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Conditions for Stabilization of the Tokamak Plasma VerticalInstability Using Only a Massless Plasma Analysis

J. William Helton Kevin J. McGown M. L. Walker

Abstract—This paper describes the problem of feedback con-trol for stabilization of the plasma vertical instability in atokamak. Such controllers are typically designed based on amodel that assumes the plasma mass m is identically zero whilein reality the mass is small but positive. The assumption thatm is zero can lead to a controller that appears to be stabilizingaccording to the massless analysis but in fact can increase theinstability of the physical system.

In this work, we consider a general class of controllers, whichcontains as a special case the type of controller most commonlyused in operating tokamaks to stabilize the vertical instability,a proportional-derivative controller. Suppose C is a controllerin this class which stabilizes the vertical instability with plasmamass assumed to be zero. We give easy-to-check necessary andsufficient conditions for C to also stabilize the physical system,in which the plasma actually has a small mass. We allow for thepossibility that the tokamak could have both superconductingand resistive conductors.

The practical implications of the results presented providesubstantial insight into some long-standing issues regarding feed-back stabilization of the vertical instability with PD controllersand also provide a rigorous foundation for the common practiceof designing controllers assuming m = 0. For controllers thatoperate only on the plasma vertical position, we settle thequestion: when are m = 0 models predictive of actual plasmabehavior?

I. INTRODUCTION

This paper considers the problem of feedback control forstabilization of the vertical instability in tokamaks. Tokamaksare torus-shaped devices designed to confine a plasma com-posed of ionized hydrogen isotopes while the plasma is heatedto initiate fusion reactions. An example is shown in Figure 1,which illustrates the KSTAR tokamak in Daejon, Korea [1].Introductory descriptions of tokamaks and associated plasmacontrol problems are provided in [2], [3].

A cross-section of the KSTAR device is shown in Figure 2,with a plasma cross-section shown in the interior. The insta-bility we consider is one in which the toroidal plasma moveseither up or down in the vacuum chamber until it meets theinterior vessel wall and is extinguished. One or more of thecontrol coils is typically connected in feedback with a mea-surement of the vertical position to provide stabilizing control.Currents induced in control coils and passive conductors by theplasma motion provide damping, but cannot actually stabilizethe instability. In KSTAR, the active control coils 1 through

J. William Helton (Fellow IEEE) is with the Faculty of Mathematics,University of California at San Diego, 9500 Gilman Drive, La Jolla, CA92093 helton@math.ucsd.edu

Kevin J. McGown is with the Department of Mathematics, Universityof California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093kmcgown@math.ucsd.edu

M. L. Walker (Member IEEE) is with General Atomics, San Diego, CA92121 walker@fusion.gat.com

14 outside of the vacuum vessel are superconducting and areused to establish the plasma equilibrium. The internal coils15 through 18 are copper, with coils 15 and 17 dedicated tovertical position (stability) control and coils 16 and 18 usedfor radial position control.

Fig. 1. Three-dimensional view illustration of the KSTAR Tokamak.

Fig. 2. Cross-section of the KSTAR Tokamak.

Feedback controllers for stabilizing the vertical instabilityin operating tokamaks are almost always designed based ona massless model of the plasma. However, the plasma doesin fact have a small positive mass, and of all the controllersC which stabilize the mass zero plasma, some stabilize them > 0 plasma (at least for small m > 0) and others do not.

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As we shall see this is a bifurcation-like phenomenon, with ourgoal being to not have C on a m > 0 destabilizing branch. Weprovide inequalities saying exactly when this does or does nothappen (see §I-C). An illustration is Example 1.1 with a PDcontroller, for which choosing an m > 0 stabilizing controlleramounts to its gains having the correct sign.

Our results here work under general hypotheses which applyto a far broader class than the PD controllers currently foundin today’s tokamaks. Thus they resolve to a significant extentthe massless vs. massive model issue when future tokamaksroutinely deploy more sophisticated control algorithms (e.g.LQG or H∞). For example, confirmation of the correctinequality required for stabilizing gains (i.e. with correct sign)using physics intuition, as is presently done for PD controllers,is much more difficult in high order multivariable controllers.

We emphasize that the primary purpose of this work is notto perform a conventional robustness analysis of a particularform of controller. Rather, the primary purpose is to defineconditions under which the zero-mass model can be used at allto develop controllers for stabilizing the vertical instability ofthe physical plasma-in-tokamak system. These conditions areof inherent theoretical interest, and also have some practicalimplications for plasma control design.

A. Background on the Model

The dynamic description of the plant comprising a tokamakconfining an assumed axisymmetric plasma is constructedfrom the basic electromagnetic equation [4]

MδI +RδI + Ψz zC + Ψr rC = UδV (1)

where M and R are the mutual inductance and resistanceof the toroidal conductors whose currents define the statesof (1), and Ψz,Ψr represent the partial derivatives of fluxvalues at those conductors with respect to vertical (zC) andradial (rC) motion of the plasma current centroid (“center ofmass” of the distributed plasma current). Toroidal currents in(respectively, voltages on) conductors are represented by thevector I (resp., V ) while δI = I − Ieq (resp., δV = V −Veq )represents a perturbation of the currents (voltages) from theirvalues defining a nominal plasma equilibrium. The vector Iincludes both currents in active control coils and in toroidalconducting vessel elements. In the following, we use thenotation δI = [δIc δIv]T to represent a partitioning of thecurrent vector into the nc active control coils and the nvpassive (vacuum vessel) currents, U = [Inc 0nc×nv ]T , whereInc

and 0nc×nvare identity and zero matrices respectively.

The motion of the current centroid for a plasma having massm can be represented by the inertial momentum equations

mzC = fzδzC + fIδI (2)

mrC = frδrC + (∂Fr/∂I)δI (3)

where δzC = zC−zC,eq, δrC = rC−rC,eq represent perturbedvalues of plasma current centroid vertical and radial coordi-nates relative to their values at the nominal plasma equilibrium,fz = ∂Fz/∂zC , fI = ∂Fz/∂I , fr = ∂Fr/∂rC , and Fz ,Fr are the total vertical and radial forces on the plasma, allquantities derived from a linearization of the plasma response

around the chosen nominal plasma equilibrium. We note thatΨz = fTI [5] [6]. The mass m > 0, which is difficult toaccurately estimate, will vary slowly relative to the dynamicsof the vertical stability, and therefore may be considered as(an unknown) constant in the analysis presented here.

The equations (1) through (3) can be combined to form theoverall plant model. From equation (1) we obtain

M#δI +RδI + Ψz zC = UδV (4)

where M# = M + Ψr(∂rC/∂I) and ∂rC/∂I is computedfrom (3) after setting m = 0. (Justification for setting m = 0for radial response but not for vertical response is discussedbelow.) Defining the variables vz = zC = d(δzC)/dt, xz =[vTz δzTC ]T , we can write (2) as(

0 1m 0

)xz +

(−1 00 −fz

)xz +

(0−fI

)δI = 0 .

Combining with (4), we obtain the matrix equation

Mx+ Rx = UδV , (5)

x =

vzδzCδI

, M =

0 1 0m 0 00 Ψz M#

, (6)

R =

−1 0 00 −fz −fI0 0 R

, U =

00U

.

