book

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

Name: Nonlinear H-Infinity Control, Hamiltonian Systems and Hamilton-Jacobi Equations
Author: M.D.S. Aliyu
ISBN: 9781351833219

by M.D.S. Aliyu

December 2017

Intermediate to advanced

408 pages

10h 47m

English

CRC Press

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Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
1 Introduction
1.1 Historical Perspective on Nonlinear H∞-Control1.2 General Set-Up for Nonlinear H∞-Control Problems1.2.1 Mixed H2-Control Problem1.2.2 Robust H∞-Control Problem1.2.3 Nonlinear H∞-Filtering1.2.4 Organization of the Book
1.3 Notations and Preliminaries
1.3.1 Notation1.3.2 Stability Concepts1.4 Notes and Bibliography
2 Basics of Differential Games
2.1 Dynamic Programming Principle2.2 Discrete-Time Nonzero-Sum Dynamic Games2.2.1 Linear-Quadratic Discrete-Time Dynamic Games2.3 Continuous-Time Nonzero-Sum Dynamic Games2.3.1 Linear-Quadratic Continuous-Time Dynamic Games2.4 Notes and Bibliography
3 Theory of Dissipative Systems
3.1 Dissipativity of Continuous-Time Nonlinear Systems3.1.1 Stability of Continuous-Time Dissipative Systems3.1.2 Stability of Continuous-Time Dissipative Feedback-Systems3.2 H∞-Gain Analysis for Continuous-Time Dissipative Systems3.3 Continuous-Time Passive Systems3.4 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 Systems3.7 Feedback-Equivalence to a Discrete-Time Lossless Nonlinear System3.8 Notes and Bibliography
4 Hamiltonian Mechanics and Hamilton-Jacobi Theory
4.1 The Hamiltonian Formulation of Mechanics4.2 Canonical Transformation4.2.1 The Transformation Generating Function4.2.2 The Hamilton-Jacobi Equation (HJE)4.2.3 Time-Independent Hamilton-Jacobi Equation and Separation of Variables4.3 The Theory of Nonlinear Lattices4.3.1 The G2 Periodic Toda Lattice4.4 The Method of Characteristics for First-Order Partial-Differential Equations4.4.1 Characteristics for Quasi-Linear Equations4.4.2 Characteristics for the General First-Order Equation4.4.3 Characteristics for the Hamilton-Jacobi Equation4.5 Legendre Transform and Hopf-Lax Formula4.5.1 Viscosity Solutions of the HJE4.6 Notes and Bibliography
5 State-Feedback Nonlinear H∞-Control for Continuous-Time Systems
5.1 State-Feedback H∞-Control for Affine Nonlinear Systems5.1.1 Dissipative Analysis5.1.2 Controller Parametrization5.2 State-Feedback Nonlinear H∞ Tracking Control
5.3 Robust Nonlinear H∞ State-Feedback Control
5.4 State-Feedback H∞-Control for Time-Varying Affine Nonlinear Systems5.5 State-Feedback H∞-Control for State-Delayed Affine Nonlinear Systems5.6 State-Feedback H∞-Control for a General Class of Nonlinear Systems5.7 Nonlinear H∞ Almost-Disturbance-Decoupling5.8 Notes and Bibliography
6 Output-Feedback Nonlinear H∞-Control for Continuous-Time Systems
6.1 Output Measurement-Feedback H∞-Control for Affine Nonlinear Systems6.1.1 Controller Parameterization6.2 Output Measurement-Feedback Nonlinear H∞ Tracking Control6.3 Robust Output Measurement-Feedback Nonlinear H∞-Control6.3.1 Reliable Robust Output-Feedback Nonlinear H∞-Control
6.4 Output Measurement-Feedback H∞-Control for a General Class of Nonlinear Systems
6.4.1 Controller Parametrization6.5 Static Output-Feedback Control for Affine Nonlinear Systems6.5.1 Static Output-Feedback Control with Disturbance-Attenuation6.6 Notes and Bibliography
7 Discrete-Time Nonlinear H∞-Control
7.1 Full-Information H∞-Control for Affine Nonlinear Discrete-Time Systems7.1.1 State-Feedback H∞-Control for Affine Nonlinear Discrete-Time Systems7.1.2 Controller Parametrization7.2 Output Measurement-Feedback Nonlinear H∞-Control for Affine Discrete-Time Systems
7.3 Extensions to a General Class of Discrete-Time Nonlinear Systems
7.3.1 Full-Information H∞-Control for a General Class of Discrete-Time Nonlinear Systems7.3.2 Output Measurement-Feedback H∞-Control for a General Class of Discrete-Time Nonlinear Systems7.4 Approximate Approach to the Discrete-Time Nonlinear H∞-Control Problem7.4.1 An Approximate Approach to the Discrete-Time State-Feedback Problem7.4.2 An Approximate Approach to the Discrete-Time Output Measurement-Feedback Problem7.5 Notes and Bibliography
