Closed-loop optimal experiment design: solution via moment ...

Closed-loop optimal experiment design: solution via moment extension Gloverfest

Cambridge, 24 September, 2013 at the occasion of Keith Glover’s next real start in life

To my surf partner in Canberra Roland Hildebrand (Université Grenoble) and Michel Gevers (UCL, Louvain la Neuve)

1

Outline

✤

Optimal experiment design for closed-loop PE identification

✤

Parametrization of the design variables with the generalized moments

✤

Moment extension problem and central extension

✤

Solution of the closed-loop design problem via moment extension

✤

Conclusions

2

A two-slide course on Prediction The context: Prediction Error Identification

True system S : yt = G0 (z)ut + H0 (z)et

et : white noise

Model set M : {G(z, θ), H(z, θ), θ ∈ Dθ ⊂ Rd } Prediction error : εt (θ) = H −1 (z, θ)[(G0 (z) − G(z, θ))ut + vt ] Criterion : θˆN = arg minθ∈Dθ

1 N

�N

2 t=1 [εt (θ)]

S ∈ M : G(z, θ0 ) = G0 (z) and H(z, θ0 ) = H0 (z) N →∞ • θˆN −→ θ0 and

√ N →∞ N (θˆN − θ0 ) −→ N (0, Pθ )

• Pθ can be estimated from the data • θ0 belongs to an ellipsoid with probability α(χ): Uθ = {θ|N (θ − θˆN )T Pθ−1 (θ − θˆN ) < χ} 3

$OZD\VVXEMHFWWRFRQVWUDLQWV Optimal experiment design in closed loop ([DPSOHV

!π

−π

Φu (ω)dω ≤ α, RU

!π

−π

Φy (ω)dω ≤ α

&RQVLGHULGHQWLӾFDWLRQLQFORVHGORRSZLWK ut = −Cid (z)yt + rt  :HFRQVLGHUWKH -RLQWRSWLPDOH[SHULPHQWGHVLJQSUREOHP PLQ

Cid (z), Φr (ω)

J

ZKHUH J FDQEH ‡ TXDOLW\FULWHULRQ ([ J =

!π

−π

‡ RUHQHUJ\FULWHULRQ ([ J = $OZD\VVXEMHFWWRFRQVWUDLQWV ([DPSOHV

!π

−π

ˆ N (ω)))dω V ar(C(G

!π

Φu (ω)dω ≤ α, RU

−π

Φr (ω)dω

!π

−π

Φy (ω)dω ≤ α

,GHQWLӾFDWLRQLQRSHQORRS RULQFORVHGORRSZLWK ut = −Cid (z)yt + rt ‡ 2SHQORRSGHVLJQ

4

7KHUHIRUH

The experiment design variables

ˆ N (ω)} ≤ γ(ω) ∀ω V ar{G ! " jω jω 1 ∂G(e ) ∂G(e ) ∗ ⇔ tr our paper is written ( ) Pθ systems, ≤ 1 but ∀ω for Note: for MIMO N γ(ω) ∂θ ∂θ

pedagogical reasons this presentation is SISO ⇔ tr{W (ω)Pθ }

≤

1 ∀ω

,QFORVHGORRSZLWK u = −Cid (z)y + r  UHIHUHQFHVSHFWUXP Φr (ω) DQGFRQWUROOHU Cid (ejω ) RUHTXLYDOHQWO\ Φu (ω) DQG Φue (ω)

Φu (ω) = λ0 |(1 + Cid G0 )−1 Cid H0 |2 +|1 + Cid G0 |−2 Φr (ω),

Φue (ω) = −λ0 (1 + Cid G0 )−1 Cid H0 . Cid

=

Φr

=

−Φue (λ0 H0 + G0 Φue )−1

2 |1 + Cid G0 |2 (Φu − λ−1 |Φ | ). ue 0

5

Connection between ZKHUH whereMθ � Pθ−1 =

1 2πλ0

! 1 1 π � π jω ∗ ∗ , θjω + + Fe (e , θjω (ejω 0 )F 0 )dω F (e , θ (e , θ0 )dω e0 )F e e 2π 2π design and model quality −πvariables −π �

