Farhan A. Faruqi, Defence Science and Technology Organisation, AustraliaProfessor Dr. Farhan A. Faruqi is the Head of the Intelligent Autonomous Systems Research Guidance and Control Group in the Defence Science and Technology Organisation in Australia. He is also an Adjunct Professor at the University of South Australia. His main areas of expertise include autonomous systems navigation; guidance and control; target tracking; and intelligent autonomous systems. He has more than twenty years' experience in the Aerospace and Defence Industry in the UK, USA, and Australia.

• Preface xiAcknowledgments xiiiAbout the Companion Website xv1 Differential Game Theory and Applications to Missile Guidance 1Nomenclature 1Abbreviations 21.1 Introduction 21.1.1 Need for Missile Guidance—Past, Present, and Future 21.2 Game Theoretic Concepts and Definitions 31.3 Game Theory Problem Examples 41.3.1 Prisoner’s Dilemma 41.3.2 The Game of Tic-Tac-Toe 61.4 Game Theory Concepts Generalized 81.4.1 Discrete-Time Game 81.4.2 Continuous-Time Differential Game 91.5 Differential Game Theory Application to Missile Guidance 101.6 Two-Party and Three-Party Pursuit-Evasion Game 111.7 Book Chapter Summaries 111.7.1 A Note on the Terminology Used In the Book 13References 142 Optimum Control and Differential Game Theory 16Nomenclature 16Abbreviations 172.1 Introduction 172.2 Calculus of Optima (Minimum or Maximum) for a Function 182.2.1 On the Existence of the Necessary and Sufficient Conditions for an Optima 182.2.2 Steady State Optimum Control Problem with Equality Constraints Utilizing Lagrange Multipliers 192.2.3 Steady State Optimum Control Problem for a Linear System with Quadratic Cost Function 222.3 Dynamic Optimum Control Problem 232.3.1 Optimal Control with Initial and Terminal Conditions Specified 232.3.2 Boundary (Transversality) Conditions 252.3.3 Sufficient Conditions for Optimality 292.3.4 Continuous Optimal Control with Fixed Initial Condition and Unspecified Final Time 302.3.5 A Further Property of the Hamiltonian 352.3.6 Continuous Optimal Control with Inequality Control Constraints— the Pontryagin’s Minimum (Maximum) Principle 362.4 Optimal Control for a Linear Dynamical System 382.4.1 The LQPI Problem—Fixed Final Time 382.5 Optimal Control Applications in Differential Game Theory 402.5.1 Two-Party Game Theoretic Guidance for Linear Dynamical Systems 412.5.2 Three-Party Game Theoretic Guidance for Linear Dynamical Systems 442.6 Extension of the Differential Game Theory to Multi-Party Engagement 502.7 Summary and Conclusions 50References 51Appendix 533 Differential Game Theory Applied to Two-Party Missile Guidance Problem 63Nomenclature 63Abbreviations 643.1 Introduction 643.2 Development of the Engagement Kinematics Model 673.2.1 Relative Engage Kinematics of n Versus m Vehicles 683.2.2 Vector/Matrix Representation 693.3 Optimum Interceptor/Target Guidance for a Two-Party Game 703.3.1 Construction of the Differential Game Performance Index 703.3.2 Weighting Matrices S, R p ,R e 723.3.3 Solution of the Differential Game Guidance Problem 733.4 Solution of the Riccati Differential Equations 753.4.1 Solution of the Matrix Riccati Differential Equations (MRDE) 753.4.2 State Feedback Guidance Gains 763.4.3 Solution of the Vector Riccati Differential Equations (VRDE) 773.4.4 Analytical Solution of the VRDE for the Special Case 783.4.5 Mechanization of the Game Theoretic Guidance 793.5 Extension of the Game Theory to Optimum Guidance 793.6 Relationship with the Proportional Navigation (PN) and the Augmented PN Guidance 813.7 Conclusions 82References 82Appendix 844 Three-Party Differential Game Theory Applied to Missile Guidance Problem 102Nomenclature 102Abbreviations 1034.1 Introduction 1034.2 Engagement Kinematics Model 1044.2.1 Three-Party Engagement Scenario 1054.3 Three-Party Differential Game Problem and Solution 1074.4 Solution of the Riccati Differential Equations 1114.4.1 Solution