We consider a nonlinear control system modeled as an ordinary differential equation subject to disturbance, with a state feedback controller parameterized as a feedforward neural network. We propose a framework for training controllers with certified robust forward invariant polytopes, where any trajectory initialized inside the polytope remains within the polytope, regardless of the disturbance. First, we parameterize a family of lifted control systems in a higher dimensional space, where the original neural controlled system evolves on an invariant subspace of each lifted system. We use interval analysis and neural network verifiers to further construct a family of lifted embedding systems, carefully capturing the knowledge of this invariant subspace. If the vector field of any lifted embedding system satisfies a sign constraint at a single point, then a certain convex polytope of the original system is robustly forward invariant. Treating the neural network controller and the lifted system parameters as variables, we propose an algorithm to train controllers with certified forward invariant polytopes in the closed-loop control system. Through two examples, we demonstrate how the simplicity of the sign constraint allows our approach to scale with system dimension to over $50$ states, and outperform state-of-the-art Lyapunov-based sampling approaches in runtime.

本研究解决了非线性控制系统在扰动下的鲁棒性问题，提出了一种利用认证鲁棒前不变多面体训练控制器的框架。通过引入高维空间中的提升控制系统和间隔分析，确保任何初始化在多面体内部的轨迹都能保持在其内，展示了该方法在处理高维系统方面的有效性，超越了当前基于Lyapunov的采样方法。

神经受控常微分方程中的认证鲁棒不变多面体训练