Adaptive Tikhonov regularization and dynamic control points for accurate shape parameter control of plasmas

Precise control of plasma shape parameters, such as elongation and triangularity is duly needed to achieve high-performance tokamak plasmas, for which we propose adaptive search schemes of (1) optimum regularization parameter for the Tikhonov regularization, and (2) control points to specify the key shape parameters such as elongation and triangularity. Many control points that exceed the number of actuator coils become an ill-conditioned problem, which is successfully resolved by Tikhonov regularization with adaptively optimized free parameters. Furthermore, we develop dynamically changed control points by using the Cauchy Condition Surface scheme, where elongation and triangularity become direct control values. By virtue of both the adaptive Tikhonov regularization scheme and the dynamic control point, we achieved accurate shape control with elongation up to 1.93 and triangularity up to 0.65 in JT-60SA, where the allowable speed for the change of the elongation is 0.1 s−1. We also verified the resilience of our developed our logics to the noises. The sequence of the result will contribute to enhance equilibrium controllability in upcoming JT-60SA experiments and provide the robust shape parameter control scheme for ITER and DEMO.


Introduction
The performance of tokamak plasmas is duly associated with plasma shape control.The stored energy of plasmas increases with the plasma current (I p ), and a highly elongated plasma shape is necessary for maximizing I p while maintaining the edge safety factor value from the magnetohydrodynamic (MHD) stability point of view.The ISO-FLUX control scheme Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. has been adopted for plasma shape control in various devices [1][2][3].As an example, when control points are set as shown in figure 1, the difference of the magnetic flux between each control point and the last closed flux surface (LCFS) is compensated.Furthermore, the ISO-FLUX scheme controls plasma shape and I p simultaneously by compensating magnetic flux with control coils at each point for shape control or all points for the I p control, which results in interference of these within limited power supply voltages that can easily be saturated in the limited power supply voltages.Here this interference is resolved by the developed adaptive voltage allocation scheme [4].Notably, voltage saturation becomes a severe problem in large superconducting tokamaks such as JT-60SA due to the limited number of coils and its large inductance.
The mission of the JT-60SA is to achieve plasmas with high I p up to 5.5 MA and high normalized pressure above an ideal no-wall beta-limit for a long time duration (typically 100 s) [5][6][7].Since H-mode plasma pedestal accounts for a large fraction of the plasma stored energy, contollability of the highly shaped plasmas under the limitation of acturators in the superconducting device is crusial to achive the JT-60SA missions.Targeted elongation κ and triangularity δ is 1.9 and 0.5, respectively [7], which should be fluently achieved with high I p condition and under severe control situration, where limited number of coils and limited rated power supply voltages.
Control of both κ and δ requires control of the uppermost, lowermost, innermost, and outermost points, separately, as shown by the blue-cross/red-circle marks in figure 1.If we control those points by static control points, at least three control points are needed for each to form the convex shape as seen in the cyan and purple circles in figure 1.In general, more control points are used such as in the XSC controller [8] to specify the plasma shape parameters and/or the gaps from the first wall with the static control points.Nevertheless, the shape parameter itself is not a direct control value.Furthermore, such numerous control points, and/or using tight control points, where the distance of neighboring reference points is close to specifying plasma shape parameters, cause ill-conditioned problem, which basically degrades controllability.
In this paper, we propose two novel control schemes.First, the ill-conditioned problem is resolved with adaptive regularization.Second, we develop dynamically changed control points by using the Cauchy Condition Surface (CCS) scheme [9,10], where the elongation and triangularity become direct control values.
This paper is organized as follows.A controller model is briefly reviewed and newly developed schemes are introduced in section 2. Effects of the developed logic are explored in section 3. Summaries and conclusions are given in section 4. The simulation results in this paper were computed using the MHD equilibrium control simulator (MECS) [3], which is a control simulator tool equipped with a free-boundary equilibrium solver, allowing for the self-consistent simulation of vertical instability [11].

