Literature

The same resources as for the previous lecture on stability.

Some more online resources, in particular for multiple (also piecewise) Lyapunov functions, are [1], [2], [3], [4]. The are all quite readable.

Besides the LMI and SOS programming/optimization, we need one particular result known as the S-procedure. Some treatment of S-procedure is in [5], pages 23 and 24, and [6], page 655.

References

[1]

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.

[2]

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.

[3]

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.

[4]

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.

[5]

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/

[6]

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/