If the equilibrium plasma boundary is sufficiently verticallyelongated, so that fz > 0, it can be shown [5] that thesystem (5) possesses a single positive real eigenvalue. Theeigenvector corresponding to the unstable root corresponds to anearly rigid vertical motion of the plasma current distribution,hence the name. Stabilization of the vertical instability requiresa feedback control loop that produces radial magnetic fieldacross the plasma in response to changes in some measureof the plasma vertical position, typically the plasma currentcentroid position zC [7]. The vertical control portion of atokamak shape and stability feedback system often takes the(PD) form

δV (t) = −Gp (zC(t)− zC,ref(t) )−GddzC(t)dt

, (7)

where δV is the additional (nonequilibrium) voltage appliedto the control coils, zC − zC,ref is the displacement of zCfrom some reference position zC,ref, and dzC/dt is the verticalvelocity of the plasma. The gains Gp and Gd are vectors whichmap the scalar errors to the set of active control coils. Undermany conditions, this feedback can completely stabilize thevertical instability [7].

B. The Issue of Mass in Plasma Models

For convenience of design, the mass of the plasma isneglected in virtually every design of stabilizing controllersfor the vertical instability [5], [8], [6], [7], [9]. The argumentthat seems to be made most frequently for this approximationis that the plasma mass is only relevant on a time scale muchfaster than that of the instability being stabilized. However, thetime scale being referred to is that of the intrinsic instability

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in the absence of passive damping by induced conductorcurrents. While this clearly would be a valid argument ifthe fast dynamics were asymptotically stable (e.g. fr < 0,the justification for setting m = 0 in (3)), the basis for thisassertion when they are unstable (fz > 0, equation (2)) is notas clear (see, e.g., [10]).

When plasma mass is neglected, the inertial momentumequation (2) is used with m = 0 to derive the algebraic relation∂zC/∂I = −fI/fz , which reduces by two the state dimensionof (5). However, the assumption of m = 0 can lead toerroneous conclusions and, in particular, to PD controllers thatappear to be stabilizing according to the massless analysis butwill not actually stabilize the physical system. Such controllerscan in fact cause the closed loop system to be far more unstablethan the original open-loop system [7], [5]. Following [5], weconsider a very low dimensional example that captures manyaspects of the problem, to illustrate the issue more clearly.

Example 1.1: Assume that there is only one toroidal con-ductor, a (resistive) control coil, other than the plasma presentin the system. Assume that only derivative gain is appliedto the vertical position error when driving this coil, i.e.δV = −Gd δzC . If m > 0 is maintained in the model,the maximum real part of closed loop eigenvalues, γCL, asa function of the feedback gain is shown in the top frame ofFigure 3, while if the mass m is set equal to zero, the mostunstable eigenvalue is shown in the bottom frame. In summary,the figure shows:

The model with m = 0 predicts that all feedback gainsU∗Gd < −1 are stabilizing. However, the model with m > 0shows that all of these gains are in fact extremely destabilizing(the physically correct finding).

Fig. 3. Low-dimensional example.

In the figure, U∗ = −fI/(fzM∗), with M∗ = M#−fIΨz/fz .Details of deriving this finding are later in Example 2.5. Theexistence of this unphysical “stabilizing” branch is a symptomof the fact that m = 0 is a bifurcation point of the systemmodel. This bifurcation is due to the fact that in (2), theforce fz > 0 is destabilizing, but if m < 0, then fz > 0becomes a stabilizing force, entirely contrary to the physicsof the problem. More generally, Figure 3 shows that no purederivative gain can stabilize the example system. In fact, [5]

shows that pure derivative gain can never stabilize any plasma-tokamak system in which m > 0. It was not known a prioriif the destabilizing behavior in this example has a counterpartfor controllers that are more complex than PD; one objectiveof this work was to determine whether this is the case.

In practice, for the simple PD controller (7) and m = 0,additional physical insight is typically used in the controldesign process to ensure a correctly stabilizing controller.Absent this insight or when designing a more sophisticatedcontroller (e.g., [11], [12]) the question remains as to howto guarantee during the design process that a stabilizingcontroller developed with the assumption m = 0 will also bestabilizing for the actual physical plant, since in reality m > 0.The recent emphasis on model-based methods for design ofplasma controllers, combined with uncertainty regarding thebehavior of plasma models around the bifurcation point m = 0provided the motivation for the present work. As illustratedby Example 1.1, designing a controller using a zero massmodel, then testing the controller in a simulation that alsohas zero mass cannot determine a priori if there will bedifficulties in controlling the physical (positive mass) system.It was not known a priori if this example had a counterpartfor more complex controllers. Although a small number ofmodel-based controllers (e.g. [13], [14], [15]) have been testedexperimentally, successful use of such an algorithm does notdemonstrate the absence of risk, only that the particular designprocess used did not choose a control solution in a problematicbranch of a bifurcated system.

In [5], stability properties of the open-loop system (1) werecharacterized and several conditions necessary for a PD con-troller of the form (7) to feedback-stabilize the physical (with-mass) system were derived. In the present work, we addressthe practical problem of characterizing when a controller thathas been designed using the standard but strictly incorrectzero plasma mass assumption is able to stabilize the physicalsystem with mass m > 0. We consider controllers whosetransfer functions have the form

δV (s) = −(sGd +Gp +D(s) ) (δzC(s)− δzC,ref (s) ), (8)

with D(s) a vector of strictly proper rational functions, thatis, D(s) → 0 as s → ∞. Here the length of the vectorD(s) equals the number of active coils. Such controllerssubstantially generalize the PD controller (7) to include allcontrollers operating on the vertical position error that can berepresented by a rational function matrix with entries havingrelative (denominator minus numerator) degree greater than orequal to 1.

C. Main Results

Our main results give necessary and sufficient conditions forsuch a controller to guarantee stability with (at least small)mass m > 0 if it stabilizes the system with m = 0. Theseresults hold for both resistive and superconducting coils. Theprimary technical result (see Theorems 2.10 and 2.20) is:

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Positive Mass Test: A controller (8), which asymptoticallystabilizes a massless plasma, also asymptotically stabilizesthe equivalent plasma with a sufficiently small positive mass(under some minor technical conditions) if and only if thefollowing two inequalities hold:

−fz + fIM−1# [Ψz + gd] > 0 (9)

fIM−1# gp − fIM−1

# RM−1# [Ψz + gd] < 0 (10)

Here the controller gain vectors gp = UGp and gd = UGdare defined to contain zeros in entries corresponding to thepassive (vessel) conductors and equal Gp and Gd in entriescorresponding to actively controlled coils.

If we specialize this result to gp = gd = 0, we obtain thefollowing practical consequence of this result (see Corollaries2.12 and 2.21).

