Robust Safety-Critical Control

List of Figures xi
List of Table xvi
About the Authors xvii
Preface xix
Notations xxiii
Acronyms xxvii
About the Companion Website xxix

1 Introduction 1
1.1 Characterization of Safety for Dynamical Systems 2
1.2 Robust Safety 9
1.3 Safety-critical Control: A Quadratic Programming Approach 12
1.4 A Practical Scenario of Robust Safety-critical Control 18
1.5 Challenges 28
1.6 Outline of this Book 33
1.7 Notes 37

2 Safety-critical Control Subject to Dynamic Uncertainties: A Nonlinear Small-gain Approach 41
2.1 Problem Formulation 42
2.2 Robust Safety-critical Controller Design 45
2.3 Interaction Between Velocity Tracking and Safety 54
2.4 Small-gain Synthesis for Safety of the Closed-loop System 58
2.5 Simulation and Experiment 66
2.6 Notes 70

3 Safety-critical Control Under Multiple Constraints: A Feasible-set Reshaping Technique 73
3.1 Problem Formulation 74
3.2 Trial of Directly Extending the Safety Margin 77
3.3 A Feasible-set Reshaping Technique for Robust Safety-critical Control 80
3.4 Interaction Between Velocity Tracking and Safety 92
3.5 Small-gain Synthesis for Safety of the Closed-loop System 97
3.6 Simulation and Experiment 101
3.7 Notes 107

4 Safety-critical Control in Cluttered Environments: Set-valued Measurement and Feasible-set Reshaping 111
4.1 Problem Formulation 112
4.2 A Continuous, Reactive Safety-critical Controller 116
4.3 Safety of the Closed-loop System 121
4.4 Special Case: An Integrator-like Mobile Robot 124
4.5 Simulation and Experiment 125
4.6 Notes 137

5 Safety-critical Control of Multi-agent Systems: Feasible-set Reshaping and Nonlinear Small-gain Synthesis 139
5.1 Problem Formulation 140
5.2 Trials with Standard Designs 143
5.3 Feasible-set Reshaping and Controller Design 146
5.4 Properties of the Proposed Design and Proofs 149
5.5 Small-gain Analysis for Safety of the Multi-agent System 162
5.6 Tuning the Safety-critical Controllers 166
5.7 Numerical Simulation and Experiment 168
5.8 Notes 171

6 Safety-critical Control of Eulerâ"Lagrange Systems: Incorporating Barrier and Energy Functions 173
6.1 Problem Formulation 174
6.2 Outer Loop: Safety-oriented Controller Design 178
6.3 Inner Loop: Velocity-tracking Controller Design 190
6.4 Safety Verification: Incorporating Barrier and Energy Functions 192
6.5 Numerical Simulation and Experiment 198
6.6 Notes 204

7 Safety-critical Control of Cascade Systems: Towards a Constructive Control Framework 207
7.1 Problem Formulation 208
7.2 Design Ingredient: Plants with Relative-degree-one 213
7.3 Design Ingredient: Refined Feasible-set Reshaping 218
7.4 Constructive Design for Plants in the Cascade Form 225
7.5 Experiment: VTOL Taking Off in a Narrow Space 234
7.6 Notes 240

A Mathematical Preliminaries 243
A.1 Real Vectors and Matrices 243
A.2 Basis and Positive Basis 245
A.3 Sets and Convexity 249
A.4 Continuity, Differentiability, and Convexity of Functions 251
A.5 Comparison Functions 255
A.6 Nonsmooth Analysis 256
A.7 Set Invariance 260

B Quadratic Programming 263
B.1 Quadratic Optimization Problems 263
B.2 Lipschitz Continuity of QP Solutions 264

C Lyapunov Stability, Input-to-state Stability, and the Nonlinear Small-gain Theorem 269
C.1 Lyapunov Stability Theory 269
C.2 Input-to-state Stability 273
C.3 The Nonlinear Small-gain Theorem 277

References 285
Index 299