Modern Navigation, Guidance, And Control Processing

Modem Navigation, Guidance, and Control Processing b y Ching-Fang L i n Modeling, Design, Analysis, Simulation, and ...

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Modem Navigation, Guidance, and Control Processing

b y Ching-Fang L i n

Modeling, Design, Analysis, Simulation, and Evaluation ( M D A S E ) Modern Navigation, Guidance, and Control Processing Advanced Control Systems Design Integrated, Adaptive, and Intelligent Navigation, Guidance, and Control Systems Design Digital Navigation, Guidance, and Control Systems Design

Modem Navigation, Guidance, and Control Processing

Ching-Fang Lin American Gh'C Corporation

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Prentice Hall, Englewood Cliffs, New Jersey 07632

L i b r a r y of C o n g r e s s Catalo~lng-ln-Publlcatlon D a t a

Lln. Chlng-Fang. M o d e r n n a v l g a t l o n , guldance. and c o n t r o l processing / by C h l n g -Fang Lln. p. CI. ( S e r l e s In a d v a n c e d n a v l g a t l o n , guldance. a n d c o n t ~ o l . a n d t h e l r a p p l l c a t l o n s : b k . 2) I n c l u d e s b l b l l o g r a p h t c a l r e f e r e n c e s and index. I S B N 0-13-596230-7 1. F l l g h t c o n t r o l . 2. G u i d e d ntsslles--Control s y s t e m s . I. T l t l e . 11. S e r l e s . TL589.4.L55 1991 90-43009 629.1--0C20 CIP

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Editoriallproduction supervision and interior design: Brendan M . Stewart Cover design: Bruce Kenselaar Manufacturing buyers: Kelly Behr and Susan Brunke Acquisitions editors: Bernard Goodwin and Michael Hays

O 1991 by Prentice-Hall, Inc. A Simpn & Schuster Company Englewood Cliffs. New Jersey 07632

Thii book can be made available to businesses and organizations at a special discount when ordered in large quantities. For more information, please eontad: PrenticeHall, Inc., Special Sales and Markets, College Division, Englewood Cliffs, NJ 07632 All rights reserved. N o part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher Printed in the United States of America 1 0 9 8 7 6 5 4 3 2 1

ISBN 0-13-59b230-7 Prenticc-Hall International (UK) Limited, Lotidon Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall Canada Inc., Toronto Prenticc-Hall Hispanoamericana. S.A.. Mexico Prentice-Hall of India Private Limited, ,Vru, Drlhi Prentice-Ha11 ofJapan. Inc.. Tokyo Simon & Schuster Asia Pte. Ltd.. Singapore Editora Prenticc-Hall do Brasil. Ltda.. Rio de Jatreiro

To my family for their love, understanding, and support throtrghout.

Contents Series Foreword Preface Acknowledgments 1 Introduction 1.1 Overview, 1 1.2 Outline/Scope,

8

2 Modeling-Design-Analysis-SimulationEvaluation (MDASE) of NGC

Processing 2.1 LinearlNonlinear Intercept NGC System, 13 2.1.1 LinearlNonlinear Intercept N G C Processing, 14

Contents

2.1.2 2.1.3 2.1.4

Modeling and Simulation, 18 Guidance System Classification, 21 Flight Control System (FCS) and FCS Sensing, 22 2.2 Target Signal Processing, 23 2.2.1 Targeting, 30 2.2.2 Kinematic/Relative Geometry, 34 2.2.3 Targeting Sensor Dynamics, 37 2.3 N G C System Design and Analysis, 45 2.3.1 Guidance Filtering/Processing, 45 2.3.2 N G C Stability and Pevfovmance Analysis, 52 2.4 Target Tracking State Modeling, 68 2.4.1 Target Noise and Target Maneuver Modeling, 68 2.4.2 Two-Dimensional Target Tracking State Modeling, 72 2.4.3 Three-Dimensional lntercept State Modeling, 74

3 Modem Multivariable Control Analysis Singular Value Analysis, 78 Sensitivity and Complementary Sensitivity Functions, 82 Design Requirements, 84 Structured Singular Value, 86 General Robustness Analysis, 89 Robustness of Real Perturbations, 90 3.6.1 State-Space Model for Additive Uncertainty, 91 3.6.2 Slate-Spare Sing~rlarI/allres, 92 3.6.3 Root Loctrs, 92 3.6.4 A Monte Carlo Comnputation, 94 3.6.5 A Monte Carlo Analysis.for SecondOrder Systetns, 95 3.6.6 Stability Margin Comttp~rtatiot~, 98 Monte Carlo Analysis, 100 Covariance Analysis, 102 Adjoint Method, 106 3.9.1 Adjoint P/~ilosophy,106 3.9.2 Applications, 108

Contents 3.10 Statistical Lzncarization, 11 1 and Tools for Statistical 3.10.1 l'c~clrtriq~rc~s L~trcartzation, 116 3.10.2 Statistical Litlearization with Adjoint hlrtlrod, 1 18 3.10.3 Applicatiot~s, 120 3.11 Qualitativc Comparison, 121 3.12 Other l'erformancc Analysts, 124

4 Modern Filtering and Estimation

Techniques 4.1

4.2

4.3

4

4.5

4.6

Linear Minimum Variance Estimation: Kalman Filter, 128 4.1.1 Discrete Kalman Filter, 129 4.1.2 Contirl~rortsKalmatr Filter, 129 Nonlinear Filtering, 133 4.2.1 Estetrded Kalmaw Filter (EKF), 133 4.2.2 Sratisrical Lit~earization Teclzrtiqrre, 135 4.2.3 ~tfrrltipleModel Estirnadon, 137 Prediction and Smoothing, 138 4.3.1 Prediction, 139 4.3.2 Strroothirrg, 140 Kalman Filter Design and Performance Analysis, 140 4.4.1 Kalrnar~Filter Desigtl Process, 142 4.4.2 Selectiort of Noise Itltensity Matrices, 142 4.4.3 Error Analysis, 146 Operational Considerations, 148 4.5.1 Filter Pevfarmance, 149 4.5.2 Computational Reqtrirements, 149 4.5.3 Absence o f A Priori Information, 153 4.5.4 Filter Divergence, 153 4.5.5 &lodeling Process Noise and Biases, 156 Combined Complementary/Kalman Filter Approach to Estimator Design, 158 4.6.1 First-Order Complert~entary/Kalman Filter, 158 4.6.2 Second-Order Complementary/Kalman Filter, 163 4.6.3 Third-Order Complernentary/Kalman Filter, 170

Contents

5 Inertial Navigation 5.1 Introduction, 176 5.2 Common Requirements for Inertial Navigators, 181 5.3 Navigation Computation and Error Modeling, 184 5.3.1 Coordinate Systems, 184 5.3.2 Position and Velocity Generation, 186 5.3.3 Data Processing, 188 5.3.4 IA'S Error Analysis and Modeling, 188 5.4 Gimballed Inertial Navigation System (INS), 192 5.4.1 Ginzbal Mechanism, 192 '5.4.2 Iitertial Setisors on the Stable PlatJorrn, 193 5.4.3 Plarfornt Aligrlriient Modes, 193 5.4.4 nhvigatioit Mode, 194 5.4.5 Systein Ititernal arid External Integaces, 195 5.4.6 Navigatioii Mechanizatiorz and Error Model, 195 5.5 Strapdown Inertial Navigation System (INS), 199 5.5.1 Generalized Strapdowti Mechai~izatiori, 199 r, 5.5.2 Strapdouwi Navixatiori ~ o r n ~ u t e 201 5.5.3 Strapdown Coniputer Requireineiits, 201 5.5.4 Strapdowrl Error Model and Attalysis, 204 5.5.5 Operation Flow Diagram, 208 Comparison and Analysis of Gimbal Versus 5.6 Strapdown, 209 5.6.1 Fcatitrc Coiiiparisoii, 209 5.6.2 h'ai~igatiori Errors Cori~parisoil,209 5.7 External Navigation Aids, 213 5.7.1 Aided liierti~l,Vavkatioir Meckatiizatior?, 213 5.7.2 Global Positioi~iiigSysterri ( G P S ) , 215 5.7.3 Tactical A i r hlavigatiorl ( T A C A N ) , 277 5.7.4 L o r ~ gRange Navigation ( L O R A N ) , 220

176

Contents

5.7.5

Terrain Contour Matching (TEHCO.M), 220 5.7.6 Doppler Radar, 225 5.7.7 Star Trackers, 225 5.7.8 Kaln~anFiltering, 226 5.7.9 Kaltnon Filtering Performance, 228 5.8 Integrated Inertial Navigation System (IINS). 229 5.8.1 I t ~ t e ~ ~ r a rSensittcq/Flight ed Control Reference Systrtn (ISFCRS), 231 Sensory Subsystetn 5.8.2 it~te~qrated ( I S S ) , 231 5.8.3 Itltecqrared Inertial Sensing Assembly ( I I S A ) , 232 5.8.4 Helicopter Integrated Inertial h'avigation Systern (HIINS), 241 5.8.5 integrated Missile Guidance Systerns, 245

6 Guidance Processing 6.1 Guidance Processors, 252 6.2 Guidance Mission and Performance, 268 6.2.1 Guidance Pevformance, 269 6.2.2 Pltases of Flight, 279 6.2.3 Operation, 280 6.3 Multiple Mode Guidance Modeling, 298 6.3.1 M~dcourseGuidance, 298 6.3.2 Terminal Guidance, 302 6.3.3 Error Analysis Model Development, 308 6.4 Guidance Algorithm, 310 6.4.1 Preset Guidance, 310 6.4.2 Direct Guidance Methods, 312 6.5 Guidance Law, 347 6.5.1 LOS An3le Guidance, 348 6.5.2 LOS Rate Guidance, 348 6.5.3 Sensitivity and Cornparison of G~ridarrceLaws, 373 6.6 Single-Mode, Dual-Mode, and Multimode Guidance, 379 6.6.1 Single-Mode Guidance, 379 6.6.2 Dual-Mode Guidance, 380 6.6.3 Multitnode Guidance Applications, 384

Contents 6.7

Defense and Offense Systems, 392 6.7.1 Performance Paratneters, 392 6.7.2 Low-Altitude Air Defense Systems, 397 6.8 Future Guidance Processing, 404 6.8.1 Areas for Teclztiological Advartces in Signal Processitig, 405 6.8.2 Future Signal Processing for Missile Guidatlce, 407

7 Navigation and Guidance Filtering

Design 7.1 Target 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5 7.1.6

State Estimation, 414 Target Tracking Filter Suiizmary, 416 The Wiener Filter, 420 Kalmatl Filter, 420 The Sitiiplified Kalnlari Filter, 420 Alpha-Beta-Canima Filter, 421 Modified Maxitn~rm-Likelihood Filter, 424 7.1.7 Two-Point Extrapolator, 425 7.1.8 Coriiparisori of Targct Trackirrg Filters, 425 7.2 Practical Navigation and Guidance Filtcr Design, 427 7.2.1 Guidance Tracking Filtcr, 427 7.2.2 Navigation and Guidance Filteringfor Position Estiriiatc, 430 7.2.3 Navigatiorr arid Cuidatice Filteritlgfor Position and Velocity Estitilate, 431 7.2.4 Navigation atrd Cuidarrcc Filterit~gfor Position, I/clociry, and Accrleratiot~ Estiriiatc, 434 7.2.5 Advatrccd Guidance Filter, 440 7.3 Radar Tracking, 442 7.4 Spacccraft Attitudc Estimation, 444 7.5 Advanccd Navigation System Design, 445 7.5.1 Global Positioning Systetir (GPS) Acc~rracyItnpro~~etttcrit, 445 7.q.2 Integrated CPSIIR'S, 457

Contents

8 Advanced Guidance System Design 8.1

8.2

8.3 8.4

8.5

8.6

Advanced Guidancc Laws, 468 8.1.1 O~itirttnlCrridatrce Law Survey, 469 8.1.2 Atrnlyticnl Solrrtiotr of Optirnal Filters and Optitnal G~ridatrceLaw, 474 Complenicntary/Kalman Filtered Proportional Navigation: Biased PNG and Complcmcntary I'NG, 481 8.2.1 Corttliitted Seeker-Czridatrce Filtering it1 a Cottrplettret~taryFilter, 483 8.2.2 Biased PAYC ( B P N G ) Algorithm, 486 8.2.3 Cort~plettret~tary PNG (CPNG) Alyoritl~trr, 492 8.2.4 Terrtr inal G~ridatlceSystem Analysis, 501 Other Terminal Guidance Laws, 510 Radome Error Calibration and Compensation, 517 8.4.1 Radorne Error Cotnpensntion, 51 7 8.4.2 Gtridowce Perforrnat~ceAnalysis with In-Fl!qht Radotne Error Calibratiotr, 520 8.4.3 Des[gtt Equations for P N C with Parasitic Feedback, 528 Command Versus Semiactive Homing Guidance System Design and Analysis, 544 8.5.1 Modeling, 544 8.5.2 Miss-Distance Analysis for Command Guidance, 550 8.5.3 Analysis of Optimal Command Ccridance Versus Optimal Semiactive Homirlg C~ridance,557 Analytical S o l ~ ~ t i oonf Optimal Trajectory Shaping for Combined Midcourse and Terminal Guidance, 562 8.6.1 Optitnal Trajectory Sllaping Guidance, 562 8.6.2 Problem Formulation, 563 8.6.3 Attalytic Optimal Gtridance Law, 566 8.6.4 Real-Time lmplemetztation and Pevformance, 578 8.6.5 Discussiotls, 583

Series Foreword The role played by modern navigation, guidance, and control (NGC) in the development and advancement of such areas as commercial and military aviation, to name only two, has continually expanded since its earliest inception in the 1950's. As this field began to grow and take on added importance, many books were written dealing mainly with the theoretical aspects of NGC, but most of these were confined to the earlier years of N G C development. Currently, although N G C system applications continue to take on an ever-increasing importance, the availability of reference books, especially textbooks suitable for graduate level and advanced undergraduate students as well as those who practice in the field, has not kept pace. This series emphasizing NGC systems and their applications is long overdue; in fact, it has been 30 years since such a series dealing with N G C systems has been written and available to the academic and ~rofessionalcommunities. Moreover. it is the first ever such series to thoroughly discuss the advanced control system design (modern multivariable control analysis; robust control; estimation; adaptive control; nonlinear control; intelligent control; etc.). It comes at a time when concern over issues such as the status of education and the decreasing number of trained, qualified professionals in this country is a t an all-time high. It is against such a background that the present series was conceived to assess state-of-the-art systems and control theories, and engineering applications of advanced N G C systems. Another purpose of the series is to develop future research agenda and at the same time encourage A

xuiii

Series Foreword

discussions in those areas that do not always find the systems and control community in complete agreement. The series provides a comprehensive coverage of the latest N G C technology as follows. The first book begins by introducing the various applications of N G C systems, after which it provides a thorough, fundamental treatment of what is considered the five most important stages in N G C system development: modeling, design, analysis, simulation, and evaluation (MDASE). The second book in the series takes up the subject of advanced estimation and guidance systems design, as well as N G C processing. The third book is concerned with the subject of advanced control system design, with particular emphasis placed on the topic of flight control system (FCS) design. The topics that constitute the fourth book include integrated, adaptive, and intelligent N G C systems design, while the fifth book is devoted completely to digital N G C systems design. Although most of the material in these five books is self-contained, there is a natural progression in the series as a whole toward more advanced topics. For example, much of the material in the second book actually serves as a prelude to the third, fourth, and fifth books. These books are the result of several years of experience gained on the part of the authorleditor both as a professor at the university level and as a practitioner in the field. It is believed that this series will provide invaluable insight and instruction to students, mainly at the graduate level but also to advanced urldcrgraduate students, as well as to those engineers who work directly or indirectly in the field of N G C system design and applications. In addition, this series is intended to provide both engineers and managers with the advanced N G C knowledge and concepts necessary to make correct decisions concerning the best N G C system design in a particular situation.

Preface It is very likely that few people who labor in any scientific discipline are unaware of the contributions of advanced navigation, guidance, and control (NGC) theory to aerospace-related programs. It is, however, equally unlikely that many are aware of the dramatic impact of this field on such diverse areas as medicine, industrial manufacturing, energy management, and chemical engineering. While its broad range of applications would at first appear to indicate that advanced N G C theory is enjoying an immense popularity in scientific and academic settings in general, this unfortunately turns out not to be the case. It is felt by many experts that many of those in the aerospace field in particular either are content to rest on the laurels surrounding the success of NGC theory developed in the 1950's and 1960's or have become so conservative in their design philosophies as to be unduly apprehensive about using advanced N G C theory. The latter appears to be especially true in the area of aviation. The author is quick to point out, however, that the NGC field itself is somewhat responsible for many of the misperceptions on the part of those w h o are not convinced of the usefulness of advanced N G C theory. More than a mere shadow of doubt has been cast on the usefulness of this theory as a result of its having taken

xx

Preface

a much too mathematically-oriented turn almost immediately after the theory was first applied in the solution of practical problems. It is the author's opinion that, while N G C theorv is built around a rather beautiful framework of mathematics. its primary emphasis'must nonetheless always bk placed on solving engineering problems of great practical importance. N G C technology has always been the focal point of aerospace engineering and automation research and development. A combination of theoretical concepts, the rapid evolution of computer and microelectronics technology, and the continued refinement of sensor and actuator technology has contributed to its advances. This book examines the role of modern N G C processing in the design of advanced N G C systems. This volume places major emphasis on the practical applications of advanced NGC systems, treating the subject more from an engineering than a mathematical perspective. Nevertheless, theoretical and mathematical concepts are introduced and adequately developed to make the book a self-sufficient source of instruction for readers. The intent of this book is to enable readers to achieve a level of competence that will permit their participation in the practical applications of modeling, design, analysis, simulation, and evaluation (MDASE) to advanced N G C systems. The book presents basic as well as advanced algorithms. A wide range of examples culled from various applications are provided to meet the needs of the different levels and types of readers, extending from issues requiring only a rudimentary knowledge to those involving avant-garde research. Morever, problems in the text span from those that concern only N G C to those that are interdisciplinary, and ultimately to those that encompass the entire systems and control field. An outline of the topics presented in this book is given in Fig. 1. Following the Introduction (Chap. I ) , the text is organized according to five principal topics: MDASE of N G C Processing (Chap. 2), Modern Multivariablc Control Analysis (Chap. 3), Design Algorithms for Advanced NGC Systems Design (Chap. 4). Fundamentals of N G C Processing (Chaps. 5 and 6), and Advanced N G C Systems Design (Chaps. 7 and 8). Each of these is in turn divided into a number of subtopics that discuss the relevant theories, algorithms, and computing tools related to their applications in advanced NGC systems. Referring to the outline organizing the five principal topics covered in this book, only those subtopics that are connected by solid lines are treated in this book. Those subtopics that are connected by broken lines are treated specifically in the books referenced under them. The numerous examples that are included in this book are supplemented by a liberal use of rcferenccs, thus making it easier for the reader to get access to a tremendous body of literature it] this field. As noted by one rcvicwer, there are n o comparable books currently on the market that prcscnt in a usable format such a complete collection of practical tools that are applicable to real world problems. Moreover, the organization of thc material coupled with comprchensive examples make this book well suited to self-teaching, bridging the gap betwccn thc theoretical and the practical.

Preface

Introduction

Modeling-Design-Analysis-Simulation-Evaluation (MDASE) of NGC Processing

-----

I

Hodern Multivariable Control Analysis

1

Performance

-----

rn Advanced

Modern Filtering and Estimation Techniques (Chap. 4)

Design for Advanced NGC Systems Design

Adaptive NGC I Systems: Detection I and Identification (Book 4)

Multivariable Control Design Techniques

1

Knowledge-Based and Neural-Network Approaches to Systems Control (Book 4)

- - I- - - - 1 --Fundamentals of NGC Processing

Guidance Processing

1

Flight Control Processing (Bogk 3)

--7

Navigation and Guidance Filtering Design (Chap. 7)

+

Advanced Guidance Systems Design

Design

+

Advanced Flight Control System Design (Book 3)

I

r

7

- - -C - - - 1

Integrated, Adaptive, and Intelligent NGC Systems Design (Book 4)

Digital NGC Systems Design (Book 5)

Figure 1. Modern Navigation, Guidance, and Control (NGC) Processing

Acknowledgments The number of people who have lent assistance in one way or another to this book is too great to allow me to thank each one individually. 1 am indebted to each person who has contributed to making this project a success, but I feel that I must acknowledge a few individuals by name for their support and effort during the countless hours spent on this project. My first acknowledgement goes out to my staff at American GNC Corporation (AGNC) who have assisted me as follows: Chun Yang and Jerry Juang have provided important technical input, while Jim Wright has spent many hours editing the manuscript. Special thanks are due to Kylie Hsu and Janet Young for their involvement in designing, planning, coordinating, and preparing the entire manuscript, including word processing and numerous original pieces of computer and manual artwork. I am also grateful to William R. Yueh for his technical advice. Several of my colleagues and friends have provided invaluable technical assistance. I specifically want to acknowledge William H. Gilbert of Martin Marietta Electronic Systems who, through his ardent support and encouragement of this project, made this impossible task appear less so at times. I am deeply touched by his unselfish giving of his time in the overall guidance and input to this project. I am

xxiv

Acknowledgments

am also indebted to J. Stanley Ausman of Litton Guidance and Control Systems Division who has greatly improved the quality of Chapter 5. Dr. Ausman is an authority on the subject of this chapter and has co-authored a book on Inertial Guidance (Wiley, 1962). I would also like to thank him for granting me permission to adopt his lecture notes in Tactical Aircraft Weapon Delivery Systems. I also want to express my gratitude to Gary Hewer of Naval Weapons Center, Hsi-Han Yeh of Wright-Patterson Flight Dynamics Lab., and Keqin Gu of Southern Illinois University for their painstaking effort in reviewing Chapter 3. T. Sen Lee of MIT Lincoln Lab., is also appreciated for his effort in reviewing Chapters 4 and 5. A very special note of thanks goes out to Bernard Goodwin, Vice-President, Professional and Technical Reference Publishing of Prentice Hall, who has taken on the task of publishing this book and has continued to support enthusiastically throughout this project. I would also like to extend my appreciation to Michael Hays, Executive Editor and Assistant Vice-President, Professional and Technical Reference Publishing of Prentice Hall, for his constructive advice and his special interest in this project. Also deserving special mention is Brendan Stewart, Production Editor of Prentice Hall, for his excellent job throughout the production process. I am equally obliged to other Prentice Hall personnel who have participated behind the scenes in the different stages of this book project. Finally, I would like to acknowledge those individuals and organizations that have permitted me to reprintladapt portions of their outstanding work in order to make this book a well-rounded source of information. They include (in alphabetical order): AACC, AGARD, AIAA, Arthur Gelb of TASC, Frederick W. Hardy of Hughes Missile Systems Group, Robert J. Heaston of GACIAC, Kaz Hiroshige formerly of General Dynamics Convair Division, IEEE, Johns Hopkins APL Tcchnical Digest, Robert J. Kelly of Allied-Signal Aerospace Co., Litton. James A. McLean of the U.S. Army Missile Command, Pergamon I'rcss, Rockwell International Collins Avionics Division, and SCS International.

Modern Navigation, Guidance, and Control Processing

Introduction This chapter provides an overview of advanced navigation, guidance, and control (NGC) design. The text is oriented to the applied rather than the theoretical aspects of the subject matter. Although advanced techniques are discussed, the contents are presented in such a manner as to also provide a simple and interesting picture of the central issues underlying both classical and advanced control theory and the practice of the NGC modeling, design, analysis, simulation, and evaluation (MDASE) process.

1.1 OVERVIEW The objective of this book is to provide both engineers and managers with the advanced NGC knowledge and concepts necessary to make sensible decisions concerning the best NGC system design in a particular situation. It will hopefully serve as a useful source of information for NGC systems designers by providing them with ideas for the solutions of current problems and future designs, as well as information about the problems encountered with microprocessor-based systems, microelectronics, aerodynamics, structures, propulsion, sensing, actuation, target acquisition, and weapon systems. In an attempt to achieve such an objective, this book demonstrates practical tools that are realistically applicable in the work area.

2

Introduction

Chap. 1

The development and applications of present-day systems and control theory were spurred on by the challenge of unsolved aerospace problems, especially by the series of events that has occurred since the late 1950s. Therefore, it is beneficial to review the development of systems and control theory since that time. The jolting success of the Soviet Union's satellite technology during that time inspired the United States to excel in aerospace technology, thus giving birth to an entire new generation of support for the field of systems and control. This in turn led to success with regard to solving urgent aerospace problems. The emergence of the Apollo program in the 1960s restored confidence in systems and control research and provided opportunities for conceptualized systems and control theories to be transformed into actual practical NGC system designs. Among the more well-established concepts that found great applicability to solving real engineering/control problems in the 1960s are the recursive minimum variance estimator, often referred to as the Kalman-Bucy filter, the LQG technique, the time domain concepts related to the fields of linear algebra and probability, and the Kelley-Bryson variational optimization procedure. Research into this last area proved particularly fruitful, as it resulted in the successful design of optimal trajectories for several space missions. In particular, the Apollo program and space shuttle flights that followed benefitted greatly from the application of optimal control theory. The contributions of Kalnlan filtering to the Apollo program represent yet another nlilestone for the emerging control theory of the late 1950s and early 1960s. The Kalman filter was implemented using a square-root algorithm in the measurement of stars taken by the astronauts with the aid of a sextant. Both optimal control theory and Kalman filtering provide tangible examples of the cruc~alrole played by control science in important programs such as the Apollo project. As a result of its early success in solving mainly aerospace-related problems, NGC theory soon found application in such diverse areas as mcdicine, industrial manufacturing, and energy management. For example, two French researchers werc able to formulate the problem of treating certain cerebral edemas or malignant brain tumors by the simultaneous administration of vasopressin and cortisone into a nonlinear rnultivariable control problem. The treatment resulting from their work is currently used in the neurosurgery clinic of the Hbpital de la Pitie in Paris [Fleming, 19881. Although examples of this type demonstrate the profound impact of control theory in general, and are not altogether rare, it remains true that the majority of progress in N G C research and development continues to find its greatest application in aerospace. The preceding historical account is provided to illustrate the emergence of advanced NGC systems and their applications. The emergence of the various N G C methodologies contributed significantly to progress achieved in the development of state-of-the-art systems and control theories in the 1960s. Unfortunately, it was not long before the development of the NGC system became too mathematically oriented in spite of the pressing need to solve many remaining practical problems. It would appear reasonable that the obligation of the NGC engineer should be first to commit to solving practical NGC problems [Ho, 19871. In the course of carrying

Sec. 1.1

Overview

3

out such an obligation, thc cngincer will oftcn discover relevant theorics that surface spontaneously. While theoretical discovcries of this type often allow thc engineer to conduct furthcr systcm research, it remains impcrativc for professional engineers to commit themselves to solving what are considered to bc rcal-world problems. Very oftcn, practicing engineers encounter difficulty in just understanding the theorics presented at conferences and in journals, not to mention applying them. What is at issue here is not a debate of the merits of theory versus those of applied technology. Rathcr, thc issue is onc of determining an effcctivc way to build a practical N G C systcm; understanding an csotcric N G C theory is of sccondary importance. While there is currently an abundance of N G C problems waiting to be solved, along with a whole assortment of tools capable of attacking these problems, diminished funding in this area hinders further development of N G C technology. In order to recapture the strong financial support for NGC research of the 1960s, the N G C engineering community must first be seriously committed to attacking those immediate ~racticalNGC ~ r o b l e m sthat remain outstanding. " In the absence of such a first step, any practical system and control theory is not likely to be developed, as evidenced by the slow progress of N G C technology in aerospace. T o highlight this fact, N G C tcchnologv used in the Apollo project was later directly applicd to the space shuttle with hardly any new development. As a result of the diminishing research effort in the NGC discipline since the Apollo era, scarcely any systems or control theories have proven valuable enough for practical use in N G C systems. The extent to which system and control theories have facilitated the clarification of various N G C problems and issues cannot be overemphasized. However, the N G C engineering field still faces several practical problems, one of which involves the question of identifying the basic reason for the gap between theory and practice and recommending a means to close this gap for the sake of advancement in the N G C field. In light of this, the impetus behind this book is an attempt to bridge the gap between theoretical arguments and the practical needs of the N G C community by providing detailed discussions and practical examples of MDASE in this area. A more detailed treatment of the MDASE cycle is given in Book 1 of the series. Because experimental and theoretical aspects of NGC system research both represent integral parts of systems and control science, it is essential to consider N G C theories and physical systems together. If an N G C system of a flight vehicle or robotic system is carefully modeled, it usually serves as a strong foundation for the design process which, in turn, leads to analysis and simulation for validation. The final step of the procedure involves evaluation. Thus, the MDASE cycle is an important process in yielding practical results. Through demonstrations of the various techniques and examples in this book, the author hopes to shed light on and unravel complicated systems and control theories, and to translate them into practice. Before they can be brought into the operational stage, advanced N G C algorithms must pass through a developmental stage involving several years of studies and experiments. Through organizing sessions, panel discussions [Lin, 1983-1985; Lin et al., 1986; Lin and Speyer, 19851, and workshops [Lin, 1987; Lin and Franklin,

4

Introduction

Chap. 1

19901 on aerospace vehicle N G C systems for the Amcrican Control Conference and the IEEE Conference on Decision and Control over a period of several years, the author has been able to keep abreast of the current trends in N G C technology and to understand its practical needs in different situations. Thus, the author is able to provide a diverse range of advanced NGC techniques in the book which the reader can then apply to various areas of NGC with competence, developing the best intuitive design. Since all flight vehicles and robotic systems share the same N G C techniques, the techniques presented in the book are applicable to these complex dynamic systems. Applications presented throughout the book and techniques for these applications are summarized in Book 1 of the series. In addition, appropriate applications are embedded throughout the text to enhance the reader's understanding of the techniques presented.

Importance of advanced NGC concepts and their impact.

Advanced N G C systems, although designed so that machines might interact effectively with the elements of nature, rely on human elements for their continued progress and success. The beginning of this chapter highlights the role played by control theory in the success of the Apollo project. The successful development of modern fly-by-wire aircraft such as the F-16 fighter jet can also be credited largely to control theory. And while modern manufacturing is becoming increasingly dependcnt on highly accurate process and machine control, it is an unfortunate circumstance that the continuous contributions that are made in N G C technology and the resulting achievements go largely unappreciated by the broader scientific and engil~ccring communities. The way it1 which N G C technology is isolated from the othcr technological disciplines in governmental, industrial, and academic settings is vcry apparent [Speyer, 19871. Consequently, the acrospacc industry has not fully rcalizcd the potctltial of N G C tech11010gy it1 acrospace systems, and progrcss in this area especially continues to be delayed. This situation will continue unabated so long as those involved fail to grasp both the importance of making progress in the area of advanced N G C technology and the relation of this tcchnology to vital functions in certain acrospace projects. For example, in certain commercial aircraft projects, advocates of new methodologies for enhancing safety and ride quality performances are for the most part ignored. While this can be partially attributed to thosc who prefer conventional dcsign approaches such as the root locus or othcr classical techniques, some of the blame must be laid on inadequately trained pcrsonncl, many of whom lack a complcte understanding of thc basic principles of modcrn and advanced control tcchniqucs. At~othcrreason for this is that thc designers of these commercial transports dcpcnd entirely on the aerodynamic dcsign to improve vehicle stability and efficiency, and totally ignorc the merits of advanccd NGC systems design. Those who are usually quick to criticize techt~ologicaladvance have always been skeptical about the complex and unreliabic nature of new technology, doubtirlg the wisdom of investing so much trust in the capabilities of ncw tcchnology. However, competition and survival dcpend on technological advatlcc. It xvould bc a grave

Sec. 1.1

Overview

5

error to halt technological advance just becausc some inconvcnicnces arise at the beginning of each new era. Despite the fact that any criticisms can generally be found to be true in the short term, the long-term positive contributions that advanced technology has to offer should be emphasized. After a new era of technology has set in, new systems perform more complex tasks and become less expensive than their archaic counterparts. The reliability factor, that is, the mean time between failures, also improves several times over that of older technology of similar complexity. The cost does not increase as much as the cr~ticsargue, as evidenced in the recent shift between the technologies of the 1970s and 1980s. Spccifically, in the weapon systems field, the gains have measured up to the cost. Therefore, critics can discard the notion that new technological concepts are not important [Deitchman, 19871. In the realm of aerospace, unlike other technologies such as aerodynamics, structures, and propulsion, advanced NGC technology has yet to distinguish itself as an essential discipline. There are two main factors that account for this predicament. First, N G C technology in aircraft development has been traditionally accorded a secondary role. This is because many of those who direct large aerospace programs are unacquainted with the important role of advanced NGC technology and its impact on aerospace. Second, no hardware or other tangible entities are produced by N G C technology; whereas, shapes are formed by aerodynamics, delivered by structures, and driven by propulsion [Speyer, 19871. Therefore, NGC technology has barely been utilized in any of the aerospace vehicle design processes with the exception of the Apollo project. According to the author's knowledge, only a few newly developed European commercial airplanes employ advanced control laws. Faulty instrumentation and inadequate computational capabilities reinforced the negative view of NGC technology in the past, but even with the rapid improvements existing in the present this view still persists. T o reap the benefits of advanced technology, timely investments in the pursuit thereof are crucial. Most people cite only the recent progress in military technology. However, new technology must be applied to the commercial arena as well if economic dominance is to be possible. For example, airlines tend to purchase ultramodern airplanes from companies employing more advanced technical sophistication. From this viewpoint, Airbus, the European company, is a commercial success story. It is actively involved in investigating N G C issues and incorporating new N G C technology into its products as well as its human resources. Thus, research and development (R&D) of advanced N G C concepts cannot be stressed enough. It is important that advanced NGC design concepts enter early in the design phase. The earlier engineering errors are detected, the less severe future costs and schedule delays will be. An example of this type of situation is brought into focus in the now famous article entitled, "Probing Boeing's Crossed Connections" by Karen Fitzgerald in the May 1989 issue of IEEE Spectrum, in which the author writes: "By the end of Uanuary], the [FAA] had received 12 more reports of crossed wires, not only on 757s, but on 737s, and not only on cargo fire-extinguishers, but on engine fire extinguishers, and one case of crossed wires from engine temperature sensors. . . . As of March 17, the FAA had received 66 more reports of bungled ~

~

Introduction

6

Chap. 1

wiring and plumbing in fire-protection systems on all four classes U737, J747, J757, J7671 of aircraft."' If it is possible to make such errors in those systems which are by n o means as complicated as an NGC system, then errors in more complicated systems will certainly result unless advanced system concepts coupled with the proper quality control techniques are utilized. As shown in Figure 1-la, the overall cost resulting from engineering error increases dramatically as time progresses. Therefore, errors detected early can drastically reduce the overall cost. Although engineering R&D design and analysis constitutes only approximately 15 percent of the entire project cost, as high as 90 percent of the overall cost of the project may rely on R&D results, as illustrated in Figure 1-lb. Therefore, inadequacy or errors in the R&D design and analysis due to ina)

Cost

Detection of )Engineering Error 1 2 (Normalized Year) Cost Resulting From Engineering Error Increases Dramatically As Time Progresses Therefore Errors Detected Early Can Drastically Cut Down The Overall Cost

b,

Cost

OPS: Operating Cost MAN: Manpower cost R&D: Research 8: Development Cost

Life Cycle CDst = OPS t MAN t R&D Figure 1-1 I

Cost lnvolvemcnt

Karen Fitzgerald. "Probing Bocing's Crosscd Connections." IEEE Spccfrurn. May 1989. p 33.

experience or other conservative factors can often lead to a drastic increase in the overall time and cost. Quoting from the same article, the author writes, "A Boeing engineer who asked that he not be identified said a too ambitious schedule for the new 747-400 aircraft has caused wiring errors so extensive that a prototype had to be completely rewired last year, a $1 million job. . . . In a technology that can tolerate few errors, the crossed connections raise the specter of undetected errors in other parts of aircraft. Each new II&D assignment presents additional challenges which will tax the skill and ingenuity of ~ ~ C - N G design C engineer and will only be solved through hard work, experimentation, and tests. Hence, each new assignment must be analyzed on its own merits, and past practices and methods must not be allowed to stifle new ideas and concepts. Both the past and the present must never forget the common denominator for all designs, which is the human element. Unfortunately, in advanced technology, undue conservatism often hinders progress. Ironically, the objective of any technology should be to go beyond the current state o f design concepts. The NGC field, although relatively new, is currently in an excellent state. I t is responsible €or substantial contributions to engineering, science, and economics as well as the standard of living in the United States and other countries. One of the main contributions of this text is a new perspective of the N G C system whereby the NGC system's interaction with other disciplines will establish the basis for truly innovative problem formulations and methodologies. Methodologies used throughout the book are practical and have been employed in the designing a n d testing of NGC systems. In the past, classical designs were generally used for NGC of unaugmented dynamic systems. Thus, some designs covered in the book are inevitably related to the classical approach. However, modern state-of-the-art N G C system design for highly augmented dynamic systems, which is rapidly gaining popularity, plays the principal role in the design methodologies used in the book. Integration of N G C designs with various engineering automation and signal processing systems are the ultimate goals of this advanced technology. The intention, therefore, of this book and Books 3-5 of the series is to meet this challenge.

"'

Advanced NGC systems. It is apparent that the usefulness of advanced NGC systems has infiltrated the modern world. In the modern aerospace field, NGC designers must be thoroughly equipped to handle N G C systems problems and able to draw on a rather sophisticated knowledge base encompassing many diverse fields; in particular, they must draw on their in-depth knowledge of modern dynamics and MDASE techniques (see Book 1 of the series). They must also understand the finer interactions between systems and components in their nonlinear operational range. Thus, it is natural for them to appreciate the fact that an NGC system should be designed as an integrated system. Additionally, they must consider the wide range of applications of modern digital, analog, and hybrid computers. Very often,

* Ibid.. p 34-35.

8

Introduction

Chap. 1

more sophisticated analytical and computational tools are required to properly model N G C systems. The advantage of new technologies can be utilized to the fullest extent through the advances of new mathematics, analysis, and computation. For example, N G C system scientists and engineers are now more actively involved with the designing and manufacturing of microprocessor/microelectronic chips and computers in addition to interfacing with t h m d y n a m i c s , structure, propulsion, sensing, actuation, target acquisition, and weapon systems.

1.2 OUTLINWSCOPE As shown in the Preface, in Figure 1 labeled Modern Navigation, Guidance, and Control (NGC) Processing, the five primary topics that make up this book are: (1) MDASE of N G C processing (Chapter 2); (2) modern multivariable control analysis (Chapter 3); (3) design algorithms for advanced N G C system design (Chapter 4); (4) fundamentals of N G C processing (Chapters 5 and 6); and (5) advanced N G C systems design (Chapters 7 and 8). O f the subtopics that appear alongside each of these five main categories in the figure, only those that are connected by solid lines are treated in this book. Those subtopics that are connected by broken lines are treated specifically in the books referenced after them.

MDASE of NGC processing. For the purpose of continuity, the first of the five main topics, as shown in Figure 1, deals with MDASE of N G C processing in Chapter 2, highlighting some ofthe key features of Book 1. In particular, Chaprer 2 summarizes the major stages in the evolution of the design and development process. Modem multivariable control analysis. Modern multivariable control analysis, which is the second of the five primary topics as shown in Figure 1, is mainly concerned with the analyses of robustness, performance, dynamics, and stability. Chapter 3 deals specifically with robustness and performance analysis. Robustness analysis is particularly important for examining multivariable control systems design. Performance analysis, which is taken up in the later part of Chapter 3, includes the various statistical methods used in the analysis and synthesis of any modern NGC system. The subject of stability and dynamics analysis, which is threaded throughout this book, is one of the principal topics in Book 1 of the series. The performance predicted by advanced control theory is rarcly achieved when designing and developing an N G C system. A detailed analysis of N G C systems must therefore include a treatment of the actual hardware equipment, citing those characteristics that tend to limit system performance. The most difficult aspects of analyzing N G C loops (at least, in missile applications) include, but are not lirnitcd to, estimating their stability, determining their accuracy, and finding the trajcctory of the pursuer vehicle along with the normal and lateral accclcrations necessary to achieve that trajcctory for varying types of targct motion. Thc theoretical and cx-

perimental techniques for treating these problems are presented in Chapters 2 and 3 in this book, and in Book 1 of the series. The methods that together constitute the third of the five main topics listed in Figure 1, design algorithms for advanced N G C systems design, arc listed with this category in the same figure. The first of these, modern filtering and estimation techniques, is covered in Chapter 4, while advanced multivariablc control systems design techniques are the subject of Book 3. The last two topics in this category, adaptive N G C systems (detection and identification) and knowledge-based and neural network approaches to system control are given special treatmcnt in Book 4 of the series. I t is worthwhile at this point to examine the hierarchical and interactive relationships among the different design techniques mentioned previously that make up the design algorithms for advanced NGC systems. These techniques provide a methodology for selecting successful N G C systems and integrating them according to the mission requirements of a particular vehicle. The primary focus of this particular treatmcnt of design algorithms is to describe present-day components, systems, and synthesis techniques from the system-integration point of view. The goal is to demonstrate how to analyze and select an N G C system to meet a set of performance requirements when the vehicle maneuver and dynamic environment are specified. The various techniques which make up the design algorithms are arranged in hierarchical order in Figure 1-2a. From the figure, it can be seen that modern

Design algorithms for advanced NGC systems.

Modern Filering and Estimation

+

Advanced Multivariable

Control Design Techniques (Book3)

(Chap. 4)

+

Adaptive NGC System Design Twhniques

CBook 4)

v v Knowledge-Eased and Neural-Network Awroacha to Svstems ConUol

Figure 1-2 Design Algorithms for Advanced NGC Systems (a) Hierarchical Relationship (b) Advanced NGC Systems

10

Introduction

Chap. 1

filtering and estimation techniques represent the starting point in the use of design algorithms. Results from the application of these techniques are then used for both advanced multivariable control and adaptive N G C system designs; however, adaptive N G C techniques are also used to iterate or improve upon the multivariable control design. The results from the application of these last two techniques serve as input to the design of a knowledge-based and neural network intelligent system. The interactive relationships between components and techniques are depicted in Figure 1-2b. This figure illustrates how the design algorithms are made more intelligent by incorporating a knowledge-based and neural network approach to N G C system design. Ultimately, an N G C system is designed to accurately control the outputs of a system whose dynamics contain significant uncertainties. This involves the following fundamental processes and their associated algorithms: (1) modeling ofthe system based on physical laws (see Book 1 ofthe series); (2) systern identification based on experimental data (presented in Book 4 of the series); (3) signal processing of the output by filtering, prediction, state estimation, and detection (modern filtering and state estimation are presented in Chapter 4, while adaptive N G C systems detection is presented in Book 4 of the series, and digital processing is presented in Book 5 of the series); and (4) synthesizing the control input and applying it to the system (advanced control theory is presented in Book 3).

NGC processing. Referring again to Figure 1 of the Preface, the fourth of the five main subjects which comprise this book deals with N G C processing. As indicated in the figure, N G C processing includes as subtopics navigation. guidance, and flight control processing. The first of these, navigation processing, is covered in Chapter 5 . Guidance processing is treated in Chapter 6, while flight control processing is presented in Book 3 of the series. In the course of the last three decades, N G C systems have evolved from a state in which they existed only in the imagination to their current state in which they are very much a part of reality. Both military and commercial aviation, as well as other fields including automation and manufacturing, owe a good deal of their success and technical advancement to the parallel advancement in the area of N G C systems. It can readily be said that the development of N G C systems and techniques, which in their ,--infancy often led to unpredictable accuracy and reliability performance, has been most successful, especially when examined in light of the often extreme environmental conditions to which the components of these systems are typically subjected. Figure 1-3 gives a block diagram representation of all the signal processing elements that are needed to perform the functions listed in the previous paragraphs. Each block in the diagram additionally lists the corresponding chapter number where the individual elements are treated in more detail. It can be seen from the figure that N G C proccssing involves several processors, each of which must function individually as well as in unison with others. Advanced NGC systems design. This constitutes the last of the five main topics covered in this book. Looking once again at Figure 1 of the Preface, it

Feedback Sensing Information

I

I

I Tracking Sensing Information

+

+

I

Target Tracking (Chap. 7) I

Guidance Algorithms (Chap. 6)

---

Advanced Navigation Guidance System 4 System (Chap. 5) (Chap. 8)

Control Signal

Reconstruction/Estimation

State Estimate

(Book 1 )

-----Vehicle Motion Estimator (Chap. 4; Books 1, 3,4, 5) Guidance Commands

I

I Navigation Information Data

Advanced

night Control System (Books 1, 3,4, 5)

b

Steering Signal and Attitude Information

Man-in-the-Loop Commands (Controls & Displays) Figure 1-3

NGC Processing

Control Surface

Deflection Commands

12

Inhroductlon

Chap. 1

can be seen that this topic is concerned with the following subjects: navigation and guidance filter design (Chapter 7); advanced guidance systems design (Chapter 8); and advanced flight control system design (Book 3). These subjects prepare the reader for the actual design of integrated, adaptive, and intelligent N G C systems design, presented in Book 4 of the series, and of digital NGC systems design, presented in Book 5 of the series.

Modeling-Design-

Analysis-SimulationEvaluation (MDASE) of NGC Processing It is brought to the reader's attention at this point that a very broad picture of the modeling-design-analysis-simulation-evaluation (MDASE) cycle of the NGC system has already been put forth previously in Book 1 of the series. In order to allow for smooth discussions of the modern NGC processing tasks and for highlighting the way in which NGC processing gets involved in the MDASE cycle, advanced NGC concepts must first be examined. The examples chosen in the following discussions serve to introduce various parameters and functions of NGC processing with respect to particular applications.

2.1 LINEARINONLINEAR INTERCEtT NGC SYSTEM The conventional approach taken by design engineers to the avionics has been to break up the avionics into three distinct and independent systems: the navigation, guidance (midcourse andlor terminal), and flight control systems.

Navigation system. The navigation system functions to provide position, velocity, and attitude of the vehicle with respect to a reference coordinate frame. Using high-accuracy gyros and accelerometers, it is conventionally configured as an inertial system in either a gimballed or strapdown mode.

14

MDASE of NGC Processing

Chap. 2

Guidance. From the perspective of a control system, guidance is a matter of finding the appropriate compensation network to place in series with the plant in order to accomplish an intercept. In order for the pursuer to impact a maneuverable target with little miss distance, guidance uses the principles of feedback control. The purpose of the guidance system is to determine appropriate pursuer flight path dynamics such that some pursuer objective might be achieved efficiently. The guidance system decides the best trajectory (physical action) for the pursuer based on its knowledge of the pursucr's capability, target capability, and desired objectives. In many applications, the guidance system is designed so that it makes use of an inertially stabilized tracker (for example, seeker) that directly measures the angular rates between the pursuer and its target in a fixed coordinate frame. The function of the guidance computer is to mathematically integrate the separate functions of navigation and the flight control system (FCS).

FCS. The function of the FCS is to control the pursucr in pitch, yaw, and roll motion. The FCS executes the guidance commands and stabilizes the pursuer in flight. The FCS, upon receiving commands from the guidance law, then issues its own commands to the appropriate aerodynamic andlor thrust controls of the pursuer so that the guidance command can be properly executed. It is usually configured as a system equipped with low-accuracy inertial components (gyros and accelerometers). Small tactical missiles and most large transport vehicles often have open-loop control instead of the more complex FCS control. 2.1.1 LinearlNonlinear Intercept NGC Processing

Guided weapons or missilcs are normally guided from shortly after launch until target intcrccption. The NGC systcrn supplics stccring commands to aerodynamic control surfaces or corrccting clemcnts of the thrust vector subsystem to maneuver the weapon to its targct and to make it possible for the weapon to intercept moving targets. The guidance process, which is ruled by signal proccssing algorithms implcmented in the N G C system, a prcset flight program, or both, is essentially a feedback control systcrn where the pursuer-target engagcmcnt is considered part of the N G C loop. Thc N G C loop consists of the guidance systcnl together with dynamic controls. Elc~ncntsof this loop include an information subsystem, control clcmcnts, an opcrator, and pursucr dynamics. The components for the N G C systcrn arc shown in Figurc 2-1 in which the ovcrall control of thc pursucr is dividcd into two or lnorc loops. The main control loop in thc diagram is thc guidancc loop, which is thc outcr loop that controls translational dcgrccs of frccdom, whilc the inner controlIFCS loop controls pursucr attitude. ,The guidancc loop contains guidancc scnsors for sensing pursucr motion, targct motion, or rclativc motion of thc targct with rcspcct to the pursucr. This information is uscd in thc guidance computer or the corrccting nctworks, togcthcr with information conccrning thc intended flight profilc, to gcncratc guidance (latcral acceleration) commands for the FCS. Thc FCS and actuation in turn direct control surface dcflcctions to altcr thc pursucr's trajcctory. The body ratcs and accclcrations arc fed back to thc incrtial sensors to closc the FCS loop.

16


Chap. 2

The guidance and control laws used in current tactical missiles are based largely on classical control design techniques. These control laws took birth in the 1950s and have evolved into fairly standard design procedures. Proportional feedback is generally used to correct missile course in the guidance loop, which is commonly referred to as proportional navigation guidance (PNG), and is quite successful against nonmaneuvering targets. The controller for a homing missile, in general, is a closedloop system known as an autopilot, which is a minor loop inside the main guidance loop. In addition to the control surface and servon~echanism,the autopilot consists of mainly the acceleron~etersand/or (rate) gyros to provide additional feedback into the missile servos for missile motion modifications. Advanced sensors may measure other variables. N o explicit state estimators are used and the signals are filtered to reject high-frequency noise. Broadly speaking, autopilots control either the motion in the pitch and yaw planes (lateral autopilots), or the motion around the missile axis (roll autopilots). In general, the roll: pitch, and yaw channels are uncoupled and are typically controlled independently of each other. All commands are amplitude or torque constrained to ensure autopilot and n~issilcstability. Classical controllers have two major advantages, simplicity in design and simplicity in implementation, but they also have several problems. A pursuer is designed to complete the basic homing loop, which requires a sensor (a seeker in the case of a guided missile) to track the targct, a noise filter to reduce the effect of noise, a guidance law to generate thc dcsircd guidance (acccleration) commands to home on the target. an FCS to rcccibc thc dcsircd accelcration commandfrom the guidance system and to generate the required acceleration capability to insure interception of the maneuvering targct, and, finally, a good understanding of the physics of the homing engagement itsclf. Figure 7-2 depicts a miss-distance analysis model conlposed of elemenrs rcprcscnting the intercept kinematics plus clclncnts affecting the intcrccpt guidancc dynamics (that is. pursuer dynamics and the NGC system), and illustrates the intcrdcpcndcncc bct~veensystcnl elements. Inasmuch as thc total system perfornlancc is affcctcd by the individual characteristics of evcry elcmcnt, the system cnginecr is grcatly conccrned with this system interaction. The kinematic part inside the lowcr left dottcd box generates the LOS angle in terms of pursucr and target motion. Based on the LOS anglc, the N G C system generates a latcral acceleration for the pursuer, whilc trying to bring the projectcd miss to zero. In the model, a rclativc position ) I , is added to a glint noisc y,, thc result of which is multiplied by the inverse of thc range R to produce a true LOS anglc corruptcd by glint noise. u, in inertial coordinates. A mcasured LOS angle u is thcn obtaincd by adding ro (5. a radomc crror coupling u,and the following angular noisc tcrms: rangc-depcndcnt (thermal) noisc o,,,, rangc-indcpcdcnt (fading) noisc a,, and cluttcr noisc o,.It thcn scrvcs as input to thc scckcr dynamics to producc a mcasured LOS ratc 6 that is itsclf corruptcd by bias u l , and g sensitivity. The output from thc scckcr u is the input to thc guidancc filter. Many signal processing tcchniqucs arc uscd to discrin~inatcthctargct from its background, other targcts, and decoys. The guidancc law functions to gcncratc midcoursc and/or


18

Chap. 2

terminal steering commands u, based on the guidance filter output. Before being sent to the autopilot, u, is limited. In general (except for coordinated turn vehicle), the methods used to generate the guidance command signals u, for guiding a pursuer in each of two mutually perpendicular planes are identical. Consequently, the guidance algorithm needs to be determined forwnly one of the planes. In the treatment presented here, this plane will be the missile s horizontal plane. In Figure 2-2, u, is normally taken to be the lateral acceleration command. As long as saturation effects are ignored, the model shown in Figure 2-2 is a linear time-varying system driven by stochastic inputs. The inclusion of acceleration saturation effects causes the model to become nonlinear.

2.1.2 Modeling and Simulation The digital simulation technique in which the equations of motion are represented by' a set of first-order nonlinear differential equations is widely recognized. The modeling vector form is given as (see Figure 2-3): x(t) = f(x(t), u(t), t) with initial conditions x(tO) = xg

(2- 1a)

where the function f(x, u ) includes all of the model equations and NGC algorithms. Equation (2-la) corresponds to Figure 2-3, in the absence of any noise inputs. Depending on the specific application, the initial conditions are given, and the terminal time tf is to be determined. For intercept flight simulation, rf is determined by either the point of closest approach of the two vehicles or the time at which the range rate passes through zero for the first time. Numerical integration techniques are then used to integrate these equations with respect to time so that the time history of the state from an initial condition x(t0) to a terminal state x(tf) can be obtained. The output of the simulation is typically the time history of the state, from which a performance criterion can be formulated as a function of the terminal state. The flight vehicle to be designed determines the performance criteria in the simulation. In the case of a guided missile, good performance is represented by short miss distance or short range between the two vehicles at t,. The overall simulation block diagram, shown in Figure 2-4, defines all of the state equations. Referring again to

Nonlinear Function

t

-

f(x(t).uo),t) Figure 2-3

Nonlinear Dynamic Systems

MDASE of N G C Processing

20

Chap. 2

Figure 2-3, the modeling vector form, when a noise input vector is considered as shown in the figure, can be written k(t) = f(x(t), 4 t h t )

+

w(t)

(2-1 b)

where w(t) usually, although not necessarily, .denotes a white-noise input vector. A more general case is considered in Figure 2-4, in which the model includes certain error sources resulting from imperfect measurements or unmodeled dynamics. Incorporating these error sources in Equation (2-la) gives k = f(x,

U)

+ b + n ~ ( x t), + m(t)

(2- lc)

where b is the constant bias term; n l (x, t) is a state-dependent random vector; and nz(t) is a state-independent random vector. Equation (2-lc) could also apply to Figure 2-3, in which case w ( t ) is simply the sum of 6, nl(x, t), and n2(t). If M I or n2 represents a colored-noise process, a shaping-filter technique can be used to model either or both so that the new state vector can be augmented to Equation (2-lc). The resulting augmented state equation is driven by the white-noise characteristics only. A Monte Carlo technique, covariance analysis method, or an adjoint method can be used to properly evaluate the selected performance criteria. These techniques are described in Chapter 3. In closed-loop control system analysis, it must be decided what level of detail will be used to represent each element in the guidance loop. If it is desired to study high-frequency system instabilities, then one must employ a representation that is accurate at those frequencies. The elements that may be required to construct a sufficiently accurate representation in this case include modeling the dynamics of the nonrigid system, knowledge of the full nonlinear aerodynamic characteristics, and detailed modeling of the seeker track loop, guidance signal processing, autopilot, and noise sources. Conversely, a complicated model of this type would not be required in the case of a preliminary parametric study of homing trends since the main effccts of interest are found at low frequencies. Rathcr, a simplified representation of the more complex system capable of incorporating the available trim aerodynamic data and the low-frequency approximations to the different subsystcms that make up the guidance kinematic loop would be more appropriate. After being validated, this simplified representation can be used to study the relative performance of various missile configurations as well as the result of varying system paramctcrs such as time constants and limits [Reichert, 19811. In problems involving dynamic modeling and controls, a state model is usually specified in addition to thc nieasuremcnt equation, relating the state and measurement vcctors as shown in Table 2-1. Bcfore implctnenting thc preceding N G C simulation, it is important to understand the roles of thc pursucr system, the targets, and thc cr~virot~ment so that appropriate modcls in Figure 2-4 can be devclopcd for a particular pursuer-targct engagcmcnt scenario. Essential guidance and control softwarc modulcs arc as follows: intcrfacc, flight path planning, strapdown device, navigation, guidance, stabilization and control, fin actuation, target tracking, and self-test. Their relationships are dcscribcd in Figure 2-5.

LinearlNonlinear Intercept NGC System

Sec. 2.1

TABLE 2-1

-

21

STATE MODELING

D t f ~ n xg ( m l ) state vcclor. y

-

(pxl) system output vcclor. z = (1x1) measured output vcetor.

u = (mxl) mnlml vmor, I, i(qil) cxogenram inpu v m o r .?(I)w d a i n s all thc variables that can be measured and arc fed back to thc controller. Thc enuies of y(t) arc the variables lo be conmllcd. up) is h c output of ihs mnuollsr. lc(l] wnsisu of thc reference inputs, the disturbance inpuu (e.g.. process noise w(t)), and the rcnror M i s s ~(1).Note that y(t) and z(t) may have some wmponenu in common. The wnlinuous-lime nonlinear system is dcscribcd by

i t

=x

I

I t

I

Z ( I ) s h(x(1). I) t

,

X

"(I),

I

=x

Q ~ I= 0. I ~ 4q1)*TI

T

I = at)b(t-r)

T Qv(t)] = 0. Qv(l) "(1) ] = R(t) b(1-z)

The continuous-time nonlinear system with d*crete.time nonlinear mcasuremcnls is repnscnled by: i(1) = f(X(1). Nt), 11 + *I) T

(la) ., (lb)

(2=4

q k ) = h[x(tkh kl + v(k); qv(kl1 = 0. Qv(k) v(k) I = Q

Gb) w h e n the parameter k is considered to be a timc t and the measurements an sampled at discrete timc intcwak. The discrete form of the state equation (la) is as follows: k =k l1 k l Qqk)l = 0, q W k ) wT(k)l = or (2c) h applying many MDASE uchniqucs. the system quations (1)need to be linearized as:

where u, w, and v arc defined in Eqs. (1). Eq.(3) is equivalently expressed as i(t) = &(I) + Bqf) t Br i(1) and y = Cx + h ~,

The state equation in Egs. (4) is generally a nonhomogenwm differential quation with forcing term ~ .t f. (where ~ ) f(l) is a command vcctor anUor an anticipated forcing function mulling from a system nonlinearity andior an estimated dinurbance. In the output quation of Eq. (4), h consisls of the measurement nonlinearilia and uncenainties. The discrete form of the system quations (3) is: x(k) = Wk.1) x(k-1) + W(k-1) qk.1) + wk-1) qk-1)

2.1.3 Guidance System Classification

There are several solutions that can be applied to the problem of guiding a missile to an eventual near miss or collision with a target using only the observed motion of the missile-target line of sight (LOS). In one of these, the missile is constantly flying . - directly toward the current target location. This method, termed simple pursuit, suffers from the fact that the missile must generally undergo rather severe maneuvers as intercept is approached. A second solution concerns a constant-bearing trajectory wherein the missile leads the target, much like the way in which a projectile is traditionally fired at a moving object. A collision invariably occurs as long as the missile flies a course that keeps the relative missile-target velocity aligned with the LOS. As shown in Figure 2-2, the missile trajectory becomes straight line when the missile speed, target speed, and course are constant. PNG is a steering law that is designed to produce a constant-bearing course even in the case of a maneuvering target. T o accomplish this, one makes the rate of change of the missile heading directly proportional to the LOS rate. While this technique presupposes that the missile is able to respond instantaneously to changes in the LOS direction,


22

Chap. 2

-

StrapdownAFtU Guidance Navigation o initialization o Midcourse & Terminal o State Estimation o Euler o Position Transformation _* 4 + o Power On Mode o Power Off Mode o Velocity o Bias Compensation o Vertical Launch o Acceleration o Coordinate E m r s o Separation Mode o Estimate Transformation

Autopilot o Rate Stabilization o Acceleration Control o Aerodynamic Coupling Stability

*

t I

Seeker

I

t Relative

Interface o Aim Point o Initialization 0 Status

4-

Self Test o Test Sequence 4-W

-

1

Right Path Planning o Stored Mission Profile o Change Profile

Actuation o Fin Rate Limitation o Fin Angle Limitation

RF Environment & Clutter ECM: Target Kinematics

Figure 2-5

Sensors o Rate Gyro o Accelerometer

,

Dynamics o Equations of Motion o Engagement o Environment

Pursuer Target Tracking Processing Software Modules

there is in practice some delay in returning to a constant-bearing trajectory subsequent to a target maneuver as a result of finite missile responsiveness and filtering introduced to diminish the effects of noise. Figure 2-6 defines the conventional PNG configuration, in which the measured LOS rate u is filtered through a noise filter to produce a smoothed LOS rate estimate u, and u, is made proportional to u A similar idea is used in modern guidance approaches based on optimization techniques. If measurement noise is to be suppressed, then consideration must be given to a low-pass noise filter. PNG systems generally employ a first-order lowpass filter, which has a transfer function as shown in Figure 2-6. The missile guidante computer, using the preceding relationships, computes steering commands. Besides noise filters, com~ensationsare introduced to offset such factors as variations in the missile's velocity and radome effects. Also, in order to maintain aerodynamic stability along with structural integrity, steering command limits are imposed [Witte and McDonald, 19811.

