modeling and nonlinear control for airship autonomous flight

UNIVERSIDADE TECNICA DE LISBOA

INSTITUTO SUPERIOR TECNICO

MODELING AND NONLINEAR CONTROL

FOR AIRSHIP AUTONOMOUS FLIGHT

Alexandra Bento Moutinho

(Mestre em Engenharia Mecanica)

Dissertacao para a obtencao doGrau de Doutor em Engenharia Mecanica

Orientador: Doutor Jose Raul Carreira Azinheira

Juri:

Presidente: Reitor da Universidade Tecnica de Lisboa

Vogais: Doutor Joao Manuel Lage de Miranda Lemos

Doutor Jose Manuel Gutierrez Sa da Costa

Doutor Felix Mora-Camino

Doutor Jorge Manuel Miranda Dias

Doutor Pedro Manuel Goncalves Lourtie

Doutor Jose Raul Carreira Azinheira

Dezembro de 2007

To all, and to a few in particular.

Resumo

O trabalho desenvolvido e apresentado nesta tese foca o projecto, validacao

e comparacao de diferentes solucoes de controlo nao-linear que permitam um

dirigıvel navegar autonomamente. De modo a atingir esta meta, e desen-

volvido um modelo nao-linear de seis graus de liberdade do dirigıvel baseado

nas equacoes de Lagrange, reproduzindo a resposta do dirigıvel a entradas

dos actuadores e a perturbacoes de vento. A linearizacao deste modelo para

diferentes condicoes de equilıbrio resulta no conhecido desacoplamento dos

movimentos longitudinal e lateral, e permite uma analise exaustiva do prob-

lema de projecto de controlo de dirigıveis em todo o envelope de voo. Existem

entao condicoes para propor solucoes alternativas de controlo nao-linear de

modo a obter uma unica lei de controlo, valida para diferentes missoes, inde-

pendente das condicoes de voo, e robusta a perturbacoes realistas de vento.

As metodologias de controlo desenvolvidas neste trabalho com vista ao voo

autonomo de dirigıveis sao o Escalonamento de Ganho, a Dinamica Inversa

e o Backstepping. Alem da analise de problemas especıficos inerentes ao pro-

jecto e implementacao de cada um dos controladores, sao tambem definidos

criterios desejados de desempenho, permitindo a comparacao das diferentes

solucoes. Esta analise, baseada em resultados de simulacao para missoes de

voo completas definidas desde a descolagem ate a aterragem, e considerando

perturbacoes de vento realistas, e importante de modo a estabelecer a viabili-

dade da implementacao dos controladores a bordo da plataforma experimental

do dirigıvel. Esta tese e parte da investigacao feita na area de controlo de

voo nao-linear de dirigıveis nos projectos AURORA e DIVA do Instituto de

Engenharia Mecanica (IDMEC) do Instituto Superior Tecnico, Universidade

Tecnica de Lisboa.

Keywords: Controlo nao-linear, Escalonamento de ganho, Dinamica inversa,

Backstepping, Dirigıvel, Modelacao, Controlo de voo.

i

Abstract

The work developed and presented in this thesis focuses on the design, val-

idation and comparison of different nonlinear control solutions allowing an

airship to navigate autonomously. To accomplish this task, a six-degrees-of-

freedom nonlinear model of the airship is developed based on the Lagrangian

equations, reproducing the airship response to actuator and wind disturbances

inputs. The linearization of this model for trim conditions over the flight enve-

lope results in the known decoupling of the longitudinal and lateral motions,

and allows a thorough analysis of the airship control design problem over the

entire aerodynamic range. The conditions are then set to propose alternative

nonlinear control solutions so as to have a single control law valid for different

missions, independent of the flight region, and robust to realistic wind dis-

turbances. The control methodologies developed in this work for the airship

autonomous flight are Gain Scheduling, Dynamic Inversion and Backstepping.

Besides the analysis of specific problems inherent to the design and implemen-

tation of each controller, desired performance criteria are also defined, allowing

the comparison of the different solutions. This assessment, based on simula-

tion results for complete flight missions defined from take-off to landing, and

considering realistic wind disturbances, is important in order to establish the

viability of the controllers implementation onboard the experimental airship

platform. This thesis is part of the research made in the area of nonlinear

flight control of airships for the AURORA and DIVA projects of the Institute

of Mechanical Engineering (IDMEC) in Instituto Superior Tecnico, Technical

University of Lisbon.

Keywords: Nonlinear control, Gain Scheduling, Dynamic inversion, Back-

stepping, Airship, Modeling, Flight Control.

iii

Acknowledgments

My first words of appreciation are undoubtedly to my supervisor, Professor

Jose Raul Azinheira. In his words, “A PhD thesis is not intended to close

doors, but to open some more”. His broad knowledge on subjects like modeling,

control, aerodynamics and instrumentation certainly opened a lot of doors for

me, being at the basis of clarifying and motivating discussions for this work.

I also want to thank Doctor Ely Carneiro de Paiva and Doctor Samuel Siqueira

Bueno of the AURORA project in CenPRA Brazil, one of the leading projects

in autonomous airships research, for the joint work developed even from such

a distance. A more recent project in this area is the Portuguese DIVA project,

whose team I also want to thank for the insight provided in the different aspects

of building an airship.

My appreciation goes also to my colleagues at GCAR, namely Miguel Pedro

Silva for always being available to discuss some of my doubts, and Mario

Mendes for all the technical support provided.

Last but definitely not least, I want to thank Carlos, my family and my friends

for being there. Bem hajam!

v

Contents

Resumo i

Abstract iii

Acknowledgments v

List of Figures xi

List of Tables xv

Notation xvii

Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

Variables description . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Introduction 1

1.1 Airships and their history . . . . . . . . . . . . . . . . . . . . . 1

1.2 The motivation - airship applications . . . . . . . . . . . . . . . 5

1.3 The control of airships flight . . . . . . . . . . . . . . . . . . . . 7

1.4 Structure of this work . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Contributions of this thesis . . . . . . . . . . . . . . . . . . . . . 13

2 The Airship Model 17

2.1 Airship platform . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Airship equations of motion . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Airship dynamics . . . . . . . . . . . . . . . . . . . . . . 20

2.2.2 Airship kinematics . . . . . . . . . . . . . . . . . . . . . 32

2.2.3 Airship simulator . . . . . . . . . . . . . . . . . . . . . . 34

vii

viii CONTENTS

2.3 Airship linearized models . . . . . . . . . . . . . . . . . . . . . . 34

2.3.1 Trim or equilibrium conditions . . . . . . . . . . . . . . . 35

2.3.2 Model linearization . . . . . . . . . . . . . . . . . . . . . 38

2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Common Concepts and Tools 45

3.1 Position errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.1 Path-following . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.2 Path-tracking . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Attitude reference and wind estimation . . . . . . . . . . . . . . 47

3.3 Controllers performance evaluation . . . . . . . . . . . . . . . . 49

3.3.1 Case-study mission . . . . . . . . . . . . . . . . . . . . . 50

3.3.2 Sensitivity and robustness . . . . . . . . . . . . . . . . . 51

4 Classical Approach: Linear Control 53

4.1 Airspeed and altitude regulation model . . . . . . . . . . . . . . 55

4.2 Lateral models . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.1 No-roll approximation . . . . . . . . . . . . . . . . . . . 56

4.2.2 Space domain approximation . . . . . . . . . . . . . . . 57

4.3 Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . . . 58

4.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.1 Airspeed and altitude regulation model . . . . . . . . . . 60

4.4.2 No-roll vs. space domain . . . . . . . . . . . . . . . . . . 62

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 Gain Scheduling 67

5.1 More linear models . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.1.1 Groundspeed and altitude regulation . . . . . . . . . . . 68

5.1.2 Complete 12-states linear model . . . . . . . . . . . . . . 69

5.2 Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3 Robustness analysis . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3.1 Performance robustness . . . . . . . . . . . . . . . . . . 75

5.3.2 Stability robustness . . . . . . . . . . . . . . . . . . . . . 80


5.4.1 Groundspeed and altitude regulation . . . . . . . . . . . 87

CONTENTS ix



5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6 Dynamic Inversion 99

6.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.1.1 Local coordinates transformation . . . . . . . . . . . . . 102

6.1.2 Exact linearization via feedback . . . . . . . . . . . . . . 105

6.1.3 Asymptotic output tracking . . . . . . . . . . . . . . . . 107

6.2 New formulation for cascaded systems . . . . . . . . . . . . . . 109

6.3 Application to airship path-tracking problem . . . . . . . . . . . 111




6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7 Backstepping 125

7.1 Wind estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.2 Backstepping design approach . . . . . . . . . . . . . . . . . . . 128

7.3 Application to the path-tracking problem . . . . . . . . . . . . . 128

7.4 Control design with saturation constraints . . . . . . . . . . . . 132

7.5 Control implementation . . . . . . . . . . . . . . . . . . . . . . 136

7.5.1 Adapted control law to deal with underactuation . . . . 136




7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8 Comparison of controllers performance 149

8.1 Performance for case-study mission . . . . . . . . . . . . . . . . 149

8.2 Sensitivity test results . . . . . . . . . . . . . . . . . . . . . . . 152

8.3 Computational effort . . . . . . . . . . . . . . . . . . . . . . . . 154

8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

9 Conclusions and Future Work 157

x CONTENTS

A Referentials 161

A.1 Frames definition . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A.1.1 Earth-Centered Inertial (ECI) frame . . . . . . . . . . . 161

A.1.2 North-East-Down (NED) or i frame . . . . . . . . . . 162

A.1.3 Aircraft-Body Centered (ABC) or l frame . . . . . . . 162

A.1.4 Aerodynamic or a frame . . . . . . . . . . . . . . . . . 163

A.2 Changing frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

B Dryden Model For Continuous Gust 165

C Differential geometry and topology 167

C.1 Lie derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

C.2 Diffeomorphisms and state transformations . . . . . . . . . . . . 169

Bibliography 171

List of Figures

1.1 The history of airships. . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Manned airships. . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Unmanned airships. . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 The AURORA airship. . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 AURORA airship sensors and actuators. . . . . . . . . . . . . . 19

2.3 Simulator block diagram for airship open-loop model. . . . . . . 34

2.4 Trim values of state and control input. . . . . . . . . . . . . . . 37

2.5 Poles of linearized longitudinal dynamics vs. airspeed. . . . . . . 40

2.6 Poles of linearized lateral dynamics vs. airspeed . . . . . . . . . 42

3.1 Path-following errors definition. . . . . . . . . . . . . . . . . . . 46

3.2 Path-tracking errors definition. . . . . . . . . . . . . . . . . . . 47

3.3 Wind and yaw reference estimation. . . . . . . . . . . . . . . . . 48

3.4 Case-study mission reference. . . . . . . . . . . . . . . . . . . . 50

4.1 Linear control block diagrams. . . . . . . . . . . . . . . . . . . . 60

4.2 Trajectory and altitude for airspeed and altitude regulation. . . . 61

4.3 Longitudinal groundspeed and airspeed for airspeed and altituderegulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4 Control action for airspeed and altitude regulation. . . . . . . . . 62

4.5 Trajectory, lateral error and yaw angle for lateral control com-parison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.6 Sideslip angle, rudder deflection and roll angle for lateral controlcomparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.1 Poles of linearized dynamics vs. airspeed. . . . . . . . . . . . . . 71

5.2 Evolution of B matrix coefficients with airspeed. . . . . . . . . . 73

xi

xii LIST OF FIGURES

5.3 Evolution of B matrix Tx and Tz coefficients with airspeed. . . . 74

5.4 Gain scheduling diagram block. . . . . . . . . . . . . . . . . . . . 74

5.5 Closed-loop nominal system. . . . . . . . . . . . . . . . . . . . . 75

5.6 Disturbed feedback control system. . . . . . . . . . . . . . . . . . 76

5.7 Singular values relations. . . . . . . . . . . . . . . . . . . . . . . 79

5.8 Frequency analysis of the MIMO nominal system. . . . . . . . . 79

5.9 Frequency-domain performance specifications - disturbance re-jection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.10 Robust stability analysis of the uncertain systems. . . . . . . . . 83

5.11 Stability robustness to plant parameter variation. . . . . . . . . . 86

5.12 Airship position coordinates and errors. . . . . . . . . . . . . . . 88

5.13 Airship ground velocity components and aerodynamic variables. . 88

5.14 Airship actuators input. . . . . . . . . . . . . . . . . . . . . . . 89


5.16 Airship north-east position and attitude. . . . . . . . . . . . . . 91



5.19 Airship north-east trajectory and attitude, and wind attitude. . . 93

5.20 Airship position errors and aerodynamic variables. . . . . . . . . 94


6.1 Normal form representation, with no internal dynamics. . . . . . 105

6.2 Closed-loop system, with new reference input υ. . . . . . . . . . 107

6.3 Closed-loop system, with model reference input . . . . . . . . . 109








7.1 Air velocity reference estimation (2D). . . . . . . . . . . . . . . 129



LIST OF FIGURES xiii






8.1 Comparison of airship 3D trajectories. . . . . . . . . . . . . . . 150

8.2 Comparison of position errors. . . . . . . . . . . . . . . . . . . . 151

8.3 Comparison of north-east trajectories with airship heading. . . . 151

8.4 Comparison of actuators request . . . . . . . . . . . . . . . . . . 153

A.1 Relationship between the different coordinate systems. . . . . . . 162

A.2 ABC and wind frames. . . . . . . . . . . . . . . . . . . . . . . . 163

B.1 Block diagram for gust generator. . . . . . . . . . . . . . . . . . 165

xiv LIST OF FIGURES

List of Tables

5.1 Robustness tests on model parameters. . . . . . . . . . . . . . . 96



8.1 Comparison of baseline results. . . . . . . . . . . . . . . . . . . 154

8.2 Computational effort comparison. . . . . . . . . . . . . . . . . . 155

8.3 Overall controllers comparison. . . . . . . . . . . . . . . . . . . 155

xv

xvi LIST OF TABLES

Notation

Acronyms

ABC Airship Body Centered

AF Aerodynamic Flight

AURORA Autonomous Unmanned Remote mOnitoring Robotic Airship

CB Center of Buoyancy

CG Center of Gravity

CV Center of Volume

DIVA Dirigıvel Instrumentado para Vigilancia Aerea

ECI Earth-Centered Inertial

GPS Global Positioning System

HF Hover Flight

IMU Inertial Measurement Unit

LQR Linear Quadratic Regulator

LTA Lighter-Than-Air

LTI Linear Time-Invariant

MIMO Multi-Input / Multi-Output

NED North-East-Down

PP Pole Placement

RMS Root Mean Square

UAV Unmanned Aerial Vehicle

xvii

xviii NOTATION

Nomenclature

Typeface

italic − scalar variables

bold − vector or matrix variables

Subscripts

a − apparent for masses; airspeed for velocities

B − buoyancy

e − equilibrium condition

h − horizontal plane (lateral mode)

v − vertical plane (longitudinal mode); virtual for masses

w − wind

Superscripts

c − regarding the CG

o − regarding the CV

Operations

× − cross-product

Others

. − variation relative to equilibrium condition or reference

NOTATION xix

Variables description

Symbol Domain Unit Definition

0i Ri×i none zero matrix

α R rad angle of attackβ R rad sideslip angleδ R m vertical position errorε R m lateral position errorδa R rad aileron deflectionδe R rad elevator deflectionδr R rad rudder deflectionδv R rad main propellers vectoring angleη R m longitudinal position errorθ R rad pitch angleΦ R

3×1 rad angular position vector relative to i framewith Euler angles, [φ, θ, ψ]T

φ R rad roll angleψ R rad yaw angleΩ R

3×3 rad/s cross-product matrix equivalent to ω×ω R

3×1 rad/s angular velocity and components in l frame,[p, q, r]T

Ω6 R6×6 rad/s see equation (2.45)

ag R3×1 m/s2 inertial gravity acceleration vector, [0, 0, g]T

A Rn×n − n-state dynamic matrix

B Rn×m − m-input matrix

C R3×1 m CG, center of mass of the airship

C3 R3×3 m cross-product matrix equivalent to OC×

d R6×1 − disturbance vector

h R m altitude, −pDIi R

i×i none identity matrixJ R

3×3 kg.m2 inertia matrix of the airshipJΦ R

6×6 none see equation (2.77)JB R

3×3 kg.m2 inertia matrix of the buoyancy air, diagonalmatrix

Jv R3×3 kg.m2 virtual inertia matrix, diagonal matrix

m R kg airship massMa R

6×6 − generalized apparent mass matrix of the airshipwith masses and inertias, Mo + Mv

mB R kg buoyancy massMB R

6×6 − generalized inertial mass matrix of the buoyancyair with masses and inertias, diag(mBI3,JB)

MBa R6×6 − generalized apparent mass matrix of the buoyancy

air with masses and inertias, MB + Mv

xx NOTATION

Symbol Domain Unit Definition

Mo R6×6 − generalized mass matrix of the airship with masses

and inertias, see equation (2.11)Mv R

3×3 kg virtual mass matrixMv R

6×6 − generalized virtual mass matrix with masses andinertias, diag(Mv,Jv)

mw R kg weighting mass, m−mB

O R3×1 m center of buoyancy CB = CV, origin of l frame

OC R3×1 m vector from CV to CG, C − O = [ax, 0, az]

T

P R6×1 − position vector, [pT ,ΦT ]T

p R3×1 m cartesian position vector and components in i

frame, [pN , pE, pD]T

p R rad/s roll rate in l framepN or N R m position northpE or E R m position eastpD or D R m position downp R

3×1 m/s linear velocity and components in i frame,[pN , pE, pD]T

q R rad/s pitch rate in l framer R rad/s yaw rate in l frameR R

3×3 none see equation (2.18)S R

3×3 none see equation (2.17)TD R N engines differential thrust, TL − TRTL R N left engine thrustTR R N right engine thrustTx R N engines longitudinal thrustTy R N tail motor thrustTz R N engines vertical thrustu R m/s forward speed in l frameu R

7×1 − input vector, [δe, TL, TR, δv, δa, δr, Ty]T

u R7×1 − perturbation input vector, u − ue

ue R7×1 − equilibrium input vector, [δee

, TLe, TRe

, δve, δae

, δre , Tye]T

V R6×1 − velocity vector, [vT ,ωT ]T

v R3×1 m/s linear velocity and components in l frame, [u, v, w]T

v R m/s lateral speed in l frameV3 R

3×3 m/s cross-product matrix equivalent to v×V6 R

6×6 m/s see equation (2.45)

Vt R m/s true airspeed,√

u2a + v2

a + w2a

w R m/s vertical speed in l frameW R J kinetic energyx R

12×1 − state vector, [vT ,ωT ,pT ,ΦT ]T

x R12×1 − perturbation state vector, x − xe

xe R12×1 − equilibrium state vector, [vTe ,ω

Te ,p

Te ,Φ

Te ]T

XT R N engines total thrust, TL + TR

Chapter 1

Introduction

Contents

1.1 Airships and their history . . . . . . . . . . . . . . 1

1.2 The motivation - airship applications . . . . . . . 5

1.3 The control of airships flight . . . . . . . . . . . . . 7

1.4 Structure of this work . . . . . . . . . . . . . . . . . 11

1.5 Contributions of this thesis . . . . . . . . . . . . . 13

Ha um ditado que ensina “o genio e uma grande paciencia”;

sem pretender ser genio, teimei em ser um grande paciente. As

invencoes sao, sobretudo, o resultado de um trabalho teimoso, em

que nao deve haver lugar para o esmorecimento.

Alberto Santos-Dumont, 19181

1.1 Airships and their history

An airship or dirigible is a buoyant lighter-than-air aircraft that can be steered

and propelled through the air. Airships were the first aircraft to do controlled,

powered flight, being widely used prior to the 1940s.

Airships were developed from the free balloon. Early balloons were not truly

navigable. Attempts to improve maneuverability included elongating the bal-

loon shape and using a powered screw to push it through the air. Credit for the

1There is a saying that teaches “the genius is a great patience”; not intending to be agenius, I insisted on being very patient. The inventions are, most of all, the result of apersistent work, where no wilting shall take place. Alberto Santos-Dumont, 1918

1

2 CHAPTER 1. INTRODUCTION

construction of the first navigable full-sized airship goes to French engineer,

Henri Giffard, who, in 1852, attached a small, steam-powered engine to a huge

propeller and chugged through the air for seventeen miles at a top speed of

five miles per hour (fig. 1.1(a)).

The first airship to demonstrate its ability to return to its starting place in

a light wind was La France (fig. 1.1(b)), developed in 1884 by the French

inventors Charles Renard and Arthur Krebs. It was propelled by an electrically

driven propeller. However, it was not until the invention of the gasoline-

powered engine in 1896 that practical airships could be built.

Count Ferdinand von Zeppelin, the German inventor, completed his first air-

ship in 1900; this airship had a rigid frame and served as the prototype of

many subsequent models. The first Zeppelin airship consisted of a row of 17

gas cells individually covered in rubberized cloth; the whole was confined in a

cylindrical framework covered with smooth-surfaced cotton cloth. It was about

128m long and 12m in diameter; the hydrogen-gas capacity totalled 11.3 mil-

lion liters. The ship was steered by rudders fore and aft and was driven by two

11kW Daimler internal-combustion engines, each turning two propellers. Pas-

sengers, crew, and engine were carried in two aluminium gondolas suspended

forward and aft. At its first trial, on July 2, 1900, the airship carried five

people (fig. 1.1(c)); it attained an altitude of 396m and flew a distance of 6km

in 17 minutes.

In 1898, the Brazilian Alberto Santos-Dumont was the first to construct and fly

a gasoline-powered airship, developing a series of 14 airships in France. In his

No. 6, he circled the Eiffel Tower in 1901 (fig. 1.1(d)). The American inventor

Thomas S. Baldwin built a 53-foot airship, the California Arrow, winning a

one-mile race in October 1904. Walter Wellman failed in an effort to cross the

Atlantic Ocean in an airship in 1910, after five unsuccessful attempts to reach

the North Pole. Although many successful flights were made before 1910, the

best engine available for use in the early airship was too heavy in proportion

to its power.

At the beginning of World War I, Germany had ten zeppelins. By 1918, 67

zeppelins had been constructed, and 16 survived the war. Those not captured

were surrendered to the Allies by the terms of the Treaty of Versailles in

1919. Airships were operated in a number of nations between the two world

wars. The major operators of rigid airships were Britain, the United States

and Germany, with Italy and France also operating a few. Italy, the Soviet

1.1. AIRSHIPS AND THEIR HISTORY 3

Union, the United States and Japan operated semi-rigid airships, while blimps

(nonrigid airships) were operated in many nations.

The war, however, disclosed the vulnerability of airships to aeroplane attack,

and caused the abandonment of the dirigible for offensive military purposes.

The United States was the only power to use airships during World War II,

and the airships played a small but important role. The Navy used them

for minesweeping, search and rescue, photographic reconnaissance, scouting,

escorting convoys, and antisubmarine patrols. Airships accompanied many

oceangoing ships, both military and civilian. Of the 89.000 ships escorted by

airships during the war, not one was lost to enemy action. The Akron and

Macon were two rigid airships built in the United States for the U.S. Navy.

They were the only airships that could launch and retrieve planes in midair

(fig. 1.1(e)).

When the various restrictions imposed by the Treaty of Versailles on Germany

were lifted, Germany was again allowed to construct airships. It built three

giant rigid airships: the LZ-127 Graf Zeppelin, LZ-l29 Hindenburg, and LZ-l30

Graf Zeppelin II. The Graf Zeppelin is considered the finest airship ever built.

It flew more miles than any airship had done to that time or would in the future.

Its first flight was on September 18, 1928. In August 1929, it circled the globe.

Its flight began with a trip from Friedrichshaften, Germany, to Lakehurst, New

Jersey, stopping only at Tokyo, Japan, Los Angeles, California, and Lakehurst.

The trip took twelve days, less time than the ocean trip from Tokyo to San

Francisco. During the ten years the Graf Zeppelin flew, it made 590 flights

including 144 ocean crossings. It flew more than 1.609.344km, visited the

United States, the Arctic, the Middle East, and South America, and carried

13.110 passengers.

When the Hindenburg was built in 1936, Zeppelins had been accepted as a

quicker and less expensive way to travel long distances than ocean liners pro-

vided. After making ten transatlantic crossings in regular commercial service

in 1936, when it was preparing to land at Lakehurst, New Jersey, its hydrogen

ignited and the airship exploded and burned (fig. 1.1(f)).

Since the destruction of the Hindenburg, airship activity has been confined

to the nonrigid type of craft. Although airships are no longer used for pas-

senger transportation, they continued to be used for other purposes such as

advertising and sightseeing.


(a) The first flight of an airship, Henri Giffard’s steamairship, 1852.

(b) La France made thefirst deliberate circlethrough the air in 1884.

(c) The first ascent of Ferdinand von Zeppelin’s LZ-1in 1900.

(d) Alberto Santos-Dumontwins the Deutsch prize in1901 for circling the EiffelTower in his No.6 airship.

(e) Akron, the world’s largest dirigible,pays its first visit to Washington, D.C.in 1931.

(f) Explosion of the Hindenburg, at Lake-hurst, New Jersey, in 1937.

Figure 1.1: The history of airships.

1.2. THE MOTIVATION - AIRSHIP APPLICATIONS 5

1.2 The motivation - airship applications

After half century of hibernation, the interest in using airships for several

different applications is increasing worldwide nowadays [1, 2].

The lift of airships is mainly aerostatic, as opposed to aerodynamic as in

airplanes and helicopters. Consequently, and comparing to other aerial vehi-

cles, airships spend most energy moving and compensating wind disturbances,

rather than trying to keep themselves on air. For this reason, they need less

powerful engines, leading to a lower energy consumption, as well as less noise

or vibrations. They possess a big load capacity and long endurance, and they

can fly at low speeds or even hover. Airships also present a slow degradation

in case of failure and are intrinsically more stable than other platforms.

Considering these characteristics, airships have a wide spectrum of applica-

tions as observation and data acquisition platforms. They can be used in

several fields related to biodiversity, ecological and climate research and mon-

itoring. Inspection oriented applications cover different areas such as mineral

and archaeological prospecting, agricultural and livestock studies, crop yield

prediction, land use surveys in rural and urban regions, fire detection and also

inspection of man-made structures such as pipelines, power transmission lines,

dams and roads.

Besides their use in advertising2 and leisure flights3, manned airships are be-

ing used in some of the above mentioned applications, among other. The

US/LTA conducts remote sensing experiments with airships since 1992. In

2000, SkyKitten maiden flight takes place in Cardiff. It is capable of landing

virtually anywhere on land or water without need of ground infrastructure and

carrying heavy payloads. In 2001, the first test of a new airship from Cargo-

Lifter, also designed to carry up heavy loads, happens in Berlin. The Russian

company RosAeroSystem commercializes the Au-30 Patrol Airship series. The

Total Pole Airship Project (fig. 1.2(a)), for instance, aims to measure the thick-

ness of the pack ice layer covering the Arctic Ocean, using one of the series.

Two others are used for surveillance of power lines in Russia and one other is

scheduled to monitor traffic conditions in Moscow. With a more humanitarian

purpose, Mineseeker (fig. 1.2(b)) is an airship-based mine detection system

with optical, electro-optical and ground penetrating radar sensors, tested in

2http://www.globalskyships.com/, http://www.airshipman.com/,http://www.lightships.com/

3http://www.zeppelinflug.de/, http://www.nac-airship.com/

http://www.globalskyships.com/

http://www.airshipman.com/

http://www.lightships.com/

http://www.zeppelinflug.de/

http://www.nac-airship.com/


Kosovo by the United Nations.

(a) Total Pole airship 4. (b) The Mineseeker airship at theKFOR base in Podujevo, Kosovo 5.

Figure 1.2: Manned airships.

High altitude flight provides a unique vantage point for scientific exploration as

well as for observation and surveillance. An airship, with its heavy lifting ca-

pacity, provides the potential for carrying certain types of payloads that would

not be practical for other types of high altitude long endurance vehicles. The

main interest in high altitude airships [3] has been for communications or wide

area surveillance. For civilian applications, high altitude airships represent a

low cost alternative to a geostationary satellite. For the military it represents

a versatile platform that can be positioned over key areas of interest quickly

and provide continuous wide area coverage for extended periods of time.

With these goals in mind, in the USA, Lockheed Martin finished a detailed

design of a high-altitude airship prototype airship in support of the Depart-

ment of Defense, in order to demonstrate, among others, launch and recovery,

station-keeping and flight control capabilities. The Japan Aerospace eXplo-

ration Agency (JAXA) [4] has developed and flight tested a 60m-class un-

manned airship successfully to the altitude of 4km in October 2004. In Ko-

rea [5], the 50m unmanned airship system KARI (fig. 1.3(a)) is developed and

4http://www.jeanlouisetienne.com/.5http://www.airforce-technology.com/projects/mineseeker/.

http://www.jeanlouisetienne.com/

http://www.airforce-technology.com/projects/mineseeker/

1.3. THE CONTROL OF AIRSHIPS FLIGHT 7

flight tested to acquire basic technologies required to develop a station keeping

electrical powered airship.

A regional navigation system using geosynchronous satellites and stratospheric

airship is proposed by Won [6], and stratospheric communications platforms

for rural applications is developed by Ilcev and Singh [7]. Hurd et al. [8]

consider airships for deep space optical communications. Rao et al. [9] argue

that unmanned airships present a unique potential in emergency management

cycle and discuss their applications for surveillance, search and rescue, and

communication.

Looking further beyond, airships are also being considered for the exploration

of planetary bodies with an atmosphere [10]. NASA [11] is already designing

and testing a robotic lighter-than-air vehicle (fig. 1.3(b)) for the exploration

of planets and moons such as Venus, Mars, Titan and the gas giants.

With such a wide spectrum of applications, and considering the quest for au-

tonomy, airships present characteristics and competitive costs when compared

to other aircrafts, certainly constituting an important option for research, de-

velopment and experimental validation in autonomous aerial robotics. More-

over, most of the solutions established for this kind of air vehicle may be

transferred or adapted for airplanes or helicopters, where the risks and costs

involved in testing new methodologies are obviously higher.

1.3 The control of airships flight

During the last decades Unmanned Aerial Vehicle (UAV) systems have evolved

into highly capable machines, used mostly for surveillance and data acquisition

purposes. For a rapid unmanned capability advancement, and from a military

perspective, the US Dept. of Defense presented a UAV roadmap in 2005 [12],

containing a survey of platforms and UAV technologies. From the civilian

side, a capabilities assessment of UAVs use in Earth observations is presented

by NASA [13], addressing the technologies and capabilities required for viable

UAV missions.

Many of the UAV applications require the capacity for autonomous flight,

involving the development of a flight control and navigation system. Several

advances made in this field have been published, applying different control

solutions to a variety of UAVs [14, 15, 16, 17, 18, 19, 20, 21, 22].


To provide airships with autonomous operation capacity is a recent focus of

investigation worldwide. Project AURORA - Autonomous Unmanned Remote

mOnitoring Robotic Airship (fig. 1.3(c)) results from a partnership between

DRVC/CenPRA in Brazil, IDMEC/IST in Portugal and ICARE/INRIA in

France [23]. It focuses on the establishment of the technologies required to

substantiate autonomous operation of unmanned robotic airships for environ-

mental monitoring and aerial inspection missions. This includes sensing and

processing infrastructures, control and guidance capabilities, and the ability to

perform mission, navigation, and sensor deployment planning and execution.

Other important researches related to outdoor autonomous airships in the

world at this moment are the Lotte Project at Germany (fig. 1.3(d)) [24], the

French projects at LAAS-CNRS (fig. 1.3(e)) [25], and LSC in Universite d’Evry

(fig. 1.3(f)) [26]. In the USA there is a partnership between the projects of

STWing-SEAS6 [27] of University of Pennsylvania and the EnviroBLIMP at

CMU7 (fig. 1.3(g)). Recently, Project DIVA - Dirigıvel Instrumentado para

Vigilancia Aerea (fig. 1.3(h)) started in Portugal8, sharing a partnership with

the AURORA Project.

Aiming at the autonomous airship goal, aerial platform positioning and path-

tracking should be assured by a control and navigation system. Such a system

needs to cope with the highly nonlinear and underactuated airship dynamics,

ranging from hover flight (defined here as a flight in low airspeed condition) to

cruise or aerodynamic flight. In addition, the abrupt and continuous dynamics

transition between the two regions, and the different use of actuators necessary

within each region, makes that a very difficult issue to be dealt with by the

control scheme.

The most common solution to the highly nonlinear airship dynamics lies in its

linearization. One important result of the linearization approach is the sepa-

ration of two independent motions: the motion in the vertical plane, named

longitudinal, and the motion in the horizontal plane, named lateral. This

decoupling allows the design of independent controllers for the two motions.

Following this approach, experimental results were obtained for the AURORA

airship for path following through a set of pre-defined points in latitude/longitude,

along with an automatic altitude control [23]. Also based on a linearized air-

ship model, Wimmer et al. [24] introduced a robust controller design method

6http://www.stwing.org/blimp/7http://www.frc.ri.cmu.edu/projects/enviroblimp/8http://paloma.isr.uc.pt/diva/

http://www.stwing.org/blimp/

http://www.frc.ri.cmu.edu/projects/enviroblimp/

http://paloma.isr.uc.pt/diva/

1.3. THE CONTROL OF AIRSHIPS FLIGHT 9

(a) Korean airship KARI. (b) NASA JPL aerobot.

