We present a systematic approach to the modeling of thermodynamic systems with phase transitions where it is assumed that in each single phase the system's dynamics can be described in terms of equilibrium thermodynamics. This permits the application of well developed methods from contact geometry to identify the system's state space manifold in each phase. The overall dynamics of such thermodynamic systems can be adequately represented within the framework of regional hybrid systems where the discrete state changes autonomously at the sub-manifold boundaries of connected domains in the continuous state space.
For the class of systems under consideration there is a natural optimal control problem wherein the increase of entropy is used as a criterion to be minimized. To illustrate these ideas a hybrid model of a simple thermodynamic system with a liquid-vapour phase transition is presented; the system-theoretic properties of this model are analyzed and a hybrid optimal control problem is formulated.
Finally, we present several results on the stability of interconnected
thermodynamic systems.
Peter Caines received the BA in mathematics from Oxford University in 1967 and the PhD in systems and control theory in 1970 from Imperial College,University of London, under the supervision of David Q. Mayne, FRS. After periods as a postdoctoral researcher and faculty member at UMIST, Stanford, UC Berkeley, Toronto and Harvard, he joined McGill University, Montreal, in 1980, where he is James McGill Professor and Macdonald Chair in the Department of Electrical and Computer Engineering. Peter Caines is a Fellow of the IEEE, SIAM and the Canadian Institute for Advanced Research, was elected to the Royal Society of Canada in 2003 and received the IEEE Control Systems Society Bode Lecture Prize in 2009; he is the author of Linear Stochastic Systems, John Wiley, 1988, and his research interests include stochastic, multi-agent and hybrid systems theory together with their links to physics, economics and biology.
