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.