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University Microfilms International 300 N 8100112 Ba tes , C arlto n Ja m es OPTIMAL AND SUBOPTIMAL TERMINAL GUIDANCE LAWS WITH PRACTICAL CONSIDERATIONS FOR A SHORT-RANGE MISSILE AGAINST AN ACCELERATING TARGET The Ohio State University Ph.D. 1980 University Microfilms International 300 N. Zeeb Road, Ann Arbor, MI 48106 PLEASE NOTE: In all cases this material has been filmed 1n the best possible way from the available copy. Problems encountered with this document have been identified here with a check mark v ' ' . 1. Glossy photographs ______ 2. Colored Illustrations ______ 3. Photographs with dark background__ _____ 4. Illustrations are poor copy ______ 5. °rint shows through as there 1s text on bothsides of page ______ 6. Indistinct, broken or small print on several pages ______ 7. Tightly bound copy with print lost in spine ______ 8. Computer printout pages with indistinct print _ 9. Page(s) ____ lacking when material received, and not available from school or author 10. Page(s) ______ seem to be missing in numbering only as text follows 11. Poor carbon copy ______ 12. Not original copy, several pages with blurred type ___ 13. Appendix pages are poor copy ______ 14. Original copy with light type ______ 15. Curling and wrinkled pages ______ 16. Other University MicixSilms Intenvitional 300 N ZEES RD.. ANN ARSOR Ml .18106 '313) 761-4700 OPTIMAL AND SUBOPTIMAL TERMINAL GUIDANCE LAWS WITH PRACTICAL CONSIDERATIONS FOR A SHORT RANGE MISSILE AGAINST AN ACCELERATING TARGET DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Carlton James Bates, B.S., M.S. ***** The Ohio State University 1980 Reading Committee: Approved By Prof. E. 0. Doebelin, Chairman Prof. D. R. Houser Prof. K. Srinivasan Adviser Department of Mechanical Engineering ACKNOWLEDGMENTS I wish to express particular appreciation to Professor Ernest 0. Doebelin for his support and encouragement given during the completion of this dissertation. I wish to thank Ralph D. Ehrich and James 0. Everly for many productive discussions and their assistance in developing the computer programs used in this study. I appreciate the understanding and encouragement shown by my super­ visors at Rockwell International during my course of study. I wish to thank Debbie M. Sacksteder for typing the original draft and Carol M. Hawes and Ruth K. Sanderson for typing the final manuscript. I especially want to thank Jack R. Brewer for the excellent job of preparing the figures. Finally, I wish to express my deep apprecia­ tion to my wife and family for their encouragement and patience which were of invaluable support. ii VITA. March 27, 1936 Born - Cleveland, Ohio 1958 B.S., The Pennsylvania State University, University Park, Pennsylvania 1960 M.S., The Pennsylvania State University, University Park, Pennsylvania 1960-1980 Member Technical Staff, Rockwell International, Columbus, Ohio PUBLICATIONS "Rocket Turbines", ASME Journal of Engineering for Power, pp 280-29.0, July 1959 (Co-author). FIELDS OF STUDY Major Field: Mechanical Engineering Studies in Measurements and Control Systems. Professors F. C. Weimer and E. 0. Doebelin Studies in Engineering Mechanics. Professor J. C. Kennedy Studies in Mathematics. Professor H. D. Colson iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS................................................ ii VITA........................................................... iii LIST OF TABLES................................................. vii LIST OF FIGURES................................................. viii LIST OF SYMBOLS................................................. xiii Chapter 1 INTRODUCTION AND ORGANIZATION...................... 1 1.1 Introduction and Statement of Problem. .......... 1 1.2 Organization................... 8 2 LITERATURE REVIEW........ 12 2.1 Introduction. ..... 12 2.2 Homing Guidance Studies.......................... 13 3 HOMING GUIDANCE MODEL................................... 23 3.1 Introduction........... 23 3.2 Engagement Geometry.............................. 24 3.3 Missile-Target Kinematic Equations.............. 25 3.4 Missile Dynamics................................ 31 3.5 Missile Velocity and Longitudinal Acceleration.................................. 34 4 OPTIMAL GUIDANCE LAW.................................... 36 4.1 Introduction....... 36 4.2 Missile-Target System Description................ 37 4.3 Review of the General Optimal Control Problem and Solution........................... 42 4.4 Terminal Control Problem......................... 46 4.5 Analytic Solution............................... 49 4.5.1 Optimal Guidance Gains................ 63 4.5.2 Preliminary Performance Evaluation.......................... 6 6 iv TABLE OF CONTENTS (Continued) Page 4.6 Numerical Integration Solution............ 