Data Assimilation and Driver Estimation for Space Weather ...morozova/MySite/papers/def.pdf · Data...

Data Assimilation and Driver Estimation forSpace Weather Models using Ensemble Filters

Alexey V. Morozov

UofM Committee:Dennis S. BernsteinAaron J. RidleyPierre T. KabambaIlya V. Kolmanovsky

NCAR Collaborators:Nancy CollinsTimothy J. HoarJeffrey L. Anderson

March 29, 2013

AV Morozov, UM Data Assimilation and Driver Estimation 1/56

Overview

1 Summary

2 Problem Statement

3 GITM

4 EAKF

5 Assimilating CHAMP Neutral Density

6 Assimilating GPS Total Electron Content

7 Conclusions and Future Work

1 SummaryMain ContributionsPublications

2 Problem Statement

3 GITM

4 EAKF

Main Contributions

Contributions relating to data assimilation are:

modified the Data Assimilation Research Testbed (DART) tointerface it with the Global Ionosphere-Thermosphere Model(GITM),

developed a novel inflation technique for the Ensemble AdjustmentKalman Filter (EAKF) as applied to GITM for purposes of dataassimilation and driver estimation, and

introduced an ability to assimilate Total Electron Content (TEC)measurements into the DART-GITM interface.

Contributions relating to adaptive control are:

described the Retrospective Cost Adaptive Control (RCAC) stabilitymargins for plants with uncertain nonminimum-phase zeros,

introduced a convex constraint on the controller pole locations toimprove RCAC transient and steady-state performance, and

modeled nonlinear system and described the achievable amplitudeand frequency control ranges.

Main Contributions

Contributions relating to data assimilation are:

modified the Data Assimilation Research Testbed (DART) tointerface it with the Global Ionosphere-Thermosphere Model(GITM),

developed a novel inflation technique for the Ensemble AdjustmentKalman Filter (EAKF) as applied to GITM for purposes of dataassimilation and driver estimation, and

introduced an ability to assimilate Total Electron Content (TEC)measurements into the DART-GITM interface.

Contributions relating to adaptive control are:

described the Retrospective Cost Adaptive Control (RCAC) stabilitymargins for plants with uncertain nonminimum-phase zeros,

introduced a convex constraint on the controller pole locations toimprove RCAC transient and steady-state performance, and

modeled nonlinear system and described the achievable amplitudeand frequency control ranges.

Publications

1 M. S. Fledderjohn, M. S. Holzel, A. V. Morozov, J. B. Hoagg, and D. S. Bernstein, “On the Accuracy ofLeast Squares Algorithms for Estimating Zeros,” Proc. Amer. Contr. Conf., Baltimore, MD, June 2010.

2 A. V. Morozov, J. B. Hoagg, and D. S. Bernstein, “A Computational Study of the Performance andRobustness Properties of Retrospective Cost Adaptive Control,” AIAA Guid. Nav. Contr. Conf., Toronto,August 2010.

3 A. V. Morozov, J. B. Hoagg, and D. S. Bernstein, “Retrospective Adaptive Control of a Planar MultilinkArm with Nonminimum-Phase Zeros,” Proc. Conf. Dec. Contr., pp. 3706–3711, Atlanta, GA, December2010.

4 A. V. Morozov, A. M. D’Amato, J. B. Hoagg, and D. S. Bernstein, “Retrospective Cost Adaptive Controlfor Nonminimum-Phase Systems with Uncertain Nonminimum-Phase Zeros Using Convex Optimization,”Proc. Amer. Contr. Conf., pp. 1188–2293, San Francisco, CA, June 2011.

5 A. M. D’Amato, E. D. Sumer, K. S. Mitchell, A. V. Morozov, J. B. Hoagg, and D. S. Bernstein, “AdaptiveOutput Feedback Control of the NASA GTM Model with Unknown Nonminimum-Phase Zeros,” AIAAGuid. Nav. Contr. Conf., Portland, OR, August 2011.

6 A. V. Morozov, A. A. Ali, A. M. D’Amato, A. J. Ridley, S. L. Kukreja, and D. S. Bernstein,“Retrospective-Cost-Based Model Refinement for System Emulation and Subsystem Identification,” Proc.Conf. Dec. Contr., pp. 2142–2147, Orlando, FL, December 2011.

7 E. D. Sumer, A. M. D’Amato, A. V. Morozov, J. B. Hoagg, and D. S. Bernstein, “Robustness ofRetrospective Cost Adaptive Control to Markov-Parameter Uncertainty,” Proc. Conf. Dec. Contr., pp.6085–6090, Orlando, FL, December 2011.

8 M. W. Isaacs, J. B. Hoagg, A. V. Morozov, and D. S. Bernstein, “A Numerical Study on Controlling aNonlinear Multilink Arm Using a Retrospective Cost Model Reference Adaptive Controller,” Proc. Conf.Dec. Contr., pp. 8008–8013, Orlando, FL, December 2011.

