Time-optimal neural feedback control of nilpotent systems as a binary classification problem
This work addresses the problem of efficient real-time control for specific linear systems, offering a novel computational approach that is incremental in combining existing methods.
The paper tackles the synthesis of time-optimal feedback control for linear nilpotent systems by formulating it as a binary classification problem, using a computational method that combines polynomial root-finding with deep neural networks, achieving accurate and robust real-time control in numerical tests on integrators of increasing dimension.
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.