70 4.7 Guidance Gain Variation Analysis................ 79 4.8 Alternative State Variables.................... 83 4.8.1 Revised Optimal Guidance Gains....... 8 8 4.9 Summary........................................ 90 5 SUBOPTIMAL GUIDANCE LAW................................ 94 5.1 Introduction............ 94 5.2 Form of the Suboptimal Law..................... 95 5.3 Approximate Navigation Gain.................... 96 5.4 Effective Closing Velocity..................... 101 5.5 Longitudinal Acceleration Guidance Command....... Ill 5.6 Target Acceleration Estimate.................... 115 5.7 Passive Range/Range-Rate Estimate............... 117 5 .8 Summary........................................ 119 6 PERFORMANCE EVALUATION................................. 123 6.1 Introduction................................... 123 6.2 Missile-Target Parameter Description.......... 124 6.3 Random Error Sources and Models................. 130 6.4 Statistical Analysis........ 135 6.4.1 Adjoint Definition.................. 135 6.4.2 Analysis Results .............. 146 6.5 Terminal Homing Analysis....................... 159 6.5.1 Guidance Law Comparison Tests........ 159 6.5.2 Launch Boundaries................... 175 6 . 6 Summary and Conclusions........................ 194 7 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS................ 200 7.1 Summary........................................ 200 7.2 Conclusions........... 202 7.3 Recommendations................................ 207 REFERENCES........ 209 APPENDIX A ANALYTIC SOLUTION OPTIMAL GAIN CURVES.............. 212 A.l Introduction....... 212 A. 2 Gain Variation With Missile Natural Frequency.................. 213 A. 3 Gain Variation With Terminal StateWeighting and Constant Missile Velocity....... 221 v TABLE OF CONTENTS (Concluded) Page A.4 Gain Variation With Missile Constant Longi­ tudinal Acceleration and Deceleration...... 227 A. 5 Effect of Unknown Final Conditions.............. 234 APPENDIX B REVISED OPTIMAL GAIN CURVES........................ 239 B.l Introduction................................... 239 B.2 Revised Gain Variation With Missile Constant Longitudinal Acceleration and Deceleration 240 B.3 Gain Variation With Missile Boost-Coast Velocity and Longitudinal Acceleration......... 247 APPENDIX C SIMULATION PROGRAMS DESCRIPTIONS AND LISTINGS....... 255 C.l Introduction................................... 255 C.2 Analytic Gains Calculation Program Listing...... 265 C.3 Numerical Integration Gains Calculation Programs Listings............. 273 C.4 Adjoint Simulation Block Diagrams and Listing 290 C.5 Terminal Homing Simulation Block Diagram and Listing...................................... 310 vi j LIST OF TABLES Table Page 1. System Parameter Values............................... 129 2. Random Error Model Values. ...... ................ 136 3. Effect of Variable Missile Velocity on SMS Miss Distance.......................................... 136 4. Summary of Final Conditions for Comparison Tests With Launch in a Tail Aspect (Condition //l)......... 165 5. Summary of Final Conditions for Comparison Tests With Launch in a Beam Aspect (Condition //2). ..... 170 6 . Summary of Final Conditions for Comparison Tests With Launch in a Head Aspect (Condition if3)......... 174 7. Summary of Final Conditions for Comparison Tests With Launch in a Tail Aspect (Condition #4)......... 176 8 . Summary of Final Conditions for Comparison Tests With Launch in a Beam Aspect (Condition #5)......... 176 9. Units of the Analytic Solution Optimal Gains .......... 213 10. Parameter Values for Figures 44 Through 49 ............ 214 11. Parameter Values for Figures 50 Through 53 ............ 222 12. Parameter Values for Figures 54 Through 58 ............ 228 13. Parameter Values for Figures 59 Through61 ............ 235 14. Parameter Values for Figures 62 Through6 6 ............ 241 15. Parameter Values for Figures 67 Through 72 ............ 248 16. Computer Programs Variable List....................... 256 vii LIST OF FIGURES Figure Page 1. Homing Loop Block Diagram............................. 2 2. Engagement Geometry................................... 26 3. Missile-Target Kinematic Model ................... 30 4. Missile-Target State Space Model ..................... 41 5. Alternative Missile-Target State Space Model .......... 85 6 . Effect of Variable Velocity in the Missile Dynamics on the Optimal Navigation Ratio (NRq )............... 87 7. Optimal Guidance Law M o d e l ............. 91 8 . Variation of RMS Miss Distance with Terminal State Weighting for the Suboptimal Form Definition........ 98 9. Variation of Optimal Navigation Ratio (NR0) With Constant Acceleration, Constant Deceleration, and Acceleration to Deceleration ................ 100 10. Missile Longitudinal
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