(c) AURORA airship. (d) Lotte airship.

(e) LAAS-CNRS airship Karma. (f) LSC airship CEMIF.

(g) STWing-SEAS/EnviroBLIMP airship. (h) DIVA airship.

Figure 1.3: Unmanned airships.


to compensate for the lack of knowledge about the Lotte airship dynamic be-

havior and model parameters. The decoupled longitudinal and lateral control

systems both consist of an inner H∞-controller for the dynamics and an outer

SISO P- or PI-controller for the remaining states. Experimental results are

shown therein for the pitch and velocity control. We remark that, as far as

we are aware, both experimental results (from Lotte and AURORA Projects)

on automatic control for outdoor airships are the only ones reported in the

literature at this moment.

Also based on linearized decoupled models of the airship, and for aerodynamic

flight, solutions for the lateral control include H∞ [28], H2/H∞ approach for

the design of a lateral PD-PI controller [29] and state feedback with integral

control [25]. Considering the control of both lateral and longitudinal motions

different solutions are also proposed, namely one-loop-at-a-time PID [30] and

PI control [31], sliding modes techniques [32, 33], vision-based [34, 35, 36, 37],

fuzzy logic [9] and fuzzy logic improved with genetic algorithms [38]. The

LAAS/CNRS autonomous blimp project [39, 40] proposes a global control

strategy including hover and aerodynamic flight. It is achieved by switching

between four sub-controllers based on linear and backstepping solutions, one

for each of the independent flight phases considered, take-off and landing, and

longitudinal and lateral navigation.

The use of linearized model dynamics restricts the validity of the controller to

trim points, or implies the scheduling between controllers. Using a nonlinear

model avoids these limitations, allowing to design an automatic control system

covering all the aerodynamic range, such that the different flight regions, from

hover to aerodynamic flight, are considered inside a sole formulation. For

security reasons, as well as simplicity and flexibility, a global nonlinear control

is more interesting than a linearized and decoupled one.

Considering a nonlinear model, but assuming a simplified case where the diri-

gible motion is limited to the horizontal plane, Bestaoui and Hima [41] propose

an input-output linearization control. Using a six-degrees-of freedom nonlinear

model of the airship, Beji et al. [26] introduce a backstepping tracking feed-

back control for ascent and descent flight maneuvers, where the objective is to

stabilize the airship engine around trimmed flight trajectories. Park et al. [42]

also propose an input-output linearization with a neural network applied to

compensate the underlying model errors, to control velocity, pitch and yaw.

Image-based solutions are also used in the control of indoor airships [43, 44, 45].

1.4. STRUCTURE OF THIS WORK 11

Recently, Guzman [46] compared different control laws that assured reference

tracking in velocity, altitude and heading during aerodynamic flight. The

compared controllers are the classic PID strategy, a generalized predictive

controller and nonlinear first order techniques.

None of these works, with the exception of the LAAS-CNRS group solution

with decoupled controllers [40, 47], presented results for complete missions

including take-off and landing, path-tracking and stabilization. Moreover, sel-

dom are the ones that consider such an important issue as robustness to wind

disturbances. This work, inserted in the AURORA and DIVA projects, aims to

accomplish this task: to develop and compare airship control solutions, valid

for the entire flight envelope and capable of executing realistic missions, while

being robust to wind input.

1.4 Structure of this work

Part of the AURORA and DIVA projects objectives lies on the development of

control solutions to provide an airship with autonomous flight capacities. The

progress in this area comprises different stages: (i) control design theoretical

development and analysis; (ii) implementation of the controller in the simula-

tor, and consequent validation; and (iii) experimental validation in autonomous

flight. This work focuses on the first two steps. We propose to design, validate

on simulation and compare different nonlinear control solutions that will allow

an airship to navigate autonomously.

The first milestone is obviously the modeling of the airship, which is a com-

plex dynamic system with six-degrees-of-freedom. Chapter 2 is dedicated to

this objective. In order to provide a general idea of this kind of aerial vehicle,

like usual sensor and actuators available, their configuration and limitations,

the AURORA prototype is described. A model description allows us to bet-

ter understand the airship behavior using a simulator, and is at the basis of

model-based control laws. The airship equations of motion are comprised of

both dynamics and kinematics, and include the wind input. The dynamics are

obtained using the Lagrangian approach, instead of the Newtonian method

used in [48]. The airship equations of motion provide a good system descrip-

tion, and therefore allow to reproduce the airship response to actuator and

wind disturbances inputs. For demonstration purposes, a simulator based on

the AURORA airship characteristics is available [29]. The complexity of the


equations, however, justifies the search for a linear version, a usual practice

in aeronautics. The linearization of the airship model around an equilibrium

condition results in the decoupling of the longitudinal and lateral modes. The

analysis of the different models obtained for trim points over the flight enve-

lope provides a good knowledge of the airship behavior at different airspeeds,

important in order to develop a controller valid for any type of mission.

Before advancing to the control design, some common concepts and tools are

described in Chapter 3. These include references and errors definition, wind

estimation, and the criteria used in the comparison of the different nonlinear

controllers presented.

Chapter 4 is justified as an introduction for the control design part. Con-

sidering only linear control, and therefore a single model and controller, the

applicability of the solutions presented is restricted to the regulation of the

state errors, assuring the validity of the linearization at the chosen trim point.

This limitation vanishes if the linear systems and controllers are not fixed but

change with the measured airspeed (which defines an equilibrium condition).

This is the idea of the Gain Scheduling technique presented in Chapter 5,

which extends the validity of the linearization approach to a range of operating

points, instead of a single one. For each linear model obtained, a control law

is designed, and the overall control synthesis is achieved by switching between

models and respective controllers as function of scheduling variables. This is

the first of the three nonlinear control solutions we propose.

This approach, however, depends mostly on the engineer knowledge of the

system for a good choice of the scheduling parameters, resulting in an iterative

and time consuming process. Moreover, its guarantees of success depend on

extensive simulations covering different possible scenarios. This leads to the

search of a single control law, more related to the airship system itself than to

the control designer experience. The Dynamic Inversion approach described in

Chapter 6 is such a methodology. By inverting the system model, a control law

is obtained that cancels existing deficient or undesirable dynamics by replacing

them with a set of desired ones.

The third and last solution is Backstepping, a Lyapunov-based control design

approach presented in Chapter 7. By formulating a scalar positive function

of the system states and then choosing a control law to make this function

decrease, it is guaranteed that the nonlinear control system thus designed will

be stable. Moreover, it will be robust to some unmatched uncertainties.

1.5. CONTRIBUTIONS OF THIS THESIS 13

Any of the three control solutions presents its advantages and disadvantages,

many of which are discussed in the respective chapters. Yet, a common com-

parison between them is important as to provide a better overview of the

different control options. In Chapter 8 this assessment is made considering

parameters such as path-tracking trajectory errors and actuators request for

a case-study complete mission, controller performance robustness in face of

model parameter uncertainty and computational effort. These factors, to-

gether with some implementation issues, are relevant to evolve to the next

phase, the experimental validation in autonomous flight.

In Chapter 9, the conclusions of this work summarize the knowledge gained,

and point the directions of our forthcoming investigation.

1.5 Contributions of this thesis

The main contributions of this thesis are the following:

− the derivation of the airship dynamic equations of motion using the La-

grangian approach;

− a thorough analysis of the airship control design problem over the entire

flight envelope;

− the proposal of alternative control solutions so as to have a single con-

trol law valid for different missions, independent of the flight region and

robust to realistic wind disturbances, their evaluation and comparison.

This covers the following control methodologies:

– Gain Scheduling, providing a linearized reference for comparison;

– Dynamic Inversion, with a new formulation for cascaded systems

described in terms of velocity and position, and where the output

of interest is the position;

– Backstepping, including input saturations.

− analysis of specific problems, as well as desired performance criteria.

Parts of the work related with this thesis have been previously published.

Related with Chapter 5, the application of gain scheduling to airship path-

tracking is presented in:


Alexandra Moutinho and Jose Raul Azinheira. A Gain-Scheduling

Approach for Airship Path-Tracking. In Proceedings of the 4th In-

ternational Conference on Informatics in Control, Automation and

Robotics, Setubal, Portugal, August 2006.

The application to hover control is described in:

Alexandra Moutinho and Jose Raul Azinheira. A Gain-Scheduling

Approach for Airship Stabilization. In Proceedings of the 7th Por-

tuguese Conference on Automatic Control, Lisbon, Portugal, Septem-

ber 2006.

Related with Chapter 6, the application of dynamic inversion to the lateral

control synthesis of the Aurora airship and its comparison with linear control

can be found in:

Alexandra Moutinho and Jose Raul Azinheira. Path control of

an autonomous airship using dynamic inversion. In Proceedings

of the 5th IFAC/EURON Symposium on Intelligent Autonomous

Vehicles, Lisbon, Portugal, July 2004.

A sensitivity and robustness analysis of the dynamic inversion controller was

first presented in:

Alexandra Moutinho and Jose Raul Azinheira. Stability and ro-

bustness analysis of the AURORA airship control system using

dynamic inversion. In Proceedings of the IEEE International Con-

ference on Robotics and Automation, Barcelona, Spain, April 2005.

The hover control based in the dynamic inversion approach is described in:

Alexandra Moutinho and Jose Raul Azinheira. Hover Stabilization

of an Airship using Dynamic Inversion. In Proceedings of the 8th

International IFAC Symposium on Robot Control, Bologna, Italy,

September 2006.

Preliminary results on the dynamic inversion and backstepping solutions are

included in this AURORA project report:

1.5. CONTRIBUTIONS OF THIS THESIS 15

Ely Carneiro de Paiva, Jose Raul Azinheira, Josue G. Ramos Jr.,

Alexandra Moutinho and Samuel Siqueira Bueno, Project AU-

RORA: Infrastructure and Flight Control Experiments for a Robotic

Airship. In Journal of Field Robotics, Vol.23 (3-4), pp. 201-222,

March/April 2006.

Related with Chapter 7, a backstepping solution using quaternions for airship

stabilization is proposed in:

Jose Raul Azinheira, Alexandra Moutinho and Ely Carneiro de

Paiva. Airship Hover Stabilization using a Backstepping Approach.

In Journal of Guidance, Control and Dynamics, Vol.29 (4), pp.

903-914, July/August 2006.

while a backstepping solution for the stabilization of a generic UAV, also using

quaternions, is described in:

Jose Raul Azinheira and Alexandra Moutinho. Hover Control of

a UAV with Backstepping Design Including Input Saturations. In

press, IEEE Transactions on Control and Systems Technology.

Chapter 2

The Airship Model

Contents

2.1 Airship platform . . . . . . . . . . . . . . . . . . . . 18

2.2 Airship equations of motion . . . . . . . . . . . . . 19

2.2.1 Airship dynamics . . . . . . . . . . . . . . . . . . . . 20

2.2.2 Airship kinematics . . . . . . . . . . . . . . . . . . . 32

2.2.3 Airship simulator . . . . . . . . . . . . . . . . . . . . 34

2.3 Airship linearized models . . . . . . . . . . . . . . . 34

2.3.1 Trim or equilibrium conditions . . . . . . . . . . . . 35

2.3.2 Model linearization . . . . . . . . . . . . . . . . . . . 38

2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 43

A good knowledge of the airship model and behavior is essential for a success-

ful control design. For this reason, this chapter presents the airship modeling.

In order to provide a general idea of this kind of aerial vehicle, like usual sen-

sor and actuators available, their configuration and limitations, the AURORA

prototype is described in Section 2.1. The airship nonlinear dynamic model

is introduced in Section 2.2, formed from both dynamic and cinematic equa-

tions. The complexity of the nonlinear model justifies the search for a linear

simplified version. Section 2.3 describes the linearization procedure that leads

to the decoupled longitudinal and lateral state-space models. For the insight

it provides on the airship behavior, the longitudinal motion of the airship is

analyzed as function of the airspeed.

Whenever necessary for demonstration purposes, the AURORA airship plat-

form characteristics and configuration are used.

17

18 CHAPTER 2. THE AIRSHIP MODEL

2.1 Airship platform

The Lighter-Than-Air (LTA) robotic prototype AURORA has been built as

an evolution of the Airspeed Airships’ AS800. It is a nonrigid airship 10.5m

long, with 3.0m diameter and 34m3 of volume. The payload capacity is ap-

proximately 10kg and maximum speed is around 50km/h (see fig. 2.1).

Figure 2.1: The AURORA airship.

The main control and navigation sensors currently used on the airship are (see

fig. 2.2(a)): a GPS with differential correction that provides the inertial po-

sition coordinates and velocity; an Inertial Measurement Unit (IMU), which

provides the roll, pitch, and yaw attitude, the angular rates and body axes

linear acceleration, serving as an inclinometer and compass as well; a Wind

sensor that measures the relative airship air speed in all three axes, the aero-

dynamic incidence angles, as well as the barometric altitude; and a Camera

that provides aerial images for vision processing algorithms.

The airship actuators are its deflection surfaces and two main propellers dis-

posed on each side of the gondola (see fig. 2.2(b)). The four deflection surfaces

at the stern, arranged in a ’×’ shape with allowable deflections in the range

[−25o,+25o], generate the equivalent rudder δr and elevator δe commands of

the classical ’+’ tail. The aileron command δa is obtained with the rota-

tion of the four deflection surfaces in the same direction. The two engines,

with a vectoring angle δv within the interval [−30o,+120o], are driven by two-

stroke engines providing total XT (within [0, 80]N) and differential TD (within

[0, 40]N) thrusts. A small lateral stern thruster Ty may also be available, per-

pendicular to the airship longitudinal axis, to introduce one extra horizontal

input actuation during hovering tasks.

2.2. AIRSHIP EQUATIONS OF MOTION 19

GPS, IMU

GPS antenna

Wind sensor

Camera

(a) Sensors.

+

+

Tδr

_Y

_

+

δ_

δv

e

TX , T

D

(b) Actuators.

Figure 2.2: AURORA airship sensors and actuators. (The aileron commandδa is obtained with the rotation of the four deflection surfaces in the samedirection.).

2.2 Airship equations of motion

The airship model is a mathematical description of the airship motion. It

is given by a set of differential equations called equations of motion, which

represent the relations between the control inputs and the state variables. A

description of the referentials mentioned herein may be found in Appendix A.

The airship nonlinear model results in the dynamic equation expressed in a

state-space form:

x = f(x,u,d) (2.1)

where:

− the state x = [vT ,ωT ,pT ,ΦT ]T includes the linear v = [u, v, w]T and

angular ω = [p, q, r]T inertial velocities of the airship expressed in the

l frame, the cartesian position p = [pN , pE, pD]T of its center of volume

in the i frame, and the attitude of the airship Φ = [φ, θ, ψ]T given by

the Euler angles;

− the input vector u = [δe, TL, TR, δv, δa, δr, TY ]T includes the elevator de-

flection δe, the left and right engines thrust TL, TR, the engines vectoring

angle δv, the aileron deflection δa, the rudder deflection δr and the lat-

eral thrust TY (since it is not yet implemented in the AURORA airship,

although mentioned, it will not be used for control);


− the disturbance vector d includes the wind input (wind velocity) ex-

pressed in the i frame with a constant term and a six components

vector modeling the atmospheric turbulence (nonconstant wind). It is

represented by linear velocity pw = [pNw, pEw

, pDw]T (a horizontal repre-

sentation in polar components is [psw, phw

]T , with wind strength pswand

heading phw) and angular velocity ωw.

In the attempt to establish a workable mathematical model of the airship flight,

a number of considerations have to be taken into account [48]:

1. the airship displaces a very large volume of air and its virtual (added)

mass and inertia properties become significant, i.e., the LTA vehicle be-

haves as if it had a mass and moments of inertia substantially higher

than those indicated by conventional physical methods;

2. three kinds of masses and inertia matrices must be considered: the mass

and inertia (m,J) of the vehicle itself; the mass and inertia (mB,JB) of

the buoyancy air, corresponding to the air displaced by the total volume

of the airship; and the virtual mass and inertia (Mv,Jv), which may be

regarded as the mass of air around the airship and displaced with the

relative motion of the airship in the air;

3. the airship mass changes in flight due to ballonet deflation or inflation.

However, fuel changes are ignored;

4. the airship is assumed to be a rigid body, and the aeroelastic effects are

neglected.

The airship model (2.1) can be described by two equations. The first one

characterizes the system dynamics with respect to the l frame, while the

second one represents the cinematic relation between the l and i frames.

The next two sections present these two equations, that together describe the

airship nonlinear dynamic model.

2.2.1 Airship dynamics

When the displaced fluid mass is not negligible, as is the case for airships, the

equations of motion are usually derived using the Lagrangian approach [49, 50].


Let the motion of the airship be described by its inertial velocity V = [vT ,ωT ]T ,

a 6D vector including the inertial linear (v) and angular (ω) velocities. Let

the surrounding air be described by an inertial wind velocity Vw = [vTw,ωTw]T .

The airship has thus a relative air velocity Va equal to the difference of the

previous two:

Va = V − Vw (2.2)

The total kinetic energy W is defined as a sum [48]:

W = W c +W oB +W o

v (2.3)

accounting for:

− the vehicle motion, expressed in the center of gravity C, with M c =

diag(mI3,J) is the generalized mass matrix:

W c =1

2VcT M cVc (2.4)

− the kinetic energy added to the buoyancy air (displaced by the airship

volume), expressed in the center of buoyancy O where the l frame is

fixed (we will drop the O to lighten the notation), and where MB =

diag(mBI3,JB) is the generalized mass matrix of the buoyancy air:

W oB = −1

2VT MBV +

1

2VTa MBVa (2.5)

− the energy due to an extra virtual mass, also expressed in O, and where

Mv = diag(Mv,Jv) is the generalized virtual mass matrix:

W ov =

1

2VTa MvVa (2.6)

Therefore we may write W as:

W =1

2VcT M cVc − 1

2VT MBV +

1

2VTa MBVa +

1

2VTa MvVa (2.7)

We can represent all terms of the kinetic energy in the l frame considering

that the linear speed of the CG (vc) is related to the linear speed of the CV


(vo = v) through the angular speed:

vc = vo + ω × OC = v − OC × ω (2.8)

From (2.8) and knowing that ωc = ωo = ω, we have:

Vc =

[

I3 −C3

03 I3

]

V (2.9)

where C3 denotes the antisymmetric cross-product matrix corresponding to

the operation OC×. 1

Substituting (2.9) into (2.7) gives:

W =1

2VT MoV − 1

2VT MBV +

1

2VTa MBVa +

1

2VTa MvVa (2.10)

with:

Mo =

[

I3 03

C3 I3

]

M c

[

I3 −C3

03 I3

]

=

[

mI3 −mC3

mC3 J −mC23

]

(2.11)

Introducing (2.2) into (2.10) we obtain:

W =1

2VT (Mo − MB)V +

1

2(V − Vw)T (MB + Mv)(V − Vw)

=1

2VT (Mo + Mv)V +

1

2VTw(MB + Mv)Vw − VT (MB + Mv)Vw (2.12)

Finally, we have the kinetic energy expressed as function of the airship and

wind inertial velocities:

W =1

2VT MaV +

1

2VTw MBaVw − VT MBaVw (2.13)

with Ma = Mo + Mv and MBa = MB + Mv.

1The antisymmetric cross-product matrix is defined for a generic vector a = [ax, ay, az]T

as:

a× =

0 −az ayaz 0 −ax−ay ax 0


The Lagrangian or Euler-Lagrange equations of motion may be given by [51]:

F(q,q) =d

dt

(∂W

∂q

)

− ∂W

∂q(2.14)

where W (q, q) is the system kinetic energy expressed as function of the gen-

eralized coordinates q vector and its time derivative q, and F(q,q) is the

generalized forces vector.

We will apply equation (2.14) to each of the three terms of equation (2.13)

separately. We start with the first term, repeating after the procedure for the

remaining two terms.

Let us define the generalized coordinates vector as:

q = [pN , pE, pD, φ, θ, ψ]T =[pT ,ΦT

]T(2.15)

whose time derivative is related with the velocity vector V by:

[

p

Φ

]

=

[

ST 03

03 R

][

v

ω

]

⇐⇒ q = JΦV (2.16)

with the orthogonal transformation matrix S defined by

S =

cosψ cos θ sinψ cos θ − sin θ

cosψ sin θ sinφ− sinψ cosφ sinψ sin θ sinφ+ cosψ cosφ cos θ sinφ

cosψ sin θ cosφ+ sinψ sinφ sinψ sin θ cosφ− cosψ sinφ cos θ cosφ

(2.17)

and the coefficient matrix R given by:

R =

1 sinφ tan θ cosφ tan θ

0 cosφ − sinφ

0 sinφ/ cos θ cosφ/ cos θ

(2.18)

The first term of the kinetic energy corresponding to the case with no wind is

then:

W1 =1

2VT MaV (2.19)


or, using equation (2.16):

W1 =1

2qTJ−T

Φ MaJ−1Φ q (2.20)

The partial derivative of the kinetic energy W1 relative to q is:

∂W1

∂q=

1

2

[(J−T

Φ MaJ−1Φ )T + (J−T

Φ MaJ−1Φ )

]q

= J−TΦ MaJ

−1Φ q (2.21)

since MT

a = Ma. Computing now its time derivative leads to:

d

dt

(∂W1

∂q

)

= ˙(J−TΦ )MaJ

−1Φ q + J−T

Φ Ma˙(J−1Φ )q + J−T

Φ MaJ−1Φ q

= J−TΦ MaV + ˙(J−T

Φ )MaV (2.22)

where we used (2.16) and the relation:

˙(J−1Φ )JΦ + J−1

Φ JΦ = 0 (2.23)

The partial derivative of the kinetic energy W1 now relative to q is:

∂W1

∂q=

1

2

∂

∂q(qTJ−T

Φ MaJ−1Φ q)

= KMaV (2.24)

with:

K =∂

∂q

(J−1

Φ q)⇐⇒

[

03 03

K1 K2

]

=

[∂∂p

(J−1

Φ q)

∂∂Φ

(J−1

Φ q)

]

(2.25)

The generalized force relative to the kinetic energy with no wind is then ob-

tained from the difference between (2.22) and (2.24) according to (2.14):

F1(q, q) = J−TΦ MaV + ˙(J−T

Φ )MaV − KMaV (2.26)

We will now proceed applying (2.14) to the second term of equation (2.13).

We start defining the wind coordinates vector as:

qw = [pNw, pEw

, pDw, φw, θw, ψw]T =

[pTw,Φ

Tw

]T(2.27)


whose time derivative is related with the wind velocity vector Vw by:

[

pw

Φw

]

=

[

ST 03

03 R

][

vw

ωw

]

⇐⇒ qw = JΦVw (2.28)

The second term of the kinetic energy (2.13) corresponds to:

W2 =1

2VTw MBaVw

=1

2qTwJ

−TΦ MBaJ

−1Φ qw (2.29)

Applying (2.14) to (2.29), with q defined in (2.15), we obviously have:

d

dt

(∂W2

∂q

)

= 0 (2.30)

The partial derivative of the kinetic energy W2 relative to q is:

∂W2

∂q=

1

2

∂

∂q(qTwJ

−TΦ MBaJ

−1Φ qw)

= KwMBaVw (2.31)

with:

Kw =∂

∂q

(J−1

Φ qw)⇐⇒

[

03 03

Kw1 Kw2

]

=

[∂∂p

(J−1

Φ qw)

∂∂Φ

(J−1

Φ qw)

]

(2.32)

The generalized force relative to W2 is then obtained from the difference be-

tween (2.30) and (2.31) according to (2.14):

F2(q, q) = −KwMBaVw (2.33)

Finally, applying the same procedure to the last term of (2.13):

W3 = −VT MBaVw (2.34)

we obtain:

F3(q, q) = −J−TΦ MBaVw −

[˙(J−1Φ )

]T

MBaVw + KwMBaV + KMBaVw

(2.35)


Summing up all the generalized forces we have:

F(q, q) =F1(q, q) + F2(q, q) + F3(q, q)

=J−TΦ MaV + ˙(J−T

Φ )MaV − KMaV − KwMBaVw

− J−TΦ MBaVw −

[˙(J−1Φ )

]T

MBaVw + KwMBaV + KMBaVw

(2.36)

Considering that:

F(V) = JTΦF(q, q) (2.37)

we have:

F(V) =MaV + (JTΦ˙(J−TΦ ) − JTΦK)MaV − JTΦKwMBaVw

− MBaVw −(

JTΦ

[˙(J−1Φ )

]T

− JTΦK

)

MBaVw + JTΦKwMBaV (2.38)

Moreover, considering the following equalities:

V3 = −RTK1 (2.39)

Ω = RT ( ˙(R−T ) − K2) (2.40)

Vw3 = −RTKw1 (2.41)

0 = RTKw2 (2.42)

where V3, Ω and Vw3 denote the antisymmetric cross-product matrices cor-

responding, respectively, to the operations v×, ω× and vw×, leads to:

JTΦ˙(J−TΦ ) − JTΦK = Ω6 + V6 (2.43)

JTΦKw = −Vw6 (2.44)

whereas:

V6 =

[

03 03

V3 03

]

, Ω6 =

[

Ω 03

03 Ω

]

, Vw6 =

[

03 03

Vw3 03

]

(2.45)

The dynamics equation of the airship in the inertial frame is then given by:

F = MaV + (Ω6 + V6)MaV − MBaVw − (Ω6 + V6)MBaVw − Vw6MBa(V − Vw)

(2.46)


in accordance with the equations derived by Thomasson in [50], using quasi-

coordinates, without the gradient terms.

Let us now deduce the dynamics equation in the air frame which we will

limit to the case of constant translation wind velocity in the inertial frame.

Considering that (see Appendix A.2):

pw =dvwdt

= vw + ω × vw (2.47)

we have:

constant wind ⇒ vw = −ω × vw (2.48)

translation wind ⇒ ωw = 0 (2.49)

which leads to:

Vw = −Ω6Vw (2.50)

Substituting (2.2), (2.50) and the matricial relation:

V6 = Va6 + Vw6 (2.51)

(Va6 is defined similarly to V6 in (2.45) but considering the airspeed velocity

va) into (2.46) results in:

F = MaVa − MaΩ6Vw + Ω6MaVa + Ω6MaVw + (Va6 + Vw6)Ma(Va + Vw)

+MBaΩ6Vw − Ω6MBaVw − (Va6 + Vw6)MBaVw − Vw6MBaVa

= MaVa + Ω6MaVa +[Ω6(Ma − MBa) − (Ma − MBa)Ω6

]Vw +

+(Va6 + Vw6)(Ma − MBa)(Va + Vw) + Va6MBaVa (2.52)

Considering that:

Ma − MBa =

[

(m−mB)I3 −mC3

mC3 J −mC23 − JB

]

(2.53)


we have the following equalities:

[Ω6(Ma − MBa) − (Ma − MBa)Ω6

]Vw =

[

0

m(ΩC3 − C3Ω)vw

]

(2.54)

(Va6 + Vw6)(Ma − MBa)(Va + Vw) =

[

0

−(Va3 + Vw3)mC3ω

]

(2.55)

Va6MBaVa =

[

0

Va3Mvva

]

(2.56)

Substituting equalities (2.54)-(2.56) into equation (2.52) and knowing that the

cross-product satisfies the Jacobi Identity2, leads to:

F = MaVa + Ω6MaVa +

[

0

Va3(Mvva −mC3ω)

]

(2.57)

Finally, since:

[

0

Va3(Mvva −mC3ω)

]

= Va6MaVa (2.58)

we obtain the dynamics equation of the airship in the air frame for a constant

translation wind:

F = MaVa + (Ω6 + Va6)MaVa (2.59)

which has the same form as equation (2.46) for the no-wind case, i.e., equa-

tion (2.46) is invariant under a steady translation, as pointed out in [52].

Let us now go back to equation (2.46). We can write it as:

MaV = Fkw + F (2.60)

with the kinematics and wind forces given by:

Fkw = −(Ω6 + V6)MaV + MBaVw + (Ω6 + V6)MBaVw + Vw6MBa(V − Vw)

(2.61)

and where F = Fg + Fa + Fp contains gravitational, aerodynamic and propul-

sion forces. The aerodynamic force contains all terms proportional to the dy-

2a × (b × c) + b × (c × a) + c × (a × b) = 0


namic pressure (12ρV 2

t , with ρ the air density), identified in wind tunnel tests.

This means all terms proportional to the square of the airspeed in Fkw are

already included in Fa and are at the moment accounted twice in the dynamic

model.

A closer look at (2.61) shows that:

FV ≡ −V6MaV + V6MBaVw + Vw6MBaVa

= −V6(Ma − MBa)V − Va6MBaVa (2.62)

According to (2.56), the term −Va6MBaVa in (2.62) is proportional to the

square of the airspeed and is already accounted for in Fa. Therefore it should

be removed from Fkw, which can now be written as:

Fkw = −Ω6MaV − V6(Ma − MBa)V + MBaVw + Ω6MBaVw (2.63)

According to [48], the gravitational force Fg, which adds the weight force

applied at the GC and the buoyancy force applied at the CV, is a function of

the transformation matrix S:

Fg =

[

S(m−mB)ag

OC × Smag

]

=

[

mwI3

mC3

]

Sag = EgSag (2.64)

with the gravity acceleration ag = [0, 0, g]T given in the i frame and the

airship weighting mass defined as the difference between its weight and its

buoyancy, mw = m−mB.

Finally, the dynamic equation of the airship in the inertial frame is given by:

MaV = −Ω6MaV − V6(Ma − MBa)V + MBaVw + Ω6MBaVw︸︷︷︸

Fkw

+EgSag︸︷︷︸

Fg

+Fa + Fp

(2.65)

while in the air frame it is given by (considering translation constant wind):

MaVa = −Ω6MaVa − Va6(Ma − MBa)Va︸︷︷︸

Fkw

+EgSag︸︷︷︸

Fg

+Fa + Fp (2.66)

where the previous search of terms already included in Fa was applied to the

term (2.58) of (2.59) and where

[

0

mVa3C3ω

]

= −Va6(Ma − MBa)Va.


2.2.1.1 Forces to actuators input

As previously mentioned, the aerodynamic force vector Fa contains all dynamic

terms proportional to the dynamic pressure, identified in wind tunnel tests.

These include any control surface deflection effects corresponding to aileron

δa, rudder δr and elevator δe, which are usually presented as a separate entity.

Therefore, if we divide the aerodynamic force such that:

Fa = Fa(Va) + Fa(δ) (2.67)

with the state only depending part Fa(Va) and the control surfaces force input

Fa(δ), we may rewrite equations (2.65)-(2.66), respectively, as:

MaV = − Ω6MaV − V6(Ma − MBa)V + MBaVw + Ω6MBaVw

+ EgSag + Fa(Va) + uf (2.68)

MaVa = − Ω6MaVa − Va6(Ma − MBa)Va + EgSag + Fa(Va) + uf (2.69)

where uf = Fa(δ)+Fp contains both the control surfaces and propulsion forces

input.

The relation between the actuators represented in fig. 2.2(b) and the control

force inputs uf depends on the flight region, as explained in more detail at the

end of Section 2.3.1:

− In the low airspeed region, the tail surfaces have reduced authority since

the action from the surface deflections is a function of the dynamic pres-

sure and varies as the square of the airspeed Vt, according to the aero-

dynamic characteristics of the airship [53]. This leaves the airship to be

mainly controlled by the propulsion force inputs. The two main pro-

pellers correspond to 3 inputs (TL, TR, δv) - left and right thrust and

vectoring angle - providing longitudinal and vertical forces, pitching and

rolling torques. If available, the tail lateral thruster adds one input (TY ),

providing a side force and a yawing torque. These force actuators are

slightly influenced by the airspeed, but may be considered as independent

on a first approach.

− In aerodynamic flight, the vectoring angle is no longer necessary, leaving

the airship with a reduced vertical force. The maneuvering is mostly

accomplished by the tail fins. The surface deflections correspond to

the three standard inputs of aileron, elevator and rudder deflections


(δa, δe, δr), which mostly correspond to torque inputs for the control of

the pitch and yaw motions, keeping the airship with reduced lateral force

input.

With the six actuators inputs (δe, TL, TR, δv, δa, δr) to control six forces (three

forces and three torques), the airship does not seem underactuated, but nu-

merous limitations severely reduce its controllability:

− no actuator is really available to oppose the aerodynamic side force;

− the main engines provide four coupled force components with only three

inputs;

− the tail surfaces depend on the airspeed and their authority vanishes in

the no-wind case, leaving the airship to be controlled by the force inputs

only;

− all the actuators have level and rate saturation limits, that cannot be

avoided;

− the force actuators, in particular, have their own dynamics, with limited

response times, that must be taken into account.

The relation between actuators and force inputs may then be established for

design purposes, neglecting the actuators dynamics, using the airspeed mea-

surement and resolving the possible redundancies according to the usual oper-

ation of the airship [23] (the airship aerodynamic angles also have their effect,

but they may be neglected on a first approach, assuming small angles):

uf = fu(u, Vt) (2.70)

where u = [δe, TL, TR, δv, δa, δr]T is the real actuators input (the lateral thrust

TY is not considered since it is not yet implemented in the AURORA airship),

Vt is the true airspeed, and uf = [fu, fv, fw, fp, fq, fr]T is the force vector,

solution of the system composed by the six equations below, in agreement


with the AURORA airship configuration:

fu = XT cos δv + k1δe (2.71a)

fv = −k2δr (2.71b)

fw = −XT sin δv + k3δe (2.71c)

fp = k2l4δa + b4 sin δvTD (2.71d)

fq = XT b3 cos δv + k5δe (2.71e)

fr = k2l6δr + b4 cos δvTD (2.71f)

where XT = TL+TR is the total thrust, TD = TL−TR is the differential thrust,

(bj, lj) are geometrical constants of the airship, and kj(Vt) are second order

polynomials expressing the airspeed depending authority of the tail deflections.

