**UFBA**

# Spacecraft and Rockets

(Parte **1** de 6)

Aerospace Series List

Basic Helicopter Aerodynamics: Third Edition Seddon and Newman July 2011

Cooperative Path Planning of Unmanned Aerial Vehicles Tsourdos et al November 2010

Principles of Flight for Pilots Swatton October 2010

Air Travel and Health: A Systems Seabridge et al September 2010 Perspective

Design and Analysis of Composite Kassapoglou September 2010

Structures: With applications to aerospace Structures

Unmanned Aircraft Systems: UAVS Design, Development and Deployment Austin April 2010

Introduction to Antenna Placement and Installations Macnamara April 2010

Principles of Flight Simulation Allerton October 2009 Aircraft Fuel Systems Langton et al May 2009 The Global Airline Industry Belobaba April 2009

Computational Modelling and Simulation of

Aircraft and the Environment: Volume 1 - Platform Kinematics and Synthetic Environment Diston April 2009

Handbook of Space Technology Ley, Wittmann April 2009 and Hallmann

Aircraft Performance Theory and Swatton August 2008 Practice for Pilots

Surrogate Modelling in Engineering Forrester, Sobester August 2008 Design: A Practical Guide and Keane

Aircraft Systems, Third Edition Moir and Seabridge March 2008

Introduction to Aircraft Aeroelasticity And Loads Wright and Cooper December 2007

Stability and Control of Aircraft Systems Langton September 2006 Military Avionics Systems Moir and Seabridge February 2006

Design and Development of Aircraft Systems Moir and Seabridge June 2004

Aircraft Loading and Structural Layout Howe May 2004 Aircraft Display Systems Jukes December 2003 Civil Avionics Systems Moir and Seabridge December 2002

Ashish Tewari

Department of Aerospace Engineering Indian Institute of Technology, Kanpur, India

Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

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Library of Congress Cataloguing-in-Publication Data Tewari, Ashish.

Advanced control of aircraft, rockets, and spacecraft / Ashish Tewari. p. cm.

Includes bibliographical references and index. ISBN 978-0-470-74563-2 (hardback) 1. Flight control. 2. Airplanes–Control systems. 3. Space vehicles–Control systems. 4. Rockets (Aeronautics)–Control systems. I. Title. TL589.4.T49 2011 629.1’1–dc22 2011009365 A catalogue record for this book is available from the British Library.

Print ISBN: 9780470745632 ePDF ISBN: 9781119971207 oBook ISBN: 9781119971191 ePub ISBN: 9781119972747 Mobi ISBN: 9781119972754

Set in 9/11pt Times by Thomson Digital, Noida, India.

The intellectual or logical man, rather than the understanding or observant man, set himself to imagine designs – to dictate purposes to God.

– Edgar Allan Poe

Contents

Series Preface xiii Preface xv

1 Introduction 1 1.1 Notation and Basic Definitions 1 1.2 Control Systems 3 1.2.1 Linear Tracking Systems 7 1.2.2 Linear Time-Invariant Tracking Systems 9 1.3 Guidance and Control of Flight Vehicles 10 1.4 Special Tracking Laws 13 1.4.1 Proportional Navigation Guidance 13 1.4.2 Cross-Product Steering 16 1.4.3 Proportional-Integral-Derivative Control 19 1.5 Digital Tracking System 24 1.6 Summary 25 Exercises 26 References 28

2 Optimal Control Techniques 29 2.1 Introduction 29 2.2 Multi-variable Optimization 31 2.3 Constrained Minimization 3 2.3.1 Equality Constraints 34 2.3.2 Inequality Constraints 38 2.4 Optimal Control of Dynamic Systems 41 2.4.1 Optimality Conditions 43 2.5 The Hamiltonian and the Minimum Principle 4 2.5.1 Hamilton–Jacobi–Bellman Equation 45 2.5.2 Linear Time-Varying System with Quadratic Performance Index 47 2.6 Optimal Control with End-Point State Equality Constraints 48 2.6.1 Euler–Lagrange Equations 50 2.6.2 Special Cases 50 2.7 Numerical Solution of Two-Point Boundary Value Problems 52 2.7.1 Shooting Method 54 2.7.2 Collocation Method 57 2.8 Optimal Terminal Control with Interior Time Constraints 61 2.8.1 Optimal Singular Control 62 2.9 Tracking Control 63 viii Contents

2.9.1 Neighboring Extremal Method and Linear Quadratic Control 64 2.10 Stochastic Processes 69 2.10.1 Stationary Random Processes 75 2.10.2 Filtering of Random Noise 7 2.1 Kalman Filter 7 2.12 Robust Linear Time-Invariant Control 81 2.12.1 LQG/LTR Method 82

