Robust control of resistive wall modes using pseudospectra · RWM control code package Controller...

RWM control code package Controller optimization concepts ITER-like test case Conclusions

Robust control of resistive wall modes usingpseudospectra

M. Sempf, P. Merkel, E. Strumberger, C. Tichmann, and S. Gunter

Max-Planck-Institut fur Plasmaphysik, EURATOM Association, Garching, Germany

GOTiT Seminar, January 2009

Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 1 / 24


Outline

1 Code package for resistive wall mode feedback stabilization

2 Controller optimization: eigenvalues vs. pseudospectra

3 ITER-like test case

4 Conclusions



STARWALL codeAn external kink mode can be stabilized by an ideally conducting wallclose to the plasma; if the wall is not superconducting, the mode growson the resistive time scale of the wall −→ resistive wall mode (RWM)

STARWALL [1, 2]:3D ideal MHD stability code specialized to RWMs (Ekin neglected)feedback coil system included to stabilize RWMscoupling between different toroidal mode numbers n (3D effect)

Inputs to STARWALL:plasma equilibrium3D wall and coil geometriessensor positions and orientationsfeedback controller logics



General controller model

Coils and sensors are grouped into toroidal arrays, respectively.

Voltage vector applied to k -th coil array:

uk =

L∑

l=1

Gklsl − Rk ik

sl magnetic field perturbation vector measured by l-th sensor arrayGkl proportional gain matrix linking coil array k to sensor array lL number of sensor arraysRk artificial additional “coil resistance” (to make coils “faster”)ik vector of coil currents in array k



Structure of the gain matrix Gkl

Gkl is strucured in such a way that the coil array produces a field withtoroidal mode number n in response to a perturbation with the same n:

Gklij =

∑

n

(

αkln cos(nϕkl

ij )︸︷︷︸

in-phase response

+ βkln sin(nϕkl

ij )︸︷︷︸

(90/n)◦ phase-shifted response

)

,

ϕklij = toroidal angle between coil i of array k and sensor j of array l .

The sum runs over all n’s to be simultaneously controlled.

Remaining free parameters defining the feedback logics:

αkln “cosine gains”

βkln “sine gains”

Rk additional coil resistances



Structure of the STARWALL equationDynamics of the plasma-wall-coils system:

Lx = Rx, R = −R0 +∑

n,k ,l

(

αkln Rkl

n,α + βkln Rkl

n,β

)

−∑

k

Rk Rk

x state vector (coil currents, wall current potentials)L inductance matrix + plasma contributionR0 resistance matrixRkl

n,α, Rkln,β, Rk matrices describing the effect of feedback

αkln , βkl

n , Rk free parameters

x(t) ∼ eγt −→ parametrized eigenvalue problem (R, L ∈� N×N ):

L−1Rxi = γixi ; stability⇐⇒ Re γi < 0 ∀i = 1, . . . , N

Stabilizing parameter set←− OPTIM code [1, 3]



Automatic controller design procedureSTARWALL

compute matrices L, R0, Rkln,α, Rkl

n,β , Rk

TRANSFORM (model reduction)construct orthogonal “pattern matrix” P (details on next slide)transform each matrix M computed by STARWALL as M = P−1MPin each M, retain only upper left Nred × Nred block (Nred � N)

OPTIM (feedback optimization in reduced system)

find {αkln , βkl

n , Rk} so that L−1R is “optimally” stable (details later)

Cross-check against full-sized system

using the optimal set {αkln , βkl

n , Rk}, compare properties of L−1R withthose of L−1R (eigenvalues, pseudospectra, . . . ; details later)



Model reduction: “isometric truncation”The robust stability concept introduced later requires orthogonality ofthe projection underlying the model reduction =⇒ newly developed“isometric truncation” procedure: projection onto leading columns ofthe pattern matrix P

Properties of the pattern matrix P = (p1 p2 . . . pN):

each column pi represents a system state (current pattern)“ohmic loss” orthogonality: pT

i R0pj = δij , i = 1, . . . , N,j = 1, . . . , Nfirst columns are unstable eigenvectors of −L−1R0 (systemwithout feedback), i., e., RWMsthe remaining columns represent physical processes in the stablesubspace of −L−1R0, ordered according to decreasingcontrollability and observability



