Theory of Chattering Control with applications to Astronautics, Robotics, Economics, and Engineering

Theory of Chattering Control with applications to Astronautics, Robotics, Economics, and Engineering / by Michail I. Zelikin, Vladimir F. Borisov.

The common experience in solving control problems shows that optimal control as a function of time proves to be piecewise analytic, having a finite number of jumps (called switches) on any finite-time interval. Meanwhile there exists an old example proposed by A.T. Fuller [1961) in which optimal con...

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Bibliographic Details
Main Authors: Zelikin, Michail I. (Author), Borisov, Vladimir F. (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser, 1994.
Edition:1st ed. 1994.
Series:Systems & Control: Foundations & Applications,
Springer eBook Collection.
Subjects:
Calculus of variations.
System theory.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 1.Introduction
  • 1.1 The Subject of the Book
  • 1.2 Hamiltonian Systems and Singular Extremals
  • 1.3 The Semi-Canonical Form of Hamiltonian Systems
  • 1.4 Integral Varieties with Chattering Arcs
  • 1.5 An Example of Designing a Lagrangian Manifold
  • 2.Fuller’s Problem
  • 2.1 Statement of Puller’s Problem
  • 2.2 Chattering Arcs
  • 2.3 Untwisted Chattering Arcs
  • 2.4 The Geometry of Trajectories of Hamiltonian Systems
  • 3.Second Order Singular Extremals and Chattering
  • 3.1 Preliminaries
  • 3.2 Manifolds with Second Order Singular Trajectories
  • 3.3 The Connection with Fuller’s Problem
  • 3.4 Resolution of the Singularity of the Poincaré Mapping
  • 3.5 The Connection with the Problem of C. Marchal
  • 3.6 Fixed Points of the Quotient Mapping
  • 3.7 The Hyperbolic Structure of the Quotient Mapping
  • 3.8 Non-Degeneracy of the Fixed Point
  • 3.9 Bundles with Chattering Arcs
  • 3.10 Lagrangian Manifolds
  • 3.11 Synthesis with Locally Optimal Chattering Arcs
  • 3.12 Regular Projection of Chattering Varieties
  • 4.The Ubiquity of Fuller’s Phenomenon
  • 4.1 Kupka’s Results
  • 4.2 Codimension of the Set of Fuller Points
  • 4.3 Structural Stability of the Optimal Synthesis in the Two-Dimensional Fuller Problem
  • 5.Higher Order Singular Extremals
  • 5.1 Conjectures Concerning Higher Order Singular Modes
  • 5.2 Problems with Linear Constraints
  • 5.3 Problems with Symmetries
  • 5.4 Bi-Constant Ratio Solutions of Fuller’s Problems
  • 5.5 Optimality of b.c.r. Solutions
  • 5.6 Numerical Verification of the Conjecture on the Number of Cycles in the Orbit Space
  • 5.7 Three-Dimensional Puller Problems
  • 6.Applications
  • 6.1 Fibrations in Three-Dimensional Space
  • 6.2 Stabilization of a Rigid Body
  • 6.3 The Resource Allocation Problem
  • 6.4 Control of Two Interdependent Oscillators
  • 6.5 Lowden’s Problem
  • 6.6 Robot Control
  • 7.Multidimensional Control and Chattering Modes
  • 7.1 Multidimensional Problems with a Polyhedral Indicatrix
  • 7.2 Multidimensional Problems with a Smooth Indicatrix
  • Epilogue
  • List of Figures.

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