Differential Game Theory with Applications to Missiles and Autonomous Systems Guidance

Contents

Preface

Acknowledgments

About the Companion Website

1 Differential Game Theory and Applications to Missile Guidance

Nomenclature

Abbreviations

1.1 Introduction

1.1.1 Need for Missile Guidance—Past, Present, and Future

1.2 Game Theoretic Concepts and Definitions

1.3 Game Theory Problem Examples

1.3.1 Prisoners Dilemma

1.3.2 The Game of Tic-Tac-Toe

1.4 Game Theory Concepts Generalized

1.4.1 Discrete-Time Game

1.4.2 Continuous-Time Differential Game

1.5 Differential Game Theory Application to Missile Guidance

1.6 Two-Party and Three-Party Pursuit-Evasion Game

1.7 Book Chapter Summaries

1.7.1 A Note on the Terminology Used In the Book

References

2 Optimum Control and Differential Game Theory

Nomenclature

Abbreviations

2.1 Introduction

2.2 Calculus of Optima (Minimum or Maximum) for a Function

2.2.1 On the Existence of the Necessary and Sufficient Conditions for an Optima

2.2.2 Steady-State Optimum Control Problem with Equality Constraints Utilizing Lagrange Multipliers

2.2.3 Steady-State Optimum Control Problem for a Linear System with Quadratic Cost Function

2.3 Dynamic Optimum Control Problem

2.3.1 Optimal Control with Initial and Terminal Conditions Specified

2.3.2 Boundary (Transversality) Conditions

2.3.3 Sufficient Conditions for Optimality

2.3.4 Continuous Optimal Control with Fixed Initial Condition and Unspecified Final Time

2.3.5 A Further Property of the Hamiltonian

2.3.6 Continuous Optimal Control with Inequality Control Constraints—the Pontryagins Minimum (Maximum) Principle

2.4 Optimal Control for a Linear Dynamical System

2.4.1 The LQPI Problem—Fixed Final Time

2.5 Optimal Control Applications in Differential Game Theory

2.5.1 Two-Party Game Theoretic Guidance for Linear Dynamical Systems

2.5.2 Three-Party Game Theoretic Guidance for Linear Dynamical Systems

2.6 Extension of the Differential Game Theory to Multi-Party Engagement

2.7 Summary and Conclusions

References

Appendix: Vector Algebra and Calculus

A2.1 A Brief Review of Matrix Algebra and Calculus

A2.2 Characteristic Equations and Eigenvalues

A2.3 Differential of Linear, Bi-Linear, and Quadratic Forms

A2.4 Partial Differentiation of Scalar Functions w.r.t. a Vector

A2.5 Partial Differentiation of Vector Functions w.r.t. a Vector

A2.6 The Hessian Matrix

A2.7 Partial Differentiation of Scalar Quadratic and Bilinear Functions w.r.t. a Vector

A2.8 First and Second Variations of Scalar Functions

A2.9 Properties of First and Second Variations for Determining the Nature (Min/Max Values) of Scalar Functions

A2.9.1 Extension to Multi-Vector Case

A2.10 Linear System Dynamical Model

3 Differential Game Theory Applied to Two-Party Missile Guidance Problem

Nomenclature

Abbreviations

3.1 Introduction

3.2 Development of the Engagement Kinematics Model

3.2.1 Relative Engage Kinematics of n Versus m Vehicles

3.2.2 Vector/Matrix Representation

3.3 Optimum Interceptor/Target Guidance for a Two-Party Game

3.3.1 Construction of the Differential Game Performance Index

3.3.2 Weighting Matrices

3.3.3 Solution of the Differential Game Guidance Problem

3.4 Solution of the Riccati Differential Equations

3.4.1 Solution of the Matrix Riccati Differential Equations (MRDE)

3.4.2 State Feedback Guidance Gains

3.4.3 Solution of the Vector Riccati Differential Equations (VRDE)

3.4.4 Analytical Solution of the VRDE for the Special Case

3.4.5 Mechanization of the Game Theoretic Guidance

3.5 Extension of the Game Theory to Optimum Guidance

3.6 Relationship with the Proportional Navigation (PN) and the Augmented PN Guidance

3.7 Conclusions

References

Appendix

A3.1 Verifying the Positive Semi-Definiteness of Matrix [S]

