Sample-based Safe Design Constraint

• SDC is a finite sample-based constraint framework that guarantees hard safety properties in dynamic, control, and learning systems.
• It employs techniques such as convex polytopic sampling, barrier functions, and scenario approaches to ensure forward invariance and robust feasibility.
• Practical implementations use adaptive heuristics, model predictive control, and data-driven error bounds to balance conservatism with computational efficiency.

A Sample-based Safe Design Constraint (SDC) is any constraint imposed using a finite set of samples (trajectories, parameters, functions, or system states) to guarantee hard safety properties of dynamical, control, or learning systems subject to sampled execution, parametric uncertainty, or limited information. SDCs generalize classical safety constraints by leveraging sampled-data, scenario-based, or simulation-driven information rather than requiring explicit analytic models or infinite-dimensional verifications. The SDC paradigm encompasses techniques for sampled-data systems, chance-constrained design, model learning, and safety filters, and provides rigorous finite-sample or practical guarantees for constraint satisfaction in continuous and discrete-time, linear and nonlinear, and even learning-based controllers.

1. SDCs in Sampled-Data Control: Viability and Barrier Approaches

Sampled-data SDCs appear prominently in constraint satisfaction for systems evolving under discrete-time control but continuous-time dynamics. In LTI sampled-data systems, the viability kernel

$$\mathcal{V} = \left\{ x_0 \in K \mid \exists u(\cdot) \in U_T \;\; \forall t \in [0,\tau]:\; x_{x_0}^u(t)\in K \right\}$$

characterizes all initial conditions from which admissible controls keep the state within safe set $K$ for a time horizon. The SDC framework under-approximates $\mathcal{V}$ via convex polytopic sampling: rays from an interior point are extended to the boundary using feasibility LPs to validate safety on each sample, incorporating inter-sampling and discretization errors by eroding $K$ accordingly. Convergence is guaranteed as the number of sampled rays grows, with volumetric error decaying as $O(N^{-2/(n-1)})$ for state dimension $n$. Heuristically biased sampling (using vMF distributions) improves exploration in high-curvature regions, further accelerating convergence. This methodology achieves scalable robust safety guarantees in high dimensions, as demonstrated on 12D flight envelope protection (Kaynama et al., 2014).

Barrier-function SDCs provide sampled-data versions of continuous-time safety-theoretic methods. The guarantee of forward invariance (i.e., safety) is modified to account for sampling effects and inter-sample behavior. Methods include:

• Continuous-time CBF-motivated SDCs, adding a sampling-dependent controller margin;
• Discrete-time barrier-function SDCs, employing a Taylor expansion and including second-derivative error terms;
• Zero-Order CBFs (ZOCBFs), defining SDCs purely by stepwise increments $\Delta h = h(x_{k+1}) - h(x_k)$ and bypassing the need for derivatives.

These SDCs are embedded as affine or convex constraints in quadratic programs or nonlinear programs solved at each sampling instant, ensuring provable forward invariance provided the SDC is satisfied at each step (Breeden et al., 2021, Tan et al., 2024).

2. SDCs in Learning and Model Predictive Control under Uncertainty

In learning-based and model-predictive control (MPC) contexts with model uncertainty, SDCs enable safe controller synthesis using finite samples from uncertainty sets. In Gaussian Process (GP) dynamics, the finite-sample SDC constructs an over-approximate reachable set by propagating candidate dynamical models sampled from the GP posterior. The sample complexity theorem guarantees (with high probability $1-\delta$) that the true system trajectory lies within the convex hull of the sampled reachable sets, provided the number of samples $$N = O(\epsilon^{-2}[n \log(1/\epsilon) + \log(1/\delta)])$$ for prescribed confidence and coverage tightness $\epsilon$. MPC constraints are then imposed not on the (unknown) true reachable set, but on the constructed sample-based tube, tightened via Lipschitz bounds, ensuring with high probability that the closed-loop trajectory never violates safety constraints $g(x) \leq 0$ (Prajapat et al., 12 May 2025).

In policy optimization, SDCs are formulated directly in parameter space when safety constraints are only available via rollout-based black-box evaluation. The SCPO method projects raw gradient steps onto a trust region defined by empirical samples and local Lipschitz bounds on the constraint metrics, formulating a convex SOCP that provably maintains the safety constraint at every iterate if the initial parameter is feasible. This safe-by-induction principle ensures recursive feasibility and constraint satisfaction throughout the learning process (Cao et al., 15 Dec 2025).

