Performance and closed-loop stability of control systems can be jeopardised by actuator faults. Actuator redundancy in combination with appropriate control laws can increase the resiliency of a system to both loss of efficiency or jamming. Passive Fault-Tolerant Control (FTC) systems aim at designing a unique control law with guaranteed stability in both nominal and faulty scenarios. In this work, a novel machine learning-based method is devised to systematically synthesise control laws for systems affected by actuator faults, whilst formally certifying the closed-loop stability. The learning architecture trains two Artificial Neural Networks, one representing the control law, and the other resembling a Control Lyapunov Function (CLF). In parallel, a Satisfiability Modulo Theory solver is employed to certify that the obtained CLF formally guarantees the Lyapunov conditions. The method is showcased for two scenarios, one encompassing the stabilisation of an inverted pendulum with redundant actuators, whilst the other covers the control of an Autonomous Underwater Vehicle. The framework is shown capable of synthesising both linear and nonlinear control laws with minimal hyperparameter tuning and within limited computational resources.

This article discusses the optimum layout of coils of a superconducting magnet system for magnetic density separation (MDS). MDS is a novel separation technology that combines a vertical magnetic field gradient with a ferrofluid to separate mixtures of non-magnetic particles based on their mass density. The MDS process can separate more than two types of particles in a single process step, thereby distinguishing it from other separation techniques using a ferrofluid. The authors are currently constructing a superconducting MDS demonstrator. Ideally, the gradient of the magnetic field magnitude should change only with the distance above the magnet but remain constant in a horizontal plane. In principle, such an ideal field profile can be generated with an infinite harmonic sheet current. In practice, edge effects appear due to the necessity of using a finite number of coils. These cause a horizontal component in the field gradient and also change the vertical component. We compare the vertical magnetic field gradient of various coil layouts to see which configuration performs best. To facilitate ease of production, the analysis is restricted to flat racetrack coils. The main result is that the specific shape of a racetrack coil has a larger influence on the vertical gradient than the number of coils. The feed particles need to be pushed through the separation chamber from the insertion to the collection point. One option to realize this is to use an MDS setup in which the magnet is inclined with respect to the horizontal plane. This tilting results in a horizontal magnetic force component, which drives feed particles through the fluid bed. We show that a three-coil layout provides the largest usable fluid bed depth for a wide range of tilt angles.