What does “optimal stability” mean?

x = Ax =⇒ x(t) = etAx(0) =

N∑

i=1

ξixieγi t here: A = L−1R

(asymptotic) stability: limt→∞ x(t) = 0⇐⇒ Re γi < 0 ∀i = 1, . . . , N

Basic objectives for stability optimizationgood asymptotic stability: Re γi < −q ∀i = 1, . . . , N, q > 0 “large”robust stability: A stable =⇒ A + E stable for any “moderate” E

With feedback, A is non-normal, i.e., the xi ’s are far from orthogonal.

Exclusive features which non-normal matrices can have:extreme eigenvalue sensitivity (affects robustness)transient growth (||etA|| � 1 for some t > 0, although A is stable)

−→ another optimization objective: keep supt>0 ||etA|| small!Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 9 / 24


Measures of asymptotic stabilityThe traditional measure of asymptotic stability:

σ(A) = maxi=1,...,N

Re γi (spectral abscissa)

Minimization of σ(A) pushes the leading eigenvalue(s) as far aspossible into the left complex halfplane.

Another stability measure, “seeing” not only the leading eigenvalues:

η(A) =N∑

i=1

exp(Re γi) (“exponential spectral function”)

Minimization of η(A) pushes all the eigenvalues as far as possible tothe left. However, neither σ(A) nor η(A) guarantee robustness ofstability, because sensitivity of eigenvalues is not taken into account.



Eigenvalue sensitivity and robust stabilityAn idea of sensitivity of the entire eigenvalue spectrum:

γε(A) is the set of z ∈ � such that z is an eigen-value of A + E for some E ∈ � N×N with ||E|| < ε

(ε-pseudospectrum)

Measure for robustness of stability:

ρ(A) = sup{ε : A + E is stable for all E ∈ � N×N with ||E|| < ε}

(complex stability radius)

Measure of stability for perturbations of “maximum allowable ε”:

σε(A) = sup{Re z : z ∈ γε(A)} (ε-pseudospectral abscissa)

In particular, σρ(A)(A) = 0.



Relationship between σε(A) and ||etA||

σε(A) and ||etA|| are related by various theorems. One of them [4] :

supt≥0||etA|| ≥ σε(A)/ε ∀ε > 0.

That means, if the eigenvalues of A are so sensitive that σε(A)/ε > 1for some ε, there must be transient growth.

Tradeoff:optimize σε(A) for small ε =⇒ good asymptotic stabilityoptimize σε(A) for large ε =⇒ temperate transient behavior

Is optimization of ρ(A) a good compromise?



OPTIM’s functionalityParallel eigenvalue optimization code for parametrized matrices A

OPTIM’s objective functionsF1 = σ(A) spectral abscissaF2 = η(A) exponential spectral functionF3 = −ρ(A) neg. complex stability radius (“two-step algorithm” [5])F4 = σε(A) ε-pseudospectral abscissa (“criss-cross algorithm” [6])

Minimization algorithm: “gradient bundle method” [7], suitable fornon-smooth, non-Lipschitz functions

Additional features for given A:computation of ε-pseudospectra boundaries (contour plots in � )computation of ||etA|| plots



ITER-like test case [3]plasma: “Scenario 4” equilibrium, β = 2.29%, n = 1 perturbations onlywall: interior wall only, simplified geometrycoils: single array, 7 port plug coils (2 coils missing due to collision with NBI)sensors: single array, 18 sensors, z orientation

2 4 6 8 10−6

−4

−2

0

2

4

6

R [m]

Z [m

] plasma

wall

sensor

coil



Two unstable RWMs

growth rates not exactly equal due to broken axisymmetrycurrent patterns (φ isolines, j = n×∇φ) almost equal, buttoroidally phase-shifted by 90◦

RWM 1 (γ1 = 21.9 s−1) RWM 2 (γ2 = 21.7 s−1)