Practical Controller Test: Suppose a massless plasma isasymptotically stabilized by a strictly proper controller. Thenassuming the conditions −fz + fIM

−1# Ψz 6= 0 and

fIM−1# RM−1

# Ψz > 0 (11)

(automatically true for resistive coils since fTI = Ψz , M# > 0and R > 0) the same plasma with sufficiently small positivemass is asymptotically stabilized by the controller if and onlyif

fz < fIM−1# Ψz. (12)

The condition defined by the equality fz = fIM−1# Ψz is

referred to by plasma physicists as the ideal stability limit;systems that exceed this limit have exponential instabilitiesso fast (a few microseconds) that they cannot be practicallyfeedback stabilized. (See [7], where this limit is defined bya different calculation and considered for a low-dimensionalsystem.) In Theorem 3 of [5], it is shown that for plasmasthat are so unstable as to exceed the ideal stability limit,i.e. fz > fIM

−1# Ψz , the massless plasma analysis actually

declares the system to be open-loop stable. The PracticalController Test says that, for a plasma that remains belowthe ideal stability limit, a strictly proper feedback controllerdesigned using the massless model will predict the closed-loop behavior with a physical (positive mass) plasma. Thisresult rigorously validates (provided the condition (12) holds)the established practice of designing a vertical stabilizationcontroller with plasma mass assumed zero if the controller isstrictly proper.

The Positive Mass Test provides caveats for controllers thatare not strictly proper. Considering this test in the contextof the low dimension case in Example 1.1, we see that withchoice of the incorrect sign of derivative gain, it is possibleto drive the system past the ideal stability limit into idealinstability (where U∗Gd < −1 in Figure 3), with the masslessanalysis predicting stability. As a side benefit, the inequalitygiven in the Practical Controller Test also provides a methodfor calculating the ideal stability limit that is intuitive andeasily computable from the plasma equilbrium and devicegeometry.

Note that power supply dynamics and control circuit fil-tering can be incorporated into the feedback model (8) and

will frequently modify the feedback dynamics so that theoverall feedback control transfer function is strictly proper.Thus, even when a nominal controller has set Gp and Gd tobe nonzero, the asymptotic expansion which includes theseexternal dynamics in (8) can have gp = gd = 0.

II. DERIVATION OF MAIN RESULTS

In this section we start by describing precisely the math-ematical problem corresponding to the mass zero versusnonzero issue. Then we state our main results in full detail.

In the following, we shall consider separately the case inwhich the diagonal matrix R is positive definite, correspondingto all coils resistive, and the case in which R is only positivesemi-definite, corresponding to some number of control coilsbeing superconducting.

In §II-B we give the solution to Problem 2.6, and in §II-Cwe translate this into the language of the tokamak and plasmasystem, thus giving the solution to Problem 2.1. In §II-Dwe treat the superconducting case, and in §II-E we outline apractical method for determining the maximum plasma massfor which the zero-mass feedback controller is stabilizing.

A. Mathematical Problem

In §I-A and §I-B, we provided a description of the physicalsystem and control problem that we wish to solve. Here(in §II-A), we transform this description into an equivalentmathematical problem, which is then solved in §II.

We first analyze, in the frequency domain, the plasma withmass model (5) with PD feedback (7) as in [5]. First Laplacetransform the matrix differential equation (5) and insert thePD feedback equation (8) (where, for the moment, we assumed(s) = 0), to obtain a set of state evolution equations definedby the matrix coefficient

s

0 1 0m 0 00 Ψz M#

+

−1 0 00 −fz −fI0 0 R

+

0 0 00 0 00 sgd + gp 0

multiplying the state x on the left hand side of the equality, andall terms on the right hand side representing only operationson the input reference signal.

Adding all these terms together, replacing the Laplacevariable s with λ, and taking the determinant of the resultingmatrix, one obtains

det

−1 λ 0λm −fz −fI0 λΨz + λgd + gp λM# +R.

. (13)

Clearly, the Laplace transformed system has poles wherethis “characteristic polynomial” (see [5]) vanishes. Thus theclosed loop system is asymptotically stable if and only if thisdeterminant is nonzero for all values of λ in the closed RHP.Using simple row and column operations, it can be shown thatthis determinant is the same as the determinant (except for thesign) of the smaller matrix

G(λ) =(

λ2m− fz −fIλΨz + λgd + gp λM# +R

). (14)

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It follows that the closed loop system is asymptotically stableif and only if det G(λ) 6= 0, or equivalently, the matrix G(λ)is invertible, for all λ in the closed RHP.

At this point we note that for all resistive coils both M#

and R are positive definite and hence [λM# +R] is invertiblefor all λ in the closed RHP [5]. Later (in §II-D) we will treatthe superconductor case where it will only be assumed that Ris positive semidefinite instead of positive definite. In order tostudy more general controllers, we let d(λ) = UD(λ) whereD(λ) is a column vector of rational functions such that D(λ)is real on the real axis and D(λ)→ 0 as λ→∞, and considerthe slightly more general matrix

G(λ) =(

λ2m− fz −fIλΨz + λgd + gp + d(λ) λM# +R

)(15)

We would like to answer the following:

Problem 2.1: Suppose one has chosen a controller(gd, gp, d(λ)) so that the system is asymptotically stable formass m = 0. What are necessary and sufficient conditions sothat the system will be asymptotically stable for all masses insome interval containing zero? Furthermore, we would like amethod for computing the maximum allowable mass.It is clear that the physical Problem 2.1 is equivalent toinvertibility of G(λ) for λ in the closed RHP. Next we willexpress this condition in terms of another function of λ.

Definition 2.2: We define a function

Sm(λ) := λ2m−fz+fI [λM#+R]−1 [λΨz+λgd+gp+d(λ)] .

The reason for defining this function is the following:

Remark 2.3: For any complex number λ where [λM# +R]is invertible, we have

G(λ) is invertible ⇐⇒ Sm(λ) 6= 0 .

Proof. This is immediate since Sm(λ) is just the Schurcomplement of the matrix G(λ). See, for example, page 21 of[16].

In addition, this function has the following useful properties:

Lemma 2.4: For all m ≥ 0,1) Sm(λ) is a rational function of λ,2) Sm(λ) = S0(λ) + λ2m,3) If M# is invertible, then S0(λ) is analytic and real

valued at ∞. In particular,

S0(∞) = −fz + fIM−1# [Ψz + gd] .

Proof. From the definition of Sm(λ), the first and second claimare obviously true. It remains to prove the last claim. We have

S0(λ) = −fz + fI [λM# +R]−1[λΨz + λgd + gp + d(λ)] .

Observe that for any nonzero λ where [λM# +R] is invertiblewe have

[λM# +R]−1[λΨz + λgd + gp + d(λ)]

= λ−1[M# + λ−1R]−1λ[Ψz + gd + λ−1gp + λ−1d(λ)]

= [M# + λ−1R]−1[Ψz + gd + λ−1gp + λ−1d(λ)] .

As λ→∞ we have

[Ψz + gd + λ−1gp + λ−1d(λ)] → [Ψz + gd]

and since M# is invertible,

[M# + λ−1R]−1 → M−1# .