2.1.4 Flight Control System (FCS) and FCS Sensing That part o f the system comprising the FCS also includes the inertial reference unit (IRU)to measure A,,,. A,,, is the acceleration of the pursuer in response to the guidance command, which is applied to the FCS and results in operation of the pursuer's control surfaces. The control surfaces can be caused by aerodynamic cross coupling, dynamics nonlinearity, and structural vibration to deflect at different orientations from the desired guidance input, thus contributing to miss distance. Au-

Target Signal Processing

Set. 2.2

vmf

Mi& velocity vector

23

--

MiSc without

, , R

Missile maneuver

~ V target ~ velocity vector

The basic proportional navlgation equation

Seeker Guidance filter

Am

a

V,,,? = AV,b

/

Autopilot

-*-

/ 1\

Gs(S)

GF(S)

GA(S)

Missile ~uidance.res~onse / \ maneuver Navigation clos/nng Line sight ratio velocity = Velocity I line of sight

,

rate

Range

g(S)

* k = Ik, Noise Ntcr bandwiith

V, Closing vebcily A Effcaive navigalioo mtw

s+t

Figure 2-6

Proportional Navigation (Courtesy o j K . Hiroshige, 1984)

topilot, actuator, structural aeroelasticity, sensors, and nonlinear airframe dynamics are part of the actual mode of an FCS. Details of the FCS are presented in Book 3 of the series. The useful models for NGC design are summarized in Figure 2-7, Table 2-2. and Table 2-3. 2.2 TARGET SIGNAL PROCESSING

A mathematical model is used to embody pursueritarget kinematics in the form of parameters of the LOS as shown in Figure 2-8, which shows what type of processing is required. A complete simulated engagement of pursuer target tracking begins with the launching of the missile, is followed by a midcourse and a terminal phase, and ends in an intercept and the scoring of missile-to-target miss distance. In active homing, after launch the missile seeker transmits radio-frequency (RF) energy to the target. The target converts this to a synthesized return and radiates it back to the seeker radar. Multiple target returns, which include different ranges, velocities, amplitudes, angular noise, and environments, can be generated simultaneously with or without ECM. The returns are tracked by the seeker. As shown in the relative geometry box in Figures 2-2 and 2-8, the homing loop is closed by calculating the missile-to-target geometry as well as using the RF array to update the relative angular

. MDASE of NGC Processing

Chap. 2

N [NORMAL FORCE1 X AXIS

(PITCH TOTAL MOMENT) REFERENCE Fx

(YAW TOTAL MOMENT1

(AXIAL FORCE) V

Z AXIS (A)

XCG XACC

-

center of gravity lacation arcelermeter location

(B)

6

-

-

c~ TOTAL CN 0 NORMAL FORCE COEFFICIENT

MAX (TRIM1

0

MOMENT COEFFICIENT

-

MACH =CONSTANT ALTITUDE = CONSTANT

k@ya MACH CONSTANT ALTITUDE =CONSTANT

.30°

OMAX ITRIM)

& = . 3 0 ° = 6MAX

6 = 00 b = -100

6 = .20°

TOTAL MOMENT COEFFICIENT CM VERSUS ANGLE OF ATTACK n (D)

Figure 2-7 (a) Horning Missile Axis System and Forces and Moments (b) Forces Due t o Angle of Attack a and Control Fin Uetlection S in a Tail-Controlled Missile (c) Normal Force Coefficient CN VS. Angle of Attack a (d) Total Moment Coefficicnr CM 1,s. Angle of Attack a (Froin [n'erlincs, 1984(a)] u,irk prnnisrio~tfi.otnAACC)

position. As shown in the endgame computation and miss-distance computation boxes, immediatcly before interception, the characteristics of the endgame including the miss distance are determined by extrapolation. Finally, the LOS rates in the pitch and yaw axes are computed and sent to the missile seeker as pitch and yawhead radar errors to fully close the guidance loop. E C M J E C C M modeling involves digital computer simulations of an operating radarlsensing system including the effects from an enen~v'sE C M . This allows the sensine " unit to be assessed under field conditions as to whether it would perform satisfactorily so as to support the tactical control system nlission. A sensing detection model is required to obtain signal-to-noise ratio data as a function of target range. The model must be able to allow these data to be obtained in both clear and noise-jamming environn~cnts. Jamming can be produced by a self-protecting or stand-offjammer operating against the radar signal in either the main lobe or the side lobe. Enemy threats may be taking placc in ascending, level, or descending flight profiles. Multiparh and terrain (terrain-masking and clutter) effects may also need to be considered.

TABLE 2-2

LINeARIZED AIRPRAME RESPONSE

Hjrmal Auxlemlion: A,

= v,?,

? = Aa + I36

"

I1ilch Anxllrr - Aoxlcrdlion: 0 = -(h-1%

6

(

6

s2

-= -

'liansfcr l'maions:

-=

G

Dl

~- I i~ -C HI^

- lib; u = 0 - y - 1*i2)

+ (A+#>&+ II E 112. 2. Multiplicative. The plant is perturbed as C(I + E) for some stable E. The system is stable if [ I + C(I + E)] or [ I + C + GE] is nonsingular or if [I + G + E ] is nonsingular or IT(I+ G -') > (1 E ((7.

+

-'

Singular value analysis is a powerful tool for checking whether the system is robustly stable. This notion leads directly to the generalization of gain and phase margins for multiloop systems. The multiloop gain margin is the pair of real numbers c1 and c2 defining the largest interval ( c l , c2) such that for all Ci(s) = C i E R in Figure 3-2 satisfying the inequalities, ci < C , < c 2 (i= 1, 2, . . . , m), the closed-loop system remains stable. On the other hand, the multiloop phase margin is the number c defining the largest interval (-c, c) such that for all C,(s) = e-'"('") in Figure 3-2 satisfying the inequalities - c < &(o)< c ( i = 1 , . . . , I N ) , the closed-loop systcm remains stable. The following arc obtained using the preceding definitions and the concept of singular values: (1) If the original system is stable and ~ ( +l C) > a , then c l 5 111 (1 - a)], c2 2 [1/(1 a)], and c r 2 sin-'[a/2]; and (2) If the nominal system is stable and ~ ( +1 G - ' ) > a , then 5 1 - a , ( 2 2 1 + a, and c 2 2 sin-'[a/2]. Clearly, the singular value plays a crucial role in determining the robustness of the system. Depending on thc typcs and characteristics of the uncertainty, therc are diffcrcnt robustness mcasures for diffcrcnt sources of unccrtaintics such as plant uncertainty or cxternal disturbanccs that have to bc considcrcd. In the next scctiotl,

+

Figure 3-2 Configuration for Multiloop C;aiti/Phasc Margin l)cfi~iiti~)~l

Sec. 3.1

Slngular Value Analysb

81

suitable robustness measures are identified and more general robustness analyses are discussed. T o show the necessity of using singular values instead of eigenvalucs, consider the following 2 x 2 matrix:

While the determinant of this matrix is always unity for any bation

E,

an additive permr-

produces a singular matrix which cannot be predicted by the eigenvalues. The singular values of M 6 defined by Equation (3-2) are the square roots of the eigenvalues [(E' + 2) t E c2 + 4112 of the matrix M*(E)M(E).Note that the singular values depend on E, while the eigenvalues and determinant do not. Note also that

s-

clearly shows that ~ ( M ( E ), ) 0 as c -t m. A more interesting example is discussed in Laub [1985]. The Ostrowski matrix is an n x n matrix whose upper triangle elements are all + 1, whose diagonal elements are all - I , and whose lower triangle elements are all zero. That is,

This matrix is nonsingular and its eigenvalues are all equal to - 1. Its determinant is ( - 1)" for any n. However, the Ostrowski matrix becomes increasingly nearer to a singular matrix as the dimension n gets larger [Laub, 19833.

In fact, the smallest singular value of the matrix P is 2-"+I and approaches zero as n + cc. Let AP be the perturbation matrix

Modem Multfuariable Control Analysls

82

Chap. 3

that is, P - h P is a singular matrix. g ( P ) is in the order of a(AP). Looking at Equation (3-j), the disturbance A D has therefore caused the closed-loop system P - AP to have zero eigenvalues even though the eigenvalues of the nominal system P are all equal to - 1 [Hewer et al., 1988(a)].

3.2 SENSITMTY AND COMPLEMENTARY SENSITMTY FUNCTIONS Consider the system depicted in Figure 3-3. In control system design, typical requirements are (1) closed-loop stability, (2) disturbance rejection, (3) tracking capability, (4) noise insensitivity, (5) robustness against plant uncertainty, and (6) functionality under component failures. It will be shown in this section that, under mild assumptions, the sensitivity and the complementary sensitivity function play crucial roles in characterizing the feedback performance. That is, the singular values of the sensitivity and the complementary sensitivity function can be measured in order to assess the robustness. First, consider the disturbance rejection property; the transmission from the disturbance source d to the output y is the sensitivity function So:

The smaller the sensitivity function, the better the disturbance rejection property. Similarly, the tracking error as a function of the reference input r is e = (1 + GK)-' r. Again, the smaller the value of (I + GK)-' is, the smaller the tracking error will be. Next, consider that the plant is perturbed as (I A)-'G for some uncertainty A. The uncertainty block matrix A can be used to represent low-frequency parameter errors or the changing number of right half-plane poles of the plant [Doyle et al., 19821. After some manipulations, it can be shown that the system remains stable if A is a stable transfer function and [I (I GK) - A] is nonsingular for all w. This implies that the sensitivity matrix (1 GK)-' must be small such that g (1 + GK) > 5 (A), which in turn means that I + G K + A is nonsingular. Since I + GK + A is nonsingular and det(1 + (1 + GK)-'A) = det(I + GK)-' det(1 + GK + A), the matrix I + (I + GK)-'A will be nonsingular. Another function that is useful in characterizing the robust stability is the complementary sensitivity function To(s) = G K ( l + GK)-'. Let the plant transfer function be disturbed as ( I + A(s))C(s) for some stable A(s) representing sensor error or ne-

+

+ + +

'

Controller K(s) Figure 3-3

Feedback System

Sec. 3.2

Sensitivity and Complementary Sensitivity Functions

83

glected high-frequency dynamics. Assume that the maximum singular value of A(s) is bounded by the inequality -d[A(jw)] < e m ( j w ) for all w 5 0. Then the robust stability in the high-frequency region is guaranteed if the nominal system is asymptotically stable and the following condition is satisfied [Doyle and Stein, 19811: -d[GK(;o){l

+ GK(;~)}-'1 < l/ern(;w)

for a l l o such that e m ( o ) B 1

(3-7)

The system remains robustly stable if Equation (3-7) is true. That is, the smaller u [ G K ( I + GK)-'(jw)] is, or the smaller the maximum singular value of the complementary sensitivity function is, the better the robust stability will be. Also, the noise effect at the output node y is y = - GK(1 GK)-'r. Hence, the smaller GK(1 + G K ) - ' is, the better the noise immunity will be. Similar arguments can be made for the sensitivity function (I KG)-' and the complementary sensitivity function K G ( 1 K G ) - ' at the plant input node. More interpretations of the sensitivity and complementary sensitivity functions can be found in Cruz and Perkins [I9641 and Kwakernaak [1985]. Hence, many feedback properties can be represented in terms of these two functions, and the robustness analysis can be done by investigating the size (singular values) of the sensitivity and the complementary sensitivity functions. For a particular class of problems, the control design technique that reduces the size of these two functions or their combinations directly leads to a robust design. There are, however, important fundamental limitations that restrict the achievable To = (I + performance. The most important one is due to the fact that So GK)-' + G K ( 1 GK)-' = I and (I KG)-' KG(1 KG)-' = I. Thus, it is impossible to make both functions small at the same time, and trade-offs have to be made. In general one requires the sensitivity function to be small in the lowfrequency range (to reject disturbance) and the complementary sensitivity function to be small in the high-frequency range (to be robust against high-frequency uncertainties). Since at low frequencies the loop gain is high, the sensitivity function in the low-frequency range can be approximated as S o ( j o ) = (I + GK)-' == ( G K ) - ' ( j o ) . Also, since at high frequencies the loop gain is low, the complementary sensitivity function in the high-frequency range can be approximated as To(jo) = GK(1 GK)-' = GK(jw). These two requirements simply show that the loop gain G K must be large at low frequencies for good tracking performance and small at high frequencies for robustness. o n e ' drawback of using the singular value in measuring robustness is its conservativeness. Conservativeness results from the structure and the nature of the uncertainty. For a repeated, block-diagonal, or real uncertainty, a less conservative measure has to be adopted. The structured singular value (to be discussed in Section 3.4) represents one attempt in reducing the conservativeness of the singular value. In the following section, another feature of singular value will be discussed, that is, how the singular value analysis of the complementary sensitivity and sensitivity functions is translated into feedback design requirements.

+

+

+

+

+

+

+

+

+

Modem Plultivariable Control Analusis

84

Chap. 3

3.3 DESIGN REQUIREMENTS Design requirements which the control system must satisfy in general are: (1) low sensitivity property, a low-frequency requirement; (2) fast response characteristic, an intermediate-frequency requirement; and (3) large stability margins, a high-frequency requirement. One of the design methods which can handle these requirements explicitly is the frequency response method of classical control theory. However, these classical techniques cannot be directly extended to multivariable control systems in general. Performance specifications and stability robustness requirements for multivariable control systems can be examined by singular value plots in the frequency domain. The plots corresponding to the maximum and minimum singular values over a given frequency range can be established as the design criteria for a multivariable control system design.

Low-frequency requirement. From Equation (3-6), the largest singular value of the sensitivity matrix can be written as

where, as defined in Equation (3-I), E and g represent the maximum and minimum singular values, respectively. In general, the upper bound p ( o ) must satisfy 0 < p ( o ) 1 to obtain a sensitivity reduction property. Using the well-known inequality o[l - + A] 2 g [ A ] - 1 in Equation (3-8), a sufficient condition of Equation (3-8) can be given as 1 dcK(jw)l> 1 f o r o < o,

dw)+

Since g [ G K ( j o ) ]rolls off with the gradient of -20 dB1dec above the maximum frequency of the open-loop poles, the upper-bound frequency in Equation (39) has to be lower than the maximum open-loop pole frequency. Following Ohta and Fujimori [1988],the bound p ( o ) is studied more carefully as follows. T o restrict p ( o ) to values less than unity means that the sensitivity and the response to disturbances are reduced to at least k ( ~x ) 100%. This also means that the maximum steady-state error will be decreased by a factor of p(0) as follows. When r(s) in Figure 3-3 is a step function given by r(s) = [I, . . . , 1IT/s = Hls, following Figure 3-3, the steady-statc crror e, can be written as 1 e , = lim SSO H = So(0)H P--=

-5

(3- 10)

The maximum singular value of e, becomes E[e,] = if[So(O)H]S F [ S o ( O ) ] ] E [ H ] . Therefore, E[e,] 5 p(O)if[H].

High-frequency requirement. The rcquircmcnt in thc high-fr~qucnc)~ region is to make a [ G K ( j w ) ]as small as possiblc. As shown in Ohta and Fujimori [1988],the upper bound of a multiplicative uncertainty of the model is assumcd to

be denoted as t',,(w). Then, the condition of stability robustness can be expressed as F[GK(jw)] < l/t,.(w) for t',,,(w) > 1. Thus, the singular value of the complementary sensitivity matrix has to be less than a given upper - - bound. This condition is effective for suppressing the effect of measurement noises as well.

Intermediate-frequency requirement or crossover frequency. The crossover frequency w, is related to the response speed of the system. The larger w, is, the faster the achievable response speed can be. However, a trade-off between the requirement in the high-frequency region and the response speed must be taken into acount. This consideration restricts the maximum crossover frequency w,,,, below the crossover frequency of t',,(w), and the minimum crossover frequency w,,,, will be determined from the settling time T,,which is an alternative measure of the response speed. Both measures ofw, and T, are closely related with the closedloop poles. As shown in Ohta and Fujimori [1988], a unity feedback system is considered with the transfer function G(s) = o:/[s(s 25w,)]. When 0 < 5 < 1, the crossover frequency and the settling time within 5 percent error of the command input are given as w, = w,,[(4i4 + 1)'12 - 2(2]1'2 and

+

T , is inversely proportional to the real part of the closed-loop pole. The smaller 5 is and the larger o,, is, the larger w, becomes. In addition, the value of w,,,, that gives a desirable settling time will be determined according to Equation (3-11). A typical requirement is shown in Figure 3-4. Design engineers then shape the loop transfer function to meet the desired requirements. There exists a fundamental limitation here due to the right half-plane poles and zeros. For the system to be stable, when the plant (or controller) has right half-plane zeros, the sensitivity function

dB

Roll-Off C.baracIcristiu

Robust Stability Requirement

Figure 3-4 GK(jw)

Singular Value Plots and Constraints for the Loop Transfer Function

86

Modem Multluariable Control Analysis

Chap. 3

must be greater than 1 over some frequency range. O n the other hand, for a plant having right half-plane poles, its complementary sensitivity function must be greater than 1 over some frequency range. These fundamental limitations set up the bottom line for design engineers in judging the achievable performance.

3.4 STRUCTURED SINGULAR VALUE It is a fact in engineering practice that any model used in the design process is at best an approximation tb the actual system. The difference between the physical svstem and its mathematical model is called plant uncertaintv. This inevitable modeling error will cause performance degradation and stability problems if plant uncertainties are not accounted for during the design process. For single-input singleoutput (SISO) systems, the standard frequency domain techniques such as Nyquist diagram, Bode plot, root locus, inverse Nyquist, and Nichol's chart can all be used effectively to ensure robust stability and certain desired performance characteristics. However, these techniques do not readily extend to multivariable systems in general. Singular value analysis discussed in Section 3.1 was introduced as a candidate for a multivariable robustness measure in the late 1970s. In principle, the stability margin of a multivariable system can be measured in terms of the maximum singular value of a certain transfer function matrix if the plant uncertainty is unstructured [Chen and Desoer, 1981; Doyle and Stein, 1981; Kimura, 1984; Lehtomaki, 19811. Unfortunately, using singular value analysis for testing stability robustness is usually too conservative because the actual plant uncertainties are more or less structured in some way. This motivates the definition of the structured singular value. In Doyle 119821, it is shown that any norm-bounded perturbation problem, regardless of structure, can be rewritten as a structured block diagonal perturbation problem. Consider, for example, a linear multivariablc plant G(3) with two mu]tiplicative perturbations appearing simultaneously a t the inputs and outputs as shown in Figure 3-5. Hcre, K(s) is a stabilizing controller for the nominal plant G(s). In a straightforward way Figure 3-5 can be equivalently redrawn as shown in Figure 3-6. In Figure 3-6, the combination of G(s) and K(s) inside the square box can be denoted by a nominally stable closed-loop system M(s), a transfer function matrix from [ d r d' vT] to [,I, T r12T e T 1. It is clear that the perturbations can then be rcprescntcd by a block diagonal feedback matrix, that is.

.

Note that the pcrturbed systcln in Figurc 3-6 is just a special casc of the gcncral structured block diagonal perturbation problem as shown in Figurc 3-7 with 11 = 2, A = block diag { A , , A,), 2 = [ d l , d2IT. Z = [n,, n2IT, 1,. = r, and = E . 11, Figure 3-7, M(s) has the following form:

Sec. 3.4

Structured Singular Value

I+A,(s)

G(s)

t-+l I+Az(s)

87

Figure 3-5 T w o Multiplicative Perturbations Appear Simultaneously a t the Plant Input and Output

The issue of robust stability becomes one of characterizing the set of allowable perturbations such that the closed-loop system remains stable. Note also the appearances of the sensitivity and the complementary sensitivity functions in the matrix M(s). It is well known that the systcm in Figure 3-7 is stable if and only if the det [I - ibl(s)A(s)]has roots with strictly negative real polynomial 'PIlnr(s)'PA(s) parts, where 'Picl(s)and 'PA($)are the characteristic polynomials of M ( s ) and A(s), respectively. Since M ( s ) is stable, all the roots of 9 m ( s ) have strictly negative real parts. Thus, if A(s) is also stable, the system is stable if and only if det [I - M ( j w ) A ( j o ) ] 0 for all A and o E R. A set of structured block diagonal perturbations is defined as

+

A(s) = block diag [ A ,( s ) , Az(s), . . . Ak(s) stable, E I A k ( j o ) ]5 6(w), for all k

(3- 13)

Then the structured singular value (SSV) [Doyle, 19821 is defined as

The following theorem is just a direct result of the definition (3-14) Theorem 3-1. an only if

The system in Figure 3-7 is stable for all A(s) E X ( 6 ( o ) ) if

p [ M ( j w ) ] < 1/6(w) for all u E R

(3- 15)

Note that the perturbation 6(w) and the structured singular value p [ M ( j w ) ] are frequency dependent rather than constants. From the preceding theorem it can be seen that the stability margin of the system M(s) can be defined by the structured singular value p [ M ( j o ) ] . While the p function is conceptually simple, its computation poses a challenge.

e

+

Figure 3-6 Equivalent System Diagram of the System Shown in Figure

Modem Multluarlable Control Analusis

88

a = [dl, 4- .... %?

n = [nl,

.

T

nz, .... bl

I

Chap. 3

Figure 3-7

Structured Block Diagonal Perturbation with A = Block Diag { A , , . . . , Am)

When complex perturbations are allowed, some earlier results by Doyle [I9821prove to be very useful:

where p(M) is the maximum modulus of the eigenvalue of M [Horn and Johnson, 19851. Moreover, let it = { U E Q I U*U = I ) and CT) = { D I D A = A D for A E Q), then p(M) = k(UM) = p(DMD-')

for all U E

and D E 9

It follows that max p(MU) I k(M) 5 inf ~ ( D M D - ' ) UEU

(3-1 7 )

DEm

It turns out that the left part of the preceding inequality is actually an equality, and this fact provides an approach to compute the p function involving multiextremum optimization. Under some restrictive conditions, namely the number of blocks m a 3, the right part of the inequality in Equation (3-17) also becomes an equality [Doyle, 19821. It should be noted, however, that when only real perturbations are allowed, the preceding relation no longer holds, and the other methods hare to be pursued. At present, there are several algorithms to compute structured singular values [Doyle, 1982; Fan and Tits, 19861. However, no single algorithm is univcrsally superior. The situation is worse where there are mixed (real and dynamic), repeated uncertainties. It must also be noted that the structured singular value is merely an analytical tool, offering no assistance for controller design if the robust stability test (3-15) fails. Recent progress in robust controller synthesis using a combination of the H" optimization technique and structured singular value has been reported in Doyle 11983, 1984). Although this so-called synthesis approach does not provide an exact solution for the robust perforn~anceproblem, it often yields a satisfactory (suboptimal) solution. Further research is thus necdcd in this area. Auothcr alternative measure of stability margin is called the block-limiting norm, proposed by Fan and Fu [1989], who extcndcd thc notion of a limiting norm [Pokrovskii, 19791 by associating with it the block structure that forms thc basis of the structurcd singular value. It is shown that the block-limiting norm is no ICSS than the structured singular value and no grcatcr than the infimum of the scalcd maximum singular value. More interestingly, where thcrc are no morc than thrcc blocks, the block-limiting norm is cqual to thc structurcd singular value. Unfortunatcly, a computational mcthod for the block-limiting norm is also unavailable to date.

See. 3.5

General Robustness Analysis

8Q

3.5 GENERAL ROBUSTNESS ANALYSIS As described in Section 3.4, the general framework of robustness analysis can be represented in Figure 3-7. Any system may be rearranged to fit in this general framework, although the interconnection structure can become quite complicated for complex systems. The uncertainty considered here may be modeled in two ways, either as external inputs or as perturbations to the nominal model. The performance of a system is measured in terms of the behavior of the outputs or errors. Robustness analysis is heavily related to the assumptions made concerning the input signal, the uncertainty, and the performance requirement. Assume that the transfer matrix from I, to can be represented in the linear fractional map

where M , , are real, rational, proper, stable transfer functions. This assumption can be made without loss of generality because nonlinear or nonrational parts can be regarded as uncertainties, and the system has to be stable before robustness analysis is carried out. Depending on the assumptions made on the input, uncertainty, and performance, diffcrent robustness measures are used as shown in Table 3-1, which is similar to Table 1 of Doyle [1983]. Thus, robustness analysis boils down to the computation of a certain "norm" of a particular transfer function. In some cases, this transfer function is a combination of the sensitivity and the complementary sensitivity functions. Also, the weighting functions used in design process are absorbed into the interconnection structure. The definitions of each "norm" follow.

where u i is the ith singular value, E is the maximum singular value, and p,(M), defined in Equation (3-14), is a measure of the amount of structured disturbance that can be tolerated without destabilizing the system. T o conclude this subsection, two important theorems from Table 3-1 are stated. Let

BA = -

{A E Q

1 ll A 112 5 1 )

This leads to the following theorems.

Theorem R S (Robust Stability). F,,(M, A) is stable for all A E only if II M22 1Iw = SUP ~ [ M 2 2 ( j w )< I 1. 0

a if and

Modern Multivarlable Control Analysk

90

Chap. 3

TABLE 3-1 VARIOUS ROBUSTNESS MEASURES

w

Perturbatiqn

P e r f o m

YIc(t) I ~ ) =I I b(t-II)

A =0

weT@)e(t)l < 1

1 2z 11% .15 ( i = 1, 2, 3) at the frequency 7.326. Note that the margin of stability occurs at the reciprocal of the peak. Using the Monte Carlo algorithm and one called the penalty function technique, Figure 3-8f lists two possible nearminimum values for the joint variation in A. Figure 3-8g shows some bounds for CL. Curve n is a plot of the minimum singular value of M, while curve b is a plot

100

Modem Multluariable Controf Analysis

Chap. 3

of the spectral norm of DMD-' using a balancing algorithm in Garbow et al. [1977]. A balancing algorithm is discussed in Osborne [1960]. An algorithm due t o Fran and Tits [I9861 was used to generate curve c, and curve d is a plot of the spectra] radius of M. Recall that the svectral radius is the maximum absolute value over the eigenvalues of M. Curves a an2 d can be computed using standard routines in arbo ow et al. [1977].

3.7 MONTE CARLO ANALYSIS

The Monte Carlo method [Hammersley and Handscomb, 19641 is the most general method available for estimating the performance of noise-driven time-varying systems, being applicable to linear as well as nonlinear systems. The method is founded on direct and repeated simulation,'involving a large number of random error factors. Because of this, much of the computational work revolves around looking a t a sufficient number of test cases plus postprocessing of the resultant data in order to determine the average system behavior. If there is a minor drawback to this method, it is the often large number of simulations needed to achieve an adequate level of confidence in the accuracy of the results. For this reason, it is primarily used as an evaluation tool. Owing to its wide range of applicability and the ease with which it can be used, however, it remains perhaps the most widely used tool for performing nonlinear statistical analysis. In the case of systems that are driven by white-noise disturbances, these disturbances are approximated by Gaussian random number generators. The standard deviation of these generators, which are used to approximate white noisc, relate to thc spccial density of the white noise being simulated by:

where R k is the covariancc of the Gaussian N(0, Rk)noise gencrator, @ is the power spectral dcnsity of the white noise in Hertz, and A t is the interval of simulation integration (time spacing bctwccn random nunlbcr calls). Equation (3-23) allows for the approximation of white noise since elements in the rcal covariance matrix Rb(t) take on infinite valucs. If the bandwidth of thc noise is much greater than the systcm bandwidth, then Equation (3-23) is accurate to thc cxtcnt that the noisc appcars white to thc systcm. Bccausc thc integration intcrval A t is usually taken much sn~allerthan the sn~allcstsystcm time constant, the noisc in fact looks white as far as thc systcnl is concerned. Statistical cstialation by the Montc Carlo tccl~niquc begins by computing thc standard dcviation of the rcsults according to

whcre y (1,) is the mcan valuc of final rcsults 2nd Aidenotes thc numbcr of simulations in the Montc Carlo proccdurc. Bccausc only a finite numbcr of simulation runs arc performed, thc standard dcviation dcrived in this mcthod is only approximatc and

Sec 3.7

Monte Carlo AnalysLs

NORMALIZED CONFIDENCE INTERVAL

I

I

I

I

I

I

0

50

100

150

200

250

NUMBER OF SAMPLES

X.200 RUN MONTE CARL0 SAMPLE SIZE

Figure 3-9 (a) Theoretical Confidence Intervals for Gaussian Distributed Random Variable (b) Acceleration Limit Study Utilizing Monte Carlo Approach (From [Zarchan, 19881 with permission from AGARD)

102

Modem Multluariable Control Analysls

Chap. 3

must itself be treated in a statistical fashion. This gives rise to expressing confidence in a particular estimate of the standard deviation. The confidence level rises as the number of simulations is made larger. As an example, for the case in which these statistics are Gaussian distributed, confidence intervals can be calculated like those shown in Figure 3-9a. In the case of 50 simulations, the confidence level can be shown to be around 95 percent, that is, the standard deviation is found to be between 0.85 u and 1.28 a. The uncertainty is reduced even more by increasing the number of simulations to 200, a sample size that would give rise to a 95 percent confidence that the standard deviation in fact lies between .91 u and 1.12 u [Zarchan, 19881. For this reason, confidence levels are usually presented in conjunction with Monte Carlo data.

Example 3-1. Following Equation (3-24), the miss-distance standard deviation is calculated from:

where jj(tf) is the miss distance from the ith run. Following Example 2-2, a 200run Monte Carlo simulation was performed of the homing loop of Figure 2-l5a(ii) and 2-15b, corresponding to the nonlinear mode. In the study conducted by Zarchan [1988], one of the parameters was the missile acceleration limit. The results of the study are displayed in Figure 3-9b, which includes the 95 percent confidencc intervals. This study highhghts the effect of the acceleration limit on miss distance. It can be seen that results approach those of the linear analysis only in the case of large acceleration limit. It is interesting to note that, even with the large number of simulation runs, the 95 percent confidence limits point out the high degree of uncertainty in the answers.

3.8 COVARIANCE AFWYSlS It is important to be able to calculate the vchicle responsc to random process modcls such as atmospheric turbulence. Calculating the response of dynamic systems to random processcs can be achievcd through the use ofthe covariance matrix equation, which is applicable to both deterministic and random excitation functions. Using the computer-implemented method of covariance analysis, only one computcr run is required to do an exact analysis of linear and nonlinear, time-varying noise-driven systems. Uy directly intcgrating a linear or nonlinear matrix differential equation, the covariance matrix of thc system state vector is propagated in time to generate exact time-dependent statistical performance projections of any state. Extensive use

Sec. 3.8

Covariance Analysls

103

of this nlcthod is found with problems involving optimal cstimators, as is shown in Chapter 4. The linear differential equations rcprcsenting the vehicle's equations of motion are first written as linear timc-invariant statc equations. The statc equation includes a term representing the random process input function. The statc vector is then augmented to include the components of the original statc vectors, the original systcnl input, plus any required auxiliary variables, that is, thc auxiliary disturbance statc variables, as shown in Equation (3) ofTable 2-1 where, for simplicity, D and E are assunled zero from this point on. and where w is a white-noise vector having PSI1 matrix Q ( t ) which gives the 1'SD function of the white-noise process a constant value of ( l I 2 n ) Q over all possible values of the circular frequency w. For example, the properties of the turbulence field are usually specified as PSD functions making the preceding representation suitable for aeronautical applications. Equation (3) of Table 2-1 can be used to analyze whole-body motion, flexible motions, and control system dynamics. Handling unsteady aerodynamic effects can be achieved through a quasi-steady assumption, through the use of approximate lift-growth functions, or by the numerical matching of unsteady aerodynamic matrices. Bias and drift, which are both types of nonwhite-noise inputs, can be handled by white noise passing through an appropriate shaping filter. It is also interesting to take note of the covariance matrix of the augmented state and output as follows: P(t) = E[(x(t) - x)(x(t) - x ) ~ ] ,

x = E[x(t)]

(3-26a)

~,(~)=E[(Y(~)-Y)(Y(~)-Y Y =) E~ []Y, ( t ) l (3-26b) For the augmented state, the covariance takes on the following matrix form:

u

u:.>

... ... ...

1

u;,,,

u;,

u?.~~,~

is used for the variances of the individual states .u,. is used to represent the covariances between different states xi and xj.

T o take it one step further, the diagonal elements of P(t) denote state variable variations when the expectations of the disturbance responses are zero. Furthermore, the off-diagonal elements denote the degree of correlation between the state variables. In the proceeding discussion, both the system states x and the forcing functions w are vectors, the elements of which are random variables. For simplicity, the deterministic inputs u are neglected here. There are two situations (continuous and discrete cases) to be analyzed. In the first situation, the state vector obeys the linear relationship Equation (3) of Table 2-1 without the input term Bu. The covariance matrix is defined as in Equation (3-26a) and the covariance matrix P satisfies

This equation is used to study the parameters that characterize the state during the vehicle's flight. Since Equation (3-27) can be integrated, the statistical properties

Modem Multluarlable Control Analysis

104

Chap. 3

of P(t) can be analyzed directly in only one computer run. The solution of Equation (3-27) gives the augmented state covariance matrix, which can be written as

Stability of the augmented dynamic matrix A guarantees that P(W) = 0.SO that, if PAr + NQNr. Using Equation (3-26a), A is stable, at steady state 0 = AP the covariance matrix Equation (3-26b) is given by P, = CPCr. In the discretestate model given by Equation (5) of Table 2-1, assuming 4' ' = 0, the equation for projecting the error covariance is given by

+

Furthermore, for a continuous nonlinear dynamic system described by Equation (1) of Table 2-1, the nonlinear variance equation is approximated by ~ ( t= ) A(x(t), t)P(t)

+ P(t)AT(x(t), t ) + N(t)Q(t)Nr(t)

(3-30)

where A(x(t), t) is the matrix with its ijth element defined by Aci(x(t),t) = [ a f , ( x ( t ) , t)]l[dxj]. In the nonlinear discrete case, similar results can be obtained.

Example 3-2.

This example, following Zarchan [1979(a)] and Example 2-2, again uses the linear homing guidance loop of Figure 2-15a(ii) and 2-15b to illustrate here the usefulness of covariance analysis. The system equation expressed in matrix form is

The state covariance P(t) is obtained by intcgrating the covariance propagation equation, Equation (3-27), with N equal to an identy matrix. With respect to Equation (3-27), the preceding equations define A ( t ) , and can be uscd to determine Q ( t ) as follows:

Fi ure 3-10a plots the RMS relative separation between the missile and target, as a function of time, which war found by intcgrating Equation (3-27). This value becomes the RMS miss distance at if, or the end of flight. That is, RMS

h,

TlME lrecl

a)

corariance Ana1y.i~ Provides RI Trajectory Profile TlME lucl

b)

cwariuc.

Malyml.

Provldoa

mm Accoloretion Protile Figure 3-10 Covariance Analysis Provides RMS Trajectory Profile (From [Zarthan, 1979(a)], 0 1979 AIAA)

Modem MultIuarlable Control Analysis

106

Chap. 3

miss distance = v ~ ( 3 3) , (,,,,. Because covariance analysis is capable of generating statistical information corresponding to each state, like the RMS acceleration of Figures 3-lob, it is possible with this technique to establish the correctness of assumptions concerning system linearity. That is, such an assumption would be validated if there were no acceleration saturation.

3.9 ADJOINT METHOD The method of adjoints [Besner and Shinar, 1979; Derusso et al., 1970; EASAM, 1969; Howe, 1965; Laning and Battin, 1956; Peterson, 1961; Zarchan, 1979(a)] represents a widely used technique in guidance system design. This adjoint technique is founded on the system impulse response, and is capable of analyzing exactly linear, time-varying noise-driven systems such as a guidance loop in one computer run. The exact performance projections of any quantity at any instant can be estimated using this method. Additionally, the information that measures the contribution of all disturbances (inputs) to the total performance projection (output) can be extracted. Although it has been used mainly in the past in missile guidance system design and analysis, the adjoint technique can certainly be applied to many other problems.

3.9.1 A4joint Philosophy The impulse response of the adjoint system I* is related to the impulse response of the original system 1 in the following manner:

I*(tf - ti, tf - to) = I(to, ti)

(3-32)

where t , and to are the impulse application and observation times of the original system, respectively. If it is desired to observe the impulse response of the original system at the final time tf after several impulses are applied at times t, ( i = 1, . . . , n), then the system response must be simulated for each of the impulse application times to generate [(I/, t,), as demonstrated in Figure 3-lla. For the adjoint system, however, if the observation time is the final time (to = t,), then only one adjoint response must be generatcd. Equation (3-32) reduces to I*(tf - I,, 0) = I((/, 1,). The impulse response of the adjoint system, except for being generatcd backwards, is identical in evcry aspect to that of the original system. Figures 3-1 la and 3-llb illustrate the way in which the two responscs are relatcd [Zarchan, 1979(a)]. The system rcsponse y(t) at time t to any input u is given by

T o dctcrminc how the method of adjoints becomes handy in this situation, Equation (3-32) is substituted into Equation (3-33) to obtain [Pctcrson, 19611

Aaolnt Method

~ c 3c.9

I

Uto, t , )

crvation Time 0

-------Impulse Application Time.t i

tO=ti

Figure 3-11 (a) Generation of I(tf, ti) in Original System (b) Impulse Responses of Original and Adjoint Systems are Related (From [Zarchan, 1979(a)], 6 1979 AIAA)

using y(t) =

ul

t

0

I*(tf

- T . tf -

Z = t - T

t)

U(T)

di

3

y(t)

=lI*(tf-t+z,y-t)u(r-z)dz

(3-34)

108

Modem Multiuariable Control Analysis

Chap. 3

If the final time t k is the time at which the response is desired, then Equation (3-34) becomes

The concept of adjoint system can be extended to account for stochastic inputs. A linear time-varying system that is driven by white noise has a mean square response [Zarchan, 1979(a)]

where 4, is the spectral density of the white-noise input in units of Hertz, and is assumed double-sided and stationary. Unfortunately, the many computer runs required to generate I(tf, t i ) render the direct simulation of Equation (3-36) impractical, as mentioned previously. Substitution of Equation (3-32) into Equation (3-36) yields

For the case in which the final time is of interest ( t = tf), Equation (3-37) can be expressed as

Therefore, the integral of the square of the output of the impulsively driven adjoint system from t = O to t = tf is equixralent to the mean-square response at time t, of a linear time-varying system driven by white noise. The RMS value of the output of the terminal time is the square root of Equation (3-38). The RMS value of the output of the original system in the presence of a white-noise input is then computed by first squaring, then integrating, and finally taking the square root of the adjoint system's impulse response.

3.9.2 Applications

From any linear system, an adjoint system can be constructed by the following stcps [Besncr and Shinar, 1979; Pctcrson, 1961; Zarchan, 1979(a)]:

1. Substitute tf - t for t in the arguments of all variable coefficients, that is, gains. 2. Reverse all signal flow, redefining branch points as summing junctions and vice versa. 3. The input to the adjoint system is a unit i~npulse6(t), whose application point corresponds to the output of interest in the original system.

Sec. 3.9

AdJolnt Method

10g

Using the systcnl adjoint together with the outputs described previously, it is possible to plot llMS miss at interception versus range a t the start o f hotning for each computer run. Because the RMS miss caused by each of the input quantities can likewise be obtained, it is also possible to isolate which factors have the greatest influence on RMS miss. Using the adjoint system technique is therefore much less time consuming than using Monte Carlo techniques and applying noise generators to the r c g ~ ~ l analog ar circuit. By going to acceleration adjoints, one can also study the RMS lateral acceleration requirements for the missile. A balanced design can then be achieved by comparing miss and acceleration statistics. It is easily seen that the adjoint technique is well suited to the problem of statistical atlalysis and optimization of linear time-varying systems. Using this technique, quick estimates of the required gains, natural frequencies and damping, acceleration limits, and ailowable radomc tolerances can be obtained for several initial geometry and other input conditions. As a preliminary design tool, the method of adjoints can significantly lower the amount of work involved in the ensuing analysis of a refined nonlinear system model [Peterson, 19611. The adjoint solution resulting from only one simulation Example 3-3. supplies a tremendous amount of information concerning system performance and behavior as shown in this esa~llplefrom Zarchan [1979(a)]. What has been presented can now be used to study thc homing loop shown in Figures 2-15a(iii) and 2-15b of Examplc 2-2. In the deterministic input, the homing loop has a step target maneuver xvhich has been converted to an impulsive input through integration. The ) to the step target maneuver. output to be considered is the miss distance y ~ ( t f due Figure 3 - 1 3 shows an adjoint model of the homing guidance loop of Figures 213a(ii) and 2-15b, constructed according to the rules laid down previously. The idea is to apply an in~pulseto the adjoint system at the location X4, corresponding to xvhich the output ofinterest in the original system is denoted by y,. In the actual simulation, however, an initial condition of unity is specified at X4 as opposed to a unit impulse o n the derivative. In the process of constructing the adjoint model, the three inputs o f the original system, target maneuver, glint noise, and fading noise, all become outputs. That is, the outputs in the adjoint system are now miss sensitivities due to target maneuver (y=(tf) for deterministic, y.ar,(tf) for stochastic), miss sensitivity due to glint noise y,, and miss sensitivity due to fading noise yf(tf).

Stochastic inputs. Because there is no dependence of the sensitivity coefficients of the adjoint system on the spectral density levels of the error sources, changes in the latter do not necessitate additional simulation runs. The total RMS miss distance, [E(yz(tf))]1'2 applying the principle of superposition, is given by

I Tgt Mvr + u:,(tf) I Glint + uf,(tf) I RIN + u;,(tf) I Range Dep.)Ii2

uy,(tf) = {u;,(tf)

yT(Q Step Target Maneuve~ Miss Distance Sensitivity

4 GLINT NOISE

4r ('t) FAD. NOISE

=m

M

= =

11s 1 I (1)

~2 AT,

2

q1( I f ) + FAD. NOISE

1' I +

9,( 1 0

GLINT NOISE

-,/+ =~ ~ Y L ? + ~ E) IIY($)I ~

+

1

I'

YI: (141

ADJOINT TIME (wcl (8)

ADJOINT TlME l u )

(C)

Figure 3-12 (a) Adjoint Model ofLinear Homing Loop (b) RMS Miss Distance Error Budgct Is Automatically Generated hy Adjoint Method (c) Adjoint IJrovides information on Performance Sensitivity l>ue to Target Mat~cuver(From /Zarchan, 197Y(a)l. Q 1979 AIAA; and /Zarrhat~,19881 ulirh prJnnissiot#.jam ACARDJ

Sec. 3.10

Statistical Linearization

111

Figurc 3-12b plots the individual RMS contributions to miss distance together with the total RMS miss distancc as functions of adjoint time. Hence, adjoint time implies cithcr timc of flight or time to go at which disturbances occur. The figure shows that the biggest contributor to miss distancc in this homing guidance loop is glint noise.

Deterministic inputs. It is, howcvcr, neither a difficult nor expensive task to compute othcr disturbance sensitivities as well. For example, thc miss distance scnsitivity due to a step in target acceleration y T(tf) was also calculated and plotted against timc in Figure 3-12c, which suggests an optimal time of 0.6 sec for the targct to maneuver in ordcr to rnaximizc miss distance before intercept. The figure also indicates that miss distance will bc small if the target maneuvcrs too soon, that is, if the adjoint timc is too large. As the adjoint time approaches infinity, y T(tf) will always approach zero if the guidance system is well designed. The actual adjoint timc required for the miss-distance scnsitivity curve to settle down is a function of the overall guidance system time constant.

3.10 STATISTlCAL LINEARIZATION When it is desired to treat nonlinearities in a stochastic guidance system, that is, acceleration saturation, Monte Carlo techniques must be employed to compute mean-square miss distance. The problem lies with the large number of trials that must be run to achieve even moderate results. Most often, system nonlinearities are dealt with through the construction of a linear model. Using suitable conditions (for example, assuming the actual trajectory of the vehicle is close to its nominal trajectory), a nonlinear function can be expanded in a Taylor series about some operating point (nominal state vector) of the nominal trajectory and only the firstorder or linear terms kept. This approximates a nonlinear function for small perturbations about the operating point and can be used to obtain small signal linearization. When an input is no longer accurately represented by a particular linearization, the input is relinearized about a new operating point. The entire trajectory can be segmented so that the small linearization approximation is valid over each segment to reduce the analysis errors. The perturbations of the system states around the nominal values can be studied using the covariance method. If changes in the variable N G C loop parameters are very slow in comparison with the rate of the associated processes, then the errors introduced by treating the time-dependent coefficients in the linearized equation as constants will be correspondingly small. Moreover, significant analysis errors are not introduced by the process of linearizing the N G C loop equations if: (1) the quantity in question is deterministic; or (2) if the 3u value of the random signal at the input to each of the elements does not exceed the

Modem Multiuarlable Control Analysis

112

Chap. 3

linear region of the element; or (3) the linearized function is differentiated. An even more subtle restriction is posed by the fact that the linearization procedure involves the system state vector itself. Thus, any results obtained will require a lot of computation. These types of restrictions therefore lead to Monte Carlo approaches. However, discontinuities in the system (for example, jamming and sudden target guidance) such as a limiter and two-level switches restrict maneuver in interce~t" these techniques and render linearization in the ordinary sense impossible [Gelb and Warren, 19731. When Monte Carlo techniques are required in the linearization procedure, or when the conditions for ordinary linearization are lacking, statistical linearization may be used. A statistical linearization that approximates a nonlinear operation, but depends on some properties of the input signal, is often referred to as a quasilinearization. This technique can produce different linear approximations to the same nonlinear function applied to different input signal forms. Quasi-linear approximation, unlike linear approximation, can be used for any range of signal magnitudes, and is dependent on the input signal amplitude [Gelb and Vander Velde, 19681. What has emerged as an immensely useful tool for the analysis of nonlinear systems having random inputs is the method of statistical linearization. When statistical linearization is used to analyze nonlinear systems with random inputs, the nonlinear element is replaced by an equivalent gain which depends on the form of the input signal. A method for determining the equivalent gain was developed by Booton [I9531 and Gelb (19741. Consider an approximation of the nonlinear function f(s) by a linear function, in the sense suggested by Figure 3-13, using statistical linearization. T h e input x in the nonlinear system can be decomposed into a mean component m plus a zero-mean illdependent random process I., giving x = PI + r. For this systcm, it is desired to replace the nonlinear element by an equivalent gain

.

Figure 3-13 Statistical Linearizatiori Approxinlation (Frorn /Grlh, 1Y80/ufirlr pcrrrriuiorl ,fnorn Tltc Arrillyti( Sricriic~ Corp. )

K,,. The error signal e(t) is defined as the difference bctwcen y = f ( x ) and the equivalent gain output, and has a mean-square value defined by

Using standard calculus, the minimum value of E[e2] can be found by differentiating this quantity with respect to K,, and setting the derivative equal to zero. When this is done, the result is

It is assumed that the signal r(t) is a zero-mean Gaussian random process whose u) where u is the RMS value probability density function is p(r) = e-'2'('"2) /(6 of r(t). The equivalent gains can therefore be written as

K,, =

1 d3

dr,

-7 utn

K, =

,.f(,,,

1 d 2 7 u~

+ r)e-r21(zo2)

dr

Now, when it is known that the signal x itself is a zero-mean random Gaussian process, m is zero, that is, x = r and the equivalent gain reduces to

In Gelb and Vander Velde [I9681 input-sensitive gains of the preceding type which approximate the transfer characteristics of the nonlinearity are called random input describing functions. These functions are tabulated for several important nonlinearities.

Example 3-4. This example, following the work of Zarchan [1979(a)] and Example 2-2, centers around the limiter shown in Figure 3-14a. The describing function given by Equation (3-43) becomes xe - ~

lim +

Lim

2 / ( 2 ~ 3

dx

m

xe -s1(2u:)

dx

+

-* 1

lim

J-lim

x2e-x21(2u:) dx (3-44)

Modem Multlvariable Control Analysls

Chap. 3

LINEAR (INFINITE LIMIT) PERFORMANCE 0

h

lb ;O do 50 MISSILE ACCELERATION LIMIT (g'sl (C)

Figure 3-14 (a) Input-Output Characteristics of a Limiter (b) Randoni Input Describing Function Approximation to Limiter (c) Acceleration Limit Study Utilizing CADET Approach (d) SLAM Model of Nonlinear Homing Loop (e) Describing Functions for Various Acceleration Limits vs. Forward Time (0 Acceleration Limit Study Utilizing SLAM Approach (g) SLAM Provides Information on Performance Sensitivity Due t o Target Maneuver (From [ Z a r than, 1979fa)). 0 1979 AIAA; and /Zarchan, 19881 u~irhprnnirsion.fiom ACARD)

60

REVERSED GAIN FROM CADET PORTION OF PROORAM

RMS

=

FAD. NOISE

A ~ Y

FORWARD TIME lkal

MISSILE ACCELERATION LIMIT Ig'sl

(E)

ADJOINT TIME lwcl

('3

Modem Multluarhble Control Analysis

116

Chap. 3

which can be evaluated to give 1

lim -x2/(20f)

dx

Finally, rewriting this last equation in terms of the probability integral yields as the describing function -.

Abramowitz and Stegun [I9641 give the following approximation to the preceding integral, which can also be found by table lookup, that is accurate to five decimal places

K,, = 1 - (21fi)e-""'~'(~":)

(0.43618360 - 0. 1201676w2

+ 0.937298w3) (3-47)

where

Figure 3-14b plots the describing function for the limiter against values of lim/u,, the ratio of the limit to the RMS value ofthe input signal. As expected, the describing function is determined by these two values only.

3.10.1 Techniques and Tools for Statistical Linearization Recent techniques involving statistical linearization of nonlinear system elements with covariance analysis allow for the direct statistical analysis of nonlinear systems and have been used on nonlinear missile guidance systems [Pricc and Warren, 19731. An important tool for analyzing and evaluating the statistical behavior of nonlinear stochastic systems is an approximate computer-implemented technique known as the covariance analysis describing function technique (CADET). It involvcs the direct statistical linearization of nonlinear systems, combining in the process clcments of covariance analysis with those of random input describing function analysis [Booton, 1953; Gelb and Vandcr Velde, 19681 to yield statistical pcrformancc projections in one cornputcr run. Thc dcrivation of the describing functions makcs usc of a Gaussian assumption, which may at first glance seem unduly restrictive. However, because lincar elements outnunlbcr their nonlinear countcrparts in the majority of dynamical systems, this turns out not to be the case. Morcover, nonlinear outputs that are not Gaussian in nature are easily converted to nearly Gaussian inputs through the use of low-pass filtering. What has been shown is that the large nurnbcr of Monte Carlo runs can be replaced by one computer run with this ncw method, and that the mean-square miss distance can be computed about as accurately. For 111any

types of nonlinear systems the CADET method [Gelb and warren, 19731 can often be used as a less expensive alternative to the Monte Carlo approach in order to obtain approximate performance projections. CADET has proved itself to be a useful and efficient tool in the preliminary evaluation of nonlinear missile guidance system performance. T o apply CADET to guidance systems, the following principal steps must be implemented. 1. Substitute the appropriate random input describing function gain for each nonlinear element in the original system, assuming a Gaussian probability density function for the input to the nonlinearity. 2. With the linear system model that results from the preceding substitution, use conventional covariance analysis techniques (Section 3.8) to propagate the statistics of the system state vector. Note that the describing function gains are themselves functions of these statistics. 3. Use the elements of the system covariance matrix to calculate the mean-square output at t,, for example, mean-square miss distance at the intercept time.

Example 3-5.

This example follows the work of Zarchan [1979(a)] and Example 2-2. It illustrates how CADET can be applied to the system of Figure 2lja(ii) and 2-15b when the latter is operating in the nonlinear mode with saturation effects. Following the previous discussion, the first step is to replace the acceleration saturation nonlinearity by a random input describing function Kl,,. The linearized system equation is then given by

118

Modem Multluariable Control Analysls

Chap. 3

At this point, P(t) is found by integrating Equation (3-27), in which A is obtained from the preceding linearized system equation and Q is still given by Equation (331). The previously derived describing function gain for the limiter is a function of the statistics of the unlimited commanded acceleration u, and the limit level nli,, and can be computed from Equation (3-47). T o compute the RMS level of input signal to the nonlinearity u:,, u, is first expressed as a function of the states. The resulting mean-square value can be written as 02, = (AV,)'P(5, 5). Example 2-2 provides the input values that were used to perform C A D E T analysis of the homing guidance loop. As in the case of the previous Monte Carlo analysis, the parameter that was selected for the simulation runs was the missile acceleration limit MI,,,. Figure 3-14c displays the results of this study, which were generated from six C A D E T runs. This can be compared with the 200 runs used to generate the Monte Carlo results, which are superimposed on the results of this study in Figure 3-14c. T h e results shown in the figure point to the high degree of accuracy that can be obtained using CADET. A study undertaken by Price and Warren [I9731 showed that CADET was capable of producing results comparable in accuracy to those obtained by Monte Carlo analysis involving several hundred simulation runs.

3.10.2 Statistical Linearization with A4joint Method

Statistical linearization has also been used in connection with adjoint techniques to produce successful results. The combination of the two methods allows for the assessment of primary factors in the overall measure of performance as well as the ability to cvaluatc the stability of the guidance system. SLAM [Zarchan, 1979(a)] in particular, is one of the approaches that have combined statistical linearization with an adjoint method, and is another example of an approximate, computerized technique that is available for the complete statistical analysis of nonlinear noisedriven systems. It basically combines CADET with an adjoint tcchnique, the result of which is capable of generating accurate statistical performance projections and producing an approximate error budget that quantifies the influence of each distrubance on the total system performance. In addition, SLAM has been successfully applied to preliminary analyses of guidance system performance. The following discussions on SLAM are based on [Zarchan, 19881. The principal steps to be followed for using SLAM are: 1. This stcp is identical to stcp 1 of the C A D E T method. 2. This step is identical to stcp 2 of thc C A D E T mcthod. 3. Store thc resulting dcscribing function gains for cach nonlinearity as a function of time.

Sec 3.10

Statistical Linearization

llD

4. Convert the linearized system modcl to an adjoint model in the following way: (a) substitute tr - t for t i n the arguments of all variable coefficients including the described function gains; and (b) reverse signal flow, causing the original systcm inputs to become adjoint system outputs. 5. Propagate the systcm in adjoint time. Steps 4 and 5 together replace the linearized system model of the original system by its adjoint modcl. As pointed out previously, the RMS miss distance derived from SLAM is identical to that derived from CADET. However, SLAM can additionally generate an approximate error budget that shows how each individual error disturbance contributes to the total RMS miss distance. The approximate nature of this error budget, which is valid only so long as the time history of the of the describing function is valid, renders suspect any estimates of miss gain K,,,,, distance for different error source input levels based on extrapolations. Such an error budget is still useful inasmuch as it serves to put in relief those error sources that have the greatest impact on the total RMS miss distance in the nonlinear system. It is also much less expensive to generate than its counterpart derived from the use of Montc Carlo or C A D E T techniques. These methods, in order to examine individual contributors to the total RMS miss distance, must rely on running simulations with one error source a t a time, the results of which are combined in some fashion to compute the total RAMSmiss distance. For nonlinear systems, this method of combining individual error sources does not always produce the same total RMS miss distance that would have resulted from running all error inputs at once. In addition to the aforementioned error budget, another product that arises from the use of statistical linearization methods are sensitivity functions. Although they cannot be used to compute miss distance, being more qualitative in nature, these functions relate system sensitivity to a step-target maneuver, and provide insight into the relative stability of the system. Finally, another meritorious feature of the SLAM concept lies in its self-checking nature. Should the adjoint and CADET portions of the program fail to yield identical RMS miss distances, then it is certain that the program is flawed in some manner, conceptually or otherwise. While it is not possible for any program to totally eliminate the possibility of undetected errors, the SLAM concept makes great strides toward this goal.

Example 3-6. Following Zarchan [1979(a)], the example here - presented . looks again at the nonlinear stochastic guidance system of Figures 2-15a(ii) and 2l j b to illustrate the usefulness of SLAM. Using the inputs of Example 2-2, the time history of the describing function Kli, is first generated from the CADET module of the program. Substituting tf - t for t as described earlier, the SLAM program then generates K*,,,, the gain of the reversed describing function, which

120

Modem Multiuarlable Control Analysls

Chap. 3

is entered into a linearized adjoint model of the original nonlinear system. This is shown in Figure 3-14d. An approximate error budget for the nonlinear system is then generated by running the adjoint module of the SLAM program. The parameter selected in this study was, as in the case of the previous examples in this chapter, the acceleration limit nli,. Figure 3-14e plots the describing function gains resulting from the C A D E T analysis as a function of forward time for different values of nl,,. It can be seen from the figure that, as long as no saturation occurs, the gains are unity, and that these decrease with increased saturation. This is, of course, as the theory predicted. Figure 3-14f plots the total RMS miss-distance error budget obtained from SLAM as a function of the missile acceleration limit. As expected, the total computed RMS miss distance corresponding to each of the acceleration limits is identical to that obtained from running the CADET program alone in Example 3-5 (see Figure 3-14c). When the system is slightly saturated, that is, ittin, = 60 g, glint noise can be seen to be the greatest contributor to the total RMS miss distance, which can also be confirmed by Figure 3-12b. This is quickly superseded by random target maneuver as saturation increases, or as nl,, decreases. Conversely, the contributions from fading and glint noise slightly decrease. This is understood by realizing that saturation acts to provide additional filtering. The sensitivity function due to a step-target maneuver is also plotted in Figure 3-149 as a function of adjoint time for various values of titi,,,. These functions represent mathematically the impulse response of the quasi-linearized system, but cannot, as mentioned before, be used to compute the miss distance due to steptarget maneuver. The sensitivity curve of Figure 3-14g corresponding to t~l,,,,= 5 , that is, the linear casc, first peaks then rapidly decreases to an asyn~ptoticvalue of zero. This situation, which is also depicted in Figure 3-12c, would be found to occur in a well-designed missile guidance system en~ployingI'NG. I n this casc. a target maneuver initiated a t a time to go of not less than ten guidance time constants should have no bearing 011 the total miss distance. Figure 3-14g demonstrates that the miss-distance sensitivity asymptotically approaches values other than zero as soon as an acceleration limit is introduced. Such a situation is typical of PNG systems whose effective navigation ratio is not sufficiently large. In the case of acceleration limits smaller than those indicated in the figure, there would be a monotonic increase in the sensitivity function, representing a system that is stable yet incapable of guiding effectively on maneuvering targets. Such information, while not of a purely quantitative nature, noncthclcss provides the guidance system designer with insight into thc details of system behavior.

3.10.3 Applications It has been shown [Zarchan, 1979(a)]that the preceding C A D E T and SLAM tncth-

ods produce the same degree of accuracy demonstrated in Monte Carlo methods applied to missile guidance system evaluation. In gcncral, though, thc missile ac-

Five to Seven Times the Acceleration Capability of the Tar et Is Needed by the Missile in Order to Intercept the Target sing PNG.

8

Two to Four Times the Acceleration Ca ability of the Tar et Is Needed b the Missile in Order to Intercept the arget Using APr$or Advanced duidance Law.

.F

Linear Analysis

I

I 10

Figure 3-15

I

I

I

I

20 30 40 50 Missile Acceleration Limit (g's)

I

60

I

70

Missile Acceleration for Target Intercept: Nonlinear System Analysis

celeration limit greatly influences the RMS miss distance. Figure 3-15 shows that, for 7-g maneuvering targets an acceleration capability five to seven times greater than that of the target is required by the missile to be able to intercept the target close to the accepted level if PNG law is used. However, the use of augmented proportional navigation (APN) guidance or modern guidance laws requires the missile to have an acceleration capability two to four times that of the target to be able to intercept the target. For further details, see Chapters 6 and 8.

3.11 QUALITATIVE COMPARISON The qualitative comparison presented in this section is based on material presented in Zarchan [1988]. The selection of a particular computerized method for performing statistical analysis is not always a straightforward matter. Depending on the system being analyzed and the type of information required, there are occasions that call

Modem Mu(tiuariab1e Control Analysis

122

Chap. 3

for the use of more than simply one method to gain a complete system understanding. Because it requires basically a random number generator and a postprocessing routine, in addition to a simulation program for the system equations, the Monte Carlo method is the most general and most easily applied o f these techniques. Inspection of a block diagram of the system is often easily translated into the system equations. When it is desired to use the adjoint technique, the modified diagram can be obtained from the original block diagram of the system, and these are also easily translated into system equations. For this reason, the adjoint technique is almost as easy to utilize as the Monte Carlo method. The same is not true of covariance analysis, which requires the system equations to first be expressed in statespace form. Covariance analysis is therefore more difficult to implement, and systems composed of both analog and digital sections must be handled very carefully. Problems that are basically linear except for the inclusion of specific nonlinearities such as saturations or dead zone lend themselves well to analysis by application of the CADET and SLAM techniques, both of which are more difficult than the Monte Carlo method t o utilize yet no more so than covariance analysis. Besides difficulty of implementation, another way in which the methods differ from one another is their C P U usage, or computer running time. The greatcr the number of differential equations needed to describe the system and the number of computer runs required to perform a statistical analysis, the higher is the cost of doing the analysis. The Monte Carlo method requires many more computer runs than any of the other methods to obtain accurate results; whereas, in the casc of linear systems, the adjoint and covariance methods only require one run because they are both exact methods. For an 11-state system, n differential cquations must be integrated for the adjoint technique compared to ti' differential equations for a covariance method. Thcrc arc no exact methods of statistical analysis for nonlinear system analysis. The accuracy obtained by using CADET and SLAM is close to that obtained from several hundred Monte Carlo runs [Gelb and Warren, 1973; Price and Warren, 1973; Zarchan, 1979(a)]. For an 11-stable system, 17 differential equations must be integrated with CADET, while ,I' + rr cquations must be integrated with SLAM. A comparison of the different statistical analysis techniques can therefore be made on the basis of cost, which is defined hcre as cost

A

number of equations x number of runs

(3-48)

Figurcs 3-16a and 3-16b comparc the costs of using these methods. The): s l ~ o \ ~ ~ that, in spite of the usually large number of computer runs required, thc Montc Carlo mcthod can be chcapcr to i~nplcmcntthan covariance analysis, CADET. or SLAM. In the previous discussions of the quantitative and qualitative aspects of thc various computerized statistical analysis mcthods, it has been dcmonstratcd for thc case of linear systems that the mcthods of adjoints and covariance analysis yield exact pcrformancc projections. In the casc of nonlinear system analysis, the Montc Carlo mcthod is easily rivalcd by the CADET and SLAM techniques.

