real-time tracking and feedback control strategies for rotating magnetic islands

12/01/2004 E.Lazzaro Plasmi04, Arcetri

Real-time tracking and feedback control strategies for rotating magnetic islands

E. Lazzarowith the contribution of

J.O.Berrino1,G.D’Antona2,S.Cirant1

1IFP “P.Caldirola”, Euratom-ENEA-CNR Association, Milano, Italy2Dip.Ingegneria Elettrotecnica, Politecnico di Milano, Italy


Outline

• Introduction• Key questions on observed processes• Relation between measured signals and state

variables• Methods and models of control theory


Introduction• In the design of a realistic tokamak reactor device, the problem of

reaching the desired performance targets and the problem of controlling various MHD instabilities have objectively a status of equal importance, scientifical and technical.

• Automatic control theory is highly developed and practically successful: in plasma physics there is ample scope of application of its principles and methods.

• The problem of control involves the basic aspects the plasma physics as well the need of a specific approach that requires a clear statement of the task, that includes:

• Development of the (simplest) models of the process to be controlled. • Definition (selection )of the appropriate state variables xi and

admissible control variables ui (i=1,…N) , measurable and controllable with definite error bounds.

• Selection of the control policy,as a subset of the control variables.• Choice of the decisional algorithms, implementation into a detection/

control device, and a suitable actuator.


Key Questions on Resistive Modes in Tokamaks• Theory predicts tearing of nested magnetic surface with rational (q=m/n ) with

formation of growing and rotating magnetic “islands” that deteriorate the confinement properties and may cause a number of dire events.Several conditions for appearance of these instabilities have been predicted and confirmed.

• But what are the actual, observed aspects of the phenomenon that we may decide to control ?

• These instabilities are mainly observed as magnetic signals, picked up externally, with a certain frequency and amplitude (generally growing ).

• • What are the footprints of these modes on the plasma properties, that

suggest (or demand ) control?• The most impressive footprints are the rapid, localized fluctuations on the

background electron temperature profile measured by ECE radiation,

• in the density profile by reflectometry and sometimes in the plasma rotation profile measured by CXS.

• Other footprints are visible in soft -X-ray signals and the decrease of the signal of monitoring the thermal energy content.

€

˙ b ϑ (t)

€

Te (r, t) ≈ Te0(rm,n,εt) + δTe (r, t)

€

βϑ (t)


R

Z

R0

X Point

O Point

TEARING MODES (classical or neoclassical)

Current perturbations alter the topology of magnetic confinement (isobaric) surface

Safety factor q is rational

q≅rBϕ

RBϑ

=mn On this surface the force line

are closed after m toroidal loops and n poloidal loops


Relation between measured signals and state variables

• For a single (m,n) mode the “theoretical“ state variables xi are the island width W(t) and rotation frequency t governed by the equations:

• The control variables ui are embodied in

• The control task would be to reduce to zero W in minimal time;• RF current drive aimed on the “O” point could provide the suppression

(“stabilizing” )effect

• Attempts of stabilization have been done with external control fields with (m,n ) helical pitch and recently with ECCD an, LHCD and ECRH applied the calculated q=m/n surface , without island tracking.

€

τR

rs

dW

dt= − rs ′ Δ + rsβθε1/ 2 Lq

Lp

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

W

W 2 + Wd2

− f (ω)W p

2

W 3

⎡

⎣ ⎢

⎤

⎦ ⎥+ rs ′ Δ control ⋅cosδφ(t)

€

dω

dt=

1

Iφ

−n Tφ em + Tφ visc( ) − ω −ωT( )dIφ

dt

⎡

⎣ ⎢

⎤

⎦ ⎥

€

′ Δ (ui)control


Questions on the localization & modulation of RF power driving current in NTM rotating islands

• Within JECCD is spread ergodically on island flux tubes 1/W2 scaling adequate to balancebalance the JBOOT destabilizing term

• Outside JECCD is spread ergodically on flux surface 1/W scaling inadequate to balanceinadequate to balance JBOOT term

• Synchronizing RF pulses to keep a constant ~0 favors the 1/W2 scaling, but it‘s very complicated.

• The RF power depends on what is needed to balance the JBOOT term, not on the RF pulse strategy

Δ”ECCD~1/W

Δ”ECCD~1/W2

A constant RF source in the lab frame appears oscillating in the moving island frame !

RF

RF

€

ζ =mϑ − nφ − ω( ′ t )d ′ t t

∫

Steering the RF deposition profile on the correct radiusis the most important task!