The result follows. Example 2.5: We illustrate the formulas above by using

them to derive analytically the assertions of Example 1.1. Notethat the vector of gains gd in this example is the scalar Gdand recall M∗ = M# − fIΨz/fz , which is negative for anopen-loop unstable plasma. Instability or marginal stability ofthe closed loop system is equivalent to Sm(λ) = 0 for some λin the closed RHP (by Remark 2.3 and the italicized sentencefollowing (14)). Multiplying Sm(λ) by the scalar (λM# +R)and dividing by fzM∗, we find that Sm(λ) = 0 is equivalentto

0 = λ+R

M∗+ λU∗Gd − λ2m(λM# +R)

fzM∗, (16)

for all m ≥ 0. From this equation, it is clear that the unstableeigenvalue of the open loop system is equal or approximatelyequal to − R

M∗ > 0 for very small m ≥ 0.A necessary condition for closed-loop asymptotic stability

is that the coefficients in the polynomial (16) are all positiveor all negative.1 The second and third order terms in (16) thatare present when m > 0 force the sign of all coefficients to bepositive for asymptotic stability (recall that M# > 0, R > 0,and M∗ < 0). Therefore any gains satisfying U∗Gd < 0 willnot asymptotically stabilize the m > 0 system. However, theydo asymptotically stabilize the m = 0 system, by the samearguments above. This is the conclusion claimed in Example1.1.

Remark 2.3 and Lemma 2.4 combine to prove that wewould solve Problem 2.1 if we could solve the following moregeneral problem:

Problem 2.6: Suppose that for all m ≥ 0,1) sm(λ) is a rational function of λ,2) sm(λ) = s0(λ) + λ2m,3) s0(λ) is analytic and real valued at ∞.

Suppose s0(λ) has no zeros in the closed RHP. What arenecessary and sufficient conditions so that there exists m∗ > 0with the property that sm(λ) has no λ zeroes in the closedRHP for all m ∈ [0,m∗). Furthermore, we would like amethod for computing the maximum choice for m∗.

Since it is merely a question concerning rational functions,Problem 2.6 constitutes a considerable abstraction of Problem2.1. For this reason, we write sm(λ) to denote a general ratio-nal function with the desired properties (as in the statement ofProblem 2.6), and write Sm(λ) to mean the specific function(which has these properties) given in Definition 2.2.

In this paper we will solve Problem 2.6, thereby solvingProblem 2.1.

1For this second order polynomial, coefficients having a single sign isactually necessary and sufficient, but the condition is necessary for arbitraryorder polynomials.

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B. Solution in Terms of Rational Functions

The solution to Problem 2.6 is given by the followingtheorem, whose proof we postpone until §III.

Theorem 2.7: Suppose that for m ≥ 0,1) sm(λ) is a rational function of λ,2) sm(λ) = s0(λ) + λ2m,3) s0(λ) is analytic and real valued at ∞.

Define a constant c by writing

s0(λ) = s0(∞) + cλ−1 + b(λ) , with λb(λ)→ 0 as λ→∞.

Suppose that s0(∞) 6= 0 and c 6= 0. Further, suppose thats0(λ) has no zeros in the closed RHP. Then there exists m∗ >0 such that sm(λ) has no zeros in the closed RHP for allm ∈ [0,m∗) if and only if s0(∞) > 0 and c < 0.

It remains to give a method for finding the largest possiblechoice of m∗. The following definition will help us define sucha method.

Definition 2.8: Let s0(λ) be a rational function. We call ω0

a critical value if the graph of s0(iω) crosses the positive realaxis at ω = ω0. Clearly there are finitely many critical values.

The following proposition, whose proof we also postponeuntil §III, gives the expression we seek.

Proposition 2.9: In Theorem 2.7, the largest choice for m∗is

m∗ = mink=1,...,`

s0(iωk)ω2k

, (17)

where ω1, . . . , ω` are the nonzero critical values. If there areno nonzero critical values, then the quantity on the righthandside of (17) is defined to be ∞.

C. Vertical Stability of the Tokamak Plasma

Here are necessary and sufficient conditions for a masslessplasma analysis to predict the vertical stability of a plasmawith small mass (thus solving Problem 2.1). That is, weformally state our Positive Mass Test.

Theorem 2.10: Consider the tokamak and plasma systemdiscussed in §I. Suppose the closed loop system is asymptot-ically stable for mass zero. Define

ξ = −fz + fIM−1# [Ψz + gd]

η = fIM−1# gp − fIM−1

# RM−1# [Ψz + gd] .

Suppose that ξ 6= 0 and η 6= 0. Then there exists m∗ > 0 suchthat the system is asymptotically stable for all m ∈ [0,m∗) ifand only if ξ > 0 and η < 0.

Based on a small example set (see §II-E) and on physicalintuition, we speculate that the maximum mass m∗ will alwaysbe so large as to impose no practical constraint beyondsatisfaction of the basic necessary and sufficient conditionsξ > 0 and η < 0 just described.

Example 2.11: For the low dimensional example discussedin Examples 1.1 and 2.5, asymptotic stability requires thatξ > 0, i.e. that −M#fz+fI [Ψz+Gd] > 0. Dividing by fz > 0and combining terms gives −M∗ + fIGd/fz > 0. Dividing

again by −M∗ > 0 gives U∗Gd > 1, clearly contradictingthe condition U∗Gd < −1 derived during the massless plasmaanalysis.

We remark that the necessity of these conditions has alreadybeen shown in [5] for a pure PD controller. If we specializethis result to gp = gd = 0, we obtain the following corollaryby noting that any strictly proper controller can be expressedas an asymptotic expansion d(λ) of the form assumed in (15).

Corollary 2.12: Consider the tokamak and plasma systemdiscussed in §I. Suppose the system with mass 0 is asymp-totically stabilized by a strictly proper controller, and supposefz 6= fIM

−1# Ψz . Then there exists m∗ > 0 such that the

close-loop system is asymptotically stable for all m ∈ [0,m∗)if and only if fz < fIM

−1# Ψz .

Proof. For a strictly proper controller, gp = gd = 0. In thiscase, equation (9) always holds because fTI = Ψz and bothM# and R are positive definite.

To prove the above results we require a lemma whichexplicitly computes the constant c as described in Theorem2.7.

Lemma 2.13: The function S0(λ) given in Definition 2.2has the following expansion in 1/λ:

S0(λ) = −fz + fIM−1# [Ψz + gd]

+fIM

−1# gp − fIM−1

# RM−1# [Ψz + gd]

λ+ . . . .

The above series converges to S0(λ) for |λ| sufficiently large.

Proof. We set M = M# and v = Ψz+gd for ease of notation.We recall that d(λ) in the definition of S0(λ) is a vector ofstrictly proper rational functions. Since each element of d(λ)is a meromorphic function which is analytic at infinity, wemay write

d(λ) =d1

λ+d2

λ2+ . . . .

Now we have

S0(λ) = −fz + fI [λM +R]−1

[λv + gp +

d1

λ+d2

λ2+ . . .

]Observe that, for |λ| sufficiently large,

[λM +R]−1 =[λM(I + (λM)−1R)

]−1

=[I + (λM)−1R

]−1(λM)−1

=[I − (λM)−1R+ ((λM)−1R)2 − . . .

](λM)−1

=M−1

λ− M−1RM−1

λ2+M−1RM−1RM−1

λ3− . . . ,

and hence

[λM +R]−1[λv + gp + d1

λ + d2λ2 + . . .