2.2.2 Airship kinematics

For control and navigation purposes, the velocity vector V, expressed in the

airship l frame, must be transformed to the i frame. This leads to the

cinematic relations.

Consider the airship position is given by its coordinates in the i frame and the

attitude is described in terms of the Euler angles (φ, θ, ψ). Then, the airship

position may simply be regarded as the integration of the inertial velocity in

the i frame:

N

E

D

= ST

u

v

w

(2.72)

where S is the orthogonal transformation matrix (2.17) that satisfies the equa-

tion:

S = −ΩS (2.73)

Similarly, the time derivatives of the Euler angles may be related to the local


angular rates (p, q, r):

φ

θ

ψ

= R

p

q

r

(2.74)

with the angular transformation matrix R given by (2.18) and satisfying:

R = −R ˙(R−1)R (2.75)

If P = [pN , pE, pD, φ, θ, ψ]T defines the 6D position and V = [u, v, w, p, q, r]T

the local velocity, the position cinematic equation is expressed by:

P = JΦV (2.76)

with:

JΦ =

[

ST 03

03 R

]

(2.77)

and:

JΦ = JΦCJ , with CJ =

[

Ω 03

03 − ˙(R−1)R

]

(2.78)

The kinematics equation may also be given considering the air relative velocity

by:

P = JΦ(Va + Vw) (2.79)

or, considering again translation constant wind:

P = JΦVa + BI pw (2.80)

with BI =

[

I3

03

]

.


2.2.3 Airship simulator

Based on the 6 degrees of freedom nonlinear model compounded by equa-

tions (2.68), (2.71) and (2.76), a MATLABr/Simulinkr-based simulator was

built, allowing the design and validation of flight control and guidance strate-

gies [29]. The simulator block diagram of the airship open-loop model is rep-

resented in fig. 2.3.

yc u

x

dwind / turbulence

input

actuators

model

airship

nonlinear model

u

Figure 2.3: Simulator block diagram for airship open-loop model.

The airship nonlinear model, as described by equation (2.1), is a function of

the state variables x, the actuators input u and wind disturbances d. These

last include both constant wind and atmospheric turbulence, modeled here by

the Dryden model (see Appendix B). The sensors described in Section 2.1

are considered ideal while the actuators model includes the propellers and

control surfaces dynamics, like delays and saturations, applied to the actuators

command input uc. The output vector y consists of interest variables to be

monitored.

2.3 Airship linearized models

The complexity of the airship nonlinear dynamic equations presented before

justifies the search for a linear version, also important in order to analyze

and evaluate the characteristics of the airship dynamics, and usual practice in

aeronautical systems.

In the following sections we will describe the procedure applied on the lin-

earization of the airship model given by equations (2.68), (2.71) and (2.76),

and analyze the airship characteristics at the different equilibrium points over

the flight envelope.

2.3. AIRSHIP LINEARIZED MODELS 35

2.3.1 Trim or equilibrium conditions

An equilibrium or trim point corresponds to a condition at which a dynamical

system is steady or ‘at rest’.

We consider here no disturbance, being the deterministic nonlinear model of

the airship given by:

x = f(x,u) (2.81)

A trim point of the system (2.81) is then a point (x,u) = (xe,ue) such that

the airship is in equilibrium, with a subset ye of its derivatives null:

xe = f(xe,ue) (2.82)

ye = Cxe = 0 (2.83)

This implies there is a balance between forces acting on the airship and the

airship will remain in that particular flight condition until some disturbance

or some control input occurs.

The first step, prior to the system linearization, is to find the solutions (xe,ue)

of (2.82)-(2.83) over the flight envelope, i.e., for varying airspeed. Due to the

complex functional dependence of the aerodynamic data, this cannot be done

analytically. A numerical way to do so is to specify a convex optimization prob-

lem, with the following constraints for the restricted case problem of a straight

level flight at a given constant altitude he = −De and constant airspeed Vte :

− for steady flight, the derivatives of the linear and angular velocities are

zero: v, ω = 0;

− for steady straight flight, the derivative of the vertical position and the

angular velocity are zero: D,ω = 0;

− for symmetric pure longitudinal flight, the sideslip and roll angles are

zero: β, φ = 0;

− from the cinematic relation (2.74), the derivatives of the Euler angles are

zero: Φ = 0;

− for still straight flight, the left and right engines thrust is equal: TL = TR;

− no need for lateral actuation: δa, δr, Ty = 0;


− for constant altitude: D = De;

− for constant airspeed:√u2 + v2 + w2 = Vte ;

− for level flight: θ = α.

The position variables N,E and ψ are not necessary to find a steady-state

condition, allowing the airship to take any straight level direction.

The minimization of the convex cost function, that fulfils the above constraints,

provides trim values of state and control input:

xe = [ue, 0, we, 0, 0, 0, ⋆, ⋆,De, 0, θe, ⋆]T (2.84)

ue = [δe, TLe, TRe

, δve, 0, 0, 0]T (2.85)

where ⋆ indicates the variables for which no direct constraint is set.

The results obtained repeating the procedure for a range of reference airspeeds

allow us to analyze the changes in the longitudinal motion behavior of the

AURORA airship over the flight envelope.

Figure 2.4 represents the equilibrium values of pitch θe, elevator δe, total thrust

XTe= TLe

+TReand vector angle δve

necessary for the AURORA airship, with

different weighting masses mw = 1, 3, 5kg, to maintain a steady straight level

flight at 50m altitude, and at different airspeeds Vt varying from 0 to 15m/s

in steps of 0.1m/s.

At low airspeeds, the propellers vectoring angle is necessary in order to com-

pensate for the loss of lift force from aerodynamics, as may be seen in fig. 2.4(a).

It can also be observed that the heavier the airship, the later (or at higher air-

speeds) it will need the propellers to be vectored.

For very low airspeeds, the vectoring angle is almost 90o and the model is

essentially that of aerostatic forces, with the weight excess being compensated

by the propellers vectoring. For very high airspeeds, the propellers vectoring

is not necessary, and the model is essentially that of aerodynamic forces. The

fast transition from low to high vectoring angles separates two flight regions,

namely cruise or aerodynamic flight (AF) and low airspeed or hover flight (HF).

In this transition there exists compensation from both aerodynamic forces and

propellers vectoring. For these reasons, this should be the most difficult region

to be corroborated in the model validation process and also for the control

design.


Vt (m/s)

δ ve

(deg

)

mw = 1kg

mw = 3kg

mw = 5kg

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15-10

0

10

20

30

40

50

60

70

80

90

100

(a) Thrusters vector angle δvevs. airspeed

Vt.

Vt (m/s)

θ e(deg

)

mw = 1kg

mw = 3kg

mw = 5kg

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15-4

-2

0

2

4

6

8

10

12

(b) Pitch angle θe vs. airspeed Vt.

Vt (m/s)

δ ee

(deg

)

mw = 1kg

mw = 3kg

mw = 5kg

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15-14

-12

-10

-8

-6

-4

-2

0

2

4

6

(c) Elevator deflection δeevs. airspeed Vt.

Vt (m/s)

XT

e(N

)

mw = 1kg

mw = 3kg

mw = 5kg

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

10

20

30

40

50

60

(d) Thrust input XTevs. airspeed Vt.

Figure 2.4: Trim values of state and control input for different weightingmasses mw over the flight envelope.

It can be seen in fig. 2.4(b) that the need for a higher pitch angle occurs in

the mid-range airspeeds, when the airship is asking for more aerodynamic lift.

At low airspeeds, as this extra up force is supplied by the propellers vectoring

and the aerodynamic efficiency decreases, this is no longer necessary.

Also observed is, the lower the weighing mass, the lower the pitch angle nec-

essary to provide the sufficient aerodynamic force to maintain the equilibrium

condition.

The necessary lift is obtained through the pitch angle, which in turn produces

a pitch rate, that must be compensated by the elevator control surface action

represented in fig. 2.4(c), which justifies the similarity of both curves. At low

airspeeds, the elevator, as all the control surfaces, has reduced authority since

its action is a function of the dynamic pressure and varies with the squared


airspeed [53].

The elevator deflection is almost the same for the three kinds of weighting

conditions, except in the transition area.

At low airspeeds, the total thrust, represented in fig. 2.4(d), is demanded basi-

cally to generate the upward force for weight compensation. At high airspeeds,

however, it is demanded mainly to compensate for the drag forces, which in-

crease with the square of the airspeed [1] and do not depend on the weighting

condition.

2.3.2 Model linearization

The linearization of the system (2.81) corresponds to the first-order term of

its Taylor expansion around the point of interest, in this case, (xe,ue):

x ≈ f(xe,ue) +∂f

∂x

∣∣∣∣x=xe,u=ue

(x − xe) +∂f

∂u


(u − ue) (2.86)

Substituting the jacobian matrices:

A ≡ ∂f

∂x


(2.87)

B ≡ ∂f

∂u


(2.88)

and the variations around the equilibrium values:

x = x − xe (2.89)

u = u − ue (2.90)

into (2.86) we obtain the airship linear model:

˙x = Ax + Bu (2.91)

in the absence of disturbances (deterministic case).

This procedure, however, is only possible if all terms in the dynamic model

are analytical. Generally, aerodynamic data is included in an aircraft model

in the form of lookup tables. Hence, the linearization must be done numeri-

cally, performing a numerical differentiation by finite difference of the nonlinear

equations. Numerical linearization is done perturbing each state or input sig-


nal at a time, and then computing the resulting accelerations. The elements

of the A and B matrices are approximated as:

Aij =∂fi∂xj

≈ fi(∆xj) − fie∆xj

(2.92)

Bij =∂fi∂uj

≈ fi(∆uj) − fie∆uj

(2.93)

where fi(∆xj), fi(∆uj) are the accelerations at the disturbed state and input,

∆xj, ∆uj are the perturbation value of the jth state and input and fie is the

value of fi at the equilibrium condition.

The linearized model (2.91), i.e., the dynamic matrix A and the input matrix

B, depend on the trim point chosen for the linearization, and in particular of

the chosen airspeed Vte and altitude he.

Note that, as seen in Section 2.2.1, the airship dynamics represented in the

inertial frame when no wind disturbance is present corresponds to the airship

dynamics represented in the air frame for constant translation wind. This

means the state vectors (2.89) may contain either v or va, as long as there is

no wind or it is a steady translation one.

As a result of the system linearization, and as usual in aeronautics, two inde-

pendent (decoupled) motions may be considered: the motion in the vertical

plane, named longitudinal, and the motion in the horizontal plane and rolling,

named lateral. The corresponding linearized models will be presented next.

The analysis of the eigenvalues of the dynamic matrices A will allow us to

make an approximate description of the airship stability modes [1].

2.3.2.1 Longitudinal model

In the longitudinal case, the state vector is xv = [u, w, q, θ]T and the input

vector is given by uv = [δe, XT , δv]T , where all the variables represent the

variations around the trim value. Therefore, the longitudinal dynamic equation

is given by:

˙u

˙w

˙q˙θ

= Av

u

w

q

θ

+ Bv

δe

XT

δv

(2.94)


where δe is the change in elevator deflection, XT = TL + TR is the change in

thrust demand and δv is the change in the vectoring angle.

When necessary, the altitude may be introduced as an additional integrating

state of the longitudinal motion, since:

˙h ≈ Vte θ − w (2.95)

Sometimes, it is interesting to consider the thrust input expressed in its carte-

sian coordinates instead of the polar ones, replacing the pair (XT , δv) by

(Tx, Tz), which are, respectively, the changes of thrust demand in the forward

and down axes.

Figure 2.5 shows the evolution of the poles of the linearized longitudinal model

with the airspeed Vt.

Real axis (rad/s)

Imagin

ary

axis

(rad/s)

2©

3©

1©

3©: longitudinal pendulum

2©: heave / pitch subsidence

1©: surge

-12 -10 -8 -6 -4 -2 0 2-1.5

-1

-0.5

0

0.5

1

1.5

Figure 2.5: Poles of linearized longitudinal dynamics vs. airspeed Vt (: 0m/s,×: 15m/s).

The surge mode is described by a real pole with a long time constant and is

associated to the forward speed u. The influence of the airspeed increase in this

mode is hardly noticeable. The other real pole, associated with the vertical


speed w (or, equivalently, the angle of attack α), describes, at hover (poles

indicated by ’’), the heave mode. As the airspeed increases (poles indicated

by ’×’), the mode becomes faster and develops into a pitch subsidence. This

mode is described by the faster of the two real poles.

The longitudinal pendulum mode corresponds to the complex pair of poles that

is associated to the pitch angle θ and the pitch rate q. In the hover condition the

damping is zero and the pendulum oscillation property in this mode becomes

evident. As the airspeed increases the frequency decreases. The damping, on

the other hand, augments, reaching its maximum value in the transition region

with a near coupling of the four modes.

All modes are stable (marginally for air-hover) over the flight envelope.

2.3.2.2 Lateral model

In the lateral case, the state vector considered for the dynamic characteristics

is xh = [v, p, r, φ]T and the input vector is given by uh = [δa, δr, TD]T . All the

variables represent the variations around the trim value, which for the lateral

case corresponds to xeh= 0 and ueh

= 0 (see Section 2.3.1).

Therefore, the lateral dynamic equation is given by:

˙v

˙p

˙r˙φ

= Ah

v

p

r

φ

+ Bh

δa

δr

TD

(2.96)

where δa is the aileron deflection, δr is the rudder deflection and TD is the

differential thrust TD = TL − TR between left and right propellers.

When necessary, the yaw angle ψ may be introduced as a supplementary inte-

grating state of the lateral motion, since:

˙ψ = r/ cos(θe) (2.97)

Figure 2.6 shows the change in the poles of the linearized lateral model with

the airspeed Vt.

In the hover condition (poles indicated by ’’), the zero damping of the complex

pair of poles characterizes the oscillatory roll mode related to the roll rate p


Real axis (rad/s)

Imagin

ary

axis

(rad/s)

1©: sideslip subsidence

2© 1©

3©

3©: roll oscillation

2©: yaw subsidence

-6 -5 -4 -3 -2 -1 0 1-3

-2

-1

0

1

2

3

Figure 2.6: Poles of linearized lateral dynamics vs. airspeed Vt (: 0m/s, ×:15m/s).

and the roll angle φ. This oscillatory rolling movement is the lateral equivalent

of the longitudinal pendulum oscillation and arises for similar reasons, i.e., the

fact that the center of gravity is located below the center of buoyancy of the

airship.

With the increase in airspeed (poles indicated by ’×’), the general stability

improves, since the damping ratio of the oscillatory mode increases and the

eigenvalues of the real modes become more negative. However, in contrast to

what happens in the longitudinal case, the frequency of the oscillatory mode

stays virtually unchanged throughout the speed range and only the damping

ratio undergoes an increase, which in turn appears to be directly proportional

to the speed variation.

Referring to the two real poles, the slow mode, usually named sideslip subsi-

dence mode, is associated with the lateral speed v (or equivalently, the sideslip

angle β), and it does not appear to be much affected by the airspeed increase.

The fast mode is the yaw subsidence mode related to the yaw rate r and

presents a time constant that decreases with the airspeed.

As in the longitudinal case, all modes are stable (marginally for air-hover) over

2.4. CONCLUSIONS 43

the flight envelope.

2.4 Conclusions

This chapter presented the airship nonlinear model, based on both dynamics

and kinematics analysis. The UAV model introduced considers all forces that

act upon it, namely, aerodynamics, gravity, propulsion, kinematics and wind.

The nonlinear model is, however, too complex to allow a system analysis.

Therefore, a linearization procedure was followed to obtain the linearized mod-

els of the decoupled airship motions, lateral and longitudinal. The observation

of the poles location of each individual system as function of airspeed, provides

a good knowledge of the airship behavior over the flight envelope. The fact, for

example, that the damping of the oscillatory roll mode approximates zero near

hover, indicates that this motion should not be overlooked at low airspeeds.

The airship is controlled by the action of two vectored propellers and control

surfaces. It was seen, however, that these actuators authority or influence is

not constant. In fact, it varies as function of the airspeed. This indicates that

the action of the different actuators shall depend not only on the goal mission

which may include, for instance, groundspeed tracking, but also on the wind

disturbances present since they have influence in the resulting airspeed.

All this information and knowledge is undoubtedly essential to the next chap-

ters, where the control of the airship will be addressed.

Chapter 3

Common Concepts and Tools

Contents

3.1 Position errors . . . . . . . . . . . . . . . . . . . . . 45

3.1.1 Path-following . . . . . . . . . . . . . . . . . . . . . 46

3.1.2 Path-tracking . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Attitude reference and wind estimation . . . . . . 47

3.3 Controllers performance evaluation . . . . . . . . 49

3.3.1 Case-study mission . . . . . . . . . . . . . . . . . . . 50

3.3.2 Sensitivity and robustness . . . . . . . . . . . . . . . 51

The design, implementation on a simulation environment and evaluation of

control laws that solve the airship path-tracking problem requires the prior

definition of some concepts and tools.

This chapter gathers all the common elements used hereafter. Section 3.1 de-

fines the position errors. Section 3.2 presents the reference attitude considered

in this work and a for wind estimation method. Section 3.3 describes the cri-

teria used as common baseline for the performance evaluation of the different

controllers.

3.1 Position errors

The linearization procedure described in Section 2.3 considers straight level

flights, independent of the NE-direction chosen as reference, as equilibrium

condition. If the dynamic state x of the linear system (2.91) included the

cartesian position error in the i frame, it could only be used with north-

45

46 CHAPTER 3. COMMON CONCEPTS AND TOOLS

aligned references. By considering the errors in the reference trajectory frame,

the linear model is valid for any horizontal direction chosen. Therefore, it is

necessary to define a coordinate transformation to the reference local frame.

This coordinate transformation depends, however, on the mission objective,

whether it is a path-following or path-tracking case.

3.1.1 Path-following

The path-following problem considers a reference trajectory defined by way-

points, not being therefore time-dependent. This means the lateral ǫ and

vertical δ position errors at a given time are defined as the closest distances

(respectively, in the horizontal and vertical planes) between the airship position

p and the reference trajectory (see fig. 3.1). This reference, given any AB

segment, defines the r frame.

p

A

B

ε

z

δ

r

yr

xr

vhvv

dh

dv

Figure 3.1: Path-following errors definition.

Therefore, the position errors are given by the projections of the vectors d

onto the lines v, perpendicular to the reference line AB:

ε = |projvhdh| = vThdh (3.1)

δ = |projvvdv| = vTv dv (3.2)

3.1.2 Path-tracking

The path-tracking problem differs from the path-following one by the fact that

the reference path is in this case time-dependent. The reference coordinates

may be obtained from given way-points if a desired velocity is also set. This is

the case of the missions considered in this work, as will be seen ahead in this

chapter.

3.2. ATTITUDE REFERENCE AND WIND ESTIMATION 47

Consider the airship is at position p(t) when it is supposed to be at the refer-

ence position pr(t). This reference point lies in the desired trajectory defined

by the given way-points A and B, which define the reference frame r (see

fig. 3.2).

εr

rN

rE

xr

xl

yl

xi

yiyr

E

N

B

rp

p(t)

(t)

A

ψ

ψη

Figure 3.2: Path-tracking errors definition (2D).

Therefore, the position errors given in the reference frame are obtained from:

Γ =

η

ε

δ

= Sr

N −Nr

E − Er

D −Dr

(3.3)

with Sr = S(Φr) defined in (2.17) and Φr = [0, θr, ψr]T corresponding to the

angles that define the transformation from i to r frames, i.e., the angles

the reference inertial velocity pr does with the i frame.

Note that when the objective is ground-hover, pr = 0 and θr, ψr are therefore

undetermined. In this case, we arbitrarily define θr = ψr = 0 for simplicity.

This leads to Sr = I and Γ = p − pr.

3.2 Attitude reference and wind estimation

The first idea is that the attitude reference shall be coincident with the ref-

erence trajectory attitude Φr. However, there are two situations when this is

not desirable: in the presence of wind disturbances (a certainty when flying

outdoors) and if the objective is ground-hover (since Φr is arbitrarily defined).

An aircraft of conventional shape must fly against the apparent wind in order

to have low drag [53]. This is also true for airships, moreover because of the


lateral underactuation. So, whenever there is wind, the airship will try to align

itself with the relative air, reducing the sideslip angle β (see Appendix A). For

this reason, the relative air attitude Φaris chosen as attitude reference.

We compute Φarfollowing these four steps (see fig. 3.3):

1. with the airship attitude Φ and the aerodynamic variables Vt, β and α

(all measured variables), compute de inertial air velocity pa:

pa = STva = STSTa

Vt

0

0

(3.4)

with Sa defined by (A.1) and S by (2.17);

2. using the measured inertial velocity p, estimate the wind inertial velocity

vector pw by:

pw = p − pa (3.5)

3. compute the relative air velocity reference as:

par= pr − pw (3.6)

4. finally, compute Φar= [0, θar

, ψar]T , which corresponds to the angles par

does with the the i frame.

ψ

aψ

yl

.pr

.pa

xl.

pw

r

.pa

pr p

aψr

.p

.pw

β

i

rψ

yi

A

B

x

Figure 3.3: Wind and yaw reference estimation.

Note that, according to the methodology described:

3.3. CONTROLLERS PERFORMANCE EVALUATION 49

− in the no-wind case, pw = 0 and Φaris coincident with the reference

trajectory attitude Φr;

− for the ground-hover objective, pr = 0 implies the reference trajectory

attitude is undefined and Φar= −Φw, reducing the lateral effort by

minimizing the drag force.

Remark that θarcomputed in this way does not consider the equilibrium pitch.

Therefore, to reduce the airship lift, a corrected value shall be used:

θ′ar= θar

+ θe (3.7)

The corrected attitude reference is then given by:

Φ′ar

= [0, θ′ar, ψar

]T (3.8)

The wind attitude Φw = [φw, θw, ψw]T may also be computed, corresponding

to the angles pw does with the i frame (with φw = 0).

3.3 Controllers performance evaluation

One of the objectives of this work is to compare the performance of the different

control methodologies used. The performance will be evaluated according to

the following three criteria:

− Airship behavior for a selected case-study mission. This mission is de-

fined to be representative of a realistic case. The controller performance

will be mainly evaluated by the airship path-tracking errors (see Sec-

tion 3.1.2) and the actuators request.

− Sensitivity and robustness to parameter uncertainty. The controller

should guarantee the stability of the closed-loop system even in the

present of wind disturbances and model parameter uncertainty.

− Computational effort. For a real-time implementation to be possible, the

computational time taken by the controller is an important measure of

its performance.


3.3.1 Case-study mission

The simulation examples presented throughout this work consider a 3kg weight-

ing mass. The controllers are implemented at 10Hz. This frequency is high

enough when compared with the frequency of the airship dynamic system,

that the continuous control design is still applicable. The fastest frequency

obtained in open-loop is approximately 2rad/s ≃ 0.32Hz for the roll oscilla-

tion pendulum.

Although other missions might occasionally be used to demonstrate the con-

trollers performance, the following airship mission, realistic and in agreement

with the airship characteristics, will be used for comparison between con-

trollers. It starts with a vertical take-off, a path-tracking with two semi-

circles, airship stabilization for ground-hover, and finally a vertical landing

(see fig. 3.4).

E (m)N (m)

h(m

)

0©

1©≡ 3© 2©

4©

-1000

100200

300

-200

-100

0

100

200

0

10

20

30

40

50

60

Figure 3.4: Case-study mission reference. North N , east E and altitude refer-ence (bold) and projections (normal).

The same initial conditions are considered, namely at the position (Ni, Ei, hi) =

(−30,−20, 1)m and with attitude (φi, θi, ψi) = (10, 1,−20)o. The airship has

then 15s to be stabilized at the initial reference point pr0 = (Nr0 , Er0 , hr0) =

(−30,−20, 5)m so as to be stable and ready to start the mission. From this

point, the vertical take-off at 1m/s climbing rate begins, finishing at the first

point of the horizontal path-tracking pr1 = (Nr1 , Er1 , hr1) = (−30,−20, 50)m.

At 7m/s groundspeed, the airship is to track a reference path provided, com-

prised of straight lines and two semi-circles of 200m diameter. Although in

this mission we do not always use a straight line reference, with a groundspeed

3.3. CONTROLLERS PERFORMANCE EVALUATION 51

of 7m/s and a 200m circle radius, the approximation is quite acceptable since

the yaw rate is fairly small. Obviously, the angular reference must be adapted

to the case. When reaching the point pr2 = (Nr2 , Er2 , hr2) = (−100, 0, 50)m,

the path-tracking gives place to the airship stabilization at the coordinates

pr3 = (Nr3 , Er3 , hr3) = (−30,−20, 50)m during 40s, preparing it for vertical

landing at pr4 = (Nr4 , Er4 , hr4) = (−30,−20, 1)m at 0.5m/s descent rate.

In order to test the controllers robustness to wind disturbances, the airship is

submitted to a 4m/s constant wind blowing from northwest at 20o, added to

a 3D 3m/s continuous turbulence.

This mission, defined to be representative and illustrative of a realistic case,

clearly represents a challenge for the automatic control system, as (i) the dy-

namics varies from air-hover to aerodynamic flight during the path-tracking,

(ii) the wind input has different incidence angles (as the trajectory is circu-

lar) and also stochastic components, and (iii) the mission includes vertical

maneuvers.

3.3.2 Sensitivity and robustness to parameter uncer-

tainty

Some of the parameters that describe the airship system are likely to be un-

certain. These parameters are mostly the aerodynamic model parameters,

obtained in wind tunnel experiments. The weighting mass or heaviness, which

represents the difference between the weight and buoyancy forces, is also con-

sidered, since the equilibrium flight is mostly affected by its value. The pa-

rameters for which some uncertainty is assumed are then:

− mw - weighting mass;

− Clp , CMq, CNr

- roll, pitch and yaw damping aerodynamic coefficients;

− CD0, CL0

- drag and lift coefficients;

− CM - pitching moment coefficient;

− CDi, CYβ

, CLα- aerodynamic force coefficient derivatives;

− Clβ , CMα, CMαβ

, CMβα, CMβ

, CNβ- aerodynamic torque coefficient deriva-

tives;

− CLδe, CYδr

, CMδe, CNδr

- aerodynamic input coefficient derivatives.


The mission considered for the evaluation of the controllers sensitivity and

robustness to parameter uncertainty corresponds to a straight line at 50m

altitude aligned with the north axis, which the airship is to follow at 8m/s

groundspeed.

The control laws are designed considering a deterministic model of the airship,

named nominal. However, the real airship system has a wind disturbance

input, since in a real flight wind disturbances are always present. The following

wind perturbation, with two components, is considered:

− constant wind blowing from west at 4m/s;

− turbulent gust, with an intensity of 3m/s, which is an intermediate value,

in a scale from 0m/s for clear air with no turbulence, to 7m/s for a

hurricane [54].

For the baseline simulation, we consider no error in the model parameters,

only wind disturbance input for the aerodynamic flight described. For selected

variables, we then compare the Root Mean Square (RMS) values obtained in

this baseline simulation with the RMS results obtained repeating the simula-

tion varying each of the above parameters. The parameters vary one at a time

in order to allow the evaluation of the influence of each one.

Chapter 4

Classical Approach: Linear

Control

Contents

4.1 Airspeed and altitude regulation model . . . . . . 55

4.2 Lateral models . . . . . . . . . . . . . . . . . . . . . 56

4.2.1 No-roll approximation . . . . . . . . . . . . . . . . . 56

4.2.2 Space domain approximation . . . . . . . . . . . . . 57

4.3 Linear Quadratic Regulator . . . . . . . . . . . . . 58

4.4 Simulation results . . . . . . . . . . . . . . . . . . . 60

4.4.1 Airspeed and altitude regulation model . . . . . . . 60

4.4.2 No-roll vs. space domain . . . . . . . . . . . . . . . 62

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 65

The airship equations of motion, derived following the Lagrangian approach,

were described in Chapter 2. These equations represent the airship dynamics

and were useful in the construction of a simulator [29], necessary to better

understand the behavior of the prototype airship and also to evaluate the

performance of potential controllers.

On the other hand, this 12-state nonlinear model is too complex to allow an

analysis of the airship dynamic characteristics, since most analysis and design

tools require a linear representation of the system. For this reason, Section 2.3

was dedicated to the linearization of the airship model, resulting in the usual

decoupling of the longitudinal and lateral motions [1, 53].

In view of control implementation, and bearing in mind that the linear models

describe the dynamics of the perturbations about a given equilibrium condi-

53

54 CHAPTER 4. CLASSICAL APPROACH: LINEAR CONTROL

tion, some variations of these models are considered in Section 4.1.

For the design of the linear controllers, several control design methodologies

are available within Classical and Modern Control theories. The latter, how-

ever, allows to base the control design directly on the state-variable model, an

important resource for Multi-Input / Multi-Output (MIMO) systems.

Pole Placement (PP) and Linear Quadratic Regulator (LQR) are among the

most popular modern controller design techniques for MIMO systems. Pole

Placement or Eigenvector Assignment [55] allows to allocate the poles of the

MIMO system to desired locations in one step by solving equations for the feed-

back gains. The airship system requirements, however, are not easily specified

in terms of eigenvalues/vectors. For aircraft design, the desirable pole locations

are usually found in flying qualities specifications, which consider for instance

the size, weight and maneuverability of the aircraft and the pilot workload.

These specifications, however, may not be suitable for an autonomous airship

flight. Moreover, the PP strategy does not confer any stability robustness to

the closed-loop system. This is an important factor since any (linearized or

even nonlinear) model of the system is an approximation of the real nonlinear

airship dynamics. Furthermore, these models are usually deterministic, not

taking into account disturbances such as wind gusts or sensor measurement

noise. Robustness to model parameters errors and to disturbances is, in fact,

a key issue in the choice of the controller.

LQR is the solution to an optimization problem that has some very attrac-

tive properties. Namely, the optimal controller automatically ensures a stable

closed-loop system, achieves guaranteed levels of stability robustness, and is

simple to compute. LQR is the control that minimizes a quadratic cost func-

tion subject to constraints imposed by the system dynamics. Typically, LQR

controllers design is carried out by choosing values for the design weights,

synthesizing the control law, evaluating how well the control law achieves the

desired robustness and performance, and iterating through the process until a

satisfactory controller is found. The design weights (state and control weight-

ing matrices) are the designer’s tools to balance the state errors against the

control effort. In the airship control case, the control weighting matrix is a

specially important tool in the sense that it allows the designer to change the

control effort of the different actuators over the flight envelope.

A restrictive aspect of LQR and PP controllers is that they are full state

feedback controllers. This means that every state that appears in the model

4.1. AIRSPEED AND ALTITUDE REGULATION MODEL 55

of the physical system must be measured by a sensor. This is not a problem

for the AURORA airship, since all the state variables may be easily measured.

Due to the clear advantages that the LQR brings for the airship control design,

this will be the technique applied as linear control methodology. Section 4.3

briefly reviews its theory. For a more in depth survey see [56, 55].

4.1 Airspeed and altitude regulation model

The longitudinal model presented in Section 2.3.2.1 considered the state xv =

[u, w, q, θ]T and the input uv = [δe, XT , δv]T .

With the purpose of maintaining the trim conditions chosen for linearization,

namely a straight level flight at a chosen airspeed Vte and altitude he, the

regulation of these two variables is an important issue.

Assuming either no wind is present or a steady translation one is, we can

substitute the groundspeeds u and w respectively by the airspeeds ua and wa

in the state vector xv.

Also, as referred in Section 2.3.2.1, the altitude may be added to the model

as an additional integrating state of the longitudinal motion. Using equa-

tion (2.95) and for D = −h:

˙D ≈ w − Vte θ (4.1)

Finally, the airspeed and altitude regulation model is:

˙ua˙wa˙q

δ˙θ

=

av11 av12 av13 0 av14

av21 av22 av23 0 av24

av31 av32 av33 0 av34

0 1 0 0 −Vteav41 av42 av43 0 av44

ua

wa

q

δ

θ

+

bv11 bv12 bv13

bv21 bv22 bv23

bv31 bv32 bv33

0 0 0

0 0 0

δe

XT

δv

(4.2)

where avijand bvij

are the coefficients of the constant matrices Av and Bv


defined in (2.94). The state variables are given by:

ua = ua − uae= Vt cosα cos β − Vte cos θe (4.3)

wa = wa − wae= Vt sinα cos β − Vte sin θe (4.4)

q = q (4.5)

θ = θ − θe (4.6)

and δ is the vertical position error measured in the trajectory reference frame

(see Section 3.1).

4.2 Lateral models

The lateral model used in Section 2.3.2.2 for the dynamic characteristics anal-

ysis considered the state xh = [v, p, r, φ]T and the input uh = [δa, δr, TD]T .

However, two different models will be used for control purposes.

