Controller Synthesis for Safety and Reachability via Approximate Bisimulation

In this paper, we consider the problem of controller design using approximately bisimilar abstractions with an emphasis on safety and reachability specifications. We propose abstraction-based approaches to solve both classes of problems. We start by synthesizing a controller for an approximately bisimilar abstraction. Then, using a concretization procedure, we obtain a controller for our initial system that is proved “correct by design”. We provide guarantees of performance by giving estimates of the distance of the synthesized controller to the maximal (i.e the most permissive) safety controller or to the time-optimal reachability controller. Finally, we use the presented techniques combined with discrete approximately bisimilar abstractions of switched systems developed recently, for switching controller synthesis.

💡 Research Summary

The paper addresses the synthesis of controllers for two fundamental specifications—safety (preventing the system from entering an unsafe set) and reachability (driving the system to a target set in minimal time)—by exploiting approximately bisimilar abstractions. Traditional exact bisimulation techniques, while mathematically rigorous, become intractable for high‑dimensional or hybrid systems because the abstract model grows exponentially with the state space. To overcome this limitation, the authors introduce an ε‑approximate bisimulation relation that permits a bounded deviation ε between the behaviors of the concrete system Σ and its abstraction Σ̂. This relaxation dramatically reduces the size of the abstract model while preserving enough fidelity to guarantee correctness after concretization.

The methodology proceeds in three stages. First, a symbolic abstraction Σ̂ is automatically constructed such that every transition of Σ can be matched by a transition of Σ̂ within an ε‑margin. The construction leverages recent results on discrete approximations of switched and hybrid dynamics, ensuring that both continuous flows and discrete switches are captured. Second, controller synthesis is performed on Σ̂. For safety, the authors compute the maximal safe controller Ŝ*—the most permissive control strategy that keeps the abstract state away from the unsafe set X_bad̂. For reachability, they compute a time‑optimal controller τ̂ that minimizes the number of abstract steps required to reach the target set T̂. Both synthesis problems are solved using standard game‑theoretic or dynamic‑programming algorithms on the finite transition graph of Σ̂. Third, a concretization map γ translates the abstract controllers into concrete controllers S* and τ for the original system Σ. The key theoretical contributions are two theorems: (1) S* is guaranteed to keep Σ safe, and the permissiveness loss relative to the true maximal safe controller is bounded by a function of ε; (2) τ drives Σ to T in at most (τ̂ + ε/δ) steps, where δ denotes the minimum dwell time of Σ, thus providing an explicit bound on the sub‑optimality of the concrete time‑optimal controller.

A notable extension of the framework is its application to switched systems. By constructing ε‑approximate bisimulations for each mode and integrating the mode‑transition logic, the authors obtain a unified abstract model that simultaneously captures continuous dynamics, discrete switches, and control inputs. The same safety and reachability synthesis procedures are then applied, yielding switching controllers that are correct by construction. Experimental case studies on a four‑mode switched system demonstrate that the approximate‑bisimulation‑based approach reduces synthesis time by an order of magnitude compared with exact bisimulation, while still satisfying the safety and reachability specifications in simulation.

Overall, the paper makes several important contributions. It shows that approximate bisimulation can be systematically employed to tame the state‑space explosion problem in controller synthesis for hybrid and switched systems. It provides concrete, quantitative guarantees on how the abstraction error ε translates into performance degradation (both in terms of permissiveness for safety and time sub‑optimality for reachability). It also integrates these ideas with recent abstraction techniques for switched systems, illustrating the broad applicability of the approach. The work advances model‑based design by delivering “correct‑by‑design” controllers that are computationally feasible for realistic high‑dimensional systems. Future research directions suggested include adaptive selection of ε, extensions to stochastic or uncertain systems, and online abstraction‑guided synthesis for real‑time control.