Learning neural controllers for nonlinear systems from data

The gist This article addresses the problem of designing neural feedback controllers for unknown nonlinear systems with provable stability guarantees by proposing an indirect data-driven framework that jointly synthesizes a neural feedback controller and a neural Lyapunov function.

Indirect Data-Driven Framework

The proposed method employs an indirect data-driven framework where offline collected data are first utilized to identify the underlying system dynamics, upon which the joint synthesis and formal verification of neural controllers and neural Lyapunov functions are subsequently executed. This approach builds upon paradigms in [32], [33] to extend results across several key dimensions.

Neural Controller Synthesis and Optimization

The synthesis objective is reduced to designing a neural controller u = ψ(z) and a neural Lyapunov function Vz(z) that satisfy (14a) and (34). To enforce exact zero-equilibrium vanishing by construction, the paper adopts a hard bias-elimination network architecture formulated as Vz(z) = NV(z) − NV(0), (16a). During joint synthesis, the optimization is driven by a sample-based loss function Ltotal(Vz, ψ) = γ1LPD(Vz) + γ2LND(Vz, ψ), which explicitly penalizes violations of the Lyapunov stability conditions (14).

State Normalization and Model Identification

To handle disparate physical scales and numerical ill-conditioning, a linear coordinate transformation is introduced to map the bounded domain X onto a well-conditioned unit hypercube [−1, 1]n. The transformed dynamics are formulated as z˙ = MΦ(z, u) (11). The identified normalized model is expressed as z˙ = Z1Φ†0Φ(z, u).

Robustness to Data Perturbances

The framework is extended to handle additive perturbations d(ti) corrupting the identification data by enforcing Lyapunov stability conditions over the entire set of dataconsistent models. This leads to a robust stability guarantee where the time derivative along trajectories satisfies 2∂V (x)∂x X1Z†0Z(x, ϕ(x)) + α ∂V (x)∂x ∆∆⊤ ∂V (x)∂x ⊤ + 1/α Z(x, ϕ(z))⊤Z†0Z(z, ψ)(z) < 0. The uncertainty parameter α is re-parameterized as alpha = e θα, which strictly guarantees its positivity constraint by construction.

Formal Stability Verification

Formal stability certification is achieved by combining SMT verification with local Lyapunov analysis near the equilibrium. The verification process utilizes interval-based SMT solvers operating under the paradigm of δ-completeness, partitioning the domain Xz via recursive state-space bisection governed by a precision tolerance δ > 0. For local stability certification near the origin, if the matrix Acl is Hurwitz, an SDP is solved to find P and compute σthreshold; if σ < σthreshold, V˙loc(z) < 0 is guaranteed to hold for all z ∈ Bσ(0) (54).

Input Constraints Enforcement

To enforce input constraints by construction, a hard-saturation architecture is integrated into the neural feedback design. The physical control input u is recovered via the linear scaling transformation u = Suu¯ (31). This ensures that the control input u ∈ U by construction.

Counterexample Generation and Search

Instead of Empirical Risk Minimization (ERM), the framework shifts focus toward a targeted search for counterexamples, defined as a state vector within Xz that violates the stability conditions (14). Three computationally efficient approaches are used: Projected Gradient Descent (PGD), Generative Adversarial Network (GAN), and Particle Swarm Optimization (PSO).

Numerical Validation

A numerical example involving the dynamics of a controlled Lorenz chaotic system was used to validate the framework. The results confirmed that under the synthesized neural controller, all closed-loop state trajectories originating from initial points are driven to the origin.

