LMIs in Control Systems : Analysis, Design and Applications book cover
1st Edition

LMIs in Control Systems
Analysis, Design and Applications

ISBN 9781466582996
Published June 17, 2013 by CRC Press
483 Pages 53 B/W Illustrations

USD $125.00

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Book Description

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.

Table of Contents

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

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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