqi

+

− 1U 00 00 0

[wδ

](3.35)



Defining the bending state xf = [q1, q1, ..., qN , qN ]T and the matrices:

Afi =

(0 1

−ω2i −2ζiωi

)Bfri =

(0

−Tϕi(ℓC)

)Cfpi =

(0 00 ϕi(ℓimu)

)Cfci =

(−σi(ℓimu) 0

)Af =

Af1 0 0

0. . . 0

0 0 AfN

Bfr =

Bfr1...

BfrN

Cfp =

(Cfp1 . . . CfpN

)Cfc =

(Cfc1 . . . CfcN

)(3.36)

the equation 3.35 can be rewritten in a more compact form:[xrxf

]=

(Ar 00 Af

)[xrxf

]+

(Brw0

)w +

(BrδBfr

)δ

yp =(Crp Cfp

)x+Drpw

yc =(Crc Cfc

) (3.37)

In the equations 3.35 the contributions of nozzle inertia are neglected.

3.3.1 Bending modes impact

The appearance of the bending modes could then lead the system to be unstable.Adding two bending modes to the rigid model defined before makes the system unstable.Nichols plot of G and GK are plotted in figure 3.30

0 0.5 1 1.5 2 2.5 3−8

−6

−4

−2

0

2

4

6

8

10x 10

12

Figure 3.29: Step Model

To solve this problem it is very usual to define a notch or a low-pass filter, to filter thebending modes effect and return the system to a rigid body behaviour. Shapes of typical



−180 −90 0 90 180−100

−80

−60

−40

−20

0

20

40

6 dB 3 dB

1 dB 0.5 dB

0.25 dB 0 dB

−1 dB

−3 dB

−6 dB

−12 dB

−20 dB

−40 dB

−60 dB

−80 dB

−100 dB

Nichols Chart


Ope

n−Lo

op G

ain

(dB

)

(a) Black G(s)

Nichols Chart


Ope

n−Lo

op G

ain

(dB

)

−360 −270 −180 −90 0 90 180−100

−80

−60

−40

−20

0

20

40

60

6 dB 3 dB

1 dB 0.5 dB

0.25 dB 0 dB

−1 dB

−3 dB

−6 dB

−12 dB

−20 dB

−40 dB

−60 dB

−80 dB

−100 dB

GtotalK*Gtotal

(b) Black GK(s)

Figure 3.30: Black plot of G(s) and GK(s)

used filters are plotted in figure 3.31. Knowing the shape it is easy to define a good filterbased in some background that the designer could have.

−140

−120

−100

−80

−60

−40

−20

0

Mag

nitu

de (

dB)

102

−90

−45

0

45

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

(a) Notch filter

−40

−30

−20

−10

0

Mag

nitu

de (

dB)

100

101

102

103

104

−90

−45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

(b) Low pass filter


However this time it is not necessary to design any specific filter. The WTV C actuator,being a second order with a static gain of one, has the same behaviour than a low-passfilter and will act as it stabilising the system. So adding the actuator in the loop thesystem will be stable again and margins will be:

Gain Margin LF Gain Margin HF Phase Margin Delay Margin

5.1833 db 16.3438 db 23.18 0.2014

The bode plot shows the resonance due to the presence of the bending modes. In afinal design it is necessary to roll-off this pic making that the first flexible modes have nota gain over -3db. This constraint is avoided because this is only a preliminary study andit is going to be used as comparison to the flexible modes. But it is important to take itinto account in the final design.



0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

Step Response

Time (sec)

Am

plitu

de

(a) Step Response having PD controller+alpha as feedback

−250

−200

−150

−100

−50

0

50

Mag

nitu

de (

dB)

10−2

100

102

104

−450

−360

−270

−180

−90

0

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

(b) Step Response having only PD con-troller

Figure 3.32: Step Response and close loop Bode plot

To check the system behaviour a simple Simulink model was created an some timesimulations were done. In this time simulations, input is always a step and parameters aretime invariant, being the numerical values of the maximum dynamic pressure. Further,when the robust control will be introduced, a non-linear time simulator will be used.However, because this is just a way to see how the system reacts it is not necessary to usethe non-linear simulator. All the results from the time simulations are in annex D


Chapter 4

Robust Control

4.1 Uncertainty Modelling

Launch Vehicles have some time variant parameters that make the controller designharder. It is necessary to check if the controller can keep stability and performances in allthe time domain and even if stability is possible, performances will suffer a big decrease.One solution, mentioned before, is the gain scheduling technique. Using this technique,some parameters are constant in some time slides increasing then the model precision.However, parameters are varying constantly and are not ever well known. As it wasshown in chapter 3 knowing the variation of a6 and k1 it is possible to keep the desiredperformances given the fact that values of Kp and Kd are adapted to those variations buta6 and k1 are not always well known.

Uncertainties or perturbations are going to modify in a random but bounded way thevalue of some parameters. Adding some uncertainties in the model will better approximatethe real system behaviour. A block-diagonal containing all the possible perturbations oruncertainties can be define as the (∆) block:

∆ =

(δa6 00 δk1

)4.1.1 Kp and Kd as a function of a6

Taking back the equation 3.18 the closed loop transfer is expressed as a function ofa6 only. That means that the value of the controller can be obtained as an expression ofa6. Just as complementary information boundaries for Kp and Kd are plotted in figure 4.1.

4.1.2 Using the PD controller defined by slides

Using the PD controller defined in section 3.1.2 it is possible to define some boundariesthat will indicate the limits of the system performance. This time, damping is not goingto be longer constant but it will vary between the two boundaries found. Figure 4.2 shows

67

CHAPTER 4. ROBUST CONTROL

0 10 20 30 40 50 60 70 80 90−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0Kp envelop

nominalboundary

(a) Kp evolution

0 10 20 30 40 50 60 70 80 90−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0Kd envelop

nominalboundary

(b) Kd evolution

Figure 4.1: Evolution and boundaries of Kp and Kd on time

all the different boundaries for each parameter.

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

nominalboundary

(a) A6 evolution

0 20 40 60 80 1004

6

8

10

12

14

16

18

20

nominalboundary

(b) K1 evolution

Figure 4.2: Evolution and boundaries of A6,W and k1 as a function of time

Looking to the roots position (Real part vs Imaginary part) like in figure 4.3 one cannotice that poles are no longer in the same damping line.

Furthermore margins change a lot depending in the uncertainty. In figure 4.4 threepossible configurations are plotted in a Nyquist plot with their gain and phase margins.

So, PD controller gives good results but without a robust response (The system can-not afford all the uncertainties defined). All the performances needs to be checked at eachpoints because there is no guarantee at all. That is why robust control is interesting. Theonly design of a robust controller will guarantee stability and will give a reference of howwell all the performances required are satisfy (γ value). In addition in a robust design itis possible to define the desired shape at some points and the uncertainty level.



−1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Real Part

Imag

Par

t


Nyquist Diagram

Real Axis

Imag

inar

y A

xis

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Figure 4.4: Nyquist

4.2 Building a new synthesis model

With the introduction of robustness a new synthesis model is required. Now it isimportant to define and to make a difference between the control and the measurementoutputs as well as between the reference and noise inputs. Using the same procedure thanin the previous chapter, a rigid body controller is first design and after bending modesand drift minimization is taken into account. The rigid body used keeps the three statesdefined but will have some uncertainties. The TVC actuation system will also be includedto increase the realism of the model.The wind input will not be considered for the design at the beginning but will be takeninto account for the drift control, where wind has a real and more important role.

The new system will look like the one in figure 4.5. The elements of this new systemare:




• The plant G modelled with two uncertainties. One in a6 and another one in k1 θ

θz

=

0 1 0a6 0 a6

V−a1 0 −a2

θ

θz

+

0−k1−a3

[δ]

θ

θz

=

[1 0 01 0 1/v

] θ

θz

+

000

[δ]

(4.1)

• TVC actuation system with an uncertainty in frequency

WTV C =e−t1

12 · ξ · wtvcwtvc2(4.2)

with wtvc having a 15% of uncertainty

• Wi Weighting functions

W1 =0.15s+ 1.5

s+ 0.01

W2 =2 · a6 + 100

a6·

1s2 +√a6 + a6

√a6s2 + (2 · a6 + 100 · √a6)s+ 2 · a6 + 100

W3 = 0.7

Walpha = 3.4 · s+ 200

s+ 1000

Wext = 0.5

Wnoise = 0.01

(4.3)

4.2.1 Weighting Functions

The weighting functions will allow to introduce the desired behaviour of the system.Forexample W1 and W2 will control the shape of the tracking and the closed loop behaviour,



being the inverse of the sensitivity function S and the complementary sensitivity T. Allthis functions reflects then how the desired system should be in order to define a controllerwhich will satisfy all this requirements.

However, to define the weighting functions is necessary to know how the system shouldbe, what kind of shape should it have. Thanks to the analysis done with a classicalcontroller all this shapes are known.

• W1

This function is used to weight the tracking error (θref − θIMU ). It has to satisfy:

∥W1S ∥inf

The final value theorem impose S(0)=0 to avoid the position error. So S needs ahigh reduction of the norm | S(jw) | at low frequencies. HenceW1, being the inverseof S, will have high gain at low frequencies having a high-pass filter shape.

• W2

A big decrease of the norm | T (jw) | at high frequencies is important to reduce theimpact of the noise in the output and to avoid having big changes in the command.A high agitation of it can saturate the actuator producing a non-linear behaviourthan can destabilise the loop. Being W2 the inverse of T, a low-pass filter shape isdesired.In addition in the study done for the classical controller ( chapter 3) the behaviour ofthe closed-loop is known and so is the function W2. One can just invert the transferfunction of 1 + GK adding a high frequency pole and adjusting the gain to have arealizable function

• W3

To avoid having huge command values, a constant boundary was defined. Anotherpossible options was to define a high-pass filter in order to have a low-pass filterresponse in the command.

• Walpha

Same as W2 thanks to the previous studies and looking to the desired behaviour ofalpha a definition of the weighting functions is very easy. The shape will be similarto the one of W2

• Wext

This functions bound an external perturbation coming before the going into theplant, in this case, into the TVC actuator.



• Wnoise

This function bound the noise added to the measurements done.

Figure 4.6 shows some possible shapes for the different weighting functions.

