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coincide exactly. We believe that the actual deviations shown in Figure 4 are small enough to be neglected and thus support the hypothesis of a rigid displacement.
The simplest verifiable prediction of the linear model is the dependence of the growth time on wall resistivity. From Equation (16), we can see that if the wall resistivity is scaled, the eigenvalues of the system are scaled by the same factor. From Table I, we can see that this is reproduced in the non- linear simulation rather well. For example, scaling the wall conductivity to 50% of its nominal value results in an insta- bility growth time that is 50.6% as fast.
The second important prediction of the linear model is the growth time of the instability. This prediction, how- ever, depends on the choice of rigid volume, and so com- paring it with the non-linear results is non-trivial. Figure 5 shows an overview of the different growth times that are predicted by the linear model for different choices of rigid volumes. Two kinds of rigid volumes were used: a set of cylinders with different radii and half-lengths (“cyl”) and a set of spheroids with different radii (“spheroid”). The vari- ation is considerable, and so that in principle, one could obtain any desired prediction by picking the right rigid vol- ume—no matter what plasma equilibrium has been used for linearization. In order for the linear model to be useful, one thus needs some mechanism to select the appropriate rigid volume. The data indicate that such a mechanism must depend on the plasma equilibrium: the rigid volume that best reproduces the observed non-linear growth rate for a perturbation at 1 ms is different from the rigid volume that produces the best match for a perturbation at 2 ms or 3 ms.
Unfortunately, there seems to be no procedure to derive the correct rigid volume from the plasma equilibrium with- out a-priori knowledge of the expected non-linear growth rates. As far as the determination of growth rates is con- cerned, the suitability of the linear model is thus limited to the computation of the upper and lower bounds by trying a variety of rigid volumes. When considered from this point of view, the conclusion that one would draw from the results shown in Figure 5 is that the characteristic growth time of the instability is somewhere between 50ms (smaller rigid volumes would cut into the separatrix) and 600ms (even larger rigid volumes do not significantly change this value).
Luckily, for the intended use-case of the linear model, this limitation does not cause problems. When used for the design of a feedback control system, the primary advantage of a linear model is the ability to quickly evaluate a large number of potential controller parameters and algorithms. The time required for the (one-time) derivation of the plasma model is thus less significant, and the need for one non- linear simulation to determine the correct rigid-volume is acceptable.
In Sec. V, we will discuss the suitability of the linear model for the design of a feedback control system for the axial separatrix location. We will assume that we have iden- tified the appropriate rigid volume by running a non-linear simulation and matching the fitted growth rate. The pertinent question is: is the assumption of rigid displacement correct, so that the right choice of rigid volume will give correct pre- dictions for stability under active control or is the instability not rigid? In the latter case, we would expect that even with a rigid volume that gives the correct uncontrolled growth rate, we would not get correct predictions when adding external driving factors like a feedback controller.
V. CLOSED-LOOP SIMULATION RESULTS
Closed-loop simulations were run with two main objec- tives: to determine the suitability of the linear, rigid plasma model for control algorithm design and to obtain a general idea of the control hardware requirements for controlling the separatrix position in an experiment.
Ideally, we would have compared the predictions of the linear model with the non-linear model in open-loop simula- tions first (i.e., by applying various (feed-forward) wave- forms to the control coils and observing the system response). Unfortunately, this is not possible because the model is unstable: attempting to run without the feedback control eventually leads to exponential growth dominating over all other dynamics. Theoretically, one could still com- pare the system response while the instability growth is slow—but in practice, this is ruled out because the instability growth time is comparable to the soak-through time of the wall. Since control coils are separated from the plasma by the resistive wall, the effect of any control coil currents takes just as long to penetrate through the walls as it takes the instability to grow. For this reason, we compared the linear and non-linear model in closed loop simulations with an active feedback to keep the system stable.
FIG. 5. Instability growth times predicted by the linear model for different rigid volumes. Colors of the horizontal bars indicate the time at which the plasma has been linearized. Vertical boxes indicate the range of values obtained by least-squares fitting of the non-linear simulations (assuming 10% uncertainty).