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Sliding Mode Control and Observation.pdf

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    Preface
    Contents
    Chapter 1 Introduction: Intuitive Theory of Sliding Mode Control
    1.1 Main Concepts of Sliding Mode Control
    1.2 Chattering Avoidance: Attenuation and Elimination
    1.2.1 Chattering Elimination: Quasi-Sliding Mode
    1.2.2 Chattering Attenuation: Asymptotic Sliding Mode
    1.3 Concept of Equivalent Control
    1.4 Sliding Mode Equations
    1.5 The Matching Condition and Insensitivity Properties
    1.6 Sliding Mode Observer/Differentiator
    1.7 Second-Order Sliding Mode
    1.8 Output Tracking: Relative Degree Approach
    1.8.1 Conventional Sliding Mode Controller Design
    1.8.2 Integral Sliding Mode Controller Design
    1.8.3 Super-Twisting Controller Design
    1.8.4 Prescribed Convergence Law Controller Design
    1.9 Notes and References
    1.10 Exercises
    Chapter 2 Conventional Sliding Modes
    2.1 Introduction
    2.1.1 Filippov Solution
    2.1.2 Concept of Equivalent Control
    2.2 State-Feedback Sliding Surface Design
    2.2.1 Regular Form
    2.2.2 Eigenvalue Placement
    2.2.3 Quadratic Minimization
    2.3 State-Feedback Relay Control Law Design
    2.3.1 Single-Input Nominal Systems
    2.3.2 Single-Input Perturbed Systems
    2.3.3 Relay Control for Multi-input Systems
    2.4 State-Feedback Unit-Vector Control
    2.4.1 Design in the Presence of Matched Uncertainty
    2.4.2 Design in the Presence of Unmatched Uncertainty
    2.5 Output Tracking with Integral Action
    2.6 Output-Based Hyperplane Design
    2.6.1 Static Output-Feedback Hyperplane Design
    2.6.2 Static Output-Feedback Control Law Development
    2.6.3 Dynamic Output-Feedback Hyperplane Design
    2.6.4 Dynamic Output-Feedback Control Law Development
    2.6.5 Case Study: Vehicle Stability in a Split-Mu Maneuver
    2.7 Integral Sliding Mode Control
    2.7.1 Problem Formulation
    2.7.2 Control Design Objective
    2.7.3 Linear Case
    2.7.4 ISM Compensation of Unmatched Disturbances
    2.8 Notes and References
    2.9 Exercises
    Chapter 3 Conventional Sliding Mode Observers
    3.1 Introduction
    3.2 A Simple Sliding Mode Observer
    3.3 Robustness Properties of Sliding Mode Observers
    3.4 A Generic Conventional Sliding Mode Observer
    3.5 A Sliding Mode Observer for Nonlinear Systems
    3.6 Fault Detection: A Simulation Example
    3.7 Notes and References
    3.8 Exercises
    Chapter 4 Second-Order Sliding Mode Controllers and Differentiators
    4.1 Introduction
    4.2 2-Sliding Mode Controllers
    4.2.1 Twisting Controller
    4.2.2 Suboptimal Algorithm
    4.2.3 Control Algorithm with Prescribed Convergence Law
    4.2.4 Quasi-Continuous Control Algorithm
    4.2.5 Accuracy of 2-Sliding Mode Controllers
    4.3 Control of Relative Degree One Systems
    4.3.1 Super-Twisting Controller
    4.3.2 First-Order Differentiator
    4.4 Differentiator-Based Output-Feedback 2-SM Control
    4.5 Chattering Attenuation
    4.6 Case Study: Pendulum Control
    4.6.1 Discontinuous Control
    4.6.2 Chattering Attenuation
    4.7 Variable-Gain Super-Twisting Control
    4.7.1 Problem Statement
    4.7.2 The Variable-Gain Super-Twisting Algorithm
    4.8 Case Study: The Mass–Spring–Damper System
    4.8.1 Model Description
    4.8.2 Problem Statement
    4.8.3 Control Design
    4.8.4 Experimental Results
    4.9 Notes and References
    4.10 Exercises
    Chapter 5 Analysis of Sliding Mode Controllers in the Frequency Domain
    5.1 Introduction
    5.2 Conventional SMC Algorithm: DF Analysis
    5.3 Twisting Algorithm: DF Analysis
    5.4 Super-Twisting Algorithm: DF Analysis
    5.4.1 DF of Super-Twisting Algorithm
    5.4.2 Existence of the Periodic Solutions
    5.4.3 Stability of Periodic Solution
    5.5 Prescribed Convergence Control Law: DF Analysis
