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

483 Pages
by CRC Press

Also available as eBook on:

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.

Introduction
What are LMIs?
General form
Standard form
Manipulations
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)
Structure
Features
Using it in courses
Exercises

PRELIMINARIES

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
Exercises

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

CONTROL SYSTEMS ANALYSIS

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

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
Exercises

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

CONTROL SYSTEMS DESIGN

Feedback Stabilization
State feedback stabilization
Case of continuous-time systems
Case of discrete-time systems
D-stabilization
H (a,B)-stabilization
D(q,r)-stabilization
General D-stabilization
Family of systems
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
Stabilization
Robust Stabilization
Stabilization of time-delay systems
Case of delay independence
Case of delay dependence
Notes and references
Exercises

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
Exercises

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
Examples
H filtering
Problems formulation
Solution to H filtering
H2 filtering
Problems formulation
Solution to H2 filtering
Notes and references
Exercises

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
Exercises

APPLICATIONS

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
Numerical results and simulation
Notes and references
Exercises

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
Exercises

APPENDICES

Proofs of Theorems

Proof of Theorem 4.1
Preliminaries
Sufficiency
Necessity

Proof of Theorem 5.1
The first step
The second step

Proof of Theorem5.2
The first step
The second step

### Biography

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