Aircraft Control Allocation

Wayne Durham, Virginia Polytechnic Institute and State University, USA.Kenneth A. Bordignon, Embry-Riddle Aeronautical University, USA.Roger Beck, Dynamic Concepts, Inc., USA.

• Dedication xiiiSeries Preface xvGlossary xviiAbout the Companion Website xxiii1 Introduction 11.1 Redundant Control Effectors 11.2 Overview 3References 52 Aircraft Control 62.1 Flight Dynamics 62.1.1 Equations of Motion 62.1.2 Linearized Equations of Motion 102.2 Control 122.2.1 General 122.2.2 Aircraft Control Effectors 132.2.3 Aircraft Control Inceptors 172.3 Afterword 18References 193 Control Laws 203.1 Flying Qualities 203.1.1 Requirements 213.1.2 Control Law Design to Satisfy Flying Qualities Requirements 213.2 Dynamic-inversion Control Laws 213.2.1 Basics 213.2.2 Types of Equations 223.2.3 The Controlled Equations 233.2.4 The Kinematic and Complementary Equations 253.3 Model-following Control Laws 273.4 ‘Conventional’ Control Laws 273.5 Afterword 28References 294 The Problem 304.1 Control Effectiveness 304.2 Constraints 314.3 Control Allocation 314.3.1 The Control Allocation Problem 324.4 Afterword 32References 335 The Geometry of Control Allocation 345.1 Admissible Controls 345.1.1 General 345.1.2 Objects 345.1.3 Intersection and Union 375.1.4 Convex Hull 395.2 Attainable Moments 395.3 The Two-moment Problem 435.3.1 Area Calculations 485.4 The Three-moment Problem 495.4.1 Determination of Φ3 495.4.2 Volume Calculations 565.5 Significance of the Maximum Set 585.5.1 As a Standard of Comparison of Different Methods 595.5.2 Maneuver Requirements 605.5.3 Control Failure Reconfiguration 625.6 Afterword 62References 646 Solutions 656.1 On-line vs. Off-line Solutions 656.1.1 On-line Solutions 656.1.2 Off-line Solutions 656.2 Optimal vs. Non-optimal Solutions 666.2.1 Maximum Capabilities 666.2.2 Maximum Volume 666.2.3 Nearest to Preferred 666.2.4 Unattainable Moments 676.3 Preferred Solutions 686.4 Ganging 686.5 Generalized Inverses 706.5.1 The General Case, and the Significance of P 2 706.5.2 Tailored Generalized Inverses 736.5.3 ‘Best’ Generalized Inverse 746.5.4 Pseudo-inverses 756.5.5 Methods that Incorporate Generalized Inverses 776.6 Direct Allocation 806.6.1 The Direct Method for the Two-moment Problem 816.6.2 The Direct Method for the Three-moment Problem 826.7 Edge and Facet Searching 846.7.1 Two-dimensional Edge Searching 856.7.2 Three-dimensional Facet Searching 886.8 Banks’ Method 906.8.1 Finding the Original Three Vertices 926.8.2 Determining a New Vertex 936.8.3 Replacing an Old Vertex 936.8.4 Terminating the Algorithm 956.9 Linear Programming 956.9.1 Casting Control Allocation as a Linear Program 966.9.2 Simplex 996.10 Moments Attainable by Various Solution Methods 1006.10.1 General Case (Three-moment Problem) 1016.10.2 Generalized Inverses (Two- and Three-moment Problems) 1026.11 Examples 1116.11.1 Generalized Inverses 1116.11.2 Direct Allocation 1196.11.3 Edge and Facet Searching 1226.11.4 Banks’ Method 1286.11.5 Linear Programming 1326.11.6 Convex-hull Volume Calculations 1346.12 Afterword 137References 1377 Frame-wise Control Allocation 1397.1 General 1397.2 Path Dependency 1417.2.1 Examples of Path Dependency 1427.3 Global vs. Local Control Effectiveness 1477.4 Restoring 1497.4.1 The Augmented B matrix 1507.4.2 Implementation 1527.4.3 Chattering 1537.4.4 Minimum-norm Restoring 1548 Control Allocation and Flight Control System Design 1618.1 Dynamic-inversion Desired Accelerations 1618.1.1 The Desired Acceleration: ẋdes 1618.1.2 Command and Regulator Examples 1638.2 The Maximum Set and Control Law Design 1688.2.1 In the Design Process 1688.2.2 In a Mature Design 1728.2.3 Non-optimal Example 174References 1779 Applications 1789.1 Lessons Learned from the Design of the X-35 Flight Control System 1789.1.1 Theory vs. Practice 1789.2 Uses of Redundancy 1799.2.1 Preferred Solutions 1799.2.2 Resolving Path-dependency Issues 1809.3 Design Constraints 1809.3.1 Axis Prioritization 1809.3.2 Structural Loads 1829.3.3 Effector Bandwidth 1839.3.4 Gain Limiting and Stability Margins 1849.4 Failure Accommodation 184References 185A Linear Programming 186A.1 Control Allocation as a Linear Program 187A.1.1 Optimality for Attainable Commands 188A.1.2 Optimality for Unattainable Commands 188A.2 Standard Forms for Linear Programming Problems 193A.2.1 Dealing with Negative Unknowns 194A.2.2 Dealing with Inequality Constraints 195A.2.3 Writing a Program for Control Allocation in Standard Form 197A.2.4 Revised Standard Form with Upper Bound 199A.3 Properties of Linear Program Solutions 201A.3.1 Basic Solutions 202A.3.2 Degenerate Basic Solutions 203A.3.3 Basic Feasible Solutions 204A.4 Allocating Feasible Commands 204A.4.1 Minimizing Error to a Preferred Solution 205A.4.2 Minimizing Maximum Errors 209A.4.3 Optimizing Linear Secondary Objectives 212A.5 Building a Control Allocator for Feasible and Infeasible Solutions 213A.5.1 Dual Branch 214A.5.2 Single-branch or Mixed Optimization 215A.5.3 Reduced Program Size without Secondary Optimization 218A.6 Solvers 219A.6.1 Preprocessing 220A.6.2 Solution Algorithms 221A.6.3 Simplex Method 222A.6.4 Initialization of the Simplex Algorithm 232A.7 Afterword 234References 235B Flight Simulation 237B.1 Introduction 237B.2 Modifications 237B.2.1 Three of the top-level blocks have been left almost completely unaltered 237B.2.2 Minor modifications consist of the new Pilot and Sensors blocks 238B.3 NDI_CLAW 238B.3.1 NDI_CLAW/Rate Transition 238B.3.2 NDI_CLAW/PILOT_Mod 238B.3.3 NDI_CLAW/INPUT 239B.3.4 NDI_CLAW/MissionManager 239B.3.5 NDI_CLAW/DynamicInversionControl 240References 246C Annotated Bibliography 247References 247Index 277

"The book is a vital reference for researchers and practitioners working in aircraft control, as well as graduate students in aerospace engineering" Expofairs, Sept 2017