B3M35HYS, BE3M35HYS - B231: Literature for LMI, SDP and SOS programming

Linear matrix inequalities (LMI)

The topic of linear matrix inequalities and the related semidefinite programming is dealt with in numerous resources, many of them available online. The following monograph (also available online for free) was one of the first systematic treatments of the topic and still offers a relevant material.

Boyd, Stephen, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan. Linear Matrix Inequalities in System and Control Theory. Studies in Applied and Numerical Mathematics. Society for Industrial and Applied Mathematics, 1994. https://doi.org/10.1137/1.9781611970777. Available online at https://web.stanford.edu/~boyd/lmibook/.

The authors also provide some shorter teaching material, tailored to their Matlab toolbox called CVX

S. Boyd. Solving semidefinite programs using cvx. Lecture notes for EE363. Downloadable at http://stanford.edu/class/ee363/notes/lmi-cvx.pdf, alternativaly, the text https://stanford.edu/class/ee363/sessions/s4notes.pdf is even richer by two pages.

Another recommendable lecture notes are also available for free:

Scherer, Carsten W., and Siep Weiland. ‘Linear Matrix Inequalities in Control’. Lecture notes, January 2015. https://www.imng.uni-stuttgart.de/mst/files/LectureNotes.pdf.

Finally, a very useful material for studying the topic can be found among the tutorials and examples for Yalmip software, which is a Matlab interface to a numerous optimization solvers:

J. Lofberg. Semidefinite programming in Yalmip. https://yalmip.github.io/tutorial/semidefiniteprogramming/

Sum-of-squares (SOS) programming

Sum-of-squares programming is a very trendy research topic in optimization and a wealth of resources are available. For our course, a we will restrict the focus to the analysis of dynamical systems. As an introduction, the following paper is recommendable

Papachristodoulou, A., and S. Prajna. ‘A Tutorial on Sum of Squares Techniques for Systems Analysis’. In Proceedings of the 2005 American Control Conference, 2686–2700 vol. 4. Portland, OR, USA: IEEE, 2005. https://doi.org/10.1109/ACC.2005.1470374.

The computational problems described in the paper can be solved in Matlab using the SOSTOOLS toolbox. Its documentation then solves as yet another tutorial:

Papachristodoulou, Antonis, James Anderson, Giorgio Valmborbida, Stephen Prajna, Peter Seiler, Pablo A. Parrilo, Matthew P. Peet, and Declan Jagt. ‘SOSTOOLS Sums of Squares Optimization Toolbox for Matlab: User’s Guide’. Matlab. University of Oxford Control Group, 14 September 2021. https://github.com/oxfordcontrol/SOSTOOLS/blob/SOSTOOLS400/docs/sostools.pdf.

Last but not least, YALMIP software contains a well-developed section on SOS

Sum-of-squares programming in Yalmip https://yalmip.github.io/tutorial/sumofsquaresprogramming/.

Naposledy změněno: úterý, 14. listopadu 2023, 22.31