of the Matrix Riccati Differential Equation (MRDE) 1114.4.2 Solution of the Vector Riccati Differential Equation (VRDE) 1124.4.3 Further Consideration of Performance Index (PI) Weightings 1154.4.4 Game Termination Criteria and Outcomes 1164.5 Discussion and Conclusions 116References 117Appendix 1185 Four Degrees-of-Freedom (DOF) Simulation Model for Missile Guidance and Control Systems 125Nomenclature 125Abbreviations 1265.1 Introduction 1265.2 Development of the Engagement Kinematics Model 1265.2.1 Translational Kinematics for Multi-Vehicle Engagement 1265.2.2 Vector/Matrix Representation 1285.2.3 Rotational Kinematics: Relative Range, Range Rates, Sightline Angles, and Rates 1285.3 Vehicle Navigation Model 1305.3.1 Application of Quaternion to Navigation 1315.4 Vehicle Body Angles and Flight Path Angles 1335.4.1 Computing Body Rates (p I ,q I ,r I) 1345.5 Vehicle Autopilot Dynamics 1355.6 Aerodynamic Considerations 1355.7 Conventional Guidance Laws 1365.7.1 Proportional Navigation (PN) Guidance 1365.7.2 Augmented Proportional Navigation (APN) Guidance 1375.7.3 Optimum Guidance and Game Theory–Based Guidance 1375.8 Overall State Space Model 1385.9 Conclusions 138References 139Appendix 1406 Three-Party Differential Game Missile Guidance Simulation Study 150Nomenclature 150Abbreviations 1506.1 Introduction 1516.2 Engagement Kinematics Model 1516.3 Game Theory Problem and the Solution 1546.4 Discussion of the Simulation Results 1576.4.1 Game Theory Guidance Demonstrator Simulation 1576.4.2 Game Theory Guidance Simulation Including Disturbance Inputs 1606.5 Conclusions 1626.5.1 Useful Future Studies 162References 163Appendix 164Addendum 165Index 189

Farhan A. Faruqi, Defence Science and Technology Organisation, AustraliaProfessor Dr. Farhan A. Faruqi is the Head of the Intelligent Autonomous Systems Research Guidance and Control Group in the Defence Science and Technology Organisation in Australia. He is also an Adjunct Professor at the University of South Australia. His main areas of expertise include autonomous systems navigation; guidance and control; target tracking; and intelligent autonomous systems. He has more than twenty years' experience in the Aerospace and Defence Industry in the UK, USA, and Australia.

• Preface xiAcknowledgments xiiiAbout the Companion Website xv1 Differential Game Theory and Applications to Missile Guidance 1Nomenclature 1Abbreviations 21.1 Introduction 21.1.1 Need for Missile Guidance—Past, Present, and Future 21.2 Game Theoretic Concepts and Definitions 31.3 Game Theory Problem Examples 41.3.1 Prisoner’s Dilemma 41.3.2 The Game of Tic-Tac-Toe 61.4 Game Theory Concepts Generalized 81.4.1 Discrete-Time Game 81.4.2 Continuous-Time Differential Game 91.5 Differential Game Theory Application to Missile Guidance 101.6 Two-Party and Three-Party Pursuit-Evasion Game 111.7 Book Chapter Summaries 111.7.1 A Note on the Terminology Used In the Book 13References 142 Optimum Control and Differential Game Theory 16Nomenclature 16Abbreviations 172.1 Introduction 172.2 Calculus of Optima (Minimum or Maximum) for a Function 182.2.1 On the Existence of the Necessary and Sufficient Conditions for an Optima 182.2.2 Steady State Optimum Control Problem with Equality Constraints Utilizing Lagrange Multipliers 192.2.3 Steady State Optimum Control Problem for a Linear System with Quadratic Cost Function 222.3 Dynamic Optimum Control Problem 232.3.1 Optimal Control with Initial and Terminal Conditions Specified 232.3.2 Boundary (Transversality) Conditions 252.3.3 Sufficient Conditions for Optimality 292.3.4 Continuous Optimal Control with Fixed Initial Condition and Unspecified Final Time 302.3.5 A Further Property of the Hamiltonian 352.3.6 Continuous Optimal Control with Inequality Control Constraints— the Pontryagin’s Minimum (Maximum) Principle 362.4 Optimal Control for a Linear Dynamical System 382.4.1 The LQPI Problem—Fixed Final Time 382.5 Optimal