    5.6 Suboptimal Algorithm: DF Analysis
    5.7 Comparisons of 2-Sliding Mode Control Algorithms
    5.8 Notes and References
    5.9 Exercises
    Chapter 6 Higher-Order Sliding Mode Controllers and Differentiators
    6.1 Introduction
    6.2 Single-Input Single-Output Regulation Problem
    6.3 Homogeneity, Finite-Time Stability, and Accuracy
    6.4 Homogeneous Sliding Modes
    6.5 Accuracy of Homogeneous 2-Sliding Modes
    6.6 Arbitrary-Order Sliding Mode Controllers
    6.6.1 Nested Sliding Controllers
    6.6.2 Quasi-continuous Sliding Controllers
    6.7 Arbitrary-Order Robust Exact Differentiation
    6.8 Output-Feedback Control
    6.9 Tuning of the Controllers
    6.9.1 Control Magnitude Tuning
    6.9.2 Parametric Tuning
    6.10 Case Study: Car Steering Control
    6.11 Case Study: Blood Glucose Regulation
    6.11.1 Introduction to Diabetes
    6.11.2 Insulin–Glucose Regulation Dynamical Model
    6.11.3 Higher-Order Sliding Mode Controller Design
    6.11.4 Simulation
    6.12 Notes and References
    6.13 Exercises
    Chapter 7 Observation and Identification via HOSM Observers
    7.1 Observation/Identification of Mechanical Systems
    7.1.1 Super-Twisting Observer
    7.1.2 Equivalent Output Injection Analysis
    7.1.3 Parameter Identification
    7.2 Observation in Single-Output Linear Systems
    7.2.1 Non-perturbed Case
    7.2.2 Perturbed Case
    7.2.3 Design of the Observer for Strongly ObservableSystems
    7.3 Observers for Single-Output Nonlinear Systems
    7.3.1 Differentiator-Based Observer
    7.3.2 Disturbance Identification
    7.4 Regulation and Tracking Controllers Driven by SM Observers
    7.4.1 Motivation
    7.4.2 Problem Statement
    7.4.3 Theoretically Exact Output-Feedback Stabilization (EOFS)
    7.4.4 Output Integral Sliding Mode Control
    7.4.5 Precision of the Observation and IdentificationProcesses
    7.5 Notes and References
    7.6 Exercises
    Chapter 8 Disturbance Observer Based Control: Aerospace Applications
    8.1 Problem Formulation
    8.1.1 Asymptotic Compensated Dynamics
    8.1.2 Finite-Time-Convergent Compensated Dynamics
    8.1.3 Sliding Variable Disturbed Dynamics
    8.1.4 Output Tracking Error Disturbed Dynamics
    8.2 Perturbation Term Reconstruction via a Disturbance Observer
    8.2.1 SMDO Based on Conventional SMC
    8.2.2 SMDO Based on Super-Twisting Control
    8.2.3 Design of the SMC Driven by the SMDO
    8.3 Case Study: Reusable Launch Vehicle Control
    8.3.1 Mathematical Model of Reusable Launch Vehicle
    8.3.2 Reusable Launch Vehicle Control Problem Formulation
    8.3.3 Multiple-Loop Asymptotic SMC/SMDO Design
    8.3.4 Flight Simulation Results and Analysis
    8.4 Case Study: Satellite Formation Control
    8.4.1 Satellite Formation Mathematical Model
    8.4.2 Satellite Formation Control in SMC/SMDO
    8.5 Simulation Study
    8.6 Notes and References
    8.7 Exercises
    Appendix A Mathematical Preliminaries
    A.1 Linear Algebra
    A.1.1 Rank and Determinant
    A.1.2 Eigenvalues and Eigenvectors
    A.1.3 QR Decomposition
    A.1.4 Norms
    A.1.5 Quadratic Forms
    Appendix B Describing Functions
    B.1 Describing Function Fundamentals
    B.1.1 Low-Pass Filter Hypothesis and Describing Function
    B.1.2 Limit Cycle Analysis Using Describing Functions
    B.1.3 Stability Analysis of the Limit Cycle
    Appendix C Linear Systems Theory
    C.1 Introduction
    C.1.1 Linear Time-Invariant Systems
    C.1.2 Controllability and Observability
    C.1.3 Invariant Zeros
    C.1.4 State Feedback Control
    C.1.5 Static Output Feedback Control
    Appendix D Lyapunov Stability
    D.1 Local Results
    D.2 Global Results
    D.2.1 Quadratic Stability
    Bibliography
    Index
