Contents

1    Introduction

1.1    Historical Perspective on Nonlinear ${\mathcal{H}}_{\infty }$-Control

1.2    General Set-Up for Nonlinear ${\mathcal{H}}_{\infty }$-Control Problems

1.2.1    Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\infty }$-Control Problem

1.2.2    Robust ${\mathcal{H}}_{\infty }$-Control Problem

1.2.3    Nonlinear ${\mathcal{H}}_{\infty }$-Filtering

1.2.4    Organization of the Book

1.3    Notations and Preliminaries

1.3.1    Notation

1.3.2    Stability Concepts

1.4    Notes and Bibliography

2    Basics of Differential Games

2.1    Dynamic Programming Principle

2.2    Discrete-Time Nonzero-Sum Dynamic Games

2.2.1    Linear-Quadratic Discrete-Time Dynamic Games

2.3    Continuous-Time Nonzero-Sum Dynamic Games

2.3.1    Linear-Quadratic Continuous-Time Dynamic Games

2.4    Notes and Bibliography

3    Theory of Dissipative Systems

3.1    Dissipativity of Continuous-Time Nonlinear Systems

3.1.1    Stability of Continuous-Time Dissipative Systems

3.1.2    Stability of Continuous-Time Dissipative Feedback-Systems

3.2    ${\mathcal{L}}_2$-Gain Analysis for Continuous-Time Dissipative Systems

3.3    Continuous-Time Passive Systems

3.4    Feedback-Equivalence to a Passive Continuous-Time Nonlinear System

3.5    Dissipativity and Passive Properties of Discrete-Time Nonlinear Systems

3.6    ℓ₂-Gain Analysis for Discrete-Time Dissipative Systems

3.7    Feedback-Equivalence to a Discrete-Time Lossless Nonlinear System

3.8    Notes and Bibliography

4    Hamiltonian Mechanics and Hamilton-Jacobi Theory

4.1    The Hamiltonian Formulation of Mechanics

4.2    Canonical Transformation

4.2.1    The Transformation Generating Function

4.2.2    The Hamilton-Jacobi Equation (HJE)

4.2.3    Time-Independent Hamilton-Jacobi Equation and Separation of Variables

4.3    The Theory of Nonlinear Lattices

4.3.1    The G₂ Periodic Toda Lattice

4.4    The Method of Characteristics for First-Order Partial-Differential Equations

4.4.1    Characteristics for Quasi-Linear Equations

4.4.2    Characteristics for the General First-Order Equation

4.4.3    Characteristics for the Hamilton-Jacobi Equation

4.5    Legendre Transform and Hopf-Lax Formula

4.5.1    Viscosity Solutions of the HJE

4.6    Notes and Bibliography

5    State-Feedback Nonlinear ${\mathcal{H}}_{\infty }$-Control for Continuous-Time Systems

5.1    State-Feedback ${\mathcal{H}}_{\infty }$-Control for Affine Nonlinear Systems

5.1.1    Dissipative Analysis

5.1.2    Controller Parametrization

5.2    State-Feedback Nonlinear ${\mathcal{H}}_{\infty }$ Tracking Control

5.3    Robust Nonlinear ${\mathcal{H}}_{\infty }$ State-Feedback Control

5.4    State-Feedback ${\mathcal{H}}_{\infty }$-Control for Time-Varying Affine Nonlinear Systems

5.5    State-Feedback ${\mathcal{H}}_{\infty }$-Control for State-Delayed Affine Nonlinear Systems

5.6    State-Feedback ${\mathcal{H}}_{\infty }$-Control for a General Class of Nonlinear Systems

5.7    Nonlinear ${\mathcal{H}}_{\infty }$ Almost-Disturbance-Decoupling

5.8    Notes and Bibliography

6    Output-Feedback Nonlinear ${\mathcal{H}}_{\infty }$-Control for Continuous-Time Systems

6.1    Output Measurement-Feedback ${\mathcal{H}}_{\infty }$-Control for Affine Nonlinear Systems

6.1.1    Controller Parameterization

6.2    Output Measurement-Feedback Nonlinear ${\mathcal{H}}_{\infty }$ Tracking Control

6.3    Robust Output Measurement-Feedback Nonlinear ${\mathcal{H}}_{\infty }$-Control

6.3.1    Reliable Robust Output-Feedback Nonlinear ${\mathcal{H}}_{\infty }$-Control

6.4    Output Measurement-Feedback ${\mathcal{H}}_{\infty }$-Control for a General Class of Nonlinear Systems

6.4.1    Controller Parametrization

6.5    Static Output-Feedback Control for Affine Nonlinear Systems

6.5.1    Static Output-Feedback Control with Disturbance-Attenuation

6.6    Notes and Bibliography

7    Discrete-Time Nonlinear ${\mathcal{H}}_{\infty }$-Control

7.1    Full-Information ${\mathcal{H}}_{\infty }$-Control for Affine Nonlinear Discrete-Time Systems

7.1.1    State-Feedback ${\mathcal{H}}_{\infty }$-Control for Affine Nonlinear Discrete-Time Systems

7.1.2    Controller Parametrization

7.2    Output Measurement-Feedback Nonlinear ${\mathcal{H}}_{\infty }$-Control for Affine Discrete-Time Systems

7.3    Extensions to a General Class of Discrete-Time Nonlinear Systems

7.3.1    Full-Information ${\mathcal{H}}_{\infty }$-Control for a General Class of Discrete-Time Nonlinear Systems

7.3.2    Output Measurement-Feedback ${\mathcal{H}}_{\infty }$-Control for a General Class of Discrete-Time Nonlinear Systems

