Workshop on MHD Control 2008 – O.Katsuro-Hopkins Computational analysis of advanced control methods applied to RWM control in tokamaks Oksana N. Katsuro-Hopkins

Workshop on MHD Control 2008 – O.Katsuro-Hopkins

Computational analysis of advanced control methods applied to RWM control in tokamaks

Oksana N. Katsuro-HopkinsColumbia University, New York, NY, USA

with S.A. Sabbagh, J.M. Bialek

US-Japan Workshop on MHD Control,

Magnetic Islands and Rotation

November 23-25, 2008 University of Texas, Austin, Texas, USA


Control theory terminology used in this talk

Advanced controller



Advanced controller

State-space based controller



Advanced controller


Linear Quadratic Gaussian Controller

(LQG)



Advanced controller



(LQG)

Optimal Controller

Optimal Observer



Advanced controller



(LQG)

Optimal Controller

Optimal Observer

Kalman Filter


LQG controller is capable to enhance resistive wall mode control system

• Motivation To improve RWM feedback control in NSTX with present

external RWM coils

• Outline Advantages of the LQG controller VALEN state space modeling with mode rotation and

control theory basics used in the design of LQG Application of the advanced controller techniques to NSTX


Limiting bn RWM in ITER can be improved with LQG controller* and external field correction coils

• Simplified ITER model includes double walled vacuum vessel 3 external control coil pairs 6 magnetic field flux sensors on the

midplane (z=0)

*Nucl. Fusion 47 (2007) 1157-1165




midplane (z=0)

• 10 Gauss sensor noise RWM

• LQG is robust for all C b < 86% with respect to bN

*Nucl. Fusion 47 (2007) 1157-1165

bN

Gro

wth

rat

e ,

g1/

sec

Passive

Ideal Wall

PD Cb = 68%

LQG Cb = 86%




midplane (z=0)

• 10 Gauss sensor noise RWM

• LQG is robust for all C b < 86% with respect to bN

*Nucl. Fusion 47 (2007) 1157-1165

bN

Gro

wth

rat

e ,

g1/

sec

MW

att

sRMS, Control Coil Power

Passive

Ideal Wall

PD Cb = 68%

LQG Cb = 86%


DIII-D LQG is robust with respect to bN and stabilizes RWM up to ideal wall limit

DIII-D with internal control coils



• LQG is robust with respect to bN and stabilize RWM up to ideal wall limit for 0.01 < torque < 0.08


Ideal Wall

Passive g, 1/s

LQG g, 1/s

Passive w, rad/s



• LQG is robust with respect to bN and stabilize RWM up to ideal wall limit for 0.01 < torque < 0.08

• LQG provides better reduction of current and voltages compared with proportional gain controller


Ideal Wall

Passive g, 1/s

LQG g, 1/s

RM

S C

ontr

ol c

oils

Vol

tage

, V

Mode phase, degrees

PID controller

PID controller & Kalman Filter

LQG controller

Passive w, rad/s


Initial results using advanced Linear Quadratic Gaussian (LQG) controller in KSTAR yield factor of 2 power reduction for white noise*

• Conducting hardware, IVCC set up in VALEN-3D* based on engineering drawings

• Conducting structures modeled Vacuum vessel with actual port structures Center stack back-plates Inner and outer divertor back-plates Passive stabilizer (PS) PS Current bridge

n=1 RWM passive stabilization currents

* IAEA FEC 2008 TH/P9-1 O. Katsuro-Hopkins





• IVCC allows active n=1 RWM stabilization near ideal wall bn limit, for proportional gain and LQG controllers



RW

M g

row

th r

ate,

1/s

106

105

104

103

102

101

100

10-1

2 3 4 5 6 7 8

bN

no wall with wall





• IVCC allows active n=1 RWM stabilization near ideal wall bn limit, for proportional gain and LQG controllers



FAST IVCC circuit L/R=1.0ms

Unloaded IVCCL/R=12.8ms

White noise (1.6-2.0G RMS)

RW

M g

row

th r

ate,

1/s

106

105

104

103

102

101

100

10-1

2 3 4 5 6 7 8

bN

no wall with wall


Tutorial on selected control theory topics


state estimate

Digital LQG controller proposed to improve stabilization performance

RWM

measured or simulated

LQG Controller €

u

€

y

€

u

LinearQuadraticEstimator

Optimal Observer

€

w disturbancesand noise

control vector(applied voltagesor current to control coils)

€

z diagnostics

measurements(magnetic fieldsensors)

LinearQuadraticRegulator

Optimal Controller

€

ˆ x


Detailed diagram of digital LQG controller

• State estimate stored in observer provides information about amplitude and phase of RWM and takes into account wall currents

