Plasma q -profile control in tokamaks using a damping assignment passivity-based approach

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Plasma q -profile control in tokamaks using a damping

assignment passivity-based approach

Ngoc Minh Trang Vu, Laurent Lefevre, Rémy Nouailletas, Federico Felici

To cite this version:

Ngoc Minh Trang Vu, Laurent Lefevre, Rémy Nouailletas, Federico Felici. Plasma q -profile control

in tokamaks using a damping assignment passivity-based approach. Control Engineering Practice,

Elsevier, 2016, 54, pp.34-45. �10.1016/j.conengprac.2016.05.003�. �hal-01624840�

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Bb  bvKK2i`B+ TQbBiBp2 b2KB@/2}MBi2 /BbbBTiBQM Ki`Bt-H (x) Bb i?2 iQiKi`Bt-H 2M2`;v 7mM+iBQM Q` >KBKi`Bt-HiQMBM M/ g (x) Bb i?2 +QMi`QH Ki`Bt r?B+? /2T2M/b QM bvbi2K bii2 xX

h?2 KBM B/2 Q7 i?2 A.@S"* K2i?Q/ Bb iQ +?QQb2 M TT`QT`Bi2 722/#+F +QMi`QH Hr u(x) bQ i?i i?2 Q`B;BMH bvbi2K UkXRV Bb TmHH2/ #+F iQ  `272`2M+2 bvbi2K rBi?  b2i Q7 /2bB`2/ T`QT2`iB2bX G2i mb /2bB;M  +HQb2/ HQQT `272`2M+2 bvbi2K, ˙x = [Jd(x) − Rd(x)]∂Hd ∂x (x) UkXkV rBi?Jd(x) = −JdT(x)- Rd(x) = RTd (x)  0 M/  bi`B+i HQ+H KBMBKmK xd 7Q` i?2 +HQb2/ HQQT >KBHiQMBM HdX h?Bb KBMBKmK xdBb  HQ+HHv bi#H2 2[mBHB#`BmK bBM+2, d dtHd= − (∂xHd) T_R d(∂xHd) ≤ 0 UkXjV

i?MFb iQ Jd(x) = −JdT(x)X h?2 biiB+ bii2 722/#+F Bb

i?2M +?Qb2M bm+? i?i i?2 +HQb2/@HQQT bvbi2K Ki+?2b i?Bb `272`2M+2 S*> bvbi2K #v mbBM; i?2 dzimMBM; T`K2i2`bǴ

Ja(x)- Ra(x)- Ha(x) bm+? i?i Jd(x) = J (x) + Ja

(x)-Rd(x) = R (x) + Ra(x) M/ bm+? i?i Hd(x) = H (x) +

Ha(x) ?b i?2 KBMBKmK i xdX h?Bb H2/b iQ  Ki+?BM;

2[miBQM 7Q` i?2 2[mBpH2M+2 Q7 UkXRV M/ UkXkV r?B+? `2/b, (J − R)∂H ∂x + gu = (Jd− Rd) ∂Hd ∂x UkX9V M/ H2/b iQ i?2 722/#+F, u=gTg−1gT  (Jd− Rd) ∂Hd ∂x − (J − R) ∂H ∂x  UkX8V

"2bB/2b- i?2 7QHHQrBM; +QM/BiBQMb `2 `2[mB`2/ 7Q` i?2 bQHm@ iBQM, BV UAMi2;`#BHBivV ∂2Hd ∂x2 (x) =  ∂2Hd ∂x2 (x) T UkXeV BBV U1[mBHB#`BmK bbB;MK2MiV ∂Hd ∂x (xd) = 0 UkXdV BBBV UGvTmMQp bi#BHBivV ∂2Hd ∂x2 (xd) > 0 UkX3V

h?2 }`bi +QM/BiBQM BKTHB2b i?2 2tBbi2M+2 Q7 i?2 b+H` 2M@ 2`;v 7mM+iBQMHd- r?BH2 i?2 irQ Qi?2`b 2Mbm`2 `2bT2+iBp2Hv

i?2 2tBbi2M+2 Q7 i?2 KBMBKmK Q7Hd i i?2 /2bB`2/ 2[mBHB#@

`BmK pHm2 xd M/ Bib bi#BHBivX

h?Bb ;2M2`H /2bB;M K2i?Q/QHQ;v T`2b2`p2b KMv /2@

;`22b Q7 7`22/QK bBM+2 i?2 +QMi`QHH2` Bb b2i QMHv QM+2Ja,Ra

M/ Ha ?p2 #22M +?Qb2MX AM i?Bb TT2`- r2 T`QTQb2 

T`iB+mH` /2bB;M K2i?Q/QHQ;v 7Q` Qm` bvbi2K BM i?2 M2ti bm#b2+iBQMbX

kXkX AMi2;`iQ` 2ti2MbBQM

M BMi2;`H +iBQM Kv #2 //2/ iQ i?2 A.@S"* +QM@ i`QH +iBQM BM Q`/2` iQ 2HBKBMi2 i?2 biiB+ 2``Q` /m2 iQ i?2 TT`QtBKiBQMb Q` /Bbim`#M+2b- r?BH2 Hrvb +QMb2`p@ BM; i?2 S*> bi`m+im`2 Q7 i?2 r?QH2 +HQb2/@HQQT bvbi2K U bBKTH2 /2bB;M 7Q` i?Bb BMi2;`H +iBQM Bb T`QTQb2/ BM (Rk-b2+Xe)VX AM i?Bb rQ`F- r2 ?p2 /Ti2/ i?Bb BMi2;`iQ` BMiQ Qm` bvbi2KX

S`QTQbBiBQM kXRX *QMbB/2` i?2 bvbi2K Q7 2[miBQM UkXRV BM +HQb2/@HQQT rBi? u= β (x) + v,  ˙x ˙ xI  =  Jd(x) − Rd(x) KI −KT I 0   ∂xW ∂xIW  UkXNV

r?2`2 xI ∈ Rk `2 i?2 2ti2M/2/ BMi2;`H p`B#H2b M/

KI ∈ Rn×ki?2 +Q``2bTQM/BM; BMi2;`H ;BMbX h?2 2ti2M/2/

>KBHiQMBM

W(x, xI)  Hd(x) + HI(xI) UkXRyV

[mHB}2b b  M2r GvTmMQp 7mM+iBQM- BM r?B+?HI Bb +QM@

bB/2`2/ b i?2 2M2`;v 7mM+iBQM bbQ+Bi2/ rBi? i?2 BMi2;`H 2z2+iX Ai Kv #2 /2bB;M2/ b i?2 mbmH [m/`iB+ 7Q`K,

HI = 1₂xTIQIxI QI = QTI ∈ Rk+×k UkXRRV h?2 BMi2;`H +QMi`B#miBQM BM i?2 +QMi`QH +iBQM Bb i?2M /2@ `Bp2/, v= −gTg−1gTKIQI ˆ t 0 K T I∂xHd UkXRkV

ai#BHBiv T`QT2`iB2b Q7 xd `2KBM T`2b2`p2/ bBM+2 i?2 m;@

K2Mi2/ bvbi2K UkXNV F22Tb i?2 +MQMB+H S*> 7Q`KX

_2K`F kXkX h?2 A.@S"* +QMi`QH Hr /2p2HQT2/ BM i?2 TT2` +M #2 +QMbB/2`2/ 2[mBpH2Mi iQ  T`QTQ`iBQMH +QM@ i`QHH2`X >Qr2p2`- i?2 +QMi`QHH2` ;BM Bb /2`Bp2/ #b2/ QM i?2 bvbi2K bi`m+im`2- r?B+? +M ;m`Mi22 i?2 bi#BHBiv M/ i?2 `Q#mbiM2bb ;BMbi i?2 bvbi2K mM+2`iBMiB2b M/ /Bbim`#M+2bX h?2 BMi2;`iQ` Bb QM2 Q7 i?2 2ti2MbBQMb 7Q` S*> bvbi2K r?B+? r2`2 T`2b2Mi2/ BM (Rk- b2+Xe)X h?2 F2v TQBMi Bb i?i i?Bb BMi2;`iQ` +M #2 Tmi `QmM/ i?2 TbbBp2 QmiTmi iQ T`2b2`p2 MQi QMHv i?2 bi#BHBiv- #mi HbQ i?2 S*> bi`m+im`2X

jX *QMi`QH T`Q#H2K M/ bi`i2;v

h?2 bBKTHB}2/ iQFKF THbK KQ/2H BM };m`2 RXR Bb  KmHiB@T?vbB+b bvbi2K- r?B+? +M #2 `2T`2b2Mi2/ BM i?2 };@ m`2 jXRX h?2 ;Qp2`M2/ 2[miBQMb `2 i?2 Jtr2HHǶb 2[m@ iBQMb BM i?2 2H2+i`QK;M2iB+ /QKBM- b r2HH b i?2 2M2`;v #HM+2 2[miBQMb BM i?2 Ki2`BH /QKBMX 6Q` i?2 bF2 Q7 bBKTHB+Biv- i?Bb rQ`F iF2b BMiQ ++QmMi i?2 [mbB@biiB+ 2[mBHB#`BmK bbmKTiBQM U+7X (j- *?TX e)V- BM r?B+? i?2 THbK Kv #2 bbmK2/ iQ ?p2 `2+?2/  K2+?MB+H bi@ iBQM`v T`Q}H2 i 2p2`v BMbiMi +QMbB/2`2/ 7Q` i?2 ?2i Q` K;M2iB+ ~mt /BzmbBQM T?2MQK2MX h?2`27Q`2- QMHv i?2 i?2`KQ/vMKB+b bbQ+Bi2/ rBi? i?2 ?2i i`MbTQ`i 2[m@ iBQM Bb +QMbB/2`2/ BM i?2 Ki2`BH /QKBMX

6B;m`2 jXR, JmHiB@T?vbB+b iQFKF bvbi2K

h?2`2 `2 bQK2 T`QT2` +imiQ`b BM 2+? /QKBM 7Q` THbK ?2iBM;X h?2 Q?KB+ ?2iBM; #v BM/m+iBp2 +m``2Mi Bb +QMi`QHH2/ #v i?2 #QmM/`v HQQT pQHi;2 Vloop Q7 i?2

+2Mi`H bQH2MQB/ U};m`2 RXRVX >Qr2p2`- r?2M i?2 i2KT2`@ im`2 Bb Qp2` R F2o - Q?KB+ ?2iBM; #2+QK2b T`+iB+HHv mb2H2bb /m2 iQ i?2 `2bBbiBpBiv /2+`2bBM; rBi? i?2 i2KT2`@ im`2X >2M+2- MQM BM/m+iBp2 ?2iBM; M/ +m``2Mi /`Bp2 K2i?Q/b r2`2 /2p2HQT2/ iQ /2H rBi? i?Bb T`Q#H2KX h?2 MQM@BM/m+iBp2 +m``2Mi JextM/ i?2 ?2i ~Qr Sheat`2 #Qi?

bmTTHB2/ #v p`BQmb Mi2MMb ?2iBM; bvbi2Kb Ui?2 `2/2` +M `272` iQ (kk) 7Q` /2iBHb Q7 /Bz2`2Mi Mi2MMb bvbi2KbVX h?2 Kbb BMD2+iBQM Bb mb2/ iQ +QMi`QH i?2 THbK /2MbBiv

#v M BM/2T2M/Mi bvbi2K M/ MQi +QMbB/2`2/ BM i?Bb rQ`FX BM i?Bb rQ`FX

hrQ bi`QM; +QmTHBM;b #2ir22M i?2 2H2+i`QK;M2iB+ M/ i?2`KH /QKBMb `2 i?2 GQ`2Mix 7Q`+2 M/ i?2 CQmH2 27@ 72+iX JQ`2 T`2+Bb2Hv- i?2b2 +QmTHBM;b `2 BM+Hm/2/ BM i?2 bvbi2K T`K2i2`b- bm+? b i?2 `2bBbiBpBiv η- i?2 i?2`KH /BzmbBQM +Q2{+B2Mi χ M/ i?2 #QQibi`T +m``2Mi Jbs- M