Basic control model
An adaptive voltage management scheme for equilibrium control has been developed for JT-60SA experiment [4], which adaptively adjusts the balance between the plasma position and shape (P/S) control and the I p control under the saturated power supply voltage condition.For the sake of completeness, let us briefly review a basic model of our control scheme [4], which is a basis for the shape parameter control.In our equilibrium controller, the P/S and the I p are controlled by ISO-FLUX scheme [3].First, we define magnetic flux vectors ψ, whose vector elements are magnetic flux at prescribed control points, such as seen in figure 1.A target shape of plasmas is lower divertor configuration in all the simulations of this paper, and the lowermost/uppermost control points are controlled to achieve a target κ X , where the κ X represents the elongation at LCFS.The magnetic flux is controlled by active coils to achieve prescribed P/S and I p references, which can be expressed by where '∆' represents a difference of variables during one control cycle, and I c represents active coil currents.M represents the green function of active coils against control points, then M is the 2D vector, whose rows and columns are the number of active coils and prescribed control points elements, respectively.Here numbers of the rows and the columns are basically different, thus ∆I c is determined to minimize the following residual error S, Here T is the stabilizer matrix, and G and λ are newly introduced control parameters, where G is the gains for control values, and the λ is the regularization parameter for the Tikhonov regularization.The gain G and the regularization parameter λ are newly introduced parameters to improve the controllability.The stabilizer matrix in this paper is the self-inductance of each coil.The coil currents are determined by a circuit equation with power supply voltages.The circuit equations can be expressed by where R c is active coil resistivity, and V c is a power supply voltage, respectively.In addition, we also define Ṽc as the coil voltage normalized by the rated voltage of the device, i.e.JT-60SA in this paper.M c denotes self-and mutual-inductance of active coils.Regarding the I p control, variations of magnetic flux at the plasma boundary can be obtained from the Poynting theorem by ignoring the resistive energy dissipation as where ψ X is the magnetic flux at the LCFS, and L i represents the plasma internal inductance from the electrical engineering standard.Regarding the P/S control, the prescribed shape control points are forced to have the same flux value at the LCFS.The control value for the P/S control can be expressed as For feedback control, we adopt a PID controller.PID gains are introduced as where G xp , G xi , G xd and G sp , G si , G sd are prescribed PID gains for the I p and the P/S control, respectively.The overall control value is The obtained magnetic flux is substituted in equation (2).From here, the effect of gain G and the regularization parameter λ is discussed in the following section.