9 A. V. Morozov, A. J. Ridley, D. S. Bernstein, N. Collins, T. J. Hoar, and J. L. Anderson, “DataAssimilation and Driver Estimation for the Global Ionosphere-Thermosphere Model Using the EnsembleAdjustment Kalman Filter”, Journal of Atmospheric and Solar-Terrestrial Physics, 2013, Submitted.

10 A. V. Morozov, A. G. Burrell, A. J. Ridley, D. S. Bernstein, “Assimilation of the Total Electron ContentMeasurements into the Global Ionosphere-Thermosphere Model”, Journal of Atmospheric andSolar-Terrestrial Physics, 2013, To be submitted.

1 Summary

2 Problem StatementBig PictureProblem Statement

3 GITM

4 EAKF

laneposition

steeringwheel angle

outputinputPlant

- output

desired

correction

Controller

Plant-output

Where does model inversion come in?

The goal is to make y = y, that is yy

= 1 or e = 0, that is ey

y = Pd

y = P (y − Cy)

y = Py − PCy(1 + PC)y = Py

1 + PC

want= 1

Pw= 1 + PC

Cw= 1− P−1

e = y − y

e = y −P

1 + PCy

e =1 + PC − P

1 + PCy

1 + PC − P1 + PC

1 + PC − P w= 0

Cw= 1− P−1

Example of inversion in frequency domain

Consider a linear plant P = 1z+0.5

, which can be described in time domain asyk = −0.5yk−1 + dk−1. Suppose the goal is for the plant output y to follow desiredcommand y (command following). RCAC converges to controller C = 0.7104

z−0.5797

0 50 100 150 200

(a) Time step [samp]

y, desired

positio

0 50 100 150 200−2

(b) Time step [samp]

e, positio

0 50 100 150 200−2

(c) Time step [samp]

rce [N

0 50 100 150 200−2

(d) Time step [samp]

ol gain

0 1 2 3−10

(e) Frequency [rad\samp]

Magnitude [d

0 1 2 3−200

−150

−100

(f) Frequency [rad\samp]

Phase [d

Plant Inverse (1−P−1

Controller (C)

Closed Loop (P/(1+PC))

Command frequency (w)

How is data assimilation on GITM similar to driving a car?

- output

desired

correction

Controller

Plant-output

states+ states-

densitydesired

F10.7+

-density

Problem Statement

We want to drive GITM output (black) to match the satellite data (red).

1 Summary

2 Problem Statement

3 GITMInputs and OutputsEquationsImplementation

4 EAKF

Inputs and Outputs

Inputs Outputs

Solar flux index F10.7 or I∞ → → Ns Neutral number densities

Cooling rates Le(X) →

→ ρ Neutral mass density

Heating efficiency ε → I → p Neutral pressure

Thermal conductivity κc →

→ T Neutral temperature normalized

Earth magnetic field simplified →

→ u Neutral velocity

or APEX → Nj Ion number densities

Interplanetary simplified → → Tj Ion temperature normalized

magnetic field or ACE data → v Ion velocity

Hemispheric simplified →power index or POES data

Initialize using simplified →or MSIS/IRI

Table 21 compiled from 1

1Ridley, A. J., Y. Deng, and G. Toth. “The Global IonosphereThermosphereModel,” Journal of Atmospheric and Solar-Terrestrial Physics 68, no. 8 (2006)

GITM Vertical Equations

∂ Ns

∂t+∂ur,s∂r

+2ur,sr

+ ur,s∂ Ns

NsSs, (continuity) (1)

∂ur,s∂t

+ ur,s∂ur,s∂r

∂ur,s∂θ

r cos(θ)

∂ur,s∂φ

∂r+k T

∂ Ns

g + Fs +u2θ + u2φ

r+ cos2(θ)Ω2r + 2 cos(θ)Ωuφ, (momentum) (2)

∂t+ ur

∂r+ (γ − 1) T

+∂ur∂r

cvρmnQ , (energy) (3)

GITM Vertical Equations and source terms

(red - outputs, green - inputs)

∂ Ns

∂t+∂ur,s∂r

+2ur,sr

+ ur,s∂ Ns

NsSs, (continuity) (4)

∂ur,s∂t

+ ur,s∂ur,s∂r

∂ur,s∂θ

r cos(θ)

∂ur,s∂φ

∂r+k T

∂ Ns

g + Fs +u2θ + u2φ

r+ cos2(θ)Ω2r + 2 cos(θ)Ωuφ, (momentum) (5)

∂t+ ur

∂r+ (γ − 1) T

+∂ur∂r

cvρmnQ , (energy) (6)

Ss =∂

(∂Ns∂r− ∂N

)]+ Cs, (7)

Fs =ρiρsνin(vr − ur,s) +

∑q 6=s

NqNDqs

(ur,q − ur,s), (8)

Q = QEUV +QNO +QO +∂

((κc + κeddy)

mi + mnνin(v − u)2.

