nonlinear design of 3-axes autopilot for … · short range skid-to-turn surface-to-surface homing...

NONLINEAR DESIGN OF 3-AXES AUTOPILOT FOR

SHORT RANGE SKID-TO-TURN

SURFACE-TO-SURFACE HOMING MISSILES

By

Abhijit Das

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE (BY RESEARCH)

IN

ELECTRICAL ENGINEERING

AT

INDIAN INSTITUTE OF TECHNOLOGY

KHARAGPUR

MAY 2006

Certificate

This is to certify that the thesis entitled “Nonlinear Design of 3-axes Autopilot for

Short Range Skid-to-Turn Surface-to-Surface Homing Missiles” submitted by

Abhijit Das for the award of the degree of Master of Science (by research) is a

record of bonafide research work carried out by him under our guidance and supervision

during the period 2003-2006. The results embodied in this thesis have not been submitted

to any other University or Institute for the award of any degree or diploma.

IIT, Kharagpur

1st May, 2006

Siddhartha Mukhopadhyay

Professor,

Department of EE

Indian Institute of Technology

Kharagpur -721 302, INDIA

Amit Patra

Professor,

Department of EE


Kharagpur -721 302, INDIA

INDIAN INSTITUTE OF TECHNOLOGY

Date: May 2006

Author: Abhijit Das

Title: Nonlinear Design of 3-Axes Autopilot for Short

Range Skid-to-Turn Surface-to-Surface Homing

Missiles

Department: Electrical Engineering

Degree: M.S. Convocation: May Year: 2007

Permission is herewith granted to Indian Institute of Technology tocirculate and to have copied for non-commercial purposes, at its discretion, theabove title upon the request of individuals or institutions.

Signature of Author

THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, ANDNEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAYBE PRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR’SWRITTEN PERMISSION.

THE AUTHOR ATTESTS THAT PERMISSION HAS BEEN OBTAINEDFOR THE USE OF ANY COPYRIGHTED MATERIAL APPEARING IN THISTHESIS (OTHER THAN BRIEF EXCERPTS REQUIRING ONLY PROPERACKNOWLEDGEMENT IN SCHOLARLY WRITING) AND THAT ALL SUCH USEIS CLEARLY ACKNOWLEDGED.

iii

To the loving memory of ’MA’.

From the album of my memory, I remember those early days of my life, a

naughty boy always tried to get rid of his mother’s domination. Now I

can understand that without Her domination, I may not be able to see

these days.

To ’BABA’, who had the arduous task of rising that incorrigible boy.

Acknowledgments

I would like to thank Prof. Siddhartha Mukhopadhyay and Prof. Amit Patra, my su-

pervisors, for their many suggestions and constant support during this research. I am

also thankful to Mr. Ranajit Das, Sc. ”C”, DRDL Hyderabad and Mr. Sourav Patra,

research scholar, Electrical Engineering, for their help through the early years of chaos

and confusion.

Abhijit Das

Systems and Information Lab

Dept. of Electrical Engineering


Kharagpur-721302

Abstract

Traditionally, missile autopilots have been designed using linear control approaches with

gain scheduling. Autopilot design is typically carried out in the frequency domain and

the plant is linearized around various operating points on the trajectory. Moreover, three

single axis autopilots are usually designed without considering the interaction among the

motion axes, i.e., the autopilots in each of the three axes are designed independently of

each other. Such designs can not handle the coupling among pitch-yaw-roll channels, es-

pecially under high angles of attack occurring in high maneuver zones. In the last decade,

design of missile autopilots has been extensively studied using modern control design

paradigms such as, robust control, feedback linearization, sliding mode control, singular

perturbation etc. But in most of these studies, realistic factors like fin saturation and lim-

itation of gimbal freedom have not been considered. One therefore cannot really evaluate

the performance and relative merits of these methods in practical applications. This work

presents a nonlinear multivariable approach to the design of an autopilot for a realistic

missile that overcomes these difficulties. At first, exact input-output (IO) feedback lin-

earization and decoupling have been carried out for the dynamic IO characteristics of the

inner rate loop of the pitch and yaw channels. In the process, the missile dynamics also

becomes largely independent of flight conditions such as missile velocity, air density etc.

This enables the design of scalar linear controllers for the inner rate loops. In this work

the superiority of the new nonlinear multi-input multi-output (MIMO) autopilot over a

traditional autopilot has been demonstrated through realistic simulation results in pres-

ence of closed loop guidance and seeker. However performance deteriorates when the plant

model is perturbed, due to aerodynamic uncertainties, from the nominal model. A robust

IO linearization technique is therefore needed to tackle the aerodynamic uncertainties in

the system. In this study H∞ and sliding mode techniques have been applied to design

a robust controller for the feedback linearized plant. A nonlinear Luenberger observer

for the missile airframe dynamics has also been designed for estimating the unmeasured

states that are required for feedback linearization. All the design and simulations have

been carried out in a realistic environment and thus the results presented in this thesis

may be useful for designing more accurate missile system in future.

Contents

1 Introduction 1

1.1 Missile basics: an overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Missile coordinate system . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Angle of attack(AoA), side-slip angle and aero-phi . . . . . . . . . . 2

1.1.3 Missile model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.4 Aerodynamic data used in 6−dof model . . . . . . . . . . . . . . . 6

1.1.5 Skid-to-turn steering law . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Missile subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.2 Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.2.1 Guidance sensors . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.2.2 Types of guidance . . . . . . . . . . . . . . . . . . . . . . 13

1.2.2.3 Main features of PN guidance . . . . . . . . . . . . . . . . 14

1.2.2.4 Design of trajectory/ guidance scheme . . . . . . . . . . . 14

1.2.2.5 Turning radius capability . . . . . . . . . . . . . . . . . . 16

1.2.2.6 Switching sight line range at PN start . . . . . . . . . . . 16

1.2.2.7 On board velocity computation . . . . . . . . . . . . . . . 16

1.2.3 Autopilot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.3.1 Lateral autopilot . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.3.2 Roll autopilot . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.4 Missile control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

CONTENTS ii

1.2.4.1 Aerodynamic . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2.4.2 Jet reaction forces . . . . . . . . . . . . . . . . . . . . . . 21

1.3 Typical problems with classical autopilots for highly maneuvering missiles . 22

1.3.1 Sources of roll disturbance . . . . . . . . . . . . . . . . . . . . . . . 22

1.3.2 Lateral and Roll aerodynamic Characteristics . . . . . . . . . . . . 24

1.3.3 Problems associated with roll rate . . . . . . . . . . . . . . . . . . . 25

1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.5 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.5.1 Feedback linearization based controller . . . . . . . . . . . . . . . . 29

1.5.2 Modern gain scheduling based controller . . . . . . . . . . . . . . . 30

1.5.3 Sliding mode controller . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.5.4 Linear and nonlinear robust controller . . . . . . . . . . . . . . . . 33

1.5.5 Model based adaptive controller . . . . . . . . . . . . . . . . . . . . 34

1.5.6 Nonlinear observer . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.5.7 Literature related to other control approaches . . . . . . . . . . . . 36

1.6 Contribution of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.7 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2 Autopilot Design with Input-Output Linearization by Feedback 39

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2 Brief theory and application to an nonlinear missile model . . . . . . . . . 40

2.2.1 Case I : Input-output linearization with q, r, p as outputs . . . . . . 42

2.2.1.1 Zero dynamics analysis . . . . . . . . . . . . . . . . . . . . 46

2.2.1.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . 47

2.2.1.3 Decoupling of three axes motion . . . . . . . . . . . . . . 49

2.2.1.4 Input-Output Linearization . . . . . . . . . . . . . . . . . 49

2.2.2 Case II: Input-output linearization with w,v and p as outputs . . . 62

2.2.2.1 Formulation of the problem for the STT missile . . . . . . 62


2.3 Nonlinear Luenberger observer design . . . . . . . . . . . . . . . . . . . . . 68

CONTENTS iii

2.3.1 Introducion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.3.2 Observer construction . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.3.3 Application to a realistic STT missile model . . . . . . . . . . . . . 72

2.3.3.1 Gain ’K’ calculation . . . . . . . . . . . . . . . . . . . . . 73

2.3.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3 H∞ Control of Feedback Linearized Inner Rate Loop Dynamics 80

3.1 Introducion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.2.1 A linear matrix inequality approach to H∞ control for designing K

in pitch plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.2.2 Design Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.2.3 Application to the Nonlinear Missile . . . . . . . . . . . . . . . . . 91

3.2.3.1 Uncertainty and disturbance modeling between y and v . . 91

3.2.3.2 Uncertainty and disturbance model for nonlinear missile . 94

3.2.3.3 Robust controller formulation . . . . . . . . . . . . . . . . 96


3.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4 Sliding Mode control after Feedback Linearization 106

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.2 Formulation of sliding mode controller . . . . . . . . . . . . . . . . . . . . 108

4.2.1 Step II: Formulation of sliding mode control law for designing v . . 108

4.2.1.1 Specifying sliding surfaces . . . . . . . . . . . . . . . . . . 109

4.2.1.2 Achieving sliding condition . . . . . . . . . . . . . . . . . 110

4.2.2 Application to the STT missile model . . . . . . . . . . . . . . . . . 111

4.2.2.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 113

4.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5 Conclusions 128

CONTENTS iv

A A Brief Theory of Feedback Linearization 131

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A.1.1 Input-output linearisation . . . . . . . . . . . . . . . . . . . . . . . 134

A.2 Multi input multi output systems . . . . . . . . . . . . . . . . . . . . . . . 139

A.2.1 Feedback Linearization of MIMO Systems . . . . . . . . . . . . . . 139

A.2.2 Zero-dynamics and control design . . . . . . . . . . . . . . . . . . . 144

B LMI Approach to H∞ Control 146

B.1 The Theory of H∞ Control based on LMI Approach . . . . . . . . . . . . . 146

B.1.1 Singular value decomposition . . . . . . . . . . . . . . . . . . . . . 146

B.1.2 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

B.1.3 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

B.1.4 Basis Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

B.1.5 L2 space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

B.1.6 L∞ space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

B.1.7 H2 space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

B.1.8 H∞ space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

B.1.9 Packed Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . 151

B.1.10 Robust Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

B.2 A Linear Matrix Inequality Approach to H∞ Control . . . . . . . . . . . . 157

B.2.1 Brief theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

B.2.2 Advantages of LMI . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

B.2.3 Basic Idea about LMI . . . . . . . . . . . . . . . . . . . . . . . . . 159

B.2.4 Matrices as variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

B.2.5 Lyapunov’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 161

B.3 Stabilizing Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

B.3.1 System Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

B.3.2 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

B.4 H∞ Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

B.4.1 Two important matrix inequalities . . . . . . . . . . . . . . . . . . 171

CONTENTS v

B.4.2 The KYP Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

B.4.3 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

B.4.4 Controller reconstruction and connections . . . . . . . . . . . . . . 190

List of Figures

1.1 Anti Tank Missile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Fixed body frame (x, y, z) and definitions . . . . . . . . . . . . . . . . . . . 3

1.3 Sideslip Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Maneuver Plane Roll Orientation; Aerophi(φ) . . . . . . . . . . . . . . . . 5

1.5 Skid to turn Steering law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Definitions of positive pitch, yaw and roll Control . . . . . . . . . . . . . . 10

1.7 Basic Steering / Roll Control . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.8 Autopilot functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.9 Missile seeker showing angular geometry . . . . . . . . . . . . . . . . . . . 13

1.10 PN guidance trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.11 General trajectory for a surface-to-surface STT missile . . . . . . . . . . . 15

1.12 Comparison of missile velocity and estimated velocity . . . . . . . . . . . . 17

1.13 Block diagram lateral autopilot . . . . . . . . . . . . . . . . . . . . . . . . 19

1.14 Block diagram of a roll autopilot . . . . . . . . . . . . . . . . . . . . . . . 21

1.15 Rolling moment coefficient characteristic . . . . . . . . . . . . . . . . . . . 24

1.16 Side force coefficient characteristic . . . . . . . . . . . . . . . . . . . . . . . 25

1.17 Yawing moment coefficient characteristic . . . . . . . . . . . . . . . . . . . 26

1.18 Rolling moment coefficient characteristic during PN maneuver . . . . . . . 27

1.19 Normal Force coefficient characteristics . . . . . . . . . . . . . . . . . . . . 28

2.1 Block diagram of the system with FBLC for outputs q, r and p . . . . . . . 43

2.2 Boundedness of u,v and w in zero dynamic condition . . . . . . . . . . . . 48

LIST OF FIGURES vii

2.3 Autopilot response with step command . . . . . . . . . . . . . . . . . . . . 50

2.4 Decoupling in the three channels . . . . . . . . . . . . . . . . . . . . . . . 51

2.5 Evidence of linearization in pitch channel . . . . . . . . . . . . . . . . . . . 52

2.6 Evidence of linearization in yaw channel . . . . . . . . . . . . . . . . . . . 53

2.7 Evidence of linearization in roll channel . . . . . . . . . . . . . . . . . . . . 54

2.8 Pitch and yaw latax profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.9 Effective pitch yaw roll deflection . . . . . . . . . . . . . . . . . . . . . . . 56

2.10 Fin deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.11 Fin deflection rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.12 Gimbal angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.13 Alpha and beta profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.14 Force and moment coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.15 Block diagram of the system with output w, v and p . . . . . . . . . . . . 63

2.16 Autopilot response with step command . . . . . . . . . . . . . . . . . . . . 66

2.17 Autopilot response with guidance command . . . . . . . . . . . . . . . . . 67

2.18 Effective Pitch yaw roll deflection . . . . . . . . . . . . . . . . . . . . . . . 68

2.19 Estimated and true α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.20 Estimated and true β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.21 Estimated and true α for full flight time . . . . . . . . . . . . . . . . . . . 77

2.22 Estimated and true β for full flight time . . . . . . . . . . . . . . . . . . . 78

3.1 Block diagram of the system representing robust feedback linearization for

outputs q, r and p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.2 Structure of robust control K in pitch plane . . . . . . . . . . . . . . . . . 83

3.3 General feedback arrangement . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.4 Mixed sensitivity configuration . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5 Complete model structure of the system and noisy signal . . . . . . . . . . 93

3.6 Characteristics of ∆pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.7 Characteristics of ∆yaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.8 Characteristics of ∆roll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

LIST OF FIGURES viii

3.9 Comparison of FBLC and robust controller in pitch plane . . . . . . . . . . 100

3.10 Comparison of FBLC and robust controller in yaw plane . . . . . . . . . . 101

3.11 Comparison of FBLC and robust controller in roll plane . . . . . . . . . . . 102

3.12 Control deflection comparison . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.13 Control deflection rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.1 Block diagram of the system representing robust feedback linearization for

outputs q, r and p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2 Linearization in pitch channel . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.3 Linearization in yaw channel . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.4 Linearization in roll channel . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.5 Control deflection comparison . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.6 Pitch, yaw and roll rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.7 α and β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.8 Gimbal angle profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.9 Pitch and yaw latax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.10 Pitch,yaw and roll deflections . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.11 Fin demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.12 Fin deflection rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.13 Force and moment Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 126

B.1 System interconnections: (a) Series connection, (b) Inversion, (c) Parallel

connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

B.2 Closed loop with loop transfer H . . . . . . . . . . . . . . . . . . . . . . . 152

B.3 Small gain stability in Nyquist space . . . . . . . . . . . . . . . . . . . . . 153

B.4 Baby small gain theorem for additive model error . . . . . . . . . . . . . . 154

B.5 Control sensitivity guards stability robustness for additive model error . . . 154

B.6 Baby small gain theorem for multiplicative model error . . . . . . . . . . . 155

B.7 Complementary sensitivity guards stability robustness for multiplicative

model error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

B.8 General feedback arrangement . . . . . . . . . . . . . . . . . . . . . . . . . 162

LIST OF FIGURES ix

B.9 Input-output stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

List of Tables

1.1 Calculations of aero-coefficients from αR and φ . . . . . . . . . . . . . . . . 7

1.2 Calculations of aero-coefficients on the basis of Table 1.1 variables . . . . . 8

1.3 Calculations of input-coefficients . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Calculations of aero-derivatives . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Variation in aerodynamic coefficients and thrusts in x− y − z directions . . 99

4.1 Variation in aerodynamic coefficients and thrusts in x− y − z directions . . 113

List of Symbols

(gx)f Gravity component in fin frame along x direction in m/s2

(gy)f Gravity component in fin frame along y direction in m/s2

(gz)f Gravity component in fin frame along z direction in m/s2

α Angle of attack in rad

β Side slip angle in rad

δω1 to δω4 Wing deflection due to misalignment in rad

δc1 to δc4 Fin deflection due to misalignment in rad

∆Xf , ∆Yf , ∆Zf Lateral C.G shift w.r.t nose of the missile in m

δp, δy, δr Control Deflection in pitch, yaw and roll respectively in rad

η1, η3 Control deflection for fins 1 and 3 in rad

φ Maneuver plane roll orientation in rad

ξ2, ξ4 Control deflection for fins 2 and 4 in rad

CL Rolling moment coefficient

CN Normal force coefficient

LIST OF TABLES xii

CS Side force coefficient

CDO Force Coefficients due to drag

CLp Rolling damping coefficient

Clzetaw Control moment coefficient for wing

Clzeta Control moment coefficient for control surface or fin

Cmq Damping moment coefficient due to pitch rate

Cmr Damping moment coefficient due to yaw rate

CmαDamping moment coefficient due to change in angle of attack

CmβDamping moment coefficient due to change in side slip

Cmpcg Pitching moment coefficient w.r.t C.G

Cnδ Control effectiveness

Cnαw Control force effectiveness for wing

Cnycg Yawing moment coefficient w.r.t C.G

CYr Damping force coefficient due to yaw rate

CYβDamping force coefficient due to β

Czq Damping force coefficient due to pitch rate

CzαDamping force coefficient due to change in angle of attack

D Diameter of the missile in m

fz, fy Lateral accelerations (latax) along z and y axes respectively m/s2

IXX Moment of inertia about x in kgm2

LIST OF TABLES xiii

IY Y Moment of inertia about y in kgm2

IZZ Moment of inertia about z in kgm2

m Mass of the missile

p, q, r Angular velocities round the x, y, z-axes respectively in rad/sec

Q Dynamic pressure of the missile in Pascal

S Surface area of the missile in m2

TX Thrust component in x direction in Newton

TX Thrust component in x direction in Newton

TY Thrust component in y direction in Newton

TZ Thrust component in z direction in Newton

Tmx Thrust misalignment along x in Newton

Tmy Thrust misalignment along y in Newton

Tmz Thrust misalignment along z in Newton

U, v, w Linear velocities along x, y and z axes respectively in m/sec

Vm Resultant velocity along trajectory in m/sec

x, y, z Body frame coordinates

Xcg C.G of missile w.r.t nose in m

Xcpωp C.P of wing w.r.t nose of the missile in m

Xcpc C.P of fin w.r.t nose of the missile in m

XLTP , XLTY Distance of hinge line from nose in elevation and azimuth in m

LIST OF TABLES xiv

CG Center of gravity

FBLCL FBLC with linear rate loop controller

FBLCR FBLC with robust rate loop controller

FBLCSM FBLC with sliding mode rate loop controller

FBLC Feedback linearizing controller

PN Proportional Navigation

LOS Line of sight

The units of aerodynamic coefficients are commensurate with the units of other

variables

Chapter 1

Introduction

A missile is a projectile that is, something thrown or otherwise propelled. The earliest

form of a missile was probably a stone that, when thrown forcefully through the air, would

follow a ballistic path. Adding gunpowder to a projectile, resulted in the rocket, the first

powered, but as yet unguided, missile [34]. Rockets were first invented in medieval China

(Circa 1044 AD) but its first practical use for serious purpose other then entertainment

took place in 1232 AD, by the Chinese against the Mongols at the siege of Kai-Feng-Fue.

Thereafter from 1750 AD to 1799 AD Haider Ali and Tipu Sultan (Sultan of Mysore, in

south India) perfected the rocket’s use for military purposes, very effectively using it in

war against British colonial armies. It was not until the early 1900s that guided missile

development was begun. In sofar as the missile as we know it today is concerned, the

impetus came primarily from World War II and in particular from German scientists.

Immediately after the war there was a rapid growth in missile activity through out the

world. Although this work is concerned with a particular type of short range missile

autopilot design.

Chapter 1. Introduction 2

1.1 Missile basics: an overview

The missile that is considered in this thesis is a short range, surface to surface, Skid-to-

Turn, homing missile and for an example some relevant parts of that kind of a missile are

shown in Figure 1.1.

Figure 1.1: Anti Tank Missile

1.1.1 Missile coordinate system

A fixed body frame coordinate system (x, y, z) is introduced and shown in Figure 1.2. Its

origin is placed at the center of gravity. The movement and position of the missile are

described in this coordinate system. The resultant speed of the missile, Vm, is given by

Vm =√

U2 + v2 + w2 where U is usually the major contributor as the x axis is generally

aligned with the thrust direction, or nearly so.

1.1.2 Angle of attack(AoA), side-slip angle and aero-phi

Angle of attack (α) is a term used in aerodynamics to describe the angle between the

missile’s body x axis and the direction of airflow wind, effectively the direction in which


Figure 1.2: Fixed body frame (x, y, z) and definitions

the Center of Gravity (C.G) of the missile is currently moving, as shown in this schematic

1.2. From the Figure the mathematical expression of α can be derived as α = tan−1(

wU

).

The sideslip angle(β) is defined by the angle between the velocity vector of the vehicle and

the longitudinal axis also shown in Figure 1.3 and from Figure 1.2, β = tan−1(

vU

). The

orientation of the tail fins with respect to the body axes for a cross or ’X’ configuration

and their conventional numbering are shown in Figure 1.4. The plane defined by the

velocity vector V and the body x- axis is called the maneuver plane. This is the plane

along which the resultant force of air acts on the missile body, in the direction opposite to

the velocity vector Vm. The term aerophi(φ) denotes orientation of the maneuver plane

w.r.t the fin axis f2-f4, the angle being measured from f2. The fin axis system is the body

axis system rotated 45 around the x−axis so that the y and z body axes align with the

fin planes. Figure 1.4 shows the aerophi and side force CS as well as yawing moment Cny

are defined 90 anticlockwise w.r.t maneuver plane (αR).

1.1.3 Missile model

The standard nonlinear differential equations relating the forces and moments acting on

a missile with its motion components are given by,


Figure 1.3: Sideslip Angle

Force equations

U = rv − qw + 1m

[TX −QSCDO] + (gx)f

v = pw − rU + QSm

Cy(α, β, Vm

Vs, δy) + TY

m+ (gy)f

w = qU − pv + QSm

Cz(α, β, Vm

Vs, δp) + TZ

m+ (gz)f

(1.1)

Moment equations

p = 1IXX

QSDCl(α, β, Vm

Vs, δr) + Tmx

IXX

q = 1IY Y

QSDCm(α, β, Vm

Vs, δp) + IZZ−IXX

IY Ypr + Tmy

IY Y

r = 1IZZ

QSDCn(α, β, Vm

Vs, δy) + IXX−IY Y

IZZpq + Tmz

IZZ

(1.2)

where, Cy and Cz are force coefficients and Cl, Cm and Cn are the moment coefficients

including aerodynamic variables and inputs. From Equation 1.1, we can subdivide the

force Euler equations in the following way: Force = coriolis component+ thrust compo-


Figure 1.4: Maneuver Plane Roll Orientation; Aerophi(φ)

nent + aerodynamic component + gravity component. For example, if we consider the

expression in Equation 1.1 of v, then, coriolis component=pw− rU , thrust component =TY

m, aerodynamic component = QS

mCy(α, β, Vm

Vs, δy), gravity component = (gy)f . Similarly

for moment equations we can write, moment = aerodynamic component + thrust com-

ponent + centrifugal component. For example, if we consider the expression in Equation

1.2 of q, aerodynamic component = 1IY Y

QSDCm(α, β, Vm

Vs, δp), thrust component = Tmy

IY Y,

centrifugal component = IZZ−IXX

IY Ypr. More detailed nonlinear 6-DOF equations which is

used for realistic simulations are given in [3], [29], [30], [13], [11], [12] and are expressed

as follows:


Force equations

U = rv − qw + 1m

[TX −QSCDO] + (gx)f

v = pw − rU + 1m

[TY + QSCNBn + CNBs + Clzie + D2Vm

(−CYββ + CYrr)]

+(gy)f + QSm

[Cnαw(δω2 − δω4) + Cnδ(δc2 − δc4)/2]

w = qU − pv + 1m

[TZ + QSCNAn + CNAs + Cleta + D2Vm

(−Czqq − Czαα)]

+(gz)f − QSm

[Cnαw(δω1 − δω3) + Cnδ(δc1 − δc3)/2]

(1.3)

Moment equations

p = 1IXX

[−IXXp + TmX + QSD D2Vm

CLpp− Clzeta

2δR + CL]

− QSD2IXX

[Clzetaw(δω1 + δω2 + δω3 + δω4) + Clzeta(δc1 + δc2 + δc3 + δc4)]

+ 1IXX

[(TY + QS(CNBn + CNBs + CLzie + D2Vm

(−CYββ + CYrr)))∆Zf

−(TZ + QS(CNAn + CNAs + Cleta + D2Vm

(−Czqq − Czαα)))∆Yf ]

q = 1IY Y

[(QSD[−Cmeta + CMAn + CMAs + D2Vm

(Cmqq + Cmαα)] + TmY )

−(IXX − IZZ)pr − IY Y q]− QSIY Y

[Cnαw(δω1 − δω3)(Xcpωp −Xcg)

+Cnδ

2(δc1 − δc3)(Xcpc −Xcg)] +

∆Zf

IY Y[−TX − (−QSCDO)]

r = 1IZZ

[(QSD[Cmzie + CMBn + CMBs + D2Vm

(Cmrr − Cmββ)]+

TmZ)− (IY Y − IXX)pq − IZZr]− QSIZZ

[Cnαw(δω2 − δω4)(Xcpωp −Xcg)

+Cnδ

2(δc2 − δc4)(Xcpc −Xcg)] +

∆Yf

IZZ[TX + (−QSCDO)]

(1.4)

1.1.4 Aerodynamic data used in 6−dof model

The aerodynamic coefficients and data used in the above six degree of freedom (i.e. a

set of six independent displacements that specify completely the displaced or deformed


position of the body or system) equations are mostly functions of angle of attack, side

slip, aero-phi and Mach number. From the wind tunnel test data, it is possible to express

these aerodynamic coefficients as polynomials in α, β etc. Some of these coefficients such

as CN , CS etc can be directly computed as a polynomial in αR and φ as shown in Table

1.1.