π

F (ejω , θ0 ) Φχ0 (ω) F ∗ (ejω , θ0 )dω

−π

F (z, θ) θ)= =[Fu[F (z, (z, θ) θ) Fe (z, F (z, Feθ)] (z, θ)] u where ∂G(z, θ) ∂H(z, θ) −1 ∂G(z, θ) ∂H(z, θ) � −1 (z, � Fu (z, θ) = H θ) , F (z, θ) = H (z, θ) −1 −1 e , Fe (z, θ) Fu (z, θ) = H (z, θ) =H ∂G(z, ∂H(z, θ) (z, θ)∂θ ∂θ θ) F (z, θ) � H −1 (z, θ) H −1 (z, θ) , ∂θ ∂θ Φχ0 (ω) �

�

∂θ � Φu (ω) Φue (ω) Φ∗ue (ω) λ0

∂θ

Mθ = DYHUDJHSHUVDPSOHLQIRUPDWLRQPDWUL[ Mθ = average per sample�information matrix: π 1 ∗ jω Mθ � Pθ−1 = Fu (e θ0 )FΦ (e , θ0 )Φu (ω)dω Mθ LVDԀQHLQWKHGHVLJQYDULDEOHV Φjω u ,DQG u ue 2πλ0 variables Mθ is a!ne in the design Φ and Φue −π u � π 1 0DLQLQJUHGLHQWVRIWKHQHZH[SHULPHQWGHVLJQUHVXOWV + Fe (ejω , θ0 )Fe∗ (ejω , θ0 )dω Main ingredients of the new2π experiment design results: −π

‡ 2EWDLQDӾQLWHGLPHQVLRQDOSDUDPHWUL]DWLRQRIWKHGHVLJQYDULDEOHV • Express the design criterion as a linear function of Mθ ‡ ([SUHVVWKHFULWHULRQDVDOLQHDUIXQFWLRQRIWKHVHGHVLJQYDULDEOHV where • Express the constraints as an LMI on Mθ ‡ ([SUHVVWKHFRQVWUDLQWVDVDQ/0, F (z, θ) = [Fu (z, θ) Fe (z, θ)] RQWKHVHGHVLJQYDULDEOHV • Obtain a finite dimensional parametrization of the design variables Fu (z, θ) = H −1 (z, θ)

∂G(z, θ) , ∂θ 6

Fe (z, θ) = H −1 (z, θ)

∂H(z, θ) ∂θ

Finite-dimensional parametrization of the design variables The partial correlation parametrization Define the 2 × 2 generalized moments: Define the 2 × 2� generalized moments: +π 1 � +π 1 jkω mk = 1 dω, k = −∞, . . . , ∞ 1jω 2 Φχ0 (ω)e jkω mk = 2π −π |d(ejω )| Φχ0 (ω)e dω, k = −∞, . . . , ∞ 2 2π −π |d(e )| where d(z) is a suitably chosen polynomial. where d(z) is a suitably chosen polynomial. �m l � • d(z) = m d z of degree m, m ≥ 0 l l l=0d z of • d(z) = l=0 degree m, m ≥ 0 l •• coe!cients arereal, real,obey obeyd d0�=�=0,0d, dm coe!cients ddl are �= �= 0 0 0

l

m

•• d(z) has all all roots rootsoutside outsideofofthe theclosed closed unit disk. d(z) has unit disk. T T . •• the moments m mkkare arereal realwith withmm = m k the moments = m . k −k−k

The d(z)isischosen chosensuch suchthat thatthe the criterion and The polynomial polynomial d(z) criterion and the dependonly onlyon ona afinite finitenumber number . ,nm the constraints constraints depend m0m , .0., .. ,.m . n. The optimization optimization is . ,. . ,. m The is then thenperformed performedw.r.t. w.r.t.mm . , nmn 0, 0