Adaptive Tikhonov regularization (ATR) and gain control
The gain G and the regularization parameter λ are newly introduced to achieve precise control of κ and δ, which is adaptively determined by monitoring the margin of power supply.Here, we fix G or λ with unity or zero while changing the other.In addition, we fixed all PID gains (G xp , G xi , G xd , G sp , G si , G sd ), which were optimized by frequency response analysis, before adaptive search of G and λ.An increase/decrease in G causes an increase/decrease in the output voltages.
Similarly, an increase/decrease in λ causes the decrease/increase in the output voltages.From this point of view, we can determine the optimum gain G and the regularization parameter λ from the power supply voltage information.Now, let R V be voltage saturation rate, which is calculated by the average of | Ṽc | as, where | Ṽc,i | is the absolute value of ith element of the normalized voltage, Ṽc , and n c represents the number of superconducting coils, respectively.We can choose λ or G so that R V converges to specified input.In other words, by changing the input parameter for the convergence, we can control the gain or the degree of the regularization.We use binary search to find optimum λ and G, where R V converges to the input value, as where R V,up is the input parameter for the R V convergence.We note that if the power supply voltage is not saturated when G = 1 or λ = 0, then the iteration loop for the R V convergence is skipped.Thus, R V,up is the upper limit for the R V .
The next question is whether the gain or the regularization control is suitable to avoid the voltage saturation and what value of R V,up should be chosen.In equation ( 2), the parameter G linearly changes the control value, while λ nonlinearly through modifying balance between the first term and the second regularization term in equation ( 2).As a results, R V converges to the input value.In figures 3 and 4, the two schemes are in comparison.First, in figure 3, R V,up is scanned from 0.1 to 1.0 with the gain control, where no regularization, i.e. fixing λ to zero.The case without the R V,up is labeled by '∞' in the figure.The shape controllability is tested by changing the plasma shape as shown in figure 2, where the following plasma parameters are fixed: I p = 5.5 MA, l i = 0.85, β p = 0.50, which is a targeted high β H-mode scenario in JT-60SA [12].Time evolves from the red crosses to the cyan crosses; thus, the upper reference points mainly move upward to increase the elongation.In figure 3, low R V cases (0.5) and high R V cases (0.9) are vertically unstable (see figure (a)).Furthermore, in the case of the mid-range of R V (0.7), although the κ X attempts to follow the reference, the reference lower divertor configuration cannot be retained, and its configuration changes back-and-forth (see figure 3(c)), which is not acceptable from the machine protection point of view.The case with no R V,up (the ∞ label) simply represents the conventional SVD case (G = 1 and λ = 0 in equation ( 2)).Clearly, the downward displacement occurs as shown in figure 3(a).Therefore, avoidance of the voltage saturation by changing the gain G is not effective.
In figure 4, the R V,up is scanned from 0.2 to 1.0 by changing λ, while fixing G to unity.The main plasma parameters are the same as in the previous case: I p = 5.5 MA, l i = 0.85, β p = 0.50.The evolution of the reference points is also the same as figure 2. Here, although control fails in the low λ cases (0.1 and 0.2), in the other cases, there is no backand-forth configuration change, and the κ X successfully follows the reference.Considering that linear suppression of the control value via the gain control cannot improve controllability, the sequence of results indicates that the resolving the ill-conditioned problem is the key to the improvement of the controllability.
Lastly, the dependence of coil current profile on regularization parameter λ is shown in figure 5.The necessary coil currents are compared to a change of +1 mWb at the lowermost reference point (the X-point), while the other reference is controlled with ±0 mWb.Selected timing of the shape is 12 s of figure 2.Here the location of the X-point remains fixed thus the dependence on the choice of timing is not so significant.λ is adaptively determined to converge the voltage saturation ratio R V to the input parameter, R V,up , which is scanned from 0.2 to 1.0.The label ∞ denotes the case with λ = 0, which indicates this is the standard SVD case without regularization.Since coils located at lower region have a strong correlation with the lowermost control points, variations of EF3-EF5, and CS4 are large in comparison with the other coils.By using the Tikhonov regularization, the variation itself is suppressed, while in the gain control, all the coil currents are suppressed with the similar balance.Taking into account the poor controllability with the gain G adjustment, juxtaposed with the successful regularization of the Tikhonov scheme, the resolving the variation through the Tikhonov scheme contributes to enhancing controllability.The upper limit for the R V is scanned from 0.5 to 0.9 with the gain control.The label '∞' indicates the case with G ≡ 1.Control fails in the all R V,up cases.

