Mathematical Control Theory

1 Path Integrals and Stability.- 1.1 Introduction.- 1.2 Path Independence.- 1.3 Positivity of Quadratic Differential Forms.- 1.4 Lyapunov Theory for High-Order Differential Equations.- 1.5 The Bezoutian.- 1.6 Dissipative Systems.- 1.7 Stability of Nonautonomous Systems.- 1.8 Conclusions.- 1.9 Appendixes.- 2 The Estimation Algebra of Nonlinear Filtering Systems.- 2.1 Introduction.- 2.2 The Filtering Model and Background.- 2.3 Starting from the Beginning.- 2.4 Early Results on the Homomorphism Principle.- 2.5 Automorphisms that Preserve Estimation Algebras.- 2.6 BM Estimation Algebra.- 2.7 Structure of Exact Estimation Algebra.- 2.8 Structure of BM Estimation Algebras.- 2.9 Connection with Metaplectic Groups.- 2.10 Wei-Norman Representation of Filters.- 2.11 Perturbation Algebra and Estimation Algebra.- 2.12 Lie-Algebraic Classification of Maximal Rank Estimation Algebras.- 2.13 Complete Characterization of Finite-Dimensional Estimation Algebras.- 2.14 Estimation Algebra of the Identification Problem.- 2.15 Solutions to the Riccati P.D.E.- 2.16 Filters with Non-Gaussian Initial Conditions.- 2.17 Back to the Beginning.- 2.18 Acknowledgement.- 3 Feedback Linearization.- 3.1 Introduction.- 3.2 Linearization of a Smooth Vector Field.- 3.3 Linearization of a Smooth Control System by Change-of-State Coordinates.- 3.4 Feedback Linearization.- 3.5 Input-Output Linearization.- 3.6 Approximate Feedback Linearization.- 3.7 Normal Forms of Control Systems.- 3.8 Observers with Linearizable Error Dynamics.- 3.9 Nonlinear Regulation and Model Matching.- 3.10 Backstepping.- 3.11 Feedback Linearization and System Inversion.- 3.12 Conclusion.- 4 On the Global Analysis of Linear Systems.- 4.1 Introduction.- 4.2 The Geometry of Rational Functions.- 4.3 Group Actions and the Geometry of Linear Systems.- 4.4 The Geometry of Inverse Eigenvalue Problems.- 4.5 Nonlinear Optimization on Spaces of Systems.- 5 Geometry and Optimal Control.- 5.1 Introduction.- 5.2 From Queen Dido to the Maximum Principle.- 5.3 Invariance, Covariance, and Lie Brackets.- 5.4 The Maximum Principle.- 5.5 The Maximum Principle as a Necessary Condition for Set Separation.- 5.6 Weakly Approximating Cones and Transversality.- 5.7 A Streamlined Version of the Classical Maximum Principle.- 5.8 Clarke’s Nonsmooth Version and the ?ojasiewicz Improvement.- 5.9 Multidifferentials, Flows, and a General Version of the Maximum Principle.- 5.10 Three Ways to Make the Maximum Principle Intrinsic on Manifolds.- 5.11 Conclusion.- 6 Languages, Behaviors, Hybrid Architectures, and Motion Control.- 6.1 Introduction.- 6.2 MDLe: A Language for Motion Control.- 6.3 Hybrid Architecture.- 6.4 Application of MDLe to Path Planning with Nonholonomic Robots.- 6.5 PNMR: Path Planner for Nonholonomic Mobile Robots.- 6.6 Conclusions.- 7 Optimal Control, Geometry, and Mechanics.- 7.1 Introduction.- 7.2 Variational Problems with Constraints and Optimal Control.- 7.3 Invariant Optimal Problems on Lie Groups.- 7.4 Sub-Riemannian Spheres—The Contact Case.- 7.5 Sub-Riemannian Systems on Lie Groups.- 7.6 Heavy Top and the Elastic Problem.- 7.7 Conclusion.- 8 Optimal Control, Optimization, and Analytical Mechanics.- 8.1 Introduction.- 8.2 Modeling Variational Problems in Mechanics and Control.- 8.3 Optimization.- 8.4 Optimal Control Problems and Integrable Systems.- 9 The Geometry of Controlled Mechanical Systems.- 9.1 Introduction.- 9.2 Second-Order Generalized Control Systems.- 9.3 Flat Systems and Systems with Flat Inputs.- 9.4 Averaging Lagrangian and Hamiltonian Systems with Oscillatory Inputs.- 9.5 Stability and Flatness in Mechanical Systems with Oscillatory Inputs.- 9.6 Concluding Remarks.