2ti` +m``2Mi /2MbBiv T`Q/m+2/ #v i?2 K;M2iQ?v/`Q/v@ MKB+ +QmTHBM; 2z2+i U+7X (kk) VX

h?2 TT2` Bb #b2/ QM T`+iB+H +QMi`QH Bbbm2b QM iQF@ KF THbK- QM2 Q7 i?2K Bb i?2 /pM+2/ +QMi`QH Q7 i?2 b72iv 7+iQ` T`Q}H2 qX h?Bb 7+iQ` Bb BM@7+i i?2 `iBQ #2@ ir22M i?2 K;M2iB+ iQ`QB/H }2H/ M/ i?2 TQHQB/H QM2-M/ Bi Bb TT`QtBKi2/ #v BMp2`bBQM Q7 i?2 THbK +m`@ `2Mi T`Q}H2X h?2 7+iQ` Bb `2HHv BKTQ`iMi 7Q` K;M2iQ@ ?v/`Q/vMKB+ bi#BHBiv M/ THbK +QM}M2K2Mic 7Q` 2t@ KTH2 Bi Bb mbmHHv /2bB`#H2 q >1 U+7X (j- *?TX e)V M/ iQ #2 F2Ti i?2 q@T`Q}H2 M2` i?2 +2Mi2` b ~i b TQbbB#H2X aQK2iBK2- i?2 T`Q}H2 Q7 q < 1 Bb HbQ bim/B2/ 7Q` Qi?2` THbK #2?pBQ`b- bm+? b i?2 br@i22i? T?2MQK2MQM U+7X (kk) VX

h?2 THMi KQ/2H 7Q` i?2 q T`Q}H2 +QMi`QH Bb rB/2Hv mb2/ b i?2 `2bBbiBp2 /BzmbBQM 2[miBQM- r?B+? Bb  S`iBH .Bz2`@ 2MiBH 1[miBQM US.1V /2`Bp2/ 7`QK i?2 Jtr2HHǶb 2[m@ iBQMbX h?2 /2iBHb Q7 i?Bb +QMi`QHH2/ KQ/2H `2 /2p2HQT2/ BM b2+iBQM 9X JQ`2Qp2`- b2+iBQM 8 `2p2Hb  +QmTH2/ +QMi`QHH2/ KQ/2H iQ iF2 BMiQ ++QmMi i?2 i?2`KH /QKBM rBi? i?2 ?2i i`MbTQ`i 2[miBQM M/ i?2 bvbi2K +QmTHBM;b 7Q` q T`Q}H2 +QMi`QH- r?B+? KF2b i?2 /pMi;2 Q7 Qm` +QM@ i`QH /2bB;M BM +QKT`BM; rBi? i?2 T`2pBQmb +QMi`QH Hrb UKQ/2H@#b2/ T`2/B+iBp2 +QMi`QH BM (R3)- +m``2Mi T`Q}H2 QT@ iBKH i`+FBM; BM (Rd)- `Q#mbi +QMi`QHH2` 7Q` i?2 TQHQB/H K;M2iB+ ~mt T`Q}H2 BM (Re)- GvTmMQp TT`Q+? BM (R) Q` bHB/BM; KQ/2 BM (d)V r?2`2 QMHv Jtr2HHǶb 2[miBQMb `2 +QMbB/2`2/X

h?2 i`;2i T`Q}H2b Q7 q `2 mbmHHv ;Bp2M #v i?2 T?vbB@ +Bbib #b2/ Km+? QM i?2 iQFKF K+?BM2- ++QKTMv rBi? i?2 +QKTH2t +H+mHmb T`Q;`Kb b r2HH b i?2 i2bi b+2M`BQX

AM i?2 b2[m2H- i?2 q T`Q}H2 +QMi`QH mbBM; A.@S"* K2i?Q/ Bb i2bi2/ rBi? i?2 _ShP_ +Q/2 /2p2HQT2/ 7Q` i?2 h*o iQFKF U+7X (e- 8)VX _ShP_ Bb  R. i`Mb@ TQ`i +Q/2 bT2+BHHv /2bB;M2/ 7Q`  7bi 2t2+miBQM +QKTi@ B#H2 rBi? i?2 M22/b 7Q` `2H@iBK2 +QMi`QH Q` 7Q` MQMHBM2` QTiBKBxiBQM b+?2K2bX h?2 _ShP_ T`K2i2`b `2 pH@ B/i2/ rBi? 2tT2`BK2MiH /iX h?2 +Q/2 Bb HbQ +QmTH2/ iQ i?2 `2H THMi Q7 h*o 7Q` bQK2 /B;MQbiB+b M/ +QM@ i`QH Tm`TQb2b- bm+? b ;BpBM; bQK2 BKTQ`iMi THbK T`Q@ }H2 2biBKiBQMb, i?2 q@T`Q}H2- i?2 `2bBbiBpBiv η- i?2 #QQi@ bi`T +m``2Mi jbs- M/ i?2 +QMi`QH T`Q}H2 fextQ` fheat- 2i+X

>2M+2- BM Qm` rQ`F- Bi Bb mb2/ iQ pHB/i2 i?2 +QMi`QH Hrb #27Q`2 BKTH2K2MiBM; QM h*o `2H THMiX

jXRX *QMi`QH T`Q#H2K

"2bB/2b i?2 Bbbm2 Q7 i?2 +QMi`QHH2` T`K2i2` +?QB+2 UJa,Ra,HaV Q7 i?2 A.@S"* K2i?Q/- i?2 MQM@HBM2`BiB2b 9

M/ +imiQ` +QMbi`BMib biBHH +QKTHB+i2 i?2 +QMi`QH bvM@ i?2bBbX _2;`/BM; i?2 /Bbi`B#mi2/ +QMi`QHb- i?2 `2H +iBQM bB;MHb `2 i?2 iQiH TQr2`b Pext(t) M/ Pheat(t) Q7 i?2

2ti2`MH +m``2Mi /`Bp2 Jext(z, t) M/ i?2 2ti2`MH ?2iBM;

bQm`+2 Sheat(z, t) `2bT2+iBp2Hv Uz /2MQi2b i?2 bTiBH +QQ`@

/BMi2VX h?2 +QMi`QH /2TQbBib g(x) u (t) `2 TT`QtBKi2/

b, _

Jext = fext(z, t) Pext(t)

Sheat = fheat(z, t) Pheat(t)

UjXRV r?2`2 fext(z, t) M/ fheat(z, t) `2 i?2 bT2+B}+ bTiBH

/2TQbBi b?T2b Q7 i?2b2 dz/Bbi`B#mi2/Ǵ +QMi`QHb UivTB+HHv :mbbBM b?T2bV- i?2 +QMi`QH Ki`Bt g(u) +M #2 MQr +QMbB/2`2/ bTiBHHv /2T2M/2Mi g(z)X

h?2b2 +imiQ`b `2 +QMbB/2`2/ BM i?2 h*o +b2 rBi? 1**. U 1H2+i`QM *v+HQi`QM ?2iBM; M/ *m``2Mi .`Bp2V bvbi2K U+7X (kj)VX hrQ 1**. +Hmbi2`b9`2 mb2/ iQ ;2M2`@ i2 #Qi? i?2 MQM@BM/m+iBp2 +m``2Mi M/ i?2 2ti2`MH ?2i@ BM; bQm`+2, _

Pext = PA− PB

Pheat = (PA+ PB)

UjXkV h?2 }`bi QM2 PA Bb mb2/ b  +Q@+m``2Mi bQm`+2 UiQ BM@

+`2b2 i?2 iQiH THbK +m``2MiV M/ i?2 b2+QM/ QM2 PB

b  +QmMi2`@+m``2Mi bQm`+2 UiQ /2+`2b2 i?2 iQiH THbK +m``2MiVX

LQiB+2 i?i BM ;2M2`H- bQK2 2ti2`MH bQm`+2 /2TQbBi T`Q}H2b HbQ /2T2M/ QM i?2 bQm`+2 iQiH TQr2`b bm+? i?i

fext:= fext(z, t, Pext) M/ fheat:= fheat(z, t, Pheat)X PM2

Kv `272` iQ i?2 GQr ?v#`B/ ?2iBM; bQm`+2 r?B+? Bb bBKTHv /2}M2/ BM (kj) 7Q` KQ`2 /2iBHX h?mb- r?2M i?2 MQM@HBM2` +QMi`QH g(x, u) /Q2b MQi +Q``2bTQM/ iQ i?2 biM/`/ S*> 7Q`K BM UkXRV- i?2 bBKTH2 722/#+F +QMi`QH /2`Bp2/ BM 2[m@ iBQM UkX8V +M MQi #2 /B`2+iHv TTHB2/X

jXkX *QMi`QH bi`i2;v

AM Q`/2` iQ /2H rBi? i?2 bvbi2K +QMi`QH T`Q#H2K-  722/@ 7Q`r`/8 _{+QMi`QH rBHH #2 /2bB;M2/ }`bi r?B+? H2/b iQ}

`2+?#H2 bi2/v bii2- i?2M i?2 +HQb2/@HQQT bi#BHBxiBQM M/ i?2 +QMp2`;2M+2 `i2 rBHH #2 BKT`Qp2/ pB M A.@ S"* 722/#+F +QMi`QHX h?Bb UmbmHV +QMi`QH bi`i2;v Bb bmKK`Bx2/ BM i?2 };m`2 jXkX

6Q` i?2 /2bB;M Q7 i?2 A.@S"* 722/#+F +QMi`QH- BM ;2M@ 2`H- r2 rBHH KF2 mb2 Q7 i?2 HBM2`BxiBQM Q7 g(x, u) `QmM/ i?2 2[mBHB#`BmK T`Q}H2(x, u)<sub>d</sub> Ur?2`2 ud/2MQi2b i?2 722/@

7Q`r`/ T`i Q7 i?2 +QMi`QH +iBQM r?B+? Bb /2bB;M2/ rBi? i?2 MQM HBM2` S*> KQ/2HVX h?Bb rBHH ?2HT mb iQ +QK@ TH2i2Hv /BbiBM;mBb? i?2 722/7Q`r`/ M/ 722/#+F 2z2+ib-b r2HH 2z2+ib-b HHQrb mb iQ TTHv i?2 dzi`/BiBQMHǴ A.@S"* 722/#+F /2bB;M /Bb+mbb2/ BM i?2 T`2pBQmb bm#b2+iBQMX

9_{h?2 /2iBHb 7Q` i?2 h*o +imiQ`b Kv HbQ #2 7QmM/ BM i?2}

r2#bBi2 ?iiT,ff+`TTX2T~X+?f`2b2`+?nh*on>2iBM;

8_{h?2 i2`K dz722/7Q`r`/Ǵ mb2/ BM i?Bb TT2` `272`b iQ i?2 biiBQM@}

`v +QMi`QH i i?2 2[mBHB#`BmKX Tokamak model Nonlinear PCH Model inversion Linear IDA-PBC feedback model IDA PBC feedback P Reference Linear IDA PBC + + Nonlinear feedforward Reachable magnetic field profile

u signals Estimated Resistivity, Bootstrap,… III _ ++ e fi Reacchable magnetic f T u signals Mesured states 6B;m`2 jXk, h?2 T`QTQb2/ +QMi`QH bi`i2;v,  MQMHBM2` 722/7Q`r`/ +QMi`QH M/  722/#+F +QMi`QH /2bB;M2/ rBi? i?2 ?2HT Q7 i?2 HBM2`Bx2/ KQ/2H