A3.2 Derivation of Riccati Differential Equations

A3.3 Solving the Matrix Riccati Differential Equation

A3.3.1 Inversion of Matrix

A3.3.2 Solution of the Inverse Matrix Riccati Differential Equation

A3.4 Solution of the Vector Riccati Deferential Equation

A3.4.1 Analytic Solution of the VRDE—Case2

A3.5 Sight Line Rates for Small Angles and Rates

4 Three-Party Differential Game Theory Applied to Missile Guidance Problem

Nomenclature

Abbreviations

4.1 Introduction

4.2 Engagement Kinematics Model

4.2.1 Three-Party Engagement Scenario

4.3 Three-Party Differential Game Problem and Solution

4.4 Solution of the Riccati Differential Equations

4.4.1 Solution of the Matrix Riccati Differential Equation (MRDE)

4.4.2 Solution of the Vector Riccati Differential Equation (VRDE)

4.4.3 Further Consideration of Performance Index (PI) Weightings

4.4.4 Game Termination Criteria and Outcomes

4.5 Discussion and Conclusions

References

Appendix

A4.1 Derivation of the Riccati Equations

A4.2 Analytical Solution for Riccati Differential Equations

A4.3 State Feedback Gains

A4.4 Disturbance Inputs

A4.5 Guidance Disturbance Inputs

5 Four Degrees-of-Freedom (DOF) Simulation Model for Missile Guidance and Control Systems

Nomenclature

Abbreviations

5.1 Introduction

5.2 Development of the Engagement Kinematics Model

5.2.1 Translational Kinematics for Multi-Vehicle Engagement

5.2.2 Vector/Matrix Representation

5.2.3 Rotational Kinematics: Relative Range, Range Rates, Sightline Angles, and Rates

5.3 Vehicle Navigation Model

5.3.1 Application of Quaternion to Navigation

5.4 Vehicle Body Angles and Flight Path Angles

5.4.1 Computing Body Rates (pi, qi, ri)

5.5 Vehicle Autopilot Dynamics

5.6 Aerodynamic Considerations

5.7 Conventional Guidance Laws

5.7.1 Proportional Navigation (PN) Guidance

5.7.2 Augmented Proportional Navigation (APN) Guidance

5.7.3 Optimum Guidance and Game Theory-Based Guidance

5.8 Overall State Space Model

5.9 Conclusions

References

Appendix

A5.1 State Space Dynamic Model

A5.2 Aerodynamic Forces and Equations of Motion

A5.2.1 Yaw-Plane Equations

A5.2.2 Pitch-Plane Kinematics Equations

A5.2.3 Calculating the Aerodynamic Forces

A5.2.4 Body Incidence

A5.3 Computing Collision Course Missile Heading Angles

A5.3.1 Computing (𝛃TS) Given (VT, 𝛙T, 𝛉T, 𝛙S, 𝛉S)

A5.3.2 Computing (𝛃MS)cc Given (VM, 𝛃TS)

A5.3.3 Computing the Closing Velocity (VC) and Time-to-Go (Tgo)

A5.3.4 Computing the Collision Course Missile (Az. and El.) Heading: (𝛉M)cc; (𝛙M)cc

A5.3.5 Example: Computing 2-DOF Collision Course Missile Heading Angles

6 Three-Party Differential Game Missile Guidance Simulation Study

Nomenclature

Abbreviations

6.1 Introduction

6.2 Engagement Kinematics Model

6.3 Game Theory Problem and the Solution

6.4 Discussion of the Simulation Results

6.4.1 Game Theory Guidance Demonstrator Simulation

6.4.2 Game Theory Guidance Simulation Including Disturbance Inputs

6.5 Conclusions

6.5.1 Useful Future Studies

References

Appendix

A6.1 Analytical Solution for Riccati Equations

Addendum

Index

EULA