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Brief paper: Controller synthesis for safety and reachability via approximate bisimulation
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This paper deals with the synthesis of state-feedback controllers using approximately bisimilar abstractions with an emphasis on safety problems. Such problems consist in synthesizing a controller that restricts the behaviors of a system so that its outputs remain in some specified safe set. One is usually interested in designing a controller that is as permissive as possible since this enables to ensure, a posteriori, secondary control objectives. Using the natural refinement technique for approximately bisimilar abstractions, a controller for a system can be synthesized from a controller for an abstraction. However, these controllers have some limitations in terms of performances, implementation cost and robustness. The main contribution of this paper is a new procedure for the synthesis of controllers for safety specifications using approximately bisimilar abstractions. Given a controller for an abstraction, we describe an approach for synthesizing a controller for the system that does not suffer from the previous limitations. Moreover, we show that if the controller of the abstraction is the maximal controller (i.e. the most permissive) then we can evaluate the distance between the synthesized controller and the maximal controller of the system. This distance can be made arbitrarily small by choosing sufficiently precise abstractions. We apply our results to synthesis problems for switched systems.