Exercises 98 References 101

3 Optimal Navigation and Control of Aircraft 103 3.1 Aircraft Navigation Plant 104 3.1.1 Wind Speed and Direction 110 3.1.2 Navigational Subsystems 112 3.2 Optimal Aircraft Navigation 115 3.2.1 Optimal Navigation Formulation 116 3.2.2 Extremal Solution of the Boundary-Value Problem: Long-Range

Flight Example 119 3.2.3 Great Circle Navigation 121 3.3 Aircraft Attitude Dynamics 128 3.3.1 Translational and Rotational Kinetics 132 3.3.2 Attitude Relative to the Velocity Vector 135 3.4 Aerodynamic Forces and Moments 136 3.5 Longitudinal Dynamics 139 3.5.1 Longitudinal Dynamics Plant 142 3.6 Optimal Multi-variable Longitudinal Control 145 3.7 Multi-input Optimal Longitudinal Control 147 3.8 Optimal Airspeed Control 148 3.8.1 LQG/LTR Design Example 149

3.8.2 H∞ Design Example 160 3.8.3 Altitude and Mach Control 166

3.9 Lateral-Directional Control Systems 173 3.9.1 Lateral-Directional Plant 173 3.9.2 Optimal Roll Control 177 3.9.3 Multi-variable Lateral-Directional Control: Heading-Hold Autopilot 180 3.10 Optimal Control of Inertia-Coupled Aircraft Rotation 183 3.1 Summary 189 Exercises 192 References 194

4 Optimal Guidance of Rockets 195 4.1 Introduction 195 4.2 Optimal Terminal Guidance of Interceptors 195 4.3 Non-planar Optimal Tracking System for Interceptors: 3DPN 199 4.4 Flight in a Vertical Plane 208 4.5 Optimal Terminal Guidance 211

Contents ix

4.6 Vertical Launch of a Rocket (Goddard’s Problem) 216 4.7 Gravity-Turn Trajectory of Launch Vehicles 219 4.7.1 Launch to Circular Orbit: Modulated Acceleration 220 4.7.2 Launch to Circular Orbit: Constant Acceleration 227 4.8 Launch of Ballistic Missiles 228 4.8.1 Gravity-Turn with Modulated Forward Acceleration 232 4.8.2 Modulated Forward and Normal Acceleration 233 4.9 Planar Tracking Guidance System 237 4.9.1 Stability, Controllability, and Observability 241 4.9.2 Nominal Plant for Tracking Gravity-Turn Trajectory 243 4.10 Robust and Adaptive Guidance 247 4.1 Guidance with State Feedback 250 4.1.1 Guidance with Normal Acceleration Input 250 4.12 Observer-Based Guidance of Gravity-Turn Launch Vehicle 254 4.12.1 Altitude-Based Observer with Normal Acceleration Input 255 4.12.2 Bi-output Observer with Normal Acceleration Input 260 4.13 Mass and Atmospheric Drag Modeling 266 4.14 Summary 274 Exercises 275 References 275

5 Attitude Control of Rockets 277 5.1 Introduction 277 5.2 Attitude Control Plant 277 5.3 Closed-Loop Attitude Control 281 5.4 Roll Control System 281 5.5 Pitch Control of Rockets 282 5.5.1 Pitch Program 282 5.5.2 Pitch Guidance and Control System 283 5.5.3 Adaptive Pitch Control System 288 5.6 Yaw Control of Rockets 294 5.7 Summary 295 Exercises 295 Reference 296

6 Spacecraft Guidance Systems 297 6.1 Introduction 297 6.2 Orbital Mechanics 297 6.2.1 Orbit Equation 298 6.2.2 Perifocal and Celestial Frames 299 6.2.3 Time Equation 301 6.2.4 Lagrange’s Coefficients 304 6.3 Spacecraft Terminal Guidance 305 6.3.1 Minimum Energy Orbital Transfer 307 6.3.2 Lambert’s Theorem 311 6.3.3 Lambert’s Problem 313 6.3.4 Lambert Guidance of Rockets 322 6.3.5 Optimal Terminal Guidance of Re-entry Vehicles 327 x Contents

6.4 General Orbital Plant for Tracking Guidance 334 6.5 Planar Orbital Regulation 339 6.6 Optimal Non-planar Orbital Regulation 345 6.7 Summary 352 Exercises 352 References 355

7 Optimal Spacecraft Attitude Control 357 7.1 Introduction 357 7.2 Terminal Control of Spacecraft Attitude 357 7.2.1 Optimal Single-Axis Rotation of Spacecraft 358 7.3 Multi-axis Rotational Maneuvers of Spacecraft 364 7.4 Spacecraft Control Torques 375 7.4.1 Rocket Thrusters 375 7.4.2 Reaction Wheels, Momentum Wheels and Control Moment Gyros 377 7.4.3 Magnetic Field Torque 378 7.5 Satellite Dynamics Plant for Tracking Control 379 7.6 Environmental Torques 380 7.6.1 Gravity-Gradient Torque 382 7.7 Multi-variable Tracking Control of Spacecraft Attitude 383 7.7.1 Active Attitude Control of Spacecraft by Reaction Wheels 385 7.8 Summary 389 Exercises 389 References 390