4.2.1 No-roll approximation

The three degrees of freedom approximation that describes the coupling be-

tween the yawing and rolling oscillations is called Dutch-roll motion in the

flight control literature [57, 53]. For the airship case we may neglect the

rolling motion, p, φ ≈ 0, and we designate the remaining side slipping and

yawing motions as no-roll mode. In this case, the aileron δa, whose function

is to regulate the roll movement, is not used for control. For the same reason,

the differential thrust TD is not used in hover flight (when δv ≈ 90o). When

in aerodynamic flight (δv ≈ 0o), the differential thrust TD and the rudder δr,

both control the yaw angle. Since at these airspeeds the control surfaces show

a high authority, the rudder will be used over the differential thrust.

Assuming the reference path aligned with north, and for position control or

guidance, this no-roll approximation can be complemented with the lateral

position E and the yaw angle ψ, whose dynamic equations are given by:

˙E ≈ Vte sin ψ ≈ Vteψ (4.7)

˙ψ = r/ cos θe (4.8)

where θe is the trim value of the pitch angle, and sin ψ ≈ ψ. For a generic

4.2. LATERAL MODELS 57

reference heading, the lateral error is given by ε, measured in the trajectory

reference frame (see Section 3.1).

The lateral model here named no-roll approximation is then:

˙va˙r

ε˙ψ

=

ah11ah13

0 0

ah31ah33

0 0

0 0 0 Vte

0 1/ cos θe 0 0

va

r

ε

ψ

+

bh12

bh32

0

0

δr (4.9)

where ahijand bhij

are the coefficients of the Ah and Bh constant matrices


va = va − vae= Vt sin β (4.10)

r = r (4.11)

ψ = ψ − ψar(4.12)

4.2.2 Space domain approximation

A simpler approach may be obtained for aerodynamic flight if we assume an

additional simplification in the lateral dynamics, considering it as a first order

system relating the yaw rate r and the rudder deflection δr:

r ≈ −kVte δr (4.13)

where the positive constant k is obtained by observing the yaw rate originated

by different values of airspeed and rudder deflection. The negative sign is

due to the convention that a positive rudder deflection leads to a negative

yaw rate. This equation results from simulation and flight observations, which

show that the yaw rate obeys an almost proportional relation with the product

of airspeed and rudder deflection, when in aerodynamic flight.

Substituting this simplified dynamics into (4.8), yields:

˙ψ ≈ −(kVte/ cos θe)δr (4.14)

The time derivative of a variable z may be written as the product z = dzdt

=∂z∂x

∂x∂t

where ∂x∂t

= u is the longitudinal groundspeed. Assuming u ≈ Vte , valid

in the no-wind case, it is possible to rewrite equations (4.7) and (4.14) now in


the space domain:

E ′ = ψ (4.15)

ψ′ = −kv δr (4.16)

where kv = k/ cos θe and z′ = dzdx

is the space derivative.

This leads to the following space domain lateral model [58]:

[

ε′

ψ′

]

=

[

0 1

0 0

][

ε

ψ

]

+

[

0

−kv

]

δr (4.17)

which, for the assumptions made earlier, is only valid when in aerodynamic

flight.

4.3 Linear Quadratic Regulator

This section briefly describes the LQR theory. The reader is referred to [56, 55]

for a more in depth survey.

Consider the linear time-invariant (LTI) dynamical system:

˙x = Ax + Bu, x(t0) = x0 (4.18)

where the n×1 vector x is the state, the m×1 vector u is the control input and

the p× 1 vector y is the measured output. If the pair (A,B) is stabilizable1,

then there exists a solution to an infinite time LQR problem. The controls will

be state feedbacks of the form (see fig. 4.1(a)):

u = −Kx (4.19)

where K is the matrix of constant feedback coefficients to be determined by

the design procedure.

The objective of state regulation of the airship is to drive any initial condition

error to zero, thus guaranteeing stability. This may be achieved by selecting

1The pair (A,B) is stabilizable if there exists a real matrix K such that A − BK isstable.

4.3. LINEAR QUADRATIC REGULATOR 59

the control input u that minimizes the quadratic cost function:

J =

∫ ∞

t0

(xTQx + uTRu)dt (4.20)

In this performance index, the size of state x is weighted relative to the effort

of the control action u through the weighting matrices Q and R, respectively

nonnegative and positive definite matrices. The minimization of J is a gener-

alized minimum energy problem. The objective is to minimize the energy in

the states without using too much control energy. A larger control-weighting

matrix R leads to a smaller control action u, while a larger state-weighting

matrix Q makes x go to zero more quickly with time.

The LQR problem with state feedback is the following: given the linear sys-

tem (4.18), find the Kalman gain matrix K in the control input (4.19) that

minimizes the value of the quadratic cost functional (4.20). This is achieved

by solving the algebraic Riccati equation for the symmetric positive definite

matrix P:

PA + ATP − PBR−1BTP + Q = 0 (4.21)

Finally, the Kalman gain K is computed by:

K = R−1BTP (4.22)

Under the assumptions made above a unique solution exists, and the closed-

loop dynamics, obtained by substituting (4.19) into (4.18):

˙x = (A − BK) x (4.23)

are guaranteed to be stable. The dynamic matrix of the closed-loop system is

given by:

Ac = A − BK (4.24)

While the design of the linear controller only involves the state and input

variations (x and u), its implementation produces and requires the complete

variables (x and u). For instance, the actuation request u has a feedback

component u and a feedforward component ue, as illustrated in fig. 4.1(b).


u

-

−K

˙x = Ax + Bux

(a) Design.

+

ΣlΣ -

?

6

--

−K

airshipx

x

+

xe

−u

u

ue

+

l

(b) Implementation.

Figure 4.1: Linear control block diagrams.

4.4 Simulation results

This section presents illustrative simulation results of the airship linear control

using the longitudinal and lateral models presented, and implemented accord-

ing to the block diagram in fig. 4.1(b).

4.4.1 Airspeed and altitude regulation model

This section focuses on the longitudinal control using model (4.2). The purpose

is airspeed and altitude regulation, so the desired operating conditions are

maintained.

The chosen equilibrium values are 10m/s for airspeed and 50m for altitude.

This corresponds to an aerodynamic flight. In order to observe only the longi-

tudinal behavior, the trajectory coincides with a straight line. The simulation

starts with no wind, and at t = 20s a 3m/s wind starts blowing from north.

Figure 4.2 shows the cartesian position variables. The motion is only made

along the longitudinal plane, as may be noticed by the trajectory and altitude

graphics. The altitude regulation is well achieved, with an error inferior to

0.6m. However, it presents a small static error, which (if considered significant)

may be canceled including an integrator state in the model.

The state Q and input R weighting matrices are set as (in SI units):

Q(ua, wa, q, δ, θ,∫δ) = diag(1, 1, 1, 1, 1, 0.1) (4.25)

R(δe, XT , δv) = diag(500, 0.1, 1000) (4.26)

The longitudinal groundspeed u and airspeed Vt are represented in fig. 4.3.

Even with the wind perturbation at t = 20s, the airspeed regulation at 10m/s

is well accomplished (not presenting static error, and therefore not needing

4.4. SIMULATION RESULTS 61

E (m)

N(m

)

Time (s)

h(m

)

0 20 40 60-200 0 20049.8

50.0

50.2

50.4

50.6

0

200

400

Figure 4.2: NE trajectory and altitude h (−.− equilibrium value, − real valuewithout integrator, −− real value with D integrator) for airspeed and altituderegulation.

integrator), to the obvious expense of the groundspeed reduction.

Time (s)

u(m/s)

Time (s)

Vt

(m/s)

0 20 40 600 20 40 609

10

11

12

13

14

6

7

8

9

10

11

Figure 4.3: Longitudinal groundspeed u and airspeed Vt (−.− equilibrium value,− real value without integrator, −− real value with D integrator) for airspeedand altitude regulation.

Finally, in fig. 4.4 it is possible to see the control action. As expected, only

the engines thrust and the elevator deflection are necessary to accomplish the

airspeed and altitude regulation objectives (the vectoring angle variation is

negligible). In face of the wind incidence, the thrust XT is reduced until

the equilibrium airspeed value is attained, which is possible by reducing the

groundspeed. The engines vectoring δv is close to zero as expected for an

aerodynamic flight. The rudder deflection is zero, since no lateral control is

used.


Time (s)

XT

(N)

Time (s)

δ v(deg

)

Time (s)

δ e(deg

)

Time (s)

δ r(deg

)

0 20 40 600 20 40 60

0 20 40 600 20 40 60

-1.0

-0.5

0.0

0.5

1.0

0

1

2

3

4

5

0.66

0.68

0.70

0.72

0.74

0.76

27

28

29

30

31

32

Figure 4.4: Control action for airspeed and altitude regulation: total thrust XT ,vectoring angle δv, elevator δe and rudder δr (−.− equilibrium value, − realvalue without integrator, −− real value with D integrator).

4.4.2 No-roll vs. space domain

The purpose of this section is to compare the performance of the lateral ap-

proximation models, no-roll approximation and space domain. The equilib-

rium conditions, necessary to the validity of the models, are guaranteed by the

longitudinal controller, regulating both airspeed and altitude.

The simulation considers an aerodynamic flight at Vte = 10m/s and De =

−50m subject to three tests:

− initial alignment on a straight line segment, with no wind incidence.

The airship starts deviated from the reference trajectory at (Ni, Ei) =

(0,−10)m and with an orientation ψi = −10o;

− reference trajectory following. The airship has to track a two-segment

trajectory in the shape of a 50o elbow, corresponding to the crossing of

a route way-point, again with no wind incidence;

− robustness to disturbances. At t = 60s wind starts blowing from north-

west at 3m/s.


For the no-roll approximation, the state Q and input R weighting matrices

are set as (in SI units):

Q(va, r, ε, ψ) = diag(1, 1, 1, 1) (4.27)

R(δr) = diag(1000) (4.28)

while for the space domain approximation they are set as:

Q(δ, ψ) = diag(1, 1) (4.29)

R(δr) = diag(1000) (4.30)

The NE trajectory, lateral position error ε and orientation ψ may be seen in

fig. 4.5. Both approximations allow a good lateral control, with the airship

E (m)

N(m

)

Time (s)

ε(m

)

Time (s)

ψ(deg

)

0 50 100

0 50 100

0 200 400 600-20

0

20

40

60

80

-20

-10

0

10

0

100

200

300

400

500

600

700

800

Figure 4.5: North-east trajectory, lateral error ε and yaw angle ψ (−.− refer-ence trajectory, − no-roll approximation, −− space domain approximation).

always being able to track the reference path, after the initial deviation and

orientation are corrected. The space domain approximation presents slightly

higher errors and slower corrections than the no-roll model, namely when re-

acting to the wind disturbance step.

Note that when the wind starts blowing at t = 60s, the reference yaw angle is

no longer the trajectory reference angle ψr but something in between it and the


blowing wind (ψw = −45o). This yaw reference corresponds to the yaw angle

the relative air inertial velocity pardoes with the i frame, ψar

, as described

in Section 3.2. Allowing the airship to align itself with the relative air reduces

the drag force and minimizes the sideslip angle β, as seen in fig. 4.6. In the

no-wind case, ψar≡ ψr.

The lateral control actions (only the rudder δr is used in both cases) are very

similar for both cases (see fig. 4.6). Although it is the no-roll approximation

which requests higher rudder deflections, both approximations reach the lower

saturation limit in the elbow curve.

Time (s)

δ r(deg

)

Time (s)

φ(deg

)

Time (s)

β(deg

)

0 50 100

0 50 1000 50 100

-20

-15

-10

-5

0

5

-5

0

5

-30

-20

-10

0

10

20

30

Figure 4.6: Sideslip angle β, rudder deflection δr and roll angle φ (− no-rollapproximation, −− space domain approximation).

In fig. 4.6 it is also possible to see the roll angle φ. For both models, the

maximum values of the roll angle φ confirm the validity of the approximation

made, neglecting the rolling motion.

The small differences between the results obtained with both approximations

and the fact that the space domain model only requires the measurement of

two variables, leads to a major advantage of this approximation over the no-roll

one. On the other hand, the latter one is valid over the entire flight envelope,

while the space domain model is only applicable when in aerodynamic flight.

4.5. CONCLUSIONS 65

4.5 Conclusions

This chapter analyzes the linear control of the AURORA airship using the

Linear Quadratic Regulator.

The use of linear control is obviously limited to the existence of linear models

of the airship. For this reason, the linearization of the nonlinear system around

a given equilibrium condition is performed, as described in Section 2.3. The

linearization, as usual in aerial systems, leads to simplified decoupled longitu-

dinal and lateral linear models of the airship motion.

In order for the linear models to be valid, the equilibrium condition for which

the linearization is performed must be guaranteed. Usually, the trim coincides

with a straight level flight with no wind incidence. Therefore, the linear control

presented in this chapter aims at the regulation of the state variables x so that

x ≡ xe. The longitudinal controller is responsible to regulate both airspeed

and altitude, while the lateral approximations are in charge of correcting the

lateral error and the yaw angle.

Simulation results demonstrate the good performance of the LQ regulator ap-

plied to the three models, even in the presence of wind disturbances. Position

errors are corrected, and so is the airspeed, to compensate for the variation of

wind.

However, linear control is limited, since the controller designed for a given

system is only valid in the vicinity of the equilibrium condition considered.

For instance, mission objectives like ground-hover that involve groundspeed

regulation are not possible, since the equilibrium airspeed would be unknown

a priori in the presence of wind.

This problem is solved if the linear systems and controllers are not fixed but

change with the measured airspeed and altitude (which define an equilibrium

condition). This is the idea of the gain scheduling technique, which extends the

validity of the linearization approach to a range of operating points, instead of

a single one. This control technique, already considered a nonlinear one, shall

be presented in the second part of this work.

Chapter 5

Gain Scheduling

Contents

5.1 More linear models . . . . . . . . . . . . . . . . . . 68

5.1.1 Groundspeed and altitude regulation . . . . . . . . . 68

5.1.2 Complete 12-states linear model . . . . . . . . . . . 69

5.2 Scheduling . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3 Robustness analysis . . . . . . . . . . . . . . . . . . 75

5.3.1 Performance robustness . . . . . . . . . . . . . . . . 75

5.3.2 Stability robustness . . . . . . . . . . . . . . . . . . 80


5.4.1 Groundspeed and altitude regulation . . . . . . . . . 87



5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 97

Today, Gain Scheduling is the prevailing flight control design methodology [59].

While the linear control presented in the previous chapter is only valid around

a single equilibrium condition, this conventional solution performs point de-

signs for a large set of trim conditions and then constructs a gain schedule by

considering gains with respect to flight conditions.

Using several linear models to describe the aircraft dynamics over the flight

envelope, allows the control designer to make use of all the classical design and

analysis tools. For this reason, gain scheduling is the obvious next step in the

AURORA airship control. It will hopefully provide a better knowledge of the

control design issues, as well of possible solutions.

Depending on the process considered for linearization, it is sometimes possible

67

68 CHAPTER 5. GAIN SCHEDULING

to find auxiliary variables that correlate well with the changes in the process

dynamics. In the airship case, these variables mostly correspond to the air-

speed and altitude. Still, the altitude influence may be disregarded for low

altitude flights where the envelope pressure is kept practically constant. For

this reason, from now on we will only consider the airspeed as trim parameter.

This chapter presents the gain scheduling approach applied to the airship path-

tracking control. In Section 5.1 two new linear models are introduced, one for

the longitudinal motion only, the other considering all the 12-states, combin-

ing the longitudinal and lateral motions in the same model. Section 5.2 de-

scribes the scheduling procedure and in Section 5.3 the closed-loop robustness

is analyzed, considering both performance and stability issues. Finally, some

simulation results are presented in Section 5.4, followed by some conclusions

in Section 5.5.

5.1 More linear models

This section introduces alternative linear models which are a natural evolution

of the previously presented ones when considering gain scheduling.

5.1.1 Groundspeed and altitude regulation

Although airspeed regulation is important to assure the linearized model is

valid, inspection oriented applications usually demand control of the ground-

speed. Also, for tracking purposes, the north or longitudinal position N should

also be considered. Assuming the reference path aligned with north, its dy-

namics may approximately be given by:

˙N ≈ cos θeu+ sin θew ≈ u+ θew (5.1)

For a generic reference heading, the longitudinal error is given by η, measured

in the trajectory reference frame (see Section 3.1).

The complete groundspeed regulation model, which in fact includes all 6 longi-

5.1. MORE LINEAR MODELS 69

tudinal variables, is then:

˙ua˙wa˙q

η

δ˙θ

=

av11 av12 av13 0 0 av14

av21 av22 av23 0 0 av24

av31 av32 av33 0 0 av34

1 θe 0 0 0 0

0 1 0 0 0 −Vteav41 av42 av43 0 0 av44

ua

wa

q

η

δ

θ

+

bv11 bv12 bv13

bv21 bv22 bv23

bv31 bv32 bv33

0 0 0

0 0 0

0 0 0

δe

XT

δv

(5.2)

where avijand bvij

are the coefficients of the constant matrices Av and Bv


ua = ua − uar(5.3)

wa = wa − war(5.4)

q = q (5.5)

θ = θ − θ′ar(5.6)

and δ is the vertical position error, measured in the trajectory reference frame

(see Section 3.1).

Given the desired groundspeed reference pr and having estimated previously

the wind velocity pw, the air velocity reference var= [uar

, var, war

]T is obtained

from (see Section 3.2):

var= Sar

(pr − pw) (5.7)

with Sar= S(Φ′

ar) and Φ′

argiven by (3.8). The equilibrium values xe and

ue, as well as the linear matrices Av and Bv, are obtained in real-time from a

lookup table, function of the measured airspeed Vt.

5.1.2 Complete 12-states linear model

The next and final linear model evolution is obviously the complete 12-states

model. Although the longitudinal and lateral motions are still decoupled (a

consequence of the linearization), the advantage of joining both models in a

single one is having also a single controller instead of two. This requires that

all state variables are measured so the full state feedback is possible. This is

the AURORA case (see Section 2.1), otherwise a state estimator may be used.


The complete 12-states linear model is then given by:

˙va˙ω

Γ˙Φ

= A

va

ω

Γ

Φ

+ B

δe

XT

δv

δa

δr

TD

(5.8)

with:

va = va − varwith var

given by (5.7) (5.9)

ω = ω − ωr with ωr = [0, 0, rr]T (5.10)

Γ = [η, ε, δ]T given by (3.3) (5.11)

Φ = Φ − Φ′ar

with Φ′ar

given by (3.8) (5.12)

Note that, although we have assumed a rectilinear reference path for the lin-

earization, the approach may also be extended to the cases where the refer-

ence path varies slowly, with negligible derivatives when compared to the state

derivative. In this case, the angular velocity reference must be adapted to the

case, reason for which we considered ωr 6= 0.

Again, the equilibrium values xe and ue, as well as the linear matrices A and

B, are obtained in real-time from a lookup table, function of the measured

airspeed Vt.

The evaluation of the longitudinal and lateral decoupling due to the lin-

earization is easily noticeable if we rearrange the A matrix considering x′ =

[xTlong, xTlat]

T . In fact, the A′ matrix can be partitioned into four distinct sub-

matrices as:

A′ =

[

Along 0

0 Alat

]

(5.13)

where all elements in the lower left submatrix are zero, while some few in the

upper right submatrix are not exactly zero but of smaller magnitude when

compared with the elements in Along and Alat.

We can also rearrange the B matrix lines considering the x′ state. The resulting

5.1. MORE LINEAR MODELS 71

B′ matrix can also be partitioned into four distinct submatrices as:

B′ =

[

Blong 0

0 Blat

]

(5.14)

The elements relative to either longitudinal and lateral position variables are

obviously null since the kinematics does not directly depend on the input. The

lower left and upper right submatrices are null. This shows the decoupling be-

tween what we already called longitudinal actuators inputs ulong = [δe, Xt, δv]T

(or ulong = [δe, Tx, Tz]T ) and the lateral states xlat = [va, p, r, ε, φ, ψ]T , as well

as of the lateral actuators input ulat = [δa, δr, TD]T and the longitudinal states

xlong = [ua, wa, q, η, δ, θ]T .

Figure 5.1 describes the evolution of the 12-states linear model poles with the

airspeed. As expected, 8 of the 12 modes match the longitudinal and lateral

ones described in sections 2.3.2.1 and 2.3.2.2, respectively.

Real axis

Imagin

ary

axis

2©long 2©lat

1©lat

1©long: surge

2©long: heave/pitch subsidence

3©long: longitudinal pendulum

1©lat: sideslip subsidence

2©lat: yaw subsidence

3©lat: roll oscillation

1©long

3©lat

3©long

-12 -10 -8 -6 -4 -2 0 2-3

-2

-1

0

1

2

3

Figure 5.1: Poles of linearized dynamics vs. airspeed Vt (: 0m/s, ×: 15m/s).

The remaining 4 modes rest in the origin over the entire flight envelope. This

is expected since they correspond to the 4 natural position integrators N,E,D


and ψ. Although unstable, the system is controllable since:

rank(B AB . . . An−1B) = 12 (5.15)

5.2 Scheduling

The linear models presented in Section 5.1 vary with the airspeed (we are

assuming the altitude influence may be disregarded for low altitude flights

where the envelope pressure is kept practically constant). The airspeed is thus

the considered scheduling variable. In applications of gain scheduling, the

practice has been that one can schedule on time-varying variables, as is the

case of the airspeed, as long as they are slow enough relative to the dynamics

of the system [60].

The gain scheduling methodology may be defined as a routine of four steps,

executed for different airspeeds over the flight envelope:

1. definition of the equilibrium condition;

2. linearization of the nonlinear system equations around the trim;

3. computation of the control gain;

4. computation of the control input obtained from full state feedback.

At each sampling time t, the true airspeed Vt is measured. The first step is

then to obtain the corresponding equilibrium variables, xe and ue, solving the

optimization problem described in Section 2.3.1. We may then proceed to the

second step, where the nonlinear system is numerically linearized about the

trim, obtaining the A and B system matrices (see Section 2.3.2). We now have

almost all variables necessary to complete the third step, where we compute

the control gain K, solution of the LQR problem described in Section 4.3.

First we have to define the state Q and input R weighting matrices.

In order to have an idea of the actuators influence on the system dynam-

ics, the evolution of the B matrix coefficients with the airspeed are shown in

fig. 5.2. The more obvious conclusion is that the longitudinal/lateral actuators

only influence the longitudinal/lateral states. In fact, observing fig. 5.2(a), we

note that the coefficients relative to the lateral states v, p, r are null, while in

fig. 5.2(b) we have the longitudinal states u, w, q coefficients null.

5.2. SCHEDULING 73

Vt (m/s)

B(δe)

Vt (m/s)

B(X

T)

Vt (m/s)

B(δv)

0 5 10 15

0 5 10 15

0 5 10 15

-1.0

-0.5

0.0

-0.05

0.00

0.05

0.10

-10

0

10

(a) Longitudinal actuators (− v, p, r, −−

u, : w, −.− q).

Vt (m/s)

B(δa)

Vt (m/s)

B(δr)

Vt (m/s)

B(TD

)

0 5 10 15

0 5 10 15

0 5 10 15

-0.05

0.00

0.05

-10

0

10

-40

-20

0

(b) Lateral actuators (− u, w, q, −− v, :

p, −.− r).

Figure 5.2: Evolution of B matrix coefficients with airspeed.

We also observe that the influence of the actuators depends on the available

airspeed. For instance, the reduced authority of the control surfaces δe, δa

and δr is noticeable at low airspeeds, where the respective coefficients are null.

From the observation of the B(XT ) coefficients evolution, we note the total

thrust XT mostly influences the longitudinal speed u in aerodynamic flight,

while at low airspeeds the influence is on the vertical speed w. This is justified

for the change of the vectoring angle δv from 0o to 90o. The differential thrust

TD makes its more significant contribution at low airspeeds, being the sole

responsible for the rolling motion control.

For ease of implementation, we have chosen the cartesian (Tx, Tz) version of

the thrust longitudinal input over the polar (XT , δv) one. This way we consider

the forces produced by the propellers and the forces produced by the control

surfaces deflection. The evolution of the respective B coefficients is represented

in fig. 5.3.

We can then see that the B matrix coefficients have the necessary information

on the actuators behavior over the flight envelope. Therefore, the R matrix

may be constant with Vt, only having to be adjusted to the different types

(units) of actuators presents, namely engines vs. control surfaces. Although

being constant, the R matrix coefficients are chosen by the designer (with

an iterative process) such that the performance of the closed-loop system can

satisfy the desired requirements.


Vt (m/s)

B(Tx)

Vt (m/s)B

(Tz)

0 5 10 15

0 5 10 15

-0.05

0.00

0.05

-0.1

0.0

0.1

Figure 5.3: Evolution of B matrix Tx and Tz coefficients with airspeed (−v, p, r, −− u, : w, −.− q).

The weighting matrix R is then defined as:

R(δe, Tx, Tz, δa, δr, TD) = diag(1000, 0.1, 0.1, 5000, 1000, 0.5) (5.16)

where the different weights between δa and δe, δr means there is a preference

of these two over the former. The same happens with TD and Tx, Tz.

The weighting matrix Q is defined as:

Q(u, v, w, p, q, r, η, ε, δ, φ, θ, ψ) = diag(1, 1, 1, 1, 1, 1, 10, 10, 10, 1, 1, 1) (5.17)

where the higher weights of η, ε and δ indicate the request of a faster correction

of these three variables over the remaining ones.

The Kalman gain K may now be computed from (4.22). Considering a real-

time implementation, the three first steps may be condensed, taking xe, ue and

K from a lookup-table that gathers all these variables for different airspeeds

over the flight envelope. Lastly, the fourth and final step, the control input u is

obtained from (4.19)-(2.90). The gain scheduling diagram block is represented

is fig. 5.4.

Vt

--?

outputu

regulator

control

airshipsignal

x

y

state

schedulingparametersxe,ue,K

regulatorconditionoperating

-

Figure 5.4: Gain scheduling diagram block.

We have seen in Section 4.3 that the optimal gains at each gain scheduling point

5.3. ROBUSTNESS ANALYSIS 75

guarantee the stability of the closed-loop system. However, they should also

guarantee robust stability and performance. This means, they should guaran-

tee stability and good performance at points near the design equilibrium point.

Such robustness can be verified after the LQR design by using multivariable

frequency-domain techniques [53]. This shall be done in the sequence.

5.3 Robustness analysis

In this section, we analyze the robustness of the closed-loop 12-states system.

Although what implementation concerns the result is the same as the full

state feedback described in Section 4.3, for the robustness analysis we will

consider the closed-loop nominal system represented in fig. 5.5. It comprises

an internal feedback look of yint = [vTa , ωT , ΦT ]T and an external feedback

loop of the cartesian position errors yp = Γ.

yint

j -

6

-

-- -

6

-0+−

xuup ypCp

Kint Cint

Gp

ΣΣ+

epx = Ax + Bu

−

Kpj

Figure 5.5: Closed-loop nominal system.

In the following, two important robustness issues will be analyzed:

− performance robustness, which is the ability to guarantee acceptable per-

formance even though the system may be subject to disturbances;

− stability robustness, which is the capacity to provide stability in spite of

modeling errors due to incorrect dynamics coefficients identification and

plant parameter variations.

5.3.1 Performance robustness

We begin with the analysis of the closed-loop system performance robustness

according to [53].

Consider the general closed-loop system represented in fig. 5.6. The plant

is G(s), and K(s) is the compensator. The plant output is z(t), the plant


control input is u(t), and the reference input, null for the regulation problem,

is r(t). The uncertainties are characterized by a disturbance d(t) acting on the

K(s) Σ

++ ?

d(s)

- - -

?

-

6

- G(s)

Σ

++ n(s)

z(s)r(s)Σ

+−

u(s)ed(s)

Figure 5.6: Disturbed feedback control system.

system (wind gusts, for instance), and sensors measurement noise n(t). The

disturbances occur typically at low frequencies, below some value ωd, while the

measurement noise has its predominant effect at high frequencies, above some

value ωn.

The regulation error is:

e(t) = −z(t) (5.18)

Due to the presence of noise n(t), e(t) may not be represented in fig. 5.6. The

signal ed(t) is given by:

ed(t) = −z(t) − n(t) = e(t) − n(t) (5.19)

In terms of Laplace transforms, we may take the following relations regarding

the closed-loop system:

z(s) = G(s)K(s)ed(s) + d(s) (5.20)

ed(s) = e(s) − n(s) (5.21)

e(s) = −z(s) (5.22)

We can rewrite z(s) and e(s) as:

z(s) = −T(s)n(s) + S(s)d(s) (5.23)

e(s) = −S(s)d(s) + T(s)n(s) (5.24)

defining the system sensitivity as:

S(s) = (I + GK)−1 (5.25)


and:

T(s) = GK(I + GK)−1 = (I + GK)−1GK (5.26)

Since:

S(s) + T(s) = I (5.27)

T(s) is called the complementary sensitivity, or cosensitivity. The loop gain is

G(s)K(s).

To ensure small regulation errors, we must have S(jω) small at those frequen-

cies ω where the disturbance d(t) is large. This will yield good disturbance

rejection. On the other hand, for satisfactory sensor noise rejection, we should

have T(jω) small at those frequencies ω where n(t) is large.

We will now make use of the singular values to obtain performance specifi-

cations in the frequency domain. In fact, for any input, the magnitude of a

transfer function H(jω) at any given frequency ω, may be bounded above by

its maximum singular value, denoted σ(H(jω)), and below by its minimum

singular value, denoted σ(H(jω)). Therefore, our results need only take into

account these two constraining values of magnitude.

Some facts we shall use in this discussion are:

σ(GK) − 1 ≤ σ(I + GK) ≤ σ(GK) + 1 (5.28)

σ(M) = 1/σ(M−1) (5.29)

σ(AB) ≤ σ(A)σ(B) (5.30)

for any matrices A, B, GK and M, with M nonsingular. Let us also define

the L2 operator gain, denoted ||H||2, as:

||H||2 = maxω

[σ(H(jω))] (5.31)

Next, we consider the low and high-frequency specifications on the singular

value plot.


Low-frequency specifications For low-frequencies, let us suppose that the

sensor noise n(t) is zero so that (5.24) becomes:

e(s) = −S(s)d(s) (5.32)

Thus, to keep the regulation norm ||e(t)||2 small, it is only necessary to ensure

that the L2 operator norm ||S||2 is small at all frequencies where d(jω) is

appreciable. This may be achieved by ensuring that, at such frequencies,

σ(S(jω)) is small. So, since at low frequencies (see fig. 5.7(a)):

σ(S) = σ[(I + GK)−1] =1

σ(I + GK)≈ 1

σ(GK)(5.33)

this may be guaranteed if we select:

σ(GK(jω)) ≫ 1, for ω ≤ ωd (5.34)

where d(s) is significant for ω ≤ ωd.

High-frequency specifications We now turn to the high-frequency perfor-

mance specifications. The sensor noise is generally appreciable at frequencies

above some known value ωn. Thus, according to (5.24), to keep the regulation

norm ||e(t)||2 small in face of measurement noise, we should ensure the oper-

ator norm ||T||2 is small at high frequencies above this value. Since at high

frequencies (see fig. 5.7(b)):

σ(T) = σ[GK(I + GK)−1] ≈ σ(GK) (5.35)

this is guaranteed if:

σ(GK(jω)) ≪ 1, for ω ≥ ωn (5.36)

Figure 5.8 represents the singular values of the closed-loop system gain of

the nominal system in fig. 5.5, together with a graphical representation of

conditions (5.34) and (5.36). The minimum σ(GK) and maximum σ(GK)

singular values, are represented respectively in figs. 5.8(a) and 5.8(b). Refer-

ring to our nominal system represented in fig. 5.5, the loop gain corresponds

to GK = GpKp, where Gp is the transfer-function matrix of the inner-loop

delimited by a dashed line. Since different equilibrium conditions, i.e., different


Frequency (rad/s)

Magnitude

(dB

)

σ(S(jω)) –

1/σ(GK(jω)) --

10−3 10−2 10−1 100 101 102 103-150

-100

-50

0

50

100

150

(a) σ(S(jω)) and 1/σ(GK(jω)).

Frequency (rad/s)

Magnitude

(dB

)

σ(T(jω)) –

σ(GK(jω)) --

10−3 10−2 10−1 100 101 102 103-150

-100

-50

0

50

100

150

(b) σ(T(jω)) and σ(GK(jω)).

Figure 5.7: Singular values relations for Vt = 15m/s.

airspeeds Vt, lead to different linear models and respective controllers, fig. 5.8

contains the singular values of the different systems over the flight envelope

range 0 ≤ Vt ≤ 15 m/s (steps of 0.5m/s).

Frequency (rad/s)

σ(G

K)

(dB

)

↑ Vt

Low-frequencycondition

10−3 10−2 10−1 100 101 102 103-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

(a) Minimum singular values, σ(GK).

σ(G

K)

(dB

)

↑ Vt

Frequency (rad/s)

High-frequencycondition

10−3 10−2 10−1 100 101 102 103-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

(b) Maximum singular values, σ(GK).

Figure 5.8: Frequency analysis (singular values) of the MIMO nominal systemover the flight envelope 0 ≤ Vt ≤ 15m/s (steps of 0.5m/s).

Observing fig. 5.8 we see that the cutting frequency of σ(GK) varies between

0.05 and 0.4rad/s, while for σ(GK) it varies between 0.3 and 0.5rad/s. Ac-

cording to conditions (5.34) and (5.36), we need to know what is the frequency

range where wind disturbances and measurement noise are significant (ωd and

ωn, respectively).