−20

0

20

40

60

Mag

nitu

de (

dB)

10−4

10−3

10−2

10−1

100

101

102

103

−90

−45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

(a) W1

−20

−10

0

10

20

30

Mag

nitu

de (

dB)

10−2

10−1

100

101

102

103

104

−45

0

45

90

Pha

se (

deg)

W2

Frequency (rad/sec)

(b) W2

−4.5

−4

−3.5

−3

−2.5

−2

Mag

nitu

de (

dB)

100

101

−1

−0.5

0

0.5

1

Pha

se (

deg)

W3

Frequency (rad/sec)

(c) W3

−45

−40

−35

−30

Mag

nitu

de (

dB)

101

102

103

104

105

−60

−30

0

Pha

se (

deg)

Walpha

Frequency (rad/sec)

(d) Walpha

Figure 4.6: W-functions

Parameters are time variant, hence some Wfunctions likeW2 must also be time variant.Because W2 adjust the closed loop behaviour it is important that it adjusts with time.For that reason its design is based in a function of parameter a6. In fact W2 reflects theideal/desired shape of the closed loop. This was defined when the classical controller wasintroduced (see section 3.1.2)



−7.5

−7

−6.5

−6

−5.5

−5

Mag

nitu

de (

dB)

100

101

−1

−0.5

0

0.5

1

Pha

se (

deg)

Wext

Frequency (rad/sec)

(a) Wext

−41

−40.5

−40

−39.5

−39

Mag

nitu

de (

dB)

100

101

−1

−0.5

0

0.5

1

Pha

se (

deg)

Wnoise

Frequency (rad/sec)

(b) Wnoise

Figure 4.7: W-functions

4.3 Robustness Analysis

A control system is robust if it is insensitive to differences between the actual systemand the model of the system which was used to design the controller. These differences arereferred to some model/plant differences or simply model uncertainty. The key idea then,in the robust control paradigm, is to check whether the design specifications are satisfiedeven for the “worst case” uncertainty. The approach is then as follows:

1. Determine the uncertainty set: find a mathematical representation of the modeluncertainty.

2. Check for RS: determine whether the closed-loop system remains stable for all plantsin the uncertainty set.

3. Check for RP: if RS is satisfied, determine whether the performance specificationsare met for all plants in the uncertainty set.

The mathematical representations of the model including the uncertainties is in figure4.8 in which the uncertain perturbations are in the block diagonal matrix,

∆ = diag{∆i} =

∆1

. . .

∆i

. . .

where each ∆i represents a specific source of uncertainty.

The P and K transfer function appearing in 4.8 have the following state space descrip-tion:

P =

A B∆ B1 B2

C∆ D∆∆ D∆1 D∆2

C1 D1∆ D11 D12

C2 D2∆ D21 D22

, K =

[AK BKCK DK

](4.4)



Figure 4.8: LFT Model

If the controller is absorbed into the loop we will end up with the so called N∆structure shown in 4.9. The N transfer function appearing in 4.9 has the following state

Figure 4.9: N∆-structure

space description:

N =

A B∆ B1

C∆ D∆∆ D∆1

C1 D1∆ D11

(4.5)



where now the matrices appearing in 4.5 have the following meaning

A =

(A+B2DK(I −D22DK)

−1C2 B2CK +B2DK(I −D22DK)−1D22CK

BK(I −D22DK)−1C2 AK +BK(I −D22DK)

−1D22CK

)

B∆ =

(B∆ +B2DK(I −D22DK)−1D2∆

BK(I −D22DK)−1D2∆

)

B1 =

(B1 +B2DK(I −D22DK)

−1D21

BK(I −D22DK)−1D21

)

C∆ =(C∆ +D∆2DK(I −D22DK)

−1C2 D∆2CK +D∆2DK(I −D22DK)−1D22CK

)D∆∆ =

(D∆∆ +D∆2DK(I −D22DK)

−1D2∆

)D∆1 =

(D∆1 +D∆2DK(I −D22DK)

−1D21

)C1 =

(C1 +D12DK(I −D22DK)−1C2 D12CK +D12DK(I −D22DK)

−1D22CK)

D1∆ =(D1∆ +D12DK(I −D22DK)

−1D2∆

)D11 =

(D11 +D12DK(I −D22DK)−1D21

)(4.6)

In order to analyse the robust stability of the whole uncertain model, we only needto consider the M∆-structure of 4.10 where M = N11 is the transfer function from theoutput to the input of the perturbations. If the nominal (∆ = 0) feedback system is stable

Figure 4.10: M∆- structure

then the stability of the system in 4.9 is equivalent to stability of the system in 4.10, where



∆ = 0 andM(s) = D∆ + C∆(sI −A∆)

−1B∆

where A∆, B∆, C∆ and D∆ are state space representation matrices of the subsystem N11

mentioned above. We now apply the Nyquist generalized stability condition to the systemin 4.10. We assume that ∆ and M are stable; the former implies that the nominal andthe uncertain open-loop transfer function must have the same unstable poles, the latter isequivalent to assuming nominal stability of the closed-loop system.

4.3.1 Robust Stability And Performance

Theorem 1 Determinant stability condition (real or complex perturbations)Assume that the nominal system M(s) and the perturbations ∆(s) are stable. Considerthe 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 4.10 is stable for allallowed perturbations (we have RS) if and only if:

Nyquist plot of det(I −M∆(s)) does not encircle the origin, ∀∆⇔det(I −M∆(s)) = 0, ∀ω,∀∆⇔λi(M∆) = 1, ∀i,∀ω,∀∆

(4.7)

The following theorem guarantees robust stability for an unstructured uncertainty,where ∆(s) is allowed to be any (full) complex transfer function matrix satisfying ∥∆∥∞ ≤1.

Theorem 2 RS for unstructured (“full”) perturbations.) Assume that the nominalsystem M(s) is stable and the perturbations ∆(s) are stable. Then the M∆-system in 4.10is stable for all perturbations ∆ satisfying ∥∆∥∞ ≤ 1 (we have RS) if and only if:

σ(M(jω)) < 1 ⇔ ∥M∥∞ < 1 (4.8)

A function useful to investigate the robust stability is the structured singular value (de-noted Mu, mu, SSV or µ), which provides a generalization of the singular value σ.

Let M be a given complex matrix and let ∆ = diag{∆i} denote a set of complexmatrices with σ(∆) ≤ 1 and with a given block-diagonal structure (in which some of theblocks may be repeated and some may be restricted to be real). The real non-negativefunction µ(M), called the structured singular value, is defined by

µ(M) , 1

min{km|det(I − kmM∆) = 0 for structured ∆, σ(∆) ≤ 1}(4.9)

If no such structured ∆ exists then µ(M) = 0.The following Theorem provide a necessaryand sufficient condition for robust stability with structured uncertainty.

Theorem 3 RS for block-diagonal perturbations (real and complex). Assumethat the nominal systemM(s) and the perturbations ∆(s) are stable. Then theM∆-systemin 4.10 is stable for all allowed perturbations with σ(∆),∀ω if and only if:

µ(M(jω)) < 1, ∀ω (4.10)



In the LFT general control configuration in 4.8 the K block represents the controller,which is a system itself, so the state vector of the N -system is

xT =

[xPxK

](4.11)

where xP and xK are state vectors of the systems P and K respectively.

Theorem 4 Robust Performance Rearrange the uncertain system into the N∆-structureof 4.9. Assume NS such that N is (internally) stable. Then

RP ⇔ ∥F∥∞ = ∥Fu(N,∆)∥∞ < 1, ∀ ∥∆∥∞ ≤ 1 (4.12)

⇔ µ∆(N(jω)) < 1, ∀ω (4.13)

where µ is computed w.r.t. the structure

∆ =

[∆ 00 ∆p

]and ∆p is a full complex perturbation with the same dimensions as F T .

4.3.2 Worst Case Performance and Skew µ

If we want to keep some of the uncertainty blocks fixed we would know how large oneparticular source of uncertainty can be before instability. We define this value as 1/µs(M),where µs is called skewed-µ. We may view the µs(M) as a generalization of µ(M).

For example, let ∆ = diag(∆1,∆2) and assume we have fixed ∥∆1∥ ≤ 1 and we wantto find how large ∆2 can be before we get instability. The solution is to select

Km =

[I 00 kmI

]and look at each frequency for the smallest value of km which makes det(I−KmM∆) = 0,and we have that skewed-µ is

µs(M) = 1/km

Assume we have a system for which the peak µ-value for RP is 1.1. The definitionof µ tells us that our RP requirements would be satisfied exactly if we reduced both theperformance requirement and the uncertainty by a factor of 1.1. So µ does not directlygive us the worst case performance (max∆ σ(F (∆))) as one might have expected.

To find worst case weighted performance for a given uncertainty, one needs to keep themagnitude of the perturbations fixed (σ(∆) ≤ 1); that is, we must compute skewed-µ ofM . We have, in this case,

maxσ(∆)≤1

σ(Fu(N,∆)(jω)) = µs(N(jω))



4.4 Rigid Body controller

With the new synthesis model defined and applying the µ-synthesis algorithm a newrigid body controller is designed. From the first one designed until the last version thereis a trial and error processing where the Wfunctions need to be tuned. In fact, even if theshape is known there still are some small gain adjustments necessaries. The shape of thecontroller design by the µ-synthesis technique is plotted in 4.11

−20

−10

0

10

20

30

40

Mag

nitu

de (

dB)

10−4

10−2

100

102

104

90

135

180

225

270

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 4.11: Rigid body controller

The nominal point of design for the controller is the maximum dynamic pressure. Withonly this controller the system stays stable all long the time domain. All the performancesare in the table below.

Time slide PM GM(LF) GM(HF) DM RS(%)

30 38.9701 7.1765 18.4129 0.2 13840 40.4746 7.2222 17.6195 0.1957 13650 42.9776 8.1476 16.1515 0.1747 15360 44.4235 9.8240 14.5609 0.1478 18270 44.0486 13.1122 12.7698 0.1169 22580 42.1874 17.2593 11.3299 0.0941 19190 38.3592 22.4296 9.5355 0.0704 201

Even if the system stays stable and margins are good, it could be interesting to seewhat happens if a gain scheduling technique is also applied. With more than one con-troller the decrease on the phase margin after the 60s can be avoided and all the marginswill follow a more standard variation. A standard variation can be defined as a parabolawere the minimum is next to the maximum dynamic pressure. With only one controllerperformances and robustness are subject to the parameters variation from the nominaldesign point. That explain the result of a parabola but with a maximum at the maximumdynamic pressure.

However, despite the results obtained it is possible to keep only one controller as itwas shown and it will guarantee both performances and stability. Thanks to the robust



approach of the controller it is no longer strictly necessary to use a Gain Schedule tech-nique since this one gives little improvements in comparison with only one controller.

Defining ten controllers, one each ten seconds of time domain (from ten to ninetyseconds) is a more homogeneous approach. These controllers will look very similar one toanother but with small differences like in bandwidth or in the static gain for example.

10−4

10−2

100

102

104 20

40

60

80

100

−10

0

10

20

30

40

time [s]

frequency [rad/s]

Mag

nitu

de [d

B]

Figure 4.12: 3D plot of the controllers

The performances achieved now are (in margins talking) very similar but the systemis better prepared to affront the desired robust stability and performances. In general the% of robust stability supported by the system increase.

