Scalable iterative Gramian synthesis for control-affine systems

TL;DR

This paper introduces a scalable, high-precision iterative Gramian-based control synthesis method for nonlinear control-affine systems, demonstrated on complex systems including neural networks.

Contribution

It offers a computationally efficient scheme that overcomes key bottlenecks, enabling practical high-dimensional nonlinear control synthesis.

Findings

Rapid convergence across diverse systems

High-precision results in 100-dimensional neural networks

Convergence depends on system properties, not dimension

Abstract

This article presents a scalable implementation of nonlinear Gramian-based control synthesis for control-affine systems, including a minimum energy control construction. These synthesis advances are achieved by addressing key computational bottlenecks inherent to iterative synthesis map formulations, yielding a computational scheme that exhibits rapid convergence and high-precision. The efficacy of this synthesis framework is demonstrated across five canonical nonlinear control systems and 100-dimensional recurrent neural network models, including underactuated systems. Empirical scaling results further indicate that convergence is primarily governed by intrinsic system properties, such as nonlinearity and controllability, rather than by state-space dimensionality. This work provides a practical, scalable computational pathway for translating rigorous nonlinear synthesis theory into high-dimensional control applications.