Aero-coefficient (force) CalculationsCN fn(φ, αR) = A2α

2R + A1αR

CS fn(φ, αR) =m∑

i=0

diαiR

Aero-coefficient (moment) CalculationsCmp fn(φ, αR) = B2α

2R + B1αR

CL fn(φ, αR) =p∑

i=0

CiαiR

Cny fn(φ, αR) =n∑

i=0

eiαiR

Table 1.1: Calculations of aero-coefficients from αR and φ

Some other aerodynamic coefficients are computed in fin frame given in Table 1.2 from

the aero-coefficients listed in Table 1.1.

Some of the coefficients which are associated with the inputs are calculated as given

in Table 1.3. These coefficients are mainly functions of Mach number.

The coefficients and aerodynamic derivatives given in Table 1.4 are taken to be con-

stant for simulations carried out in this thesis. Although these are also function of Mach

number. Since Vm is assumed constant, these are also assumed constants.

1.1.5 Skid-to-turn steering law

Two principal steering policies have been used in the design of tactical missile systems,

namely, skid-to-turn (STT) control and bank-to-turn (BTT) control. The former has

been the customary choice for most missiles in the past and the same steering law has

been used for the missile that we have considered in this thesis.


Aero-coefficient(force) in fin frame CalculationsCNAn −CN × cos(φ)CNAs −CS × sin(φ)CNBn −CN × sin(φ)CNBs CS × cos(φ)Aero-coefficient(moment) in fin frame Calculations w.r.t c.g.

Cmpcg Cmp + CN × Xcg

D

Cnycg Cny + CS × Xcg

D

CMAn Cmpcg × cos(φ)CMBn −Cmpcg × sin(φ)CMAs Cnycg × sin(φ)CMBs Cnycg × cos(φ)

Table 1.2: Calculations of aero-coefficients on the basis of Table 1.1 variables

Input-coefficient in fin frame CalculationsCnδ fn (Mach no), value=2.43Cleta Cnδ × δp

Clzie Cnδ × δy

CmetaCleta×(XLTP−XCG)

D

Cmzie−Clzie×(XLTY −XCG)

D

Table 1.3: Calculations of input-coefficients

Aerodynamic derivatives and Clzeta ValueClzeta fn (Mach no), value=1.0Czq 85.0Czα

52.0Cmq −420.0Cmα

−220.0CLp −10.0

Table 1.4: Calculations of aero-derivatives


Figure 1.5: Skid to turn Steering law

For an STT system as shown in Figure 1.5, the missile is roll stabilized in space and

performs aerodynamic maneuvers in each of two orthogonal planes, pitch and yaw, to

achieve a resultant maneuver in any desired planes. Generally these types of missile are

made to be roll stabilized and its four control surfaces are placed in a cruciform pattern as

shown in Figure 1.6. In this case, all of the four control surfaces are used simultaneously

for pitch, yaw and roll. If the surface deflections are considered to be positive when

the panel is rotated clockwise as one looks down the hinge line toward the body, then

pitch,yaw and roll control for the two cases can be defined in terms of individual surface

deflections by the formulas for δp = −δ1+δ2+δ3−δ44

, δy = δ1+δ2−δ3−δ44

and δr = δ1+δ2+δ3+δ44


Figure 1.6: Definitions of positive pitch, yaw and roll Control

1.2 Missile subsystems

A missile consists of several subsystems. The most closely related subsystems are shown

in the block diagrams of Figures 1.7 and 1.8, which describe the principal function of each

subsystem and list the principal elements of each. A short description of each subsystem

follows

Figure 1.7: Basic Steering / Roll Control


1.2.1 Navigation

When the missile is launched, its position, attitude, speed, acceleration and rotation are

to be known. The navigation subsystem updates these variables during the flight. This

is done by using sensor data and strap down navigation algorithms. The variables are

supplied to the guidance subsystems.

1.2.2 Guidance

Although we have not designed the guidance law for the missile, but for the sake of com-

pleteness, in the following sections, some relevant terms are described. The guidance

subsystem computes the error between the missile’s actual and desired courses, computes

the corrections necessary to reduce or nullify the error according to a chosen guidance law,

and gives commands to the autopilot to activate the controls to achieve the corrections.

These commands may be for lateral accelerations, angular rates, etc. The navigation

system contains sensors that provide information on the missile’s actual and desired tra-

jectories, noise filters and a computer in guidance system to process the information into

the commands to the autopilot.

Figure 1.8: Autopilot functions


1.2.2.1 Guidance sensors

The guidance system operates based on the relative target-pursuer kinematics, which are

defined by the following variables:

1. Line of sight (LOS) angles in azimuth and elevation

2. LOS rates

3. Range between the target and the interceptor

4. Range rate etc

These variables can be measured by different sensors based on indirect measurements.

One of the important sensors is seeker. This sensor is mounted on the radome of an

interceptor (in case of homing engagement) and detects the relative position vector of the

target with respect to the platform where it is mounted.

The line of sight(LOS), λ, is defined as the angle between a line from the center of the

seeker antenna to the target, and some arbitrary non-rotating(e.g. inertial) reference line.

LOS is generally defined for both azimuth and elevation planes. Figure 1.9 shows the

LOS in elevation plane only. Commonly, it is convenient to select this reference equal to

the LOS position at the beginning of the homing guidance phase. Consequently, λ(t) at

time t is the total change in the angular position of the LOS relative to the initial LOS.

Here θm is defined as the pitch angle in elevation. Furthermore the angular position of

the antenna centerline is defined by the gimbal angle θh in elevation. Therefore, the LOS

angle in elevation (λ) is given by

λ = θm + θh + ε

where, ε is the true boresight error, that is, the error between antenna centerline and line

of sight to the target. Same treatment can also be applied for azimuth plane.


Figure 1.9: Missile seeker showing angular geometry

1.2.2.2 Types of guidance

Many different guidance laws have been developed and used depending upon the capa-

bilities of the interceptor and the target and the engagement scenario. The examples of

some typical guidance laws are as follows:

1. LOS angle guidance

2. LOS rate guidance

3. Pursuit guidance

4. Attitude Pursuit guidance

5. Proportional navigation Guidance

6. Optimal guidance


Here, in this thesis we are concerned about the Proportional Navigation (PN) guidance

only as it is used for surface-to-surface STT missile that we have considered here.

1.2.2.3 Main features of PN guidance

1. Sightline angle (and rate) vary at the start of engagement

2. Later the sightline moves parallel to itself so the sightline rate → 0 and a shrinking

interception triangle is produced

Figure 1.10: PN guidance trajectory

1.2.2.4 Design of trajectory/ guidance scheme

For a typical short range surface-to-surface missile trajectory, two distinct trajectory/guidance

policies may be evolved based on target range. Up to a certain small target range; im-

pact angle optimized trajectory/ guidance policy has been considered, which maximizes

impact angle in the presence of seeker gimbal angle limit, airframe turning capability and


subsystem’s lags and is known as attitude hold phase. For target range above that

specified range, maximum height constraint on the trajectory is to be satisfied through

a ’Height limited trajectory/guidance policy’. However, during initial phase, to optimize

the impact angle, missile trajectory is made to pitch up as sharply as seeker gimbal angle

limit allows, through a gimbal angle hold(GAH) phase. The trajectory in elevation

plane is depicted in Figure 1.11. Switching from gimbal angle hold phase to PN guid-

ance phase is done as soon as the required sight line range or equivalently sight line rate

is reached. As the switching sight line range RSW is obtained, the missile jumps into

unsaturated PN phase from saturated PN phase correcting the heading error (gimbal

angle at switching) between sight line and flight path with minimum radius of turn ca-

pability. Required homing distance (dH) is available for settling the errors/transients in

Figure 1.11: General trajectory for a surface-to-surface STT missile

the final homing phase. Minimum requirement of homing distance is established with

normalized simulation of seeker based homing guidance loop for achieving the required

miss distance. The required RSW is derived from the known value of maximum gimbal

angle freedom θ, minimum radius of turn of the configuration R and the required hom-


ing distance dH. Therefore,Rsw is obtained as a fundamental parameter independent of

target range, prospective impact angle and velocity profile. In general for a surface-to-

surface STT missile, R is constrained by a tight limit on configuration size arising out of

tube launching requirement. For example, the finally achieved design value corresponding

to a configuration of a typical short range surface-to-surface STT missile may be raised

up to 5.5g maximum latax for a base velocity of 200m/s in one plane which is taken as

elevation and the demand in azimuth may be limited to 2g during maximum elevation

demand period.

1.2.2.5 Turning radius capability

The minimum turning capability of missile configuration R which is a function of Cnα,

maximum angle of attack permissible and air density is obtained for different range and

temperature. The maximum latax limit is calculated based on αmax(≈ 18) and low-

est velocity condition for each range and temperature and stored on OBC into a two

dimensional look up table as a function of range and temperature. The turning radius

R(R = U2m

latax) during SPN phase is obtained for each range and temperature.

1.2.2.6 Switching sight line range at PN start

A range independent relation for switching sight line range Rsw at PN start is obtained

from geometry so that the heading error θg is corrected with turning radius R, keeping

homing distance dH for settling of errors. Rsw is a function of turning radius R, dH and

gimbal angle at PN switching θg.

1.2.2.7 On board velocity computation

From cost and size consideration, a typical short range surface-to-surface homing missile

has no velocity sensor on board. Based on 6-DOF simulations, considering thrust versus

time profiles for each temperature and PN event, empirical formula based velocity algo-

rithm is developed on board for each range and temperature. The average acceleration


during booster and sustainer phase is stored on OBC into 2−D look up table as a func-

tion of range and temperature. The estimated velocity is matched with missile nominal

velocity at each range and temperature in Figure 1.12. This estimated velocity closely

matches the missile forward velocity u and is assumed to be known during flight.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

40

60

80

100

120

140

160

180

200

Normalised time

Res

ulta

nt v

eloc

ity (

Vm

)and

Est

imat

ed v

eloc

ity (

Ues

t)

Vm

Uest

Typical surface−to−surface missile

Figure 1.12: Comparison of missile velocity and estimated velocity

1.2.3 Autopilot

The autopilot receives commands from the guidance computer and processes them into

commands to the controls such as deflections or rates of deflection of control surfaces or

jet controls through action of servomechanisms. To provide the deflection at a desired

rate, the servomechanism motors must contend with the inertia of the control device

and the torque about its shaft. Since the autopilot will convert guidance commands of

acceleration or angular rate into control commands, it must have a way to determine


if the accelerations and angular rates provided by deflecting the controls are meeting

the guidance commands. Thus this subsystem will have accelerometers for measuring

the achieved accelerations and gyroscopes for measuring the angular positions or rates.

Depending on where these instruments are placed, the autopilot may have to provide

corrections to the instrument readings to obtain the true accelerations of the missile’s

center of gravity (CG) and true angular rates about its principal axes. Classically, the

missile autopilot comprises three independent autopilots, one for each lateral direction,

namely pitch and yaw, and one for roll.

1.2.3.1 Lateral autopilot

The missile autopilot controls the acceleration normal to the missile body. In this case

study, the autopilot structure is a three-loop design using measurements from an ac-

celerometer located ahead of the missile’s center of gravity and from a rate gyro to

provide additional damping. Figure 1.13 shows the classical configuration of an autopi-

lot. The controller gains are scheduled based on Mach number and tuned for robust

performance at an altitude of around 4000 meters (10000 feet) in general for small range

surface-to-surface STT missiles. According to Figure 1.13, there are three feedback

loops present. Since control of acceleration is required, the outermost loop is closed

by an accelerometer. This outer accelerometer loop has the lowest bandwidth of the

three loops. The innermost rate-damping loop is required to damp the response of

the bare airframe, which has an under-damped resonance in the stable case. In addi-

tion, the innermost rate-damping loop has a wide bandwidth for damping the poles

of airframe. The last one, known as synthetic stability loop, improves the high frequency

poles of the autopilot if the airframe is stable, and enables the autopilot to tolerate some

instability of the airframe. Furthermore the synthetic stability loop in Figure 1.13 effec-

tively feeds incremental pitch angle back to the fin servos, thereby moving the autopilot

closed-loop poles, corresponding to the bare airframe poles. Three important parameters

of the 3-Loop Autopilot (Figure 1.13) i.e., system damping, time constant of desired Latax

demand transfer function fz

fyand open-loop cross over frequency of innermost loop need

to be tightly controlled. Controlling system damping ensures that guidance system is not


Figure 1.13: Block diagram lateral autopilot

sensitive to body rate coupling. Selecting and controlling system time constant to the

specified value as per guidance loop requirement tightly means that adequate perfor-

mance in terms of miss distance can be achieved. Finally, controlling open-loop cross

over frequency means that we will have a robust design, which is not overly sensitive

to un-modeled high frequency dynamics.

1.2.3.2 Roll autopilot

The basic function of the roll autopilot is to make the missile roll stabilized, that

is, to provide missile stabilization of roll attitude about the longitudinal axis. This is

accomplished by sensing roll rate, and using the fins (or wings) defection by an amount

sufficient to counteract roll disturbances. Moreover, the response of the system must

be sufficiently fast to prevent the accumulation of significant roll angles. A block diagram

of the roll autopilot is shown in Figure 1.14. One common type of roll autopilot utilizes

a spring-restrained rate gyroscope for measurement of roll rate, in conjunction with

proportional-plus-integral (PI) compensation in the autopilot amplifier, in order to give


the approximate equivalent of roll-rate plus roll-angle feedback. Other roll autopilot de-

signs utilize a free vertical gyroscope as an attitude reference. That is, in order to maintain

a desired roll angle, an attitude reference must be used. A simple PI controller is used

for controlling the missile roll dynamics. To overcome the uncertainties during high angle

of attack, a roll autopilot design concept based on on-board disturbance estimation

has been formulated. The roll autopilot (Figure 1.14) estimates the disturbance torque on

board based on flight measurements and generates the required feed-forward command to

the actuator for correcting the effect of disturbances. The expression for disturbance

torque (Td) is obtained from the equation of motion in roll as:

Td = IXX p + 2QSDCLζδfb

where, δfb = effective fin deflection for roll obtained from control surface deflections

measured. Q = dynamic pressure. IXX = moment of inertia (roll axis). The required

feed-forward command δff to cancel the disturbance torque is obtained from the torque

balance equation as: Feed-forward control torque= disturbance torque. Or,

2QSDCLζδff = IXX p + 2QSDCLζδfb

∴ δff = δfb + IXX

2QSDCLζ× p

, (1.5)

Therefore, the required feed-forward command is obtained from effective fin deflection for

roll δfb measured in flight and roll angular acceleration measured from flight measure-

ment of p. In order to cater for estimation/ measurement error, 80% of the feed-forward

command as obtained in (1.14), is applied in the proposed design .

1.2.4 Missile control

The choice of control system will depend on the use of the missile, flight path and the

height at which it will operate. There are two main types of control system used:


Figure 1.14: Block diagram of a roll autopilot

1.2.4.1 Aerodynamic

Aerodynamic change of flight path is obtained by using a wing at incidence to the direction

of flight. The wing is set at incidence in one of two ways:

• By using a rotatable wing.

• By using a fixed wing, but a movable control surface to set the whole missile at

incidence. The control surfaces may be at the front or rear of the missile.

1.2.4.2 Jet reaction forces

Change of flight path by jet reaction forces is achieved in one of two ways:

• By altering the direction of thrust of the propulsion unit, either by swivelling the

whole unit, or by deflecting the gas stream by vanes or similar devices.


• By the use of separate auxiliary jet reaction units.

1.3 Typical problems with classical autopilots for highly

maneuvering missiles

The classical autopilots suffer from the problems of instability due to their highly nonlin-

ear and uncertain aerodynamic characteristics. At large angles of attack (AoA) or at high

maneuvering zones, missile flight dynamics become highly nonlinear, due to significant

amount of cross-coupling between three axes. Also, almost all missiles have significant

nonlinearities, associated with limitations in the movement of aerodynamic control sur-

faces. The other forms of uncertainties are namely as model variations in mass, inertia

and center of gravity positions, aerodynamic tolerances, air data system tolerances, struc-

tural modes, failure cases, etc. Due to the issues outlined above, all flight control laws

are required to undergo a rigorous certification (or clearance) process before being evalu-

ated in flight tests. The search for worst-case control inputs is an important part of this

process.

Generally, missiles with clean configuration (no protrusions) exhibit smooth aerodynamic

characteristics. However, in many missiles due to functional requirements, some ad-

ditional objects such as wire tunnels, lifting lugs, nozzles, antenna are required for a

complete configuration. The existence of these devices modifies the flow field around the

lifting surfaces and in turn longitudinal and lateral aerodynamic characteristics of the

configuration. That is why the wind tunnel data shows undesired lateral aerodynamic

characteristics (side force, yawing moment and rolling moment) for a small surface to

surface missile configuration particularly at higher angles of attack.

1.3.1 Sources of roll disturbance

For low moment of inertia about roll axis of the missile, the effect of aerodynamic asym-

metry on performance is highly appreciable due to proportionately higher cross coupling

forces / moments with respect to control forces/ moments. With simultaneous pitch and


yaw maneuvers present, appreciable cross coupling roll torque at around lateral frequency

occurs mainly due to aerodynamic asymmetry. Several flight trial data show high roll rate

oscillation during both the maneuver phases. This roll motion is detrimental to the mis-

sion of the missile. Hence, it is imperative to control this undesired rolling motion within

the permissible limits. Flight roll rate recorded during SPN phase shows the recorded roll

rate profile of a flight test data during saturated PN phase where the angles of attack

is quite high. It shows that, during sustained ’PN’ phase; roll oscillation builds up from

lower value to a large value as 100 /sec at end. The FFT of this roll rate shows that the

frequencies of oscillation are mainly ≈ 3Hz (Roll-lateral frequency) and 5/6 Hz (roll fre-

quency). Yaw latax and rate also oscillate at frequency ≈ 3Hz. The oscillation gradually

builds up from lower value to higher value and has been seen in flight roll rate and yaw

dynamics profiles during the SPN phase. From the flight test data, it can be observed

that with desired pitch down maneuver, angles of attack builds up to higher value. Due

to uncertainty of rolling moment at higher angles of attack, these roll disturbance torque

can not be controlled accurately by the roll autopilot as the autopilot BW is limited by

the hardware. During SPN phase, roll rate increases. As fy = v + Ur − pw, therefore,

fy increases as w increases in maneuver to a high value. Hence, fy increases with high

fz leads to increase in missile maneuver plane roll orientation φ from desired orientation.

Aerodynamic characteristic shows that rolling moment coefficient CL increases with roll

orientation variation from desired values. This higher value of CL leads to higher roll

rate. Therefore, it is a vicious cycle of cross coupling of p ↑→ fy ↑→ φ ↑→ p ↑. Hence it

can be remarked that high frequency roll disturbance torque (3 to 7Hz) is found during

maneuver phases of several flight records. The roll autopilot design can not cater to this

disturbance torque as the autopilot stiffness at that frequency can not be improved due

to low stability. This disturbance torque leads to high roll rate oscillation during the

maneuver phases. Secondly, the onboard roll disturbance torque estimation scheme gives

inaccuracy due to high frequency of disturbance torque and actuator/computation delay.

This leads to inaccuracy in control torque generated. The unbalanced torque leads to

high roll oscillation in flight.


1.3.2 Lateral and Roll aerodynamic Characteristics

Aerodynamic forces and moments are important inputs for the controller design. The

wind tunnel test data obtained (used in 6 − DOF simulation) are thoroughly analyzed

and presented here for different roll orientations and resultant angles of attack. Figure

1.15 represents the rolling moment coefficient for different angles of attack during up

and down maneuver with roll orientation variation [52]. It is noticed that roll moment

uncertainty increases for angles of attack more than 15 . Also rolling moment value is

more for negative angles of attack (pitch down maneuver).

Figure 1.15: Rolling moment coefficient characteristic

Figure 1.16 represents the side force coefficient and Figure 1.17 represents the yawing

moment coefficient for different angles of attack during up and down maneuver with

±5 roll orientation variation. Side force and Yawing moment uncertainty with small

roll orientation variations increase with higher angle of attack. Therefore control force

requirement to cancel the disturbances will be more for higher angle of attack.


Figure 1.16: Side force coefficient characteristic

Figure 1.18 represents the rolling moment coefficient during pitch down maneuver for

different angle of attack. It is noticed that the rolling moment variation with respect

to small roll orientation is large for a high angle of attack. This is the reason for high

frequency roll disturbance at high angle of attack. The magnitude of roll disturbance

increases with the increase of angle of attack. Figure 1.19 shows the characteristics of

the normal force coefficients. The normal force coefficient increases almost proportionally

with the angle of attack specially in high angle of attack zones.

1.3.3 Problems associated with roll rate

For a high maneuvering missile with a small moment of inertia about roll axis, the roll

rate needs to be well controlled with in the specified limit. The problems associated with

unwanted roll disturbances are listed out after detailed investigations through 6 −DOF


Figure 1.17: Yawing moment coefficient characteristic

simulations. These are :

• The roll-pitch-yaw cross coupling effect. The roll induced yaw rate helps to build

up azimuth gimbal angle ( gimbal angle freedom in azimuth plane is only ±12 ).

• The lateral and roll aerodynamics force and moment characteristics show that with

roll rate, roll orientation changes and side force disturbances increase. To control

the yaw disturbances, yaw control requirement increases. And hence, with desired

pitch requirement, control goes to saturation level (mainly one pair of control fin)

with high yaw and roll disturbances.

• The roll rate and roll induced yaw rate are coupled to guidance demand through the

seeker body rate coupling. This causes high miss distance as guidance commands

are erroneous.


Figure 1.18: Rolling moment coefficient characteristic during PN maneuver

• The turning radius during SPN phase is limited with the limit of angle of attack. It

is already mentioned that with high angle of attack, the uncertainty of roll distur-

bances increases. The early switching and shallow turning deteriorates the terminal

performances in terms of impact angle and miss distance. Hence minimum range

increases in order to keep the angle of attack within permissible low value.

1.4 Motivation

In summary, the traditional three-loop autopilot which is being widely used in many

aerospace industry for most missile autopilot designs, suffers from the following problems.

• Coupling between the pitch-roll axes tends to cause high roll rates whenever a sharp

pitch maneuver is commanded.


Figure 1.19: Normal Force coefficient characteristics

• Due to high roll rate an induced yaw rate is generated as a result of roll-yaw coupling.

• To correct for the undesirable yaw rate, control deflections are commanded by the

yaw autopilot. This results in saturation of the deflection of one or more fins in the

presence of a simultaneous high pitch and roll channel input.

• Further, increased yaw rate causes the side slip angle to increase, causing gimbal

angle saturation in the azimuth plane. This can result in track loss of the missile if

the target goes out of the view of the seeker.

• Aerodynamic coefficients, especially side force coefficients (CS) and yawing moment

coefficient (Cny), increase in value, which in turn effectively increase the values of

yaw rate and yaw deflections. Thus the effect feeds on the cause and an instability

mechanism is set up.

The main motivation of this work is to achieve superior performance in terms of impact

angle, roll rates and miss distance over the traditional three-loop autopilot. In addition

to this some other performance criteria are to be achieved. For examples, roll angle has


to be kept within specified bounds due to asymmetric look angle freedom of the seeker in

elevation plane and image rotation vis-a-vis pixel addresses.

1.5 Literature survey

During the last decade, a significant research effort has been contributed in the area of

nonlinear missile autopilot design. The nonlinear design approaches found during survey

are briefly classified into following methods.

1. Feedback linearization based controller

2. Modern gain scheduling based controller

3. Sliding mode controller

4. Robust controller

5. Model based adaptive controller.

6. Nonlinear observer

7. Other approaches

1.5.1 Feedback linearization based controller

In [53] a new nonlinear controller has been proposed considering the coupling effect on a

more or less generalized missile model. In the paper it has also been sshown that through

a kind of partial linearization along with singular perturbation techniques, it is possible

to transform the non-minimum phase missile dynamics into an approximate input-output

feedback linearizable system and thereby achieving a linear input-output dynamics, which

is decoupled as well as independent of flight conditions. An almost similar methodology

has been developed in [52]. However, some aspects have not been considered in these two


papers like actuator saturation and robust stability of the system. Another related paper

[62] demonstrates some nonlinear methodology like feedback linearization approach along

with a linear controller on a short range air-to-air missile. Here the superiority of the non-

linear autopilot over the traditional autopilot based on gain scheduling technique has also

been shown through simulation studies. The example of robust feedback linearization has

been given in the [35]. In this paper a HAVE DASH missile system is being linearized with

input-output linearization technique and then a sliding mode control has been designed

for robustness. This paper presents a novel approach to design robust feedback lineariza-

tion with sliding mode control. The states are assumed to be known. In [89] input-output

approximate linearization of a non-linear sixth order system has been studied. The order

of the zero dynamics has been made forcefully zero by increasing the relative degree of

the system. The design procedure may not be applicable for complicated missile systems

where the aerodynamic coefficients can not be represented as a function of state variables

of the system. The paper [84], presents a novel systematic approach for the autopilot

design of STT missiles. First, the nonlinear model of a STT missile is partially linearized

via functional inversion techniques and then, the additional set-point tracking controller

is designed by the well-known LMI approach. The stabilization conditions are given in

terms of LMI’s. Feedback Linearization is the way of converting a nonlinear system into

a linear one and thus may be considered as the best choice for designing the nonlinear

missile autopilot. The above review revealed that the problem of decoupling has not yet

been well addressed with all practical considerations like fin saturation, exceeding of az-

imuth gimbal angle limitations etc. The other performance criteria like minimizing miss

distance, maximizing the impact angle are also not taken into account. Moreover in most

of the cases the missile or system model taken is not a realistic one.

1.5.2 Modern gain scheduling based controller

The paper [90] deals with gain scheduled control system synthesis, applied to a missile

autopilot design problem. Here low order linear autopilots designed at discrete operating

points using classical control techniques are gain scheduled. Also, the Mach number,


modeled as a measured, time-varying exogenous signal, is viewed as a disturbance to be

rejected in the linear design phase, leading to improved steady-state tracking performance.

[73] presents two nonlinear controller designs for a bank-to-turn, air-to-air missile. The

first controller is a gain-scheduled H∞ design and the second is a nonlinear dynamic

inversion design using a two time scale separation. Comparisons in simulation results

for the two methods are also given. Different linear to nonlinear approaches on missile

autopilot design are reported in [16] and the comparisons among the simulation results

of the various controller has been also been presented. Another paper [71] presented

sequentially the tuning procedures for two schemes of lateral acceleration autopilots and

for one roll autopilot. They also presented the design relationship between plant and

controller. [55] has designed the missile autopilot based on gain scheduling with sequential

tuning for lateral as well as roll autopilot. [62] presents linear and nonlinear procedures

for designing the missile autopilot via feedback linearization as well as gain scheduling

techniques. [60] shows how the time-scale separation helps to improve the robustness

of feedback linearized autopilots by simplifying the feedback linearization maps, and by

permitting the design of low-order controllers. This paper presents the development of

three distinct time-scale separation schemes for the design of feedback linearized missile

autopilots. Gain scheduling based controller will be more effective for the systems where

the order of nonlinearity and uncertainties is not that much of concern. And so the

direct application of this method specially to a aerodynamic missile may not provide

effective results in practice. In those cases adaptive gain scheduling methods may be

useful depending upon complexity of aerodynamic uncertainties. In this thesis we have

not discussed about this method.