Appendix A: Linear Systems 391

A.1 Definition 391 A.2 Linearization 392 A.3 Solution to Linear State Equations 392

A.3.1 Homogeneous Solution 393 A.3.2 General Solution 393

A.4 Linear Time-Invariant System 394 A.5 Linear Time-Invariant Stability Criteria 395 A.6 Controllability of Linear Time-Invariant Systems 395 A.7 Observability of Linear Time-Invariant Systems 395 A.8 Transfer Matrix 396 A.9 Singular Value Decomposition 396 A.10 Linear Time-Invariant Control Design 397

A.10.1 Regulator Design by Eigenstructure Assignment 397 A.10.2 Regulator Design by Linear Optimal Control 398 A.10.3 Linear Observers and Output Feedback Compensators 398 References 400

Appendix B: Stability 401

B.1 Preliminaries 401 B.2 Stability in the Sense of Lagrange 402 B.3 Stability in the Sense of Lyapunov 404 B.3.1 Asymptotic Stability 406

Contents xi

B.3.2 Global Asymptotic Stability 406 B.3.3 Lyapunov’s Theorem 407 B.3.4 Krasovski’s Theorem 408 B.3.5 Lyapunov Stability of Linear Systems 408 References 408

Appendix C: Control of Underactuated Flight Systems 409

C.1 Adaptive Rocket Guidance with Forward Acceleration Input 409 C.2 Thrust Saturation and Rate Limits (Increased Underactuation) 415 C.3 Single- and Bi-output Observers with Forward Acceleration Input 417 References 432

Index 433

Series Preface

The field of aerospace is wide ranging and multi-disciplinary, covering a large variety of products, disciplines and domains, not merely in engineering but in many related supporting activities. These combine to enable the aerospace industry to produce exciting and technologically advanced products. The wealth of knowledge and experience that has been gained by expert practitioners in the various aerospace fields needs to be passed onto others working in the industry, including those just entering from University.

The Aerospace Series aims to be practical and topical series of books aimed at engineering professionals, operators, users and allied professions such as commercial and legal executives in the aerospace industry. The range of topics is intended to be wide ranging, covering design and development, manufacture, operation and support of aircraft as well as topics such as infrastructure operations and developments in research and technology. The intention is to provide a source of relevant information that will be of interest and benefit to all those people working in aerospace.

Flight Control Systems are an essential element of all modern aerospace vehicles, for instance: civil aircraft use a flight management system to track optimal flight trajectories, autopilots are used to follow a setcourseortolandtheaircraft,gustandmanoeuvreloadalleviationsystemsareoftenemployed,military aircraft are often designed for enhanced manoeuvrability with carefree handling in flight regimes that have statically unstable open loop behaviour, Unmanned Autonomous Vehicles (UAVs) are able to follow a pre-defined task without any human intervention, rockets require guidance and control strategies that are able to deal with rapidly time varying parameters whilst ensuring that rocket’s orientation follows the desired flight path curvature, and spacecraft need to be guided accurately on missions that may last many years whilst also maintaining the desired attitude throughout.

This book, Advanced Control of Aircraft, Spacecraft and Rockets, considers the application of optimal control theory for the guidance, navigation and control of aircraft, spacecraft and rockets. A uniform approach is taken in which a range of modern control techniques are described and then applied to a number of non-trivial multi-variable problems relevant to each type of vehicle. The book is seen as a valuable addition to the Aerospace Series.

Peter Belobaba, Jonathan Cooper, Roy Langton and Allan Seabridge

Preface

There are many good textbooks on design of flight control systems for aircraft, spacecraft, and rockets, but there is none that covers all of them in a similar manner. The objective of this book is to show that modern control techniques are applicable to all flight vehicles without distinction. The main focus of the presentation is on applications of optimal control theory for guidance, navigation, and control of aircraft, spacecraft, and rockets. Emphasis is placed upon non-trivial, multi-variable applications, rather than those that are handled by single-variable methods. While the treatment can be considered advanced and requires a basic course in control systems, the topics are covered in a self-contained manner, without the need frequently to refer to control theory textbooks. The spectrum of flight control topics covered in the book is rather broad, ranging from the optimal roll control of aircraft and missiles to multi-variable, adaptive guidance of gravity-turn rockets, and two-point boundary value, optimal terminal control of aircraft, spacecraft, and rockets. However, rotorcraft and flexible dynamics have been excluded for want of space. The task of designing an optimal flight controller is best explained by the parable of a pilgrim (cf.

(Parte **1** de 6)