Provably-Stable Neural Network-Based Control of Nonlinear Systems

TL;DR

The paper tackles the lack of stability guarantees in neural-network–based control for nonlinear systems by introducing a provably-stable framework that couples a novel one-step-ahead predictive controller with a Lyapunov-based NN training pipeline. The core method trains an NN to imitate the optimal one-step policy, and then proves stability and bounded tracking error for the closed-loop system in the presence of NN approximation errors, with a tunable proximity to the target equilibrium controlled by a design parameter $\theta$. Key contributions include recursive feasibility and stability proofs for the predictor, a data-driven offline training strategy to approximate the policy, and a RoA estimation procedure, all validated through simulations on an inverted pendulum and experiments on a Parrot Bebop 2 drone. The approach enables real-time NN-based control with formal guarantees, applicable to affine nonlinear systems and capable of incorporating arbitrary objective functions via the predictive control formulation. This has practical impact for safety-critical or performance-constrained nonlinear control tasks where stability guarantees are essential.

Abstract

In recent years, Neural Networks (NNs) have been employed to control nonlinear systems due to their potential capability in dealing with situations that might be difficult for conventional nonlinear control schemes. However, to the best of our knowledge, the current literature on NN-based control lacks theoretical guarantees for stability and tracking performance. This precludes the application of NN-based control schemes to systems where stringent stability and performance guarantees are required. To address this gap, this paper proposes a systematic and comprehensive methodology to design provably-stable NN-based control schemes for affine nonlinear systems. Rigorous analysis is provided to show that the proposed approach guarantees stability of the closed-loop system with the NN in the loop. Also, it is shown that the resulting NN-based control scheme ensures that system states asymptotically converge to a neighborhood around the desired equilibrium point, with a tunable proximity threshold. The proposed methodology is validated and evaluated via simulation studies on an inverted pendulum and experimental studies on a Parrot Bebop 2 drone.

Figures (10)

• Figure 1: General structure of proposed methodology.
• Figure 2: The NN trained to approximate the solution of \ref{['eq:OptimizationProblemMain']}, with $s$ hidden layers and $l_i$ neurons in the $i$th hidden layer.
• Figure 3: Geometric illustration of the falsification procedure for determining the RoA $\Phi(r)$.
• Figure 4: Inverted pendulum system.
• Figure 5: The obtained RoA with $\theta=0.01$ and $r=0$; the blue and yellow represent $\mathcal{X}$ and $\Phi(0)$, respectively.

Theorems & Definitions (14)

• Remark 1
• Remark 2
• Remark 3
• Theorem 1: Recursive Feasibility
• proof
• Theorem 2: Stability and Convergence
• proof
• Remark 4
• Remark 5
• Remark 6