A New Paradigm in Optimal Missile Guidance Item Type text; Electronic Dissertation Authors Morgan, Robert W. Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 24/09/2021 22:40:12 Link to Item http://hdl.handle.net/10150/194121 A New Paradigm in Optimal Missile Guidance by Robert W. Morgan A Dissertation Submitted to the Faculty of the Department of Electrical & Computer Engineering In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy In the Graduate College The University of Arizona 2007 2 The University of Arizona Graduate College As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Robert W. Morgan entitled A New Paradigm in Optimal Missile Guidance and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy Date: 04/05/2007 Dr. Hal Tharp Date: 04/05/2007 Dr. Jeffrey J. Rodriguez Date: 04/05/2007 Dr. Jerzy W. Rozenblit Date: 04/05/2007 Dr. Thomas L. Vincent Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final coppies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. Date: 04/05/2007 Dissertation Director: Dr. Hal Tharp 3 Statement by Author This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. Signed: Robert W. Morgan 4 Acknowledgments The author wishes to express his appreciation to Professors J.J. Rodriguez, J.W. Rozenblit, H.S. Tharp, and T.L. Vincent for their service on his doctoral committee. Special acknowledgement is due the committee chairman, Professor H.S. Tharp, for his advice and efforts to review and scrutinize the content and quality of this dissertation. The author would also like to express his appreciation for the inspiration and encouragement given him by Professor T.L. Vincent throughout the author’s academic career. This work was made possible by the generous fellowship awarded to the author by Raytheon Missile Systems. Several people at Raytheon have been instrumental to the author’s ability to participate in the program. Ron Reid, the director of the program, has been a source of encouragement from the very beginning. The author’s acceptance into and desire to participate in the program is, with great thanks, due to Martin Ulehla. Special acknowledgement is also due Chris Poage, who has been an exemplar professional and a friend to the author while participating in the program. The author would also like to thank his wife, Angela, and his children for their support and encouragement these past four years. 5 Ta b l e o f Co n t e n t s List of Figures ................................. 11 List of Tables .................................. 14 Abstract ..................................... 15 Chapter 1. Introduction .......................... 16 1.1.BackgroundandScope.......................... 16 1.2.HistoryandStateoftheArt....................... 18 1.3. Organization ............................... 25 1.4.Notation.................................. 30 Chapter 2. Estimation ........................... 32 2.1.GeneralConceptsinEstimation..................... 32 2.1.1. Estimation Defined........................ 32 2.1.2.MarkovModel........................... 33 2.1.3.OptimalEstimationandRisk.................. 34 2.2. Bayesian Estimation........................... 38 2.3. Multiple Model .............................. 40 2.3.1.InteractiveMultipleModel.................... 43 2.4. Linear Estimation ............................ 43 2.4.1.UnbiasedEstimation....................... 44 2.4.2.BLUEEstimation......................... 45 2.4.3.LeastSquareEstimation..................... 49 2.4.4.MaximumLikelihoodEstimation................ 50 2.4.5.FullRankModels ........................ 51 2.4.6. A Priori Estimates and Rank DeficientModels......... 52 Chapter 3. Estimation in Linear Sampled Data Systems ...... 55 3.1. Bayes’ Estimator ............................. 57 3.1.1.FirstMeasurement........................ 58 3.1.2. kth MeasurementandFinalResultbyInduction........ 64 3.2. Estimates and Confidence Regions (Error Ellipsoids) for the Bayes’ Estimator................................. 67 3.3.TheWhiteNoiseAssumptionandBayesianEstimation........ 68 3.4. The Deterministic Input Assumption and Bayesian Estimation . 70 3.5.BayesianEstimationBetweenMeasurements.............. 