Constructive Nonlinear Control

1 Introduction.- 1.1 Passivity, Optimality, and Stability.- 1.1.1 From absolute stability to passivity.- 1.1.2 Passivity as a phase characteristic.- 1.1.3 Optimal control and stability margins.- 1.2 Feedback Passivation.- 1.2.1 Limitations of feedback linearization.- 1.2.2 Feedback passivation and forwarding.- 1.3 Cascade Designs.- 1.3.1 Passivation with composite Lyapunov functions.- 1.3.2 A structural obstacle: peaking.- 1.4 Lyapunov Constructions.- 1.4.1 Construction of the cross-term.- 1.4.2 A benchmark example.- 1.4.3 Adaptive control.- 1.5 Recursive Designs.- 1.5.1 Obstacles to passivation.- 1.5.2 Removing the relative degree obstacle.- 1.5.3 Removing the minimum phase obstacle.- 1.5.4 System structures.- 1.5.5 Approximate asymptotic designs.- 1.6 Book Style and Notation.- 1.6.1 Style.- 1.6.2 Notation and acronyms.- 2 Passivity Concepts as Design Tools.- 2.1 Dissipativity and Passivity.- 2.1.1 Classes of systems.- 2.1.2 Basic concepts.- 2.2 Interconnections of Passive Systems.- 2.2.1 Parallel and feedback interconnections.- 2.2.2 Excess and shortage of passivity.- 2.3 Lyapunov Stability and Passivity.- 2.3.1 Stability and convergence theorems.- 2.3.2 Stability with semidefinite Lyapunov functions.- 2.3.3 Stability of passive systems.- 2.3.4 Stability of feedback interconnections.- 2.3.5 Absolute stability.- 2.3.6 Characterization of afRne dissipative systems.- 2.4 Feedback Passivity.- 2.4.1 Passivity: a tool for stabilization.- 2.4.2 Feedback passive linear systems.- 2.4.3 Feedback passive nonlinear systems.- 2.4.4 Output feedback passivity.- 2.5 Summary.- 2.6 Notes and References.- 3 Stability Margins and Optimality.- 3.1 Stability Margins for Linear Systems.- 3.1.1 Classical gain and phase margins.- 3.1.2 Sector and disk margins.- 3.1.3 Disk margin and output feedback passivity.- 3.2 Input Uncertainties.- 3.2.1 Static and dynamic uncertainties.- 3.2.2 Stability margins for nonlinear feedback systems.- 3.2.3 Stability with fast unmodeled dynamics.- 3.3 Optimality, Stability, and Passivity.- 3.3.1 Optimal stabilizing control.- 3.3.2 Optimality and passivity.- 3.4 Stability Margins of Optimal Systems.- 3.4.1 Disk margin for R(x) =/.- 3.4.2 Sector margin for diagonal R(x)/I.- 3.4.3 Achieving a disk margin by domination.- 3.5 Inverse Optimal Design.- 3.5.1 Inverse optimality.- 3.5.2 Damping control for stable systems.- 3.5.3 CLF for inverse optimal control.- 3.6 Summary.- 3.7 Notes and References.- 4 Cascade Designs.- 4.1 Cascade Systems.- 4.1.1 TORA system.- 4.1.2 Types of cascades.- 4.2 Partial-State Feedback Designs.- 4.2.1 Local stabilization.- 4.2.2 Growth restrictions for global stabilization.- 4.2.3 ISS condition for global stabilization.- 4.2.4 Stability margins: partial-state feedback.- 4.3 Feedback Passivation of Cascades.- 4.4 Designs for the TORA System.- 4.4.1 TORA models.- 4.4.2 Two preliminary designs.- 4.4.3 Controllers with gain margin.- 4.4.4 A redesign to improve performance.- 4.5 Output Peaking: an Obstacle to Global Stabilization.- 4.5.1 The peaking phenomenon.- 4.5.2 Nonpeaking linear systems.- 4.5.3 Peaking and semiglobal stabilization of cascades.- 4.6 Summary.- 4.7 Notes and References.- 5 Construction of Lyapunov functions.- 5.1 Composite Lyapunov functions for cascade systems.- 5.1.1 Benchmark system.- 5.1.2 Cascade structure.- 5.1.3 Composite Lyapunov functions.- 5.2 Lyapunov Construction with a Cross-Term.- 5.2.1 The construction of the cross-term.- 5.2.2 Differentiability of the function *.- 5.2.3 Computing the cross-term.- 5.3 Relaxed Constructions.- 5.3.1 Geometric interpretation of the cross-term.- 5.3.2 Relaxed change of coordinates.- 5.3.3 Lyapunov functions with relaxed cross-term.- 5.4 Stabilization of Augmented Cascades.- 5.4.1 Design of the stabilizing feedback laws.- 5.4.2 A structural condition for GAS and LES.- 5.4.3 Ball-and-beam example.- 5.5 Lyapunov functions for adaptive control.- 5.5.1 Parametric Lyapunov Functions.- 5.5.2 Control with known 6.- 5.5.3 Adaptive Controller Design.- 5.6 Summary.- 5.7 Notes and references.- 6 Recursive designs.- 6.1 Backstepping.- 6.1.1 Introductory example.- 6.1.2 Backstepping procedure.- 6.1.3 Nested high-gain designs.- 6.2 Forwarding.- 6.2.1 Introductory example.- 6.2.2 Forwarding procedure.- 6.2.3 Removing the weak minimum phase obstacle.- 6.2.4 Geometric properties of forwarding.- 6.2.5 Designs with saturation.- 6.2.6 Trade-offs in saturation designs.- 6.3 Interlaced Systems.- 6.3.1 Introductory example.- 6.3.2 Non-affine systems.- 6.3.3 Structural conditions for global stabilization.- 6.4 Summary and Perspectives.- 6.5 Notes and References.- A Basic geometric concepts.- A.1 Relative Degree.- A.2 Normal Form.- A.3 The Zero Dynamics.- A.4 Right-Invertibility.- A.5 Geometric properties.- B Proofs of Theorems 3.18 and 4.35.- B.1 Proof of Theorem 3.18.- B.2 Proof of Theorem 4.35.