References The lMii lab Background and terminology Overview of the lmi lab 8-6 Specifying a System of LMIs A Simple Example 8-9 setlmis and getlmis Imilar Limiter The lmi Editor lmiedit How It all work 8-18 Retrieving Information 8-21 Imiinfo 8-21 Iminbr and matnbr 8-21 LMI Solvers From Decision to Matrix Variables and vice versa 8-28 Validating Results Modifying a System of LMIs delli 8-3 setmvar Advanced Topics 8-83 Structured matrix variables 8-33 Complex-Valued LMIs Specifying c" Objectives for mincx Feasibility radius 8-39
v References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-15 8 The LMI Lab Background and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3 Overview of the LMI Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6 Specifying a System of LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9 setlmis and getlmis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11 lmivar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11 lmiterm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13 The LMI Editor lmiedit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-16 How It All Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18 Retrieving Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-21 lmiinfo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-21 lminbr and matnbr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-21 LMI Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-22 From Decision to Matrix Variables and Vice Versa . . . . . . 8-28 Validating Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-29 Modifying a System of LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-30 dellmi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-30 dellmi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-30 setmvar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-31 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-33 Structured Matrix Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-33 Complex-Valued LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-35 Specifying cTx Objectives for mincx . . . . . . . . . . . . . . . . . . . . . 8-38 Feasibility Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-39
Well-Posedness issues Semi-Definite B(x)in gevp Problems Efficiency and Complexity Issues 4红2 Solving M PTXQ+QTXTP<0 References 8-44 Command reference List of functions 93 Hoo Control and Loop Shaping II Lab: Specifying and Solving LMIs 97 LMI Lab: Additional Facilities vI Contents
vi Contents Well-Posedness Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-40 Semi-Definite B(x) in gevp Problems . . . . . . . . . . . . . . . . . . . . 8-41 Efficiency and Complexity Issues . . . . . . . . . . . . . . . . . . . . . . . 8-41 Solving M + PTXQ + QTXTP < 0 . . . . . . . . . . . . . . . . . . . . . . . . 8-42 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-44 9 Command Reference List of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3 H∞ Control and Loop Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 9-6 LMI Lab: Specifying and Solving LMIs . . . . . . . . . . . . . . . . . . 9-7 LMI Lab: Additional Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . 9-8
Prefe Ab。 ut the authors Dr Pascal Gahinet is a reserach fellow at INRIA Rocquencourt, France. His research interests include robust control theory, linear matrix inequalities numerical linear algebra, and numerical software for control Prof Arkadi Nemirovski is with the Faculty of Industrial Engineering and Management at Technion, Haifa, Israel. His research interests include convex optimization, complexity theory, and non-parametric statistics Prof Alan J. Laub is with the Electrical and Computer engineering Department of the University of California at Santa Barbara, USA. His research interests are in numerical analysis, mathematical software scientific computation, computer-aided control system design, and linear and large-scale trol and filtering Mahmoud Chilali is completing his Ph D at INRIA Rocquencourt, france His thesis is on the theory and applications of linear matrix inequalities
Preface viii About the Authors Dr. Pascal Gahinet is a reserach fellow at INRIA Rocquencourt, France. His research interests include robust control theory, linear matrix inequalities, numerical linear algebra, and numerical software for control. Prof. Arkadi Nemirovski is with the Faculty of Industrial Engineering and Management at Technion, Haifa, Israel. His research interests include convex optimization, complexity theory, and non-parametric statistics. Prof. Alan J. Laub is with the Electrical and Computer Engineering Department of the University of California at Santa Barbara, USA. His research interests are in numerical analysis, mathematical software, scientific computation, computer-aided control system design, and linear and large-scale control and filtering theory. Mahmoud Chilali is completing his Ph.D. at INRIA Rocquencourt, France. His thesis is on the theory and applications of linear matrix inequalities in control
Acknowledgments Acknowledgments The authors wish to express their gratitude to all colleagues who directly or indirectly contributed to the making of the LMI Control Toolbox. Special chanks to Pierre Apkarian, gregory Becker, Hiroyuki Kajiwara, and Anca Ignat for their help and contribution. Many thanks also to those who tested and helped refine the software, including bobby bodenheimer, Markus Lu, Roy Lurie, Jason Ly, John Morris, Ravi Prasanth, Michael Safonov bo Brandstetter. Eric Feron, K.C. Goh. Anders Helmersson. Ted Iwasaki. Ji Carsten Scherer, Andy Sparks, Mario Rotea, Matthew Lamont Tyler, Jim Tung, and John Wen. apologies, finally to those we may have omitted The work of Pascal Gahinet was supported in part by INRIA
Acknowledgments ix Acknowledgments The authors wish to express their gratitude to all colleagues who directly or indirectly contributed to the making of the LMI Control Toolbox. Special thanks to Pierre Apkarian, Gregory Becker, Hiroyuki Kajiwara, and Anca Ignat for their help and contribution. Many thanks also to those who tested and helped refine the software, including Bobby Bodenheimer, Markus Brandstetter, Eric Feron, K.C. Goh, Anders Helmersson, Ted Iwasaki, Jianbo Lu, Roy Lurie, Jason Ly, John Morris, Ravi Prasanth, Michael Safonov, Carsten Scherer, Andy Sparks, Mario Rotea, Matthew Lamont Tyler, Jim Tung, and John Wen. Apologies, finally, to those we may have omitted. The work of Pascal Gahinet was supported in part by INRIA
Introduction Linear Matrix Inequalities Toolbox Features LMIIs and lmi Problems 1-4 The Three Generic LMi Problems Further Mathematical Background References 1-10
1 Introduction Linear Matrix Inequalities . . . . . . . . . . . . . 1-2 Toolbox Features . . . . . . . . . . . . . . . . . . 1-3 LMIs and LMI Problems . . . . . . . . . . . . . . . 1-4 The Three Generic LMI Problems . . . . . . . . . . . . 1-5 Further Mathematical Background . . . . . . . . . 1-9 References . . . . . . . . . . . . . . . . . . . . . 1-10