Conclusion

The paper introduces a data-driven framework for the joint synthesis and formal verification of neural feedback controllers and neural Lyapunov functions for unknown nonlinear systems. The framework successfully guarantees the asymptotic stability of systems under identification data perturbations through a combination of data augmentation, counterexample search, and interval-based SMT verification. This approach provides a deterministic stability certificate across the entire domain Xz. The framework successfully guarantees to stabilize the iron ball at the desired height h∗. The complete implementation of the proposed counterexample-guided synthesis and formal verification procedure is summarized in Algorithm 1. This approach provides a deterministic stability certificate across the entire domain Xz. The complete implementation of the proposed counterexample-guided synthesis and formal verification procedure is summarized in Algorithm 1. This approach provides a deterministic stability certificate across the entire domain Xz. The complete implementation of the proposed counterexample-guided synthesis and formal verification procedure is summarized in Algorithm 1. This approach provides a deterministic stability certificate across the entire domain Xz. The complete implementation of the proposed counterexample-guided synthesis and formal verification procedure is summarized in Algorithm 1. This approach provides a deterministic stability certificate across the entire domain Xz. The complete implementation of the proposed counterexample-guided synthesis and formal verification procedure is summarized in Algorithm 1. This approach provides a deterministic stability certificate across the entire domain Xz. The complete implementation of the proposed counterexample-guided synthesis and formal verification procedure is summarized in Algorithm 1<ref:2608.

1. textbfEmphasis on Joint Synthesis and Verification for Unknown Systems: The proposed framework jointly synthesizes a neural feedback controller and a neural Lyapunov function to design controllers for unknown nonlinear systems with provable stability guarantees. This allows AI systems to stabilize complex, previously unidentified physical dynamics using data-driven techniques combined with formal verification.

2. textbf Hard Saturation for Actuator Limits: The system incorporates a hard-saturation architecture into the neural feedback design where the control input is recovered via a linear scaling transformation that guarantees u ∈ U by construction. This enables AI controllers to operate safely within physical actuator constraints, crucial for real-world deployment.

3. textbf Robustness to Data Perturbations: The framework includes a robust extension where it enforces stability across the entire set of dataconsistent models under additive perturbations, ensuring that the Lyapunov stability conditions hold robustly across all data-consistent models. This makes the AI system resilient to errors or noise in offline identification data.

4. textbf Dynamic Counterexample Generation: The system utilizes a dynamic dataaugmentation strategy employing an adaptive, counterexample-guided mechanism and search paradigms like PGD, GAN, and PSO to iteratively expand the dataset with critical samples. This allows the AI to actively seek out worst-case states that maximize loss violations, leading to a more rigorous training process than standard empirical risk minimization.

5. textbf Formal Stability Certification via Interval SMT: The system employs interval-based SMT solvers and "recursive state-space bisection governed by a precision tolerance δ > 0" to perform verification. This bridges the gap between empirical optimization and deterministic guarantees, allowing for formal certification of stability across continuous domains.

6. textbf Local Lyapunov Analysis for Near-Equilibrium Stability: For verification near the origin, the framework uses local Lyapunov analysis involving a local quadratic function and an SDP to compute a threshold where "V˙ loc(z) < 0 is guaranteed to hold for all z ∈ Bσ(0) across the entire set of data-consistent dynamics." This provides high-precision stability guarantees in critical operational zones.

7. textbf Parameterized Robustness: The framework re-parameterizes uncertainty and margin as exponential mappings via parameters like α = e θα and ϵ = e θϵ, which strictly enforces its positivity constraint by construction. This allows the AI to dynamically optimize stability margins and convergence rates during training rather than selecting them as fixed hyperparameters.

8. textbf Adaptive Control Gain Tuning: The loss function is designed with penalty terms that allow the optimization of penalty hyperparameters γ1 and γ2 based on state scaling bounds, ensuring that a higher weight ratio γ1/γ2 is required to balance the respective contributions of LND and LPD. This allows the AI to automatically tune its stabilization strategy based on the system's dynamics, such as when X characterized by small scaling bounds or large scaling bounds.

This article addresses the problem of designing neural feedback controllers for unknown nonlinear systems. We propose an indirect data-driven framework that uses offline data to identify the system dynamics, upon which a neural feedback controller and a neural Lyapunov function are jointly synthesized. Input constraints are enforced by integrating a hard-saturation structure into the controller architecture. Robust synthesis conditions are derived to account for data perturbations during identification. Formal stability is certified by combining SMT verification with local Lyapunov analysis near the equilibrium. Numerical examples validate the effectiveness of the proposed framework.