Model reduction: stable subspace patterns 1-4









Stability optimization

Only n = 1 plasma perturbations accounted for, single coil andsensor array =⇒ 3 free parameters: α, β, R

Full model dimension: N = 5190; reduced model dimension:Nred = 58

Optimization of F1 = σ(A), F2 = η(A), F3 = −ρ(A), F4 = σε(A),where ε = 2ρopt with ρopt being the optimal value of ρ(A) obtainedafter optimizing F3

Very good agreement between reduced and full model [3]



Pseudospectra after stability optimizationboundaries of γε(A), ε indicated by contour labels, dots = eigenvalues

growth rate [1/s]

angu

lar f

requ

ency

[1/s

]

15

107

5

3.5

2.5

1.5

1

0.7

0.5

−150 −100 −50 0 50

0

20

40

60

80

100F1, full model

growth rate [1/s]

angu

lar f

requ

ency

[1/s

]

0.5 0.7

1

0.7

0.5

1.5

2.5

3.5 5

7

0.5

0.7

1

−150 −100 −50 0 50

0

20

40

60

80

100F3, full model

growth rate [1/s]

angu

lar f

requ

ency

[1/s

]

1

0.7

0.5

0.7

0.5

1

1.5 2.

5

3.5

5

7

−150 −100 −50 0 50

0

20

40

60

80

100F2, full model

growth rate [1/s]

angu

lar f

requ

ency

[1/s

]

1

0.7

0.5

1

0.7

0.5

1.5

2.5

3.5

5

7

101

−150 −100 −50 0 50

0

20

40

60

80

100F4, full model



Transient amplification (||etA|| curves)

0 0.25 0.5 0.75 10

40

80

120

160

t [10−1 s]

||etA

||

F1

F2

F3

F4



Conclusions

Optimization of F1 = σ(A) or F2 = η(A) does not produce veryrobust stability, compared to optimization of F3 = −ρ(A) orF3 = σε(A)

Optimization of F2 gives a catastrophic transient peak

Even for the F3- and F4-optimal solutions, the transient behavioris not entirely satisfactory, but there are strategies to improve thisfurther [3]

Robustness and transient peaks might be an issue for ITER

Robustness and transient behavior should generally be taken intoaccount when designing and rating plasma scenarios to be RWMstabilized



Plans for the near future

Improvement of the approximation used when taking a time delaybetween sensors and actuators into account

Include the voltage loss in the busbar (this is simple)

Realistic ITER modeling including the double wall with portextensions and blanket support, the blanket modules, andin-vessel coils

AUG modeling



ReferencesP. Merkel and M. Sempf.Feedback stabilization of resistive wall modes in the presence of multiply-connected wall structures.21st IAEA Fusion Energy Conference 2006, Chengdu, China, paper TH/P3-8, 2006.

E. Strumberger, P. Merkel, M. Sempf, and S. Gunter.On fully three-dimensional resistive wall mode and feedback stabilization studies.Phys. Plasmas, 15:056110, 2008. DOI:10.1063/1.2884579.

M. Sempf, P. Merkel, E. Strumberger, C. Tichmann, and S. Gunter.Robust control of resistive wall modes using pseudospectra.New J. Phys., 2009, submitted.

L. N. Trefethen and M. Embree.Spectra and Pseudospectra - The Behavior of Nonnormal Matrices and Operators.Princeton University Press, Princeton and Oxford, 2005, 606 pp.

N. A. Bruinsma and M. Steinbuch.A fast algorithm to compute the H∞-norm of a transfer function matrix.Syst. Contr. Lett., 14:287–293, 1990.

J. V. Burke, A. S. Lewis, and M. L. Overton.Robust stability and a criss-cross algorithm for pseudospectra.IMA J. Numer. Anal., 23:359–375, 2003.

J. V. Burke, A. S. Lewis, and M. L. Overton.Two numerical methods for optimizing matrix stability.Linear Algebra Appl., 351-352:117–145, 2002.


Date post:	04-Oct-2020
Category:	Documents
Upload:	others
View:	2 times
Download:	0 times

Robust control of resistive wall modes using pseudospectra · RWM control code package Controller...

Documents