Relation between measured signals and state variables• We change the point of view and choose as state variables the ECE

fluctuations (radiative Temperature) just as they are measured in amplitude, phase and frequency.

• The typical ECE signal is the superposition of a slowly varying component related to the equilibrium electron temperature,

• plus noise and coherent fluctuations due for instance to• the magnetic islands

• The temperature fluctuation• associated with a finite size island is generally expected to be

negative on the inner island edge and positive on the outer edge

• Therefore the fluctuation amplitude should be Te=0 on the island “O” and “X” stagnation points.

• The relative phase of two neighbouring ECE channels changes smoothly except if between them a rational surface q=m/n with an island is located: here a phase jump close to occurs.

€

T0 = Te (Ψ, t)

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Ψ* = Ψ0 +ψ m,n cos(mϑ − nϕ − Δφ(t))

€

Te (Ψ*,t) = Te (Ψ*,t) − Te (Ψ0, t)


Tr

0 r

Effects of the MHD perturbations on the Temperature profile

q=m/n


First step of the control problem: mode location & tracking

• We concentrate here in the development of an island tracking device based on ECE. Eventually the control variables ui shall be ECW steering angle, ECW power, pulse timing

• The rotating island frequency and phase are identified by a special Digital phase-locked loop that captures the ECE perturbation frequency within few cycles

• The island radial location is tracked by a fast algorithm that identifies the maximum phase change between of two neighbouring ECE channels

Second step: designing optimal control strategy

The basic “power cost” of the “stabilizing” action is due to the low JECCD efficiency

Formal control theory helps also understanding the physics of the controlled

system and the best strategies can then be found


A robust control technique for active MHD control with EC waves requires the real-time measurement of the location of both the island and the EC deposition layer.

The most appropriate signals related to the state variables come from a multichannel EC emission (ECE) diagnostic.

Te oscillations caused by periodic flux perturbations reveal rotating modes. The location is identified by a channel number.

The algorithm described can be used to localize Tearing Modes, sawteeth and the EC deposition layer.

The algorithm has been tested off-line on FTU 12-channels polychromator data, on a variety of different MHD combinations.

The algorithm is implemented on DSP modules for on-line action.

DSP modules will be used for data acquisition, data processing and feedback control (ECH/ECCD is the actuator).


4 GYROTRONS400 KW 140 GHz

R

B

R0

ECE

I(ω) =ω2

8π3c2 Tr(ω)Bϕ =B0R0

R

CONTROL

EXPERIMENT SCHEME

Microwave heterodyne radiometers


Mirror

Gyrotron

(B) dep.

R

It is possible to change Rdep by changing the the angle of the launcher mirror


oi =ECEi − ECEi T1

Ai = oi2

T2oi

norm=oi

Ai

Pij = oinormoj

norm

T3

Corr(x,y) =x−x ⋅ y−y

x−x ( )2 ⋅ y−y ( )2

Sampling frequency = 20 KHz

(0.05 msec)

T1 = 2 x 10-3 sec 500Hz

T2 = 4 x 10-3 sec 250Hz

T3 = 20 x 10-3 sec 50HzEquivalent to cross-correlation:

Radial Identification Algorithm: creating the sensitive parameter


First steps in signal processing

0.49 0.495 0.5 0.505 0.51 0.515 0.52 0.525 0.53

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

sparo 14979Te channel 12

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2channel 11

Low-pass filter (<ECEi>)

Oscillation (oi) and modulus (Ai)


Decisional algorithmPij≈ 1 if both i and j are on the same side with respect to the island O-point.

Pij≈ -1 if on opposite sides.

A positive concavity in the Pij sequence locates the island.

channels

1

-1

0

Pi j


11,...,2 )(1 ,11, =−= −+ iPPiD iiii

10,...,2 )(1)1(12

12 =−+=⎟

⎠⎞

⎜⎝⎛ + iiDiDiD

Island FoundBetween ch i and ch i+1

1+=ii)1(1)(1

)1()1(1)(1

+++⋅++⋅

=iDiD

iiDiiDRIsland

3.02

12 IF >⎟

⎠⎞

⎜⎝⎛ +iD

N

Y

10 IF <iY

N

Next time step


PC

VME BUS

COMM PORT

ETHERNET

FTU CLOCK

12 FTU ECECHANNELS

FTU GATES

PROGRAMSDATA

PARAMETERS

DATAPROGRAM

PARAM.

PROGRAMDATAPARAM.