3. Scenario and Chance-Constrained Optimization SDCs

The scenario approach defines SDCs by requiring feasibility for all sampled “scenarios” of parametric uncertainty. Solutions to scenario programs correspond to SDCs that are $ε$-feasible for the original chance-constraint with high confidence, provided enough samples $N \geq (2/ε)(\log(1/β) + n)$. Under regular-variation conditions, constraint scaling (shrinking the right-hand-side by a factor $s \geq 1$) yields an exponential reduction in sample size while guaranteeing violation probabilities $P(\text{violation}) \leq ε^{1+o(1)}$, making SDCs far more computationally tractable for stringent safety requirements (Choi et al., 2024).

4. SDCs for Nonlinear, Unknown, and High-Relative-Degree Systems

For nonlinear and unknown systems, SDCs must address unknown drift and inter-sample deviations. In the CBF approach for unknown sampled-data systems, Lipschitz and data-driven error bounds provide a feasible “margin” $E(\Delta t,u_k)$ ensuring that the CBF condition holds not just at sampling instants but throughout the inter-sample interval, guaranteeing safety even under model uncertainty. The optimization is generally nonconvex but can be decomposed into a two-stage procedure: convex inner approximation followed by projection onto a feasible control set. Margin terms can be adaptively tightened as more data is collected, reducing conservatism (Niu et al., 2021).

The SACBF (Sampling-Aware CBF) framework extends this to high-relative-degree or multi-objective safety and reachability by using Taylor approximations between sample times, introducing explicit bounds on higher derivatives in the SDC. Relaxed SACBFs introduce slack variables penalized in the cost, ensuring feasibility under potentially conflicting constraints while maintaining forward invariance (Liu et al., 14 Nov 2025).

For sampled-data systems with state- and input-dependent safety constraints or arbitrary relative-degree, ZOCBFs generalize by imposing SDCs via difference-based increments, dispensing with derivative calculations. Enforcement can be via linearization, numerical integration, or parallel simulation over candidate controls. Provided a robustness margin is included, this methodology is provably equivalent to continuous-time CBFs in the small-sampling-period limit and achieves efficient real-time execution in complex safety-regimes (Tan et al., 2024).

5. Theoretical Guarantees and Quantitative Conservatism

The SDC paradigm yields both finite-sample and asymptotic safety guarantees, with conservatism and computational cost depending critically on the form of the constraint, margin, and the number and placement of samples. Analytical bounds are provided for:

• Controller margin: quantifying the additional conservatism vs. continuous-time constraints;
• Physical (set) margin: describing the contraction of the safe set due to sampling;
• Volumetric convergence: rate of under-approximating the viability kernel with polytopic samples.

Discrete-time–motivated SDCs, such as those based on Taylor expansions, can reduce both controller and physical margin by up to a factor of two compared to continuous-time margined bounds. Numerical evidence indicates substantial computational gains (e.g., 8× reduction in sample size for constraint scaling SDCs at modest scaling factors) and improved feasibility for real-time control applications (Breeden et al., 2021, Choi et al., 2024).

6. Implementation Considerations and Scalability

Practical deployment of SDCs requires careful selection of sample size, sampling strategy, model approximation (e.g., Runge–Kutta order), and computational tractability of the SDC enforcement (QP/SOCP/NLP). Memory and computation scale linearly with the number of samples for polytopic SDCs and can be managed even in dimensions $\gg 10$. Biasing and adaptive tightening heuristics further improve efficiency in high-dimensional or high-curvature regimes. The SDC design pipeline typically consists of constraint identification, numerical enforcement choice, explicit margin computation, and real-time optimization at each sampling instant or learning iteration (Kaynama et al., 2014, Tan et al., 2024).

7. Applications, Numerical Studies, and Limitations

SDCs have been demonstrated in high-dimensional flight-envelope protection, robust pole assignment, recursive safe learning in policy optimization, unicycle obstacle avoidance, collision and rollover prevention, and nonlinear inverted pendulum stabilization. In all cases, SDCs achieved hard safety at all times—even in the presence of uncertainty, unknown dynamics, or adversarial learning—whereas non-SDC baselines failed or violated safety.

A plausible implication is that SDC methods provide a unifying architecture for robust, scalable safety enforcement in both model-based and learning-based systems. However, trade-offs between conservatism and sample effort, error-reset mechanisms for higher-order and time-varying constraints, and scalability for very large-scale or black-box systems remain active research topics (Cao et al., 15 Dec 2025, Liu et al., 14 Nov 2025).

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References:

(Kaynama et al., 2014, Breeden et al., 2021, Niu et al., 2021, Taylor et al., 2022, Choi et al., 2024, Tan et al., 2024, Prajapat et al., 12 May 2025, Liu et al., 14 Nov 2025, Cao et al., 15 Dec 2025)