Constructive Nonlinear Control

Constructive Nonlinear Control presents a broad repertoire of constructive nonlinear designs not available in other works by widening the class of systems and design tools. Several streams of nonlinear control theory are merged and directed towards a constructive solution of the feedback stabilization problem. Analysis, geometric and asymptotic concepts are assembled as design tools for a wide variety of nonlinear phenomena and structures. Geometry serves as a guide for the construction of design procedures whilst analysis provides the robustness which geometry lacks. New recursive designs remove earlier restrictions on feedback passivation. Recursive Lyapunov designs for feedback, feedforward and interlaced structures result in feedback systems with optimality properties and stability margins. The design-oriented approach will make this work a valuable tool for all those who have an interest in control theory.

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.