(9)AV Morozov, UM Data Assimilation and Driver Estimation 18/56

Implementation

In the CHAMP simulations, 5 resolution in longitude and latitude is used.

Variable resolution in altitude (from about 2km to about 18km) is used to spanthe range between about 100km and 660km.

The atmosphere is broken up into 32 (8 in longitude, 4 in latitude) blocks toallow for parallel computation.

A typical EAKF run (20 ensemble members) requires 640 CPUs (cores) andabout 8 wall hours per 24 simulated hours on NASA Pleiades supercomputer.

(a) Longitude [deg]

Latitu

de [deg]

0 60 120 180 240 300 360−90

Mass D

ensity [kg m

Horizontal resolution.

0 60 120 180 240 300 360100

(b) Longitude [deg]

Altitude (

Mass D

ensity [kg m

−3])

Vertical resolution.

1 Summary

2 Problem Statement

3 GITM

4 EAKFBig PictureKalman FilterExtended Kalman Filter (EKF)Unscented Kalman Filter (UKF)Ensemble Kalman Filter (EnKF)Ensemble Adjustment Kalman Filter (EAKF)

7 Conclusions and Future WorkAV Morozov, UM Data Assimilation and Driver Estimation 20/56

Pros and Cons of These Filters

KF → EKF → UKF→→

EnKFEAKF

KF → EKF The advantage of EKF over KF is that it allows the system tobe nonlinear.EKF → UKF The advantage of UKF over EKF is that it is more accurate

in propagating means and covariances through the nonlinear system anddoes not require linearization.UKF → EnKF The advantage of EnKF over UKF is that it does not

require 2n ensemble members (sigma points, particles), and allows the userto pick N , the number of ensemble members. The newest versions ofEnKF allow for localization, which improves computational performanceeven more than just reduction in number of ensemble members.UKF → EAKF EAKF is similar to EnKF in being an ensemble filter, but

utilizes different update equations. Some advantages of EAKF over UKFare addition of localization functionality and decreased computational load.

EnKFEAKF

KF → EKF The advantage of EKF over KF is that it allows the system tobe nonlinear.

EKF → UKF The advantage of UKF over EKF is that it is more accuratein propagating means and covariances through the nonlinear system anddoes not require linearization.UKF → EnKF The advantage of EnKF over UKF is that it does not

EnKFEAKF

in propagating means and covariances through the nonlinear system anddoes not require linearization.

UKF → EnKF The advantage of EnKF over UKF is that it does notrequire 2n ensemble members (sigma points, particles), and allows the userto pick N , the number of ensemble members. The newest versions ofEnKF allow for localization, which improves computational performanceeven more than just reduction in number of ensemble members.UKF → EAKF EAKF is similar to EnKF in being an ensemble filter, but

EnKFEAKF

require 2n ensemble members (sigma points, particles), and allows the userto pick N , the number of ensemble members. The newest versions ofEnKF allow for localization, which improves computational performanceeven more than just reduction in number of ensemble members.

UKF → EAKF EAKF is similar to EnKF in being an ensemble filter, bututilizes different update equations. Some advantages of EAKF over UKFare addition of localization functionality and decreased computational load.

EnKFEAKF

Kalman Filter

Consider the linear discrete-time system given by

xk = Fk−1xk−1 +Gk−1uk−1 + wk−1, wk ∼ N(0, Qk), (10)

yk = Hkxk + vk, vk ∼ N(0, Rk), (11)

where xk ∈ Rn, yk ∈ Rm, uk ∈ Rp, wk is the process noise, vk is the measurementnoise, and wk and vk are uncorrelated (i.e. E(wkv

Tj ) = 0).

If Fk, Gk, Hk, Qk, Rk, are known, the discrete time Kalman Filter2 is given by

x−k = Fk−1x+k−1 +Gk−1uk−1, (prior estimate) (12)

P−k = Fk−1P+k−1F

Tk−1 +Qk−1, (prior error covariance) (13)

Kk = P−k HTk (HkP

−k H

Tk +Rk)−1, (estimator gain) (14)

P+k = (I −KkHk)P−k , (posterior error covariance) (15)

x+k = x−k +Kk(yk −Hkx−k ). (posterior estimate) (16)

2Simon, Dan. Optimal state estimation: Kalman, H infinity, and nonlinearapproaches. Wiley-Interscience, 2006.

