Data-based Low-conservative Nonlinear Safe Control Learning

Abstract: This paper develops a data-driven safe control framework for nonlinear discrete-time systems with parametric uncertainty and additive disturbances. The proposed approach constructs a data-consistent closed-loop representation that enables controller synthesis and safety certification directly from data. Unlike existing methods that treat unmodeled nonlinearities as global worst-case uncertainties using Lipschitz bounds, the proposed approach embeds nonlinear terms directly into the invariance conditions via a geometry-aware difference-of-convex formulation. This enables facet- and direction-specific convexification, avoiding both nonlinearity cancellation and the excessive conservatism induced by uniform global bounds. We further propose a vertex-dependent controller construction that enforces convexity and contractivity conditions locally on the active facets associated with each vertex, thereby enlarging the class of certifiable invariant sets. For systems subject to additive disturbances, disturbance effects are embedded directly into the verification conditions through optimized, geometry-dependent bounds, rather than via uniform margin inflation, yielding less conservative robust safety guarantees. As a result, the proposed methods can certify substantially larger safe sets, naturally accommodate joint state and input constraints, and provide data-driven safety guarantees. The simulation results show a significant improvement in both nonlinearity tolerance and the size of the certified safe set.

• The paper introduces a direct data-driven safe controller synthesis framework that leverages geometry-aware, difference-of-convex formulation for nonlinear systems.
• It contrasts conventional Lipschitz-based methods by embedding nonlinear dynamics into facet-specific invariance conditions, substantially reducing control conservatism.
• Empirical evaluations demonstrate up to 433× improvement in admissible nonlinearity and enlarged invariant sets under practical state, input, and disturbance constraints.

Data-based Low-conservative Nonlinear Safe Control Learning: Technical Analysis

Problem Context and Limitations of Existing Approaches

Certification of safety in nonlinear discrete-time systems under uncertainty is a core challenge in control theory, particularly as deployed systems face parametric uncertainty and additive disturbances. Prevailing paradigms—control barrier functions (CBFs), control merging, and model predictive control (MPC)—address safety but introduce either computational intractability, excessive conservatism, or reliance on indirect model learning with limited efficacy for nonlinear, uncertain systems. Conventional CBF extensions to discrete-time systems suffer significant nonconvexities, while control merging approaches are often infeasible for high-dimensional nonlinear systems subject to disturbances. MPC frameworks, though systematic, resort to terminal set design and conservative over-approximations, especially when coupled with linear controllers for nonlinear dynamics.

Data-driven approaches bifurcate into indirect (system identification plus control) and direct (controller synthesis sans explicit model identification) methodologies. The latter promises tighter integration with control objectives but, in the nonlinear setting, has been predominantly limited to either norm minimization of nonlinear terms (effectively disturbance attenuation) or conservative absorption of nonlinearity via global Lipschitz bounds. These treatments inevitably result in restricted invariant sets and poor exploitation of nonlinear dynamics' structure.

Formulation: Difference-of-Convex Synthesis and Geometry-aware Control

The focal contribution is a direct, data-driven safe controller synthesis framework for nonlinear discrete-time systems:

$x(t+1) = A_1 x(t) + A_2 S(x(t)) + Bu(t) + w(t),$

where nonlinearities $S(\cdot)$ are known up to parameterization and $A_1, A_2, B$ are unknown. The only accessible data are input-state trajectories. The method addresses the certification of a robust invariant polytopic safe set defined as $\mathcal{S}(F, g) = \{x: Fx \le g\}$ with explicit state and input constraints.

Distinctively, the controller parameterization is constructed to avoid the two main sources of conservatism in the literature:

1. Nonlinearity Minimization via Lipschitz Bounds: Previous direct data-driven control methods [e.g., "Learning controllers for nonlinear systems from data" [Data4]] treat unmodeled nonlinear terms as disturbances, enforced via global Lipschitz norm constraints. The closed-loop is designed by canceling nonlinearities or minimizing their norm, often resulting in unnecessarily conservative invariant sets, especially when nonlinearity is beneficial for safety. The authors rigorously illustrate, through motivating examples, scenarios where standard approaches force the cancellation of nonlinearity that is actually safety-promoting or where global bounds demolish any geometric specificity in the verification process.
2. Geometry-aware, Difference-of-Convex (DC) Formulation: Rather than canceling or worst-case bounding nonlinear terms, nonlinear effects are directly embedded into the invariance (contractivity) conditions using a difference-of-convex analytic formulation of each facet map in the polyhedral safe set. Specifically, facet-specific epigraph conditions are enforced, leveraging both the geometry (sign-symmetric facets and active face structures) and the local convexifiability of nonlinear terms. Direction-dependent convexification is introduced, enabling the controller to certify larger and less conservative invariant sets while retaining the tractability of convex optimization.

Technical Developments

Data-based Closed-loop Representation

A data-rich representation is established, exploiting Willems' fundamental lemma and input-state persistency-of-excitation. The construction guarantees that the closed-loop—under a parametrized controller—is expressed as a linear combination of collected trajectories (with known nonlinearity structure), parameterizing both the nominal and nonlinear feedback terms. Key assumptions include knowledge of nonlinearity structure, full-rank data excitation, bounded disturbances, and polyhedral safe sets.