S`QTQbBiBQM jXRX bbmKBM;  ;QQ/ ++m`+v Q7 i?2 KQ/2H-i?2 722/#+F 2z2+i +QMi`B#miBQM Bb +QMbB/2`2/ rB/2Hv r2F2` i?M i?2 722/7Q`r`/ QM2 BMiQ i?2 iQiH +QMi`QH Hr g(x, u)uX h?mb i?2 722/7Q`r`/ T`i Q7 i?2 +QMi`QHH2` Bb +QKTmi2/ mb@ BM; MQM@HBM2` 2tT`2bbBQM Q7 i?2 bvbi2K M/  HBM2`BxiBQM `QmM/ i?2 2[mBHB#`BmK TQBMi +M #2 /QM2 7Q` i?2 +QKTmi@ iBQM Q7 i?2 722/#+F T`i,

g(x, u)u = g (xd, ud+ δu) (ud+ δu)

≈ gdud+ gdδu

UjXjV

r?2`2 gd = g (xd, ud)- ud Bb i?2 722/7Q`r`/ M/ δu Bb 

722/#+F +QMi`QHX

jXjX _Q#mbiM2bb MHvbBb

AM i?Bb bm#b2+iBQM- i?2 `Q#mbiM2bb Q7 i?2 +QMi`QHH2` ;BMbi irQ FBM/b Q7 mM+2`iBMiB2b Bb bim/B2/X h?2 }`bi QM2 Bb QM i?2 bvbi2K /BbbBTiBQMR +QKBM; 7`QK  #/ 2b@ iBKiBQM Q7 i?2 bvbi2K T`K2i2`b Ubm+? b i?2 THbK `2@ bBbiBpBiv η M/ i?2 i?2`KH /BzmbBQM +Q2{+B2Mi χ b22 (kj)VX h?2 /Bbim`#M+2 /2}M2/ pB i?2 mM+2`iBMiv QM i?2 `2bBb@ iBpBiv δη M/ QM i?2 i?2`KH /BzmbBQM +Q2{+B2Mi δχ H2/b iQ M mM+2`iBMiv δR = δRT <sub>TTHB2/ QMRX h?2 /Bbim`#2/</sub>

bvbi2K +M #2 /2}M2/ b,

˙x = [Jd− (Rd+ δR)] ∂xHd UjX9V

qBi? i?Bb FBM/ Q7 /Bbim`#M+2- i?2 bvbi2K Bb biBHH bi#H2 b HQM; b i?2 iQiH /BbbBTiBQM Bb TQbBiBp2- BX2X[Rd+ δR] > 0X

h?2 b2+QM/ QM2 +QM+2`Mb i?2 mM+2`iBMiv QM i?2 +im@ iQ` T`Q}H2b UJext, Sheat b r2HH b  #/ FMQrH2/;2 Q7 i?2

#QQibi`T +m``2Mi JbsVX Ai +M HbQ BM+Hm/2 i?2 TT`QtBK@

iBQM 2``Q` /m`BM; i?2 HBM2`BxiBQM BM UjXjV,

g(x, u) u = gdud+ gdδu+ δg (u_{ } d+ δu) ζ

UjX8V

h?Bb FBM/ Q7 mM+2`iBMiv Bb MQi i`BpBH iQ bi#BHBx2 #v i?2 +?QB+2 Q7 +QMi`QH T`K2i2`b Rd M/ HdX h?Bb T`Q#H2K

Kv #2 ?M/H2/ +b2@#v@+b2 #v i?2 bKHH ;BM i?2Q`2K U+7X (3)VX

9X 6B`bi +b2, i?2 S*> `2bBbiBp2 /BzmbBQM KQ/2H

h?2 T`QTQb2/ A.@S"* +QMi`QHH2` HHQrb iQ bi#BHBx2 i?2 b72iv 7+iQ` T`Q}H2 i i?2 /2bB`2/ `272`2M+2 mbBM; irQ +QMi`QHH2/ Ub+H`V p`B#H2b, i?2 HQQT pQHi;2 Vloop(t) M/

i?2 2ti2`MH +m``2Mi TQr2` bQm`+2 Pext(t)X h?2 722/7Q`@

r`/ +QMi`QH Bb /2bB;M2/ #v +?QQbBM; i?2 U`2+?#H2V /2@ bB`2/ bi2/v bii2X AM i?Bb 2tKTH2-  bBKTH2 +QMi`QHH2` imMBM; 7Q` i?2 A.@S"* T`K2i2`b Bb T`QTQb2/X h?Bb imMBM; Bb 2[mBpH2Mi iQ /KTBM; bbB;MK2Mi M/ 2M2`;v b?TBM;X h?2 bBKmHiBQMb +?Qb2M iQ BHHmbi`i2 i?2 TT`Q+? `2 #b2/ QM h*o +QM};m`iBQMX

9XRX h?2 `2bBbiBp2 /BzmbBQM +QMi`QH KQ/2H

PM2 Q#iBMb BM (ky)- mbBM; Qm` bvKTH2+iB+ :H2`FBM b+?2K2- i?2 7QHHQrBM; iBK2@/2T2M/2Mi /BbbBTiBp2 SQ`i@ *QMi`QHH2/ >KBHiQMBM KQ/2H 2[mBpH2Mi iQ i?2 `2bBbiBp2 /BzmbBQM 2[miBQM- /2`Bp2/ 7`QK i?2 Jtr2HHǶb 2[miBQMb BM 2H2+i`QK;M2iB+ /QKBM, ∂t  / #     ∂tx = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣  0 −J1 −J2 0     J − R−1(z, t) 0 0 0     R ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦  Gel 0 0 Gmg   / #     ∂xHEM + ⎛ ⎝ −fext −PextCbs 0 0 J4 ⎞ ⎠    g(z)  _Pext Vloop     u(t) U9XRV

h?2 #QH/ p`B#H2b /, #- CbsM/ Cext`2 i?2 iBK2@p`vBM;

+Q2{+B2Mib Q7 i?2 2tTMbBQMb BM i?2 +?Qb2M TT`QtBKiBQM #b2b Q7 `2bT2+iBp2Hv i?2 2H2+i`B+ i?2 K;M2iB+ }2H/-i?2 #QQibi`T +m``2Mi /2MbBiv M/ }2H/-i?2 2ti2`MH +m``2Mi bQm`+2 /2MbBivX LQi2 i?i i?2 #QmM/`v +QMi`QH Vloop(t)

Bb MQr 2K#2//2/ BM i?2 }MBi2 /BK2MbBQM bii2 2[miBQMX h?2 Ki`B+2b J₁, J₂rBi? J₁= −JT

2 `2 Q#iBM2/ 7`QK i?2

/Bb+`2iBxiBQM UM/ `2/m+iBQMV Q7 i?2 bTiBH /2`BpiBQM ∂z

BM i?2 2zQ`i TT`QtBKiBQM #bBb r?BH2 J₄Bb `2Hi2/ iQ i?2 #QmM/`v +Q2{+B2Mi Q7 i?Bb TT`QtBKiBQMX h?2 /BbbBT@ iBQM Ki`Bt R Bb /2i2`KBM2/ #v mbBM; i?2 `2bBbiBpBiv T@ T`QtBKiBQM BM i?2 bK2 TT`QtBKiBQM #b2b η(z, t)X h?2 iQiH UTT`QtBKi2V 2M2`;vHEM Bb /2}M2/ b  [m/`iB+

7mM+iBQM, HEM =1 2  /TGel/+ #TGmg#  =1₂xTQx r?2`2Q =  Gel 0 0 Gmg  - i?2 Ki`B+2b GelM/ Gmg `2

bvKK2i`B+ TQbBiBp2 M/ `2/m+2/ iQ- `2bT2+iBp2Hv- i?2 2H2+@ i`B+ M/ K;M2iB+ T2`K2#BHBiv- 1

0C3 M/ C2

μ0- BM i?2 bBK@

TH2 MBbQi`QTB+ +b2X h?2 +Q2{+B2Mib C2, C3`2 i?2 iQ`B+

+QQ`/BMi2 +Q2{+B2Mib /2}M2/ #v i?2 THbK [mbBbiiB+ 2[mBHB#`BmK U+7X (j- *?TX e)VX h?2 2ti2`MH +m``2Mi bQm`+2 Bb bbmK2/ 2[mH iQ Cext = fext(z) Pext(t) r?2`2 fext(z)

Bb i?2 2ti2`MH +m``2Mi bQm`+2 bTiBH /Bbi`B#miBQMX

h?Bb `2/m+2/ bvbi2K Bb i?mb /2}M2/ /B`2+iHv BM i?2 mbmH 2tTHB+Bi HBM2` S*> 7Q`K UkXRVX L2p2`i?2H2bb- bBM+2 i?2 K@

i`Bt g(z) Bb MQi 7mHH `MF- r2 ?p2 ?2`2 M mM/2`@+imi2/

bvbi2K iQ /2H rBi?X h?2`27Q`2-  `2+?#H2 2[mBHB#`BmK b r2HH b i?2 +Q``2bTQM/BM; 722/7Q`r`/ Bb M2+2bb`v iQ #2 /2i2`KBM2/ BM i?2 b2[m2HX

9XkX _272`2M+2 bii2 ;2M2`iBQM 7Q` i?2 `2bBbiBp2 /BzmbBQM KQ/2H

h?2 b2iiBM; Q7 i?2 2[mBHB#`BmK TQBMi Bb `2bi`B+i2/ #2+mb2 Q7 i?2 +imiQ` HBKBiiBQMb T`2pBQmbHv /Bb+mbb2/X hQ T`Q@ pB/2 iQ i?2 +QMi`QHH2`  `2+?#H2 2[mBHB#`BmK T`Q}H2- 7`QK irQ TQBMib Q7 i?2 `272`2M+2 qref mb2/ #v i?2 T?vbB+Bbi- i?2

bi2/v bii2 xd = (/, #)Td Bb +QKTmi2/ UrBi? #d /2/m+2/

7`QK i?2 BMp2`b2 Q7 i?2 b72iv 7+iQ`V- M/ i?2 +Q``2bTQM/@ BM; 722/7Q`r`/ +QMi`QH ud= (Pext, Vloop)T_d Bb /2`Bp2/X h?2

THbK `2bBbiBpBiv Bb bmTTQb2/ iQ #2 FMQrM b r2HH b Qi?2` bvbi2K T`K2i2`bX h?2 2[mBHB#`BmK T`Q}H2b xd rBHH #2

/Ti2/ QMHBM2 iQ iF2 BMiQ ++QmMi i?2 p`BiBQMb Q7 i?2 +imiQ`b T`K2i2`b fext.d M/ i?2 `272`2M+2 qrefX lbBM;

i?2 ;2QK2i`B+ `2/m+iBQM M/ i?2 bvKTH2+iB+ /Bb+`2iBxiBQM K2i?Q/ BM (ky)- i?2 `2HiBQM #2ir22M i?2 q@T`Q}H2 M/ i?2 K;M2iB+ }2H/ Bθ U+7X (j- TXk88)V #2+QK2b BM i?2 +?Qb2M

TT`QtBKiBQM bT+2b,

Bθ(zi, t) = rf(zi) #d(t) = 2πBφ0a2z_qi

i

U9XkV r?2`2 qi/2MQi2b i?2 pHm2b Q7 q i i?2 TQbBiBQM zi- Bφ0i?2

iQ`QB/H K;M2iB+ }2H/ BMi2MbBiv i z = 0- a Bb i?2 bKHH THbK `/Bmb M/ rf_(z

i) =

wf₁(zi) ... wfN−1(zi)

 i?2 N− 1 "2bb2H TT`QtBKiBQM 7mM+iBQMbX PM i?2 Qi?2` ?M/- i i?2 2[mBHB#`BmK ˙xd = 0- i?2 2[miBQM U9XRV #2@

+QK2b,  J1Gmg#d+ R−1Gel/d+ Cext.d+ Cbs.d = 0 −J2Gel/d+ J4Vloop.d = 0 ⇒ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ /d= 0 (J2Gel)−1_J 4  Pext Vloop  d #d= − (J1Gmg)−1  fext.d R(t)−1J₂−1J4     C(t) ud − (J1Gmg)−1Cbs.d U9XjV