Kalman Filter and Discrete Algebraic Riccati Equation

As an aside, we demonstrate that Discrete Algebraic Riccati Equation (DARE) can bederived from the update equations derived so far.Substituting (16) into (12), we realize that prior state estimate can be updated directlywithout computation of the posterior estimate as in

x−k+1 = Fk(I −KkHk)x+k−1 + FkKkykGkuk. (17)

Similarly, by substituting (15) and (14) into (13), it can be shown that prior errorcovariance matrix can also be updated in one step, as given by

P−k+1 = Fk(P−k −[P−k H

Tk (HkP

−k H

Tk +Rk)−1

−k )FTk +Qk (18)

= FkP−k F

Tk − FkP

−k H

Tk (HkP

−k H

Tk +Rk)−1HkP

−k F

Tk +Qk, (19)

which is the Discrete Algebraic Riccati Equation.

Extended Kalman Filter (EKF)

Now consider the nonlinear discrete-time system given by

xk = fk−1(xk−1, uk−1, wk−1), wk ∼ N(0, Qk), (20)

yk = hk(xk, vk), vk ∼ N(0, Rk), (21)

where functions fk(·) and hk(·) are known explicitly and hence can be linearized via

Fk−1 =∂fk−1

∣∣∣x+k−1

, Lk−1 =∂fk−1

∣∣∣x+k−1

, (22)

Hk =∂hk

∣∣∣x−k

, Mk =∂hk

∣∣∣x−k

. (23)

Next, EKF3 can be updated via

x−k = fk−1 (x+k−1, uk−1, 0), (24)

P−k = Fk−1P+k−1F

Tk−1 + Lk−1Qk−1L

Tk−1, (25)

Kk = P−k HTk (HkP

−k H

Tk +MkRkM

Tk )−1, (26)

P+k = (I −KkHk)P−k , (27)

x+k = x−k +Kk

[yk − hk (x−k , 0)

]. (28)

3Simon, Dan. Optimal state estimation: Kalman, H infinity, and nonlinearapproaches. Wiley-Interscience, 2006.

Unscented Kalman Filter (UKF)

Consider the nonlinear discrete-time system with additive noise given by

xk = fk−1(xk−1, uk−1) + wk−1, (29)

yk = hk(xk) + vk. (30)

UKF propagates 2n realizations (sigma points, x(i)) of the state’s probability densityfunction, as defined and propagated by

x(i)k−1 = x+

k−1 + (−1)bi−1nc(√

nP+k−1

)Ti, (31)

x(i)k = fk−1(x

(i)k−1, uk−1) , y

(i)k = hk(x

(i)k ) . (32)

Accordingly, UKF can be updated via

x−k =1

(i)k , y−k =

(i)k , (33)

P−k =∑2ni=1(x

(i)k − x

−k )(x

(i)k − x

−k )T /(2n) +Qk−1 , (34)

Kk = PxyP−1y , Pxy =

(i)k − x

−k )(y

(i)k − y

−k )T /(2n) (35)

P+k = P−k −KkPyK

Tk , Py =

(i)k − y

−k )(y

(i)k − y

−k )T /(2n) +Rk (36)

x+k = x−k +Kk(yk − yk). (37)

Ensemble Kalman Filter (EnKF)

EnKF generates N vectors of initial conditions (x+1 ) and feeds these into the model

given by (21). Model started from different initial conditions is referred to as differentensemble members.

Mean states estimate is calculated via µx−k =∑Ni=1

x−k,i

hk(·) still needs to be known explicitly and its linearization Hk needs to be computedabout ensemble mean.Accordingly, N EnKF ensemble members can be updated via4

x−k,i = fk−1(x+k−1,i, uk−1, 0), (38)

P−k =∑Ni=1(x−k,i − µx

−k )(x−k,i − µx

−k )T /(N − 1) , (39)

Kk = P−k HTk (HkP

−k H

Tk +MkRkM

Tk )−1, (40)

P+k = (I −KkHk)P−k , (41)

x+k,i = x−k,i +Kk

[yk − hk(x−k,i, 0)

]. (42)

4Evensen, Geir. “Sequential data assimilation with a nonlinear quasi-geostrophicmodel using Monte Carlo methods to forecast error statistics.” J. Geophys. Res., - 99(1994): 10-10.

Ensemble Adjustment Kalman Filter (EAKF)

First, define joint5 state-observation vector as

]. (43)

Recall that xk ∈ Rn, yk ∈ Rm. We then define H ∈ Rm×(n+m) such that yk = Hzk,i.e. H = [ 0m×n Im×m ].Accordingly, N EAKF ensembles can be updated via6

z−k,i = [fk−1(x+k−1,i, uk−1, 0); hk(x+

k−1,i, 0)], (44)

P−k =∑N

i=1(z−k,i − µz

−k )(z−k,i − µz

−k )T /(N − 1), (45)

Ak = (FTk )−1GTk (UTk )−1BTk (GTk )−1FTk , (46)

P+k = [(P−k )−1 +HTR−1

k H]−1 , µz+k = P+

k [(P−k )−1µz−k +HTR−1k yk] , (47)

z+k,i = ATk (z−k,i − µz

−k ) + µz+

k . (48)

5Tarantola, Albert. Inverse problem theory and methods for model parameterestimation. Society for Industrial Mathematics, 2005.