1.5.3 Sliding mode controller

A comparison via Monte-Carlo simulations is made of Sliding Mode Observers versus

Kalman Filter in the homing missile guidance system using different guidance laws has

been given in [80]. It has been shown that Sliding mode observer shows more accurate

results in terms of miss distance than the Kalman Filter due to its noise. The paper [74]


presents an optimal sliding mode control design method for a nonlinear system with a

cascade or two-loop structure.The control design method is demonstrated on a nonlinear

model of an F-16 aircraft. But the aircraft model has been taken to be much simpler

than any other practical model. [82], [33], [81], [59], [50] and [43] are the examples of

time varying sliding mode control, traditional high order sliding modes, dynamic sliding

modes, terminal sliding modes and sliding mode observer with gain adaptation. Simu-

lation results have been presented along with a reusable launch vehicle model, satellite

formation as well as integrated guidance and control application of aerodynamic missiles.

A sliding mode controller is shown in [78] for an integrated missile autopilot and guidance

loop. Motivated by a differential game formulation of the guidance problem, a single slid-

ing surface, defined as the zero-effort miss distance, is used. [96] presents a good example

of second order sliding mode control along with backstepping approach application on a

aerodynamic missile. Another three papers [42], [94] and [36] give the example of applica-

tion of Fuzzy-Neural and adaptive fuzzy sliding-mode control and traditional sliding mode

approach to a BTT missile. The example of robust feedback linearization has been given

in [35]. In this paper a HAVE DASH missile system is being linearized with input-output

linearization technique and then a sliding mode control has been designed for robustness.

This paper presents a novel approach to design robust feedback linearization with sliding

mode control. The states are assumed to be known. [4] shows the robust smooth MIMO

sliding mode controller with finite reaching time. A sliding mode estimator is also pro-

vided to eliminate the effects of unmodeled, bounded disturbances and uncertainties. In

[79], an integrated two-loop guidance and flight control system is designed to incorporate

a variety of guidance strategies and robustly enforce them regardless of target maneuvers,

atmospheric disturbances, and dynamic uncertainty of airframe actuator. Backstepping

approach and sliding mode approach have been introduced simultaneously for designing

the controller. Aerodynamic coefficients of most of the missiles in general express their

uncertain characteristics during flight in atmosphere. The application of sliding mode con-

trol may be one of the solutions to get rid of this problem. The literature survey shows

that the only problem regarding the implementation of sliding mode control is chattering.

In this thesis we have used this method, to cater to aerodynamic uncertainties. [35] shows


the application of sliding mode control for a HAVE DASH II missile and here we have

tested in a more realistic skid-to-turn homing missile model.

1.5.4 Linear and nonlinear robust controller

[57], [31] and [24] present a robust multivariable autopilot design for missile system. In the

first case, a canonical robust control design formulation is introduced followed by three

robust autopilot designs. The next one shows the gain scheduled control performance

along with H∞ control application. The third and last one deals with the H∞ as well as

µ synthesis approaches successfully. The simulation shows that the designs achieve good

response against significant kinematic and inertia couplings and aerodynamic parameter

variations. In both cases it is assumed that a linear model for the nonlinear missile can

be found properly. In [37] a design method is proposed namely, dynamic robust recursive

control, in order to obtain output tracking performance for a general class of nonlinear

systems. Application of this method was then demonstrated for the design of a robust

autopilot controller for a nonlinear missile model with structured uncertainty. Simulation

results for this system demonstrated that the dynamic robust recursive design was able

to achieve significant performance improvements over a more conventional I/O linearized

controller design. But all the simulation results has been given using a second order

example. [68] applies a result on output feedback guaranteed cost control of stochastic

uncertain systems to the problem of designing a missile autopilot. Missile autopilot model

has been taken as a simple one. A total least squares approach is used to fit the data

to a norm bounded uncertain system model. [56] presents a robust H2 and H∞ control

design for a HAVE DASH II missile system using a generalized Hamiltonian formulation.

The design endures significant kinematic and inertia couplings, aerodynamic parameter

variations and high frequency flexible effects. [64] presents a roll-yaw autopilot for a non-

axis symmetric missile model to robustly decouple roll tracking from yaw regulation. The

controller structure exhibits considerable inherent robustness and decoupling capability

without high actuator activity, providing a useful framework for dealing with truly NLTV

problems. The time-varying closed-loop PD-spectrum allows real-time adjustment of


bandwidth, thereby achieving in flight tradeoffs among performance, energy consumption,

robustness and other operation concerns. [65] also deals with the robustness analysis

of the missile autopilot. A different approach has been pointed out for designing the

nonlinear robust controller although simulation results has been given on the basis of a

second order nonlinear missile model. In the paper [57] robust multivariable autopilot

designs are examined through a canonical robust control design formulation. Although

the simulation results are given for HAVE DASH missile system, coupling effect has not

been taken fully into account. Another example of application of state feedback control

has been given in [87]. In this paper two different control laws for roll rate and acceleration

of a air-to-air missile are shown along with simulation results. It is also shown that the

results are applicable for both STT and BTT missile models with time varying flight

conditions. The estimator for α and β have been left for future study. Robust H∞ control

is one of the most popular approaches from 1992 for tackling the uncertainties. But as

the application of this method directly to a nonlinear system is not as easy as for a linear

one. Literature survey shows that direct application of H∞ control to a realistic short

range missile model considering different types of uncertainties and disturbances in full

amount is rare. We have applied the same method in this thesis to a feedback linearized

missile model.

1.5.5 Model based adaptive controller

Applications of Adaptive control and Neural Network in missile autopilot design can be

found in the literature surveys [72]. Some research work is also found in the field of opti-

mal/classical approach of missile autopilot design [66]. Model based adaptive controller is

effective for system with uncertainties. We have not considered this method in our design.

1.5.6 Nonlinear observer

[22] presents a nonlinear approach of observer design to Inverted Pendulum model in the

form of a extended Kalman Filter. But the observer gains are restricted for stability rea-

sons. [5] gives a high-gain observer for a large class of nonlinear systems. This observer is


designed to work for systems with bounded state. [10] presents the traditional approach of

designing the nonlinear state observer. Stability of the observer is also proved along with

the drift-observability property shown to be a sufficient condition for existence of the expo-

nential observer. A new approach of designing the nonlinear observer has been proposed

in [15]. Carleman linearization technique has been used to get exact input-output lin-

earization and the simulation results have been given as an example of good performance

with a simple nonlinear system. [88], [17], [45] and [77] give the idea of designing non-

linear observer using backstepping, high gain extended Kalman Filter, and constant gain

exponential approach. [86] is a geometric study of finding general exponential observers

for nonlinear systems. Using center manifold theory, it derives necessary and sufficient

conditions for general exponential observers for Lyapunov stable nonlinear systems. [44]

presents the nonlinear state observer based on input-output linearization. Assumption

has been made that the internal dynamics of the system is globally stable. [49] describes

a linearizing feedback adaptive control structure which guarantees high quality regulation

of the output error in the face of unknown parameters. The simulation results have been

given based on a CSTR without considering the jacket dynamics. [47] presents a nonlinear

reduced order Luenberger observer for estimating the states of a nonlinear system. [39]

and [40] present a reduced order nonlinear observer with application to a permanent mag-

net synchronous motor. The authors show through simulation results that the proposed

observer can estimate the rotor position and velocity very accurately. Similar papers [25],

[85] apply the same reduced order observer with different formulation on chaotic systems

as well as reactors. [23], [58] and [2] give a different observer model for a large class of

MIMO nonlinear systems. The gain of the observer can be tuned with a single parameter

variation. Observer has been also tested in continuous tank reactors and bio-reactors. In

[69], the problem of robust state observation is tackled. A high-gain observer is employed

to carry out the state estimation of a continuous time uncertain nonlinear system subject

to external perturbations of stochastic nature. The papers [93] and [18] give two different

types of observer design. The first one with sliding mode observer and the second one

gives a large overview with Extended Luenberger and Extended Kalman filter approach

and along with their application to a bio-chemical process. Most of the linear and non-


linear feedback control requires state feedback for computing the plant input. Therefore

if all of the system states are not measurable then a state observer is required. The liter-

ature survey shows that implementation of nonlinear observer is common to most of the

nonlinear systems. But, often the measured outputs of the system are such that they can

not be easily expressed in terms of system states although they are directly or indirectly

functions of states. Observer design for these types of system is rare. In this thesis we

have designed a nonlinear Luenberger observer for these types of systems.

1.5.7 Literature related to other control approaches

Some more approaches to designing the missile autopilots have been reported in [75]

another different approach has been taken for autopilot deign using Linear Parameter

Varying (LPV) techniques. Here firstly the missile dynamics has been brought into an

LPV form via a state transformation rather than the usual coordinate transformation.

And finally an inner-outer loop decomposition is applied. This paper presents a good

method for outer loop design of the missile autopilot keeping the robustness of the inner

loop. But only 10% variation in aerodynamic coefficients has been taken into account

which in practical cases can be almost 100%. [97] and [98] present a missile autopilot

design using extended mean assignment (EMA) control technique for linear time vary-

ing (LTV) systems. The EMA control technique is based on a new series D-eigenvalue

(SD eigenvalue) concept in a way similar to the conventional pole placement design for

linear time invariant (LTI) systems. The nonlinear dynamics of the missile is rendered

into a linear one that is tractable by the EMA control technique via the classical lin-

earization along a nominal normal acceleration profile, followed by a linear coordinate

transformation. Simulation results are also presented for the zero input stabilization.

But the classical linearization technique may not be applicable for all complex nonlinear

missile model. [67] presents plant input mapping digital redesign methods for the dig-

ital implementation of missile autopilots. The authors show along with the simulation

results that the new controller response is good enough. But the coupling effect and the

model uncertainties have not been considered in full. The paper [7] presents a novel ap-


proach of designing nonlinear autopilot design for an air-to-air missile. The authors have

considered actuator saturations and show good tracking performance through simulation

results. [76] presents an example of system inversion and manifold invariance applied for

designing missile autopilot. It has been shown that the classical notions of invariance and

inversion, together with tools from the nonlinear regulator theory can be used to design

globally stabilizing control laws for general nonlinear systems. An approach to integrated

guidance/autopilot design for missiles is proposed in [95]. Integrated guidance/autopilot

design has been formulated based on variable structure techniques and simulation results

have been given for an anti-vessel missile.

1.6 Contribution of the thesis

The thesis has proposed a nonlinear 3-axes control structure that uses robust linear and

nonlinear control designs. The controller has been tested both within an ideal simulation

environment built using MATLAB, as well as a highly detailed and realistic simulation

environment coded in FORTRAN. The performance of the proposed controller has the

following features:

1. Good decoupling between the three axes during pitch maneuver.

2. Minimized roll rate, yaw deflection, rate of change of aerodynamic coefficients even

during high angle of attack maneuver.

3. Estimation of the unmeasured states of the system using a nonlinear observer re-

quired for the computation of nonlinear feedback.

4. Design of a robust H∞ control law that retains performance even with aerodynamic

uncertainties.

5. Design of a robust sliding mode controller to tolerate the uncertainties caused due

to aerodynamic coefficients.


1.7 Organization of the thesis

This thesis has been organized into five chapters.

Chapter 1 provides an introduction to guidance and control system of a missile. It also

presents typical trajectories, guidance schemes and structure of autopilots. Finally,

this section presents the limitation of the existing autopilot and thereby motivates

the work done here. This chapter also carries a literature survey for autopilot design

methods.

Chapter 2 presents input-output feedback linearization technique for designing rate loop

control law.

Chapter 3 presents robust feedback linearizing control law with H∞ technique for de-

signing the inner rate loop of the autopilot.

Chapter 4 presents the design of inner rate loop of the autopilot based on sliding mode

technique.

Chapter 5 draws the final conclusions and outlines the scope for future study.

Chapter 2

Autopilot Design with Input-Output

Linearization by Feedback

2.1 Introduction

Typical short range tactical missiles suffer from the high roll rates due to uncertain aero-

dynamic coefficients with increasing angle of attack and coupling among pitch, yaw and

roll. The effort with three loop autopilot has been seen to be inefficient particularly

when decoupling is the point of attraction as in this conventional case of three separate

decoupled and linear autopilots, a maneuver in one axis causes disturbances in the oth-

ers and effectively causes fin saturation for high lateral accelerations demand. To tackle

the decoupling problem the use of nonlinear multivariable method such as input-output

linearization technique may be effective. Feedback linearization approach to nonlinear

control design has attracted a great deal of research interest in recent years. The central

idea is to algebraically transform a nonlinear system dynamics into a (fully or partially)

linear one, so that linear control techniques can be applied. This chapter provides an

application of input-output linearization technique to a small range surface-to-surface

skid-to-turn homing missile to show the decoupling among pitch-yaw-roll axes along with

its effect on missile overall performance such as control deflections, side slip angle, azimuth

Chapter 2. Autopilot Design with Input-Output Linearization by Feedback40

gimbal angle etc.

2.2 Brief theory and application to an nonlinear mis-

sile model

The following discussion of input/output linearization for multi-input/multi-output sys-

tems is adapted from [48],[83],[61],[32],[8] and Appendix A. Consider the square system

x = f(x) + g1(x)u1 + ..... + gm(x)um

y1 = h1(x)

· · ·ym = hm(x)

(2.1)

Where x is the state vector,uis(i = 1, ....., m) are control inputs, yjs(j = 1, ....., m) are

outputs, and , f and gis are the smooth vector fields, and hjs are smooth scalar functions.

To obtain an input-output relation of the system 2.1, we differentiate the jth output, for

m = 1,

yj =∂yj

∂xx =

∂yj

∂x[f(x) + g(x)u] = Lfh1 + Lgh1u (2.2)

and for MIMO case,

yj = Lfhj +m∑

i=1

(Lgihj)ui, (2.3)

Where, Lfhj is defined as Lie derivative of hj w.r.t f(x) and is equal to∂hj

∂xf(x). If

Lgihj(x) = 0 ∀i ,then the inputs do not appear and we have to differentiate again. Assume

that rj is the smallest integer such that at least one of the inputs appear in y(rj)j , then

y(rj)j = L

rj

f hj +m∑

i=1

LgiL

rj−1f hjui, (2.4)


With LgiL

rj−1f hj(x) 6= 0 for at least one i. If we perform the above procedure for each

output yj, we can obtain a total of m equations in the above form, which can be written

compactly as

y(r1)1

· · ·· · ·y

(rm)m

=

Lr1f h1

· · ·· · ·Lrm

f hm

+ E(x)

u1

· · ·· · ·um

(2.5)

where the m×m matrix E is defined as

E(x) =

Lg1Lr1−1f h1 · · · · · · LgmLr1−1

f h1

.... . .

......

. . ....

Lg1Lrm−1f hm · · · · · · LgmLrm−1

f hm

(2.6)

The matrix E(x) is called the decoupling matrix for the MIMO system. If the decoupling

matrix is non-singular around a point x0, then the input transformation

u = −E−1

Lr1f h1

· · ·· · ·Lrm

f hm

+ E−1

v1

· · ·· · ·vm

(2.7)

yields a linear differential relation between the output y and the new input v

y(r1)1

· · ·· · ·y

(rm)m

=

v1

· · ·· · ·vm

(2.8)

Note that the above input-output relation is decoupled, in addition to being linear. If

the new inputs v1, v2, v3 are designed in such a way that the control law only affects the


corresponding yjs and not the others, then such a control law of the form 2.7 is called a

decoupling control law, or non-interacting control law. As a result of the decoupling, one

can apply SISO linear control laws on each y− v channel in the above decoupled dynamics

to construct tracking or stabilization controllers. The relative degree of the system can

be computed as ρ = r1 + r2 + .... + rj + .... + rm. If ρ = n, i.e. if the relative degree

of the system is equal to the order of the system, then that system is known as a full

input-output linearizable system. On the other hand, if ρ < n, then it is possible to make

the system partially input-output linearized, as all of the states may not appear in the

linearized form. These remaining states appear as internal dynamics of the system. This

internal dynamics may not be ’visible’ from the linearized plant but the whole system is

stable only if the internal dynamics is. This aspect has been discussed later and in detail

in Appendix A.

2.2.1 Case I : Input-output linearization with q, r, p as outputs

Figure 2.1 describes the schematic block diagram of the system with feedback linearizing

controller or FBLC. fzd, fyd and φd are demanded lateral accelerations and roll angle

respectively while fz, fy, φ are the sensed ones. qd, rd and pd are the demanded rates

computed from error dynamics of lateral accelerations and roll angle. FBLC is the feed-

back linearizing controller which computes u with the help of a new input vector v. This

v is being computed from rate error dynamics. Now from the theory of feedback lin-

earization, it can be shown that the relation between v and rate dynamics could be made

linear and decoupled as shown in the dotted box. Beside that v is to be computed in

such a way, that good tracking performance in the inner rate loop as well as in outer

acceleration loop is maintained. In Figure 2.1, the block ’LC’ stands for ’Linear Con-

troller’, that is used to compute v. Let the state, output and the input vectors be defined

as, v = [v1, v2, v3]T , x = [U, v, w, p, q, r]T , u = [δp, δy, δr]

T . Now the whole system can be

represented as,

x = f(x) + g(x)u

where f(x) and g(x) are described as follows:


Figure 2.1: Block diagram of the system with FBLC for outputs q, r and p

f(x) =

(rvm−qwm+TX−QSCD0−Q11GRm)

m(√

2TY +√

2QSCNBn+√

2QSCNBs−√

2QS D2Vm

Cyβ

β+√

2QS D2Vm

rCyr+√

2pwm−√2rUm−GRmQ21

−GRmQ31+√

2FMY Q) √2m

−(−√2qUm+√

2pvm−GRmQ21+GRmQ31−√

2TZ−√

2QSCNAn−√

2QSCNAs+√

2QS D2Vm

qCZq

+√

2QS D2Vm

αCzα ) √2m

(−IXXp+Tmx+QSDCL+QSD D2Vm

CLP p)

IXX−(−QSDCMAnD−QSDCMAsD−QSD D

2VmDCmq q−QSD D

2VmDCmα α−TmyD+prDIXX−prDIZZ+IY Y qD)

D×IY Y(QSDCMBnD+QSDCMBsD+QSD D

2VmDrCnr−QSD D

2VmDCm

ββ+TmzD−qpDIY Y +qpDIXX−IZZrD)

DIZZ


g(x) =

0 0 0

0QSCnδ

m0

−QSCnδ

m0 0

0 0 −QSDClzeta

IXX(−QSDCnδ

XLTP +QSDCnδXCG)

DIY Y0 0

0 − (QSDCnδXLTY −QSDCnδ

XCG)

DIZZ0

For easy reference we can write, f(x) = [fi(x)]T

i=1→6

and g(x) = [gij(x)]i=1→6,j=1→3

.

From the theory of input-output linearization we obtain, with reference to the equa-

tions A.37, A.38 and A.39, that,

y1

y2

y3

= M(x) + E(x)

δp

δq

δr

where,

M(x) =

Lfh1(x)

Lfh2(x)

Lfh3(x)

and

E(x) =

Lg1h1 Lg2h1 Lg3h1

Lg1h2 Lg2h2 Lg3h2

Lg1h3 Lg2h3 Lg3h3

Note that, as r1 = 1, r2 = 1, r3 = 1, the relative degree of the system obtained in our case

is ρ = 3. It can be seen from the above equations that to get the desired input output

relation we need to differentiate the outputs only once and thus the E(x) matrix obtained


is non-singular for the whole flight time. In this particular case

M(x) =

f5

f6

f4

and E(x) =

g62 0 0

0 g51 0

0 0 g42

It has been seen from numerical simulations that, diagonal terms of the g(x) matrix

remain non-zero for all typical flight trajectories and over the whole flight time. Thus

the E(x) matrix is non-singular for the whole flight time. As the off-diagonal terms are

all zero, the outputs become decoupled with each others and coupled only one of the

inputs. So we see that with the outputs chosen as q, r and p, the input-output relation

of the nonlinear missile model can made input-decoupled for every instant of the flight by

providing appropriate inputs. The design of the control law will be as follows

u = −E−1M + E−1

v1

v2

v3

(2.9)

to linearize the system leading to

y1

y2

y3

=

v1

v2

v3

(2.10)

A tracking controller for this single-integrator relation can be designed using linear control

techniques. For this particular case, defining the tracking error

e =

y1 − y1d

y2 − y2d

y3 − y3d

=

q − qd

r − rd

p− pd

(2.11)


and choosing the new input v1, v2 and v3 as

v1 = y1d + k1(y1d − y1) = qd + k1(qd − q)

v2 = y2d + k2(y2d − y2) = rd + k2(rd − r)

v3 = y3d + k3(y3d − y3) = pd + k3(pd − p)

(2.12)

with k1, k2 and k3 being positive constants, the tracking error dynamics of the closed loop

system is given by

e1 + k1e1 = 0

e2 + k2e2 = 0

e3 + k3e3 = 0

which is exponentially stable and e(t) converges to zero exponentially. Of course,the

stability of internal zero dynamics needs to be an internally stable closed-loop system.

2.2.1.1 Zero dynamics analysis

The basic theory of zero dynamics has also been adopted from Appendix A and [1]. The

design of linear inputs v1,v2 and v3 could be done in a straightforward manner using a

classical linear control strategy such as pole placement / optimal control (LQR) provided

the zero dynamics of the system are asymptotically stable, i.e., the system is minimum-

phase.

To compute the zero dynamic stability, first we define the internal dynamics as

U = f1

v = f2 + g2u2

w = f3 + g3u3

(2.13)

, where the functions fi and gi have been defined earlier. The stability of the internal

dynamics has been studied with the help of zero dynamic analysis through simulation.

That means, we have to investigate the characteristics of zero dynamic state variables


such as U, v, w have been studied giving an input u0 =

δp0

δy0

δr0

to the plant dynamics

such that the outputs q, r, p and their derivatives remain zero. The Figure 2.2 represents

the characteristics of U ,v and w with the application of u0. u0 can be computed as

q = f5 + g51δp0 = 0

r = f6 + g62δy0 = 0

p = f4 + g43δr0 = 0

or

δp0 = − f5

g51

δy0 = − f6

g62

δr0 = − f4

g43

Simulations have been performed in various cases of flight conditions to show zero

dynamics stability for all cases. Figure 2.2 is one of the test results for a particular flight

condition. It can be seen from Figure 2.2 that all the three states of the internal dynamics

are bounded. Same scenario can be seen for any other flight conditions also. Thus it can

be concluded that the overall system can be stabilized with a stable linear controller as

the internal dynamics are asymptotically stable.

2.2.1.2 Simulation results

A program is used to simulate the full scale 6-DOF model of the missile dynamics. A con-

ventional three loop configuration lateral autopilot (pitch and yaw channels are identical)

and a roll autopilot in PI configuration are also considered for performance comparison.

The linear controller gains are scheduled as a function of velocity throughout the flight

trajectory. The 6-DOF simulation platform includes a seeker based homing guidance

(nonlinear Proportional Navigation law) loop to present the performances of the nonlin-


Figure 2.2: Boundedness of u,v and w in zero dynamic condition

ear controller under closed loop guidance. The objective of this subsection is to bring out

the salient features of the nonlinear controller through 6-DOF simulations. The plant is

considered to be time varying. A ”boost-sustain-coast” thrust profile is considered. All

the aerodynamic parameters are used in the form of look up tables. Nonlinearities of

actuation system such as dead zone, command saturation and rate saturation are con-

sidered. All the numerical results given in this thesis have been normalized based on the

results obtained from conventional three loop configuration as the data and tables used

for simulations need to be kept confidential.

As stated in the Chapter 1, when the PN switch over has taken place, the missile tries

to maneuver at its maximum capability (depending mainly on the structure and strength

of the missile). During this PN maneuver the latax demand reaches its maximum value

and when the missile tries to achieve this demand the coupling among pitch-yaw-roll

axes, large variations in yaw deflection, side slip angle, gimbal angle and aerodynamic

coefficients can be observed. In view of these phenomenon, to compare the performances


between FBLC and conventional three loop autopilot, simulation results have been orga-

nized in the following way:

1. Decoupling of the FBLC.

2. Input-output linearization in three channels.

3. Decreased yaw deflection during PN switch over.

4. Reduced side slip, gimbal angle and CS and Cny during PN switch over.

In the normalized time axis of the simulation results, the PN switch over can be observed

during t = 0.75 to t = 0.9.In comparative simulation results we have compared the

autopilot performances between FBLC (new nonlinear autopilot) and the existing one

(three loop autopilot).

2.2.1.3 Decoupling of three axes motion

Figure 2.3 illustrates the decoupling phenomenon by applying step body rate demands in

the three channels. Step commands have been given in three different instants such as

t = 0.25, t = 0.5 and t = 0.75 respectively for pitch,yaw and roll channels. Now from the

figure it can be seen that step command in one channel does not affect the other channels.

Figure 2.4 shows the comparative autopilot performance with rate demand as per

closed loop guidance requirements. The comparative results of classical three loop au-

topilot and the feedback linearizing control (FBLC) show decoupling property of the

latter. During pitch maneuver at t = 0.46 and t = 0.67, FBLC shows almost no effect

on roll rate as well as on yaw rate. But for the conventional one, during pitch maneuver

simultaneous effects on roll rate and yaw rate can be seen.

2.2.1.4 Input-Output Linearization

Figures 2.5, 2.6 and 2.7 describe the input-output linearization in the pitch-yaw-roll chan-

nel respectively applying the guidance commands. Referring to equation 2.10, Figure 2.5,

2.6, 2.7 shows the plots of q and v1, r and v2, p and v3 respectively. It can be seen that for


Figure 2.3: Autopilot response with step command

each channel, the plotted parameters are almost superimposed to each other, revealing

that very good linearization is achieved in all the three channels. Figure 2.8 shows the

comparative performance of the autopilot and illustrates how the latax demands in pitch

and yaw planes are satisfied. Almost similar latax demand and sensed profiles can be

seen for the FBLC and the classical autopilots. For classical one the yaw latax has been


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1−1

−0.5

0

0.5

Normalised Time

Nor

mal

ised

Pitc

h R

ate

Comparative Performance in 6−DOF Simulation

FBLCExisting

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−3

−2

−1

0

1

Normalised Time

Nor

mal

ised

Yaw

Rat

e

FBLCExisting

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−1

0

1

Normalised Time

Nor

mal

ised

Rol

l Rat

e

FBLCExisting

Figure 2.4: Decoupling in the three channels

increased due to the effect of coupling during PN switch over. As a result yaw deflection

is seen to increase in Figure 2.9. Figure 2.9 illustrates the input requirement in terms

of fin deflection in the three channels. It can be seen from the comparative simulation

results that during PN maneuver, input demand in yaw channel is greater for existing

classical controller over the FBLC. This happens due to the induced yaw rate caused by

the increased roll rate during PN switch over for the classical controller. The fin require-

ment is also shown in Figure 2.10, where we can see that the control requirements in the

three planes have been distributed to the four fins.