COS NUMBER OF RUNS

COVARIANCE ANALYSIS 100 10

100 NUMBER OF STATES

a)

1000

coat Cmparlmn lor Linear Syatema

COST

NUMBER OF STATES

b)

Comt C-11-

lor Nonlinear Syatema

Figure 3-16 (a) Cost Comparison for Linear Systems (b) Cost Comparison for Nonlinear Systems (From [Zarchat~,19881 with permission from ACARD)

124

Modem Muitiuarhble Control AnalysIs

Chap. 3

3.12 OTHER PERFORMANCE ANALYSES Stochastic disturbances that greatly affect NGC performance must be identified and accounted for in the NGC system synthesis. The turbulence and gust velocities produce incremental aerodynamic forces and moments acting on the vehicle. The turbulence and gust disturbances are applied to the aerodynamic terms in the equations of motion. but not to the inertial or acceleration terms. Similarly. . . , their effects are included for the aerodynamic sensors but not for the inertial sensors. For example, a 30-ftlsec horizontal crosswind hitting a flight vehicle with a 300-ftlsec forward speed will create a 5.7-deg aerodynamic sideslip while the inertial sideslip still remains zero. Turbulence and gust are responsible for random variations in airspeed VtOt,r,in the angle of attack a, and in the sideslip angle P. The model accounts for turbulence and gust through gust components of a, P, Vt,,,l as a = ai + a,, Q = Q i + and Vt,,,l = V i+ V,, respectively, where If,, ol,, and p, are the increments of VtOtal, a, and p, respectively, due to the turbulence and gust. The dynamic model includes three channels of independently generated Dryden or von Karman model turbulence. Dynamic loads due to atmospheric turbulence, discrete gusts, and discrete control inputs must be calculated in order to analyze vehicle's flexibility effects. In the dynamic load analysis, the following data are needed: (1) externally produced structural data composed of mode shapes, lumped masses, generalized masses, and generalized stiffness; (2) vehicle geometry; (3) control law; and (4) information of flight condition, for example, velocity, altitude, Mach number, and mass distribution (center-of-gravity location). The maximum expected load varies as a function of the atmospheric turbulencc environment and the maneuver level of the vehicle. It is the sum of the steady-state load and the predicted peak incremental value. Results of a dynamic load analysis are rcpresentkd in terms of statistical quantities (for example, RMS responses), PSD response, time response of the dynamic loads (for instance, torsion, bending moment, acceleration, and both deflection and rate control surface activity), and singular value response. Recall that spectral (PSD) analysis is used to obtain vehicle responses to random disturbances. ~ h k r e s u l t of s PSD analysis can then be used to estimate fatigue damage rates, maximum expected loads, stability information, and crew compartment acceleration environmcnts. In gcncral, vehicle response is constraincd by aerodynamic and structural restrictions. Control surfacc activity is limited by actuator and hinge momcnts. As an cxamplc, the closed-loop system RMS control surface rate must be significantly below lld2 times the rate limit, sincc the boundary value (maximum and minimum valuc) is about times the RMS valuc. Generalized harmonic analysis is an cxtcnsion of thc spectral analysis in obtaining a more comprehensivc vehicle's dynamic response to atmospheric turbulence. In generalized harmonic analysis, the theoretical frequency response functions are derived from the nonlincar ,

f&,

~

-

Sec. 3.12

Other Performance Analyses

125

flexible dynamic equations of motion. Letting the gust input be a unit impulse and taking the Fourier transform of the flexible dynamic equations of motion yield the frequency response equations. The response equations are then solved algebraically to obtain complex motion frequency response functions. Summing the resulting airload and inertial load responses allows the load and stress frequency responses to be obtained. Consequently, the complex load and motion frequency responses are computed.

Modem Filtering and Estimation Techniques As the caption of Figure 4-1 suggests, this chapter is concerned with modern filtering and estimation techniques. As can be seen from the figure, the fundamental (purely classical) aspects of filtering and estimation are covered extensively in Book 1 of the series, which also includes discussions on signal reconstruction and observers. Filtering and estimation techniques can be divided into two approaches, deterministic and stochastic, the latter of which is applied to systems with noisy inputs. The deterministic approach leads to the design of the Luenberger observer, treated in Book 1 of the series. Both deterministic and stochastic methods are related to the classical elements of complementary filtering, also covered in Book 1 of the serics. The bulk of the current chapter deals with optimal estimation techniques, which make up Sections 4.1 to 4.5. These are stochastic methods including linear minimum variance estimation, nonlinear filtering, prediction and smoothing, Kalman filtcr analysis, and operational considerations. Finally, in Section 4.6, a combined complementary1Kalman filter approach is presented to close the gap between the hcuristic classical dcsign and the analytical modern estimator dcsign. Many dynamic modeling and control applications require at some time during operation estimates ofunknown variables in the equations and/or state of the system. Ultimately, the values of these quantities depend on measurements and the assumcd model structure, and on statistics related to measurements, the model, and inputs to the system. This section develops techniques designed to produce such estimates,

Chap. 4

Modem Fllterlng and Btlmatlon Techniques

Filtering & Estimation (Book 1)

Deterministic Luenberger Observer ( B o ~ k1)

Stochastic Complementary Filtering (Bopk 1)

+

Optimal Estimation (Kalman Filter) (Secs. $1-4.5)

Combined ComplementaryiKalman Filter Approach to Estimator Desig (Sec. 4.6)

4

Adaptive and Digital Applications (Books 4 & 5 )

Figure 4-1 Modern Filtering and Estirnatlon Techniques

which can be applied in either a batch or a recursive mode. In a batch mode, the estimate is determined after all of the measurements are recorded, whereas in a recursive mode, an updated estimate is obtained after each measurement is recorded. Recursive techniques take on special importance in the case of on-line applications involving digital computers. Modeling of a time-varying dynamic system for which there are uncertainties associated with inputs, outputs, or the system itself, is formulated through state vectors. Modern estimation methods make use of known relationships to compute the desired information from measured information, taking into account measurement errors. the effects of disturbances and control actions on the system, and prior knowledge of these variables for which the new information is desired. Filters designed from modern estimation methods for noisy data reduce the effects of drift with regard to the desired information. State estimates of a dynamic system are not confined solely to being estimates of the current state. In fact, there are occasions where it is desirable to obtain an estimate of the state at a specified future time. Such a problem arises in the application of intercept guidance. This and other types of similar problems are known collectively as the prediction problem, and estimation techniques can easily be modified to treat these. The smoothing problem represents yet another extension of basic estimation techniques. It is desired in this problem to determine the state at some previous time based on measurements recorded up to the current time. Both the prediction and smoothing problems are covered in subsequent sections. Kalman filtering is a method for estimating the state variables of a dynamic

Modem Nlterlng and Estimation Techniques

128

Chap. 4

system recursively from noise-contaminated measurements. The filter determines the system's present state by optimally combining theoretical estimates with measurement noise characteristics, based on a knowledge of the system model. In contrast to methods that -process measured data simultaneously and require substantial data-storage capability, such as least-squares methods, the Kalman filter processes measurements sesuentiallv and thus reauires much less data to be stored. Each new measurement results in a new state estimate. In addition to the estimate itself, a covariance matrix of estimation errors representing uncertainties in the estimate is also generated by the Kalman filter. For linear systems, the actual, measured data are not required to generate the covariance matrix, so that the latter can be calculated in advance based o n different models or measurement arrangements. Consequently, estimation errors are known before the dynamic system is actually implemented, allowing a determination to be made of the effect of additional state variables on estimation accuracy. This in turn permits a prediction of improvements in accuracy that can be achieved by using additional sensors. The Apollo space program is generally accepted as being the first known application of the Kalman filter. Since then, Kalman filtering has proved eminently useful in such applications as the determination of spacecraft orbits, satellite tracking and navigation, digital image processing, economic forecasting, industrial processes, and nuclear power-plant instrumentations. Some particular applications of the use of Kalman filter theory include ascent guidance at Cape Kennedy and orbital determinations for Voyager 1 and Voyager 2. In this last problem, spacecraft-based optical observations were combined with earth-based radiometric observations to accurately determine the orbit during the Jupiter encounter approach phase. In the case of nuclear powcrplant instrumentation, on-line instrument failures are detected and fault-tolerant computer control systems are designed, each by making use of Kalman filter theory [Baheti, 19871. As powerful and useful as Kalman filter theory is in so many diverse areas of application, it still cannot circumvent unacceptable results arising from poor system modeling and improper conditioning of the covariance equations. Modeling is primarily concerned with the definition of state variables, selection of the coordinate system to be used, the method of linearization, and the effects of bias, disturbance, and modeling errors. Even in the same area of application, the dynamic model developed for a particular problem may no longer be suitable if changes or new restrictions are introduced into the problem. The following sections provide a revicw of approaches and advances in Kalman filtering. The reader who is also intcrcsted in a chronological account of the development of this mathematical filtering thcory and its widespread engineering acceptance should consult Schmidt [I9811

.

- -

4.1 LINEAR MINIMUM V A M C E ESTIMATION: W M A I Y FILTER The Kalman filter technique is summarized in this section and is shown in Figure 4-2.

Sec. 4.1

Linear Minimum Variance Estimation: Kalman Filter

a) Syucm Model and Discrete Kalrnan Filter w(t.1)

i(k,+

b) Linear Continuas Kalrnan Filter

, Figure 4-2 Filter

Optimal Stale Gnimalion

(a) System Model and Discrete Kalman Filter (b) Linear Continuous Kalman

4.1.1 Discrete Kalman Nlter Table 4-1 defines the well-known Kalman filter [Kalman, 19601. A historical treatment of this subject is documented [Gelb, 19741. The Kalman filter is the optimal state estimator that provides a systematic scheme for computing the estimator gain matrix Kk.A flow chart is drawn in Figure 4-2a. Table 4-2 defines the multiplerate Kalman filter.

4.1.2 Continuous Kalman Nlter In Gelb [I9741 and numerous other places the Kalman filter is derived for a linear system with a continuous measurement equation given in Equations (3) of Table 2-1 by performing a limiting process involving Atk = tk+, - tk on the discrete filter solution. The result is a natural generalization of the discrete case and only the I result will be presented here [Powers, 19781.

1. Given the state equations (3) of Table 2-1 and the initial values E[x(to)]= 20 and E[(x(to)- io)(x(to)- 2 0 ) ~=] Po. The matrix R ( t )must exist at each t. 2. Integrate the matrix differential equation using P(to) = Po as follows

-'

Modem Nltering and Estfmation Techniques

130

Chap. 4

TABLE 4-1 DISCRETE KALMAN FILTER STEP-BY-STEPPROCEDURE

Step I : Problem Formulation

Given Eq. (5) of Table 2-1, no measurement is made at time to, and k(0)' represent the estimate of x at to with known covariance

P: = E[x(O) - i(0)') (x(0) - k ( ~ ) + ) ~ ]

is used to (1)

Application of the exbectation operator to Eq. (1) produces an apriori estimate of x(1) which is denoted by %(I)'. The associated covariance P; is obtained in a similar manner. The notation [ ](k) - represents the value of [ ] at 1, before the measurement z(k) is performed, while I ](k) + represents the value of [ ] at t, after the measurement. The latter is called the apmeriori estimate. The equations for fr(k)+ and P: are functions of %(k)-PI, and estimate of i(k)'. In z(k), and represent the linear unbiased minimum variance this recursive approach, each time a new measurement is performed, immedialely afterwards an updated estimate of x is obtained. Step 2: Time Update i Since E[w(k-l)]=O, then given %+(k-I),and fj , the expectation operation on Eq. (Sa) of k-1 Table 2- 1 gives i(k)- = Q(k-1) 8(k-1)' + Y(k-1) u(k-I) (2) Based on the definition. P; = E{[x(k) - i(k).] [x(k) - i(k)-lT) (3) The following equafon is derived: Pi = @(k-l)~;., QT(k-1) + N(k-1) Q,., liT(k-I) (4)

(LUMV)

..

Step 3: Meastrrenrerlr Update Given $,(k)- and P: ,calculate

P; = [I - K, H(k)]PL i(kf = %(k)-+ K,[z(k)

(6)

- H(k) k(ky]

(7)

Step 4: If measurements are exhausted, stop; otherwise, set k=k+l and return to step 2. -

-

-

It should be noted that in many cases, Eq. (6) is replaced - by P; = [I - K, H(k)] P i [I - K, ~ ( k ) ] '+ K, R, K: (8) because of its superior numerical properties. That is, Eq. (6) may result in bad subtraction, whereas Eq. (8) involves the sum of two positive definite matrices (which will keep P: positive definite).

-

Linear MInlmwn Varknce Estlmatlom Kalman Fflter

Sec 4.1 TABLE 4-2

Dl

MULTIPLE-RATE MLMAN rlLTER

S~epI : Problem Formulation Given Eq.(5) of Table 2-1,the problem formulation is the same as that in step 1 of Table 4-1. Let the measurement vector be defined as zf = fast measurement available every Tf sec. = q = slow measurement available every T sec. L

J

and zfis available 1times as often as q, i.e.. T, = Tf. Time is defined by the index (k, i), where k refers to the basic time period T, and i refers to the short time period Tf. At time index (k, 0) both q and z, are available giving the multi-rate measurement equation. Hp (k, 0) + v, (k, 0) v, (k,0) N(0, r,) & - H,x (k, 0) t v, (k, 0) v, (k, 0) N(0, r,) At the remainder of the time instants only q is available and the corresponding measurement equation is H?(k,i)tv,(k,i) i = 1 , 2 , ..., 11-1 =[ 0 vf (k, i) N(0, rf)

[-I'[=.

-

I

.=[a\

-

Step 2: Time Updare

Repeat for i = 0, 1, 2,

..., 1-1.

i(k, i+l)' = @(k, i) k(k, i) + Y(k, i) u(k, i)

Step 3: Measurement Update For i = 0. P(k, 0)'

= k(k,

For i = 1, 2,

.

0)-+ Kf[zAk, U) - H&(k, O).]

-1,

S(k, i)' = i(k, i)'

+ &[q(k, 0) - H,k(k, O r ]

+ K,[z&k, i) - ~ $ ( k ,i)']

Step 4:

If the measurements are exhausted, stop; otherwise, set k = k + l and return to step 2. The above results can also be applied to multiple sampling rates. Following [Lewis, 19861, let the continuous system be discretized using a period T = Ti, where Tf is much less than the data sampling period T,,where T, = k Tt. No data arrive between kT,, so during these intervals only the time update (step 2) should be performed. For this update, use the system dynamics discretized with period TI. When data are received, at times kT,, the measurement update portion (step 3 but treat K, = 0) should be performed. For this update, the measurement noise covariance R' = R,T, is used.

and define rhe Kalman gain ~ ( t =) ~ ( t ) ~ ( t ) ~ ~ - l ( t )

(4- 1 b)

The state estimate is defined by integrating

i ( t ) = A(t)?(t) + B ( t ) u ( t )+ K ( t ) [ r ( t )- H ( t ) i ( t ) ] ,

$(to) = i o

(4-lc)

132

Modem Filtering and Estimation Techniques

Chap. q

The estimation is corrected by the residual term x - H2 through K. Figure 4-2b presents the linear continuous Kalman filter. The Bu(t) term includes all the known inputs to the model dynamics so that the filter can use this information to improve its estimate. If E [ w ( t ) v ( ~ ) ~=] S(t)8(t - T) is a white-noise process, then only Equation (4-lb) is modified to give

'

K(t) = [ p ( ~ ) H ( t+) ~N(t)S(t)]R- (t)

(4- 1d)

In many cases the filter solution is studied in two phases: the transient phase and the steady-state phase, with the steady-state characteristics defined by the solution as t approaches infinity. In classical linear differential equation terminology, these correspond exactly to the homogeneous (steady-state) and particular (transient) solutions. There exists a steady-state Kalman filter gain which is the most useful for control problem applications. Supposing a t this point that the matrices A, B, N, H, Q, R, and S in Equations (4-1) are all constants, it can be shown that the limit of the covariance matrix P(t) as t approaches infinity, exists, for example, P,, and furthermore that it is defined by the algebraic Riccati equation

This last equation is nothing more than Equation (4-la) with Kalman filter gain is the constant matrix

P = 0. The resultant

and thus the steady-state estimate by Equation (4-lc) is

Given that these arc the statc equations o i thc stcady-state Kalnlan estimator, thc stability of this last differential cquation is strictly a function of the matrix A K,H. A great deal of information about the bchavior of Equation (4-4) can be obtaincd by analyzing the roots of the characteristic cquation corresponding to det[sl - (A - K,H)] = 0. A sufficient condition for the steady-state filter estimate to be asymptotically stable is that thc system is observable. Assuming Q 2 0 (positive sernidcfinite) and R > 0 (positive definite'), the system is observable if and only if the 11 x r ~ pmatrix [ H T I A ~ i (AT)'HT H ~ i ... i ( A ~ ) " - ' H ~has ] a rank 11.

Example 4-1: First-order altitude rate Kalman filter. As in Book 1 of thc scrics, thc altitudc rate h can bc infcrrcd from two mcasurcmcnts, a bar-

omctric tncasurctncnt iln and an accclcromctcr mcasurcmcnt /;:. It is worth\vhilc here to apply the stcady-statc Kal~nanfiltcr to thc samc problem. First, is trcntcd as a stochastic input with known noisc charactcristic w(t), that is, ?(I) = 1 ) = /;,(I) t u f ( t ) , whcrc ul(r) is a zero-mean whitc-noise process with variance q. Sccondly, /IB is trcatcd as a nlcasurclncnt z = = 11 + v = x v where v(r) is a

I;,

+

zcro-mean white-noise proccss with variancc r , and r is a constant. With rcgard to Equations (4-1). A([) = 0, B(r) = 0, N(r) = 1 , H(r) = 1, Q = q, and R = r. Applying the steady-state Kalman filter produces the algebraic Riccati cquation - ~ ' l+ r q = 0 whose solution is P, = d :.i The stcady-state Kalman filter gain is K I = d& and the estimate equation is h = I;:, + g& ( i l H - i). Comparing i this solution with the complementary filter solution of Book 1 of thc series, h = ;1

+ -T1 ( I.l l ,

-

jl),

shows that

T =

VG. If T E 12, 61, then, physically, the h B ( t )

measurement is about two to six times noisier than thc I;:, measurement, with respect to variance [Powers, 19781.

4.2 NONLINEAR FILTERING The linear filtering problems of the previous sections are basically solved by a single approach. In many applications, the system dynamics and the measurement models are nonlinear. Nonlinear problems in general require more problem-oriented tcchniques, along with considerably more knowledge of stochastic process theory. Fortunately, one of the most successful techniques for treating nonlinear problems developed to date is a natural extension of the linear-oriented Kalman filter known as the extended Kalman filter (EKF).

4.2.1 Extended Kalman Filter (Em) The Kalman filter is a standard filter for state estimation in linear systems. For systems with nonlinear dynamics, a natural extension is to linearize the system around the current state estimate 2(t) and apply the Kalman filter to the resulting linear, time-varying system. This is the EKF which has been described in detail in Jazwinski [1970]. This filter has also been widely used for the combined (nonlinear) problem of estimating th'e state and the parameters of a linear system. Discussions of the latter application are presented in Book 4 of the series. The EKF is presented in this section along with the procedure for treating the continuous dynamic model, discrete measurement case. The continuous nonlinear system is described by Equation (1) of Table 2-1. The main idea is to use expansions of the nonlinear functions f and h about the estimate 2 whenever linear forms are necessary in the development. There is not a great deal of difference in the structures of the covariance and gain computation equations for the Kalman filter and for the EKF, and these are summarized as follows. The initial condition is also assumed to have a normal distribution, E[x(to)] = 20, E[(x(to) - 20)(x(t0) - 20)T] = PO.


134

1. Define A[i(t), t]

=

-

H[?(t), t] =

Chap. q

(4-5a)

2. Integrate the coupled (in general) differential equations

where K[2(t), t]

= p(t)HT[2(t), t ] ~ - ' ( t )

(4-5d)

The EKF yields very nearly optimal estimates if +e linearization is accurate. The EKF involves the integration of n + [n(n + 1)/2] '(since P is symmetric) coupled differential equations in the variables 2 and P with the initial conditions given by Equations (4-5b and 4-5c). As mentioned previously, the idea behind the EKF is that of linearizing the nonlinear equations about the current state estimate 2(t), a technique which is termed relinearization. In utilizing such a procedure, the approximation remains linear because the estimated state is as often as not closer to the actual state. The structure of the algorithm for the EKF is illustrated in Figure 4-3. Because of the linearization procedure, the EKF covariance (and hence the gains) depends on the current state estimate and the measurement data. Consequently, it is not possible to compute the EKF covariance and gain matrix as a function of time off line as in the linear case of Section 4.1 since the state estimates of the EKF are no longer optimal. O n the other hand, if an a priori nominal trajectory, for example, x*(t) replaces 2(t) as the base trajectory in the expansion procedure, then Equations (4-5a to 4-5d) will involve known functions of time (for example, A[x*(t), t], H[x*(t), t ] , and so on) and the Kalman gain can be computed a priori. However, this procedure has not met with the same success as the EKF in actual applications. Note that the standard KF guarantees stability while the EKF does not.

E~s. (4-k&d) Eqs. (4-5a) Calculation of

Initial Estimation Linearization of Estimation Error Error Covariance -b Covariance p(t) and 4- Model About State Estimate P(t,) Kalman Gain K(t) State Variable Model A K(t) of Nonlinear System Measurement Data Initial State Variable Estimate

+Computation of

Figure 4-3

State Estimate Eq.(4-Sb)

*

State +Variable Estimates

Structure of an Extended Kalman Filter

Sec 4.2

Nonlinear Flltertng

135

TABLE 4-3 CONTINUOUS DYNAMICS, DISCRETE MEASUREMENTS: EXTENDED KALMAN PILTER

Given Eqs. (2a&b) of Table 2-1, and E[x(to)]= Step I: Defme

4,

Step 2: Given a(!;), Pi, integrate forward to ti+, = qa(t),u(I), ti, = ~ i ( t11)P. + PA[$(^). 1 Set k = k+ 1and go to step 3. Step 3: Given $(ti), Pi, z(k), deternine = $(ti) + K,{z(k) h[!!(t;),

8

P

&ti)

-

E[(x(to) $,)

-

1+~Q(0.

(~(1,)

- to)T] = PO.

P(!,) = P(~;)

kl)

P: = [I - KrHk[%(t;)]]P; where Go to step 2.

= Pi ~:[ft(ti)l{Hk[a(ti)lPi H:[s(~~)]+ %I-'

Note that if to < I,, then the definitions %(t:) = ko, P; = Po are made, and step 2 is implemented as the first step. If a measurement is given at to, i.e., I, = to, then the definitions $(ti) = 9,. P; = Po are made and step 3 is implemented as the first step.

Several modified versions of the EKF have been proposed and implemented to deal with nonlinearities in the model. Some of the more successful variations of the EKF are represented by the second-order filter, the iterated EKF, and the Jadaptive filter Uazwinski, 19701. The algorithms developed are capable of yielding significant benefits only in certain applications and are not of a very general nature. For this reason, the application of any of these algorithms must be carefully examined and acccomvanied bv a substantial amount of simulation studies. The EKF filter for the continuous-time nonlinear system with discrete-time nonlinear measurements (see Equations (2a & b) of Table 2-1) is presented in Table 4-3, from which the linear case of continuous dynamics, discrete measurements can be easily deduced.

4.2.2 Statistical Linearization Technique A linear approximation is desired for a vector function f(x, t) of a vector random variable x, having probability density function p(x). Following the statistical approximation technique, a statistically linearized filter is given in Table 4-4 in which it should be observed that the structu2e is similar to the EKF in Table 4-3; however, now K, and K h have replaced A and\ H k . The computational requirements of the statistically linearized filter may be greater than for EKF derived by taking Taylor series expansions of the nonlinearities because the expectations f , h[x(tk), k], and so on, must be performed over the assumed Gaussian density for x. However, the A

..


136

Chap. q

CONTINUOUS DYNAMICS, DISCRETE MEASUREMENTS: STATISTICALLY LINEARIZED FILTER

TABLE 4-4

Given Eqs. (2aBrb) of Bble 2-1, and vx(to)] = kO, E[(x(to) It is reasonable to approximate f(x,t) by the linear expression

- 9,) (~(t,)- 9,) T] = Po.

f(x, t) m a(t) + x(t) (1) where a(t) and K,(t) are a vector and a matrix to be determined. Defining the error e f(x, I) - a(t) - %(I) x(t) (2) it is desired to choose a(t) and K, such that the quantity J = ~ [We] e ~ (3) is minimized for some symmetric positive semi-definite matrix W. Substituting Eq. (2) into Eq.(3)and setting the partial derivative of J with respect to the elements of a(t) and K, to zero, gives a({) = ?(XI - yk, K, = {[fxT],,, - ?kT} P-I (4) To derive approximate minimum variance filtering algorithms, Eq. (4) is substituted into Eq. (1) yielding f(x, 1) m kx) + &(x - 8). h[x(tk). k] h[k(tk). k] + I$,(k)[~(t,) 9(ti)l

-

-

(5)

This formulation necessitates computing f, hi, &, and Kh, each of which depends on the probability density function for x, a quantity that is generally not readily available. Consequently, an approximation is needed for p(x) that permits the above quantities to be calculated. For this purpose, it is frequently assumed that x is Gaussian. Since the probability density function for a Gaussian random variable is completely defined by its mean k and its covariance matrix P, both of which are already computed in any filtering algorithm, it is possible to compute all the averages in Eqs. (5). Assuming that the right sides of Eqs. (5) are calculated by making the Gaussian approximation for x, the statistical linearization for f and h can be used for the nonlinear filter. The step by step procedure of the statistically linearized filter is summarized as follows: Srep 1: Definitions [fxT],,, ?, Et are expectations calculated assuming x -N($, P). k, k(ti) and [h xT(t;)],,, are

-

expectations calculated assuming ~ ( t k ) N [$(ti), pi].

Step 2: Given kcti),

P;, integrate forward to ti+, Et = REt(t), u(t), I], P = K$P + Q I ) , P(1,) = PO;)

PC+

= {[fxqC,,- ?kT} p-l(r) Set k = k+l and no to step 3. Step 3: Given a(&). Pi, a k ) , detlrmine %(ti) = k(tk) + Kk { ~ ( k )h[k(tk), k]} &(I)

-

-

-

= [I $K,,(k)] PC where

Kk= Pi K L ( ~() q ( k ) Pi ~ l ( k+) RI,)-'

From [Gclb. 19801 with prrmission from The Analytic Scie~lccsCorp.

Sec. 4.2

Nonlinear Mitering

W7

performance advantages offered by statistical linearization may make the additional computation worthwhile. A detailed derivation can be found in Gelb [1974].

4.2.3 Multiple Model Estimation A multiple model approach considered here is to assume two or more levels of process noise. A filter is set up for each model and, based on their likelihood functions, the probability for each model being correct is obtained [Bar-Shalom and Luh, 19861. Let Ae ( e = 1, . . . , m) bc the event that model e is correct with prior probabilities P{Ae} = pe. The likelihood function of measurements up to time k under the assumption of model Ae is Le(k) = P ( z k 1 Ae) =

n P [ v e ( i ) ]where k

zk

,=I

= { z ( l )... z ( k ) }and, under the Gaussian assumption, the probability density func-

tion of the innovation from model

e is

Using Bayesian rule, the a posteriori probability that model E is correct at time k is

PIAe I z k } = P { A ~ } P { ( zAe} ~ j=l

P{A,}P{zk I A,} = L e ( k ) p e

j= 1

(4-6) It is assumed in Equation (4-6) that the same model has been in effect from the

initial time. However, this is obviously not true if the model changes at some time during the interval [ I , k ] . In view of this, a "sliding-window" likelihood function has to be used. It is now assumed that the model changes at time k - N, so that one considers only the likelihood functions of the measurements from k - N + 1 to k. Each filter starts from the same initial condition at k - N . The likelihood function corresponding to model e is then

Le(k - N , k) = P [ z ( k - N

+ I), . . . , ~ ( kI Ae, ) ~

( -k N ) , . . . z ( l ) I

probability of model P{Ae I z k } and t at time k:

Modem N1terin.q and Estimation Techniques

138

Chap. q

The state estimate is a weighted average of the model-conditioned estimates with the preceding probabilities as weights: 2,

E{x(k)I zk} =

E { x ( k ) 1 A,, z k } P { A j 1 z k )

(4-8)

j=1

A "fading memory" approach to the computation of probabilities in a multiple model situation is now considered. The discounted (fading-memory) average norm of innovations with discount factor 0 5 a-,5 1 ( j = 1, . . . , m ) is k-e

- 1)

=

P.;(K)

+ ~ ; r ( k ) s ; l ( k ) ~ , ( k=) 2 a ~ - C ~ ~ ( t ) ~ . ; l ( e ) (4-9) ~j(e) E=O

where v j ( k ) is the innovation from the filter based on model j. The "effective" memory length is (1 - a j ) - I . Let S j ( k ) = S, be the steady-state covariance of the innovations from filter j. Then, it is possible to replace Equation (4-7) with 1

P_I ( k )

=

~ , e - ' ~ " ~P;(O) '

1 ,P 0, the estimator is stable with a pole at - kh. Therefore, if the open-loop model dynamics have a pole on the jw-axis, it must be disturbed with a process noise in order to have a stable estimator. For a zero input, u = 0, and a constant initial state x(O), at steady state, x = x(0). From the state estimate equation, 2 = x + v l h . If v = 0, then 2 = x = k(0). QR-'-, m. In this case, all measurement covariances R go to zero (that is, ideal sensors with excellent measurements) or the process covariance Q goes to infinity. As R goes to zero or as Q approaches infinity, the estimator gains increase drastically, and the bandwidth of the resulting Kalman filter increases very rapidly. When all the transmission zeros in the model dynamics are in the left half-plane, that is, minimum phase, the closed-loop poles asymptotically approach the transmission zeros, and the residues of the slow model which is created by the poles approaching zero have small values in 2 ( t ) = x ( t ) - 2(f). However, the residues of these modes in K[z(t) - Hk = s k k , r r l = All + I/c;ycsls = hls, rr = 0, and b = k. Applying Equation (4-36) yiclds This equation verifies that the problem formulation is corrcct. If thcrc cxists an acceleration bias A h in z , then z = hlk + Alak = 1;ik + sAj11k. Substituting the new z into Equation (4-36) yiclds

As mentioned previously, the acceleration bias Al;?k can be implemented in the position bias term by adding A hlk to u r as ul = hls + A hlk. Substituting thc new u l into Equation (4-36) also yields Equation (4-38). This verifies that an alternative implementation can be done.

Case 2: Rate bias, b = 1. If the rate input contains a bias error jCb, that is, u = j: + i b , then the rate bias error jCb can be removed by a position correction input u l = xh after the integrator as in the position bias case. Since the rate input u has a unit forward loop gain, u l should also have a unit forward loop gain, that is, b = 1. Example 4-10: Mrst-ordera estimator using inertial a rate at IRU. In this case, the C G normal acceleration estimate is not used in the inertial a rate computation, and the problem is formulated as follows: x = a = angle of attack, z = x + v = a~ = a1 + a~ = measured or derived air mass a, a r = inertial a, X D = ac = turbulencelgust a, and u = &I,,L, = inertial a rate at IRU. Also, aIIRU contains an error due to pitch acceleration, Q, from the inertial a rate at the actual C G location iurC,, as discussed in Book 1 of the series. The correct terms are &rcc = + dlb and & b = LxQ/LTAs One way to obtain arc, information at and then integrate the sum of the CG would be to add the error term & b to iYI, , these two terms. Unfortunately, Q, and hence & b , are not available. A better scheme and then add a pitch rate correction term u1 = a b = L,QI is to integrate &I,R,, VrAS with b = 1 after the integrator to obtain accurate arcG. Example 4-11: First-order sideslip estimator using inertial sideslip rate at IRU. The problem is formulated the same manner as in Example 4-10 as follows: x = P '= sideslip, z = x + v = PA = PI + PC = measured or derived air mass P, pr = inertial P, x~ = PC = turbulence/gust P, u = pl,,, = inertial sideslip rate at IRU, U I = P b = LxRIVTAs, and b = 1. Case 3: NO bias, b = 01. In this case, b = 0. Hence, u l = 0 and the estimate is 2 = u + k(z - 2 ) . If there exists a disturbance x D in the x measurement,


162

Chap. q

+

then z ( t ) = x ( t ) x D ( t ) and the rate input is u = i.hn the Laplace domain, Ul(s) = 0, U ( s ) = s X ( s ) , and Z(s) = X ( s ) +I X D ( S ) .Substituting the preceding result into Equation (4-36),

Example 4-12: Rate estimator. The problem is formulated such that x = x = rate, z = x + v = measured/der,ived X , u = ji = measuredlderived ul = 0, and 6 = 0. The estimate equation is k ( t ) = ~ ( t +) k ( X ( t ) - ~ ( t ) )Applying . Equation (4-36), ~ ( s )= [ X ( s ) + k ~ ( s ) ] l (+s k ) = ~ ( s )This . verifies that the formulation is correct. The first-order altitude rate estimator in Example 4-1 can be solved by this formulation with x = h , and x = ha. The estimate equation is the same as in Example 4-1.

x,

Example 4-13: First-order a estimator using inertial a rate at CG. Following Example 4-10, if the CG normal acceleration estimator is used in the inertial a rate computation, then b = 0, 1 4 1 = 0, and ir = &I,, = inertial rate at CG. Here, & tracks mainly the air mass a.4 a t low frequencies so that the complementary a rate A & = k ( a A - &) is zero a t low frequencies. Applying Equation (4-39) yields &(s) = a l ( s ) + k a c ( s ) l ( s + k ) . Thus, a~ and &I,, are operated on the filter to produce an a that contains pure c r l plus filtered &c. Example 4-lk First-order sideslip estimator using inertial side. slip rate at CG. Following Example 4-1 1, if the C G lateral acceleration cstimator is used in the inertial sidesiip rate computation, then 6 = 0, i d l = 0, and PI,, = inertial sideslip rate at CG. fi tracks mainly the air mass sideslip P A a t low frequencies so that complcn~entarysideslip rate A @ = k ( P A is zero a t low frequency. Applying Equation (4-39) yields fi(s) = Pl(s) + kPc(s)l(s + k ) . Thus P A and OI,, are operated on the filter to produce a fi that contains pure inertial sideslip P I plus filtered fit.

p)

Example 4-15: First-order velocity estimator. mulated as follows:

x = I~T.AS 2

= x

+

=

air data vclocity, x ~

=

Thc problem is for-

Vc = turbulence/gust vclocity

v = V I - V c = air data velocity measurement, V I = inertial velocity

u = V I = inertial acceleration,

ul = 0

Applying Equation (4-39) yields ~ T A S ( S = ) c l ( s ) - K V C ( S ) I ( Sf k ) . Thus, V r ~ s and 01are operated on the filter to produce a velocity estimate 6 ' ~ that ~ s contains pure V I plus filtered 6 ' ~ .

ComplementarylKalman Nlter Approach to Estimator Design

Sec. 4.6

Example 4-16: First-order altitude estimator.

la3

The problc~nis for-

mulated as follows: x = It = altitudc,

z = ItR

= x

+

u

= /I = altitude ratc via INS and FCS

v = barometric altitudc,

The resulting altitude estimator is = it + k(k important in INS and FCS altitudc tracking.

11,

=

0

- I;). The altitudc estimate is very

Example 4-17: First-order Euler attitude estimator.

The problem

is formulated as follows: x =

4

z =

IJJ,

=

attitude,

tr =

r/r

= attitude ratc transformed from body rate

rr, = 0

= INS attitude,

The resulting Euler attitude estimator is shown in Figure 4-9.

4.6.2 Second-Order ComplementarylKalman Filter Figure 4-10 shows a generalized second-order complcmentarytKalman filter. Consider the model for the second-order dynamic system. Applications for this filter can be classified into three major categories: (Formtrlntiorr 1) Position and Rate Estimators with Position and Acceleration

Measurements: Given measuredlderived position x and measuredtderived acceleration 2 , it is desired to obtain the best estimate of the position and rate. (Forrrrrrlntion 2) Position and Rate Estimators with Position and Rate Measurements: Given measuredlderived position x and measuredlderived velocity k, it is desired to obtain the best estimate of the position and rate. (Formulation 3) Position and Rate Estimators with Rate and Acceleration Measurements: Given rneasuredlderived rate k and measuredtderived acceleration x, it is desired to obtain the best estimate of the position and rate.

Second-order complementarylKalman filter: formulation 1. This formulation is used to design position and rate estimators with position and accel-

Roll Rate Pitch Rate Yaw Rate

Body to

Euler Axis Transformation

Figure 4-9

Euler Attitude Estimator

Modern Filtering and Estimation Techniques

164

Chap. 4

Kalman a l t e r

L.

Complementary Filter

High Pass

4 =Fl -nT Low Pass

n-v

z

s*

+ 2 j wn s +a);;

Low Pass

G(s)

Comparison

el + bu2

L

z = r + v = n l + b%+v

z = s+v. v = high frequency noise

JJu &dl + Jul

y = x+n, n = low frequency noise

k/kI Figure 4-10

d

-

2j!w,,

-

T

-

= complementary filter time constant

Second-Order ComplementaryiKalman Filter

eration measurements. Consider the model for the second-order dynamics system in which x l = x = position, xz = i = rate, and 14 = x = acceleration. T h e equation o f motion in state-space form is i l ( t ) = ~ ~ ( and 1 ) i , ( t ) = . ( I ) , with position x as an output, and z ( t ) = x, ( I ) . If the acceleration measurement ii contains a bias crror A x , that is, ii = x + A x , then the acceleration bias error A x can be removed by a rate correction input u l = A i after the rate integrator so that i l ( t ) = x 2 ( t ) + I I I ( ~ ) + w l ( t ) where w l is a zero-mean white process with variance q l . ril can be a rate correction equal to the rate difference between the actual rate measurement and the sensor rate measurement. The measured/derived acceleration ji is treated as a control w Z ( t ) ,wherc w z input u with known noise characteristics such that i 2 ( 1 ) = U ( I )

+

Sec. 4.6

ComplementaryIKaiman Filter Approach to Estimator Deslgn

165

is a zero-mean white process with variance q 2 . The preceding model dynamics are illustrated in Figure 4 - 1 0 which includes the optimal state estimator. A comparison of a second-order complementary filter and Kalman filter is also shown in the figure. As in the first-order filtcr formulation, the measured/dcrived position x is treated as a measurement z with bias error bu2 and a measurement noise v , and z = xl + bir? + v , where v is a zero-mean white process with variance r. With regard to Kalman filter formulation,

where K is the Kalman gain matrix, and the state estimate equations are

where i = .?

+ u l ( t ) + k [ z ( t ) - i l ( t ) - bu2(t)] (4-40) i2(t) = ~ ( t+ ) k I [ z ( t ) - . t l ( t ) - 6112(t)] = .GI + bit2 = position estimate, .t2(()= rate estimate, and i ( t ) =

.??(I)

=

il(t)

+ irl(t)

=

22(t)

calibrated rate. Applying Equation ( 4 - 1 8 ) yields

where gains k and k l have been expressed in terms of the second-order parameters. Therefore,

i(s)

=

2(s)

=

kl(s)

+ bU2(s)

+ s U ~ ( S +) b s 2 U r ( s ) + ( k s + k J z ( s ) ] l D ( ~ ) k ( s ) = k2(s) + UI(S) = [(s + k ) U ( s ) + (s2 + k s ) U l ( s ) - bkIsU2(s) + s k I Z ( s ) ] l D ( ~ )

(4-42a)

= [U(S)

(4-42b)

All the transfer functions from either the control inputs u , u l , and 242, or the measurement z to the state estimates are low-pass filters. The estimator gains k and kr


166

Chap. q

can be related to the estimator poles sl and s2 by the following formulae:

The location of the estimator poles sl and s2 are chosen based on applications and are different for different applications. The transfer functions from the measured z ( t ) t o k and 4 are simply the last column of Equation (4-41a), k ( s ) = (ks

+ kl)Z(s)lD(s),

~ ( s =) klsZ(s)lD(s)

(4-44)

The steady-state values for the elements of P, and K , are found to be

L

If ql = q2 = 0, then in the steady state the elements of the error covariance matrices all vanish, that is, P, -+ 0, and the Kalman filter gain vanishes. Since the perfect model and the filter need only to find the missing initial condition of the state, the processing being observed is nonstationary and the estimate converges on the true value of the states which is purely the double integral of the illput Case (a).

U.

Case ( b ) .

If q, = 0, qz # 0, then in the steady state k

=

~ ' Z ( ~ ~ l r ) ' / k1 ~ ,=

Using Equation (4-41 b), w,, =

(4-46)

k 6= (q2/r)114and 5 = - 0.707. Thus, the 2%

damping ratio is a constant and the undamped natural frequency is determined to be a function of only the ratio of errorlnoise in the rneasuredlderived acceleration x to the measuredlderived position x . Furthermore, w,, depends only on the relative noise between q2 and r, not on the absolute noise levels. and kl If q 2 = 0 and ql # 0, then in the steady state k = = 0. From Equations (4-40) k1 = i 2+ U , + k [ -~ 2 , - bu2] and i2= u. Since qz = 0, a perfect accelerometer model is used and the filter needs only to find the missing initial condition on the rate state, the process being observed is nonstationary, and the rate converges on the true value of the rate states that are Case (c).

Sec. 4.6

Complementaryllblman Filter Approach to Btlmator Deslgn

167

the integral of the accelerometer input. The position estimate is the first-order completnentary/Kaltria~~ filter position estitr~ate. If the Kalman gain K is assumed zero where the forcing function in the filter design is not considered, the covariance equations (4-23) are employed here with K = 0 since they hold for an arbitrary equation K-matrix.

The error covariance will grow according to Equation (4-47) cvcn if there is only a small amount of white noise of spectral dcnsity q l forcing the .ul state or a small amount of white noise of spectral dcnsity q- forcing the xz state.

Example 4-18: Navigation filtering for position and velocity estimate. The navigation filtering for position and velocity estimate presented in Chapter 7 falls cxactly into the previous category, with the following definitions: x =

X, Y, Z position; z = X, Y, Z position measurements;

i= I!

=

A, Y , z velocity X , V , z acceleration;

til

=

A X , A Y , A Z velocity corrcction; u2

=

0

Following Example 4-1 and defining x l = h = altitude and x2 = h = altitude rate, this example can then be applied to the design of an altitude and altitude rate estiftlator where z = ha.. = h + v = X I + v = Z vosition measurements. The .. acceleration bias error sail is assumed to exist so that h = I;, s h h + w = u j t r l + w , and the state equation is obtained as $1 = x? + ul and i z = u + w. Thus, this is the same problem as that in Case (a) wherein q l = 0. q, = q, and r = r. The Kalman gain is given by Equation ( 4 - 4 3 , and the state estimate eqyations are given by Equation (4-40) where i = i1 = h = altitude estimate and h = 22 = altitude rate estimate. The calibrated altitude rate estimate is h , = 22 + u l . The transfer function from z ( t ) = h R ( t ) to h^ is h^lhR = ( k s k,)lD(s) = s + V & ) l ( s 2 + V'2Vqlr s + d&)and it is a low-pass filter. In the Laplace domain, from Equation (4-42) the altitude estimate and the altitude rate estimate in terms of u = hlRS, u 1 = A h , and z = h are h^(s) = ( U ( s ) + s U l ( s ) f (o: 2<w,s)Z(s)]/ D ( s ) = [ h ( s ) s h h ( s ) + (w: + 2<w,,s)h(s)]lD(s)and = [ ( s + 2<w,,)h(s) + (s2 + ~ < w , , s ) A / L (+x )SW: h ( s ) ] / D ( s ) The . problem of estimating gear altitude and altitude rate is formulated as follows:

+

+

A

-

+

x = gear altitude,

(a

+

i(j)

+

z = gear altitude measurement,

u = inertial altitude acceleration measurement, u2 = 0 u l = inertial altitude rate correction,

Example 4-19: Position error and rate error estimator with POsition error and acceleration measurements. Trajectory errors in terms

Modem ~iiteringand Estimation Techniques

168

Chap. 4

of position error and rate error are important parameters for NGC and hence need to be estimated. The problem of estimating trajectory error is formulated as follows: x = AX = trajectory position error, z = x

u = ul

x

+v

= trajectory position error measurement

= accelerometer measurement

= A X = trajectory rate correction

Applying EAquation(4-42) yields A ~ ( S )= [ ~ ( s + ) SAX(S) + (ks + kl)AX(s)]/ D(s) and AX(^) = [(s + k ) ~ ( s + ) (s2 ks) AX(S)+ sk~AX(s)]/D(s).In estimating the glideslope deviation, the problem is formulated as AX = A h = altitude error, A X = ~h = altitude rate error, and x = h = vertical acceleration. In estimating the localizer deviation, the problem is formulated as AX = A y = lateral position error, A X = A j = lateral velocity error, and x = j = lateral acceleration.

+

Second-order complementaryIKalman filter: Formulation 2. This formulation is used to design position and rate estimators with position and rate measurements. The following are defined: x = position (same as before),

z

=

x

+v

=

measured position

u(t) = Ax(t) = acceleration bias error (if it exists) instead of x in Formulation 1

rate measurement (regarded as a control input) instead of A i in Formulation 1

ul(t) = i ( t ) =

xl(r)

=

position (same as before), x2(t) = A i = rate measurcment disturbance instead of i in Formulation 1

2 , ( t ) = position estimate (same as before), turbance estimate i(t) =

?(t) =

i(t) = i2(t)

.G2(t) =

rate measurement dis-

+ bu*(t) = position estimate (same as before)

+ u l ( f ) = rate estimate (same as before)

The state and measurcment equations are the same as those in Formulation 1 , that is, i 1 = x2 + U I + U ' , . i 2 = u + ul2, and z = x, buz o. The results of the second-order filter (Formulation 1) can all be applied here with a slight revision as follows. If there exists a disturbance X D in the x measurement, then z(t) = >-(I) + x ~ ( t )In . the Laplace domain, U(s) = 0, Ul(s) = sX(s), and Z(s) = X(s) + X D ( ~ ) . Applying Equation (4-42) yields

+

+

Sec. 4.6

CompiernentaryIKalman Niter Approech to Estimator Design

169

Thus, z ( r ) and r r , ( 1 ) arc operated on by the filter to produce: ( 1 ) a position estimate i ( t ) that consists of pure position .u(t) and a filtcrcd i I > ( t )arid ; (2) a rate estimate ; ( I ) that consists of a pure ratc i ( r ) and a filtered io(r). The preceding results assume the acceleration bias rr = A x to be zero. If it is nonzero, then a third state needs to be modeled. Therefore, a third-order complcmcntaryIKaln~anfilter such as the one presented in Section 4.6.3 is required.

Example 4-20: Second-order a estimator. This example follows the first-order a estimator problem formulation in Example 4-10. Case 1: Using inertial a rate at ZRU. In this case, u l = k = &I = &, = inertial a ratc at IRU, b = 1, and u = 0. A means of removing offsets in the accelerometer used to compute the inertial a rate is to bias the inertial a rate by the integral of the error between a.+and ti. This also removes computational errors. As in Example 4-10, a better scheme of correcting ah is to integrate &I,,,, and then add a pitch rate correction u 2 = a b = L,Ql VTAS after the position integrator (see Figure 4-10) to obtain an accurate a at C G information. The estimates are given bv & = i t + 6rr2 and & = .?? + 1 4 , . Note that & has the same formulation as that i n the first-order estimator, while & is not provided by the first-order estimator but is provided here by this second-order estimator. Case 2: Using inertial a rate at C G . In this case, u l = jr = Cul = &I,, = inertial a ratc at C G , 6 = 0, tr = 0, and 1.12 = 0.The solution of this case is simpler than that in previous Case I . Applying Equation (4-48) yields & ( s ) = ads)

+

(ks

+

kl)ac(s)/D(s),

&(s) =

&l,,(s)

+ klsaG(s)lD(s)

Thus, a.4 and &l,:,; are operated on by the filter to produce: (1) an & that consists of a pure a 1and a filtered tic; and (2) an & that consists of a pure &I,, and a filtered s

kc. Example 4-21: Second-order sideslip estimator. This example follows the first-order sideslip estimator problem formulation in Example 4-11. Case 1: Using inertial sideslip rate at IRU. In this case, U I = 2 = PI,,, = inertial sideslip rate a t IRU, b = 1 , u = 0 , and u2 = Pb 7 LxRIVTAS.The state estimate is given by = 2 1 + bu2 = sideslip estimate and P = i2 ul = sideslip rate estimate. *Note that p for the same formulation as that in the first-order estimator, while 6 is not provided by the first-order estimator but is provided here by this second-order estimator.

+

p

Case 2: Using inertial sideslip rate of C G . In this case, u l = jr = P I = Dl,, = inertial p rate at CG, b = 0, u = -0, and u 2 = 0. The solution of this case is simpler than that ip previous Case 1. The state estimates are now P = 21 = sideslip estimate and 6 = 322 u1 = sideslip rate estimate. Applying Equation (440) yields

+

fi = ?2 + 0lcC + k(PA -~fi),

i 2

= k l ( P ~-

Cj)

Modem Filtering

170

Applying Equations (4-42) and using yield

and Estimatton Techniques

@I($)

=

s P ~ ( s and )

PA(s)

=

Chap. g

PI(s) + PC(S)

+ ( k s + k r ) P ~ ( s ) ] f D ( =s ) PI(^) + (ks + k ~ ) P c ( s ) l D ( s ) b ( 3 ) = [(s2 + k s ) b ~ ( s + ) sk~P~(s)]lD= ( s )b l ( s ) + k ~ s P c ( s ) l D ( s ) Applying Equations (4-48) also yields the same equation. Thus, P A and PI,, are = [X@I(~)

fi that consists qf pure P I and a filtered fiG; and (2)

oppated on to produce: (1) a a 3( that consists of a pure @I,,

and a filtered

bc.

4.6.3 Third-Order Complementary/Kalman Filter Figure 4-11 shows a generalized third-order complementary/Kalman filter.

Third-order complementary/Kalman filter: Formulation 1. Consider the model for the third-order dynamic system in which x = position, i =

U

4

a) Formulation 1

U

-

,A

Wi3

b) Formulation 2 Figure 4-11

Third-Order Complementary/Kalrnan Filter

Comp(ementarylKa1manNlter Approach to Estimator Design

Sec. 4.6

171

rate, and x = acceleration, and the equations of motion in state-space form are as follows: XI

=

where x = is

X?

s l

+

111

+

LVI,

E \ W I ( ~ )= ] 0.

E [ I v ~ ( ~ ) w ~ =( Tqlb(t )] -

T)

+ 6 1 1 2 ,i = x 2 + l r l , and .t = x.3 + rr, and the measurement equation

The preceding model dynamics are illustrated in Figure 4-lla which includes the optimal state estimator. With regard to Kalman filter formulation,

where K is the Kalman gain matrix, and the state estimate equations are

r

and the estimator outputs are i = 2 = it + b112, x = i 2 + U I , and x = 23+,,. The solution of the second-order filters is identical to that of the third-order filter presented in this section, with the exception of k, being set to zero. This results in the elimination of the last state x 3 in the third-order filter. The transfer function from the control inputs u ( t ) , i r l ( t ) , and u 2 ( t ) , and the measurements z ( t )to the state ) estimate vector ~ ( t are

-

A

Modem Nlterlng and Estimation Techniques

Chap. q

where d ( s ) = O is the characteristic equation. Therefore, Z(S) =

A($)= kl(s)+ bU2(s) =

[sU(s) + s 2 U l ( s )

+ bs3U2(s) + (ks2 + kls + k l ) Z ( s ) ] l d ( s )

The transfer functions from z ( t ) to ~ ( trepresent ) the last column of Equation (451) as

As shown in Equation (4-52), all the transfer functions from either the control inputs u, u l , and u2. or the measurement z , to the state estimates and the estimator outputs, are low-pass filters. The algebraic Riccati equation is

and the steady-state Kalman gain is

In this case, 43 = O, q l # 0, and q2 # 0, and Equation (4-55) with kl = O has been found to be the same as the second-order filter gain Equation (445). Therefore, when q3 = 0, that is, the process noise intensity on the third-state is zero, the second-order filters employed in Section 4.6.2 are identical to the thirdorder filter presented. Case 1.

Case 2.

In this case, ql = 0, q2 = 0, q3 # 0, and Equation (4-55) has been

found to be k = 2(q3/r)I1",

kl = 2(q3/r)'I3,

k l = (q3/r)1'2

(4-56)

Sec. 4.6

ComplementaryIKaImanFilter Approach to Estimator Design

173.

Case 3. If there exists a disturbance x n in the x measurement, then r = x XI,. If ~ c ( t )= 0, u l ( t ) = jc = rate, and u2(t) = 0, then in the Laplace domain, U ( s ) = 0, U l ( s ) = s X ( s ) , and Z ( s ) = X ( s ) X D ( S ) .Applying Equation (4-52)

+

+

yields

?(s) = X ( s ) A

X(S)

= X(S)

X(5) =

X(S)

+ (ks2 + k r s + k l ) X n ( s ) l d ( s ) , + (kls2 + k l s ) X ~ ( s ) l d ( s ) + kls2X~(s)/d(s)

(4-57)

By setting k , = 0, Equation (4-37) is reduced to the second-order filter Equation (4-48). If a gain h is added to the measurement input, then z' = h r = h x , + hbuz + hv and v' = hv is the new measurement noise with zero mean and variance r' = h2r. The new state estimate equations and Kalman gains are Case 4.

21 = 22

+

141

+ k ' ( ~ -' h.?] - hbL42)

j3= k ; ( z ' - hG1 - hbuz),

k'

=

klh,

k ; = krllt,

ki = kllh

The error covariance P is the same as that in the previous formulation except that r is replaced by r'lhZ.

Example 4-22: Trajectoly estimation.

In a three-dimensional flight, the trajectory estimation problem involves finding the best estimate of position, velocity, and acceleration in the X , Y, Z inertial coordinates, given the position, velocity, andlor acceleration measurements. The problem is defined by

x = X , Y , Z position,

u = X, Y , z acceleration

z

=

X, Y, Z position measurement,

ul

2

=

X, Y,

z velocity,

=

AX, AY, A Z velocity correction

u2 = AX, A Y , A Z position correction

Note that the problem formulation is similar to the navigation filtering for position and velocity estimate presented in Example 4-18. However, uz is nonzero here and the acceleration estimates are provided as the estimator outputs which were not available in that section. In some cases, X, Y , Z acceleration measurements may not be available and hence need to be estimated. Then, u = AX, A Y , A Z acceleration correction and x j = unknown acceleration to be estimated. In altitude, altitude rate, and altitude acceleration estimator design, the problem is formulated as

x = h = altitude

+

v =

+ bu2 = altitude.measurement

z =

x

u

h = altitude acceleration

=

XI


Chap. 4

ul = ~ h = altitude rate correction u2 = A h = altitude correction Terminal guidance tracking filter (presented in Chapter 7) is another example of the trajectory estimation problem discussed in this section.

Third-order wmplementary1Kalman filter: Formulation 2. The problem is defined the same way as in the third-order complementary/Kalman filter (Formulation I), except x 3 here is assumed to be filtered white noise, that is, the state Equations (4-49a) now become and the state estimate equations are

i1= 22 + u1 + k(z - 21 43

= -2317

- bu2)

+ k l ( z - 21 - bu2)

Applying Equation (4-la),

Equations (4-61) are solved nun~ericallyfor P ( t ) to generate gains k, kl, and k ~ , which mechanize the third-order filter depicted in Figure 4-11b. The steady-state Kalman gain is

k = Plllr,

k,

= P12/r,

k, = P l J r

(4-62)

If a gain h is added to the nIcasurcment input, then z' = Irz = h x l + I~brrz + Ifv and v ' = hv is the new measurement with zero mean and variance r' = h'r. The new state estimate equations and Kalman filter gains are

ComplementaryIKalman Filter Approach to Estimator Deslgn

Sec. 4.6

175

The error covariance P i s the same as that in the previous case except that r is replaced by r'lh2. Assuming measurement noise v in Equation (4-49b) to be a colored noite, then following Equations (4-28) to (4-31) yields the new measurement equations z = x v, ir = F v G v l , = ~ H HA - FH = [ - F , 1,0], E [ f i ( t ) ~ ( ~=) ~ ] [ H N Q N T H T GrGT]6(t - 7 ) = r,S(t - 7), andS(t) = Q ( I ) N ( ~ ) ~ HApplying (~)~. Equations (4-31) yields the new equations for state estimates, covariance matrix, and Kalman filter gain matrix. If F and C are scalar parameters, then Equations (431) yield the following simple form of new covariance and Kalman gain equations:

+

-~ J ' I-I dt

+

- 2P12

+

+ q l - kfr,,

+

-PI-2 dr

-

P22 +

PI3

- k,krcr,

Inertial Navigation' The block diagram of Figure 5-1 is provided to illustrate how the different parts o f the subject matter presented in this chapter are related t o one another. For instance, the discussions of common requirements, navigation con~putation,and coordinate systems together form the framework for the design of a self-contained INS. The t w o types o f self-contained INS are the gimballed INS and the strapdown INS. Following discussions of each of these two types o f INS, a comparison o f the two is presented. Combining INS with other navigation aids introduces the subject of filtering, and finally extending INS uses to provide flight control outputs leads to an integrated INS dcsign.

5.1 INTRODUCTION Inertial navigation is thc process of measuring accclcration onboard a vchiclr 2nd then integrating that accclcration to dctcrminc the vehicle's vclocity and position relative to a known starting point. Accclcrotnctcrs arc uscd to sense thc magnitude of thc accclcration, but acceleration is a vector quantity having direction as wcll as

'

This chapter has becn technically cditcd and improvcd by I l r . J. Stanley Ausrnan of Litton Guidance and Control Systcms.

176

Sec 5.1

lntroductlon

Common Requiremenu Coordinate Systems (Sec. 5.3)

Self-contained INS (Secs. 5.4 & 5.5)

Navigation Computation (Sec. 5.3)

Comparison of Gimballed INS & Strapdown INS I (Sec. 5.6)

I External Navigation Aids (Sec. 5.7)

Figure 5-1

Inertial Navigation

magnitude. For this reason a set ofgyroscopes, or simply gyros, are used to maintain the accelerometers in a known orientation with respect to a fixed, nonrotating coordinate system, commonly referred to as inertial space. This does not mean that the accelerometers themselves are kept parallel to the axes of this nonrotating coordinate system, although some systems are mechanized this way. Instead, the accelerometers may be kept parallel to a set of earth fixed axes by causing the gyroscopes to precess at earth rate. O r they may be maintained locally level by precessing the gyros at earth rate plus the angular rate of the vehicle over the earth. In most of the modern aircraft inertial navigation systems today, the gyros are precessed to rotate with the aircraft itself, resulting in what is known as the strapdown system. One characteristic inherent in accelerometers is that they do not measure gravitational acceleration. For example, an accelerometer in free fall will measure zero acceleration (0 g), but it is really accelerating downward at the acceleration of gravity. For another case in point, consider an accelerometer sitting on a stationary table. It will measure 1 g due to the gravitational reaction force of the table pushing upward on the accelerometer, but the accelerometer is not really going anywhere, at least with respect to the earth. In order to overcome this deficiency of accelerometers, it is necessary to add in the effect of gravity based on a gravity model. A computer uses this model to calculate gravity as a function of position, which in turn is determined from the double integration of the measured acceleration. The most common gravity model employed is one which accounts for both the latitudinal and altitudinal variations in gravity. This model is symmetric about the earth's polar axis and also about the equator. For most navigational purposes, this ellipsoidal model is sufficient, but highly accurate systems must in addition account for gravity anomalies which are deviations from this simpler model.

