Robust control of resistive wall modes using pseudospectra

Robust control of resistive wall modes using pseudospectra M. Sempf, P. Merkel, E. Strumberger, C. Tichmann, and S. Günter Max-Planck-Institut für 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 Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 2 / 24

STARWALL code An external kink mode can be stabilized by an ideally conducting wall close to the plasma; if the wall is not superconducting, the mode grows on the resistive time scale of the wall resistive wall mode (RWM) STARWALL [1, 2]: 3D ideal MHD stability code specialized to RWMs (E kin neglected) feedback coil system included to stabilize RWMs coupling between different toroidal mode numbers n (3D effect) Inputs to STARWALL: plasma equilibrium 3D wall and coil geometries sensor positions and orientations feedback controller logics Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 3 / 24

General controller model Coils and sensors are grouped into toroidal arrays, respectively. Voltage vector applied to k-th coil array: s l G kl L R k i k L u k = G kl s l R k i k l=1 magnetic field perturbation vector measured by l-th sensor array proportional gain matrix linking coil array k to sensor array l number of sensor arrays artificial additional coil resistance (to make coils faster ) vector of coil currents in array k Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 4 / 24

Structure of the gain matrix G kl G kl is strucured in such a way that the coil array produces a field with toroidal mode number n in response to a perturbation with the same n: G kl ij = ( ) α kl n cos(nϕkl ij ) + βn kl sin(nϕkl ij ), n } {{ } in-phase response }{{} (90/n) phase-shifted response ϕ kl ij = 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: α kl n βn kl R k cosine gains sine gains additional coil resistances Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 5 / 24

Structure of the STARWALL equation Dynamics of the plasma-wall-coils system: Lẋ = Rx, R = R 0 + ( ) α kl n Rkl n,α + βkl n Rkl n,β n,k,l k R k Rk x L R 0 R kl n,α, R kl n,β, R k α kl n, β kl n, R k state vector (coil currents, wall current potentials) inductance matrix + plasma contribution resistance matrix matrices describing the effect of feedback free parameters x(t) e γt parametrized eigenvalue problem (R, L N N ): L 1 Rx i = γ i x i ; stability Re γ i < 0 i = 1,..., N Stabilizing parameter set OPTIM code [1, 3] Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 6 / 24

Automatic controller design procedure STARWALL compute matrices L, R 0, R kl n,α, Rkl n,β, R k TRANSFORM (model reduction) construct orthogonal pattern matrix P (details on next slide) transform each matrix M computed by STARWALL as M = P 1 MP in each M, retain only upper left N red N red block (N red N) OPTIM (feedback optimization in reduced system) find {α kl n, βkl n, R k } so that L 1 R is optimally stable (details later) Cross-check against full-sized system using the optimal set {α kl n, βkl n, R k }, compare properties of L 1 R with those of L 1 R (eigenvalues, pseudospectra,... ; details later) Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 7 / 24

Model reduction: isometric truncation The robust stability concept introduced later requires orthogonality of the projection underlying the model reduction = newly developed isometric truncation procedure: projection onto leading columns of the pattern matrix P Properties of the pattern matrix P = (p 1 p 2... p N ): each column p i represents a system state (current pattern) ohmic loss orthogonality: p T i R 0p j = δ ij, i = 1,..., N, j = 1,..., N first columns are unstable eigenvectors of L 1 R 0 (system without feedback), i., e., RWMs the remaining columns represent physical processes in the stable subspace of L 1 R 0, ordered according to decreasing controllability and observability Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 8 / 24

What does optimal stability mean? ẋ = Ax = x(t) = e ta x(0) = N ξ i x i e γ it i=1 here: A = L 1 R (asymptotic) stability: lim t x(t) = 0 Re γ i < 0 i = 1,..., N Basic objectives for stability optimization good asymptotic stability: Re γ i < 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 x i s are far from orthogonal. Exclusive features which non-normal matrices can have: extreme eigenvalue sensitivity (affects robustness) transient growth ( e ta 1 for some t > 0, although A is stable) another optimization objective: keep sup t>0 e ta small! Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 9 / 24

Measures of asymptotic stability The traditional measure of asymptotic stability: σ(a) = max i=1,...,n Re γ i (spectral abscissa) Minimization of σ(a) pushes the leading eigenvalue(s) as far as possible into the left complex halfplane. Another stability measure, seeing not only the leading eigenvalues: η(a) = N exp(re γ i ) i=1 ( exponential spectral function ) Minimization of η(a) pushes all the eigenvalues as far as possible to the left. However, neither σ(a) nor η(a) guarantee robustness of stability, because sensitivity of eigenvalues is not taken into account. Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 10 / 24

Eigenvalue sensitivity and robust stability An idea of sensitivity of the entire eigenvalue spectrum: γ ɛ (A) is the set of z such that z is an eigenvalue of A + E for some E N N with E < ɛ (ɛ-pseudospectrum) Measure for robustness of stability: ρ(a) = sup{ɛ : A + E is stable for all E N N with E < ɛ} (complex stability radius) Measure of stability for perturbations of maximum allowable ɛ : σ ɛ (A) = sup{re z : z γ ɛ (A)} (ɛ-pseudospectral abscissa) In particular, σ ρ(a) (A) = 0. Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 11 / 24