We start by determining ωd. In order to represent the perturbations introduced

in the airship system by the nonhomogeneous properties of the surrounding air,


a turbulence model is used. The model chosen to represent the atmospheric

turbulence is the spectral model of Dryden [54]. The perturbation introduced

is represented as gust velocity components constant in time but spatially dis-

tributed. Appendix B describes how to obtain the turbulence velocity from

filtered white noise. Observing the filters transfer-functions (B.2)-(B.4), we

notice that the higher cut-off frequency is the one from Gw, which leads to:

ωd ≈ ωcGw=Vth

(5.37)

Notice that this is a conservative value 1 that depends on the airspeed and

altitude of the airship. So one solution is to evaluate the value of σ(GK(jω))

at ω = ωd. The magnitude of σ(GK(jωd)) corresponds to the factor by which

wind disturbances will be attenuated (if positive) or amplified (if negative).

Figure 5.9(a) describes the variation of σ(GK(jωd)) with airspeed Vt and alti-

tude h, which we see is always positive. We also see that, the higher the alti-

tude, the higher the wind disturbances attenuation. The magnitudes of σ(GK)

and of the gust transfer functions (B.2)-(B.4) are represented in fig. 5.9(b) for

Vt = 0.5m/s and in fig. 5.9(c) for Vt = 15m/s, both for h = 50m. These

figures show us that, in fact, σ(GK)(jω) ≫ 1 for ω < ωd, which we may

consider here as ω : 20 log10 |Gi(jω)| > 0. In face of this analysis, we conclude

the closed-loop system is robust to wind disturbances, i.e., condition (5.34) is

verified.

Measurement noise is well attenuated, since usually ωn ≫ 1 rad/s. At frequen-

cies ω > ωn, σ(GK(ω)) ≪ 0 dB (see fig. 5.8), which validates condition (5.36).

So far we analyzed the performance robustness of the closed-loop system. We

now proceed to the evaluation of its stability robustness.

5.3.2 Stability robustness

It is unusual for the plant model to be exactly known. Two basic sorts of

modeling errors are incorrect dynamics coefficients identification and plant

parameter variation. It is therefore important to determine if the closed-loop

system remains stable in the case these errors occur, i.e., if the system is

robustly stable.

1Considering (5.37) valid beyond the ground boundary layer, we reasonably assume h≫ 0so that ωd 9 ∞.


Vt (m/s)

σ(G

K(jωd))

(dB

)

h = 50 → 500m

0 5 10 150

5

10

15

20

25

30

35

(a) Wind disturbance attenuation.

Frequency (rad/s)

Magnitude

(dB

)

20 log10 |Gu| :

20 log10

|Gv| – –

20 log10

|Gw| –.–20 log10 σ(GK) –

10−3 10−2 10−1 100 101 102 103-160

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

(b) Vt = 0.5m/s and h = 50m.

Frequency (rad/s)

Magnitude

(dB

)

20 log10 |Gu| :

20 log10

|Gv| – –

20 log10

|Gw| –.–20 log10 σ(GK) –

10−3 10−2 10−1 100 101 102 103-160

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

(c) Vt = 15m/s and h = 50m.

Figure 5.9: Frequency-domain performance specifications - disturbance rejec-tion.

Stability robustness to incorrect dynamics coefficients identification

The airship motion is described by nonlinear dynamic and cinematic equations.

While the kinematics is quite exact, in the dynamics model case this is not so

straightforward. Some model parameters are not so easy to measure, namely

the weighting mass (difference between airship weight and buoyancy) and the

aerodynamic coefficients, usually identified in wind tunnel tests. Obviously,

an incorrect parameter identification will lead to an incorrect dynamic model.

While the closed-loop system is surely stable for the nominal plant, what

happens if the actual plant is different?

We analyze here the closed-loop system stability robustness to incorrect dy-

namics coefficients identification. We shall do so using the Matlabr Robust

Control Toolbox [61].


We consider here that the airship nominal linear model (5.8) has a parame-

ter uncertainty of 20%, i.e., each nonzero coefficient of the A and B matrices

has 20% uncertainty over its nominal value. Although having the same maxi-

mum value, the uncertainty of each coefficient is defined independently of the

uncertainty of the remaining coefficients.

The performance of a nominally stable uncertain system will generally degrade

for specific values of its uncertain elements. Moreover, the maximum possible

degradation increases as the uncertain elements are allowed to further and

further deviate from their nominal values.

The robust stability margin is the size of the smallest deviation from nomi-

nal of the uncertain elements that leads to system instability, and allows us

to evaluate the stability robustness of uncertain systems. A nominally stable

uncertain system is generally unstable for specific values of its uncertain ele-

ments. If the uncertain system is stable for all values of uncertain elements

within their allowable ranges, the uncertain system is robustly stable. Con-

versely, if there is a combination of element values that cause instability, and

all lie within their allowable ranges, then the uncertain system is not robustly

stable. A robust stability margin greater than one means that the uncertain

system is stable for all values of its modeled uncertainty. A robust stability

margin less than one implies that certain allowable values of the uncertain

elements, within their specified ranges, lead to instability.

As with other uncertain-system analysis tools, only bounds on the exact stabil-

ity margin are computed. The precise value is guaranteed to lie between these

upper and lower bounds. Figure 5.10 expresses the evolution of the lower and

upper bounds of the stability margin with the airspeed Vt in steps of 0.25m/s.

In the analysis we consider the worst scenario described by the lower bound.

We can see that for airspeeds between 2 and 4.5m/s and above 8.25m/s the

20% uncertain system is robustly stable since both bounds are higher than 1.

A margin of 1.3, for example, implies that the uncertain system remains stable

for all values of uncertain elements up to 30% outside their modeled uncertain

ranges (i.e., in this case the system is robustly stable for an uncertainty of 26%

over all matrices coefficients).

For the remaining airspeeds the stability margin is less than one, which means

the system is not robustly stable for some values of uncertainty. A margin

of 0.5, for instance, implies the uncertain system remains stable for all values


Vt (m/s)

Upper bound

∗ Lower bound

Sta

bility

Marg

in

0 5 10 150.0

0.5

1.0

1.5

2.0

2.5

3.0

Figure 5.10: Robust stability analysis of the uncertain systems.

of uncertain elements that are less than 0.5 normalized units away from their

nominal values (which corresponds to an uncertainty of less than 10% in our

case) and, there is a collection of uncertain elements that are more than or

equal to 0.5 normalized units away from their nominal values that results in

instability (which corresponds to an uncertainty between 10% and 20%).

We can see that the stability margin reduces for low airspeeds and during the

transition to higher ones. These are the more problematic regions, one for the

lack of controllability from the control surfaces, the other due to the change of

the actuators control action. Considering that usually Vt > 2m/s, we observe

that a 10% error in all coefficients assures a robustly stable closed-loop system

whatever the airspeed. This, however, is still a low error margin.

Stability robustness to plant parameter variations

Finally, we consider the closed-loop system stability robustness to model pa-

rameter variations due to changes in the linearization equilibrium point of the

nonlinear model. This is a low-frequency phenomenon, which we will analyze

according to [53].

It is important for the control gains K in (4.19) to stabilize the system at

all points near the design operating point for gain scheduling to be effective.

However, in passing from operating point to operating point, the parameters

of the state variable model vary.

Consider again the nominal model (4.18) used for design, which has the transfer


function:

G(s) = C(sI − A)−1B (5.38)

However, due to operating point changes, the actual system is described by:

˙x = (A + ∆A)x + (B + ∆B)u (5.39)

y = (C + ∆C)x (5.40)

where the plant parameter variation matrices are ∆A, ∆B and ∆C. This

results in the transfer function:

G∆(s) = G(s) + ∆G(s) (5.41)

with:

∆G(s) = C(sI − A)−1∆B + ∆C(sI − A)−1B − C(sI − A)−1∆A(sI − A)−1B

(5.42)

where second-order effects have been neglected. A state-space realization of

∆G is given by:

x∆ =

[

A −∆A

0 A

]

x∆ +

[

∆B

B

]

u∆ (5.43)

y∆ =[

C ∆C]

x∆ (5.44)

Since we may write the additive uncertainty equation (5.41) in the multiplica-

tive form:

G∆(jω) = [I + ∆G(jω)G−1(jω)]G(jω) ≡ [I + M(jω)]G(jω) (5.45)

(where we substituted s = jω), we shall proceed considering multiplicative

uncertainties in the form (5.45), where the unknown discrepancy satisfies the

bound:

σ(M(jω)) < m(ω) (5.46)

with m(ω) known for all ω.

So suppose we have designed a controller K so that the nominal closed-loop


system (4.23) is stable. A frequency-domain condition that guarantees the

stability of the actual closed-loop system, which contains not G(s) but G∆(s)

satisfying (5.45)-(5.46), will now be derived using the multivariable Nyquist

condition:

Theorem 5.1 (Generalized (MIMO) Nyquist Theorem). Let Pol denote the

number of open-loop unstable poles in GK. The closed-loop system with loop

transfer function GK and negative feedback is stable if and only if the image

of det(I + GK) as s goes clockwise around the Nyquist D-contour (right-half

of the s-plane including the jω axis)

1. makes Pol anti-clockwise encirclements of the origin, and

2. does not pass through the origin.

Proof. See [62], pp. 146.

So it is required that the encirclement count of the map det(I+G∆K) be equal

to the negative number of unstable open-loop poles of G∆K. By assumption,

this number is the same as that of GK. Thus, the number of encirclements of

det(I+G∆K) must remain unchanged for all G∆ allowed by (5.46). This is as-

sured if and only if det(I+G∆K) remains nonzero as G is warped continuously

toward G∆, or equivalently:

0 < σ[I + [I + ǫM(s)]G(s)K(s)] (5.47)

for all 0 ≤ ǫ ≤ 1, all M(s) satisfying (5.46), and all s on the standard Nyquist

contour.

Since G∆ vanishes on the infinite radius segment of the Nyquist contour, and

assuming for simplicity that no indentations are required along the jω-axis

portion, this reduces to the following condition:

σ(GK(I + GK)−1) = σ(T(jω)) <1

m(ω)(5.48)

for all 0 ≤ ω < ∞. Thus, stability robustness in face of parameter variations

∆A, ∆B and ∆C translates into a requirement that the cosensitivity T(jω)

be bounded above by the reciprocal of the multiplicative modeling discrepancy

bound m(ω).


The m(ω) bound for robustness to gain scheduling model parameter variation

is obtained the following way. Define ∆Gij(s) as the transfer function matrix

of the state-space system (5.43)-(5.44) with ∆A = Ai − Aj, ∆B = Bi − Bj

and ∆C = Ci − Cj, and i and j representing two consecutive linear system

models. From (5.45) we have that:

M(jω) = ∆G(jω)G−1(jω) (5.49)

The bound m(ω) is obtained from:

σ(M(jω)) ≤ σ(∆G)σ(G−1) = σ(∆G)1

σ(G)= m(ω) (5.50)

where we used relations (5.29)-(5.30).

Figure 5.11 compares the reciprocal of the multiplicative modeling discrep-

ancy bound, 1/m(ω), with the maximum singular value of the cosensitivity,

σ(T(jω)), for the entire flight envelope. Figure 5.11(a) considers a different

system every 0.01m/s, while for fig. 5.11(b) the scheduling is made for Vt

steps of 0.1m/s. While condition (5.48) is always fulfilled in the first case,

Frequency (rad/s)

Magnitude

(dB

)

σ(T(jω))

1/m(ω)

10−3 10−2 10−1 100 101 102 103-140

-120

-100

-80

-60

-40

-20

0

20

40

60

(a) For different systems every 0.01m/ssteps in Vt.

Frequency (rad/s)

Magnitude

(dB

)

σ(T(jω))

1/m(ω)

10−3 10−2 10−1 100 101 102 103-140

-120

-100

-80

-60

-40

-20

0

20

40

(b) For different systems every 0.1m/ssteps in Vt.

Figure 5.11: Stability robustness to plant parameter variation (darker:σ(T(jω)), lighter: 1/m(ω)).

thus demonstrating the stability robustness of the scheduled system, for the

second case it is not always the case. In fact, if a new system is considered

every 0.1m/s, the robustness to plant parameter variations is only assured for

the systems in the intervals [2.90, 5.70]m/s and [7.40, 15]m/s. This means a

thinner schedule should be used in the remaining flight envelope, correspond-


ing to the lower airspeed and transition regions, clearly where the dynamics

more rapidly change.

If there is no problem in keeping and accessing a big dimension lookup ta-

ble in real-time, the 0.01m/s grid should be used for system scheduling. In

practice, this corresponds to consider a new dynamic system at each airspeed

measurement (assuming at least 0.01m/s resolution of the wind sensor).


This section presents illustrative simulation results of the airship gain schedul-

ing control.

5.4.1 Groundspeed and altitude regulation

This section focuses on the longitudinal control using the model (5.2). The

purpose is to follow groundspeed and altitude profiles. The groundspeed profile

is given in terms of velocity along the reference path, i.e., vr = [ur, 0, 0]T m/s.

In order to observe only the longitudinal behavior, the trajectory coincides

with a straight line. The groundspeed profile starts at 10m/s and at tu1= 10s

accelerates to 12m/s, with a limit of 1m/s2. At tu2= 65s goes back to 10m/s,

with a rate limit of −0.5m/s2. Concerning the altitude profile, it starts at 50m

and at th1= 40s it goes up to 60m. At th2

= 100s goes down again to 50m.

The ascent and descent rates are ±1m/s. The simulation considers constant

wind incidence from north at 3m/s.

The NED trajectory, longitudinal (η) and vertical (δ) position errors, and the

altitude profile are shown in fig. 5.12. As expected, there is only motion in the

vertical plane, as may be noticed by the perfect following of the straight refer-

ence segment. The position errors oscillate around zero, with η and δ inferiors

to 2m. The η and δ errors correspond respectively to the transient response

of the speed and altitude profiles following. Although with an overshoot and

a small delay, the altitude profile is well followed.

The groundspeed profile and output are represented in fig. 5.13, together with

the aerodynamic variables. The groundspeed components v = [u, v, w]T are

described in fig. 5.13(a) and the airspeed Vt, the sideslip angle β and the

angle of attack α may be seen in fig. 5.13. Notice that the difference between


E (m)N (m)

h(m

)

-200

-100

0

100

200

0

500

1000

150040

45

50

55

60

65

(a) Comparison of airship north N , east Eand altitude h position (−) and projections(−.−) with reference (−−).

Time (s)

η(m

)

Time (s)

δ(m

)

Time (s)

h(m

)

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

45

50

55

60

65

-2

-1

0

1

2

-2

-1

0

1

2

(b) Longitudinal (η) and vertical (δ) errorsand altitude profile (− output, −− refer-ence).

Figure 5.12: Airship position coordinates and errors.

Time (s)

u(m/s)

Time (s)

v(m/s)

Time (s)

w(m/s)

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

-2

-1

0

1

2

-1

0

1

9

10

11

12

13

(a) Groundspeed: longitudinal u, lateral vand vertical w (−− reference, − output).

Vt

(m/s)

Time (s)

β(deg

)

Time (s)

α(deg

)

Time (s)

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

-10

-5

0

5

10

-1

0

1

12

13

14

15

16

(b) Airspeed Vt, sideslip angle β and angleof attack α.

Figure 5.13: Airship ground velocity components and aerodynamic variables.

the u and Vt (approximately equal to ua) lies on the 3m/s north wind. The

longitudinal speed presents also an overshoot and a small delay, but in the

overall a good regulation is achieved. The lateral speed and sideslip angle

are null, as expected in this case. The vertical groundspeed reference is null,

indicating the airship is requested to follow the reference path at ur = 10m/s

at all times, even during the altitude profile. This request does not take

into account the angle of attack (pitch) necessary to provide lift at a given

airspeed, reason for which the output w presents an offset (probably close to

the equilibrium value we not used as reference).


The actuators input is represented in fig.5.14. The groundspeed and altitude

regulation is possible through the action of the longitudinal actuators, which

control effort is represented in fig. 5.14(a). The engines thrust XT is mainly

responsible for the airspeed control, while the altitude is responsibility of the el-

evator deflection δe. The variation of the two control variables is not, however,

independent. A change in the groundspeed (altitude) reference, and hence

output, provokes a variation in the altitude (groundspeed) output, corrected

by the elevator (engines thrust). This may be observed comparing the respec-

tive figures. The engines vectoring δv is small during the entire simulation,

as expected for an aerodynamic flight. The lateral actuation, represented in

Time (s)

δ e(deg

)

Time (s)

XT

(N)

Time (s)

δ v(deg

)

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

-6

-3

0

3

6

0

20

40

60

80

-20

-10

0

10

20

(a) Longitudinal actuators: elevator δe, to-tal thrust XT and vectoring δv.

Time (s)δ a

(deg

)

Time (s)

δ r(deg

)

Time (s)

TD

(N)

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

-1

0

1

-1

0

1

-1

0

1

(b) Lateral actuators: aileron δa, rudder δrand differential thrust TD.

Figure 5.14: Airship actuators input.

fig. 5.14(b) for completeness, is obviously null for all three inputs, aileron δa,

rudder δr and differential thrust TD.


This section demonstrates the performance of the gain scheduling approach

using the complete 12-states model (5.8) executing the case-study mission

described in Section 3.3.1. The scheduling is made between airspeed steps

of 0.1m/s. The mission covers a wide range of the flight envelope, with the

airspeed Vt varying between 3 and 12m/s.

The airship position coordinates and errors are represented in fig. 5.15. The

vertical take-off and landing are well perceived in fig. 5.15(a), as well as the

path-tracking performance. Figure 5.15(b) displays the longitudinal η, lateral ε


and vertical δ errors, allowing to identify the more problematic mission points,

namely when the wind is aft the airship, at the end of the first half-circle

and the transition from the vertical ascent and the horizontal path-tracking

(see fig. 5.16(a)). The remaining noticeable errors correspond to instantaneous

references changes before the second stabilization, which the airship smoothly

corrects. In order to avoid saturation of the propellers, probable when the

controller tries to rapidly correct the longitudinal position, the error η is lim-

ited. This approaches the idea of Teel [63] which will be better explored in

Section 7.4. The limitation of η can be noticed by the constant rate at which

the north position is corrected in fig. 5.15(b). Note that the existence of in-

stant position errors might be caused by a transition in the mission objectives,

namely from path-tracking to stabilization, but also from a discontinuity in

the position provided by the GPS when the available satellites change.

E (m)N (m)

h(m

)

-100

0

100

200

300

-200

-100

0

100

200

0

10

20

30

40

50

60

(a) Airship north N , east E and altitude hposition (bold) and projections (normal).

Time (s)

η(m

)

Time (s)

ε(m

)

Time (s)

δ(m

)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-5

0

5

-30

-20

-10

0

10

20

-80

-60

-40

-20

0

(b) Longitudinal (η), lateral (ε) and verti-cal (δ) errors.


Figure 5.16 represents the airship horizontal trajectory and its attitude during

the mission. The airship north-east coordinates and heading during the mission

are described in fig. 5.16(a). The preferential alignment with the wind during

take-off and landing is well recognized, while in maneuvers at low airspeeds the

airship appears as slightly crabbing. The Euler angles evolution is displayed

in fig. 5.16(b). The roll angle (φ), with a null reference, presents a higher

oscillation in the two above mentioned problematic parts. The pitch (θ) and

yaw (ψ) angles try to follow the respective references described in Section 3.2.


The pitch angle shows higher errors during take-off and landing, as well as

during stabilization.

E (m)

N(m

)

windheading

-100 0 100 200 300-250

-200

-150

-100

-50

0

50

100

150

200

250

(a) North-east position with airshipheading (−− reference, − output).

Time (s)

φ(deg

)

Time (s)

θ(deg

)

Time (s)

ψ(deg

)0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-90

0

90

180

270

360

450

-20

0

20

-10

0

10

(b) Roll φ, pitch θ and yaw ψ (−− refer-ence, − output).

Figure 5.16: Airship north-east position and attitude.

The airship ground velocity and the aerodynamic variables are depicted in

fig. 5.17. The ground velocity components are represented in fig. 5.17(a).

The longitudinal groundspeed u mostly follows the reference that varies be-

tween 0m/s for stabilization, take-off and landing, and 7m/s during the path-

tracking. Along the circular segments, the errors are more noticeable due to

the change of the wind incidence angle while the airship is turning. The er-

ror is higher when the wind is aft the airship. The lateral velocity v is also

mostly influenced by the circular segments and during the tail wind segment.

The vertical velocity w follows its reference, with the −1 and 0.5m/s steps

corresponding to the take-off and landing vertical motion. The airspeed and

aerodynamic angles can be seen in fig. 5.17(b). During the whole mission, the

airspeed Vt varies significantly, from values around 3m/s up to 12m/s. The

airship covers a wide flight envelope, from hover to the aerodynamic flight,

crossing the troublesome transition region between the two. The sideslip an-

gle β and the angle of attack α vary between ±20o. Although more directly

related with wa, the behavior of α is also clearly correlated with w.

The actuators input is described in fig. 5.18, with the longitudinal actuators

elevator δe, total thrust XT and vectoring angle δv in fig. 5.18(a) and the

lateral ones, aileron δa, rudder δr and differential thrust TD, in fig. 5.18(b).


Time (s)

u(m/s)

Time (s)

v(m/s)

Time (s)

w(m/s)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-2

-1

0

1

2

-6

-4

-2

0

2

4

0

2

4

6

8


Vt

(m/s)

Time (s)

β(deg

)

Time (s)

α(deg

)

Time (s)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-30

-20

-10

0

10

20

-20

-10

0

10

20

2

4

6

8

10

12

14



The elevator δe action corresponds to the correction of the vertical error δ,

while the rudder δr has a higher command with tail wind, due to the reduced

authority at lower airspeeds. The aileron δa is responsible for the control of

the roll angle. When the control surface loses authority at low airspeeds, this

function is assumed by the differential thrust TD. The vectoring angle δv is

Time (s)

δ e(deg

)

Time (s)

XT

(N)

Time (s)

δ v(deg

)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-30

0

30

60

90

120

0

20

40

60

80

-30

-15

0

15

30


Time (s)

δ a(deg

)

Time (s)

δ r(deg

)

Time (s)

TD

(N)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-20

-10

0

10

20

-30

-15

0

15

30

-30

-15

0

15

30



responsible for the airship lift when the airspeed Vt is too low to provide the


necessary aerodynamic lift. The correlation between these two variables is

obvious comparing the graphics of δv and Vt.


tainty

Although the closed-loop system robustness has been analyzed in Section 5.3,

we present here the results of the sensitivity and robustness test described in

Section 3.3.2 as a tool of comparison between controllers performance.

For the baseline simulation, we consider no variation of the model parameters,

only wind disturbance input for the aerodynamic flight at 8m/s groundspeed

and 50m altitude. Figure 5.19(a) shows the airship north-east position and

heading when following the straight line reference aligned with north, while

subject to the 4m/s constant wind blowing from west, plus 3m/s turbulent

gust. We notice the airship is able to follow the reference, although with an

E (m)

N(m

) headingwind

-100 -50 0 50

0

200

400

600

800

1000

1200

(a) Airship north-east position withairship heading (−− reference,− output).2

φ(deg

)θ

(deg

)ψ

(deg

)θ w

(deg

)ψw

(deg

)

Time (s)

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

80

90

100

-10

-5

0

5

-35

-30

-25

-20

-5

0

5

10

-10

0

10

(b) Airship attitude : roll φ, pitch θ andyaw ψ (−− reference, − output), and es-timated wind attitude, θw, ψw.

Figure 5.19: Airship north-east trajectory and attitude, and wind attitude.

2The different scales might give a wrong idea of the airship heading (ψ ≈ −27o).


orientation that helps it minimize the drag force produced by the lateral wind.

Figure 5.19(b) represents the airship attitude references (see Section 3.2) and

output, as well as the wind estimated attitude. Notice that neither θw nor ψw

are constant, since they echo both constant and gust wind components. The

excitation of the signals is due to the wind turbulence.

As may be expected, the airship position errors oscillate around zero instead

of converging, due to the wind turbulence input, as may be seen in fig. 5.20(a).

The aerodynamic variables are represented in fig. 5.20(b). The around 9m/s

airspeed corresponds to the relative air speed between the 8m/s groundspeed

heading north and the 4m/s wind speed from west. The sideslip angle β is

close to zero, showing the airship is aligned with the relative airspeed.

Time (s)

η(m

)

Time (s)

ε(m

)

Time (s)

δ(m

)

0 50 100 150

0 50 100 150

0 50 100 150

-1

0

1

2

-1

0

1

-1

0

1

(a) Airship longitudinal (η), lateral (ε) andvertical (δ) errors.

Vt

(m/s)

time (s)

β(deg

)

time (s)

α(deg

)

time (s)

0 50 100 150

0 50 100 150

0 50 100 150

-5

0

5

10

-5

0

5

8

9

10


Figure 5.20: Airship position errors and aerodynamic variables.

The actuators input applied to the AURORA airship is represented in fig. 5.21,

with the longitudinal actuators action given in fig. 5.21(a), and the lateral

actuation in fig. 5.21(b). The vectoring angle δv is near zero in aerodynamic

flight, as is the differential thrust TD.

Table 5.1 shows the RMS values of selected variables. They are the airship

positions errors, namely longitudinal η, lateral ε and vertical δ errors, and the

true airspeed Vt, the angle of attack α and the sideslip angle β, together with

the groundspeed error eu relative to the 8m/s reference. The first row has

the RMS values obtained for the baseline case, and is to serve as reference


Time (s)

δ e(deg

)

Time (s)

XT

(N)

Time (s)

δ v(deg

)

0 50 100 150

0 50 100 150

0 50 100 150

-30

0

30

60

90

120

0

20

40

60

80

-5

0

5

10

(a) Longitudinal actuators input: elevatorδe, total thrust XT and vectoring δv.

Time (s)

δ a(deg

)

Time (s)

δ r(deg

)

Time (s)

TD

(N)

0 50 100 150

0 50 100 150

0 50 100 150

-1

0

1

-10

0

10

-2

0

2

(b) Lateral actuators input: aileron δa,rudder δr and differential thrust TD.


for the remaining lines where each of the listed coefficients is varied one at a

time. For a parameter uncertainty of ±70%, the controlled airship performed

qualitatively like the baseline case. We then increased the parameter uncer-

tainty, and present here the results obtained for ±90% uncertainty around the

nominal value.

Only for some parameters the uncertainty leads to significant deviations from

the baseline RMS values, either because the control action is insufficient due

to the actuators saturation (the gain scheduling demands a too high control

input) and/or just because the tested coefficient appears to be a more sensitive

model parameter. These cases are represented in bold in table 5.1.

With noticeable deviations from the baseline RMS values, for CLαand CD0

(both for −90% uncertainty) the gain scheduling still controls the AURORA

airship within acceptable bounds. CLαis the lift coefficient derivative due to

angle of attack; and CD0is maybe the most important aerodynamic coefficient,

expressing the drag suffered by the airship envelope at zero lift.

For the remaining coefficients, CMδe, CNδr

, CMqand CNr

(all for −90% un-

certainty), either saturation of the actuators occurred or the uncertainty of

the model parameter is too important for the gain scheduling controller to

overcome it. These parameters correspond respectively to the authority of the

elevator and rudder as pitching and yawing control inputs, and the pitch and

yaw damping derivatives. While the gain scheduling demonstrates to be ro-


Table 5.1: Robustness tests on model parameters (RMS values of selected vari-ables).

η (m) ε (m) δ (m) Vt (m/s) α (deg) β (deg) eu (m/s)

Baseline 0.24 0.33 0.45 9.01 3.41 1.17 0.16

Clβ−90% 0.20 0.28 0.45 9.01 3.47 0.93 0.12+90% 0.34 0.45 0.45 9.02 3.22 1.72 0.25

CM0

−90% 0.24 0.33 0.46 9.01 3.47 1.17 0.16+90% 0.24 0.33 0.45 9.01 3.35 1.17 0.16

CMα

−90% 0.33 0.35 1.07 9.01 4.09 1.20 0.16+90% 0.25 0.33 0.69 9.01 3.12 1.15 0.15

CMαβ

−90% 0.24 0.33 0.45 9.01 3.40 1.17 0.16+90% 0.24 0.33 0.45 9.01 3.42 1.17 0.16

CMβ

−90% 0.24 0.33 0.45 9.01 3.35 1.17 0.16+90% 0.24 0.34 0.46 9.01 3.48 1.17 0.16

CMβα

−90% 0.24 0.33 0.45 9.01 3.40 1.17 0.16+90% 0.24 0.33 0.45 9.01 3.42 1.17 0.16

CMδe

−90% 166.29 91.32 100.08 13.64 2.87 1.97 7.30

+90% 0.23 0.34 0.48 9.01 3.61 1.18 0.15

CNβ

−90% 0.21 0.29 0.45 9.01 3.44 1.15 0.12+90% 0.25 0.35 0.45 9.01 3.40 1.22 0.19

CNδr

−90% 23.75 9.07 2.57 9.28 4.10 7.71 1.37

+90% 0.16 0.12 0.44 9.01 3.43 1.05 0.11

CYβ

−90% 0.38 0.56 0.51 9.02 3.48 2.56 0.22+90% 0.21 0.30 0.45 9.01 3.42 0.85 0.14

CYδr

−90% 0.21 0.28 0.44 9.01 3.44 1.22 0.13+90% 0.26 0.37 0.45 9.01 3.39 1.53 0.19

CD0

−90% 1.78 0.51 0.66 9.02 4.26 1.24 0.27

+90% 0.78 0.32 0.65 9.00 2.75 1.12 0.14

CDi

−90% 0.31 0.33 0.47 9.01 3.49 1.17 0.17+90% 0.24 0.34 0.44 9.01 3.35 1.17 0.15

CL0

−90% 0.24 0.33 0.45 9.01 3.33 1.17 0.16+90% 0.24 0.33 0.45 9.01 3.50 1.17 0.16

CLα

−90% 6.49 0.41 7.18 9.05 19.78 1.57 0.36

+90% 0.24 0.34 0.59 9.01 2.00 1.12 0.15

CLδe

−90% 0.23 0.33 0.43 9.01 4.08 1.20 0.16+90% 0.26 0.33 0.68 9.01 3.16 1.15 0.17

Clp−90% 0.26 0.33 0.51 9.01 3.39 1.29 0.18+90% 0.24 0.34 0.45 9.01 3.41 1.15 0.16

CMq

−90% 401.88 102.32 11.96 8.55 20.31 13.76 2.81

+90% 0.43 0.37 1.37 9.03 4.44 1.19 0.21

CNr

−90% 626.99 281.01 7.55 3.76 39.40 75.92 3.96

+90% 0.43 0.79 0.48 9.01 3.43 1.43 0.22

mw−90% 0.28 0.32 1.33 9.01 1.67 1.03 0.16+90% 0.23 0.34 1.45 9.01 6.54 1.31 0.16

5.5. CONCLUSIONS 97

bust to a ±90% uncertainty in the remaining parameters, for these four cases,

the mismatch between the airship system and the model considered in the gain

scheduling controller design is too significant for the control action to overcome

it.

In any case, the gain scheduling controller may be considered robust to wind

disturbances and plant uncertainties. Among the list selected, these six pa-

rameters CLα, CD0

, CMδe, CNδr

, CMqand CNr

(and specially the last four) are

in fact the model parameters for which a more careful identification or deter-

mination should take place, though the required precision could merely remain

inside a 70% margin.

5.5 Conclusions

This chapter covers the analysis and results obtained applying a gain scheduled

state-feedback optimal controller to the airship path-tracking control problem

over the entire flight envelope. Considering the 12-states model, and for each

equilibrium condition considered, the control law provides in a single action

actuator commands to regulate both lateral and longitudinal motions.

Although at each equilibrium point the closed-loop system is guaranteed to

be stable by the optimal controller, it is important to analyze the robustness

to input disturbances as well as model uncertainties and parameter variation.

Doing so we have come to the following conclusions about the closed-loop

system:

− its performance is robust to wind disturbances;

− its performance is robust to measurement noise;

− it is robustly stable to model parameters uncertainties up to 10% for

airspeeds over 2m/s. A 20% uncertainty in the parameters still leads to

stable systems, except for very low airspeeds and in the transition region.

This indicates that a better identification of the model should be made

for these airspeeds, namely of the aerodynamic coefficients;

− it is robustly stable to parameter variations for a 0.01m/s scheduling.

This means the closed-loop system remains stable even if the actual

system does not correspond to the equilibrium condition considered. For

a 0.1m/s grid, the parameter variation still leads to stable systems, again


except for very low airspeeds and in the transition region. If such a

scheduling is necessary, one should try to avoid missions that induce the

airship to fly at such airspeeds for significant periods of time.

These robustness properties, together with (many) satisfactory simulation re-

sults and implementation simplicity, indicate the gain scheduling control is a

possible solution to the AURORA airship path-tracking problem.

However, is it the best? The control solution is optimal for given weighting

matrices, which are still obtained empirically. Moreover, we are only consider-

ing a linearized version of the airship, eventually discarding important features

of the system. These issues do not guarantee an optimal overall result, reason

for which other nonlinear solutions will be considered.

