Topological Feasibility Guarantees for Differentiable Predictive Control

This paper establishes deterministic feasibility guarantees for differentiable predictive control (DPC) using a novel topological analysis of the induced reachable safe set, without requiring online safety filters. The authors exploit the inherent model-based nature of DPC, in which differentiable system dynamics are embedded directly into the computational graph, to analyze the properties of the learned control policies and the corresponding system states from topological and geometric perspectives.

The paper proposes a self-supervised offline policy learning strategy that utilizes a proxy loss with Control Barrier Functions (CBFs). These properties not only significantly improve policy training but also enable the derivation of strict, deterministic feasibility guarantees from a finite number of training samples. Extensive closed-loop simulations validate the theoretical findings, demonstrating that the empirical constraint violations monotonically decrease to zero as the training sample size increases.

The main contributions are:

1. A novel topological framework for analyzing the feasibility of DPC policies, establishing deterministic closed-loop feasibility guarantees from a finite number of offline training samples under mild assumptions.

2. A self-supervised offline learning strategy based on a CBF proxy loss that constructs neural control policies satisfying feasibility by design, eliminating the need for online safety filters or optimization-based corrective actions.

3. Demonstration of the effectiveness of the proposed framework on several nonlinear control benchmarks, with theoretical guarantees corroborated by empirical results showing that constraint violations vanish as the number of training samples increases.

The paper proves that the set of reachable safe state sequences can be covered by finitely many open sets (Theorem 3.10), and that with the increase of the number of training data samples, the probability of inferred controls and corresponding states violating feasibility conditions is non-increasing and converges to 0 with a finite number of samples (Theorem 3.11). The authors also derive an explicit feasibility radius (Lemma 3.12) and a maximal required sample size (Proposition 3.13).

The proposed CBF-proxy DPC algorithm uses a two-stage homotopy training scheme: Stage 1 involves warm-start via soft penalties, and Stage 2 involves offline CBF-proxy fine-tuning. The paper proves deterministic closed-loop feasibility (Proposition 4.1) under the condition that the training dataset is sufficiently dense such that the induced open feasibility balls cover the reachable safe set at each time step.

Numerical results on three problems—nonholonomic mobile robot navigation, unstable linear system stabilization, and constrained quadcopter system stabilization—validate the theoretical findings, showing that the empirical constraint violations monotonically decrease to zero as the training sample size increases.

Improvements to AI Systems:

1. Guaranteed-Safe Learning-Based Controllers: AI systems for autonomous control (e.g., drones, robots, self-driving cars) can be trained offline to produce policies that are deterministically feasible—meaning they never violate safety constraints during deployment—without needing a real-time safety filter. This eliminates the computational overhead and failure modes of online optimization layers.

2. Finite-Sample Feasibility Certification: AI systems can now certify safety from a finite number of training samples. Given a dense-enough dataset, the system can provably guarantee that all inferred control actions and resulting states stay within safe bounds, with a computable maximal sample size needed for a desired safety margin.

3. Self-Supervised Safety Fine-Tuning: The CBF-proxy loss enables AI systems to fine-tune pre-trained neural controllers in a self-supervised manner (no labeled data or environment interaction). This improves training convergence and ensures the final policy is feasible by construction, even for unstable or highly nonlinear dynamics.

4. Topological Coverage for Exploration: The novel topological framework (covering reachable safe sets with finitely many open balls) allows AI systems to systematically sample training data that guarantees full coverage of the safe state space, rather than relying on random or heuristic exploration. This reduces data requirements and improves generalization.

5. Monotonic Violation Reduction: The improved AI system can monitor its own training progress—empirical constraint violations monotonically decrease to zero as sample size grows. This provides a reliable stopping criterion and a quantitative measure of safety during learning, useful for adaptive or continual learning settings.

6. Filter-Free Deployment: The resulting AI system can be deployed directly on hardware without any external safety layer, reducing latency and complexity. It is particularly beneficial for resource-constrained platforms (e.g., embedded quadcopters) where running a safety filter in real-time is infeasible.

7. Provably Safe Stabilization of Unstable Systems: The system can learn to stabilize unstable dynamics (e.g., inverted pendulum, quadcopter) while respecting state/input constraints, with formal guarantees that the closed-loop trajectory never leaves the safe set—a capability that current RL or imitation learning approaches lack.

8. Sample-Efficient Safety Verification: Instead of exhaustive simulation or formal verification, the AI system can use the derived feasibility radius (Lemma 3.12) to verify safety with a minimal number of test points, reducing verification cost for safety-critical applications.