• Dimensions of LQG matrices depends on State estimate (reduced balanced VALEN states)~10-20 Number of control coils ~3 Number of sensors ~ 12-24

• All matrixes in LQG calculated in advance using VALEN state-space for particular 3-D tokamak geometry, fixed plasma mode amplitude and rotation speed

€

ˆ x i+1

LQG Controller

Optimal Observer

€

ˆ x i+1 = Aˆ x i + Bui + K f y i − ˆ y i( )

ˆ y i = Cˆ x i

Optimal Controller

€

ui+1 = −Kcˆ x i+1

€

ui+1

€

y i

state process control filterestimate matrix matrix gain

measure- measurement ment matrix

controllergain


Balanced truncation significantly reduces VALEN state space

Balancingtransformation

€

ˆ x =t T

r x

What is VALEN state-space?

How to find balancing transformation ?

How to determine number of states to keep ?

~3000 states

€

rx =

r I w

r I f Id( )

T;

r u =

r V f ;

t A = −

t L −1 ⋅

t R ;

t B =

t L −1 ⋅

t I cc;

r y =

r Φ ;

t C =

t M

sensor mutualfluxes inductance

matrix

Full VALEN computed wall currents:

…

RWMfixed phase

2999 Balanced states:

€

ˆ x 1

€

ˆ x 2

€

ˆ x 3

€

ˆ x 3000

Truncation€

Balanced realization:

State reduction:

€

˜ A 11˜ B 1

˜ C 1 0

⎡

⎣ ⎢

⎤

⎦ ⎥≡

A B

C 0

⎡

⎣ ⎢

⎤

⎦ ⎥

€

tT

t A

t T −1

t T

t B

t C

t T −1

t 0

⎡

⎣ ⎢

⎤

⎦ ⎥=

˜ A ˜ B ˜ C 0

⎡

⎣ ⎢

⎤

⎦ ⎥=

˜ A 11˜ A 12

˜ B 1˜ A 21

˜ A 22˜ B 2

˜ C 1 ˜ C 2 0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

~3-20 states

€

tT II

I

III

III

II

I


State-space control approach may allow superior feedback performance

• VALEN circuit equations after including unstable plasma mode. Fluxes at the wall, feedback coils and plasma are

• Equations for system evolution

• In the state-space form

where

& measurements are sensor fluxes. State-space dimension ~1000 elements!

• Classical control law with proportional gain defined as

€

rΦ w =

t L ww ⋅

r I w +

t L wf ⋅

r I f +

t L wp ⋅ Id

r Φ f =

t L fw ⋅

r I w +

t L ff ⋅

r I f +

t L fp ⋅ Id

Φ p =t L pw ⋅

r I w +

t L pf ⋅

r I f +

t L pp ⋅ Id

€

t Lww

t L wf

t L wpt

L fwt L ff

t L fpt

L pw

t L pf

t L pp

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟⋅

d

dt

r I wr I fId

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟=

t R w 0 0

0t R f 0

0 0t R d

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟⋅

r I wr I fId

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟+

r 0 r V f0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

r x =t A

r x +

t B

r u

r y = C

r x

€

rx =

r I w

r I f Id( )

T;

t A = −

t L −1 ⋅

t R ;

t B =

t L −1 ⋅

t I cc;

r u =

r V f

€

ru = −

t G p

r y

€

ry =

r Φ s

I


VALEN formulation with rotation allows inclusion of mode phase into state space*

• VALEN uses two copies of a single unstable mode with p/2 toroidal displacement of these two modes

• The VALEN uses two dimensionless parameter normalized torque “a” and normalized energy “s”

• The VALEN parameters ‘s’ and ‘a’ together determine growth rate g and rotation W of the plasma mode

• LQG is optimized off line for best stability region with respect to ‘s’ and ‘ ’a parameters.

*Boozer PoP Vol6, No. 8, 3190 (1999)

€

s = −δW

LB Ib2 /2

= −energy required with plasma

energy required WITHOUT plasma

€

α =torque

LB Ib2 /2

=torque on mode by plasma

energy required WITHOUT plasma


Measure of system controllability and observability is given by controllability and observability grammians

• Given stable Linear Time-Invariant (LTI) Systems

• Observability grammian, , can be found be solvingcontinuous-time Lyapunov equation, , provides measure of output energy:

• defines an “observability ellipsoid” in the state space with the longest principalaxes along the most observable directions

• Controllability grammian, , can be found be solving continuous-time Lyapunov equation, , provides measure of input(control) energy:

• defines a “controllability ellipsoid” in the state space with the longest principalaxes along the most controllable directions

€

Γo = eA Tτ CTCeAτ

0

∞

∫ dτ

€

ATΓo + ΓoA + CTC = 0

€

y2

2= x0

TΓO x0

€

Γc = eAτ BBT

0

∞

∫ eA Tτ dτ

€

AΓc + Γc AT + BBT = 0

€

u2

2= x0

TΓC−1x0

€

ΓO = UΛOUT

€

ΓC = VΛCV T

€

ΓO

€

ΓC

€

A B

C 0

⎡

⎣ ⎢

⎤

⎦ ⎥

most observable

most controllable

II


• Controllable, observable & stable system called balanced if

where

- Hankel Singular Values• Balancing similarity transformation

transforms observability and controllabilityellipsoids to an identical ellipsoid aligned withthe principle axes along the coordinate axes.

