Literature

The MLD description of discrete-time hybrid systems was originally introduced in [1], but perhaps even more accessible introduction is in Chapter 16 of the freely downloadable book [2]. In our text here we followed their expositions.

Just in case some issues are still unclear, in particular those related to the connection between the constraints (inequalities) imposed on continuous (aka real) variables and logical conditions imposed on binary variable, you may like the little bit more formal discussion in Section 2.2 of the thesis [3]. Strictly speaking, this use of binary (0-1 integer) variables to encode some constraints on real variables is standard in optimization and is described elsewhere – search for indicator variables or indicator constraints. A recommendable general resource is the book (unfortunately not available online) [4], in particular its section 9.1.3 on Indicator variables.

All the theoretical concepts and procedures introduced in this lecture (and in those corresponding papers and books) are straightforward but rather tedious to actually implement. There is a HYSDEL language for modelling hybrid systems (discrete hybrid automata as considered in this lecture) that automates these procedures. The HYSDEL language is described not only in the documentation but also in the dedicated paper [5], which can also serve as a learning resource for the topic.

Case studies

Batch evaporator [6] and [7].

References

[1]

A. Bemporad and M. Morari, “Control of systems integrating logic, dynamics, and constraints,” Automatica, vol. 35, no. 3, pp. 407–427, Mar. 1999, doi: 10.1016/S0005-1098(98)00178-2.

[2]

F. Borrelli, A. Bemporad, and M. Morari, Predictive Control for Linear and Hybrid Systems. Cambridge, New York: Cambridge University Press, 2017. Available: http://cse.lab.imtlucca.it/~bemporad/publications/papers/BBMbook.pdf

[3]

D. Mignone, “Control and estimation of hybrid systems with mathematical optimization,” Doctoral {{Thesis}}, ETH Zurich, 2002. doi: 10.3929/ethz-a-004279802.

[4]

H. P. Williams, Model Building in Mathematical Programming, 5th ed. Hoboken, N.J: Wiley, 2013.

[5]

F. D. Torrisi and A. Bemporad, “HYSDEL—a tool for generating computational hybrid models for analysis and synthesis problems,” IEEE Transactions on Control Systems Technology, vol. 12, no. 2, pp. 235–249, Mar. 2004, doi: 10.1109/TCST.2004.824309.

[6]

A. Bemporad, F. D. Torrisi, and M. Morari, “Discrete-time Hybrid Modeling and Verification of the Batch Evaporator Process Benchmark,” European Journal of Control, vol. 7, no. 4, pp. 382–399, Jan. 2001, doi: 10.3166/ejc.7.382-399.

[7]

S. Kowalewski and O. Stursberg, “The Batch Evaporator: A Benchmark Example for Safety Analysis of Processing Systems under Logic Control,” in Proceedings 4th Int. Workshop on Discrete Event Systems (WODES’98), Cagliari, Italy: IEE, London, Aug. 1998, pp. 302–307.