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Computational analysis of advanced control methods applied to RWM control in tokamaks. Oksana N. Katsuro-Hopkins Columbia University, New York, NY, USA w ith S.A. Sabbagh, J.M. Bialek. US-Japan Workshop on MHD Control, Magnetic Islands and Rotation - PowerPoint PPT Presentation
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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
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
Control theory terminology used in this talk
Advanced controller
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
Control theory terminology used in this talk
Advanced controller
State-space based controller
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
Control theory terminology used in this talk
Advanced controller
State-space based controller
Linear Quadratic Gaussian Controller
(LQG)
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
Control theory terminology used in this talk
Advanced controller
State-space based controller
Linear Quadratic Gaussian Controller
(LQG)
Optimal Controller
Optimal Observer
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
Control theory terminology used in this talk
Advanced controller
State-space based controller
Linear Quadratic Gaussian Controller
(LQG)
Optimal Controller
Optimal Observer
Kalman Filter
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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)
• 10 Gauss sensor noise RWM
• LQG is robust for all Cb < 86% with respect to bN
*Nucl. Fusion 47 (2007) 1157-1165
bN
Gro
wth
rate
g,
1/se
c
Passive
Ideal Wall
PD Cb = 68%
LQG Cb = 86%
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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)
• 10 Gauss sensor noise RWM
• LQG is robust for all Cb < 86% with respect to bN
*Nucl. Fusion 47 (2007) 1157-1165
bN
Gro
wth
rate
g,
1/se
c
MW
atts
RMS, Control Coil Power
Passive
Ideal Wall
PD Cb = 68%
LQG Cb = 86%
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
DIII-D LQG is robust with respect to bN and stabilizes RWM up to ideal wall limit
DIII-D with internal control coils
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
DIII-D LQG is robust with respect to bN and stabilizes RWM up to ideal wall limit
• LQG is robust with respect to bN and stabilize RWM up to ideal wall limit for 0.01 < torque < 0.08
DIII-D with internal control coils
Ideal Wall
Passive g, 1/s
LQG g, 1/s
Passive w, rad/s
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
DIII-D LQG is robust with respect to bN and stabilizes RWM up to ideal wall limit
• 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
DIII-D with internal control coils
Ideal Wall
Passive g, 1/s
LQG g, 1/s
RM
S C
ontro
l coi
ls V
olta
ge,
V
Mode phase, degrees
PID controller
PID controller & Kalman Filter
LQG controller
Passive w, rad/s
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
• IVCC allows active n=1 RWM stabilization near ideal wall bn limit, for proportional gain and LQG controllers
n=1 RWM passive stabilization currents
* IAEA FEC 2008 TH/P9-1 O. Katsuro-Hopkins
RW
M g
row
th ra
te, 1
/s
106
105
104
103
102
101
100
10-1
2 3 4 5 6 7 8bN
no wall with wall
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
• IVCC allows active n=1 RWM stabilization near ideal wall bn limit, for proportional gain and LQG controllers
n=1 RWM passive stabilization currents
* IAEA FEC 2008 TH/P9-1 O. Katsuro-Hopkins
FAST IVCC circuit L/R=1.0ms
Unloaded IVCCL/R=12.8ms
White noise (1.6-2.0G RMS)
RW
M g
row
th ra
te, 1
/s
106
105
104
103
102
101
100
10-1
2 3 4 5 6 7 8bN
no wall with wall
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
Tutorial on selected control theory topics
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
state estimate
Digital LQG controller proposed to improve stabilization performance
RWMmeasured
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
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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 = −Kc ˆ x i+1
€
ui+1
€
y istate process control filterestimate matrix matrix gain
measure- measurement ment matrix
controllergain
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
€
r x =