• The balanced transformation can be defined in two steps: Start with SVD of controllability grammian

and define the first transformation as Perform SVD of observability grammian in the new basis:

the second transformation defined as The final transformation matrix is given by:

Balanced realization exists for every stable controllable and observable system

€

˜ Γ C = ˜ Γ O =

σ 1 0 0

0 O 0

0 0 σ n

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟= Σ,

€

ΓO

€

ΓC

€

(T−1)T ΓOT−1

= TΓCTT = Σ

€

T€

A B

C 0

⎡

⎣ ⎢

⎤

⎦ ⎥

€

TAT −1 TB

CT−1 0

⎡

⎣ ⎢

⎤

⎦ ⎥

€

σ i > σ j for i > j

€

Γc = VScVT

€

˜ Γ o = T1TΓoT1 = USoU

T

€

T1 := VSc1/ 2

€

T2 := USo−1/ 4

€

T := T1T2 = VSc1/ 2USo

−1/ 4

€

T

€

σ i

II


Determination of optimal controller gain for the dynamic system

For given dynamic process:

Find the matrix such that control law:

minimizes Performance Index:

where tuning parameters are presented by - state and control

weighting matrixes,

Controller gain for the steady-state can be calculated as

Where is solution of the controller Riccati matrix equation

III

€

r x =t A

r x +

t B

r u

€

tK c

€

ru = −

t K c

r x

€

J = (r x '(τ )

t Q r (τ )

r x (τ ) +r u '(τ )

t R r(τ )

r u (τ ))dτ → min

t

T

∫

€

tQ r,

t R r

Solution:

€

tK C =

t R −1

t B r

Tt S

€

tS

€

tS

t A r +

t A r

Tt S -

t S

t B r

t R r

−1t B r

Tt S +

t Q r = 0


Determination of optimal observer gain for the dynamic system

For given stochastic dynamic process:

with measurements:

Find the matrix such that observer equation

minimizes error covariance:

where tuning parameters are presented by - plant and measurement noise

covariance matrix

Observer gain for the steady-state can be calculated as

Where is solution of the observer Riccati matrix equation

III

€

r x =t A r

r x +

t B r

r u +

r ν

€

ry =

t C r

r x +

r ω

€

tK f

€

r ˙ x =t A r

r x +t B r

r u +

t K f

r y − Cr

r x ( )

€

E (r x −

r x )(r x −

r x )T r y (τ ),τ ≤ t{ } → min

€

tV ,

t W

€

tK f =

t P

t C '

t W −1

€

tP

€

tA r

t P +

t P

t A r

T −t P

t C r

Tt

W −1t C r

t P +

t V T = 0

Solution:


Closed system equations with optimal controller and optimal observer based on reduced order model

Measurement noise

Full order VALEN model

Optimal observer

Optimal controller

Ü Closed loop continuous systemallows to Test if Optimal controller and observer

stabilizes original full order model and defined number of states in the LQG controller

Verify robustness with respect to bn

Estimate RMS of steady-state currents, voltages and power

III

€

˙ x = Ax + Bu

y = Cx + ω

€

ˆ ˙ x = Arˆ x + Bru + K f (y − ˆ y )

ˆ y = Crˆ x

€

u = −Kcˆ x

€

ω

€

y

€

ˆ x

€

u

€

u

€

˙ x

ˆ ˙ x

⎛

⎝ ⎜

⎞

⎠ ⎟=

A −BKc

K f C F

⎛

⎝ ⎜

⎞

⎠ ⎟x

ˆ x

⎛

⎝ ⎜

⎞

⎠ ⎟+

0

K f

⎛

⎝ ⎜

⎞

⎠ ⎟ω

F = Ar − K f Cr − BrKc


Advanced controller methods planned to be tested on NSTX with future application to KSTAR

• VALEN NSTX Model includes Stabilizer plates

External mid-plane control coils closely coupled to vacuum vessel

Upper and lower Bp sensors in actual locations

Compensation of control field from sensors

Experimental Equilibrium reconstruction (including MSE data)

• Present control system on NSTX uses Proportional Gain

RWM active stabilization coils

RWM sensors (Bp)

Stabilizerplates


Advanced control techniques suggest significant feedback performance improvement for NSTX up to = 95%

•Classical proportional feedback methods VALEN modeling of feedback

systems agrees with experimental results

RWM was stabilized up to bn = 5.6 in experiment.