6`QK U9XkV M/ U9XjV- i?2 722/7Q`r`/ +QMi`QH Bb i?2`27Q`2,

 Pext Vloop  d =  rf_(z1) rf_(z2)  C(t) −1 × 2πBφ0a2  z_1/q₁ z2/q2  U9X9V 9XjX A.@S"* bBKTH2 +?QB+2, 2M2`;v b?TBM; M/ /KTBM; bbB;MK2Mi AM i?2 b2[m2H-  MQM@HBM2` dzT`QTQ`iBQMHǴ 722/#+F +QM@ i`QH Bb T`QTQb2/X h?2 /2bB`2/ 2M2`;v Hd = H + Ha Bb bbmK2/ iQ #2 i?2 mbmH  [m/`iB+ 7Q`K, Hd=1 2XTQdX =1 2XT(Q + Qa) X U9X8V

rBi? X = x − xd i?2 bii2 2``Q` M/ Qd = QTd > 0 UBM

bm+?  rv i?iHd biBb}2b i?2 BMi2;`#BHBiv- 2[mBHB#`BmK

M/ bi#BHBiv +QM/BiBQMb UkXe@kX3VVX

q2 T`QTQb2 ?2`27i2`  bBKTH2 T`K2i2` imMBM; iQ KQ/@ B7v QMHv i?2 /KTBM; M/ i?2 2M2`;v Q7 i?2 bvbi2KX q2 +?QQb2 Ja = 0 bBM+2 i?2 BMi2`+QMM2+iBQM bi`m+im`2 Jd

/Q2bMǶi KQ/B7v i?2 +QMp2`;2M+2 bT22/ Q7 i?2 bvbi2K iQ@ iH 2M2`;v Hd BM 2[miBQM UkXjVX h?2 Ki`Bt Qa b?T2b

i?2 iQiH 2M2`;v biQ`;2 7mM+iBQM Hd Q7 i?2 +HQb2/ HQQT

bvbi2K- r?BH2 Ra THvb i?2 `QH2 Q7 //BM; /BbbBTiBQMX

aBM+2 i?2 BMBiBH bvbi2K ?b /BbbBTiBQM R QMHv QM i?2 2H2+i`B+ }2H/ /QKBM UHBMF2/ iQ i?2 }`bi p`B#H2 /V- BiǶb TQbbB#H2 iQ // /BbbBTiBQM QM i?2 Qi?2` }2H/ UHBMF2/ iQ i?2 b2+QM/ K;M2iB+ /QKBM r?Qb2 bii2 Bb /2b+`B#2/ #v

i?2 p`B#H2 #V- pB i?2 Ki`Bt Ra #v mbBM; i?2 7Q`K

Ra = RTa =  0 0 0 Ra  ,0  Ra ∈ Rn×n r?2`2 Ra Bb

+?Qb2M iQ #2  /B;QMH Ki`Bt UQMHv QM i?2 r?QH2 bvbi2K bii2V Q` 2p2MimHHv QMHv ?b bQK2 MQM x2`Q /B;QMH pH@

m2b +Q``2bTQM/BM; iQ i?2 irQ `272`2M+2 TQbBiBQMb z1M/ z2X

h?2 ?B;?2` i?2 pHm2 Q7 Ra Bb- i?2 7bi2` i?2 bvbi2K rBHH

+QMp2`;2X >Qr2p2`- i?2 BMTmi TQr2` HBKBiiBQM QM i?2 HQQT

pQHi;2 Vloop /Q2bMǶi HHQr iQ b2i  ?m;2 pHm2 7Q`RaX

h?2 +?QB+2 Q7Qa Bb KQ`2 +QKTHB+i2/X h?2 QMHv +QM/B@

iBQM iQ #2 biBb}2/ Bb Qd = Q + Qa > 0X PM2 +M `272`

iQ b2+iBQM j BM (R8) 7Q`  /2iBH2/ /Bb+mbbBQM #Qmi ?Qr iQ +?QQb2 Qa bQ@i?i i?2 Ki+?BM; 2[miBQM +QmH/ #2 U7mHHv

Q` T`iBHHvV biBb}2/X M `#Bi``v +?QB+2 Q7Qa rBHH Mv@

rv ;Bp2 M 2[mBpH2Mi +QMi`QH bB;MH #v TTHvBM; UkX8VX >Qr2p2`- i?2 +QMi`QH Hr BM i?Bb +b2 /Q2bMǶi 7mH}HH i?2 Ki+?BM; 2[miBQM UkX9VX h?Bb HbQ K2Mb i?i i?2 biiB+ 722/#+F i`Mb7Q`Kb i?2 Q`B;BMH bvbi2K UkXRV BMiQ i?2 /2@ bB`2/ QM2 UkXkV rBi? M mMFMQrM 2``Q` +QKBM; 7`QK i?2 Ki+?BM; 2[miBQM 2``Q`X h?mb - MQi?BM; ;m`Mi22b i?2 2tBbi2M+2 Q7  +QMi`QH Hr ;`22BM; rBi?  T`iB+mH` +?QB+2 Q7 A.@S"* T`K2i2`bX

PM2 Q7 i?2 TT`QtBKiBM; bQHmiBQMb T`QTQb2/ BM (R8) Bb iQ T`2@KmHiBTHv i?2 Ki+?BM; 2[miBQM U9XeV rBi? i?2 H27i MMB?BHiBQM g⊥ Q7 gd UBX2X bm+? i?i gd⊥gd = 0V- M/ iQ

+?QQb2Qa b i?2 bQHmiBQM Q7 i?2 dzHBM2`Ǵ 2[miBQM,

0 = g⊥

dgdδu = g⊥([J − (R + Ra)] QaX− RaQX)

U9XeV PM2 b?QmH/ `2K`F i?i Qa /2}M2/ BM i?Bb rv Bb iBK2@

/2T2M/2Mi /m2 iQ Bib /2T2M/2M+2 rBi?R (z, t)X

9X9X aBKmHiBQM `2bmHib

AM i?2 bBKmHiBQMb-i?2 2H2+i`QMB+ i2KT2`im`2 T`Q}H2 Q7 i?2 THbK Bb +QKTmi2/ mbBM;  i?2`KH /BzmbBQM KQ/2H BM i?2 _ShP_fh*oX h?2 THbK `2bBbiBpBiv KBMHv /2@ T2M/b QM i?Bb i2KT2`im`2 r?2`2b i?2 #QQibi`T +m``2Mi Bb  7mM+iBQM Q7 Bib ;`/B2MiX

PMHv BM i?Bb i2bi b+2M`BQ- i?2 iQiH ?2iBM; bQm`+2 Pheat

Bb F2Ti mM+?M;2/ /m`BM; i?2 THbK /Bb+?`;2X q2 HbQ +QMbB/2` i?i HH i?2 bii2b `2 K2bm`#H2 Q` +QKTmi#H2 7`QK K2bm`2K2MibX PM2 +M dz+Q``2+iHvǴ 2biBKi2 i?2 r?QH2 T`Q}H2 Q7 q BM h*oX hrQ `272`2M+2 pHm2b 7Q` i?2 b72iv 7+iQ` q `2 /2}M2/ i i?2 `/BH `2HiBp2 +QQ`/B@ Mi2b z<sub>1</sub> = 0.1 M/ z<sub>2</sub> = 0.4 r?B+? `2 BM i?2 BMi2`2biBM; xQM2X h?2 722/7Q`r`/ +H+mHmb ;Bp2b udb r2HH b i?2 `27@

2`2M+2 T`Q}H2 qdU+Q``2bTQM/BM; iQ i?2 irQ `272`2M+2 pHm2b

M/ iFBM; BMiQ ++QmMi +imiQ` HBKBibVX h?2 A.@S"* +QMi`QH /2i2`KBM2b i?2 722/#+F bB;MH δu 7`QK UjXjV iQ +Q``2+i i?2 2``Q` XX

_2K`F 9XRX AM i?2 h*o +b2- i?2 QmiTmi q@T`Q}H2 /2i2`@

KBM2/ #v _ShP_ Bb #b2/ QM  }MBi2 2H2K2Mi /Bb+`2iBx@ iBQM K2i?Q/ Q7 i?2 `2bBbiBp2 /BzmbBQM KQ/2H- r?BH2 i?2 QM2 BM Qm` KQ/2H +QK2b 7`QK  bvKTH2+iB+ ;2QK2i`B+ K2i?Q/ U(ky)VX q?BH2 i?2 qref Bb /2i2`KBM2/ 7`QK Qm` S*> KQ/2H

BM U9XRVX h?Bb 2tTHBMb i?2 MQM@T2`72+i Ki+? Q7 irQ q@ T`Q}H2b b?Qr2/ BM i?2 };m`2b 9XR@9XjX AM Qi?2` rQ`/b- #v BM+`2bBM; i?2 MmK#2` Q7 TT`QtBKiBQM #b2 7mM+iBQMb-i?2 2``Q` +M #2 /2+`2b2/ M/ 7mM+iBQMb-i?2 +QKTH2tBiv M/ 7mM+iBQMb-i?2 +H@ +mHiBM; iBK2 BM+`2b2X h?2 };m`2 9X9 BHHmbi`i2b i?2 KQ/2H rBi? ky #b2 7mM+iBQMb BMbi2/ Q7 8 BM i?2 };m`2 9XjX h?2 T`2+BbBQM Bb Km+? BKT`Qp2/ i i?2 THbK 2/;2 M/ KQbiHv mM+?M;2/ i i?2 +2Mi2`X h?2 +QMi`QHH2` Q#D2+iBp2 Bb ?Qr@ 2p2` QMHv iQ `2;mHi2 q i bQK2 +?Qb2M TQbBiBQMb- mbmHHv M2` i?2 +2Mi2` UM/ bHB;?iHv `2Ht2b 2Hb2r?2`2VX h?2`27Q`2-i?2 TT`QtBKiBQM +?QB+2 Bb  +QKT`QKBb2 #2ir22M T`2+B@ bBQM M/ +QKTH2tBiv- 2bT2+BHHv `2;`/BM; `2H@iBK2 +QMi`QH r?2`2 i?2 +QKTH2tBiv Bb MQi T`QKQi2/X >2M+2- i?2 bKHH MmK#2` Q7 #b2 7mM+iBQMb Bb mb2/ BM i?2 b2[m2HX

AM i?2 i2bi +b2- i?2 `272`2M+2 q@T`Q}H2 Bb b2i rBi? i?2 pHm2b i i?2 irQ TQBMib z= 0.1, 0.4 b qa = (0.85, 1.25)X

h?2 ?2iBM; TQr2` Bb brBi+?2/ QM i t= 0.2s- i?2 722/#+F +QMi`QH bi`ib i t= 0.45sX h?2M i t = 1.2s- i?2 `272`2M+2 Bb +?M;2/ iQ i?2 M2r /2bB`2/ T`Q}H2 qb = (1, 1.25)X q2

/2MQi2 error1, 2 i?2 ;Tb #2ir22M q M/ qref T`Q}H2b i

irQ +QMbB/2`2/ TQbBiBQMbX

aBM+2 i?2 bvbi2K Bb Mim`HHv /BbbBTiBp2- i?2 722/7Q`r`/ BM };m`2 9XRX b?Qrb i?2 +QMp2`;2M+2 Q7 i?2 QT2M2/@HQQT bvbi2KX h?MFb iQ i?2 `2@+QKTmiiBQM Q7 i?2 2[mBHB#`BmK T`Q}H2 b BM U9XjV i 2+? bi2T iBK2- i?2 q@T`Q}H2 `2+?2b  bi2/v bii2 r?B+? Bb [mBi2 +HQb2/ iQ i?2 `272`2M+2 pHm2bX AM };m`2 9Xk M/ 9Xj- i?2 `2bTQMb2 iBK2 Bb /2+`2b2/ #v i?2 722/#+F 2z2+i pB i?2Ra/KTBM; M/ i?2Qa2M2`;v

b?TBM;X

AiǶb BKTQ`iMi iQ MQi2 i?i- BM ;2M2`H i?2 722/#+F 2z2+i /Q2b MQi bB;MB}+MiHv +ib QM i?2 +QMbB/2`2/ T`iB+mH` TQ@ bBiBQMbX Ai /Q2b K2HBQ`i2 i?2 r?QH2 q@ T`Q}H2 r?2M KQ`2 /KTBM; bbB;MK2Mi Bb //2/ Ub22 };m`2 9Xk M/ 9XjVX 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 2 3 4x 10 5 t [s] 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.05 0.1 X: 1.49 Y: 0.06974 t [s] qerrors X: 1.72 Y: 0.104 X: 0.84 Y: 0.05991 X: 0.945 Y: 0.004089 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −5 0 5 10 z qprofile at t=1.995s q qref 5*qerror err1 err2 PA (co) PB (ctr) Pheat/2 −3.105<sub>V</sub> loop 6B;m`2 9XR, PT2M@HQQT `2bTQMb2 rBi? 722/7Q`r`/ +QMi`QH d

