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
內容試閱：
Chapter 1 Introduction
1.1 Problem Description
Hypersonic glide vehicle (HGV) is generally a near-space vehicle that can fly at a velocity of more than Mach number 5. It is characterized by high velocity， long flight distance and high heating rate. Therefore， through traditional methods， it is difficult to obtain the optimal trajectory satisfying various constraints for HGV. Reference [1] drew the conclusion that the maximum range flight scheme can be approximated by the maximum lift-to-drag ratio gliding flight. Hence， simulations of the maximum lift-to-drag ratio flight are carried out and the trajectory and heating rate profiles are shown in Figs. 1.1 and 1.2 respectively， where the initial altitude is 94 km， initial Mach number is 20.4 and initial flight-path angle is 0.
As shown in Figs. 1.1 and 1.2， the maximum lift-to-drag ratio flight trajectory is a skipping gliding trajectory with a large range. Due to the severe oscillations of trajectory， the heating rate at the stagnation point cannot be controlled and the maximum heating rate may far exceed the endurance of the vehicle， thus leading to destructive damage to the vehicle structure. It can be seen from Fig. 1.2 that the maximum heating rate reaches 1400 W/cm2， which is far beyond the allowable peak heating rate of 650 W/cm2. The influences of the initial altitude and the initial Mach number on the maximum heating rate for the maximum lift-to-drag ratio flight are presented in Fig. 1.3.
Fig. 1.3 shows the poility that the maximum heating rate can be controlled below 650 W/cm2 at a certain initial height and Mach number. However， due to the constraints of each phase and the requirement of the maximum range， the initial altitude and the initial Mach number of the entry flight may not satisfy the heating rate constraint. This means that the constrained maximum range problem cannot be ressed by adopting the maximum lift-to-drag ratio gliding flight scheme directly. As a result， it is necessary to further study the optimal control problem under various constraints and propose a fast on-line guidance algorithm.
Fig. 1.1 Trajectory of maximum lift-to-drag ratio flight
Fig. 1.2 Heating rate of maximum lift-to-drag ratio flight
1.2Research Significance
Owing to advance in long-range， high velocity， near-space flight， low detectability and strong maneuverability， HGV haeen developed vigorously by many countries.
With characteristics of high speed， wide range of speed variation and long range in entry phase， HGV shows strong maneuverability and wide maneuvering range. Because of its advantages in fast arrival and maneuverability， hypersonic glider is considered to be a re-entry vehicle with wide application prospect， which can achieve long-distance， fast and precise strike and force delivery.
Thiook introduces the entry dynamic characteristics and various entry guidance methods for unpowered hypersonic gliding vehicle. The trajectory of hypersonic gliding vehicle can be generally divided into the ascending phase， transition phase， gliding phase and descending phase.
Fig. 1.3 Profiles of maximum heating rate， initial altitude and initial Mach number corresponding to the maximum lift-to-drag ratio flight
The ascending phase is similar to that of ballistic flight vehicle and carrier rocket， where rocket is used as the booster and has a comparatively mature technology. The transition phase has characteristics of high altitude and weak control ability， thus large angle of attack (AOA) and small bank angle control method are adopted here to guide the vehicle to the starting point of the gliding phase. And the flight mission of hypersonic gliding vehicle is mainly realized by the flight control in gliding phase and descending phase.
During entry flight of hypersonic vehicle， the flight distance is long， the airspace is large， and the flight Mach number varies widely. Due to the continuous increase of heat during the long-range flight，the structure of the vehicle will be under great pressure. Therefore， the peak and accumulation value of heating rate should be fully considered in trajectory design and entry guidance， thus it is important to find a reasonable guidance method which can reduce the peak heating rate and total heat flux during the research. At the same time， the control of dynamic pressure is also very imperative. When the vehicle mainly flies in near space with small air density， if the dynamic pressure is too small， the rer will not have the capacity to provide enough moment to ilize the attitude of the vehicle. Moreover， in order to accomplish some specific missions， the gliding vehicle needs to drop down rapidly at the end of the descending phase，which may lead to large dynamic pressure and load factor. Large dynamic pressure will produce too much hinge moment for rer structure to sustain， and large load factor will cause damage to some weak parts of the vehicle ……
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