Dynamic control point (DCP)
The ISO-FLUX scheme controls the plasma position and shape with control points/gaps.If we control the shape parameter with 'static' control points, numerous control points are needed.Separate control of the uppermost, lowermost, innermost, and outermost points, is at least needed, and the minimum number of control points is 12 to form the convex shape (see figure 1, 12 pts.case).Actually, numerous gaps are needed as discussed in [13], where 85 gaps are considered at most.Control with many control points basically degrades the controllability from the least-squares point of view, for which we develop the dynamically determined control points (DCP).
Here we search the control points to connect the residual magnetic flux ∆ψ s and the shape parameter.The schematic view to obtain the dynamic control point for the uppermost control point is shown in figure 1 with the device conditions of JT-60SA.First, CCS scheme [9,10] detects the uppermost point on the LCFS, and calculates the intersections of LCFS and a circle, whose origin is the uppermost point shown by the upper blue cross-mark.Here, the radius of the circle is denoted by r DCP , whose effects for the controllability are discussed at the end of this section.ISO-FLUX control is applied against the shifted intersecting points of the circle, whose shift is calculated from the reference of the uppermost point.The position of the uppermost point is explicitly determined to satisfy references of κ X and δ X .Other feature points such as for the lowermost, the innermost, and the outermost, are also calculated via the same procedures.Here we define two cases, where κ X , δ Xu , and δ Xl are controlled in the six points case (blue + red dots), while κ Xu and κ Xl are controlled separately in the eight points case (blue + purple dots) in figure 1.Here the upper and lower elongation is denoted by κ Xu or κ Xl , respectively, and the upper and lower triangularity by δ Xu and δ Xl , respectively.
In figure 6, control with the conventional SVD case, the dynamic point (DCP) scheme, the adaptive Tikhonov scheme (ATR) with R V,up = 0.8, and its combination (ATR+DCP) are in comparison.Here we control all four characteristic control points, thus the κ Xu , κ Xl , δ Xu , and δ Xl can be controlled separately.The temporal evolution of these parameters is shown in the figures 6(b)-(e), where its Here, the required coil currents are in comparison to a change of +1 mWb at the lowermost reference points (the X-point) while the other reference is controlled with ±0 mWb.The used equilibrium is that at 12 s in figure 2. λ is adaptively determined to converge the voltage saturation ratio R V to the input parameter, R V,up , which is scanned from 0.2 to 1.0.The label ∞ is the case with λ = 0. Large variation of the coil currents is suppressed by using the finite R V,up value.
reference is shown by the dashed line.In the DCP case, the ATR scheme is not used (λ = 0), while in the ATR case, the DCP scheme is not used (prescribed control points).In the SVD case, both the ATR and the DCP are not utilized, while both are applied in the ATR+DCP case.Control in both SVD and DCP cases fails due to the voltage saturation as shown in figure 6(g), which indicates that the ATR scheme is at least required for the κ and δ control.By introducing the DCP scheme to the ATR scheme, all four parameters are precisely controlled while holding the I p to the reference value.Here, the voltage utilization ratio is held with the fixed level (R V = 0.8 in this case) in the ATR scheme, which indicates that even with a saturation of a few coils (CS4 in this case), other coils have a margin in the voltage to contribute the position and shape control.Therefore, despite the short period saturation of the CS4, the ATR and the ATR+DCP cases avoid the failure of the control.
Next, the temporal evolution of dynamically changed reference points in the DCP scheme is in comparison with the static control points in figure 7. Here, R aux and Z aux represent the R and Z position of auxiliary points on the inboard side for the uppermost control points (denoted by 'aux' in figure 1).Clearly, dynamic control points do not correspond to the static control points, which are adjusted for the change in the shape of plasmas.On the other hand, since the position of the control points depends on the reconstructed LCFS, the variation of the reconstructed LCFS affects the controllability.Therefore, a relatively large variation can be seen in the control voltage (see figure 6 , (e) lower triangularity (δ Xl ), (f ) vertical position of plasmas, and (g) power supply voltages of CS4 coil.The CS4 location can be seen in figure 1.By introducing the DCP scheme to the ATR scheme (ATR+DCP case), all four parameters are precisely controlled while holding the Ip to the reference value.the ATR case.Improvement of the DCP scheme, such as the removal of reconstruction noises, is our future work.

(g)) of the ATR+DCP case in comparison with
Lastly, the optimum value of r DCP is explored in figure 8, where r DCP /a p is scanned from 0.09 to 1.15.Here a p is the minor radius of plasmas.Although the controllability is not sensitive against r DCP , upper and lower limits exist, e.g.0.09 and 1.15, respectively.Each limit is determined by different causes.The upper limit simply indicates that the r DCP /a p is too long for the size of the plasma shape (above unity), since circles at different origins, such as the uppermost and innermost points, will come closer together or even overlap.The lower limit stems from the ill-conditioned problem.If neighboring control points are close, the variation of required coil currents becomes large.The acceptable r DCP /a p is from 0.4 to 0.8.In this paper, we set r DCP /a p as 0.44 throughout this paper.Temporal evolution of dynamically changed reference in comparison with the static control points.The dynamically changed reference is in the ATR + DCP scheme, while the static reference is in the ATR scheme.Here, Raux and Zaux represent the R and Z position of auxiliary points on the inboard side for the uppermost control points, which is denoted by 'aux' in figure 1.

Demonstration of various control examples utilizing the developed schemes
In this section, how the ATR and the DCP scheme work in the realistic sequence is presented by using the MECS simulation.The regularization parameter λ for the ATR scheme is set with R V,up = 0.8, and r DCP /a p = 0.44.

