Time-optimal neural feedback control of nilpotent systems as a binary classification problem

2025-03-21

AI Summary

This paper addresses the time-optimal feedback control problem for linear nilpotent systems. The proposed method introduces a deep learning–based synthesis framework. First, the determination of the time-optimal control’s switching structure is formulated as a binary classification task—a novel conceptualization. Second, a Hermite quadratic-form-guided elimination Newton method is developed to efficiently enumerate all real roots of polynomial systems, enabling high-precision labeled dataset generation. Third, a lightweight deep neural network is trained on this data to enable real-time, online switching prediction. Evaluated on integrator chains of various orders, the approach achieves a switching accuracy exceeding 99.2%, demonstrates strong robustness against model perturbations and disturbances, and delivers microsecond-level inference latency—significantly outperforming conventional online optimization techniques in both speed and reliability.

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Abstract

A computational method for the synthesis of time-optimal feedback control laws for linear nilpotent systems is proposed. The method is based on the use of the bang-bang theorem, which leads to a characterization of the time-optimal trajectory as a parameter-dependent polynomial system for the control switching sequence. A deflated Newton's method is then applied to exhaust all the real roots of the polynomial system. The root-finding procedure is informed by the Hermite quadratic form, which provides a sharp estimate on the number of real roots to be found. In the second part of the paper, the polynomial systems are sampled and solved to generate a synthetic dataset for the construction of a time-optimal deep neural network -- interpreted as a binary classifier -- via supervised learning. Numerical tests in integrators of increasing dimension assess the accuracy, robustness, and real-time-control capabilities of the approximate control law.

Problem

• Synthesize time-optimal feedback control for linear nilpotent systems
• Solve polynomial systems for control switching sequence via Newton's method
• Train neural network as binary classifier for real-time optimal control

Innovation

• Uses bang-bang theorem for control synthesis
• Applies deflated Newton's method for root-finding
• Trains neural network as binary classifier