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 - January 2007

Venue and dates

The course is given at the Czech Technical University, Charles Square, down-town Prague (Karlovo Namesti 13, 12135 Praha 2) from Monday January 29, 2007 to Friday February 2, 2007.

The course consists of six two-hour lectures (10am to 12am everyday, 2pm to 4pm on Wednesday, in room K24 of building E) and two two-hour labs (2pm to 4pm on Thursday and Friday, 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.

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.

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 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)

    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:

  • 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)
  • 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 (2h)

    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.


    Last updated on 21 January 2008.