O'Reilly logo

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

Nonlinear H-Infinity Control, Hamiltonian Systems and Hamilton-Jacobi Equations

Book Description

A comprehensive overview of nonlinear H∞ control theory for both continuous-time and discrete-time systems, Nonlinear H∞-Control, Hamiltonian Systems and Hamilton-Jacobi Equations covers topics as diverse as singular nonlinear H∞-control, nonlinear H -filtering, mixed H2/ H∞-nonlinear control and filtering, nonlinear H∞-almost-disturbance-decoupling,

and algorithms for solving the ubiquitous Hamilton-Jacobi-Isaacs equations. The link between the subject and analytical mechanics as well as the theory of partial differential equations is also elegantly summarized in a single chapter.

Recent progress in developing computational schemes for solving the Hamilton-Jacobi equation (HJE) has facilitated the application of Hamilton-Jacobi theory in both mechanics and control. As there is currently no efficient systematic analytical or numerical approach for solving them, the biggest bottle-neck to the practical application of the nonlinear equivalent of the H∞-control theory has been the difficulty in solving the Hamilton-Jacobi-Isaacs partial differential-equations (or inequalities). In light of this challenge, the author hopes to inspire continuing research and discussion on this topic via examples and simulations, as well as helpful notes and a rich bibliography.