Controllable parameter space for elongation and triangularity
Here, we explore the accessible parameter spaces of elongation and triangularity by using the ATR and the DCP scheme.The targeted elongation and triangularity are widely scanned from 1.4 to 1.9 for κ X and 0.1 to 0.65 for δ X .The temporal evolution of κ/δ scan is shown in figures 9 and 10, and these are summarized in figure 11.The targeted I p is all 5.5 MA and the targeted δ X is 0.5 for the κ scan, while the targeted κ X is 1.8 except the δ X = 0.1 case, where κ X is changed to 1.7 not to contact the stabilization plate.In all cases, both I p and shape are well controlled and successfully follow the reference.In figure 11, both κ and δ scans are summarized.The achieved shape parameter and its reference are shown in the horizontal and vertical axis.All data points are arranged on the diagonal dashed line with small variations, which indicates that the developed ATR+DCP scheme assures a wide range of κ and δ controllability for the future JT-60SA experiment.
Lastly, for the realistic scenario applicaiton, the elongation is increased with the I p ramp-up in figure 12 for sustaining the q 95 level.Here, during the I p ramp-up from 3.0 to 4.5 MA, the elongation is increased from 1.4 to 1.85, which sustains the q 95 above 3.0.As shown in the figure (b) and (c), both κ and δ are precisely controlled even during the ramp-up.For further high κ control during the I p ramp-up, the cooperation with the in-vessel coil is essential, which will be our future work.

Trackability
In this subsection, the trackability of the shape parameter is explored.Since the plasma elongation restricts the achievable I p due to the safety factor requirement from the MHD stability, I p ramp-up rate is restricted by the achievable trackability of κ.In figure 13, the temporal evolution of (a) I p , (b) elongation, and (c) triangularity is scanned by varying change rates of κ.Targeted κ increases from 1.5 to 1.9 with various change rates: 2 s (red), 4 s (blue), and 6 s (green).Each reference is shown by a dashed line, while the achieved value is shown by a solid line.The 6 s case (green line) is the original case, where the achieved κ and δ precisely follow a reference from around 12 s.For a further fast change of κ, such as the 4 s case (blue line), although the change rate is improved, κ overshoots around 15 s.In the fastest 2 s case, κ no longer follows its reference, and the change rate itself is degraded.Therefore, we conclude acceptable speed of κ change is 0.1 [1/s] in our ATR+DCP scheme.The change speed of the κ is basically limited by the L/R time of the superconducting coils, while our ATR scheme suppresses the change rate of the coil current through the regularization term.Thus, a large discrepancy between the actual shape and its reference could make the regularization term dominant, which basically degrades the controllability and can be seen as the larger discrepancy from the reference in the I p and the κ X control (see the 2 s case).
For further acceleration, the use of an in-vessel coil is an option in the JT-60SA, while we focus on the superconducting coil control in this paper from the viewpoint of DEMO application.On the other hand, as discussed in [14], active use  of in-vessel coils significantly enlarges the operational space and/or gives robustness against unknown disturbances, and also is essential for access of highly elongated shape during the I p ramp-up, where the interference between the I p control and the shape control strong degradation of controllability.Therefore, further optimization for the JT-60SA experiment with in-vessel coils will be discussed in another paper.We note that operating frequency is significantly different between the in-vessel and the superconducting coils, thus special logic to resolve it is needed.