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 2 3 4x 10 5 t [s] 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.04 −0.02 0 0.02 0.04 0.06 0.08 X: 1.45 Y: 0.04142 t [s] X: 0.93 Y: −0.01258 X: 0.765 Y: 0.03651 X: 1.525 Y: 0.08376 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −5 0 5 10 z qprofile at t=1.995s q qref 5*qerror err₁ err2 PA (co) PB (ctr) Pheat/2 −3.105_V loop

6B;m`2 9Xk, *HQb2/ HQQT bvbi2K rBi? M A.@S"* +QMi`QHH2` M/  bKHH /KTBM;Ra 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 2.5 3 3.5 4x 10 5 t [s] 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.05 0 0.05 0.1 X: 1.65 Y: 0.05548 t [s] X: 0.85 Y: −0.03063 X: 0.97 Y: 0.001151 X: 1.495 Y: −0.01682 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 z qprofile at t=1.995s q qref 5*q_error err1 err2 PA (co) P_B (ctr) Pheat/2 −3.105_V loop

6B;m`2 9Xj, *HQb2/ HQQT bvbi2K rBi? M A.@S"* +QMi`QHH2` M/  H`;2` /KTBM;Ra 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1.5 2 2.5 3 3.5 4 x 105 t [s] 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.1 −0.05 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 z q_profile at t=1.995s q qref 5*qerror err1 err2 PA (co)[W] PB (ctr)[W] Pheat/2[W] −3.105_V loop[V]

6B;m`2 9X9, aBKmHiBQM rBi? ky #b2 7mM+iBQMb M/ H`;2` /KTBM;

Ra

aBM+2 i?2`2 `2 Hrvb biiB+ 2``Q`b +mb2/ #v i?2 T@ T`QtBKiBQMb BM i?2 HBM2`BxiBQM Ub22 bm#b2+iBQM jXkV Q7

i?2 +imiQ` /Bbi`B#miBQM T`Q}H2 M/ BM i?2 bii2 2biBK@ iBQM BM UjXkV- M BMi2;`iQ` Bb //2/ iQ i?2 bim/B2/ b+2M`BQ M/ `2bmHib `2 b?QrM BM };m`2 9X8X

_2K`F 9XkX h?2 2``Q`b QM i?2 q@T`Q}H2 i i?2 irQ +QMbB/@

2`2/ TQbBiBQMb `2 iQiHHv 2HBKBMi2/ 7i2`0.5sX >Qr2p2`-Bi Bb dz2tT2MbBp2Ǵ BM i2`K Q7 +QMi`QHH2` 2M2`;v iQ +QKT2Mbi2 i?2 ;T QM i?2 irQ pHm2b Q7 i?2 q@T`Q}H2 #2HQr0.05X 6m`@ i?2`KQ`2- PA +MǶi #2 i2+?MB+HHv b2i mM/2` i?2 TQr2` Q7

150 KW M/ +QMb2[m2MiHv i?2 #bQHmi2 +QMp2`;2M+2 b22M BM };m`2 9X8 Bb mM`2+?#H2 BM i?2 `2HBivX PM i?2 Qi?2` ?M/-  bKHH 2``Q` QM q@T`Q}H2 /Q2bMǶi H2/ iQ  bB;MB}@ +Mi +?M;2 BM i?2 #2?pBQ` Q7 i?2 r?QH2 bvbi2KX

AM i?2 };m`2 9Xe- irQ /Bbim`#M+2b `2 //2/X M 2``Q` Q70.5η Bb //2/ QM i?2 K2bm`2/ `2bBbiBpBiv i t = 0.65s M/ M 2``Q` Q70.5fextBb //2/ QM i?2 2biBKi2/ +imiQ`

T`Q}H2 i t= 0.9sX h?2b2 T2`im`#iBQMb `2 2z2+iBp2Hv `2@ D2+i2/X b 2tT2+i2/- i?2 HQQT pQHi;2 VloopT`2/QKBMMiHv

+Q``2+ib i?2 T2`im`#iBQM QM `2bBbiBpBiv r?BH2 i?2 2ti2`MH +m``2Mi bQm`+2 Bb mb2/ iQ +QKT2Mbi2 i?2 /Bbi`B#mi2/ 2``Q` QM i?2 +m``2Mi T`Q}H2X 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8x 10 5 t [s] 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.1 −0.05 0 0.05 0.1 t [s] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −5 0 5 10 z qprofile at t=1.995s q qref 5*q_error err₁ err2 PA (co) PB (ctr) Pheat/2 −3.105_V loop

6B;m`2 9X8, *HQb2/ HQQT bvbi2K rBi? A.@S"* +QMi`QHH2` rBi? BMi2@ ;`iQ` 2z2+i

9X8X 1tT2`BK2MiH `2bmHi

h?2 A.@S"* +QMi`QH rBi? i?2 bBKTH2 /KTBM; M/ 2M@ 2`;v b?TBM; BM bm#b2+iBQM 9Xj rb BKTH2K2Mi2/ QM h*o hQFKF BM Bib Hbi +KTB;M #27Q`2 Bib `2+QMbi`m+iBQM M/ KBMi2MM+2 BM kyRjX h?2 `2H 2tT2`BK2MiH /i `2 +QK@ T`2/ iQ i?2 bBKmHiBQM QM2 BM i?2 };m`2 9XdX h?2 i2bi b+2M`BQ Bb b2i b i?2 bK2 b Qm` T`2pBQmb bBKmHiBQM BM i?2 T`2pBQmb bm#b2+iBQMX

h?2 +QMi`QHH2` `2+i2/ b T`2/B+i2/ BM i?2 }`bi T2`BQ/ `QmM/0.45s−0.8sX h?2 p2`;2 +QMi`QH pHm2b `2 +QMbBb@ i2Mi rBi? i?2 bBKmHiBQM `2bmHi 7Q` HH PA- PB- M/ VloopX

lM7Q`imMi2Hv QM2 Q7 i?2 +Hmbi2` Ui?2 +Q@+m``2Mi bQm`+2-+Hmbi2` AV biQTT2/ rQ`FBM; i t = 0.8sX 6`QK i?2M r2 HQbi i?2 +QMi`QH 2{+B2M+v b i?2 +QMi`QHH2` rbMǶi BM7Q`K2/ #Qmi i?Bb 2p2MiX >Qr2p2`- i?2 +HQb2/ HQQT +QMi`QH i H2bi +ib BM i?2 dz;QQ/ /B`2+iBQMǴ UmbBM; +Hmbi2` B QMHvV r?2M 3

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 2 3 4 5 6 x 105 t [s] 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.05 0 0.05 0.1 t[s] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 z qprofile at t=1.995s q qref 5*qerror err₁ err₂ PA (co) PB (ctr) Pheat/2 −3.105_V loop

6B;m`2 9Xe, *HQb2/ HQQT bvbi2K rBi? A.@S"* +QMi`QHH2` UrBi? i?2 BMi2;`iQ`V rBi? T2`im`#iBQMb QM i?2 `2bBbiBpBiv M/ QM i?2 +imiQ` bTiBH /Bbi`B#miBQM

qref T`Q}H2 `272`2M+2 +?M;2/ mMiBH i?2 2M/X q2 +M `2@

K`F i t = 0.8s, r?2M PA rb +mi Qz- i?2 bBKmHiBQM

`2HBx2/ i?2 p`BiBQM Q7 q@T`Q}H2- i?2M `2+i2/ QM PB.sim

-r?BHbi BM T`+iB+2- Bi b22K2/ i?i i?2 +QMi`QHH2` /B/MǶi };@ m`2 Qmi i?2 +?M;2X Ai KBMiBM2/ PB mMiBH i?2 `272`2M+2

KQp2/ iQ i?2 M2r T`Q}H2X PM2 Q7 i?2 mM2tTHBM2/ Q#b2`p@ iBQM Bb i?i i?2 `2H +QMi`QHH2` PB `2bTQM/2/ KQ`2 bHQrHv

i?M BM bBKmHiBQM- 2p2M BM +QKT`BM; iQ i?2 #2;BMMBM; Q7 i?2 722/#+F i t= 0.45sX

h?2 q@T`Q}H2 i irQ +QMbB/2`2/ TQbBiBQMb `2 HbQ b?Qr2/ BM i?2 };m`2 9XdX h?2 722/#+F +QMi`QH Bb `2@bBKmHi2/ #b2/ QM i?2 2tT2`BK2MiH p2`;2 THbK /2MbBiv n 7`QK i?2 b?Q+F 9N8R9Xh?2 2tT2`BK2MiH +QMi`QH bB;MHb `2 2[mBpH2Mi iQ i?2 bBKmHi2/ QM2- i?2 };m`2 9Xd b?Qrb MQ /Bz2`2M+2b #mi i?2 K2bm`2 MQBb2 #2ir22M i?2KX 6m`i?2`@ KQ`2- i?2 K2bm`2/ pHm2b Q7 i?2 q@T`Q}H2b /Q Ki+? [mBi2 r2HH i?2 bBKmHiBQM QM2bX

8X a2+QM/ +b2, hJ>. *QmTH2/ bvbi2K

+imHHv- i?2 `2bBbiBp2 /BzmbBQM KQ/2H mb2/ BM i?2 T`2pB@ Qmb b2+iBQM Bb 2[mBpH2Mi iQ i?2 `2bBbiBp2 /BzmbBQM 2[miBQM r?B+? Bb rB/2Hv mb2/ b  +QMi`QH KQ/2H 7Q` i?2 THbK Ur?2M QM2 i`B2b iQ `2;mHi2 i?2 b72iv 7+iQ` T`Q}H2- 7Q` BMbiM+2VX >Qr2p2`- b i?2 BM~m2M+2 Q7 i?2 i2KT2`im`2

T QM +2`iBM T`K2i2`b BM 2H2+i`QK;M2iB+ /QKBM- bm+? b i?2 `2bBbiBpBiv η(T ) M/ i?2 #QQibi`T MQM@BM/m+iBp2 +m``2Mi Jbt(∂zT)- Bb MQi M2;HB;B#H2X h?2 J>. +QmTHBM;b

#2ir22M i?2 2H2+i`QK;M2iB+ M/ i?2`KH /QKBMb `2 T`2@ b2Mi2/ BM (kR) BM i?2 B``2p2`bB#H2 2Mi`QTv bQm`+2 i2`K M/ BM i?2 #QQibi`T +m``2Mi bQm`+2X

h?2`27Q`2-  +QmTH2/ +QMi`QH KQ/2H K/2 rBi? i?2 `2@ bBbiBp2 /BzmbBQM bm#KQ/2H M/ i?2 i?2`KH /BzmbBQM bm#@ KQ/2H- Bb BMp2biB;i2/ ?2`27i2`X h?Bb }MBi2 /BK2MbBQMH +QMi`QH KQ/2H Bb Q#iBM2/ rBi? i?2 +QmTHBM; Q7 i?2 irQ }MBi2 /BK2MbBQMH TT`QtBKiBQMb 7`QK i?2 irQ /BzmbBQM KQ/2Hb- mbBM; irQ ;2QK2i`B+ `2/m+iBQM b+?2K2bX ;BM- 