178

Inertlal Haulgation

Chap. 5

It is difficult to navigate a flight vehicle at night or in bad weather, even with the availability of radio and radar navigational aids. Similarly, smaller, faster, more maneuverable surface vehicles armed with guided missiles, torpedos, and cannons meet with the same difficulty. Consequently, the tremendous need for improved navigation gave rise to the necessity for onboard systems that would supply the position, velocity, and attitude of the vehicle on which they are mounted. The first inertial navigation systems (INS) in operation were those in the German V-2 rockets of World War 11. Since then, INS have become standard equipment in ballistic missiles and in all modern fighter aircraft, airliners, and naval ships. what makes INS particularly suitable for ballistic missile guidance is their autonomous mode of operation during the boost phase. This feature yields a jam-proof guidance system [Bar-Itzhack and Berman, 19881. Much of the progress in the design and use of inertial systems can be traced to the strategic missile programs of the 1950s and, specifically, to the work done at the MIT Instrumentation Laboratory (currently the Draper Laboratory). It was during this time that the need for high accuracy at ranges exceeding 1,000 k m using autonomous systems became more and more apparent. As a result, the U.S. Air Force requested the laboratory to develop inertial systems for the Thor and Titan missiles, while the U.S. Navy requested the same laboratory to develop an inertial system for the Polaris missile. The practicability of a self-contained all-inertial navigation system (INS) for aircraft had just been demonstrated by the laboratory in a series of flight tests using a system known as Spire [Draper, 19811. With the MIT Instrumentation Lab providing the spark, several privately-owned companies took up the challenge and developed and produced their own INS for long-range cruise missiles (North American Aviation for the Navaho missile and Northrop Aircraft for the Snark missile), for strategic bombers (Sperry Gyro Corp. for the B-38), and for ballistic missiles (GE for Polaris and American Bosch Arnla for the ATLAS missile). The remarkable succcss of those early missile and aircraft programs paved the way for applications in aircraft, ships, missiles, and spacecraft. Today, inertial systems are standard equipment in both commercial and military applications. The types of gyros currently found in inertial navigation systems (INS) include the floated rate integrating gyro, the electrostatically supported gyro (ESG), the tuned rotor gyro (TRG), and the ring laser gyro (RLG). The first three are mechanical in nature, relying on a spinning mass to produce a large angular momentum which tends to stay fixcd in inertial space. The floated and tuned rotor gyro suspend the spinning mass in mechanical gimbals, while the ESG suspends the spinning mass, generally a spinning ball, with electrostatic forces so as to prevent any mcchanical contact between the spinning ball and the outside case [Schmidt, 19871. The RLG is a closed optical cavity containing two beams of single-frequency light traveling in opposite directions around the closed cavity. The two beams interfere with one another to form a standing wave pattern around the cavity. This pattern of bright and dark spots tends to stay fixed in inertial space, so that an observer or light detector fixcd to the cavity can sense rotation of the cavity as he (or it) rotates relative to this fixed pattern. A more complete dcscription of the four

types of gyros listed earlier, together with accelerometers and error models for each can be found in Savage 11978, 1984(a)]. Inertial navigation systems are either gimballed or strapdown. In a gimballed system the gyros and accelerometers are isolated from the rotations of the vehicle so that they can be maintained in a specific orientation relative to the earth or inertial space. This greatly simplifies the computations of velocity and position and also greatly reduces the requirements on the gyros which would otherwise have to measure very high angular rates. In a strapdown INS the gyros measure the full rotation rates of the aircraft and keep trace of the instantaneous orientation of the accelerometers in order to properly integrate the accelerations into velocity and position. I t was the advent of small, high-speed digital computers that made strapdown INS possible. In general, the three basic functions of an INS are sensing, computing, and outputting. The accelerometers and gyros perform the sensing function and send their measurements of acceleration and angle rate to the computer. T h e computer uses these data to generate velocity, position, attitude, attitude rate, heading, altitude, range and bearing to destination. If true air speed is available from an air data system, the INS can also compute wind speed and direction as xvell as drift angle. The outputting function simply means sending the appropriate data to thc flight control system, fire control system, reconnaissance sensors, or the control and display unit (CDU) as required for the particular mission. Accordingly, the significance of the navigation system to the flight mission becomes obvious, as the former provides dynamic parameters for flight and path control, assists in terrain-following flight, allows avoidance of enemy defenses via corridor flight, and reduces operator load. The ability of a navigation system to accomplish these tasks greatly enhances mission success. Particularly desirable features in a navigation system include: self-containment; security; worldwide capability; bounded navigation errors for position, speed, attitude and heading; no attitude, weather, or terrain limitations; minimum airframe structural impact; minimum size, weight, power, and cost; and high availability. If aided by other external navigation devices, the INS is called an aided INS. The so-called external navigation devices can be position aids or velocity aids. The former consists of GPS, MLS, MLS-GPS, LORAN, OMEGA, T A C A N and barometer (see the following for definition). The latter consists of Doppler, odometer, and laser velocity meter. The following list covers several important navigation devices and the corresponding section in which each one is covered in some detail: global positioning system (GPS), Section 5.7.2; tactical air navigation (TACAN), Section 5.7.3; long-range navigation (LORAN), Section 5.7.4; terrain contour matching (TERCOM), Section 5.7.5; Doppler radar, Section 5.7.6; and star trackers, Section 5.7.7. An aided INS composed of the INS and external navigation devices is depicted in Figure 5-2a. The evolution of these flight vehicle navigation devices is given in Figure 5-2b. It is seen that a GPS is the most up-to-date navigation device, offering the highest accuracy for flight vehicle navigation. The operational characteristics of

Inertial Nauigation

180

1950

1960

1970

INERTIAUDOPPLER

1 MILE ABSOLUTE

198)

Chap. 5

Sec. 5.2

181

Common Requirements for lnertlal Navigators

CHARACTERISTIC

L REOUlRES REMOTE COOPERATING TRANSMIllER

8. ANTENNA LOCATION I S IMPORTAN1

2.

POLAR REGION LIMITATIONS

9.

TIME FRAME DfPENMNT

3.

HEAVILY SMOOTHED SPEED I S AVAllABLE

10.

DOPPLER SUBJECT TO D m C l l O N

- POST 198)

4 WIND VELOCITY VECTOR REQUIED

11 STORMS AND SUN SPOT ACTIVITY

5.

U. U.

CERTAIN WEATHR LIMITATIONS

6 LIFE OF S I C H I LIMITED

7.

LIMITED OVER WATER AND HOMOGENEOUS LAND I.E.

MANEWERS M G R A M ACCURACY ACCURACY LIMITED TO MAG COMPASS

DESERT (C)

Figure 5-2 (a) Navigation Parameters Available from Sensors (b) Navigation Technology Evaluation (c) Operational Characteristics of Navigation Sensors ( F r o ~ n[.21izrri11, 19861 w i t h pennissio~zfrom Litton)

these navigation devices are given in Figure 5-2c. Generally speaking, the accuracy required of a navigation system for most applications is on the order of 1 nautical mph of flight. More stringency may be placed on the required accuracy in certain phases of flight, such as ground attack for example, in which the accuracy must be in the neighborhood of a few meters. In these cases, an external reference is necessary to update the INS in order to achieve the required accuracy.

5.2 COMMON REQUIREMENTS FOR INERTIAL NAVIGATORS

Vehicle and weapon systems. The requirements of a navigation system for weapon systems can be categorized as follows. In the case of piloted vehicles, medium accuracy (1 nautical mph) is mainly what is required of the navigation

Inertial Navigation

182

Chap. 5

system for tactical aircraft, whereas landfall position fix, velocity, and nuclear hardness are all required of the navigation system for long-range strategic bombers. Furthermore, low accuracy or simple attitude reference is the basic navigation requirement for observation aircrafts, while size and weight are critical t o helicopteri. In the case of short-range missiles and tanks, size, weight, and low cost figure predominantly in the requirements, whereas medium-range missiles require somewhat better navigation. Reliability is a key requirement for intercontinental ballistic missiles and tanks. Submarines and ships require extremely high accuracy over long time periods. For military use, the ideal navigation system should be fully autonomous, undetectable, unjammable, usable worldwide in all weathers, and of vanishingly low weight and infinitesimal cost. These aims have driven development for the past thirty years. Systems are now sufficiently reliable and accurate for virtually all modern fixed-wing military aircraft to be equipped with an inertial navigator [Barnes, 19871.

Navigation requirements for weapon delivery.

Navigation accuracy is a key factor for successfully carrying out low-level attacks against targets at known locations. First, it gets the attacking aircraft to the right position to find the target. Once the target is identified, the INS must provide an attitude reference for the targeting sensors (optical sight, laser range finder, radar, IR sensor, or TV sight). After the target position is located with respect to the aircraft, that position is continually updated by the INS as the aircraft moves into its weapon launch position. For the weapon delivery itself, the INS provides initial position, velocity, and attitude data for the weapon. Stand-off weapons require the same set of funct~ons,but the missile itself takes over the final target acquisition and terminal homing. For such weapons, the INS in the launch aircraft simply has to provide initial conditions for the missile, which may also have a small INS for midcourse guidance, good enough to steer the n~issile into a target acquisition basket.

Stand-off weapon systems.

The following discussion on navigation requirements for precision-guided stand-off weapon (PGSOW) delivery are based on detailed results shown in Shapiro (19861. The successful delivery of a PGSOW depends, in general, on three key elements: knowledge of launch position parameters, knowledge of target position, and accuracy of navigation between these two positions. The basic PGSOW mission scenario is shown in Figure 5-3 which depicts the flight legs from aircraft takeoff to weapon launch point, then the weapon flight to target arca window, and finally to the impact point. The focus hcrc is on the choices for providing accurate navigation to the lowcost, short-range (20-mile) RPV or powered glide bomb which cannot sustairl the size, weight, or cost penalties of a high-precision navigation system. In order to ascertain the accuracy requirement for the PGSOW, it is necessary to estimate the capability of the terminal seeker which, in the final analysis, determines the sizc of the target window or search arca. The burden placed on the terminal seeker incrcnscs

Sec. 5.2

Common Requirements for Inerthi Naulgators

183 TARGET

/SEARCH - ~

TARGET

LAUNCHING AIRCRAFT

FLIGHT ORIGINATION POINT Figure 5-3

PGSOW Mission Scenario ( F r o ~ n[Shapiro, 19861 w~idtpennisrionjom ACARD)

with the size of the required search area imposed by uncertainty in relative weapon and target positions. In the case of an 1R seeker, for example, the window may be relatively narrow for a given angular field of view because of the limited range due to atmospheric attenuation. An active millimeter wave radar can provide longerrange operation in bad weather (several kilometers); however, the limited aperture available for the antenna and the consequent limited angular resolution creates problems with false alarms and poor detection probability as the search area is increased. Therefore, the PGSOW must navigate to the target zone with sufficient crosstrack accuracy to insure that the target lies within the swath described by its forward motion and the scanning limits of the terminal seeker. Based on the foregoing types of considerations. it can be concluded that a vractical search area width is about 1 km. The corresponding navigation cross-track error budget should be limited to one half of this, or 0.5 km. The along-track error is dependent on the allowable false alarm and desired detection probability criteria. Obviously, as the along-track extent of the search decreases, these probabilities become more favorable. The alongtrack navigation error can be specified, however, as being the same as the crosstrack error, namely 0.5 km. T o achieve the preceding performance objectives, the navigation system for the PGSOW must have a heading error over the 20 miles of better than 1 deg and an along-track error ofless than 1.5 percent of distance traveled. The future will see an increased demand for more agile missiles for both airand ground-launched applications. Gun-launched missiles involving high accelerations at launch will contain guidance systems. Inasmuch as strapdown guidance systems have been included in missiles that are subject to high roll rates and high

184

Inertial Navigation

Chap. 5

acceleration, the need for more rugged navigation instruments having a dynamic range greater than what conventional gyros offer is brought to the forefront. A fiber-optic gyro is described [Kay, 19873 that, although it is still in the laboratory stage, has demonstrated good performance and, at the same time, ruggedness, quick starting, and low cost.

Visual attack systems. Following Barnes [1987], in an effort to expose the aircraft to as little ground-based radar as possible, much of the development subsequent to World War 11 has been focused on low-altitude flight. Another goal related to this effort was, until the development of look-down Doppler radars, to make it more difficult for a defensive fighter aircraft to identify an incoming raid. Considering that the total time of an attack pass may be just a few seconds, target acquisition is rendered more difficult by low-altitude attacks. In these few seconds, the pilot of an attacking aircraft armed with unguided weapons must maneuver on to an attack heading, stabilize the aircraft path, and release the weapon. Precise navigation and an accurate calculation of weapon release are two elements that are absolutely essential for a successful attack. The former ensures that target acquisition occurs as early as possible, while the latter minimizes delivery error. Regarding attacks that can be conducted under visual conditions, the requirements for precise navigation with an acceptable workload, especially at high speed and low altitude, have been met by the introduction of inertial platforms and moving map displays. Although such developn~entssimplify the task of acquiring preplanned, fixed-position targets early, they do little for acquisition of targets of opportunity. Similarly, two cle~nentshave combined to meet thc rcquirements of accurate calculation of weapon release. The first involves the ability to obtain data on aircraft altitude, heading, and speed from the inertial platform. Thc second rclates to the ability to digitally process in the aircraft range to target obtained from a ranging systenl. The ability to carry out high-speed, low-altitude attacks a t night and in poor visibility has also bcen enhanced owing to other dcvclopmcnts. One of the first of thcsc was ground-mapping radar, which enabled pilots ro attack targets giving discrete radar returns and targets whose position was known relative to some observable and recognizable feature on the radar screen. However, radar resolution is poor relative to visual wavclcngths and forward-looking infrared (FLIR) scnsors greatly cnhancc the ability of the aircrew to identify both targets at known Iocatiol~s as wcll as targets of opportunity. 5.3 NAVIGATION COMPUTATION AND ERROR MODELING

5.3.1 Coordinate S y s t e m s I n navigation computations, thrcc colnponct~tsof accclcration scr~scdby a n orthogonal triad of accclcro~nctcrshave to be transfornlcd onto a rcfcrencc coorditlntc systcm. Thc vclocity and position of thc vchiclc arc then computcd with rcspcct to

Sec. 5.3

Naufgatlon Computation and Error Modeling

Figure 5-4

(a) Earth-Fixed Coordinate (b) Geographic Coordinate

this reference coordinate system (RCS). Because the reference coordinate system affects both the computation time and accuracy, a suitably selected RCS can minimize the processing errors. Beginning with an earth-fixed set of coordinates (Figure 5-4), we can, by a series of two rotations first about the z-axis through the longitude angle and then about the negative y-axis through the latitude angle plus 90 deg, transform from the earth-fixed coordinates into a navigation coordinate system. In this case the navigation coordinate system is defined as one in which the x-axis is north, the yaxis is east, and the z-axis is down. The transformation is depicted graphically in Figure 5-5. Figure 5-6 derives the resulting direction cosine matrix. Such a transformation is known as an Euler angle transformation. A similar transformation will take us from the navigation coordinates into the aircraft bodyaxis coordinate system. The first rotation is about z (down) through the heading

Figure 5-5 mation

(a) [Z] Transformation (b)

[YI] Transformation (c) [Yz] Transfor-

Inertial Navigation

186

X Y Z

NAV

X

X Y

w s h sin h 0 -sinhwsh 0 0 0 1

-sin @ 0 - a s @ 0 1 0 ms @ 0 -sin cP

Z

Chap. 5

Z

FAR'IH

FARlH

- sin@ms h - sin@sin h - cosQ 29 dB), thc rcccivcr can phaselock onto the carricr. Thc Doppler shift observed 1s uscd to dctcrm~ncthe linc-of-sight (LOS) velocity betwccn the satcllltc and uscr. This mcasurcn~cntis implcmcnted as an integrated Doppler known as the

Sec. 5.7

External Naulgatlon AIds

217

pseudo-delta-range (I'DR), which is a measure o f the range change during the integration interval.

Performance.

Table 5-1 lists the error budget for the PR measurement, assuming that the dual-frequency ionospheric co~npensationtechnique is used. As shown, a measurement error of 5 m is predicted. The test results indicate that the high-frequency fluctuation of the error is less than 1 m. The error was dominated by a bias component that ranged from 7-10 m [McGowan, 19871.

INS updating.

Figure 5-34 illustrates a block diagram of a typical implementation using the GPS system as an update aid for an inertial system. Note that the filter estimates receiver errors as well as inertial navigation errors. However, realistic modeling of all error sources usually results in a state vector dimension that prohibits full implementation in the vehicle computer. Trade-offs to determine the appropriate suboptimal filter to implement are then required [Schmidt, 1976; Setterlund, 19861. From these performance trade-off analyses, a set of system performance specifications can be developed prior to implementation and integration [Schmidt, 19871.

5.7.3 Tactical Air Navigation (TACAN) T A C A N is a two-dimensional navigation system used for piloted aircraft and is based on generating azimuth and distance to a fixed ground station for the aircraft. The mechanization of TACAN centers around an airborne-interrogator and ground transponder. The flight vehicle is provided with range and bearing lines of position TABLE 5-1 GPS PSEUDORANGE ERROR BUDGET Segment Space Segment

Control Segment

User Segment

Error mechanism Clock Stability Track Perturbations Other Segment RSS Ephermeris Prediction and Modeling Other Segment RSS Ionospheric Delay Tropospheric Delay Receiver Noise/Resolution Multipath Interference Other Segment RSS System RSS

From [McGowan, 19871 with permission from AGARD

System budget (1) meters

Inertial Navlgatlon

Chap. 3

Uncorrected

nOviVion vorlo ICS

~ovigationcorrections

pseudoronge and pseudo delta range

Figure 5-34 Aided Inertial-GPS Mechanization (Reprinted with permission jot11 (Schmidt, Systems fi Control Enrydopedio), Copyrishr (1987), Pergemort Press P L C )

by a ground transponder which uses a nearly pulse type of transmission for range1 bearing loci measurements. It operates on a frequency in the L-band (1.0-GHz) range, and has an operational range of 150 N M (LOS) at an altitude of 15,000 ft. It has an accuracy that is on the order of 200-1,500 ft in range/(bounded) radial position error and 0.5 deg-1.0 deg (100 ft/NM) in bearing/(bounded) angular position error. The resolution is approximately 100 ft in range and 0.5 deg in heading. It is the bandwidth that limits the range resolution to 100 ft, while the L-band frequency and the beamwidth of the transponder antenna limits the bearing resolution. In Figure 5-35a, it is seen that the angular position error is 10,000 ft at 100 NM, corresponding to the bearing accuracy mentioned earlier. O n e way to overcome the large angular position error is to go to a dual-TACAN mechanization. By implementing a ~ U ~ ~ - T A Csystem A N or range-range mechanization, the position error is reduced to 500 ft (CEP), as illustrated in Figure 5-35b [Martin, 19861.

TACAN advantages and limitations. Advantages of T A C A N include excellent enroutelhoming navigation aid, and accurate range and bearing over short distances. Furthermore, it is amenable to fixed and mobile (that is, ships) ground stations. One of the limitations of TACAN is that it is not a navigation system and thus requires a heading device. In addition, it is not sclf-contained because it requires a radiating ground station. At long ranges, its cross-track errors are large. There are also limitations on its LOS, which require aircraft to fly high in mountainous regions and for long-range measurements. There is a disadvantage in the fact that radiation from the aircraft can be detected during military OPS. T A C A N is also subject to natural and enemy environments as well as to ground maintenance ell-

Sez. 5.7

External Navigation Aids

218

OPERATIONAL RANGE

150 NM (LO9 AlRCRAFl AT 15,000 FT ALT.

RADIAL POSITION ERROR (BOUNDED)

200

ANGULAR POSITION ERROR (BOUNDED)

LO0 ( - 100 FTINM)

- 1500 FT

I

TACAN STATION

ACAN STATION R

Figure 5-35 (a) TACAN Characteristics (b) Dual TACAN Capability Reduces Large TACAN Cross Track Position Errors (From [Martin, 19861 with permission from Litton)

220

Inertial Navigation

Chap. 5

vironment, Due to the earth's curvature, as the range to the T A C A N station in, creases, so must the altitude of the flight vehicle increase. A relationship between the TACAN range and the altitude of the flight vehicle is given in Figure 5-36.

5.7.4 Long-Range Navigation (LORAN) LORAN is a time-difference of arrival system that requires one master and at least two slave ground stations that transmit radiation to the flight vehicle. Like TACAN it is a two-dimensional system. The transmission of the ground station is approximately in the form of 8 or 16 pulsed (groups) at 10 KHz PRF, entailing measurements of hyperbolic loci of position provided by pulse envelope and phase difference. It has an operating frequency of nearly 100 KHz and an accuracy that is on the order of 50-500 ft or more, contingent upon the geometric dilution of precision (GDOP). The resolution of the transmission is approximately 0.01 psec X GDOP.

LOBAR characteristics. LORAN characteristics are depicted in Figure 5-37a, which include an operational range of 600-1,000 N M on land and 1,0001,500 N M over water and a radial position error of approximately 300-500 ft in the primary zone. The radial position error is the region of interception of the M-S#l line of position (LOP) (see Figure 5-37b), and the M-S#2 LOP as given in Figure 5-37a. M-S#1 and M-S#2 LOP are hyperbolic curves between M-S#l and M-S#2, respectively. The hyperbolic LOP is constructed based on the receiving time (that is, range) difference measurements between the master station and the slave station. LORAN advantages and limitations. Advantages of LORAN include simplicity and reliability of airborne units, accurate medium-range navigation aid, and nonradiation of passive receiver. There are some limitations and disadvantages associated with LORAN, one of which is that radiating ground stations are required for LORAN. In addition. LORAN, like TACAN, is subiect to natural and enemy " environments as well as to ground maintenance environments. Also, like TACAN, LORAN is not a full navigation system and needs to be augmented with heading, velocity, and altitude. Its performance is reduced by the GDOP. The long smoothing circuits of LORAN require low dynamics during measurement or rate aid to maintain accuracy/track.

5.7.5 Terrain Contour Matching (TERCOM) TERCOM is a form of correlation guidance based on a comparison between the measured and the prestored features ofthe profile of the ground (terrain) over which a missile or aircraft is flying. Generally, terrain height forms the basis of this comparison. Obtaining the reference data requires prior measurement of the ground contours of interest. This type of guidance is used for updating a midcourse inertial guidance system on a periodic basis, and has been applicd to the guidance of cruise

Inertial Naulgatlon

222

Chap. 5

missiles, which usually fly at subsonic speeds and fairly constant altitude [Heaston and Smoots, 19831. A TERCOM system uses an airborne altimeter and a data processor to correlate the measured terrain contours with the prestored contours to obtain the best estimate of position. The transmission characteristics of the airborne altimeter include an operating frequency of approximately 4.4 GHz (incidental to operation) and a transmission type that is pulsed or CW. As the missile flies, the radar altimeter first measures the variations in the ground's ~rofile.These measured variations are then digitized and processed for input to a correlator for comparison with stored data. High-density digital storage permits large quantities of data to be stored, and modern microprocessors allow the comparison to be done sufficiently and quickly [Barnes, 19871. Through this process, the missile can determine its position and correct any errors that have developed since the previous update. Accuracy is 100-1,000 ft and resolution is 100-200 ft. Figure 5-38 illustrates the basic concept employed by TERCOM. The terminal guidance stage may bebased on the final TERCOM update and a preprogrammed course relying on the inertial system, or a separate terminal homing seeker may be employed that can recognize the target and provide the final guidance commands [Heaston and Smoots, 19831. The block diagram shown in Figure 5-39 shows the mechanization of the TERCOM system. OPERATIONAL RANGE RADIAL POSITION ERROR SLAVE I1, STAT ION

@XI- 1OOO N M (LAND) lOCKl - 1500 NM (WATER) -300 - 500 FT (PRIMARY ZONE)

SLAVE X2 STATION

MASTER STAT l ON (A)

Figure 5-37 (a) LORAN Characteristics (b) LORAN Lines o f Position ( a ) j o m [Martin, 19861 with permission j o m Liftot~( b ) courfesy 0jF.U'. Hardy, 1986)

Sec. 5.7

external Navlgatlon Alds

(B)

Figure 5-37

(Continued)

TERCOM advantages and limitations.

The advantages of TERCOM include total automation of its operation, accurate position location with a good heading device, and self-contained operation requiring no external support. The

Inertial Navigation

224

Cruise

Cruise Missile

Terrain

Chap. 5

Elevation Map Stored In Onboard Computer

Terrain Mapped Previously

Guidance

Update

T A R G E T AREA

OPERATING RANGE RADIAL POSITION ERROR

- UNLIMITED (COMPUTfR - ZW - % FEnI

CAPACITY1

Figure 5-38 (a) Terrain Corltour Matching (b) T E R C O M ~haracteristics ((aJjom /Heaston and Smoors, 198.71 lr'irh prrmissiot~j o m GACIAC (b)jottr /hlnr.ritlt 19861 with prrmission j o m Lirrorr)

Sez. 5.7

External Navigation Aids

Data Cornlation

4

Recommended Flight Path

I

h

Figure 5-39

AutopiSol

TERCOM Mechanization

limitations of TERCOM are as follows. TERCOM requires varying-altitudc tcrrain and a radiating altimeter. Moreover, thc prestored maps must be obtained, generated, and programmed in advance. In order for the updating tcchnique described earlier to perform properly, there must be sufficient variation in the terrain to generate a usable signal. Obviously, such a system will not work over water. It may have difficulty in the midwestern region of the United States where the ground is very flat. Therefore, consideration must be given to the terrain surrounding the target when employing this type of guidance. The data collected for the reference map will provide the information necessary to predetermine if the system will work in the areas of interest [Heaston and Smoots, 19831. 5.7.6 Doppler Radar

Doppler radar on an aircraft provides a measure of the aircraft's velocity relative to the earth. It usually consists of four beams, one forward, one aft, one to port, and one to starboard. They are all angled downward at a 45-deg to 60-deg angle. The fore and aft beams measure velocity along the track heading of the aircraft, while the left and right beams measure cross-heading velocity. All four beams measure vertical velocity. The Doppler set measures velocity in aircraft axes. The INS, or sometimes an AHRS, provides the reference coordinate system for resolving the Doppler velocity components onto along-track and cross-track axes or onto earthfixed axes.

Doppler advantages and limitations. Doppler radar has the advantage of being autonomous; it does not require any ground-based or space-based facilities. Because it does not have its own attitude reference, it must always be used as an adjunct to an INS or an AHRS. The Doppler return signal over water is poor due to more specular and less diffuse reflection off the surface. In addition, the scattering cells on the water surface may be moving due to currents or wind-blown spray. For this reason Doppler radars usually have two modes, one for land and one for over water. This landlsea switch can be used to reduce the amount of feedback gains to the INS during over-water operations. 5.7.7 Star Trackers

Astro-inertial systems consist of a telescope mounted in two gimbals, azimuth1 elevation, which in turn are mounted directly onto an inertial system. The systems in operational use today are integrated into a !gimballed inertial system. This means

Inertial Navigation

226

Chap. 5

that the whole system must include at least five gimbals: the pitch, roll, and azimuth gimbals of the inertial system plus the azimuth and elevation gimbals of the stellar telescope. Consequently, these systems tend to be quite large and expensive. Conceptually, the gimballed inertial system could be replaced by a strapdown inertial. thereby reducing the size of the overall system. T o date, however, this development is still in the early stages of design. The basic idea of the astro-inertial system is as follows. Given the position of the aircraft from the inertial system, the computer looks up the ephemeris data for a particular star from the star catalog contained in its memory, and calculates the azimuth and elevation angles for that star at that time. It then drives the telescope to those angles and looks for the star. If the star is off the axis of the telescope, the star tracker provides correction signals to center the star on the telescope axis. These same azimuth and elevation correction signals go the inertial system as psi-angle corrections (see Figure 5-10). One such star fix can update two attitude error components. For a full threeaxis update, the telescope is driven to another star, widely separated in angle from the first, and the process is repeated.

Star tracker advantages and limitations.

Like inertial and Doppler systems, the star tracker is autonomous, requiring no outside facilities, except of course for the stars whlch are always there. Cloud cover is an obvious limitation for star trackers, and this relegates their usefulness to high-altitude flight. Because it is a psi-angle measuring device, it basically updates the gyros and not the accelerometers. (Recall that the psi equations are independent of the acceleration equations.) This means that acceleration errors can grow without bound. The astro-inertial system will simply apply a tilt error to compensate for the position error; it cannot distinguish between the two. However, with an occasional position fix or with Doppler radar updating, the integrated system can produce a navigation system with bounded errors over very long periods of time. 5.7.8 Kalman Filtering

In order to combine the INS data with one or more of the foregoing augmentations, most modern avionics systems use a Kalman filter (KF). The KF is a linear feedback system with time-varying gains. The gains are varied in an optimal or nearly optilnal fashion so as to take account of the relative accuracy of the INS and the updating measurements as well as the geometry of the measurements. Because the accuracy of the INS varies with time, the KF must include an error model of the INS. This error model includes a number of error states, the dynamic coupling betwccn these states, and the noise which drives these states. A similar but simpler error model for the measurement noise allows the KF to properly weight the feedback gains to the INS. The feedback corrections themselves may be applied in two different ways, One is called open-loop correction, and the other is called closed-loop correctioll,

External Naulgatlon Aids

Sec. 5.7

227

In the first, the corrcctions are added to the INS outputs and do not affect the operation of the INS itself. Figure 5-40 illustrates such a mechanization. The advantage of this type of mechanization is that spurious measurements affect only the corrected INS outputs and not the INS itself. If the KF diverges as a result o f these spurious updates, the INS is still operational, and the system can recover by reinitializing the KF. The disadvantage of the open-loop KF is that the state errors in the error model arc not driven to zcro and can become large enough for nonlinearity effects to disrupt the system operation. In thc closed-loop feedback system, the feedback corrcctions actually correct the INS itself so as to maintain the INS state crrors near zcro. This eliminates the nonlinearity concerns. O n the other hand, spurious updates can drive the INS so far off that it may not be able to recover.

Kalman filtering approach. A standard Kalman filter minimizes the state-vector variance and estimates the state errors. The Kalrnan filter is constructed based on an error model, as mentioned previously. First, an INS model and a measurement model arc given as jc = Z =

.4.u

Hs

+

+

E[~(~)Iu(T = ) Q6(t ] - 7) E[v(I)v(T)] = R6(t - T)

(INS) E[ru(t)] = 0, v (measurement) E[v(t)] = 0,

lu

INCREMENTAL ERROR ESTIMATES

I

I

SENSOR SUBSYSTEM

SENSOR SUBSYSTEM

-c

tl

SUBSYSTEM ERROR ESTIMATES

ERROR

-

SUBSYSTEM CORRECTED OUTPUT

1 MEASUREMENT ERROR VARIABLE DIFFERENCES

SENSOR SUBSYSTEM

MATRIX

SUBSYSTEM OUTPUT RATIONAL SUBSYSTEM ERROR

SUBSYSTEM CORRECTED OUTPUT

ERROR

Figure 5-40 Generalized Mechanization of Filtering Open-Loop Error Control System (From ['Martin, 19861 with permissionfrom Lirton)

Inertial Navigation

228

Chap. 5

where x is the error state with X E R ~ ,and w and v are white noises. The measurement z can be a combination of attitude, velocity, o r position measurements. If position measurement is available, position error 6, must be included in the state vector x, An example application of error-covariance analysis to inertial platform errors is constructed as illustrated in Figure 5-41. As presented in Chapter 4 (Sections 4.1 and 4.2), the fiow diagram in Figure 5-41 [Tucker and Stern, 19861 can be con.. structed, with v = a as the input. From this figure, it is seen that there are nine error state variables: three position errors, one along each of the coordinate axes; three corresponding velocity errors; and three misalignment errors, one about each of the coordinate axes. The velocity errors are obtained by integrating the acceleration error inputs, and the position errors are obtained by a second integration. In addition, the misalignment error about any axis leads to an additional acceleration input along each of the other two axes. The resulting dynamics matrix (A) from Figure 5-41 is as follows:

I

0 -k? 0 0 0 0 0 0 0

-

1 0 0 0 0 0 0 k 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 0

kz 0 0 0

0 0

0 0 1 0 0 0 0

0 0 0 a47

0 aj7

0 0 0

0

0

a28

a29

0 0 0 ass 0 0 0

0 049

0 0 0 0 0 *

Examination of the A matrix and the flov,. diagram in Figurc 5-41, howcver, indicates that the systcm can bc decoupled into three indcpcndcnt subsystems, cach of dimension four. For example, states 1, 2, 8, and 9 form a subsystem. It IS noted that cach of thc misalignment error states 7, 8, and 9 feed into two of the three subsystems, but this does not cause any direct coupling among the subsystcms, since there are no feedback inputs into the misalignrncnt states. T h e four-dimensional problem for states 1, 2, 8, and 9 is of the form

5.7.9 Kalman Filtering Performance

For conventional filtering, the position error (in ft) ncvcr converges to zcro, as shown in Figure 5-42a. For Kalman filtcring (optimal filtcring), on the other hand, the position error (in ft) convcrgcs to, but ncvcr rcachcs, zcro, as shown in Figure 5-4%. Figure 5-42c displays a table showing thc relationship of mechanization and performance.

Sec 5.8

Integrated Inertial Navigation System (IINS)

9 n3 n5

n2

-

x velocity

n7 ' misalignment about x-axis

= y Position

n4 = y velocity

nB = misalignment about y-axis

= z position

n6

ng

x Position

Figure 5-41 AACC)

= z velocity

-

misalignment about z-axis

Flow Diagram (From [Tucker and Stern, 19861 with permission from

INTEGRATED INERTIAL NAVIGATION SYSTEM (IINS) When inertial systems were first included as part of the avionics package on aircraft circa 1960, they performed only the navigation function. Typically, there would also be an AHRS for driving the flight instruments, and there would be rate gyros

Inertial Naulgatlon

230

CONVENT lONAL FlLTER

OPTlMAL (KALMAN) FILTER

(A)

'HARS

- BOUNDED loHCADING,

Chap. 3

(B)

DG

PIHRFRfE.

VG 1 8 1 ~ ~

Navigation PerFigure 5-42 (a) & (b) Integrated Performance (c) Augme~~red formance Summary (From /Manin, 19861 ulith pcrmir~ionJiom Lirrotr)

Sec. 5.8

Integrated Inertlal Naulgatlon System (IINS)

231

for flight control sensors. Gradually, as the INS became more reliable they began to take over the AHRS functions. Then as the strapdown INS came into being, their use for flight control became a real possibility. In order for the INS to be part of the flight control system (FCS), it must be made ultra reliable. This leads to the design of multiplcly redundant INS. Such systems, properly updatcd, can providc reliable and accurate position, velocity, and attitudc for the navigation function, as well as body-axis accelerations and attitudc rates for the FCS. Such systems are called intcgratcd inertial navigation systems (IINS). Only strapdown INS arc candidates for IINS, because the gimballcd INS cannot providc the attitude rates for the FCS with a wide enough bandwith.

5.8.1 Integrated SensinglFIight Control Reference System (ISFCRS) One way to improve the reliability of the INS is to employ redundant sensors, as in the case of an integrated sensinglflight control reference system (ISFCRS). The ISFCRS uses six ring laser gyros and accelerometers in an arrangement of two orthogonal triads, onc of which has its axes aligned with the flight vehicle's body axis. The geometry is depicted in Figure 5-43. The sensors are numbered from 1 to 6 with 1, 2, and 3 corresponding to the x-, y-, and z-axes of the body aligned unit, and 4, 5, and 6 corresponding to the x-, y-, and z-axes of the skewed unit. A functional block diagram of an ISFCRS is presented in Figure 5-44. The ISFCRS contains signal selection, lever arm compensation, and parity equations. The ISFCRS was proposed for the MFCRS and MIRA systems that provide all required incrtial reference and air data in advanced transport and fighter aircraft for flight control, navigation, weapon delivery and targeting, terminal area functions, automatic TFITA, sensorltracker stabilization, flight instruments, and displays, as illustrated in Figure 5-45. Figure 5-46 shows a conventional approach of comparison with MIRA [Boggess, 1986; Young et al., 19831.

5.8.2 Integrated Sensory Subsystem (ISS) Detailed discussions of the integrated sensory subsystem (ISS) presented in this example can be found in Toolan and Grobert [1983]. The ISS is composed of three major elements. The first element consists of the various sensors/sensory data and preprocessing electronics. The second element is a redundant I 1 0 system which provides data transfer between the sensors/transducers and redundant computers. The third ISS element is the data handling system (DHS) contained within the redundant digital computers in which redundancy data management and output parameter calculations are performed. The ISS sensor set consists of redundant skewed arrays of strapdown integrating rate gyros and accelerometers, redundant air data probes and transducers, magnetic heading reference sensors, radio aids, and command inputlsurface position transducers. A system-level block diagram of the

Ineriial Navigation

232

Figure 5-43

Chap. 5

ISFCRS Geometry

ISS DHS is shown in Figure 3-47a. The skewed and dispersed gyros and accelerometers supply raw input data to the DHS. The DHS removes the vehicle bending and kinematic effects and provides orthogonal rate and acceleration data to the SAHRS (strapdown AHRS) algorithms, the FCS, and the user systems. The six gyros and six accelerometers are configured in a symmetrical conical array and provide a 2-fail operational sensor capability. In addition, the sensors are dispersed to assure a survivable system. The ISS design philosophy for inertial sensor packaging and dispersion for survivability permits installation o f gyros and accclerometers in less than ideal locations with respect to bending nodeslantinodes and the aircraft center of gravity. Without appropriate compensation generated by the use o f a state estimator, the sensor locations can result in FCSIbending-mode coupling andlor nuisance trips of the RDM failure detection routines. The configuration of the gyro and accclerometer RDM routines and the state estimator is shown in Figure 5-47b.

5.8.3 Integrated Inertial Sensing Assembly (IISA) Another type o f IINS is the IISA. This is an integrated inertial sensing assct1lbJ~ (IISA) that uses six ring-laser gyros and six inertial-grade accelcromcters in two, separated clustcrs. It is a system that is well suited to the combined navigatioll and flight control sensing needs of modern high-performance fly-by-wire vchiclcs. The background of integrated navigationlflight control sensors is given by Ebncr and Wei [1984]. Krasnjanski and Ebncr [1983], and Jar~kovitzct al. 119871 includillg a spectrum of possible system configurations as a function of avionics redundancy

Inertial Navigation

234

?-={

I

I

I

I

iI

MILSTD.1553A

I I I

L

---

Chap. 5

USERS Right Control Navigation Weapon Fire Control Delivery & Trajectory Sensorhcker Stabilization NAV Axis Cockpit Display

I

-l

Figure 5-45

MlRA System

requirements. Within each inertial navigation assembly (INA), sensor axes are orthogonal but skewed relative to the vehicle yaw axis, as shown in Figure 5-482, One accelerometer and one gyro in an INA are oriented along each skewed axis. Each of the three channels is largely independent, as shown in Figure 5-48b. Figure 5-48a also depicts the orientation of axes when the INA is installed into the left equipment bay of the vehicle. When an identical INA is installed into the right equipment bay, with 180-deg rotation about yaw relative to the left INA, the six sensor axes are then distributed uniformly about a 54.7-deg half-angle cone. No two axes are coincident, nor are three in the same plane. Thus, any three sensors may be used to derive three-axis outputs in vehicle axes after suitable computer transformation. The IISA development uses high-reliability laser gyro and accelerometer sensors packaged in a strapdown system configuration to provide a common, efficient source of aircraft body rates, attitude, and accelerations. These measurements provide the essential inertial data inputs for all core and mission avionic functions including stability control and stability augmentation system (CSAS). The IISA (Figure 5-48c) consists of five assemblies: two identical INAs containing the inertial sensors and navigation computers (shown in Figure 5-48d); two identical digital computer assemblies (DCAs) each containing dual redundant flight control redundancy management and sensor selection logic computers; and a multifunction

L

k Air Data

User

Figure 5-46 Conventional ~ p p r o a c ~ to Sensing and Flight Control Referellce System

ORTHWWAL RAT1b ACCIL DATA

OW111 UtR FIST#MS PITCH L ROLL

IU OAT A HANDLINO SYSTEM

DELIVERY

INTEORATID N A V SVSTEM CONTROL SYSTEM

-- -

SKEWLD OVROS ACCEL8

I

IEXTRAPOLATED EENDlNG ACCELERATIONS1

I

I

USE EXTRAPOLATED ESTIMATES TO REMOVE BENDINO ACCELERATIONS

SKEWED ACCELERWETER

AIRCRAFT DYNAMICS

USE GYRO RDMS OUTPUT 6 EXTRAPOLATED ROTATIONAL ACCELERATION ESTIMATES TO REMOVE KINEMATIC ACCELERATIONS

p-r-4 ACCELEROMETER

7

ESTIMATOR -TE

t+

TO SAHRS 6 CONTROL L A M PITCH 6 ROLL ATTITUDE FROM SAHRS

ELBP USE EXTRAPOLATED ESTIMATES TO REMOVE BENDINO RATES

I

1

1

IEXTRAPOLATED BENDING RATE ESTIMATES1

(B)

Figure 5-47 (a) Inertial System Block Diagram (b) Inertial Sensor RDM (From [Toolan and Groberf, 19831, 0 1983 IEEE)

Inertial Navigation

Chap. 5

control display unit (CDU) for displaying IISA data and providing the operator interface for initializatior~and mode selection.

Configuration of sensor locations.

Table 5-2 is a comparison of configuration and attitude/velocity redundancy for the IISA. T h e four different system configurations with variable levels of attitude/velocity redundancy are described as follows [Ebncr and Wci, 19841. (1) Configuration 1 assumes that neither attitude nor velocity is required for flight safety. The INS can be nonskewed relative to aircraft pitchlrolllyaw axes for minimum size. A companion unit would be required

Integrated Inertial Naulgatfon System (IINS)

Sec 5.8 INA

--

MILStD-IS538

NAY ILICIRONIC~

1014 Hx

3 RLOS 3 ACCtU 3 HVPS 3 fC ILICTRONICS

N A V TO * I C

- -=-

rcro AIC(MH.,

-

d,

PRoClssoR

+-(I

PC T O A I C (80 Ha)

DCA INA

A

---

-

3 RLOS 3 ACCIU 3 HVPS 3 PC I L I C I R O N I C S

-

-- -

PROCESSOR

NAV tLtCIRONIU

DCA

a

-

-

-

N

-

-,I

-

rc TO AIC

(80 HZ)

-I,

c rcc TO AIC POHZ) A

V

TO AIC

(C)

INSTALL

Figure 5-48 (a) IISA Sensor Orientation (b) INA Functional Block Diagram (c) IISA Functional Block Diagram (d) INA Installation Configuration (a) G. ( c ) j o m [Krasnjanski and Ebner, 19831, O 1983 IEEE (b) G. ( d ) j o t n uankovitz yr al., 19871 with permissionjotn AGARD)

238

Inertial Navigation

TABLE 5-2

CONP'IGURATION VS. ATTITUDEWELOCITY REDUNDANCY

Config

Attitude redundancy

Velocity redundancy

None Dual Dual Quad

None None Dual Quad

1 2 3 4

-

Unit 1 INS INS INS INS

(nonskew) (nonskew) (skew) (3NS+lS)

Chap, 5

Unit 2

-

Sensors only (skew) SD AHRS (skew)

-

INS (3 N S + IS)

From [Ebner and Wei, 19841, O 1984 AlAA

containing three gyros and three accelerometers, skewed relative to the INS axes s o that n o t w o are coincident and three are not in the same plane. (2) Configuration 2 would be virtually identical to configuration 1 except that a computer is added to unit 2 to solve the strapdown equations in order t o derive the aircraft attitude and heading. (3) Configuration 3 consists basically o f t w o identical INS (skew) to provide the required fail-operationlfail-operationifail-safe (FO-FO-FS) angular rate and acceleration outputs, that is, dual attitude and navigation outputs. (4) FO-FO-FS navigation can be achieved by adding one gyro and one accelerometer to each INS. Since the four sensors within an INS are bolted together rigidly, with proper geometry good navigation can be achieved using any three of thc four. Among the four configurations, configuration 3 is commonly used for redundancy. For this configuration, thc sensors are contained in t w o INA, each of which provides full, independent incrtial navigation outputs.

Effect of gyro dither.

Basically, IISA utilizes ring laser gyros that dcpcnd on a small amount of angular mechanical dither motion to avoid lock-in between C W and C C W lascrs to achieve full navigation accuracy. This mcchanical dither produces accelerations sensed by the acccleronlcters. If nonlinearitics occur somewhere in the proccss, diffcrent frcqucncics bctwectl gyro dithers o r aliasing with the sampling frequcncy can causc low-frequency beats to occur on acccleration outputs that could enter the FCS and causc wander or actuator flutter. The acccleration output from thc actual operational IlSA hardware with gyro dithcring is random with a noise arnplitudc of 0.05 ftlsec' (less than 2 milli - g) [Ebner and Wei, 19841.

Redundancy management.

Redundancy managcmcnt providcs FOFO-FS opcration through the usc of parity cquations which lincarly conlbinc the sensor outputs to dctcrminc thc residual crrors. Thc opcration scqucncc of rcdundancy managcmcnt is givcn in Figure 5-49a. Thc six axcs of skcwcd angular rate and accclcration arc sent to an cxtcrnal computcr for redundancy managcmcnt pro~

-

Figure 5-49 (a) Itcdundancy Management Operation (b) Trajrrtory Flow Chart and System Response Sinlulator (c) I'arity Flow Chart and llesign Equation Sinlulator (a) .font Oankovirz rr a/.. 1Y87/ luirh prn~~issiotl . f i ~ n A C A R D ( h ) G (0fro111 /Ebttrr and Wri, 19841, 0 I984 A I A A )

SlWSql DATA

OYRO AND A C C U R O U T S B SELF-TfST

0

o I/O A V A I L U I L R I ,o D Y X L X C E I 0 N A B L I : I E S S

I

1

i s taurrrons

o

o ISOLATE 2 S x n a t A N E o u s FAILORES

mon UNFILTERED omms FILTERED OUTPUTS MR P E R W N I FAILSELF-ADJOSTING TRIP E E L S

o RAPID DETECTION 0 O

( SENSOR SELECTION

+I

1

o SENSOR PERCORCUNCL INDEX RESULTS USE BEST 3 OR 4 SENSORS

o -0

o 29 COI4BINATIOll WJATIONS 0

LEAST SQULRE CSTItiATE FOR I) SEXSORS

1

I

h I G 8 T CONTROL OUTPUTS (A)

1 I

INA-1 TRAJECTORY

I

SYSTEM DYNAMICS MODEL

I

u G EFFECT

a GYRO DESIGN EQUATION

I I ? I l I I I LEVER ARM COMPENSATION

ERRORMODEL

ACCELEROMETER PARITY

QUANTIZATION FILTER ANDSAMPLER

I

ACCELEROMETER DESIGN EQUATION

I

Inertial Navigation

240

Chap.

cessing to derive body-axis rates and acceleration, free from the effects of any tx~, hard or soft sensor failures. The redundancy management consists of parity equations, sensor selection logic, and design equations. The parity equations ensure detection of up to three sensor failures by comparison of redundant sensor data. Sensor errors above a predetermined level (threshold) are defined as failures. Because of information limitations, a third sensor failure of the same type can be detected but not compensated. Sensor selection logic considers the state of all the parity equations and determines which sensors are to be used to derive angular rate and acceleration outputs. Based on selected sensors, design equations are used in Ebner and Wei [I9841 to derive the required outputs, removing skew angles and accounting for redundant data where applicable. Citing Jankovitz et al. [1987], realistic simulations have been performed, to evaluate the effects of factors such as vibration isolators, antialiasing filters, and misalignments on the redundancy management process. The redundancy management simulator consists of the following three programs: trajectory simulator, system response simulation, and parity and design equation simulator. The trajectory and system response simulator for trajectory and system response simulation are illustrated in Figure 5-49b. The system dynamic model in this figure is based on the sensor assembly model that results from rigid body dynamics, including effects of vibration isolators and gyro dither mechanisms. The input excitation is obtained from the trajectory sin~ulator[Ebner and Wei, 19841. The parity and design equation simulator is flowcharted in Figure 5-49c.

IlSA performance. requiremcnts arc:

Following Jankovitz et al. [1987], the performance

Radial position error rate Velocity errors, per axis Reaction time

1 nmiihr (1.852 kmlhr) (CEP) 3 ftlsec (91.44 cmlsec) (rms) 5 min

The specified accuracy of outputs to the FCS follows. Actual accuracy will be significantly better since the outputs are derived from inertial navigation grade sensors:

Scale factor Bias Alignn~etit llcsolutioi~ Range

Error sources.

Angular rate

Accclcration

4(X) drciscc

20 g

The primary sources of error associated with llSA follow. (1) Axis alignt~~cr~r emor: Achieving the 1-2 arc scc axis alignment stability needed, if a significant portion of flights is to contain tcrrain avoidance and cvasivc n~ancuvcring, rcquircs very careful design. (2) Atcelcratint~scale factor stability: Skewing 0f

Sec 5.8

Integrated Inertlal Naulgatlon System (IINS)

241

accelerometer axes requires that accelerometer scale factor stability be signifcantly better than for a nonskewcd configuration. (3) It~putaxis bending: Input axis bending is the major error source for strapdown navigators in a vibration environment. (4) Data synchronization: 1mportant.considerations for flight control are the time delays and synchronization of data from the IISA when used as part of a digital FCS controlling the states of an aircraft in real time. Data sampling and processing time delays in the sensor clement have a destabilizing effect in an aircraft control system and must be carefully selected. (3) Noise andfilterittg: Modern FCSs are digital, and sensor data are sampled at some fixed frequency. for example, 80 Hz for modern fighter aircraft. Sensor noise or vibration inputs at high frequencies can be aliased by the sampling process to a frequency within the flight-control bandwidth, causing control surface flutter or pilot discomfort. In light of the preceding error sources, extensive laboratory testing of the system must be undertaken to insure that IISA is suitable for installation and flight test in a flight vehicle. These tests are described in detail Uankovitz et a]., 19871, which examines IISA system performance for navigation and flight control.

5.8.4 Helicopter Integrated Inertial Navigation System (HIINS) Fingerote et al. [I9871 present the subject of HIINS, including the following discussions on accuracy requirements as they relate to configuration 1 shown in Figure 5-50. The roles of the military maritime helicopter include search and rescue, antisurface surveillance and targeting (ASST), antisubmarine warfare (ASW), and antiship missile defense (ASMD). Many of the missions must be carried out at ultralow altitudes under all weather and visibility conditions. The increased range, speed, and accuracy of modern weapon systems impose stringent accuracy and reliability requirements on the aircraft navigation system.

Accuracy requirements. In view of the often dangerous missions that must be carried out, the small crew of the helicopter must not be burdened with monitoring the functioning of, or updating, the navigation system. Consideration of these factors has led to the following accuracy requirements. 1. Radial position error (95 percent), with external aids = 2.0 nautical mi (nm), and without external aids = 1.5 nmlhr. 2. Radial velocity error (95 percent), with external aids = 3.0 ftlsec, and without external aids = 4.0 ftlsec. 3. Attitude error (95 percent), with or without external aids = 0.5 deg. 4. Heading error (95 percent), with o r without external aids = 0.5 deg. Another similar requirement proposed by Hassenpflug and Baumker [I9871 is listed in Table 5-3. Comparing with the preceding requirements, it is seen that the ac-

Inertial Nauigation

Chap. 5

Inertial Navigation

+

MIL-STD-1553B Data Bus

Reference

(h4ILSTD-1750A)

Figure 5-50 HIINS Configuration 1 (Front [Fingerore et a l . , 19871 with permission jom ACARD)

TABLE 5-3

PERFORMANCE PARAMETERS

Parameter

Range

f90°

Refreshrate [HZ 1 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 b.25 6.25 6.25

Accuracy (95

.5* .So .5%+.25kt .5X+ .25kt .6%+.2 kt .6%+.2 kt .5%+.25kt .Olg .Olg .Olg .25*ls .25@/s .25*ls 2% 300m/l/b h 1.

O++15Okm/h f 90. -25++100mls f l4mls f l5mls -45tr70aC 480t1100mb OIZSOOf t f90*/f180. 0 + 360' t5Okmlh flOO* t30' 10.1s

6.25 6.25 12.5 12.5 12.5 6.25 6.25 50 12.5 6.25 6.25 6.25 6.25 12.5

1.2m1s 1 Zmls 2mIs Imls 2*C*:T 1100: 3mE .5m 0.5% 0.5nm 1 I km 1 0.1. 0.6'16

INS kquircmcnts ........................................................................................ Pitch 8 -30 + 45. .So Roll 6 f 90. .5 * Heading True Heading Velocity along Velocity across Velocity vertical geographic vertical Ground speed Acceleration Acceleration Acceleration Angular rates Position(Enr0ute) Position(N0E) Drift Wind Direction TAS

:. Vx v Y z v v g

ax a Y *z P 9

r P.P

P. P 6 vW *W u v w

T e m p e r a t u r e static Static pressure Height above ground Target Desired T r a c k XTrack T r a c k An6le Error Roll commanded

WPT DTK XTK TKE

Turnrate

dqldt

-

*e

360. 360. -60++400kmIh f50kmlh f l5mls tl5mls -60++400km/h t. 5g f.5g -.5g++3.5g 100~1s 60n/s LOO*ls

From [Hassenpflug and Baumker. 1987) with pcrniission from AGAHI)

%)

HWS .25* .25O .5* .5'

.5%+.2kt .5%+.2kt .Z%+.lkt TED .5%+.25kt .01g .Olg .Olg .2*ls .2*/s .2*ls 1.5% 25Om/1/4h .5 1.2mIs 1'

2mIs 2mIs lm/s 2 * C + : T 11001 3mE .5m 0 .5X 0.5nm 1' I km 1 0.1' 0.6'1s

Sec. 5.8

Integrated Inertial Navlgatlon System (IINS)

243

curacy requirements given by this table have the same attitude and heading accuracy requirements and slightly different position accuracy requirements.

Configuration 1. Figure 5-50 illustrates HllNS configuration 1. The recomnlendcd system shown in this figure comprises an airborne processor to which four primary subsystem/sensors are interfaced by means of a MIL-STD-l553B serial multiplex data bus: (1) inertial navigation system (INS); ( 2 ) NAVSTARIglobal positioning system (GPS) receiver; (3) Doppler radar velocity sensor; and (4) magnetic heading reference. Also interfaced to the proccssor arc a radar altimeter, a T A C A N receiver, and an air data subsystem which supplies barometric altitude and true-air spccd information. T h e self-contained INS is a standard form-fit function (F3) inertial navigation employing ring laser gyros (RLGs).

Configuration 2.

Configuration 2 of HIINS (based on Hassenpflug and Baumker, 1987) and its 110 parameter are shown respectively in Figures 5-51. The HIINS is a heading- and velocity-augmented INS system, providing three-dimensional navigation information in conjunction with a radar altimeter and calculating the wind vector by means of a TAS system for the entire speed regime of the helicopter. The latitude range is ?80 deg ( U T M range). Angular rates and linear acceleration in the body-frame coordinate system for flight control and weapon delivery purposes are supplied by the INS. The autopilot functions are supported by the following signals: Radar altitude h~ Doppler vertical velocity v , ~ Magnetic heading +,L, Attitude +, 0 Velocities in the navigation frame

Inertial altitude h, Inertial vertical velocity v , True heading Body velocities v,, v,, v,

+

VE, VN,

bv

Besides calculating the present position coordinates, the following navigation functions are available: (1) bearing and distance to the selected waypoint; (2) time to go to this waypoint based on the momentary speed; (3) optimal steering information to the selected waypoint; (4) targets of opportunity; (5) position update by flying over known landmarks whereby the position coordinates of these landmarks are: (a) already stored, (b) read from the map and manually inserted after freezing the position flown over, and (c) gathered and inserted by means of a map display after freezing the position flown over. Figure 5-52a is another view of an integrated NAVJFCS for helicopters, a detailed discussion of which appears in Osder [1986]. It is a double fail-operative (fail-op2) system in which the navigation function achieves the same level of failop2 performance as the flight control computers. Each flight control computer (FCC) contains its o w n internal redundancy structures and techniques (hardware and software) needed to detect its own faults, and achieve the required reconfiguration to support the system's fail-op2 specifications. The emphasis here is on the

Inertial Navigation

------------

r----------

1 HUNS 1I

7I

I I I I

I

I

II I

I L

I

DVS RDN 808

I

II I

Chap. 5

t

Q-

mu RADM ALTII(E1ER

-

I I

I

-

1I

--

-

I I

I I

I

,

INS

I

I

, ucs

I

I

*

I

L

---------- -------

fl

-----_I

II

I

DUAL MIL-BUS lSS3B

Figure 5-51 (a) HIINS Configuration 2 (b) Configuration 2 110 Parameter (Fr*m [HasscnpJug and Baumker, 19871 with permission j o m AGARD)

Sec. 5.8

integrated Inertial Nauigatlon System (IINS)

245

system architecture advantages obtained by making the navigationlflight control sensor function an autonomous navigation subsystem by virtue of the consolidation of sensor measurements, state estimation algorithms and self-contained reversion1 recollfiguration stratcgics. Figure 5-52c illustrates the data flow interfaces between the integrated navigationlsensor assembly and the flight control computers and external sensors. The broadcast bus transmissions of flight critical data to the FCCs are summarized in Figure 5-52d. The tight coupling of data from various measurements is implied in Figure 5-32b by the position vector output which contains the four states of latitude, longitude, altitude, and GI's clock bias ( C R ) For . accurate navigation, the GI's clock bias as well as its first and second derivatives are included in the navigation filter.

5.8.5 Integrated Missile Guidance Systems This section discusses the design of integrated navigation, guidance, and control (NGC) systems for tactical missiles using optimum control and estimation theory based on Williams et al. [1983]. Sensors used in this design consist of strapdown accelerometers and rate gyros and a strapdown homing seeker. Guidance and control algorithms, in addition to performance studies, is presented in Book 3 of the series. The overall system is shown in Figure 3-53a. The design uses a passive strapdown seeker and a set of strapdown inertial sensors (gyros and accelerometers). These sensors provide the two LOS angles between the missile and target as measured in the missile pitch and yaw planes, and the missile body rates (P, Q, R) and accelerations (A,, A,, A,) along the three principal body axes of the missile. T h e LOS angles, plus information about the missile body rates and accelerations as provided by the inertial sensors, are inputs to the navigation and FCS. The integrated N G C system is shown schematically in Figure 5-33b. Since the guidance algorithm requires knowledge of the inertial LOS rate, which cannot be directly obtained from the output of a strapdown seeker, an algorithm by which these rates can be estimated given the angular inputs is essential in order to achieve good performance. In the design of Figure 5-53b, these rates are not obtained by a finite difference algorithm which could produce serious errors in the presence of noisy o r infrequent data, but instead an extended Kalman filter (EKF) includes these rates as state variables. The EKFs are part of the navigation filters as shown in the figure. Since both channels have identical dynamics, the navigation filters for each channel are also identical and have the structure shown in Figure 5-53c. The block diagram is intended to show that the output of the filter u, as well as the state estimates 8 and 4 are continuous-time variables, whereas the inputs to the filter are discrete-time variables. (In actual implementation, the continuous-time variables are also discrete-time variables, but the sampling interval is much smaller.) The navigation filter automatically extrapolates the state estimates from one observation until the next in accordance with the missile dynamics. This is very important when the seeker-sampling interval T, is relatively long compared with the dynamic behavior of the missile. It also automatically solves the problem

Inertial Navigation

Chap. 3

Sec. 5.8

Integrated Inertlal Naulgatlon System (IINS) Parameter

DesCrIDtlon body angular rates body I l n e a r accel. P l t c h , R o l l . Iltadlng

vn. vc,

L.A.

Yz

h l m d other coordlnatc systea equivalents)

North. East. Dan Veloclty Latitude, Longitude, A l t l t v d e

Baro A l t l t u d c Calibrated Alrspeed True Airspeed A t r Dtnslty A l r Temperature ('=)

El .

Figure 5-52 (a) Integrated NavigationlFlight Control Overview Block Diagram (b) Flight ControllNavigation Interfaces (c) Summary (d) Typical Strapdown Block Diagram for Low Cost/Performance IMU with Kalman Filter Updating (From [Osder, 19861, O 1986 IEEE)

Inertial Nauigatlon

Chap. 5

Sight Angler

--

.-PI

*rr.L-

C001101NATE TIIANSFORUTIO*

YziT.la

-

A"TOPILOT

".A

b

----C

CO*I"OL SURFACE DEFLECTIONS

. 8

0

V

DIIECTIOY

UKI*E

UlC"

I O U "A.

UIlllX l*TEt"AT,O*

Q

tf'

P R

Figure 5-53 (a) Overall System (b) Integrated Guidance and Control System Schematic (c) Navigation Filter Schematic (From /Mrilliatns rt al., 19831, 0 1983 A I A A )

Sec. 5.8

Integrated inertial Naulgation System (IINS)

MEASURED INERTIAL LINE-OF-SIGHT

of seeker blinding which occurs when the target conlpletely fills the field of view of the seeker and the latter can no longer provide useful directional signals. When this happens, the switches shown in Figure 5-53c remain open and the missile continues by dead reckoning to the interception.

Another example of an integrated seeker-guidance navigation system. In a classical design, a weapon guidance system includes two distinct inertial sensor sections. Figure 3-34a showsthe separate flight control using a reference gyro for yaw and pitch corrections. Also shown in this figure is the additional seeker equipment, which in this case is an optical device including a tracker, that relies on a second reference gyro system for LOS stabilization. Figure 5-54b(i), however. shows one stra~downreference svstem made UD of the reference unit and the strapdown computer. This single unit is capable of referencing both the flight control and the seeker system. Figure 5-54b(ii) shows how future weapon system guidance and seeker-processing tasks will be integrated into one processor. Figure 5-54c illustrates a structure of the navigation software development system (NSD). Figure 5-54d depicts a block diagram for the LOS control. The gimbal control including the controlled element (gimbal and torquer) together with a high-bandwidth angle transducer are both contained in the inner loop. The nonlinear characteristics shown in the figure represent the effects of friction found with all gimbal

.

Inertial Naulgation

(B)

Figure 5-54 (a) Functional Diagram ofa Classical IS.~S lr.>~lu.>p! I I V ..>IIPI,~ 10111111.1 .11311!5 e JOJ SI!M .1.1.>11 IIMOIIS III.>ISAS ~11.1,. J ~ I ~ I I I O J ? ) J ~ I I I ! 3111uue~s r pas11 ((0~61) ~ . > ~ ! ~ . > a ~ S t ~ !soleL r u o q "13 2 1 1 (q) ~ J ~ I ~ C I I O I ~ J J (c) W I I€-9 a~n%!a

Guidance Processing

336

Chap. 6

where o is the microwave frequency and 6 is the electrical phase difference at the antennas. A , and A2 denote the amplitudes, which could be different. The phase of X2 is advanced by an amount @(t) by the scanning phase shifter according to @(t) = o,t (P where o,is the scan frequency and (P is the initial phase shift. X, can then be expressed as

+

X3 = A2 sin

o

nd . + o, + 2 uL cos UL A

Adding XI to X3 gives X4, which can be expressed as

X4 = A1 sin wt

+ A2 sin

o

7 ~ d. + o, + 2 UL cos UL A

This last result shows that X4 is a carrier signal at w and is amplitude modulated a t nd . o, + 2 - UL cos

U'.

