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

LMIs in Control Systems

Analysis, Design and Applications, 1st Edition

By Guang-Ren Duan, Hai-Hua Yu

CRC Press

483 pages | 53 B/W Illus.

Purchasing Options:$ = USD
Hardback: 9781466582996
pub: 2013-06-17
eBook (VitalSource) : 9780429190254
pub: 2013-06-17
from $60.00

FREE Standard Shipping!


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.


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

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

About the Authors

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.

Subject Categories

BISAC Subject Codes/Headings:
TECHNOLOGY & ENGINEERING / Engineering (General)