Resilience to noises in diagnostics
In this subsection, the resilience of our control logic to noises in diagnostics is explored.We added the realistic noises to the MECS simulation that will be expected in the JT-60SA.Although the condition of the noise is the same as discussed in [4], let us introduce the precise condition for the sake of completeness.First, we assume Gaussian noises as a disturbance of the magnetic diagnostics.The maximum noise for the magnetic field diagnostics is 1 G and that for the flux loop diagnostics is 0.5 mWb [15], respectively, which is empirically determined from the JT-60U experiment and will be validated through the JT-60SA integrated commissioning.In the simulation, we applied the Gaussian noises at each control time-step (250 µs) with a variance σ = 1/3 G for the magnetic field, and 1/6 mWb for the magnetic flux.On the other hand, we neglect biased errors that are such as offset/drift of signals due to the integrator, and low-frequency noises such as those reported in [16], which could have non-negligible impacts on the P/S and the I p control.The estimation of the effect of those errors on the control will be our future work through the integrated commissioning.Regarding the control coil current measurement, we assume white noises with 0.15% of the coil current at each time step (250 µs) since the analog/digital conversion is the dominant error source.Regarding the control coil systems, the rated coil currents and the rated power supply voltages both for thyristor converters and booster converters are based on Figure 14.Temporal evolution of (a) plasma current, (b) elongation, and (c) triangularity.Targeted κ increases from 1.5 to 1.9.The red/blue curve shows the case with and without the noise, respectively.The developed ATR+DCP scheme shows resilience to the noises.[17,18].Here, the slew rate is 100 kV s −1 for the thyristor converters, and 500 kV s −1 for the booster converters, respectively.Since the slew rates depend on the ripple characteristics of the power supply voltages, the above values are estimated rated slew rates, which also will be validated through the integrated commissioning.Regarding the control cycle, the controller time step in the simulation is 250 µ s, which is the same as the real experiment.The calculation time of the equilibrium control including the newly developed scheme in this paper is around 100 µs, which has a sufficient buffer for the controller time step, 250 µs.We note that the equilibrium controller uses the LCFS reconstruction results, which are calculated in 2 ms in JT-60SA.
In figure 14, the temporal evolution of (a) I p , (b) κ, and (c) δ is shown for the case with (red) and without (blue) the above noises, respectively.Here the targeted κ increases from 1.5 to 1.9 while fixing δ with 0.5.We note that the initial perturbation of I p of both cases is caused by a discrepancy between the initial equilibrium and the reference.Although we could eliminate the discrepancy by adjusting the reference precisely, in order to reproduce the realistic state of the simulation, the excessive adjusting of the reference to the initial equilibrium is not tried in this case.Excluding the initial perturbation, there are no significant discrepancies with and without noises, which indicates that the developed logic shows robustness against noises.We note that the developed logic will be tested against a further realistic noise environment, which will be obtained from the integrated commissioning of JT-60SA.

Summary and conclusion
We have developed a novel control scheme for the shape parameters, such as κ and δ.Control of both κ and δ requires control of the uppermost, lowermost, innermost, and outermost points, separately.If we these points with the static control points, at least three control points are needed for each to form the convex shape.In the superconducting tokamaks, the number of coils is limited from the point of view.Under the condition, the control matrix, which is calculated from the green function between the reference points and the coil currents, becomes ill-conditioned.The control with the ill-conditioned matrix results in a large variation of coil currents to compensate the residual magnetic fluxes at reference points, which easily causes voltage saturation in the real device.The Tikhonov regularization scheme is frequently utilized to resolve the singular matrix, while free regularization parameters exist.We have developed an adaptive search scheme of the regularization parameter for the Tikhonov scheme.The voltage saturation rate R V is newly introduced, and the regularization parameter λ is adaptively determined to converge the voltage saturation ratio R V to the input parameter.The developed ATR scheme improves controllability in comparison with the conventional SVD scheme.Furthermore, although voltage saturation can be avoided by suppressing the control values itself, we find that such a linear suppression is not effective in comparison with the adaptive Tikhonov regularization, which indicates that resolving the ill-conditioned matrix is the key to cope with the voltage saturation.
Although the ATR scheme resolves the degradation of controllability due to the voltage saturation, which does not directly indicate that the ATR scheme can achieve precise control of κ and δ.Although the ATR scheme controls the LCFS to the specified 'static' reference points under the ISO-FLUX control framework, κ and δ values themselves are not the direct control value.We have developed the dynamic control point (DCP) scheme, which directly connects the shaper parameters κ and δ with the reference points by using the CCS scheme under the ISO-FLUX framework.Finally, by virtue of both the DCP scheme and the ATR scheme, we achieve precise control of κ and δ.The acceptable speed of κ change is 0.1 [1/s] in the developed scheme, and for further acceleration, the use of an in-vessel coil is an option in JT-60SA, which will be our future work.The resilience against realistic noises in JT-60SA is also confirmed, where the κ X from 1.5 to 1.9 with δ = 0.5 is precisely controlled in the presence of the noise.
In conclusion, the project mission of JT-60SA is to contribute to early realization of fusion energy by supporting exploitation of ITER and by complementing ITER with resolving key physics and engineering issues for DEMO reactors.Controllability of the highly shaped plasma is the key to enhance the plasma performance via maximizing I p with sustaining the edge safety factor and controlling the pedestal.The novel ATR and the DCP schemes will enhance the both controllability, which will provide the robust control scheme of the highly shaped plasma not only for upcoming JT-60SA experiments, but also for ITER and DEMO.