Although the carrier frcquency was changed by the heterodyne process in the receiver from microwave to a lower frequency, this did not alter the basic u L information since the nlodulation was unaffected. Moreover, since thc desired information was the frequency ofthe n~odulation.the nleasurcment ofuL was not directly affected by changes in the amplitude of the signal. A body-mounted rate gyroscope was used to decouple body motion to provide the LOS rate measurement u,and the gyro output frequcncy modulated an osc~llatorto produce a carrier frequency wo with a deviation proport~onalto the missile-body turning rate 0. Because a fixed value of K,xvhose term in the gyro channel is an esti~natcof cos DL, for all flight conditions turned out not to be satisfactory. Talos relied on a sclcction of two values based on thc crossing component of the target speed. This value was sct into the missile at laut~chtime. The scanning ~ h a s shiftcr e frcquency o,\vas eliminated fro111 the final term by the use of double modulation, which allowcd a frcquency discriminator tuned to w,, to providc an output that was proportional to (2ndlA) ( ~ -8bL cos uL). Ignoring biases, the discriminator output was a good approximation proportional to the dcsircd u.

Stable platform phase follow-up system (STAPFUS). The first T a l a homing system suffcrcd from t\tro critical problcms, the first of xt~hichwas that a bias resulted from anv oifsct bct\vccn the voltarc-controlled oscillator and the discriminator center frcqucncy. The second major problem was that it \tias difficult to maintain a stablc gyro gain factor equal to (2ndlA)h!. The solution to both of these problems came in the way of thc stablc platform phase follow-up system (STA1'FUS). Figurc 6-31c shows the scannit~gintcrfcromctcr using STAI'FUS. AS writh thc initial systcnl, a motor \vai used to drive the phase shiftcr and a rcfcrcncc crator. I'hasc corrections for nlissilc body motion were llladc by a singlc-dcgrccof-frccdotn frcc gyro coupled to a synchro-resolver. The gear ratio bctwccl1 the gyro and rcsolvcr providcd a constant gain factor for thc gyro coupling. The difc,

Guidance Processfng

Chap. 6

Range

(A)

Figure 6-32 (a) Pulse radar signals from a moving target can be obscured by large, stationary reflecting surfaces such as the sea or land masses. (b) Continuous wave (CW) radar signals from a moving target can be detected by filtering at the Doppler frequency. (c) The rear reference receiver used a phase-locked klystron and a groundaided acquisition (GAA) loop. The klystron frequency was coarsely corrected by the discriminator to the frequency of the illuniinator signal, followed by exact frequency control by the phase-locked loop. The voltage-controlled oscillator frequency was offset from the intermediate frequency by the j ~ a . 4loop and caused the klystron frequency to be offset precisely for the target signal entering the front receiver (see Fig. d). (d) A target signal having a Doppler frequency equal to.f~fcs.< would pass through the amplifiers and narrow-bandpass quartz crystal filters and be detected by the acquisition circuit. Upon detection, the discriminator output readjusted the frequency of the voltage-controlled oscillator to keep the target signal in the center of the receiver bandpass. The narrow-bandpass filters removed the land and sea clutter signals entering the antennas. (e) The monopulse seeker was a refinement of the original C W seeker. The target andlor jammer signals were processed through a two-channel receiver (per guidance plane). Narrowbanding occurred alniost immediately following the microwave mixers. The intermediate frequency (IF) amplifiers \r.ere hard limited on receiver noise. The Doppler tracking e oscillator and the first mixer. Predicted loop was closed through the n ~ i c r o \ ~ a vlocal Doppler inforniation provided by the ship was used to aid target search. The homeon-jamming niode eniployed the same narrow bandwidth and angle processing circuits used for tracking the target echo. (0Guidance-control logic used to ilnplement guidance switchover from midcourse to homing, to gate the steering information once homing began, and to control Doppler-search routines was based on the coherency of the angle data. The signal used for that was obtained from a phase detector that monitored the STAPFUS phase-tracking loop. (g) Multiple jammer flight tests were conductcd to demonstrate dichotomous angle-tracking capabilities inherent in the Talos interferometer guidance system. Briefly, if signals froin tiiultiple sources were present siniultaneously, the n~issilewould track one of them if a power differential of 2 decihcls or greater cxistrd within the narrow bandwidth of the receiver. In practice. the niissile ufouldnearly always intercept one ofthe targets. For thc riiultiplejanin~ertest illustrated here, the primary target was a P-4Y aircraft with a rioise jammer. After achieving a near miss on the aircraft, the missile successfully intercepted the westernmost jammer on the ground. (h) The effect of selfprotection electronic counternieasures (ECM) o t ~riiiss distance is illustrated. The curves were derived from miss distance data obtained froni laboratory ECM tests conductcd a t the Naval Ordnance Laboratory. Corona. Calif.. froni the monopulse flight tests against jamming targets, and froni historic test data of Talos flight tests against nonjamming targets. Note that the pcrforniancc against the janiming targets was superior t o that against the tionjamming targcts. (0 1982 b y Joh~tsHirl>kittrAPL Techni(a1 Digc~.sr)

-

7 To STAPFUS measurement system

Target tracking loop

Bandwidth = 1600 Hz >

Target -C

F i wnptifii with Mixer

namM

VJ*

2.5 MHz_ crystal filter

fihu

Interferometer antenna system

2 3 MHz

-

Rear antenna

F i g 10

Figure 6-32

(Continued)

64

Guidance Algorithm

-

T a m reaiver Doppler rcquirition m d mek

Reference receiver -Predicted Dopplw (GA.4)

Rear

1

!

antenna

-TO

autopilot pier search control Switchover. midcourse to homing

(F)

Figure 6-32

(Continued)

Guidance Processing

0.4

V"#K&&

Laboratov ECM tests

--

Flight tests (monopulre) with ECM. 9 tests. 6 direct hits Flight tests without ECM (Tala history)

0.2

0.1 Miss distance

(H)

Figure 6-32

(Continued)

6.4

Guidance Algorithm

343

Figure 6-33c shows a block diagram of the phase-locked klystron loop and the ground-aided acquisition loop. Just before launch, the klystron was electrically pretuned to the approximate frequency of the illuminator. Approximately 15 sec prior to intercept, the ship-based CW illuminator began radiating, and automatic frequency control pull-in occurred. Using radar data, the target Doppler frequency, fD, was computed and then transmitted to the missile by a 400-kHz plus fD frequency modulation on thc illuminator frequency. The ground-aided acquisition loop used this estimate of the Doppler to establish the initial frequency for the voltage-controlled oscillator. By doing this, the klystron frequency was not very far off from the correct value, and the search required to acquire the target was minimal. The front receiver with its narrowband quartz-crystal filters and discriminators, together with the Doppler-tracking loop, is shown in Figure 6-32d. The function of the loop was to control the voltage-controlled oscillator and, therefore, the klystron so as to keep the target signal in the center of the receiver's narrow bandpass. The SAMN-6cl missile with its CW homing receiver was well suited to a countermeasures environment. First, the narrowband characteristics of the seeker and the short (610 sec) homing time made for limited exposure to hostile environments. Second, the high-speed phase-tracking loop in STAPFUS provided a nominal angle-tracking sensitivity (volts per deg per sec) when it was tracking, independent of the SIN. Finally, the Talos phase-interferometer angle processing was intrinsically capable of resolving signals from two or more separate sources such as multiple noise jammers if there was a small power difference ( 5 2 dB) between them. There was, however, some uncertainty as to whether the CW guidance system would perform well against surface targets. In this application, the target signal and the sea clutter signals do not have a useful Doppler frequency separation. However, 'the combination of the 800-Hz and 100-Hz filter bandwidths maintained a signal-to-clutter ratio suitable for guidance, and the Talos system with CW interferometer guidance performed well against surface ships and boats.

Monopulse Homing System. The performance of the original SAM-N-6cl missile was exceptionally good against several types of jamming. Unfortunately, the missile was limited in its capabilities against certain types of deception countermeasures owing to its RF sequential lobing (scanning) angle processing, its slow automatic gain control and its time-consuming Doppler search routines. One of the foremost objectives of the rnonopulse seeker design was to enable the missile to win any o~e-on-oneencounter with an aircraft, regardless of the form of AM andlor FM noise or deception jamming the latter might employ. The result of this effort was the development by the mid-1960s of a seeker that was virtually unjammable by an attacking aircraft. One requirement for the Talos rnonopulse seeker was that it be compatible with the existing STAPFUS, which, as described previously, re(2adlA)sin uL)signal. As shown in quired a scan reference (w,) and a scan (w, Figure 6-32e, the Talos monopulse receiver was designed to provide these two signals. The receiver is somewhat similar to the scanning receiver, with the exception that a pseudo-scan was introduced at IF following narrowband filtering, which

+

344

Guidance Processing

Chap. g

involved offsetting the frequencies of the second IF amplifiers by an amount equal t o the pseudo-scan. By choosing the scan reference to be greater than the bandwidth o f the narrowband input filters, it was made certain that the pseudo-scan would be invulnerable to ECM. A deficiency of the Doppler search routines in the original C W seeker was that target reacquisition was slow or in some cases could be prevented altogether. T w o factors that played a significant role in this deficiency were the relatively slow sweep-repetition rate and the complete reliance on memory for positioning the sweep. The design of the original search patterns was based on providing a high first-look acquisition probability under minimum signal-level conditions. This imposed a limitation on the maximum search speed that opposed the ECM requirement for fast reacquisition. The reason for this last requirement is that a jammer can cause the seeker to lose acquisition repeatedly throughout the homing phase. For the monopulse seeker, a compromise was pursued in which the sweep speed was increased such that the single-look probability for the small signal case was lower than previous but, after 1 or 2 sec, was comparable as a consequence of the increased number of looks. Late in the homing phase, the probability of single-look acquisition would be essentially unity because of the relatively high target signal-to-noise ratio and, consequently, the desired fast reacquisition was achicved. Also, taking advantage of the predicted Doppler information provided by the ship during the homing phase helped solve the problem of positioning the Doppler search center. The control of critical guidance functions was based on the coherency of the angle data, which made it possible to resolve a s~gnalemanating from essentially a point source forward of the missile (targct skin ccho or a jammer) from other signals such as sea clutter, scattered chaff, reccivcr noise, and so on. An HOJ served to delay guidance switchover from midcoursc to homing in cases where acquisition of the targct ccho was not achieved immcdiatcly. This averted immediate HOJ on a stand-off jammer. Figurc 6-33f sholvs the guidance control logic.

Tests of Continuous-Wave and Monopulse Homing System. The results of cxtcnsivc laboratory tcsting of the CW and monopulse homing systcms strongly indicated that the Talos missilc with the tnonopulsc sccker could not be bcatcn by virtually any sclf-protcction noise or dcccption jammer. Twenty-fivc of the 26 valid flight tcsts of thc Unificd Missile wcrc SUCCCSS~UI. Ninc of thcsc 25 were with the ~nonopulscscckcrs aqainst -. a wide sclcction of ECM typcs and paramctcrs. Two of the successful tcsts \trcrc against multiplc jammers, one of \vhich is shown in F i ~ u r c 6-32s. Talos' pcrforn~anccagainst both jamming and nonjalnming targcts is illustrated in Figurc 0-32h, in which it can bc sccn that Talos pcrformcd bcttcr against jamlning targcts. This is not altogether surprising \vhcn onc considers that a jam111cr generally provides point-sourcc cnhanccmcnt of the targct. 111 othcr words, the janlmcr is literally providing a beacon on u~hichthc missilc can homc. Dec~elopmentof Antiradiation Hotninx ( A R H ) Missile Guidance. As disC U S S C ~in Gulick ct al. 11982], it wras not long aftcr the Victnatn conflict brokc out that thc nccd for an cficctivc long-rangc nntiradiation missilc (AIIM) to suppress

Sec. 6.4

Guidance Algorithm

345

enemy radars became evident. Because of its long-range capabilities and the ease with which it could be adoptcd for new missions, Talos was seen as both a logical and dcsirablc choicc for radar supprcssion. The Talos ARM program rcquired the development of a unique operational concept because the ship could not conceivably track the targct and implement missile guidance as it did for the cngagcment of air targets. The techniques for providing ARM with the appropriatc targct information cot~sistingof !geographic location and RF emission characteristics (ircqucncy, pulsercpctition frecluency, and so on) were as follows. First, several navigational techniqucs, including the Navy Navigation Satcllitc System, were used to establish ship coordinates. The ship then directed the missilc to thc vicinity of the targct using the beam-rider midcourse guidance. Upon approaching the targct, the missile was put into a dive and the homing system was activated. The missile approached the targct in one of two possible terminal geometries, either an approximate 45-deg dive or a near-vertical dive. Because the low pulse-repetition frequency of some target radars was not compatible with the scan frequency used by the carlicr semiactivc scckcrs, it was required foremost of the ARM homing system that it be monopulse. It was also desired to have a system with a high sensitivity so as to provide continuous guidance on the low sidelobc levels from thc targct. Finally, thc ARM guidance system had to be compatible with the existing Talos airframe. Figure 6-33a shows a simplified block diagram of the ARM seeker. The receiver used two parallel IF amplifiers (per guidance plane) with subsequent in-phase and quadrature processing. By using limiting amplifiers, a very large instantaneous dynamic range greater than 120 dB was achieved. Circuits were carefully designed so as to maintain the differential phase shift between these amplifiers at a low level, and good phase tracking was made possible with input peak power levels greater than 10 watts. For very low-frequency targets (L band), it was considered undesirable to have a guidance error associated with even 10 electrical degrees differential phase shift between the receiver channels. By adding microwave and IF transfer switches that, on a pulse-to-pulse basis, allowed radar pulses to be processed alternately through one channel and then the other. the error was all but eliminated. With these switches. the internally generated errors would average zero. Figure 6-33b shows the measured angle error arising from differential phase shift plotted against the signal power level. The two curves in the figure show the errors with and without the transfer switches operating. The acquisition and discrimination channel was placed in parallel with the angle channels, and the receiver was self-gating. A signal that met the radiation and pulse repetition frequency requirements designated at launch would provide a gate to the angle channel. The seeker could discriminate between two signals as long as these differed in their operating frequencies by at least 3 MHz. In order to minimize the effects of multipath reflections near the target, leading-edge gating was used. The outputs from the angle phase-comparator were multiplied by a 400-Hz signal to provide an amplitude-modulated signal to drive the STAPFUS resolvers. The seeker measuring the LOS rate to the target at the time of acquisition provided geometric discrimination. The ship, utilizing the midcourse beam-riding guidance, caused the missile to dive toward a point approximately four miles beyond

Guidance Processing -Guidance -Gating -Frequency

Ckp. 6

signals and switching control

B channel guidance plane

..,

~

.

I

ating ignal

Target reject from angle rate renror

I

-", 0.8 u

'0

f

0.4

I

-

-1 5

I

I

I

I

I

I

I

T

400 Hz

I

Transfer switches out ---Transfer switches in

-25

-35

-45

-55

-65

Input Power (decibel meters) (B)

Figure 6-33 Talos A l l H Missile Guidance (a) T h e Talos Antiradiation Missile (AIIM) seeker used t w o parallel IF amplifiers per guidance plane with subsequent in-phase and quadrature processing. A signal that satisfied the prelaunch frequency and pulse repctitiot~frequency requirements provided the gating signal to the angle channel. (h) Angle bias resulting from diffcrential phase shift bct\vcen the recciver channcls was virtually eliminated with the use of radio frequcncy and of IF transfer switches that operated on a pulse-to-pulse basis. 1C 1982 by Jokns Hopkit~sAPL T r ( h ~ i i ( aDixcst) l

Sec. 6.5

Guidance Law

347

the intended target. The resulting missile-to-target LOS rate was in the down direction. In the cvent of a failure to detect that downward angular rate during the target acquisition process, the signal was rejected and the seeker continued to search for another target. If the target was accepted, the missile executed a down maneuver to the target. In this way, the elevation angle at which the missilc would intercept the target was nearly 90 deg. According to Gartcn and Dean 119821, although specific details of Vietnam combat applications cannot be released, the radiation-seeking capability of the Talos ARM (RGM-8H) was demonstrated by the fact that shore batterics were silenced by missiles fired from ships placed in the Gulf of Tonkin. Because of its target discrimination capabilities, Talos was also able to effect long-range kills of MIG aircraft over land. The long-range intercept capacity put enemy air tactics in check whenever a Talos ship was nearby. The opportunity to use the surface-to-surface capability never presented itself.

6.5 GUIDANCE LAW

In order to intercept the target, a missile must constantly travel in the proper direction. The direction in which a missile travels is dictated by an algorithm built into the guidance processor known as the guidance law. Many different guidance laws have been developed over the years, and with the advent of highly maneuverable airborne targets, research on improved guidance laws is continuing [Heaston and Smoots, 19831. Guidance laws currently in use on existing and fielded missiles may be inadequate in the battlefield environments envisioned for the 1990s and beyond. Performance criteria will probably require applications of newly developing theories. which in turn will necessitate a large " comvutation capabilitv relative to classical guidance technology. Until very recently this task has been relegated to large computers, in many cases ground based. However, advances in microprocessors will allow increased use of onboard computation [Fraser, 19811. Geometry and other parameters for homing guidance are drawn in Figure 2-10. Currently, several guidance laws of the two-point (target and missile) or three-point (target, missile, and station) type are implemented in homing missiles. As shown in Figure 6-1, current guidance laws, whether of the two-point or three-point type, can be classified as LOS guidance (Section 6.5.1), LOS rate guidance (Section 6.5.2), or advanced guidance (Chapter 8). It must be pointed out that a guidance law design, to be made complete, requires guidance filter techniques such as those presented in Chapter 7. Also, sensitivity analysis and a comparison of these guidance laws will be presented in Section 6.5.3 and in Chapter 8. Figure 2-10 shows the basic geometry. In command guidance, the target and missile dynamics are separated because the radar tracks them separately. The command guidance geometry is described in Figure 6-23c for the head-on case while the semiactive homing using the standard PNG is described in Figure 2-10. The dynamics are linearized about the LOS vector at the start of the terminal portion of guidance.

Guidance Processing

348

Chap. 6

6.5.1 LOS Angle Guidance The guidance law definition used by Locke [I9551 is followed here and depicted in Figure 6-34. The LOS guidance (either CLOS or beam-rider system) in this figure is designated as a three-point guidance system. As shown in Figure 6-23a, a beamriding missile generates its own commands internally whereas a CLOS missile re, ceives its commands from a remote station. These two schemes, however, have essentially the same lateral acceleration command mechanism. From Figure 2-10,

=

- vmsin ~ l ~ < d / R ,

or

sin ul,,d

;=

- R,uIV,

(6-5 b)

Differentiating Equation (6-5b) gives Since A, = Vm(6 - ul,,d),using Equations (6-5a), (6-Sb), and (6-5c) A, becomes

A,

=

2bVm + R,ulcos ulecrd+ Vmtan

UI,,~

(6-5d)

which represents the lateral acceleration required by the missile to stay on the LOS. Combining Equations (6-ja), (6-5b), and (6-5d) yields the kinematic transfer function

where 0, replaces u to emphasize that this is the angle subtended by the missile (see Figure 6-23b) [East, 19841. The LOS guidance loop which is a two-integrator system becomes unstable under small perturbations. A stabilizing compensator is thus required [Durieux, 1984; East, 19841

A LOS guidance loop is shown in Figure 6-35a. The loop gain is maintained constant by incorporating the missile range R, in the feedback loop. This in turn ensures a sufficient loop bandwidth. The beam stiffness K, is adjusted to trade-off the tracking response and stability against sensor noises. Finally, the filter G(s)in Figure 6-3% is used to provide cnough phasc stability. The inverse kinematics in a feedforward fashion (Figure 6-35a) arc includcd to allcviatc the need of high loop bandwidth One example implc~ncntationbased on Equation (6-5d) was donc in Equation (6l b ) whcrc 0r replaces u.

6.5.2 LOS Rate Guidance The homing guidancc systcm, which contrasts with the LOS guidance, is designated as a two-point guidancc system and is implemented mostly as LOS rate guidance.

Sec. 6.5

GufdanceLaw

Figure 6-34 AGARD)

Guidance Law Types (Froin /Goodirein, 1972(b)/ wirh p e n n i s i i o ~ ~ j r o ~ n

Figure 6-35b depicts a conventional LOS rate guidance loop for both command and homing systems [Alpert, 1988; Durieux, 19841. For easy comparison, the LOS rate guidance in this figure is arranged correspondingly to the LOS guidance in Figure 6-35a. In other words, a derivative network is cast in the guidance computer. . Since there is only one pole at the origin, the loop stability is not likely to be a problem as in the case of the LOS guidance, at least until near interception when l / ( V c t g* ) 30. The trade-off between tracking performance and high frequency stability is thus alleviated. However, it is still highly desirable to keep the LOS rate guidance loop bandwidth as small as possible [East, 19841. If K, = A t k = A/ ,,t ki = 0, then the LOS guidance in Figure 6-35a is identical to the command guidance using a synthetic PNG in Figure 6-33b(i). These gains are not sufficient to keep the missile within the tracker beam when t, is large. Similar to dynamic lead guidance to be presented later, it is thus necessary to reserve the synthetic P N G (see Figure 6-35b(i)) for the intercept phase and to use LOS guidance during the initial phase [Durieux, 19841. LOS rate guidance is defined by

where y, is the commanded flight path angle, h is the navigation gain, and C is the constant or initial flight path angle yo. The guidance information is illustrated in Figure 6-36a. As shown in the figure, a body-fixed or strapdown seeker senses the look angle, UL; a gimballed, inertially stabilized seeker senses the LOS rate, u;and an airstream stabilized seeker senses the lead angle, ale,&Radar seekers can be used to sense range and range rate (closing velocity), as long as there is amitable waveform and reference signal information. Other information, such as that dealing with missile airframe motion, may be required to implement a guidance law. There are

,--------------

1I

Guidance Computer

I

I

I Range Stiffness L Estimate -------------A r--------1

em

I

Kinematics

I

L--------J

(A)

r - - - - - - - - - - - - I I

Guidance Computer Derivative Network G(s) s

1 I I

-

-

t = Y (Approximate for Figure (A)) Qr = el.f\T - Om 8 = R e c o n s r m c t e ~ v e dLQS Angle For Synthetic PNG, K, = M, = u, = A V, 8

(i) Command Guidance r---_-___1 ----

!

Guidance Computer

I

I

a = Geomebical LaSAngle For Standard PNG, Kg= &'$ = u, = A V, b Independent of R

(ii) Homing Guidance

Airframe Dvnamics -8.

or q,,,

-

-

0-7

I

Target Motion

'Feedback with Body Pitch Angle 8: Anirude Pursuit "Feedback with Flight Path Angle y: Velocily Pursuit

(C) Figure 6-35 (a) LOS Guidance (b) Conventional LOS Ratc Guidance (c) Pursuit Guidance ( ( a ) from [East, 19841 with permission from ACARD; ( b ) from [Durieux, 19841 with permissiotl from ACARD)

Sec. 6.5

Guidance Law

(iuidancx: Law

With l'erfect Response

C:ummond to 1- (hidance (&.am R i k r o r CUX)

r =O

1 Atlitude Pursuit

Guidance Equation Lateral h l e r a t i o n Command U, = Vm f c I(g G(s)lR, €1

look angle oL = 0, u = 0, A = 1. C = 0

= 0, o = y. k = 1. C = O

Vclwily Pursuit

lead angle u

Ckvialed Pursuit

lead angle olead = preset v n l u olCs4

$'"=

ft off the beam $(u-8)

$ole,,= s ( o - Y )

- ole.4)

('onslant b a r i n g Proportional Navigation tiiicbnce (PW,)

o = cro = constant. A = q C = 0

Kg ( t -~ u,)

; = 0, = A h A =

V, cos yo A

C o ~ ~ s rPNG al

A=O

A V, & 1cos uL

Extended PNG Velocity Compensated PNG

&=O

AV,~Icoso,,-V,tanoL

&=o

A V, &

(vcpffi)

constant

;o r

A V, &

- i~,,, s i n atead

Figure 6-36 (a) Geometry and Parameters for Homing Missile (b) Basic Homing Guidance Law ( ( a ) reprinted w i t h p e r m i s s i o n ~ o m(Vriends, Systems ?t Control Encyclopedia), Copyright /1987), Pergamon Press PLC)

several useful guidances which are popularly used in missile-guidance systems, as shown in Figure 6-36b. Many guidance laws currently implemented in homing missiles are of the two-point (target and missile) type, and these can be based on pursuit, proportional navigation, or both [Vriends, 19871. Pursuit guidance and PNG can be implemented in a guided-missile system using only LOS angle and angle rate information, respectively. However, an additional ground station is required for a command to LOS guidance scheme where the missile attempts to follow an EO beam directed at the target (see Figure 623c(i)). These three classical guidance schemes are shown in Figures 6-35 in. the

Guidance Processing

352

Chap. 6

form of loop flow charts, and are compared through simulation in Table 6-1. Both command to LOS and pursuit guidance laws are shown to have limited capability to engage maneuvering targets. O n the other hand, P N G guidance can be implemented fairly easily. PNG and its variants reduce the maneuver requirements and can produce good miss-distance performance [Hiroshige, 19861. It has a long history of acceptable performance and has been used successfully in several fielded-missile weapon systems. Nevertheless, it also has its limitation in engaging targets with a target-to-pursuer velocity ratio that is significantly greater than unity, or targets with higher maneuvers. The maximum normal acceleration force required by PNG is much less compared to that required for turning a pursuit or command to LOS guided missile under any combination of disturbances. If no filter is provided in the PNG, more drag and control command saturation can occur in the case where the guidance system responds to seeker noise than occurs in the case where it responds to true geometric guidance information. Both pursuit (attitude and velocity) schemes are sensitive to biases. The miss distance is proportional to bias errors. However, when there is no lag, that is, the guidance system time constant T, = 0, the sensitivity to biases theoretically goes to zero for the PNG law since differentiation of the constant bias of the LOS yields zero. The performance is limited by the control lag T, in practice in any general LOS rate guidance system [Durieux, 1984; Neslines and Lin, 19871. For a sampled data system where the last command is held for an interval of time, decreasing the miss distance does not zero out the bias error [Leistikow et al., 19671. Table 6-2 summarizes the advantages and disadvantages of the different guidance laws when they are used in combination with classical low-pass noise filters. The state-of-the-art pneumatic actuator weight increases with sideTABLE 6-1 COMPARlSOtl OP GUIDANCE LAWS POR SHORT-RANGE TACTICAL MISSILES Pursuit Ability to engage Urgnr Armracy (It CEP) Mmncuvmb'~lity A d d i d d criteria[ Yd(f,) l ~

~

~.~~ -~. ~.

State required~ On-board gyro (re0 Gimbal mshanization (seeker) On-boud clstronics ~

~

Cost (on-bwd) S m r requirements

Airfrunelpopulsion requirements

T a i u l considerations "nre and forget"

From [Pastrick

et

21..

CLOS

Beamrider

Attitude

Velocity

PNG

Optirml

M

Low No

Low No

V T . The only indeterminacy left is the choice for y , which can only be one of two possible choices since V,. sin(u - y ) is specified. The choices f

360

Guidance Processing

Chap. 6

are yl or a - yl since yt = sin(a - y,). This amounts to choosing the pursuer velocity vector so as to close the gap along the range vector (pursuit) as opposed to widening the gap (evasion). If the target does not maneuver, then the sit;atiO" shown in Figure 6-37b occurs. The reason is that the components of velocity perpendicular to the LOS are equal so that the two missiles are going the same Speed . . in the ~erpendiculardirection. However, at the same time, the radial distance is decreasing continuously and eventually goes to zero. This is also apparent from the equation for R (even for the maneuverable case). That is, suppose V , cos(a - y) > VT COS(O.- YT),

and the integrand is always negative (which implies Ro - I $ [ ] d t I should eventually equal zero, that is, a definite interception). Missiles that possess sufficient maneuverability permit the launch aircraft to be guided with a constant or programmed bearing angle [Maksimov and Gorgonov, 19881. The missile will have to perform postlaunch maneuvers in the case of a nonzero bearing angle, and some of its fuel resources will be depleted in an effort to get on the required trajectory. It appears that constant-bearing guidance might be the solution to all of the problems since it will intercept a maneuvering target and is not as wasteful as pursuit guidance. However, constant-bearing guidance requires the pursuer to be able to detect and correct instantaneously any changes in the LOS direction. Realistically, this would be asking too much of any guidance system. Since constant-bearing guidance has so many good qualities, it seems feasible to seek a slight modification of the scheme \x.hich will make it easier to implement while preserving its good intercept characteristics. O n e possibility is to treat u as an input error signal which the pursuer drlves to zero by adjusting y. A simple way of doing this is to require j = X u since y = constant when a = constant and y changes until jr = 0. Actually, such a guidance scheme is called PNG and, indeed, it is the most popular homing guidance scheme. The development of PNG was a major breakthrough in homing missile guidance. The fact that a constant-bearing course results in a collision course led to the development of the PNG law. PNG is one in which the missile heading rate is made proportional to the LOS rate from the missile to the target. The purpose of such a course is to counter the tendency for the LOS to rotate and, hence, to approximate a constantbearing course. The initial turning rate of a missile launched on a collision course and using PNG is zero. For a nonmaneuvering target, if the missile and target velocities remain unchanged, the missile will continue on a collision course, having achieved constant-bearing navigation. If it happens that the missile is not launched on a collision course, there will be some curvature associated with the resultillg trajectory that depends on the navigation constant. For a small navigation constant A, the missile corrections are small early in the flight, yet may become quite Pronounced as the missile nears intercept. The situation is reversed for larger values f

Proportional navigation guidance (PNG)and its variations.

Sec. 6.5

Guidance Law

361

A, where the collision course errors are corrected early on in flight, and maneuvers conscqucntly are kept at reasonable levels in the terminal phase. For values of A 5 2 (A = 1 is pursuit guidance), there is an infinite accclcration required in the region of terminal flight. A lower limit therefore corresponds to A = 2. For valucs of A close to 8, the missilc steers in response to very high-frequency noise as well as to lower-frequency signals. This situation is characterized by gross oversteering with a constant dithcr o f the controls and high drag [Trottier. 1987). A reasonable value of h based on experience is usually set at between 2 and 4. 'The trajectory of a pursuer operating with a PNG law is also plotted in Figure 6-37b. A purser turning rate of three times that of the LOS rotation rate was used to plot the trajectory. As in the Figure 6-37a, the missilc was launched in thc direction ofthe target. T h e rotation of the LOS is measured by the seeker, which causes conlmands to be generated to turn the missile in the proper direction. With PNG, the proper direction o f flight is established shortly after launch, and the missilc then flies on a constant-bearing (collision) course to intercept the target [Heaston and Smoots, 19831. If the LOS does not rotate (in inertial space), a collision will eventually occur, as shown in Figure 6-39a [Fossier, 19841. Substituting j = A u t o Equation (2-4) yields

Ru

+ [?R + =

V,,, cos(a - y)A] I?

V,,,sin(u - y)

+

YT cos(u - y ~ j~ ) -

vT sin(o - 7,)

(6- 12)

If the pursuer velocity is constant and the target is not maneuvering, the right-hand side of Equation (6-12) vanishes, and the behavior of u(t) is governed by a homogeneous differential equation. If the gain factor h(t) is set as A = - A,,R~[v,,, cos(u - y)] = A,,, where A,, > 2, and substituted into Equation (6-12). then R u - R[.A,, - 216 = 0, which can be solved analytically, giving .A,, - 2

b =

(k)

6,

and hence j = jo

(6- 13)

The solution o f this differential equation tends to zero for the pursuer-target closing, that is, d ( t ) < 0. Equation (6-13) shows that b(t), which is maximum at the beginning o f the flight, decreases linearly to zero for A, = 3 and approaches the value of zero asymptotically for A,, > 3. The collision course condition of u(t) = 0 is satisfied exactly at the final point R = 0 with a vanishing turning rate j = 0. Hence the disadvantages of pursuit guidance are avoided by thc P N G law with the gain factor A(t). As mentioned earlier, u(t) for P N G tends to the collision-course condition u(t) = 0 for increasing values of A,,. Therefore, the constant-bearing guidance is viewed as a special case of P N G in this approach, which is reached in the limiting case of A, -* m.

Example 6-13. P N G is selected as the steering law for the T G S M of Example 6-10. For the sake of simplicity, a linear model is considered for the homing head (perfect information) and all TGSM characteristics. Such simplification still makes it possible to obtain useful qualitative results. In the case of a fixed target, it

INTERCEPT

MISSILE LlNE OF SIGHT

\ UNDERSHOOT

MISSILE &TARGET L.O.S.

A

OVERSHOOT

MISSILE

L.O.S.

MISSILE TRUECTORY BASED ON w (DECELERATION PHASE)

TARGET

CONSTANT VELOCITY TARGET TRAJECTORY

ENGINE BURNOUT

A'

MORE DIRECT MISSILE PATH

NOTE: IN THE DIRECT PATH. LINE OF SIGHT RATE IS POSITIVE BEFORE BURNOUT AND NEGATIVE FOLLOWING BURNOW

Figure 6-39 Line-of-Sight Motion o f lntcrccpt ((aj.fio111[Fossirr, 19841, 8 1984 A I A A (~).FoIII[COIIZ~IPZ, IY7YIh/] ulifl1 pcrr~lissio~l.ho~s AGARD)

Sec. 6.5

Guldance Law

363

is possible to analytically solve the PNG equations to find siniplc relations bctwccn the initial altitude and look-down angle, and bctwccri the maximum acceleration and impact anglc [Trottier, 19871. From Equation (6-13). the heading rate and, consequently, acceleration for A,, ? 2 are maximum at the start of flight, diminishing to zero at its end where R = 0. Equating Equations (6-13) and (2-2b), one has for the end of the flight y, = - y,,l(A,, - 1). From the target's perspective, the final heading o f the pursuer is always at an angle y,,l(i\,, - 1) above the initial LOS. Figurc 6-40a is a diagram of pursuer altitude i t versus the down range x of the target when the terminal phase is initiated. As sho\vn in the figure for :I,,= 1 and C;, = I00 miscc, curvcs of constant impact angle arc straight lines emanating from the origin, and curves of m a x i n ~ u macceleration form circles that are tangent to the origin. Curvcs of constant impact angle are given for 20, 40, 60, and 80 dcgrees, and curvcs of constant maximum acceleration correspond to 3, 3, 10, and 15 g. Figurc 6-4Ob isolates that portion of Figure 6-40a in the pursuer's range of opcration where the thick line with the arrow at an altitude of 130 m shows the trajectory traced out by the pursuer during the search phase. Impact angles for this trajectory can be anywhere between 30 and 60 dcgrees, depending on how much time is required for acquisition. Thc acceleration requirement for this trajectory ranges from 9.8 to 13.6 g. The PNG eq~lationspresented here are valid for a stationary target and provide the pursuer's required acceleration at the initiation of the terminal guidance phasc in order to effect an intercept without acceleration saturation. This is not to say that the occurrence of accelcration saturation eliminates the possibility o f realizing a hit. Also, similar to the existence of a maximum acceleration capability required to hit a stationary target at x = 150 m from an altitude of 130 m without acceleration saturation, there also exists a minimum acceleration below which intercept is not possible a t all. The former corresponds to 13.6 g, while the latter corresponds to 6.8 g, at which acceleration the pursuer will fly in a circular path o f 130-m radius.

PNG System Design and Analysis. The preceding discussion of P N G assumed constant missile speed and a nonmaneuvering target. Upon removing these assumptions, Equation (6-12) is no longer a homogeneous differential equation. The solution will not tend to zero unless the guidance law of PNG is extended to compensate the driving functions o f Equation (6-12) as follows:

This type of guidance law is denoted by extended PNG (see Figure 6-36b). T h e first term in the brackets represents the well-known drag compensation, whereas the other terms provide target tnaneuver compensation [Goodstein, 1972(b)]. In most missiles, normal acceleration is commanded instead of turning rate. Since-A,,, = V,,cos y o +, the guidance Equation (6-6) is expressed as

u , = A,,,, = Vm cos yo A6 = lateral acceleration command

(6-15)

Guidance Processing

2m

I

1

1

I

400

ba)

800

lax,

Chap. g

DOWNRANGE (m) (A)

260

A L T I 1

u

2CX)

160

D E

(m) 60

60

100

260

Figure 6-40 (a) Curves of Constant Impact Angles and Constant Lateral Accelerations (b) Zoonl of Figure (a) in the Neighborhood of the Target (From [Trorrier, 19871 u8irh prnnissinn j o t n AGAKD)

Guidance Law

Sec. 6.5

385

The navigation ratio k;, used in Figure 6-35b(ii) is an inappropriate gain term. The guidance gain should be kept srllall for most of the flight to maintain stability and large a t the end of the flight to achieve a small miss distance. Assuming the overall navigation loop gain K, = All, ( i l = effective navigation ratio) in Figurc 6-35b(ii), the guidance equation becomes rr,

=

A liu

= ilI.., u and similarly in vertical plane

rr,,.

= AV,

6,

(6-16)

1,

The missile's lateral accclcration history is generally invariant. The invariance is not with A but with A = A I*',,, cos y,, 1 l,',, which is obtained by comparing Equations (6-15) and (6-16). This indicates that the value selected for A should be proportional to the missile-target closing rate. The rcsponsivcncss of the missile in correcting for LOS rotation is thus determined by A. That is, a higher closing rate necessitates a more responsive missile. Equation (6-16) shows that the missile velocity vector's required lead angle results when the accelerations A,,,,. and A,,,: of the missile reach rr,. and rri,,. In order to employ PNG, it is required to measure V,. as well as A,,,, and A,,,:. An automatic target tracking system is used to measure 6 and u,. A is varied to make the missile attain the degrce of rcsponsivcncss which is compatible with its response, tracker noise, targcr s~gnaln o w , and target maneuvering capability. One of the important features of PNG is the generation of a nearly straight-line flight path for the missile to a slo\vly moving target. Another very important feature of I'NG lies in its ability to cffectivelv deal with all attack altitude and aspects. That the missile flies along a straight path with constant velocity, given that the target also travels uniformly along a straight line, is most easily seen when A is very large. For i l 1, the LOS moves parallel to itself during the guidance process as shown in Figure 6-3Ya. This type of LOS motion can occur only if the projections of the missile and target velocity vectors V,,,and VTonto the normal to the LOS are equal to each other. It can be shown through using elementary geometrical constructions that these projections are only equal in the case of straight-line missile flight, with V,,,equal to a constant and with uniform, straight-line target motion. In practice, A 2 3 results in a nearly straight-line missile trajectory. It is not recommended to increase A too much, in order ro straighten the missile's trajectory, since guidance accuracy degrades as A is increased. This is a result of the manner in which the effects of internal perturbations are made more severe as A is increased. Because the missile control surfaces, used for steering in the horizontal plane, for example, can be deflected by any angle required to make Amyequal to i r , , PNG can be used to attack at all attitudes. Higher altitudes require greater deflections to produce the required value of A,, for a given value of u,. Therefore, PNG-based guidance systems are able to adapt to varying external conditions. Owing to the presence of the closing velocity V, in Equation (6-16), P N G possesses the ability to attack at all aspects. This can be illustrated by assuming that the missile is steered in the horizontal plane, and using Figure -3-11b. The system may be kept stable for any attack aspect i f the guidance signal t i i generated by the guidance algorithm Equation (6-

*

Guidance Processing

366

Chap. 6

16) is a linear function of the closing velocity V( [Maksimov and Gorgonov, 19881, Applying Equation (2-6) to Equation (6-16) yields

The expression in parentheses in Equation (6-17) represents the miss distance that would occur, lacking any target maneuver, if the missile underwent no further Yr -=-A A

tZ =Y

A ( A - 2)

tit, (A- l)(A- 2)

+

A(A-l)(A-2)

I

e

A -2

A-2 0

A=3 A=4

1.O

0

Illf (B)

Figure 6-41 (a) Normalized Relative Trajectory (b) Missile Acceleration Required to Overcome Step Function Target Maneuver (c) Missile Acceleration Required to Overcome Initial Heading (d) Miss Distance Due to Initial I'osition Error y , (c) Miss Distance Due to Target Maneuver ((b)-(r)jotn [Fosrirr, 19841, 0 1984 AIAA)

Sec 8.5

GuIdance Law

From Eq. (2-6), $=initial LOS rate = V ~ Y , ( O ) / R * (+~ Y,(O]/R(O) ) = -v, HVNO). assuming y,(O) = 0, yr(0) = -V,HE Ax2

0

0.6

1.0

t/tf (C)

(E)

(Continued)

corrective accelerations. This miss distance is termed the zero effort miss perpendicular to the LOS. Thus, PNG can be considered as a guidance law in which the commanded accelerations are inversely proportional to the time-to-go squared and directly proportional to the zero effort miss [Nesline and Zarchan, 19811. The normalized form for the miss distance due to a step maneuvering target is shown in Figure 6-41. A normalized relative trajectory y, for different A is shown

368

Guidance Processing

Chap. 6

in Figure 6-41a in which the ~ c r f e c sccker t and noise filter are assumed, and a zero lag dynamic system is also assumed, that is, T , ~ = 0 in Equation (2-35) Uerger, 19601. As shown here, a smaller position error will result from increasing A during flight. The normalized form of the acceleration required by the P N G law in order to intercept a step maneuvering target is shown in Equation (2) of Figure 6-41, while its trajectory is illustrated in Figure 6-41b. It is shown that increasing thevalues of A vl,ill decrease the missile nlaxi~numacceleration requirements for target intercept. Parasitic effects and noise considerations impose a practical upper limit on the maximum value of A. From Equation (2) of Figure 6-41b, the maximum missile acceleration required can be obtained at the intercept point as (II,),,,,, = I. AT,,l(A - 2). Thus, three times the target maneuver acceleration at A = 3 is required by the missile. More acceleration capability of the missile is generally required when other noise inputs plus system dynamics are included, as shown further on in Figure 6-43a. The normalized form of the acceleration required by the PNG law in order to intercept a targct with an initial heading error is shown in Equation (3) of Figure 6-41, while its trajectory is illustrated in Figurc 6-41c. The examples of Figures 641b and 6-41c show that, for a target maneuver and an initial heading error, thc lateral accelcratiori history is proportional to the disturbaticc. Further~norc.the shape of this curvc is dctcrnlincd strictly by A. Using a simplified two-time lag control system A,,,/e = Alf,r/(l + 7,,5/2)' it is possible to analytically dctcrrnine miss distance for thcse cases as shown in Figures 6-41d and 6-41c. Following Travcrs [1965], Figure 6-42 dcfincs thc normalizations associated with several coniponcnts of the error budget. Only one deterministic miss coniponent is listcd, the miss duc to a step of cross velocity. It is a function of normalizcd flight time rfl~,,.The remaining nor~iializedadjoints arc stochastic and depend on the signal (targct accclcration) and noise I'SD, 4. Thc dil~~cnsionlcss normalizcd adjoint cocfficicnrs in Figurc 6-32 arc indcpcndcnt of 7, and I]g v~cuutn rubes a t;zih hlc *t that r h r wcr: Inre9 {and miirophtmic) {Fossjcr, t988\,

Atr dg.f:enw horn{- ~3;sC:em. C3i1c nfrhc adv;lnriti;cs c3f 2ii active ~ccker is is aabiiity t~ operate atzt~tr~rltuusly stscic~u l o sec) required, for example, for decision integration, there can be a significant variation of the range-profile structure, as shown in Figure 650b. Although image frame times for SAR may be about 1 sec, target motion may blur the detailed RCS structure of the desired targct. Moreover, look times for inverse SAR requircd to capture the best reprcsentative image of a target can easily exceed 10 sec. An effective classifier should bc able to deal with such variations in targct signature for any conceivable seeker type within the appropriate look time, without a large measure of pcrforrnancc degradation. The features, thcrcforc, selected to enable classification (that is, the input to thc classificr) should alwrays be separable in feature space to allow the output decision to bc made with a high dcgrce of confidcncc. A design goal to which considerable attention nccds to bc given is optimization of classifier performance. The clcmcnts that combine to detcrminc performance are thc spatial resolution of thc sensor, the fcaturcs cxtractcd from thc signatures, the n~athernaticalstructurc of thc classifier, and the type of dccision-integration schell1e uscd. When rclying on range-only profiles, pcrformancc vcrsus resolution is Particularly important ro targct aspect. For fcaturc extraction, onc gcncrally errlp]oYs

Sec. 6.8

411

Future Guidance Processing

Fourier harmonics of the range profile since they rcveal how rapidly the RCS of the target varies and which spectral components of the radar are dominant. With neural networks, important issues have to do with optimizing the network structure in terms of the learning algorithm (for example, back propagation) and the number of connections. Optimization with respect to the choice of a decision-integration rule is desirable when dealing with long dwells on the intended target. A higher level of confidence can be obtained by integrating sequential output decisions. As described in Boone and Steinberg [1988], the neural network al*yorithm is based on a metaphor of the human brain's capacity to process information (see [Roth and Jenkins, 19881). While the subject of neural networks encompasses many options, one of those pursued by Rumelhart et al. (19861 focuses on the multilayer perceptron, or back-propagation-based algorithm. Although such algorithms require longer times to train, the time to classify is relatively short, which can be an important tactical advantage for a missile in a typical war-at-sea engagement, where there may be little time to classify multiple targets. Figure 6-50c illustrates the implementation of a multilayer perceptron via back propagation for the recognition of ship radar signatures. After the ship signature is received, it is conditioned, digitized, and then fast Fourier transformed. The first few harmonics of the transform are chosen by reason of the sensor resolution and SIN. The number of input layers to the perceptron equals the number of harmonics. The number of nodes of the hidden layer will typically be threefold that of the input layer, enough to allow for effective separation of the classes in feature space. The number of output nodes corresponds to the number of classes, and the connections between nodes are weighted. At every node, a summation of the weighted inputs is subjected to a sigmoidal nonlinearity. In the course of training, a heuristic rule derived from the classical Widrow-Hoff technique [Widrow and Hoff, 19601 is used to adapt the weights, and the weight changes are updated, working backward from the output layer to the input layer. Afterxvards, when classification occurs in real time onboard the missile, the particular output node that coincides with the input class will be activated. ,

Comparing neural network classifiers. It is important when comparing neural-network classifiers with conventional ones to consider a particular neural-net algorithm in conjunction with its proper conventional counterpart. Lippman [I9871 submits that the single-layer perceptron is analogous to the Bayesian classifier, and the multilayer perceptron is analogous to the k-nearest-neighbor classifier. The customary basis on which classifiers are evaluated is a single-look probability of correct classification. This number can be derived from a confirsion matrix, which might look like Testing Target 1 2 3 21'

p32

I3l3

Training Target

412

Guidance Processing

Chap. 6

T h e diagonal elements of this matrix are the probabilities of correctly classifying, while the off-diagonal elements give the probabilities of misclassifying. The goal is, of course, to achieve unity diagonal terms (correct decisions) and zero off-diagonal terms (incorrect decisions) with high confidence. In actuality, one attempts to maximize the trace of the matrix, while minimizing the sum of the off-diagonal elements, where different situations may call for the use of a particular cost function to weight the off-diagonal terms. Typical radar-range profiles for a decommissioned U.S. naval combatant are shown in Figure 6-50b, and a representative set of features is shown as an input to the schematic of the neural network of Figure 6-50c. When a number of other combatants together with this one are used for training and testing, the average probability of correct classification is derived from the single-look confusion matrices. Subsequently, a comparison is made of results for Bayesian, nearestneighbor, and back-propagation-based algorithms a t various aspects, and the degradation of those single-look results versus distance from the training point is determined. While the results indicate a good performance on the part of neural networks in some cases, the fact that the algorithm, features, and decision-integration rule have not been optimized means that no firm conclusions can be drawn. The algorithms described so far have been applied to a limited closed-set classification problem and, consequently, a broader range of data sets must be used for training. A synthetic radar-signatures simulation model developed by Georgia Tech. Research Institute [Tuley et al., 19831 provides a means of generating such data sets. This tool makes it possible to accomplish extensive optimization of classical as well as neural-network algorithms. Applications of these algorithms will eventually include SAR, inverse SAR, and rnonopulse imaging radar.

Extended target detection and segmentation. Besides RF seckers, consideration is also being given to IR seekers for current and future missile-guidance roles that will include both antisurface and land-strike warfare. Some of the advantages inherent in IR technology for antiship missile application include passive operation, good resistance to jamming, and high spatial resolution. T h e last ofthese, high spatial resolution, is needed in connection with potential operational requirements for target classification. Given that there is an adequate SIN, high classification accuracy requires high spatial resolution, regardless of whether the imagery is intcrpreted by a human operator [Rosell and Wilson, 19731 or processed by a computer [Ricdel and White, 19831. The primary drawback to IR antiship missile seekers may lic in their lin~itedrange performance when conditions of degraded atmospheric visibility csist. The signal processor dcscribed as follows [Steinberg and Rivera, 19871 is designed to optimize the detection range of 1It sensors against ship targets. A single-framc signal proccssing approach is dcvcloped for maximizing II\ scnsor SIN, which is conlplernentary to earlier approaches for SIN optimizations, such as waveband optimization, advanced I11 detector dcvclopmcnts, and multiframc image proccssing. The idea of a signal processing concept bascd on a l l ~ r r n n r j r ~ i s i n r isysrcrn 111cldcl followcd a study \vhosc results indicated that ranges obtained by human observers

Sec. 6.8

Future Guidance Processing

413

of visual displays could in many instances and under very diverse conditions substantially surpass ranges obtained by a hot-spot detection algorithm. The performance of ship-detection algorithms could therefore be judged against the predicted performance of human observers. The acquisition range against ship targets and classification accuracy will be to a large extent be determined by seeker spatial resolution, or pixel size. The SIN (and, hence, detection range) is maximized by matching the pixel sizc to the target size. However, assuming adequate SIN, high classification accuracy demands that the pixel size be made much smallcr than the target sizc. Therefore, selecting the pixel size to maximize detection range means that the resolution will be inadequate for classification, while making the pixel size as small as possible to facilitate accurate classification means that the initial detection range will be very poor. Resolving the conflicting spatial resolution requirements for detection and those for classification is a matter of employing image processing methods based on a model of the human vision system used largely in the past by EO engineers to predict the performance of human operators of thermal imaging equipment and televisions [Rosell and Wilson, 19731. In this model, there is an infinite-dimensional bank of spatial filters, each of which corresponds to a possible target shape. Every conceivable shape is represented in the filter bank, plus all variants thereof that can be formed by the process of translation, rotation, and scaling. Although the original human vision system model did not lend itself directly to digital realization, MRSI was developed that approximates the human vision system in performing detections of targets seen against uncomplicated (uncluttered) backgrounds, such as are likely to be found at sea [Gasparovic, 19821. MRSI, like the human vision system model that preceded it, consists of a bank of spatial filters tuned for maximum response to objects of differing sizes and shapes. Given that the human model, because it is infinite-dimensional, is also nonrealizable, a particular challenge in designing the MRSI was to be able to achieve detection performance akin to the human system, using a low-dimensional filter bank. The noise-reducing characteristics of MRSI cannot generally be calculated analytically, and consequently a computer program was designed to numerically evaluate the effectiveness of the processor. Results of initial computer simulations measuring ship imagery as input were encouraging, even if they were not always easy to interpret. This difficulty was due to the fact that the ship images were recorded under uncertain conditions. Computer simulation was then used to determine the probability of detection ( P D ) versus SIN for three ship profiles. It was found that the SIN improvement provided by MRSI processing could be estimated as G = 0.8 qx,where A is the area of the target in pixels.

Navigation and Guidance Filtering Design The order of presentation of those elements that constitute navigation and guidance filtering design is shown in Figure 7-1. Chapter 2 has presented examples of typical target trackit~gsensors, such as IR, radar, laser sensor, TV, and tracker modeling. In this chapter, however, emphasis is placed on guidance filtering and tracking as applied to target tracking. The subjects of uncertainty in measurements due to clutter, target acquisitionldetection and recognition are presented in Book 4 of the series. For target tracking, many signal processing techniques are used to discriminate the target from its background, other targets and decoys, while filters are used to extract the geometric and kinematic variables involved in target tracking and guidance (see Sections 7.1 to 7.5).

7.1 TARGET STATE ESTIMATION As depicted in Figure 7-1, target state estimation involves stochastic filtering based on target acceleration modeling for the purpose of target maneuvcr detection and target tracking. In this context, the term lrackit{q has often been used to mean accurate estimation of target states, without consideration for the antenna pointing/control aspect of the problem [Cloutier et al., 19881. This section reviews the various filtering techniqucs.

Sec. 7.1

Target State /%timation

Target Tracking Sensor (Chap. 2.2) I Target ~ a n e u v J Modeling r Target state' Estimation Autonomous~a!get Acquisition: (Chap. 2.4) Detection and Recognition Navigation ahd Guidance Filter Design (Sec. 7.2)

Radar ?racking (Sec. 7.3) I

1

Advanced ~aviga4onSystem Design (Sec. 7.5 & Book 4) Figure 7-1

Spacecraft ~ttiiudeEstimation (Sec. 7.4)

I I

Tracking in Clutter/ ~ultitargetTracking (Book 4)

Navigation and Guidance Filtering Design

Target tracking fllter.

The tracking filter functions mainly to receive the noise radar position data and to provide smoothed target position, velocity, and acceleration estimates. These are subsequently used to generate a one-scan prediction of target position for track correlation purposes. Tracking filters can generally be classified according to memory features into the following three groups: the fixed memory filter, the expanding memory filter, and the fading memory filter. The fixed memory filter, such as the maximum-likelihood filter, typically demands the storage of all data reports which take place within the time corresponding to the memory's window. The expanding memory filter (for example, classical linear regression filter) incorporates all past data samples at the time of a new observation to form a new estimate. This characteristic is disadvantageous for estimating maneuvering targets because of the substantial fluctuations of the target state during the estimation period. Thus, fading memory filters are often favored, where the old data are used but are forgotten at an exponential rate [Schleher, 19801. Six tracking filters that are widely used in systems are the Kalman filter, the simplified Kalman filter, the modified maximum-likelihood filter, the a-P-y filter, the Wiener filt'r, and the two-point extrapolator. These filters are typically appropriate for implementation with track-while-scan and other nearly constant data rate tracking sepsors. Other types of filters include least-squares filters, polynomial filters, and adaptive filters, but these are not presented explicitly in this section. Book 4 of the series presents the topic of adaptive filters. It is assumed that the gain vectors of the first three filters (Kalman filter, simplified Kalman filter, and maximum-likelihood filter) can be calculated in real time. This enables the filters to be adapted to varying tactical environments. It also permits optimal tracking in the presence of missed data points. An implementation of the stored-gain versions of these filters can be achieved at significant computational savings. Examples of stored-gain filters are the last three filters considered (a-P-y filter, Wiener filter, and the two-point extrapolator). The first five filters just described are examples of recursive filters "with memory." Figure 7-2 illustrates


Chap. 7

Signal Processing Sensor

and

Data Association

Predicted Grget State

Figure 7-2

Block Diagram Representation of the Recursive Filter A

a general block diagram representation of the recursive filters considered. The differences in filter formulations are found in the system models and the way in which the filter gain vector is calculated. Because it requires so little memory, the filter that is easiest to implement is the two-point extrapolator.

7.1.1 Target Tracking Filter Summary The following, adapted from Cloutier et al., [1988, 19891, summarizes various types o f stochastic filters that havc been employed to estimate target motion. A comparative evaluation o f some maneuvering target t r a r k i n g a l g o r i t h m s is p r e s e n t e d in Lin and Shafroth [1983(c)]. Single-model adaptive Kalman filters can be split into t w o groups-classical and reinitializing. Classical adaptive filters (for example, Greenwell ct al. [1983]), when applicd to thc air-to-air problem, rely on a continuous modulation of the filtcr bandwidth in response to targct maneuvers (implicit maneuver dctcction). Rcinitializing-type filtcrs arc based on explicit targct maneuver detection. When target motion differs from that assumed in the model, a bias appears in the innovations process. This bias is detected b y employing statistical hypothcsis testing. During the detection process, the filter is in a nonadaptive mode. Once dctection occurs, the filter's biased-state estimates are instantaneously adjusted according to the CStimated input and the filtcr is rcinitialized. [Bullock and Sangsuk-Iam, 1984; Chan et al., 19791 assumed constant acceleration over small time intervals in developing reinitializing filters. Dowdlc ct al. [I9821 and Tang ct al. 119841 assumcd constant velocity. Lin and Shafroth [1983(b)]avoided input estin~ationby dcvcloping a rcinitializing filtcr bascd on concatcnatcd mcasurcmcnts. T h e sequential tracker cnlploys batched mcasurcmcnts and is reinitialized by a local cstimatc whcncvcr the diffcrcllce between the two cxcceds a predetermined value. Thcrc arc drawbacks to both types of single-model adaptivc Kalman filters. The classical adaptivc filtcr suffers primarily from an inhcrent lag, which is sufficicntly pronounccd as to rcsult in a significant loss o f pcrformancc. In particular, by the timc the filtcr bandwidth cxpands, the target may have transitioncd into nonmancuvcring flight. Thc primary difficulty in reinitializing filtcrs is sctting thc dctcction threshold. Thc dcsirc to havc thc filtcr

Sec. 7.1

Target State Estlmatlon

41 7

rapidly respond to maneuvers runs contrary to the goal of a low probability of false detection. As shown in Chapter 4.2.3, the multiniodcl Kalman filtcr consists of a bank of filters and is ideally suited for the cstiniatioti of systems with parametric variations. Each indrvidual filtcr from the bank is optinlally designed for a discrete parameter level. As shown in Book 4 of the series, the addptivc state estimate is obtairicd either from the conditioned probabilitylwcightcd average of the bank mcrrlbcrs or fro111 the single member which displays maximum a posteriori likelihood [Yuch arid Liri. 1984, 1985(a)l.In the case ofa semi-Markov model, switching within the bank occurs according to a Markov transition. While swift adaptation can be achieved with the multimodcl filtcr, its exponentially expanding memory requirements must be limited to irnplcmcntation purposes. This can be done in a variety of ways. Maybeck and Hentz [I9871 employed a moving bank tcchnique and recommended decision logic for moving the bank. In a nonlinear application, Verriest and Haddad [1986, 19881 used a consistency test based on the original linear regions of the nonlinear system. Gholson and Moose [I9771 made certain statistical assumptions concerning the individual filtcrs \vhich reduced the bank to a single Kalman filter structure augmented by a recursive learning term. The tcchnique employed by Blom and Bar-Shalom [I9891 is one of hypothesrs merging wherein a single Gaussian probability density function (PDF) is used to approximate the mixture of assumed Gaussian PDFs by moment matching. In the case of tracking filter design of the air-to-air dynamics1measurements structure, most target state estimation models consist only of the linear kinematic equations even though the actual target dynamics are nonlinear. As a practical matter, it makes more sense to erroneously estimate target acceleration with a linear model than with a nonlinear model. Also, in the small time interval between seeker updates, a linear acceleration model yields nearly the same target motion as a nonlinear model derived from similar assumptions. Seeker measurements, by contrast, are spherical in nature (range and angle) and are therefore nonlinear functions of the state in Cartesian coordinates. Thus, one cannot avoid a nonlinear structure here, either that of linear dynamicslnonlinear measurements in Cartesian coordinates, Figure 7-3a, or that of nonlinear dynamicsllinear measurements in spherical coordinates, Figure 7-3b, where the nonlinear dynamics are a result of the nonlinear transformation of Cartesian linear dynamics to the spherical frame. One is looking at the coordinate systems in Figure 7-3 from below the x-y plane and from the right of the y-z plane. Using" Cartesian coordinates. several researchers have transformed the nonlinear measurements into linear pseudo-measurements [Dowdle et al., 1982; Fitts, 1974; Lin and Shafroth, 1983(a);~ & etg al., 19841. Speyer and Song [I9811 compared pseudo-measurement and extended Kalman observers and found that the former was biased. They later confronted the nonlinear measurements by developing the modified gain extended Kalman filter [Song and Speyer, 19851. In this extended filter, the total variation of the nonlinear measurement function is not approximated by its first variation, but rather identically replaced by a linear structure which is a function of both the state estimate and the actual measurement. Verriest and Haddad


418

I

Chap. 7

Taraet Target ( r ! -0, - $ )

Figure 7-3 Air-to-Air DynamicslMeasurement Models /From [Cloutirr et a / . , 19891, O 1989 IEEE)

[I9881 constructed a piecewise linear function to approximate the nonlinear measurement function and, based on the linear segments, developed a semi-Markov multimodel Kalman filter. Sammons et al. 119791 and Balakrishnan and Speyer [I9861 have developed hybrid filters which take advantage of the linearity in both coordinate systems. I'ropagation is performed in the Cartesian coordinate framc while updating is performed in the spherical (or polar) coordinate frame. Just before the update, the state statistics for these filters must be transformed nonlinearly into the second coordinate frame, and the inverse transformation must be performed just after.