The table below sum up the margins achieved with these news controllers


30 44.08 7.1765 18.4129 0.1885 16240 42.26 7.2222 17.6195 0.1797 15650 42.97 8.1476 16.1515 0.1797 15360 46.12 9.8240 14.5609 0.1797 16170 49.29 13.1122 12.7698 0.195 17680 52.10 17.2593 11.3299 0.2185 21890 54.25 22.4296 9.5355 0.2582 253

One can find the complete report of robust stability and performances for each con-troller in annex E and all the plots of the time simulation in annex D. It is important toremark that despite in the PD controller θ and θ was used here it is not longer necessary tointroduce both feedbacks. However, introducing θ as new feedback will make the resolu-tion to find a controller satisfying all the requirements easier (we are adding a new degreeof freedom) and the robust stability can also be increased. However, the controller will bemore complex. Just to give an example, defining a controller for the time slide t=50s theperformances obtained in stability margins are similar, but the robust stability is around400%. More than the double obtained by only having θ as feedback. Nevertheless, duringall this study only a θ feedback will be considered, knowing that adding a feedback in θcould improved the performances.



−50

−40

−30

−20

−10

0

10From: r To: [+r−Grigidn(1)]

Mag

nitu

de (

dB)

10−4

10−2

100

102

104

−90

0

90

180

270

360

450

Pha

se (

deg)

S

Frequency (rad/sec)

(a) Sensitivity plot (S)

−400

−300

−200

−100

0

100From: r To: [+Grigidn(1)]

Mag

nitu

de (

dB)

10−2

100

102

104

−540

−360

−180

0

180

Pha

se (

deg)

T

Frequency (rad/sec)

(b) Complementary sensitivity (T)

−30

−20

−10

0

10

20

30From: r To: [+Ksec]

Mag

nitu

de (

dB)

10−4

10−2

100

102

104

0

180

360

540

Pha

se (

deg)

KS

Frequency (rad/sec)

(c) KS

−250

−200

−150

−100

−50

0

From: r To: [+Grigidn(3)]

Mag

nitu

de (

dB)

10−2

100

102

104

−540

−360

−180

0

180

Pha

se (

deg)

alpha output

Frequency (rad/sec)

(d) Alpha

Figure 4.13: S,T,KS,and Alpha Bode plots

4.5 Bending Modes

Bending modes will now include some uncertainty in frequency (20%) and damping(10%) creating different possible behaviours.

The new terms added to the ∆ matrix due to the introduction of the bending modes are:

∆ =

δw1 0 0 0 0 0

0. . . 0 0 0 0

0 0 δwi 0 0 00 0 0 δξ1 0

0 0 0 0. . . 0

0 0 0 0 0 δξi

(4.14)

It is important to considered the structure of ∆. As it was already mentioned ∆ willhave block diagonal structure in which the generic block can be a scalar real parameter,a repeated real parameter, a scalar complex perturbation or a full complex m x n block.



20 30 40 50 60 70 80 9042

44

46

48

50

52

54

56

(a) Phase Margin

20 30 40 50 60 70 80 9016

16.5

17

17.5

18

18.5

19

19.5

20

20.5

(b) High Frequency Gain Margin

20 30 40 50 60 70 80 908

8.5

9

9.5

10

10.5

11

11.5

12

12.5

(c) Low Frequency Gain Margin

20 30 40 50 60 70 80 900.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

0.26

(d) Delay Margin

Figure 4.14: Margins

From a computational time it is important to avoid having large repeated real parametersblocks.

Hence, instead of having two times the number of bending modes introduced as newparameters and having an uncertainty in the parameters, a filter (Wflex) which magnitudeapproximates the maximum gap between the nominal behaviour and the different onescreated by the uncertainties is designed.

Sysflex = Gflex · (1 +Wflexδ) (4.15)

Being Gflex the state space model for the bending modes defined in section 3.3 and δthe complex uncertainty value (δ ≤ |1|)

Because nominal value of frequency and other parameters are time variant parameters,Wflex

must also be time variant. Looking to the difference between the behaviours and the nom-inal one, the order of the filter was adjusted. In figure 4.15 one can see the different shapesof the filter for different time slides and how it covers the uncertainty introduced.



10−2

10−1

100

101

102

103

104

−80

−60

−40

−20

0

20

40

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

(a) t=30s

10−1

100

101

102

103

104

105

−80

−60

−40

−20

0

20

40

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

(b) t=60s

10−1

100

101

102

103

104

105

−80

−60

−40

−20

0

20

40

60

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

(c) t=90s

Figure 4.15: Shapes of Wflex filter

Figure 4.16: Rigid body with bending modes system

As in the classical controller when adding the bending modes to the structure the rigidbody controller cannot guarantee stability and system becomes unstable so a filter is againnecessary. Despite the way it was define the first time, now the main purpose is to obtaina filter given by the system, i.e that can guarantee stability and performances by design.To find the rigid body controller a synthesis model was defined applying µ − synthesis



algorithm, one can get a controller that tries to satisfy the desired performances in thebest way and that guarantee stability also. The principle now is to use the same approachbut adding the bending modes and the rigid controller in the loop.

System behaviour has changed because of the bending mode so the new controllercreated by using the µ − synthesis has to guarantee the stability and try to reach theperformances but because all the performances were achieved by the rigid controller forthe rigid body the only thing needed is a filter for the bending modes and that is exactlywhat the µ − synthesis algorithm will provide. Despite what happens with the classicalcontroller this time, the TVC actuator is not enough to stabilise the system because it wasconsidered for the rigid body controller design. Now the filter is created by the system toguarantee stability taking into account the different uncertainties of the model and givena robust solution.

10−2

100

102

104

106

108

−70

−60

−50

−40

−30

−20

−10

0

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

Figure 4.17: Shape of a filter designed using µ− synthesis

Because this filter was design using some uncertainties on frequency and damping ofthe bending modes, it can be used for all the time domain even if the bending modes fre-quencies change. However, performances decrease because the filter was not only designedfor a nominal bending mode frequency but also for a specific rigid body controller. Hence,it is good to design as many filters as controllers to maximize performances. Figure 4.18shows a 3D plot of the different filters used and the product of the rigid body controllerand the filter.

The table below sum up the margins for the different time slides.



10−4

10−2

100

102

104 30

40

50

60

70

80

90

−60

−40

−20

0

20

time [s]

frequency [rad/s]

Mag

nitu

de [d

B]

(a) 3D Plot of the notch filter

10−4

10−2

100

102

104 20

40

60

80

100

−60

−40

−20

0

20

40

time [s]

3D plot Krigid*Notchfilter

frequency [rad/s]

Mag

nitu

de [d

B]

(b) 3D plot of Krigid*Wflex

Figure 4.18: 3D plot of the notch filter and the complete controller


30 18.4 2.83 5.88 0.1795 98.540 30.4 4.9 10.1 0.2090 98.550 22 3.4 7.64 0.1869 89.660 32.4 5.16 9.55 0.2041 91.970 23.7 3.71 7.57 0.2098 9280 36.4 6 13.9 0.2837 12090 34.1 6.17 12.1 0.3485 115

Even if with this configurations 100% of robustness is not achieved it is just a matterof modifying the weighting functions until the system reach the performances. Robustnessand performances are opposite and it is hard to combine both. If robustness is not satisfythat means that the performances asked or the uncertainty level introduced is very bigand the system cannot satisfy all the restrictions. However, it is important to remark thatthe % of uncertainty tolerated is almost 100% and using only θ as feedback.

4.6 Drift control

When introducing the wind into the system, the launcher drift and the angle of attackis increased. It is important to keep both values between the boundaries to guaranteethe tracking and the structural safety. After making a first simulation to see the values,one can notice than the drift is not so important (466m. for a maximum allowed value of1000m) but the angle of attack cross the allowed limits (4.22° for a maximum of 3° al-lowed).

The main point of this new feedback will be then to reduce the angle of attack, spe-cially at the maximum dynamic pressure so it seams logical to make a feedback on theangle of attack instead of the drift and the drift speed. Remembering the study done insection 3.2.1 a feedback in alpha will reduce the angle of attack and will also introduce a



−40

−30

−20

−10

0

10

From: r To: [+r−Grigidn(1)−Gflexn]

Mag

nitu

de (

dB)

10−4

10−2

100

102

−180

0

180

360

540

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)


−400

−300

−200

−100

0

From: r To: [+Grigidn(1)+Gflexn]

Mag

nitu

de (

dB)

10−2

100

102

104

−720

−360

0

360

720

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)


−100

−80

−60

−40

−20

0

20

From: r To: [+Ksec]

Mag

nitu

de (

dB)

10−2

100

102

104

−360

0

360

720

1080

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

(c) KS

−300

−200

−100

0

From: r To: [+Grigidn(3)]

Mag

nitu

de (

dB)

10−2

100

102

104

−720

−360

0

360

720

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

(d) Alpha


little reduction on the drift, but the tracking error will increase. However, even if makinga feedback on the drift and the drift speed the angle of attack can also be decreased,tracking is very affected and because the drift value still between boundaries, making afeedback on alpha seams the best option.The principle used to get the controller is the same that the one used to design the notchfilter for the bending modes. The synthesis model will be very similar, keeping the rigidcontroller in the loop, but this time opening the command input will be added to theoutput of the rigid controller and the measurement output for the feedback will be theangle of attack coming for the rigid body plant.

This time is not possible to avoid the wind effect in the synthesis model. Because theangle of attack (and the drift) are very affected by it, it is important to include it sincethe beginning. Without wind, the angle of attack is very small and did not exceed themaximum value but it does when the wind appears. Avoiding the wind into the synthesismodel will create more than a controller a filter for alpha, in order to filter the feedback.The system was already satisfying the requirements and adding this feedback will onlymakes things worst, so the system creates a filter to avoid its effect. Nevertheless, by



Figure 4.20: Synthesis Model

introducing the wind the feedback becomes useful and the system can create a controllerto contribute to satisfy all the requirement. This new input will need a weight like allthe others ”perturbations” defined. The weight defined will have the same shape than alow-pass filter because the main effect of the wind will be at low frequencies 4.21

10−2

10−1

100

101

102

103

−50

−40

−30

−20

−10

0

10

20

30

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

Figure 4.21: Weighting function for the wind input

For the drift control design is not necessary to include the bending modes in. Thebending modes are not affected by the wind and they affect mainly the pitch angle. It istrue that they are going to have an impact on the angle of attack because the pitch anglehas an impact on it, but to reduce the number of states and the complexity of the synthesismodel, it is better to avoid include them. Once the drift control is done, it is important to



verify that even including the bending modes all the requirements are satisfy. If there isany problem in the angle of attack for example, it suffice to modify the weighting functionsto make it more constraint and increase the reduction in the rigid body to compensate thepossible variation introduced by the flexible modes. Of course, with powerful computersit is possible to make the drift control including the bending modes.