Nonlinear H∞-Control, Hamiltonian Systems and Hamilton-Jacobi Equations was written for practicing professionals, educators, researchers and graduate students in electrical, computer, mechanical, aeronautical, chemical, instrumentation, industrial and systems engineering, as well as applied mathematics, economics and management.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright Page
  5. Dedication
  6. Table of Contents
  7. 1 Introduction
    1. 1.1 Historical Perspective on Nonlinear H∞-Control
    2. 1.2 General Set-Up for Nonlinear H∞-Control Problems
      1. 1.2.1 Mixed H2-Control Problem
      2. 1.2.2 Robust H∞-Control Problem
      3. 1.2.3 Nonlinear H∞-Filtering
      4. 1.2.4 Organization of the Book
    3. 1.3 Notations and Preliminaries
      1. 1.3.1 Notation
      2. 1.3.2 Stability Concepts
    4. 1.4 Notes and Bibliography
  8. 2 Basics of Differential Games
    1. 2.1 Dynamic Programming Principle
    2. 2.2 Discrete-Time Nonzero-Sum Dynamic Games
      1. 2.2.1 Linear-Quadratic Discrete-Time Dynamic Games
    3. 2.3 Continuous-Time Nonzero-Sum Dynamic Games
      1. 2.3.1 Linear-Quadratic Continuous-Time Dynamic Games
    4. 2.4 Notes and Bibliography
  9. 3 Theory of Dissipative Systems
    1. 3.1 Dissipativity of Continuous-Time Nonlinear Systems
      1. 3.1.1 Stability of Continuous-Time Dissipative Systems
      2. 3.1.2 Stability of Continuous-Time Dissipative Feedback-Systems
    2. 3.2 H∞-Gain Analysis for Continuous-Time Dissipative Systems
    3. 3.3 Continuous-Time Passive Systems
    4. 3.4 Feedback-Equivalence to a Passive Continuous-Time Nonlinear System
    5. 3.5 Dissipativity and Passive Properties of Discrete-Time Nonlinear Systems
    6. 3.6 ℓ2-Gain Analysis for Discrete-Time Dissipative Systems
    7. 3.7 Feedback-Equivalence to a Discrete-Time Lossless Nonlinear System
    8. 3.8 Notes and Bibliography
  10. 4 Hamiltonian Mechanics and Hamilton-Jacobi Theory
    1. 4.1 The Hamiltonian Formulation of Mechanics
    2. 4.2 Canonical Transformation
      1. 4.2.1 The Transformation Generating Function
      2. 4.2.2 The Hamilton-Jacobi Equation (HJE)
      3. 4.2.3 Time-Independent Hamilton-Jacobi Equation and Separation of Variables
    3. 4.3 The Theory of Nonlinear Lattices
      1. 4.3.1 The G2 Periodic Toda Lattice
    4. 4.4 The Method of Characteristics for First-Order Partial-Differential Equations
      1. 4.4.1 Characteristics for Quasi-Linear Equations
      2. 4.4.2 Characteristics for the General First-Order Equation
      3. 4.4.3 Characteristics for the Hamilton-Jacobi Equation
    5. 4.5 Legendre Transform and Hopf-Lax Formula
      1. 4.5.1 Viscosity Solutions of the HJE
    6. 4.6 Notes and Bibliography
  11. 5 State-Feedback Nonlinear H∞-Control for Continuous-Time Systems
    1. 5.1 State-Feedback H∞-Control for Affine Nonlinear Systems
      1. 5.1.1 Dissipative Analysis
      2. 5.1.2 Controller Parametrization
    2. 5.2 State-Feedback Nonlinear H∞ Tracking Control
    3. 5.3 Robust Nonlinear H∞ State-Feedback Control
    4. 5.4 State-Feedback H∞-Control for Time-Varying Affine Nonlinear Systems
    5. 5.5 State-Feedback H∞-Control for State-Delayed Affine Nonlinear Systems
    6. 5.6 State-Feedback H∞-Control for a General Class of Nonlinear Systems
    7. 5.7 Nonlinear H∞ Almost-Disturbance-Decoupling
    8. 5.8 Notes and Bibliography
  12. 6 Output-Feedback Nonlinear H∞-Control for Continuous-Time Systems
    1. 6.1 Output Measurement-Feedback H∞-Control for Affine Nonlinear Systems
      1. 6.1.1 Controller Parameterization
    2. 6.2 Output Measurement-Feedback Nonlinear H∞ Tracking Control
    3. 6.3 Robust Output Measurement-Feedback Nonlinear H∞-Control
      1. 6.3.1 Reliable Robust Output-Feedback Nonlinear H∞-Control
    4. 6.4 Output Measurement-Feedback H∞-Control for a General Class of Nonlinear Systems
      1. 6.4.1 Controller Parametrization
    5. 6.5 Static Output-Feedback Control for Affine Nonlinear Systems
      1. 6.5.1 Static Output-Feedback Control with Disturbance-Attenuation
    6. 6.6 Notes and Bibliography
  13. 7 Discrete-Time Nonlinear H∞-Control
    1. 7.1 Full-Information H∞-Control for Affine Nonlinear Discrete-Time Systems