7.4    Approximate Approach to the Discrete-Time Nonlinear ${\mathcal{H}}_{\infty }$-Control Problem

7.4.1    An Approximate Approach to the Discrete-Time State-Feedback Problem

7.4.2    An Approximate Approach to the Discrete-Time Output Measurement-Feedback Problem

7.5    Notes and Bibliography

8    Nonlinear ${\mathcal{H}}_{\infty }$-Filtering

8.1    Continuous-Time Nonlinear ${\mathcal{H}}_{\infty }$-Filtering

8.1.1    Infinite-Horizon Continuous-Time Nonlinear ${\mathcal{H}}_{\infty }$-Filtering

8.1.2    The Linearized Filter

8.2    Continuous-Time Robust Nonlinear ${\mathcal{H}}_{\infty }$-Filtering

8.3    Certainty-Equivalent Filters(CEFs)

8.3.1    2-DOFCertainty-Equivalent Filters

8.4    Discrete-Time Nonlinear ${\mathcal{H}}_{\infty }$-Filtering

8.4.1    Infinite-Horizon Discrete-Time Nonlinear ${\mathcal{H}}_{\infty }$-Filtering

8.4.2    Approximate and Explicit Solution

8.5    Discrete-Time Certainty-Equivalent Filters(CEFs)

8.5.1    2-DOF Proportional-Derivative (PD) CEFs

8.5.2    Approximate and Explicit Solution

8.6    Robust Discrete-Time Nonlinear ${\mathcal{H}}_{\infty }$-Filtering

8.7    Notes and Bibliography

9    Singular Nonlinear ${\mathcal{H}}_{\infty }$-Control and ${\mathcal{H}}_{\infty }$-Control for Singularly-Perturbed Nonlinear Systems

9.1    Singular Nonlinear ${\mathcal{H}}_{\infty }$-Control with State-Feedback

9.1.1    State-Feedback Singular Nonlinear ${\mathcal{H}}_{\infty }$-Control Using High-Gain Feedback

9.2    Output Measurement-Feedback Singular Nonlinear ${\mathcal{H}}_{\infty }$-Control

9.3    Singular Nonlinear ${\mathcal{H}}_{\infty }$-Control with Static Output-Feedback

9.4    Singular Nonlinear ${\mathcal{H}}_{\infty }$-Control for Cascaded Nonlinear Systems

9.5    ${\mathcal{H}}_{\infty }$-Control for Singularly-Perturbed Nonlinear Systems

9.6    Notes and Bibliography

10  ${\mathcal{H}}_{\infty }$-Filtering for Singularly-Perturbed Nonlinear Systems

10.1  Problem Definition and Preliminaries

10.2  Decomposition Filters

10.3  Aggregate Filters

10.4  Examples

10.5  Notes and Bibliography

11  Mixed $\mathscr{H}$₂/$\mathscr{H}$_∞ Nonlinear Control

11.1  Continuous-Time Mixed $\mathscr{H}$₂/$\mathscr{H}$_∞ Nonlinear Control

11.1.1  The Infinite-Horizon Problem

11.1.2  Extension to a General Class of Nonlinear Systems

11.2  Discrete-Time Mixed $\mathscr{H}$₂/$\mathscr{H}$_∞ Nonlinear Control

11.2.1  The Infinite-Horizon Problem

11.3  Extension to a General Class of Discrete-Time Nonlinear Systems

11.4  Notes and Bibliography

12  Mixed $\mathscr{H}$₂/$\mathscr{H}$_∞ Nonlinear Filtering

12.1  Continuous-Time Mixed $\mathscr{H}$₂/$\mathscr{H}$_∞ Nonlinear Filtering

12.1.1  Solution to the Finite-Horizon Mixed $\mathscr{H}$₂/$\mathscr{H}$_∞ Nonlinear Filtering Problem

12.1.2  Solution to the Infinite-Horizon Mixed $\mathscr{H}$₂/$\mathscr{H}$_∞ Nonlinear Filtering

12.1.3  Certainty-Equivalent Filters(CEFs)

12.2  Discrete-Time Mixed $\mathscr{H}$₂/$\mathscr{H}$_∞ Nonlinear Filtering

12.2.1  Solution to the Finite-Horizon Discrete-Time Mixed $\mathscr{H}$₂/$\mathscr{H}$_∞ Nonlinear Filtering Problem

12.2.2  Solution to the Infinite-Horizon Discrete-Time Mixed $\mathscr{H}$₂/$\mathscr{H}$_∞ Non-linear Filtering Problem

12.2.3  Approximate and Explicit Solution to the Infinite-Horizon Discrete-Time Mixed $\mathscr{H}$₂/$\mathscr{H}$_∞ Nonlinear Filtering Problem

12.2.4  Discrete-Time Certainty-Equivalent Filters(CEFs)

12.3  Example

12.4  Notes and Bibliography

13  Solving the Hamilton-Jacobi Equation

13.1  Review of Some Approaches for Solving the HJBE/HJIE

13.1.1  Solving the HJIE/HJBE Using Polynomial Expansion and Basis Functions

13.2  A Factorization Approach for Solving the HJIE

13.2.1  Worked Examples

13.3  Solving the Hamilton-Jacobi Equation for Mechanical Systems and Application to the Toda Lattice

13.3.1  Solving the Hamilton-Jacobi Equation

13.3.2  Solving the Hamilton-Jacobi Equation for the ${\mathcal{A}}_2$-Toda System

13.4  Notes and Bibliography

A Proof of Theorem 5.7.1

B Proof of Theorem 8.2.2

Bibliography

Index