Figure 2.5: Evidence of linearization in pitch channel

The rate of fin deflections are also shown in Figure 2.11 which indicate that the rate

is below the maximum rate limit (normalized to 1).

Figures 2.12 and 2.13 describe comparative results for gimbal angle, α and β. From

these figures it can be seen that the gimbal angle in the azimuth plane and β rises higher

value for classical controller over the FBLC. This increase is prominent during PN switch

over where coupling effect is maximum for the existing controller.

Figure 2.14 shows the variations in the aerodynamic coefficients CS, Cny and CL

during flight for the classical controller. During PN switch over, due to the effect of


Figure 2.6: Evidence of linearization in yaw channel

increased β, the values of CS and Cny increase.


Figure 2.7: Evidence of linearization in roll channel


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−0.5

0

0.5

Normalised Time

Nor

mal

ised

Pitc

h la

tax


FBLCExisting

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−1

0

1

2

Normalised Time

Nor

mal

ised

Yaw

lata

x


FBLCExisting

Figure 2.8: Pitch and yaw latax profiles


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−3

−2

−1

0

1

Normalised Time

Eff.

pitc

h de

fln (

δ PB

)


FBLCExisting

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−4

−2

0

2

4

Normalised Time

Eff.

yaw

def

ln (

δ YB

)

FBLCExisting

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−3

−2

−1

0

1

Normalised Time

Rol

l def

ln (

δ R )

FBLCExisting

Figure 2.9: Effective pitch yaw roll deflection


5 10 15 20

−2

−1

0

1

Normalised Time

Nor

mal

ised

Fin

−1 d

efl

n

FBLCExisting

0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5

0

0.5

1

1.5

Normalised Time

Nor

mal

ised

Fin

−2 d

efl

n

FBLCExisting

0.2 0.4 0.6 0.8 1−3

−2

−1

0

1

Normalised Time

Nor

mal

ised

Fin

−3 d

efl

n

FBLCExisting

0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

Normalised Time

Nor

mal

ised

Fin

−4 d

efl

n FBLCExisting

Figure 2.10: Fin deflections


0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

Time (Normalized)

Fin

−1 d

efln

rat

e in

deg

/sec

FBLCExisting

0.2 0.4 0.6 0.8 1

−0.5

0

0.5

Time (Normalized)

Fin

−2 d

efln

rat

e in

deg

/sec

FBLCExisting

0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

Time (Normalized)

Fin

−3 d

efln

rat

e in

deg

/sec

FBLCExisting

0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

Time (Normalized)

Fin

−4 d

efln

rat

e in

deg

/sec

FBLCExisting

Figure 2.11: Fin deflection rate


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Normalised Time

Nor

mal

ised

Gim

bal A

ngle

(El)


FBLCExisting

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−4

−2

0

2

4

Normalised Time

Nor

mal

ised

Gim

bal A

ngle

(Az) FBLC

Existing

Figure 2.12: Gimbal angle


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1−2

−1.5

−1

−0.5

0

0.5

1

1.5

Normalised Time

Nor

mal

ised

Alp

ha (

Bod

y)


FBLCExisting

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−2

−1

0

1

Normalised Time

Nor

mal

ised

Bet

a (B

ody)


FBLCExisting

Figure 2.13: Alpha and beta profiles


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−0.3

−0.2

−0.1

0

0.1

A Comparative Performances in 6−DOF Simulation

Rol

ling

mom

ent c

oeffi

cien

t CL

Normalised Time

FBLCExisting

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−0.5

0

0.5

1

Sid

e fo

rce

coef

ficie

nt C

S

Normalised Time

FBLCExisting

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−8

−6

−4

−2

0

2

Yaw

ing

mom

ent c

oeffi

cien

t Cny

Normalised Time

FBLCExisting

Figure 2.14: Force and moment coefficients


2.2.2 Case II: Input-output linearization with w,v and p as out-

puts

In Section 2.2.1, feedback linearization of the inner rate loop has been discussed consid-

ering the output of the system as q, r and p. But in that case, a part of the plant has

been linearized that does not include the rate to latax dynamics.In other words only the

inner rate loop of the plant was linearized. The outer loop has been designed based on

the conventional approach. Now in this section IO linearization is applied considering the

inner and the outer loops as described in the block diagram given in Figure 2.15. As a

result the plant dynamics from wd, vd, pd to w, v, p can be made linearized and decoupled.

Another advantage is that there is no extra effort required for designing the outer loop

controller. To obtain these conditions, we have chosen the outputs as w,v and p. One

important property of the feedback linearization is that, for different choice of outputs

the relative degree of the same system would become different and at times the system

may not be linearizable for some particular sets of outputs. That is why the choice of

proper outputs is important in the case of feedback linearization. From Figure 2.15, it

can be seen that the new input v can be computed from reference inputs wd, vd and pd as

per guidance (if possible) and v to w, v and p have been linearized. Note that outer loop

controller like Figure 2.1 is absent here and this is the main advantage of linearizing the

whole plant rather than partial linearizing. The aim of this thesis is to track the missile

lateral acceleration demand in both the pitch and yaw planes as well as the roll rate in the

roll plane, using the missile fins; hence yielding a system with 3 inputs and 3 controlled

outputs.

2.2.2.1 Formulation of the problem for the STT missile

The state-space form of the non-linear system of the homing missile can now be written

in a compact parametric format, as: 1.3 and 1.4 or in matrix form as described in Section

2.2.1. State vectors and inputs are the same as considered in Case I. The only difference

is in output vector and here it is taken as y = [w, v, p]T . Now the whole system can be


Figure 2.15: Block diagram of the system with output w, v and p

represented as,

x = f(x) + g(x)u

where f(x) and g(x) are the same as in Section 2.2.1. The FBLC is similar to that in

Case I, however, since the outputs are the derivative of the states, there is no need for

further differentiation since the input-output model is the same as the state space model.

Only the output p needs to be differentiated for linearization of the roll channel [51].Thus

we have,

y1

y2

y3

= M(x) + E(x)

δp

δq

δr


where,

M(x) =

h1(x)

h2(x)

Lfh3(x)

i.e, M(x) =

f3

f2

f4

and, E(x) =

g31 0 0

0 g22 0

0 0 g43

From M(x) and E(x) matrices it can be observed that by choosing the outputs as w,v

and p, inputs are already decoupled from each other as in the previous Section 2.2.1. it

can be verified similar to the case is decoupling matrix E(x) is nonsingular for the whole

flight time so we can now derive the control law similar to Equation as given below:

u = −E−1M + E−1

v1

v2

v3

Note that the above input-output relation is, again, linear and decoupled. For this par-

ticular case, the tracking error is given as,

e =

e1

e2

e3

=

y1 − y1d

y2 − y2d

y3 − y3d

=

w − wd

v − vd

p− pd

(2.14)

Now, choosing the new input v1, v2 and v3 as

v1 = y1d

v2 = y2d

v3 = y3d + k3(y3d − y3)

(2.15)


with k1, k2 and k3 being positive constants, the tracking error dynamics of the closed loop

system is reduced to,

e1 = 0

e2 = 0

e3 + k3e3 = 0

Simulations are given below to show the system performance along with this FBLC con-

troller.


Figure 2.16 illustrates the autopilot performance for step commands. Refereing to Figure

2.16, the step commands have been given at three different time instants in the three

channels, namely, t = 0.25, t = 0.5 and t = 0.75 in pitch, yaw and roll channels re-

spectively. No significant disturbance in the other channels are seen when each of these

channels are excited by a step command.


Figure 2.16: Autopilot response with step command


Figure 2.17 describes the linearization in the pitch-yaw-roll channels, with seeker based

guidance commands. One can see from this figure that the plots v1 and q are matching

very well with each other which demonstrates linearization in pitch plane dynamics. The

same phenomenon can be seen in yaw and roll channels also. Figure 2.18 illustrates the

Figure 2.17: Autopilot response with guidance command

effective pitch, yaw and roll control fin deflections.


Figure 2.18: Effective Pitch yaw roll deflection

.

2.3 Nonlinear Luenberger observer design

The control inputs computed by the FBLC are functions of state variables. Since all

the state variables of the missile can not be measured through sensors, an observer is

essential to estimate the unmeasured states of the system for control computation. This

chapter presents an explicit form of a nonlinear observer for a class of multi-input multi-

output systems. Observer construction [46] for multi-input nonlinear systems is not a


trivial extension of single-output case, especially when the global error convergence is of

interest. In this section, a nonlinear reduced order Luenberger observer for a STT short

range missile system is presented.

2.3.1 Introducion

The observer has been designed with guaranteed stability of the estimation error dynamics

for the nominal plant. For a missile, measured state variables are q, r and p, while the

unmeasured state variables which are to be estimated are U , v and w. The forward velocity

assumed to be known from predetermined thrust profile of the rocket motor. Thus, only

v and w need to be estimated. The input to the observer are the measurements of lateral

accelerations fz and fy from the accelerometers of the missile. In this work, instead of

estimating w and v, the angle of attack α and sideslip angle β have been estimated as fz

and fy are the sole functions of α and β and it will be more convenient to express those

measurement variables in terms of α and β rather than U, v, w. As a result mathematical

computations such as derivative, integration will become more easier for fz and fy if they

are expressed in terms of α and β. For example fz and fy are expressed as

fz = QSm

[CNAn + CNAs − Cleta + D2Vm


fy = QSm

[CNBn + CNBs + Clzie + D2Vm

(Cyrr − Czββ)](2.16)

Now as we know that aerodynamic coefficients such as CNAn, CNAs etc are directly related

to α and β where α and β are related to U , v and w as

α = tan−1(w

U

)

β = tan−1( v

U

) (2.17)

Thus one can imagine how complicated it will be if fz and fy are expressed in terms of

U, v and w.


2.3.2 Observer construction

We now proceed to introduce the observer. The observer has the following properties:

• The construction of the observer, based on [6], doest not require a preliminary non-

linear change of coordinates. Thus help us to avoid differentiations and integrations

with fz and fy.

• The calculation of gain is straightforward as the observer gain is constant through

out the whole flight time and can be chosen by trial and error.

• It is computationally heavy but easily implementable.

Although the theory is applicable up to forced/autonomus multiple-input/multiple-output

nonlinear system, here we shall concentrate on a general MIMO systems of the form:

·x = f (x, u)

y = h (x, u), (2.18)

which is similar to a typical nonlinear missile model described in Section 2.2 with state

x(t) ∈ Rn, h(t) ∈ Rm . In what follows, we also denote by Q(x) the so-called observability

matrix of equation 2.18 [38] as,

Q (x) = DΦ (x) =dΦ (x)

dx(2.19)


where,

Φ(x) =

h1(x)

Lfh1(x)...

Lλ1−1f h1(x)

h2(x)...

Lλ2−1f h2(x)

...

Lλm−1f hm(x)

(2.20)

where the integer numbers λ1, ···, λm are such that∑m

i=1 λi = n and that Φ(x) : X → Φ(x)

is a diffeomorphism. in a linear case, i.e., h(x) = Cx and f(x) = Ax, the matrix (2.19)

reduces to the well known observability matrix. If Q(x) has the full rank for all x ∈ Rn

then the error dynamics of the proposed observer equations

˙x(t) = f(x(t)) + [Q(x(t))]−1K[y(t)− h(x(t))]

x(0) = x ∈ Rn

(2.21)

will be stable as reported in [9] and [10]. The gain vector K, can be chosen in the following

manner:x(t)− ˙x(t) = e(t) = f(x, u)− f(x, u)−G(x, u))(y − y)

where

G(x, u) = Q−1K

or

e = (A−Q−1KC)e

where

A = ∂f∂x

∣∣x=x

andC = ∂h∂x

∣∣x=x

, (2.22)


For MIMO systems

A(x(t), u) =

∂f1

∂x1

∂f1

∂x2......... ∂f1

∂xn...

... ............

∂fr

∂x1

∂fr

∂x2......... ∂fr

∂xn...

... ............

∂fn

∂x1

∂fn

∂x2......... ∂fn

∂xn

∣∣∣∣∣∣∣∣∣∣∣∣∣x=x

and

C(x(t), u) =

∂h1

∂x1

∂h1

∂x2......... ∂h1

∂xn...

... ............

∂hr

∂x1

∂hr

∂x2......... ∂hr

∂xn...

... ............

∂hm

∂x1

∂hm

∂x2......... ∂hm

∂xn

∣∣∣∣∣∣∣∣∣∣∣∣∣x=x

where

x =

x1

x2

...

xr

...

xn

, y(x(t), u) =

h1(x(t), u)

h2(x(t), u)...

hr(x(t), u)...

hm(x(t), u)

and u =

δP

δY

δR

From equation 2.22 it is clear that the value of K can easily be found out using pole

placement technique.

2.3.3 Application to a realistic STT missile model

As we already discussed that the input of the observer or h(x) functions are taken as

lateral accelerations of the missile and it is very complicated task to represent those

parameters as a function of known and unknown states. So to overcome this difficulty

we have formulated the problem with new state variables, namely, angle of attack α and


sideslip angle β. Now the parametric model of the missile for the nonlinear observer can

be written as

α =wU − Uw

U2(1 + tan2(α))= fα(α, β, δp)

β =vU − Uv

U2(1 + tan2(β))= fβ(α, β, δy)

(2.23)

and U, v, w, U , v, w are taken from the system equations 1.3. The output equations can

be written as

y1 = fz = QSm

[CNAn + CNAs − Cleta + D2Vm


y2 = fy = QSm

[CNBn + CNBs + Clzie + D2Vm

(Cyrr − Czββ)]

y3 = QSDIXX

CL

y4 = QSDIY Y

(CMAn + CMAs) + QSD2

2VmIY YCmαα

y5 = QSDIZZ

(CMBn + CMBs) + QSD2

2VmIZZCmββ

(2.24)

As the order of the system has been reduced to 2, the observability matrix can be con-

structed with two of the outputs as

Q(x) =dΦ(x)

dx=

(∂y1

∂α∂y2

∂α∂y1

∂β∂y2

∂β

)(2.25)

It can be shown that the Q(x) matrix will be nonsingular for the whole flight time. So

we can now proceed towards the next step of the observer formulation.

2.3.3.1 Gain ’K’ calculation

Our observer equation is given as,

(˙α˙β

)=

(fα(α(t), β(t), δp)

fβ(α(t), β(t), δy)

)+ Q−1(x(t), δp, δy)K

(y1(x(t))− y1(x(t))

y2(x(t))− y2(x(t))

)(2.26)


and the error equation becomes

e =

(e1

e2

)= (A(x(t))−Q−1(x(t))KC)

(e1

e2

)(2.27)

where,

A(x(t)) =

(∂f1

∂α∂f1

∂β∂f2

∂α∂f2

∂β

)∣∣∣∣∣α=α,β=β

and C(x(t)) =

(∂h1

∂α∂h1

∂β∂h2

∂α∂h2

∂β

)∣∣∣∣∣α=α,β=β

The value of K will be such that the eigenvalues of the system 2.27 will lie in the

negative half of s plane. In this case K matrix is of the form

K =

(k11 k12

k21 k22

)

Now scalar elements k11 = −16.0, k12 = 1.0, k21 = −60.0, k22 = 0.0 can be found trial and

error basis in view of the desired eigen values of the error dynamics lie in left half of s

plane and for this case they are placed at −10.0 and −6.0 respectively. The simulation

results in the next section show the tracking performance of the proposed observer.

2.3.4 Simulation results

Figures 2.19 and 2.20 show the observer performance for a limited time period. For both α

and β, the true initial condition is zero but in the simulation we have taken some nonzero

initial conditions of about 2− 3 . The observer performance has been highlighted in the

high maneuvering zone of the missile i.e., PN switch over period. As we know that inputs

to the observer fz and fy are functions of uncertain aerodynamic coefficients which varies

with its magnitude as well as sign abnormally during PN switch over, there is maximum

possibility of divergence in error dynamics of the observer in this period. Note that

observer has been constructed based on a nominal missile model and measurement noise

has been considered during simulation. Figure 2.19 shows good tracking performance

in high maneuver zone. Figure 2.20 also depicts the same. Note that, the gain (K) of


the observer is not adaptive one and remains fixed for whole flight time. So it may be

necessary to manipulate the gain when ever the flight condition has been changed or if

there is huge change in initial condition.

0.5 0.55 0.6 0.65 0.7 0.75

−1

−0.5

0

0.5

1

Normalised Time

No

rmal

ised

alp

& a

lph in

Bo

dy

Fra

me

alp and alphat

alph

alp

Figure 2.19: Estimated and true α

Figure 2.21 and 2.22 shows the true and estimated α and β for the full envelope. One

can see from these figures that α and β tracks α and β satisfactorily.


0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1−0.1

−0.05

0

0.05

0.1

0.15

Normalised Time

No

rmal

ised

bet

a h &

bet

a in

Bo

dy

Fra

me

Normalised beta and betahat

betah

beta

Figure 2.20: Estimated and true β


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

Normalised Time

No

rmal

ised

alp

& a

lph in

Bo

dy

Fra

me

alp and alphat

alph

alp

Figure 2.21: Estimated and true α for full flight time


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Normalised Time

No

rmal

ised

bet

a h &

bet

a in

Bo

dy

Fra

me

Normalised beta and betahat

betah

beta

Figure 2.22: Estimated and true β for full flight time


2.4 Comments

The above simulation results show the linearization and decoupling in pitch yaw and roll

channels. It may be also observed that the performance of the nonlinear controller is

clearly better than the existing linear one in terms of roll rate, angle of attack and impact

angle. The main achievement of this work is to achieve the decoupling among yaw roll

channels which improves the gimble angle limitation and fin saturation. A nonlinear Lu-

enberger observer has been presented along with the estimation of α and β. It can be seen

from the plots that the proposed observer can estimate the α and β quite accurately.This

is the foundation on which subsequent chapters will be built.

Chapter 3

H∞ Control of Feedback Linearized

Inner Rate Loop Dynamics

3.1 Introducion

Although input-output linearization technique may be effective to meet the desired re-

quirements of the flight control system of a missile, it fails to handle uncertainties in the

missile model which is a natural factor during real flight conditions. Input-output lineariz-

ing controller computes plant inputs based on nominal plant and that is why it fails to

cancel the nonlinearity on a largely perturbed plant. So, in order to handle the perturbed

plant, in this chapter a robust control law has been incorporated with linearizing control

law so that the perturbed plant model can still be linearized and decoupled. Here an

H∞ controller has been synthesized based on Linear Matrix Inequality (LMI) approach

to tackle aerodynamic uncertainties along with the various disturbances. So in practical

sense this robust controller will be effective for most of uncertainties and disturbances

which are not trivial specially during high angle of attack. This chapter presents some

mathematical evidences of the fact that for a high range of uncertainty, a robust control

structure exists with which the system can be stabilized.

Chapter 3. H∞ Control of Feedback Linearized Inner Rate Loop Dynamics81

3.2 Problem formulation

Figure 3.1 represents the robust feedback linearization approach based on H∞ control.

One can observe that the block diagram represented in 3.1 is almost similar to Figure

2.1. The only difference is that here we are using a robust control block ’RC’ instead of

the linear control block ’LC’. That means the computation of the new inputs v has been

modified for handling the model uncertainties and other disturbances. The inclusion of

the block ’RC’ makes the FBLC control law u robust for the inner rate loop.

Figure 3.1: Block diagram of the system representing robust feedback linearization foroutputs q, r and p

Note that feedback linearization approach presented in Sections 2.2.1, 2.2.2 is exact

only for the nominal model. To retain good performance even with the plant perturbation

due to aerodynamic uncertainties, a robust controller needs to be designed around the

nominally feedback linearized plant. In this section, the design of an H∞ controller


synthesized using the Linear Matrix Inequality (LMI) approach for the linearized plant

after feedback linearization is presented. Let input-output(IO) dynamics for the nominal

plant be written as

y(ρ)n = Mn(x) + En(x)u (3.1)

where ρ is the relative degree of the system for the nth output. Now, the perturbed plant

(3.1) can be written as,

y(ρ)p = Mn(x) + ∆M(x) + (En(x) + ∆E(x))u

y(ρ)n + ∆yn = [Mn(x) + En(x)u] + [∆M(x) + ∆E(x)u]

(3.2)

where yn and yp are the nominal and perturbed plant outputs, and ∆M(x) and ∆E(x)

are the perturbations in M(x) and E(x), respectively. These perturbations occur due to

the variations in aerodynamic coefficients. For example, variations around the nominal

roll moment coefficient CL0 can be expressed as CL = CL0 + ∆CL. Similarly all other

aerodynamic coefficients can be expressed with their variations around the nominal value.

∆M(x) and ∆E(x) are then given as functions of these variations in the aerodynamic

coefficients. So the total variation in the plant is given by d = ∆M(x) + ∆E(x) and

is modeled as an exogenous input to the nominal plant. Here the term robust, means

that even in the presence of aerodynamic uncertainties and various disturbances like wind

gust, fin misalignment, wing misalignment, shifting in C.G and C.P etc. the input-output

linearization can be performed and the plant dynamics can be made decoupled. It can be

shown that it is possible to formulate a robust control law over the feedback linearized

plant to achieve the desired robustness. For example let us consider only the pitch channel.

After input-output linearization and decoupling, the new input v1 and pitch rate q will

be related theoretically as 1s. Referring the Figure 3.2, G = 1

sand let K be the robust

control law. d is modeled as exogenous input which indicates the effective uncertainties

and disturbances in the pitch plane model. In this case, for nominal plant the feedback

linearized input-output relation in each of the three channels becomes G = 1s

and for

the perturbed plant let us consider that the input-output relation becomes G. Now the

notation d indicates G−G i.e., effective deviation of feedback linearized IO relation from


Figure 3.2: Structure of robust control K in pitch plane

the nominal IO one. Here we have formulated the problem for pitch channel only. v1 is

the new input for the pitch plane generated by some robust control law. Similar control

structures can be formulated for yaw and roll plane model to compute v2 and v3. Now,

our aim is to design the robust control law v = [v1, v2, v3]T in such a way that the overall

control input to the nonlinear plant u = −E−1M + E−1

v1

v2

v3

as depicted in (2.9) will

be a robust one. In the following sections we will only discuss about computation of v1

in the pitch plane as similar computations are applicable to yaw and roll channels.

3.2.1 A linear matrix inequality approach to H∞ control for de-

signing K in pitch plane

In this section we will consider continuous H∞ control problem which is solved via ele-

mentary manipulations on linear matrix inequalities (LMI). The whole design procedure

can be divided into two major parts. The first is that of checking of solvability condi-

tions, and the second one involves an LMI based parametrization of the H∞-suboptimal

controller. The solvability condition involves Riccati inequalities rather than the usual in-

definite Riccati equations. Alternatively, these equations can be expressed as a system of

three LMI’s. Efficient convex optimization techniques are available in MATLAB to solve


this system of LMI’s. Its solutions parameterize the H∞ controller. The robust control

structure has finally been developed from the Kalman-Yakubovich-Popov Lemma(KYP)

lemma, otherwise known as the Bounded Real Lemma[41]. The detailed derivations and

discussions are presented in Appendix B. For the sake of continuity in this section, we

will discuss and mention only the relevant points of the theory. The synthesis problem

given in this work is concerned with the familiar feedback arrangement shown in Figure

3.3. As depicted here, the so-called closed-loop system has one external input and one

output, given by d and z respectively. We assume that this connection is well-posed. It

is convenient to introduce the following notation for the transfer function d to z in the

diagram.

Figure 3.3: General feedback arrangement

The uncertainties and various disturbances can be seen in the vehicles with aerody-

namic controls due to the change of aerodynamic coefficients, drag, thrust mis-alignments,

CG (center of gravity) shift etc. The primary function of the robust controller designed

for the inner rate loop is to minimize the impact of these disturbances on the overall

system. Figure 3.4 displays the block diagram robust control structure in the standard

mixed sensitivity configuration where G(s) is the open-loop plant which is 1s

here, K(s)

is the controller to be designed, and W1(s) and W3(s) are weights for shaping the char-

acteristics of the open-loop plant. The design objective is to minimize a weighted mix of


Figure 3.4: Mixed sensitivity configuration

the transfer function

S(G,K) := G11 + G12K(I − G22K)−1G21

where G =

[G11 G12

G21 G22

]and thus handling the robustness issues as well as the stability

and performance of the system. This is sometimes called the star-product between transfer

functions G and K, or equivalently their linear fractional transformation. This mixed

sensitivity design objective is represented as

∥∥∥∥∥

[W1(s)S(s)

W3(s)T (s)

]∥∥∥∥∥∞

< 1

where, S(s) = (I−G(s)K(s))−1 and T (s) = G(s)K(s)(I−G(s)K(s))−1 are sensitivity and

complementary sensitivity functions. The state space description of the augmented-plant

is given by

xp

z

y1

=

A B1 B2

C1 D11 D12

C2 D21 0

xp

d

v1

(3.3)


where xp is the state variable vector of the plant G(s) and weights (W1(s),W2(s)) com-

bined, d is the disturbance input (contains the effect of plant uncertainties and various

disturbances), v1 is the plant input, y1 is the measured signals including disturbances and

z is the regulated output. The state space representation of the controller is given by

xk = Akxk + Bkeq

v1 = Ckxk + Dkeq

where xk represents the controller states; v1 is the controller output. The transfer matrix

between d and z is given by

Gzw(s) =

[W1(s)S(s)

W3(s)T (s)

]= CL(sI − AL)−1BL + DL

where,

AL =

[A + B2DkC2 B2Ck

BkC2 Ak

]

BL =

[B1 + B2DkD21

BkD21

]

CL =[

C1 + D12DkC2 D12Ck

]

DL = D11 + D12DkD21

(3.4)

We can parameterize the closed-loop relation in terms of the controller realization as

follows. First we make the following definitions.

