Contents 1 Introduction 1 1.1 Problem Description 1 1.2 Research Significance 2 1.3 Research Progress 4 References 8 2 Mathematical Fundamentals 11 2.1 Regular Perturbation Method 11 2.2 Singular Perturbation Method 13 2.3 Spectral Decomposition Method 16 2.3.1 Idempotent Matrix 16 2.3.2 Spectral Decomposition Theorem 16 2.3.3 Inference 17 2.3.4 Example 19 2.4 Pseudospectral Method 19 2.4.1 Introduction of Method 19 2.4.2 Pseudospectral Discrete Process 23 2.5 Linear Gauss Pseudospectral Model Predictive Control 33 References 38 3 Mathematical Modeling for Hypersonic Glide Problem 41 3.1 The Coordinate System Adopted in This Book 41 3.1.1 Geocentric Inertial Coordinate System (I) 41 3.1.2 Geographic Coordinate System (T) 41 3.1.3 Orientation Coordinate System (O) 42 3.1.4 Velocity Coordinate System (V) 42 3.1.5 Half-Velocity Coordinate System (H) 42 3.1.6 Body Coordinate System (B) 43 3.2 Transformation Between Coordinate Systems 43 3.2.1 Transformation Between the Orientation Coordinate System and the Half-Velocity Coordinate System 43 3.2.2 Transformation Between the Velocity Coordinate System and the Half-Velocity Coordinate System 43 3.2.3 Transformation Between the Velocity Coordinate System and the Body Coordinate System 44 3.2.4 Transformation Between the Body Coordinate System and the Half-Velocity Coordinate System 45 3.3 Dynamic Equations of Hypersonic Vehicle in Half-Velocity Coordinate System 45 3.3.1 Dynamics Equations of the Center of Mass in Half-Velocity Coordinate System 45 3.3.2 The Dynamic Equations of the Center of Mass of the Vehicle 48 3.3.3 Dynamic Equations of Hypersonic Gliding Vehicle Based on BTT Control 48 3.3.4 Dynamic Equations of Hypersonic Vehicle in Vertical Plane 49 3.3.5 Atmospheric Model 50 3.3.6 Aerodynamic Model 50 3.3.7 The Stagnation Point Heat Flow，Overload and Dynamic Pressure 50 4 Mathematical Description of Glide-Trajectory Optimization Problem 53 4.1 Mathematical Description for Optimal Control Problem 53 4.1.1 Performance Index of Optimal Control Problem 53 4.1.2 Description of Optimal Control Problem 54 4.1.3 The Minimum Principle 55 4.1.4 Final Value Performance Index of Time-Invariant Systems 56 4.1.5 Integral Performance Index of Time-Invariant Systems 57 4.1.6 Optimal Control Problem with Inequality Constraints 58 4.1.7 Methods for Solving Optimal Control Problems 58 4.2 Mathematical Description of Optimal Control Problem for Hypersonic Vehicle Entry Glide 61 4.2.1 Maximum Final Speed Problem 61 4.2.2 Maximum Range Problem 62 4.2.3 Shortest Time Problem 62 4.2.4 Optimal Trajectory Problem with Heating Rate Constraint 63 4.2.5 Optimal Trajectory Problem with Heating Rate and Load Factor Constraints 64 5 Indirect Approach to the Optimal Glide Trajectory Problem 65 5.1 Combined Optimization Strategy for Solving the Optimal Gliding Trajectory of Hypersonic Aircraft 67 5.1.1 Mathematical Model of Hypersonic Gliding 67 5.1.2 Necessary Conditions for Optimal Gliding Trajectory 68 5.1.3 Solving Two-Point Boundary Value Problem by Combination Optimization Strategy 69 5.1.4 Numerical Calculation Results 70 5.1.5 Conclusion 73 5.2 Trajectory Optimization of Transition Section of Gliding Hypersonic Flight Vehicle 74 5.2.1 Aerodynamic Data for