Filter implementation. Concerning filter imple~nentationand the corresponding coordinate system selection, spherical coordinates should be given more consideration. Examples of totally non-Cartesian filter implementations are found in Moose et al. [I9791 and Gholson and Moose [I9771 w h o selected spherical coordinates in implementing their semi-Markov models, both of which produced simpler and more accurate trackers than their Cartesian counterparts, and Aidala and Hammel [I9831 who used modified polar coordinates in a bearings-only tracker. The filtering may be performed in polar coordinates with range, azimuth. and elevation angles, and their derivatives, as the state variables. In polar coordinates the measurement system model is linear and uncoupled. However, evcn for a target moving in a straight line at constant speed, the dynamic model is nonlinear, and the linearization leads to large errors. In reality, the problem structure in Cartesian coordinates is nonlinear dynamicslnonlinear measurements, linear dynamics appear only as a result of modeling (for example, assuming constant-speed target). There is a natural tendency to avoid spherical implcmcntations, in spite of their linear

Sec. 7.1

Target State Estimation

419

~ncasuren~cnts, because thcy transform thc linearly modeled. Cartesian dynamics into nonlinear sphcrical dynamics. Cotisidcring that target dynamics arc nonlinear in any coordinate systcm, it sho~tldbe true that linearizing nonlinearly modeled spherical dynamics is no more detrimental than linearly niodcling Cartcsian dy-. namics. Such was shown to be the casc in Moose ct al. [I9791 and Cholson and Moose [19771. Finally, thcre is no compelling reason to model target acceleration in Cartcsian coordinates. Lincar modcls of inertial radical and angular accclcration could be devclopcd directly in sphcrical coordinates. In this way, thc entire filtering problem could be pcrformed linearly in that frame. This would rcquirc the nonlinear transformation of inertial strapdown outputs to sphcrical coordinatcs, and the inverse nonlinear transformation of the filter's output to Cartcsian coordinates. In dual-control applications, the nonlinear transformation of state statistics could be avoided if thc guidance law were formulated in sphcrical coordinates. This type of filtering should inspire a whole new rcsearch cffort concerning what is thc best way to model radial and angular acccleration, and what is the simplest way to incorporate target flying characteristics into such a tnodel [Clouticr et al., 19881.

Adaptive filtering. Although extended Kaltnan and various other nonlinear filters have demonstrated improvement over thc standard Kalman filter, these have exhibited only limited effcctivencss as trackers in thc air-to-air encounter. This can be blamed for the most part on inadequate target acceleration modcls. Even though much progress has been made in acceleration modeling, no single set of model statistics can accurately represent the huge set of diverse maneuvers capable of being performed by a modern tactical fighter. This fact neccssitates the use of some type of adaptive filtering for best tracking performance. Such a filter must be able to quickly respond to a rapidly changing target motion. With the proper design, the multimodel adaptive Kalman filter could be more effective than its single model counterpart, although filter complexity is increased [Cloutier et al., 19881. This performance superiority was demonstrated in Lin and Shafroth [1983(c)] in a comparison of several advanced tracking filters. In addition, Bar-Shalom et al. [I9891 employed a hypothesis-merging technique in which the mixture of assumed Gaussian PDFs is approximated by a single Gaussian PDF via moment matching. A new approach to the maneuvering target tracking problem such as adaptive autoregressive (AR) target model identification is presented in Speakman [1986]. The primary focus of this work is on the method used to update the system dynamic model and the process noise covariance. The procedure described herein assumes only that the target accel'eration is exponentially correlated. The need for heuristics in determining the target time constant and driving noise covariance is removed due to the adaptive nature of the method. A nonlinear six degree-of-freedom digital simulation of an air-to-air engagement is used to demonstrate the technique. Results indicate that much more accurate estimates of target states can be obtained using the adaptive method rather than the conventional Gauss-Markov model. Book 4 of the series presents further adaptive filter design.

420


Chap. 7

7.1.2 The Wiener Filter T h e Wiener filter is a fading-memory, constant-gain filter that is used in trackwhile-scan systems. The gain vector employed in the Wiener filter, which represents the steady-state gain vector o f the regular Kalman filter, is computed off line and stored in the computer. When steady state is reached quickly by the Kalman filter, the Wiener and the Kalman filters, exhibit similar performance as shown in previous sections. Except for the fact that no covariance elements are computed initially in the case ofthe Wiener filter, both the Wiener filter and the Kalman filter are initialized identically. Possessing constant gain, the Wiener filter does not need to solve any auxiliary equations, and requires very little computer storage. Furthermore, because its gain is derived from the Kalman filter, accounting for target maneuver statistics directly, it can be adapted to a variety of vehicles and possesses the ability to effectively track both maneuvering and nonmaneuvering vehicles [Singer and 13ehnke, 19711.

7.1.3 Kalman Filter The Kalman filter, a recursive, fading-memory filter en~ployedin track-ivhile-scan systems, is the optimal filtcr for tracking when target motion equations are known. The estimated quantities such as position, velocity, and acceleration are known as states and are embodied in the state vcctor. The statc equation is given as a difference equation for the targct dynamics in terms of state vectors. The Kalman filter makes usc of dcrcrrr~inisricrargct dynamics plus a random process that accoullts for the inexactness of the difference equations used to express the dynamics and for other error sources. An observation equation, corruptcd by measurement noisc. is employed to observe system states. A product of the Kalman filter algorithm is the optimum linear n1inimun1-error-variance unbiased statc estimate. Thc Kalman filter algorithm also cstimates thc covariance matrix of the errors included in the estimate, and therefore provides a technique for adapting the filter to changing targct dynamics. However, if the Kalman filter is not "tuned" to the appropriate target dynamics, differences in the filter output and the true cstinlatc may causc it to become unstable. This property of thc Kalman filter, prcsentcd in Chaptcr 4.5.4, is termed di~~ergcr~tr and must be considcrcd in any application of the Kalman filter algorithln [Schlehcr. 19801. T o achieve ~vhitecxcitation (~nancuvcr)noisc which is rcquircd for thc filter's optimality, the Kalman filrer e~nploysthc augmcntcd versior~of targct dynamics model. O f all thc filters studied, thc Kalman filtcr is the ]nost sophisticated, the tnost accurate, and the most cxpcnsive to implement.

7.1.4 The Simplified Kalman Filter

By simplifying thc maneuver model used in tllc Kalman filtcr, the statc vector 2nd the number of indcpendcnt components of thc covariancc matrix can be reduced. O n e way in which the nlodcl is simplified, for instance, is by incorrectly assuming

Sec. 6.7

Defense and OIfense Systems

401

for targets if they arc in the parts ofthc Doppler spectrum of interest and are dctcctcd. Since they arc internally generated, they are obviously of no value to missilc guidance. Fundamental implementation problems began to be understood in the 1940s, but the state of the art in hardware was not conducive to satisfactory solutions. The microwave sources of that timc were primarily magnetrons and were quite noisy (although pcrfcctly satisfactory for conventional noncohcrcnt pulse radar opcration). Also, thc receiving vacuum tubes available at that timc were large (and microphonic) [Fossicr, 19881. Air defense homing system. One of the advantages of an active seeker is its ability to operate autonomously subsequent to missilc launch. Taking a semiactive approach was the result of the difficulty of achieving the necessary isolation ofthe recciver from the transmitccr in an active CW. In this approach, the transmitter remains at the launch point and only the receiver is flown in the missile [Fossier, 19841. One of the early examples of a semiactive air defense system is the Sperry Sparrow I with radar beam-rider guidance, which became the Navy's first operational missilc. The Douglas Sparrow II is equipped with active pulse radar guidance. In the Raythcon Sparrow Ill, the transmitter was removed from the missilc and left in the launching aircraft. By separating the transmitter from the rcccivcr by miles rather than inches (and removing comnlon sections of waveguide through which both transmitted and received signals must flow), the feedthrough and associated noise problems were reduced by orders of magnitude. This increased the maximum transmitter power level that could be tolerated in the active radar (a few watts) by orders of magnitude, and, combined with the much larger antenna in the aircraft used to illuminate the target, provided sutlicicnt tracking range to permit homing all the way a t the required flight ranges. The Sparrow 111 was Lnally selected for operation. Even more important than its low-altitude capability, it was a homing missile that worked. It was about this time, noted Fossier [1988], that a new opportunity arose to advance the state of the art in low-altitude air defense. The U.S. Army wanted to develop the technology needed to provide a battlefield SAM to protect friendly troops from attack by low-flying aircraft. From this desire was born Project Hawk. In a symposium on low-altitude guidance in early 1953, it was shown how clutter and image (or multipath) problems could be overcome with a semiactive C W radar homing missile. According to Fossier [1981], the problem of seeing moving targets hidden by ground clutter could be uniquely solved with CW radar in the Hawk system. Even when this was proposed, however, it did not allay fears that the effect on guidance accuracy of radar reflections from the earth, termed the image or multipath problem, would be severe enough to cause the missile to home somewhere between the target and its image and thus consistently collide with the ground before reaching the target (see Figure 6-490. At the time the problem was being researched, the design team of the Hawk system learned that horizontally polarized microwave energy can reflect almost completely from a smooth earth in the forward scatter direction at the grazing angles of interest, thus increasing the likelihood of the serious image

422


Chap. 7

over position measurement uncertainty, and is defined by the three fundamental target tracking modeling parameters: track period, target maneuverability, and measurement noise

F 2 T~Q:/~/R:/~

(7-4)

where T is the time period between state updates. The optimal steady-state filter is

Y,(k + 1 )

=

Y , ( k ) + a ( k ) [ z ( k )- Y r ( k ) l

(7-5)

Under steady-state conditions, F and a ( k ) are given respectively by

F2 = 4a2(k)l[l- a ( k ) ] ,

a ( k ) = ( - F2

+

+

1 6 ~ ~ ) / 8 (7-6)

The optimal tracking performance of the first-order filter is o;,= a(k)Rk.

Sewnd-order target tracker: a-p tracker.

The mathematical model

for the second-order target tracking problem is

The optimal steady-state Kalman gain is conveniently represented in the a-P tracker problem as

The steady-state values of the components of the state estimation covariance matrix are denoted by lim PC = (P,,] lrrr

The components of the prediction covariance arc dcnotcd by lim P r =

[tni.,]

k-=

The filtcr gain given by Equation (5) of Table 4-1 thus is

The covariance update equation from Equation (6) of Table 4-1 becomes, using Equation (7-1 I ) ,

Sec 7.1

Target State Estlmatlon

423

Substituting Equations (7-7) and (7-10) into Equation (4) of Table 4-1 produces

Equating the terms of Equations (7-12) and (7-13) yields, after cancelling out some terms, am11

= 2Tm12 - T2m22

+ T'4

-Qk,

T3 2

anr12 = Tm22 - -Qk,

P m l z = T3Qk (7- 14)

From Equation (7-ll), I

= R ( 1

- a)

(7-1 5)

rn12 = R k ( P / r ) / ( l - a)

and from Equation (7-14),

Using Equations (7-15) and (7-16) in Equation (7-14), gives

Comparing Equations (7-14) and (7-15), with respect to the quantity

m12

implies

where the tracking index is obtained by using Equation (7-4). The optimal position and velocity tracking performances are obtained from Equations (7-13). (7-IS), and (7-16) as a by-product of the analysis and are given by

The resulting a-fi tracker is given by the Kalman filter equation defined in Table 4-1 as follows:

f,(k) = f,(k - 1)

+

~ { , ( k - 1)

+ a(k)[z(k) - f,(k

-

1)

-

~ { " ( k- I)] (7-20a)

Navlgatfon and Guidance Filterfng Desfgn

424

Chap. 7

Example 7-1. The following example illustrates the second-order tracking problem. Given that T = 1 sec, Q:l2 = 3 g, and R:'~ = 1,000 ft, the preceding method can be used to show that the tracking index parameter is F = 0.0966 and the steady-state Kalman gains are a = 0.3551 and P = 0.776. The detailed derivation can be found in Kalata [1984]. Third-order target tracker. As with the analysis governing the optima] second-order filter, a target modeled by position, velocity, and acceleration states yields a solution also characterized by the tracking index parameter F. The mathematic model for the third-order target tracking problem is + 1) [j~"(~ y,(k + 1 ) Y

Y

(

~

+

z(k) = [ I

' ] ; [;:::;I r] 1 +

0 0

=

W(k),

~"(k)

0 0]

The optimal steady-state Kalman gain is

p = 2 ( 2 - ( ~ ) - 4 6 = 2

a2

-

a

(7-22)

The tracking index is F L yy"/[4(1 - a ) ]while the error covariance of state estimates are at, = aRk,

a;, =

8aP

+ y(P

- 2a - 4) Rk, 8Ty"(1- a )

2 = 0..

"

Y ( ~ P- Y ) R k 4T4(1 - a )

The filter equation is then given by the equations defined in Table 4-1. The methodology formulated here can be used to improve the performance of midcourse guidance accuracy. Applying the optimal filtering theory to the target tracking problem, the tracking index is found to have a fundamental role. This is true not only in the optimal steady-state solution of the stochastic regulation tracking problcm, but also in the tracking initiation process.

7.1.6 Modified Maximum-Likelihood Filter Here, it is assumed that the predicted and measured variables are independent and Gaussian. Instead of jointly considering the target position and velocity variables, which would be equivalent to the Kalman filter, position and velocity are considered

Sec. 7.1

Target State EstImatlon

425

separately to reduce the storage requirement. This technique is called the modified maximunt-likelihoodf;Itrr.The following derivation can be found in Trunk and Wilson (19761. The joint density of the predicted position X, and measured position X, is

where K, and K,,, represent the predicted-position and measured-position covariance matrices, respectively. The maximum-likelihood estimate of the position p. represents that value of p which maximizes Equation (7-23). The partial derivative of the log of Equation (7-23) with respect to p is: K;'(Xp

-

p)

+ Kil(Xm-

(7-24)

~1.)

When the preceding partial derivative is set equal to 0 and solved for p, the maximum-likelihood estimate is obtained as

i.

= (K;'

+ KG')-'

(Ki'X,

+ Kz'X,.)

In a straightforward way, it can be shown that the covariance of

(7-23) is

A new velocity estimate can be acquired by employing the new position estimate (i and the old smoothed position. An equation with the identical form as Equation (7-25) can be employed to combine the new velocity estimate with the old velocity estimate. 7.1.7 Two-Point Extrapolator

An extremely simple filter that can be implemented with very little memory requirement is the two-point extrapolator. This filter employs the last data point to obtain vehicle range and bearing, and the last two data points are utilized to determine target range rate and bearing rate. Because this filter possesses practically no memory, previous data points do not . prejudice predictions. Thus, the maneuvering and nonkaneuvering vehicles previously covered are tracked equally well (badly) [Singer and Behnke, 19711. -

7.1.8 Comparison of Target Tracking Filters Comparison of tracking accuracy. There are two ways in which data can be entered into the tracking computer: automatic and manual. As the first classification suggests, the track data are entered into the computer by someone who views the video on a screen and specifies it for entry. In the automatic mode, no such person is involved. With manual loading, only a limited number of tracks can

426


Chap. 7

be updated by the operator in a certain time interval. This limits the effective system data rate. Also, as the operator will incorrectly tag some o f the track returns, independent additive errors in range and bearing are included in the single-look mcasurements. In an analysis conducted by Singer and Behnke [1971], the steady-stat, one-sample-ahead prediction accuracies of each of the filters described up to this point were compared on a percentage basis to that of the Kalman filter. The minimum tag time per targct for manual data entry is assumed to be 3.0 sec, and the tag position accuracy is assumed to be uniformly distributed with maximum error of 0.03 in. on a 5.0 in. display scrcen. In the case of automatic data entry, the system data rate eauals the sensor data rate and no additional measurement errors are included because of the automatic entry. The final results were obtained by determining the average of the experimentally acquired percentage degradations in each of the range, bearing, course, and speed coordinates. This occurred because the transient responses of these filters were short-lived relative to the tracking period, and because the correlation coefficient p was typically small for the data rates and vehicle types evaluated. On the average, the performance of the a-f3 filter is about 50 worse than the Kalman filter, maneuvering vehicles being responsible for the largest degradation. Because the a-P filter significantly resembles the leastsquarcs filtcr, its gain vector quickly becomes too small to correct for the large estimation errors that are produced from targct maneuvers (see Book 3 of the series). The unifornl performance of the t~vo-pointextrapolator is more than 70 percent worse than the Kallnan filtcr. Since sensor and vehicle attributes can~lotbe considered, this filter cannot be "tuned" as the others can. Both thc Wicncr and simplified Kalman filters lverc no more than 20 percent worse than the Kalman filtcr, while the modified nlasimum-likelihood filtcr \vas n o more than 10 percent \verse than thc Kalman filtcr [Singcr and Hchnke. 19711.

Comparison of computer requirements.

The conlputer time and computer storage rcquircmcnts \verc examined for the initialization and main-loop phases of cach filtcr algorithm. The results wcre normalized t o the computer requircmcnts of the Kalman filter. Thc filter implementation requirements in ascending order are as follo\vs: t~vo-pointextrapolator, Wicncr filter, a-p filter, simplified Kalman filter, Kalman filtcr. Furthermorc, the complexity of cach successive filter increases by about a factor of 2. Thesc numbers depend largely on the computer hardkvarc bcing uscd. Howcver. the relative computcr storage rcquircmcnts should not dcpc11d substantially on thc computer hardware bcing cmploycd. The rclati~~c computation times \vcrc cstablishcd for a typical computer uscd in tactical system applications. Thc times to complctc an add, a subtract, and a store wcre assumed to be equal; the timc to complctc a multiply was assumed to be five times that of an add; and thc timc to complctc a divide was assumed to be nine timcs that of an add ISingcr and Behnke, 19711. Although it has acceptable performance, thc 1110dified maximum-likelihood filtcr obtained by arbitrarily dccoupling the position 2nd velocity csti~natcsis almost as complicated as thc Kalnlan filtcr and thus \vould not be employed [Trunk and Wilson. 19761.

Sec. 7.2

Practical Navigation and Guidance Filter Design

427

In many practical systems. the tracking accuracy generated by the constantgain Wiener filter is equivalent to that of the more sophisticated Kalman filter at one third the computational cost. It isbeneficial to implement the simplified Kalman filter arid modified maximum-likelihood filter if high accuracy is demanded and the length of the transient period app~oachesthat of the tracking interval. In addition, the Kalman class of tracking filters is uniquely able to supply accurate measures of tracking error statistics, even if missed data points exist. Ncvcrthclcss, in certain situations, the implementation of highly simple filters such as the two-point cxtrapolation may bejustified by the nature ofthc systcm rccluircmcnts and the impact of tracking pcrforrnancc on thcsc rcquircmcncs.

7.2 PRACTICAL NAVIGATION AND GUIDANCE FILTER DESIGN A guidance tracking filter provides target state estimation, which can be of trcmendous aid in predicting target position and improving the accuracy of an intercept NGC systcm. Low-pass filtcrs attenuate the noise component of the sensor signal, given the frequency characteristics of target signal and noise. Optimal estimators such as Kalman filters optimally separate the target signal from the noise by using information about target (and missile) dynamics and noise covariances. 7.2.1 Guidance Tracking Filter

Function and requirements. The primary components making up guidance processing are the guidance filters and guidance law algorithms. Guidance filters, which act as estimators and track the target, provide target range, bearing, and rate. Figure 2-2 shows a guidance tracking filter whose function is to remove as many nonstationary noises as possible and to preserve at the same time signal integrity in order to achieve accurate guidance. O n e of the main functions of guidance filtering is to smooth the noise in computing the pursuer-to-target LOS rate for guidance and to separate the relative target motion. In conventional PNG, they are used to estimate LOS rate ir, whereas in advanced guidance laws they are used both to measure LOS rate u and to estimate target maneuvers AT,. An advanced guidance tracking algorithm is needed to provide estimates of the target's relative position, velocity, and acceleration to support the advanced guidance law. Guidance filtering also acts to stabilize any residual missile body-motion coupling from the seeker head space stabilization control or . from radome error type distortions at the antenna. The choice of bandwidth for such a filter will be shown to reflect a trade-off between noise transmission and performance. During flight the noise and bodymotion requirements on the guidance filtering vary. Consequently, the guidance computer is programmed in such a way that it can adapt correspondingly. The missile's flight can only be smoothed if the guidance filtering is sufficiently heavy. It must be able to counter unstable motion that is the result of large random ac-


428

Chap. 7

celerations, as it responds to the noise. Such accelerations give rise to very high angles of attack and, consequently, induced drag, all of which have a depleting effect on the kinetic energy of the missile. Still, the guidance filtering must be sufficiently light as to permit quick missile response as necessary to correct for headingerrors o r to pursue a maneuvering target. It must filter out noise to the extent required to prevent FCS problems during certain flight conditions. After clipping spikes and transients, the filtering should not remain overly heavy. Moreover, the filtering should minimize the effects of quantization and variable data rate [Yueh, 1983(a)].

Design approach.

Although there appears to be a large number o f demands on the guidance filtering, these vary during flight. In flight, tracking error noise is measured by the guidance computer, which then uses information from the FCS and inertial reference unit (IRU) to ascertain which demand should be given the highest priority. This demand is then applied to the variable filtering. During the late 1960s and early 1970s, a few missile designers did take a cursory look at applying the modern control theory developed during the late 1950s and early 1960s to tactical missiles. Basically, as shown in Figure 2-17b, such an approach would replace the classically designed low-pass filter with an optimal estimator such as the Kalman filter. In theory, this would allow one to "optimally" separate the signal from the noise by using information about the missile dynamics and noise covariances rather than filtering bascd on frequency content only. In addition, missileltarget states other than LOS rate could be estimated, even if not measured, providcd they \vere mathematically obscr\~able[Gonzalez, 1979(a)]. This, in turn, \vould allo\v one to design more advanccd guidancc laws based on optimal control theory, because such theory usually requires complete information concer~li~lg the n~issilcstates. The tracking filter is bascd on bbth the laws of motion and a stochastic acceleration n~odel,evas~vemancuvcrs, and other disturbances being modeled as pecturbations upon the constant-velocity trajectory. The resulting optimal guidancc tracking filter uses a Kalman state estimator with several comDonent state vectors in which stable and accurate estimates of the relative position, relative velocity, and target acceleration are providcd to thc flight vehicle's fire control system or the missile's guidance and control systcm. Consider the guidance filter formulation sho\vn in Fig. 7-4. The objective is to find G I ( $ )\vhosc output f r ( t ) tracks y . with ~ thc least possiblc error c ( t ) dcfincd as (1) = ( 1 ) -

( 1 )(s) = ( ( 5

-

) H ( )I

S )

+

( ) I ( )

(7-27)

Figure 7-4 sum~narizcsthe optimal solutions of C,(s) that minimize txvo diffcrcnt forms of the performance index which arc functions of the l'S1) of the position cstimatc error c.(r), +,.(LO), dcfincd as + c ( ~ )

= + 2 x

Application to the infrared homing short-range AAM. Following Baba et al. [1988], SOG and PNG are applied to the model of a fictitious IR homing short-range AAM. Currently existing missiles such as the AIM-9 are used to estimate aerodynamic coefficients and geometrical and inertial data, the former of which are functions of the Mach number. The simulation consists of a mathematical model to describe the equations of motion of a missilc, the aerodynamic coefficients, and nonlinear mathematical models of major missile subsystems, including the seeker, noise filter, autopilot, and propulsion systems. The overall system is the one shown in Figure 8-25a with the same autopilot parameters as in the previous linear study. Here, again, the altitude is 2,000 m , velocity is Mach 2, the time elapsed after launch is 2.8 sec, and the angular velocity of the rolleron's rotor is 2,400 radisec. The cutoff frequency is selected to be 33 radisec, and the target model is a point mass having 3 degrees of frcedom. The initial launch conditions consist of: (1) both missile and target having vclocitics of .9 Mach; (2) both nlissilc and target at altitude 5.000 m; and (3) an initial off-borcsight angle of 0 dcg. Thc simulation rcsults for the AAM intercepting a targct which is turtling at 7 g in the horizontal plane are shown in Figure 8-26a. The simulations, consisting of a Monte Carlo technique with 100 runs, result in the ~ 0 d : f o rthc case of no noise, achieving smaller misses than those with PNG. If noisc is added, however, the broadside attack and tail-chase scenarios have S O G producing largcr misses than PNG. 111thcse cases, noise contributes most significantly to thc miss. I'NG produces a much largcr miss than SOG for the headon attack. Again, although SOG is shown by these rcsults to be more sensitive to noisc than I'NG, it is also nlorc cffcctive against a highly mancuvcrable target. For a fighter attempting to cvadc a missilc, one of thc most cffcctivc maneuvers it call undergo is the high-g barrcl roll (HGB). For this rcason, simulations arc pcrformcd for the AAM against a targct ycrforming a 7-g barrcl roll. The flight pattern of a targct undergoing a HGB nlancuvcr is s h o ~ v nin Figure 8-2Ob, together with the initial position of thc missilc. Thc cngagcnlcnt gcotnetry is assumed to be hcad on. With an initial rclatiyc range of 5,000 m , thc target flies straight until thc point at which it begins the HGU. Miss rcsults vcrsus barrcl-roll rate arc sho\vn in Figure 8-26c for a targct that initintcs thc HGB whcn it is at a distancc of 3,000 m from thc AAM. Assuming 3 nl to bc

Sec. 8.2

ComplementaryIKalman filtered Proportional Navlgatlon

loo0

180 (Head-on)

0 . 13

0. 88

15. 18

509

3 . 27

3. LO

18. 19

MISSILE

TIME-TO-GO ( s e c )

-

6-

- 5E

W

U

2

L-

m 2 E

3-

2 2s'Oo'

/ LO

8 2u Q: ?i 3 0 - -

u SOG

o

8

7

-'20-

1

We

2 (rad/sec)

(C)

3

0

1

2

3

4

5

TIME-TO-GO (sec) (E)

Figure 8-26 (a) Miss and Acceleration of the Missile Intercepting a 7-g ConstantTurn Target (b) HGB Flight Pattern and Nomenclature for Coordinate Axes (c) Miss vs. Barrel-Roll Rate (d) Miss vs. Time to Go (COB = 3 radlsec) (e) Load Factor vs. Time to Go (WB = 3 radlsec) (From [Baba et a l . , 19881, 0 1988 AIAA)

6

510

Advanced Guidance System Design

Chap. 8

the effective miss for target kill, then the PNG-guided missile fails to intercept the target if the latter performs the HGB with a barrel-roll rate o f greater than 1.2 rad/ sec. ';The SOG-guided missile, on the other hand, intercepts the target for all barrelroll kates shown in the figure. Miss versus 1, when the target initiates a barrel roll with a rate of 3 radlsec is illustrated in Figure 8-26d. Gravity acts to cause the miss to change in a sinusoidal fashion. Modifying the autopilot to compensate for gravity eliminates this sinusoidal variation in miss, producing instead the broken lines shown in Figure 8-26d. In this case the miss does not depend on the maneuver initiation time. In each case the maximum load factor is shown in Figure 8-26e. Here it can be seen that the load factor of the SOG-guided missile is smaller than that of PNG.

8.3 OTHER TERMINAL GUIDANCE LAWS Some of the guidance laws that have not been covered so far have their applications in special areas such as optimization theory or, particularly, differential games for nonlinear dynamic systems. Some involve very simple and straightfonvard implementations of ad hoc controllers. H o et al. [I9651 present one of the earlier papers on the application of differential game theory to the short-range missile problem. In this paper, variational techniques were used to solve pursuit-evasion problems and an example was provided that proves that under certain conditions PNG was optimal. Pfeiffer [I9671 constructed near-optin~umguidance laws using a method of successive approximations similar to quasi-linearization. Fivc polynomial-type approximations for nominal trajectory and sensitivity matrix cor~lputationswere studied [Gemin, 19691, with particularly useful results for prcprogrammed perturbation navigation schemes. An iterative technique for the near-optimum solution of a nonlinear differential game is developed [Anderson, 19741 based on successivc linearizations of a two-point boundary value problem. This scheme was later applied [Poulter and Anderson, 19761 to an AAM guidance problern. Vastly improved S~IIIUlation results were obtained over those of a PNG law; however, these came at a cost of increased computational expense. Still later, Anderson [I9811 compared differential game and L Q optimal guidance laws. It was concluded that differential game formulations are not as sensitive to errors in targct accclcration estimation as optimal control algorithms. Most guidancc schemes making use of high-fidelity models demand excessive computational effort, drawing the attention of many invcstigators. In Liu and Han 119751 continued fractions wcrc proposcd to rcducc the modcl of a high-order nonlinear rocket. In Sridhar and Gupta [I9791 singular perturbation methods wcrc applied to an AAM in a n attcmpt to rcducc the computational cffort required. Fast mode was assumcd for thc rotational dynamics of the missile whilc slow rnodcs wcrc assumcd for the position and vclocity translation equations. Simulation results showed improvcmcnt over PNG. As pointcd out [Clouticr ct a]., 1988; Evcrs ct al., 19881, optimal guidancc of pulse motor missilc is just one arca still in need of rescarch, having bccn addressed recently by Chcng ct al. [I9871 and Katzir et al. [1988]. O n the other hand, the

Sec. 8.3

Other Terminal Guidance Laws

511

fundatnental theoretical problems associated with boost-sustain midcourse guidance appear to have been solved. Other issucs that need to be addressed include algorithm implementation and problenl formulationlsolution. Those issucs associated with algorithm inlplcnlcntation include: software design methodology selection, highorder language run-time characteristics, cross-compiler efficiencies, and hardware throughput/memory limitations [Evers et al., 19881. T h e issue of problem form~~lation/solution involves a choice between closed-form solution to an approximate optim'll formulation (LQG, LQR) or an approximate solution to an exact nonlinear optimal formulation, which entails a nonlinear two-point boundary value problem. Moreover, a numerical solution of the exact nonlinear problem is impractical for implementation reasons. With terminal and endgame guidance, there are usually problems associated with degradation in seeker measurements, resulting in a lack of target information and the need for a stochastic endgame approach. As previously mentioned, homing guidance also reduces the information available from the measurements, which suggests dual-control techniques. Additional work on guidance lawlautopilot interactions, especially with respect to the acceleration limits of the airframe and the autopilot's finite time response, should be pursued. Embedding the autopilot models in the guidance algorithm derivation is another possibly re\varding approach. Several guidance algorithms have been developed which, in some way, attempt to deal with autopilot/airframe performance constraints [Lin and Yueh, 1984(a)-(c); Yueh and Lin, 1984, 1985(a)]. A modified PNG was derived which adjusts the autopilot gains to minimize a penalty function on control effort and time to go. Other works [Aggarwal and Moore, 1984; Caughlin and Bullock, 1984; Lin, 1983(a); Lin and Yueh, 1984(a); Stallard, 19831 derived guidance laws to deal specifically with the problems of BTT control. In Caughlin and Bullock [I9881 a reachable set theory is applied to the design of guidance laws which deal explicitly xvith hard limits on airframe achievable acceleration. The endgame part of the intercept has received only limited attention in the guidance and control literature. Dowdle et al. [1982], Dowdle et al. [1983], and Lin and Shafroth [1983(a)] generalized the LQG regulator and, after appropriate model linearization via pseudomeasurements, estimated the target state with a reinitializing Kalman filter. A generalized likelihood ratio approach was applied to the innovations process as a target maneuver detector. In a more fundamental look at the endgame, Looze et al. [I9871 used CramerRao lower-bound analysis to investigate the quality of target acceleration information available from the seeker measurements. It was found that target acceleration was accurately estimated for the maneuvers considered, but that such estimates were poorly utilized by the modified PNG law. The guidance law was subsequently altered with lead compensation of the roll command to yield improved miss-distance performance. Finally, in a departure from these approaches, Forte and Shinar 119871 formulated a planar air-to-air intercept problem as a mixed-strategy, zero-sum, stochastic differential game. The cost functional of the min-max problem was singleshot-kill-probability. The optimal pursuer's strategy is in the form of a parametric guidance lawlstate estimator which demonstrates increased single-shot-kill-proba-

-


512

Chap. 8

bility against any frequency of target maneuvers when compared with any single strategy guidance algorithm. This approach has recently been expanded to a 3-D encounter [Forte and Shinar, 19881.

Example 8-3: Tenninal guidance law based on a perturbation technique. Optimization problems have always been important in the NGC field. For example, a complete trajectory can be optimized by using the calculus of variations. However, this leads to a two-point boundary value problem requiring iterative methods to arrive at the solution. Another problem in the NGC field is the perturbation technique which has been used to derive control laws for several aircraft and missile trajectory optimization problems. All the system dynamics can be optimized by these techniques by separating the system dynamics into slow and fast states. The solution obtained is then reduced in order, after which it is combined systematically to achieve a near-optimal solution to the full-order problem. Perturbation techniques are used in this example based on Aggarwal and Moore [I9841 to obtain nonlinear feedback control solutions to the terminal guidance problem of a skid-to-turn (STT) or a long-range ramjet-powered bank-to-turn (BTT) missile. T o achieve slow and fast states separation, artificially small perturbed parameters are generally contained in the system dynamics of these problems. In the perturbation technique, Taylor's expansion is used to expand a nonlinear dynamic system solution f(x, E) about the small perturbed parameters as follows:

f (x, E) = f (x, 0)

+ af + a€ €

where the perturbed parameters are represented by E and the state variables by x. O n the right-hand side of Equation (8-53). the first term is referred to as the zeroorder solution and the second term the first-order solution. The perturbation procedure is carried out in a stepwise manner as follows. The first step involves arranging the dynamics according to slow and fast states. In the second step, the zero-order solution in Equation (8-53) is obtained by setting the perturbed parameters t to zero in the state and then solving the rcduced-order problem. The initial or terminal condition of states with E = O serves as the boundary conditions for the zero-order solution in this step. The perturbed parameter is dominant in the first-ordcr solution of Equation (8-53). With the boundary conditions derivcd in the zero-order solution and the conditions that arc not satisfied in the zero-order solution, the last step of the procedure is to obtain the first-order solution using optimality conditions with fast states.

Problem formulation.

The problenl formulation for the beginning of terminal guidance is in a nonrotating coordinate frame aligned with the seeker frame A simple planar representation of the intercept geometry of terminal guidance is found in Figure 2-10 where x, is replaced by z, in this section, and the relative

Sec. 8.3

Other Termlnal Guldance Laws

513

motion has the following dynamic model:

d . - yr = 1 4 ~ "- A,,,, = AT" - A,. cos dl

+

d - i , = AT, - A,,,: = A T , - A , s i n + dr

(8-54)

(8-55)

where the missile acceleration command in the missile and target plane is represented by A,, is the bank angle, and the missile roll-rate command is denoted by $,. Zero-lag acceleration and roll-rate autopilots are assumed for the missile. It is desirable to find an optimal choice for the controls A, and 4, such that the following performance index is minimized:

+

Optimal conditions. The Hamiltonian of Equation (8-39) with respect to Equations (8-34) through (8-58) is as follows: H

=

+ A;,(-A, cos $ + AT?) + A,4, + 1 Xi,(-A, sin + + AT,) + A+$< + - (K,Az + K&:) 2

A,y,

(8-60)

Due to the independence of the Hamiltonian from y, and z,,

A,= 0,

A, = 0

(8-61)

The other three costate variables are as follows:

A.Y ,

=

- A ,,

A;,=

-A

Z 1

A+ = A,

[A;, cos

+ - A;, sin $1

(8-62)

where the boundary conditions are

Mtf) = ~"(tf).

A=(tf) =

If d is assumed to be the terminal miss distance and final bank angle, then y,(tf) = d cos

d

=

5,

dy:(tf) + zt(tf),

(8-63)

6is assumed to be the terminal

z, ( t f ) = d

-

~ ~ ( f f )

sin

5,

+ = tan-'

~"(tf)

rr(tf)

(8-64)

Advanced Guldance System Design

514

Chap. 8

The following are obtained by substituting Equations (8-64) into Equations (8-61) and (8-62): X,(t)

= d cos

-+ = constant,

A,(t) = d sin

= constant

(8-65)

The optimal controller is expressed as

+ A;,

A;, cos

sin

+ - K.A,

=

0

(8-67)

and = - AmlK*

Substituting Equations (8-66) into Equation (8-67), the optimal acceleration command is expressed as

A, =

d

- cos (& K,

(8-69)

+)tn

Using Equations (8-62), (8-66), and (8-69), A+ is obtained as

where A*(ff) = 0.As discussed previously, the optimal problem including Equations (8-54) through (8-59), (8-h4), (8-68), and (8-70), is now represented as a twopoint boundary value problem.

Optimal solution. This two-point boundary value problem can be solved by many methods. A solution for Equations (8-68) and (8-69) must be found by numerical methods because of the nonlinear dynamics. It is necessary to obtain closed-form solutions for the control commands with respect to the current values of the system states, so that practical implementation is possible. Zero-Order Solution. Since the dynamic response of a missile to the rollrate command is much faster than that of the rcst of the system states, the following assumption can be made: $Jr

= 0 for

t

> to

(8-71)

This implies that the roll-rate command here impulses at to and will bring the rolling angle $J to 6 for tirnc t > I,,. Hence Equations (8-57) and (8-70) become A:(()

=

a,

L$,(t)

and Equation (8-69) bccomcs A:' = dr,lK,

=

o

(8-72)

Sec. 8.3

Other Termfnaf Gufdance Laws

515

where the superscript o denotes the zero-order solution. The following is obtained by substituting Equation (8-73) into Equation (8-54) at 4 =

6

Equation (8-74) is first integrated from t to twice and then compared to the boundary condition in Equation (8-64) to obtain

d cos

6=

[3Kd/(3K,+

(1)

+ y r ( t ) t, + A T , t,:/2]

(8-75)

From Equations (8-54), (8-73), and (8-83), the acceleration command in the Y direction is obtained as

A,, = A:'cos$ = (Alt,:)(y,+ y,tg

+ AT,t,:/2), A = 3t,i/(t: + 3K,)

(8-76)

A similar expression can be obtained for the acceleration command in the z direction by substituting Equations (8-73) and (8-75) into Equation (8-55) as follows:

+ 4,t,, + AT,t,;/2) = V A ?+~A?: and + = tan-'

A,, = A: sin 5 = (A/t,;)(z,

(8-77)

From Equations (8-76) and (8-77), A!! (A,,IA,,). The preceding zero-order solution assumed stabilized roll. It is exactly the same as the optimal solution applied using conventional linear theory to the STT missile with zero lag FCS in Equation (8-5) with C = 1, R = K,, and A = 0. First-Order Solution. The missile is assumed nonrolling at t > to in Equations (8-72). However, it is in reality rolled near the initial time to. The transformation T = ( I - to)/K&can be made for analyzing the dynamic behavior near to. The results in the boundary layer equations can then be obtained for the missile bank angle transition at the initiation of terminal guidance. The assumption of Km + 0 leads to the roll angle being changed from = +(to) at T = 0 to $I = at T = m, while maintaining the other states in this time range. Thus

+

6

The Hamiltonian is expressed as

H I = hby,

+ A L i , + At, ( - A ? cos 4 +

AT^) (8-79)

which remains constant for an autonomous system. Hence, H I (T) = H(t,) in Equation (8-60). The following is obtained by comparing Equations (8-60) and (8-79)

Advanced Guidance Sustem Design

Chap. 8

INITIAL CONDITIONS, TERMINAL TlME 4,

-

UPDATE INTERVAL

IFtf.to~tg SET 1 = t, . t o

8

tg

, ZERO-ORDER OUTER SOLUTION

q = ~ e e s = ( ~ : ) ( y ~ + f ~ t Y ~$ 1+2% )

3

SYSTEM DYNAMICS

3

A = 3 tg / (tg

I

= tan"

+ 3 K,)

(\/A+)

I

ZERO-ORDER INNER SOLUTION

t

INTEGRATE SYSTEM DYNAMICS OVER TlME INTERVAL tg

-

I

FIRST-ORDER OUTER SOLUTION

Figure 8-27 Scqucncc of Perturbation Solution C o m p ~ ~ t a t i o tFor ~ s Siniulating MTT Missilc Trajcctorics (Irro~n/A.@ar~~,alotld .\loort, 19841 ir~irkyri*rtr~isrio~r ,fiotn AACC)

I

Sec. 8.4

Radorne Error Calibration and Compensation

and using Equations (8-71), (8-72). and (8-78):

-

A&&

1 +; - Km 6; = h;,(t,,)A!'(cos $ - cos 4) + h;,(tll)(sin $ - sin $)

Since K+ is very small, the second term in the left-hand side can bc neglected. The following is obtained by substituting Equations (8-66) at t = t,, into Equation (880): - A

= 4

- 1

- cos(6 -

+)I

(8-81)

Substituting Equations (8-68) and (8-73) into Equation (8-81) will yield the rollrate command &, equation as: Km &f = 2 ~ 1 [I' -~ cos ~(

- +)]

or

In addition, from Equations (8-69) and (8-73), it is seen that

A, = At! cos(& - I$)

(8-83)

The near-optimal guidance law for BTT missiles is due to the fact that the acceleration command A, in the target plane and the roll-rate command 4,are given by Equations (8-83) and (8-82), respectively. Figure 8-27 shows the simulation flow chart of this BTT optimal guidance law.

8.4 RADOME ERROR CALIBRATION AND COMPENSATION Missile homing performance can be seriously degraded by boresight radome errors whose slopes generally show large variations. Thus, to increase homing performance and target intercept capability, in-flight radome error calibration and compensation is essential to overcome the restrictions imposed by radome errors [Lin and Lee, 1983; Lin and Yueh, 1984(a),(b);Yueh and Lin, 1984, 1985(a), (b); Yost et al., 19801.

8.4.1 R a d o m e Error C o m p e n s a t i o n This section first describes what has been done in the past to increase radome error tolerance for guidance system and autopilot design, and the approach this section proposes to use. T o improve STT and BTT system performance, it is essential to increase the radome error tolerance. Considerable effort has been devoted to modifying guidance system and autopilot designs to achieve this goal. Chapter 2 discusses radome error and its severity on guidance and control responsiveness of a homing missile. As the speed and intercept altitude increase, the limitation imposed by the

518


Chap. 8

radome error on guidance responsiveness also increases. This is also true for broadband homing systems. Large pitch and yalv motions occur at high altitudes causing the direction of the radar waves to change through the radome. This change and the refraction produce an apparent target motion. The effect o f a given radome slope o n the miss distance is determined by the aerodynamic turning rate time T,, which is small at lo\+,altitude and does not have much effect on the radome slope. However, as T, increases with altitude, the per~nissibleradome slope range reduces which can also lead to stability problems. This radorne error-induced stability problem can cause more severe degradation for the modern guidance system than the PNG. The randomly switched radome refraction slope cannot be easily modeled and the optimal guidance system (OGS) derived sut-fers more stringent component tolerances against unmodeled errors. In fact, the allox+-ableradome slope range is one important measure used in guidance system design to specify manufacturing tolerances on the radome. When component tolerances are met, the O G S proves to be the optimal design in terms of miss-distance performance index and acceleration command history. Thus, a design that overcomes the restrictions imposed by r a d o n ~ eerror is necessary to allow increased homing missile perforn~anceand target intercept capability for a more robust guidance system optimization. In the conventional approach, the most widcly irnplc~~lcnred tactical missile systems are based on linear control theory where heavy guidance filtering is used to increase body-motion coupling stability margins. The FCS bandwidth o has a great effect on the system radome sensitivity. As o increases, system performance becomcs more sensitive to radome slope error. This leads to stricter radomc specifications. One method of stabilizing the radome-coupling effects is to introduce a lag-lead filtcr conipensation. The lag-lead filter provides the proper stability margins at low frcqucncics without causing high-frequcncy effccts sincc radomc effects are essentially a loxv-frequency problem. Some care, however, 171ustbe taken in selecting the low-frequency corner since, at low frequcncics, a large changc in the timc constant produces only a relatively small change in attenuation near the frequencies of interest. For the parasitic loop of 0, the amount of lag-lead filtering required will depend on the flight conditions. The fact that the n~issilclag-lead nctwork must be varied with flight conditions represents one disadvantage, whilc another onc is that thc timc lag induced in thc missile guidancc loop duc to thc filtering network is also critical. T o alleviate the severe radome stability problcni for a high-altitude engagement scenario, one of thc commonly uscd engineering "fixcs" is to introduce pitch-rate compensation K, with gain K, fed back to cithcr bcforc or after the guidance filtering nctwork. I t can somcwhat syni~nctrizcthe positive and ncgativc slope stability region at the cost of slowing dolvn the autopilot rcsponsc. Hence, an a]ternative nlethod of radome compcnsation is accomplished by adding a 0 feedback loop with conipensator gain K, to the guidancc coniputcr as shown in Figure 821c. This attenuates the effects of radomc slopc so that the required stability margins arc obtained for both plus and minus radorne errors. The inain disadvantagc of this compcnsation is that thc rnissilc systcm time constant is dclibcratcly rcduccd. This can be seen from tlic previous discussion involving Equation (8-41). T h e conipcll-

Sec. 8.4

Radome Error Calibration and Compensation

510

sator gain K, from Figurc 8-21c with largc stability gain can be approximated by K, = r. Since the rndome slope error is assumed validly to be constant over small regions of the look angle, thc radome compcnstor gain K, is approxi~natelyconstant. A guidancc loop that combined seeker-guidance filtering in a complementary LOS ratc cstinlator shown in Figure 8-21c is more interesting. The radomc compensation gain that is used in this guidancc loop is independent of the seeker tracking time constant. If a conventional seeker, as shown in Fig. 2-13b, is used in Fig. 8-21c instead of a combined seeker-guidance filter, thc compensator gain K, with small tracking-loop time constant and largc stability gain can be approximated by K, = r/(1 - r). Pitch-rate compensation, which biases the radomc crror toward the positive slope region, can somewhat symmetrize the positive and negative slope stability region at the cost of slowing down the autopilot response. For the hypersonic interceptor with speed beyond Mach 6, the required guidance system time constant \vill bc larger than 4 sec, which is too sluggish for a classical homing missile. Thus the simple, conventional "fix" cannot be extended to the next generation, hypersonic interceptor. Another proposed approach (although never implemented in the past) has been to estimate the radome error slope by requiring a missile body oscillation in order to correlate the LOS ratc measurement with the known dither "test" signal. The approach taken in this study is similar to the error identification concept but removes the need for dithering. Rather, the proposed optimal guidance and control technique relies on missile pitch angular velocity due to guidance commands and explicitly accounts for the actual LOS rate contained in the radome corrupted boresight data. Figure 8-28 shows an integrated NGC system with radome error compensation.

Targets

f Kinematics

-

Adaptivc Radome Error Eslimalor

(Chaps. 4, 6-81

(SC. 8.4.2)

4

4L

Intcgratcd Control System ( ~ ~ )3-51 ~ ) k ~-+ Airframe & Propulsion

4

Gain Adjustment

-

.f

Onboard Navigation System (Chap. 5)

Figure 8-28

Airframe Filter

FCS Sensors

Integrated NGC System with Radome Error Compensation

520


Chap. 8

Integrated modified PNG and optimal controller with radome compensation Modified PNG with Radome Comnpensation. The radome compensation technique applied in Section 8.2.3 requires estimation of the radome slope error to correct the LOS rate measurement through missile body rate. A radolne slope error estimate has been incorporated in a modified PNG to generate a guidance Command. The guidance acceleration command becomes

where & and 0 are smoothed values of 6 and 8, respectively. A detailed derivation of Equation (8-84) is presented in Book 3 of the series. Adding radome compensator with gain K, to the guidance formulation in Equation (8-84) yields

Besides the small-trim aerodynamic correction term that is proportional to Z,/,\I, (< 0.01), there is also the pitch-rate compcnsation term to account for rhe radome error. K, is introduced in the pitch acceleration command by n~ultiplyingthe pitch rate by P - , the probability of radome error in the negative region, to be defined later. The use of P- to scale the pitch-rate compensation for enlarging the radome stability region has also dcmonstratcd reduced cxcitation in acceleration commands.

Integrated Modified PNG and Optimal Controller. The opti~nalcontroller derived in Book 3 of the scrics is related to a modified I'NG system \vith navigation ratio .A. According to the author's kno\vledgc, Equation (8-85) offers the first analytical expression for a rheorctical basis to introducc pitch-ratc compensation according to thc optimal control law provided that thc radonle error slope can be estimatcd.

8.4.2 Guidance Performance Analysis with In-Flight Radome Error Calibration Adaptive radome estimation design. An adaptive Kalma1-i filtering bank in the s\vitching environment is used to realizc the changing processes in a radome slopc. The rate of s\vitching is assumed to be cotlsiderably slo\vcr than that of thc boresight crror (observation) sampling rate. That is, by assuming the rate of positive-negative radomc slope switching to be snlallcr than thc data sampling rate, the esti~nationschcnie can be rcduccd to the design of a bank of Kalman filters, each matched to a ccrtain radomc slope configuration. Actually, follo\virlg thc adaptive filtcr formulation [Chang and Athans, 1978; Moose, 1975; Moose and Wang. 19731, thc random switching of thc unreliable plant (thc rado~ncslope) is modeled by a semi-Markov process. A scmi-Markov process is a probabilistic system that makes its state transitions according to thc transition probability matrix of a conventional

Sec. 8.4

Radome Error Callbration and Compensation

Estimated LOS Look Angles

Filter Weights

A Posteriori Hypothesis Probabilities Calculated

Figure 8-29 Adaptive Radome Error Slope Estimator (One-State, Three-Bank Kalman Filter)

Markov process. However, the amount of time spent in each state before the next transition to a different state is a random variable. It is this property of a random switching time that distinguishes the more general semi-Markov process from a Markov process. The multiple states mentioned earlier are actually chosen to typify the radome slope in the positive, zero, or negative slope regions. Thus, by proper choice of the state and modeling of the transition processes, it is desired to realize the variations of the radome slope as a randomly switching, semi-Markov process. A general adaptive filter, as shown in Figure 8-29, essentially consists of a bank of three Kalman filters, each matched to a possible plant configuration U, ( i = 1 , 2, 3). The filter outputs are weighted by a time-varying a posteriori probability to obtain the radome error slope estimate. The bank of three Kalman filters has the deterministic inputs U1 = U + = 0.02 degldeg, U2 = U" = 0.0 degldeg, and U3 = U - = -0.02 degldeg to characterize the three plant configurations in the positive, zero, and negative slope regions, respectively. That is, the filter estimates from each filter in the bank are further weighted by calculating the a posteriori hypothesis probabilities. It is the probability of a given hypothesis that the radome slope is around a certain U ,value, being true conditioned upon the past measurements. Since during actual flight the unknown plant configuration (due to polarization effect) might randomly switch at random times, the random switching of the radome slope parameters must be mddeled as a semi-Markov process. The a posteriori hypothesis testing involves the three conditional probabilities


522

Chap. 8

where Zk+ denotes the collective events of all the past measurement history up to time t k + 1, that is, Equation (8-86) actually states that for a new boresight datum z k + I that just comes in, a global hypothesis needs to be tested to see whether the datum indicates that the plant configuration, or the radome slope, is near the positive, zero, or negative slope region. It is a global instead of local hypothesis because it depends on the time history of the data, as shown in Equation (8-87). Thus, the a posteriori conditional probabilities should be calculated recursively to reflect all the past data dependence. Detailed formulations are given [Moose, 1975; Moose and Wang, 19731 and are shown below for i = 1, 2, 3, as

X

normalizing factor x

2 [_Pi,&x

Xii ( t h + ~- rk)]

j=l

in terms of the rccursive time index k. The normalizing factor is the inverse of the sum of the three a posteriori conditional probabilities. T h e transition matrix Xv from state Ui to Ui is symmetric and is based on the semi-Markov statistics. This renders a very simple form for in-flight, real-time computation and eliminates the computer storage problem that grows with increasing time in the conventional Markov process approach. The first term on the right-hand side o f Equation (888) furnishcs the adaptive learning feature of the system. It actually tests the new borcsight data r k + l to check the probability of being within each plant condition, that is, whcther it corresponds more to the positive slope or negative slope region. This conditional probability can be approximated by a Gaussian density for those cases in which thc probability of a transition occurring between any two adjacent sampling timcs is very small. As pointed out [Moose and Wang, 19731, the actual density is not Gaussian, but is in fact a weighted sum ofGaussian densities. However, it was determined expcrimcntally from computer simulation that even when the plant randomly switched as often as the duration of several systcm response ~ ~ I I I C S , the Gaussian approxinlation was quite good. The mean and variancc of the Gaussian dcnsity arc rccursivcly dctcrn~incdby the bank of Kalman filtcr estimates and the associated filtcr covariancc. The following discusscs the linear state dynamic equation in Kalman filter modeling bascd on the seeker boresight data mcasurcmcnt. Assume that or is the radome abcrration crror with explicit look-angle dcpendence. Thus, the time derivative of the mcasured LOS [see Equation (2-10)] to the first-order linearization approximation can be obtaincd as shown in Equation (2-15b). Assuming type I radio-frequency tracker, i + w,c = 6 . The measurement equation can be formalized through thc LOS ratc measurement modcl [scc Equation (2-15b)l as

Sec. 8.4

Radorne Error Calbratlon and Compensatlon

523

whcrc 2 is thc smoothcd LOS rate from thc guidance filtering (of a t least sccond ordcr) o ~ ~ t p uThe t . nlcasuremcnt partial H is defincd as

H = - 0

(8-90)

whcrc 6 is the s~noothcdvalue of 0 passing through the same guidance filtcr. In Equation (8-89) the mcasuremcnt noisc il is thc zero-mcan whitc-noise Gaussian process with the spectral density equal to the LOS measurement noisc spectral density N,that is, the boresight angular data noisc sigma value. For small look angles whcrc 1 ~ r . < l 10°, the sequential correlations of the radomc error slope can bc modelcd simply as a random walk process (the correlations arc not consistent enough to bc modeled),

where T = t k + l - tk and w is a Gaussian white-noise process with plant noise variance defined as E[w(t) . w(T)] = Q(I) S(r - T), whcrc Q(t) is given by the associated standard deviation from table look-up, if available, and may be time varying. In Equation (8-91), drldt = d(da,/au,)ldt is of higher-order dependence on 0 but is neglected in the linear filter approach. The first-order pitch rate dependent term is already included in Equation (2-13b). For actual flight the auto correlation function for the radome error slope tcnds to stay constant, rather than exponentially correlated such as that modeled by the Markov process. This is because the typical radomc error dependence on look angle is sinusoidal. Thus, the radome error slope can best be modeled as a random walk sk- I = xk + tub. The weighted plant input

.

actually represents a very smoothed version of the radome slope estimate and is used in a feedback version [that is, 5 in Equation (8-92) is used in Equation (893)] to generate a "prediction term" so that Equation (8-91) is modified to become

The prediction term accompanied by the system noise variance is derived based on UL = u - 0 at zero LOS, and hence drldt = measurement rcsidual/uL, where the measurement residual is equal to z H. This additional term has demonstrated improvements in the estimation procedure. The plant noise variance Q is generally determined on line with radome mapping table look-up. However, for simplicity it is assumed that Q is identical for the three-plant configurations and can be calculated in the first cut as the table look-up value with respect to the zero-mean radome slope. Thus, with the same boresight measurenlent noise variance R for the three filters, it can easily be shown that the three Kalman gains are identical if one starts with the same initial filter variances. Using the a posteriori probability as the weighting function to weight the estimator outputs, and assuming that the weighting

z.


524

Chap. 8

coefficients change very little from sample to sample, that is, P,, h i - I * P,,L (this assumption is not well justified if the slope is rapidly changing from sample to sample), the following can be obtained [Moose, 19751: ikil

= @

' ?k

+ ??b + Kb+l

..

'

[&&+I

f

0 '@

'

vk

+

0

-

' ub]

(8-94)

where indicates the operation of radome mapping process and becomes unity in case of n o available satisfactory mapping data, that is, without the prediction term introduced in Equation (8-93). N o w , the bank o f three filters is essentially reduced to a single filter with weighted plant input 5 given by Equation (8-92).

Closed-loopanalysis for adaptive radome estimator. The adaptive estimator results for terminal flight based on an open-loop guidance simulation are plotted in Figures 8-30a through 8-30c for a sawtooth radome error slope model. The sawtooth radome error slope model corresponds to the radome error model as a triangular wave function of uL,P that is, the radome error varies with the LOS. In Figure 8-30a, the broken curve 8/10 represents the sinusoidal body pitch-rate dependence introduced in the simulation. The dotted curve i . H is the reconstructed radome error and follows very closely the boresight error (actual data) z shown bv the solid curve, despite the large noise level chosen (at 0.5"isec 1 u level). I t has the tendency of cutting the corncrs when z goes through sharp spiking behavior. Figure 8-30b compares the actual r a d o ~ n eslope in the solid curve v with the estimated radonle slope in the broken curve i. A response lag of approximately 1 sec can be identified. The cstimatcd slopc also tends to overshoot when the actual slope changes sign abruptly. The rcason that the estimates degrade near 3 and 8 sec is a nulling c of the missile body pitch ratc, which characterizes the radonlc crror s l o ~ sensitivity and provides cscitation for thc estimator. The wcighted plant input C, as shown by the dots, follows thc general trend of the slope estimate and can be denoted as a further smoothed-out rcrsion of the estimatcd slopc. The a postcriori probabilities for each plant are shown in Figure 8-30c. In Equation (8-90), P+ + Po + P- = 1 . In actual flight only thc negative and positive slopes need calibration, and hcncc in actual implcmcntation of thc in-plane radome crror calibration scheme only the positivc slope probability P , and the negativc slopc probability P - arc used. Therefore, the rcsults herc arc normalized so that P- + P- = I . Thc zcro slope probability Po is still shown hcrc to indicatc the trend. Ucsidcs the 1-scc rcsponsc lag. the probabilitics yicld strong indications of thc polarity of the slope. For instancc, Preaches 98 perccnt a t about 1.2 scc after the slopc switchcs at 0.8 scc. It starts to dcgradc bccausc thc scnsitivity paramctcr 0 is gctting smaller and thus reduces the variance or confidcncc lcvcl (which is related to the a postcriori probability) in the changing cnvironmcnt. TO check thc adaptivc filter perfor~nanceagainst rcalistic data, hybrid simulation runs wcrc pcrformcd with all thc rcicvant angular information for the specific pla~lcrccordcd. A hllOF sin~ulationwas cmploycd with thc hcad-on cngagemcnts choscn a t 1-dcg heading error. Thcrc is a 0.5 dcglscc 1 u noise lcvcl on the LOS ratc. Thc ratc gyro bias of 0.05 dcg/scc appcarcd on the LOS ratc as ~ b , ~ The , .

Sec. 8.4

Radome Error Callbratlon and Compensation

525

radomc crror slopc was a fast-changing process with the signals switching several times. The bpdy pitch rate manifests more flailing behavior as can bc seen from thc broken line 6/10 in Figure 8-31. The reconstructed radonic crror curve i . H, as shown by the dots, still follows vcry closely the boresight crror (the solid curve Z ) . The comparison of the estimated slopc and the actual one was vcry difficult due to the anibiguity in the average and local slopc calculations during the hybrid run. Not estimating the cross-plane slopc simultancously makes the missile roll attitude contribution inseparable. After imbedding this adaptive estimator into a terminal guici'~nccforward siniulation program to perform the closed-loop analysis, encouraging results have been obtnincd. llcpcnding on the actual radon& crror and the approaching trajectory, the improvement in the miss-distance performance index ranges from 70 percent to 50 perccnt. In Yuch and Lin [1984) the estimated radome error slopc is used in guidance and autopilot commands to provide an optinial pitch rate compensation scheme for a modified proportional navigation system and an optimal controller [Yuch and Lin, 198j(a)l. Using P+ and P - , the probabilities of radome crror being in the positive and negative regions, respectively, to scale the

Figure 8-30 (a) Comparison fo Estimated and Actual Radome Errors after Matched Guidance Filtering for Sawtooth Model (b) Estimated and Weightrd Radome Error Slopes (c) A Posteriori Probability for Adaptive Radome Error Estimation


(C)

Figure 6-30

(Continued)

Chap. 8

Sec. 8.4


527

pitch-rate con~pensationfor enlarging the radomc stability region, has also demonstrated rcduccd cxcitation in acceleration comn~ands.

Discussion.

A Kalman filter bank is dcsigned to enhance the dynamic response timc for the radomc error slope estimate with compensation for the seeker dynamic lag. In calculating the critical weighting coefficients (a posteriori probabilities), a measure-predict-measure techniqi~eis used when the semi-Markov statistics of a random starting proccss are used to make the intermediate predictive step. That is, the resulting estimated radomc slope paramctcr is the statistical average weighted by time-varying, a posteriori hypothesis probability, which is calculated concurrently with the recursive filter scheme by using Bayesian rule. T o reduce the computational burden, the Kalman filter bank is digitally simulated and dcsigned by tuning the noise processes, including the measurement and plant noise, to allow a one-time calculation of the Kalman filter gain. The simple one-state filter described previously can be modified to include a correlation parameter for studying the crossplane errors. The bank of Kalman filters can be increased to 3 to enlarge the dynamic range while reducing the system response time simultaneously, and to estimate the cross-plane radome error slope simultaneously for three-dimensional engagement. The adaptive radome estimator design is intended to be an add-on compensation

Figure 8-31 Data

Comparison of Estimated and Actual Radome Errors for Realistic


528

Chap. 8

network that is independent of guidance computer and autopilot design. The objective is to permit relaxation of missile bandwidth requirements by reducing error due to radome at the guidance computer output, thus enhancing missile performance.

8.4.3 Design Equations for PNG with Parasitic Feedback Following the excellent paper by Travers [1986], this section presents the design equations for PNG with parasitic feedback. In general, a good portion of guidance and control system design is taken up with iterative analysis surrounding a series o f dynamic models o f increasing complexity. T o initialize this process for PNG with parasitic feedback, algebraic equations have been developed that incorporate certain elements o f synthesis and analytic design [Newton et al., 19573. The canonic model is a seventh-order system that includes radome aberration and imperfect seeker body motion isolation. The model, the miss budget as a function of radome aberration, and the stochastic commanded acceleration and control surface rate due to range independent noise (RIN) are described by the 15 parameters that result from the design equations. Additional indicators provided to guide the design include the rate-loop gain and the lo\\.-frequency phase margin of the autopilot. Preliminary design should be very conservative fro171a stability margin standpoint, and system performance levels should be conservatively established to accommodate attrition during the design period oxving to increases in the ordcr of the dynamics.

