Overlapping decompositions, expansions, contractions, and stability of hybrid systems

Inclusion and extension principles are defined and studied for hybrid systems. The controller design problem for hybrid systems is then considered within the framework of the extension principle. It is shown that if the extension principle is used, then any controller designed in the expanded spaces is contractible to the original spaces for implementation. The stability of hybrid systems is also considered. A number of definitions of stability for hybrid systems are given and it is shown that if a system, which includes another system, is stable, then the included system is also stable. Furthermore, it is shown that, if a controller designed for the expanded system achieves stability, then the contracted controller also achieves stability for the original system. Decentralized controller design with overlapping decompositions is also discussed within the framework of extension. An example is provided to illustrate the proposed approach.