A Guaranteed-Stable Neural Network Approach for Optimal Control of Nonlinear Systems
Anran Li, John P. Swensen, Mehdi Hosseinzadeh
TL;DR
The work tackles the challenge of achieving stable, reference-tracking control for nonlinear systems when online optimization is computationally prohibitive. It introduces a Neural Optimization Machine (NOM) that recasts a nonlinear, nonconvex, one-step-ahead optimization into a neural network training problem, enabling a NN-based controller with Lyapunov-based stability guarantees. By generating a comprehensive NOM-based training set across the operating region and training a deep feedforward NN, the approach yields a closed-loop system with bounded tracking error and an attractive invariant set for sufficiently large contraction parameter $\theta$. The methodology is validated through simulations and Parrot Bebop 2 drone experiments, showing favorable performance and robustness, with the open-source data repository ensuring reproducibility and broader applicability.
Abstract
A promising approach to optimal control of nonlinear systems involves iteratively linearizing the system and solving an optimization problem at each time instant to determine the optimal control input. Since this approach relies on online optimization, it can be computationally expensive, and thus unrealistic for systems with limited computing resources. One potential solution to this issue is to incorporate a Neural Network (NN) into the control loop to emulate the behavior of the optimal control scheme. Ensuring stability and reference tracking in the resulting NN-based closed-loop system requires modifications to the primary optimization problem. These modifications often introduce non-convexity and nonlinearity with respect to the decision variables, which may surpass the capabilities of existing solvers and complicate the generation of the training dataset. To address this issue, this paper develops a Neural Optimization Machine (NOM) to solve the resulting optimization problems. The central concept of a NOM is to transform the optimization challenges into the problem of training a NN. Rigorous proofs demonstrate that when a NN trained on data generated by the NOM is used in the control loop, all signals remain bounded and the system states asymptotically converge to a neighborhood around the desired equilibrium point, with a tunable proximity threshold. Simulation and experimental studies are provided to illustrate the effectiveness of the proposed methodology.