1st Edition

LMIs in Control Systems Analysis, Design and Applications

By Guang-Ren Duan, Hai-Hua Yu Copyright 2013
    483 Pages 53 B/W Illustrations
    by CRC Press

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    Although LMI has emerged as a powerful tool with applications across the major domains of systems and control, there has been a need for a textbook that provides an accessible introduction to LMIs in control systems analysis and design. Filling this need, LMIs in Control Systems: Analysis, Design and Applications focuses on the basic analysis and design problems of both continuous- and discrete-time linear systems based on LMI methods.

    Providing a broad and systematic introduction to the rich content of LMI-based control systems analysis and design with applications, this book is suitable for use as a textbook for LMI related courses for senior undergraduate and postgraduate students in the fields of control systems theory and applications.

    Key Features:

    • Contains four well-structured parts: Preliminaries, Control Systems Analysis, Control Systems Design, and Applications, as well as an introduction chapter and two appendices
    • Summarizes most of the technical lemmas used in the book in one preliminary chapter, and classifies them systematically into different groups
    • Includes many examples, exercises, and practical application backgrounds
    • Summarizes most of the important results in the last section of each chapter, in a clear table format
    • Contains an application part composed of two chapters that respectively deal with missile and satellite attitude control using LMI techniques
    • Provides a brief and clear introduction to the use of the LMI Lab in the MATLAB® Robust Control Toolbox
    • Supplies detailed proofs for all main results, with lengthy ones clearly divided into different subsections or steps—using elementary mathematics whenever possible
    • Uses a pole assignment Benchmark problem, in support of the numerical reliability of LMI techniques, where numerical unreliability could result in a solution to a problem that is far from the true one

    A Solutions Manual and MATLAB® codes for the computational exercise problems and examples are available upon qualified course adoption.

    What are LMIs?
         General form 
         Standard form
    A few examples involving LMIs 
         Eigenvalue minimization 
         Matrix norm minimization 
         A key step in μ-analysis 
         Schur stabilization
    A brief history
         The seed planted (1890)
         The rooting period (1940-1970) 
         The growing period (1970-2000) 
         The Nourishing period (2000-present)
    About the book 
         Using it in courses


    Technical Lemmas
    Generalized square inequalities
         The restriction-free inequality 
         Inequalities with restriction
         The variable elimination lemma
    Schur complement lemma
         Schur complements 
         Matrix inversion lemma 
         Schur complement lemma
    Elimination of variables 
         Variable elimination in a partitioned matrix
         The projection lemma 
         The reciprocal projection lemma
    Some other useful results 
         Trace of an LMI 
         The maximum modulus principle 
         The Parseval lemma
    Notes and references

    Review of Optimization Theory
    Convex sets 
         Definitions and properties
         Hyperplanes, halfspaces, polyhedrons and polytopes
    Convex functions 
         Definition and properties
    Mathematical optimization 
         Least squares programming
         Linear programming
         Quadratic programming
    Convex optimization 
         The problem 
         Local and global optima
    The LMI problem
         The extreme result 
         Standard problems
    Notes and references
         About this chapter 
         The open source software CVX
         A counter example for numerical reliability


    Stability Analysis
    Hurwitz and Schur stability 
         Hurwitz stability 
         Schur stability
         Special cases 
         GeneralLMI regions 
         Generalized Lyapunov theorem
    Quadratic stability 
         Familyof systems
         Quadratic Hurwitz stability 
         QuadraticSchur stability
    Quadratic D-stability 
         Definition and main results 
         Some special cases
    Time-delay systems 
         The delay independent condition
         The delay dependent condition
    Notes and references 
         Summary and references 
         Affine quadratic stability

    H/H2 Performance
    H and H2 indices 
         H index 
         H2 index
         Equivalent definitions
    LMI conditions for H index 
         Thebasic conditions 
         Deduced conditions
    LMI conditions for H2 index
         Basic conditions 
         Deduced conditions
    Notes and references

    Property Analysis
    Hurwitz stabilizability and detectability 
         Hurwitz stabilizability 
         Hurwitz detectability
    Schur stabilizability and detectability 
         Schur stabilizability 
         Schur detectability
         Equivalent conditions
    Passivity and positive-realness 
         The positive-real lemma
         The LMI condition
    Non expansivity and bounded-realness 
         The bounded-real lemma 
         The LMI conditions
    Notes and references


