A symbolic approach to decentralized supervisory control of hybrid systems

Abstract—We suggest a synthetic symbolic approach to decen-tralized supervisory control for the class of hybrid dynamical sys-tems that can be modelled as hybrid state machines with a finiteexternal signal space. The decentralized computational schemerepresents a conjunction of a finite number of subsupervisors,which are invoked by a decomposition of the external signalspace. On this basis, we derive sufficiency conditions for theexistence of the solution to the decentralized supervisory controlproblem, which provide a suitable initial point to a constructiveapproach of appropriate decompositions of the signal space.Index Terms—decentralized control, hybrid dynamical sys-tems, supervisory control, decomposition I. I NTRODUCTION Hybrid dynamical control systems arise at heterogeneoussystems that exhibit a coupled time-triggered and event-triggered dynamics. A variety of control problems for suchsystems have been receiving extensive attention (see e.g. [5],[4], etc.). A popular approach has been the abstraction of con-tinuous dynamics in attempting to remove the aforementionedheterogeneity (e.g. see [8]). Such approaches lead necessarilyto a discrete event systems framework, where a rich toolsetfor control and verification exist [2], [7], etc. In this paper, wefollow a similar approach in investigating the decentralizedsupervisory control for the class of systems that can becaptured by a hybrid state machine with a finite external signalspace [6], [1], etc.Decentralized control is typically motivated by the inher-ent distribution of many complex systems, by computationaltime/space complexity reduction and robustness. Therefore, weinitially introduce a decomposition of the original signal spaceinto a finite number of aggregate signal spaces of a lowercardinality by means of, what we call here, the aggregationmaps. Note that this concept is different, though closelyrelated to the natural projections [3]. Thereby, each of theintroduced aggregate signal spaces invokes a state machineby relabeling the symbols of the original one. Intuitively, thismay be interpreted as a setup with a finite number of “coarse”sensors and actuators. Effectively, the information collectedfrom the different discrete-valued sensors is complete, yetcoarsely quantized, which represents a subtle difference to thesensory set with partial observations. In this article we derivea few of sufficient conditions which guarantee the existence ofthe solution to the supervisory decentralized control problemwithin our framework and also relate them to conditions ofsignal space decomposition.The remainder of the paper is organized as follows. InSection II the reader is made familiar with the used notationand the basic preliminary concepts on systems and represen-tation in the context of the behavior theory [9]. Section IIIrepresents the core of the work. We define here the con-junctive decentralized supervisory control scheme, and discussthoroughly the concepts of coobservability, non-conflictnessand implementability. The main results are summarized inSectionIII-E. Plenty of examples are elaborated to clarify andillustrate the concepts and statements.II. P