A =

[A 0

0 0

], B =

[B1

0

], C =

[C1 0

], C =

[0 I

C2 0

]

B =

[0 B2

I 0

], D12 =

[0 D12

], D21 =

[0

D21

] (3.5)


which are entirely in terms of the state space matrices for G. Then we have

AL = A + BJpC, BL = B + BJpD21

CL = C + D12JpC, DL = D11 + D12JpD21

(3.6)

where

Jp =

[Ak Bk

Ck Dk

]

The crucial point here is that the parametrization of the closed loop state space matrices

is affine in the controller matrix Jp.

Now we formulate the H∞ control problem based on LMI approach in the following steps.

Any further details along with the proofs have been given in Appendix B.

• Suppose ML(s) = CL(Is − AL)−1BL + DL. Then the following are equivalent con-

ditions.

1. The matrix AL is Hurwitz and∥∥∥ML

∥∥∥∞

< γ

2. There exists a symmetric positive definite matrix XL such that

A∗LXL + XLAL XLBL C∗

L

B∗LXL −γI D∗

L

CL DL −γI

< 0

• The above inequality is equivalent to

HXL+ Q∗J∗pPXL

+ P ∗XL

JQ < 0 (3.7)

where,

HXL=

A∗XL + XLA XLB C∗

B∗XL −γI D∗11

C D11 −γI

,


PXL=

[B∗XL 0 D∗

12

]

and Q =[

C D21 0]

• Considering the above two cases there exists a controller synthesis K if and only if

there exists a symmetric matrix XL > 0 such that

W ∗PXL

HXLWPXL

< 0 and W ∗QHXL

WQ < 0 (3.8)

where ImW ∗PXL

= KerPXLand ImW ∗

Q = KerQ

However the above inequality is not an LMI condition as the variable XL appears

in both HXLand PXL

. So now we will convert it to an LMI condition.

• Given XL > 0;

W ∗PXL

HXLWPXL

< 0, if and only if, W ∗P TXL

WP < 0

where,

TXL=

AX−1L + X−1

L A∗ B X−1L C∗

B∗ −γI D∗11

CX−1L D11 −γI

(3.9)

and

P =[

B∗ 0 D∗12

]

Recall that XL is a real and symmetric (n + nK)× (n + nK) matrix; here n and nk

are state dimensions of G and K. Let us now define the matrices X and Y which

are submatrices of XL and X−1L , by

XL =:

[X X2

X∗2 X3

]and X−1

L =:

[Y Y2

Y ∗2 Y3

](3.10)

• Suppose X and Y are symmetric, positive definite matrices in Rn×n; and nk is a

positive integer. Then there exist matrices X2, Y2 ∈ Rn×nK and symmetric matrices


X3, Y3 ∈ RnK×nK , satisfying

[X X2

X∗2 X3

]> 0 and

[X X2

X∗2 X3

]−1

=

[Y Y2

Y ∗2 Y3

](3.11)

if and only if [X I

I Y

]≥ 0 and rank

[X I

I Y

]≤ n + nK (3.12)

• Suppose XL is a positive definite (n+nK)× (n+nK) matrix and X and Y are n×n

matrices satisfy 3.12. Then

W ∗P TXL

WP < 0 and W ∗QHXL

WQ < 0

If and only if, the following two matrix inequalities are satisfied

1.[

NX 0

0 I

]∗

A∗X + XA XB1 C∗1

B∗1X −I D∗

11

C1 D11 −I

[NX 0

0 I

]< 0 (3.13)

2.[

NY 0

0 I

]∗

AY + Y A∗ Y C1 B1

C1Y −I D11

B∗1 D∗

11 −I

[NY 0

0 I

]< 0 (3.14)

Where Nx and Ny are full-rank matrices whose images satisfy

ImNX = ker[

C2 D21

]

ImNY = ker[

B∗2 D∗

12

] (3.15)

• Comments: The above steps provide us with an explicit way to determine

whether a synthesis exists which solves the H∞ problem. Another point to be

mentioned is that the above synthesis exists if and only if nk ≥ n as well as rank(X−


Y −1) ≤ nk. But if nk ≤ n then the γ-suboptimal controller of order nk (reduced

order) exist if and only if (3.13)-(3.15) hold for some X and Y which further satisfy:

Rank(I −RS) ≤ nk

3.2.2 Design Steps

1. To compute X and Y from the system of LMI’s (3.13)-(3.15) and that

Rank(I −RS) ≤ nk

2. To construct an H∞ controller from this data, we can recall that X,Y are related by

(3.11) to solution XL of the Bounded Real Lemma inequality. We therefore begin by

computing a positive definite matrix XL ∈ R(n×nk)×(n×nk) compatible with (3.11).

To this end we have to compute two full-column-rank matrices X2, Y2 ∈ R(n×nk)

such that

X2YT2 = I −XY (3.16)

XL is then obtained as the unique solution of the linear equation:

[Y I

Y T2 0

]= XL

[I X

0 XT2

](3.17)

Note that (3.17) is always solvable when Y ≥ 0 and X2 has full column rank.

3. It can be proved that (X,Y) can solve (3.13) to (3.15) if and only if XL is given by

(3.11) is positive definite and satisfies (3.8).

4. This guarantees the existence of a solution Jp =

[Ak Bk

Ck Dk

]to the Bounded Real

Lemma inequality (3.7). And from Bounded Real Lemma K(s) = Dk + Ck(sI −Ak)

−1Bk is then a γ−suboptimal controller.


5. In summary, from the given X and Y first we will compute XL ≥ 0 by step 1 and

step 2. Then we will write the Bounded Real Lemma inequality for this XL as,


L

B∗LXL −γI D∗

L

CL DL −γI

= HXL

+ Q∗J∗pPXL+ P ∗

XLJQ < 0 (3.18)

and solve this inequality (3.18) for the controller parameters Jp =

[Ak Bk

Ck Dk

]

Similarly, robust control laws can also be found out for yaw (Jy) and roll (Jr) channel.

3.2.3 Application to the Nonlinear Missile

After obtaining the desired robust control law v1, we can summarize the whole process in

four steps as given below:

• Feedback linearize the proposed nonlinear system

• Get the Linear Fractional Transformation (LFT) of the linear model obtained through

feedback linearization for the each channel.

• Get the disturbance model (d)

• Find the robust stabilizing controller v1 using LMI approach as discussed in the

Section B.4.3

So, from the list it is clear that we need to obtain disturbance model d which is given in

the next section.

3.2.3.1 Uncertainty and disturbance modeling between y and v

Let, P be the nominal feedback linearized plant transfer function, from (2.10) which is a

diagonal matrix with 1s

as the diagonal elements. Let, due to the variation in aerodynamic


coefficients, the nominal model P be perturbed to P which is also assumed diagonal. P is

estimated through system identification methods available in the MATLAB identification

toolbox using realistic simulation data obtained from FORTRAN simulation. Details of

the simulation data as well as the procedure of obtaining P have been given in the thesis.

Let ∆P denote the unmodelled dynamics for the nominal plant P .

∆P = P − P (3.19)

A brief discussion on constructing the ARMAX model has been given next.

Small theoretical background for constructing ARMAX model

It is well known that a linear, time invariant, causal system can be described by its impulse

response g(τ).

y(t) =

t∫

0

g(τ)u(t− τ)dτ

Though most processes are of continuous type, it is advantageous to choose a discrete time

model structure, compatible for parameter estimation simulation using digital computers.

Assuming a sampling interval of one time unit:

y(t) =∑

k=1

g(k)u(t− k)t = 0, 1, 2......

Now introducing the backward shift operator q−1 , we can write:

q−1u(t) = u(t− 1)

y(t) = G(q)u(t)

where, G(q) =∑

g(k)q−k known as the transfer operator of the linear system. From

Figure 3.5 the output error signal e(t) can be defined as e(t) = y(t) − ym(t) and it is

assumed that e(t) and em(t) are the gaussian and color noise respectively. H(q) is the


Figure 3.5: Complete model structure of the system and noisy signal

filter transfer function. To be able to estimate the functions G and H they have to be

parameterized as rational functions in the delay operator q−1. Assuming a basic input-

output configuration as depicted in Figure 3.5, we can write:

y(t) = G(q)u(t) + H(q)e(t)

A much used parametric model is the extended auto-regressive (ARX) that corresponds

to:G(q) = q−nk B(q)

A(q), H(q) = 1

A(q)

A(q) = 1 + a1q−1 + ................ + anaq

−na

B(q) = b1 + b2q−1 + ............... + anbq

−nb

where, the numbers na and nb are the orders of the respective polynomials, nk is the

number of delays from input to output. Another very common model structure is the

extended auto-regressive, moving average model (ARMAX):

G(q) = q−nk B(q)A(q)

, H(q) = C(q)A(q)

C(q) = 1 + c1q−1 + ................ + cncq

−nc

The prediction error can be computed having observed input-output data:

e(t) = H−1(q) [y(t)−G(q)u(t− nk)]


3.2.3.2 Uncertainty and disturbance model for nonlinear missile

Let assume that the disturbed plant or system is described as P and corresponding to

the theory discussed in Section 3.2.3.1, we get P as:

P = P + ∆P

where, P = 1s

and it is same for the all three channels. The values of P can be obtained

directly by using the MATLAB system identification toolbox. Now the ∆P ′s can be

written as:

∆pitch = −7.455e−006s6−s5+3.568s4−4.871s3+3.02s2−0.7227s+3.928e−005s6−3.568s5+4.871s4−3.02s3+0.7227s2−3.928e−005s

∆yaw = 2.455e−006s5−s4+2.985s3−3.182s2+1.34s−0.1435s5−2.985s4+3.182s3−1.34s2+0.1435s

∆roll = −0.007201s5+0.005238s4+0.001804s3−4.29e−005s2+0.0002332s+2.292e−018s5−3.051s4+3.913s3−2.642s2+0.923s−0.1419

Figures 3.6, 3.7, 3.8 shows that the maximum gain of the uncertainties and disturbances

Figure 3.6: Characteristics of ∆pitch


Figure 3.7: Characteristics of ∆yaw

Figure 3.8: Characteristics of ∆roll

can be reached up to a maximum of 20 dB. So the upper limit for γ can be obtained as

γ = 1‖∆‖∞ = 0.1


3.2.3.3 Robust controller formulation

With reference to the previous Section 2.2.1, the linearized input-output relation of the

missile model can be rewritten as

[y1 y2 y3

]T

=[

v1 v2 v3

]T

(3.20)

Now it is possible to find out three separate robust control law for designing v1, v2 and v3

for the linearized inner rate loop as discussed above. Now the remaining part of the design

is mainly concerned with the choice of weighting functions namely W1 and W3 here. In

a Riccati-based approach, the standard practice is to choose the weight W1(s) as a high

gain low-pass filter to adjust the tracking error of the system in low frequencies. The

weight W3(s) should be a high pass filter to shape the uncertainties in high frequencies.

After several trial and error effort it became possible to find the weights for which the

γ−optimal controller exists. The weights W1(s) and W3(s) are given by,

W1(s) = 100s+100

W2(s) = 0.01 s+0.04s+0.005

Note that same weights have been used for pitch, yaw roll channels. The multiobjective

(disturbance rejection and performance) feature of LMI was accessed through suitably

defined objective in the argument of the function hinflmi of the LMI Toolbox in MATLAB.

The order of the controller (nk) obtained from the LMI solution is 3. Some numerical

results have been given below for pitch plane only. Calculations for yaw and roll plane

will follow a similar pattern. According to the design steps described in the Section 3.2.2,

the values of X and Y are as follows:

X =

80877952.345 717.086 −805.697

717.086 0.0481 −0.0541

−805.697 −0.0541 2871.522

, Y =

89341574.796 −3.2181 −93147.491

−3.2181 705647.315 0.0160

−93147.491 0.016 465.7378


The value of γ is 0.08. Note that the value of γ is less than the upper limit of 0.1. So, in

this way the solvability condition has been satisfied with X,Y as positive definite matri-

ces. The augmented system matrices are given by:

A =

0 0 0

−1 −100 0

1 0 −0.005

, B1 =

0

1

0

, B2 =

1

0

0

C1 =

[0 100 0

0.01 0 0.00035

], C2 =

[−1 0 0

], D11 =

[0

0

], D12 =

[0

0

]

D21 = 1, D22 = 0

Now assuming

X1 =

1 0 0

0 1 0

0 0 1

, we can get the value of X2 as:

X2 =

−7.225e15 −245732514.558 7533578762612.71

−64065632132.52 −31695.417 66794823.456

72249781331.65 35595.128 −76386100.332

and from the equation 3.16, the value of XL obtained as

XL =

89341574.796 −3.2181 −93147.491 0.999 −2.123e− 10 −4.845e− 12

−3.2181 705647.315 0.016 2.392e− 21 1 2.31e− 18

−93147.4916 0.0160 465.737 1.483e− 17 −4.721e− 13 1

1 0 0 1.414e− 8 1.561e− 15 2.828e− 6

0 1 0 1.565e− 15 1.417e− 6 −4.745e− 11

0 0 1 2.828e− 6 −4.745e− 11 0.0027


and the value of HXLis obtained as:

HXL=

−186288.54 −705325.48 931.45 1.48e− 17 −1 1 −3.21 0 0.01

−705325.48 −141129463.1 −1.60 −2.39e− 19 −100 −2.31e− 16 705647.31 100 0

931.45 −1.60 −4.65 −7.41e− 20 2.36e− 15 −0.005 0.016 0 0.00035

0 0 0 0 0 0 0 0 0

−1 −100 0 0 0 0 1 0 0

1 0 −0.005 0 0 0 0 0 0

−3.2181 705647.31 0.016 2.39e− 21 1 2.31e− 18 −0.08 0 0

0 100 0 0 0 0 0 −0.08 0

0.01 0 0.00035 0 0 0 0 0 −0.08

Now from (3.18) it is possible to find out the robust controller as:

Jp =

[Ak Bk

Ck Dk

]=

−1658.461 −20.183 1089742.141 0.0003

−4.014 −0.0669 3519.74 −0.00042

1.9781 0.0277 −1480.52 3.8149

−1658.467 −20.183 1089747.94 0


The simulation results given in this section in order to show comparative performance

of FBLC with linear control (FBLCL) and FBLC with robust control (FBLCR) under

the following disturbance conditions shown in following Table 3.1. Note that, simulation

results have not yet been verified with seeker based closed loop guidance. Results shown

in this chapter are obtained from detailed 6-DOF model with open loop guidance. From

(2.10) it is evident that if the system is properly feedback linearized then (2.10) should be

satisfied in all the three channels pitch, yaw and roll. Here y1 = q, y2 = r, y3 = p. Figure

3.9 illustrates the comparison of FBLC with a linear rate loop controller (FBLCL) and

FBLC with a robust rate loop controller (FBLCR) in pitch plane. It can be seen that

the linearization for FBLCl is very poor. On the other hand the FBLCR shows better

performance in terms of linearization. From the figure it can be observed that maneuvers

for the missile take place around t = 0.5 unit and t = 0.8 unit. So, at these zones of flight,

the variations in aerodynamic coefficients and magnitude of disturbances become higher.

Figure 3.9 shows that the reference v1 and q have been overlapped each other when robust


Aerodynamic coefficients variation in %CL ±80CS ±10CN ±10Cm ±10Cn ±10Clζ ±10Clη ±10Cnδ

±10Cnη ±12Cmη ±12Thrust misalignment variation in %TmX

±5TmY

±2TmZ

±3

Table 3.1: Variation in aerodynamic coefficients and thrusts in x− y − z directions

control law has been applied. The same scenario can be seen for yaw and roll channel in

Figures 3.10 and 3.11 respectively. The last three Figures in 3.12 describe the control

input requirements in pitch,yaw and roll channels for robust control law. We can see that

the control input requirement is higher for FBLC and it varies rapidly. Figure 3.13 shows

the rate of control surface deflection and we can observe that except manuevering zone

the rate is well below the maximum limit (normalized to 1).

Note that we have not implemented this H∞ control law in closed loop guidance and

seeker. One main reason is that finding suitable weighting functions W1 and W3 requires

extensive efforts. We have left this part as future work.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time(Normalised)

qd

ot

and

V1 (

No

rmal

ised

)

FBLC Controller

qdot

v1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.05

0

0.05

0.1

Time (Normalised)

qd

ot

and

V1 (

No

rmal

ised

)

Robust Controller

qdot

v1

Figure 3.9: Comparison of FBLC and robust controller in pitch plane


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time (Normalised)

rdo

t an

d V

2 (N

orm

alis

ed)

FBLC Controller

rdot

v2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.05

0

0.05

Time (Normalised)

rdo

t an

d V

2 (N

orm

alis

ed)

Robust Controller

rdot

v2

Figure 3.10: Comparison of FBLC and robust controller in yaw plane


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

Time (Normalised)

pd

ot

and

V3 (

No

rmal

ised

)

FBLC Controller

pdot

v3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time (Normalised)

pd

ot

and

V3 (

No

rmal

ised

)

Robust Controller

pdot

v3

Figure 3.11: Comparison of FBLC and robust controller in roll plane


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−5

0

5

10

Time (Normalised)

Pit

ch D

efle

ctio

n

RCFBLC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−5

0

5

Time (Normalised)

Yaw

Def

lect

ion RC

FBLC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

0

10

20

Time (Normalised)

Ro

ll D

efle

ctio

n RCFBLC

Figure 3.12: Control deflection comparison


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

Time (Normalised)No

rmal

ized

Pit

ch D

efle

ctio

n R

ate

RCFBLC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−1

0

1

Time (Normalised)No

rmal

ized

Yaw

Def

lect

ion

Rat

e

RCFBLC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

Time (Normalised)No

rmal

ized

Ro

ll D

efle

ctio

n R

ate

RCFBLC

Figure 3.13: Control deflection rate


3.3 Comments

This chapter has demonstrated the use of robust feedback linearization for tackling the

aerodynamic uncertainties through a short range surface-to-surface homing missile. The

centralized design of the multivariable controller has been formulated as a multiobjective

optimization problem in the LMI framework. The solution is numerically sought through

the LMI solver. The performance robustness of the designed controller has been verified

through a realistic and practical 6-DOF simulation platform. Feedback linearization along

with a robust control law shows pretty good performance over a nominally designed feed-

back linearizing controller and thus it presents a robust feedback linearization approach.

Chapter 4

Sliding Mode control after Feedback

Linearization

4.1 Introduction

In this chapter, a robust control structure known as sliding mode control or variable

structure control, has been applied for designing rate loop controller. A variable structure

system is one whose structure can be changed or switched abruptly according to a certain

switching logic whose aim is to produce a desired overall behavior of the system. The

simplest example of variable structure systems are relay or on-off systems, in which the

control input can have only two values, on or off. Similar to the previous Chapter 3, sliding

mode controller has been formulated for the feedback linearized plant. We have already

discussed that the success of feedback linearization approach is hinged on the availability of

the accurate description of the model [14]. Indeed severe model uncertainty mainly due to

the aero-coefficients may degrade the performance of the feedback linearization approach.

In this regard, some robust scheme [28] such as sliding mode control is required. Figure

4.1 describes the proposed control structure and one can observe that this structure is

similar to the control structure proposed in Figures 2.1 and 3.1. The only difference is

that instead of linear controller block ’LC’ in Figure 2.1 and robust control block ’RC’ in

Chapter 4. Sliding Mode control after Feedback Linearization 107

Figure 4.1: Block diagram of the system representing robust feedback linearization foroutputs q, r and p

Figure 3.1, a new sliding mode control block ’SMC’ has been introduced. In Section 3.2 we

have discussed how uncertainties and various disturbances affect the nominally feedback

linearized plant. From (2.10) and (3.20), we can write the linearized input-output relation

of the plant model with q, r and p as outputs, as

y1

y2

y3

=

v1

v2

v3

(4.1)

In Chapter 3 we have designed H∞ robust control law for the new inputs v = [v1, v2, v3]T

to tolerate the model uncertainties and disturbances. Here in next sections we will discuss

about the formulation of sliding mode control law instead of H∞ control law for designing

v.


4.2 Formulation of sliding mode controller

In this section a very short discussion on sliding modes which is a particular approach to

the design of variable structure systems, has been introduced. Developed in the Soviet

Union more than 40 years ago, sliding mode controllers differ from simpler relay controllers

in that they rely on extremely high speed switching among the control values. As discussed

already total design has been carried out in two steps.

• Performing input-output linearization of the nominal plant.

• Formulation of robust sliding mode control law for that feedback linearized plant.

We consider the system described by

·x (t) = f (x (t)) + g (x (t)) u (t)

y (t) = h (x (t)), (4.2)

where x(t) is the n-dimensional plant state, u is m-dimensional plant input, y is m-

dimensional plant output, f : Rn → Rn and g : Rn → Rn ×Rm and h : Rn → Rm are

smooth functions. Sliding mode controller for system 4.2 can be designed by the following

steps.

• STEP I : Performing input-output feedback linearization

• STEP II: Formulation of sliding mode control law for designing v

STEP I has been already done in Section 2.2.1 and the relevant equations that we will

demonstrate in this chapter will be exactly the same as Section 2.2.1.

4.2.1 Step II: Formulation of sliding mode control law for de-

signing v

The main idea behind sliding mode control is to choose a suitable surface in state space,

typically a linear hypersurface, called the switching surface, and switch the control input


on this surface. The control input is then chosen to guarantee that the trajectories near

the sliding surface are directed toward the surface. Ideally then, any control input will

suffice so long as the resulting trajectories are pointing toward the surface. Once the

system is trapped on the surface, the closed loop dynamics are completely governed by

the equations that define the surface. In this way, since the parameters defining the surface

are chosen by the designer, the closed loop dynamics of the system will be independent

of perturbations in the parameters of the system and robustness is achieved. The design

of sliding mode control can be broken down into two steps:

• Specifying a suitable sliding surface

• Achieving the sliding condition and designing system dynamics on the surface

4.2.1.1 Specifying sliding surfaces

Let ei = yi − ri with ri the reference trajectories, be the tracking error for the output yi

and let

ei = yi − ri =[

ei ei ... ... eri−1i

]T

be the tracking error vector. Furthermore, let us define a time-varying surface Si(t) in

the state space Rri−1 by the scalar equation si(yi; t) = 0, where

s(yi; t) = (d

dt+ k)n−1ei (4.3)

and k is strictly positive constant. Or we can write (4.3) as

si (t) = e(ri−1)i + ki(ri−1)e

(ri−2)i + · · ·+ ki2e

(1)i + ki1e

(0)i + ki0

∫eidt (4.4)

In this way we can define m sliding surfaces si, i= 1, · · · ,m, based on the input-output

linearized system given in (4.1) ki(r−1), · · · , ki0 are such that

λri + ki(ri−1)λri−1 + · · ·+ ki1λ + ki0 (4.5)


is Hurwitz polynomial. For the tracking task to be achievable using a finite control law

v, the initial state ri(0) must be such that

ri(0) = yi(0) (4.6)

Given initial condition (4.6), the problem of tracking ri = yi is equivalent to that of

remaining on the surface Si(t)∀t ≥ 0; indeed si ≡ 0 represents a linear differential equation

whose unique solution is ei = 0, given initial condition (4.6). Thus the problem of tracking

the n-dimensional vector ri can be reduced to that of keeping the scalar quantity s at

zero.

4.2.1.2 Achieving sliding condition

The closed-loop system is said to satisfy the sliding condition if the following applies [27].

1

2

ds2i

dt≤ −ηi |si| , (ηi > 0) (4.7)

where ηi, i = 1, · · · ,m are positive numbers. Note that sliding condition will make si (t) =

0 and si (t) = 0 in a finite time. Since is a stable differential equation, satisfaction of

si (t) = 0 by ei(t) in turn leads to asymptotic tracking.

Let

·s

def=

·s1

...·

sm

, y(ρ) =

y(r1)1...

y(rm)m

(4.8)

sgn (s) = [sgn (s1) , · · · , sgn (sm)]

where sgn(·) is the signum function. Now it has been reported in [36] that a control law

that achieves the sliding condition (4.7) is given by

u = E−1(( ·

s−y(ρ))−M − λsgn (s)

)(4.9)


where λ = diag [λ1, · · · , λm] with λi a positive number greater than the given positive

number ηi. Note that

( ·si−yri

i

)= −rri

i + ki(ri−1)e(ri−1)i + · · ·+ ki2e

(2)i + ki1e

(1)i + ki0ei

so( ·s−y(ρ)

)does not depend on u. The integral term in (4.4) can be omitted by setting

ki0 = 0. Since the sliding condition also implies si(t) = 0, the asymptotical tracking can

still be achieved by the control law (4.9) as long as, for i = 1, · · · ,m, ki(r−1), · · · , ki1 are

such that λri−1 + ki(ri−1)λri−2 + · · · + ki1 are Hurwitz. In practice the implementation of

variable structure controllers results in control chattering. The ideal behavior of sliding

mode controller is achieved in the theoretical limit as the switching frequency becomes

infinite. In practice the small, but nonzero delay in control switching will cause the tra-

jectory to slightly overshoot or undershoot the switching surface each time the control is

switched. This is known as chattering. It has been observe that more the model uncer-

tainties, the chattering becomes severe. The approach taken in this thesis to overcome

the undesirable chattering is to introduce what is known as boundary layer around the

sliding surface and approximate the switching control law by a continuous control inside

this boundary layer. Thus the discontinuous control law sgn(si) is often replaced by the

saturation function sat( si

εi) where

sat (x) = x, if |x| ≤ 1

sat (x) = sgn (x) , if |x| ≥ 1(4.10)

4.2.2 Application to the STT missile model

As we have already tried input-output feedback linearization approach with nominal mis-

sile model in Section 2.2.1, let us start from the linearized input-output relation obtained

in 2.10,

y1

y2

y3

=

v1

v2

v3

(4.11)


and we can easily observe that

r1 = 1, r2 = 1, r3 = 1. So, referring to equation (4.4) we can say that in this particular

case there will be three sliding surfaces namely

s1 = e1, s2 = e2, s3 = e3 where

e1 = q − qd

e2 = r − rd

e1 = p− pd

(4.12)

As discussed earlier qd, rd, pd can be obtained from outer loop or lateral acceleration error

dynamics. So from (4.8) we can write,

(s− y(ρ)) =

s1 − q

s2 − r

s3 − p

=

−k11qd

−k21rd

−k31pd

Now the control input can be easily found out from (4.9) with a boundary layer vector

ε =

ε1

ε2

ε3

=

0.1

0.2

0.2

The design parameters are given by k11, k21 and k31. These parameters have been chosen

by trial and error method. It has been observed for this particular case that using higher

values of kijs, the system becomes unstable and shows more chattering. For lower values

of kijs, the system performance shows a few second delay with respect to the nominal

FBLC. Thus we have chosen these kij values where the system performance is more or

less satisfactory. The 6-DOF simulation results have been shown in the next section to

check the system performance with sliding condition.


