Linear Systems

Linear Systems: Optimal and Robust Control

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Features

  • Provides the material for two graduate-level courses: one in linear systems and optimal control and the other in robust control
  • Includes H and sliding mode methods together for the first time in book form
  • Develops mathematical analyses, including the derivation of H, to avoid learning additional mathematical tools
  • Highlights the use of MATLAB® software to solve practical problems via the computer
  • Contains end-of-chapter exercises and offers a solutions manual with qualifying course adoptions
  • Summary

    Balancing rigorous theory with practical applications, Linear Systems: Optimal and Robust Control explains the concepts behind linear systems, optimal control, and robust control and illustrates these concepts with concrete examples and problems.

    Developed as a two-course book, this self-contained text first discusses linear systems, including controllability, observability, and matrix fraction description. Within this framework, the author develops the ideas of state feedback control and observers. He then examines optimal control, stochastic optimal control, and the lack of robustness of linear quadratic Gaussian (LQG) control. The book subsequently presents robust control techniques and derives H control theory from the first principle, followed by a discussion of the sliding mode control of a linear system. In addition, it shows how a blend of sliding mode control and H methods can enhance the robustness of a linear system.

    By learning the theories and algorithms as well as exploring the examples in Linear Systems: Optimal and Robust Control, students will be able to better understand and ultimately better manage engineering processes and systems.

    Table of Contents

    Introduction
    Overview
    Contents of the Book
    State Space Description of a Linear System
    Transfer Function of a Single Input/Single Output (SISO) System
    State Space Realizations of a SISO System
    SISO Transfer Function from a State Space Realization
    Solution of State Space Equations
    Observability and Controllability of a SISO System
    Some Important Similarity Transformations
    Simultaneous Controllability and Observability
    Multiinput/Multioutput (MIMO) Systems
    State Space Realizations of a Transfer Function Matrix
    Controllability and Observability of a MIMO System
    Matrix-Fraction Description (MFD)
    MFD of a Transfer Function Matrix for the Minimal Order of a State Space Realization
    Controller Form Realization from a Right MFD
    Poles and Zeros of a MIMO Transfer Function Matrix
    Stability Analysis
    State Feedback Control and Optimization
    State Variable Feedback for a Single Input System
    Computation of State Feedback Gain Matrix for a Multiinput System
    State Feedback Gain Matrix for a Multiinput System for Desired Eigenvalues and Eigenvectors
    Fundamentals of Optimal Control Theory
    Linear Quadratic Regulator (LQR) Problem
    Solution of LQR Problem via Root Locus Plot: SISO Case
    Linear Quadratic Trajectory Control
    Frequency-Shaped LQ Control
    Minimum-Time Control of a Linear Time-Invariant System
    Control with Estimated States
    Open-Loop Observer
    Closed-Loop Observer
    Combined Observer–CONTROLLER
    Reduced-Order Observer
    Response of a Linear Continuous-Time System to White Noise
    Kalman Filter: Optimal State Estimation
    Stochastic Optimal Regulator in Steady State
    Linear Quadratic Gaussian (LQG) Control
    Impact of Modeling Errors on Observer-Based Control
    Robust Control: Fundamental Concepts and H2, H, and μ Techniques
    Important Aspects of Singular Value Analysis
    Robustness: Sensitivity and Complementary Sensitivity
    Robustness of LQR and Kalman Filter (KF) Feedback Loops
    LQG/LTR Control
    H2 and HNorms
    H2 Control
    Well-Posedness, Internal Stability, and Small Gain Theorem
    Formulation of Some Robust Control Problems with Unstructured Uncertainties
    Formulation of Robust Control Problems with Structured Uncertainties
    HControl
    Loop Shaping
    Controller Based on μ Analysis
    Robust Control: Sliding Mode Methods
    Basic Concepts of Sliding Modes
    Sliding Mode Control of a Linear System with Full State Feedback
    Sliding Mode Control of an Uncertain Linear System with Full State Feedback: Blending Hand Sliding Mode Methods
    Sliding Mode Control of a Linear System with Estimated States
    Optimal Sliding Mode Gaussian (OSG) Control
    REFERENCES
    Appendix A: Linear Algebraic Equations, Eigenvalues/Eigenvectors, and Matrix Inversion Lemma
    System of Linear Algebraic Equations
    Eigenvalues and Eigenvectors
    Matrix Inversion Lemma
    Appendix B: Quadratic Functions, Important Derivatives, Fourier Integrals, and Parseval’s Relation
    Quadratic Functions
    Derivative of a Quadratic Function
    Derivative of a Linear Function
    Fourier Integrals and Parseval’s Theorem
    Appendix C: Norms, Singular Values, Supremum, and Infinimum
    Vector Norms
    Matrix Norms
    Singular Values of a Matrix
    Singular Value Decomposition (SVD)
    Properties of Singular Values
    Supremum and Infinimum
    Appendix D: Stochastic Processes
    Stationary Stochastic Process
    Power Spectrum or Power Spectral Density (PSD)
    White Noise: A Special Stationary Stochastic Process
    Response of a SISO Linear and Time-Invariant System Subjected to a Stationary Stochastic Process
    Vector Stationary Stochastic Processes
    Appendix E: Optimization of a Scalar Function with Constraints in the Form of a Symmetric Real Matrix Equal to Zero
    Appendix F: A Flexible Tetrahedral Truss Structure
    Appendix G: Space Shuttle Dynamics during Reentry
    INDEX
    Exercises appear at the end of each chapter.

     
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