The rigid body controller was designed again but this time, including the wind as aninput just to compare with the other controller. Looking to figure 4.22 one realises thatboth controllers are very similar, and even in their performances (time simulation andmargins). For this reason it is justified to keep the same controller defined in section 4.4

10−4

10−2

100

102

104

−10

0

10

20

30

40

50

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

KrigiddriftKrigid

Figure 4.22: Comparison between both rigid body controllers

10−4

10−2

100

102

104

−20

0

20

40

60

80

100

120

140

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

(a) Bode plot for a Kdrift controller

10−4 10

−2 100 10

2 104

20

40

60

80

100−50

0

50

100

150

frequency [rad/s]

3D plot Kdrift

time [s]

Mag

nitu

de [d

B]

(b) 3D plot of the Kdrift controller

Figure 4.23: Shape of the Kdrift controller



Adding this controller into the system the angle of attack get reduced in a 30% andtracking is even improved. However, there is a small increase in the drift but the finalvalue achieved stays under the boundary. The table below sum up the results achieved:

alpha Drift Max θerror Max (degrees)

Nominal 4.22 446 1.34Alpha Feedback 2.94 407.6 0.8

Now the bending modes are introduced together with the notch filter designed in 4.5.As it was expected, the system still stable and the angle of attack still under 3°.

alpha Drift Max θerror Max (degrees)

Nominal 5.22 636 3Alpha Feedback 2.69 474 3

4.7 One controller for all the Pay Loads

All the data where obtained for a Pay Load of 1200kg. One could think that for eachPay Load is necessary to do again all this study even if a lot of things can be used toreduce the amount of work. It is true that with the Pay Load, data changes and of coursevalues of the EOM are not going to be the same, but because of the robust approachit is possible that the system (more technically talking the controller) could manage thedifference.

Comparing the numerical values of the most important parameters from the rigid body(a6 and k1) together with the frequency of the bending modes one can see that even for300kg or 2200kg, values are included in the interval defined by the 20% of uncertainty ofthe design point.

Pay Load a6 k1 w1 w2 w3

300kg 3.21 6.74 27.08 60.06 82.141200kg 3.23 7.07 24.66 53.93 74.152200kg 3.26 7.42 22.85 50.43 70.06

So it seams possible to use the same controller for all the different Pay Loads. However,there is an important different between the rigid body behaviour and the introduction ofthe bending modes. When looking to parameters a6 and k1 one can see that the differenceat each Pay Load is not very important, so it seams obvious that the rigid body controllerhaving a robust stability value bigger than a hundred percent will be able to manage it.However, for the bending modes is not so easy. Difference in frequency between 300kgand 1200kg are almost the double than the ones between 1200kg and 2200kg. Adding thefact that the robust stability when adding the bending modes decrease in an import wayperhaps the system will be able to stabilise the system for all the Pay Loads, but marginsat low Pay Loads will be very poor. For that reason, it can be interesting to define twokinds of notch filter for the bending modes. One for lows Pay Loads and the other onefor the bigger ones. Of course the rigid body controller used will be the same, reducingconsiderably the amount of work.



4.8 Time simulation

All the controllers presented where tested in a non-linear simulator. This simulatorhas two main inputs: the desired pitch angle and a wind profile C.2. Outputs are theachieved pitch angle, the angle of attack, the error and the load (Qα).All the plots for each controller are in D.


Chapter 5

Multi-axis controller

Now that the robust control in one axis is done, it is interesting to study what happenswhen introducing the others axis. In fact everything is coupled in a LV when the roll rateappears, and there are also some control problems due to the presence of the roll and thepitch and yaw axis alternation. Making a multi-axis control is similar to duplicating thework done for one axis. The total number of states considered will be six instead of threeand two control inputs will be considered instead of one. A new wind profile could bedesign for the yaw, but from now on the wind is not going to be considered.

5.1 Problems due to the coupling

As it was introduced before, there is a coupling between pitch and yaw due to the rollrate. This coupling is more and more important when the roll rate also increases. Forsmall levels of roll rate coupling, the system is still stable using the rigid body controllerdefined in section 4.2 but some bumps and oscillations appear. If the roll rate is progres-sively increased, the number of bumps in the pitch and in the yaw angles will increaseuntil instability is reached. Figures 5.1 and 5.2 show the pitch and yaw response usingthe rigid body controller from the fifty seconds time slide. One can notice the number ofoscillations that appears in both axes.

Due to this coupling, and to the possibility to reach instability it is interesting to designa new rigid body controller taking the coupling into account. This new controller mustmanage the control of the rigid body (of each axis) and the coupling. So in fact, there isgoing to be two controllers for each axis, making a total of four. The controller will havetwo inputs (pitch error and yaw error) and two outputs (δp and δy) and can be expressedlike [

K11 K12

K21 K22

](5.1)

Where K11 and K22 will manage the rigid body behaviour and K12 together with K21

will manage the coupling. Hence K11 and K22 will have a very similar shape to Krigid andK12 together with K21 will filter the roll rate frequency to avoid the coupling. It is similarto the flexible modes case but for the roll rate.

91

CHAPTER 5. MULTI-AXIS CONTROLLER

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

(a) p = 0°/sec

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

(b) p = 2°/sec

Figure 5.1: Pitch response for different roll rates (p)

0 10 20 30 40 50−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(a) p = 0°/sec

0 10 20 30 40 50−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

(b) p = 2°/sec

Figure 5.2: Yaw response for different roll rates (p)

5.2 Building the system model

The system defined in section 4.2 is no longer valid since the coupling between axesis considered. The considered model take the roll as an input, defined by the user. Rolland roll rate are known and their evolution is fixed by the user, not by the system. Themodel will have then five degrees of freedom. The purpose of this section is to studythe system stability and behaviour when introducing a coupling between two axis. Theevolution of this coupling is not interesting by the moment so keeping the roll as an inputis justified. In addition for a stability study, the roll rate is the key parameter. The rollis more necessary when studying the trajectory because pitch and yaw will be switchingfrom attitude to lateral drift according to the roll angle but for stability purpose it can befixed to zero.

The complete equations for this new system are:



θ = pψ + q

ψ = −pθ + r

q = −k1δp + a6α+Izz − IxxIyy

pr

r = k1δy + a6β +Ixx − Iyy

Izzpq

x =T −D

m− ψ

(−Tδyb + Lββb

m

)+

(−Tδpb − Lααb

m

)y = −T −D

mψ +

(Lβ∆β − T∆δy

m

)+ pz

z = −T −D

mθ +

(−Lα∆α− T∆δp

m

)− py

(5.2)

Rewriting these equations in a state space form:

θq

ψr

Z

Y

=

0 1 p 0 0 0a6 0 0 p · In a6

v 0−p 0 0 1 0 00 −p · In a7 0 0 −a7

v−a1 0 0 0 −a2 −p0 0 a4 0 p −a5

θqψr

Z

Y

+

0 0

−k1 00 k1

−a3 00 −a3

[δpδy

]

[θψ

]=

[1 0 0 0 0 00 0 1 0 0 0

]

θqψr

Z

Y

+

[0 00 0

] [δpδy

](5.3)

With:

a1 = Lα+T−Dm a2 = Lα

mv

a3 = Tm a6 = LαℓGA

Iyy

Lα = qSCNα k1 = TℓCGIyy

a4 =Lβ+T−D

m a5 =Lβ

mv

a7 =LβℓGA

IyyLβ = qSCNβ

In = 1− IxxIyy

(5.4)

In figure 5.2 only the system was drawn. To avoid overbook the figure, the weightingfunctions were not included, but they still be included in the design.

5.3 Designing a new controller

Similar to the one axis controller, it is necessary to define the weighting functionsto design a robust controller. However, due to the launcher’s axis symmetry, the same



Figure 5.3: Multi-axes rigid body model

weighting functions can be used in both axis. In fact, the weighting functions defined insection 4.2 were thought in a way such as to have a desired launcher behaviour satisfyingthe requirements. Nothing has changed from this point, i.e the launcher has to satisfy thedesired requirement independently of the number of axes considered. That is why it ispossible to reutilise the weighting functions without introducing, a priori, any modification.

−20

−10

0

10

20

30

40From: In(1)

To:

Out

(1)

100

−20

−10

0

10

20

30

40

To:

Out

(2)

From: In(2)

100

Bode Diagram

Frequency (rad/sec)

Mag

nitu

de (

dB)

Figure 5.4: Multi-axis rigid body model

After following the same procedure than for the rigid controller, the controller designedlooks like the one plotted in figure 5.3. One can notice that there is an important peak atthe same frequency of the roll rate. This peak appears due to the roll rate coupling andincreases its value according to the roll rate value. 5.5.

The old rigid body controller was very focused on only one axis and cannot guarantee



−60

−40

−20

0

20

40From: In(1)

To:

Out

(1)

10−5

100

−60

−40

−20

0

20

40

60

80

To:

Out

(2)

From: In(2)

10−5

100

Bode Diagram

Frequency (rad/sec)

Mag

nitu

de (

dB)

(a) p=0

−20

−10

0

10

20

30

40From: In(1)

To:

Out

(1)

100

−20

−10

0

10

20

30

40

To:

Out

(2)

From: In(2)

100

Bode Diagram

Frequency (rad/sec)

Mag

nitu

de (

dB)

(b) p=45

Figure 5.5: Controller shape for different roll rates

30 40 50 60 70 80 9010

15

20

25

30

35

40Roll rate profile

Time

roll

rate

Figure 5.6: Roll rate profile

Time p (°/s)30 1040 1050 2760 2770 2780 3890 38

any stability when introducing some coupling. This controller however, can manage thiscoupling and it tolerates a variation in the roll rate. Nevertheless, as in the one axis case, itis more interesting to define more than only one controller because this time the couplingwill vary if the roll rate evolves.

The shape of this new controller is shown in figure 5.7. One can notice that as pre-dicted at the beginning of this chapter, the shape of the controller between θerror and δpand between ψerror and δy looks like the first robust rigid body controller and the crossedterms, are a filter acting in the roll rate frequency.

The table below sums up the robust stability (so the level of uncertainty allowed beforereaching instability) together with the roll rate(p).