Synthesis Certificates

A structured hierarchy of safety certificates is developed:

• Theorem 1 (Lipschitz-based, Nonlinearity Minimization): Constructs a convex program minimizing an explicit slack on the Lipschitz-bounded nonlinear remainder, subject to data-consistent closed-loop realization and facet-wise dual constraints. Solutions are always at least as conservative (and typically more so) than those exploiting system geometry.
• Theorem 2 (Pure Geometry-based DC Convexification): Introduces a facet-wise, difference-of-convex majorization, allowing verification via maximization at polytope vertices. Relaxes global convexity by introducing minimal curvature slack $\varepsilon$, directly penalized in the optimization, thus mitigating the effects of unavoidable sign changes in curvature. Dual multipliers and slack variables exploit the specific polytope geometry to reduce conservatism.
• Proposition 1 (Hybrid Geometry–Lipschitz Certificate): For highly nonconvex nonlinearities, facet maps are split into exactly convexifiable and residual Lipschitz-bounded terms. This hybridization allows for substantially less tightening than pure Lipschitz envelopes, with proofs showing strict improvement in invariant set size under the DC certificate.
• Theorem 3 (Face-supported/Vertex-dependent Synthesis): Further reduces conservatism by enforcing contractivity only along active faces at each vertex, exploiting the fact that, locally, curvature constraints are less severe than when enforced globally. Controller parameterizations become vertex- or face-dependent, interpolated in the online control application via convex coefficients. This further enlarges certifiable invariant sets and provides additional flexibility for input constraint handling.
• Theorem 4 (Disturbance-aware Convexification): For additive disturbances, disturbance sensitivity terms are directly embedded into the geometry-aware facet verifications, rather than inflating the safe set margin globally. This produces less conservative robust invariant sets, especially in scenarios where disturbance directionality interacts nonuniformly with polytopic geometry.

Implementation and Complexity

All certificate constructions are posed as convex programs (LPs, SOCPs), often with the main computational load arising from vertex-based epigraph constraints. For hyper-rectangular sets, complexity is linear in both dimension and number of active facets. The vertex-interpolation induced by the face-supported approach is computationally lightweight during online execution. The scenario-based verification step, using Monte Carlo sampling, is invoked to guarantee high-probability satisfaction in cases where strict global convexification is infeasible, achieving practical certification with modest computational overhead.

Empirical Evaluation

Simulation benchmarks are conducted on representative third-order nonlinear discrete-time systems with cubic nonlinearities and box-constrained invariant sets. Multiple parameter sweeps are performed:

• Admissible Nonlinearity Magnitude: The maximum values of nonlinear coefficients allowed while preserving invariance are reported for various certificate types.
• Invariant Set Enlargement: For fixed system parameters, the maximal radius $r$ of the certified invariant set is computed.

Results indicate that the pure Lipschitz-based approach (Theorem 1) yields significantly smaller admissible sets and allowable nonlinearities, as demonstrated by concrete numerical examples:

• Admissible $|e_1|$ increases from $0.04$ under Theorem 1 to $0.25$ (structured Theorem 3), a factor of $6.25\times$ improvement.
• For $S(\cdot)$0, the improvement is more dramatic, from $S(\cdot)$1 to $S(\cdot)$2 ($S(\cdot)$3).
• Certified set radius enlarges from $S(\cdot)$4 (Theorem 1) to $S(\cdot)$5 with scenario-enhanced face-restricted DC synthesis, over $S(\cdot)$6 improvement.
• These enlargements are achieved without violation of state or input constraints and with tractable computation (on the order of hundreds of milliseconds per synthesis).

The largest gains are observed with scenario-based face-restricted DC synthesis, which achieves up to $S(\cdot)$7 and $S(\cdot)$8 increases in nonlinearity tolerance for the two parameter sweeps, respectively.

Theoretical and Practical Implications

This work establishes that the geometry-aware, difference-of-convex synthesis discipline substantially outperforms methods relying on global Lipschitz bounds or structural cancellation of nonlinearities, both in achievable safety set size and admissible nonlinearity. The explicit embedding of nonlinear effects into the safety certificates, facet- and direction-specific convexification, and vertex-dependent control law interpolation offer a formal reduction in conservatism and a pathway to scalable synthesis for high-dimensional systems. Input constraints are naturally incorporated without recourse to additional conservatism. Robustness to structured additive disturbances is guaranteed with only minimal performance degradation compared to the disturbance-free case, and the method is directly extensible to safe terminal set design in MPC.

From a theoretical standpoint, the synthesis mechanisms described provide new tools that bridge set-theoretic methods, scenario-based optimization, and direct data-driven control. Their applicability extends to both polynomial and general nonlinear basis functions. All results require only input-state data (persistently exciting and full rank), aligning with practical constraints in real-world systems.

Future Research Directions

The authors suggest further directions, including:

• Automated enlargement of invariant sets via facet-wise or local geometric refinement, leveraging observed regions of curvature slack activity.
• Integration with basis function learning: learning nonlinear dictionaries that conform to convexifiability conditions.
• Adaptive, scenario-based incremental synthesis, optimally targeting regions where contractivity is not initially certified.
• Extension to noisy-data settings and formal treatment of measurement uncertainties.
• Application to safety-critical MPC with explicit robust or scenario-based terminal sets and costs.

Conclusion

This work provides a rigorous, low-conservatism, data-driven safe control framework for nonlinear discrete-time systems. By exploiting geometry-aware difference-of-convex synthesis and eschewing cancellation or worst-case bounding of nonlinearities, it certifies substantially enlarged invariant sets and admissible nonlinear dynamics under explicit state and input constraints. Both theoretical guarantees and extensive simulations support its efficacy, positioning it as a scalable tool for practical deployment in safety-critical nonlinear systems under uncertainty and disturbance.

Reference:

"Data-based Low-conservative Nonlinear Safe Control Learning" (2604.01156)