6Anderson, Jeffrey L. “An ensemble adjustment Kalman filter for dataassimilation.” Monthly Weather Review 129.12 (2001): 2884-2903.

EAKF: intuition

0 1 2 3 4 5 6 7−1.5

−0.5

x−[1]

x[k]y[k]

x−i [k ]

x+i [k ]

Filter Divergence

Consider linear system

xk = 0.5xk−1 + uk−1, uk = 1.0 + sin(0.5k), (49)

yk = xk + vk, vk ∼ N(0, 0.2). (50)

If the driver varies with time but we assume it is constant, ensemble mightcollapse.

1 5 10 15 20 25 30 35 40 45−6

(a) Time k

True state s k

Measurement yk

EAKF mean µ [ s k]EAKF spread µ [ s k] ± σ [ s k]EAKF ini t i al p df ( s+

1 5 10 15 20 25 30 35 40 45−3

(b) Time k

Inputuk

True input uk

EAKF mean µ [ uk]EAKF spread µ [ uk] ± σ [ uk]EAKF ini t i al p df ( u+

Ensemble Inflation

Ensemble inflation given by

x−k =√λ(x−k − µ[x−k ]) + µ[x−k ], (51)

with λ = 2.0 alleviates filter divergence.

1 5 10 15 20 25 30 35 40 45−6

(a) Time k

True state s k

Measurement yk

EAKF mean µ [ s k]EAKF spread µ [ s k] ± σ [ s k]EAKF ini t i al p df ( s+

1 5 10 15 20 25 30 35 40 45−3

(b) Time k

Inputuk

True input uk

EAKF mean µ [ uk]EAKF spread µ [ uk] ± σ [ uk]EAKF ini t i al p df ( u+

Differential Inflation

Inflating driver by

u−k =

√σ2i√

σ2[u−k ](u−k − µ[u−k ]) + µ[u−k ], (52)

with σ2i = 0.12 removes the lag in the driver estimate.

1 5 10 15 20 25 30 35 40 45−6

True state s k

Measu rem ent y k

EAKF mean µ [ s k]EAKF sp re ad µ [ s k] ± σ [ s k]EAKF in i t ial p df ( s+

1 5 10 15 20 25 30 35 40 45−3

T im e k

Inputu

True inpu t u k

EAKF mean µ [ u k]EAKF sp re ad µ [ u k] ± σ [ u k]EAKF in i t ial p df ( u+

1 Summary

2 Problem Statement

3 GITM

4 EAKF

5 Assimilating CHAMP Neutral DensitySimulated CHAMP dataReal CHAMP data

CHAMP and GRACE orbits

Sim: Does actual density track the desired density?

Measurement source: ρ from GITM truth simulation (F10.7 = 148) isinterpolated to the CHAMP location

Ensemble members: 20 GITM instances with F10.7 = N(130, 25)

Same plot, but orbit averages

Sim: Does error at the measurement location go to zero?

Define RMS percentage error RMSPE4=

√(ρ−ρ)2√ρ2

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480

(a) Hours since 00UT 01/12/2002

RMSPE (2nd day) along CHAMP path = 2%

GITM without EAKFEAKF posterior

Sim: Does error at the validation location go to zero?

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480

(b) Hours since 00UT 01/12/2002

RMSPE (2nd day) along GRACE path = 4%

Sim: Does F10.7 estimate converge to the true value?

Real: Does actual density track the desired density?

Measurement source: real CHAMP ρ with real CHAMP uncertainty

Ensemble members: 20 GITM instances with F10.7 = N(130, 25)

Real: Does error at the measurement location go to zero?

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 450

(c) Hours since 00UT 01/12/2002

Real: Does error at the validation location go to zero?

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 450

(d) Hours since 00UT 01/12/2002

Real: Does F10.7 estimate converge to the NOAA value?