0 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 9x 10 5 t [s] [W] [V] P_ATCV P_BTCV −3.105_V loopTCV P_Asim P_Bsim −3.105_V loopsim 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 t [s] q₁TCV q₂TCV q₁sim q₂sim 6B;m`2 9Xd, h*o b?Q+F 9N8R9

722/7Q`r`/ +QMi`QH rBHH #2 mb2/ iQ ;m`Mi22 i?2 +QKTiB@ #BHBiv Q7 i?2 /2bB;M2/ +QMi`QH rBi? i?2 +imiQ` +QMbi`BMibX h?2M- M A.@S"* 722/#+F Hr rBHH #2 bvMi?2bBx2/ mbBM; i?2 HBM2`Bx2/ 2``Q` bvbi2KX

b i?2 `2bmHi Q7 //BM; i?2 i?2`KH /BzmbBQM bm#KQ/2H- i?B`/ bm#KQ/2H-+imbm#KQ/2H-iQ`- i?2 2ti2`Mbm#KQ/2H-H ?2bm#KQ/2H-iBM; TQr2` bQm`+2 Pheat

-rBHH #2 BMpQHp2/X h?2M- /m2 iQ i?2 mM/2`@+imiBQM Q7 i?2 bim/B2/ bvbi2K- QMHv i?2 pHm2b Q7 i?2 b72iv 7+iQ`b i i?`22 `272`2M+2 TQBMib z<sub>1</sub>, z2M/ z3- /2MQi2/ `2bT2+iBp2Hv q1, q2M/ q3 rBHH #2 bbB;M2/ rBi? i?2 ?2HT Q7 i?2 i?`22

pBH#H2 +QMi`QH p`B#H2b Vloop- Pext M/ PheatX

>Qr2p2`-i?2 +Q``2bTQM/BM; 7mHH `/BH T`Q}H2 `272`2M+2 7Q` q@T`Q}H2 U?2M+2 7Q` i?2 bii2 p`B#H2b Q7 Qm` KQ/2HV Bb `2[mB`2/ BM Q`/2` iQ /2}M2 Qm` A.@S"* bii2 722/#+FX h?Bb T`Q@ }H2 rBHH #2 +QKTmi2/ #v iFBM; BMiQ ++QmMi i i?2 bK2 iBK2 i?2 +imiBQM +QMbi`BMib M/ i?2 hJ>. +QmTHBM;bX h?Bb T`Q+2/m`2 H2/b iQ M +?B2p#H2 bi2/v bii2 7Q` i?2 722/#+F /2bB;M M/- QM i?2 Qi?2` ?M/- i`Mb7Q`Kb i?2 722/#+F /2bB;M BMiQ  HBM2`Bx2/ A.@S"* 722/#+F +QM@ i`QH T`Q#H2KX h?2 Q#iBM2/ MmK2`B+H `2bmHib `2 pHB/i2/ rBi? i?2 _ShP_ +Q/2X

8XRX hJ>. +QMi`QH bvbi2K

h?2 }`bi `2bBbiBp2 /BzmbBQM bm#KQ/2H Bb `2+HH2/ BM 2[m@ iBQM U9XRV BM i?2 T`2pBQmb b2+iBQMX h?2 b2+QM/ bm#KQ/2H Bb i?2 /Bb+`2i2 i?2`KH KQ/2H /2`Bp2/ 7`QK i?2 R. TQ`i@ >KBHiQMBM i?2`KH /BzmbBQM KQ/2H U+7X (kR)V- mbBM; i?2 N

bK2 /Bb+`2iBxiBQM K2i?Q/QHQ;v b 7Q` i?2 `2bBbiBp2 /B7@ 7mbBQM bm#KQ/2HX h?Bb }MBi2@/BK2MbBQMH TT`QtBKiBQM `2/b,  ∂t2ex 0  = 0 JT1 JT2 0  − 0_{0 R}0−1 T   QT2ex 7q  +  S JT4T1  HT= 1 22 T exQT2ex U8XRV

r?2`2 2ex,7q ∈ RN×1 `2 `2bT2+iBp2Hv i?2 iBK2 /2T2M/2Mi

+Q2{+B2Mib Q7 T Dts M/ Q7 i?2 ?2i ~mt fqX h?2 K@

i`B+2b JT1, JT2 ∈ RN×N rBi? JT1 = −JTT2 `2 Q#iBM2/

7`QK i?2 `2/m+iBQM Q7 i?2 bTiBH /2`BpiBQM QT2`iQ` ∂zBM

i?2 +?Qb2M }MBi2 /BK2MbBQMH bTiBH TT`QtBKiBQM #b2bX

QT ∈ RN×N Bb i?2 TQbBiBp2 /2}MBi2 Ki`Bt Q#iBM2/ 7`QK

i?2 /Bb+`2iBxiBQM Q7 i?2 +QMbiBimiBp2 `2HiBQM 7Q` i?2 2M2`;v UBX2X #2ir22M eex M/ T V M/ RT ∈ RN×N Bb i?2 i?2`KH

`2bBbiBpBiv TT`QtBKiBQM r?B+? /2T2M/b QM i?2 i?2`KH /BzmbBQM +Q2{+B2Mi χX h?Bb i?2`KH /BzmbBpBiv +Q2{+B2Mi UBM Qm` KQ/2HV Bb mb2/ iQ r`Bi2 i?2 /BbbBTiBp2 +QMbiBimiBp2 `2HiBQM #2ir22M i?2 i?2`KH 7Q`+2 F M/ i?2 ?2i ~mt fqX

JT4 ∈ RN×1 Bb `2Hi2/ iQ i?2 #QmM/`v +Q2{+B2Mi- T1 Bb

i?2 }t2/ pHm2 Q7 i?2 p2`;2 i2KT2`im`2 i i?2 #QmM/`v M/HT Bb i?2 i?2`KH 2M2`;vX

LQi2 i?i i?2 S*> KQ/2H 7Q` i?2 i?2`KH /BzmbBQM U8XRV Bb ;Bp2M BM BKTHB+Bi 7Q`K QMHv- bBM+2 i?2`2Ƕb QMHv QM2 #H@ M+2 2[miBQM 7Q` i?2 2Mi`QTvX h?2 b2+QM/ 2[miBQM Umb2/ iQ +HQb2 i?2 +QMbiBimiBp2 2[miBQM rBi? i?2 bF2r@/DQBMi QT2`iQ` iQ 7Q`K i?2 .B`+ bi`m+im`2V Bb i?2 /BbbBTiBp2 `2HiBQM ;BpBM; i?2 i?2`KH 7Q`+2X

h?2 ;;`2;iBQM Q7 i?2 `2bBbiBp2 /BzmbBQM M/ i?2 ?2i i`MbTQ`i bm#KQ/2Hb `2bmHib BM i?2 }MBi2 /BK2MbBQMH hJ>. KQ/2H r?B+? Bb r`Bii2M BM i?2 UBKTHB+BiV S*>@HBF2 7Q`K rBi? x= (/, #, 2ex)T, ˙  x 0  = [J (x) − R(x)]  ∂xH 7q  (x) + gu U8XkV r?2`2 J = −JT = ⎡ ⎢ ⎢ ⎣ − 0 J1 J2 0  0 0  0 JT1 JT2 0  ⎤ ⎥ ⎥ ⎦ R = RT ₌ ⎡ ⎢ ⎢ ⎣  R−1 0 0 0  0 0  0 0 0 R−1T  ⎤ ⎥ ⎥ ⎦  0

J Bb  bF2r@bvKK2i`B+ BMi2`+QMM2+iBQM Ki`Bt /2}MBM;  +Q``2bTQM/BM; .B`+ bi`m+im`2 r?BH2R Bb  bvKK2i`B+ TQb@ BiBp2 b2KB@/2}MBi2 /BbbBTiBQM Ki`Bt r?B+? Bb MQMHBM2`Hv /2T2M/BM; QM i?2 bii2 p`B#H2bX h?2 iQiH 2M2`;v 7mM+@ iBQM Q` >KBHiQMBM Bb bBKTHv i?2 bmK Q7 2H2+i`QK;M2iB+ M/ i?2`KH 2M2`;v, H = HEM+ HTX b  +QMb2[m2M+2

Q7 i?Bb TQ`i@>KBHiQMBM `2T`2b2MiiBQM 7Q` i?2 hJ>. KQ/2H- i?2 A.@S"* TT`Q+? 7Q` MQMHBM2` +QMi`QH Kv #2 TTHB2/ iQ i?2 r?QH2 BMi2`+QMM2+i2/ KQ/2H b Bi rb

H`2/v i?2 +b2 7Q` i?2 2H2+i`QK;M2iB+ T`i Q7 Bi U2[mBp@ H2Mi iQ i?2 `2bBbiBp2 /BzmbBQM KQ/2HVX

h?2 +QMi`QH Q7 i?2 BMi2`+QMM2+i2/ bvbi2K U`2bBbiBp2 /B7@ 7mbBQM M/ i?2`KH /BzmbBQM 2[miBQMbV Bb 2tT2+i2/ iQ iF2 /pMi;2 Q7 i?2 2tTHB+Bi bii2 bT+2 `2T`2b2MiiBQM Q7 i?2 hJ>. +QmTHBM; MHvbBb iQ BKT`Qp2 i?2 +QMi`QH T2`7Q`KM+2 i?`Qm;?  #2ii2` T`K2i2` 2biBKiBQMb 7Q` i?2 `2bBbiBpBiv R UpB  ;QQ/ TT`QtBKiBQM Q7 η(Te) M/

Q7 i?2 #QQibi`T +m``2Mi Jbs(Te, ∂zTe)V- b r2HH b 7Q`

i?2 i?2`KH `2bBbiBpBiv RT UpB i?2 /BzmbBQM +Q2{+B2Mi

χ(∂zT, B)VX

"2bB/2b- i?2 +QMi`QH +iBQMb `2 bbmK2/ iQ biBb7v bT2@ +B}+ b?T2b U`/BH /Bbi`B#miBQMV pB i?2 7mM+iBQMb fextM/

fheatUMK2Hv :mbbBM /Bbi`B#miBQMb BM i?2 bim/B2/ +b2VX

h?2`27Q`2 +QMi`QH p`B#H2b `2 i?2 b+H` iQiH 2ti2`MH +m``2Mi TQr2` PextM/ ?2iBM; TQr2` PheatX b /Bb+mbb2/

BM bm#b2+iBQM jXk- i?Bb BKTHB2b i?i QMHv  HBKBi2/ b2i Q7 2[mBHB#`BmK bii2b xd = (/, #, 2ex)T<sub>d</sub> `2 `2+?#H2X h?mb

 722/7Q`r`/ +QMi`QH rBHH #2 /2bB;M2/ }`bi r?B+? H2/b iQ  `2+?#H2 bi2/v bii2 7Q` r?B+? i?2 +HQb2/ HQQT +QMp2`@ ;2M+2 Q7 i?2 722/#+F /BzmbBQM bvbi2K Kv #2 Q#iBM2/ pB M A.@S"* +QMi`QHH2`X S`2pBQmbHv-  722/7Q`r`/ +QMi`QH (Pext, Vloop)T<sub>d</sub> ?b #22M T`QTQb2/ iQ +?B2p2 i?2 `2;mH@

iBQM 7Q` irQ `272`2M+2 TQBMib Q7 i?2 b72iv 7+iQ` T`Q}H2, i i?2 +2Mi2` M/ i i?2 #QmM/`vX >2`2- mbBM; i?2 BM@ i2`+QMM2+i2/ hJ>. KQ/2H rBHH HHQrb mb iQ //  i?B`/ `272`2M+2 TQBMi Q7 q #v i?2 mb2 Q7 i?2 M2r +QMi`QH +iBQM

PheatX

8XkX ai2/v bii2 ;2M2`iBQM 7Q` +QmTH2/ hJ>. KQ/2H

h?2 bi2/v bii2 xd Q7 U8XkV biBb}2b,

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ J₁Gmg#d+ R−1Gel/d+ fextPext+ Cbs = 0 −J2Gel/d+ J4Vloop.d = 0 JT17qd+ S = 0 JT2GT2ex− R_T−17qd+ JT4T1 = 0 U8XjV h?2 7QHHQrBM; TQBMib +QmH/ #2 MQiB+2/ ,