    Control Engineering Series Editor William S. Levine Department of Electrical and Computer Engineering University of Maryland College Park, MD USA Editorial Advisory Board Okko Bosgra Delft University The Netherlands Graham Goodwin University of Newcastle Australia Iori Hashimoto Kyoto University Japan Petar Kokotovi´c University of California Santa Barbara, CA USA Manfred Morari ETH Z¨urich Switzerland William Powers Ford Motor Company (retired) Detroit, MI USA Mark Spong University of Illinois Urbana-Champaign USA For further volumes: http://www.springer.com/series/4988
    Yuri Shtessel Christopher Edwards Leonid Fridman Arie Levant Sliding Mode Control and Observation
    Y. Shtessel Department of Electrical and Computer Engineering University of Alabama in Huntsville Huntsville, AL, USA C. Edwards College of Engineering, Mathematics and Physical Science University of Exeter Exeter, UK L. Fridman Department of Control Division of Electrical Engineering Faculty of Engineering National Autonomous University of Mexico Mexico A. Levant Department of Applied Mathematics School of Mathematical Sciences Tel-Aviv University Israel ISBN 978-0-8176-4892-3 DOI 10.1007/978-0-8176-4893-0 Springer New York Heidelberg Dordrecht London ISBN 978-0-8176-4893-0 (eBook) Library of Congress Control Number: 2013934106 Mathematics Subject Classifications (2010): 93B12, 93C10, 93B05, 93B07, 93B51, 93B52, 93D25 © Springer Science+Business Media New York 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com)
    We dedicate this book with love and gratitude to Yuri’s wife Nina, Chris’ parents Shirley and Cyril, Leonid’s wife Millie, Arie’s wife Irena.
    Preface Control in the presence of uncertainty is one of the main topics of modern control theory. In the formulation of any control problem there is always a discrepancy between the actual plant dynamics and its mathematical model used for the controller design. These discrepancies (or mismatches) mostly come from external disturbances, unknown plant parameters, and parasitic dynamics. Designing control laws that provide the desired closed-loop system performance in the presence of these disturbances/uncertainties is a very challenging task for a control engineer. This has led to intense interest in the development of the so-called robust control methods, which are supposed to solve this problem. In spite of the extensive and successful development of robust adaptive control [159], H1 control [48], and backstepping [121] techniques, sliding mode control (SMC) remains, probably, the most successful approach in handling bounded uncertainties/disturbances and parasitic dynamics [67, 182, 186]. Historically sliding modes were discovered as a special mode in variable structure systems (VSS). These systems comprise a variety of structures, with rules for switching between structures in real time to achieve suitable system performance, whereas using a single fixed structure could be unstable. The result is VSS, which may be regarded as a combination of subsystems where each subsystem has a fixed control structure and is valid for specified regions of system behavior. It appeared that the closed-loop system may be designed to possess new properties not present in any of the constituent substructures alone. Furthermore, in a special mode, named a sliding mode, these properties include insensitivity to certain (so-called matched) external disturbances and model uncertainties as well as robustness to parasitic dynamics. Achieving reduced-order dynamics of the compensated system in a sliding mode (termed partial dynamical collapse) is also a very important useful property of sliding modes. One of the first books in English to be published on this subject is [85]. The development of these novel ideas began in the Soviet Union in the late 1950s. The idea of SMC is based on the introduction of a “custom-designed” function, named the sliding variable. As soon as the properly designed sliding variable becomes equal to zero, it defines the sliding manifold (or the sliding surface). The vii
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