the Transition Section 74 5.2.2 Unconstrained Trajectory of Maximum Terminal Velocity 75 5.2.3 Heat Flow Constrained Trajectory of Maximum Terminal Velocity 76 5.2.4 Solving the Two-Point Boundary Value Problem for the Transition Section 77 5.2.5 Optimizing the Transition Trajectory with Direct Method 77 5.2.6 Steps for Solving the Optimal Transition Trajectory 78 5.2.7 Transitional Trajectory Obtained by Indirect Method 81 5.3 The Maximum Range Gliding Trajectory of the Hypersonic Aircraft 84 5.3.1 Guess Initial Values for Optimal Control Problem by Direct Method 84 5.3.2 Indirect Method for Solving Optimal Control Problems 89 5.3.3 The Maximum Range Gliding Trajectory of the Hypersonic Aircraft 94 References 101 6 Direct Method for Gliding Trajectory Optimization Problem 103 6.1 Direct Method for Solving Optimal Control Problems 103 6.2 Direct Shooting Method 104 6.2.1 Direct Multiple Shooting Method 104 6.2.2 Direct Method of Discrete Control 105 6.2.3 Gradual Subdividing Optimization Strategy 106 6.3 Direct Collocation Method 107 6.3.1 General Form of Direct Collocation Method 107 6.3.2 Direct Transcription 108 6.3.3 Implicit Integral Method 109 6.3.4 Solving Optimal Trajectory Problems with NLP 110 6.4 Direct Collocating Method for Trajectory with Maximum Gliding Cross Range of Hypersonic Aircraft 111 6.4.1 Mathematical Model 111 6.4.2 Re-entry Flight Control Law with Given Angle of Attack Profile 113 6.4.3 Solution of Maximum Cross Range Problem by Direct Collocation Method 113 6.4.4 Optimization Example 116 6.4.5 Summary 118 6.5 Pseudospectral Method for the Optimal Trajectory of the Hypersonic Vehicle with the Longest Cross-Range 119 6.5.1 Introduction of Pseudospectral Method 119 6.5.2 Optimization Examples and Results 122 7 Concept of Steady Glide Reentry Trajectory and Stability of Its Regular Perturbation Solutions 125 7.1 Introduction 125 7.2 Kinetic Equations 126 7.3 Definition of the Steady Glide Trajectory 127 7.4 Effects of Control Variable on SGT 128 7.5 Effects of Initial Value on SGT 129 7.6 Analytical Solution of SGT 129 7.6.1 Altitude Dynamic Differential Equation 129 7.6.2 Analytical Steady Glide Altitude 131 7.6.3 Analytical Solutions of Flight-Path Angle and Vertical Acceleration 134 7.7 Dynamic Characteristics of SGT 135 7.7.1 Stability Analysis 135 7.7.2 Natural Frequency and Damping 137 7.8 Feedback Control of SGT 140 7.8.1 Feedback Design 140 7.8.2 Fixed-Damping Differential Feedback Method 144 7.9 Conclusions 147 References 147 8 Analytical Solutions of Steady Glide Reentry Trajectory in Three Dimensions and Their Application to Trajectory Planning 149 8.1 Introduction 149 8.2 Mathematical Model 150 8.2.1 Definition of Coordinate Frame 150 8.2.2 Kinematic Equations 150 8.2.3 Decoupling of Equations 152 8.3 Analytical Solution of Glide Trajectory 153 8.3.1 Analytical Solution of Altitude 153 8.3.2 Analytical Solution of Range 154 8.3.3 Analytical Solution of Heading Angle 154 8.3.4 Analytical Solution of Longitude and Latitude 155 8.3.5 Analytical Solution of Velocity 156 8.3.6 Optimal Initial Glide Angle 157 8.4 Simulation 157 8.4.1 Comparison Between Analytical Solution and Numerical Integral 157 8.4.2 Comparison with Bell Analytical Solution 157 