4.2.2.1 Simulation Results

The simulation results shown for the sliding mode control is based on the detailed 6-DOF

given by (1.3) and (1.4). First we have tried with the same variations in aerodynamic

coefficients given in Table 3.1. But the control law (4.9) failed to handle that much

of variations and the system became unstable. A huge chattering was present in that

case. In the second stage we have decreased the variations in aerodynamic coefficients.

The simulations have been performed for open loop guidance as well as for closed loop

Aerodynamic coefficients variation in %CL ±30CS ±5CN ±5Cm ±5Cn ±5Clζ ±5Cnη ±5Thrust misalignment variation in %TmX

±1

Table 4.1: Variation in aerodynamic coefficients and thrusts in x− y − z directions

guidance with seeker. Open loop simulations have been performed to compare with the

H∞ robust control performance presented in Chapter 3. Figures 4.2,4.3 and 4.4 show

the input-output linearization with the sliding mode controller (FBLCSM) in comparison

with FBLC with linear controller (FBLCL) with guidance and seeker in open loop as done

before for the H∞ controller. But it can be seen that much improvement has not been

noticed for FBLCSM.

Figure 4.5 shows the input deflections for sliding mode and FBLC. Here also we can

see that in some cases sliding mode demands more input for almost the same performance.

So if we compare the open loop performances of SMC with H∞ control, we can see that

the latter one can handle more aerodynamic variations than the former one. As stated

before sliding mode control law has also been tested in FORTRAN 6-DOF platform with


closed loop guidance and seeker. Some of these results have been explained below. We

will see that openloop and closed loop simulation results gives almost same performance

for sliding mode controller in comparison with FBLC. Figure 4.6 shows the pitch,yaw and

roll rates for FBLC and sliding mode controller. Here we can see that FBLC and the

sliding mode controller behave similarly.

Figures 4.7 and 4.8 show almost similar response for α, β and gimbal angle for both

FBLCL and FBLCSM. Figure 4.9 shows the pitch and yaw latax profiles from which

it can be concluded that performance is similar for small aerodynamic perturbations as

shown in Table 4.1. For heavy perturbation in aerodynamic coefficients, the FBLCL as

well as sliding mode controller become unstable. Figures 4.10 and 4.11 show pitch-yaw-

roll channel deflections and the fin distributions respectively. As expected, the sliding

mode controller takes more fin than the FBLCL. The rate of fin deflections are shown in

Figure 4.12 and the rate is below the maximum level.

Figure 4.13 represents the variation of aerodynamic coefficients and for small pertur-

bations in aerodynamic coefficients the response for FBLC and the sliding mode control

remain more or less the same. Thus we have seen that sliding mode control implemented

here can tolerate very small perturbations. Performance wise FBLCL and FBLCSM are

more or less similar and we have not found any significant improvements that is claimed

in [36].


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−5

0

5

x 10−3

Time(Normalised)

qd

ot

and

V1 (

No

rmal

ised

)

FBLC Controller

qdot

v1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−0.05

0

0.05

0.1

Time (Normalised)

qd

ot

and

V1 (

No

rmal

ised

)

SMC Controller

qdot

v1

Figure 4.2: Linearization in pitch channel


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−1

−0.5

0

0.5

1

x 10−3

Time (Normalised)

rdo

t an

d V

2 (N

orm

alis

ed)

FBLC Controller

rdot

v2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time (Normalised)

rdo

t an

d V

2 (N

orm

alis

ed)

SMC Controller

rdot

v2

Figure 4.3: Linearization in yaw channel


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−2

−1

0

1

2x 10

−3

Time (Normalised)

pd

ot

and

V3 (

No

rmal

ised

)

FBLC Controller

pdot

v3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.04

−0.02

0

0.02

0.04

Time (Normalised)

pd

ot

and

V3 (

No

rmal

ised

)

SMC Controller

pdot

v3

Figure 4.4: Linearization in roll channel


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−4

−2

0

2

Time (Normalised)

Pit

ch D

efle

ctio

n

SMCFBLC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−10

−5

0

Time (Normalised)

Yaw

Def

lect

ion

SMCFBLC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−10

−5

0

Time (Normalised)

Ro

ll D

efle

ctio

n

SMCFBLC

Figure 4.5: Control deflection comparison


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1−1

−0.5

0

0.5

1

Normalised Time

Nor

mal

ised

Pitc

h R

ate


SlidingFBLC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1−4

−2

0

2

Normalised Time

Nor

mal

ised

Yaw

Rat

e

SlidingFBLC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−1

0

1

Normalised Time

Nor

mal

ised

Rol

l Rat

e

SlidingFBLC

Figure 4.6: Pitch, yaw and roll rates


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−1

−0.5

0

0.5

1

Normalised Time

Nor

mal

ised

Alp

ha (

Bod

y)


SlidingFBLC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−2

−1

0

1

Normalised Time

Nor

mal

ised

Bet

a (B

ody)


SlidingFBLC

Figure 4.7: α and β


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Normalised Time

Nor

mal

ised

Gim

bal A

ngle

(El)


SlidingSliding

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−2

0

2

4

Normalised Time

Nor

mal

ised

Gim

bal A

ngle

(Az) Sliding

Sliding

Figure 4.8: Gimbal angle profile


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1−1

−0.5

0

0.5

Normalised Time

Nor

mal

ised

Pitc

h la

tax


SlidingFBLC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−1

0

1

2

Normalised Time

Nor

mal

ised

Yaw

lata

x


SlidingFBLC

Figure 4.9: Pitch and yaw latax


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−2

−1

0

1

Normalised Time

Eff.

pitc

h de

fln (

δ PB

)


SlidingFBLC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−2

0

2

Normalised Time

Eff.

yaw

def

ln (

δ YB

) SlidingFBLC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−3

−2

−1

0

Normalised Time

Rol

l def

ln (

δ R )

SlidingFBLC

Figure 4.10: Pitch,yaw and roll deflections


5 10 15 20

−2

−1

0

1

2

Normalised Time

Nor

mal

ised

Fin

−1 d

efl

n SlidingFBLC

0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5

0

0.5

1

1.5

Normalised Time

Nor

mal

ised

Fin

−2 d

efl

n

SlidingFBLC

0.2 0.4 0.6 0.8 1

−2

−1

0

1

Normalised Time

Nor

mal

ised

Fin

−3 d

efl

n

SlidingFBLC

0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5

0

0.5

1

1.5

Normalised Time

Nor

mal

ised

Fin

−4 d

efl

n SlidingFBLC

Figure 4.11: Fin demands


5 10 15 20

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Normalised Time

Nor

mal

ised

Fin

−1 d

efl

n r

ate Sliding

FBLC

0.2 0.4 0.6 0.8 1

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Normalised Time

Nor

mal

ised

Fin

−2 d

efl

n r

ate Sliding

FBLC

0.2 0.4 0.6 0.8 1

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Normalised Time

Nor

mal

ised

Fin

−3 d

efl

n r

ate Sliding

FBLC

0.2 0.4 0.6 0.8 1

−0.5

0

0.5

Normalised Time

Nor

mal

ised

Fin

−4 d

efl

n r

ate Sliding

FBLC

Figure 4.12: Fin deflection rate


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1−0.3

−0.2

−0.1

0


Rol

ling

mom

ent c

oeffi

cien

t CL

Normalised Time

SlidingFBLC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

0

0.5

1

Sid

e fo

rce

coef

ficie

nt C

S

Normalised Time

SlidingFBLC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−8

−6

−4

−2

0

2

Yaw

ing

mom

ent c

oeffi

cien

t Cny

Normalised Time

SlidingFBLC

Figure 4.13: Force and moment Coefficients


4.3 Comments

The above simulation results describe the performance of sliding mode controller for a

feedback linearized skid-to-turn homing missile. The main idea in designing sliding mode

controller is to tackle model uncertainties which are inherently present in the system. The

simulation results presented here constitute a case study of robust feedback linearization

where the parametric uncertainties are small enough. In comparison with the Chapter 3,

one can say that sliding mode control law, does not give a very satisfactory performance

even in small aerodynamic perturbations shown in Table 4.1. However it has been tested

successfully in closed loop seeker based detailed 6-DOF model. On the other hand, H∞robust control law, gives satisfactory performance with huge aerodynamic uncertainties

shown in Table 3.1, has been tested in open loop guidance only.

Chapter 5

Conclusions

In this thesis, an approach has been presented for controlling a highly maneuverable short

range surface-to-surface missile. Our choice has been oriented by the industrial as well

as academic context. The necessity of increasing missile performances with reduced flight

range has motivated new investigations in order to estimate not only the potential of

recent theoretical control methods, but also their application level, in order to meet the

industrial demands.

In this thesis, the application of input-output linearization has been presented for a short

range surface-to-surface skid-to-turn homing missile. Here, a state dependent nonlinear

feedback control law has been designed for improving the overall missile performance in

view of reduced range, less control effort etc. Aerodynamic missiles generally suffer from

coupling effect among pitch-yaw-roll axes. This coupling effect results in large variations

in aerodynamic coefficients, side-slip, control effort in yaw etc. These in turn may cause

fin saturation and gimbal angle limit violation which cause the missile to lose its actual

track. The conventional three loop autopilot generally fails to solve this coupling prob-

lem. To get rid of this problem a nonlinear multivariable approach has been proposed.

In this approach, the whole missile model has been feedback linearized. Then a linear

proportional controller has been designed for obtaining the tracking performance in inner

rate loop. Thus the pitch-yaw-roll dynamics, specifically rate dynamics, of the missile

becomes linear and decoupled which are the most essential requirement for most of the

Chapter 5. Conclusions 129

aerodynamic missiles. The simulation results show the evidence of linearization and de-

coupling and the improved performance of the Feedback Linearizing Control (FLC) law

over the traditional three loop autopilot.

Although input-output linearization is successful in decoupling and linearizing nonlinear

dynamics, it fails to handle the same problem when plant dynamics become uncertain.

Generally short range aerodynamic missiles, in which boost time is small enough, suffers

from the problem of uncertainties due to aerodynamic coefficients. This problem becomes

more acute when the missile tries to turn rapidly to track the target. In view of this prob-

lem, two different robust control laws have been formulated. In both cases the nominal

missile dynamics has been input-output linearized as before. In the first case, a linear

matrix inequality approach to H∞ control has been presented along with its application to

uncertain missile dynamics. It has been shown through open loop MATLAB simulation

results that upto 100% uncertainty in aerodynamic coefficients can be tolerated through

this method.

In the second case, another approach of sliding mode control has been considered. Here

also, sliding mode control law has been applied to an input-output linearized plant. Sim-

ulation results have been presented with closed loop guidance and seeker and it has been

seen that this method is not so useful when the system poses very high uncertainties in

terms of aerodynamic coefficients.

All of the above control laws presented in this thesis, require whole state measurements.

But only some of the states are measured (rates in three channels) and the rest are un-

measured (velocity components in three channels). That means the unmeasured states

are to be estimated to realize the control laws practically. For that purpose, a nonlinear

Luenberger observer has been used. The forward velocity of the missile is assumed to

be known from thrust profile and observer equations have been formulated in terms of

α and β as input to the observer i.e., the output of the system are direct functions of α

and β. Thus α and β have been first estimated and then velocity components are derived

easily from standard mathematical relations. Simulation results show good tracking per-

formances in presence of sensor noise.

Chapter 5. Conclusions 130

The main contributions of the thesis can be summarized as follows:

1. Minimized roll rate even during high angle of attack maneuver.

2. Good decoupling among the three axes during pitch maneuver.

3. Estimation of the unmeasured states of the system using a nonlinear observer re-

quired for the computation of nonlinear feedback.

4. Design of a robust H∞ control law that retains performance even with aerodynamic

uncertainties.

5. Design of a robust sliding mode controller to tolerate the uncertainties caused due

to aerodynamic coefficients.

Some recommendations can be drawn in view of the overall work done in this thesis.

• Outer loop feedback linearization along with the rate loop feedback linearization

will make the whole plant dynamics linear and decoupled.

• Some more advanced robust control laws such as µ−synthesis, H∞ loop shaping etc,

are to be tested in order to cater more aerodynamic uncertainties and disturbances.

• Observer gain has to be adaptive and robust irrespective of different flight condi-

tions.

The analysis and numerical results presented in this thesis amply demonstrate the fea-

sibility of designing nonlinear control systems for the next generation high performance

missile autopilots. Nonlinear design methods have the potential for enhancing missile

performance while simplifying the design process. This can result in a lighter and more

accurate missile system.

Appendix A

A Brief Theory of Feedback

Linearization

A.1 Introduction

A single input-single output control affine system obeys the following state and output

equation x = f(x) + g(x)u

y = h(x), (A.1)

x ∈ Rn is a vector, u is a scalar input, y is a scalar output,f : Rn → Rn and g : Rn → Rn

are vector fields and h : Rn → R is a scalar function.Suppose that f(·), g(·) and h(·) are

differentiable function.

Obviously y does not depend explicitly on u since u is not an argument of function

h. If u is changed instantaneously, there will be no immediate change in y. The change

comes gradually via x. To check the corresponding behavior of ywe can differentiate the

Chapter A. A Brief Theory of Feedback Linearization 132

output equation [54]:

y = ∂h∂x

x = ∂h∂x

(f(x) + g(x)u)

= ∂h∂x

f(x) + ∂h∂x

g(x)u

(A.2)

This shows that y depends directly on u if and only if [∂h/∂x] g(x) 6= 0 we will say that

the system has relative degree 1 if [∂h/∂x] g(x) 6= 0

Now, let us assume that [∂h/∂x] g(x) = 0 differentiating the output y once more, we

obtain:

y =∂

∂x

[∂h

∂x

]f(x) +

∂

∂x

[∂h

∂xf(x)

]g(x)u (A.3)

The system is said to have relative degree 2 if:

∂h

∂xg(x) = 0 and

∂

∂x

[∂h

∂xf(x)

]g(x) 6= 0 (A.4)

The idea of relative degree can be easily generalized with the help of the following notation.

Definition 1 (Lie Derivative) Let λ : Rn → R be a differentiable function and f a

vector field, both defined on an open subset U of Rn.The derivative of λ along f or Lie

derivative of λ along f is given by the inner product

⟨∂λ

∂x, f(x)

⟩=

∂λ

∂xf(x) (A.5)

The lie derivative of λ along f is usually denoted as Lfλ, so that:

Lfλ(x) =n∑

i=1

∂λ

∂xi

fi(x) (A.6)


If Lfλ is again differentiated along another vector field g, the following is obtained:

LgLfλ(x) =∂ (Lfλ)

∂xg(x) (A.7)

This operation could be used recursively along the same vector field.Lkfλ indicates that λ

is being differentiated k times along f such that

Lk

fλ(x) =∂(Lk−1

f λ)∂x

f(x)

L0fλ(x) = λ(x)

(A.8)

Definition 2 (Relative Degree) The SISO system given by Equation A.1 is said to

have a relative degree r at a point xo if

1. LgLk−1f h(x) = 0 ∀ x in a neighborhood of xok = 1, ..., r − 1

2. LgLr−1f h(xo) 6= 0

The following formula describing the time derivatives of the output is an immediate

consequences of Definition 2 :

dky

dtk= y(k) =

Lk

fh(x), k = 1, ..., r − 1

Lkfh(x) + LgL

k−1f h(x)u, k = r

, (A.9)

Remark 1 For SISO linear systems:

x = Ax + Bu

y = Cx(A.10)


The relative degree is the difference in degree between the nominator and denominator of

its equivalent transfer function:

G(s) = C(sI − A)−1B (A.11)

Equivalently, the relative degree of a SISO linear system is the positive integer r such

that:

LgLr−1f h = CAr−1B 6= 0 (A.12)

Definition 3 (Strong Relative Degree) A system is said to have a strong relative de-

gree if the relative degree is r for all xo ∈ Rn

A.1.1 Input-output linearisation

Consider a system with a strong relative degree r.Then we have that the rth derivative of

the output is given by:

y(r) = Lrfh + LgL

r−1f hu, LgL

r−1f h 6= 0 (A.13)

There is an interesting possibility. Introduce the feedback law:

u =1

LgLr−1f h

(v − Lr

fh), (A.14)

Where v is a reference signal The resulting relationship is between v and y(r) is:

y(r) = v (A.15)


Which is a linear system. Taking the Laplace transform:

y(s) =1

snv(s) (A.16)

By using the feedback law A.14, we have obtained a system that is linear from the reference

signal v to the output y.

Proposition 1 Suppose a system of the form of equation A.1 has relative degree r at a

point xo. Define,

ϕ1(x) = h(x)

...

ϕr(x) = Lr−1f h(x)

, (A.17)

If r is strictly less than n, it is always possible to find n−r arbitrary functions ϕr+1(x), ..., ϕn(x)

such that the mapping

Φ(x) = [ϕ1(x), ..., ϕr(x), ..., ϕn(x)]T (A.18)

has a jacobian matrix which is nonsingular at xo and therefore represents a local coordi-

nates transformation in a neighborhood of xo. Moreover, it is always possible to choose

ϕr+1(x), ..., ϕn(x) in such a way that

Lgϕi(x) = 0, r + 1 ≤ i ≤ n, (A.19)


Differentiating Equation A.17 with respect to time and using Equation A.9, we obtain:

dφ1(x)dt

= dh(x)dt

= Lfh(x) = φ2(x)...dφr−1(x)

dt= Lr−1

f h(x) = φr(x)dφr(x)

dt= Lr

fh(x) + LgLr−1f h(x)u

, (A.20)

By introducing the variables:

ξ =

ξ1

...

ξr

=

φ1(x)...

φr(x)

(A.21)

η =

η1

...

ηn−r

=

φr+1(x)...

φn(x)

(A.22)

Then Equation A.20 can be written as follows:

dξ1dt

= ξ2(t)...dξr−1

dt= ξr(t)

dξr

dt= b(ξ, η) + a(ξ, η)u

, (A.23)

where, y = ξ1 a(ξ, η) = LgL

r−1f h (Φ−1(ξ, η))

b(ξ, η) = Lrfh (Φ−1(ξ, η))

, (A.24)

Notice that

x = Φ−1(ξ, η), (A.25)


As far as the other new co-ordinates are concerned, corresponding to η = [η1, ..., ηn−r]T ,

we cannot expect any special structure in the corresponding equations. However, if

φr+1, ..., φn have been chosen such that Lgφr+i = 0, i = 1, ..., (n− r), then

dηi

dt= Lfφr+i (x(t)) , i = 1, ..., (n− r) (A.26)

By defining qi (ξ, η) = Lfφr+i (Φ−1 (ξ, η)) , then we have that the derivatives of ηi can be

written as follows:dηi

dt= qi (ξ, η) , i = 1, ..., (n− r) (A.27)

or, in vector form:dη

dt= q (ξ, η) (A.28)

In summary, the normal form of a control affine SISO nonlinear system of Equation A.1,

with relative degree r around a point xo, is given by:

ξ1 = ξ2

ξ2 = ξ3

...

ξr−1 = ξr

ξr = b(ξ, η) + a(ξ, η)u

η = q(ξ, η)

y = ξ1

(A.29)

Notice that if we choose the feedback law:

u = 1a(ξ,η)

(v − b(ξ, η))

= 1LgLr−1

f h(Φ−1(ξ,η))

(v − Lr

fh(Φ−1(ξ, η)))

= 1LgLr−1

f h(x)

(v − Lr

fh(x)) (A.30)


Then the resulting system from the reference input v to the output y = ξ1 is linear. The

normal form is given by:

ξ1 = ξ2

ξ2 = ξ3

...

ξr−1 = ξr

ξr = v

η = q(ξ, η)

y = ξ1

(A.31)

Notice, however, that the internal dynamics η = q(ξ, η) are possibly nonlinear, so that

the system has not been fully linearised by the feedback law.

If the feedback law is changed to:

u =1

a (ξ, η)(v − b (ξ, η)− λ0ξ1 − ...− λr−1ξr) (A.32)

Then the normal form is given by:

ξ1 = ξ2

ξ2 = ξ3

...

ξr−1 = ξr

ξr = −λ0ξ1 − ...− λr−1ξr + v

η = q(ξ, η)

y = ξ1

(A.33)

and the transfer function from input to output is:

y(s)

v(s)=

1

λr−1sr + ... + λ0

(A.34)


It is easy to see that the unforced internal dynamics (or zero dynamics) which are defined

for ξ = 0 (η = −η3) , are asymptotically stable around η = 0, so that the control signal

should be bounded for bounded v and ξ. Notice that ξ is bounded if v is bounded since

with a > 0 the transfer function y(s)/v(s) is stable.

A.2 Multi input multi output systems

The concepts used in the above section for SISO systems, such as input-state linearization,

input-output linearization, normal forms, zero-dynamics, and so on, can be extended to

MIMO systems. For the MIMO case, we consider square systems (i.e., systems with the

same numbers of inputs and outputs) of the following form

x = f(x) + g1(x)u1 + ..... + gm(x)um

y1 = h1(x)

· · ·ym = hm(x)

(A.35)

Where is the state vector,u′is(i = 1, ....., m) are control inputs, y′js(j = 1, ....., m) are

outputs, and , f and g′is are the smooth vector fields, and h′js are smooth scalar functions.

If we collect the control inputs ui into a vector u, the corresponding vectors into a matrix

G, and the outputs into a vector y, the system’s equations can then be compactly written

asx = f(x) + G(x)u

y = h(x)(A.36)

A.2.1 Feedback Linearization of MIMO Systems

The approach to obtain the input-output linearization of the MIMO systems is again to

differentiate the outputs yj of the system until the inputs appear, similarly to the SISO


case. To start with,

yj = Lfhj +m∑

i=1

(Lgihj)ui, (A.37)

If Lgihj(x) = 0 ∀i ,then the inputs do not appear and we have to differentiate again.

Assume that rjis the smallest integer such that at least one of the inputs appear in y(rj)j ,

then

y(rj)j = L

rj

f hj +m∑

i=1

LgiL

rj−1f hjui, (A.38)

With LgiL

rj−1f hj(x) 6= 0 for at least one i, ∀x ∈ Ω.If we perform the above procedure for

each output yj, we can obtain a total of m equations in the above form, which can be

written compactly as

y(r1)1

· · ·· · ·y

(rm)m

=

Lr1f h1

· · ·· · ·Lrm

f hm

+ E(x)

u1

· · ·· · ·um

, (A.39)

Where the m×m matrix E is defined as

E(x) =

Lg1Lr1−1f h1........LgmLr1−1

f h1

................

................

Lg1Lrm−1f hm........LgmLrm−1

f h1m

(A.40)


The matrix E(x) is called the decoupling matrix for the MIMO system. If the decoupling

matrix is non-singular in a region Ω around a point x0, then the input transformation

u = −E−1

Lr1f h1

· · ·· · ·Lrm

f hm

+ E−1

v1

· · ·· · ·vm

, (A.41)

yields a linear differential relation between the output y and the new input v

y(r1)1

· · ·· · ·y

(rm)m

=

v1

· · ·· · ·vm

(A.42)

Note that the above input-output relation is decoupled, in addition to being linear. Since

only affects the corresponding output yj , but not the others, a control law of the form

A.41 is called a decoupling control law, or non-interacting control law. As a result of

the decoupling, one can use SISO design on each y − v channel in the above decoupled

dynamics to construct tracking or stabilization controllers. It is useful to formalize the

concept of relative degree for MIMO systems at this point. Since there is a relative degree

associated with each output, the relative degree of the MIMO system is defined by m

integers.

Definition 4 The system A.35 is said to have relative degree (r1, · · · , rm) at x0 if there

exists a neighborhood Ω of x0 such that ∀x ∈ Ω,

• LgiLk

fhj (x) = 00 ≤ k ≤ ri − 11 ≤ i, j ≤ m

• E(x) is non-singular


The total relative degree of the system is defined by

r = r1 + · · ·+ rm

Let us consider the case of r < n first. A normal forma can be obtained for the system in

a manner similar to the SISO case, as we now show. First, choose as coordinates

ζ11 = h1 (x) ζ1

2 = Lfh1 (x) · · · ζ11 = Lr1−1

f h1 (x)

· · ·ζm1 = hm (x) ζm

2 = Lfhm (x) · · · ζmm = Lrm−1

f hm (x)

(A.43)

These are simply the m outputs yj and their derivatives up to order rj.

Similarly to the SISO case, the r coordinates ζji (j = 1, . . . , m; i = 1, . . . , rj) are in-

dependent and can be used as a partial set of a new state vector. This is because the

gradient vectors

Lifhj (x) 0 ≤ i ≤ rj − 1 1 ≤ j ≤ m

are linearly independent, as can be shown in a manner analogous to SISO case, using the

non-singularity of the decoupling matrix E. Now, let us complete the choice of the new

state vector by choosing n− r more functions η1, . . . , ηn−r (x) which are independent with

respect to each other and to the r coordinates chosen earlier. This can always be done,

based on the Frobenius theorem. However, unlike the SISO case, it is no longer possible

to guarantee that

∀x ∈ Ω Lgiηk (x) = 0 1 ≤ i ≤ m 1 ≤ k ≤ n− r

unless the vector fields g1, . . . , gm are involutive on Ω . As a result, the state equations

for these n− r coordinates will have the input vector u appearing.


With (ζ, η) as coordinates, the system equations can also be put into a ”normal form”.

Specifically, the external dynamics is

ζj1 = ζj

2

· · ·ζjrj

= aj (ζ, η) +m∑

i=1

bij (ζ, η) ui

where j = 1, 2, . . . ,m, and

aj (ζ, η) = Lrj

f hj (x)

bij (ζ, η) = Lgi

Lfr−1j hj (x)

The internal dynamics is·η = w (ζ, η) + P (ζ, η) u

with (k = 1, . . . , n− r) and (i = 1, . . . ,m)

wk (ζ, η) = Lfηk (x)

Pki (ζ, η) = Lgiηk (x)

Note that P ∈ R(n−r)×m and w ∈ Rn−r. As in the SISO case, feedback law of A.41

renders the (n− r) states η unobservable.

An interesting case of the above input-output linearization corresponds to the total

relative degree being n,i.e.,m∑

j=1

rj = n

In this case, there is no internal dynamics. With the control law in the form of A.41, we

obtain an input-state linearization of the original nonlinear system. With the equivalent

inputs vi designed as in the SISO case, both stabilization and tracking can then be achieved

for the system without any worry about the stability of the internal dynamics. We remark


that the necessary and sufficient conditions for input-state linearization of multi-input

nonlinear systems are similar to and more complex than those for single input systems.

A.2.2 Zero-dynamics and control design

When designing controllers based on the linear input-output relation in (A.39)[63], one

has to be concerned with the stability of the internal dynamics. It is therefore of interest to

study the stability of the zero-dynamics, an extent case of internal dynamics with output

being exactly zero.Similarly to the SISO case, the zero-dynamics of a MIMO system is

obtained by constraining the output to zero [70],[19].