1 Summary

2 Problem Statement

3 GITM

4 EAKF

6 Assimilating GPS Total Electron ContentSimulated TEC dataReal TEC data

Real TEC data

Simulated TEC data

0 60 120 180 240 300 360Longitude [deg]

2002-12-01 00:30:00 0

Sim: True and Estimated TEC

Sim: TEC error

Sim: average error and driver

0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002

10Average VTEC error [TECU]

Average VTEC error

Average VTEC spread (± SD)

0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002

.7 [SFU

F10.7 measured

F10.7 estimated

F10.7 estimated ± SD

Sim: average error without driver estimation

0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002

Average VTEC error

0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002

.7 [SFU

F10.7 measured

F10.7 estimated

Real: True and Estimated TEC

Real: TEC error

Real: average error and driver

0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002

Average VTEC error

0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002

.7 [SFU

F10.7 estimated

Real: average error without driver estimation

0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002

Average VTEC error

0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002

.7 [SFU

F10.7 estimated

1 Summary

2 Problem Statement

3 GITM

4 EAKF

Conclusions

1 DART-GITM interface: This dissertation developed the frameworkfor data assimilation of satellite data into a space weather model. Inparticular, it introduced driver estimation and demonstrated thatdrivers need to be inflated differently from the model states.

2 Differential inflation: The driver estimate was inflated differentlyfrom the rest of the variables, that is its spread was set to beconstant to allow for continuous updating.

3 CHAMP density assimilation: The data assimilation techniquewas first demonstrated by using sparse thermospheric measurements.

4 GPS TEC assimilation: The interface was then modified to handlemore global (ionospheric) measurements coming from the GPSsatellites. It was found that the driver estimation in this more globalcase was not needed.

Future Work

1 F10.7 Localization: F10.7 estimate currently affects both theday-side and the night-side. It would be interesting to see the effectof localizing F10.7.

2 Slant TEC: Only vertical total electron content measurements wereconsidered so far. Implementing slant TEC would allow us to solve abroader class of problems.

3 Heating efficiency: Heating efficiency is a stronger driver for TEC,so estimating it using TEC data should be easier than estimatingF10.7.

4 Geomagnetic storms: This study only considered geomagneticquiet times, so performing data assimilation during geomagneticstorms is subject of future work.

Questions?

RCAC Review

Consider the multi-input, multi-output discrete-time system

x(k + 1) = Ax(k) + Bu(k) +D1w(k), (53)

y(k) = Cx(k) +Du(k) +D2w(k), (54)

z(k) = E1x(k) + E2u(k) + E0w(k). (55)

Our goal is to develop an adaptive controller that generates a control signal u that minimizes the performance z inthe presence of the exogenous signal w. For this presentation we consider SISO plants with no direct feedthroughand z(k) = y(k) in command following context (i.e. E1 = C, E2 = D = 0, E0 = D2 6= 0, D1 = 0). We

define Markov parameters as Hi4= CAi−1B for i > 0.

z(k) =n∑

−αiz(k − i) +n∑

βiu(k − i) +n∑

γiw(k − i), (56)

u(k)4=

nc∑i=1

Mi(k)u(k−i) +

nc∑i=1

Ni(k)y(k−i), u(k) = θ(k)φ(k), u(k) = θ(k)φ(k),

Z(k)4=

z(k − 1):

z(k − pc)

, U(k)4=

u(k − 1):

u(k − pc)

, U(k)4=

u(k − 1):

u(k − pc)

, (58)

Bzu4= [ 0d Hd × poly(NMPz) ], or Bzu = [ 0d Hd Hd+1 ], (59)

Z(θ(k), k)4= Z(k) − Bzu

(U(k) − U(k)

), (60)

where green highlighting represents known variables, yellow - measured variables, and red - variables to be found.

RCAC Review continued

We now consider the cost function

J(θ, k)4= Z

T(θ, k)Z(θ, k) + ζ(k)tr

[(θ−θ)T(θ−θ)

], (61)

where the positive scalar ζ(k) is the learning rate. Substituting (60) into (61), the cost function can be written asthe quadratic form

J(θ, k) =(vec θ

)TA(k) vec θ + B(k)T vec θ + C(k) , (62)

whereD(k)

pc∑i=1

(k − i)⊗ (BzuLi),

f(k)4= Z(k)− BzuU(k),

A(k)4= D

T(k)D(k) + ζ(k)Inclu(lu+ly),

B(k)4= 2D

T(k)f(k)− 2ζ(k)vec θ(k),

C(k)4= f(k)

Tf(k) + ζ(k)tr

(k)θ(k)]. (63)

Since A(k) is positive definite, J(θ, k) has the strict global minimizer

θ(k) = − 12

vec−1(A(k)−1B(k)). (64)

The controller gain update law is θ(k + 1) = θ(k).

Computational Study of RCAC Robustness

Consider the discrete-time system

G(z) =z − 1.4

(z − .5)(z − .6)(z − .7). (65)

With NMP zero location known exactlyRCAC achieves transient performance ofabout 1.6. Transient and steady stateperformances are defined as

ztr = maxk|z(k)|, (66)

zss = maxk=900:1000

|z(k)|. (67)

200 400 600 800 1000−2

Perform

200 400 600 800 1000

−0.5

Controlu(k)

200 400 600 800 1000

−0.5

Controller

-sθ(k)

k [steps]

Computational Study of RCAC Robustness continued

When location of the nonminimum phase zero is uncertain, transient performance canbecome unbounded

(a) Plant NMP−zero locations

locations

2 4 6 8

(b) Plant NMP−zero locations

locations

2 4 6 8

In this plot, the true nonminimum phase zero (x-axis) and zero estimate (y-axis) arevaried from 1.1 to 8.1. We conclude that RCAC is more robust to overestimating thelocation of the nonminimum phase zero than to underestimating it. Additionally, thecases with NMP zeros further out on the real axis result in greater stability marginsthan those with NMP zeros closer to 1.

