Course on LMI optimization with applications in control
(Didier Henrion, LAAS-CNRS, Toulouse, France)
Czech Technical University,
Prague, Czech Republic - November 2003
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 November 10, 2003 to Friday November 14, 2003.
It consists of five two-hour lectures (10am to 12am) and three
two-hour labs (2pm to 4pm).
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 and Slides
In the first part of the course, historical developments of
LMIs and SDP are surveyed. Convex sets that can be represented with
LMIs are classified and studied. LMI relaxations are introduced to
solve non-convex polynomial optimization problems. Finally,
interior-point algorithms are described to solve LMI problems and latest
achievements in software and solvers are reported.
The second part 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 Hinf 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.
See a more recent version of the course
for downloadable lecture slides.
>
Part I - LMI optimization
I.0 - General introduction:
course outline and material (0h10)
I.1 -
What is an LMI ?:
history, convexity, cones, duality, semidefinite programming and
applications (1h20)
I.2 -
Geometry of LMI sets:
convex semialgebraic sets, conic and LMI representable sets,
lift and project techniques (1h)
I.3 -
LMI relaxations:
Shor's relaxation, polynomial moments and sum-of-squares,
hierarchy of LMI relaxations (1h30)
I.4 -
Solving LMIs:
interior-point methods, solvers and interfaces (1h)
Part II - LMIs in control
II.1 -
State-space methods - stability analysis:
Lyapunov stability, pole placement in LMI regions,
uncertain systems and robustness analysis (1h30)
II.2 -
State-space methods - control design:
H2, Hinf, robust state-feedback and output-feedback design (1h)
II.3 -
Polynomial methods - stability analysis:
polynomials in control, robust stability of polynomials (1h)
II.4 -
Polynomial methods - control design:
robust fixed-order controller design (1h30)
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.
Acknowledgements
Thanks to (in alphabetical order) Denis Arzelier, Miguel Bernal, Pavel
Burget, Martin Cech, Marcin Cychowski, Tiziano Fiorenzani, Michal
Kocvara, Mahbub Gani, Ivan Havel, Ales Kruczek, Zdenek Hanzalek,
Zdenek Hurak, Jiri Mertl, Arturo Molina-Cristobal, Volkan Nalbantoglu,
Sorin Olaru, Wojciech Paszke, Dimitri Peaucelle, Jaroslav Pekar,
Stanislaw Pietrzko, Jiri Roubal, Burak Seymen, Jaroslav Sobota, Jan
Stecha, Jos Sturm, Petr Urban and Lieven Vandenberghe for their
feedback.
Last updated on 13 December 2006