Literature

The lecture was partly built upon the chapter 4 of the textbook [1], which in turn was (to a large extent) built upon chapter 2 of the research monograph [2]. None of these two is available online, but fortunately, the latter has a free shorter online version [3]. The chapter 4 (pages 20 through 27) give the necessary material. Possibly, the chapter 3 can serve with some recap of Lyapunov analysis of stability.

The lecture was also partly inspired by the sections 8.2 and 8.3 (pages 158–168) of the text [4], which used to be available online, but has resently dissapeared – most probably it is about to be publised as a textbook.

Some more online resources, in particular for multiple (also piecewise) Lyapunov functions, are [5], [6], [7], [8]. The are all quite readable.

Linear matrix inequalities

The topic of linear matrix inequalities and the related semidefinite programming, which we used for analysis of stability, is dealt with in numerous resources, many of them available online. The monograph [9] was one of the first systematic treatments of the topic and still offers a relevant material. The authors also provide some shorter teaching material [10], tailored to their Matlab toolbox called CVX. Alternatively, the text [11] is even richer by two pages. Another recommendable lecture notes are also available for free: [12]. Finally, a section on Semidefinite programming in the documentation for Yalmip software can also serve as learning resource.

S-procedure

Some treatment of S-procedure is in [9], pages 23 and 24, and [13], page 655.

Sum-of-squares programming

The topic of sum-of-squares programming, which we also relied upon in analysis of stability, is a trending topic in optimization and a wealth of resources are available. As an introduction, the paper [14] is recommendable. The computational problems described in the paper can be solved in Matlab using the SOSTOOLS toolbox. Its documentation [15] can serve as yet another tutorial. Last but not least, YALMIP software contains a well-developed section on Sum-of-squares programming.

References

[1]

H. Lin and P. J. Antsaklis, Hybrid Dynamical Systems: Fundamentals and Methods. in Advanced Textbooks in Control and Signal Processing. Cham: Springer, 2022. Accessed: Jul. 09, 2022. [Online]. Available: https://doi.org/10.1007/978-3-030-78731-8

[2]

D. Liberzon, Switching in Systems and Control. in Systems & Control: Foundations & Applications. Boston, MA: Birkhäuser, 2003. Available: https://doi.org/10.1007/978-1-4612-0017-8

[3]

D. Liberzon, “Switched Systems: Stability Analysis and Control Synthesis,” Lecture {{Notes}}, 2007. Available: http://liberzon.csl.illinois.edu/teaching/Liberzon-LectureNotes.pdf

[4]

J. Lygeros, S. Sastry, and C. Tomlin, “Hybrid Systems: Foundations, advanced topics and applications,” Jan. 2020. Available: https://www-inst.eecs.berkeley.edu/~ee291e/sp21/handouts/hybridSystems_monograph.pdf

[5]

R. A. Decarlo, M. S. Branicky, S. Pettersson, and B. Lennartson, “Perspectives and results on the stability and stabilizability of hybrid systems,” Proceedings of the IEEE, vol. 88, no. 7, pp. 1069–1082, Jul. 2000, doi: 10.1109/5.871309.

[6]

M. Johansson and A. Rantzer, “Computation of piecewise quadratic Lyapunov functions for hybrid systems,” IEEE Transactions on Automatic Control, vol. 43, no. 4, pp. 555–559, Apr. 1998, doi: 10.1109/9.664157.

[7]

S. Pettersson and B. Lennartson, “Hybrid system stability and robustness verification using linear matrix inequalities,” International Journal of Control, vol. 75, no. 16–17, pp. 1335–1355, Jan. 2002, doi: 10.1080/0020717021000023762.

[8]

A. Hassibi and S. Boyd, “Quadratic stabilization and control of piecewise-linear systems,” in Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207), Jun. 1998, pp. 3659–3664 vol.6. doi: 10.1109/ACC.1998.703296.

[9]

S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. in Studies in Applied and Numerical Mathematics. Society for Industrial and Applied Mathematics, 1994. Accessed: Apr. 16, 2021. [Online]. Available: https://web.stanford.edu/~boyd/lmibook/

[10]

S. Boyd, “Solving semidefinite programs using cvx,” Stanford University, Stanford, CA, Lecture Notes for {{EE363}}, 2008. Accessed: Aug. 22, 2024. [Online]. Available: https://stanford.edu/class/ee363/notes/lmi-cvx.pdf

[11]

S. Boyd, “EE363 Review Session 4: Linear Matrix Inequalities,” Stanford University, Stanford, CA, Lecture Notes for {{EE363}}, 2008. Accessed: Aug. 22, 2024. [Online]. Available: https://stanford.edu/class/ee363/sessions/s4notes.pdf

[12]

C. W. Scherer and S. Weiland, “Linear matrix inequalities in control,” Jan. 2015. Accessed: Apr. 16, 2021. [Online]. Available: https://www.imng.uni-stuttgart.de/mst/files/LectureNotes.pdf

[13]

S. Boyd and L. Vandenberghe, Convex Optimization, Seventh printing with corrections 2009. Cambridge, UK: Cambridge University Press, 2004. Available: https://web.stanford.edu/~boyd/cvxbook/

[14]

A. Papachristodoulou and S. Prajna, “A tutorial on sum of squares techniques for systems analysis,” in Proceedings of the 2005 American Control Conference, Portland, OR, USA: IEEE, Jun. 2005, pp. 2686–2700 vol. 4. doi: 10.1109/ACC.2005.1470374.

[15]

A. Papachristodoulou et al., “SOSTOOLS Sums of Squares Optimization Toolbox for Matlab: User’s Guide.” University of Oxford Control Group, Sep. 2021. Available: https://github.com/oxfordcontrol/SOSTOOLS/blob/SOSTOOLS400/docs/sostools.pdf