Time p (°/s) RS(%) RP(%)

30 10 166 0.473840 10 169 0.475250 27 129 0.398660 27 163 0.345370 27 135 0.374280 38 114 0.286190 38 209 0.4186



10−5

100

105

20

40

60

80

100−40

−20

0

20

40

frequency [rad/s]

3D Krigid(1,1)

time [s]

Mag

nitu

de [d

B]

(a) From θerror to δp

10−5

100

105

20

40

60

80

100−40

−20

0

20

40

frequency [rad/s]

3D Krigid(1,2)

time [s]

Mag

nitu

de [d

B]

(b) From ψerror to δp

10−4

10−2

100

102

104 20

40

60

80

100

−40

−20

0

20

40

time [s]

3D Krigid(2,1)

frequency [rad/s]

Mag

nitu

de [d

B]

(c) From θerror to δy

10−5

100

105 20

40

60

80

100

−20

0

20

40

time [s]

3D Krigid(2,2)

frequency [rad/s]

Mag

nitu

de [d

B]

(d) From ψerror to δy


−50

−40

−30

−20

−10

0

10From: r

To:

[+r−

Grig

idn(

1)]

10−4

10−2

100

102

104

−400

−300

−200

−100

0

To:

[+t−

Grig

idn(

2)]

S

Frequency (rad/sec)

Mag

nitu

de (

dB)


−400

−300

−200

−100

0

100From: r

To:

[+G

rigid

n(1)

]

10−4

10−2

100

102

104

−400

−300

−200

−100

0

To:

[+G

rigid

n(2)

]

T

Frequency (rad/sec)

Mag

nitu

de (

dB)


Figure 5.8: S and T Bode plot from θref



−60

−40

−20

0

20

40From: r

To:

[+K

sec(

1)]

10−4

10−2

100

102

104

−60

−40

−20

0

20

To:

[+K

sec(

2)]

KS

Frequency (rad/sec)

Mag

nitu

de (

dB)

(a) KS

10−4

10−2

100

102

104

−300

−250

−200

−150

−100

−50

0


Mag

nitu

de (

dB)

alpha output

Frequency (rad/sec)

(b) Alpha

10−4

10−2

100

102

104

−300

−250

−200

−150

−100

−50

0


Mag

nitu

de (

dB)

beta output

Frequency (rad/sec)

(c) Beta

Figure 5.9: KS, Alpha and Beta bode plot from θref

−400

−300

−200

−100

0From: t

To:

[+r−

Grig

idn(

1)]

10−5

100

105

−60

−40

−20

0

20

To:

[+t−

Grig

idn(

2)]

S

Frequency (rad/sec)

Mag

nitu

de (

dB)


−400

−300

−200

−100

0From: t

To:

[+G

rigid

n(1)

]

10−5

100

105

−400

−300

−200

−100

0

100

To:

[+G

rigid

n(2)

]

T

Frequency (rad/sec)

Mag

nitu

de (

dB)


Figure 5.10: S and T Bode plot from ψref



−60

−40

−20

0

20

40From: t

To:

[+K

sec(

1)]

10−4

10−2

100

102

104

−60

−40

−20

0

20

40

To:

[+K

sec(

2)]

KS

Frequency (rad/sec)

Mag

nitu

de (

dB)

(a) KS

10−4

10−2

100

102

104

−300

−250

−200

−150

−100

−50

0

50From: t To: [+Grigidn(5)]

Mag

nitu

de (

dB)

alpha output

Frequency (rad/sec)

(b) Alpha

10−4

10−2

100

102

104

−300

−250

−200

−150

−100

−50

0


Mag

nitu

de (

dB)

beta output

Frequency (rad/sec)

(c) Beta

Figure 5.11: S,T,KS,and Alpha bode from ψref

5.4 Bending modes and drift control

5.4.1 Bending modes

With the bending modes in the system, as in the one axis case, the system becomesunstable. It is necessary to introduce the bending modes filter. One of the hypothesistested says that there is no roll coupling in the bending modes and they are independentfor each axis, so it is possible to use the filter designed in section 4.5.

When introducing them, as it was expected, system becomes stable again. In fact thefilter is introduced twice, one for each axis. Figure 5.4.1 reflects the new scheme of themodel with all its elements without the weighting functions.



−1

−0.5

0

0.5

1x 10

13 From: r

To:

[+G

rigid

n(1)

+G

flexn

1]

0 10 20 30 40−5

0

5

10x 10

11

To:

[+G

rigid

n(2)

+G

flexn

2]

From: t

0 10 20 30 40

Step Response

Time (sec)

Am

plitu

de

(a) Without any filter

−0.5

0

0.5

1

1.5

2

2.5

3From: r

To:

[+G

rigid

n(1)

+G

flexn

1]

0 20 40 60 80 100−0.5

0

0.5

1

1.5

2

2.5

3

To:

[+G

rigid

n(2)

+G

flexn

2]

From: t

0 20 40 60 80 100

Step Response

Time (sec)

Am

plitu

de

(b) Using the bending modes filter

Figure 5.12: Step response with and without filter

Figure 5.13: Multi-axis model

Time p °/s RS(%) RP(%)

30 10 30 0.109840 10 59 0.246950 27 59 0.144960 27 66 0.197170 27 35 0.106480 38 36 0.058390 38 77 0.1102



−60

−40

−20

0

20From: r

To:

[+r−

Grig

idn(

1)]

10−5

100

105

−400

−300

−200

−100

0

To:

[+t−

Grig

idn(

2)]

S

Frequency (rad/sec)

Mag

nitu

de (

dB)


−250

−200

−150

−100

−50

0

50

From: r

To:

[+G

rigid

n(1)

+G

flexn

1]

10−4

10−2

100

102

104

106

−600

−400

−200

0

200

To:

[+G

rigid

n(2)

+G

flexn

2]

T

Frequency (rad/sec)

Mag

nitu

de (

dB)


−60

−40

−20

0

20

40From: r

To:

[+K

sec(

1)]

10−4

10−2

100

102

104

−60

−40

−20

0

20

40

To:

[+K

sec(

2)]

KS

Frequency (rad/sec)

Mag

nitu

de (

dB)

(c) KS

Figure 5.14: S,T and KS from θref

−400

−300

−200

−100

0From: t

To:

[+r−

Grig

idn(

1)]

10−6

10−4

10−2

100

102

104

−150

−100

−50

0

50

To:

[+t−

Grig

idn(

2)]

S

Frequency (rad/sec)

Mag

nitu

de (

dB)


−800

−600

−400

−200

0

200From: t

To:

[+G

rigid

n(1)

+G

flexn

1]

10−4

10−2

100

102

104

106

108

−800

−600

−400

−200

0

200

To:

[+G

rigid

n(2)

+G

flexn

2]

T

Frequency (rad/sec)

Mag

nitu

de (

dB)


Figure 5.15: S and T bode plot from θref



−60

−40

−20

0

20

40From: t

To:

[+K

sec(

1)]

10−4

10−2

100

102

104

−60

−40

−20

0

20

40

To:

[+K

sec(

2)]

KS

Frequency (rad/sec)

Mag

nitu

de (

dB)

(a) KS

Figure 5.16: KS bode plot from ψref

5.5 Limitations of this approach

This approach is interesting to study and to see what is the effect of the roll rate in thesystem and how a controller can be designed to solve the coupling. However, it is necessaryto know the roll rate , what is always not always known in principle, and the controllersare optimized for one roll rate only. It could be interesting to design the controller in asuch way that the roll rate was an input so it can adapt itself to different roll rates orpossible variations in a more accurate way instead of just playing with the robustness ofthe controller.


Conclusions

The main conclusion of this study is that it is possible to design a complete controller(rigid body controller, bending modes filter and drift controller) using a robust techniqueand taking the desired requirements into account since the beginning. The time simula-tions confirmed that the system was stable for all the situations and that the structuralrequirements (drift and angle of attack) were satisfied. However other conclusions can beextracted.

A classical controller (such a PD controller) will guarantee stability at least while thesynthesis model stays very simple (two or three states at maximum). When having a simplemodel, the PD controller can assume the role of stabiliser and performances guarantor butwith its own limitations (design point, low uncertainty level...). Between these limitationsthere is the need of using a filter for the bending modes and something to reduce the driftor the angle of attack. In addition to adapt the controller to the time variant parameters,it was necessary to introduce the gain scheduling technique, defining several controllers fordifferent points. This technique had as drawback some implementation problems in timesimulations like bumps appearance in the switching process or even instability in a worstcase.

The classical controller was designed using the desired closed loop performances of thesystem and considering that parameters are time invariant and well known. The role ofthe robust controller is to design a controller taking into account the desired performancesbut also introducing some perturbations or uncertainties in the parameters. Using theµ − synthesis technique a robust controller was successfully designed for the rigid body.This controller presented similar performances in gain and phase margin in front of theclassical controller and had bigger robustness.

Another important difference lies in the design method. Taking the bending modes asexample, the filter is designed by knowing the shape it should have without consideringthe requirements of the system, so the filter obtained can stabilise the system but will notconsider to minimize the error, to have a good close-loop behaviour or any of the desiredrequirements. However, when the filter is designed using the µ− synthesis technique, thefilter is fruit of the system together with the requirements. It keeps the desired shapebut filtering the necessary frequencies and because it was designed by a robust technique,stability is guarantee in advance and there is an indicator of how well the desired perfor-mances are satisfied. Results shown that it was possible to define a filter and to keep a

103


robust system, while it was almost impossible by doing it manually or using the desiredshape.

The advantages of using a robust technique can be extrapolated to any situation wherea new controller or an improvement of an old one could be necessary. For example, tominimize the angle of attack it was necessary to add a new feedback and to introduce acontroller. The introduction of the controller reduced the angle of attack from 5.2°to 2.8°.As for the bending modes, instead of using a classical gain (that could have worked) arobust controller was used. Acting in the same way that in the other cases and definingthe desired behaviour of the system, the µ− synthesis technique brings a controller sat-isfying in the best possible way all the desired behaviour and guaranteeing stability. Thistechnique can be used for complex situations like multi-axis problem. A controller takingthe coupling between the different axes into account was successfully designed, showingthat the robust control is a good choice for high complexity problems

Keeping in mind that the robust controller was successfully designed next steps wouldbe to achieve similar results for the multi-axis scenario. The rigid body controller wasalready defined and it was shown that it was not necessary to introduce any new filter butit is still necessary to realise a complete time simulation in a non-linear simulator. Afterthat it could be interesting to define the roll rate and the roll as states, to synthesise anew controller and to implement it in a 6-DOF time simulation.

It is also important to remark the difficulties found during the designing process. Themain drawback was the non-convexity of the algorithm. That means there is no way toguarantee an optimal result. The second problem was the definition of the weighting func-tions, used for the synthesis model of the robust controller. There was also an importantcomputational constraint and some implementations problems. As it is explained in annexD when using a gain scheduling techniques the controllers used need to be similar and havenot to present important peak differences.

This study may be a good basis and a reference for a further study using LPV tech-niques. The robust controller can be designed using LPV techniques where controllers canbe designed in one shot over the entire envelope. The rate for the multi axis system canbecome an LPV parameter that can be scheduled making a controller for pitch, yaw androll possible.


References

[1] Arthur L. Greensite. Control Theory Volume II : Analysis and Design of Space VehicleFlight Control Systems. Spartan Books, 1971

[2] Ph. Saunois. Comparative Analysis of Architectures for the Control Loop of LaunchVehicles during Atmospheric Flight. EADS Space Transportation, France

[3] LT. Andrew Allen Martin. Model Predictive Control for Ascent Load Management ofa Reusable Launch Vehicle. Massachusetts Institute of Technology, June 2002.

