Non-linear Model Predictive Control for cooling strings of superconducting magnets using superfluid helium

In each of eight arcs of the 27 km circumference Large Hadron Collider (LHC), 2.5 km long strings of super-conducting magnets are cooled with superfluid Helium II at 1.9 K. The temperature stabilisation is a challenging control problem due to complex non-linear dynamics of the magnets temperature and presence of multiple operational constraints. Strong nonlinearities and variable dead-times of the dynamics originate at strongly heat-flux dependent effective heat conductivity of superfluid that varies three orders of magnitude over the range of possible operational conditions. In order to improve the temperature stabilisation, a proof of concept on-line economic output-feedback Non-linear Model Predictive Controller (NMPC) is presented in this thesis. The controller is based on a novel complex first-principles distributed parameters numerical model of the temperature dynamics over a 214 m long sub-sector of the LHC that is characterized by very low computational cost of simulation needed in real-time optimization based advanced process control. I present a thorough analysis of the thermohydraulic physical processes governing the temperature dynamics, reviewing the related R&D work. The analysis explains the key characteristics of the temperature dynamics and has been the starting point for development of the model and the control strategy. Experimental setups used to identify dynamics and model parameters of the unique superfluid cryogenic system are also described. Through the thesis, I highlight the practical methods used to achieve real-time feasibility of the controller, including: 1) approximations, handling of stiffness and algebraic equations in modelling and simulation, 2) application of Hybrid Luenberger Observer - Moving Horizon Estimation approach enabling output feedback control at very low computing cost, 3) parametrisation of the optimized variables trajectories that strongly reduces the number of optimized variables, 4) approximation of the original non-linear optimization problem with inequality constraints using one with box inequality constraints that is much easier to solve and 5) application of a single step Quasi-Newton solver to the specific box-constrained optimization problem that is repeatedly solved in an optimization-based controller. Two NMPC configurations are presented stabilizing the magnets temperature over a 214 m long LHC sub-sector: 1) using two control valves as manipulates variables, experimentally tested at the LHC, and 2) manipulating the two valves and 12 electric heaters, tested in simulations. Both setups are real-time feasible and exhibit excellent robust performance in wide range of operating conditions, thus validating both the distributed-parameters model and the on-line output-feedback NMPC based on a complex first-principles model.