•Advanced feedback control may improve feedback performance Optimized state-space

controller can stabilize up to Cb=87% for upper Bp sensors and up to Cb=95% for mid-plane sensors

Uses only 7 modes for LQG design

Gro

wth

rat

e (1

/s)

bN

DCONno-wall

limit

Experimental (control off)( b collapse) With-wall limit

active control

passivegrowth

activefeedback

Experimental (control on)

AdvancedFeedback

Bp sensors

AdvancedFeedback

M-P sensors€

βn βnwall


LQG with mode phase needs only 9 modes using Bp upper and lower sensors sums and differences

• Fixed mode (fixed phase)

• 7 states (1 unstable mode + 6 stable balanced states)

• LQG(bN=6.7,N=7)

• Rotating mode (phase included)

• 9 states (2 unstable modes + 7 stable balanced states)

• LQG(a=0,bN=6.7,N=9)

StableUnstableStableUnstable


LQG(0,6.7,9) stabilizes slow rotating RWM mode up to bN<6.7

RWM unstableRWM stable

Ideal Wall

4.5 5.0 5.5 6.0 6.5 7.0 7.5

= 0.06a

bN

Mo

de

gro

wth

ra

te,

Re

(g),

1/s

103

102

101

100

10-1

10-2




Ideal Wall

4.5 5.0 5.5 6.0 6.5 7.0 7.5

= 0.06a

bN

Mo

de

gro

wth

ra

te,

Re

(g),

1/s

103

102

101

100

10-1

10-2

0 100 200 300 400 500-200

-100

0

100

200

300

400

500

Mode rotation, rad/s

Mo

de

gro

wth

ra

te,

1/s

bN=6.7

bN=6.65

bN=6.4

bN=6.1

bN=5.9

RW

M

sta

ble

aincreasing

RW

M u

nsta

ble



0 100 200 300 400 500-200

-100

0

100

200

300

400

500


Mo

de

gro

wth

ra

te,

1/s

bN=6.7

bN=6.65

bN=6.4

bN=6.1

bN=5.9

RW

M u

nsta

ble

RW

M

sta

ble

4.5 5.0 5.5 6.0 6.5 7.0 7.5

= 0.06a

bN

Mo

de

gro

wth

ra

te,

Re

(g),

1/s

103

102

101

100

10-1

10-2

LQG unstable


Ideal Wall

aincreasing

LQG unstable



0 100 200 300 400 500-200

-100

0

100

200

300

400

500


Mo

de

gro

wth

ra

te,

1/s

LQG unstable

bN=6.7

bN=6.65

bN=6.4

bN=6.1

bN=5.9

RW

M u

nsta

ble

RW

M

sta

ble

4.5 5.0 5.5 6.0 6.5 7.0 7.5

= 0.06a

bN

Mo

de

gro

wth

ra

te,

Re

(g),

1/s

103

102

101

100

10-1

10-2

LQG unstable


Ideal Wall

aincreasing

LQG unstableLQG

unstable


>90% power reduction for white noise driven time evolution of the controlled RWM

RMS values Peak Values

Cb ICC, A VCC, V P, Watts ICC, A VCC, V P, Watts

10% 70% 84% 94% 66% 84% 93%

20% 73% 85% 95% 68% 84% 94%

30% 77% 86% 96% 73% 85% 95%

40% 85% 90% 98% 82% 89% 98%

0 10 20 30 40 50 60 70 80 90

Cb, %

RM

S P

ow

er,

Wa

tts

10-3

10-4

10-5

10-6

LQG

• White noise 7 gauss in amplitude with 5kHz sampling frequency

• Power is proportional to white noise amplitude2 and sampling frequency-1

• No filters on Bp sensors in proportional controller was used in this study

Proportional gain only


Advanced controller study continuing…

• Conclusions NSTX advanced controller with mode phase has been tested

numerically, 9 modes needed, large stability region for slow rotating mode. (RWM is usually locked in the NSTX experiments.)

• Next Steps Study LQG with different torque, to improve robustness with

respect to bN and mode rotation speed. Off line comparison for mode phase and amplitude calculated with

reduced order optimal observer and the present measured RWM sensor signal data SVD evaluation of n = 1 amplitude and phase from the experimental data

Redesign state-space for control coil current controller input Analyze time delay effect on LQG performance


Thank you!

Documents

Workshop on MHD Control 2008 – O.Katsuro-Hopkins Computational analysis of advanced control methods applied to RWM control in tokamaks Oksana N. Katsuro-Hopkins