••

7

‡ &DUDWKpRGRU\)HMHUWKHRUHPDOORZVH[WHQVLRQRI {m0 , . . . , mn } WRLQӾQLWHVHTXHQFHFRUUHVSRQGLQJWRDSRVLWLYHVSHFWUXP min J EXWRQO\LIWKHUHDUHQRFRQVWUDLQWVRQWKLVH[WHQVLRQ Objective: subject to constraints Cid (z), Φr (ω)

Assumption: inear criterion in $VVXPSWLRQV

‡ &ULWHULRQFDQEHZULWWHQDV

−1 Pθ

J (m0 , m1 , . . . , mn , x1 , . . . , xN ) =

n $

"Ck , mk # +

k=0

N $

c l xl

l=1

‡ &RQVWUDLQWVFDQEHZULWWHQDV

Pθ ) ≤ γ I Pθ−1

�

A(m0 , m1 , . . . , mn , x1 , . . . , xN ) $ 0 ZKHUH Ck DUHӾ[HGPDWULFHV cl DUHӾ[HGUHDOV

≥ "A, 0 B# = tr(AB T ) DQG x

1 , . . . , xN

n variable Φχ0

DUHDX[LOLDU\YDULDEOHV

Integral criteria and constraints can always be written this way

8

Φχ 0 =

−λ0 H0∗ M ∗ (U + M Q)∗

λ0

using the correlation approach 2EVHUYHWKDW ΦχSolution 0 LVDԀQHLQ Φu DQG Q 1RZXVHDӾQLWHGLPHQVLRQDOSDUDPHWUL]DWLRQIRU Φu DQG Q ‡ 2SWLPL]DWLRQRYHU Φu (ω), Φue (ω) LVUHSODFHGE\DVHPLGHӾQLWHSURJUDP RSWLPL]DWLRQRYHU DӾQLWHVHW {m0 , . . . , mn } ‡ 7KLVRSWLPDOVHTXHQFH {m0 , . . . , mn } FDQJHQHUDWHDQLQӾQLWHVHWRI SRVVLEOHH[WHQVLRQV mn+1 , mn+2 , . . .  DQGKHQFHDQLQӾQLWHVHWRIVSHFWUD ‡ &DUDWKpRGRU\)HMHUWKHRUHP {m0 , . . . , mn } LVH[WHQGDEOH WRWKHPRPHQWVRIDSRZHUVSHFWUXPLӽ



m0

   m1 Tn =      

mn

mT 1 m0   mn−1







mT n−1

mT n





mT n−2

mT n−1







m1

m0



   "0   

‡ %XWUHVXOWKROGVRQO\LIQRFRQVWUDLQWVDUHLPSRVHGRQWKHH[WHQVLRQ 7KLVSUHYHQWHGWKHXVHRIWKHPRPHQWPHWKRGIRUFORVHGORRSH[SGHVLJQ

9 ‡ &DUDWKpRGRU\)HMHUWKHRUHPDOORZVH[WHQVLRQRI {m0 , . . . , mn }

The extension problem of closed-loop experiment design ‡ &DUDWKpRGRU\)HMHUWKHRUHPDOORZVH[WHQVLRQRI {m0 , . . . , mn } WRLQӾQLWHVHTXHQFHFRUUHVSRQGLQJWRDSRVLWLYHVSHFWUXP EXWRQO\LIWKHUHDUHQRFRQVWUDLQWVRQWKLVH[WHQVLRQ ‡ %XW LQRXUH[SHULPHQWGHVLJQSUREOHP WKHUHDUHFRQVWUDLQWVRQ Φχ0 (ω) +HQFHWKHUHDUHFRQVWUDLQWVRQWKHDGPLVVLEOHH[WHQVLRQVRI {m0 , . . . , mn } ‡ ,QSDUWLFXODU

mk,12 Φue mk,22

=

1 2π

!