Definition 5 The zero-dynamics of the MIMO nonlinear system is the dynamics of the

system when the outputs are constrained to be identically zero.

since the constraint that the output identically equal to zero implies that all the

derivatives of the output are zero, we have

ζ(t) ≡ 0

In order to keep the outputs identically zero, the control inputs must be chosen as

u(t) = −E−1(0, η)a(0, η)

where η(t) is the soltion of the differential equations.

η(t) = w(0, η)− P (0, η)E−1(0, η)a(0, η)

with η(0) arbitary. In the original x coordinates, when the system operates in zero-

dynamics, the system states x evolve on the surface.

M∗ =

x ∈ Ω|hj(x) = Lfhj(x) = ..... = Lrj−1f hj(x) = 0, 1 ≤ j ≤ m


Of course, the initial states x(0) must be chosen to b on this surface. In terms of x(0)

the constraining control inputs u are

u∗(x) = −E−1(x)

Lr1f h(x)

.....

.....

Lrmf hm(x)

(A.44)

The zero-dynamics is given by the equation

x = f(x) + g(x)u∗(x) (A.45)

with the states constrained on the surface M∗

Similarly to the SISO case, we can define the notation of minimum phase systems.

Definition 6 The MIMO nonlinear system A.35 is said to be asymtotically minimum

phase if the zero-dynamics is locally asymtotically stable.

The definition of exponentially minimum phase is similar.

For minimum phase systems, the control design results in section 2.2 can easily ex-

tended to MIMO case.

Appendix B

LMI Approach to H∞ Control

B.1 The Theory of H∞ Control based on LMI Ap-

proach

A brief introduction to the robust control has been given in the Chapter 3. Here we have

given more detailed analysis of robust control formulation.

B.1.1 Singular value decomposition

In classical control theory, Bode magnitude plot gives the gain of a SISO system at

different frequencies. But if the system is a MIMO one then the Bode plots of all the

elements of the transfer matrix give little idea about the gain of the system since it

undermines the interactions among the elements. Eigen values of the transfer matrix can

be an answer but then they are not a very good measure of the gain or size of a matrix.

Moreover, eigen values are limited to square transfer matrices (equal no. of inputs and

outputs) only. Therefore, a more general entity is required and singular values have been

found suitable.

Chapter B. LMI Approach to H∞ Control 147

The singular values of a complex valued matrix A are defined as the positive square

roots of the eigen values of A × A. Singular values are always real and non-negative.

They are generally denoted by σ The same way as a square matrix can be digitalized by

a similarity transformation with the modal matrix, similarly a non-square matrix can be

digitalized by a method called singular value decomposition. Let A be an m × n matrix

with rank r. Then it can be decomposed into singular values as,

A = U∑

V ∗

where, U is an m×m unitary matrix,V is an n× n unitary matrix and

∑=

σ1 0 .... 0

0 : ... 0

: : σr :

0 0 ... 0

is an m× n matrix.

B.1.2 Norms

Norms can be viewed as the measure of the size of a matrix or a vector. A generalized

p− norm of a vector x ∈ Cn is defined as,

‖x‖p

∆=

(n∑

i=1

|xi|p)1/p

, for 1 ≤ p ≤ ∞


In particular, 1-norm, 2-norm and ∞-norm are most commonly used. They are denoted

and defined in the following way:

‖x‖1

∆=

(n∑

i=1

|xi|)

‖x‖2

∆=

(n∑

i=1

|xi|2)1/2

‖x‖∞∆= max

i|xi|

The following are some properties of any of the vector norms:

• ‖x‖ ≥ 0

• ‖x‖ = 0 if and only if x = 0

• ‖αx‖ = |α| ‖x‖ for any scalar α

• ‖x + y‖ ≤ ‖x‖+ ‖y‖

Matrix norms are the extension of the concept of length in three-dimensional space to

higher dimensional hyperspace. On the basis of the vector norm, matrix norms or operator

norms of matrices are defined as,

‖A‖p

∆= sup

x 6=0

‖Ax‖p

‖x‖p

where A ∈ Cm×n

In particular, operator 2-norm can be computed as, ‖A‖2 = σ (A) Apart form the prop-

erties of the vector norms, any of the matrix norms obey the following relations:

• ρ (A) ≤ ‖A‖ where ρ denotes spectral radius

• ‖AB‖ ≤ ‖A‖ . ‖B‖

• ‖UAV ‖ = ‖A‖ for any appropriately dimensioned unitary matrices U and V


B.1.3 Vector Spaces

It is a set whose elements are real or complex valued vectors and where vector addition

and multiplication of vectors by a scalar are defined i.e. if v1, v2 ∈ Cn where Cn denotes

an n-dimensional vector space then,

(α1v1 + α2v2) ∈ Cn, for any α1 and α2 and

βv1, βv2 ∈ Cn for any β

B.1.4 Basis Vector

It is a set of linearly independent vectors that span the vector space (i.e. linear combina-

tions of the basis vectors can produce any vector that belongs to that vector space). A

set of basis vectors u1, u2....un is said to be orthogonal if the inner product of any two

of them is zero i.e. 〈ui, uj〉 = 0∀i, j and i 6= j An orthogonal set of unit vectors is called

an orthonormal set (i.e. in addition to 〈ui, uj〉 = 0, 〈ui, ui〉 = 1,∀i)

B.1.5 L2 space

Let f(t) and g(t) be two matrix valued time functions. An L2[or, L2(−∞,∞)] space is

defined as the space of real matrix valued time functions with finite inner products given

by,

〈f(t), g(t)〉 =

∞∫

−∞

tr[fT (t)× g(t)

]dt (B.1)

The norm induced by the inner product is given by,

‖f(t)‖22 =

∞∫

−∞

tr[fT (t)× f(t)

]dt (B.2)


This is how the 2-norm of signals is defined. If this norm exists then it is said that,f(t) ∈L2(−∞,∞). For causal signals (i.e.f(t) = 0 for t < 0) if the 2-norm exists then, f(t) ∈L2 [0,∞).

B.1.6 L∞ space

A matrix valued time function f(t) is said to belong to L∞ Space if it has a finite ∞-norm

defined as ‖f(t)‖∞ = supt

σ [f(t)]. If f(t) is a scalar function then the ∞-norm is defined

as,‖f(t)‖∞ = supt|f(t)|

B.1.7 H2 space

H2 space is nothing but the frequency-domain version of the space. In other words, it is

the space of complex matrix functions having bounded 2-norms or H2-norms i.e.

‖F (jω)‖22 =

1

2π

∞∫

−∞

tr [F ∗(jω).F (jω)]dω < ∞ (B.3)

It may be noted that the above 2-norm will exist for strictly proper functions with no

pole on the imaginary axis. The real rational subspace of H2 which consists of all strictly

proper real rational stable transfer matrices is denoted by RH2

B.1.8 H∞ space

Simply stating H∞ space is the space of frequency dependent matrix functions having

bounded H∞-norm,which is defined as

‖F‖∞ = supω

σ [F (jω)] (B.4)


For scalar functions, the H∞-norm is nothing but the peak value of the Bode magnitude

plot. H∞-norm exists for proper transfer functions with no pole on the imaginary axis.

RH∞ is a subspace of H∞ space consisting of real rational proper and stable transfer

matrices.

B.1.9 Packed Matrix Notation

A transfer matrix can be found from the A, B, C, D matrices as, G(s) = C(sI−A)−1B+D

In packed matrix notation, G(s) is represented in terms of A,B,C, D matrices as,

G =

[A B

C D

]

It is to be kept in mind that G written in the above form, is not a matrix in the

original sense but only a notation, which gives some computational advantages. The

following three formulae regarding the packed matrix notation of different interconnection

of systems are very useful and will be used frequently in later sections.

Figure B.1: System interconnections: (a) Series connection, (b) Inversion, (c) Parallelconnection


Series Connection Let there be two systems connected in series as shown in Fig. If,

G1 =

[A1 B1

C1 D1

]and G2 =

[A2 B2

C2 D2

]then G1G2 =

A1 0 B1

B2C1 A2 B2D1

D2C2 C2 D2D1

Inversion If ,G =

[A B

C D

]then G−1 =

[A−BD−1C BD−1

−D−1C D−1

],provided D is non-

singular.

Parallel Connection If G1 =

[A1 B1

C1 D1

]and G2 =

[A2 B2

C2 D2

]then it can be shown

that G1 + G2 =

A1 0 B1

0 A2 B2

C1 C2 D1 + D2

B.1.10 Robust Stability

Robustness of the stability in the face of model errors will be treated briefly[41],[20].The

whole concept is based on the so called small gain theorem which trivially applies to the

situation sketched in fig B.2.

Figure B.2: Closed loop with loop transfer H

The stable stable transfer H represents the total looptransfer in a closed loop. If we

require that the modulus (amplitude) of H is less than 1 for all frequencies it is clear

from fig B.6 that the polar curve cannot encompass the point −1 and thus we know


Figure B.3: Small gain stability in Nyquist space

from nyquist criterion that the loop will always constitute a stable system.So stability is

guaranteed as long as

‖H‖∞def= sup

ω|H(jω)| < 1 (B.5)

sup stands for supremum which effectively indicates the maximum.(only in the case that

the supremum is approached at within any small distance but never really reached it is

not allowed to speak of a maximum.) Notice that no information concerning phase angle

has been used which is typically H∞. In the above formula we get the first taste of H∞by simultaneous definition of the infinity norm which has been discussed already.For the

MIMO system the small gain condition is given by

‖H‖∞def= sup

ωσ (H(jω)) < 1 (B.6)

Where σ denotes the maximum singular value (always real) of the transfer H (for the

ω under consideration).All together, these conditions may seem somewhat exaggerated,

because transfers, less than one, are not so common. The actual application is therefore


somewhat ”nested” and very depictively indicated in the literature as ”the baby small gain

Figure B.4: Baby small gain theorem for additive model error

theorem” illustrated in fig . In the upper block scheme all relevant elements of fig B.2 have

been displayed in case we have to deal with an additive model error ∆P . We now consider

the ”baby” loop as indicated containing ∆P explicitly. The lower transfer between the

Figure B.5: Control sensitivity guards stability robustness for additive model error

output and the input of ∆p, as once again illustrated in fig , can be evaluated and happens

to be equal to the control sensitivity R as shown in the lower block scheme. (Actually we

get a minus sign that can be joined to ∆P .Because we only consider absolute values in the

small gain theorem, this minus sign is irrelevant: it just causes a phase shift of 180 which

leaves the conditions unaltered.) Now it is easy to apply small gain theorem to the total

loop transfer H = R∆P . The infinity norm will appear to be an induced operator norm

in the mapping between identical signal spaces L and such it follows Schwartz inequality


so that we may write:

‖R∆P‖∞ ≤ ‖R‖∞ ‖∆P‖∞ (B.7)

Ergo,if we can guarantee that:

‖R‖∞ ≤ 1

α(B.8)

a sufficient condition for stability is :

‖R‖∞ ≤ α (B.9)

If all we require from ∆P is stated in equation B.7 then it is easy to prove that the

condition on R is also necessary condition. till this is rather crude condition but it can

be defined by weighting over the frequency axis. Once again from fig we recognize that

the robustness stability constraint effectively limits the feedback from the point, where

both the disturbance and the output of the model error block ∆P enter, and the point

of the plant such that the loop transfer is less than one. The smaller error bound 1α

the

greater the feedback α can be and vice versa! We so analyzed the effect of additive model

error ∆P . Similarly we can study the effect of multiplicative error which is very easy if

we take:

Ptrue = P + ∆P = (I + ∆) P (B.10)

where obviously ∆ is the bounded multiplicative model error. ( Together with P it

evidently constitutes the additive model error ∆P .) In similar blockschemes we now get

Figure B.6: Baby small gain theorem for multiplicative model error


figures . The ”baby” -loop now contains ∆ explicitly and we notice that transfer P is

somewhat ”displaced” out of the additive perturbation block. The result is that ∆ sees

Figure B.7: Complementary sensitivity guards stability robustness for multiplicativemodel error

itself fed back by (minus) the complementary sensitivity T . (The P has, so to speak ,

been taken out of ∆P and adjoined to R yielding T .) If we require that:

‖∆‖∞ ≤ 1

β(B.11)

the robust stability follows from:

‖T∆‖∞ ≤ ‖T‖∞ ‖∆‖∞ ≤ 1 (B.12)

yielding as final condition:

‖T‖∞ ≤ β (B.13)

Again proper weighting may refine the condition.


B.2 A Linear Matrix Inequality Approach to H∞ Con-

trol

The continuous and discrete-time H∞ control problems are solved via elementary ma-

nipulation on linear matrix inequalities (LMI)[26]. Two interesting new features emerge

through this approach: solvability conditions valid for both regular and singular problems,

and an LMI-based parametrization of all H∞-suboptimal controllers, including reduced-

order controllers.The solvability conditions involve Riccati inequalities rather than the

usual indefinite Riccati equations. Alternatively, these conditions can be expressed as a

system of three LMI’s. Efficient convex optimization techniques are available to solve this

system. Moreover, its solutions parameterize the set of H∞ controllers and bear impor-

tant connections with the controller order and the closed loop Lyapunov functions. In

this thesis after some brief introduction on LMI, the H∞ synthesis has been described in

the view of LMI.

B.2.1 Brief theory

The history of LMIs in the analysis of dynamical systems goes back more than 100 years.

The story begins in about 1890, when Lyapunov published his seminal work introducing

what we now call Lyapunov theory. It has been reported in his seminal work that the

differential equation,d

dxx(t) = Ax(t) (B.14)

is stable (i.e., all trajectories converge to zero) if and only if there exists a positive definite

matrix P such that

AT P + PA < 0 (B.15)

The requirement ,P > 0, AT P + PA < 0 is what we now call a Lyapunov inequality on

P , which is a special form of an LMI. Lyapunov also showed that this first LMI could

be explicitly solved. Indeed, we can pick any Q = QT > 0 and then solve the linear


equation AT P + PA = −Q for the matrix P , which is guaranteed to be positive-definite

if the system B.14 is stable. In summary, the first LMI used to analyze stability of a

dynamical system was the Lyapunov inequality B.15, which can be solved analytically

(by solving a set of linear equations).The important role of LMIs in control theory was

already recognized in the early 1960’s, especially by Yakubovich . The Positive-Real

lemma and extensions were intensively studied in the latter half of the 1960s. By 1970, it

was known that the LMI appearing in the Positive-Real lemma could be solved not only

by graphical means, but also by solving a certain Algebraic Riccati Equation (ARE). In

1971, a paper on quadratic optimal control by J. C. Willems led to the LMI

[AT P + PA + Q PB + CT

BT P + C R

]≥ 0 (B.16)

and pointed out that it can be solved by studying the symmetric solutions of the ARE

AT P + PA− (PB + CT )R−1(BT P + C) + Q = 0 (B.17)

B.2.2 Advantages of LMI

Linear matrix inequalities (LMIs) and LMI techniques have emerged as powerful tools

in areas ranging from control engineering to system identification and structural design.

Three factors make LMI techniques appealing:

• A variety of design specifications and constraints can be expressed as LMIs.

• Once formulated in terms of LMIs, a problem can be solved exactly by efficient

convex optimization algorithms.

• While most problems with multiple constraints or objectives lack analytical solutions

in terms of matrix equations, they often remain tractable in the LMI framework.

This makes LMI based design a valuable alternative to classical analytical method.


• iv. If the system is strictly proper, the solution of H∞ control problem is not possible

using DGKF method but with LMI technique, it is possible to solve sub-optimal

H∞ control using Bounded Real Lemma

B.2.3 Basic Idea about LMI

A linear matrix inequality (LMI) has the form

F (η) = F0 +m∑

i=1

ηiFi > 0 (B.18)

Where η ∈ <m the variable and the symmetric matrices are Fi = F Ti ∈ <n×n, i =

0, 1..........,m given. The inequality symbol in B.18 means that F (η) is positive definite,

i.e.,uT F (η)u > 0 for all nonzero u ∈ <n. Of course, the LMI B.18 is equivalent to a set of

n polynomial inequalities in η , i.e., the leading principal minors of F (η) must be positive.

We will also encounter non-strict LMIs, which have the form

F (η) ≥ 0 (B.19)

The LMI B.18 is a convex constraint on η, i.e., the set η|F (η) > 0 is convex. Al-

though the LMI B.18 may seem to have a specialized form, it can represent a wide

variety of convex constraints on η. In particular, linear inequalities, quadratic inequal-

ities, matrix norm inequalities, and constraints that arise in control theory, such as

Lyapunov and convex quadratic matrix inequalities, can all be cast in the form of an

LMI. Multiple LMIs F (1)(η) > 0, ......., F (η)(p) > 0 can be expressed as the single LMI

diag(F (1)(η), ....., F (p)(η)) > 0. Therefore we will make no distinction between a set of

LMIs and a single LMI, i.e., ”the LMI F (1)(η) > 0, ......., F (η)(p) > 0” will mean ”the LMI

diag(F (1)(η), ....., F (p)(η)) > 0”. When the matrices Fiare diagonal, the LMI F (η) > 0is

just a set of linear inequalities. Nonlinear (convex) inequalities are converted to LMI form


using Schur complements. The basic idea is as follows: The LMI

[Q(η) S(η)

S(η)T R(η)

]> 0 (B.20)

Where Q(η) = Q(η)T , R(η) = R(η)T and depend affinely on ’η’, is equivalent to, R(η) > 0,

Q(η)− S(η)R(η)−1S(η)T > 0

B.2.4 Matrices as variable

We will often encounter problems in which the variables are matrices, e.g., the Lyapunov

inequality AT P + PA < 0. Where A ∈ <n×n is given and P = P T is the variable. In this

case we will not write out the LMI explicitly in the form F (η) > 0, but instead make clear

which matrices are the variables. The phrase the LMI AT P + PA < 0 in P means that

the matrix P is a variable. As another related example, consider the quadratic matrix

inequality

AT P + PA + PBR−1BT P + Q < 0 (B.21)

Where A,B, Q = QT , R = RT > 0 are given matrices of appropriate sizes, and P = P T

is the variable. Note that this is a quadratic matrix inequality of the variable P. It can

be expressed as the linear matrix inequality using the Schur complement as,

[−AT P − PA−Q PB

BT P R

]> 0


B.2.5 Lyapunov’s Inequality

We have already mentioned the Linear Matrix Inequality Problem (LMIP) associated

with Lyapunov’s inequality, i.e.

P > 0 , AT P + PA < 0

Where P is a variable and A ∈ <n×n is given. It can be shown that this LMI is feasible

if and only if the matrix A is stable, i.e., all trajectories of x = Ax converge to zero as

t →∞, or equivalently, all eigen values of P must have negative real part. To solve this

LMIP, we pick any Q > 0 and solve the Lyapunov equation AT P + PA = −Q, which is

nothing but a set of n(n+1)2

linear equations for the n(n+1)2

scalar variables in P . This set

of linear equations will be solvable and result in P > 0 if and only if the LMI is feasible.

In fact this procedure not only finds a solution when the LMI is feasible, it parameterizes

all solutions as Q varies over the positive-definite cone.

B.3 Stabilizing Controllers

A necessary feature of any feedback system is that it be stable in some appropriate sense.

In this chapter we introduce the feedback arrangement we will be studying for the rest of

the course. Once introduced, our main objective is to precisely define feedback stability

and then to parametrize all controllers that stabilize the feedback system. The general

feedback setup we are concerned with is shown above. As depicted the so-called closed-

loop system has one external input and one output, given by w and z respectively. The

signal or function w captures the effects of the environment on the feedback system; for

instance noise, disturbances and commands. The signal z contains all characteristics

of the feedback system that are to be controlled. The maps G and K represent linear

subsystems where G is a given ”plant” which is fixed, and K is the controller or control

law whose aim is to ensure that the mapping from w to z has the desired characteristics.

To accomplish this task the control law utilizes signal y, and chooses an action u which


Figure B.8: General feedback arrangement

directly affects the behavior of G.

Here G and K are state space systems, with G evolving according to

x(t) = Ax(t) +[

B1 B2

] [w(t)

u(t)

],

[z(t)

y(t)

]=

[C1

C2

]x(t) +

[D11 D12

D21 D22

][w(t)

u(t)

],

(B.22)

and K being described by

xK(t) = AKxK(t) + BKy(t)

u(t) = CKxK(t) + DKy(t)(B.23)

Throughout this section we have the standing assumption that the matrix triples (Ak, Bk, Ck)

and are both stabilizable and detectable

As shown in the figure, G is naturally partitioned with respect to its two inputs and two


outputs. We therefore partition the transfer function of G as

G(s) =

A B1 B2

C1 D11 D12

C2 D21 D22

=

[G11(s) G12(s)

G21(s) G22(s)

](B.24)

so that we can later refer to these constituent transfer functions.

At first we must determine under what conditions this interconnection of components

makes sense. That is, we need to know when these equations have a solution for an

arbitrary input w.

The system of Figure B.8 is well-posed if unique solutions exist x(t), xK(t), y(t) and u(t),

for all input functions w(t), and all initial conditions x(0), xK(0).

Proposition 2 The connection of G and K in Figure B.8 is well-posed, if and only if,

I −D22DK is nonsingular.

Proof : The proof of this result amounts to simply writing out the system state

equations. So we have

x(t) = Ax(t) + B1w(t) + B2u(t)

xK(t) = AKxK(t) + BKy(t),(B.25)

and

[I −DK

−D22 I

][u(t)

y(t)

]=

[0 CK

C2 0

][x(t)

xK(t)

]+

[0

D21

]w(t) (B.26)

Now it is easily seen that the left hand side matrix is invertible if and only if I −D22DK

is nonsingular. If this holds, clearly one can substitute u, y into B.25 and find a unique

solution to the state equations. Conversely if this does not hold, from B.26 we can find

a linear combination of x(t), xK(t), and w(t) which must be zero, which means that,

x(0), xK(0), w(0) cannot be chosen arbitrarily.


We have the following result which is frequently used.

Corollary 1 If either D22 = 0 or Dk = 0, then the interconnection in Figure B.8 is

well-posed.

We are now ready to talk about stability. From now on we tacitly assume that our

feedback system is well-posed.

B.3.1 System Stability

In this section we introduce the notion of internal stability, and discuss its relation to the

boundedness of input-output maps [91].

Definition 7 The system in Figure B.8 is internally stable if for every initial condition

x(0) of G, and xk(0) of K, the limits

x(t), xK(t)t→∞→ 0

hold

When w = 0

The following is an immediate test for internal stability.

Proposition 3 Suppose that the system of Figure B.8 is well-posed. Then the system is

internally stable if and only if

A =

[A 0

0 AK

]+

[B2 0

0 BK

][I −DK

−D22 I

]−1 [0 CK

C2 0

](B.27)


is Hurwitz

Proof : This is easily seen by noting that is the A-matrix of the closed-loop; this follows

from B.25 and B.26.

As defined, internal stability refers to the autonomous system dynamics in the absence of

an input w; in this regard it coincides with the standard notion of asymptotic stability of

dynamical systems. However it has immediate implications on the input-output properties

of the system.

In particular, the transfer function from w to z, denoted T (s), will have as poles a subset

of the eigenvalues of A; for example, when Dk = 0 we have

T (s) =[

C1 D12CK

](Is− A)−1

[B1

BKD21

]+ D11

If A is Hurwitz, this function has all its poles in the left half plane of C. An important

consequence is that w 7→ z defines a bounded operator on L2 [0,∞); this is termed input-

output stability.

The question immediately arises as to whether the two notions are interchangeable, i.e.

whether the boundedness of w 7→ z implies internal stability; clearly, the answer is nega-

tive: an extreme example would be to have C1, D11, D12 be all zero which gives T (s) = 0

but clearly says nothing about A. In other words, the internal dynamics need not be

reflected in the external map.

There is, however, a way to characterize internal stability in terms of the boundedness of

an input-output operator, by considering the map between injected interconnection noise

in the feedback loop, to the interconnection variables. The relevant diagram is given in

Figure B.9, where the controller K has the same description as in Figure B.8. The system

G22 is the lower block of G, described by the state space equations

x22(t) = Ax22(t) + B2v1(t)

v2(t) = C2x22(t) + D22v1(t)


Figure B.9: Input-output stability

where (C2, A, B2, D22) are the same matrices as in the state space description of G. We

have also included the external inputs d1 and d2 at the interconnection between G22 and

K.

As with our more general system, we say that this new system is well-posed if there

exist unique solutions for x22(t), xK(t), v1 and v2 for all inputs d1(t) and d2(t) and initial

conditions x22(0), xK(0). We say it is internally stable if it is well posed and for di = 0

x(t), xK(t)t→∞→ 0 holds for every initial condition.

It is an easy exercise to see that the system is well-posed, if and only if, I − D22DK

is nonsingular; this is the same well-posedness condition we have for Figure B.8. Also

noticing that all the states in the description of G are included in the equations for G22,

it follows immediately that internal stability of one is equivalent to internal stability of

the other.

Lemma 1 Given a controller K, Figure B.8 is internally stable, if and only if, Figure

B.9 is internally stable.

The next result shows that with this new set of inputs, internal stability can be charac-

terized by the boundedness of an input-output map.

Lemma 2 Suppose that (C2, A, B2) is stabilizable and detectable. Then Figure B.9 is


internally stable if and only if the transfer function of

[d1

d2

]7→

[v1

v2

]

has no poles in the closed right half plane of C.

Proof : We begin by finding an expression for the transfer function. For convenience

denote

D =

[I −DK

−D22 I

],

then routine calculations lead to the following relationship:

[v1(s)

v2(s)

]=

D−1

[0 CK

C2 0

](Is− A

)−1

[B2 0

0 BK

]D−1 + D−1 +

[0 0

0 −I

][d(s)

d2(s)

]

where is the closed loop matrix from B.27. Therefore the ”only if” direction follows

immediately, since the poles of this transfer function are a subset of the eigenvalues of A

, which is by assumption Hurwitz ; see Proposition 3 and Lemma 1.

To prove ”if”: assume that the transfer function has no poles in C+, therefore the same

is true of [0 CK

C2 0

]

︸︷︷︸C

(Is− A

)−1

[B2 0

0 BK

]

︸︷︷︸B

We need to show that A is Hurwitz ; it is therefore sufficient to show that(C, A, B

)is a

stabilizable and detectable realization. Let

F =

[F 0

0 FK

]− D−1

[0 CK

C2 0

]


Where F and Fk are chosen so that A + B2F and AK + BKFK are both Hurwitz. It is

routine to show that

A + BF =

[A + B2F 0

0 AK + BKFK

]

and thus(A, B

)is stabilizable.

A formally similar argument shows that(C, A

)is detectable.

B.3.2 Stabilization

In the previous section we have discussed the analysis of stability of a given feedback

configuration; we now turn to the question of design of a stabilizing controller. The

following result explains when this can be achieved.