GYROTRON I

GYROTRON II

GYROTRON III

GYROTRON IV

ON/OFF

COMM PORT

HARDWARE DEVELOPED BY IFP FOR FTU


17

6

1 −− zKK

17

6

1 −− zKK

)sin(•

ω0

×

111

−−z

17

6

1 −− zKK

ππ 2 )2( −•=•>•if

MEM

Digital Phase Locked Loop to track rotating island phase and frequency

Implemented on DSP B

Sampling rate

25 KHz


Identification of locking island: the red DPLL trace tracks the physical signal

DPLL performance: identification of mode frequency

Shot 21742 ch7


0.5

1

1.5

2

2.5

3

1 2 3 4 5 6 7 8 9 10 11 12

keV

ECE channel

#18004

t=0.5904÷0.5906 s

Te profile during one cycle

correlation between nearby channels

Case1 -ECRH boosts TMsif rdep < risland

1

1.5

2

2.5

3

3.5

keV

#18004ch.4

ch.8

430 kW860 kW

430 kW

-100

0

100

0.5 0.6 0.7 0.8 0.9

T/s

t(s)

#18004

no data

Te,ECE

Mirnovcoil

-0.4

-0.2

0

0.2

0.4

0.6

2 4 6 8 10 12

Pi,i+1

ich

#180040.49 s


oscillation frequency at minimum Pij

m=1,n=1

axis

minimum correlation (negative maximum) = island position

(1,1) and (2,1) are coupled

(2,1)


TM modes with different m - order coexist

Te & Te profiles, fast ECE, showing presence of even and odd TM.

All modes are tightly coupled <--> same frequency at all radii

Even mode:m=2, n=1.

1

1.5

2

2.5

3

3.5#18015

<Te>

1 cycle

-0.1

-0.05

0

0.05

0.1

-0.1 0 0.1 0.2-R R

0 ( )m

Te

-0.04

0

0.04

0.08.49658 s.49668 s

Te

even m order

-0.08

-0.04

0

0.04

0.08

-0.1 0 0.1 0.2-R R

0 ( )m

Te

odd m order

IAEA, Sorrento, October 4-10, 2000ASSOCIAZIONE EURATOM/ENEA/CNR


0

0.5

1

1.5

2

2.5

1 2 3 4 5 6 7 8 9 10 11 12

keV

ECE channel

#18015

t=0.49898÷0.49920 s

1

1.5

2

2.5

3

3.5

keV

#18015

ch.4

ch.8430 kW

860 kW

-80

-40

0

40

0.5 0.6 0.7 0.8 0.9

T/s

t(s)

#18015

no data

(2,1) stabilized

(1,1) evolves in s.t.sawteeth are stabilized

Case2- ECRH suppresses MHDif rdep ≈ risland and/or rdep ≥ rs.t. inversion

(2,1) & (1,1)

Te profile during one cycle

correlation between nearby channels

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 2 4 6 8 10 12

18015_0.49 s18015_0.59 s

18015_0.80 s

Pi,i+1

ich


oscillation frequency at minimum Pij

m=1,n=1

axis

inversion radius

(2,1) stabilized

frequency jump of dominant oscillation: from (1,1) mode to (1,1) reconnection (s.t. crash)

Sawteeth and TM dynamics simultaneously detected

s.t. stabilized


axis

sawteeth

(2,1)

(2,1) locks

(1,1)

(3,2)

(1,1) & (2,1) & (3,2) are coupled

Coupled TMs simultaneously detected


Formal aspects of the control problem

• The physical objective is to reduce the ECE fluctuation to zero in minimal time using ECRH /ECCD on the position q=m/n identified by the phase jump method

• The TM control problem in the extended Rutherfprd form, belongs to a general class known in in the theory of multistage decision processes [*] . In a linearized form the governing equation for the state variable x(t) is

• with the initial condition x(0)=x0, and a control variable (steering function) u(t).