+π −π

Φue (ω) |d(ejω )|

jkω e dω ZKHUH 2

= −λ0 (1 + Cid G0 )−1 Cid H0 PXVWEHVWDEOH ! +π 1 λ0 jkω = e dω 2π −π |d(ejω )|2

$VVXPSWLRQV ‡ &ULWHULRQFDQEHZULWWHQDV

J (m0 , m1 , . . . , mn , x1 , . . . , xN ) = 10

n "

"Ck , mk # +

k=0

N " l=1

c l xl

Moment Extensions 1. Extension by one &RQVLGHU {m0 , . . . , mn } ZLWK Tn ! 0 :HӾUVWSDUDPHWHUL]HWKHVHWRIDOOH[WHQVLRQVRIWKHӾQLWHVHTXHQFH m0 , . . . , mn E\ RQHDGGLWLRQDOPDWUL[ mn+1 VXFKWKDW Tn+1 ! 0    T mn m1   †    mn+1 =    Tn−1   

mn m1     T 1/2  T  1/2 mn mn m1 m1       †       †      + m0 −    Tn−1    ∆n+1 m0 −    Tn−1     m1 m1 mn mn

† ZLWK ∆n+1 ∈ R2×2 VDWLVI\LQJ σPD[ (∆n+1 ) ≤ 1 DQG Tn−1 = SVHXGRLQYHUVHRI Tn−1 

∆n+1 LVFDOOHG 9HUEOXQVN\FRHԀFLHQW HTXLYDOHQWWRD6FKXURU6]HJ|FRHԀFLHQW 7KLVSDUDPHWUL]HVWKHVHWRIDOO Tn+1 ! 0

Keith gave us a key technical hint for the proof of this result ! ‡ 2QHFDQFRQVWUXFWVXFFHVVLYHH[WHQVLRQV mn+1 , mn+2 , . . . , mn+k E\UHSHDWLQJWKHSURFHGXUH

11

Keith’s restaurant Theorem:

12

∆n+1 LVFDOOHG 9HUEOXQVN\FRHԀFLHQW HTXLYDOHQWWRD6FKXURU6]HJ|FRHԀFLHQW

7KLVSDUDPHWUL]HVWKHVHWRIDOO Tn+1 !and 0 2. Successive extensions

infinite extensions

‡ 2QHFDQFRQVWUXFWVXFFHVVLYHH[WHQVLRQV mn+1 , mn+2 , . . . , mn+k E\UHSHDWLQJWKHSURFHGXUH  &KRRVHDFRQWUDFWLYHPDWUL[ ∆n+1 DQGFDOFXODWH mn+1   7KHQFKRRVH ∆n+2 DQGFRPSXWH mn+2  LWGHSHQGVDOVRRQ ∆n+1 YLD mn+1   $QGVRRQXQWLOWKHӾQDOFKRLFHRI ∆n+k ZKLFKGHWHUPLQHV mn+k  ‡ $QLQӾQLWHH[WHQVLRQLVGHWHUPLQHGE\DQLQӾQLWHVHTXHQFH ∆n+1 , ∆n+2 , . . . ‡ 7KHVHWRIDOOLQӾQLWHH[WHQVLRQVLVSDUDPHWUL]HGE\DOOVXFKVHTXHQFHV ‡ ,IWKHEORFN7|SOLW]PDWULFHV Tk DUHGHJHQHUDWH GLӽHUHQWFKRLFHV RIWKH9HUEOXQVN\PDWULFHV ∆k FDQOHDGWRWKHVDPHH[WHQVLRQ

13

Central Extension ‡ 7KH FHQWUDOH[WHQVLRQ LVREWDLQHGE\WDNLQJ ∆k = 0 IRUDOO k ≥ n + 1 ‡ ,QWKHUHJXODUFDVHLH Tn " 0 LWKDVD FORVHGIRUPVROXWLRQ LWZDVDFWXDOO\GHӾQHGWKLVZD\LQ*HQLQ .DPS 'HOVDUWH  ‡ 'HӾQHWKH 2 × 2(n + 1) SRO\QRPLDOPDWUL[

! Un (z) = z n I2

z n−1 I2

···

I2

WKH 2 × 2 SRO\QRPLDOPDWUL[

An (z) =

Un (z)Tn−1 UnT (0)

=

n #

"

Akn z k .

k=0

DQGWKHVSHFWUXP

Φ(ω) = An (ejω )−∗ An (0)An (ejω )−1 7KHQ Φ(ω) GHӾQHVWKHFHQWUDOH[WHQVLRQYLD mk =

1 2π

$ +π −π

Φ(ω)ejkω dω.