    Feedback Stabilization
    State feedback stabilization
         Case of continuous-time systems
         Case of discrete-time systems
         H (a,B)-stabilization 
         General D-stabilization
    Quadratic stabilization 
         Family of systems
         Quadratic Hurwitz stabilization 
         Quadratic Schur stabilization
    Quadratic D-stabilization
         Problem formulation 
         The solution 
         Special cases
    Insensitive region design 
         Sensitivity of matrix eigen values 
         Insensitive strip region design 
         Insensitive disk region design
    Robust stabilization of second-order systems 
         Robust Stabilization
    Stabilization of time-delay systems 
         Case of delay independence 
         Case of delay dependence
    Notes and references

    H/H2 Control
    H state feedback control
         The problem 
         The solution 
         Other conditions
    H2 state feedback control
         The problem 
         The solution 
         Other conditions
    Robust H/H2 state feedback control 
         The problems
         Solution to the robust H control problem 
         Solution to the robust H2 control problem
    LQ regulation via H2 control 
         Problem description 
         Relation with H2 performance
         The solution
    Notes and references 
         Summary and references
         Dissipative, passive, and non-expansive control

    State Observation and Filtering
    Full- and reduced-order state observers 
          Full-order state observers 
         Reduced-order state observer design
    Full-order H/H2 state observers 
         Problems formulation 
         Solutions to problems 
    H filtering 
         Problems formulation 
         Solution to H filtering
    H2 filtering
         Problems formulation 
         Solution to H2 filtering
    Notes and references

    Multiple Objective Designs
    Insensitive region designs with minimum gains 
         Insensitive strip region designs with minimum gains 
         Insensitive disk region designs with minimum gains
    Mixed H/H2 designs with desired LMI pole regions 
         The problem 
         Solutions to the problem
    Mixed robust H/H2 designs with desired LMI pole regions 
         The problem 
         Solutions to the problem
    Notes and references 
         Summary of main results
         Further remarks


    Missile Attitude Control
    The dynamical model
         Models for non-rotating missiles 
         Models for BTT missiles
    Attitude control of non-rotating missiles 
         The problem 
         The solution
    Attitude control of BTT missiles 
         The problem 
         Quadratic stabilization 
         Numerical results and simulation
    Notes and references

    Satellite Control
    System modelling 
         The second-order system form 
         The state space form
    H2 and H feedback control 
         H control 
         H2 control
    Mixed H2/ H feedback control 
         The problem 
         Numerical and simulation results
    Notes and references


    Proofs of Theorems

    Proof of Theorem 4.1 

    Proof of Theorem 5.1 
         The first step 
         The second step

    Proof of Theorem5.2
         The first step
         The second step


    Guang-Ren Duan received his BSc. degree in Applied Mathematics, and both his MSc and PhD degrees in Control Systems Theory. From 1989 to 1991, he was a post-doctoral researcher at Harbin Institute of Technology, where he became full professor of control systems theory in the end of 1991. Prof. Duan visited the University of Hull, UK, and the University of Sheffield, UK from December 1996 to October 1998, and worked as a lecturer at the Queen's University of Belfast, UK from October 1998 to October 2002. Since August 2000, he has been elected Specially Employed Professor at Harbin Institute of Technology sponsored by the Cheung Kong Scholars Program of the Chinese government. He is currently the Director of the Center for Control Theory and Guidance Technology at Harbin Institute of Technology.

    He is the author and co-author of 3 books and more than 180 SCI indexed publications. Particularly, he has published with Springer a book entitled Analysis and Design of Descriptor Linear Systems, and has published over 30 papers in IEEE Transactions. His main research interests include parametric robust control systems design, LMI-based control systems analysis and design, descriptor systems, flight control and magnetic bearing control.

    He has taught quite a few courses both at Harbin Institute of Technology, China, and at the Queen’s University of Belfast, UK. Particularly, he has lectured at Harbin Institute of Technology the graduate course "Linear Matrix Inequalities in Control Systems Analysis and Design", based on this set of lecture notes.

    LMIs play the same central role in the postmodern theory as Lyapunov and Riccati equations played in the modern, and in turn various graphical techniques such as Bode, Nyquist, and Nichols plots played in the classical.
    —J. Doyle, A. Packet, and K. M. Zhou