[Paper Review] Robust Decidability of Sampled-Data Control of Nonlinear Systems with Temporal Logic Specifications

Jun Liu|arXiv (Cornell University)|Mar 15, 2019

Advanced Control Systems Optimization23 references4 citations

TL;DR

This paper establishes the robust decidability of sampled-data control for nonlinear systems with temporal logic specifications by proving that if a robust control strategy exists for a perturbed system, it can be algorithmically constructed via robustly complete finite abstractions. Using a validated forward Euler scheme with quantified error bounds, the method ensures correctness for inter-sampling behaviors through labeling function strengthening, enabling finite-state synthesis for continuous-time systems under mild Lipschitz conditions.

ABSTRACT

This paper explores the theoretical limits of using discrete abstractions for nonlinear control synthesis. More specifically, we consider the problem of deciding continuous-time control with temporal logic specifications. We prove that sampled-data control of nonlinear systems with temporal logic specifications is robustly decidable in the sense that, given a continuous-time nonlinear control system and a temporal logic formula, one can algorithmically decide whether there exists a robust sampled-data control strategy to realize this specification when the right-hand side of the system is slightly perturbed by a small disturbance. If the answer is positive, one can then construct a (potentially less) robust sampled-data control strategy that realizes the same specification. The result is proved by constructing a robustly complete abstraction of the original continuous-time control system using sufficiently small discretization parameters. We illustrate the result with a nonlinear control example.

Motivation & Objective

• To determine whether a computational procedure exists to decide if a control strategy exists for a given formal specification in nonlinear systems.
• To address the lack of completeness guarantees in abstraction-based control synthesis, especially for continuous-time nonlinear systems.
• To establish that robust sampled-data control strategies can be algorithmically constructed when they exist, even under small system perturbations.
• To account for inter-sampling behaviors in control synthesis while preserving correctness in continuous-time semantics.
• To provide a theoretical foundation for scalable, sound, and complete controller synthesis using finite abstractions.

Proposed method

• Constructs a robustly complete abstraction of the sampled-data control system using a validated forward Euler numerical scheme with quantified error bounds.
• Uses a time-discretized transition system that under-approximates the control space and over-approximates reachable sets to ensure soundness and completeness.
• Applies a labeling function strengthening (ε-strengthening) to account for inter-sampling behaviors and ensure correctness under continuous-time semantics.
• Employs feedback refinement relations and simulation relations (e.g., pre-approximation and alternating simulation) to relate the abstract system to the original system.
• Leverages local Lipschitz continuity of the vector field and bounds on system dynamics (L and M) to derive error bounds for the discretization.
• Uses a hierarchy of abstractions indexed by sampling period τ and perturbation levels δ₁ < δ₂ to enable decidability via finite-state search.

Experimental results

Research questions

• RQ1 Can we algorithmically decide whether a robust sampled-data control strategy exists for a continuous-time nonlinear system to satisfy a given temporal logic specification?
• RQ2 Under what conditions can a finite abstraction of a nonlinear sampled-data system be both sound and complete for control synthesis?
• RQ3 How can inter-sampling behaviors be formally accounted for in abstraction-based control synthesis without losing correctness?
• RQ4 Is it possible to construct a control strategy that remains valid under small system perturbations, ensuring robustness?
• RQ5 Can the decidability of control synthesis be guaranteed for any temporal logic specification using finite abstractions of nonlinear systems?

Key findings

• Robust decidability is established: if a robust sampled-data control strategy exists for a system under small perturbations, it can be algorithmically constructed.
• A robustly complete abstraction of the sampled-data system is constructed using a validated forward Euler scheme with error bounds derived from Lipschitz continuity and system dynamics bounds.
• For τ = 0.2, the method achieves a robust abstraction with δ = 0.1 and ε = 0.02, enabling decidability for any temporal logic specification.
• Inter-sampling behaviors are formally accounted for by strengthening the labeling function by ε, ensuring correctness in continuous-time semantics.
• The construction of a finite transition system allows for algorithmic controller synthesis via finite-state search, even for complex nonlinear systems.
• The result extends prior work by proving robust completeness for continuous-time systems, filling a gap left open in earlier studies on discrete-time systems.