 :HVKRZWKDWLI {m0 , . . . , mn } VDWLVӾHVWKHFRQVWUDLQWVDQG Tn ≥ 0 14 WKHQ ∃{mn+1 , mn+2 , . . .} ZLWK Tn+k ≥ 0 ∀k > 0

Φ(ω) = An (ejω )−∗ An (0)An (ejω )−1

Moment extensions for closed-loop experiment design 7KHQ Φ(ω) GHӾQHVWKHFHQWUDOH[WHQVLRQYLD mk =

Back to closed-loop experiment design

1 2π

$ +π

5HPHPEHUWKHVSHFLDOIRUPRIWKHPRPHQWV & ' % +π 1 1 Φu (ω) Φue (ω) jkω mk = e dω ∗ jω 2 Φ (ω) λ 0 2π −π |d(e )| ue ( )* + Φχ0 (ω)

ZKHUH fue (ejω ) ! Φue (ω) PXVWEHVWDEOH

7KLVLPSOLHVWKHIROORZLQJFRQVWUDLQWVRQWKHPRPHQWV ‡ Tk $ 0 IRUDOO k ≥ 0 $ +π 1 0 ‡ mk,22 = 2π −π |d(eλjω ejkω dω IRUDOO k ∈ Z )|2 ‡ mk,12 =

1 2π

$ +π

fue (ejω ) jkω 1 e −π d(ejω ) d(e−jω )

15

dω IRUDOO k ∈ Z

−π

Φ(ω)ejkω

Moment extensions for closed-loop experiment design

Result 1 $QLQӾQLWHVHTXHQFH m0 , . . . , mn , . . . LVJHQHUDWHGE\ # " ! +π 1 1 Φu (ω) Φue (ω) jkω mk = e dω ∗ jω 2 Φ (ω) λ 0 2π −π |d(e )| ue ,) $1' 21/< ,)

 Tk ! 0 IRUDOO k ≥ 0 $ +π 1 jkω 0  mk,22 = 2π −π |d(eλjω e dω IRUDOO k ∈ Z 2 )| %s  i=0 di mk−i,21 = 0 IRUDOO k > 0

ZKHUH s = deg(d(z))

 :HVKRZWKDWLI {m0 , . . . , mn } VDWLVӾHVWKHFRQVWUDLQWVDQG Tn ≥ 0 WKHQ ∃{mn+1 , mn+2 , . . .} ZLWK Tn+k ≥ 0 ∀k > 0 DQGWKDWDOVRVDWLVI\WKHFRQVWUDLQWV 7KLV\LHOGVDYDOLGRSWLPDO Φχ0  .H\WHFKQLFDOWRRO 7KHSDUWLDOSRVLWLYHGHӾQLWHPDWUL[FRPSOHWLRQWKH 16

 Tk ! 0 IRUDOO k ≥ 0 $ +π 1 λ0 jkω  mk,22extensions = 2π −π |d(e dω IRUDOOexperiment k∈Z Moment for jω )|closed-loop 2e 

%s

i=0

design

di mk−i,21 = 0 IRUDOO k > 0

Result 2 deg(d(z)) ZKHUH s =

&RQVLGHUD ӾQLWHVHTXHQFH (m0 , . . . , mn ) WKDWREH\V  Tn ! 0  mk,22 = 

%s

i=0

1 2π

$ +π

λ0 ejkω −π |d(ejω )|2

dω IRU k = 0, . . . , n

di mk−i,21 = 0 IRU k = 1, . . . , n

7KHQWKH FHQWUDOH[WHQVLRQ RI (m0 , . . . , mn ) VDWLVӾHV WKHVDPHFRQGLWLRQVIRUDOO k ≥ n

 :HVKRZWKDWLI {m0 , . . . , mn } VDWLVӾHVWKHFRQVWUDLQWVDQG Tn ≥ 0 WKHQ ∃{mn+1 , mn+2 , . . .} ZLWK Tn+k ≥ 0 ∀k > 0 DQGWKDWDOVRVDWLVI\WKHFRQVWUDLQWV 7KLV\LHOGVDYDOLGRSWLPDO Φχ0 