Ç h?2 BMTmi bB;MH T1Bb bbmK2/ +QMbiMi BM i?2b2 2[mB@

HB#`BmK 2[miBQMb UT1= 0 7Q` BMbiMiV

Ç h?2 bQm`+2 i2`K S BM+Hm/2b i?2 CQmH2 2z2+i SJoule=

ηJtot(Jtot− Jni) Ur?2`2 JtotBb i?2 iQiH +m``2Mi /2M@

bBivV M/ i?2 2ti2`MH ?2iBM; bQm`+2 Sheat r?B+? Bb

+QMi`QHH2/ #v i?2 ?2iBM; TQr2` PheatX

Ç h?2 hJ>. +QmTHBM;b @ BM+Hm/BM; i?2 i2`Kb

Jbs(Bθ, T, ∂zT)- η (T ) , M/ χ (∂zT, Bθ) @ `2 2biB@

Ki2/ #v 2KTB`B+H MHviB+ 2tT`2bbBQMb ;Bp2M BM (kj)X qBi? i?2 bbmKTiBQM i?i i?2 i?2`KH bi2/v bii2 7Q` T M/ ∂zT `2 [mB+FHv 2bi#HBb?2/ M/ HbQ b@

bmKBM;  +QMbiMi T`iB+H2 /2MbBiv n- r2 +M /2/m+2 7`QK i?2b2 2tT`2bbBQMb i?2 7QHHQrBM; pHm2b 7Q`,

Ĝ i?2 `2bBbiBpBiv +Q2{+B2Mi ,

η (T ) = Cη(#) T−3/2 Ry

Ĝ i?2 i?2`KH /BzmbBQM +Q2{+B2Mi,

χ (∂zT, Bθ) = Cχ(#) ∂zT

Ĝ i?2 /Bb+`2i2 #QQibi`T +m``2Mi,

Cbs= 1 #  β12ex+ β2R−1T 7q  = 1 #  β1(JT2GT)−1+ β2(JT1RT)−1S U8X9V = 1 #  β1(JT2GT)−1+ β2(JT1RT)−1 × (SJoule+ Sheat) r?2`2 Cη(#)- Cχ(#) - β1 M/ β2 `2 i?2 +Q2{@

+B2Mib /2T2M/BM; QM i?2 bvbi2K bii2 U+7X (kj)VX h?2 CQmH2 2z2+i SJoule Kv #2 +QMbB/2`2/ b 

K2bm`#H2 QmiTmi bbmKBM; i?i i?2 iQiH +m`@ `2Mi M/ i?2 2ti2`MH MQM@BM/m+iBp2 +m``2Mi `2 FMQrMX AiǶb HbQ QM2 Q7 i?2 J>. +QmTHBM;b #mi BM T`+iB+2- Bi Bb M2;HB;B#H2 r?2M +QKT`2/ iQ i?2 2ti2`MH ?2iBM; bQm`+2X Sheat Bb ;Bp2M #v i?2

MHviB+ 2tT`2bbBQM fheatPheat r?2`2 fheat Bb 

+?Qb2M UFMQrMV :mbbBM 7mM+iBQM Q7 z r?B+? Bb  +?`+i2`BbiB+ Q7 i?2 mb2/ +imiQ`X

.m2 iQ i?2 /Bz2`2Mi Q`/2`b Q7 K;MBim/2 #2ir22M T M/

# M/ iQ i?2 7+i i?i QMHv bKHH p`BiBQMb Q7 i?2 K;@

M2iB+ }2H/ `2 +QMbB/2`2/- i?2 /2T2M/2M+2 Q7 i?2 #QQibi`T +m``2Mi- i?2 THbK `2bBbiBpBiv M/ i?2 i?2`KH /BzmbBQM +Q2{+B2Mi rBi? i?2 K;M2iB+ }2H/ Kv #2 M2;H2+i2/X h?Bb HHQrb  HBM2`BxiBQM 7Q` i?2 722/7Q`r`/ +QKTmiiBQM- mb@ BM;- 7Q` i?2 +QKTmiiBQM Q7 i?2b2 [mMiBiB2b- i?2 K2bm`2@ K2Mi Q7 i?2 K;M2iB+ }2H/ BMbi2/ Q7 i?2 7Q`2b22M `272`2M+2

#/X Ai ?b iQ #2 MQiB+2/ ?Qr2p2` i?i /QBM; bQ- i?2 bi@

#BHBxiBQM rBi? i?2 bii2 722/#+F +QMi`QH rBHH #2 Q#iBM2/ QMHv HQ+HHv- r?2M i?2 `2[m2bi2/ `272`2M+2b rBHH #2 +HQb2 2MQm;? 7`QK i?2 bvbi2K BMBiBH bii2 pHm2bX

_2K`F 8XRX PM2 +M `2K`F i?i QMHv i?2 BM~m2M+2 7`QK

i?2 i?2`KH /QKBM iQ i?2 2H2+i`QK;M2iB+ QM2 Bb `2T`2@ b2Mi2/ M/ MQi i?2 QTTQbBi2 b2Mb2X h?Bb +?QB+2 ?Qr2p2` Bb DmbiB}2/ #v i?2 bbmKTiBQM i?i i?2 /BzmbBQM iBK2@b+H2b Q7 2+? /QKBM `2 b2T`i2/ #v b2p2`H Q`/2`b Q7 K;MB@ im/2X h?2 i2KT2`im`2 2bi#HBb?2b i?QmbM/ iBK2b 7bi2` i?M i?2 K;M2iB+ }2H/ /Q2bX AM +QMb2[m2M+2- i?2 K;M2iB+ }2H/ +M #2 +QMbB/2`2/ b dzbiiB+Ǵ BM i?2 +QKTmiiBQM Q7 i?2 J>. +QmTHBM;bX

6BMHHv- i?2 722/7Q`r`/ Bb /2/m+2/ 7`QK i?2 bi2/v bii2 2[miBQM U8XjV mbBM; i?2 `2HiBQM #2ir22M i?2 b72iv 7+iQ`

qM/ i?2 K;M2iB+ bii2 # BM 2[miBQM U9XkVX h?2 Q#iBM2/ 722/7Q`r`/ ud Bb, ⎛ ⎜ ⎝ Pext  Pheat3/2Vloop Pheat ⎞ ⎟ ⎠ d = ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ wf_(z1) wf_(z2) wf_(z3) ⎞ ⎟ ⎠ C ⎞ ⎟ ⎠ −1 × ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 2π_Bφ0a2  z1 q1d, z2 q2d, z3 q3d T + ⎛ ⎜ ⎝ wf_(z1) wf_(z2) wf_(z3) ⎞ ⎟ ⎠J1Gmg−1Cbs.dSJoule ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ U8X8V r?2`2 C = − (J1Gmg)−1 fext CRJ₂−1J4 Cbsfheat  c

CR= R/ (Pheat)3/2_d - r?BH2 z1, z2, z3 `2 i?2 i?`22 TQbBiBQMb Q7 i?2 i?`22 `272`2M+2b(q1, q2, q3)d 7Q` i?2 b72iv 7+iQ`X

h?2 722/7Q`r`/ +QMi`QH Bb i?mb /2`Bp2/ 7`QK i?2 bi2/v bii2 7Q` i?2 bvbi2K Q#iBM2/ #v i?2 HBM2`BxiBQM i 2+? iBK2 bi2T Q7 i?2 MQM@HBM2` T`K2i2`b R, RT, GT M/

JbsX h?2 722/#+F +QMi`QH Bb i?2M `2[mB`2/ MQi QMHv iQ

BM+`2b2 i?2 +QMp2`;2M+2 bT22/ #mi HbQ iQ Qp2`+QK2 i?2 2``Q`b +mb2/ #v i?2 HBM2`BxiBQM bbmKTiBQMbX

8XjX *QMi`QHH2` imMBM;

h?2 BMi2`+QMM2+i2/ bvbi2K Mim`HHv +QMp2`;2 iQ Bib 2[mBHB#`BmK i?MFb iQ i?2 irQ /BbbBTiBQMb `2T`2b2Mi2/ #v i?2 /BbbBTiBQM Ki`B+2b R M/ RTX q2 /2+B/2 iQ T`2@

b2`p2 i?2 BMi2`+QMM2+iBQM bi`m+im`2J Q7 i?2 Q`B;BMH bvb@ i2K- ?2M+2 MQi KQ/B7vBM; i?2 BMi2`MH /vMKB+H +QmTHBM;b UJa = 0VX Pm` +QMi`QH /2bB;M +QMbBbib BM b2iiBM; Ra +QM@

biMi bm+? i?i Rd = RTd > 0 Ui?2 /2bB`2/ /BbbBTiBQM

`i2V M/ i?2M iQ /2i2`KBM2 i?2 b?T2/ >KBHiQMBM Hd

M/ i?2 722/#+F bB;MH δu rBi? i?2 ?2HT Q7 i?2 Ki+?@ BM; 2[miBQM U9XeV M/ i?2 BMi2;`#BHBiv- 2[mBHB#`BmK M/ bi#BHBiv +QM/BiBQMb UkXe@kX3VX

h?2 `Q#mbiM2bb Q7 i?2 +QMi`QHH2` rBi? `2bT2+i iQ irQ FBM/b Q7 mM+2`iBMiB2b Bb bim/B2/X h?2 }`bi FBM/ Q7 mM@ +2`iBMiB2b `2 i?Qb2 QM i?2 bvbi2K /BbbBTiBQMR `2bmHiBM; 7`QK TQQ` 2biBKiBQMb Q7 i?2 THbK `2bBbiBpBiv η M/ i?2 i?2`KH /BzmbBQM +Q2{+B2Mi χX h?2 b2+QM/ FBM/ Q7 mM+2`@ iBMiB2b `2 `2Hi2/ iQ i?2 HBM2`BxiBQM bbmKTiBQMb K/2 BM i?2 /2`BpiBQM i?2 722/7Q`r`/ +QMi`QH M/ BM i?2 T@ T`QtBKiBQM Q7 i?2 #QQibi`T +m``2Mi CbsX

"`B2~v- rBi?  +?QB+2 Q7 bmTTH2K2Mi`v /BbbBTiBQM Ra

bm+? i?iRd Bb bm{+B2MiHv H`;2- r2 +M ?M/H2 i?2b2 mM@

+2`iBMiB2b Ub22 bm#b2+iBQM jXjVX P7 +Qm`b2- i?2 +imiQ` TQr2` bim`iBQM rBHH T`2p2Mi mb iQ +QKT2Mbi2 p2`v H`;2 T2`im`#iBQMbX "2bB/2b- i?2 /2bB;M2/ +QMi`QHH2` #2BM; #@ bB+HHv  T`QTQ`iBQMH +QMi`QHH2`- i?2 +?QB+2 Q7 H`;2 pHm2b 7Q` i?2 T`QTQ`iBQMH ;BM Kv +`2i2 mM/2bB`2/ Qb+BHHiBQMb M/ BMbi#BHBiv 7Q` i?2 +HQb2/ HQQT bvbi2KX AM 7+i- i?2 HBM2`BxiBQM +QM/BiBQM BM UjXjV BbMǶi biBb}2/ MvKQ`2X