We now generate 50 stable second order plants with random poles and NMP zero at 2and test RCAC with NMP zero estimates varying from 1.4 to 10.

2 3 4 5 6 7 8 9 100

Estimate of the nonminimum phase zero

For these 50 random plants, the stability of the closed-loop system is less sensitive tooverestimating the location of the NMP zero than it is to underestimating the locationof the NMP zero. However, none of the 50 plants have an upward margin in excess of10, which corresponds to 5 times the true value of the NMP zero.

We can extend these findings to plants with higher order and bigger relative degree.

Plant order

lative d

1.5 2 2.51.5 2 2.51.5 2 2.5

Estimated nonminimum−phase zero

1.5 2 2.51.5 2 2.5 0

ent of pla

with u

sed−

loop r

esponse

We find that robustness to the estimate of the NMP zero increases in rows from leftto right (with decreasing order), and in columns from bottom to top (with increasingrelative degree).

Convex Constraint on Pole Locations

The denominator coefficients of the controller (60) are given by

den(θ(k))4= [1 −M1 −M2 · · · −Mnc ]. (68)

We modify the problem of minimizing (62) by imposing a maximum singular valueconstraint on the companion-form matrix

M1 M2 . . . Mnc−1 Mnc

1 0 . . . 0 00 1 . . . 0 0...

.... . .

......

0 0 . . . 1 0

, (69)

σmax(K) ≤ γ, γ > 1. (70)

−1 −0.5 0 0.5 1

−0.5

Real Axis

Imagin

Convex Constraint on Pole Locations continued

Convex constraint improves transient performance

200 400 600 800 1000−5

Perform

200 400 600 800 1000−10

Controlu(k)

200 400 600 800 1000−2

Controller

-sθ(k)

k [steps]

200 400 600 800 1000−2

Perform

200 400 600 800 1000

−0.5

Controlu(k)

200 400 600 800 1000

−0.5

Controller

-sθ(k)

k [steps]

CC-RCAC

Convex Constraint on Pole Locations continued

Evolution of the controller poles is shown as a function of time in terms of color.CC-RCAC poles settle twice as fast.

−1 −0.5 0 0.5 1 1.5−1

−0.5

Imaginary

Real Axis

k[steps]

−1 −0.5 0 0.5 1 1.5−1

−0.5

Imaginary

Real Axis

k[steps]

CC-RCAC

Nonlinear Multilink Arm

Consider planar multilink arm

θ1•p1

θ2•p2

•q2•p3

7ıB1Z

···

θN•pN

3+m2l21

m2l1l22

cos(θ1 − θ2)m2l1l2

2cos(θ1 − θ2)

[ θ1θ2

[m2l1l2

2sin(θ1 − θ2)θ2

−m2l1l22

sin(θ1 − θ2)θ21

[c1 + c2 −c2−c2 c2

] [θ1θ2

[k1 + k2 −k2

−k2 k2

] [θ1θ2

Nonlinear Multilink Arm continued

Nonlinear system can be controlled for small command magnitudes and frequencies.

0 10 20 30 40 50−0.5

Perform

0 10 20 30 40 50

Controlu(k)

Time (sec)

RCAC performance at empiricallyfound maximum commandamplitude. Region plot for RC-MRAC, taken from [1]

[1] M. W. Isaacs, J. B. Hoagg, A. V. Morozov, and D. S. Bernstein, “A Numerical Study on Controlling a

Nonlinear Multilink Arm Using a Retrospective Cost Model Reference Adaptive Controller,” Proc. Conf. Dec.

Contr., pp. 8008–8013, Orlando, FL, December 2011.

Vertical Energy Equation

∂t+ ur

∂r+ (γ − 1)T

+∂ur∂r

cv ρ mnQ , (72)

Q = QEUV +QNO+QO+∂

((κc+κeddy)

mi + mnνin(v − u)2,

QEUV =∑s

[Ns(z) I∞(λ) e−sec(χ)

∑sNs(z)σ

as (λ)Hs(z)

], (73)

I∞(λ) = f(λ)

1 + a(λ)

F10.7 +⟨F10.7

⟩81d

, (74)

where equation (73) is a combination of equation (9.17)7 and notes from AOSS495. Equation (74) is an empirical model.8

7Schunk, R. W., and A. F. Nagy. “Ionospheres: Physics, Plasma Physics, andChemistry.” Cambridge University Press, 2004.