]=[M−1

λ − M−1RM−1

λ2 + M−1RM−1RM−1

λ3 − . . .]

·[λv + gp + d1

λ + d2λ2 + . . .

]= M−1v + M−1gp−M−1RM−1v

λ

+ M−1RM−1RM−1v−M−1RM−1gp+M−1d1λ2 + . . .

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and the result follows. We remark in passing that the above also gives an alternate

proof to part 3 of Lemma 2.4.

Proof of Theorem 2.10. We apply Theorem 2.7 (which willbe proved in §III) to the function Sm(λ) given in Definition2.2 and take into account the explicit computations of S0(∞)and c given in Lemmas 2.4 and 2.13.

D. Superconducting Case

We recall from §II-A the matrix G given in (15) whichgoverns closed loop stability:

G(λ) =(

λ2m− fz −fIλΨz + λgd + gp + d(λ) λM# +R

)In the superconducting case, though R is positive semidefinite,it is not invertible. In fact, we have

R =(

0 00 Rr

)with Rr invertible. We also have the partitions

fI =(fsI frI

), gp =

(gspgrp

), d(λ) =

(ds(λ)dr(λ)

),

where superscripts s and r stand for superconducting andresistive conductors, respectively. We observe that λM# +Ris invertible for all λ in the closed RHP except λ = 0. Here wecannot hope for the system to be asymptotically stable becausewe will always have a problem at λ = 0. This motivates theintroduction of the following terminology.

Definition 2.14: We call the tokamak and plasma systemalmost asymptotically stable if G(λ) is invertible for all λ inthe closed RHP except possibly λ = 0.

Note that as the terminology suggests, almost asymptoticallystable is a stronger condition than stable, but weaker thanasymptotically stable. The condition that G(λ) is almostasymptotically stable means that the physical system has beenstabilized to the point that all closed loop eigenvalues are inthe open left half plane (asymptotic stability) except for thosefew eigenvalues at λ = 0 due to the superconducting coils.We wish to answer the following:

Problem 2.15: Suppose one has chosen a controller(gd, gp, d(λ)) so that the system is almost asymptoticallystable for mass m = 0. What are necessary and sufficientconditions so that the system will be almost asymptoticallystable for all masses in some interval containing zero? Further-more, we would like a method for computing the maximumallowable mass.

Since R is not invertible in this setting, we introduce thestandard generalization of the inverse of a matrix.

Definition 2.16: The Moore-Penrose pseudoinverse A+ ofa matrix A is the unique matrix satisfying:

1) AA+A = A and A+AA+ = A+

2) (AA+)∗ = AA+ and (A+A)∗ = A+A

It follows that if A−1 exists, then A+ = A−1, and that forany two matrices A and B we have (AB)+ = B+A+.

Definition 2.17: We define the following function

Sm(λ) := λ2m−fz+fI [λM#+R]+ [λΨz+λgd+gp+d(λ)] .

The preceding definition is analogous to the function Sm(λ)given in Definition 2.2. In fact, for all nonzero λ in the closedRHP, Sm(λ) = Sm(λ) since [λM# +R]+ = [λM# +R]−1.

Remark 2.18: For any complex number λ where [λM#+R]invertible, we have

G(λ) is invertible ⇐⇒ Sm(λ) 6= 0 .

Proof. This is the same as the proof of Remark 2.3. If the closed loop tokamak and plasma system with super-

conductors is almost asymptotically stable at mass zero, thenby the preceding remark, we know S0(λ) 6= 0 for all nonzeroλ in the closed RHP. In order to apply Theorem 2.7 to thissituation we will have to impose the additional hypothesis thatS0(0) 6= 0. In order to explicitly describe this extra condition,we now evaluate S0 at zero.

Lemma 2.19: If Sm(λ) is the function in Definition 2.17,then

S0(0) = −fz + frIR−1r [grp + dr(0)] .

Proof. We compute

S0(0) = −fz + fIR+[gp + d(0)]

= −fz +(fsI frI

)(0 00 R−1

r

)(gsp + ds(0)grp + dr(0)

)= −fz + frIR

−1r [grp + dr(0)] ,

which is the desired conclusion. We apply Theorem 2.7 to obtain:

Theorem 2.20: Consider the tokamak and plasma systemwith superconductors discussed above. Suppose the closedloop system (6) and (8) is almost asymptotically stable formass zero. Further, suppose that

−fz + frIR−1r [grp + dr(0)] 6= 0 .

Define

ξ = −fz + fIM−1# [Ψz + gd]

η = fIM−1# gp − fIM−1

# RM−1# [Ψz + gd] .

Suppose that ξ 6= 0 and η 6= 0. Then there exists m∗ > 0 suchthat the closed loop system is almost asymptotically stable forall m ∈ [0,m∗) if and only if ξ > 0 and η < 0. Furthermore,the multiplicity of the eigenvalue at the origin in the systemwith m > 0 is the same as in the m = 0 system.

Proof. First we verify that the functions Sm(λ) satisfy all ofthe hypotheses of Theorem 2.7. Since Sm(λ) = Sm(λ) forall λ in the closed RHP except λ = 0, it is clear that Sm(λ)satisfies properties (1),(2),(3) from Theorem 2.7. In fact, thecomputations of S0(∞) and c we did in Lemmas 2.4 and2.13, respectively, apply here as well. The assumption that thesystem is almost asymptotically stable at mass zero gives usthat S0(λ) is nonzero for all λ in the closed RHP except forpossibly λ = 0. But we have taken S0(0) 6= 0 as an additionalhypothesis and hence S0(λ) is nonzero for all λ in the closed

8

RHP. Therefore the functions Sm(λ) satisfy all the hypothesesof Theorem 2.7.

Since the situation here is a little more subtle, we proveeach implication of the theorem separately. First suppose thatξ > 0 and η < 0. Theorem 2.7 tells us that there exists m∗ > 0such that Sm(λ) has no zeros in the closed RHP for all m ∈(0,m∗), which implies the system is almost asymptoticallystable by Remark 2.18.

Now we treat the converse. By way of contradiction supposethere exists m∗ > 0 such that the system is almost asymptot-ically stable for all m ∈ (0,m∗). This implies that Sm(λ)is nonzero for all λ in the open RHP. If ξ > 0 and η > 0,then Corollary 3.3 says that for m small enough the functionSm(λ) always has a zero in the open RHP. If ξ < 0, thenProposition 3.4 says that for some m ∈ (0,m∗), the functionSm(λ) has a zero in the open RHP. In either case, we havecontradicted the existence of m∗.

Finally, to see that the multiplicity of the eigenvalue at theorigin is the same in either system, we merely notice that thematrix G(0) is independent of m.

We specialize the result in Theorem 2.20 to the case ofstrictly proper controllers to obtain the following Corollary.

Corollary 2.21: Consider the tokamak and plasma systemdiscussed in §I. Suppose the system with mass 0 is asymp-totically stabilized by a strictly proper controller, and supposefz 6= fIM

−1# Ψz and fz 6= frIR

−1r dr(0). Then there exists

m∗ > 0 such that the close-loop system is asymptoticallystable for all m ∈ [0,m∗) if and only if fz < fIM

−1# Ψz .