.H\WHFKQLFDOWRRO 7KHSDUWLDOSRVLWLYHGHӾQLWHPDWUL[FRPSOHWLRQWKHR 17



s i=0

di mk−i,21 = 0 IRU k = 1, . . . , n

Moment extensions for closed-loop experiment design 7KHQWKH FHQWUDOH[WHQVLRQ RI (m0 , . . . , mn ) VDWLVӾHV

WKHVDPHFRQGLWLRQVIRUDOO k ≥ n Result 3 $ ӾQLWHVHTXHQFH (m0 , . . . , mn ) LVH[WHQGDEOHWR DQ LQӾQLWHVHTXHQFH (m0 , . . . , mn , . . .) VDWLVI\LQJ  Tk ! 0 IRUDOO k ≥ 0 $ +π 1 jkω 0  mk,22 = 2π −π |d(eλjω e dω IRUDOO k ∈ Z 2 )| 

%s

i=0

di mk−i,21 = 0 IRUDOO k > 0

,) $1' 21/< ,) WKHӾQLWHVHTXHQFHVDWLVӾHV  Tn ! 0  mk,22 = 

%s

i=0

1 2π

$ +π

λ0 ejkω −π |d(ejω )|2

dω IRU k = 0, . . . , n

di mk−i,21 = 0 IRU k = 1, . . . , n

18

Closed-loop experiment design: explicit solution in the regular case

Result 4

/HW (m0 , . . . , mn ) VDWLVI\  Tn ! 0  mk,22 = 

"s

i=0

1 2π

! +π

λ0 ejkω −π |d(ejω )|2

dω IRU k = 0, . . . , n

di mk−i,21 = 0 IRU k = 1, . . . , n

'HӾQH

# ‡ Un (z) ! z n I2

z n−1 I2

···

‡ An (z) ! Un (z)Tn−1 UnT (0) =

I2 "n

$

k=0

Akn z k

‡ Φχ0 (ω) ! |d(ejω )|2 · An (ejω )−∗ An (0)An (ejω )−1 7KHQ Φχ0 (ω) KDVWKHUHTXLUHGIRUP % & Φu (ω) Φue (ω) Φχ0 (ω) = Φ∗ue (ω) λ0 ZLWK Φue VWDEOH DQGLWUHSURGXFHV (m0 , . . . , mn ) /HW (mis . . , mcentral 0 , .the n ) VDWLVI\extension solution This  Tn ! 0 1

! +π

19 λ0

jkω

Closed-loop experiment design: parametrization of all feasible extensions &RQVLGHUDӾQLWHRSWLPDOVHTXHQFHVDWLVI\LQJ  Tn ! 0  mk,22 = 

"s

i=0

1 2π

! +π

λ0 ejkω −π |d(ejω )|2

dω IRU k = 0, . . . , n

di mk−i,21 = 0 IRU k = 1, . . . , n

:HFDQFRQVWUXFWWKHVHWRIDOO mn+1 VDWLVI\LQJWKHDERYHFRQVWUDLQWV

ˆ n+1  ZLWK n UHSODFHGE\ n + 1 XVLQJD UHVWULFWHG9HUEOXQVN\SDUDPHWHU ∆ ˆ n+1 , ∆ ˆ n+2 , . . . :HWKHQSURFHHGVWHSE\VWHS ∆ 6HHSDSHUIRUGHWDLOV 7KHGHVLJQSUREOHPLVH[SUHVVHGDVD VHPLGHӾQLWHSURJUDP # n % N $ $ PLQ "Ck , mk # + ck xk mk ,xk