Proposition 4 A necessary and sufficient condition for the existence of an internally

stabilizing K for Figure B.8, is that (C2, A, B2) is stabilizable and detectable. In that

case, one such controller is given by

K(s) =

A + B2F + LC2 + LD22F −L

F 0

Where F and L are matrices such that A + B2F and A + LC2 are Hurwitz.

Proof : If the stabilizability or detectability of (C2, A, B2) is violated, we can choose an

initial condition which excites the unstable hidden mode. It is not difficult to show that

the state will diverge to infinity regardless of the controller. Details are left as an exercise.

Consequently no internally stabilizing K exists, which proves necessity. For the sufficiency

side, it is enough to verify that the given controller is indeed internally stabilizing. Start


by noting that Dk = 0 and so the configuration is well-posed. Now substitute the state

space for the controller into the expression for A given in Proposition 3.

A =

[A B2F

−LC2 A + B2F + LC2

]

Let

T =

[I 0

I I

]

and notice

T−1AT =

[A + B2F B2F

0 A + LC2

]

Since the eigenvalues of A are therefore given by those of A + B2F and A + LC2 we see

is A Hurwitz.

B.4 H∞ Synthesis

Now we consider optimal synthesis with respect to the H∞ norm introduced already[21].

Again we are concerned with the feedback arrangement of figure where we have two state

space systems G and K, each having their familiar role. We will pursue the answer to the

following question: does there exist a state space controller K such that

• The closed loop system is internally stable;

• The closed loop performance satisfies

∥∥∥∥S

(∧G,

∧K

)∥∥∥∥∞

< 1


Thus we only plan to consider the problem of making the closed loop contractive in the

sense of H∞ . It is clear, however, that determining whether there exists a stabilizing

controller so that

∥∥∥∥S

(∧G,

∧K

)∥∥∥∥∞

< γ, for some constant γ , can be achieved by rescaling

the γ dependent problem to arrive at the contractive version given above. Furthermore,

by searching over γ , our approach will allow us to get as close to the minimal H∞ norm as

we desire, but in contrast to our work on H2 optimal control, we will not seek a controller

that exactly optimizes the H∞ norm. There are many approaches for solving the H∞

control problem. Probably the most celebrated solution is in terms of Riccati equations.

Here we will present a solution based entirely on linear matrix inequalities, which has the

main advantage that it can be obtained with relatively straightforward matrix tools, and

without any restrictions on the problem data. In fact Riccati equations and LMIs are

intimately related, an issue we will explain when proving the Kalman-Yakubovich-Popov

lemma concerning the analysis of the H∞ norm of a system, which will be key to the

subsequent synthesis solution.

Before getting into the details of the problem, we make a few comments about the

motivation for this optimization.

As we know already that the H∞ norm is the L2-induced norm of a causal, stable,

linear-time invariant system. More precisely, given a causal linear time-invariant operator

G : L2(−∞,∞) → L2(−∞,∞), the corresponding operator in the isomorphic space∧L2

(jR) is a multiplication operator M∧G

for a certain∧G (s) ∈ H∞, and

‖G‖L2→L2= ‖MG‖L2→L2

=∥∥∥G

∥∥∥∞

The motivation for minimizing such an induced norm lies in the philosophy making

error signal small (z), we are minimizing the maximum ”gain” of the system in the energy

or L2 sense. Equivalently, the excitation w is considered to be an arbitrary L2 signal and

we wish to minimize its worst-case effect on the energy of z.


B.4.1 Two important matrix inequalities

The entire synthesis approach of the chapter revolves around the two technical results

presented here. The first of these is a result purely about matrices; the second is an

important systems theory result and is frequently called the Kalman-Yacubovich-Popov

lemma, or KYP lemma for short. We begin by stating the following which the reader can

prove as an exercise.

Lemma 3 Suppose P and Q are matrices satisfying ker P = 0 and ker Q = 0. Then for

every matrix Y there exists a solution J to

P ∗JQ = Y

The above lemma is used to prove the next one which is one of the two major technical

results of this section

Lemma 4 Suppose,

1. P , Q and H are matrices and that H is symmetric;

2. The matrices WP and WQ are full rank matrices satisfying ImWP = kerP and

ImWQ = kerQ then there exists a matrix J such that

H + P ∗J∗Q + Q∗JP < 0 (B.28)

if and only if, the inequalities

W ∗P HWP < 0 and W ∗

QHWQ < 0


both hold.

Observe that when the kernels of P and Q are not both nonzero the result does not apply

as stated. However it is readily seen from Lemma 1, that if both of the kernels are zero

then there is always a solution J . if for example only kerP = 0 then W ∗QHWQ < 0

is a necessary and sufficient condition for a solution to lemma 1 to exist, as follows by

simplified version of the following proof.

Proof : We will show the equivalence of the conditions directly by construction. To

begin define V1 to be a matrix such that

ImV1 = ker P ∩ ker Q,

and V2 and V3 such that

Im [V1V2] = ker P and Im [V1V3] = ker Q

without loss of generality we assume that V1, V2 and V3 have full column rank and define

V4 so that

V =[

V1 V2 V3 V4

]

is square and nonsingular. Therefore the LMI B.21 above holds, if and only if

V ∗HV + V ∗P ∗J∗QV + V ∗Q∗JPV < 0 (B.29)

does. Now PV and QV are simply the matrices P and Q on the domain basis defined by

V ; therefore they have the form

PV =[

0 0 P1 P2

]and QV =

[0 Q1 0 Q2

];


we also define the block components

V ∗HV =:

H11 H12 H13 H14

H∗12 H22 H23 H24

H∗13 H∗

23 H33 H34

H∗14 H∗

24 H∗34 H44

Further define the variable Y by

Y =

[Y11 Y12

Y21 Y22

]=

[P ∗

1

P ∗2

]= J∗

[Q1 Q2

]

from their definitions ker[

P1 P2

]= 0 and ker

[Q1 Q2

]= 0, and so by Lemma

1 we see that Y is freely assignable by choosing an appropriate matrix J . Writing out

inequality B.29 using the above definitions we get

H11 H12 H13 H14

H∗12 H22 H23 + Y ∗

11 H24 + Y ∗21

H∗13 H∗

23 + Y11 H33 H34 + Y12

H∗14 H∗

24 + Y21 H∗34 + Y ∗

12 H44 + Y22 + Y ∗22

< 0

Apply the Schur complement formula to the upper 3 × 3 block, and we see the above

holds, if and only if, the two following inequalities are met.

−H :=

H11 H12 H13

H∗12 H22 H23 + Y ∗

11

H∗13 H∗

23 + Y11 H33

< 0


and H44 + Y22 + Y ∗22 −

H14

H24 + Y ∗21

H34 + Y12

∗

H−1

H14

H24 + Y ∗21

H34 + Y12

< 0

as already noted above Y is freely assignable and so we see that provided the first in-

equality can be achieved by choosing Y11, the second can always be met by appropriate

choice of Y12, Y21 and Y22. That is the above two inequalities can be achieved, if and only

if,−H < 0 holds for some Y11. Now applying a Schur complement on

−H with respect to

H11, we obtain

H11 0 0

0 H22 − Y ∗12Y

−111 H12 Y ∗

11 + X∗

0 Y11 + X H33 −H∗13H

−111 H13

< 0,

where X = H∗23 −H∗

13H−111 H12. Now since Y11 is freely assignable we see readily that the

last condition can be satisfied, if and only if, the diagonal entries of the lift hand matrix

are all negative definite. Using the Schur complement result twice these three conditions

can be converted to the equivalent conditions

[H11 H12

H∗12 H22

]< 0 and

[H11 H13

H∗13 H33

]< 0

by the choice of our basis we see that these hold, if and only if, W ∗P HWP < 0 and W ∗

QHWQ <

0 are both met. Having proved this matrix result we move on to our second result, the

KYP lemma.

B.4.2 The KYP Lemma

There are many versions of this result, which establishes the equivalence between a fre-

quency domain inequality and a state-space condition in terms of either a Riccati equation


or an LMI. The version given below turns an H∞ norm condition into an LMI. Being able

to do this is very helpful for attaining our goal of controller synthesis, however it is equally

important simply as a finite dimensional analysis test for transfer functions.

Lemma 5 Suppose∧

M (s) = C (Is− A)−1 B + D . Then the following are equivalent

conditions.

1. The matrix A is Hurwitz and ∥∥∥∥∧

M

∥∥∥∥∞

< 1;

2. ii. There exists a symmetric positive definite matrix X such that

C∗

D∗

[C D

]+

A∗X + XA XB

B∗X −I

< 0 (B.30)

the condition in (ii) is clearly an LMI and gives us a very convenient way to evaluate the H

norm of a transfer function. In the proof below we see proving that condition (ii) implies

that (i) holds is reasonably straightforward, and involves showing the direct connection

between the above LMI and the state space equations that describe M . proving the

converse is considerably harder; fortunately we will be able to exploit the Riccati equation

techniques. An alternative proof, which employs only matrix arguments is beyond the

scope. Proof :We begin by showing (ii) implies (i). The top left block in B.30 states that

A∗X + XA + C∗C < 0. Since X > 0 we see that A must be Hurwitz. It remains to

show contractiveness which we do by employing a system-theoretic argument based on

the state equations for M . using the strict inequality B.30 choose 1 > ε > 0 such that

[C∗

D∗

] [C D

]+

[A∗ + XA XB

B∗X − (1− ∈) I

]< 0 (B.31)


holds. Let ω ∈ L2 [0,∞] and realize that in order to show that M is contractive, it is

sufficient to show that ‖z‖2 ≤ (1− ∈) ‖ω‖2 , where z := Mω. The state space equations

relating ω and z are·x (t) = Ax (t) + Bω (t) , x (0) = 0,

z (t) = Cx (t) + Dω (t) .

Now multiplying inequality B.31 on the left by [x∗(t)w∗(t)] and on the right by the adjoint

we have

|z(t)|22 + x∗(t)X(Ax(t) + Bw(t)) + (Ax(t) + Bw(t))∗Xx(t)− (1− ∈) |w(t)|22 ≤ 0

By introducing the storage function V : Rn → R , defined by V (x(t) = x∗(t)Xx(t), we

arrive at the so-called dissipation inequality

V + |z(t)|22 ≤ (1− ∈) |w(t)|22

Integrating on an interval [0, T ], recalling that x(0) = 0, gives

x(T )∗Xx(T ) +

∫ T

0

|z(t)|22 dt ≤ (1− ∈)

∫ T

0

|w(t)|22 dt

Since X > 0, we can suppress the first term above; now taking the limit as T → ∞, we

find that

||z||22 ≤ (1− ∈) ||w||22which completes this direction of the proof. We now tackle the direction (i) implies (ii).

To simplify the expressions we will write the derivation in the special case D = 0, but an

analogous argument applies to the general case. Starting from

M(s) =

[A B

C 0

]


and from the definition of para-Hermitian conjugate, if M∼(s) :=

M∼(−s∗)∗

we derive

the state-space representation

I − M∼(s)M(s) =

A 0 −B

−C∗C −A∗ 0

0 B∗ I

It is easy to verify that

[I − M∼(s)M(s)

]−1

=

A BB∗ B

−C∗C −A∗ 0

0 B∗ I

(B.32)

Since∥∥∥M

∥∥∥∞

< 1 by hypothesis, we conclude that[I − M∼(s)M(s)

]−1

has no poles on

the imaginary axis. Furthermore we now show, using the PBH test, that the realization

B.32 has no unobservable eigen values that are purely imaginary. Suppose that

jω0I − A −BB∗

C∗C jω0I + A∗

0 B∗

[x1

x2

]= 0

for some vectors x1 and x2. Then we have the following chain of implications

B ∗ x2 = 0 ⇒ (jω0I − A) x1 = 0,

therefore x1 = 0 since A is Hurwitz;

this means (jω0I − A∗) x2 = 0;

which implies x2 = 0 again because A is Hurwitz.

We conclude that B.32 has no un-observable eigenvalues on the imaginary axis; an anal-

ogous argument shows the absence of uncontrollable eigenvalues. This means that the


matrix

H =

[A BB∗

−C∗C −A∗

]

has no purely imaginary eigenvalues. According to the lemma

Lemma 6 Suppose that H is a Hamiltonian matrix and

• H has no purely imaginary axis eigen value;

• R is either positive or negative semidefinite;

• (A,R) is a stabilizable matrix pair

Then H is in the domain of the Riccati operator

Here notice thatBB∗ ≥ 0 and (A,BB∗) is stabilizable since A is Hurwitz. Hence H is in

the domain of the Riccati operator, and we can define X0 = Ric(H) satisfying

A∗X0 + X0A + C∗C + X0BB∗X0 = 0 (B.33)

and A + BB∗X0 Hurwitz. Also note that B.33 implies A∗X0 + X0A ≤ 0, therefore from

our work on Lyapunov equations we see that

X0 ≥ 0

since A is Hurwitz. To obtain the LMI characterization of (ii) we must slightly strengthen

the previous relationships. For this purpose define X to be the solution of the Lyapunov

equation

(A + BB∗X0)∗X + X(A + BB∗X0) = −I (B.34)


Since (A + BB∗X0) is Hurwitz we have X > 0. Now let X = X0+ ∈ X > 0, which is

positive definite for all ε > 0. Using B.33 and B.34 we have

A∗X + XA + C∗C + XBB∗X = − ∈ I+ ∈2 XBB∗X

Choose ε > 0 sufficiently small so that this equation is negative definite. Hence we have

found X > 0 satisfying the strict Riccati Inequality

A∗X + XA + C∗C + XBB∗X = 0

Now applying a Schur complement operation, this inequality is equivalent to

[A∗X + XA + C∗C XB

B∗X −I

]< 0

which B.30 for the special case D = 0.

The preceding proof illustrates some of the deepest relationships of linear systems

theory. We have seen that frequency domain inequalities are associated with dissipativity

of storage functions in the time domain, and also the connection between LMIs (linked

to dissipativity) and Riccati equations (which arise in quadratic optimization).

In fact this latter connection extends as well to problems of H∞ synthesis, where both

Riccati equations and LMIs can be used to solve the suboptimal control problem.


B.4.3 Synthesis

Lets start with the state space realizations that describe the systems G and K [? ]:

G(s) =

A B1 B2

C1 D11 D12

C2 D21 0

, K(s) =

[AK BK

CK DK

]

Notice that we have assumed D22 = 0. Removing this assumption leads to more compli-

cated formulae, but the technique is identical. We make no other assumptions about the

state space systems. The state dimensions of the nominal system and controller will be

important: A ∈ Rn×n, Ak ∈ Rnk×nk

Our first step is to combine these two state space realizations into one which describes

the map from w to z. We obtain

S(G,K) =

[AL BL

CL DL

]=

A + B2DKC2 B2CK B1 + B2DKC21

BKC2 AK BKD21

C1 + D12DKC2 D12CK D11 + D12DKD21

Now define the matrix

J =

[AK BK

CK DK

]

which collects the representation for K into one matrix. We can parameterize the closed-

loop relation in terms of the controller realization as follows. First make the following


definitions.

A =

[A 0

0 0

], B =

[B1

0

]

C =[

C1 0], C =

[0 I

C2 0

]

B =

[0 B2

I 0

], D12 =

[0 D12

]

D21 =

[0

D21

]

(B.35)

which are entirely in terms of the state space matrices for G. Then we have

AL = A + BJC, BL = B + BJD21

CL = C + D12JC, DL = D11 + D12JD21

(B.36)

The crucial point here is that the parametrization of the closed loop state space matrices

is affine in the controller matrix J .

Now we are looking for a controller K such that the closed loop is contractive and inter-

nally stable. The following form of the KYP lemma will help us.

Corollary 2 Suppose ML(s) = CL(Is−AL)−1BL+DL. Then the following are equivalent

conditions.

1. The matrix AL is Hurwitz and∥∥∥ML

∥∥∥∞

< 1


2. There exists a symmetric positive definite matrix XL such that


L

B∗LXL −I D∗

L

CL DL −I

< 0

This result is readily proved from Lemma 5 by applying the Schur complement formula.

Notice that the matrix inequality in (b) is affine in XL and J individually, but it is not

jointly affine in both variables. The main task now is to obtain a characterization where

we do have a convex problem.

Now define the matricesPXL

=[

B∗X 0 D∗12

]

Q =[

C D21 0]

And further

HXL=

A∗XL + XLA XLB C∗

B∗XL −I D∗11

C D11 −I

It follows that the inequality in (b) above is exactly

HXL+ Q∗J∗PXL

+ P ∗XL

JQ < 0

Lemma 7 Given the above definitions there exists a controller synthesis K if and only if

there exists a symmetric matrix XL > 0 such that

W ∗PXL

HXLWPXL

< 0 and W ∗QHXL

WQ < 0


Where are as defined in Lemma 4

Proof : From the discussion above we see that a controller K exists if and only if there

exists XL > 0 satisfying

HXL+ Q∗J∗PXL

+ P ∗XL

JQ < 0

Now invoke Lemma 4.

This lemma says that a controller exists if and only if the two matrix inequalities can be

satisfied. Each of the inequalities is given in terms of the state space matrices of G and

the variable XL. However we must realize that since XL appears in both HXLand WPXL

,

that these are not LMI conditions. Converting to an LMI formulation is our next goal,

and will require a number of steps. Given a matrix XL > 0 define the related matrix

TXL=

AX−1L + X−1

L A∗ B X−1L C∗

B∗ −I D∗11

CX−1L D11 −I

(B.37)

And the matrix

P =[

B∗ 0 D∗12

](B.38)

Which only depends on the state space realization of G.the next lemma converts one of

the two matrix inequalities of the lemma, involving HXL, to one in terms of TXL

.

Lemma 8 Suppose XL > 0. Then

W ∗PXL

HXLWPXL

< 0 , if and only if, W ∗P TXL

WP < 0

Proof :Start by observing that

PXL= PS where,

S =

XL 0 0

0 I 0

0 0 I


Therefore we have ker PXL= S−1 ker P

Then using the definitions of WPXLand WP we can set

WPXL= S−1WP

Finally we have that W ∗PXL

HXLWPXL

< 0 if and only if

W ∗P (S−1)∗HXL

S−1WP < 0

And it is routine to verify (S−1)∗HXLS−1 = TXL

Combining the last two lemmas we see

that there exists a controller of state dimension nk if and only if there exists a symmetric

matrix XL > 0 such that

W ∗P TXL


WQ < 0 (B.39)

The first of these inequalities is an LMI in the matrix variable XL, where as the second

is an LMI in terms of XL. However the system of both inequalities is not an LMI. Our

intent is to convert these seemingly non-convex conditions into an LMI condition. Recall

that XL is a real and symmetric (n + nK) × (n + nK) matrix; here n and nk are state

dimensions of G and K. Let us now define the matrices X and Y which are submatrices

of XL and X−1L , by

XL =:

[X X2

X∗2 X3

]and X−1

L =:

[Y Y2

Y ∗2 Y3

](B.40)

We now show that the two inequality conditions listed in B.39, only constrain the sub-

matrices X and Y

Lemma 9 Suppose XL is a positive definite (n + nK) × (n + nK) matrix and X and Y


are n× n matrices defined as in B.40. Then

W ∗P TXL

WP < 0andW ∗QHXL

WQ < 0

If and only if, the following two matrix inequalities are satisfied

1.

NX 0

0 I

∗


B∗1X −I D∗

11

C1 D11 −I

NX 0

0 I

< 0

2.

NY 0

0 I

∗

AY + Y A∗ Y C1 B1

C1Y −I D11

B∗1 D∗

11 −I

NY 0

0 I

< 0

Where Nx and Ny are full-rank matrices whose images satisfy

ImNX = ker[

C2 D21

]

ImNY = ker[

B∗2 D∗

12

]

Proof : The proof amounts to writing out the definitions and removing redundant con-

straints. Let us show that W ∗P TXL

WP < 0 is equivalent to the LMI in (ii). From the


definitions of TXL in B.37, and A, B and C in B.35 we get

TXL=

AY + Y A∗ AY2 B1 Y C∗1

Y ∗2 A∗ 0 0 Y ∗

2 C∗1

B∗1 0 −I D∗

11

C1Y C1Y2 D11 −I

Also recalling the definition of P in B.38, and substituting for B and D12 from B.35 yields

P =

[0 I 0 0

B∗2 0 0 D∗

12

]

Thus the kernel of P is the image of

WP =

V1 0

0 0

0 I

V2 0

Where

[V1

V2

]= NY

spans the kernel of [B∗2D

∗12] as defined above. Notice that the second block row of WP is

exactly zero, and therefore the second block-row and block-column of TXL, as explained

above, do not enter into the constraint W ∗P TXL

WP < 0. Namely this inequality is

V1 0

0 I

V2 0

∗

AY + Y A∗ B1 Y C∗1

B∗1 −I D∗

11

C1Y D11 −I

V1 0

0 I

V2 0

< 0


By applying the permutation

V1 0

0 I

V2 0

=

I 0 0

0 0 I

0 I 0

[NY 0

0 I

]

We arrive at (ii). Using a nearly identical argument, we can readily show that W ∗QHXL

WQ <

0 is equivalent to LMI (i)in the theorem statement.

What we have shown is that a controller synthesis exists if and only if there exists an

(n + nK) × (n + nK) matrix XL that satisfies conditions (i) and (ii) of the last lemma.

These latter two conditions only involve X and Y , which are submatrices of respectively

XL and X−1L . Our next result tell us under what conditions, given arbitrary matrices X

and Y , it is possible to find a positive definite matrix that satisfies B.40.

Lemma 10 Suppose X and Y are symmetric, positive definite matrices in Rn×n; and nk

is a positive integer. Then there exist matrices X2, Y2 ∈ Rn×nK and symmetric matrices

X3, Y3 ∈ RnK×nK , satisfying

[X X2

X∗2 X3

]> 0and

[X X2

X∗2 X3

]−1

=

[Y Y2

Y ∗2 Y3

]

if and only if [X I

I Y

]≥ 0 and rank

[X I

I Y

]≤ n + nK (B.41)

Proof : First we prove that the first two conditions imply the second two. From

[X X2

X∗2 X3

][Y Y2

Y ∗2 Y3

]= I (B.42)


It is routine to verify that

0 ≤[

I 0

Y Y2

][X X2

X∗2 X3

][I Y

0 Y ∗2

]=

[X I

I Y

]

Also the schur component relationship

[X I

I Y

]=

[I Y −1

0 I

][X − Y −1 0

0 Y

][I 0

Y −1 I

](B.43)

implies that

rank

[X I

I Y

]= n + rank(X − Y −1) = n + rank(XY − I) ≤ n + nK

where the last inequality follows from B.42 : I −XY = X2Y∗2 andX2 ∈ Rn×nK . To prove

”if” we start with the assumption that B.41 holds; therefore B.43 gives

X − Y −1 ≥ 0andrankX − Y −1 ≤ nK

These conditions ensure that there exists a matrix X2 ∈ Rn×nK so that

X − Y −1 = X2X∗2 ≥ 0

From this and the Schur component argument we see that

[X X2

X∗2 I

]> 0


Also [X X2

X∗2 I

]−1

=

[Y −Y X2

−X∗2Y X∗

2Y X2 + I

]

and so we set X3 = I

The lemma states that a matrix XL in R(n+nK)×(n+nK), satisfying B.40, can be constructed

from X and Y exactly when the LMI and rank conditions in B.41 are satisfied. The rank

condition is not in general an LMI, but notice that

rank

[X I

I Y

]≤ 2n

Therefore we see that if nK ≥ n in the lemma, the rank condition becomes vacuous and

we are left with only the LMI condition. We can now prove the synthesis theorem.

Theorem 1 A synthesis exists to the H∞ problem, if and only if there exist symmetric

matrices X > 0 and Y > 0 such that

1.

NX 0

0 I

∗


B∗1X −I D∗

11

C1 D11 −I

NX 0

0 I

< 0

2.

NY 0

0 I

∗

AY + Y A∗ Y C1 B1

C1Y −I D11

B∗1 D∗

11 −I

NY 0

0 I

< 0

3.

X I

I Y

≥ 0


Where NX and NY are full-rank matrices whose images satisfy

ImNX = ker[

C2 D21

]

ImNY = ker[

B∗2 D∗

12

]

Proof : Suppose a controller exists, then by Lemma 9 a controller exists if and only if the

inequalities

W ∗P TXL


WQ < 0

hold for some symmetric, positive definite matrix XL in R(n+nK)×(n+nK). By Lemma 9

these LMIs being satisfied imply that (1) and (2) are met. Also invoking Lemma 10 we see

that (3) is satisfied. Showing that (1-3) imply the existence of a synthesis is essentially

the reverse process. Since nk ≥ n we have by Lemma 10 that there exists an X2 in

R(n+nK)×(n+nK) which satisfies B.40.

The proof is now completed by XL using and (1-2) together with Lemma 9. This theorem

gives us exact conditions under which a solution exists to our H∞ synthesis problem.

Notice that the conditions are totally independent of the controller state dimensions nk,

provided that nk is not smaller than the system state dimension n. This clearly means

that a synthesis exists if and only if one exists with state dimension nk = n.

Theorem 1 gives us necessary and sufficient conditions for the existence of a synthesis,

however we still need to trace these conditions backwards to explicitly construct such a

controller.

B.4.4 Controller reconstruction and connections

The results of the last section provide us with an explicit way to determine whether a

synthesis exists which solves the H∞ problem. Implicit in our development is a method

to construct controllers when the conditions of Theorem 1 are met. We now outline this

procedure, which simply retraces our steps so far [92].


Suppose X and Y have been found satisfying Theorem 1 then by Lemma 10 there exists

a matrix satisfying

XL =

[X ?

? ?

]and X−1

L =

[Y ?

? ?

]

From the proof of the lemma we can construct XL by finding a matrix X2 ∈ Rn×nK such

that X − Y −1 = X2X∗2 .Then

XL =

[X X∗

2

X2 I

]

has the properties desired above. Notice that there is some freedom in choosing the

controller state dimension nk. We typically would like to select nk as small as possible.

Immediately we know that need be no larger than n, and can be chosen to be the rank

of X − Y −1 Next by Lemma 4 we know that there exists a solution to

HXL+ Q∗J∗PXL

+ P ∗XL

JQ < 0

and that any such solution J provides the state space realization for a feasible controller

K. The solution of this LMI can be accomplished using standard techniques, and there

is clearly an open set of solutions J .

Finally let us observe that we can extend Theorem 1 to provide conditions for controllers

of state dimension nk less than n. To do this we add the constraint that

rank(X − Y −1) ≤ nK

to the list (1) to (3).Unfortunately this constraint is not convex in general when nk < n,

however it does provide an explicit condition which can be exploited in certain situations

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