Optimal Control Simplified Example for NI HTS Magnets

A demonstration for Optimal Control applied to Non-Insulated HTS Magnets and Coils, research activity presented at MT28 conference

A demonstration for Optimal Control applied to Non-Insulated HTS Magnets and Coils

For any question, doubt, feedback, please contact [e-mail]

Running time on i7 laptop: 180 s

Paper Link: https://ieeexplore.ieee.org/document/10444007

Cite as: L. Balconi, E. De Matteis, L. Rossi, C. Santini, S. Sorti and M. Statera, "Nonlinear Optimal Control of No-Insulation and Controlled-Insulation HTS Coils and Magnets," in IEEE Transactions on Applied Superconductivity, vol. 34, no. 5, pp. 1-5, Aug. 2024, Art no. 4604405, doi: 10.1109/TASC.2024.3369009. (https://www.mathworks.com/matlabcentral/fileexchange/135031), MATLAB Central File Exchange.

Nonlinear optimal control is a robust method for managing intricate dynamic systems while accommodating constraints and goals. It relies on a Hamiltonian that combines system dynamics and cost functions, alongside an adjoint vector capturing system sensitivities. The Pontryagin Minimum Principle guides the determination of control inputs that minimize the Hamiltonian. These principles facilitate the optimization of nonlinear dynamics across diverse fields, from engineering to biology.

The core equations involve integrating state dynamics, propagating the adjoint vector backward in time, applying Pontryagin's Minimum Principle, and employing numerical and optimization techniques due to nonlinearity. The transversality condition dictates the final adjoint vector value.

In our specific context, we employ a simplified lumped element model with one block per coil, representing radial resistance and coil inductance in parallel (assuming I < Ic). These blocks are inductively linked. Additionally, we incorporate a heating/cooling equation considering source terms such as Joule heating (a nonlinear term) from radial current and resistive joints, and cryocooler cooling.

Our primary goal is to minimize heat generation during charging while ensuring a reasonable charging time. To achieve this, we utilize quadratic functions for objectives and implement soft constraints using exponential functions

A gradient-based method is implemented in MATLAB. A pseudo-code in MATLAB-like syntax is given. The main features of the method are:

1. We introduce gradient of Hamiltonian dH/du, to be the gradient of our gradient-based method

2. For-cycle driven by max iterations or convergence

3. Fully numeric implementation of all the functions, solved by built-in MATLAB functions

4. Control update by gradient dH/du, with a weight for the sake of convergence

This code allows for great flexibility, but it means that many iterations (102-103) may be needed.