      1. 7.1.1 State-Feedback H∞-Control for Affine Nonlinear Discrete-Time Systems
      2. 7.1.2 Controller Parametrization
    2. 7.2 Output Measurement-Feedback Nonlinear H∞-Control for Affine Discrete-Time Systems
    3. 7.3 Extensions to a General Class of Discrete-Time Nonlinear Systems
      1. 7.3.1 Full-Information H∞-Control for a General Class of Discrete-Time Nonlinear Systems
      2. 7.3.2 Output Measurement-Feedback H∞-Control for a General Class of Discrete-Time Nonlinear Systems
    4. 7.4 Approximate Approach to the Discrete-Time Nonlinear H∞-Control Problem
      1. 7.4.1 An Approximate Approach to the Discrete-Time State-Feedback Problem
      2. 7.4.2 An Approximate Approach to the Discrete-Time Output Measurement-Feedback Problem
    5. 7.5 Notes and Bibliography
  14. 8 Nonlinear H∞-Filtering
    1. 8.1 Continuous-Time Nonlinear H∞-Filtering
      1. 8.1.1 Infinite-Horizon Continuous-Time Nonlinear H∞-Filtering
      2. 8.1.2 The Linearized Filter
    2. 8.2 Continuous-Time Robust Nonlinear H∞-Filtering
    3. 8.3 Certainty-Equivalent Filters(CEFs)
      1. 8.3.1 2-DOFCertainty-Equivalent Filters
    4. 8.4 Discrete-Time Nonlinear H∞-Filtering
      1. 8.4.1 Infinite-Horizon Discrete-Time Nonlinear H∞-Filtering
      2. 8.4.2 Approximate and Explicit Solution
    5. 8.5 Discrete-Time Certainty-Equivalent Filters(CEFs)
      1. 8.5.1 2-DOF Proportional-Derivative (PD) CEFs
      2. 8.5.2 Approximate and Explicit Solution
    6. 8.6 Robust Discrete-Time Nonlinear H∞-Filtering
    7. 8.7 Notes and Bibliography
  15. 9 Singular Nonlinear H∞-Control and H∞-Control for Singularly-Perturbed Nonlinear Systems
    1. 9.1 Singular Nonlinear H∞-Control with State-Feedback
      1. 9.1.1 State-Feedback Singular Nonlinear H∞-Control Using High-Gain Feedback
    2. 9.2 Output Measurement-Feedback Singular Nonlinear H∞-Control
    3. 9.3 Singular Nonlinear H∞-Control with Static Output-Feedback
    4. 9.4 Singular Nonlinear H∞-Control for Cascaded Nonlinear Systems
    5. 9.5 H∞-Control for Singularly-Perturbed Nonlinear Systems
    6. 9.6 Notes and Bibliography
  16. 10 H∞-Filtering for Singularly-Perturbed Nonlinear Systems
    1. 10.1 Problem Definition and Preliminaries
    2. 10.2 Decomposition Filters
    3. 10.3 Aggregate Filters
    4. 10.4 Examples
    5. 10.5 Notes and Bibliography
  17. 11 Mixed H∞2/H∞∞ Nonlinear Control
    1. 11.1 Continuous-Time Mixed H∞2/H∞∞ Nonlinear Control
      1. 11.1.1 The Infinite-Horizon Problem
      2. 11.1.2 Extension to a General Class of Nonlinear Systems
    2. 11.2 Discrete-Time Mixed H∞2/H∞∞ Nonlinear Control
      1. 11.2.1 The Infinite-Horizon Problem
    3. 11.3 Extension to a General Class of Discrete-Time Nonlinear Systems
    4. 11.4 Notes and Bibliography
  18. 12 Mixed H∞2/H∞∞ Nonlinear Filtering
    1. 12.1 Continuous-Time Mixed H∞2/H∞∞ Nonlinear Filtering
      1. 12.1.1 Solution to the Finite-Horizon Mixed H∞2/H∞∞ Nonlinear Filtering Problem
      2. 12.1.2 Solution to the Infinite-Horizon Mixed H∞2/H∞∞ Nonlinear Filtering
      3. 12.1.3 Certainty-Equivalent Filters(CEFs)
    2. 12.2 Discrete-Time Mixed H∞2/H∞∞ Nonlinear Filtering
      1. 12.2.1 Solution to the Finite-Horizon Discrete-Time Mixed H∞2/H∞∞ Nonlinear Filtering Problem
      2. 12.2.2 Solution to the Infinite-Horizon Discrete-Time Mixed H∞2/H∞∞ Non-linear Filtering Problem
      3. 12.2.3 Approximate and Explicit Solution to the Infinite-Horizon Discrete-Time Mixed H∞2/H∞∞ Nonlinear Filtering Problem
      4. 12.2.4 Discrete-Time Certainty-Equivalent Filters(CEFs)
    3. 12.3 Example
    4. 12.4 Notes and Bibliography
  19. 13 Solving the Hamilton-Jacobi Equation
    1. 13.1 Review of Some Approaches for Solving the HJBE/HJIE
      1. 13.1.1 Solving the HJIE/HJBE Using Polynomial Expansion and Basis Functions
    2. 13.2 A Factorization Approach for Solving the HJIE
      1. 13.2.1 Worked Examples
    3. 13.3 Solving the Hamilton-Jacobi Equation for Mechanical Systems and Application to the Toda Lattice
      1. 13.3.1 Solving the Hamilton-Jacobi Equation
      2. 13.3.2 Solving the Hamilton-Jacobi Equation for the A2-Toda System
    4. 13.4 Notes and Bibliography
  20. A Proof of Theorem 5.7.1
  21. B Proof of Theorem 8.2.2
  22. Bibliography
  23. Index