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 Colloca