Optimal Tuning of Tokamak Plasma Equilibrium Controllers in the Presence of Time Delays Eugenio Schuster, David Sondak, Reza Arastoo, Michael L. Walker and David A. Humphreys Abstract— When designing the control loops for tokamaks, it is important to acknowledge the effects of time delays. An assumption sometimes made for tokamaks having supercon- ducting coils is that these extra time delays will not have any undesirable effects on control. In fact, introducing extra delays into the axisymmetric control loops of certain superconducting tokamaks can have significant detrimental consequences. Aside from qualitative observations, the detrimental effects of extra time delays in tokamak control loops are not always well understood outside the control community. This study exposes and quantifies the detrimental effects imposed by time delays in the control loop in superconducting tokamaks, by focusing on plasma current control and radial position control in a vertically stable circular plasma in the KSTAR tokamak. Delays in the power supplies, data acquisition, and vessel structure are taken into account. Extremum-seeking-based optimal tuning of PID controllers is proposed as a possible method for remediating the negative effects of time delays. The Nyquist dual locus technique is employed to assess stability of the optimally tuned closed-loop system in the presence of time delays. I. I NTRODUCTION With the introduction of fully superconducting tokamaks comes the need to understand how to operate and control plasmas within these devices, given new constraints imposed by superconducting PF coils. There is a concern about AC losses triggering coil quench. The minimum distance of coils from the plasma is increased due to cryogenic insulation requirements. There is a greater emphasis on minimizing the number of control coils due to cost. Passive structures are often more conductive, due to requirements for increased structural strength, multiple conducting walls, or intentional placement of highly conductive passive conductors near the plasma to reduce the growth rate of instabilities. All of these changes from present devices tend to change the plasma shape control properties, several of them nega- tively because of increased delays in responding to plasma disturbances. One response to worries about AC losses is to impose limits on the speed of response of the coils. (Contrary to sometimes-stated opinion, the natural response of superconducting coils are not intrinsically slower than conductive coils of the same cross-section and number of turns, since the relevant response time is determined by the coil inductance, which is defined purely by coil geometry.) Larger distances between coils and plasma mean larger changes in coil currents are needed to accomplish the same This work was supported in part by the NSF CAREER award pro- gram (ECCS-0645086), and DoE contract number DE-FC02-04ER54698. E. Schuster (schuster@lehigh.edu), D. Sondak and R. Arastoo are with the Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA. M. L. Walker and D. A. Humphreys are with the DIII-D tokamak, General Atomics, San Diego, California, USA. Fig. 1. KSTAR tokamak. change in shape-controlling magnetic field at the plasma. Fewer control coils means fewer controllable degrees of freedom, or conversely, less redundancy in controlling the most critical degrees of freedom. Increased conductivity in passive structures implies longer delays in magnetic field penetrating these structures and affecting the plasma. It is sometimes assumed in the fusion community that, because superconducting tokamaks already have significant intrinsic or imposed sources of control delay, introducing extra delays into the axisymmetric control loops will have negligible detrimental impact on the plasma control. Since it is not obvious what constitutes an acceptable amount of delay, we have begun a study of this issue in an attempt to provide guidance to designers of external systems (primar- ily power supplies, control computers, and communication networks) regarding acceptable pure delays (and also phase lags) contributed by these systems. This study has been car- ried out using models of the KSTAR (Korea Superconducting Tokamak Advanced Research) tokamak, which has recently begun operation in Daejon, Korea [1]. A cross-section of the KSTAR Tokamak is shown in Fig. 1. The active control coils outside of the vacuum vessel are superconducting and are used to establish the plasma equilibrium. In this work we consider a vertically stable circular plasma. Two PID controllers are proposed for plasma cur- rent and radial position control. Extremum seeking [2] is proposed for optimal tuning of the PID gains in presence of time delays. Extremum seeking, which is a nonmodel- based method, iteratively modifies the arguments of a cost function (in this application, the PID parameters) so that the tracking error is minimized [3] (see references therein for alternative PID tuning methods). The stability analysis of the closed-loop system is carried out using the dual-locus