8.4.3 Application of Analytic Solutions in Trajectory Planning 160 8.5 Summary 164 References 164 9 Trajectory Damping Control Technique for Hypersonic Glide Reentry 167 9.1 Introduction 167 9.2 Guidance Scheme 168 9.2.1 Mathematical Proof 168 9.2.2 Command Flight-Path Angle for L/Dmax 170 9.2.3 Guidance Scheme for Range Maximization and Trajectory Damping Control 172 9.2.4 Extended Guidance Scheme for Glide Range Control 173 9.3 Hypersonic Vehicle Model 174 9.4 Results and Discussion 176 9.4.1 Performance of Guidance Scheme 176 9.4.2 Application of the Extended Guidance Scheme 183 9.5 Conclusions 189 References 189 10 Steady Glide Dynamic Modeling and Trajectory Optimization for High Lift-To-Drag Ratio Reentry Vehicle 191 10.1 Introduction 191 10.2 Dynamics and Vehicle Description 193 10.2.1 Entry Dynamics 193 10.2.2 Entry Trajectory Constraints 194 10.2.3 Vehicle Description and Model Assumption 194 10.3 Trajectory-Oscillation Suppressing Scheme 195 10.3.1 Generic Theory for the Oscillation Suppressing Scheme 195 10.3.2 Performance of the Trajectory-Oscillation Suppressing Scheme 197 10.4 Steady Glide Dynamic Modeling and Trajectory Optimization 198 10.4.1 Steady Glide Dynamic Modeling 199 10.4.2 Hp-Adaptive Gaussian Quadrature Collocation Method 200 10.4.3 Numerical Example of Trajectory Optimization Without Bank Reversal 201 10.4.4 Numerical Example of Trajectory Optimization with Bank Reversal 205 10.4.5 Verification of Feasibility for the Pseudospectral Solution 206 10.5 Conclusion 209 References 210 11 Singular Perturbation Guidance of Hypersonic Glide Reentry 213 11.1 Singular Perturbation Guidance for Range Maximization of a Hypersonic Glider 213 11.1.1 Problem Formulation (Dimensionless Model) 213 11.1.2 Reduced-Order System Solutions 215 11.1.3 Slow-Boundary Layer Solutions 216 11.1.4 Fast-Boundary Layer Solutions 218 11.1.5 Simulation Results 220 11.1.6 Comparison and Analysis 221 11.2 Improved Singular Perturbation Guidance for Maximum Glide Range 225 11.2.1 Dynamic Model and Solutions to the Reduced-Order System 226 11.2.2 Boundary Layer Correction 227 11.2.3 Slow Boundary-Layer Correction 227 11.2.4 Fast Boundary-Layer Correction 228 11.2.5 Guidance Law Derivation 228 11.2.6 Simulation Results and Analyses 229 11.3 Summary 232 References 232 12 3-D Reentry Guidance with Real-Time Planning of Reference Using New Analytical Solutions Based on Spectral Decomposition Method 233 12.1 Introduction 233 12.2 Equations of Motion 235 12.3 Entry Trajectory Constraints 237 12.3.1 Path Constraints 237 12.3.2 Terminal Conditions 237 12.4 Analytical Solutions to Hypersonic Gliding Problem 237 12.4.1 Auxiliary Geocentric Inertial (AGI) Frame 237 12.4.2 Linearization of the Equations of Motion 239 12.4.3 Analytical Solutions 241 12.4.4 Example for Accuracy Verification 245 12.5 Entry Guidance 248 12.5.1 Descent Phase 248 12.5.2 Quasi-Equilibrium Glide Phase 249 12.5.3 Altitude Adjustment Phase 260 12.5.4 Results and Discussion 262 12.5.5 Nominal Cases 262 12.6 Conclusions 273 Appendix 273 References 275 13 Omnidirecdonal Autonomous Reentry Guidance Based on 3-D Analytical Glide Formulae Considering Influence of Earth’s Rotation 277 13.1 Introduction 277 13.2 Entry Guidance Problem 