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A tutorial on inversion-based shape control with design application to NSTX-U

J. T. Wai, M. D. Boyer, D. J. Battaglia, F. Carpanese, F. Felici, W. P. Wehner, A. S. Welander, E. Kolemen

Abstract

One of the most common designs for magnetic control in tokamaks is to ``linearize an equilibrium'' to obtain a sensitivity mapping, then invert this mapping in order to determine the feedback control currents or voltages. For this work, we refer to this broad class of methods as inversion-based shape control (IBSC). In this work we describe the IBSC framework in a comprehensive manner and show how variations such as using dynamic voltage mappings and quadratic-program constrained control fit naturally into the framework. Despite the prevalence of IBSC, some of these extensions and the challenges associated with specific variations of IBSC are less widely known. We pay special attention to the challenge of decoupling the interaction between shape control and vertical control, which was a source of degraded vertical control performance on NSTX-U. This work is intended to provide a systematic overview of IBSC, and to that end, we have provided background material, proposed design procedures, and tutorials on the magnetic control design process within the appendices. Applying the systematic design procedure to NSTX-U, we find that the vertical control bobble on NSTX-U can be removed via decoupling, and that the vertical control phase margin can be improved by 6 degrees just by including PF1A and PF2 as vertical actuators.

A tutorial on inversion-based shape control with design application to NSTX-U

Abstract

One of the most common designs for magnetic control in tokamaks is to ``linearize an equilibrium'' to obtain a sensitivity mapping, then invert this mapping in order to determine the feedback control currents or voltages. For this work, we refer to this broad class of methods as inversion-based shape control (IBSC). In this work we describe the IBSC framework in a comprehensive manner and show how variations such as using dynamic voltage mappings and quadratic-program constrained control fit naturally into the framework. Despite the prevalence of IBSC, some of these extensions and the challenges associated with specific variations of IBSC are less widely known. We pay special attention to the challenge of decoupling the interaction between shape control and vertical control, which was a source of degraded vertical control performance on NSTX-U. This work is intended to provide a systematic overview of IBSC, and to that end, we have provided background material, proposed design procedures, and tutorials on the magnetic control design process within the appendices. Applying the systematic design procedure to NSTX-U, we find that the vertical control bobble on NSTX-U can be removed via decoupling, and that the vertical control phase margin can be improved by 6 degrees just by including PF1A and PF2 as vertical actuators.
Paper Structure (63 sections, 78 equations, 42 figures, 3 tables)

This paper contains 63 sections, 78 equations, 42 figures, 3 tables.

Table of Contents

  1. Introduction and outline
  2. Inversion-based shape control (IBSC) framework
  3. Principles of IBSC
  4. Current map vs voltage map
  5. Static map vs dynamic map
  6. Static map
  7. Dynamic map
  8. Plasma response model
  9. Control inversion methods
  10. Pseudoinverse decoupling PID control
  11. SVD Control
  12. Quadratic program constrained control
  13. Static plasma response maps and implications for vertical decoupling
  14. Proposed design procedure
  15. NSTX-U shape control: summary of main findings
  16. ...and 48 more sections

Figures (42)

  • Figure 1: A typical shape control design based on response maps to coil currents. The vertical controller is often designed separately with a dedicated control loop. The shape errors are measured and mapped to coil current targets, and a coil current controller works to track these targets.
  • Figure 2: A typical shape control design based on response maps to coil power supply voltages. The shape errors are mapped directly to voltage commands.
  • Figure 3: Depiction of $P_{vs,Ic}$, the system transfer function from coil current target requests to shape parameters. This system can be "inverted" to determine the dynamic shape-to-current response map. This is a natural choice of system for this task, since it is equivalent to the controller design of \ref{['fig:ibsc_block_diag_current']} but without the shape-to-current map.
  • Figure 4: Comparison of several flux response maps, to PF2U current perturbations. It is considered that the nonrigid dynamic response map is the highest fidelity map since this includes both nonlinearities of the equilibrium linearization and dynamics of the system. Surprisingly, the static vacuum response map is in better agreement with this than the static nonrigid response map. This behavior is generally consistent across different coils.
  • Figure 5: Step response of the plasma vertical position with a proportional-derivative vertical controller applied to the (PF3U-PF3L) coil combination. The long-term, steady-state solution to move the plasma upwards requires the current in PF3U to decrease. However, to achieve the vertical motion PF3U must transiently increase (by a relatively much larger amount!) before it decreases.
  • ...and 37 more figures