Chapter 6

Dynamic Inversion

Contents

6.1 General theory . . . . . . . . . . . . . . . . . . . . . 101

6.1.1 Local coordinates transformation . . . . . . . . . . . 102

6.1.2 Exact linearization via feedback . . . . . . . . . . . . 105

6.1.3 Asymptotic output tracking . . . . . . . . . . . . . . 107

6.2 New formulation for cascaded systems . . . . . . . 109

6.3 Application to airship path-tracking problem . . 111




6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 121

As in most complex engineering problems we are facing nowadays, flight con-

trol design is generally based in a divide-and-conquer approach. First, the

nonlinear equations of motion of the air vehicle are linearized about selected

operating points over the flight envelope. The tools of linear control theory can

then be used to design individual compensators to satisfy closed-loop specifi-

cations. Finally, a gain scheduled control is obtained by switching between the

individual compensators according to predefined scheduling variables. This

approach, for its relevance in flight control, was outlined in Chapter 5.

However, the gain scheduling procedure is time consuming, costly to iterate,

and still relies substantially on the engineer’s knowhow. An alternate method-

ology to flight control design, that avoids this iterative tuning process and

directly considers the nonlinear nature of the problem, is Dynamic Inversion.

Dynamic Inversion is a methodology to design closed-loop control laws for

99

100 CHAPTER 6. DYNAMIC INVERSION

nonlinear systems [64, 65]. As opposed to gain scheduling, dynamic inversion

searches for a global controller from a single global nonlinear model of the

plant. Its application in flight control [66] is justified since it can explicitly

address the nonlinearities in the aircraft or airship dynamics and provides a

control law which is valid for the whole flight envelope.

With dynamic inversion, the set of existing deficient or undesirable dynamics

are canceled out and replaced by a designer selected set of favored dynamics.

This is accomplished by careful algebraic selection of a feedback function,

reason for which the dynamic inversion methodology is also called Feedback

Linearization.

The feedback linearization is viewed as a generalization of pole placement for

linear systems. Its basic idea is to first transform a nonlinear system into

a (fully or partially) linear one, and then use the well-known and powerful

linear design techniques to complete the control design. However, it does not

guarantee robustness in face of parameter uncertainty or disturbances.

Two main assumptions are made in the dynamic inversion methodology, (i) the

plant dynamics is perfectly modeled, and (ii) the system states are measured

or estimated accurately.

In practice, neither of these assumptions is realistic, and the robustness of

the closed-loop system must be secured in order to suppress any undesirable

behavior. The use of an outer-loop controller to improve a dynamic inversion

inner-loop controller robustness has been reported in some works. A Linear

Quadratic Gaussian outer-loop is addressed by Ito et. al [67], and µ-synthesis

is used by Reiner et. al [68] and Bennani and Looye [69]. Yedavalli et. al [70]

present a stability robustness analysis of dynamic inversion based control laws

used for flight control with uncertainties in model data.

Before advancing to a complete nonlinear control design, we took an inter-

mediate step between linear and nonlinear control. It is an hybrid approach,

that considers the linearized dynamic models presented in Chapter 4, and the

nonlinear kinematics of the airship. The lateral and longitudinal motions are

decoupled, reason for which the kinematics, though nonlinear, allows to keep

the independence of the controllers. This was applied to the lateral control of

the airship and compared with LQR proportional feedback in [71].

A fully, yet simplified, nonlinear airship model was used in the hover stabiliza-

tion of the AURORA airship using dynamic inversion [72]. At hover conditions

6.1. GENERAL THEORY 101

the relative wind dynamics may be insignificant, downsizing the aerodynamic

forces when compared with the kinematic, gravity and propulsion ones. The

control law is therefore obtained by inversion of a simplified dynamic model,

discarding the aerodynamic forces at low airspeeds.

In this chapter, the dynamic inversion methodology will be applied to the

path-tracking problem of the AURORA airship, by inversion of the 12-state

complete nonlinear dynamic system deduced in Chapter 2. Although the clas-

sical approach is reviewed in Section 6.1, a new formulation, applicable to

cascaded systems described in terms of velocity and position, and where the

output of interest is the position, is developed in Section 6.2.

The performance of the dynamic inversion closed-loop system applied to the

airship path-tracking problem is analyzed in Section 6.4. We present the sim-

ulation results obtained for the case-study mission, and examine the controller

sensitivity and robustness to parameter uncertainty. Finally, Section 6.5 closes

the chapter with some final remarks.

6.1 General theory

This section presents the classical dynamic inversion theory, and mostly follows

reference [64].

The objective of the dynamic inversion approach is to change a nonlinear

system into a linear and controllable one by means of feedback and coordinates

transformation. With this in mind, and for MIMO (multi-input multi-output)

square systems (i.e., n-states systems with the same number m of inputs and

outputs) in a neighborhood of a point xo, we first describe a suitable change of

coordinates in state space that allows us to represent the system in a normal

form of special interest. It is then based on this normal form that we obtain

a state feedback control law which applied yields a linear and controllable

closed-loop system. Finally, we consider the problem of asymptotic output

tracking.


6.1.1 Local coordinates transformation

Consider the affine or linear in control system:

x = f(x) +m∑

i=1

gi(x)ui (6.1a)

y1 = h1(x)

· · · (6.1b)

ym = hm(x)

where x is the state n-vector, u is the control input m-vector (of components

ui) and y is the outputm-vector (of components yi); f(x) and h(x) are smooth1

n- and m-vector fields respectively, and g(x) is an n×m matrix whose columns

are smooth vector fields gi. A more condensed form of (6.1) is:

x = f(x) + g(x)u (6.2a)

y = h(x) (6.2b)

We will start our development with the multivariable version of relative degree

(see Appendix C for the Lie derivative Lkf h(x) used notation). A multivariable

nonlinear system of the form (6.1) has a (vector) relative degree r1, . . . , rmat a point xo if:

(i)

LgjLkf hi(x) = 0 (6.3)

for all 1 ≤ j ≤ m, for all 1 ≤ i ≤ m, for all k < ri − 1, and for all x in a

neighborhood of xo;

(ii) the m×m matrix:

A(x) =

Lg1Lr1−1

f h1(x) · · · LgmLr1−1

f h1(x)

Lg1Lr2−1

f h2(x) · · · LgmLr2−1

f h2(x)

· · · · · · · · ·Lg1

Lrm−1f hm(x) · · · Lgm

Lrm−1f hm(x)

(6.4)

is nonsingular at x = xo.

1A vector field f(x) is considered smooth if it has continuous partial derivatives of anyrequired order.


Note that ri is exactly the number of times one has to differentiate the i-th

output yi(t) at t = to in order to have at least one component of the input

vector u(to) explicitly appearing.

Proposition 6.1. Suppose a system has a (vector) relative degree r1, · · · , rmat xo. Then:

r1 + . . .+ rm ≤ n (6.5)

Set, for 1 ≤ i ≤ m:

φi1(x) = hi(x)

φi2(x) = Lfhi(x)

· · ·φiri(x) = Lri−1

f hi(x)

(6.6)

If r = r1 + . . . + rm is strictly less than n, it is always possible to find n − r

more functions φr+1(x), . . . , φn(x) such that the mapping:

Φ(x) = [φ11(x), . . . , φ1

r1(x), . . . , φm1 (x), . . . , φmrm(x), φr+1(x), . . . , φn(x)]T (6.7)

has a jacobian matrix which is nonsingular at xo and therefore qualifies as a

local coordinates transformation in a neighborhood of xo.


In the remaining of this section, we will restrict our description to the systems

where the sum r = r1 + r2 + . . . + rm is exactly equal to the dimension n of

the state space. In this case, the set of functions:

φik(x) = Lk−1f hi(x) for 1 ≤ k ≤ ri, 1 ≤ i ≤ m (6.8)

defines completely a local coordinates transformation at xo. Differentiating


with respect to time, we obtain:

dφi1dt

= φi2(t)

· · ·dφiri−1

dt= φiri(t)

dφiridt

= Lrif hi(x(t)) +m∑

j=1

LgjLri−1

f hi(x(t))uj(t)

(6.9)

for all 1 ≤ i ≤ m. Note that the coefficient that multiplies uj(t) in the latter

equation is exactly equal to the (i, j) entry of the matrix A(x) in (6.4).

Set now:

ξi =

ξi1

ξi2

· · ·ξiri

=

φi1(x)

φi2(x)

· · ·φiri(x)

for 1 ≤ i ≤ m (6.10)

ξ = [ξ1, . . . , ξm]T (6.11)

Then, the equations in question can be rewritten as:

ξi1 = ξi2

· · ·ξiri−1 = ξiri

ξiri = bi(ξ) +m∑

j=1

aij(ξ)uj(t)

yi = ξi1

(6.12)

for 1 ≤ i ≤ m and with:

aij(ξ) = LgjLri−1

f hi(Φ−1(ξ)) for 1 ≤ i, j ≤ m (6.13)

bi(ξ) = Lrif hi(Φ−1(ξ)) for 1 ≤ i ≤ m (6.14)

The structure of equations (6.12) characterizes the normal form of the equa-

tions (see fig. 6.1) describing (locally around a point xo) a nonlinear system,

with m inputs and m outputs, having a (vector) relative degree r1, . . . , rmat xo, with r = r1 + . . . + rm exactly equal to the dimension n of the state


space. In this case, when r = n, the system has no internal (zero) dynamics,

and so it is minimum phase by default. Note that in (6.12) the coefficients

· · ·

∫ ∫- - - -

666

-ξi

riξi

riξi2 ξi

1 = yi

· · ·bi(ξ) +∑m

j=1aij(ξ)uj

uj

Figure 6.1: Normal form representation, with no internal dynamics.

aij(ξ) are exactly the entries of matrix (6.4), with x replaced by Φ−1(ξ), and

the coefficients bi(ξ) are the entries of a vector:

b(x) =

Lr1f h1(x)

Lr2f h2(x)

· · ·Lrmf hm(x)

(6.15)

again with x replaced by Φ−1(ξ).

6.1.2 Exact linearization via feedback

The main problem dealt with in this section is that of using feedback and

coordinates transformation to the purpose of changing a nonlinear system into

a linear and controllable one. Formally, the problem in question can be stated

the following way:

State-space exact linearization problem Given a set of vector fields f(x)

and g1(x), . . . ,gm(x) and an initial state xo, find (if possible), a neighborhood

U of xo, a pair of feedback functions α(x) and β(x) defined on U , a coordinates

transformation z = Φ(x) also defined in U , a matrix A ∈ Rn×n and a matrix

B ∈ Rn×m, such that:

[∂Φ

∂x(f(x) + g(x)α(x))

]

x=Φ−1(z)

= Az (6.16)

[∂Φ

∂x(g(x)β(x))

]

x=Φ−1(z)

= B (6.17)


and:

rank(B AB · · · An−1

B) = n (6.18)

Our point of departure will be the normal form developed in the previous

section. Recall that in a neighborhood of the point ξo = Φ−1(xo) the matrix

A(ξ) is nonsingular and therefore the equations:

υ = b(ξ) + A(ξ)u (6.19)

with υ = [υ1, υ2, · · · , υm]T the new reference input, can be solved for u. The

input u solving these equations has the form of a state feedback:

u = A−1(ξ)(−b(ξ) + υ) (6.20)

Imposing this feedback yields a closed-loop system characterized by the m sets

of equations:

ξi1 = ξi2

· · ·ξiri−1 = ξiri

ξiri = υi

(6.21)

for 1 ≤ i ≤ m, which is clearly linear and controllable.

In terms of the original description of the system, the linearizing feedback has

the form:

u = α(x) + β(x)υ (6.22)

with α(x) and β(x) given by:

α(x) = −A−1(x)b(x) (6.23)

β(x) = A−1(x) (6.24)

and A(x) and b(x) as in (6.4) and (6.15). The linearizing coordinates are

defined as:

ξik(x) = Lk−1f hi(x) for 1 ≤ k ≤ ri, 1 ≤ i ≤ m (6.25)


The closed-loop system obtained applying the control law (6.22) to the sys-

tem (6.2) is (see fig. 6.2):

x = f(x) + g(x)α(x) + g(x)β(x)υ (6.26a)

y = h(x) (6.26b)

y = h(x)- -

6

uυα(x) + β(x)υ

y

x

x = f(x) + g(x)u-

Figure 6.2: Closed-loop system, with new reference input υ.

The conditions that the system, for some choice of output functions h1(x), . . . ,

hm(x), has a (vector) relative degree r1, . . . , rm at xo, and that r1+. . .+rm =

n, imply the existence of a coordinates transformation and a state feedback,

defined locally around xo, which solve the state space exact linearization prob-

lem. The following Lemma shows that these conditions are also necessary.

Lemma 6.1. Suppose the matrix g(xo) has rank m. Then, the state space

exact linearization problem is solvable if and only if there exists a neighborhood

U of xo and m real-valued functions h1(x), . . . , hm(x) defined on U , such that

the system (6.2) has some (vector) relative degree r1, . . . , rm at xo and r1 +

. . .+ rm = n.


6.1.3 Asymptotic output tracking

In this section we consider the problem of tracking the output of a reference

model, which in turn is subject to some input (t). Consider the linear model,

described by:

ζ = Aζ + B (6.27a)

yr = Cζ (6.27b)

The asymptotic model matching problem is solved by finding a feedback control

which causes, irrespectively of what the initial states of the system and of the


model are and for every input (t) to the model, an output y(t) asymptotically

converging to the corresponding output yr(t) produced by the model under the

effect of (t).

Suppose the model has a (vector) relative degree equal to the (vector) relative

degree r1, . . . , rm of the system and r1 = . . . = rm = r. In this case, since:

CB = CAB = . . . = CAr−2

B = 0 (6.28)

we have that:

y(i)r (t) = CA

iζ(t) for all 0 ≤ i ≤ r − 1 (6.29a)

y(r)r (t) = CA

rζ(t) + CAr−1

B(t) (6.29b)

Consider again the normal form (6.12) and choose the control input as:

u = A−1(x)(−b(x) + y(r)

r − q)

(6.30)

with A(x) and b(x) as in (6.4) and (6.15) and q a column vector with elements

qj =∑r

i=1 ci−1(ξji − y

(i−1)rj ) for 1 ≤ j ≤ m where c0, . . . , cr−1 are real numbers.

Define an error e(t) as the difference between the real output y(t) and the

model output yr(t):

e(t) = y(t) − yr(t) (6.31)

Since, by construction, ξji = y(i−1)j = Li−1

f hj(x) for 1 ≤ i ≤ r, we may write

qj =∑r

i=1 ci−1e(i−1)j . Substituting (6.29) into (6.30) leads to:

u = A−1

−b + CA

rζ + CAr−1

B −

∑ri=1 ci−1(L

i−1f h1 − C1A

i−1ζ)

· · ·∑r

i=1 ci−1(Li−1f hm − CmA

i−1ζ)

(6.32)

Note that imposing the input (6.32) implies:

ξjr = y(r)j = y(r)

rj− cr−1e

(r−1)j − . . .− c1ej − c0ej (6.33)

i.e.:

e(r)j + cr−1e

(r−1)j + . . .+ c1ej + c0ej = 0 (6.34)

6.2. NEW FORMULATION FOR CASCADED SYSTEMS 109

for 1 ≤ j ≤ m. The error functions ej(t) satisfy a linear differential equation

of order r whose coefficients can be arbitrarily preset.

Thus, by construction, the system (6.2), subject to an input of the form (6.32)

with coefficients c0, . . . , cr−1 appropriately chosen, will produce an output

asymptotically converging to the output yr(t) of the model.

Note that the input (6.32) depends explicitly on the state x(t) of the system,

on the input (t) of the model, and on the state ζ(t) of the model, which in

turn obeys the differential equation (6.27) (see fig. 6.3). This represents a more

y = h(x)- -

?

-

-

ζ

x

ζ = Aζ + B

α(ζ,x) + β(ζ,x)u yx = f(x) + g(x)u

6

Figure 6.3: Closed-loop system, with model reference input .

general form of state feedback, in that includes also an internal dynamics. A

feedback of this form is called a dynamic state feedback.

6.2 New formulation for cascaded systems

Section 6.1 presents the general theory of the dynamic inversion approach for

MIMO systems with the same number m of inputs and outputs, and for which

the sum of the relative degrees ri, for 1 ≤ i ≤ m, equals the dimension n of

the state space. A new formulation for dynamic systems which respect these

assumptions will now be presented here.

Consider an affine in control system whose description is given by the following

dynamics, kinematics and output equations:

V = fv(V,P) + gv(V,P)u (6.35a)

P = fp(V,P) (6.35b)

y = h(V,P) = P (6.35c)


where the n-state vector x = [VT ,PT ]T contains the velocity V and position

P components and the m-vectors u and y are the input and output vectors,

respectively; fv(V,P) and fp(V,P) are smooth m-vector fields and gv(V,P)

is an m ×m matrix with rank m for x = xo and whose columns are smooth

vector fields gvi(V,P).

This new dynamic inversion formulation simplifies the dynamic inversion im-

plementation for systems with the cascaded form (6.35a)-(6.35b), and for which

the output variable of interest is the position vector, usual in path-tracking

problems.

Let us differentiate the output (6.35c) in order to have the input u explicitly

appearing. We then have:

y = P = fp(V,P) (6.36)

y = P =∂fp∂V

fv +∂fp∂P

fp +∂fp∂V

gvu

= ∇fp(V)fv + ∇fp(P)fp + ∇fp(V)gvu (6.37)

It is easy to verify that every output i of this system has relative degree ri = 2

at a point x = xo where the m×m matrix ∇fp(V)gv is nonsingular, and that∑m

i=1 ri = n. Therefore, according to the theory reviewed in Section 6.1, there

is a local coordinates transformation:

Φ(x) = [PT1 , P

T1 , . . . ,P

Tm, P

Tm]T (6.38)

such that we may represent (locally around xo) the nonlinear system (6.35) in

the normal form:

P = fp(V,P) (6.39)

P = ∇fp(V)fv + ∇fp(P)fp + ∇fp(V)gvu (6.40)

y = P (6.41)

Moreover, applying the state feedback:

u = (∇fp(V)gv)−1 (ν −∇fp(V)fv −∇fp(P)fp) (6.42)

where ν = [ν1, . . . , νm]T is a new reference input, we obtain the linear and

6.3. APPLICATION TO AIRSHIP PATH-TRACKING PROBLEM 111

controllable closed-loop system:

P = fp(V,P) (6.43)

P = ν (6.44)

In order to guarantee an asymptotic output tracking, the reference input ν

may have the form:

ν = Pr − C1(P − Pr) − C0(P − Pr) (6.45)

where Pr may be either a time position reference Pr(t) or a dynamic model

position output Pm(t). The diagonal matrices C0 and C1 define the roots of

the characteristic equation:

e + C1e + C0e = 0 (6.46)

where e = P − Pr is the position error.

In the next section we apply this dynamic inversion formulation to the AU-

RORA airship path-tracking problem.

6.3 Application to the airship path-tracking

problem

This section applies the dynamic inversion approach to the airship path-tracking

problem, using the formulation described in the previous section.

Consider the deterministic no-wind case where the inertial and air velocities

are equal (V = Va). Moreover, consider the position is given, not relative to

the inertial frame, but to the reference trajectory. In this scenario, we may

represent the airship equations of motion as:

Va = −M−1

a (Ω6MaVa + Va6(Ma − MBa)Va − EgSag − Fa) + M−1

a uf

(6.47a)

˙P = JΦVa − Vr (6.47b)

where P = [ΓT , ΦT ]T corresponds to the position error relative to the reference

trajectory. The transformation matrix between the local and the reference


trajectory frames is given by JΦ = J(Φ), with Φ = Φ − Φr the attitude

between frames. Vr corresponds to the reference groundspeed given in the

reference trajectory frame, as is assumed constant.

We may represent (6.47a)-(6.47b) in the general form (6.35a)-(6.35b) as 2:

Va = fv(Va, P) + gvuf (6.48a)

˙P = fp(Va, P) (6.48b)

with:

fv(Va, P) = −M−1

a (Ω6MaVa + Va6(Ma − MBa)Va − EgSag − Fa) (6.49)

fp(Va, P) = JΦVa − Vr (6.50)

gv = M−1

a (6.51)

Take the position relative to the reference trajectory as the output variables

of interest:

y = P (6.52)

According to (6.42), applying the force control input:

uf = (∇fp(Va)gv)−1

(

ν −∇fp(Va)fv −∇fp(P)fp

)

(6.53)

with:

∇fp(Va) =

cψcθ cψsθsφ − sψcφ cψsθcφ + sψsφ 0 0 0

sψcθ sψsθsφ + cψcφ sψsθcφ − cψsφ 0 0 0

−sθ cθsφ cθcφ 0 0 0

0 0 0 1sθsφ

cθ

sθcφcθ

0 0 0 0 cφ −sφ0 0 0 0

sφ

cθ

cφcθ

(6.54)

2We may write the transformation matrix in the gravity force as S = S(Φ) = S(Φ+Φr)and assume Φr to be constant.

6.3. APPLICATION TO AIRSHIP PATH-TRACKING PROBLEM 113

∇fp(P) =

0 0 0 (cψsθcφ + sψsφ) v + (−cψsθsφ + sψcφ)w

0 0 0 (sψsθcφ − cψsφ) v + (−sψsθsφ − cψcφ)w

0 0 0 cθcφv − cθsφw

0 0 0sθcφq

cθ− sθsφr

cθ

0 0 0 −sφq − cφr

0 0 0cφq

cθ− sφr

cθ

· · · (6.55)

−cψsθu+ cψcθsφv + cψcθcφw −sψcθu+ (−sψsθsφ − cψcφ) v + (−sψsθcφ + cψsφ)w

−sψsθu+ sψcθsφv + sψcθcφw cψcθu+ (cψsθsφ − sψcφ) v + (cψsθcφ + sψsφ)w

−cθu− sθsφv − sθcφw 0

sφq +(sθ)2sφq

(cθ)2+ cφr +

(sθ)2cφr

(cθ)20

0 0

sφqsθ

(cθ)2+

cφrsθ

(cθ)20

using the condensed notation cos(θ) = cθ and sin(θ) = sθ, will lead to an

asymptotic output tracking if the new reference input ν has the form (6.45).

In the airship case we verified a time reference excites unmodelled dynamics.

To solve this problem, and in order to provide some robustness to the dynamic

inversion solution, we consider tracking a model dynamics, i.e., Pr ≡ Pm.

The obvious choice is the linear 12-states system with the LQR state feedback

control described in Chapter 5, which, besides robust, takes into account the

real limitations of the airship system. The model dynamics is described by:

˙xm = (A − BK)xm = Acxm (6.56)

ym = Cxm = Pm (6.57)

with the position as output. We then have:

ym = CAcxm (6.58)

ym = CA2c xm (6.59)

and the new input ν in (6.53) is:

ν = CA2c xm − C1(JΦVa − CAcxm) − C0(P − Cxm) (6.60)

The control input (6.53) exists if ∇fp(V)gv is nonsingular, i.e, as long as

φ − θ 6= ±π/2, a reasonable assumption in the case of the stable airship


platform.

The control input, however, is a force input, which cannot be directly fed to

the airship. The control law (6.53) considers an input uf that includes both

forces and torques. However, the real inputs of an airship are its actuators.

For this reason, a conversion from forces to actuators inputs is necessary for

the proper implementation of the attained controller.

The actuators input u = [δe, TL, TR, δv, δa, δr]T is obtained solving the equa-

tions system (2.71). As referred in Section 2.2.1.1, although we have six ac-

tuators inputs to control the six forces, several limitations lead to the under-

actuation of the airship. Moreover, the system of equations (2.71) is not di-

rectly invertible, which implies an empiric allocation in some situations.


This last section demonstrates the performance of the dynamic inversion ap-

proach applied to the airship path-tracking problem. We present the results

obtained for the case-study mission and to the sensitivity and robustness to

parameter uncertainty test.


The first results we present concern the case-study mission described in Sec-

tion 3.3.1. The mission covers a wide range of the flight envelope, with the

airspeed varying from 3 to 14m/s.



path-tracking performance. Figure 6.4(b) displays the longitudinal η, lateral

ε and vertical δ errors. The end of the first half-circle, when the wind is aft

the airship, and the transition between the ascent and the horizontal tracking,

remain problematic mission points, with the last one showing higher position

errors than in the gain scheduling case. But other regions also rise difficulties:

the beginning of the second curve and the stabilization prior to the descent.

In those occasions we notice an increase of the position errors. In order to

avoid saturation of the propellers, probable when the controller tries to rapidly

correct the longitudinal position, the error η is limited to 2m, as was in the


gain scheduling approach.

E (m)N (m)

h(m

)

-1000

100200

300

-200

-100

0

100

200

0

10

20

30

40

50

60


Time (s)

η(m

)

Time (s)

ε(m

)

Time (s)

δ(m

)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-10

-5

0

5

10

-30

-15

0

15

30

-80

-60

-40

-20

0

20



Comparing the airship trajectory executed with the dynamic inversion con-

troller (fig. 6.4(a)) with the one controlled by the gain scheduling approach

(fig. 5.15(a)), we observe that both control laws lead to the accomplishment of

the mission within acceptable deviations from the trajectory reference. How-

ever, the dynamic inversion resulting trajectory is more erroneous.




take-off and landing is again well recognized. Just as in the gain scheduling

case, the airship appears as slightly crabbing during the maneuvers at low

airspeeds. The Euler angles evolution is displayed in fig. 6.5(b). The roll angle

(φ) presents higher oscillations in the transition from vertical to horizontal

tracking and when the wind is aft, same as in the gain scheduling case but

with higher amplitude. The pitch (θ) and yaw (ψ) angles try to follow the

respective references described in Section 3.2.


fig. 6.6 and are somewhat similar to the ones obtained in the gain scheduling

case (see fig. 5.17). The ground velocity components are described in fig.

6.6(a). The longitudinal groundspeed u mostly follows the reference that varies

between 0m/s for stabilization, take-off and landing, and 7m/s during the


E (m)

N(m

)windheading

-100 0 100 200 300-250

-200

-150

-100

-50

0

50

100

150

200

250


Time (s)

φ(deg

)

Time (s)

θ(deg

)

Time (s)

ψ(deg

)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0

90

180

270

360

-20

-10

0

10

20

30

-30

-15

0

15

30



path-tracking. Along the circular segments, the errors are more noticeable due

to the change of the wind incidence angle while the airship is turning. The

error is higher when the wind is aft the airship. The lateral velocity v is also

mostly influenced by the circular segments and during the tail wind segment.

The vertical velocity w follows its reference, with the −1 and 0.5m/s steps

corresponding to the take-off and landing vertical motion. The airspeed and

Time (s)

u(m/s)

Time (s)

v(m/s)

Time (s)

w(m/s)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-3

-2

-1

0

1

2

-6

-4

-2

0

2

4

0

24

68

10


Vt

(m/s)

Time (s)

β(deg

)

Time (s)

α(deg

)

Time (s)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-30

-20

-10

0

10

20

-20

-10

0

10

20

2468

101214



aerodynamic angles can be seen in fig. 6.6(b). During the whole mission, the

airspeed Vt varies significantly, from values around 3m/s up to 14m/s. The


airship covers a wide flight envelope, from hover to the aerodynamic flight,

crossing the troublesome transition region between the two. Here, the behavior

of α shows also correlation with w.


elevator δe, total thrust XT and vectoring angle δv in fig. 6.7(a) and the lateral

input, aileron δa, rudder δr and differential thrust TD, in fig. 6.7(b). The

elevator δe, while responsible for the altitude and pitch control, shows a more

constant demand during the vertical displacements. The rudder δr, mainly

responsible for the lateral position and airship yaw, has a higher command

with tail wind, due to the reduced authority at lower airspeeds. The airship

roll φ is controlled by the aileron δa at higher airspeeds, and by the differential

thrust TD when the control surface loses authority (which usually corresponds

to a vectoring angle close to 90o, allowing TD to effectively control the roll and

not the yaw). This actuator has a negligible action in aerodynamic flight (the

control surfaces authority is sufficient for the rudder δr to control the airship

yaw ψ), an option taken when converting the forces input to an actuators

request.

Time (s)

δ e(deg

)

Time (s)

XT

(N)

Time (s)

δ v(deg

)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-300

306090

120

0

20

40

60

80

-30

-15

0

15

30


Time (s)

δ a(deg

)

Time (s)

δ r(deg

)

Time (s)

TD

(N)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-10

0

10

-30

-15

0

15

30

-30

-15

0

15

30



The vectoring angle δv is responsible for the airship lift when the airspeed Vt

is too low to provide the necessary aerodynamic lift. The correlation between

these two variables, although not as obvious as in the gain scheduling case,

is visible when comparing the graphics of δv and Vt. At higher airspeeds, the

dynamic inversion control law chooses to use this actuator to help the elevator


on the altitude and pitch control, unlike the gain scheduling controller.

As a remark, note that the engines inputs are coupled, as are the tail surfaces

due to the ’×’-shape. Note, for instance, that when the rudder δr is saturated,

the other two control surfaces inputs, elevator δe and aileron δa, are zero.

In the overall, the airship under dynamic inversion control executed the mission

satisfactorily, although with a more erroneous behavior than under the gain

scheduling control.


tainty

A fundamental assumption in the dynamic inversion methodology is that the

plant dynamics can be perfectly modeled and may be canceled exactly. In

practice, this assumption is obviously not realistic, and the robustness of the

closed-loop dynamics must be secured, in order to suppress any undesired

behavior due to plant uncertainties and wind disturbances.

This section evaluates the stability and performance robustness of the dynamic

inversion control methodology when solving the path-tracking control problem

of the AURORA airship. Due to the nonlinearity of both system and dynamic

inversion control law, we cannot make use of the analysis tools used in the gain

scheduling robustness analysis. Therefore, we limit our analysis of the dynamic

inversion closed-loop system robustness to the test described in Section 3.3.2.








Figure 6.8(b) represents the airship attitude references (see Section 3.2) and

output, as well as the wind estimated attitude. Notice that neither θw nor ψw

are constant, since they echo both constant and gust wind components. The

excitation of the signals is due to the wind turbulence.




E (m)

N(m

) headingwind

-100 -50 0 50

0

200

400

600

800

1000

1200


φ(deg

)θ

(deg

)ψ

(deg

)θ w

(deg

)ψw

(deg

)

Time (s)

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

80

90

100

-10

-5

0

5

-35

-30

-25

-20

-5

0

5

10

-10

0

10

(b) Airship attitude : roll φ, pitch θ andyaw ψ (−− reference, − output), and es-timated wind attitude, θw, ψw.



airspeed corresponds to the relative airspeed between the 8m/s groundspeed




with the longitudinal actuators action given in fig. 6.10(a), while the lateral

actuation is in fig. 6.10(b). The vectoring angle δv, which is expected to have

a negligible action in aerodynamic flight, is clearly helping the elevator to

cancel the altitude error δ. The differential thrust TD, represented here for

completeness, has a negligible control action in aerodynamic flight, an option

taken when converting from forces to actuators.






Time (s)

η(m

)

Time (s)

ε(m

)

Time (s)

δ(m

)

0 50 100 150

0 50 100 150

0 50 100 150

-3-2-10123

-2

-1

0

1

2

-1

0

1


Vt

(m/s)

time (s)

β(deg

)

time (s)

α(deg

)

time (s)

0 50 100 150

0 50 100 150

0 50 100 150

-5

0

5

10

-5

0

5

8

9

10



Time (s)

δ e(deg

)

Time (s)

XT

(N)

Time (s)

δ v(deg

)

0 50 100 150

0 50 100 150

0 50 100 150

-30

0

30

60

90

120

0

20

40

60

80

-10

0

10

20


Time (s)

δ a(deg

)

Time (s)

δ r(deg

)

Time (s)

TD

(N)

0 50 100 150

0 50 100 150

0 50 100 150

-1

0

1

-20

-10

0

10

20

-5

0

5



the groundspeed error eu relative to the 8m/s reference. The first row has the

RMS values obtained for the baseline case, and is to serve as reference for the

remaining lines where each of the listed coefficients is varied one at a time.

For a parameter uncertainty of ±70%, the controlled airship still performed

qualitatively like the baseline case, except for a few cases. For the coefficients

CMδeand CMq

, a +30 and −30% uncertainty respectively already influenced

the controller performance, while for CLαthis influenced appeared for −50%.

The dynamic inversion controller appeared to be affected only by the increase

6.5. CONCLUSIONS 121

of parameter uncertainty of these three longitudinal coefficients, respectively

the authority of the elevator as pitching control input, the pitch damping

derivative and the lift coefficient derivative due to angle of attack.

To see how the dynamic inversion controller behaved for higher levels of un-

certainty, we then increased the parameter uncertainty to ±90% around the

nominal value. We present the results obtained in table 6.1, with the cases

which lead to significant deviations from the reference RMS values represented

in bold.

For the previous coefficients, CMδe(±90%), CMq

(−90%) and CLα(−90%),

the uncertainty still leads to an inefficient control action, either because it is

insufficient due to actuators saturation (the dynamic inversion demands a too

high control input) and/or just because the tested coefficient appears to be

a more sensitive model parameter. Besides these three parameters, we now

notice the influence of the uncertainty in lateral coefficients, CNδrand CNr

(both for −90%), respectively the authority of the rudder as yawing control

input, and the yaw damping derivative.

While the dynamic inversion demonstrates to be robust to a ±90% uncertainty

in the remaining parameters, for these five parameters, the mismatch between

the airship system and the model considered in the dynamic inversion controller

design is too significant for the control action to overcome it.

