NMPC for Stiff, Distributed Parameter System: Semi-Automatic Code Generation and Optimality Condition Evaluation

AutoGenU is a Mathematica program to automatically generate simulation programs for Nonlinear Model Predictive Control (NMPC). It analytically evaluates the Jacobians necessary to calculate the optimality condition in the NMPC realized using Continuation/Generalized Minimum Residual (C/GMRES) method. However, in the case of the LHC Superfluid Helium Cryogenic System, which is distributed parameter system, these Jacobians, expressed directly in terms of inputs, states and co-states become complex expressions due to cascading relations between internal variables of the circuit's model. A semi-automatic code generation procedure based on AutoGenU is presented, where intermediate variables are introduced and the chain rule is applied to evaluate the Jacobians, thereby avoiding complex expressions. In addition, the ODE set describing the system state dynamics is stiff, thus the dynamics time integration step is small. The intermediate variables are available at each step and are used to evaluate the optimality condition more precisely at low additional computing cost. The observed computational cost of the semi-automatically generated code is slightly lower than that of automatically generated and the controller performance is similar in both cases. However, the generation of semi-automatic code requires significantly less memory, and is much faster, widening the applicability of code generation for complex systems.