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 BC 0 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
t T
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
€
t T II
I
III
III
II
I
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
€
r Φ 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 L ww
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 00
t 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
€
r x =
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
€
r u = −
t G p
r y
€
r y =
r Φ s
I
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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 = − δWLB Ib
2 /2= − energy required with plasma
energy required WITHOUT plasma
€
a =torqueLB Ib
2 /2= torque on mode by plasma
energy required WITHOUT plasma
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
€
y 2
2 = x0TΓO x0
€
Γc = eAτ BBT
0
∞∫ eA T τ dτ
€
AΓc + Γc AT + BBT = 0
€
u 2
2 = x0TΓC
−1x0
€
ΓO = UΛOUT
€
ΓC = VΛCV T
€
ΓO
€
ΓC
€
A BC 0 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
most observable
most controllable
II
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
• 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 00 O 00 0 σ n
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟= Σ,
€
ΓO
€
ΓC
€
(T−1)T ΓOT−1
= TΓCTT = Σ
€
T€
A BC 0 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
TAT −1 TBCT−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
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
€
t K c
€
r u = −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
∫
€
t Q r,
t R r
Solution:
€
t K C =
t R −1
t B r
Tt S
€
t S
€
t S
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
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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 noisecovariance 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
€
r y =t C r
r x + r ω
€
t K f
€
r ˆ ˙ x =t A r
r ̂ x +t B r
r u +t K f
r y − Crr ̂ x ( )
€
E ( r x − r ̂ x )( r x − r ̂ x )T r y (τ ),τ ≤ t{ } → min
€
t V ,
t W
€
t K f =
t P
t C '
t W −1
€
t P
€
t A 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:
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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 + Buy = Cx + ω
€
ˆ ˙ x = Ar ˆ x + Bru + K f (y − ˆ y )ˆ y = Cr ˆ x
€
u = −Kc ˆ x
€
w
€
y
€
ˆ x
€
u
€
u
€
˙ x ˆ ˙ x ⎛ ⎝ ⎜
⎞ ⎠ ⎟=
A −BKc
K f C F ⎛
⎝ ⎜
⎞
⎠ ⎟xˆ x ⎛ ⎝ ⎜
⎞ ⎠ ⎟+
0K f
⎛
⎝ ⎜
⎞
⎠ ⎟ω
F = Ar − K f Cr − BrKc
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
rate
(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€
bn βnwall
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
a = 0.06
bN
Mod
e gr
owth
rate
, Re(
g), 1
/s
103
102
101
100
10-1
10-2
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
a = 0.06
bN
Mod
e gr
owth
rate
, 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
Mod
e gr
owth
rate
, 1/s
bN=6.7
bN=6.65
bN=6.4
bN=6.1
bN=5.9
RW
M
stab
le
aincreasing
RW
M u
nsta
ble
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
LQG(0,6.7,9) stabilizes slow rotating RWM mode up to bN<6.7
0 100 200 300 400 500-200
-100
0
100
200
300
400
500
Mode rotation, rad/s
Mod
e gr
owth
rate
, 1/s
bN=6.7
bN=6.65
bN=6.4
bN=6.1
bN=5.9
RW
M u
nsta
ble
RW
M
stab
le
4.5 5.0 5.5 6.0 6.5 7.0 7.5
a = 0.06
bN
Mod
e gr
owth
rate
, Re(
g), 1
/s
103
102
101
100
10-1
10-2LQG unstable
RWM unstableRWM stable
Ideal Wall
aincreasing
LQG unstable
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
LQG(0,6.7,9) stabilizes slow rotating RWM mode up to bN<6.7
0 100 200 300 400 500-200
-100
0
100
200
300
400
500
Mode rotation, rad/s
Mod
e gr
owth
rate
, 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
stab
le
4.5 5.0 5.5 6.0 6.5 7.0 7.5
a = 0.06
bN
Mod
e gr
owth
rate
, Re(
g), 1
/s
103
102
101
100
10-1
10-2LQG unstable
RWM unstableRWM stable
Ideal Wall
aincreasing
LQG unstableLQG
unstable
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
>90% power reduction for white noise driven time evolution of the controlled RWM
RMS values Peak ValuesCb 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
ower
, Wat
ts
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
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
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
Workshop on MHD Control 2008 – O.Katsuro-Hopkins
Thank you!