71 3.6.NoAPrioriInformationandBayesianEstimation .......... 74 6 Table of Contents–Continued 3.7. The Kalman Filter ............................ 77 3.7.1.CovarianceSimulations...................... 79 3.7.2.ScalarSystemEstimationExample............... 79 3.8. Multiple Model .............................. 83 Chapter 4. Stochastic Motion Models ................. 91 4.1. Markov Models ............................. 91 4.1.1.PrincipleofInertia........................ 92 4.2.ProcessNoiseModels........................... 93 4.3.RandomWalk............................... 95 4.3.1.ContinuousTimeRandomWalk................. 96 4.4. White Acceleration............................ 98 4.4.1.DiscreteEquivalent........................ 99 4.5.CorrelatedAcceleration.......................... 101 4.5.1.DiscreteEquivalent........................ 101 Chapter 5. Optimization and Control Theory ............ 104 5.1.BasicControlTheoryConcepts..................... 104 5.2.ParametricOptimization ........................ 105 5.2.1. Constraints . ........................... 106 5.2.2. Necessary Conditions for a Local Minimum ........... 107 5.3.LyapunovControlTheory........................ 111 5.3.1.QuickestDescentControl..................... 113 5.3.2.QuickestDescentwithMinimumIncrementalCost....... 115 5.4.OptimalControlTheory......................... 117 5.4.1.OptimalReturnFunction.................... 118 5.4.2. The Augmented State Vector and the Augmented State Space 119 5.4.3.TemporalBoundaryConditions................. 121 5.4.4. The Optimal Control H Function................ 123 5.4.5.StateIndependentControlConstraints............. 124 5.4.6.TheOptimalControlMinimumPrinciple............ 126 5.4.7.LinearQuadraticRegulator(LQR)............... 127 5.4.8.LinearSystemswithProcessNoise............... 133 5.4.9. Linear Systems with Process Noise and Measurement Noise . 134 5.5. DifferentialGameTheory........................ 140 5.5.1. System Definition......................... 140 5.5.2.ControlConstraints........................ 141 5.5.3.TerminalSet........................... 141 5.5.4. The Payoff ............................. 141 5.5.5.GamesofKindandGamesofDegree.............. 142 7 Table of Contents–Continued 5.5.6.Min-MaxPrinciple........................ 143 5.5.7. Min-Max Necessary Conditions . ................ 144 Chapter 6. Airframe and Autopilot Modeling ........... 145 6.1. Introduction ................................ 145 6.2.AirframeModelingofTailControlledMissiles............. 151 6.2.1.AerodynamicForces....................... 151 6.2.2.AirframeDynamics........................ 156 6.2.3.AirframeTransferFunction................... 158 6.3.Three-LoopAutopilot.......................... 162 6.3.1.HighBandwidthActuator.................... 165 6.3.2.3-LoopSummary......................... 166 6.3.3.3-LoopParameters........................ 166 6.3.4.3-LoopPerformance....................... 167 6.4.FirstOrder(Pole)ApproximationofFlightControlSystem..... 168 6.5.Pole-ZeroApproximationofFlightControlSystem.......... 170 6.6.BinomialApproximationofFlightControlSystem........... 173 Chapter 7. Optimal Guidance ....................... 176 7.1.EngagementGeometryandDynamics ................. 176 7.2.Miss.................................... 178 7.2.1.AccelerationDynamics...................... 179 7.3. Zero EffortMiss(ZEM)......................... 181 7.3.1.AccelerationDynamics...................... 182 7.4. Heading Error .............................. 183 7.5.BasicModel ............................... 185 7.5.1.AugmentedProportionalNavigationGuidance......... 188 7.5.2.ProportionalNavigationGuidance............... 188 7.6.AccelerationDynamics.......................... 189 7.6.1.DecoupledDynamics....................... 192 7.7. Single Pole Flight Control System Model ............... 193 7.8. Optimal Evasion ............................. 198 7.9.MagnitudeConstraints(Saturation) .................. 203 7.9.1.SinglePoleFlightControlSystem................ 205 7.10.DirectionalConstraints ......................... 207 7.10.1. Simulation ............................. 212 7.10.2. Engagement Configuration...................