[4] James A. Frosch, Donald P. Vallely. Saturn AS-501/S-IC Flight Control System De-sign. J.Spacecraft Vol 4 NAº 8, August 1967

[5] Cristophe R. Roux, Irene Cruciani. Roll Coupling Effects on the Stability Margins forVega Launcher. Guidance, Navigation and Control Conference and Exhibit, August2007. AIAA 2007-6630

[6] Bong Wie, Wei Du, Mark Whorton. Analysis and Design of Launch Vehicle FlightControl Systems. Guidance, Navigation and Control Conference and Exhibit, August2008. AIAA 2008-6291

[7] Wei Du, Bong Wie. Ascent Flight Control of Ares-I Creq Launch Vehicle in the Eventof Uncontrolled Roll Drift. Guidance, Navigation and Control Conference and Exhibit,August 2009. AIAA 2009-5957

[8] Yasuhiro Morita. An Idea of Applying µ-Synthesis to Launcher Attitude and VibrationControl Design. Journal of Vibration and Control, 2004, Vol. 10, Nr. 9, p.1243-1254

[9] Dale F. Enns. Rocket Stabilization as a Structured Singluar Value Synthesis DesignExample, June 1991.

[10] Miguel A. de Virgilio, Darren K. Kamimoto. Practical Applications of Modern Con-trols for Booster Autopilot Design. The Aerospace Corporation, 1993.

[11] Choong-Seok Oh, Hyochoong Bang, Chang-Su Park. Attitude control of a fexiblelaunch vehicle using an adaptative notch filter: Ground experiment. Control Engi-neerign PRactice, 2008, Vol. 16 Nr. 1, p 30-42

[12] Lazzenec,H., Pilotage des Missiles et des Vehicules Spatiaux, Dunod, Paris, 1966.

105

REFERENCES

[13] Michael A. Creagh,Rick Lind, H-infinity Control for Attitude Manoeuvres of a Spin-ning Asymmetric Vehicle.Guidance, Navigation, and Control Conference, August2009. AIAA 2009-5641

[14] Renato Lafranconi, Miguel Lopez, The Small Launcher for Europe

[15] M.Gauvrit, P.Apkarian, Commande robuste des systemes lineaires, SUPAERO (1994)


Annex A

Complete rigid body dynamic

A.1 Rigid body dynamic

The equations of motion are most readily derived relative to an inertial frame. Con-sidering a generic mass particle, mi the total moment about the origin of the body frameis

Mo = Σρi ×d

dt(miRi) (A.1)

the time rate of change of the momentum, miRi is given by

d

dt(miRi) = miRi − mici (A.2)

where the effect of variation in mass is accounted for. Noting

Ri = Ro + ρi

and taking the derivative respect to time

Ri = Ro + ρ+ w × ρ

A further differentiation yields

Ri = Ro + w × ρi + w × (w × ρi) + ρi + 2w × ρi (A.3)

So equation A.2 can be rewritten

Mo = mρc×Ro+d

dt(I ·w)−Σρi×mi(w×ρi)−Σρ×mici+w×Σ(ρi×miρi)+Σρi×miρi (A.4)

The equation for the linear acceleration follows directly from Newton’s law

F = m[Ro + w × (w × ρc) + w × ρc + 2w × ρc + ρc

]− Σmici (A.5)

where

107

ANNEX A. COMPLETE RIGID BODY DYNAMIC

F ≡ applied force

mRo ≡ d’Alembert force2m(w × ρc) ≡ Coriolis forcem(w × ρc) ≡ Euler forcemw × (w × ρc) ≡ centrifugal force

Σmici ≡ thrust due to variation in mass ≡ FT

For purpose of writing the trajectory equations, the terms in equations A.1 and A.2that are due to bending,sloshing and jet damping are negligibly small and will be neglected.In addition, for most large booster vehicles, the motion of the mass center relative to afixed point on the body is small, and so is the rotation vector w. Consequently termsinvolving products of w,ρ and ρ will be dropped.Applying all this simplifications, equations A.1 and A.2 are reduce to

F + FT = mRo +mw × ρc

Mo +MT =d

dt(I · w) +mρc × (Ro + ρc)

(A.6)

F External forces

FT Thrust term ΣmiciMo External moments

MT r Thrust deflection Σρi × mici

Now, replacing Ro by its expression and in expanded form, equation A.6 may be writtenas

FA + Fg + FT = m[U + w × U + w × ρc

]MA +Mg +MT = I · w + I · w + w × (I · w) +mρc ×

[U + w × U + ρc

] (A.7)

Reducing the equations of motion to scalar form:

m

u+ qw − rv + zcg q − ycg rv + ru− pw + xcg r − zcgpw + pv − qu+ ycgp− xcg q

= FA + Fg + FT (A.8)

Ixxp+ ˙Ixxp+ (Izz − Iyy)qr +mycg(w + pv − qu+ zcg)−mzcg(v + ru− pw + ycg)

Iyy q + ˙Iyyq + (Ixx − Izz)pr +mzcg(u+ qw − rv + xcg)−mxcg(w + pv − qu+ zcg)

Izz r + ˙Izzr + (Ixx − Iyy)pq +mxcg(v + ru− pw + ycg)−mycg(u+ qw − rv + xcg

(A.9)

=MA +Mg +MT

During this study two main hypothesis have been used:

1. The mass center coincides with the origin of the reference system



2. ˙Ixx = ˙Iyy = ˙Izz = 0

Introducing these simplifications the system is reduced to

m

u+ qw − rvv + ru− pww + pv − qu

= FA + Fg + FT (A.10)

Ixxp+ (Izz − Iyy)qrIyy q + (Ixx − Izz)prIzz r + (Ixx − Iyy)pq

(A.11)

=MA +Mg +MT

A.2 External Forces and Moments

The external forces and moments in the above equations are due to gravity, thrust,aerodynamics, propellant sloshing and engine inertia.

ΣFx = Fxg + FxT + Fxα + Fxδ + FxE

ΣFy = Fyg + FyT + Fyα + Fyδ + FyE

ΣFz = Fzg + FzT + Fzα + Fzδ + FzE

ΣMx =Mxg +MxT +Mxα +Mxδ +MxE

ΣMy =Myg +MyT +Myα +Myδ +MyE

ΣMz =Mzg +MzT +Mzα +Mzδ +MzE

(A.12)

However the contribution done by the propellant sloshing together with the nozzleinertia will not be considered in front of the others parameters.

A.2.1 Gravity

Gravity is an external force that need to be transform to the body axis reference. Usingas external reference a fixed Earth reference gravity can be expressed like:xEyE

zE

=

00g

(A.13)

To express the gravity in the body axis reference, it is necessary to use the transformationmatrix [T ]bT . This matrix will be presented further in the section A.3.xbyb

zb

= [T ]bT ·

00g

xbybzb

=

− sin θ · gsinφ cos θ · gcosφ cos θ · g

(A.14)

And the force suffered will be equal to the mass multiplied by the gravity.



A.2.2 Thrust

(a) Pitch plane (b) Yaw plane

Figure A.1: Thrust Configuration

The thrust due to the rocket engines is one of the major forces acting on the LV duringits flight. Swivelling of the thrust vector is also the primary means by which the LVattitude is controlled.

The forces and torque generated are given by:

FTx = T (t) cos(δp)

FTy = −T (t) sin(δy)FTz = −T (t) sin(δp)MTy = −T (t)ℓCG sin(δp)

MTz = T (t)ℓCG sin(δy)

(A.15)

where ℓCG is the distance between the nozzle swivel point C and the center of gravity G.Equation A.15 emphasize that the thrust is a function of time.

Aerodynamic Forces and Torques The interaction between the LV and the atmo-sphere in which it flies generates aerodynamic forces and torques. This interaction is onlysignificant during the early stages of flight and typically has a destabilizing effect on theLV dynamics.

The forces and torques due to aerodynamic loads will be derived using quasi-steady-



state aerodynamic theory. The aerodynamic components of interest are given by:

FAx = −q(t)SCAFAy = q(t)SCY

FAz = −q(t)SCNMAy = q(t)SCNℓGA(t)

MAz = q(t)SCY ℓGA(t)

(A.16)

where it was assumed that FAx and FAz act in the negative xb and zb directions respectivelyas shown in figure 1.4. It is important to emphasize that the dynamic pressure q and thedistance ℓGA between the aerodynamic center and the COG both depend strongly on time

(a) Pitch plane (b) Yaw plane

Figure A.2: Aerodynamic Configuration

S reference surfaceCA coefficient of x-aerodynamic forceCN coefficient of z-aerodynamic force

Since the LV is not a lifting body there are no intrinsic aerodynamic torques, so thatthe expression of the aerodynamic torques in equation A.16 has the very simple expressionof a force times a length. Furthermore, FAx is simply the aerodynamic drag which is essen-tially independent from perturbations. Instead, the CN coefficient appearing in equationA.16 is typically a function of the angle of attack α, its rate α and the pitch rate θ. Alinear dependence of CN with respect to some steady state values of the aforementionedvariables, can be obtained by classical Taylor series, where the coefficients of the seriesare the classical stability derivatives. For LV having little or no lifting surfaces the only



stability derivative of real importance is typically the one associated with α, which wedenote with CNα = ∂CN/∂α. This stability derivative is, however, a function of Mach fortransonic and supersonic speeds. Furthermore, in the case of long slender LV, CNα is afunction of position along the vehicle, CNα(x) is therefore written that way to emphasizethis fact.

In order to completely define FAz one can start by introducing the local angle of attackfor the rigid/flexible LV,

αloc(x) = α+(ℓOG − x)

Uθ − ∂ξ(x, t)

∂x− ξ(x, t)

U(A.17)

where α = arctan((zb −W )xb), ℓOG is the distance between the origin and the center ofgravity of the LV and ξ(x, t) is the displacement due to the bending modes.

Since both CNα(x) and αloc(x) depend on x the hole expression must be integrate onthe whole length L of the LV in order to obtain the desired expression of the aerodynamicforce. This gives:

FAz = −1

2ρU2S

∫ L

0CNα(x)αloc(x) dx

= −1

2ρU2S

(∫ L

0CNα(x) dxα+

1

U

∫ L

0CNα(x)(ℓOG − x) dx θ

+∑i

∫ L

0CNασi(x) dx qi(t)−

∑i

1

U

∫ L

0CNαϕi(x) dx qi(t)

) (A.18)

The last two terms in equation A.18 represent the aeroelastic terms. Everything done forFAz, α and My is equivalent for FAy, β and Mz due to the axis-symmetry of the launcher.