Relationship between σ ɛ (A) and e ta σ ɛ (A) and e ta are related by various theorems. One of them [4] : sup e ta σ ɛ (A)/ɛ ɛ > 0. t 0 That means, if the eigenvalues of A are so sensitive that σ ɛ (A)/ɛ > 1 for some ɛ, there must be transient growth. Tradeoff: optimize σ ɛ (A) for small ɛ = good asymptotic stability optimize σ ɛ (A) for large ɛ = temperate transient behavior Is optimization of ρ(a) a good compromise? Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 12 / 24

OPTIM s functionality Parallel eigenvalue optimization code for parametrized matrices A OPTIM s objective functions F 1 = σ(a) spectral abscissa F 2 = η(a) exponential spectral function F 3 = ρ(a) neg. complex stability radius ( two-step algorithm [5]) F 4 = σ ɛ (A) ɛ-pseudospectral abscissa ( criss-cross algorithm [6]) Minimization algorithm: gradient bundle method [7], suitable for non-smooth, non-lipschitz functions Additional features for given A: computation of ɛ-pseudospectra boundaries (contour plots in ) computation of e ta plots Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 13 / 24

ITER-like test case [3] plasma: Scenario 4 equilibrium, β = 2.29%, n = 1 perturbations only wall: interior wall only, simplified geometry coils: single array, 7 port plug coils (2 coils missing due to collision with NBI) sensors: single array, 18 sensors, z orientation 6 4 Z [m] 2 0 2 plasma sensor coil 4 wall 6 2 4 6 8 10 R [m] Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 14 / 24

Two unstable RWMs growth rates not exactly equal due to broken axisymmetry current patterns (φ isolines, j = n φ) almost equal, but toroidally phase-shifted by 90 RWM 1 (γ 1 = 21.9 s 1 ) RWM 2 (γ 2 = 21.7 s 1 ) Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 15 / 24

Model reduction: stable subspace patterns 1-4 Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 16 / 24

Model reduction: stable subspace patterns 5-8 Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 17 / 24

Model reduction: stable subspace patterns 9-12 Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 18 / 24

Stability optimization Only n = 1 plasma perturbations accounted for, single coil and sensor array = 3 free parameters: α, β, R Full model dimension: N = 5190; reduced model dimension: N red = 58 Optimization of F 1 = σ(a), F 2 = η(a), F 3 = ρ(a), F 4 = σ ɛ (A), where ɛ = 2ρ opt with ρ opt being the optimal value of ρ(a) obtained after optimizing F 3 Very good agreement between reduced and full model [3] Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 19 / 24

1 7 10 5 7 Pseudospectra after stability optimization boundaries of γ ɛ (A), ɛ indicated by contour labels, dots = eigenvalues angular frequency [1/s] angular frequency [1/s] 100 80 60 40 20 0 1.5 2.5 0.7 150 100 50 0 50 growth rate [1/s] 100 80 60 40 20 0 0.5 1 0.7 0.7 0.5 3.5 0.5 F 1, full model 150 100 50 0 50 growth rate [1/s] 1 1.5 0.5 0.7 2.5 5 3.5 5 7 15 F 3, full model angular frequency [1/s] angular frequency [1/s] 100 80 60 40 20 0 10.7 0.5 150 100 50 0 50 growth rate [1/s] 100 80 60 40 20 0.5 1 0.7 1 0.5 0.7 1.5 0.7 1 0.5 F 2, full model 0 150 100 50 0 50 growth rate [1/s] 1 3.5 2.5 1.5 2.5 7 3.5 5 F 4, full model 10 Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 20 / 24

Transient amplification ( e ta curves) e ta 160 120 80 F 1 F 2 F 3 F 4 40 0 0 0.25 0.5 0.75 1 t [10 1 s] Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 21 / 24

Conclusions Optimization of F 1 = σ(a) or F 2 = η(a) does not produce very robust stability, compared to optimization of F 3 = ρ(a) or F 3 = σ ɛ (A) Optimization of F 2 gives a catastrophic transient peak Even for the F 3 - and F 4 -optimal solutions, the transient behavior is not entirely satisfactory, but there are strategies to improve this further [3] Robustness and transient peaks might be an issue for ITER Robustness and transient behavior should generally be taken into account when designing and rating plasma scenarios to be RWM stabilized Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 22 / 24

Plans for the near future Improvement of the approximation used when taking a time delay between sensors and actuators into account Include the voltage loss in the busbar (this is simple) Realistic ITER modeling including the double wall with port extensions and blanket support, the blanket modules, and in-vessel coils AUG modeling Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 23 / 24

References P. 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. Günter. 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. Günter. 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. Mario Sempf (IPP Garching) Robust RWM control using pseudospectra GOTiT Seminar, January 2009 24 / 24