Problem definition. Figure 8-32a. u~hichis rcdra~l-nfro171 Figure ., 2-ljb, is a block diagram of the homing process. The details of the homing controller of Figure 8-32a arc displayed in Figure 8-32b. The seeker with its target tracking and body-motion isolation subsystcn~s,as shown in Figurc 7-13b, is reprcscntcd by the multiloop section o f Figure 8-32b. Those autopilot signals of particular relevance arc 6, a,and 0 . T w o additional signals that provide a measure of the cffccts of noise on control activity arc the RMS values of u, and 8 due to RIN. The problem consists of defining all the parameters of Figurc 8-32b so that the result of a simulation will mirror thc miss distancc predicted by the stochastic inputs and the guidancc systell1 dynamics. A secondary aspect of the problem concerns computing some relevant indicators of system pcrformancc. T o this end, consider the sin~plcrdiagram of Figure 8 - 3 3 . The forward path is a fifth-ordcr system \vith the high-frcquc~lcy poles and zcros of the antenna gyro loop ncglcctcd o\\ving to the fact that they arc usually found \vcll outside thc guidance band\j,idth. The polcs of the antenna gyroloop transfer function arc likeivisc ncglcctcd in the feedback path, but the zeros (Tzs)' arc not sincc these providc thc representation of body-motion leakage. Ncglccting thc high-frequency polcs of the antcnna gyro loop thercforc allows the ordcr of thc systcm to be reduced from scvcnth to fifth. The parasitic elements in Figurc 8 - 3 2 arc a sum of the following thrcc elements: thc radolnc slope bias ( 1 . 1 ~ ) the radornc slope (r), and the leakage in the seeker body-motion isolation loop ( ' 1 . 2 ) . >'

Advanced Guidance S p t e m Design

530

Chap. 8

These elelnents are in cascade with the aerodynamic transfer function from achieved acceleration to pitch rate 0. The manner in which the parasitic elements are studied separately is as follows. First, it is assumed that v b , the value of which is not known at this point, does exist in the parasitic feedback path. For that system the effects of v 7 are investigated, following which the appropriate value of vt, is established to ensure that the best system performance is obtained when the actual radome slope is zero. Finally, one treats the effect of ( T ~ in S )the ~ presence of rr, where the signs of these coniponents are permitted to vary independently. Generally, the sign of (T>s)* in Figure 8-32c is positive. However, in the case of utilizing sight line reconstruction INesline and Zarchan, 19851 to extend the virtual bandwidth of the seeker gyro loop, the system becomes dependent on the scale factor errors of the seeker and the rate gyro. It can then happen that the sign of the parasitic feedback loop becomes negative. The zeros of the tail-controlied airframe shown in Figure 8-33c in the forward path are cancelled by the corresponding poles in the feedback path, so that they are nonexistent to the extent that feedback is concerned. However, these zeros still exist in the closed-loop response, but are neglected for the sake of simplicity in the analysis presented here.

Design equations.

In this section, the subscript 0 denotes the forward path in the absence ofany parasitic feedback, while the subscript b denotes the closedloop respo~isein the presence of r.1,. The subscript + is used for the closed-loop response in the presence of eithcr rT/2 or - I . = / - . 7 O r d e r Reduction by Truncation.

In Figure 8%32c, thc homing controller has

the following transfer functiot~:

G.~(s)=

A,,,

-) 11,

-

=

K ( 1 + 4 + 1) T ~ , , . c+. ~a,.,T:,,c2 + P.4 T.4,,s+ 1 7,. = effective autopilot time constant

-- P.4T.4,, (8-95a)

where the polcs of the autopilot transfer function arc shown norlnalized with a charactcristic tinlc constant ( T . , , ) the , cube root of the s3 tcrrn, and two-dimensionless coefficients (a, and P . 4 ) . Similar to Equation (2-42a), studies of autopilots had used this for111of the cubic because of the plot of Siljac [lY(,Y], which showed the loci of the ratio of the frequencies of the poles and the damping ratio of the quadratic in thc a, f3 planc. The guidance transfcr function \vithout parasitic effects is givcn by

A ,,, (3) u

=

A C', s C ,4 (s) (1 + ?.~)(1 +

7,,.c)

It was thought that the order of this transfcr function could be reduced from five to three to limit further the number of variables it1 characterizing the guidance dynamics, atid that the cffccrs of parasitic coupling might be plotted in the a, P plane. It happens, llowevcr, that the cubic, although sufficient for thc effects of Ya

Sec. 8.4 7'-LE

Radome Error Calfbration and Compensation

531

8-3 TRUNCATION ERRORS, 5 -+ 3

= percent error due to truncation

:\ = 3.6

From [Travcrs, 19861 with permission from AACC

and r ~ is, too low an ordcr for consideration of T2. Truncating Equation (8-9510) results in

'guidance time constant = PIjTlj= ~vherethe normalized u coefficient Po is given by T~

T,

+ T, + T~

(8-95c)

In an effort to analyze the errors arising from truncation, a comparison was performed of normalized stochastic miss adjoints prior and subsequent to truncation. Inasmuch as these adjoints are normalized to the s coefficient, which is not altered by truncation, a comparison of adjoints is equivalent to a comparison of miss distances. Some of the results ofthis study are listed in Table 8-3. The errors in two stochastic adjoints are shown for combinations of cubic coefficients and values of the ratio ( T ~+ T , ) / T ~ , which is denoted by R. The RIN miss is consistently more sensitive to truncation than the target maneuver miss. The most severe truncation errors occur for a combination of low autopilot damping ratios and low P.4 when the quadratic is the dominant term in the cubic. It was concluded that truncation errors do not generally exceed 10 percent when the damping ratio of the autopilot

Inouts Outouts Analysis:

fit

Pb

B-

Rs Kb

K+ K.

A

B+ P-

Synthesis:

Eauations

pb

R,

K+

Kb

K.

A

Note that the symbol + used in Fisure A and in subsequent expressions means that all

operations may conditionally be e~ther+ or -. The closed-loop response for the configuraration in Fisurure A is given by Kbvc s l [lt(Vc,.li,)K,(ry'-))]

4, - -

[:'~tab:)~+[~b*(~C~m)~b(~T,~)(~,~b)~b]~~t(~c~m)Kb 1 (1) For notational ease, the follo\ving paramelers are defined:

.. "c

R

q = R , K h D ; A-T,,'T, (2)

This allotvs Equation (1) to be more compactly expressed as

A" = [V,K,s] 5

where '=

=

'I

q!h

-

7

i - t CI= 7 ;

+ p,

7,

+ 11

(3) -1 3

T+s ; K r = K ~ ( I + , ~ );- 'T+ = T h ( l = q ) -1 3 ,

(ltll)

,

(

-mt h( I )

.I 3

:

B,

r

(ill, t I ~ A(l+tl)" )

The ratio of K to fl as a function of y, and 11 can be obtained from Equation (4). IK*/P*l, 1 ;11, (1-1,'V) 13

Here, 111 is defined as the value associated tvith yl and unity.

(4)

(-5)

Sec. 8.4


533

is above 0.5. It is pointed out that the airframe zeros are neglected in this study. The truncation errors are recognized as being a component of the miss scale factor, S,,, which serve to bring closer together the predicted miss distance of the equations and that resulting from numerical integration.

Guidance Loop Transfer Function with Radome Slope Bias Effect and Without Radome Slope Effect. The closed-loop guidance transfer function that results from the existence of rl, in the feedback path is obtained from Equation (8-95c) as follows:

A,,, u

Y (3)

T$

-Y?

+

Al/; s I + rhA V,/ V,,, allY? + [Po + rhA(V,l V,,,)(T,I T,,)]Y,, 1 + rhA V,l V,,,

effective guidance time constant

+

PbTh

=

r y [l

(8-96a)

+ YIJ\(V,IV,/,,,)(T,IT~)] 1

+

ri,A V, I VpfZ

(8-96b)

\vhcre 7,; has the same formula as that in Equation (3-28). The relationship between the quantities subscripted with 0 and those with b is

Attitude-Loop Parameters ( A L P ) . The study of the attitude loop begins with Equation (8-96). Having represented that configuration in the forward path by the subscript b, consideration is now given to the effect of +rT12 in the feedback path. This attitude loop is shown in Figure 8-33a. The closed-loop response for this configuration is given by Equation (1) of Figure 8-33. Equation (4) of Figure 8-33 can be considered as the basic analytical equation of ALP. From the specified parameters of the open loop, K b ,A, R,, and P b , the closed-loop parameters can be derived. These are K , and P , . Those parameters that have as a subscript b can be regarded as defining the center of the parasitic zone, while those parameters that have as a subscript f can be regarded as defining the extent of the zone. From a practical standpoint, Equation (4) of Figure 8-33 becomes more useful when inverted, as shown in Figure 8-33b. Then the center of the zone can be derived from the specified limits to the extent of the zone. It should be noted that the center of the zone is not merely the mean of the zone extremes. Disregarding 2 as a variable in ALP is a consequence of the following factors. First, the normalized adjoints discussed in the next section appear to vary only slightly with a.Second, as a result Figure 8-33 (a) Cubic Attitude Loop with Radome Effects (b) Attitude Loop Parameters (Frorn [Traverr, 19861 with p e n n i r s i o t ~ j o mAACC)

534

Sec. 8.4

Radome Error Calibration and ~ ~ t n p e n s a t i o ~

of the truncation to the cubic, there was a tendency for a3 to be very close to 2 for many practical applications. Specific values for R, and A arise from the use of the synthesis form for ALP. However, it may be advantageous to depart from using these values as a matter of practical utility in design. A well-known rule of thumb is that, when the product of R, and A remains constant, their effect can be considered as remaining invariant. Letting $, which is greater than unity, be a multiplier on A and a divisor on R,, there is then a means of identifying the condition of constant product. The ratio of K to P is an indication of the incremental variability of the normalized miss adjoints. With the introduction of $, the new value for K - is smaller, as is the ncn7 value of p- . In the next section, it will be seen that the normalized adjoints vary in a counterbalancing fashion when the introduction of $ (greater than unity) results in new values for K, and p,.

Cubic Miss Adjoints (CUBAN). The cubic miss adjoints are summarized in Table 8-4. For any set of ALP, a corresponding set of normalized adjoints can be obtained from Tablc 8-4a and the renormalizations of Equation (4) of Table 84 are defined in Equation (5) of the same table. Cubic Synthesis (CUSYN). Glint is eliniinatcd from consideration for synthesis, owing to a number of reasons. First, the approximation of glint by white additive noise is one for which the miss distance nlay not cvcn be proportional to 7 [Garnell, 19801. Inasmuch as it tends to be in the direction of target length, glint miss may not have the samc impact on lethality as the othcr con~ponents.On the other hand, one presupposes that the level of glint gets vcry small as rhc target becomes very small. If glint is rctaincd in the synthcsis problcm. it \rill alxvays be a factor in determining thc optimum bandwidth of the systcm because it is thc only disturbance for which thc miss dccrcases as thc systcm becomes slo~ver.Finally, glint is removed from the synthcsis problem in order to bring to the forefront clearly the critical issues of trading the speed of responsc (improving accuracy, except perhaps for glint) with control activity and parasitic rcquirenlcnts. Retaining the glint adjoints is done so that glint as an input option can be treated in thc miss budgets after performing synthcsis. For synthesis it is required for simplicity that any dcsigll bascd on parasitic fccdback should be characterized by symmetrical pcrforrnancc at the cxtrcmcs of thc zonc. This is a consequcncc of thc fact that thc radomc slopes in practice arc gcncrally disposcd in symrnctrical fashion about zcro, so simplc cfficicncy would rcquirc that the pcrformance zonc bc sy~nn~ctrical. In this way, the cffectivc guidance timc constant that rcalizcs this symmctry can bc detcrmincd from a specification of the stochastic inputs to thc problcnl, targct n~ancuvcr,and R1N. T h c t w o componcnts of miss distance arc cquatcd as i i l , , ~ - = MGj- ~,+;./'7,:~"' = nr~,+ = Ad&+ 4 . ~ ' 2 7 ~ " 'to yield thc timc c o n s t a ~ ~ that t cffccts this equality I/'

,;/2

This cquation is suhscqucntly substitutcd into u,,,,. = I(hlTJI,b,7 x . )

'

+

Sec. 8.4


535

a) Normalized Cubic Aaolnts

TABLE 8--4

a = ?

K = 3

p

Myg

M

K = 3.5 MTJ

Myg

K = 4.5

K = 4.0

M,r

MTJ

Myg

M,,i

MTJ

Myg

Mtri

MTJ

b) Adjoints and ALP Conditions

For the general cubic controller, the normalized miss adjoints are a function of three variables: K , a,and p identified in the transfer function for the controller, A, u

- = [KV, s]/[y3+

ay'

+

Py

+

11; y

= Ts; PT

=

T,I

(1)

These adjoints are defined for the inputs of glint, constant angle noise, and stochastic target maneuver,

which have the same formulae as in Figure 6-42. The z adjoints must be renormalized to the center of the performance zone; that is to 7;. This renormalization takes into account the ratios of P, and T f as given in the parameter q:

The renormalized adjoints, designated by a superscript asterisk, are defined as

By making iterative comparisons between ALP and typical system performance as obtained by the use of Eq. (4) in Eq. (2), it will be possible to converge on a fixed set of values in Eq. (4) for use in miss budgets. It is possible to convert the ALP zone to a performance zone using the renormalized adjoints,

and such a vrocedure lends itself to analvsis From [Travers. 19861 with permission from AACC


536

Chap.

( M ~ ~ ~ v , + ) which ~ ~ Tgives ~ ~the~ miss ~ ) with ~ ] zero ~ ~ actual ~ radome, and the result is

which gives the R M S miss in the center of the zone. Equations (8-97) determine 7; and u,,, from the inputs v&f.I2and +:I2. This gives a total of four parameters, from which any two may serve as the input parameters for synthesis. Also, from Equations (8-97) the other five equations can all be determined. The six equations involving two stochastic inputs and the RMS miss distance and 7; all constitute the synthesis equations knon-n as CUSYN that are used to initiate the design sequence. The equation for the ratio of the miss at the zone extreme to that at the center is

The importance of this ratio lies in its role in selecting the appropriate ALP, and iteration must be employed if the value resulting from the application of Equation (8-98) is not deemed satisfactory. The limited degrees of freedom associated with this simple synthesis technique may produce unexpected consequences. Experience in using this procedure has demonstrated that iteration in the C U S Y N technique may be employed to obtain reasonable values of the four parameters. A worst-case scenario, however, is one in which the synthesis procedure is so restrictive that dummy values for the stochastic inputs may have to be carried along, inserting corrections after the final step, the assembly of the miss budget. In this case the level of control activity \vould also require modification, but any modification is a simple scaling operation. Missile Siring. One of the results of C U S Y N is the identification of the stochastic target maneuver and 7;. It is required that the guidance system operate linearly. As a means of satisfying this requirement, the first step consists of identifying the step of target maneuver which is uniformly distributed over the last 7; of the intercept. Such a definition for the targct maneuver yields AT = 1 2 ( 1 ~117-/32.2(measured ) in g). For linear opcration, the maximum acceleration of the interceptor must be greater than that of the targct by a factor S , specified by the dcsigncr. This factor, termed the rnarleuvcr sfale-fanor, is gcnerally between 3.6 and 6.0, and its use defines the maximum required missile acceleration. At this point, the maximun~design angle of attack, which is a function of the missile outboard profile, must be specified by the designer. The ~nininlunlvalue of Ai, = VmZa that satisfies the requirclncnt of linear operation is given by the ratio of maximum acceleration to maximum angle of attack. From the aerodynamic definition of T= (see Table 2-2) together with an approximation, its corresponding maximum value can also be determined as T, = - 1.05/Z, and t

-

(T,),,,

5

1.05 V,,,a,, /[I845 S,q+.!'2(10 7,)t 1121

(8-99)

Sec. 8.4


The ALP together with the result of Equation (8-0')) describes

537

provide thc equation that

Since RIN is also specified from CUSYN, in addition to closing velocity, it is possible to cstimate u,,,based on cstin~atesof T, and T., as fractions of 7,:that results from C U S Y N . An early cstimate of a,,,is obtained by arbitrarily setting AIKA at 4 and T, = T,, = ~,;/8.Part of the explanation for assigning these values to T; and T, is that the effect o f r ~ is , assumed to account for 318 of T,;, while the remainder is equivalent to five equal time constants. O f these, three are in the autopilot, leaving 218 to bc divided between T , and T.,. An exact expression is given for a,,,, which will be utilized a t a point further on in the design procedure, and an approximation which is used at this point in the procedure. (est) a,,.=

v ~ + ~ K,+ ~ A / z= 1.405 V,+)/" 32.2 [~T,T,,(T, + T , ) ] " ~- 7i3I2 where AIKA = 4

(8-101)

Following C U S Y N , the missile-sizing equations can be employed by the designer to judge the reasonableness of the requirements on the airframe and the radome as well as the impact of RIN on the RMS level of commanded acceleration. Some options which could be exercised a t this point include iteration through CUSYN or the consideration of pseudo-values for the stochastic inputs. Homing Controller Expansion ( H C E ) . Before expanding to the fifth order, the necessary steps must be taken to go from the center of the parasitic zone to the truncated forward path. In order to do this, however, A and r h must be determined Equation (8-96c) is substituted into (8-96b) to get

In this equation, the solution for AlKb does not depend on 7,;. Next, one approximates the cube root of A/Kb by the cube root of 1 + E, where E is much less than unity. The solution for A is then given by

Given that K b and PJ, were determined from ALP, and that 7,; is known from C U S Y N and T, from missile sizing, A is determined here from Po defined in Equation (8-95d). Also, Equation (8-96c) can then be used to determine rb

The additional lag in the expression for 7,: is a consequence of the combination of rb and T,. The required value for T , is obtained from Equations (8-96b) and (8104). (required) T, = ~ i [ ( A l K ~ ) ( l T,/T':)

+ (T,/T~)]

(8-105)

538


Chap. 8

However, the assumed dynamics to get result in an actual 7, that may not equal the required value for T, in Equation (8-105). In this event, it is a matter of scaling the values for T,, T,, and T,, and iterating the process of determining A, r b , and the required T, until the value of 7, that is obtained is that required in Equation (8105). This process may be done manually or in an automated fashion. The rule for assigning values to the time constants in the forward path may conveniently be taken as the one given under missile sizing for the estimation of u,,, although it may have to be altered if u,, is too large. Moreover, local requirements of seeker tracking may dominate in setting 7,.Since u,, is dependent on T ~where / ~ T =7 - T,, T should be made larger if possible. This completes the expansion of the order of the controller in the forward path, and determines the closed-loop response of the autopilot and T,, T,, and A. Also determined in the feedback path are r ~ T,. , and r b

Seeker Gyro-Loop Bandwidth. Because the effect of TZis greatest on the s3 coefficient of the closed-loop response, the order of the forward path should not be less than 4 so that its effects on the dynamics and hence system performance are adequately represented. In fact, a value of T2 was derived by the use of a thirdorder system for the instance in which the sign of TZ feedback was always positive. After completing the fourth-order analysis, it was discovered that the derived expression for TZcompared favorably with that derived from the third-order case. The fourth-order analysis necessitated further simplifications to avoid increases in complexity. First, only a single set of nominal normalized coefficients for the quadratic polynomial was used. Also, the case of large $ was postulated, giving large values of T,and small values of r . ~ It. is possible in this case to neglect the variation with 1.7of A and the normalizing of To. It \\.as also determined that the effect of TZon the .r2 term was sufficiently small as to bc reasonably ncglected. Then Tr influenced the 3' term and r r , the s term of the closed-loop character~sticpolynomial. It became possible to plot these adjoints in the plane of the third- and first-order coefficicnts. Additionally, these adjoints were normalized to T4 as opposed to the sum of the time constants. o r the s coefficient. As a result, the miss distances were made directly proportional to the values of the adjoints in the coefficient plane. One axis in the plane of the adjoints represented the cffcct of r T and the other, the effect of T2.It then became an easy tnattcr to examine effects of independent variation of the txvo parameters. Assuming forward path A,,,/E = YKs/[+% a443+ ~ 4 4 ' + y 4 b + I ] with 6 E 1 ; 5 , and the feedback path H,,,/,4,,, = [ t r + (&r)'](l + T,s)/( l,',,,~),thcn thc closed-loop response is

Having simplified the problc~nin thc manncr spccificd, the adjoints in the plane of the a and 6 coefficicnts wcrc normalized again rclativc to thc values at the point 0f

Sec. 8.4

Radome Error Callbratlon and Compensatfon

539

zero parasitics. Then the pernlissible excursions of Aa and A y were established empirically by analyzing a number of practical cases. Using Equation (8-106), Aa ?= ( I .:/V,,,)KT,T i / T i , then substituting T bfor T4yields

It was found that the empirically determined value of ha of 0.4 increased the miss distance on the order of 10 percent to 15 percent.

Autopilot and Airframe. The calculation of u s requires the airframe gains, and even if thcse are known a prior^, a number of restrictive aspects of the autopilot design must be given consideration. Included among those to be examined arc the rate damping loop gain and the low-frequency gain margin (LFGM) of the loop transmission. There are many instances in which the aerodynamic gains are not available a t this juncture in the problem. However, a means of deriving the gains is provided for these cases, which is based on the fact that certain factors involved in determining the airframe parameters are more easily estimated owing to the fact that they are dimensionless ratios. The ratio of Z 8 and Z , can be either remembered from experience or estimated. This is also true for the ratio of the maximum values of 6 to ct a t trim. Finally, the ratio of .21s to - V,Zs can be approximated from the length of the missile. Length e is the single physical factor involved in the ratio of mass to pitch moment of inertia for a solid cylinder. With these three inputs and V,,, and M,, the stability derivatives are obtained

The characteristic polynomial for the autopilot can be derived from Figure 8-32b and equated to the normalized cubic used in Equation (8-95a).

By equating coefficients of like order and transposing terms that do not involve the control-loop gains, three simultaneous equations are obtained, which are expressed as follows in matrix form.

+ Za

~ A O A O

(8- 109) Here, bMs denotes the rate damping loop gain, acMs denotes the synthetic stability loop gain, and a p denotes the accelerometer loop gain. Gains a, b, and c are derived from these loop gains which are designed in Book 3 of the series. The loop transmission of the autopilot is one of a conditionally stable system. At low frequencies, the phase margin due to high attenuation rates must be large enough to accommodate the phase shifts due to high-frequency lags yet to be mod-


540

Chap. 8

elcd. T h e LFPM is =90 - tan-'(ac~4l\l(bMa))~.For the purpose of computing u6 due to RIN, the transfer function of the fifth-order system in Figure 8-32b is identified as

Because of the complicated form of the definite integral over frequency o f the magnitude squared of the transfer function of KA = (1 + CIV,)-' from minus to plus infinity, the tabulated formula of Newton [I9571 is used to compute this function numerically. However, an approximate literal solution of lower order maintains the dimensionality of the results and is beneficial for showing the sensitivities to some of the parameters. Such a solution is derived by defining the low- and high-frequency asymptotes of Equation (8-1 10) and then integrating under these asymptotes using the third-order noise integral formula. This gives (approx.) us = 10.433 hV,Q;"

) h4, ] ] ~ I ~ ( T , T , ) ~ ' ~ ( T A O (8-111) )~~*]

In Equation (8-1 1I), each of the time constants in the denominator is some fraction of the guidance time constant, so that a5 can be shown to be inversely proportional to T;(2;~6) . Given that the high-frequency asymptote cannot continue to infinity with a sGpe of - 1. a multiplicative correction factor is constructed by integrating from a fixed frequency to infinity under the high-frequency asymptote and then subtracting this result from the result of applying Equation (8-1 1I). This correction factor, denoted by CF, is then In the preceding equation for CF, the fixed frequency is arbitrarily given a value that is five timcs the rate-loop gain, and a typical value for C F i s 0.9. Thc preliminary design procedure should concludc with the ability to roughly size up the control surface actuator. It is not possible to estimate the hinge moment, but the product of wi and .tls can provide a relativc reference relating to the size of the actuator.

Applications.

At this point, many of the preceding cquations are used ro obtain the numerical valucs for the parameters required of simulation. The tables containing numerical valucs that follow also havc refcrcnccs to the pertinent equations, and the order in which thc cquations arc bcing uscd can be identificd from thcir scqucntial listing in thc tables. The pararnctcrs in the tablcs are uscd in a sirnulation that makcs usc of thc adjoint method to gcncratc nliss distanccs. Thcn, a comparison is made bctwecn these miss distanccs resulting from simulation and those produccd by using the design cquations. Prior to invoking the design equations, the ALP constants and C U B A N adjoints arc uscd to obtain the constants for the CUSYN cquations. For thc examples that follow, a listing of the ALP constarlts and C U B A N adjoints is givcn in Table 8-5a. For the ALI' constants, the zorlal extremes wcrc uscd to dcfinc the operating points of the centcr of thc zone. Using a value of 3.86 for the ratio a/Qa corresponds to a valuc of unity for $. The miss

Sec. 8.4

Radome Error Callbration and Compematlon

TABLE 8-5

(Prom [Travers, 19861 wlth permission from AACC) a) Prwrarnrned Constants for Examples

541

ALP (Fig. 8-33b) 5.00 Kb = 3.43 P- 1.75 P b = 3.52 3.0 A = 13.6 4.0 11. = 1/12 q = 117 CUBAN (Table 8-42 and Eq. (4) of Table 8-4) M$+ = 1.21 M,,th = 1.87 Mti- = 6.59

p*

= = K' = K- =

= 1.24 Myyb = 1.70 M = 3.43 M .

M = 1.71 M ~ j b = 0.77 M*,.. = 0.48

b) CUSYN Equations a n d Design Point for Examples Shortened notation: J

5

6;'; L

E

vC+/-JZ; M umb; T

I

'rg

CUSYN Equations (Table 8-4 and Eqs. (8-97)) Inputs J.T: L,T: M,T: M. L: M, J: L. J:

L J L J L T

= JT'i3.854 = 3.854L/T2 = M/(3.505T1") = 581.8L51M'

;M ;M ;J ;T

= 0.9096JT5" = 3.505LT1"

1.0994MlT5" (MIL)'/12.29 = ~~"J~'~13.572 ; T = 1.039(M/~)~" = 1.963(LiJ)11' ; M = 4.911(Li"lJ"4) CUSYN Design Point (AT is defined in Missile Sizing section) Inputs: M = 12.5 ft. T = 0.65s. V, = 5000 ftls Outputs: L = 4.42 ftlsiHzl" J = 40.3 ftls3/Hz112 = 3.2 g, + : / I = RSD of RIN = 885 x 1 0 - " a d / ~ z ~ ~ ~ AT MISSILE SIZING Maneuver Inputs: scale Sg = 4, a,,, = I5 deg (cases I & Ic), V, = 3500 ftls factor Missile e = 17 ft, a,,, = 30 deg (cases I1 & IIc) leneth " Outputs: see lines 2. 3, and 4 in Table 8-5c =

=

c) Parameters for Examples (and Comparisons with Simulations)

Line Number

Example Parameter

Un~ts

I -

1 2 3 4 5 6

amax

Tmmax rTmln (est)~,, aA

PA

d% sec

62

-

la --

15 2.34 0.0625 11.86 2.5 2.3

15 2.34 0.0625 11.86 1.8 1.8

IIa

11 -

-

30 4.68 0.0313 11.86 2.5 2.3

Ref.lEq. #

-

30 4.68 0.0313 11.86 1.8 1.8

(8-99) (8-100) (8-101) (8-95d)

Advanced Guidance Sljstem Design

542

Chap. 8

(Continued)

TABLE 8-5c

Example

Line Number

Parameter

Units

1

la

11

Ila

Ref./Eq. #

sec sec

-

KA Vrn)

22

deg

LFPM w, =

rad/sec2

u6bMa U",

radlsec3 g DESIGN RMS MISS

rT/2

Sm

+

1.3

31.1

0

1.3 1.3

16.3 31.7

-

*The adjoints M,r+ arc divergent.

-'

(1

+ C!

Low frequency phase margin (8- 107) 15(8-I 12)

V ~ I T ~sec- '

mi

5

RMS MISS FROM SIMULATION

Sec. 8.4


543

adjoints listed undcr CUBAN in Tablc 8-5a arc from Tablc 8-4a. Substituting these values for the adjoints in Equation (8-100), the ratio of maximum to minimum miss is expected to be 1.9. Table 8-5b lists the six CUSYN equations based on the adjoints of Table 8-5a. Now, in starting the examples, it is noted that any two of the four system parameters can be used as inputs. A step of target acceleration of 3.2 g with a random uniform distribution over the 6.5-sic interval before intercept is listed in Tablc 8-4b. From Equation (2-20d) T, = 0.1 sec, and from Tablc 8-5b r)' (an RMS value of RIN) = 2 millirad. The remaining portion of Tablc 8-5b deals with thc inputs for sizing the missile. Two values of a,,,.,,= 15 and 31)deg are used to highlight design sensitivities to this factor. The results of using the sizing equations are given in Table 8-5c, in which 35 numbered lines of data are arranged in columns that correspond to the examples and subexamples. The first line of data is for a,,, specified for the two designs. Lines 2-4 are the data that result from using Equations (8-99) through (8-101) to obtain (T,),,,, r ~ , , , , ,and (est) ow,,respectively. The decision is made at this point to design to the indicated values for (T,),,,, which is equivalent to placing the easiest requirements on the missile. That is, the missile is required to simply meet the specified maximum acceleration with no excess. The value of 11.9 g for a,,in line 4, for a missile whose capability is by definition only 12.8 g (4 x 3.2), clearly reflects a level of commanded acceleration due to noise that is not consistent with the requirement that the missile operate linearly. As a result, the two examples are further subdivided into additional cases, and, for each of the two missiles, different dynamics are now entered for the forward path. In lines 5 and 6, a* and PA, which are the cubic operating points of the autopilot, are equal to 2.5 and 2.3, respectively. The quadratic roots of the autopilot have a damping ratio 5.4 = 0.67 whereas for the other values 5.4 = 0.4. In lines 7 , and T, are determined in an iterative process which assures that the and 8, T ~ T,, equality of the actual .i, is equal to that required in Equation (8-105). They follow the rule of 118 closely, while 7, and T, are deliberately made larger in column Ia to reduce the value of a,, in line 26. The preceding remarks also hold for the other two columns respectively. Thus, the subdivision into four columns is motivated by the need to do something about (est) D,,. As a result, A and r b are determined. In columns I and 11, (est) a,, agrees well with the actual a,,.The only way to increase T, and T, would be to make the autopilot faster. However, increasing the speed of the autopilot by scaling TA"means that faster servos would be required and the value of a* would also increase. In columns Ia and IIa, these effects have been circumvented temporarily by reducing the quadratic damping ratio when the autopilot was made faster. In spite of this, the lower damping ratio in the autopilot will only make it harder to increase the order of the autopilot with the dynamics of the actuators, instruments, and structure, and so on. The final solution may involve reducing the amount of noise or increasing the time constant of the system. In lines 11 and 12, (Zs/Z,),, and (6/a),, are inputs used to estimate the stability - V,Zs, and I*. in lines 13-17. The differences in derivatives, M a , Ma, - V

544


Chap. 8

values across the columns are connected with the different outboard profiles of missiles designed for two different a,,,,,. As a result (L varies by a factor of three. Lines 18-23 are attributes of the autopilot, and line 23 in particular lists the gain crossover frequency of the seeker loop gyro, which is related to T2. Lines 24-26 are related to the computed levels of control activity, while lines 27-29 list the RMs miss distance for positive, zero, and negative radome slopes as designed and as simulated for the four cases. By examining the values across the columns in line 29, it can be seen that the largest errors occur with negative radome slopes and with higher a.Also, the simulation results in lines 30-32 show that decreasing T2 fourfold t o approximate the effects of perfect body-motion isolation causes a reduction in miss distance, which is greatest for negative radome slope. A divergent increase in miss distance can be seen in line 30, columns Ia and IIa. This divergence is brought about by the onset of high-frequency instability at positive radome slope, a wellknown phenomenon that is discussed in Nesline and Zarchan [1984(b)]. The last group of simulations concerns approximating the effect of neglecting the airframe .. zeros. To this end, Z s was made zero and the miss distance consequently decreases as compared to the design values with no miss scale factor, as shown in lines 3335. Here, the largest percent reduction occurs with positive radome slope.

8.5 COMMAND VERSUS SEMIACTIVE HOMING GUIDANCE SYSTEM DESIGN AND ANALYSIS Comparison of command versus homing guidance is presented in this sccrion. Further details appear in [Alpcrt, 1988; Durieux, 1984; East, 1984; Neslines, 19861.

8.5.1 Modeling As notcd in Chapter 6, semiactive homing missile systems and command guided missile systems both continue to be in operation today. The analysis in this section assumes perfectly known ranges and closing velocity. Figure 8-34a is the standard linearized system block diagram of the missile intercept problem that uses synthetic PNG in the command guidance mode, which rcquires range estimate R to obtain the LOS angle. As shown in Figure 6-35b, synthetic PNG differs from standard PNG only in that the former rcquires LOS from the missile to thc targct to be mathematically cor~structcd bascd on fire-control radar mcasurcmcnts. Conscqucntly, i t contains the radar's mcasuremcnt errors of the missile and target. It is not rcquired to know thc rangc with homing systcms because thc geometrical LOS is obtained strictly from the physics of the situation. Therefore, homing systcrns do not have to perform any rangc measurements as is necessary in the casc of command guidance. Boresight error, which is proportional to the LOS ratc, is measurcd by the homing seeker. As command guidance docs not employ a scckcr, however, the LOS ratc must bc reconstructed from thc derivative nctwork (see Figurc 6-35b(i)). Figure 8-34a contains the firc-control radar noises prcsent in a

Sec. 8.5

545

Command versus Semiactive Homing Guidance System

MISSILE TARGET TRACKING NOISES TRACKING NOISES RANGE DEPENDENT NOISES

RANGE

b-pa&E NOISES

1 " , ' P O

-

AVC8 h T

I1

+

11s

IMPULSE ACCELERATION

Figure 8-34 (a) Command Guidance System Diagram with Noises (b) Adjoint Diagram of Command Guidance System with Range-Dependent Noise (c) Equivalent Adjoint Diagram of Command Guidance System with Range-Dependent Noise (Frotrr [Alpert, 19881, O 1988 AIAA)

command guidance system. Again, the system is assumed to be fifth order. All the noises are input at the point just before the division by the range estimate. The rate f, at which the fire-control radar tracks the missile and the target is assumed to be high enough so that the track loop can be considered continuous, and not sampled data. Before entering the guidance loop, all fire-control radar noises, which are in angular units, are multiplied by the range from the radar to the object being tracked (that is, R T for the target and R , for the missile). The adjoint system diagram (see Figure 8-34b) from the linearized system diagram, often called the forward system


546

Chap. 8

diagram (see Figure 8-34a). A multiplying factor of R'% is used for the track of the target (or R: for the track of the missile) between the point at which the signal leaves the adjoint of the homing loop as ~ C L Tand the place where it enters the integrator. System analysis of the dependence of range to intercept of Command guidance system performance is discussed in Section 8.5.2 where c l o ~ e d - f o r ~ steady-state adjoint solutions for statistical miss distance are derived for Command guidance systems that use PNG. Command.versus semiactive guidance system performance at specific ranges to intercept is discussed in Section 8.5.3.

Semiactive homing guidance dynamics. The guidance filter of Figure 7-6b is shown in discrete domain in Figure 8-35a. As in Chapter 7, y, is the input signal to the Kalman filter, which is followed by the guidance law and FCS to generate missile acceleration A,,. Figures 8-35b shows the complete homing loop dynamics for the semiactive system employing optimal guidance system (OGS). LOS angle u is sampled at a 100-Hz rate and then is converted to y, by multiplying by R. Since l? is assumed to be known perfectly, the seeker model becomes a measurement of y, corrupted by noise.

Command guidance dynamics. Figure 8-3% shows the complete command guidance loop dynamics employing OGS. The ground antenna and receiver

I

I

I I

I I I UALMAN F I L T E R

--

I------

(A)

Figure 8-35 (a) Discrete Kalman Filter and Optimal Guidance Law (b) Semiactive Homing Loop (Forward Time) (c) Command Guided System (Forward Time) (From /Nerlincr, 19861 with permirsio~ifnrm AACC)

Sec. 8.5

Command versus Semiactive Homlng Guldance System GLINT CORRELATION

547

-

ITS = SAMPLING INTERVAL. TN CORRELATION TIME CONSTANT)

GLINT NOlSE I (1 GL11121 I

9-

RANGE INDEPENDENT NOlSE

Of MISSILE FLIGHT CONTROL SYSTEM

DISCRETE ANGLE NOISE

LSGAGL+\T

GEOMETRY

SEEKER

Yr klml

100 HI SAMPLER

krml

J-% FILTER AND GUIDANCE LAW

- 4, \

-

100 HZ SAMPLER

ISEE FIG 2Al

ATMOSPHERIC NOlSE ROLLOFF FILTERS

THERMAL YOlSE

4"

-J I

RANGE INDEPENDENT NOISE

ATMOSPHERIC NOlSE I 0 ATM 1 4 I

Or DISCRETE ANGLE NOlSE

MISSILE FLIGHT CONTROL SYSTEM 1

E\GAGEUL\T GEOMEIRY

SAMPLER

GROUND RADAR

KALMAN FILTER AND GUIDANCE LAW

Ym

SAMPLER

(C)

Figure 8-35

(Continued)

I T ~ S + ~ , ~

-

Phq

-

-

100 HZ SAMPLER

TRACKER

-

ENGAGEMENT GEOMETRY


548

Chap. 8

measure O,,, and O T . Unlike homing, the radar must also measure various range infor~nationR T , R,,,, and R in order to convert Or and O,,, to y coordinates by y T = RTeT, >I,,, = ~ , , , 0 , , ,and , y,. = f T - f,,, to obtain the LOS angle u = y , . l ~(these measurements are referred to as R T , R,,,, and R ) . In the command guidance system, separate noise terms corrupt the measurements o f 0 T and O,,,. It is assumed that this missile has a transponder and therefore the measurement of O,, is perfect co~npared to that of 0 T .

Noise inputs. In addition to glint noise ~ , ~ i i , , ,semiactive , systems must deal with range-independent noise (RIN) af and range-dependent (or thermal) receiver noise v,.,, sources. The major noise sources (on the measurement of Or) in the command guidance system are atmospheric fluctuation of the signal from ground radar to the target u , ~ , RIN af on the ground radar, and range-dependent noise or,,on the ground radar. Thus, there are six major noise sources to be considered, three for the homing systeln and three for the command guidance system, each with different variance and frequency spectral content. The measurements (and hence the noise on them) are multiplied by R in the semiactive honling systeln and R T in the command guidance system to obtain y,, for input to the guidance loop. In the case o f the semiactive homing system. R, and hence the noise on y,, go to zero at the end of flight. However, the noise on 11, in the command guidance system, because o f its dependency on R T , does not fall off to zero as the missile approaches the target. The larger the distance alvay from the radar a t which intercepts occur, the larger the value of R7- and, hence. the more noise that enters the system. 111 this last respect, thcre is an advantage associated with the semiactive homing system. It is shown that the range dependence of the noisc sets basic limitations on the effcctivencss of a command guidance system. Glint Noise a,li.,. The 1'SIl +,, of the target glint )I, is givcn by Equation (2-20b). )I,, is divided by R in the senliactive homing case or R T in the command guided case to create u,,,,,,,.Uccause glint noise is quite small in the command guidance since the receiver never gcts close to the target (that is, R T is always large). this noise can be neglected in command guided systems. Range-Independent Noise ( R I N ) a,. The semiactive RIN I'SD is givcn by Equation (?-?(Id). In c o n ~ c r t i n ga to 11 the It1N noisc is multiplied by R and 11cncc approaches zero a t the end of the flight. Uccausc the ground rcccivcr is usually c'r bcttcr quality than the missilc-borne rcccivcr, the I'S1) of v, i n the command guidance is smaller than that of thc semiactive systcm. Thc I'S11 of or in con111la11d guidance is 6, = ~ , , ~ . , . l\rlicrc f, rloy = (B,,.lBs,c)', f , is the data rate in Hz, B,,, is thc bcamwidtll of radar aperture, and BsH is the heam-splitting ratio for large signalto-noise ratio (SIN). Range-Dependent Noise a,.. and Ward, 19841 as r,, =

The variance of a,,can bc computed \Barton

(B,.)'I[~K;,(SIN)],

K,,, = figure of merit

Sec. 8.5

Command versus Semfactiue Homing Guidance System

549

Since SIN is a function of range, thermal noise is likewise range dependent. SIN can be calculated from the semiactive radar range equation as follows:

wherc K is the SIN a t the reference ranges RTI)and Ro. Rn, is the target-to-illuminator reference range, and Ro is the target-to-missile reference range. Substituting Equation (8-114) into Equation (8-113) yields the same form of variance as in Equation (2-300 as r,,,.q,., = r ,,,, [ ~ r ~ l ( R m ~ =r ,r,,,,,(~lR,,)' )]~ wherc r,,,,,,, is the variance of thermal noises at R T = Rn, and R = R,,. I11 this section R m = 124,000 ft, Ro = 16,000 ft, r,,,,,,, = 10'' rad'. Then the semiactive receiver noise PSD $r,,S;I is given by Equation (2-20e). The command guided S/N is

where K, is the SIN at the reference target to illuminator range RREF. The reference range is determined from radar system parameters. Substituting Equation (8-1 15a) for the SIN into Equations (8-113) and (2-20e) gives the command guided rangedependent noise variance and PSD (variance) r,,,, = r,,, ,,*,,, (RTIRREF)~.

(8- 115b)

where

and 2 T,,, = llf,, and o r is the radar cross section in m2. Before being converted to a positional noise, the command guidance angular thermal noise, which is already a function of the square of the range from the radar to the target, is multiplied once ~ ~missile ). carries a beacon, more by the same range $,nc, = @ m ~ R E F ( ~ $ / RIf $the where $ R D ~ Jand ~ ~ RREF ~ are defined for the then $mC = beacon. Another factor that causes deviations of the apAtmospheric Noise UAN. parent target position as seen by the radar tracker is atmospheric perturbation of the radar signal. Atmospheric noise statistics are taken from experimental tracking tests from which empirical formulae are derived [Barton and Ward, 19841. This effect, while very small in the semiactive system, may be quite significant in the command guidance system. The former is true because of the short missile-to-target where @ A N , ~ F = distance. One way of computing $AN is $AN = $AN,&* ~ T A . V I A Nf~,~/~~ " R* ~ , is the path length in m to tracked object in lower 5 k m of atmosphere, w is the antenna aperture in m, (standard deviation of U A N ) rykner:=

550

Chap. 8


.44 x lo-' rad, T A N is the atmospheric noise correlation time (0.6 sec), and f, is the correlated noise coefficient (0.4, see Alpert [I9881 on correlated noise effects).

Total Angular Noise PSD 4,. The total angular noise PSD 4, for the semiactive homing guidance is given in Equation (2-23), while that for the Command guidance is given by

8.5.2 Miss-Distance Analysis for Command Guidance

The treatment described here, based on the excellent work of Alpert [1988], derives new analytical equations based on adjoint theory for statistical miss distance caused by target maneuver, range-dependent, servo, glint, and atmospheric noises for command guidance. An optimal total guidance system time constant T ,is also derived which yields the minimum statistical miss distance, taking into account realistic constraints on the minimum achievable T,. The steady-state adjoint solution for miss distance is developed here only for the RIN. Looking at Figure 8-34a, using the binomial theorem,

Using Equation (8-117), Figure 8-34c can be developed from Figure 8-34b. The output of the integrator with 1,: as a factor in the integrand reaches a steady-state value a t :,,, that is, K,,T,;;-', where K,, are the normalized adjoint coefficients given by Kt =

,

( )

t+=o

I

*

)

d *

t* = t, is the adjoint time

(8-118)

Consequently, the steady-state adjoint mean-square (MS) miss distance due to range dependence is:

This same procedure call bc used to derive the miss distances for atrnosphcric I I O ~ S C and R1N. When this is done, the steady-statc adjoint solution for MS miss distance caused by target maneuver is o ~ , , ~ . ~ ,=, , b,K,\I , . ~ 7,: whcrc K.u = 7 ,'$' 7.

I*==('

[J;

C.(J;

, ~ , ; ~ ~ ( tdl* *)

+

)

1 dl* dl*

I

dl*

Table 8-6a contains the values of normalized adjoint coefficients K,, through and K M for A = 3.0, 3.5, and 4.0. Note that KO = M:, KA, = M$,, and K2 = M:,-whcrc M,, il4i-,. and M,,-arc dcfincd in Figurc 6-42. Thc stcady-statc adjoint

Sec. 8.5

Command versus Semlactlue HomIng Guldance System

551

equations for the mean-square command guidance miss distance due to each noise source arc given in Table 8-6b together with those for target maneuver. Table 86b also lists the corresponding equations for the scmiactivc homing guidance case. It 1s assumed from here on that only RIN and atmospheric noise need be considered. This could be the case, for example, if the missile carries a beacon. The variance of the total miss dist'lnce is the sum of the variances of the miss distances duc to each noise sotlrcc and due to target mancuvcr. That is,

A good approximation to Equation (8-120) that keeps the largest terms of each polynomial is given by

xx ++

where can be identified as the sum of spectral noise densities a t the intercept range. corresponds to +,, in Equation (2-23) for homing guidance. Equation (8-121) shows that the variance of the miss due to the noises is approximately inversely proportional to T , ~and , the variance of the miss due to target maneuver is proportional to the fifth power of T,?. The optimal time constant for minimum RMS miss distance can be found by setting the partial derivative of the expression with respect to T~ equal to zero. This gives Optimal time constant:

Minimum RMS miss distance:

(T,~),,,

+ (K"C K.LI+~ ) [ ( ox 4 1.25 'I6

= 0.2

U . U I ~ ~ ~= ,~\.

.

I

)]1'2

(8-123)

For this optimal condition, maneuver miss distance accounts for 41 percent of the total RMS miss distances. Note that (7JOPT in Equation (8-122) and UMISS.~IJ.V in Equation (8-123) have the same formulae as in Equation (7-34) for T and (7-35) for a,, (9, respectively, for homing guidance.

Aduanced Guldance System Design

552 TABLE 8-6

Chap. 8

(From [Alpert 19881,O 1988 AIAA)

-

a) Normalized steady-state m i s s d i s t a n c e adjoint C) Parameter values u s e d in t h e examples

coemcients I

1 I

KI K2 K3

4.4 9.2 25.9 89.7 365.0 1707.0 3.77

K4

Ks K6 K.w

6.7 15.6 47.6 176.0 742.0 3750.0 2.38

10.0 25.7 84.8 336.0 1541.0 7979.0 1.98

b) Steady-state adjoint m i s s distance e q u a t i o n s Command guidatire .for farget fvatka

2

~ M I S S Y I . Y= ( b r K . ~ ~ ~ ;

Homing guidai~teb

BW

= 0.035 rad (2 deg) S/.YKEF = 100 (20 dB)

Beamwidth Reference S/.V a t reference range RREF RREF = 50,000 m Reference range u .r = 100, 10, 1, 0.1 m2 Radar-crosssection f s = 40 Hz Data rate BSR = 80 Beam split ratio for large S/,V U', = 10m Wing span of target TGLT = 0.08 s Glint noise correlation time UI = I m Antenna aperture 7.4= 0.6 s Atmospheric noise correlation time" Navigation ratio A = 3 v . ~ = 1500 m/s Missile \relocity VT = 450 mls Target velocity (positive for incoming target) 11 T = 19.6 mls' (2 g) Target maneuver level T,tr = 2.5 s Target maneuver timeb ( T ~ ) . \ I I . ~ = 0.5 Minimum achievable system time constant' U,,C~:LIM = 98 nils2 (10 g) Maximum allowable missile acceleration causcd by noised #

' For missile track MI' replaces I/.,.. Rb = R I ~ ~ ~ R MtheT product , , ~ ~ of the illuminator-to-target range and missile-to-target range at which thcrc is S/&cc~ on a 1 m 2 target.

' Assunied target altitude is helow 5 km, hence R' = It. For a Poisson-distributed step targrt nlancu\'er with average time between maneuvers TI., use T.rf = 0.25 T p ; here T p = 10 s. ' Used only in numerical Examples 2 and 3. * Used only in numerical Example 3.

Sec. 8.5 TABLE 8-6

Command versus Semiactive Homing Guidance System

(Continued) d) Normalized steady-state acceleration ad. joint coefIlclents K;(nr) for a flfth-ordersystem with equally distributed time constants

Asymptotic system performance. At this point (following discussions in Alpert [1988]), a set of equations is developed to describe asymptotic miss-distance performance, each of which represents U.~rlssLrr, which could be achieved if only one class of guidance noise were present. Sources of guidance noise influence mindefined by Equation (8-121). Since, howimum RMS miss distance through ever, the relative magnitudes of the terms in 2 4 depend explicitly on intercept range, at any particular intercept range one noise term will dominate. The u . t ~ l s s . , , , , that would be expected if only this dominant noise terni were considered along with the maneuver term is what defines asymptotic miss distance. At the shortest intercept range, glint dominates and miss distance is independent of intercept range. Using the notation (CLT) to indicate that the glint noise dominates,

x+

As intercept range is increased, the RIN miss distance eventually increases to be equal to that of glint noise miss distance, at a range of

Beyond this range the asymptotic miss is 1.25[K~+,(2Ko+f)5 11 / 1 2 ~r 5 / 6

~ M I S S M ~ X ~=R ~ ~ ,

(8-124c)

In this region, where the RIN dominates, the asymptotic miss distance is proportional to the five-sixths power of intercept range. In the next region, where the intercept range is larger than Rlz =

(8-124d)

+fl+~~nap

the atmospheric noise dominates and the asymptotic miss distance is ~ M I S S M I N ( A ~\)= >

5 1/12

514

~ . ~ ~ [ K M + S ( ~ K O + A] N RREI F )

(8-124e)


554

Chap. 8

The miss distance is proportional to the fire-fourths power of intercept range. T h e RIN dominates at intercept ranges larger than 113

R13 = ( 2 s T + A . V ~ E ~ - R ~ E F ~ ~ R D N H U F )

(8-124~

The asymptotic miss distance is

The miss distance is proportional to intercept range to the five-halves power. T o draw asymptotic miss distance on log-log paper, it is only necessary to assign values to the various system parameters, evaluate the critical range points of Equations (8124b, d, and Q, and compute the miss distances of Equations (8-124a, c, e, and g). Between regions, only the slope of the miss distance curve with respect to intercept range varies. These miss distance curves (Figures 8-36a, b, d, e) are asymptotes of the minimum RMS miss distance that contain maneuver and the largest noise contributor (that is, glint, range-independent, atmospheric, or RIN noise).

Numerical results presented here illustrate the methodology presented earlier. Table 8-6c contains the numerical values of system variables where 4, = r~$iT.,,is the target acceleration PSD and r 7 and ~ T,,, are defined in Table 86c. Figure 8-36a is the plot of the asymptotic miss distances for the example, while the cxact solutions are presented in Figure 8-36b as curves with dots over a copy o f Figure 8-36a. The cxact solution is calculated from Equation (8-120). It can be seen that the asymptotes and the exact solution agree well ovcr most of the curves except near breakpoints in the asymptotes. Figure 8-36c contains thc time constants, ( 7 . c ) ~[SCC ~ ' ~Equation (8-122)], corresponding to thc csact solutions of Figure 836b. Also displayed are the asymptotes of ( T ~ ) ~ ) I >calculated T from the mancuver spectral dcnsity and thc donlinatlt noise spectral density.

Example 8-4.

System time constant and missile acceleration limits.

The rcsults ofthe prcvious section arc in rcality subjcct to limitations on the minimum achievable total T,? and niaxit~lumachicvablc tnissilc accelcration. Thc dcvelopmcnt so far implicitly assumed that therc was no bound on the achievable T , nor on the achicvable nlissilc accelcration. 111 real physical systems, missile autopilots and airframcs hecomc difficult, if not impossible, to dcsign as thc autopilot-airframe response-time constant requircd bccoincs very short. This irr~poscsa lolvcr bound on the achievable (T,~),,,,,,. (7,C)c)1J-1. defined prc\,iously is only \.slid as long as it escecds (T,),,,,,,at cach

Figure 8-36 (a) A s y n t ~ t o r i cMiss 1)istartccs o f l>ornit~antNoise atid Matieuvcr k)r Examplc 1 (b) I)o~~litianrNoisc and Mancuvcr Miss Ilista~iccAsymptotcs and Exact S o l u t i o ~for ~ Examplc 1 (c) Optimal Timc Constatits and Asymptotcs for E x : ~ n ~ p 1l r (d) Noisc-Mar~cuvcrAsymptotcs \\.it11 Minimum Tirne Constant Constralllt and Exact S o l u t ~ o n ,Example 2 (c) Noise-Mancuvcr Asymptotcs with and witliout Acrclrrati(~nLimits nnd Exact Solution, Exaniplr 3 (1:rd111 /Alprrr, 191181, 0 19x8 .41.4.4J

Sec. 8.5

555

Command versus Semiactive Homlng Guidance System

\ '~lss.,.,,,,,

2 1

I,,

3

5

,

, , ,

10

20

30 40 50

INTERCEPT RANGE (km) INTERCEPT RANGE Ikm) (A)

--

(B)

25 20

I-

2' a

15

z

0 10 U

EI-

08

,0 6

2

O5

I-

n 04 0 03

< D

0

C

02 3

5

10

20

30

4 0 50

100

INTERCEPT RANGE Ikm) (D)

INTERCEPT RANGE Ikm) (C)

-

1

3

"WIS~MINI~IN,

6

10

20

304060

INTERCEPT RANOE Ikm)

(E)

100

100


556

Chap. 8

operation point (that is, intercept range). Equation (8-121) shows that, for 7 , greater than (7,)0pT, the miss distance is determined mainly by the maneuver miss. consequently, the asymptotic miss for the minimum achievable 7, is approximately u ~ ~ ~ ~ ~ < ~ = ~ ( (T + ,q~ ~. , ,, ) w~ ' ,~ ~( 7~~ )where 2 : ~ the notation MIN[(T,),,,,] indicates that the minimum miss distance is set by the minimum time constant constraint.

Example 8-5.

The asymptotic miss corresponding to (7,),,,;,, = 0.3 is plotted in Figure 8-36d, in which it appears as the horizontal line at the 4.2-m level, The only difference between this example and Example 8-4 is that the time constant limitation is now included. Exact solutions [Equation (8-120)] are represented as curves with dots, and it can be seen that there is good agreement bet\veen the asymptotes and the exact solutions. Comparison with Figure 8-36b indicates that the minimum 7, limitation increases miss distance only at short ranges. Another realistic limitation that must be taken into account, as discussed in Alpert [1988], is the maximum achievable missile accleration. If the guidance system were to achieve more acceleration than the system design limit, the resulting miss distance would probably be very large due to component failure. If internal limits imposed on the acceleration command are exceeded by noise-induced acceleration commands, then no acccleration capability is available to counter target maneuver. This would also producc large miss distance. Good design practice, therefore, avoids acceleration saturation as shown previously. Here, it is shown that acceleration saturation due to noises can be avoidcd by i~nposinganother minimum timc constant constraint that is depcndent on system parameters and intcrccpt range. The adjoint technique can be used to develop cquations for the RMS accelerations induccd by noises. Using the same adjoint systcm as for the miss distance. the impulse is instcad applied at thc point corrcspo~~ding to commanded accclcration in the basic fortvard system diagram. Thc adjoint rcsults then bccornc accclcrations instcad of miss distances. Follo\ving the previous dcvclopn~cnt,which deals with the steady-state adjoint SOlutions, equatiolls can be dcvclopcd for IIMS accelerations that are similar to the equations of Table 8-6b. A ncw set of normalized steady-state adjoint coefficients K;, arc found by applying the impulse at A, whcre A is the r, bcforc intercept at which the acceleration occurs. Thesc coefficicnts increase with dccrcasing (,, or as thc point of intcrccpt is approached. Thc rcsulting RMS comtnandcd accclcration is writtcn as a polynomial cxprcssion in tcrnls of missile and target vclocity and missilc T , ~ .Once again, keeping only the dominant terms, thc variance of the cornmandcd accelcratiorl duc to noise alone can bc approximated as uXcc:(t,v) = 4)K;,(/,,.)7, Tto, achieve minimum miss distance. As happens with longer ranges, the timc constant must son~etirnesbe constrained by accclcration limits. In this case, miss distancc will be determined mainly by the maneuver RMS miss distancc, which is uL1,ss,,,l.,z~..,,:, :,,,,,, = ( ~ b ) b 3 ~ . + , ~ X ( t y ) l u ~ ( : ~ : ~where . ~ . , ~the notation ..ICCLl.LI indicates that thc minimum miss is set by the acceleration limit. Conscclucntly, by requiring the missile to remain below the acceleration limit. the noise level determines the maneuver miss which dominates the total miss distance.

Example 8-6. The curve with dots in Figure 8-36e shows the exact adjoint solution for the 10 dBsm target of Example 8-5. For this curve, T ~ c c L l n is l used as T whenever T.A.(:CLI.LI > ( T , ~ ) Oand ~ T (T,),,,,,, for the case where UACCLIM = 98 m l sec' at !,= T,,, the acceleration command due to noise is limited at one time constant before intercept. The plotted solid line asymptotes are the same as in Figure 8-36d, and the dashed lines are the asymptotes of maneuver RMS miss distance, ( T ~ ~ ,,, r., .~ , 2 iS , , ~ r .,,.,,,r, presented in Example 8-3. It can be seen that the exact solution is close to the maximum of the dashed and solid line asymptotes. 8.5.3 Analysis of Optimal Command Guidance Versus Optimal Semiactive Homing Guidance Using adjoint theory, this section presents an analysis of optimal command guidance versus optimal semiactive missile guidance based on an excellent paper by Neslines [1986]. The Kalman filter and optimal control law gains are both generated in forward time and read back in reverse time for the adjoint run. The command acceleration M, is an input to an analog third-order FCS which generates the achieved missile acceleration A l W yUniformly . distributed target maneuver shown in Figure 2-21d is assumed in this section. The target acceleration PSD +,is given by Equation (2-39).

Noise inputs. The noise inputs for the analysis in this section have been discussed previously except the following. All noises are considered to be on the discrete measurement o f u or 6T, as indicated in Figures 8-35b and 8-35c, the discrete measurements are made at .O1 sec intervals.

+,

G l i n t Noise. Glint noise PSD is obtained by applying Equation (2-20b) assuming a medium-size airplane target with YCLT = 25 ft2 and TCLT = O.j/Hz. Range-lndependent Noise ( R I N ) . sampled at a 100-Hz rate. Range-Dependent Noise.

ft.

+f

is assumed to be 2.5 x lo-'

Y,,,~,,,. = 4 X

rad2 and RREF= 1.28

rad2, X

10'


Chap. g,

Atmospheric Noise. Using empirical data presented in Barton and Ward [19841, the variance of atmospheric noise for a target at 20-deg angle above the earth is

where L , is the length o f radar beam in lower troposphere (15,000 m), and Wl is the radar aperture in meters (3.05 m). Therefore from Equation (8-125), r,!& = . o l mr. The atmospheric noise PSD $A.v rolls off sharply with frequency. This rolloff may be approximated by two first-order low-pass filters with frequency breaks a t f,.,which can be calculated using where f,,is the cut-off frequency of two first-order filters and V Ais the transverse velocity of radar beam through atmosphere (measured in mls). The break frequency f, calculated from Equation (8-126) for a roughly stationary radar beam and a normal atmospheric drift of about 3 mlsec is 0.46 Hz.

Kalman filter analysis. The square root spectral density &A''(defined in Equation (2-23) for homing guidancc and in Equation (8-116) for command guidancc) is plotted in Figure 8-37a for the short-range intercept case and in Figure 837b for the long-range intercept using different scales. For reasonably stationary signal and noise statistics, the transfer function from the input y, to the output y, o f thc Kalman filter is given by Equation (7-34) whose F is plotted in Figures 837c and 8-37d for the short- and long-range intercept, respectively. As already mentioncd, the homing +:I2 dcpends on missile-target range, and the variation in this quantity during the flight, for the exarnplc here, is shown in Figure 8-37a. Hcrc, 4;;' is grcatesr at the beginning of flight because of the thermal noise and dccrcases monotonically throughout the flight's duration. Thcre is an increase in angular glint noise at the end of the flight but is small compared to the thermal noisc. Because +L'%s low for short-range intcrcept in both systcms; cach has rather fast Kalman filter. Still, thc filter in the command guidance is faster. Owing to the largc amount of thcrmal noise, F of the homing is small at the beginning o f flight but then slo\vly increases as a result of the decrease in the noisc as the missilc draws nearcr to thc targct. F of the homing is, in fact, a function of thc SIN. While the state cstimatc is poor a t the beginning of flight \vhcrc F is small, it bccomcs quite good as thc missilc approaches thc targct. O n e rcason for which thc Kalman filter ~ O S S C S S C Sthc bcst pcrformancc is that, a t thc bcginning of thc flight, its large time constant for dealing with thermal noisc docs not influcncc thc miss due to target nlancuver at thc end of t11c flight. For long-range intcrccpt, 4;' in the scrniactive homing guidancc is at first larger than that in the command guidance but later bccomcs smaller. This is a result of thc forrncr's dcpcndcncc on R, which as stated bcforc, goes to zcro. F of thc homing in thc semiactive systcm increases to approxinlatcly thc same valuc as in the short-range casc, whilc F in thc command guidance remains low throughout thc flight bccausc thc I'SD o f thc rncasurcnlcllt

Sec. 8.5

Command versus Semiactive Homfng Guidance System

20

I

18

!