Proof. For a strictly proper controller, gp = gd = 0 andinequality (9) holds by assumption.

E. Outline of the Method with an Example

Earlier in Example 1.1 we illustrated the main problemwhich arises. Here we outline a practical method for comput-ing the maximum mass stabilizable using gains chosen basedon the massless plasma approximation.

1) Choose a controller (gd, gp, d(λ)) so that the system isasymptotically stable for mass zero (or almost asymp-totically stable for the superconducting case).

2) Perform the Positive Mass Test. That is, check that theconditions given in Theorem 2.10 hold. (For the super-conducting case, this means checking the conditions ofTheorem 2.20.)

3) Plot the graph S0(iω) : −∞ < ω <∞ and estimatethe critical values ω1, . . . , ω` (see definitions 2.2 and2.8). In the superconducting case, we use S0(λ) insteadof S0(λ) (see Definition 2.16).

4) Compute the maximum allowable mass using Proposi-tion 2.9.

As an illustration, we describe an application of this methodto a model of the KSTAR Tokamak [1], shown in figures 1and 2. We will omit the excess details. The aim here is togive the reader an idea as to how the method is carried out.To achieve vertical stability of the plasma we set the followinggains:

gd(15)=gd(17)=0; gp(15)=-gp(17)=800;

Again, we do not layout the details of the model, so we willomit the details as to how the gains were chosen, units, etc.

Step 1: Our software verifies that these gain values stabilizethe closed loop system at mass zero.

Step 2: We perform the Positive Mass Test with the data ofour model. As required by Theorem 2.10, we compute thevalues of ξ and η:

ξ = 6.2 · 106 , η = −208.0 · 106

Since ξ > 0 and η < 0, we know that the closed loop systemis asymptotically stable for sufficiently small mass. (Thereforeit makes sense to compute the maximum allowable mass.)Important Note: If the chosen gain values were to fail thePositive Mass Test, they would be unusable! This step iscrucial.

Step 3: We obtain the plot in Figure 4 for S0(iω). The graphcrosses the real axis at 4 values of ω. The (ω, S0(iω)) pairsare (−11, 2.8e6), (11, 2.8e6), (0, 2.5e7), and (∞, 6.2e6). Thecrossings at ω1 = −11, ω2 = +11 yield nonzero criticalvalues, since wk ∈ R \ 0 and S0(iωk) > 0 for k = 1, 2.

Fig. 4. Plot of S0(iω) : −∞ < ω <∞

Step 4: For k = 1, 2 we compute S0(iωk)/ω2k = 2.3 · 104.

(It was expected that these two values would be equal sinceω1 = −ω2. In general, the nonzero critical values will come inpairs and the above computation only needs to be performedonce for each pair.) The maximum allowable mass m∗ (inkg) is the minimum of the values we just computed (seeProposition 2.9). In this simple example there is only onevalue; indeed, we conclude that the maximum mass whichcan be handled by this controller is m∗ = 2.3 ·104 kg. This isa quantity far greater than the plasma masses occurring in thetokamak, which are on the order of a milligram. This leads usto speculate that in practical situations Step 3 and Step 4 willnot be important, while Step 2 of course is very important.

III. PROOFS

In this section we provide the proofs of the results statedin §II-B. Together, §III-A, §III-B, and §III-C make up theproof of Theorem 2.7, which is the meat of the situation andunderlies all our results. Then in §III-D, we give the proofof Proposition 2.9. Throughout this section we will use basicresults from complex function theory. One reference, amongmany possible references, is [17].

9

A. Counting the zeros of sm(λ)

We want to understand the behavior of the zeros of sm(λ)as we vary the parameter m. We start with a lemma.

Lemma 3.1: Suppose that for m ≥ 0,

1) sm(λ) is a rational function of λ,2) sm(λ) = s0(λ) + λ2m,3) s0(λ) is analytic and real valued at ∞.

Suppose s0(∞) 6= 0. Then we have the following:

1) If s0(λ) has k zeros counting multiplicity, then sm(λ)has k + 2 zeros counting multiplicity for all m > 0.Moreover, all of these zeros move continuously in m.

2) We may write the zeros of s0(λ) as α10, . . . , α

k0

and for fixed m > 0 the zeros of sm(λ) asα1

m, . . . , αkm, β

1m, β

2m so that m 7→ αim gives a con-

tinuous function [0,∞) → C for i = 1, . . . , k, andm 7→ βim gives a continuous function (0,∞) → C fori = 1, 2. We call β1

m and β2m the excess zeros.

3) As m→ 0 we have βim →∞ for i = 1, 2.

Proof. Since s0 is rational with s0(∞) 6= 0, we can writes0 = p/q for coprime polynomials of equal degree; hence

sm(λ) =p(λ) + λ2mq(λ)

q(λ). (18)

When m > 0, the numerator of (18) is a polynomial of degreek + 2; furthermore, (18) is in lowest terms since a commonzero of p(λ) +λ2mq(λ) and q(λ) would be a zero of p(λ). Itfollows that sm(λ) has k + 2 zeros counting multiplicity.

It is straightforward to show that the zeros move continu-ously in m. This proves Part (1) of the lemma and Part (2)follows immediately from it.

It remains to show that βim → ∞ as m → 0. Let βmdenote either of the βim. If it is not the case that βm → ∞as m → 0, then there is a sequence of nonzero real numbersmn → 0 such that βmn

converges to a complex number β0,which must be a zero of s0(λ); that is, β0 = αj0 for some j.We seek a contradiction. Setting m = 0 and letting ε > 0 besufficiently small as in the above argument (where now wehave r = k and γim = αi0) we can choose δ > 0 so thatN(αi0; m) = N(αi0; 0) when 0 ≤ m < δ. By part (1) of thelemma, the total number of zeros of s0(λ) is equal to k. Hencefor 0 ≤ m < δ, we have

′∑N(αi0; m) =

′∑N(αi0; 0) = k

where the sum is taken over a set of representatives for thecollection of distinct ε-balls. However, if we choose N largeenough so that |mN | < δ , |βmN

− αj0| < ε , then we findthat

α1mN

, . . . , αkmN, βmN

∈k⋃i=1

Bε(αi0)

are all zeros of sm(λ), and hence the value of∑′

N(αi0; m)is at least k + 1, a contradiction.

B. The asymptotics of the excess zeros

If we assume that α10, . . . , α

k0 lie in the open LHP, then for

small m > 0, we know that α1m, . . . , α

km also lie in the open

LHP. We want to know which half-plane the βim lie in whenm > 0 is small.

Proposition 3.2: Suppose that for m ≥ 0,1) sm(λ) is a rational function of λ,2) sm(λ) = s0(λ) + λ2m,3) s0(λ) is analytic and real valued at ∞.

Define a constant c by writing s0(λ) = s0(∞) + cλ−1 + b(λ),with λb(λ)→ 0 as λ→∞. Let βm denote either of the excesszeros (as in Lemma 3.1). If s0(∞) > 0, then as m → 0 wehave

<(βm)→ c

2s0(∞), |=(βm)| → ∞ .

