Course on LMI optimization with applications in control
(Didier Henrion, LAAS-CNRS, Toulouse, France and FEL-CVUT, Prague, Czech Republic)

Czech Technical University, Prague, Czech Republic - March and April 2008

Venue and dates

The course is given at the Czech Technical University, Charles Square, down-town Prague (Karlovo Namesti 13, 12135 Praha 2).

The course consists of ten 90-minute lectures on Thursdays March 27, April 3, April 10, April 17 and April 24, from 9:15 to 10:45 and from 11:00 to 12:30, in room K2 of building E.

Registration

The course is primarily aimed at students from the Czech Technical University in Prague, yet external participants are welcome. There is no registration fee. Please note that the Czech Technical University will not provide assistance regarding traveling and accomodation in Prague.

Contact Didier Henrion if you are interested in attending the course.

Description

This is a course for graduate students or researchers with a background in linear control systems, linear algebra and convex optimization.

The focus is on semidefinite programming (SDP), or optimization over linear matrix inequalities (LMIs), an extension of linear programming to the cone of positive semidefinite matrices. Since the 1990s, LMI methods have found numerous applications mostly in combinatorial optimization, systems control and signal processing.

Outline

See a more recent version of the course for downloadable lecture slides.

On Monday February 2, the course starts with fundamental mathematical features of linear matrix inequalities:

  • Part I.0: general introduction, course outline and material (0h30)
  • Part I.1 historical developments of LMIs, convexity, cones, duality, semidefinite programming (2h30)

    Then on Tuesday February 3, we cover latest achievements in semidefinite programming and real algebraic geometry:

  • Part I.2: classification of convex semialgebraic sets that can be represented with LMIs, lift and project techniques (1h)
  • Part I.2b: non-convex BMIs (0h30)
  • Part I.3: Shor's relaxation, polynomial moments and sum-of-squares, Lasserre's hierarchy of LMI relaxations to solve non-convex polynomial optimization problems, including BMIs (1h30)

    On Thursday April 10, we may start computer experiments. For the labs, we use the YALMIP interface, SeDuMi and PENBMI to define and solve LMI and BMI problems under the Matlab environment (1h30). We also briefly survey recent developments in semidefinite solvers and software packages:

  • Part I.4: basics of interior-point algorithms, latest achievements in software and solvers for LMIs and BMIs (1h30)

    The end of the course focuses on the application of LMI techniques to solve several control problems traditionally deemed as difficult, such as robustness analysis of linear and nonlinear systems, or design of fixed-order robust controllers with H-infinity specifications. The originality of the approach is in the simultaneous use of algebraic or polynomial techniques (as opposed to classical state-space methods) and modern convex optimization techniques. On Thursday April 17 we will cover:

  • Part II.1 and Part II.2: State-space methods - stability analysis: Lyapunov stability, pole placement in LMI regions, uncertain systems and robustness analysis - controller design: H2, Hinf, robust state-feedback and output-feedback design (3h)

    Finally, on Thursday April 24, we close up the course with

  • Part II.3 and Part II.4: Polynomial methods - stability analysis: polynomials in control, robust stability of polynomials - controller design: robust fixed-order controller design (3h)

    Homeworks and Labs

    Homeworks are handed out during the course. Some of this material is used during the labs. Full written solutions to the homeworks and labs (with Matlab scripts) are available on request.

    For the labs we use the Polynomial Toolbox and the YALMIP interface to define and solve LMI problems under the Matlab environment.

    References

    Very good references on convex optimization (including LMIs) are the textbooks:

  • 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.

    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.

    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, Merida, Venezuela, 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.

    Finally, the software used for the labs are YALMIP by Johan Lofberg, PENOPT by Michal Kocvara and Michael Stingl (contact the authors for a free developer version), SeDuMi by Jos Sturm and the Advanced Optimization Lab at McMaster University, and GloptiPoly by Didier Henrion and Jean-Bernard Lasserre.

    Last updated on 13 December 2006