559

I

1

i

12

~

8 4 0

1

COMMAND

0

1

2 3 TlME (SEC)

4

E TlME (SECI

(A)

(B)

. n

2-

32

! 2.8

!

I

2.4

!

z Y a

0

-

-1

1.6 1.2

I

0.8

COMMANO

K

u 0.4 k

2

2o

'

I

I

I

0

TlME (SECI

HOMING

!

3

O O

2

4

6

8

TIME (SEC)

(C)

Figure 8-37 (a) Command Noise Is Less than Homing Noise at Short Target-toIlluminator Ranges (b) Homing Noise Becomes Less than Command Noise at Long Target-to-Illuminator Ranges (c) Commmand Characteristic Frequency Is Higher than Homing at Short Target-to-Illuminator Ranges (d) Homing Characteristic Frequency Becomes Higher than Command at Long Target-to-llluminator Ranges (From [Nerliner, 19861 with permission from AACC)

noise is a function of R r , which remains large. If the filter is slower than, say the guidance law and FCS, then its speed of response will limit the overall system speed of response. Consequently, the miss distance and required accelerations against maneuvering targets will also increase. If the filter is much faster than the other components of the system, system response will be for the most part determined by the slower components. In this case, filter dynamics will have little impact on performance.

Performance analysis.

A reasonable measure of miss distance and acceleration requirement due to a maneuvering target can be obtained by assuming a l-g target acceleration, which can occur at any time throughout the flight with

10

560


Chap. 8

unifor~nprobability. Because the autocorrelation function of such a model is the same as that of integrated white noise, adjoint theory can therefore be used to calculate RMS statistics due to this target maneuver model [Fitzgerald and Zarchan, 19781. The RSS of the RMS miss distance and acceleration requirements for all sources are used as measurements of performance. This results of the short-range intercept scenario, in which intercept occurs 15,000 ft from the illuminator, are summarized in Table 8-7a. The largest contributor to miss distance and acceleration requirement for the semiactive system is glint noise. Because of the target-dependent nature of glint noise, it is not likely that the contribution to total miss distance and acceleration requirement from this quantity could be improved dramatically by improving the system components or the Kalman filter and guidance law. Random target maneuver is second to glint noise in its contribution to miss distance and high acceleration requirement. Thermal noise, which is dependent on ( R r ) * ( R ) ,represents the smallest contribution. However, the effect of RIN is also very small. Total RSS miss distance and acceleration requirement are roughly 6 ft and 10 g, respectively. The total RSS miss distance and acceleration requirement for the command guidance in the short-range case compare closely to that of the semiactive system. In this casc, total RSS miss distance is between 3 and 6 ft, and the total RSS acceleration requirement is between 5 and 7 g, depending on the atmospheric noise components. The table shows that these components have the largest impact on total RSS miss distance and acceleration requirement. Since they are determined by the intcrcept scenario, they do not fall under the control of the missile systcm designer. Although RIN and random target rnaneuvcr also contribute to miss distance and accclcration rcquircment, their values are rather small. The results of thc long-range intercept arc shown in Tablc 8-7b. This interccpt occurs 130.000 ft from thc illuminator. The overall pcrformancc of the semiactive system at the long-range intcrccpt docs not diffcr radically from thc short-range intercept. Thc contributions from glint noise, RIN, and random target maneuver to performance basically follow the same pattern as in the short-range intercept case. The target mancuvcr portion is very nearly the same because the filter and other system components arc not much different from those found in thc short-range intcrcept casc. Howcvcr, thcrmal noisc beconlcs a bigger contributor to both miss distancc and accclcration rcquircnicnt in the long-range intcrccpt casc. The increase in each is a result of thcir dcpcndencc of S T , thc target-to-illu~ninator range. If the scckcr is of poorcr quality, thermal noisc may bc an inlportant contributor to total miss distancc at long-rangc intcrccpt rangcs. For t11c command guided system, thcrnial noisc is thc most significant contributor to high miss distance and largc g's sincc from Equation (8-1 15) command guidancc thcrnlal noisc miss and accclcratjon are dcpcndent on R$- which is still largc ncar intcrccpt. Long launch rangcs suffer from corlsidcrablc pcrformancc dcgradatiou. If thc radar bcani to thc targct is stationary, thcn miss distancc and accclcration rcquircmcnt duc to atmospheric noise can also bc quitc largc. As with scn~iactivcsystcms, thc missilc systcm dcsigncr docs not havc control ovcr this componcnt. Unlike the short-rangc interccpt case, RIN is thc lcast important contributor. Icccrnbcr 1086, pp. 704-700. BANK>.I). S.. ' L C ~ t i t iWJVC n ~ ~(CW) ~ ~ Radar," E.4SCOM '75 R~cordirqs,IEEE Pub. '75 C H O 008-5 EASCON, 1975. B.+IL 1. FIt., . . M. ATIIANS, S. W. GULLY,ANI) A. S. WILLSKY, "An Optitnal Control and Estinlation Algorirhn~for Missilc Endgamc Guidancc." IEEE Cor!f: or) D r ~ i ~ i oatrd i r Cor!rr.ol. 1)cccmber 1982. pp. 1128-1 1.12. DOU~I)LII, J . R.. S. W. GULLY, A N I ) A. S. WILLSKY, "Endgan~cGuidancc S t ~ d y , "AFATLTI C . W. SMOOTS,lirtrodlrctior~to P~.~risiorr Cnidrd Allrrririorrs. Chicaso. IL: GACIAC HB-83-01, Prepared by the Tactical Weapon Guidance and Control Iiiforn~arion Analysis Center (GACIAC) under Contract DLA900-80-C-2855 with the 1)efensc Logistics Agency. May 1983. HELLER, B. J., "Adapting an Alpha-Beta Tracker in a Maneuvering Target Environment, U.S. Naval Weapons Center, Tech. Note 304-154, July 1967. HELLER,W. G., "Models for Aided Inertial Navigation System Sensor Errors," Reading, MA: TASC Tech. Rept. TR-313-3, February 1975. HEWEII,G. A , , ET AL.,"A State-Space Theory of Structurcd Uncertainty with Examples. N W C TI' 6971. China Lake. CA: Naval Weapons Ccntcr, March 1989. HEWEII. G. A,. IE. A N I ) C . KENNEY, "A Structured Sil~gularValuc Approach to Missile Autopilot Ailalysis 11," Proc. 1988 Atnericarr Cot~rrolCon[, June 1988(b), pp. 323327. H E W E IG. ~ , A,, C. KENNEY, A N I ) R. K L A I I U N I"A ~ E State , Space Model of Structurcd Singular Valucs," 27th IEEE Con[ otr Dcrisiotl atrd Control. IJcccrnbcr 1088(c), pp. 2144-2147. HILBOIIN. C. G. A N I ) 1). G . L A I N I O . ~"Optinla1 IS, Estimation in I'resencc of Unkno\\.n I'd, ramcters. 'I'rans. S)$sr.Sti. Cyhn.n., vol. SSC-5. January lYh9, pp. 38-43. 3

3

..

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640

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NFSI.INE.F. W . A N I I M. L. NESLINE. "llhase VS. G i n Stabilizatiori of Structural Feedback Oscillations in l i o m i ~ ~Missile g Autopilots." I'ror. ilrrrrrirat~Cc~rrtn~l Car!/:. June 1985, pp. 323-32'). NESLINE F.. W. A N I I M. L. NI;SL.INE, "HOW Autopilot IIeq~lirenle11tsConstrain the Aerovol. 2. June dynamic 1)csign of Homing Missiles," Proc. ofrl~eAnlericat~Co11trl11Cotl/i~rr~r~r, 11)84(n).pp. 710-730. NESLINE. F. W. A N I I M. L. N I ~ S L I N''tlotl~i~ig E, Missile Autopilot Ilcsponse Sensitivity to Stability I>erivative V~riations."IE1:'E Cclr~f1111 Vriisio~r11t111 Corltrol. I>cccrnbcr I1)X4(b). NESLINI:, F. W.. M. L. NEWNI:.A N I ) C. F. L I K . "Modern Guid;uncc and Control for Homing Missilcs," Gui~icdMissile Worksllop Notcs. .4iil(,rinli1C ~ ~ t ~ rCot!/:, r o l June 10XO. NESLINE. F. W., ti. H. WELLS,,AN!) P. ZAIN:II.\N, "Co~llbinedOptimaliClassical Approach to Robust Missile Autopilot Ilcsign," J. G r i d . atrd Cc~r~rrol, vol. 4, no. 3, May-June 1981, pp. 316-322. NESLINE, F. W. A N I I 1'. ZAIICIIAN. "Line-of-Sight R e c o t i ~ t r ~ ~ c tfor i o r Faster ~ Homing Guidvol. 8, no. 1. January-February 1985, pp. 3-8. ance," J. C~rid. J.. L. U . WE(:KESSEII. A N D It. C. MALLALIEU, "Tech~iology Si~rvcyof Radomes for Anti-Air Honiing Missiles." APLIJHU FS-80-022. March 1980. YC)CNC;, E. A N I ) L. N. C O I ~ M I E"Missil~ II, Range and Trajectory Analysis (MRATA): Quantifying Missile Pcrforrn;cncc to Aid in llecision Making." 1985 SCSC, pp. 631-636. YOCNC;, J . T.. J . M. I'EIII)ZOC:K. I>. L. SEBIIING, A N I ) L. EIIINGEII, "Dcsig~iand llcvclopment of the Multifi~nctionFlight Control Rcfcrence System," AGARIl-C1'-349. August 1983, pp. 12-1 to 12-10. YOCN(;III:N, F. R.. LLMitiinii~il~g Boresight Errors in Acrodynaniic Radomcs," Elecrro~ricDeS I , ~ H . I l e c ~ ~ i i b 20. e r 1001, pp. 152-157. YUE,A. A N I ) I . I ' O S T L E T I I W A I ~ ~ E"Hz-Optinial . Ilcsig~lfor Helicopter Control." Proc. A~rreri e t l ~ lCorrrrol Car!\:. Junc 1988. pp. 1679-1684. Y C E ~W. I , R., "Fourth Order Predictive. Augmented Proportional Navigation System Terminal Guidancc Ilcsign with MissileiTarget Decoupling," U.S. Patent No. 4,494,202, Jan. 15, 1985. Y L E I I W. . R., personal communication, 1983(a). YCEII,W. R.. "Simplified Covariance Analysis for Midcourse Simulation," Proc. Strmrner Cc~rrrpirterSirrrrrlatio~rCurd, l983(b), pp. 309-313. Y C E I IW. , R. A N L ) C . F. LIN,"Optimal Controller for Homing Missile," AIAA J . G~rida~rce, Corrrrol, (trrd Dy~l'lr~rics, vol. 8. no. 3. May-June 1985(a), pp. 408-411. Y L E I IW. . R. A N L ) C . F. L I N ,"Guidance Performance Analysis with In-Flight Radome Error Calibration," .+I.+AJ . Crridu~rce,Corltrol, n ~ r dDyrratrrics, vol. 8, no. 5, September-October 1985(b), pp. 666-660. Y C E I IW. , R. A N I ) C . F. LIN, "Bank-to-Turn Guidancc Performance Analysis with In-Flight Radome Error compensation," Paper No. 81-1889, Proc. oftlre AIAA Gtridatlce arrd Corrlrol Confhretrce, August 1984, pp. 715-722. ZIMES,G., "Fredback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seniinorms, and Approximate Inverses," IEEE Tra~rs.on A~ltomuticControl, vol. AC-26, no. 2, 1981, pp. 301-320. ZARCI-IAN, P., "Con~parisonof Statistical Digital Simulation Methods," AGARD-AG-273, 1988. ZARCHAN, P., "Complete Statistical Analysis of Nonlinear Missile Guidance SystemsSLAM," AIAA J . Giridarrce and Control, vol. 2, no. 1, January-February 1979(a), pp. 7178. ZAIICHAN, P., "Representation of Realistic Evasive Maneuvers by the Use of Shaping Filters, J . Guidarrce arid Corrtrol, vol. 2, no. 4, July-August 1979(b), pp. 290-295. Z ~ o u H. , A N D K. S. P. KUMAR,"An Adaptive Algorithm for Estimating the Acceleration of Highly Maneuvering Targets," IEEE CDC, 1982, pp. 1133-1134. 3,

Index AAM. See Air-to-air missile (AAM) Accelerating capability, 254 Acceleration comvcnsation. 310 Acceleration estimate, updating of, 608-9 Accelerometers, 14, 176-77, 193 Active homing guidance, 324-26 Adaptive radome estimation design, 520-24 Adjoint method, 106-1 1 applications, 108-1 1 adjoint model of linear hotning loop (illlts.), 110 deterministic inputs, 111 performance sensitivity due to target maneuver (illus.), 110 philosophy, 106-8 (ill~rs.),107 RMS miss distance error budget automatically generated by (illll~.),110 steps to construct adjoint systcm, 108

stochastic inputs, 109. 11 1 Advanced air-to-air missile (A.4M) navigation, guidancc, and control technology, 467-68 Advanced guidance and control s!.stem design, 50 Advanced guidance filter, 440-42 extended Kalman filter (EKF) approach, 441-42 statistical linearization approach, 442 Advanced guidance law in~plemcntation issues, 613-15 in~plemcntationproblems, 614-15 Advanced gutdance laws, 408-81 analytical solution of optimal filters and optimal guidance law. 474-81 definition, 469 optimal ,guidance law survey, 469-74 Advanccd guidancc system design, 465 analytical solution of optimal trajectory shaping for combined nlidcourse and terminal guidancc. 562

command versus semiactive homing, design and analysis, 544-62 complcmcntary1Kalman filtered proportional navigation: biased1 complementary PNG. 481-510 guidance filter design trade-off, 591 -600 improved time-to-go estimator, 606-9 jamming and clutter effects on guided intcrceptor. 609-13 multimode guidance system design, 613-19 ocher advanced midcourse guidance schemes, 583-91 other terminal guidance laws. 510-17 pulsed rocket control, 600-608 radome error calibration and compensation, 517-44 Advanced midcourse guidance schemes, 583-91 midcourse guidance for boost-sustain propulsion, 587-91 near-optimal trajectories, 590-91 singular perturbation, 584-87 Advanced missile guidance system against very high-speed target (example) 276, 278-79 Advanced multivariable control system design, 49-50 Advanced navigation, guidance, and control: concepts, 4-7 systems, 7-8 systems design, 10, 12 Advanced navigation system design, 445-64 global positioning system accuracy improvement, 445-57 integrated GPSIIN navigation system, 457-64 Advanced terminal guidance system (table), 470 AEGIS combat system, 266, 299, 329 AHRS. See Attitude and heading reference system (AHRS) Aided inertial navigation mechanization, 213-15 advantage of augmented positionvelocity inertial, 215 cost and performance improvement, 213 (illus.), 214, 215

effect of augmented inertial mechanization, 213 error improvement, 214, 215 (illus.) Aided inertial navigation systems, 179 inertial navigation systems and external navigation devices (illus.), 180-81 Air defense homing system, 401-3 Air-to-air dynamicslmeasurement models (illus.), 418 Air-to-air missile (AAM) multiple mode guidance, (illus.) 273, 281, 283 Air-to-surface missiles (ASM), 283-84 Air-to-surface roles and missions, 284-87 close air support, 284 deep tactical strike. 285 defense suppression, 285-87 tactical interdiction. 284 Air-to-surface weapons, 283-84 ALP. See Attitude-loop parameters (ALP) Alpha-beta-gamma filter, 421-24 first-order target tracker, 421-22 second-order target tracker: a-0 tracker, 422-24 third-order target tracker, 424 Alpha tracker, alpha-beta tracker, and alpha-beta-gamma tracker, 443 Altimeter aiding, 197 (illus.), 200 Altitude rate estimator (example), 161 Altitude sensors, 32-33 American Control Conference, 3 Analysis of optimal command guidance versus optimal semiactive homing guidance, 557-62 Kalman filter analysis, 558-59 noise inputs, 557-58 performance analysis, 559-62 Analytic optimal guidance law, 566-78 general solutions of optimal trajectory-shaping guidance law, 574-75 horizontal plane guidance, 575-76 mathematically probing the analytical solution of the ootimal trajectory-shaping guidance, 576-78 . . . optimal solution for air-breathing engine or the power-on stage of a solid rocket engine, 571-73 optimal solution for the power-off

Analytic optin~alguidance law (cotit.) stage of solid rocket engine, 569-71 vertical plane guidance, 567-69 Analytical solution of optimal trajectory shaping for combined midcourse and terminal guidance, 562-83 analytic optimal guidance law, 566-78 discussions. 583 optimal trajectory shaping guidance, 562-63 problem forn~ulatioil,563-65 real-time implenlentation and performance, 578-83 Analytical three-diniensional optimal guidance law, 474-75 ANISPY-1A radar, 266, 299 Antiradiation homing (ARH), 29, 331 missile guidance, 344-47 Antiradiation projectile (ARP), 331 Antisatellite (ASAT) missile. 434 Antiship niissiles. 296 Antisubmarine lvarfare (ASW), 294-96 Antitank missiles, 296-97 APN. Srr Augmented Proportional Navigation (AI'N) Apollo program, 2, 3 Kalman filter, 128, 151 A priori information, absence of. 153 Area defense. 287-88, 292 cxaniple, 202-93 ARH. Sre Antiradiation homing (AIIH) ARP. Src Antiradiation projectile (AIII') ASAT. S F PA~ltisatellitc(ASAT) ASW. Scr Antisubmarine warfare (ASW) Asymptotic system performance. 553-54 ( l l l l l ~ . ) ,555 ATLAS missile, 178 Atmospheric noise, 558 U A ~ Mj49-5(J + Attitudc and heading rcfcrcncc systctii (AHIIS), 311 Attitudc-loop paralnctcrs (All'). 531. 533-34 Attitudc pursuit (direct guidancc), 357-58 Augmented inertial n~cchanization, effect of. 213-15 advantages of augnicntcd positionvelocity inertial, 215

cost and performance improvement, 213-15 error improvement, 213 Augmented proportional navigation (APN), 475, 476, 488, 501-2 law, 477 Autopilot, 16 Autopilot and airframe, 539-40 application, 540-44 Autopilot response characteristics necessary to achieve homing, 65-66

Ballistic missile threat. 293 Batch process in^, 454 Bayesian formulation. 454 Bayesian rule. 137 Bcam-rider guidancc, 319-20 Talos beam-riding midcourse pidance system (cxaniple), 321 Best unbiased linear estiinator (BULE). 456 Bias: m o d c l i n ~ .156-58 no, 11 = 0. 161-02 position. 159-00 rate, h = I , I01 Biascd I'NG (BI'NG) algorithm, 480-87 aug~nc~lted proportioiial navigation (ANI'), 488 implementation. 488 optimal, 487-88 simplified suboptimal BI'NG algorithm: \vithout pursucr accclcration fccdback. 492 \\.it11 pursucr nccclcration feedback. 49 1

suboptimal BI'NG algorithm: without pursucr accclcration frcdback [suboptimal guidaticc (SOG)]. 489-91 with pursuer acceleration fccdback. 489 Birdies, 400 BI'NG. Set Biased I'NG (BIJNG)

BULE. Sct. Best unbiased linear estimator (I3ULE)

C.41lET. Scv Covariance analysis describing fi~nctiontechnique (CADET) Monte Carlo and statistical linearization with adjoint method qiialitative con~parison,120-22 cost comparison for linear and nonlinear systems (illrrs.), 123 nonlinear stochastic systems analyzing and evaluating tool, 116 CCI3. Src, Charge coupled devices (CCW C D U . See Control and display unit (CUU) CEP. Seij Circul.ir error probability (CEP) Center-of-gravity (CG) latcral acceleration, estimation (example), 160 Center-of-gravity (CG) normal acceleration, estimation (example), 160 CG. See Center-of-gravity (CG) CHAPARRAL guidance (example), 331, 332 Charge coupled devices (CCD), 262 Circular error probability (CEP), 261 CIWS. See Close-in-weapon system Classical versus modern terminal guidance and control (illus.), 50 CLOS. See Command-to-line-of-sight (CLOS) Close-in-weapon (CIWS) threat. 294 Closed-loop analysis for adaptive radome estimator, 524-27 discussion, 527-28 Close-loop stability and performance, 77 Clutter, 610-13 effects on guided intercepter, 609-13 Colored-measurement noise, 157 Colored noise, 156 Colored-process noise, 137 Combined seeker-guidance filtering: classifications of complementary/ Kalman filtered PNG, 486

tilter. 483-86 in con~plerr~cntary in complementary LOS anglc and LOS ratc estimator, 486-88 in coniplctiicntary LOS rate cstinlator, 483-84 conventiotial LING algorithm. 486 Command guidance, 313-16 multimodc guidance using, 389 Command gi~idancedynamics, 546-48 Comniand guided system (forward time) (ill~rs.),546 Command midcourse guidance. 290 Command to line-of-sight (CLOS) guidance 299, 312-16, 318-19, 348 Command versus semiactive homing guidance design and analysis, 544-62 analysis of optimal command guidance versus optimal scniiactive homing guidance, 357-62 miss-distance analysis for command guidance, 550-57 modeling, 51-1-50 Comparison of statistical digital simulation methods (example), 53 Comparison of suboptimal guidance (SOG) and PNG, 505-10 CompIementarylKalman filter approach to estimator design, 1 3 - 7 3 first-order, 158-63 second-order, 163-70 third-order, 170-75 CompIementarylKalman filtered proportional navigation: biased/ complementary PNG, 481-83 biased PNG (BPNG) algorithm, 486-92 combined seeker-guidance filtering in complementary filter, 484-86 complementary PNG (CPNG) algorithm, 492-501 terminal guidance system analysis, 501-10 Complementary PNG (CPNG) algorithm, 492-501 fourth-order optimal CPNG algorithm: with decoupling feature, 496-98 with target acceleration bias estimate, 498

Complementary P N G (CPNG) (cont.) optimal, with complementary: LOS angle and LOS rate estimator, 495 LOS rate estimator, 493 second-order optimal CPNG algorithm, 495 simplified suboptimal C P N G algorithm, 500-501 suboptimal C P N G algorithm, 498-500 third-order optimal C P N G algorithm, 495-96 Computer frame (psi-angle) approach, 189 Computer requirements, Kalman filter, 149-53 software, 152-53 square-root filter, 151-52 suboptimal filter, 152 Computer unit (CU), 199 Computing, incrtial navigation systems function, 179 Constant-bearing guidance, 359-60 Continuous dynamics, discrete measurements: extended Kalman filter (tahlc), 135 statistically linearized filter (table), 136 Continuous Kalman filter, 129. 131-32 first-order altitude rate Kalman filter (example). 132-33 Continuous-\vavc interferomctcr homing system, 337-43 test of, 344 Control and display unit (CDU), 179 Conventional PNG algorithm, 486 Correlated acceleration process, 70-71 Covariance analysis, 102-6 RMS trajectory profile (ill~rs.),105 Covariance analysis describing function technique (CADET). 78 CI'NG. Scc Complcmcntary I'NG (CI'N C;) Crossover frequency, 85-86 C U . Scr Computer unit (CU) CUBAN. Scr Cubic miss adjoints (CUBAN) Cubic miss adjoints (CUBAN), 534 Cubic synthcsis (CUSYN), 534-36 CUSYN. Scr Cubic synthesis (CUSYN)

Data-decoupling algorithm, 462-63 tests and results, 463-64 Data handling system (DHS), 231 (illus.), 235 Data processing, 188 generalized navigation computer (illus.), 188 Defense and offense systems, 392-404 low-altitude air defense systems, 397-404 performance parameters, 392-97 De Gaston-Safonov algorithm, 95 Design algorithms for advanced navigation, guidance, and control systems, 9-10 Designators (laser-based systems), 328-29 Design equations for PNG u i t h parasitic feedback, 528-44 applications, 540-44 design equation, 530-40 problcm definition, 528-30 Design rcquiremcnts, 84-86 Deterministic inputs, adjoints for, 111 Deviated-pursuitlfised-lead guidance, 358-59 DHS. Sec Data handling system (DHS) Discretc Kalman filter, 129 n~ultiple-rate(tahlr), 131 and optimal guidance law (illrts.), 546 step-by-step procedure (table), 130 Discrete-time system, 431 Divergence Kalman filter, 420 DME, 33 Doppler aiding, 195 (ill~ts.),199 (illlrs.) Dopplcr radar, 179, 225 advantages and limitations, 225 Doyle bounds, 95 Do):lc's inequality. 94 Drag c o n ~ p c n s a t i o ~363 ~. Draper Laboratory, 180 Dual-mode guidance, 380 concepts. 381 intcgratcd navigation and guidance system, 381, 384 midcourse plus terminal guidance. 380 Dynamic lead guidance, 372

Earth-fixedlgeographic coordinate ( i l l s ) , 185 transformation (illus.), 185 Earth-to-NAV direction cosine matrix (illus.), 186 ECM. See Electronic countermeasures (ECM) Effective slope and effective noise, 56-57 (ill~ds.),58-59 derived by minimizing mean square error (illus.), 55 Effective navigation ratio (illus.), 64 EKF. See Extended Kalman Filter (EKF) Electronic countermeasures (ECM), 260 Electronic countermeasures (ECM)/ E C C M modeling, 24 Electronic unit (EU), 199 Electro-optical seekers, 7-3 Electro-optical (EO) guided weapons, 33 1 .Electrostatically supported gyro (ESG), 178 Empirical radome slope calculation, 61-62 Environment sensing guidance/ correlation matching, 389-90 EO. See Electro-optical (EO) Error analysis, 146-48 Error analysis model development, 308-10 midcourse guidance system analysis, 308-9 ESG. See Electrostatically supported gyro (ESG) Estimator-controller, combined: optimal filters and optimal guidance law, 476-77 optimal guidance system (OGS), 478-81 Estimator gains, (table) 441 EU. See Electronic unit (EU) Euler attitude estimator (illus.), 163 Expanding memory filter, 415 Extended Kalman Filter (EKF), 133-35, 441-42, 444 continuous dynamics, discrete measurements (table), 135 Extended target detection and segmentation, 412-13

External disturbances, 77 External interface. 195 External navigation aids. 213-29 aided inertial navigation mechanization, 213-15 doppler radar, 225 global positioning system (GPS), 215-16 Kalman filtering, 226-28 performance, 228 long-range navigation (LORAN). 220 star tracker, 225-26 tactical air navigation (TACAN), 217-18, 221 terrain contour matching (TERCOM), 220, 222, 225 External references, 33-34

~

Fading memory filter, 138, 415 Kalman filter, 420 Wiener filter, 420 Fail-operationifail-operationifail-safe (FO-FO-FS), 238 Fan-Tits algorithm, 95, 100 FCS. See Flight control system (FCS) Fiber-optics guidance (FOG) (example), 303 Field of view (FOV), 255, 260 Filter computational requirements, 149-53 Kalman filter design software, 152-53 square-root filter, 151-52 suboptimal filter, 152 Filter divergence, 153-35 and suboptimal design, 155 Filtering: and estimation techniques, modern, 126-28 Filter performance, 149, (illus.) 150, (illus.) 152 First-order a estimator using inertial a rate at center-of-gravity (example), 164 First-order a estimator using inertial a rate at IRU (example), 161 First-order altitude estimator, (example), 163

Index First-order altitude rate Kalman filter (example), 132-33 First-order complementary/Kalman filter, 158-63 altitude rate estimator (example), 161 estimation of center-of-gravity lateral acceleration (example), 160 estimation of center-of-gravity normal acceleration (example), 160 first-order a estimator using inertial a rate at center-of-gravity (example). 162 first-order a estimator using inertial a rate at IRU (example), 161 first-order altitude estimator (example), 163 first-order Euler attitude estimator (example). 163 first-order sideslip estinlator using inertial sideslip rate at cetlter-ofgravity (example), 162 first-order sideslip estin~atorusing inertial sidcslip ratc at IRU (exan~plc).161 first-order velocity estimator (example), 162 n o bias, b = 0, 161-63position bias. 159-60 rate bias, ii = 1 , 161 ratc estin~ator.162 turbulencelgust anglc-of-attack rate estimate (csamplc), 160 turbuletlce/gust t.elocity cstinlator (example), 160 First-order Euler attitude estimator (example), 163 First-order sideslip estimator using incrtial sideslip rate a t C G (example), 162 First-ordcr sidcslip estimator using inertial sideslip rntc at IRU (cxatnplc). 101-02 First-order solution, 515, 517 First-order targct tracker, 43-1-22 First-order velocity cstinlator (cxatnple), 162 Fixed memory filter. 415 Flight control systctn (FCS). 14 and sensing, 22-23 timc constant (ill~cs.),64 Flight phases, 279-80 launch, 270

midcourse, 279-80 terminal, 280 typical laser-guided weapon mission (example), 280 FLIR. s e e ~oiward-lookinginfrared (FLIR) FO-FO~FS. ;see Fail-operationlfailoperationlfail-safe (FO-FO-FS) FOG. See Fiber-optics guidance (FOG) Footprint, 393 Forward-looking infrared (FLIR), 30, 184 Four-gimbal mechanism (ill~rs.),193 Fourth-order optimal C P N G algorithm: with decoupling feature, 496-98 with target acceleration bias estimate, 498 FOV. See Field of view Future guidance processing, 401-13 missile guidance, signal processing, 407-13 technological advances in signal processing. 40.5-6

Gauss-Markov: models. 70, 71 process. 157 GI>OP. See Gconlctric dilution of position (GIIOP) Generalized least squares (GLS), 451 Generalized strapdown mechanization, 199, 201 (ill~ts.),204 (il111.c.).205 Geometric dilution of position (GIIOI'), 33, 220 bias and variancc shrinkage, 454-57 Gimbal anglc tnotion, 56, 60 cffccti\re slope and cffcctive noise, dctcrminants (iil~r$.), 5.5 targct maneuver and n o w causc (1111~s.). 55 Gimballed incrtial navigation system. 179, 186, 192-99 incrtial sensors on stable platform. 193 tncchanism, 192 navigation mechanization and crror model, 195. 197-99 na\.igation nlodc, 194-05. 196-98 platform alignment modes, 193-94

system internal and external interlaces. 105, I97 Gimbal nicchanisrn, 102, 103 (;imbal/strapdown navigation systems, diffcrcncc between, 199 See a l j o Gimbal versus strapdown comparison and analysis Gimbal versus strapdowti comparison arid analysis. 200-13 ieaturc comparisoti, 200 navigation errors comparison. 209-13 Glint noisc. 47. 75. 557 Glint noise uyi,,,,, 548 Global positioning system (GI's), 179, 215-17 acctlracy irnprovcmcnt. 445-47 bias and variance shrinkage, 454-57 position-fix navigation, 446-52 recursive estimation and Kalman filter, 452-53 ridge regression, 5-11-42 inertial navigation systems updating, 217 aided inertial-global positioni~lg system mechanization (illrrs.), 218 measurement principles, 216-17 range, 316 range rate. 216-17 time, 216 overview, 213-16 performance, 217 pseudo-rangc error budget (table), 217 requiremctlts, 215 Globdl positioning systemlinertial navigation system Kalman filter systcm design, 457-64 Global positioning system pseudo-range error budget (table), 217 GLS. See Generalized least squares (GLS) GPS. See Global positioning system (G S-') Gravity anomalies, 177 Guidance, 14 computer function, 14 Guidance algorithm, 252, 310-47 direct guidance methods, 312-47 preset guidance, 310-12 Guidance filter design trade-off, 591-600 missile midcourse guidance system analysis, 593-600

optimal guidance filter design analysis, 592-93 Guidancc filtcringlprocessi~ig,45-51 advanced multivariablc control system design. 49-50 classical versus niodcrn guidance and control, 50-51 guidance kinematic loop. 45 navigation, guidance, and control systeni design rcquirenicnts and design considerations, 48-40 noisc i~iputs.45-48 nottnoisc inputs. 48 simplified pursuer dynamics, 45 Guidance kinematic loop, 45 Guidancc kinematic loop stability analysis (example), 52-53 Guidance law, 347-79 classification (illlrs.), 469 comparisoli of (table), 353 line-of-sight (LOS) angle guidance, 348 line-of-sight (LOS) rate guidance, 348-19. 35 1-73 sensitivity and comparison, 373-79 simultation, 502-5 survcy, 469-74 Guidance loop (illrrs.), 37 Guidance loop transfer function with radomc slope bias effect and without radome slope effect, 533 Guidance mission and performance, 268-98 operation, 280-98 performance, 369-79 phases of flight, 279-80 Guidance performance analysis with inflight radome error calibration, 520-28 adaptive radome estimation design, 520-26 closed-loop analysis for adaptive radome estimator, 524-28 Guidance processing, 252 algorithm, 310-47 defense and offense systems, 392-404 future processing, 404-13 law, 347-79 mission and performance, 268-98 multiple mode guidance modeling, 298-310 processors, 252, 256-68 single-mode, dual-mode, multimode guidance. 379-92

Guidance processors, 252, 254-70 precision requirements and achievements, 260-62 standard missile guidance system development (example), 262-67 Talos guidance system (example), 267-70 Guidance system classification, 21-22 Guidance tracking filter, 252, 427-30 design approach, 428-30 function and requirements, 427-28 Guided interceptor, 609-13 clutter, 610-13 deception repeater jammer, 613 IF seeker in weather condition, 611 jamming and clutter, 607 Guided weapons, 280-81 air-to-air, 281-83 air-to-surface, 283-84 air-to-surface roles and missions, 284-87 surface-to-air, 287 surface-to-surface, 294 Gyros. See Gyroscopes Gyroscopes, 14, 177-78, 192-95 ~ y r sensitivity o gain design (example), 67-68

Handover, definition, 299 HCE. See Homing controller expansion Helicopter integrated inertial navigation system (HIINS), 241-45 accuracy requirements, 241-43 configuration 1 and 2, 243-45 (illus.), 246-49 Hermite matrix for third-order perturbed polynomial, 93 Higher order system analysis, 62-63 High-frequency dynamics (table), 26 High-frequency requirement, 85-86 High-value target (HVT), 261 HIINS. See Helicopter integrated inertial navigation system (HIINS) HOJ. See Home-on-jamming (HOJ) Home-all-the-way guidance, 379-80 Home-on-jamming (HOJ), 331, 337, 344

Homing controller expansion (HCE), ' 537-38 Homing guidance, active, 324-26 Homing missile controller, 16 Homing missile guidance and control analysis (example), 63-65 Homing of aerospace vehicles, 331, 337, 344 modes of homing, 324 Homing problems, 397-401 Homing system performance: desired/actual effective navigation ratio, determinant of, 52 guidance system time constant, 32 HVT. See High-value target (HVT)

IC. See Integrated circuits (IC) IEEE conference on decision and control, 4 IINS. See Integrated inertial navigation system (IINS) IISA. See Integrated inertial sensing assembly (IISA) Illuminator (radar devices), 328 IMU. See Inertial measurement unit (IMU) Inertial guidance, multimode guidance using, 388 Inertial measurement unit (IMU), 195 (illus.). 198 Inertial midcourse guidance, 299, 301 Inertial navigation, 176-81 basic functions, 179 common requirements, 181-84 comparison and analysis of gimballed versus strapdown, 209-13 error analysis, 188-89 error models, 188. 189-92 external navigation aids, 213-29 gimballed inertial navigation system, 192-99 integrated inertial navigation system, 229-51 navigation computation and error modeling, 184-92 strapdown inertial navigation system. 199-209 updating, 217, (illus.) 220

Inertial navigation requirements. 181-84 stand-off weapon systems, 182-84 vehicle and weapon systems, 181-82 visual attack systcms, 184 weapon dclivcry. 182 lncrtial position. 186-87 Inertial reference unit (IRU), 428 primary ti~nction,299 Inertial sensors on stable platforn~,193 alignment nlodes, 193-94 three-gyrolacccleromctcr (illus.), 104 Inertial space, 177 Inertial system nlechanization (illus.), 198 Infrared (IR): detection, 309 sensor, 28 Infrared (IR) guidance. 330 Infrared (1R)-guided missiles, advantages, 281 Infrared homing short-range AAM, 508-10 Infrared (IR) image processor, 310 Infrared (IR) seeker in weather condition, 613 Integrated circuits (IC), 261 Integrated Doppler, 216-17 See also Pseudo-delta-range (PDR) Integrated GPSIIN navigation system, 457-64 data-decoupling algorithm, 462-63 GPSIINS Kalman filter system design, 458-62 tests and results, 463-64 Integrated inertial navigation system (IINS), 229, 231-51 helicopter integrated inertia! navigation system (HIINS), 241-45 (illus.), 246-49 integrated inertial sensing assembly (IISA), 232, 234, 236, 238-43 integrated missile guidance systems, 245, 249-51 integrated sensinglflight control reference system (ISFCRS), 231 (illus.), 232 (illus.), 233 (illus.), 234 integrated sensory subsystem (ISS), 231-32 (illus.), 235

Integrated inertial sensing assembly (IISA). 232. 234. 236-41 configuration of sensor locations, 236, 238 configuration versus attitude1 velocity redundancy (table), 238 effect of gyro dither. 238 error sources, 240-41 acceleration scale factor stability, 240-41 axis alignment errors, 240 data synchronization. 241 input axis bending, 241 noise and filtering, 241 performance, 240 redundancy management, 238-40 Integrated missile guidance systems, 245, 249-51 integrated seeker-guidance navigation system, 249-51 lntegrated modified PNG and optimal controller, 520 with radome compensation, 520 Integrated navigation and guidance system, 381, 384 Integrated sensinglflight control reference system (ISFCRS), 231 (illus.), 233 (illus.), 234 geometry (illus.), 232 MlRA system (illus.), 234 Integrated sensory subsystem (ISS), 231-32 Intelligent knowledge-based systems techniques, 332 Intelligent TVIimage, 330 Intercept guidance, relative geometry and parameters for (illus.), 35 Interface information for gimballed inertial navigation systems, 197 (illus.), 197 Intermediate-frequency requirement or crossover frequency, 85 Internal interface, 195 IR. See Infrared (IR) IRU. See Inertia! reference unit (IRU) ISFCRS. See Integrated sensing/flight control reference system (ISFCRS) ISS. See Integrated sensory subsystem (ISS)

Jamming and clutter effects on guided interceptor, 609-13 clutter, 610-12 deception repeater jammer, 609-10 IR seeker in weather condition, 613 Jam-to-signal ratio USR), 609 Jet Propulsion Laboratory: NASA software contract of, 152 JSR. See Jam-to-signal ratio USR)

Kalman-Bucy filter, 2 See also Recursive minimum variance estimator Kalman filter (KF), 2, 71, 420 analysis, 558-59 continuous, 131-32 design and perforn~anceanalysis, 140-48 design process, 142 error analysis, 146-48 noise intensity matrices, selection of, 142-46 discrete, 130-31, (table) 141 cstirnation techniques. 126-28 operational considcrations. 148-58 absetice of a priori inforn~ation,153 coniputational requirements, 149-53 filtcr divcrgencc, 153-58 filter performance, 149 modeling process noise and biases, 156-58 simplified, 420-21 Kalman filtering, 226-28 approach, 227-28 pcrformancc, 228. (illris.) 230 Kclley-Bryson variational optin~ization procedure, 2 KF. Srr Kalnian filtcr (KF) Kincniatic equations (illrrs.), 37 Kinematic/rclati\~cgeometry, 34-37

Laplacc domain. 162 Largc-scale integrated (LSI), 261

Laser guidance (example), 303 Laser-guided weapon mission, typical, (example), 280 Linearized airframe response (table), 25 Linear discrete/continuous smoother (table), 141 Linear miniinum variance estimation: Kalman filter, 128-33 continuous Kalman Filter, 129, 132-33 discrete Kalman filter, 129 Linearlnonlit~earintercept navigation. guidance, and control processing, 14-18 Linearlnonlinear intercept navigation, guidance, and control system, 13-23 flight control system (FCS) arid sensing, 22-23 guidance system classification. 21-22 modeling and simulation, 18-21 processing, 14 Linear radome miss prediction. 57-59 Line-of-sight (LOS): angular rate sensors, 33 guidance, 312-13, 349 system seeker model, 43 sensors. 32 LOAL. See Lock-on after launch (LOAL) LOUL. Sce Lock-on before launch (LOUL) Lock-on after launch (LOAL). 255 Lock-on bcfore launch (LOBL), 254 Long-range navigation (LORAN), 34, 179, 220 advantages and limitations, 220 characteristics, 220 Look-down Doppler radar, 184 e LORAN. Scr ~ b n ~ - r a t i gnavigation (LORAN) LOS. Scc Line-of-sight (10s) Low-altitude air defense systems. 397-404 air defense homing system, 401-3 further missile dcvclopnicnts, 403-4 homing problems, 397-401 Low-altitude cruise tiiissilc threat. 293 Low-altitude target cngagcmcnt with grazing atiglc control (example). 505-00 Low-frequency rcquirctncnt, 84 LQG tcchniquc, 2, 470, 471, 472, 473 LSI. Src Largc-scalc integrated (LSI)

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Maneuver scale factor. 536 Markov process. 71 Matched models versus minimum MSE models, 456 Maximum-likelihood filter, 415 MDASE. See Modeling, design, analysis, simulation and evaluation (MDASE) Measurement errors. 77 Medium-scale integrated (MSI), 261 Midcourse guidance, 298-302 boost-sustain propulsion, 587-91 computational scheme, 587-90 near-optimal trajectories, 590-91 command, 299 command to line-of-sight, 299 (illus.). 300 inertial, 299, 301 Midcourse euidance for stand-off " tactical weapons (example), 301-2 Midcourse guidance system analysis, 308-9 Midcourse phase navigation, guidance, and control processing (illus.), 308 ...

Midcourse plus terminal guidance, 380-81 Millimeter wave (MMW): sensors, 29 Miss-distance analysis for command guidance, 550-57 asymptotic system performance, 553-54 system time constant and missile acceleration limits, 554, 556-57 Miss distance (MD) and missile lateral acceleration due to launch error,

-5n5 --

Miss distance and missile lateral acceleration due to step-target acceleration, 505 Miss due to radome slope, 54-56 Missile, 178, 262-67 developments, 403-4 range and function, 297-98 Missile attitude, 396 Missile footprint, 396 Missile guidance: comparing neural network classifiers, 411-12

extended target detection and segmentation, 412-13 future signal processing for, 407-13 Missile midcourse guidance system analysis, 593-97 error analysis, 397-98 results, 598-600 Missile nonalignment, 607 Missile range, 396 Missile sizing, 536 Missile speed, 396 MIT Instrumentation Laboratory, 178 MMW. See Millimeter wave (MMW) Modeling, design, analysis, simulation and evaluation (MDASE): cycle, 3 linear/nonlinear intercept navigation, guidance, and control system, 13-23 of navigation, guidance, and control processing, 13 navigation, guidance and control system design and analysis, 45-68 process, 1 target signal processing, 23-45 target tracking state modeling, 68-76 Modeling biases, 156-58 Modeling errors, 76, 146-48 Modeling process noise and biases, 156-58 Modeling and simulation, 18-21 computer simulation block diagram (iil~is.),19 Model with a priori information, 452 Modern filtering and estimation techniques, 126-28 combined complementary/Kalman filter approach to estimator design, 158-75 Kalman filter design and performance analysis, 140-48 linear minimum variance estimation: Kalman filter, 128-33 nonlinear filtering, 133-38 operational considerations, 148-58 absence of a priori information, 153 computational requirements, 149-53 filter divergence, 153-55 filter performance, 149, (illus.) 150-51 modeling process noise and biases, 156-58 prediction and smoothing, 138-40

-

Modern multivariable control analysis, 8-9 Modified maximum-likelihood filter, 424-25 definition, 425 Modified P N G with radome compensation, 520 Monopulse homing system, 343 tests, 344 Monte Carlo: CADET and SLAM qualitative comparison, 122 cost comparison for linear and nonlinear systems (illus.), 123 simulations, 142 Monte Carlo analysis, 100-102, 474 acceleration limit study utilizing Monte Carlo approach (illus.), 101 for second-order systems, 95 theoretical confidence intervals for Gaussian distributed random variable (illus.), 101 Monte Carlo technique, 20, 78, 92, 111 algorithm, 94, 99 robustness margins, 94 single-input single-output autopilot, 95 for third-order system, 94 MRSI. See Multiresolution spatial integration (MRSI) MSI. See Medium-scale integrated (MSI) circuits Multiniode guidance: applications, 384-92 developnient of a niultimode/ multiband HOJ system (example), 391-92 environment sensing guidance/ correlation matching, 389-90 radar and infrared, 385-88 STANDARD missile-2 upgrade program (cxamplr). 390 using coniniand guidance, 389 using inertial guidance, 388 Multiniode guidance analysis, 618-19 Multimode guidance systcrtl design, 613-19 analysis, 61 8-1 9 Multimodcln~ultiba~~d HOI- system, . dc\~cloprncntoT (exaniple), 39 1-92 Multiple guidance system using optimal

C P N G algorithm with LOS angle and LOS rate estimator (illus.), 614 Multiple mode guidance modeling, 298 error analysis model development, 308-10 midcourse guidance, 298-310 terminal guidance, 302-8 Multiple model estimation, 137-38 Multiple-rate Kalman filter (table), 131 for radar tracking, 443 Multiresolution spatial integration (MRSI), 405 Multivariable control analysis. modern, 77 adjoint method, 106-11 covariance analysis, 102-6 design requirements, 84-86 general robustness analysis, 89-90 Monte Carlo analysis, 100-102 other performance analysis. 124-25 performance analysis, 78 qualitative comparison, 121-23 robustness analysis, 77-78 robustness of real perturbations, 90- 100 sensitivity and complementary sensitivity functions. 82-83 singular value analysis, 78-87 statistical linearization, 113-21 structured singular value, 86-88 Multivariable Nyquist stability, 79

Navaho missile, 187 Navigation aids, external. See External navigation aids Navigation computation and error modeling, 181-92 coordinate systems. 184-86 data processing, 188 inertial navigation systems error analysis and modeling, 188-92 position and velocity generation. 186-88 Navigation filtering for position and velocity estimate (exaniple), 167 Navigation, guidance, and control (NGC): application to:

aerospace, 2 energy management. 2 industrial nianufacturing, 2 medicine, 2 design overview, 1-8 advanced, systems, 7-8 inlportance of advanced, concepts and their impact, 4-7 history, 1-4 outlitic and scope, 8-1 1 advanced systems design, 10 (ill~rs,),11, 12 design algorithms for advanced systems, 9-10 MDASE of navigation, guidance, and control processing, 8 modern multivariable control analysis, 8-9 processing, 10 theory, 2 Navigation, guidance, and control stability and performance analysis, 52-68 autopilot response characteristics necessary to achieve homing, 65-66 comparison of statistical digital simulation methods (example), 53 effective slope and effective noise, 56-57 empirical radome slope calculation, 61-62 guidance kinematic loop stability analysis (example), 52-53 gyro sensitivity gain design (example), 66-67 higher order system analysis, 62-63 homing missile guidance and control analysis (example), 63-65 linear radome miss prediction, 57-59 miss due to radome slope, 54-56 miss less with periodic radome, 56 noise and target maneuver miss, 60-61 radome error compensation, 66-67 radome error-induced miss-distance predictions (example), 53-54 radome error slope in guidance response time, 66 range-independent noise (RIN) miss, 60 Navigation, guidance, and control system design and analysis,

45-68 guidance tiltcri~iglprocessin~, 45-51 rcquirenictits and design considerations. 48 stability and performance analysis, 52-68 Navigation and guidance filtering design, 414 advanced system design, 445-64 practical filter design, 427-42 radar tracking, 442-44 spacecraft attitude estimation, 444-45 target state estimation. 414-27 Navigation and guidance for position estimate, 430-31 Navigation and guidance for position and velocity estimate, 431-34 complementary filtering, 431-33 differentiation of position information, 431 inertial navigation systems with periodic radar position updates, 433-34 Navigation and, guidance filtering for position, velocity, and acceleration estimate, 434-40 poisson jinking maneuver, 71, 438-39 practical estimator gain design, 439-40 uniformly distributed target maneuver, 434-38 Navigation mechanization and error model, 195, 197-99 error models, 199 (illtrs.), 202-3 (illus.), 204 mechanization, 195, 197-98 performance trade-off analysis, 199 Navigation mode, 194-95 (illus.), 196 (ill~rs.),197 Navigation requirements: precision-guided stand-off weapon (PGSOW), 182-84 visual attack system, 184 weapon delivery, 182 Navigation signal time and range (NAVSTAR), 215 Navigation system, 13 NAVSTAR. See Navigation signal time and range (NAVSTAR) NAV-to-body direction cosine matrix (iiius.), 186

Neutrally stable first-order filter (example), 145-46 NEV. See North, east, and vertical (NEW Nine-state guidance filtering equation, 75-76 N o bias, b = 0, 161-62 Noise: glint, 47, 75 receiver, 47, 75 Noise and target maneuver miss, 60-61 Noise-driven time-varying systems, 100 Noise inputs, 45-48 atmospheric noise, 558 atmospheric noise U.am4,549-50 glint noise, 557 glint noise a,,,,,, 548 , range-dependent noise, 557 range-dependent noise u,,, 548-49 range-independent noise (RIN), 557 range-independent noise (RIN) u,, 548 total angular noise PSD Q,, 550 Noise intensity matrices, selection of, 142-46 neutrally stable first-order filter, (example), 145-46 QR-' -, x , 146 QR-' -, 0, 143 stable, first-order filter (example), 143-45 unstable first-order filter (example), 145 unstable open-loop model dynamics, 145 Nonlinear dynamic systems (illirs.), 18 Nonlinear filtering, 133-38 extended Kalman filter (EKF), 133-35 multiple model estimation, 137-38 statistical linearization technique, 135-37 Nonli~~carities. 77 Nonlinear stochastic systems: analyzing and evaluating tool, 116 Monte Carlo versus CAIIET, 116, 118 steps to apply CAIIET to guidancc systems, 117 Nonnoisc inputs. 48 Normalized radomc slope (illlrs.), 64 North, cast, and vertical (NEV) coordinates. 193

OGS. See Optimal guidance system Operational considerations, 148-58 absence of a priori information, 155 computational requirements, 149-53 filter divergence, 155-57 filter performance, 149 modeling process noise and biases, 156-58 Optimal BPNG algorithm, 487-88 implementation, 488 Optimal control theory, 2 Optimal CPNG algorithm with complementary: LOS angle and LOS rate estimator, 495 LOS rate estimator, 493 Optimal filters and optimal guidance law, analytical solution of, 474-81 analytical three-dimensional optimal guidance law, 474-76 combined estimator-controller: optimal filters and optimal guidance law, 476-77 optimal guidance system (OGS), 378-81 Optimal guidance gain time variation (ill~rs.),477 Optimal guidancc law, 476-77 analytical three-din~ensional, 474-76 Optimal guidance system (OGS), 478 Optimallnonoptin1a1 trajectory, 581 Optimal shaping guidance law structure (ill~rs.),580 Optin~alsolution for air-breathing enginc or thc power-on stagc of a solid rocket engine, 571-73 Optimal solution for the pon'er-off stagc of solid rockct engine, 500 Optimal trajcctory shaping guidancc, 562-63 law, general solutions of, 574-75 mathematically probing analytical solution of, 576-78 problcn~formulation, 3 3 - 6 5 Ordcr reductiot~by truncation, 530-31 Ostrowski matrix. 81 Outputting, INS function, 179

Parasitic fcc~lbacks.43-45 Passive hornins. 330-31 CHAI'AIIRAL guidance (c?tample), 33 1 1'1111. Src I'scudo-delta-rangc (PIIR) PGM. SLY l'rccision-guided munitions [I'GM) Penalty function technique, 99 Pert'orrn'ince analysis, 78 Performance comparison in linear homing systems, 503 Periodic radonic, less miss with, 54 (i//ll~.), 35 Perturbation (true frame) approach, 189 PGSOW. Srr Precision-guided stand-off weapon (PGSOW) Phoney data interpretation, 454 Platform aligiinicnt niodes, 193-94 PNG. S r r Proportional navigation guidnnce (I'NG) Point defense. 293 Poisson jinking maneuver, 71. 438-39 Poisson process, 71 Position and rate estimators with position and acceleration measurements (Formulation I), 163-67 Position and rate estimators with position and rate measurements (Formulation 2), 163, 168-69 Position and rate estimators with rate and acceleration measurements (Formulation 3), 163 Position bias, 159-60 Position error and rate error estimator with position error and acceleration measurements (example), 167-68 Position-fix navigation, 446-52 Position and velocity generation. 186-87 Position sensing, 310 PPS. See Precise positioning service (PPS) Precise positioning service (PPS), 216 Precision-guided munitions (PGM), 252 Precision-guided stand-off weapon (PGSOW), 182-83 mission scenario, 182 (illus.), 183, 183-84 Prediction and smoothing, 138-40

-

PRN. See Pseudorandom binary noise "Probing Bocing's Crossed Connections" (Karen Fitzgcrald),

5 Proportional navigation (illtrs.), 23 Proportional navigation guidance (L'NG), 16, 360-63, 468 pursuit plus I'NG, 371 system design and analysis, 363 velocity conipcnsated L'NG (VCPNC;). 371 Proportional na\~igationguidance (PNG) law. 477 Pseudo-dcltd-range (PI>11), 217 Pseudorandom binary noise (PRN), 216 Psi-angle (computer frame) approach, I89 Pulsed rocket control, 600-606 Pursuer Parget tracking processing soft~vareniodulcs (illlrs.), 23 Pursuit guidance, 353-38 attitude, 357-58 plus proportional navigation ~ u i d a n c e (PNG), 360

Radar and infrared, multimode guidance, 385 Radar command to line-of-sight guidance (illus.), 300 Radar cross section (RCS), 396 Radar-guided missiles, 281 Radar search mode, 309 Radar tracking. 442-44 a tracker, a-P-y tracker, 443 discrete Kalman filter, 442-43 extended Kalman filter, 444 multiple-rate Kalman filter for, 443 Radiometer and MMW guidance, 331 Radome boresight error slope on miss distance, effect of (illus.), 55

Radome coupling and compensation loops, 38-41 Radome error calibration and compensation, 517-43 compensation, 66-67, 517-20 design equations for P N G with parasitic feedback, 530-46 guidance performance analysis with in-flight radome error calibration, 520-28 Radome error compensation, 517-20 example, 66-67 Radome error distorts boresight error (illus.), 38 Radome error-induced miss-distance predictions (example), 53-54 Radome error slope in guidance response time (example), 66-67 Radome slope, miss increases with (illus.), 53 Range-dependent noise, 557 or,,548-49 Range-independent noise (RIM), 557 miss, 60 ,'rc 550 Range measurement, 216 Range rate (closing velocity), 33 Range rate measurement, 216-17 Range sensors, 32 Range-tracking servo. 309 Rate bias, h = 1. 161 Rate estimator (example), 162 Rate-sensing feedback, 310 RCS. See Radar cross section (RCS) Real-time implementation and performance, 578-79 comparison between optimal trajectory and nonoptimal trajectory, 581-82 implementation and simulation of three-dimensional flight, 579-80 midcoursc phase. 580 switching phase and terminal phase, 581 vertical launch of tactical nlissiles, 582-83 Receiver noise, 75 Recursive cstinlatiot~and Kalman filtcr, 452-53 Recursi\rc Kalman filtcr. 420 Recursi\*c filter, block diagram (illtrs.), 416

Recursive minimum variance estimator. See Kalman-Bucy filter Relinearization, 134 Repeater jammer, deception, 609-10 Residual data block, 458 RF detection, 309 Riccati equation, 152 Ridge regression, 453-54 RIM. See Range-independent noise (RIM) Ring laser gyro (RLG), 178 RLG. See Ring laser gyro (RLG) RMS miss distances, 56, 57 Robust integrated control law design process (illtrs.), 49 Robustness analysis, 77-78 general framework, 89 Robustness measures, various (table), 90 Robustness of real perturbations, 90 (illtrs.), 97 Root locus, 92 and SSV, 92

SAM. See Surface-to-air missiles (SAM) SAR. See Synthetic aperture radar^ (SAW SDF. &e ~ingle-degree-of-freedom (SDF) sea-skimming missile, 296 Search mode, 309 Search patterns: raster scan, 309 sector scan, 309 spiral scan, 309 Second-order a estiri~ator(example), 169 using inertial a rate at center-ofgravity, 169 using inertial a rate at IRU, 169 Second-ordcr c o n l p l c n i e n t a r y / K a l n ~ a ~ ~ filtcr, 163 for~rlulation1, 163-67 navigation filtering for position and velocity estimate (example), 166 position error and rate error estimator with position error and acceleration measurements (example), 167-68 formulation 2, 168-69

second-order a cstimator (exanlple). 169 second-order sideslip cstimator (example), 169-70 Second-order optinlal C P N G algorithm, 493 Second-order sideslip cstimator (cxaniplc). 169-70 using inertial sideslip rate of centerof-gravity, 169-70 using inertial sideslip rate of IRU, 169 second-order target tracker: a-p tracker, 422-23 -- -Seeker block diaeram. 38 Seeker gyro-loop bandwidth, 538-39 Seeker and optimal guidance system block diagram, (illus.) 480 Semiactive (or semipassive) homing, 326-27 guidance dynamics, 546 loop (forward time) (illus.), 5-17 pros and cons of. 339 Semiactive laser homing, 327-29 Semi-Markov filter, 417, 418 Sensing, INS function, 179 Sensitivity and comparison of guidance law, 373-79 Sensitivity and complementary sensitivity functions, 82 Sequentially correlated noise, 156 Servo angular positions sensor, 309 Short-range tactical missiles, comparison of guidance laws for, (table) 352 SIDEWINDER missiles, 330 Signal and noise curve magnitude bode plot (illus.), 429 Signal processing: autonomous acquisitionlimage processing, 406-7 missile guidance, 407-8 technological advances in, 405-6 Simplified Kalman filter, 420-21 Simplified optimal guidance law, 477 Simplified pursuer dynamics, 45 Simplified suboptimal BPNG algorithm: without pursuer acceleration feedback, 492 with pursuer acceleration feedback, 49 1 Simplified suboptimal C P N G algorithm, 500-501 Singer model, 71

Single-degree-of-freedom (SDF),193 Single-input single-output (SISO), 86 feedback control system, 78 Single-mode guidance, 379 home-all-the-way guidance, 379-80 Singular perturbation for advanced midcoursc guidance, 584 fast dynamic approach, 587 medium dynamic approach, 586 midcoursc guidance algorithm, 587 near-optimal solution using singlepcrturbation technique, 585 problem formulation, 584 slow dynamic approach, 585-86 Singular value analysis, 78-82 maneuver., 433 -. SISO. See Single-input single-output (SISO) SLAM. See Statistical linearization with adjoint method (SLAM) Monte Carlo and CADET qualitative comparison, 121-23 cost comparison for linear and nonlinear systems (illus.), 123 Smoothing, 140 linear discrete1continuous smoother (table), 141 Software: Kalman filter design, 1 2 - 3 3 SOG. See Suboptimal guidance (SOG) SONAR. See Sound navigation and ranging (SONAR) Sound navigation and ranging (SONAR), 294-95 Spacecraft attitude estimation, 444-45 Space-fixed navigation computation, 186 (illus.), 187 SPS. See Standard positioning service (SF'S) square-root' filter, 151-52 SSM. See Surface-to-surface missiles (SSM) Stable, first-order filter (example), 143-45 Stable platform, inertial sensors on, 193 Stable platform phase follow-up system (STAPFUS), 337 Stability margin computation, 98-100 Standard missile guidance system development (example), 262-67 STANDARD missile. 329 ~

STANDARD missile-2 upgrade program (example), 390-92 Standard missile-2 terminal guidance (example), 305 Standard positioning service (SPS), 216 Stand-off weapon systems, 182 STAPFUS. Scc Stable platform phase follow-up system (STAPFUS) Star trackers, 179, 225-26 advantages and limitations, 226 State modeling (mblc), 21 State-space model for additive uncertainty, 91-92 State-space singular values, 92 Statistical linearization, 111-16 applications, 120-21 missile acceleration for target intercept: nonlinear systcm analysis (illlrs.), 121 approsiniation (illris.), 112 statistical linearization with adjoint mcthod (SLAM), 118-19 steps for using SLAM. 118-19 technique. 135 and tools for. 116-17 Statistical lincarization approach, 442 Statistically linearized filter, 135 continuous dynamics, discrete mcasuremcnts (mhlr), 136 Statistical lincarization with adjoint nicthod ISLAM). 78. 118-20 Stochastic inputs. adjoints for, 109. 11 1 Strapdo~vn.dcfinition, 177 StrapdoLvn computcr requirements, 201, 204 Strapdown crror model and analysis, 2U4 computational errors. 207 crror model. 208 (ill~rs.),210 instrument errors, 204, 206 StrapdoLvn inertial navigarion systcm, 170. 180, I')'J-20

Modern Navigation, Guidance, And Control Processing

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Missile Guidance and Control Systems

Spatial Processing in Navigation, Imagery and Perception

Spatial Processing in Navigation, Imagery and Perception

Mobile Robots: Navigation, Control and Remote Sensing

Spacecraft Formation Flying: Dynamics, control and navigation

Fault Tolerant Flight Control And Guidance Systems

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Introduction to Modern Navigation Systems

Modern structural analysis: modelling process and guidance

Modern Control Engineering (Control Engineering)

Practical Navigation for the Modern Boat Owner

Missile guidance and pursuit: kinematics, dynamics and control

Modern Signal Processing

Modern Signal Processing

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Signal Processing for Active Control (Signal Processing and its Applications)

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Model Predictive Control (Advanced Textbooks in Control and Signal Processing)

Statistical Methods in Control and Signal Processing

Thermal Processing of Foods: Control and Automation

Statistical Methods in Control and Signal Processing

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