Control Applications in Differential Game Theory 402.5.1 Two-Party Game Theoretic Guidance for Linear Dynamical Systems 412.5.2 Three-Party Game Theoretic Guidance for Linear Dynamical Systems 442.6 Extension of the Differential Game Theory to Multi-Party Engagement 502.7 Summary and Conclusions 50References 51Appendix 533 Differential Game Theory Applied to Two-Party Missile Guidance Problem 63Nomenclature 63Abbreviations 643.1 Introduction 643.2 Development of the Engagement Kinematics Model 673.2.1 Relative Engage Kinematics of n Versus m Vehicles 683.2.2 Vector/Matrix Representation 693.3 Optimum Interceptor/Target Guidance for a Two-Party Game 703.3.1 Construction of the Differential Game Performance Index 703.3.2 Weighting Matrices S, R p ,R e 723.3.3 Solution of the Differential Game Guidance Problem 733.4 Solution of the Riccati Differential Equations 753.4.1 Solution of the Matrix Riccati Differential Equations (MRDE) 753.4.2 State Feedback Guidance Gains 763.4.3 Solution of the Vector Riccati Differential Equations (VRDE) 773.4.4 Analytical Solution of the VRDE for the Special Case 783.4.5 Mechanization of the Game Theoretic Guidance 793.5 Extension of the Game Theory to Optimum Guidance 793.6 Relationship with the Proportional Navigation (PN) and the Augmented PN Guidance 813.7 Conclusions 82References 82Appendix 844 Three-Party Differential Game Theory Applied to Missile Guidance Problem 102Nomenclature 102Abbreviations 1034.1 Introduction 1034.2 Engagement Kinematics Model 1044.2.1 Three-Party Engagement Scenario 1054.3 Three-Party Differential Game Problem and Solution 1074.4 Solution of the Riccati Differential Equations 1114.4.1 Solution of the Matrix Riccati Differential Equation (MRDE) 1114.4.2 Solution of the Vector Riccati Differential Equation (VRDE) 1124.4.3 Further Consideration of Performance Index (PI) Weightings 1154.4.4 Game Termination Criteria and Outcomes 1164.5 Discussion and Conclusions 116References 117Appendix 1185 Four Degrees-of-Freedom (DOF) Simulation Model for Missile Guidance and Control Systems 125Nomenclature 125Abbreviations 1265.1 Introduction 1265.2 Development of the Engagement Kinematics Model 1265.2.1 Translational Kinematics for Multi-Vehicle Engagement 1265.2.2 Vector/Matrix Representation 1285.2.3 Rotational Kinematics: Relative Range, Range Rates, Sightline Angles, and Rates 1285.3 Vehicle Navigation Model 1305.3.1 Application of Quaternion to Navigation 1315.4 Vehicle Body Angles and Flight Path Angles 1335.4.1 Computing Body Rates (p I ,q I ,r I) 1345.5 Vehicle Autopilot Dynamics 1355.6 Aerodynamic Considerations 1355.7 Conventional Guidance Laws 1365.7.1 Proportional Navigation (PN) Guidance 1365.7.2 Augmented Proportional Navigation (APN) Guidance 1375.7.3 Optimum Guidance and Game Theory–Based Guidance 1375.8 Overall State Space Model 1385.9 Conclusions 138References 139Appendix 1406 Three-Party Differential Game Missile Guidance Simulation Study 150Nomenclature 150Abbreviations 1506.1 Introduction 1516.2 Engagement Kinematics Model 1516.3 Game Theory Problem and the Solution 1546.4 Discussion of the Simulation Results 1576.4.1 Game Theory Guidance Demonstrator Simulation 1576.4.2 Game Theory Guidance Simulation Including Disturbance Inputs 1606.5 Conclusions 1626.5.1 Useful Future Studies 162References 163Appendix 164Addendum 165Index 189

Farhan A. Faruqi, Defence Science and Technology Organisation, AustraliaProfessor Dr. Farhan A. Faruqi is the Head of the Intelligent Autonomous Systems Research Guidance and Control Group in the Defence Science and Technology Organisation in Australia. He is also an Adjunct Professor at the University of South Australia. His main areas of expertise include autonomous systems navigation; guidance and control; target tracking; and intelligent autonomous systems. He has more than twenty years' experience in the Aerospace and Defence Industry in the UK, USA, and Australia.