8Richards, P. G., J. A. Fennelly, and D. G. Torr, “EUVAC: A Solar EUV FluxModel for Aeronomic Calculations”, J. Geophys. Res., 99(A5), (1994)

EAKF Implementation notes

First, some notes on matrices that were not defined above.1 Fk comes from SVD of P−k = FkDkFTk .2 Gk is a square root of Dk, as in Gk = D

1/2k .

3 Uk comes from SVD of GTk FTk HTR−1HFkGk = UkJkUTk .

4 Bk is a square root of I + Jk, as in Bk = (In+m + Jk)−1/2.

Second, here are some notes

The EAKF procedure presented so far is not exactly what is implemented inDART.The procedure presented here is the closest EAKF representation to otherfilters, but does not incorporate localization and is not optimized forparallel implementation.The version that is implemented in DART is described in 9 and 10.

9Anderson, Jeffrey L. “A local least squares framework for ensemble filtering.”Monthly Weather Review 131.4 (2003): 634-642.

10Anderson, Jeffrey L., and Nancy Collins. “Scalable implementations of ensemblefilter algorithms for data assimilation.” Journal of Atmospheric and OceanicTechnology 24.8 (2007): 1452-1463

8 ResultsSimulated measurement above Ann ArborSimulated measurement at Subsolar PointSimulated measurement at CHAMP locationReal measurement from CHAMPReal measurement from CHAMP with advanced IMF and HPImodels

Simulated measurement above Ann Arbor

Measurement source: GITM truth simulation with an input ofF10.7 fixed at about 150. ρ at (82.5W, 42.5N, 394km) is

recorded with associated uncertainty (R) of 2.6× 10−12kg/m3.

Ensemble members: 20 GITM instances prespun for 2 days prior toDec 01 with F10.7 values coming from normal distribution∼ N(130, 25). F−10.7 is inflated using Pi = 49.

F+10.7F−10.7

F10.7 GITM Truth

GITM Ensemble

x−k x+k+1

ρ above Ann Arbor

EAKF estimates of ρ are brought within the uncertainty bounds of GITMtruth simulation.

Localization

The effect of measurement assimilation can be restricted to a region to avoid updatinguncorrelated states.11

Correlation function with horizontal cutoffof 30 is shown to the right and below,and vertical cutoff of 100km is shownbottom right.

11Gaspari, G., and S. E. Cohn. “Construction of correlation functions in two andthree dimensions.” Quarterly Journal of the Royal Meteorological Society 125.554(2006): 723-757.

ρ at subsolar point

Note, ρ at subsolar point does not vary too much since F10.7 is relativelyconstant. The localized AA measurement is tripled in intensity.

ρ at CHAMP location

The localized AA measurement is tripled in intensity.

ρ at CHAMP location RMSPE

We define RMS percentage error RMSPE4=

√(ρ−ρ)2√ρ2

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480

Time (hrs) since 2002−12−01 00:00:00 UTC

Simulated measurement above Ann Arbor Summary

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480

F10.7 estimate AA

So what we learned is that it is hard to get a good estimate of F10.7

based on measurement that is fixed in longitude.

Simulated measurement at Subsolar Point

Measurement source: ρ at subsolar point is recorded with associated

uncertainty (R) of 2.6× 10−12kg/m3.

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480

F10.7 estimate SP

Conclusion here is that ρ at subsolar point is more closely related to F10.7

than ρ at any point fixed in longitude (for example, Ann Arbor).

Simulated measurement at CHAMP location

Measurement source: ρ at CHAMP location is recorded with associated

uncertainty (R) of 2.6× 10−12kg/m3.

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480

(a) Hours since 00UT 01/12/2002

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480

(b) Hours since 00UT 01/12/2002

F10.7 estimate CL

We conclude that this case is better conditioned than AA or SP ones.

Real measurement from CHAMP

Measurement source: ρ from real CHAMP has associated uncertainty (Rk)

that varies, but has a mean of 2.6× 10−12kg/m3.

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 450

F10.7 estimate CR

We conclude that EAKF F10.7 estimate did not converge to thecommonly accepted value in order to compensate for model bias.

Real measurement from CHAMP with IMF and HPI

Measurement source: ρ from real CHAMP has associated uncertainty (Rk)

that varies, but has a mean of 2.6× 10−12kg/m3.

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480

GITM with F10.7 from NOAAGITM without EAKFEAKF posteriorPercent of observations used

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480

GITM with F10.7 from NOAAGITM without EAKFEAKF posterior

F10.7 estimate CR

We conclude that EAKF F10.7 estimate did not converge to thecommonly accepted value in order to compensate for model bias, evenmore advanced IMF and HPI models are used.

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