Neural networks have shown great success in many machine learning related tasks, due to their ability to act as general function approximators. Recent work has demonstrated the effectiveness of neural networks in control systems (known as neural feedback loops), most notably by using a neural network as a controller. However, one of the big challenges of this approach is that neural networks have been shown to be sensitive to adversarial attacks. This means that, unless they are designed properly, they are not an ideal candidate for controllers due to issues with robustness and uncertainty, which are pivotal aspects of control systems. There has been initial work on robustness to both analyse and design dynamical systems with neural network controllers. However, one prominent issue with these methods is that they use existing neural network architectures tailored for traditional machine learning tasks. These structures may not be appropriate for neural network controllers and it is important to consider alternative architectures. This paper considers rational neural networks and presents novel rational activation functions, which can be used effectively in robustness problems for neural feedback loops. Rational activation functions are replaced by a general rational neural network structure, which is convex in the neural network's parameters. A method is proposed to recover a stabilising controller from a Sum of Squares feasibility test. This approach is then applied to a refined rational neural network which is more compatible with Sum of Squares programming. Numerical examples show that this method can successfully recover stabilising rational neural network controllers for neural feedback loops with non-linear plants with noise and parametric uncertainty.

翻译：神经网络因其作为通用函数逼近器的能力，在众多机器学习相关任务中展现出巨大成功。近期工作证明了神经网络在控制系统（称为神经反馈回路）中的有效性，特别是将神经网络用作控制器。然而，这种方法的一大挑战是神经网络已被证明对对抗性攻击敏感。这意味着，除非经过适当设计，否则由于鲁棒性和不确定性（这是控制系统的关键方面）问题，它们并非理想的控制器候选方案。已有初步工作针对鲁棒性展开研究，以分析并设计具有神经网络控制器的动态系统。但这些方法的一个突出问题是，它们使用了针对传统机器学习任务量身定制的现有神经网络架构。这些结构可能并不适用于神经网络控制器，因此考虑替代架构至关重要。本文研究了有理神经网络，并提出了新型有理激活函数，这些函数可有效用于神经反馈回路的鲁棒性问题。有理激活函数被替换为一种通用的有理神经网络结构，该结构在神经网络参数上是凸的。我们提出了一种方法，通过平方和可行性检验来恢复稳定控制器。随后，该方法被应用于一种与平方和编程更兼容的精炼有理神经网络。数值示例表明，该方法能够成功恢复针对具有噪声和参数不确定性的非线性被控对象神经反馈回路的稳定有理神经网络控制器。