A.3 Building the synthesis model

Combining the last two sections the complete rigid body dynamic is defined:

q =−TℓCGIyy

sin δp +LαℓGAIyy

α− Ixx − IzzIyy

pr

r =TℓCGIzz

sin δy +LβℓGAIzz

β − Iyy − IxxIzz

pq

u =Gu + T −D

m+ rv − qw

v =Gv − T sin δy + Lββ

m+ pw − ru

w =Gw − T sin δp − Lαα

m+ qu− pv

(A.19)

One may note that all the equations presented before were obtained using the vehiclebody axes. However, for control synthesis design is better to obtain the motions about



the reference trajectory. Denoting Sb as the vehicle body axes and ST as the trajectoryaxes, the relation between them can be expressed by the three Euler angles (ψ, θ and φ)as follows:

• Rotate Sb about the Zb axis by an angle psi in the positive direction

• Then rotate about the YP axis by an angle θ in the positive direction

• Finally, rotate about the XP axis by an angle φ in the positive direction

This brings Sb into ST . Defining [T ]bT as the transformation matrix

[T ]bT =

1 0 00 cφ sφ0 −sφ cφ

cθ 0 −sθ0 1 0sθ 0 cθ

cψ sψ 0−sψ cψ 00 0 1

(A.20)

Note that inverse of the matrix [T ]bT is equal to its transverse. Assuming small angles(cos θ ≃ 1 ; sin θ ≃ θ) for θ and ψ, [T ]Tb may be expressed like

[T ]Tb =

1 −ψ θψ cosφ − sinφ−θ sinφ cosφ

(A.21)

So using the matrix [T ]Tb it is easy to find the relation between the body angular ratesand the trajectory ones

wYT ≡ θ = p · ψ + q cosφ− r sinφ

wZT≡ ψ = −p · θ + r cosφ+ q sinφ

(A.22)

For control synthesis purposes it is necessary to linearise the rigid body motion of theLV about a trajectory fixed reference frame and to obtain a perturbed motion about thereference trajectory, given by a time scheduled look-up table.

The perturbed equation of motion in body axes can be obtained as:∆x∆y

∆z

= [T ]LO

xbybzb

−

xTyTzT

(A.23)

And writing the body states in term of reference variables as

qo = qL +∆q

qo =≃ ∆q

ro = rL +∆r

ro =≃ ∆r

zo = zL +∆z

yo = yL +∆y

αo = αL +∆α

βo = βL +∆β

δpb = δpL +∆δ

δyb = δyL +∆δ

(A.24)



Then playing with the equations the result for little deviation from the reference be-come:

∆θ = p∆ψ +∆q cosφ−∆r sinφ

∆ψ = −p∆θ +∆q sinφ+ r cosφ

∆q = −k1δp + a6∆α+Izz − IxxIyy

p∆r

∆r = k1δy + a6∆β +Ixx − Iyy

Izzp∆q

∆x =T −D

m− ψ

(−Tδyb + Lββb

m

)+ θ

(−Tδpb − Lααb

m

)∆y = −T −D

m∆ψ + cosφ

(Lβ∆β − T∆δy

m

)− sinφ


m

)+ pz

∆z = −T −D

mθ + cosφ


m

)− sinφ

(−Lβ∆β − T∆δy

m

)− py

(A.25)

With this it is very easy now to build a state space model. Choosing the desired levelof states different models can be defined. For example, the system built in 3.2 can beeasily obtained from this equations imposing φ = β = p = 0.


Annex B

Tuning a controller

Until now the different values ofKp andKd were found just comparing the denominatorof the transfer function to the denominator of a second order system (s2 + 2ξw + w2).Comparing both expression one can fix the desired damping and frequency. However,normally systems are not as easy as a second order or as easily identifiable as these one.For that reason it is necessary to find other methods to find out the values of the controller.The engineering software MATLAB offer a tool that allow to calculate an optimal valuefor the controller satisfying some constraint in time or frequency domain.

B.1 Introducing time domain constraint

Some times it is interesting to having a time response of some kind. More than goingon into the theoretical expressions to find out what the controller values are, it can beinteresting to impose some boundaries or some restrictions in the system time response.This is a good solution an a very easy one to set up some parameters for models withtime invariant parameters. It is also very useful to get some fast results and check someaspects of the plant like for example how it would act in front of a perturbation one timeit is stable. It is important to check the system frequency response after making this kindof set up. When affronting a time variant parameters two different approaches exist:

1. One can only impose one time restriction to be satisfied by the system even if pa-rameters change with time. That means that even if parameters change the timeperformance to reach is always the same and the controller is going to vary to reachthis performance.

2. Different time restrictions are defined because the system is going to be different ateach time slide and time requirement are also going to be different. The controllerwill be adapted at each slide of time for the specifics time requirement so we areoptimizing all the time domain.

Even if the second approach could look more efficient that the first one, we can getsome good results with the first approach because the final time response of the systemis going to be satisfactory (boundaries and constraint are going to be satisfied) however,

115

ANNEX B. TUNING A CONTROLLER

values of the controller parameters could be higher or more restrictive in some time slidesthat needed.

Margins need to be checked a posteriori because this tool only provides values thatsatisfy (if possible) the constraint presented in the time response. There is no guaranteeeven of stability. For example taking back the three states LV model presented in 3

Hcl(s) =−(Kp +Kds)(a3

a6v + a2k1 + k1s)

s3 +B2s2 +B1s+B0

B2 = a2 − k1Kd


B0 = a1a6v

− a2a6 − a3a6v

− a2k1Kp

(B.1)

and making one new tuning of Kp and Kd the time response seams to be improved.This new tuning produces three real poles so there is no more overshoot in a step response.

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

Step Response

Time (sec)

Am

plitu

de

(a) Old PD controller

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Am

plitu

de

(b) New PD controller

Figure B.1: Comparison of the step response before and after using the PD optimizationtool

Then, margins need to be checked:

Number of States Gain Margin Phase Margin Delay Margin

Old 5.3634db 28.3123 0.2405New 33.6db 87.2 0.0264

Thanks to this new controller Phase and Gain margin increase but Delay margindecrease. One needs to check what is important for the system and if the new response isacceptable or not.

B.2 Introducing frequency domain constraint

Another way to proceed is defining boundaries and constraint on the frequency planedirectly. We can directly impose a minimum damping or frequency value. It is another


ANNEX B. TUNING A CONTROLLER

way to control overshoots, time response and pole position with the advantage that onecan guarantee stability at least for the nominal case but the drawback that time responseis not so intuitive like in the previous case. Imposing some temporary boundaries willproduce a controller that satisfies these boundaries but only for the time simulation. Thesystem could have one unstable root (very slow) and be unstable for higher time values.Working directly with the system’s roots avoid this problem and control time response.Nevertheless as it was said before, the problem of time variant parameters force to finddifferent restrictions for each time slide.

Perhaps one important difference rest in the fact that time domain optimisation iseasier to implement in a loop due to the fact that can be defined by a command while thefrequency optimisation needs to be done with the GUI.


Annex C

Getting the controller in one step

Another option to manage the problem of a controller design when a lot of parametersappears is trying to get the controller in one step. For example, instead of designing a rigidbody controller and after that a notch filter, why not design it in one step?. Theoreticallythe algorithm will calculate the controller necessary to satisfy the requirements so pre-senting the complete synthesis model, i.e rigid body system, bending modes and weightingfunctions, a very similar solution should be given. The problem lies in the non-convexityof the problem. When a solution is found there is no guarantee of optimal solution. In-stead of having a big and complex system and trying to find a solution directly, making atwo steps approach a first problem easier is solved and then even if the performances aredegraded and system becomes unstable adding the flexible modes, the starting point forthis situation is very different from the other one when systems starts from zero. In thissection a one step controller for the rigid body and the bending modes is designed for eachtime slides in order to compare the results achieved.

10−5

100

105 20

40

60

80−40

−20

0

20

40

time [s]

3D plot for the one step controller

frequency [rad/s]

Mag

nitu

de [d

B]

Figure C.1: Rigid body and bending modes filter

119

ANNEX C. GETTING THE CONTROLLER IN ONE STEP


30 19.73 3.217 6.627 0.1856 58.740 23.77 4.143 8.123 0.1853 89.450 19.59 3.402 6.843 0.1661 62.360 21.92 3.881 7.673 0.1710 70.270 22.92 4.025 7.211 0.1887 72.280 30.42 6.496 8.422 0.2128 12090 39.75 9.897 12.09 0.2690 79.8

The results obtained are poor that the ones obtained with the two step controllerespecially for the robust stability where the % of uncertainty tolerated is fewer. So, itseams that making the two steps approach, the algorithm can find a better solution thanmaking only one step. The same can be done for the drift control. Instead of keeping therigid body controller it is possible to try to solve the problem in one step. However, evenif the algorithm can find a solution and the closed loop is stable, when making the timesimulation results and introducing the real wind profile, the system oscillates a lot, beingnear to instability.

20 30 40 50 60 70 80 90 10020

25

30

35

40

45

50

55

60

65

70

Figure C.2: Time simulation with drift control

Perhaps the complexity of finding a solution when trying to solve the problem in shot,makes necessary to reduce the time slides and to introduce more controllers. However,because that is a comparison between both controllers, when using the same number andin the same situation, the two step controllers give not only better results, but also lesscomputer time consumption (solution is founded faster).ss


Annex D

Time Simulation

D.1 Interpolation of Controllers

When using a Gain Scheduling technique, controllers are going to be interpolatedbetween them to cover the complete time domain. The result is a linear controller usefulfor the complete flight. However, the interpolation process is an important point. Whenimplementing a Gain Schedule technique two factors become crucial for the successfulimplementation. One factor is the duration of the transition between controllers and theother is the shape of the controller itself.

20 30 40 50 60 70 80 90 10020

25

30

35

40

45

50

55

60

65

70

Figure D.1: Theta response with 5s of transition duration and 0.1s

When interpolating two controllers a transition duration is defined to switch from oneto the other. Between this time controllers are interpolated. When time is over, one ofthe designed controllers is used until the next transition point appears. If the duration isvery small and the difference between controllers is important then the system will bumpwhen switching from one controller to the other. In figure D.1 one can see the differencebetween using o.1s or 5s as duration for the transition period. When using 5s bumps aresmaller than with 0.1s. However, time can not be put as big as the time interval betweentwo designed points. Making a very big duration will make the system switch a lot of time

121

ANNEX D. TIME SIMULATION

between different controllers, and this can produce important oscillations in very smalltime periods which can produce instability in some cases.

In the other hand, if the time evolution of the controllers is very rough that meansthat the difference to cover in the interpolation is bigger, and when switching from onecontroller to the other bumps can appear again.

10−5

100

105

30 40 50 60 70 80 90

−60

−50

−40

−30

−20

−10

0

10

20

30

time [s]frequency [rad/s]

Mag

nitu

de [d

B]

(a) Controller 1

10−5

100

105

30 40 50 60 70 80 90

−20

−10

0

10

20

30

40

time [s]frequency [rad/s]

Mag

nitu

de [d

B]

(b) Controller 2

Figure D.2: 3D shape of two different controllers

Figure D.2 shows a controller where which has a max difference between two designpoints of 30dB and a second one with less than 20dB of difference. When making a timesimulation with the same transition duration one can see how bumps appear in the simu-lation of the first controller (figure D.3.