• The formal problem consists in to reducing the state x(t) to zero in minimal time by a suitable choice of the steering function u(t)

• A number of interesting properties of this problem have been studied [*]

• [*] J.P. LaSalle, Proc. Nat. Acad. Of Sciences 45, 573-577 (1959); R.Bellman ,I. Glicksberg O.Gross, “On the bang-bang control problem” Q. Appl. Math.14 11-18 (1956)

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dx

dt= A(t)x(t) + B(t)u(t)


Formal aspects of the control problem

• An admissible (piecewise measurable in a set Ω ) steering function u* is optimal if for some t*>0 x(t*,u*) =0 and if x(t,u)≠0 for 0<t<t* for all u(t) Ω

• It can be proved [*] that :• “ Anything that can be done by an admissible steering

function can also be done by a bang-bang function”• This leads to the theorem:• “If for the control problem there is a steering function

u(t) Ω such that x(t,u)=0, for t>0, then there is an optimal steering function u* in Ω. Moreover all optimal steering functions u* are of the bang-bang form”

• Thus the only way of reaching the objective in minimum time is by using properly all the power available

• Steering times can be chosen testing ||x(t|| <

u(t)

t


CONCLUSIONS An algorithm performing real-time data analysis for locating the

island has been developed and tested .

A multichannel EC emission diagnostic is used for generating input signals to the diagnostic/control unit.

The algorithm described can be used to localize Tearing Modes, sawteeth and surface where q is rational.

Tearing mode control based on ECE+ECCD/ECRH is possible with:

A high space resolution (many channels) ECE polychromator

Power requirements depend essentially on ratio JECCD /JBOOT, and the radial location of RF profile

If the basic physical process is efficient, formal control theory helps finding the best strategy (on-off switching of full RF power at selected times is most likely )


THE END


PC

DSP A

DSP B

VGX

M66(DIGITAL IO)

InternalClock

(ANALOG INPUT)

Parameters

Measurements

Measurements/Status

Service Commands

.

-

-

4

off

4

Cut off Freq.

CutFreq.

Init.

Init.

Parameters

Measurem.

GATE PRERUN

FTU CLOCK

PRERUN

GATE FSC

4

On/off4 On/off GYROTRON

msg B-PC

msg A-PC

12ECE signals

1


axisinversion

radius

sawteeth are

stabilized

s.t inversion radius = minimum correlation (negative maximum)


0

0,5

1

1,5

2

2,5

3

3,5

4

1 2 3 4 5 6 7 8 9 10 11 12

keV

n. channel

t=0.557 s

0,5

1

1,5

2

2,5

3

3,5

4

0,5 0,6 0,7 0,8

keV

t(s)

#14979ch.9 (axis)

ch.5

800 kW ECRH

ch.3

2,35

2,45

2,55ch.8

ch.7

0,5

0,7

0,9

0,49 0,494 0,498t(s)

ch.3ch.4

1,46

1,5

1,54ch.5

2,9

3

3,1

3,2

ch.7

0,8

1

1,2

1,4

0,61 0,62 0,63t(s)

ch.3

ch.4

1,6

1,8

2

2,2

ch.5

Case 4 ECRH affects mode coupling

3,15

3,25

3,35ch.7

ch.8

0,9

1

1,1

0,68 0,684 0,688t(s)

ch.3ch.4

1,8

1,84

1,88

ch.5


1.2

1.4

1.6

1.8

2

2.2

2.4

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

keV

t(s)

800 kW ECRH

#18290

ch.4

ch.5

1.4

1.5

1.6

1.7

1.8

1.9

0.78 0.785 0.79 0.795 0.8

keV

t(s)

ch.4

ch.51.6

1.8

2

2.2

0.88 0.885 0.89 0.895 0.9

keV

t(s)

ch.4

ch.5

Case 3 - ECRH suppresses s.t.if rdep ≥ rs.t. inversion

2

2.1

2.2

2.3

0.98 0.99 1 1.01 1.02

keV

t(s)

ch.4

ch.3

0

0.5

1

1.5

2

2.5

2 4 6 8 10 12

Te

(keV)

nch

#18290t=0.9111÷0.9194 s

chdep

profile evolution during one s.t. cycle


Formal aspects of the control problem• The physical objective is to reduce the ECE fluctuation to zero in minimal time using

ECRH /ECCD on the position q=m/n identified by the phase jump method• The TM control problem in the extended Rutherford form, belongs to a general class known

in in the theory of multistage decision processes [*] . In a linearized form the governing equation for the state variable x(t) is

• The formal problem consists in to reducing the state x(t) to zero in minimal time by a suitable choice of the steering function u(t).

• If τ(x0) represents the minimum time as function of the initial state, the optimality principle [**] stipulates that

• At each point in phase space of the state vectors x0 the control vector u is chosen to maximize g.grad τ

• Optimization of with respect to u at each x,

€

dx

dt= g(x,u) x(0) = x0

€

τ(x0)

−1 = min(g ⋅∇τ (x0))u∈Ω

€

g(x,u) ⋅∇τ (x0)

real-time tracking and feedback control strategies for rotating magnetic islands

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