Here, <(z) and =(z) denote the real and imaginary parts ofa complex number z, respectively.

Proof. We write sm(λ) = s0(∞) + cλ−1 + b(λ) + λ2m withλb(λ)→ 0 as λ→∞. Let βm be either of the excess zeros.We see that βm satisfies

s0(∞) + cβ−1m + b(βm) + β2

mm = 0

and hence

βms0(∞) + c+ βmb(βm) + β3mm = 0 .

As m → 0, we have βm → ∞ by Lemma 3.1 and henceβmb(βm)→ 0. Therefore, as m→ 0, we have

βms0(∞) + c+ β3mm→ 0 . (19)

We write βm in terms of its real and imaginary parts as βm =rm + jm; for ease of notation we will write r = rm andj = jm, the latter being a pure imaginary number. Note thatβ3m = r3 + 3rj2 + 3r2j + j3 which has real part r3 + 3rj2

and imaginary part 3r2j + j3. Now we decompose (19) intoreal and imaginary parts to find:

c+ r[s0(∞) + (r2 + 3j2)m] → 0 (20)

j[s0(∞) + (3r2 + j2)m] → 0 (21)

as m→ 0. The proof of the proposition will follow from ouranalysis of (20) and (21). First we establish two easily provedclaims.

Claim: For m > 0 small, jm is bounded away from 0.

If this is not the case, then there exists mn → 0 such thatjmn→ 0. For ease of notation, we will write r = rmn

, j =jmn

, and m = mn. In this case we must have |r| → ∞ sinceβm →∞ by Lemma 3.1; hence (20) gives

s0(∞) + (r2 + 3j2)m→ 0 .

Finally, since 3j2m→ 0 this implies s0(∞) + r2m→ 0; thatis, r2m → −s0(∞). This requires s0(∞) ≤ 0 in order to bepossible, which contradicts our hypothesis that s0(∞) > 0.

Claim: rm is bounded

If this is not the case, then there is a sequence mn → 0such that |rmn

| → ∞. We seek a contradiction. For ease of

10

notation, we will write r = rmn, j = jmn

and m = mn. FromEquation (20) we have s0(∞)+(r2+3j2)m→ 0, which gives(r2 + 3j2)m → −s0(∞), and from (21) and the fact that |j|is bounded away from 0, we have (3r2 + j2)m → −s0(∞).Combining these yields

(8j2)m→ −2s0(∞) , (8r2)m→ −2s0(∞) .

Subtracting, we obtain (|j|2 + r2)m → 0 , so r2m → 0which implies s0(∞) = 0, contradicting s0(∞) > 0. Havingsuccessfully proven the claim that rm is bounded, we dispensewith our subsequences.

In light of the fact that rm is bounded, we now know that|jm| → ∞ since |βm| → ∞ by Lemma 3.1. It remains toshow that the limit of rm exists and find its limit. Equation(21) gives

s0(∞) + (3r2 + j2)m→ 0 ;

since r is bounded, this implies s0(∞)+j2m→ 0, and hencej2m→ −s0(∞). Since r is bounded, Equation (20) yields

c+ r[s0(∞) + 3j2m] → 0 .

Further, since j2m → −s0(∞), this implies c +r[−2s0(∞)] → 0 , and therefore r → c/2s0(∞) . Thiscompletes the proof of the proposition.

Now we completely understand the case s0(∞) > 0:Corollary 3.3: Under the hypotheses of Proposition 3.2

plus the assumption that s0(λ) has no zeros in the closedRHP, we have:

1) If s0(∞) > 0 and c < 0, then there exists m∗ > 0such that sm(λ) has no zeros in the closed RHP for allm ∈ (0,m∗).

2) If s0(∞) > 0 and c > 0, then there exists m∗ > 0 suchthat sm(λ) has at least one zero in the open RHP for allm ∈ (0,m∗).

Proof. This follows immediately from the continuity of thezeros given by Lemma 3.1, and the asymptotics given inProposition 3.2.

C. s0(∞) < 0 begets instability

From the analysis given up until this point, it is not clearwhat happens to the βim as m→ 0 in the case where s0(∞) <0; indeed, it is not even clear whether such a limit will exist.The following proposition gives a more delicate analysis ofthe case where s0(∞) < 0.

Proposition 3.4: Suppose that for m ≥ 0,1) sm(λ) is a rational function of λ,2) sm(λ) = s0(λ) + λ2m,3) s0(λ) is analytic and real valued at ∞.

Suppose s0(∞) < 0, and let δ > 0 be given. Then there existsm ∈ (0, δ) such that sm(λ) = 0 for some λ in the open RHP.

Proof. For any m ≥ 0, we have sm(λ) = 0 if and only ifs0(λ) + λ2m = 0. Making the substitution λ = 1/z we findthat this is true exactly when

Fm(z) := m+ z2s0(1/z) = 0

for some z 6= 0. Since s0(1/z) is analytic at z = 0 we maywrite

s0(1/z) = s0(∞) + zg(z)

where g(z) is analytic at z = 0. By taking δ small enough, wemay assume, without loss of generality, that g(z) is analyticand this expression for s0(1/z) holds for all z ∈ B2δ(0). Thisyields

Fm(z) = fm(z) + z3g(z)

where we define fm(z) := m+ z2s0(∞) . Choose ε > 0 suchthat

ε < δ ,ε2|s0(∞)|

2< δ ,

and |z| ≤ ε implies

|zg(z)| < |s0(∞)|2

; (22)

it is possible to accomplish the latter condition (22) since g(z)is analytic at z = 0 and |s0(∞)| > 0 by hypothesis. Fix

m :=ε2|s0(∞)|

2(23)

and observe that m ∈ (0, δ). Define

Ω := z ∈ C | <z > 0, |z| < ε and z0 :=√

m

|s0(∞)|.

Since s0(∞) < 0, we have fm(z0) = 0. Furthermore,

z20 =

m

|s0(∞)|<

2m|s0(∞)|

= ε2 ,

and hence z0 ∈ (0, ε); that is, z0 ∈ Ω. Clearly, the only otherzero of fm(z) is −z0 /∈ Ω. Also, being a polynomial, fm(z)has no poles. Since g(z) is analytic in B2δ(0), we know thatFm(z) has no poles in B2δ(0) ⊇ Ω.

We will shortly apply Rouche’s Theorem (see, for example,[17]) to conclude that Fm(z) has exactly one zero in Ω. Inorder to satisfy the appropriate hypotheses, we must show that|z3g(z)| < |fm(z)| for all z ∈ ∂Ω. In particular, this will alsodemonstrate that Fm(z) has no zeros on ∂Ω. Since |z| ≤ ε forall z ∈ Ω, taking into account (22), it suffices to show that

|fm(z)||z2|

≥ |s0(∞)|2

for all z ∈ ∂Ω. First we consider the portion of ∂Ω that lieson the imaginary axis. If z = iω with ω real and |z| ≤ ε, thenwe have

|fm(z)||z2|

=m

ω2+ |s0(∞)| > |s0(∞)|

2.