k=0

k=1

20

ˆ n+1  n UHSODFHGE\ n + 1 XVLQJDUHVWULFWHG6FKXUSDUDPHWHU ZLWK n UHSODFHGE\ n ZLWK + 1 XVLQJDUHVWULFWHG6FKXUSDUDPHWHU

∆

ˆ n+1 , ∆ ˆ n+2 , . . . E\VWHS ∆

Closed-loop design: ˆ n+1experiment ˆ n+2 , . . . ˆ n+1 , ∆ ˆ n+2 , . . . :HWKHQSURFHHGVWHSE\VWHS ∆ ,∆ :HWKHQSURFHHGVWHSE\VWHS ∆ 6HHSDSHUIRUGHWDLOV

solution algorithm

6HHSDSHUIRUGHWDLOV H[SUHVVHGDVD VHPLGHӾQLWHSURJUDP 7KHGHVLJQSUREOHPLVH[SUHVVHGDVD VHPLGHӾQLWHSURJUDP % 7KHGHVLJQSUREOHPLVH[SUHVVHGDVD VHPLGHӾQLWHSURJUDP N # n % $ N % ck xk PLQ $ "C , m # + $ c # k# + n N k k k xk $ $ mk ,xk k=1 k=0 k=1 PLQ "Ck , mk # + ck xk mk ,xk k=0 k=1 VWUDLQWV ZLWKUHVSHFWWRWKH FRQVWUDLQWV

, mn ) 

ZLWKUHVSHFWWRWKH FRQVWUDLQWV A(m0 , m1 , . . . , mn , x1 , x2 , . . .A(m , xN0), m !1 ,0,. . . , mn , x1 , x2 , . . . , xN ) ! 0, & +π & +π 1 λ0 Ip jkω A(m0 , m1 , . . . , m 1 λ0 jkω m = e dω, k = 0, . . . , n, k,22 mk,22 = e dω, k = 0, . . . , n, jω )|2 & +π 2π |d(e −π 2π −π |d(ejω )|2 1 λ0 I s m = $ k,22 s j $ −π |d(e di mk−i,21 = 0, k = 1, .2π . . , n, di mk−i,21 = 0, k = 1, . .i=0 . , n, s $ i=0   di mk−i,21 =  T T  m m m 0 Tn ! 0 1 n  i=0   m1 Tn =      n ) 7KLV\LHOGV (m0 , m1 , . . . , m mn 21

m0  

mn−1















m0

mT n−1 

   ! 0,   

T

Closed-loop experiment design: solution algorithm (cont’d) ‡ ,I Tn ! 0

H[SOLFLWIRUPXODIRU Φχ0 

⇒

Φχ0 (ω) = |d(ejω )|2 · An (ejω )−∗ An (0)An (ejω )−1 ! " Φu (ω) Φue (ω) = Φ∗ue (ω) λ0 ‡ ,I Tn # 0 EXWVLQJXODU DOWHUQDWLYHIRUPXOD

XVLQJWKH&DUDWKpRGRU\IXQFWLRQVHHSDSHU 

F (z) =

1 2π

#

π −π

ejω + z ejω

−z

Φχ0 (ω)dω

Finally: Cid Φr

= −Φue (λ0 H0 + G0 Φue )−1

2 = |1 + Cid G0 |2 (Φu − λ−1 0 |Φue | ).

22

Conclusions

★ Optimal experiment design is reformulated as a semi-definite program ★ Optimal solution = finite set of moments ★ Carathéodory extension: yes, but there are constraints ★ Central extension: always satisfies the constraints ★ We have parametrized all extensions that satisfy the constraints ★ Our method yields an explicit solution of the joint design problem

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Ceci n’est pas Keith Glover

Keith Glover Playwright, actor, musician, and director of many plays: • Dancing On Moonlight, produced by the New York Shakespeare Festival • Thunder Knocking On The Door, winner of the Osborn Award given by the American Critic's Association • In Walks Ed, nominated for the Pulitzer Prize. On film, Keith Glover appeared in Jacknife with Robert DeNiro.