标题： 基于神经网络的控制器的改进平方和稳定性验证 显示英文标题

标题： Improved Sum-of-Squares Stability Verification of Neural-Network-Based Controllers

摘要： 这项工作提出了对使用代数集和凸半定规划的闭环稳定性验证框架的多项改进，以检查基于神经网络的控制系统调节非线性动态系统。 首先，该框架的实用性得到了极大扩展：介绍了两种模仿常见平滑激活函数的代数函数，并确立了与包含递归平衡网络（RENs）从而递归神经网络（RNNs）的控制系统兼容性。 其次，通过另一种证明确立了该框架最先进的稳定性分析的有效性。 第三，基于此证明，提出了两个新的优化问题，以简化局部稳定性特性的分析。 为了简化闭环系统的吸引域（RoA）的分析，第一个问题显式参数化了一个比之前工作中更大的候选李雅普诺夫函数类。 第二个问题利用不变性条件下的独特保证，进一步扩展了候选李雅普诺夫函数的集合，并直接确定一个不变集是否属于系统的吸引域。 这些贡献在两个数值示例中成功得到演示，并提供了未来研究的建议。 显示英文摘要

摘要： This work presents several improvements to the closed-loop stability verification framework using semialgebraic sets and convex semidefinite programming to examine neural-network-based control systems regulating nonlinear dynamical systems. First, the utility of the framework is greatly expanded: two semialgebraic functions mimicking common, smooth activation functions are presented and compatibility with control systems incorporating Recurrent Equilibrium Networks (RENs) and thereby Recurrent Neural Networks (RNNs) is established. Second, the validity of the framework's state-of-the-art stability analyses is established via an alternate proof. Third, based on this proof, two new optimization problems simplifying the analysis of local stability properties are presented. To simplify the analysis of a closed-loop system's Region of Attraction (RoA), the first problem explicitly parameterizes a class of candidate Lyapunov functions larger than in previous works. The second problem utilizes the unique guarantees available under the condition of invariance to further expand the set of candidate Lyapunov functions and directly determine whether an invariant set forms part of the system's RoA. These contributions are successfully demonstrated in two numerical examples and suggestions for future research are provided.