  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.
