Control-Law Synthesis for Linear Hybrid Systems

The behavior of hybrid systems is described by means of continuous and discrete variables. A number of di erent philosophies have been pursued in order to establish a mathematical framework that is able to handle both, continuous and discrete processes. These range from the construction of hybrid automata [1] and structures [2] over the de nition of interfaces between the continuous and the discrete part [3] to more complex mathematical frameworks [4]. Other approaches use Petri Nets [5], bond-graphs [6] or a synchronuous approach [7], [8], just to name a few. The present work is based on the hybrid automata model introduced by [1]. There a hybrid system is interpreted as consisting of a discrete program (controller) within an analog environment (process). Hybrid automata then are generalized nite state machines where discrete transitions transfer the system between a nite number of control states or locations. The global states of a system can change continuously with time according to the laws of physics and these changes are governed for each control location by a set of di erential equations. Furthermore, each location is labeled with an invariant condition that must hold while the control resides at the location, and each transition is labeled with a guarded set of assignments. This framework has been developed with the purpose to describe linear hybrid systems and to verify that control-laws and speci cations are in harmony with each other, i.e. that the system satis es all invariants when running. In this paper we have reformulated the problem from the veri cation problem `Does the system have a run with the given speci cations and control-laws?' discussed in [1], to a control problem `With the given speci cations, do there exist control-laws that ensure that the hybrid system has correct runs between certain (marked) states?'. In other words, we are asking whether or not the speci cations and restrictions are reasonable, and if yes, a set of control-laws is generated that transfer the system between a number of marked states. In this paper we de ne the notion of controllability for hybrid systems and propose an algorithm that decides for a restricted class of hybrid systems whether or not the system is controllable. If the system is found to be controllable, the algorithm generates