In any case, the dynamic inversion controller may be considered robust to

wind disturbances and plant uncertainties. Among the list selected, these

five parameters are in fact the model parameters for which a more careful

identification or determination should take place.

6.5 Conclusions

This chapter describes the general theory of the dynamic inversion approach.

Bearing in mind systems like the airship whose dynamic equations are given

by a cascaded description, we formulate a more straightforward procedure to

obtain the dynamic inversion control law for this type of systems.

The next step is obviously its application to the AURORA airship path-

tracking problem, where the controlled system shows a satisfactory perfor-

mance in the execution of realistic missions.

The dynamic inversion approach is based on the cancelation of the system




Baseline 0.60 0.45 0.78 9.01 3.26 1.21 0.14

Clβ−90% 0.59 0.41 0.78 9.01 3.31 0.95 0.12+90% 0.62 0.51 0.74 9.01 3.03 1.80 0.19

CM0

−90% 0.58 0.45 0.79 9.01 3.32 1.22 0.14+90% 0.61 0.45 0.76 9.01 3.20 1.21 0.14

CMα

−90% 0.48 0.44 1.31 9.01 3.96 1.24 0.14+90% 0.65 0.46 0.73 9.01 2.80 1.20 0.14

CMαβ

−90% 0.60 0.45 0.77 9.01 3.25 1.21 0.14+90% 0.60 0.45 0.78 9.01 3.28 1.21 0.14

CMβ

−90% 0.62 0.45 0.76 9.01 3.19 1.21 0.14+90% 0.58 0.45 0.80 9.01 3.34 1.22 0.14

CMβα

−90% 0.60 0.45 0.77 9.01 3.25 1.21 0.14+90% 0.60 0.45 0.78 9.01 3.27 1.21 0.14

CMδe

−90% 19.40 2.66 6.79 8.81 2.83 1.47 0.44

+90% 62.17 1.84 2.05 9.91 4.11 1.47 1.44

CNβ

−90% 0.59 0.43 0.76 9.01 3.28 1.33 0.12+90% 0.60 0.45 0.78 9.01 3.25 1.13 0.15

CNδr

−90% 19.98 16.05 6.61 9.44 4.47 5.77 1.98

+90% 0.49 0.34 0.79 9.01 3.31 1.35 0.19

CYβ

−90% 0.55 0.72 0.80 9.02 3.30 2.93 0.17+90% 0.61 0.41 0.78 9.01 3.27 0.88 0.13

CYδr

−90% 0.58 0.44 0.77 9.01 3.33 1.25 0.12+90% 0.61 0.45 0.77 9.01 3.19 1.50 0.16

CD0

−90% 0.53 0.49 1.02 9.01 4.13 1.26 0.18+90% 1.07 0.46 0.98 9.01 2.60 1.17 0.14

CDi

−90% 0.55 0.45 0.81 9.01 3.36 1.21 0.15+90% 0.64 0.45 0.75 9.01 3.18 1.21 0.13

CL0

−90% 0.60 0.45 0.77 9.01 3.18 1.21 0.14+90% 0.59 0.45 0.78 9.01 3.34 1.22 0.14

CLα

−90% 140.23 42.03 87.44 9.38 27.25 8.11 3.68

+90% 0.69 0.46 0.60 9.01 1.87 1.17 0.13

CLδe

−90% 0.56 0.43 0.72 9.01 3.82 1.24 0.13+90% 0.63 0.46 1.04 9.01 3.10 1.20 0.15

Clp−90% 0.61 0.45 0.79 9.01 3.18 1.40 0.15+90% 0.60 0.46 0.77 9.01 3.28 1.18 0.14

CMq

−90% 265.78 3.12 1.52 12.79 3.41 1.33 4.65

+90% 0.78 0.52 2.09 9.04 4.72 1.21 0.24

CNr

−90% 597.96 189.82 6.01 4.19 36.24 73.23 4.81

+90% 0.64 0.77 0.80 9.01 3.23 1.40 0.16

mw−90% 0.67 0.48 1.29 9.02 1.52 1.52 0.21+90% 0.33 0.58 1.45 9.02 3.92 1.37 0.16


nonlinearities by inverting the dynamic model, resulting in a stable and linear

closed-loop system. However, this procedure assumes a complete knowledge of

the system. Therefore, the analysis of the controlled system performance and

stability robustness in the presence of wind disturbances and model parameter

uncertainties is very important. The dynamic inversion controller, robust to

wind disturbances, shows to be tolerant to uncertainties in most of the model

parameters tested. However, for some aerodynamic coefficients, namely CMδe,

CMq, CLα

, CNδrand CNr

, a more careful identification or determination should

take place.

The dynamic inversion controller requires a time reference or a model dynamics

is given as reference. In the airship case we realized a time reference excites

unmodeled dynamics. In fact, the dynamic inversion of the airship nonlinear

model results in a forces input, which is assumed by the controller to be fully

available. This is however not the case. The airship has serious actuation

constraints regarding the lateral force and, although not so severe, with the

downward force as well. An obvious choice for reference is then the closed-loop

gain scheduling system, which we have seen to be robustly stable within certain

limits, and that intrinsically provides information on the actuation constraints.

However, giving this model as reference, can we expect the dynamic inversion

to outperform it? If it is not the case, what is then the advantage of the

dynamic inversion solution over the gain scheduling one?

With the purpose of avoiding this type of problems, an in the absence at the

time of a better reference for the dynamic inversion controller, we considered

a different nature of nonlinear control, which we will describe in the sequence.

Chapter 7

Backstepping

Contents

7.1 Wind estimator . . . . . . . . . . . . . . . . . . . . 126

7.2 Backstepping design approach . . . . . . . . . . . . 128

7.3 Application to the path-tracking problem . . . . . 128

7.4 Control design with saturation constraints . . . . 132

7.5 Control implementation . . . . . . . . . . . . . . . 136

7.5.1 Adapted control law to deal with underactuation . . 136




7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 146

Although Lyapunov’s direct method is originally a method of stability anal-

ysis, an important application is the design of nonlinear controllers. Back-

stepping [60] is a recursive procedure that interlaces the choice of a Lyapunov

function with the design of feedback control. In the backstepping approach,

by formulating a scalar positive function of the system states and then consid-

ering a control law that makes this function decrease, we have the guarantee

that the nonlinear control system thus designed will be asymptotically stable,

and still robust to some unmatched uncertainties.

Several successful applications of the backstepping approach for UAV control

have been reported [73, 15, 74, 75, 76].

Backstepping as also been used in the specific case of airships. A backstep-

ping technique has been proposed by the LAAS/CNRS autonomous blimp

project [39, 40]. The global control strategy studied is obtained by switching

125

126 CHAPTER 7. BACKSTEPPING

between four sub-controllers, one for each of the flight phases considered. Each

controller is however still based on linearized models of the airship, what leads

to the separate control of the longitudinal and lateral motions. An image-

based control solution for airship tracking is based by Fukao et al. [43] on

backstepping techniques for underactuated vehicles. The airship model con-

sidered is built from the kinematics between the camera used as only sensor

and the target. Robustness issues like wind disturbance rejection are still to

be improved.

In this chapter we propose a backstepping based control solution to the airship

path-tracking problem. Based on the six-degrees-of-freedom nonlinear model

of the airship, it is valid for missions over the entire flight envelope.

Before going into the backstepping airship path-tracking control design, we

first describe in Section 7.1 a wind estimator. The interest of using this esti-

mator instead of the wind estimation method described in Section 3.2 lies, not

in a better estimation, but in the fact that it provides useful bounds, as we

will later see. We then present a general backstepping control approach in Sec-

tion 7.2, applying it to the path-tracking problem in Section 7.3. Saturation

limits are included in the control design in Section 7.4 and important imple-

mentation issues are discussed in Section 7.5.1. The controller performance is

evaluated in Section 7.6, with simulation results concerning the case-study mis-

sion and the sensitivity and robustness to parameter uncertainty test. Finally,

the conclusions are drawn in Section 7.7.

7.1 Wind estimator

The dynamics equations (2.66) or (2.69) and the kinematics equation (2.80),

expressed in the air frame, assume a constant translation wind. However, the

wind disturbance is unknown, being necessary to build an estimator based

on (2.48)-(2.49) and (2.80). Since the wind input is not affecting the angular

position part in (2.80), only the cartesian position p of the airship should be

considered:

p = STva + pw (7.1)

7.1. WIND ESTIMATOR 127

The estimator states may then be (p, ˆpw), and its dynamics may be chosen as:

[ˆp

ˆpw

]

=

[

STva

0

]

+

[

Lp I3

Lpw03

][

p − p

ˆpw

]

(7.2)

leading to an estimation error vector ǫ dynamics obtained from (7.1)-(7.2) and

given by:

ǫ =

[

−Lp I3

−Lpw03

][

p − p

pw − ˆpw

]

= Aǫǫ (7.3)

where the two constant matrices (Lp,Lpw) are chosen so that Aǫ be Hurwitz.1

Therefore, the origin of (7.3) is asymptotically stable and there exists a Lya-

punov function:

We = ǫTPǫǫ (7.4)

with a symmetric positive definite matrix Pǫ > 0 whose time derivative is

given by:

We = ǫTPǫǫ + ǫTPǫǫ = ǫT (PǫAǫ + ATǫ Pǫ)ǫ = −ǫTQǫǫ (7.5)

Choosing the symmetric positive definite matrix Qǫ with a block diagonal

form, the Pǫ matrix will then be the solution of the Riccati equation:

PǫAǫ + ATǫ Pǫ = −Qǫ = −

[

Qp 03

03 Qpw

]

(7.6)

with Qp and Qpwdiagonal matrices.

Defining the estimation errors as:

p = p − p (7.7)

˜pw = pw − ˆpw (7.8)

the estimator Lyapunov function derivative (7.5) may be rewritten as:

We = −pTQpp − ˜pTwQpw˜p (7.9)

1For instance, taking Lp = aI3 and Lpw= a2

bI3 leads to three pairs of poles at −a

2(1±σi)

with σ =√

4−b2

b, i.e., with a damping factor ξ = 1

2

√b.


7.2 Backstepping design approach

Let us consider a generic control problem with output y. We first define two

auxiliary outputs involving the output y and its derivative y:

y1 = ay + y

y2 = y⇒

y1 = ay + y

y2 = y(7.10)

where a is a positive scalar to be used as design parameter. It is easily seen

that when both auxiliary outputs are taken to the origin, the regulation of the

main output y is then achieved.

A candidate Lyapunov function may be:

W0 =1

2yT1 y1 +

1

2yT2 y2 (7.11)

Its derivative is:

W0 = yT1 y1+yT2 y2 = (ay + y)T (ay + y)+yT y = (ay + 2y)T (ay + y)−ayT y(7.12)

If the control is chosen in order to give:

ay + y = −Λ (ay + 2y) (7.13)

where Λ = ΛT is a positive definite matrix, then the derivative:

W0 = − (ay + 2y)T Λ (ay + 2y) − ayT y (7.14)

will clearly be negative definite and the system will be globally asymptotically

stable.

7.3 Application to the path-tracking problem

We shall now proceed applying the control design described in the previous

section to the path-tracking problem.

Let us assume a point pr with a constant ground velocity vr is to be tracked

with constant attitude along a rectilinear path AB (see figure 7.1):

pr = STr vr (7.15)

7.3. APPLICATION TO THE PATH-TRACKING PROBLEM 129

where Sr = S(Φr) given by (2.17) is the constant transformation matrix from

the inertial frame to the reference path.

B

A

Sar

vr

var

p

v

r

w

Figure 7.1: Air velocity reference estimation (2D).

As the wind velocity vw is considered, the desired air velocity varmay be

deduced. Moreover, since the airship is being aligned with this air velocity

we have a reference for the attitude given by the transformation Sarfrom the

inertial frame to the air velocity var, which is described by the desired attitude

vector Φar. This leads to the reference position:

Pr =

[

pr

Φar

]

(7.16)

The derivative of this reference position is:

Pr =

[

STr vr

0

]

= JΦrVr (7.17)

where the reference velocity state is Vr =

[

vr

0

]

and JΦr=

[

STr 03

03 Rr

]

,

with Rr = R(Φr) given by (2.18).

Note that, although we have assumed a rectilinear reference path, the approach

may also be extended to the cases where the reference path varies slowly, with

negligible derivatives when compared to the state derivative.

Let us now consider a Lyapunov function candidate similar to (7.11):

Wt =1

2yT1 y1 +

1

2yT2 y2 (7.18)

where the output auxiliary variables y1 and y2 are again derived from the


output y and its derivative y, but where y = P − Pr is the position tracking

error:

y1 = ay + y = a(P − Pr) + P − Pr

y2 = y = P − Pr

(7.19)

Using equations (2.80) and (7.17) we have:

y1 = a(P − Pr) + JΦVa + BI pw − JΦrVr

y2 = JΦVa + BIpw − JΦrVr

(7.20)

The derivative of the Lyapunov function candidate (7.18) is:

Wt = yT1 y1 + yT2 y2

= (ay + y)T (ay + y) + yT y

= (ay + 2y)T (ay + y) − ayT y (7.21)

If the control is chosen as:

ay + y = −Λ(ay + 2y) (7.22)

or:

a(JΦVa + BIpw − JΦrVr) + JΦCJVa + JΦVa =

− Λ(a(P − Pr) + 2(JΦVa + BIpw − JΦrVr)) (7.23)

where we used:

y = JΦVa + JΦCJVa (7.24)

this leads to the control law:

JΦVa = −aΛ(P−Pr)−JΦCJVa−(aI6 +2Λ)(JΦVa+BI pw−JΦrVr) (7.25)

As wind is estimated, the suggested control law is:

JΦVa = −aΛ(P−Pr)−JΦCJVa−(aI6 +2Λ)(JΦVa+BIˆpw−JΦr

Vr) (7.26)

and (7.22) should be rewritten as:

ay + y = −Λ(ay + 2y) + (aI6 + 2Λ)BI˜pw (7.27)

7.3. APPLICATION TO THE PATH-TRACKING PROBLEM 131

Introducing y0 = ay + 2y, and defining G = (a2Λ−1 + I6)BI , the tentative

Lyapunov derivative appears as:

Wt = −ΛyT0 (y0 − 2G˜pw) − ayT2 y2 (7.28)

or, completing the squares2:

Wt = −Λ(y0 − G˜pw)T (y0 − G˜pw) + Λ˜pTwGTG˜pw − ayT2 y2 (7.29)

If we now consider a corrected tentative Lyapunov function with the wind

estimator from Section 7.1:

W = Wt +We (7.30)

the derivative W may be written using (7.29) and (7.9) as:

W = −Λ(y0 − G˜pw)T (y0 − G˜pw) − ayT2 y2 − pTQpp − ˜pTw(Qpw− ΛGTG)˜pw

(7.31)

which is negative definite if:

Qpw− ΛGTG > 0 (7.32)

The control law may be deduced from equations (2.69) and (7.26), leading to:

uf = Ma(Va − KVa) − EgSag − Fa (7.33)

Va = −aJ−1Φ Λ(P − Pr) − CJVa − J−1

Φ Λ22(JΦVa + BI

ˆpw − JΦrVr) (7.34)

where K = −M−1

a (Ω6Ma + Va6(Ma − MBa)) and Λ22 = (aI6 + 2Λ).

The force control input is then given by:

uf = −Ma

(

A1(JΦVa + BIˆpw − JΦr

Vr) + B1(P − Pr) + C1Va

)

−EgSag−Fa

(7.35)

with A1 = J−1Φ Λ2

2, B1 = aJ−1Φ Λ and C1 = CJ + K, resulting in an asymptoti-

cally stable closed-loop system.

However, the force control input, as is it, may result in excessively high de-

2From the expansion of a square:

(y0−G˜pw)T (y0−G˜pw) = yT0(y0−G˜pw)−(G˜pw)T (y0−G˜pw) = yT

0(y0−2G˜pw)+(G˜pw)TG˜pw

it is easily deduced that: yT0(y0 − 2G˜pw) = (y0 − G˜pw)T (y0 − G˜pw) − ˜pTwGTG˜pw


mands for a real system subject to input constraints. In the next section the

control solution (7.35) will be adapted to deal with this matter.

7.4 Control design with saturation constraints

In order to include saturation limits into the control design, let us rewrite

equation (7.22), corresponding to a second derivative demand:

y = −ay − Λ(ay + 2y) = −(aI6 + 2Λ)y − aΛy = −Λ22y − Λ1y (7.36)

with Λ1 = aΛ and Λ22 as defined in the previous section.

Defining the second Lyapunov function as:

W2 =1

2yT2 y2 (7.37)

with, as before, y2 = y, its derivative may be expressed as:

W2 = yT y = −yT (Λ22y + Λ1y) = −zT2 (z2 + z1) (7.38)

where z1 = Λ−12 Λ1y and z2 = Λ2y. Writing (7.36) as function of z1 and z2

yields:

y = −Λ2(z2 + z1) (7.39)

Before proceeding, we will now define linear saturation as well as its properties,

and provide an important theorem used in the proof of stability of the saturated

control.

Definition 7.1. As a particular case and extension of the linear saturation

definition proposed by Teel [63], let us introduce the elementwise nondecreasing

saturation function σ : Rn → R

n, defined by a vector m of n positive values

mi, with mi > r > 0, and such that:

∀ z ∈ Rn, σ[z] = Σz (7.40)

7.4. CONTROL DESIGN WITH SATURATION CONSTRAINTS 133

where the diagonal matrix Σ is defined by:

|zi| < mi ⇒ Σi = 1

|zi| ≥ mi ⇒ Σi =mi

|zi|(7.41)

Properties 7.1. It may easily be verified that the definition yields the following

properties [63]:

∀z ∈ Rn ; zTσ[z] > 0

∀z ∈ Rn ; |σ[z]| ≤ R

|z| < r ⇒ σ[z] = z

(7.42)

where |z| =√

zTz is the norm of vector z as defined in Rn and R2 =

∑ni=1m

2i .

Theorem 7.1. If two saturations σ1 and σ2 are defined, such that R1 <12r2,

then:

∀(z1, z2) ∈ Rn , |z2| >

1

2r2 ⇒ zT2 σ2 [z2 + σ1 [z1]] > 0 (7.43)

Proof. Since |z2| > 12r2 and |σ1[z1]| ≤ R1 <

12r2, one can write the orthogonal

projection of the saturated vector σ1[z1] on z2 as:

σ1 [z1] = λ1z2 + v1 (7.44)

where |λ1| < 1, zT2 v1 = 0, and |λ1z2 + v1| < 12r2.

Then:

zT2 σ2 [z2 + σ1 [z1]] = zT2 σ2 [(1 + λ1) z2 + v1]

= zT2 Σ2 ((1 + λ1) z2 + v1)

= (1 + λ1) zT2 Σ2z2 > 0

(7.45)

We can now proceed and introduce the second derivative (7.39) saturated

demand:

ys = −Λ2σ2 [z2 + σ1 [z1]] (7.46)

From Theorem 7.1, if |z2| > 12r2, then W2 = −zT2 σ2 [z2 + σ1 [z1]] will be

negative definite for saturations σ1 such that |σ1 [z1]| ≤ R1 <12r2.

Since the saturated system is asymptotically stable, after a time T2 the variable

z2 will enter the linear zone of its saturation and remain inside of it, and namely


with |z2| < 12r2.

After time T2 the saturated demand will be equal to:

ys = −Λ2 (z2 + σ1 [z1]) (7.47)

Introducing (7.47) into (7.21) yields:

Wts = (ay + 2y)T (ay + ys) − ayT y

= (ay + 2y)T (ay − Λ2(z2 + σ1 [z1])) − ayT y

= (ay + 2y)T (ay − Λ2z2 − Λ2σ1 [z1]) − ayT y

= (ay + 2y)T (ay − (aI6 + 2Λ)y − Λ2σ1 [z1]) − ayT y

= −(ay + 2y)TΛ(2y + Λ−1Λ2σ1 [z1]) − ayT y (7.48)

Using the definition of the saturation σ1 [z1] = Σz1 = aΣΛ−12 Λy, from (7.48)

we get:

Wts = − (2y + ay)T Λ(2y + Λ−1Λ2Σ1Λ

−12 aΛy

)− ayT y (7.49)

Two scenarios are now possible: (i) z1 is not saturated, in which case Σ1 = I,

resulting in Wts < 0; or (ii) z1 is saturated and Σ1i=

m1i

|z1i|≤ 1. Let us further

analyze this case.

Taking z0 = 2Λ1/2y, s = aΛ1/2Σ1y, and Z = Σ−11 , we have:

Wts = − (z0 + Zs)T (z0 + s) − ayT y (7.50)

If we consider the decomposition of the vectors in their components z0 = [zi]

and s = [si] and also the diagonal matrix Z = [λi] with elements λi =|z1i

|

m1i

≥ 1,

then:

(z0 + s)T (z0 + Zs) =∑

i

(zi + si)(zi + λisi) (7.51)

Noting that s and z0 have behaviors similar to, respectively, z1 and z2, and

that z2 is in its linear zone and converging, we have that after some time

7.4. CONTROL DESIGN WITH SATURATION CONSTRAINTS 135

|zi| < |si|,∀i and then zi = µisi with |µi| < 1, so that:

(z0 + s)T (z0 + Zs) =∑

i

(µisi + si)(µisi + λisi) (7.52)

=∑

i

(si)2(µi + 1)(µi + λi) (7.53)

which shows that each term is positive, making the result of the sum also

positive. Therefore, Wts is negative definite.

To include the input forces limitations into the control law design, let us con-

sider the desired demand is a saturated one, y = ys. From (7.24) and (7.47)

we obtain:

JΦVa + JΦCJVa =

− Λ2σ2

[Λ2(JΦVa + BI pw − JΦr

Vr) + σ1[Λ−12 Λ1(P − Pr)]

](7.54)

or, solving for Va:

Va = −J−1Φ Λ2σ2


Vr) + σ1[Λ−12 Λ1(P − Pr)]

]− CJVa

(7.55)

Substituting now (7.55) into (7.33) leads to the control law:

ufs = − MaJ−1Φ Λ2σ2


Vr) + σ1[Λ−12 Λ1(P − Pr)]

]

− MaC1Va − EgSag − Fa (7.56)

Again, as the wind is estimated, the control law that considers the force input

saturations is finally given by:

ufs = − MaJ−1Φ Λ2σ2

[

Λ2(JΦVa + BIˆpw − JΦr

Vr) + σ1[Λ−12 Λ1(P − Pr)]

]

− MaC1Va − EgSag − Fa (7.57)

where σ1 and σ2 are the velocity saturation matrices obtained from (7.57)

with ufs corresponding to the input force maximum values related to the

actuators limits (see sections 2.1 and 2.2.1.1), and that satisfy the condition

R1 <12r2 < |z2|. This control law will lead to an asymptotically stable closed-

loop system as long as the estimation error is bounded according to (7.31).


7.5 Control implementation

The control law (7.57) solves the airship path-tracking problem in the presence

of constant translational wind while taking into account the limitations of the

demanded forces input. However, this control law cannot be directly fed into

the system, and needs to be adapted.

Although the control law assumes 3 forces and 3 torques are fully available, this

isn’t really the case, since the airship is an underactuated vehicle, as described

in Section 2.2.1.1. The available actuation results in weak lateral and vertical

forces responses, and therefore, the position and velocity references used in

the backstepping control law should be shaped to deal with this scenario.

This matter is analyzed in the next section.

As in the dynamic inversion controller case, the backstepping control input is a

force input, which cannot be directly fed to the airship. The control law (7.57)

considers an input ufs that includes both forces and torques. However, the

real inputs of an airship are its actuators. For this reason, a conversion from

forces to actuators inputs is necessary for the proper implementation of the

attained controller.

The actuators input u = [δe, TL, TR, δv, δa, δr]T is obtained solving the equa-

tions system (2.71). As referred in Section 2.2.1.1, although we have six ac-

tuators inputs to control the six forces, several limitations lead to the under-

actuation of the airship. Moreover, the system of equations (2.71) is not di-

rectly invertible, which implies an empiric allocation in some situations.

7.5.1 Adapted control law to deal with underactuation

As referred, the present configuration of the AURORA airship actuators results

in an underactuated system. At very low airspeeds we reach the worst-case

scenario, with the airship being uncontrollable due to the lack of authority from

the control surfaces. The implementation of the proposed control law assumes

this situation is not reached, therefore requiring that the true airspeed does

not drop below a minimum, Vt > Vtmin= 2m/s (note that it is quite realistic

in outdoor conditions to assume a wind intensity above this level).

Even if this limit is respected, the airship may still be underactuated as the

transversal forces available are too weak. This means the controllers force

request for a straightforward correction of eventual lateral and vertical position

7.5. CONTROL IMPLEMENTATION 137

errors might find insufficient response on the actuators side. In the following,

we adapt the control law (7.57) to deal with this scenario, obtaining a faster

error correction with smoother input requests.

Consider the approximated kinematic relations:

E ≃ Vtψ (7.58)

D ≃ −Vtθ (7.59)

where ψ and θ are the pitch and yaw Euler angles [53] that describe the airship

orientation. Equations (7.58)-(7.59) allow us to relate the airship orientation

with its lateral and vertical positions.

Consider now the airship is to track a rectilinear path with orientation (ψr, θr)

and has lateral and vertical errors respectively y and z. The angular errors are

defined as:

∆ψ = ψ − ψr (7.60)

∆θ = θ − θr (7.61)

Due to the airship underactuation, if we try to independently correct the posi-

tion and angular errors, depending on their magnitude, we will probably have

input saturation. However, if we consider the relation between position and

attitude, we may consider instead the following expressions:

∆ψ′ = ψ − ψr − kyy (7.62)

∆θ′ = θ − θr − kzz (7.63)

where the constants ky and kz are dependent of the airspeed Vt. This means

we will postpone the angular corrections and use them to annulate the position

errors first. The angular references used in Pr in the control law (7.57) will

then be:

ψ′r = ψr + kyy (7.64)

θ′r = θr + kzz (7.65)



This last section demonstrates the performance of the backstepping approach

applied to the airship path-tracking problem. We present the results obtained

for the case-study mission and to the sensitivity and robustness to parameter

uncertainty test.


The first results we present concern the case-study mission described in Sec-

tion 3.3.1.



path-tracking performance. Figure 7.2(b) displays the longitudinal η, lateral ε

and vertical δ errors. Like in the gain scheduling and dynamic inversion cases,

the airship deviates from the reference trajectory when the wind is at the rear,

at the end of the first half-circle, and in the transition from ascent to horizontal

tracking (see fig. 7.3(a)). Other problematic mission point, shared with the

dynamic inversion but showing higher errors, is the stabilization prior to the

descent. The correction of the longitudinal η and lateral ε position errors due

to the instantaneous reference change induces a significant vertical error. With

the backstepping solution, the stabilization is more slowly achieved. Remember

that in this solution the controller has no prior information of the actuation

available (the gain scheduling considers the actuators input and the dynamic

inversion indirectly knows the actuation limitation through the gain scheduling

model used as reference), and therefore the errors correction is not optimized.

Comparing the airship trajectory executed with the backstepping controller

(see fig. 7.2(a)) with the ones controlled by the gain scheduling (see fig. 5.15(a))

and dynamic inversion (see fig. 6.4(a)) approaches, we observe that all control

laws lead to the accomplishment of the mission within acceptable deviations

from the trajectory reference. The backstepping resulting trajectory is less

erroneous than the dynamic inversion one, but shows higher vertical errors

than the gain scheduling.




E (m)N (m)

h(m

)

-1000

100200

300

-200

-100

0

100

200

0

10

20

30

40

50

60


Time (s)

η(m

)

Time (s)

ε(m

)

Time (s)

δ(m

)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-8

-4

0

4

-30

-20

-10

0

10

20

-80

-60

-40

-20

0




take-off and landing is again well recognized. During the maneuvers at low

airspeeds the airship motion is smoother than in the previous two cases, due to

the references shaping described in Section 7.5.1. The Euler angles evolution

is displayed in fig. 7.3(b). The roll angle (φ), with a null reference, has a

significant amplitude in the transition from vertical to horizontal tracking,

which is then corrected. During the descent, the roll is well controlled. The

pitch (θ) and yaw (ψ) angles approximately follow the respective references

described in Section 7.5.1.


fig. 7.4 and are somewhat similar to the ones obtained in the previous cases.

The ground velocity components are described in fig. 7.4(a). The longitudi-

nal groundspeed u mostly follows the reference that varies between 0m/s for

stabilization, take-off and landing, and 7m/s during the path-tracking. Along

the circular segments, the errors are more noticeable due to the change of the

wind incidence angle while the airship is turning. The error is higher when

the wind is aft the airship. The lateral velocity v is also mostly influenced by

the circular segments and during the tail wind segment. The vertical velocity

w follows its reference, with the −1 and 0.5m/s steps corresponding to the

take-off and landing vertical motion. The airspeed and aerodynamic angles

can be seen in fig. 7.4(b). During the whole mission, the airspeed Vt varies

significantly, from values around 2.5m/s up to 12.5m/s. The airship covers a


E (m)

N(m

)windheading

-100 0 100 200 300-250

-200

-150

-100

-50

0

50

100

150

200

250


Time (s)

φ(deg

)

Time (s)

θ(deg

)

Time (s)

ψ(deg

)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-900

90180270360450

-10

-5

0

5

10

15

-30-15

015304560



Time (s)

u(m/s)

Time (s)

v(m/s)

Time (s)

w(m/s)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-2

-1

0

1

2

-6

-4

-2

0

2

4

-202468

10


Vt

(m/s)

Time (s)

β(deg

)

Time (s)

α(deg

)

Time (s)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-30

-20

-10

0

10

20

-20

-10

0

10

20

2468

101214



wide flight envelope, from hover to the aerodynamic flight, crossing the trou-

blesome transition region between the two. Here, the behavior of α shows also

correlation with w.


elevator δe, total thrust XT and vectoring angle δv in fig. 7.5(a) and the lat-

eral input, aileron δa, rudder δr and differential thrust TD, in fig. 7.5(b). The

elevator δe, while responsible for the altitude and pitch control, shows a more

constant demand during the ascent and stabilization phases. The rudder δr,


mainly responsible for the lateral position and airship yaw, has a higher com-

mand with tail wind, due to the reduced authority at lower airspeeds. The

airship roll φ is controlled by the aileron δa at higher airspeeds, and by the

differential thrust TD when the control surface loses authority (which usually

corresponds to a vectoring angle close to 90o, allowing TD to effectively control

the roll and not the yaw). This actuator has a negligible action in aerodynamic

flight (the control surfaces authority is sufficient for the rudder δr to control

the airship yaw ψ), an option taken when converting the forces input to an

actuators request.

Time (s)

δ e(deg

)

Time (s)

XT

(N)

Time (s)

δ v(deg

)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-300

306090

120

0

20

40

60

80

-30

-15

0

15

30


Time (s)

δ a(deg

)

Time (s)

δ r(deg

)

Time (s)

TD

(N)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-30

-15

0

15

30

-30

-15

0

15

30

-30

-15

0

15

30



The vectoring angle δv is responsible for the airship lift when the airspeed Vt

is too low to provide the necessary aerodynamic lift. The correlation between

these two variables, although not as obvious as in the gain scheduling case,

is visible when comparing the graphics of δv and Vt. At higher airspeeds, the

backstepping control law chooses to use this actuator to help the elevator on

the altitude and pitch control, unlike the gain scheduling controller.

In the overall, the airship under backstepping control executed the mission

satisfactorily.



tainty

This section evaluates the stability and performance robustness of the back-

stepping control methodology when solving the path-tracking control problem

of the AURORA airship. Due to the nonlinearity of both system and back-

stepping control law, we cannot make use of the analysis tools used in the gain

scheduling robustness analysis. Therefore, we limit our analysis of the back-

stepping closed-loop system robustness to the test described in Section 3.3.2.




E (m)

N(m

) headingwind

-100 -50 0 50

0

200

400

600

800

1000

1200


φ(deg

)θ

(deg

)ψ

(deg

)θ w

(deg

)ψw

(deg

)

Time (s)

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

80

90

100

-10

-5

0

5

-35

-30

-25

-20

-5

0

5

10

-10

0

10

(b) Airship attitude : roll φ, pitch θ andyaw ψ (−− reference, − output), and windattitude, θw, ψw.








Figure 7.6(b) represents the airship attitude references (see Section 7.5.1) and

output, as well as the wind estimated attitude. The excitation of the signals

is due to the wind turbulence.




airspeed corresponds to the relative air speed between the 8m/s groundspeed



Time (s)

η(m

)

Time (s)

ε(m

)

Time (s)

δ(m

)

0 50 100 150

0 50 100 150

0 50 100 150

-1

0

1

-1

0

1

-1

0

1


Vt

(m/s)

time (s)

β(deg

)

time (s)

α(deg

)

time (s)

0 50 100 150

0 50 100 150

0 50 100 150

-5

0

5

10

-5

0

5

8

9

10




with the longitudinal actuators action given in fig. 7.8(a), while the lateral

actuation is in fig. 7.8(b). The vectoring angle δv, which is expected to have

a negligible action in aerodynamic flight, is clearly involved in the altitude

error δ and pitch θ control. The differential thrust TD, represented here for

completeness, has a negligible control action in aerodynamic flight, an option

taken when converting from forces to actuators.




the groundspeed error eu relative to the 8m/s reference. The first row has


Time (s)

δ e(deg

)

Time (s)

XT

(N)

Time (s)

δ v(deg

)

0 50 100 150

0 50 100 150

0 50 100 150

-30

0

30

60

90

120

0

20

40

60

80

-20

-10

0

10

20


Time (s)

δ a(deg

)

Time (s)

δ r(deg

)

Time (s)

TD

(N)

0 50 100 150

0 50 100 150

0 50 100 150

-1

0

1

-20

-10

0

10

20

-20

-10

0

10

20



the RMS values obtained for the baseline case, and is to serve as reference

for the remaining lines where each of the listed coefficients is varied one at

a time. For a parameter uncertainty of ±70%, the controlled airship still

performed qualitatively like the baseline case. We then increased the parameter

uncertainty, and present here the results obtained for ±90% uncertainty around

the nominal value.

