Linear Matrix Inequalities

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Control Engineering

Synopsis:

Semidefinite programming or optimization over linear matrix inequalities

(LMIs) is an extension of linear programming to the cone of positive

semidefinite matrices. LMI methods are an important modern tool in systems

control and signal processing.

Theory: Convex sets represented via LMIs; LMI relaxations for solution of

non-convex polynomial optimization problems; Interior-point algorithms to

solve LMI problems; Solvers and software; LMIs for polynomial mehods in

control.

Control applications: robustness analysis of linear and nonlinear systems;

design of fixed-order robust controllers with H-infinity specifications.

For more information, see http://www.laas.fr/~henrion/courses/lmi

Requirements:

Syllabus of lectures:

Syllabus of tutorials:

Study Objective:

Study materials:

# S. Boyd, L. Vandenberghe. Convex Optimization, Cambridge University

Press, 2005

# A. Ben-Tal, A. Nemirovskii. Lectures on modern convex optimization:

analysis, algorithms and engineering applications. SIAM, Philadelphia,

2001. Most of the material there can be found in various lecture notes and

slides available at A. Nemirovksii's webpage at Georgia Tech.

LMI representation of semialgebraic sets and lift-and-project techniques

are described in:

# A. Ben-Tal, A. Nemirovskii. Lectures on modern convex optimization:

analysis, algorithms and engineering applications. SIAM, Philadelphia,

2001

# P. A. Parrilo, S. Lall. SDP Relaxations and Algebraic Optimization in

Control. ECC'03 and CDC'03 workshops, whose slides are available at P. A.

Parrilo's webpage at MIT.

Modern state-space LMI methods in control are nicely surveyed in:

# C. Scherer, S. Weiland. LMIs in Control, Lecture Notes at Delft

University of Technology and Eindhoven University of Technology, 2005.

Polynomials methods for robustness analysis are well described in

# B. R. Barmish. New tools for robustness of linear systems. MacMillan,

1994.

Polynomial methods and LMI optimization for fixed-order robust controller

design are described in parts III and IV of:

# D. Henrion. Course on polynomial methods for robust control, LAAS-CNRS

Toulouse, 2001

as well as in the papers

# D. Henrion, M. Sebek, V. Kucera. Positive Polynomials and Robust

Stabilization with Fixed-Order Controllers, IEEE Transactions on Automatic

Control, Vol. 48, No. 7, pp. 1178-1186, July 2003

# D. Henrion, D. Arzelier, D. Peaucelle. Positive Polynomial Matrices and

Improved LMI Robustness Conditions, Automatica, Vol. 39, No. 8, pp.

1479-1485, August 2003.

Note:

Further information:

http://dce.fel.cvut.cz/studium/linearni-maticove-nerovnosti

No time-table has been prepared for this course

The course is a part of the following study plans:

• Doctoral studies, daily studies (compulsory elective course)
• Doctoral studies, combined studies (compulsory elective course)
• Doctoral studies, structured daily studies (compulsory elective course)
• Doctoral studies, structured combined studies (compulsory elective course)