8X9X aBKmHiBQM

h?`22 `272`2M+2 pHm2b 7Q` i?2 b72iv 7+iQ` q `2 /2@ }M2/ `2bT2+iBp2Hv i i?2 `/BH `2HiBp2 +QQ`/BMi2b z = 0.1, 0.3, M/ 0.4X h?2 722/7Q`r`/ +H+mHmb ;Bp2b ud- i?2

r?QH2 `272`2M+2 T`Q}H2 qd M/ i?2 p2`;2 i2KT2`im`2

T`Q}H2 Td- +Q``2bTQM/BM; iQ i?2b2 i?`22 `272`2M+2b M/ iF@

BM; BMiQ ++QmMi i?2 +imiQ`b +QMbi`BMibX h?2 A.@S"* +QMi`QH /2i2`KBM2b i?2 722/#+F bB;MH δu 7`QK Ki+?BM; 2[miBQM UkX9V iQ +Q``2+i i?2 2``Q` XX h?2 A.@S"* T@ `K2i2`b `2 /2bB;M2/ b /Bb+mbb2/ BM i?2 T`2pBQmb bm#b2+@ iBQM- rBi? i?2 +?QB+2 Q7Ja= 0 M/ Ra bm+? b,

Ra = ⎡ ⎢ ⎢ ⎣  0 0 0 Ra1  0 0  Ra2 0 0 0  ⎤ ⎥ ⎥ ⎦ U8XeV RR

r?2`2 i?2 TQbBiBp2 /B;QMH Ki`Bt Ra1 M/ Ra2 ++QmMi

7Q` i?2 /KTBM;b //2/ BM 2H2+i`QK;M2iB+ /QKBM M/ i?2`KH /QKBM `2bT2+iBp2HvX b  T`iB+mH` +b2 QM2 +M b2i QMHv i?`22 /B;QMH pHm2b 7Q` i?2 Ki`B+2b Ra1 M/ Ra2r?B+? +Q``2bTQM/ iQ i?`22 +?Qb2M `272`2M+2 TQbBiBQMbX

h?2 T`QTQb2/ A.@S"* +QMi`QHH2` Bb i2bi2/ QM i?2 _S@ hP_ +Q/2 rBi? i?2 h*o +QM};m`iBQMX h?2 bBKmHiBQM `2bmHib `2 b?Qr2/ b #2HQrX 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 2.5 3 3.5 4 4.5x 10 5 t [s] 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.1 0.2 t [s] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 2 4 6 8 z qprofiles at t=1.995s q qref 5*qerror error 1 error 2 error 3 P_A (co) P_B (ctr) Pheat/2 −3.105_V loop 6B;m`2 8XR, 622/7Q`r`/ +QMi`QH Q7 +QmTH2/ bvbi2K 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 4 6 8x 10 5 t [s] 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.05 0 0.05 0.1 0.15 t[s] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 2 4 6 8 z q_profiles at t=1.995s q qref 5*q_error error 1 error 2 error 3 PA (co) PB (ctr) Pheat/2 −3.105_V loop 6B;m`2 8Xk, 622/#+F +QMi`QH Q7 +QmTH2/ bvbi2K

6B;m`2b 8XR M/ 8Xk b?Qr `2bT2+iBp2Hv i?2 `2bmHib Q#@ iBM2/ rBi? i?2 722/7Q`r`/ M/ i?2 722/#+F +QMi`QHbX h?2 ?2iBM; +imiQ` bi`ib i t = 0.2s rBi? i?2 BMBiBH pHm2b (Pext, Vloop, Pheat)_init= (40kW, −0.6V, 300kW

)-r?2`2b i z= (0.1, 0.3, 0.7) i?2 `272`2M+2 q@T`Q}H2 Bb b2i b qa = (0.85, 1.1, 2.4)X h?2M i t = 0.8s- i?2 `272`2M+2 Bb

+?M;2/ iQ qb= (1, 1.4, 3)X

h?2 722/7Q`r`/ /Q2b #`BM; i?2 q@T`Q}H2 +HQb2 iQ i?2 `272`2M+2 pHm2b #mi i?2 +imiQ` pHm2b b r2HH b i?2 q@ T`Q}H2 Qb+BHHi2 `QmM/ i?2 2[mBHB#`BmK /m2 iQ i?2 T`K@ 2i2` HBM2`BxiBQM M/ TT`QtBKiBQMX h?2 722/#+F ?Qr@ 2p2` KF2b 2zQ`i iQ BKT`Qp2 i?2 `2bmHi #v +QMiBMmBM; iQ `2+i bB;MB}+MiHv QM PheatX >Qr2p2`- Bi /Q2bMǶi bm++22/ BM

bi#BHBxBM; i?2 KB//H2 `272`2M+2 pHm2 7Q` i?2 q@T`Q}H2X Ai

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 4 6 8 x 105 t [s] 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.05 0 0.05 0.1 0.15 t [s] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 2 4 6 8 z q_profiles at t=1.995s q qref 5*q_error error 1 error 2 error 3 PA (co) PB (ctr) Pheat/2 −3.105_V loop

6B;m`2 8Xj, 622/#+F +QMi`QH Q7 +QmTH2/ bvbi2K rBi?  bmTTH2K2Mi`v BMi2;`iQ` X 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 z T [keV] Tref T #X 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 z T [keV] Tref T 6B;m`2 8X9,T T`Q}H2b i t = 0.7s UV M/ t = 1.5s U#V

Bb KBMHv `2;mHi2/ #v Pheat- r?B+? +QMiBMm2b iQ BM+`2b2

iBHH i?2 2M/X M BMi2;`iQ` Bb BKTHMi2/ 7Q` i?2 bBKmHiBQM T`2b2Mi2/ BM };m`2 8XjX  bB;MB}+iBQM BKT`Qp2K2Mi Bb Q#@ iBM2/ 7Q` i?2 }`bi M/ i?2 i?B`/ TQBMib- r?B+? `2 /B`2+iHv z2+i2/ #v Pext M/ Vloop- r?BH2 Pheat KF2b BKT+i BM@

/B`2+iHv iQ i?2 b2+QM/ TQBMi pB i?2 hJ>. +QmTHBM;b JbsX

h?2 };m`2 8X9 b?Qrb i?2 i2KT2`im`2 T`Q}H2 pb i?2 `272`@ 2M+2 /2}M2/ #v i?2 722/7Q`r`/ +QMi`QH i t = 0.7s M/

t= 1.5sX Hi?Qm;? i?2b2 irQ T`Q}H2b `2 #mBHi #v irQ /B7@

72`2Mi /Bb+`2iBxiBQM K2i?Q/b- i?2v biBHH Ki+? [mBi2 r2HH iQ 2+? Qi?2`X PM i?2 Qi?2` ?M/- i?2b2 `2bmHib H2/ iQ  /Bb+mbbBQM #Qmi i?2 `272`2M+2 +?QB+2, b?QmH/ r2 iF2 irQ `272`2M+2 TQBMib 7Q` i?2 q@T`Q}H2 M/ QM2 7Q` i?2 T T`Q}H2- i?2 Hii2` #2BM; /B`2+iHv z2+i2/ #v i?2 ?2iBM; bQm`+2 SheatX

eX *QM+HmbBQM

AM i?Bb TT2`- M A.@S"* 722/#+F Hr Bb T`2b2Mi2/ 7Q` i?2 `2bBbiBp2 /BzmbBQM Q7 i?2 K;M2iB+ ~mt M/ i?2M 7Q` Rk

i?2 +QmTH2/ hJ>. KQ/2H @  b2i Q7 irQ BMi2`+QMM2+i2/ KQ/2Hb 7Q` i?2 `2bBbiBp2 M/ i?2`KH /BzmbBQMb BMbB/2 i?2 iQFKF THbKX h?2 +QMi`QH K2i?Q/QHQ;v Bb #b2/ QM A.@S"* M/ i?2 +QMi`QH Hr Bb /2`Bp2/ 7`QK  S*> +QM@ i`QH KQ/2H Q#iBM2/ 7`QK i?2 ;2QK2i`B+f bvKTH2+iB+ /Bb@ +`2iBxiBQM Q7 i?2 +Q``2bTQM/BM; +QmTH2/ S.1bX M BMi2@ ;`iQ` Bb HbQ bm++2bb7mHHv mb2/ BM Q`/2` iQ 2HBKBMi2 i?2 biiB+ 2``Q`bX h?2 +imiQ` +QMbi`BMib M/ HBKBiiBQMb `2 iF2M BMiQ ++QmMi BM i?2 2[mBHB#`BmK +QKTmiiBQM 7Q` i?2 722/7Q`r`/ +QMi`QH +iBQMX h?2 i2KT2`im`2 T`Q}H2 M/ Bib BM~m2M+2 QM i?2 `2bBbiBpBiv +Q2{+B2Mib `2 BMi2;`i2/ BMiQ i?2 +QMi`QH Hr pB i?2 J;M2iQ@>v/`Q.vMKKB+b +QmTHBM;bX h?2 +QMi`QHH2`b ?p2 #22M i2bi2/ BM bBKmHiBQMb rBi? i?2 _ShP_ +Q/2- M/ +QKT`2/ ;BMbi i?2 2t@ T2`BK2MiH /iX  +QMp2`;2M+2 ?b #22M Q#b2`p2/ rBi? i?2 +QKTmi2/ 722/7Q`r`/ M/ 722/#+F +QMi`QHbX AM 7m@ im`2 rQ`Fb- r2 +M BM+`2b2 T2`7Q`KM+2 mbBM; bBKmHiM2@ Qmb +QMi`QH Q7 i?2 i?2`KH M/ K;M2iB+ KQ/2HbX "2bB/2b-i?2 Ki+?BM; B/2 rBHH #2 mb2/ iQ T2`7Q`K #QmM/`v M/ /Bbi`B#mi2/ U}MBi2 `MFV 722/#+F +QMi`QH /B`2+iHv QM i?2 BM}MBi2 /BK2MbBQM TQ`i >KBHiQMBM KQ/2HX

"2bB/2 i?2 p2`v 2{+B2Mi +QMi`QHH2` /2bB;M- QM2 Q7 i?2 /B{+mHiB2b Q7 A.@S"* Bb ?Qr iQ +?QQb2 i?2 T`K2i2`b

Jd,Rd,Hd`2;`/BM; i?2 bvbi2K M/ +imiQ` +QMbi`BMibX

M QTiBKH A.@S"* +QMi`QH +M #2 +QMbB/2`2/ b QM2 Q7 i?2 bQHmiBQMb 7Q` i?Bb T`Q#H2K- r?B+? Bb MQi?2` Q#D2+iBp2 Q7 Qm` 7mim`2 rQ`FbX

+FMQrH2/;K2Mi

h?2 mi?Q`b rQmH/ HBF2 iQ i?MF aX "_úJPL.- PX alh1_- >X"X G1 M/ i?2 h*o i2K Q7 1S6G Uú+QH2 SQHvi2+?MB[m2 6û/û`H2 /2 GmbMM2V- arBix2`HM/ 7Q` i?2B` bmTTQ`i iQ i?2 _ShP_ bBKmHiBQMb M/ h*o 2t@ T2`BK2MibX

_272`2M+2b

(R) 6X"X `;QK2/Q- *X S`B2m`- 1X qBi`Mi- M/ aX "`ûKQM/X  bi`B+i +QMi`QH HvTmMQp 7mM+iBQM 7Q`  /BzmbBQM 2[miBQM rBi? iBK2@p`vBM; /Bbi`B#mi2/ +Q2{+B2MibX A111 h`Mb+iBQMb QM

miQKiB+ *QMi`QH UkyRkV- kyRkX

(k) JX `BQH M/ X SB`QMiBX J;M2iB+ *QMi`QH Q7 hQFKF SHb@

KbX /pM+2b BM AM/mbi`BH *QMi`QHX aT`BM;2` o2`H;-

GQM/QM-kyy3X

(j) CX "HmKX LmK2`B+H aBKmHiBQM M/ PTiBKH *QMi`QH BM SHbK

S?vbB+bX :mi?B2`@oBHH`b- RN3NX

(9) JX.X "Qv2`- CX "`iQM- 1X a+?mbi2`- M/ 2i HX 6B`bi@T`BM+BTH2b@ /`Bp2M KQ/2H@#b2/ +m``2Mi T`Q}H2 +QMi`QH 7Q` i?2 /BBB@/ iQFKF pB H[B QTiBKH +QMi`QHX SHbK S?vbB+b M/ *QMi`QHH2/ 6mbBQM-U88V- kyRjX

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