Figure 1 .
Figure 1.Device conditions of JT-60SA used in the simulation and schematic view of dynamic control point scheme for the uppermost control point.

Figure 2 .
Figure 2. Temporal evolution of the reference points and the last closed flux surface with an increase in elongation.Time evolves from red to cyan curves.

Figure 3 .
Figure 3. Temporal evolution of (a) vertical position of the plasma, (b) elongation, (c) position of limiter or X-point, and (d) gain G.The upper limit for the R V is scanned from 0.5 to 0.9 with the gain control.The label '∞' indicates the case with G ≡ 1.Control fails in the all R V,up cases.

Figure 4 .
Figure 4. Temporal evolution of (a) vertical position of the plasma, (b) elongation, (c) position of limiter or X-point, and (d) λ. λ is adaptively determined to converge the voltage saturation ratio R V to the input parameter, R V,up , which is scanned from 0.2 to 1.0.The legends show the R V,up value.Controllability is improved from all the gain cases in figure 3.

Figure 5 .
Figure 5. Dependence of coil current profile on regularization parameter λ.Here, the required coil currents are in comparison to a change of +1 mWb at the lowermost reference points (the X-point) while the other reference is controlled with ±0 mWb.The used equilibrium is that at 12 s in figure2.λ is adaptively determined to converge the voltage saturation ratio R V to the input parameter, R V,up , which is scanned from 0.2 to 1.0.The label ∞ is the case with λ = 0. Large variation of the coil currents is suppressed by using the finite R V,up value.

Figure 6 .
Figure 6.Temporal evolution of (a) plasma current, (b) upper elongation (κ Xu ), (c) lower elongation (κ Xl ), (d) upper triangularity (δ Xu ), (e) lower triangularity (δ Xl ), (f ) vertical position of plasmas, and (g) power supply voltages of CS4 coil.The CS4 location can be seen in figure1.By introducing the DCP scheme to the ATR scheme (ATR+DCP case), all four parameters are precisely controlled while holding the Ip to the reference value.

Figure 7 .
Figure 7. Temporal evolution of dynamically changed reference in comparison with the static control points.The dynamically changed reference is in the ATR + DCP scheme, while the static reference is in the ATR scheme.Here, Raux and Zaux represent the R and Z position of auxiliary points on the inboard side for the uppermost control points, which is denoted by 'aux' in figure1.

Figure 8 .
Figure 8. Dependence of the controllability of the DCP scheme on r DCP /ap, which is scanned from 0.09 to 1.15.The label on the top of the figure represents r DCP value, which is normalized by the minor radius, ap.Control successes with a mid-range of r DCP , such as from 0.4 to 0.8.

Figure 9 . 10 .
Figure 9. Temporal evolution of (a) plasma current, (b) elongation, and (c) triangularity during the elongation scan.The achieved κ X successfully follows its reference.

Figure 11 .
Figure 11.Summary of elongation and triangularity scan.The achieved κ X and δ X successfully follows its reference.

Figure 13 .
Figure 13.Temporal evolution of (a) plasma current, (b) elongation, and (c) triangularity.Targeted κ increases from 1.5 to 1.9 with various change rates: 2 s (red), 4 s (blue), and 6 s (green).Each reference is shown by a dashed line, while the achieved value is shown by a solid line.Allowable speed for the change of κ X is 0.1 s −1 .