280 13.2.1 Equations of Motion 280 13.2.2 Path Constraints 281 13.2.3 Terminal Conditions 282 13.3 Omnidirectional Autonomous Entry Guidance 282 13.3.1 Overview 282 13.3.2 Descent Phase 285 13.33 Steady Glide Phase 286 13.4 Altitude Adjustment Phase 300 13.4.1 Correction of Baseline AOA Profile and Second Bank Reversal 300 13.4.2 Baseline Bank Angle in AAP 304 13.4.3 AOA and Bank Angle Commands in AAP 305 13.5 Results and Discussion 306 13.5.1 Nominal Cases 306 13.5.2 Monte Carlo Simulations 309 13.6 Conclusions 314 Appendix 1: Generalized States of Motion 315 Appendix 2: Generalized Aerodynamic Forces 318 References 319 14 Analytical Steady-Gliding Guidance Employing Pseudo-Aerodynamic Profiles 323 14.1 Introduction 323 14.2 Entry Guidance Problem 325 14.2.1 Equations of Motion 325 14.2.2 Path Constraints 326 14.2.3 Terminal Conditions 327 14.3 Analytical Entry Guidance Design 327 14.3.1 Descent Phase 328 14.3.2 Steady Glide Phase 328 14.3.3 Altitude Adjustment Phase 344 14.4 Results and Discussion 349 14.4.1 Nominal Cases 349 14.4.2 Monte Carlo Simulations 354 14.5 Conclusions 361 References 364 15 Linear Pseudospectral Guidance Method for Eliminating General Nominal Effort Miss Distance 365 15.1 Introduction 365 15.2 Generic Theory of LGPMPC 366 15.2.1 Linearization of Nonlinear Dynamic System and Formulation of Linear Optimal Control Problem 367 15.2.2 Linear Gauss Pseudospectral Method 369 15.2.3 Singularity of Differential Approximation Matrices for Different Pseudospctral Methods 374 15.2.4 Boundary Control of Linear Gauss Pseudospctral Method 374 15.2.5 Implementation of LGPMPC 375 15.3 Application to Terminal Guidance 377 15.3.1 Terminal Guidance Problem and Three-Dimensional Mode 377 15.3.2 Initial Guess and Target Model 379 15.3.3 Cases for Target with Straight-Line Movements 380 15.3.4 Comparison with Adaptive Terminal Guidance 384 15.4 Conclusion 386 Appendix 387 References 388 16 Linear Pseudospectral Reentry Guidance with Adaptive Flight Phase Segmentation and Eliminating General Nominal Effort Miss Distance 389 16.1 Introduction 389 16.2 Entry Dynamics, Entry Trajectory Constraints and Vehicle Description 391 16.2.1 Entry Dynamics 391 16.2.2 Entry Trajectory Constraints 392 16.2.3 Vehicle Description and Model Assumption 393 16.2.4 Auxiliary Geocentric Inertial Frame and Emotion Dynamics 393 16.3 Linear Pseudospectral Model Predictive Entry Guidance 394 16.3.1 Descent Phase Guidance 395 16.3.2 Glide Phase Entry Guidance 395 16.3.3 Terminal Adjustment Phase 411 16.3.4 Implementation of the Proposed Method 416 16.4 Numeric Results and Discussion 417 16.4.1 Normal Cases for Various Destinations 417 16.4.2 Monte Carlo Simulations 423 16.5 Conclusion 430 References 431 17 Trajectory-shaping Guidance with Final Speed and Load Factor Constraints 433 17.1 Introduction 433 17.2 Equations of Motion 435 17.3 Guidance Law Overview 437 17.4 Trajectory Shaping Guidance 437 17.4.1 Guidance Form 437 17.4.2 Generalized Closed Form Solutions for TSG 438 17.4.3 Stability Domain of Guidance Coefficients 448 17.5 Final Speed Control Scheme 452 17.6 Model of CAV-H 453 11.1 Results and Discussion 454 17.8 Conclusions 460 References 460
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