20 30 40 50 60 70 80 90 10020

25

30

35

40

45

50

55

60

65

70

(a) Time simulation for the first controller

20 30 40 50 60 70 80 90 10020

25

30

35

40

45

50

55

60

65

70

(b) Time simulation for the second controller

Figure D.3: Time simulation for the different controllers

As it was shown, when doing a time simulation it is important to check either theshape and the transition duration. If the shape is very rough perhaps a high transitionduration is necessary.



D.2 Time Simulation for the classical controller

All the time simulation have been done with for a Pay Load of 1200kg.

D.2.1 Time simulation with α as feedback

Time simulation having alpha as feedback and with Kα = 0.5

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

(a) θ response

0 10 20 30 40 50−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

4

(b) z response

0 10 20 30 40 50−1000

−800

−600

−400

−200

0

200

(c) z response

0 10 20 30 40 50−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

(d) α response

Figure D.4: Time simulation for θref = 1 and αw = 0



0 10 20 30 40 500

0.01

0.02

0.03

0.04

0.05

0.06

(a) θ response

0 10 20 30 40 500

2000

4000

6000

8000

10000

12000

14000

16000

18000

(b) z response

0 10 20 30 40 500

50

100

150

200

250

300

350

400

450

500

(c) z response

0 10 20 30 40 50−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

(d) α response


0 10 20 30 40 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

(a) θ response

0 10 20 30 40 50−2500

−2000

−1500

−1000

−500

0

500

(b) z response

0 10 20 30 40 50−70

−60

−50

−40

−30

−20

−10

0

10

(c) z response

0 10 20 30 40 50−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

(d) α response

Figure D.6: Time simulation for θref = 0.08 and αw = 0.03

Time simulation having alpha as feedback and with Kα = −0.3



0 10 20 30 40 500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

(a) θ response

0 10 20 30 40 50−3

−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

4

(b) z response

0 10 20 30 40 50−700

−600

−500

−400

−300

−200

−100

0

100

(c) z response

0 10 20 30 40 50−2

−1

0

1

2

3

4

(d) α response


0 10 20 30 40 50−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

(a) θ response

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3x 10

4

(b) z response

0 10 20 30 40 500

100

200

300

400

500

600

700

(c) z response

0 10 20 30 40 50−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

(d) α response




0 10 20 30 40 500

0.05

0.1

0.15

0.2

0.25

(a) θ response

0 10 20 30 40 50−1600

−1400

−1200

−1000

−800

−600

−400

−200

0

200

(b) z response

0 10 20 30 40 50−40

−35

−30

−25

−20

−15

−10

−5

0

5

(c) z response

0 10 20 30 40 50−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

(d) α response


D.2.2 Time simulation with z as feedback

Time simulation having Z as feedback and with Kz = −0.005

0 10 20 30 40 50−0.5

0

0.5

1

1.5

2

(a) θ response

0 10 20 30 40 50−8000

−7000

−6000

−5000

−4000

−3000

−2000

−1000

0

1000

(b) z response




0 10 20 30 40 50−160

−140

−120

−100

−80

−60

−40

−20

0

20

(a) z response

0 10 20 30 40 50−0.5

0

0.5

1

1.5

2

(b) α response


0 10 20 30 40 50−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

(a) θ response

0 10 20 30 40 500

1000

2000

3000

4000

5000

6000

(b) z response

0 10 20 30 40 500

20

40

60

80

100

120

(c) z response

0 10 20 30 40 50−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

(d) α response




0 10 20 30 40 500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

(a) θ response

0 10 20 30 40 50−450

−400

−350

−300

−250

−200

−150

−100

−50

0

50

(b) z response

0 10 20 30 40 50−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

(c) z response

0 10 20 30 40 50−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

(d) α response


D.2.3 Time simulation with z as feedback

Time simulation having Z as feedback and with Kz = −0.00138

0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

(a) θ response

0 20 40 60 80 100−1600

−1400

−1200

−1000

−800

−600

−400

−200

0

200

(b) z response




0 20 40 60 80 100−300

−250

−200

−150

−100

−50

0

50

100

150

200

(a) z response

0 20 40 60 80 100−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(b) α response


0 20 40 60 80 100−1

−0.5

0

0.5

1

1.5

2

(a) θ response

0 20 40 60 80 1000

200

400

600

800

1000

1200

(b) z response

0 20 40 60 80 100−150

−100

−50

0

50

100

150

200

250

(c) z response

0 20 40 60 80 100−2

−1.5

−1

−0.5

0

0.5

1

(d) α response




0 20 40 60 80 100−0.1

−0.05

0

0.05

0.1

0.15

(a) θ response

0 20 40 60 80 100−90

−80

−70

−60

−50

−40

−30

−20

−10

0

10

(b) z response

0 20 40 60 80 100−20

−15

−10

−5

0

5

10

15

(c) z response

0 20 40 60 80 100−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

(d) α response


D.2.4 Time simulation with z + z as feedback

Time simulation having Z and Z as feedback and with KZ = 0.00304 and Kz = 0.005

0 10 20 30 40 50−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

(a) θ response

0 10 20 30 40 50−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

(b) z response

Figure D.18: Time simulation for θref = 0.08 and αw = 0



0 10 20 30 40 50−2

−1.5

−1

−0.5

0

0.5

1

1.5

(a) z response

0 10 20 30 40 50−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

(b) α response

Figure D.19: Time simulation for θref = 0.08 and αw = 0

0 10 20 30 40 50−5

0

5

10

15

20x 10

−3

(a) θ response

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(b) z response

0 10 20 30 40 50−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

(c) z response

0 10 20 30 40 50−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

(d) α response

Figure D.20: Time simulation for θref = 0 and αw = 0.03



0 10 20 30 40 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

(a) θ response

0 10 20 30 40 50−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

(b) z response

0 10 20 30 40 50−2

−1.5

−1

−0.5

0

0.5

1

1.5

(c) z response

0 10 20 30 40 50−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

(d) α response


D.3 Time Simulation for the robust controller

D.3.1 Time simulation for the rigid body controller

20 30 40 50 60 70 80 90 10020

25

30

35

40

45

50

55

60

65

70

(a) θIMU

20 30 40 50 60 70 80 90 100−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

(b) θref -θIMU



20 30 40 50 60 70 80 90 100−5

−4

−3

−2

−1

0

1

2

(c) α

20 30 40 50 60 70 80 90 100−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

5

(d) Qα

20 30 40 50 60 70 80 90 100−200

−100

0

100

200

300

400

500

(e) Drift

Figure D.22: Time simulation for the rigid body

D.3.2 Time simulation including the bending modes

20 30 40 50 60 70 80 90 10020

25

30

35

40

45

50

55

60

65

70

(a) θIMU

20 30 40 50 60 70 80 90 100−2

−1

0

1

2

3

4

(b) θref -θIMU

20 30 40 50 60 70 80 90 100−6

−5

−4

−3

−2

−1

0

1

2

(c) α

20 30 40 50 60 70 80 90 100−4

−3

−2

−1

0

1

2

3x 10

5

(d) Qα



20 30 40 50 60 70 80 90 100−200

−100

0

100

200

300

400

500

600

700

(e) Drift

Figure D.23: Time simulation for the bending modes

D.3.3 Time simulation for the drift control

20 30 40 50 60 70 80 90 10020

25

30

35

40

45

50

55

60

65

70

(a) θIMU

20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

(b) θref -θIMU

20 30 40 50 60 70 80 90 100−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

(c) α

20 30 40 50 60 70 80 90 100−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

5

(d) Qα

20 30 40 50 60 70 80 90 100−100

0

100

200

300

400

500

(e) Drift

Figure D.24: Time simulation for the drift control


Annex E

Robustness

This annex compile all the robust stability and performances details for each controllerused and for each slide of time.

E.1 Robustness of the classical controller

Here are the robust stability and performances reports for the rigid and flexible bodyusing a PD controller. Each PD controller is optimized for the correspondent time slide

Rigid body

Robust Stability

Design Point (%) TolerateInstability Sensitivity

Uncert Freq a6 (%) k1 (%) Wtvc (%)

30 91.3 – – 23 39 4240 92 – – 19 39 4250 96.6 667 0.0238 28 39 4260 95.4 – – 20 39 26170 106 667 0.0206 67 90 10380 99.5 667 0.0060 43 40 4390 137 667 0.0114 25 40 38

Robust Performances

Design Point RPSensitivity

a6 (%) k1 (%) Wtvc (%)

30 0.2636 6 11 1640 0.2705 6 10 1550 0.2736 6 10 1560 0.2783 6 10 1570 0.2864 6 10 2280 0.2943 6 10 1490 0.2980 6 10 14

135

ANNEX E. ROBUSTNESS

Flexible body

Robust Stability


Uncert Freq Bend. Modes a6 (%) k1 (%) Wtvc (%)

30 24 24.1 53.2 99 0 0 540 25 25.1 53.2 99 0 0 550 26.7 26.7 53.3 99 0 0 560 43.7 43.9 22.1 91 0 1 970 41.1 41.3 32.4 91 0 0 1280 12.9 12.9 27.6 98 0 0 390 54.1 54.4 31.2 92 0 1 12

Robust performances


Bend. Modes a6 (%) k1 (%) Wtvc (%)

30 0.1804 78 0 1 340 0.1871 77 0 0 350 0.1984 75 0 0 960 0.2723 48 6 10 1470 0.2824 68 6 10 1880 0.1130 93 0 0 390 0.2967 52 6 10 14

E.2 Robustness of the µ− synthesis controller

Rigid body

Robust Stability


Uncert Freq a6 (%) k1 (%) Wtvc (%)

30 162 442 1.57 49 47 3840 156 449 1.64 15 47 4050 153 442 1.52 15 48 3960 161 – – 15 48 3870 176 – – 17 87 3880 218 667 0.0118 17 59 2790 253 – – 14 92 36


ANNEX E. ROBUSTNESS

Robust Performances


a6 (%) k1 (%) Wtvc (%)

30 0.4606 4 9 1140 0.4449 4 9 1150 0.4445 4 9 1260 0.4650 4 9 1170 0.4946 4 9 1080 0.5624 3 8 990 0.6012 3 8 9

Flexible Body

Robust Stability


Uncert Freq Bend. Modes a6 (%) k1 (%) Wtvc (%)

30 98.5 113 3.59 24 28 49 5940 98.5 115 24.7 70 10 33 2950 89.6 122 5.15 14 60 66 2260 91.9 113 0.60 1 19 39 4170 66.8 73.1 0.51 1 20 37 4380 120 120 27 76 21 36 2790 115 219 5.13 1 28 56 41

Robust Performances


Bend. Modes a6 (%) k1 (%) Wtvc (%)

30 0.1824 1 7 11 1540 0.3004 1 6 10 1450 0.2142 2 7 11 1560 0.3137 1 6 10 1470 0.2328 1 7 11 1580 0.3571 1 6 10 1390 0.3610 0 5 10 13


design of a robust controller for the vega tvc using the

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