8 Nonlinear H∞-Filtering
8.1 Continuous-Time Nonlinear H∞-Filtering8.1.1 Infinite-Horizon Continuous-Time Nonlinear H∞-Filtering8.1.2 The Linearized Filter8.2 Continuous-Time Robust Nonlinear H∞-Filtering
8.3 Certainty-Equivalent Filters(CEFs)
8.3.1 2-DOFCertainty-Equivalent Filters8.4 Discrete-Time Nonlinear H∞-Filtering8.4.1 Infinite-Horizon Discrete-Time Nonlinear H∞-Filtering8.4.2 Approximate and Explicit Solution
8.5 Discrete-Time Certainty-Equivalent Filters(CEFs)
8.5.1 2-DOF Proportional-Derivative (PD) CEFs8.5.2 Approximate and Explicit Solution8.6 Robust Discrete-Time Nonlinear H∞-Filtering8.7 Notes and Bibliography
9 Singular Nonlinear H∞-Control and H∞-Control for Singularly-Perturbed Nonlinear Systems
9.1 Singular Nonlinear H∞-Control with State-Feedback9.1.1 State-Feedback Singular Nonlinear H∞-Control Using High-Gain Feedback9.2 Output Measurement-Feedback Singular Nonlinear H∞-Control9.3 Singular Nonlinear H∞-Control with Static Output-Feedback9.4 Singular Nonlinear H∞-Control for Cascaded Nonlinear Systems9.5 H∞-Control for Singularly-Perturbed Nonlinear Systems9.6 Notes and Bibliography
10 H∞-Filtering for Singularly-Perturbed Nonlinear Systems
10.1 Problem Definition and Preliminaries10.2 Decomposition Filters
10.3 Aggregate Filters
10.4 Examples10.5 Notes and Bibliography
11 Mixed H∞2/H∞∞ Nonlinear Control
11.1 Continuous-Time Mixed H∞2/H∞∞ Nonlinear Control11.1.1 The Infinite-Horizon Problem11.1.2 Extension to a General Class of Nonlinear Systems
11.2 Discrete-Time Mixed H∞2/H∞∞ Nonlinear Control
11.2.1 The Infinite-Horizon Problem11.3 Extension to a General Class of Discrete-Time Nonlinear Systems11.4 Notes and Bibliography
12 Mixed H∞2/H∞∞ Nonlinear Filtering
12.1 Continuous-Time Mixed H∞2/H∞∞ Nonlinear Filtering12.1.1 Solution to the Finite-Horizon Mixed H∞2/H∞∞ Nonlinear Filtering Problem12.1.2 Solution to the Infinite-Horizon Mixed H∞2/H∞∞ Nonlinear Filtering12.1.3 Certainty-Equivalent Filters(CEFs)12.2 Discrete-Time Mixed H∞2/H∞∞ Nonlinear Filtering
12.2.1 Solution to the Finite-Horizon Discrete-Time Mixed H∞2/H∞∞ Nonlinear Filtering Problem
12.2.2 Solution to the Infinite-Horizon Discrete-Time Mixed H∞2/H∞∞ Non-linear Filtering Problem12.2.3 Approximate and Explicit Solution to the Infinite-Horizon Discrete-Time Mixed H∞2/H∞∞ Nonlinear Filtering Problem12.2.4 Discrete-Time Certainty-Equivalent Filters(CEFs)12.3 Example12.4 Notes and Bibliography
13 Solving the Hamilton-Jacobi Equation
13.1 Review of Some Approaches for Solving the HJBE/HJIE13.1.1 Solving the HJIE/HJBE Using Polynomial Expansion and Basis Functions
13.2 A Factorization Approach for Solving the HJIE
13.2.1 Worked Examples13.3 Solving the Hamilton-Jacobi Equation for Mechanical Systems and Application to the Toda Lattice13.3.1 Solving the Hamilton-Jacobi Equation13.3.2 Solving the Hamilton-Jacobi Equation for the A2-Toda System13.4 Notes and Bibliography
A Proof of Theorem 5.7.1
B Proof of Theorem 8.2.2
Bibliography
Index

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

11.2 Discrete-Time Mixed $H_{2}$ / $H_{∞}$ Nonlinear Control

In this section, we discuss the state-feedback mixed $H_{2}$ / $H_{∞}$ -control problem for discrete-time nonlinear systems. We begin with an affine discrete-time state-space model defined on $X \subset ℜ^{n}$ in coordinates (x₁,…,x_n) :

$\sum^{d a} : {\begin{cases} {\dot{x}}_{k + 1} = f (x_{k}) + g_{1} (x_{k}) w_{k} + g_{2} (x_{k}) u_{k}; x (k_{0}) = x^{0} \\ z_{k} = h_{1} (x_{k}) + k_{12} (x_{k}) u_{k} \\ y_{k} = x_{k}, \end{cases}$

(11.29)

where all the variables and system matrices have their previous meanings and dimensions respectively. We assume similarly that the system has a unique equilibrium at x = 0, and is such that f(0) = 0 and h₁0) = 0. For simplicity, we similarly also assume the following hold for the system matrices.

Assumption 11.2.1 The system matrices are such that

$\begin{array}{l} h_{1}^{T} \end{array}$

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

by M.D.S. Aliyu

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.