(Note that the above relied again on the fact that s0(∞) < 0.)Now we treat the remaining portion of ∂Ω. There |z| = ε and

|fm(z)||z2|

=∣∣∣mz2

+ s0(∞)∣∣∣

≥ |s0(∞)| − m

ε2=|s0(∞)|

2.

The last equality following from (23).Having satisfied the necessary hypotheses, we invoke

Rouche’s Theorem to conclude that Fm(z) has a zero, say

11

z0 ∈ Ω. It follows that γ := 1/z0 is in the open RHP andsm(γ) = 0.

Now the proof of Theorem 2.7 follows immediately fromfrom Corollary 3.3 and Proposition 3.4.

D. The largest value for m∗Having completed the proof of Theorem 2.7, we are now in

a position to prove Proposition 2.9. First we need a preliminarylemma.

Lemma 3.5: Suppose that for m ≥ 0,1) sm(λ) is a rational function of λ,2) sm(λ) = s0(λ) + λ2m.If ω is a nonzero critical value (see Definition 2.8), then

for m = s0(iω)/ω2 , the function sm(λ) has a zero on theimaginary axis.

Proof. By definition, ω being a nonzero critical value im-plies that s0(iω) is a positive real number and hence m =s0(iω)/ω2 , is a positive real number with s0(iω)−ω2m = 0.Substituting λ = iω into the equation sm(λ) = s0(λ) + λ2myields sm(iω) = s0(iω) − ω2m = 0. That is, sm(λ) has azero on the imaginary axis.

Proof of Proposition 2.9. Suppose that the hypotheses ofTheorem 2.7 are satisfied, and further suppose that s0(∞) > 0and c < 0. In this case, Theorem 2.7 says that there existsm∗ > 0 so that sm(λ) has no zeros in the closed RHP forall m ∈ [0,m∗). Choose the largest possible value for m∗,allowing the possibility that m∗ =∞. Now define

m := mink=ω1,...,ω`

s0(iωk)ω2k

,

where ω1, . . . , ω` are the critical values (see Definition 2.8),taking the convention that m = ∞ if there are no criticalvalues. We must show that m = m∗. From Lemma 3.5 wealready know m∗ ≤ m. To show that m∗ ≥ m, which wouldcomplete the proof, we must show that for all m ∈ (0, m),the function sm(λ) has no zeros in the closed RHP.

By way of contradiction, suppose that there exists m ∈(0, m) such that sm(λ) has a zero in the closed RHP. By Corol-lary 3.3, sm(λ) has no zeros in the closed RHP when m > 0is sufficiently small. Moreover, Lemma 3.1 tells us that whenm ∈ (0,∞), all of the zeros of sm(λ) move continuouslyin m. Thus there is some intermediate value of m for whichsm(λ) has a zero on the imaginary axis. That is, sm(iω) = 0for some ω ∈ R. We note that ω 6= 0, since if ω was equalto zero, we would have s0(0) = sm(0) = 0, violating thehypothesis that s0(λ) has no zeros in the closed RHP. Sincesm(iω) = s0(iω) − ω2m, we have s0(iω) = ω2m ∈ (0,∞)and hence the graph of s0(iω) crosses the positive real axisat this particular value of ω. Thus ω is a critical value and weconclude that

m =s0(iω)ω2

≥ m ,

which contradicts the assumption that m ∈ (0, m). An alternative approach to our analysis that has been

suggested is to use methods based on Kharitonov’s Theorem.These methods could determine stability of the polynomial

produced by multiplying the rational function Sm(λ) by itsdenominator. for a parameter m in an interval [0, b], but onemust check stability of Sb(λ) at the end point b. It is naturalto apply a Routh-Hurwitz test to do this. Indeed, Humphreysand Walker have attempted this with limited success, as theresulting symbolic (i.e. non-numerical) expressions became fartoo complex to analyze [5].

IV. CONCLUSIONS

In this paper, we have derived necessary and sufficientconditions for a controller of a particular form to guaranteestability with plasma mass m > 0 if it stabilizes the tokamakplasma vertical instability with plasma mass assumed to bezero. We analyze a fundamental bifurcation that occurs at m =0. The result of this analysis provides the rigorous foundationfor the common practice of designing controllers assumingm = 0. The class of controllers considered significantlygeneralizes the PD controller that is used to represent verticalstabilization algorithms on existing tokamaks, but is limited tocontrollers that operate on a measured plasma vertical positionerror signal. The conditions for m > 0 stabilization are on thecharacteristic parameters of the open-loop system and on thetwo leading terms of the generalized controller, correspondingto the gains Gp and Gd of the PD controller (7).

Note that the main issue addressed in this work is not todetermine how large the mass can be for the closed loop toremain stable. Rather, we determine when there exists anym > 0 for which the closed loop is stable. If one choosesa controller with gains violating either of our Positive MassTest inequalities (9),(10), then no plasma with positive massis stabilizable, notwithstanding the fact that the m = 0 closedloop is stable. The results given plus a conjecture based onphysical intuition and a few examples such as the one in §II-Esuggest that if the closed loop is stable for some m > 0, thenit is stable for all possible physically relevant plasma masses.

In previous analyses (e.g. [7]) of plasma vertical stabiliza-tion control, the use of pure proportional plus pure derivativegain as in equation (7) has been assumed to be representativeof the main features of the problem. We have found howeverthat this simplification is in fact not completely representativeof practical feedback controllers. One obtains strikingly dif-ferent conclusions about closed loop behavior depending onwhat one assumes about the controller’s asymptotic behavior atinfinity. When external control circuits and power supplies areaccounted for, practical controllers tend to be strictly properand therefore our Practical Controller Test, based on theCorollaries 2.12 and 2.21, is more relevant. These corollariesshow that for equilibria that are not too unstable relative to thepassive stabilization of the device, i.e., when fz < fIM

−1# Ψz ,

controllers that stabilize the massless model will also stabilizethe physical (m > 0) plasma. Conversely, when the inequalityis reversed, a massless model analysis will not suffice topredict closed-loop stability of the physical system.

Even though this paper concerns a branching type phe-nomenon, our results may alternatively be viewed as controlrobustness results, but only with respect to the uncertainplasma mass. If the two inequalities (9),(10) (see also Theorem

12

2.20) are satisfied by a controller of the form given in (8) thatis stabilizing for m = 0, we conjecture that this controllerrobustly stabilizes for all (practical) m > 0. Uncertaintiesother than plasma mass (e.g. in the coupling between plasmaand conductors, plasma resistivity, plasma current distribution,pure delays, etc.) can also have an impact on the controllerperformance, but none of these uncertainties is as fundamentalas a branching in the basic system response. Additionalpractical constraints can potentially lead to a less stable closedloop system than the ideal case analyzed here, so obviously acontroller design might not be robust to the effects we havenot analyzed.

V. ACKNOWLEDGEMENTS

Professor Helton was partially funded by NSF grant DMS0700758 and from the Ford Motor Company and KevinMcGown by the same NSF grant. Mike Walker was supportedby the U.S. Department of Energy under Contract DE-FC02-04ER54698.

We gratefully acknowledge the National Fusion ResearchInstitute of Korea for allowing us to use models of the KSTARTokamak for the practical examples.

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