• Preface xiAcknowledgments xiiiAbout the Companion Website xv1 Differential Game Theory and Applications to Missile Guidance 1Nomenclature 1Abbreviations 21.1 Introduction 21.1.1 Need for Missile Guidance—Past, Present, and Future 21.2 Game Theoretic Concepts and Definitions 31.3 Game Theory Problem Examples 41.3.1 Prisoner’s Dilemma 41.3.2 The Game of Tic-Tac-Toe 61.4 Game Theory Concepts Generalized 81.4.1 Discrete-Time Game 81.4.2 Continuous-Time Differential Game 91.5 Differential Game Theory Application to Missile Guidance 101.6 Two-Party and Three-Party Pursuit-Evasion Game 111.7 Book Chapter Summaries 111.7.1 A Note on the Terminology Used In the Book 13References 142 Optimum Control and Differential Game Theory 16Nomenclature 16Abbreviations 172.1 Introduction 172.2 Calculus of Optima (Minimum or Maximum) for a Function 182.2.1 On the Existence of the Necessary and Sufficient Conditions for an Optima 182.2.2 Steady State Optimum Control Problem with Equality Constraints Utilizing Lagrange Multipliers 192.2.3 Steady State Optimum Control Problem for a Linear System with Quadratic Cost Function 222.3 Dynamic Optimum Control Problem 232.3.1 Optimal Control with Initial and Terminal Conditions Specified 232.3.2 Boundary (Transversality) Conditions 252.3.3 Sufficient Conditions for Optimality 292.3.4 Continuous Optimal Control with Fixed Initial Condition and Unspecified Final Time 302.3.5 A Further Property of the Hamiltonian 352.3.6 Continuous Optimal Control with Inequality Control Constraints— the Pontryagin’s Minimum (Maximum) Principle 362.4 Optimal Control for a Linear Dynamical System 382.4.1 The LQPI Problem—Fixed Final Time 382.5 Optimal Control Applications in Differential Game Theory 402.5.1 Two-Party Game Theoretic Guidance for Linear Dynamical Systems 412.5.2 Three-Party Game Theoretic Guidance for Linear Dynamical Systems 442.6 Extension of the Differential Game Theory to Multi-Party Engagement 502.7 Summary and Conclusions 50References 51Appendix 533 Differential Game Theory Applied to Two-Party Missile Guidance Problem 63Nomenclature 63Abbreviations 643.1 Introduction 643.2 Development of the Engagement Kinematics Model 673.2.1 Relative Engage Kinematics of n Versus m Vehicles 683.2.2 Vector/Matrix Representation 693.3 Optimum Interceptor/Target Guidance for a Two-Party Game 703.3.1 Construction of the Differential Game Performance Index 703.3.2 Weighting Matrices S, R p ,R e 723.3.3 Solution of the Differential Game Guidance Problem 733.4 Solution of the Riccati Differential Equations 753.4.1 Solution of the Matrix Riccati Differential Equations (MRDE) 753.4.2 State Feedback Guidance Gains 763.4.3 Solution of the Vector Riccati Differential Equations (VRDE) 773.4.4 Analytical Solution of the VRDE for the Special Case 783.4.5 Mechanization of the Game Theoretic Guidance 793.5 Extension of the Game Theory to Optimum Guidance 793.6 Relationship with the Proportional Navigation (PN) and the Augmented PN Guidance 813.7 Conclusions 82References 82Appendix 844 Three-Party Differential Game Theory Applied to Missile Guidance Problem 102Nomenclature 102Abbreviations 1034.1 Introduction 1034.2 Engagement Kinematics Model 1044.2.1 Three-Party Engagement Scenario 1054.3 Three-Party Differential Game Problem and Solution 1074.4 Solution of the Riccati Differential Equations 1114.4.1 Solution of the Matrix Riccati Differential Equation (MRDE) 1114.4.2 Solution of the Vector Riccati Differential Equation (VRDE) 1124.4.3 Further Consideration of Performance Index (PI) Weightings 1154.4.4 Game Termination Criteria and Outcomes 1164.5 Discussion and Conclusions 116References 117Appendix 1185 Four Degrees-of-Freedom (DOF) Simulation Model for Missile Guidance and Control Systems 125Nomenclature 125Abbreviations 1265.1 Introduction 1265.2 Development of the Engagement Kinematics Model 1265.2.1 Translational Kinematics for Multi-Vehicle Engagement 1265.2.2 Vector/Matrix Representation 1285.2.3 Rotational Kinematics: Relative Range, Range Rates, Sightline Angles, and Rates 1285.3 Vehicle Navigation Model 1305.3.1 Application of Quaternion to Navigation 1315.4 Vehicle Body Angles and Flight Path Angles 1335.4.1 Computing Body Rates (p I ,q I ,r I) 1345.5 Vehicle Autopilot Dynamics 1355.6 Aerodynamic Considerations 1355.7 Conventional Guidance Laws 1365.7.1 Proportional Navigation (PN) Guidance 1365.7.2 Augmented Proportional Navigation (APN) Guidance 1375.7.3 Optimum Guidance and Game Theory–Based Guidance 1375.8 Overall State Space Model 1385.9 Conclusions 138References 139Appendix 1406 Three-Party Differential Game Missile Guidance Simulation Study 150Nomenclature 150Abbreviations 1506.1 Introduction 1516.2 Engagement Kinematics Model 1516.3 Game Theory Problem and the Solution 1546.4 Discussion of the Simulation Results 1576.4.1 Game Theory Guidance Demonstrator Simulation 1576.4.2 Game Theory Guidance Simulation Including Disturbance Inputs 1606.5 Conclusions 1626.5.1 Useful Future Studies 162References 163Appendix 164Addendum 165Index 189