Only for four parameters the uncertainty leads to an less efficient or inefficient

control action, either because it is insufficient due to actuators saturation (the

backstepping demands a too high control input) and/or just because the tested

coefficient appears to be a more sensitive model parameter. The cases which

lead to significant deviations from the reference RMS values are represented

in bold in table 7.1.

With noticeable deviations from the baseline RMS values, for CNδr(−90%

uncertainty) the backstepping still controls the AURORA airship within ac-

ceptable bounds. CNδrcorresponds to the authority of the rudder control

surface as yawing control input.

For the remaining coefficients, CMδe, CLα

and CMq(all for −90% uncertainty),

either saturation of the actuators occurred or the uncertainty of the model

parameter is too important for the backstepping controller to overcome it.

These parameters correspond respectively to the authority of the elevator as

pitching control input, the lift coefficient derivative due to angle of attack

and the pitch damping derivative. While the backstepping demonstrates to




Baseline 0.40 0.23 0.29 9.00 2.87 1.13 0.12

Clβ−90% 0.37 0.22 0.29 9.00 2.91 0.99 0.11+90% 0.45 0.24 0.28 9.00 2.78 1.33 0.13

CM0

−90% 0.40 0.23 0.27 9.00 2.90 1.13 0.12+90% 0.39 0.24 0.31 9.00 2.82 1.15 0.12

CMα

−90% 0.34 0.20 0.12 9.00 3.19 1.08 0.11+90% 0.36 0.25 0.48 9.00 2.53 1.18 0.17

CMαβ

−90% 0.39 0.23 0.29 9.00 2.85 1.13 0.12+90% 0.40 0.22 0.29 9.00 2.87 1.12 0.12

CMβ

−90% 0.39 0.24 0.31 9.00 2.82 1.14 0.12+90% 0.40 0.22 0.27 9.00 2.90 1.12 0.12

CMβα

−90% 0.39 0.23 0.29 9.00 2.86 1.13 0.12+90% 0.40 0.23 0.29 9.00 2.87 1.13 0.12

CMδe

−90% 1.51 6.16 27.06 8.99 2.18 1.56 0.29

+90% 0.38 0.20 0.15 9.00 3.14 1.08 0.10

CNβ

−90% 0.37 0.23 0.28 9.00 2.89 1.17 0.11+90% 0.40 0.27 0.30 9.00 2.81 1.19 0.13

CNδr

−90% 3.56 6.78 3.16 9.16 4.16 3.88 0.77

+90% 0.38 0.08 0.27 9.00 2.91 1.01 0.09

CYβ

−90% 0.39 0.60 0.34 9.01 2.83 3.13 0.20+90% 0.39 0.21 0.29 9.00 2.87 0.87 0.11

CYδr

−90% 0.38 0.09 0.27 9.00 2.91 1.13 0.09+90% 0.59 1.04 0.67 9.03 2.63 3.76 0.39

CD0

−90% 1.36 0.43 0.52 9.02 3.18 1.54 0.33+90% 0.86 0.22 0.34 9.00 2.68 1.13 0.13

CDi

−90% 0.36 0.23 0.28 9.00 2.90 1.13 0.12+90% 0.43 0.23 0.30 9.00 2.83 1.12 0.12

CL0

−90% 0.41 0.23 0.30 9.00 2.76 1.13 0.12+90% 0.38 0.23 0.28 9.00 2.97 1.13 0.12

CLα

−90% 49.86 0.74 10.84 9.72 19.32 1.52 1.06

+90% 0.47 0.27 0.44 9.00 1.52 1.18 0.13

CLδe

−90% 0.39 0.19 0.10 9.00 4.18 1.08 0.11+90% 0.42 0.29 0.54 9.00 2.10 1.22 0.15

Clp−90% 0.42 0.23 0.29 9.00 2.76 1.33 0.13+90% 0.38 0.24 0.29 9.00 2.87 1.10 0.12

CMq

−90% 265.78 3.12 1.52 12.79 3.41 1.33 4.65

+90% 0.41 0.30 0.75 9.01 2.74 1.21 0.20

CNr

−90% 0.38 0.07 0.28 9.00 2.88 0.98 0.10+90% 0.79 1.52 0.90 9.05 2.91 2.84 0.38

mw−90% 0.46 0.37 0.95 9.02 1.99 1.16 0.25+90% 0.23 0.30 0.32 9.01 3.01 1.31 0.13


be robust to a ±90% uncertainty in the remaining parameters, for these three

cases, the mismatch between the airship system and the model considered in

the backstepping controller design is too significant for the control action to

overcome it.

In any case, the backstepping controller may be considered robust to wind

disturbances and plant uncertainties. Among the list selected, these four pa-

rameters CNδr, CMδe

, CLαand CMq

(and specially the last three) are in fact

the model parameters for which a more careful identification or determination

should take place, though the required precision could merely remain inside a

±70% margin.

7.7 Conclusions

This chapter introduces a backstepping approach for the airship path-tracking

problem. The asymptotically stable backstepping controller is designed for-

mulating a scalar positive function of the system states and then choosing

a control law based on the airship six-degrees-of-freedom nonlinear model to

make this function decrease.

Some practical issues have to be addressed, and the control law is improved

to take into account input saturations and wind disturbances, maintaining its

asymptotic stability for a bounded wind estimation error. Prior to implemen-

tation, further issues are considered, namely control allocation and reference

shaping to deal with the airship underactuation. Reference shaping is vital for

the control law implementation on a underactuated airship. Remember the

control law considers six forces inputs are fully available, being blind to the

control allocation problem (changing from forces to actuators request), and

therefore to the available actuation. This is a step that, for now, is executed

after the controller design.

The application of the proposed backstepping solution to the AURORA airship

path-tracking problem resulted in a satisfactory performance in the execution

of missions including different phases as take-off and landing, path-tracking

and stabilization, even in the presence of realistic wind disturbances.

The backstepping approach is based on the six-degrees-of-freedom nonlinear

airship model with constant translation wind input. However, a real wind dis-

turbance is stochastic and the real airship parameters might differ from the


ones of the model. Therefore, the analysis of the controlled system perfor-

mance and stability robustness in the presence of realistic wind disturbances

and model parameter uncertainties is very important. The backstepping con-

troller, robust to wind disturbances, shows to be tolerant to uncertainties in

most of the model parameters tested. However, for some aerodynamic coef-

ficients, namely CNδr, CMδe

, CLαand CMq

, a more careful identification or

determination should take place.

Chapter 8

Comparison of controllers

performance

Any of the three control solutions described in the previous chapters presents

its advantages and disadvantages, many of which are discussed in the respective

chapters. Yet, an overall comparison between them is important as to provide a

better overview of the different control options. In this chapter this assessment

is made considering parameters such as path-tracking performance for a case-

study complete mission (Section 8.1), robustness in face of model parameter

uncertainty (Section 8.2) and computational effort (Section 8.3). These factors,

together with some implementation issues, are relevant to evolve to the next

phase, the experimental validation in autonomous flight.

8.1 Performance for case-study mission

The case-study mission described in Section 3.3.1 was used to test the path-

tracking performance of each of the controllers. It considers important phases

of a generic mission, like take-off and landing, path-tracking and stabilization.

The reference trajectory considered, as well as the resulting trajectories for

each of the three controllers are gathered in fig. 8.1. Although all three con-

trollers are able to accomplish the entire mission, the different performances

are noticeable. The gain scheduling and backstepping solutions both show

smaller deviations from the reference trajectory, with the backstepping con-

troller presenting more difficulties in the vertical positioning control during the

transition from vertical ascent to horizontal tracking and during the stabiliza-

149

150 CHAPTER 8. COMPARISON OF CONTROLLERS PERFORMANCE

E (m)N (m)

h(m

)

0©

1©≡ 3©2©

4©

-1000

100200

300

-200

-100

0

100

200

0

10

20

30

40

50

60

(a) Reference.

E (m)N (m)

h(m

)

-100

0

100

200

300

-200

-100

0

100

200

0

10

20

30

40

50

60

(b) Gain scheduling.

E (m)N (m)

h(m

)

-1000

100200

300

-200

-100

0

100

200

0

10

20

30

40

50

60

(c) Dynamic inversion.

E (m)N (m)

h(m

)

-1000

100200

300

-200

-100

0

100

200

0

10

20

30

40

50

60

(d) Backstepping.

Figure 8.1: Comparison of airship 3D trajectories.

tion. Notice that these two points represent nonsmooth tracking references.

The dynamic inversion visibly results in a more erroneous tracking.

In order to better quantify the position errors obtained with each of the con-

trollers, the longitudinal η, lateral ε and vertical δ local errors are represented

in fig. 8.2, together with an indication of the mission phase being executed.

Observing with this detail, it is obvious that none of the controllers is better

(or worse) during the entire mission. Dynamic inversion, for instance, although

usually having the higher deviations presents a smoother longitudinal stabi-

lization.

Figure 8.3 allows us to compare the horizontal path-tracking results. Both

gain scheduling and dynamic inversion solutions lead to a higher crabbing of

8.1. PERFORMANCE FOR CASE-STUDY MISSION 151

Time (s)

η(m

)

ascentdescent

stabilizationhorizontal path-tracking

Time (s)

ε(m

)

Time (s)

δ(m

)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-10

-5

0

5

-30

-20

-100

10

20

-80

-60

-40

-20

0

20

Figure 8.2: Comparison of position errors (− gain scheduling, −− dynamicinversion, −.− backstepping).

the airship during the tail wind periods. The backstepping control results in

a smoother overall trajectory, consequence of the references shaping to deal

with the airship underactuation.

E (m)

N(m

)

windheading

-100 0 100 200 300-250

-200

-150

-100

-50

0

50

100

150

200

250

(a) Gain scheduling.

E (m)

N(m

)

windheading

-100 0 100 200 300-250

-200

-150

-100

-50

0

50

100

150

200

250

(b) Dynamic inversion.

E (m)

N(m

)

windheading

-100 0 100 200 300-250

-200

-150

-100

-50

0

50

100

150

200

250

(c) Backstepping.

Figure 8.3: Comparison of north-east trajectories with airship heading.


The UAV capabilities report [13] analyzes 53 proposed missions in the Earth

observation scope, and 16 capabilities required. One of the requirements refers

to the precision of trajectories. They define four levels of accuracy, namely

level 5, where the trajectory is to be based on a position accuracy better than

±5m; level 3, that requires a position accuracy between ±5m and ±50m; level

1, where the mission requires some sensitivity to vehicle trajectory, absolute

or relative, but position accuracy can be less than ±50m; and level 0 for

missions that do not involve a precision trajectory. Observing these limits, and

considering only the continuous path-tracking part of the mission (neglecting

the second stabilization, at the end of the second curve), we observe that the

three controllers, with position errors below 30m, respect the level 3 limits.

The control solutions presented here are therefore appropriate, what trajectory

precision concerns, for missions such as topographic mapping and topographic

change, river discharge and urban management.

For long endurance applications, energy management is an important auton-

omy issue. A lower fuel and batteries consumption requires a reduced control

effort. Figure 8.4 allows us to compare the actuators request made by each

of the three controllers during the execution of the case-study mission. The

dynamic inversion and backstepping controllers show higher and more oscilla-

tory requests, denoting the nonlinearity of the control laws and of the control

allocation procedure. The smooth gain scheduling control effort is a direct

consequence of the linearization procedure, resulting in a linear airship model

with actuators input instead of forces.

8.2 Sensitivity test results comparison

The sensitivity and robustness to parameter uncertainty test described in Sec-

tion 3.3.2 investigates the stability of the closed-loop systems even in the

present of wind disturbances and model parameter uncertainty.

The baseline simulation considered the nominal model of the airship subject to

wind disturbances. The baseline results obtained for the particular mission of

a north-aligned straight line tracking with lateral wind from the three control

solutions are gathered in table 8.1. The dynamic inversion controller leads to

the higher position errors, while the backstepping solution results in smaller

ground velocity and lateral and vertical position errors.

Regarding the robustness to model parameter uncertainty, both gain schedul-

8.2. SENSITIVITY TEST RESULTS 153

Time (s)

δ e(deg

)

Time (s)

XT

(N)

Time (s)

δ v(deg

)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-30

0

30

60

90

120

0

20

40

60

80

-30

-15

0

15

30

Time (s)

δ a(deg

)

Time (s)

δ r(deg

)

Time (s)

TD

(N)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-20

-10

0

10

20

-30

-15

0

15

30

-30

-15

0

15

30

(a) Gain scheduling actuators request.

Time (s)

δ e(deg

)

Time (s)

XT

(N)

Time (s)

δ v(deg

)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-300

306090

120

0

20

40

60

80

-30

-15

0

15

30

Time (s)

δ a(deg

)

Time (s)

δ r(deg

)

Time (s)

TD

(N)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-10

0

10

-30

-15

0

15

30

-30

-15

0

15

30

(b) Dynamic inversion actuators request.

Time (s)

δ e(deg

)

Time (s)

XT

(N)

Time (s)

δ v(deg

)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-300

306090

120

0

20

40

60

80

-30

-15

0

15

30

Time (s)

δ a(deg

)

Time (s)

δ r(deg

)

Time (s)

TD

(N)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

-30

-15

0

15

30

-30

-15

0

15

30

-30

-15

0

15

30

(c) Backstepping actuators request.

Figure 8.4: Comparison of actuators request (elevator δe, total thrust XT ,vectoring δv, aileron δa, rudder δr and differential thrust TD).


Table 8.1: Comparison of baseline results (RMS values).

η (m) ε (m) δ (m) eu (m/s)

Gain Scheduling 0.24 0.33 0.45 0.16

Dynamic Inversion 0.60 0.45 0.78 0.14

Backstepping 0.40 0.23 0.29 0.12

ing and backstepping controllers demonstrated robustness to a ±70% uncer-

tainty in all analyzed parameters (changing one at a time), while the dynamic

inversion presented control problems for lower values of uncertainties for some

aerodynamic coefficients. For uncertainties up to ±90%, the aerodynamic co-

efficients that resulted in an inefficient control, namely CLα, CD0

, CMδe, CNδr

,

CMqand CNr

, are the ones for which a more careful identification or determi-

nation should take place.

8.3 Computational effort

For a real-time implementation to be possible, the computational time taken

by the controller is an important measure of its performance. The controller

is to be implemented onboard the airship platform (as was in the simulator)

at 10Hz.

Table 8.2 represents the computational time taken by each of the three con-

trollers tested. The computational time obviously depends on the character-

istics of the machine. Therefore, not only absolute time is presented but also

the relative time between controllers, based on the higher computational time.

This measure provides a better comparison of the computational effort.

The approximate values obtained for the gain scheduling and dynamic inver-

sion controllers is justified since the dynamic inversion execution code also runs

the gain scheduling in order to obtain the model reference used. Although with

a more complex control law, the time taken to execute the backstepping con-

troller code is almost 50% less than the time taken by the gain scheduling

controller. This may be justified by the fact that the gain matrix K used in

the gain scheduling is being computed online rather than being obtained from

a lookup table. The computational effort is obviously also a function of the

code optimization.


Table 8.2: Computational effort comparison.

Total time1 (s) Relative time (%)

Gain Scheduling 0.0475 95.4

Dynamic Inversion 0.0498 100.0

Backstepping 0.0271 54.4

8.4 Conclusions

This chapter provides an overview of the advantages and disadvantages of

the three control solutions considered in this work, namely gain scheduling,

dynamic inversion and backstepping, when applied to the path-tracking airship

problem.

In the previous sections we compared the results obtained for each controller

regarding a case-study mission, a sensitivity and robustness test for model pa-

rameter uncertainty, and the computational effort. In order to better visualize

the relative results, table 8.3 presents a qualitative overall comparison between

controllers.

Table 8.3: Overall controllers comparison (+ good, average, − poor).

Gain Scheduling Dynamic Inversion Backstepping

Path-tracking performance:Path-tracking errors + − +Tracking smoothness +Requested control effort + − −Robustness to:Wind disturbances + + +Parameters uncertainty + +

Implementation issues:Computational effort +Code simplicity +

Others:Design parameters tuning − −Possible evolution − + +

The table is divided into four parts, three of which, path-tracking performance,

controllers robustness and implementation issues, were already analyzed in the

previous sections. The last one contains more subjective, designer experience

1Computational time measured in a Pentium IV with 512MB at 2.8GHz, runningMATLABr R2006a.


related, but also relevant issues, namely design parameters tuning and possible

evolution.

The evaluation of the design parameters tuning provides a comparative idea of

the necessary effort of the designer to correctly tune the controllers parameters.

While the dynamic inversion, using a model reference, has the decision of which

model to follow, the gain scheduling requires the adjustment of the state and

input control matrices parameters. The backstepping performance depends of

a proper choice of the reference shaping parameters.

The last item refers to the possible evolution of each solution. This evaluation

is merely based on the knowledge acquired throughout this work, and serves

as an indication of the future work yet to be developed for each of the control

solutions (see next chapter).

Chapter 9

Conclusions and Future Work

Considering their particular features, airships have a wide spectrum of ap-

plications as observation and data acquisition platforms. If we also consider

the quest for autonomy, airships present characteristics and competitive costs

when compared to other aircrafts, certainly constituting an important option

for research, development and experimental validation in autonomous aerial

robotics. Moreover, most of the solutions established for this kind of air vehi-

cle may be transferred or adapted for airplanes or helicopters, where the risks

and costs involved in testing new methodologies are obviously higher.

The role of airships as UAVs depends, however, of their autonomous flight

capacity. This implies the development of control solutions for the airship

autonomous flight, that allow the execution of different missions even in the

presence of wind disturbances.

So far, a global control solution as not yet been presented for airships, with the

exception of the LAAS-CNRS group solution with decoupled controllers [40,

47], for complete missions including take-off and landing, path-tracking and

stabilization. Moreover, seldom are the ones that consider such an important

issue as robustness to wind disturbances. This work, inserted in the AURORA

and DIVA projects, made a breakthrough in this topic, developing and com-

paring airship control solutions, valid for the entire flight envelope, and capable

of executing realistic missions, while being robust to wind input.

An airship is an highly nonlinear system. The dynamics when in hover or aero-

dynamic flight varies greatly, with different combinations of actuators available,

which leads to a problematic transition region between the two. The successful

development of an overall control solution depends therefore on a good knowl-

157

158 CHAPTER 9. CONCLUSIONS AND FUTURE WORK

edge of the system behavior over the flight envelope, and on a good model of

the airship. With this in mind, a six-degrees-of-freedom nonlinear model was

developed based on the Lagrangian approach. The linearization of this model

over the aerodynamic range lead to the usual decoupling of the longitudinal

and lateral motions. A detailed analysis of these linear models provided the

necessary insight of the airship behavior characteristics, allowing the design of

the first control solution.

The Gain Scheduling approach is based on the linear description of the air-

ship. In order for the solution to be valid over the entire flight envelope, for

each linear model obtained (one for each equilibrium condition defined), an

optimal state feedback control law is designed. The overall control synthesis

is achieved by switching between models and respective controllers as function

of the scheduling variable airspeed. The main advantages of this method are

the simplicity of both linear model and controller, allowing to use the classic

control tools, and the fact that the model inputs are the airship actuators, a

result of the linearization procedure. A disadvantage is the time consuming

tuning of the control design parameters, namely the state and input control

matrices.

The Dynamic Inversion solution results of the inversion of the six-degrees-of-

freedom nonlinear airship model, obtaining a control law that cancels existing

deficient or undesirable dynamics by replacing them with a set of desired ones.

For systems, like the airship, described in a cascaded form based on dynamics

and kinematics, a new dynamic inversion formulation is presented, allowing an

easier implementation. However, a control solution based on the nonlinear de-

scription of the system presents a disadvantage if the system is underactuated,

as is the airship. The nonlinear model considers forces as input. Therefore,

the controller, obtained by inversion of this model, computes a forces request

considering all six forces are fully available. This is however not the case.

The airship has serious actuation constraints regarding the lateral force and,

although not so severe, with the downward force as well. If this information is

not provided a priori to the controller, the resulting forces request will demon-

strate to be inadequate. A first solution to this problem was found by providing

the gain scheduling closed-loop dynamics as reference to the dynamic inversion

controller, instead of the reference trajectory dynamics, since the linear model

provides indirect information on the airship actuation limits. This solution,

however, limits the performance of the dynamic inversion controller to that of

the gain scheduling provided as model.

159

The Backstepping controller is designed formulating a scalar positive function

of the system states and then choosing a control law based on the airship six-

degrees-of-freedom nonlinear model to make this function decrease, therefore

guarantying the asymptotic stability of the controller. Some practical issues

are addressed, and the control law is improved to take into account input

saturations and wind disturbances, maintaining its asymptotic stability for a

bounded wind estimation error. The control law implementation again raises

the problem of the airship underactuation. This time, the solution found in

the dynamic inversion case is not applicable. The answer to the problem was

then to provide the controller with shaped attitude references. The idea is to

delay the attitude rectification in benefit of the correction of the transversal

lateral and vertical errors.

All three control solution, Gain Scheduling, Dynamic Inversion and Backstep-

ping, proved to be capable of executing complete missions considering take-off

and landing, path-tracking and stabilization, in the presence of realistic wind

disturbances. An assessment of the advantages and disadvantages of each con-

troller, as well as a comparison between them, was also made, providing an

overall insight of the autonomous airship control problem and of the solutions

proposed.

These solutions, having already demonstrated their value, have still issues to

evolve. Variations of the proposed solutions, namely the ones based on the non-

linear airship model, are already under development. One is the evaluation of

the reference shaping solution used in the Backstepping as an alternative to us-

ing the Gain Scheduling as dynamic model in the Dynamic Inversion solution.

Other evolution pertains the control allocation from forces to actuators, which

is not a straightforward procedure since the relation between forces and actu-

ators is not invertible, requiring sometimes an empiric solution. Moreover, the

allocation process is, till now, executed after the control request computation,

leaving the controller blind to the actuators limitations, reason for which an

attitude references shaping is necessary. The inclusion of an improved control

allocation, even keeping the references shaping to minimize the airship under-

actuation effects, into the controller design is a key factor in obtaining a more

satisfying overall nonlinear control solution.

160 CHAPTER 9. CONCLUSIONS AND FUTURE WORK

Appendix A

Referentials

Contents

A.1 Frames definition . . . . . . . . . . . . . . . . . . . . 161

A.1.1 Earth-Centered Inertial (ECI) frame . . . . . . . . . 161

A.1.2 North-East-Down (NED) or i frame . . . . . . . . 162

A.1.3 Aircraft-Body Centered (ABC) or l frame . . . . . 162

A.1.4 Aerodynamic or a frame . . . . . . . . . . . . . . 163

A.2 Changing frame . . . . . . . . . . . . . . . . . . . . 163

In this chapter we define the referentials used in the description of the airship

dynamics and kinematics, and how they are related.

A.1 Frames definition

To describe the dynamics of an airship, one needs to set up a coordinate frame.

Different coordinate frames may be used to describe the airship motion. The

following summarizes some of the coordinate frames [1, 53] used while modeling

the airship motion. Figure A.1 shows the spatial relationship between the

coordinate systems.

A.1.1 Earth-Centered Inertial (ECI) frame

The ECI frame is centered at the origin of the Earth. The z-axis coincides

with the Earth’s spin axis, pointing to the north Pole and the x-axis points in

the direction of the vernal equinox (the vernal equinox is an imaginary point

161

162 APPENDIX A. REFERENTIALS

ECI frame

y

lz l

x

iz

iy

ix

z

y

ABC frame

Equatorx

ωE

NED frame

l

Figure A.1: Relationship between the different coordinate systems.

in space which lies along the line representing the intersection of the Earth’s

equatorial plane and the plane of the Earth’s orbit around the Sun or the

ecliptic). Finally, to complete an orthogonal right handed system, the y-axis

is perpendicular to the xz-plane.

A.1.2 North-East-Down (NED) or i frame

The NED frame is centered on the Earth’s surface at the point vertically below

the airship’s Center of Gravity (CG), at its initial location, where it is fixed.

The xy-plane is tangent to the Earth’s surface. The x-axis points in the north

direction, the y-axis to the east and the z-axis is normal to the Earth’s surface,

pointing inward.

In this work the Earth will be assumed as flat and taken as an inertial frame.

It will be referenced as i frame.

A.1.3 Aircraft-Body Centered (ABC) or l frame

The ABC or local frame (referenced as l frame) is a right handed orthogonal

axis system fixed to the air vehicle. In order to accommodate the constantly

changing CG, the ABC frame is centered at the airship’s Center of Volume

(CV), assumed to be also the Center of Buoyancy (CB), and constrained to

move with it. The x-axis is coincident with the axis of symmetry of the en-

velope and the xz-plane coincides with the longitudinal plane of symmetry of

the airship (see fig. A.2). It is reasonable to assume both the CV and the CG

A.2. CHANGING FRAME 163

lie on the axis of symmetry of the envelope.

w

t

xl

xa

zl

yl

r,ψ

p,φ

q,θ uv

Vrelativeair ( )

α

β

Figure A.2: ABC and wind frames.

A.1.4 Aerodynamic or a frame

The Aerodynamic frame considers the relative aerodynamic incidence angles.

It is obtained from the l frame with two rotations (see fig. A.2):

1. rotation about the yl-axis for the angle of attack α,

2. rotation about the resulting z-axis for the side-slip angle β.

This makes the x-axis coincide with the direction of the total relative air

velocity Vt. The angles α and β are known as the aerodynamic angles and are

needed to specify the aerodynamic forces and moments.

The complete transformation from the body l frame to the aerodynamic aframe is then given by the Sa matrix, expressed as function of the aerodynamic

angles α and β, and given by:

Sa =

cosα cos β sin β sinα cos β

− cosα sin β cos β − sinα sin β

− sinα 0 cosα

(A.1)

A.2 Changing frame

The time derivative is defined in the inertial frame. The time derivative from

inertial i to local l frame introduces the Coriolis acceleration:

dv

dt i=

dv

dt l+ ω × v = v + ω × v (A.2)

164 APPENDIX A. REFERENTIALS

Following the assumption of a rigid body, the linear velocity of the CG (point

C) is related to the linear velocity of the CV (O) through the angular velocity:

vc = v0 + ω × OC = v − OC × ω (A.3)

The transformation from the inertial reference to the local frame is achieved

by the following sequence of rotations:

1. rotation about the zi-axis (positive yaw angle);

2. rotation about the resulting y-axis (positive pitch angle);

3. rotation about the resulting x-axis (positive roll angle).

where the roll φ, pitch θ and yaw ψ angles are commonly referred to as Euler

angles (see fig. A.2).

The complete transformation from the inertial i to the local l frame is

then given by the S matrix (often called Direction Cosine Matrix), expressed

as function of the Euler angles Φ = [φ, θ, ψ]T and given by (2.17).

Appendix B

Dryden Model For Continuous

Gust

This Appendix mostly follows reference [54].

To generate the gust signals with the required intensity, scale lengths and

power spectral density (PSD) functions for some given velocity and height,

a white-noise source with a PSD function ΦN(ω) = 1 is used to provide the

input signal to a linear filter, chosen such that it has an appropriate frequency

response so that the output signal from the filter will have a PSD function

Φi(ω). The scheme is represented in the block diagram shown in fig. B.1.

iG (s)white noisegenerator

linear filter

Φ (ω)Φ (ω)Ni

Figure B.1: Block diagram for gust generator.

The relation of the PSD function of the output signal to the PSD function of

the input signal is given by:

Φi(ω) = |Gi(s)|2s=jωΦN(ω) (B.1)

The filters needed to generate the appropriate spectral densities for the trans-

165

166 APPENDIX B. DRYDEN MODEL FOR CONTINUOUS GUST

lational gust velocities are:

Gu(s) =

√Ku

s+ λu(B.2)

Gv(s) =√

Kvs+ βv

(s+ λv)2(B.3)

Gw(s) =√

Kws+ βw

(s+ λw)2(B.4)

where

Ku =2Vtσ

2u

Luπ, Kv =

3Vtσ2v

Lvπ, Kw =

3Vtσ2w

Lwπ(B.5)

βv =Vt√3Lv

, βw =Vt√3Lw

(B.6)

λu = Vt/Lu, λv = Vt/Lv, λw = Vt/Lw (B.7)

The turbulence intensity σ reaches its maximum value of 7 m/s in a thunder-

storm scenario. The turbulence scale length varies with height. The depen-

dence of scale length on height is defined in this manner:

h > ho ⇒ Lu = Lv = Lw = ho (B.8)

h ≤ ho ⇒ Lu = Lv =√

hoh, Lw = h (B.9)

where ho = 533 m and h is the height of the airship encountering the turbu-

lence.

Appendix C

Differential geometry and

topology

Contents

C.1 Lie derivatives . . . . . . . . . . . . . . . . . . . . . 168

C.2 Diffeomorphisms and state transformations . . . . 169

The purpose of this appendix is to introduce some mathematical tools from

differential geometry and topology, in the context of nonlinear dynamical sys-

tems [65, 64].

A vector function f : Rn → Rn is called a vector field in Rn. Only smooth

vector fields shall be considered, which means the function f(x) has continuous

partial derivatives of any required order.

Given a smooth scalar function h(x) : Rn → R of the state x, the gradient of

h(x) is defined as:

∇h(x) =∂h

∂x(C.1)

The gradient is represented by a row-vector of elements (∇h)j = ∂h/∂xj.

Similarly, given a vector field f(x), the Jacobian of f is defined as:

∇f(x) =∂f

∂x(C.2)

and is represented by a n× n matrix of elements (∇f)ij = ∂fi/∂xj.

167

168 APPENDIX C. DIFFERENTIAL GEOMETRY AND TOPOLOGY

C.1 Lie derivatives

Given a scalar function h(x) and a vector field f(x), a new scalar function

Lfh(x) is defined, called the Lie derivative of h with respect to f .

Definition C.1 (Lie Derivative). Let h : Rn → R be a smooth scalar function,

and f : Rn → R

n be a smooth vector field on Rn. Then the Lie derivative of h

with respect to f is a scalar function defined by Lfh = ∇h f .

Thus, the Lie derivative Lfh(x) is simply the directional derivative of h(x)

along the direction of the vector f(x).

If h is being differentiated k times along f , the notation Lkf h is used; in other

words, the function Lkf h satisfies the recursion

L0fh(x) = h(x) (C.3)

Lkf h(x) = Lf (Lk−1f h) = ∇(Lk−1

f h) f for k = 1, 2, ... (C.4)

Similarly, if g(x) is another vector field, then the scalar function LgLfh(x) is

LgLf h(x) = ∇(Lf h) g(x) (C.5)

This simple example will show the relevance of Lie derivatives to dynamic

systems. Consider the following single-output system:

x = f(x) (C.6)

y = h(x) (C.7)

The derivatives of the output are

y =∂h

∂xx = Lfh(x) (C.8)

y =∂(Lfh)

∂xx = L2

fh(x) (C.9)

and so on.

C.2. DIFFEOMORPHISMS AND STATE TRANSFORMATIONS 169

C.2 Diffeomorphisms and state transformations

The concept of diffeomorphism can be viewed as a generalization of the familiar

concept of coordinate transformation. It is formally defined as follows:

Definition C.2 (Diffeomorphism). A function φ : Rn → R

n, defined in a

region Ω, is called a diffeomorphism if it is smooth, and if its inverse φ−1

exists and is smooth.

If the region Ω is the whole space Rn, then φ(x) is called a global diffeomor-

phism. Global diffeomorphisms are rare, and therefore one often looks for local

diffeomorphisms, i.e., for transformations defined only in a finite neighborhood

of a given point. Given a nonlinear function φ(x), it is easy to check whether

it is a local diffeomorphism by using the following lemma:

Lemma C.1. Let φ(x) be a smooth function defined in a region Ω in Rn. If

the Jacobian matrix ∇φ is nonsingular at a point x = x0 of Ω, then φ(x)

defines a local diffeomorphism in a subregion of Ω.

A diffeomorphism can be used to transform a nonlinear system into another

nonlinear system in terms of a new set of states, similarly to what is commonly

done in the analysis of linear systems. Consider the dynamic system described

by

x = f(x) + g(x)u (C.10)

y = h(x) (C.11)

and let a new set of states be defined by

z = φ(x) . (C.12)

Differentiation of z yields

z =∂φ

∂xx =

∂φ

∂x(f(x) + g(x)) . (C.13)

One can easily write the new state-space representation as

z = f∗(z) + g∗(z)u (C.14)

y = h∗(z) (C.15)

170 APPENDIX C. DIFFERENTIAL GEOMETRY AND TOPOLOGY

where x = φ−1(z) has been used, and the functions f∗, g∗ and h∗ are defined

obviously.

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modeling and nonlinear control for airship autonomous flight

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