Max-Min-Plus-Scaling Neural Networks to Approximate Continuous Piecewise Affine Model Predictive Control Laws

Abstract

This thesis extensively examines the influential factors affecting the performance of approximations of Model Predictive Control (MPC) control laws using neural networks. MPC is a control strategy that solves an optimization problem at each timestep. This problem can be computationally complex and could be too slow to compute for online control. Sometimes an explicit solution for MPC exists, but this can become very large in memory and is not always available. That is why approximations with neural networks might offer a benefit. Under certain conditions, the explicit solution yields a piecewise affine (PWA) control law. A PWA model class is equivalent to the so-called Max-Min-Plus-Scaling (MMPS) model class, which is a generalization of max-plus and min-plus algebra. Neural networks are made up of neurons, which make use of activation functions. A feed-forward neural network with some specific activation function can yield an MMPS function. This inspires us to research the use of different activation functions in approximating MPC control laws. Additionally, we investigate different sampling strategies and the use of max-plus and min-plus layers in neural networks.

We do this by setting up different PWA and non-PWA control laws for two inverted pendulum systems and training several neural networks to approximate these control laws. We first observe a significantly better performance in approximating the PWA control laws compared to the non-PWA control laws. When varying the activation functions of the neural networks we find that for PWA control laws a MMPS activation function can offer a better performance, but it is not guaranteed for all MMPS functions. We also find that networks with custom max-plus layers can offer a similar performance on approximating control laws compared to networks with traditional layers. When investigating what sampling strategy is most beneficial we find comparable performance with a stratified sampling strategy and a uniform sampling strategy. Depending on what areas of the control law you want to capture with the most detail, you can choose the most viable sampling strategy. With this, we have researched various factors that influence the performance of approximations of MPC control laws. The thesis ends with a recommendation to research even more factors that might offer even better approximations.