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

Katholieke Universiteit Leuven, Belgium, 1-2 June 2006

Venue and dates

This course is organized by the ICCoS (Identification and Control of Complex Systems) Scientific Research Network of the Research Foundation - Flanders (FWO - Vlaanderen), and the K.U.Leuven-BOF EF/05/006 Center-of-Excellence on Optimization in Engineering.

The course takes place at the Department of Mechanical Engineering, Katholieke Universiteit Leuven, Celestijnenlaan 300B, 3001 Heverlee, seminar room C (room 03.042), on Thursday, June 1 and Friday, June 2, 2006.

It consists of six 90-minute lectures (9:00-10:30, 11:00-12:30, 14:00-15:30, 16:00-17:30 on June 1, 9:00-10:30, 11:00-12:30 on June 2) and one 3-hour lab (14:00-17:00 on June 2).

Participation is free of charge for doctoral students and participants from Belgian academic institutions. To register, email to Prof. Jan Swevers. The number of places for the lab-session is limited since it takes place in a PC-room. You will be accepted for the lab-session as long as capacity of this PC-room is not exceeded, and this will be decided on a first-come-first-serve basis.

Outline

The course starts with basic mathematical features of linear and bilinear matrix inequalities:

  • Part I.0: general introduction, course outline and material
  • Part I.1: historical developments of LMIs and BMIs, convexity, cones, duality, semidefinite programming (1h30)
  • Part I.2 and Part I.2b: classification of convex semialgebraic sets that can be represented with LMIs, lift and project techniques, BMIs (1h30)
  • 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)
  • Part I.4: basics of interior-point algorithms, latest achievements in software and solvers for LMIs and BMIs (1h30)

    Then 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 (1h30)
  • 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 (1h30)

    For the labs we use the YALMIP interface, SeDuMi and PENBMI to define and solve LMI and BMI problems under the Matlab environment (3h).

    Last updated on 13 December 2006