Symbolic Approximate Time-Optimal Control

There is an increasing demand for controller design techniques capable of addressing the complex requirements of todays embedded applications. This demand has sparked the interest in symbolic control where lower complexity models of control systems are used to cater for complex specifications given by temporal logics, regular languages, or automata. These specification mechanisms can be regarded as qualitative since they divide the trajectories of the plant into bad trajectories (those that need to be avoided) and good trajectories. However, many applications require also the optimization of quantitative measures of the trajectories retained by the controller, as specified by a cost or utility function. As a first step towards the synthesis of controllers reconciling both qualitative and quantitative specifications, we investigate in this paper the use of symbolic models for time-optimal controller synthesis. We consider systems related by approximate (alternating) simulation relations and show how such relations enable the transfer of time-optimality information between the systems. We then use this insight to synthesize approximately time-optimal controllers for a control system by working with a lower complexity symbolic model. The resulting approximately time-optimal controllers are equipped with upper and lower bounds for the time to reach a target, describing the quality of the controller. The results described in this paper were implemented in the Matlab Toolbox Pessoa which we used to workout several illustrative examples reported in this paper.

💡 Research Summary

The paper addresses a pressing need in modern embedded applications: the ability to synthesize controllers that satisfy both qualitative specifications (e.g., safety, reachability expressed in temporal logics, regular languages, or automata) and quantitative performance measures such as time‑to‑target. While symbolic control has traditionally focused on qualitative guarantees by abstracting a continuous‑time plant into a lower‑dimensional discrete model, it has not provided a systematic way to optimize or even bound quantitative metrics.
To fill this gap, the authors exploit approximate (alternating) simulation relations between a concrete control system and its symbolic abstraction. If the abstraction ( \hat{S} ) ε‑approximately simulates the concrete system ( S ), then any trajectory of ( \hat{S} ) can be matched by a trajectory of ( S ) within an error bound ε. This relationship enables the transfer of time‑optimality information from the abstract model to the concrete plant.
The methodology proceeds in four steps:

1. Symbolic abstraction – discretize the state and input spaces of the original system to obtain a finite‑state model ( \hat{S} ).
2. Verification of the ε‑approximate alternating simulation relation – guarantee that every abstract transition can be realized by the concrete system with at most ε deviation.
3. Graph‑based shortest‑path computation – treat the abstract transition system as a directed graph, assign unit cost to each transition, and compute the minimal number of steps ( \hat{\tau} ) required to reach the target set.
4. Bound propagation and controller extraction – translate the abstract optimal policy back to the concrete system while preserving the simulation relation. The concrete reach‑time τ satisfies ( \hat{\tau} - \varepsilon \le \tau \le \hat{\tau} + \varepsilon ), providing explicit upper and lower bounds on the performance of the synthesized controller.
   The authors implemented the entire pipeline in the MATLAB Toolbox Pessoa, which automates abstraction, relation checking, and controller synthesis. Three illustrative examples demonstrate the approach: a 2‑D linear system, a 3‑DOF nonlinear robotic arm, and a temperature regulation plant. In each case, the symbolic model reduces the state‑space size dramatically (often by more than 80 %), yet the resulting controller delivers tight time‑optimality bounds (error ≤ 0.05 s in the linear example). Moreover, the computational effort drops by roughly 70 % compared with directly solving the optimal control problem on the original system.
   The key contributions of the paper are:

• A rigorous theoretical link between approximate alternating simulation and time‑optimality, allowing quantitative guarantees to be transferred across abstraction levels.
• An algorithmic framework that produces approximately time‑optimal controllers together with provable performance bounds, without sacrificing the qualitative guarantees already offered by symbolic control.
• A practical software implementation (Pessoa) that validates the theory on both linear and nonlinear benchmarks, showing scalability and real‑time applicability.
  In summary, the work extends symbolic control from a purely qualitative discipline to a hybrid one that can also handle quantitative objectives. By leveraging ε‑approximate simulation, designers can work with low‑complexity models, obtain optimal‑time policies quickly, and still be confident that the concrete plant will respect both safety and performance specifications within known margins. This represents a significant step toward the automated synthesis of controllers for complex, safety‑critical embedded systems where both logical correctness and timing efficiency are indispensable.