Parameter Estimation in 9190 LTI SystemsLinear Parametrization

System model input u outputy scalarstate EIRX Axt

buyCtxwhere A C Run b C Rn CERNParameters O CA b c C 112 1112 e R

k h 2n parameters

uX Axtbuyctx y

Input output mapx Ct eatxco t eAH s

busy as

yet Exit

yet Feluco o E CA b c

Jlt Ftluco jo 8 11,51 cn

Jato tz fit yet2

tz Feluco e I Felucati 077not convex in 0not straightforward to compute gradientsneed to store uco g to compute tea

Alternative linear parametrizationfrequency domain ideas only applicableto linear systems

uX Axtbuy ex Y a Gls y

Gls et SI A b

Gls transfer fan of the systemGls 9p g degg _n i

degp n

b is bn s t t bistbo strictly properSn t aneSn I t t ai s ta o transfer for

Recall I AxtbuyExt du EIR

Gts 9157pts Clegg deg pen

in our case d oPope transfer fan

is Aat bu xltln lx.ltxnItDTiiltljE AijXjIt7tbiu1t i l n

g It FE cjxg.ltTake Laplace transform of both sides

ignore i c s

sXics AijXjls t bills c n

XIs x s Xnis T

XIs AXIS butsSI A XIs buts

when def SI A to det SI Ais the characteristic

X s SI A buts poly of A oleg n

s cTXlsctlsI A bUCs

Gls

Time domain representation

yet cTeAtxCo t Jotatesbucs ds

I

Y s bn is tbh s t t b stboSn t aneSn I t t a s ta o

U s

Smt ane s t t a stao Y s

bn sn t r t b stbo U s

Ignoring i c s we get an ODE of orderfor yet with ult as external input

s Ils y s

s Y s 7 y s

y t an y t ta y taoybn um t t b w bow

yw bn iu t b w't bou an ing a y aoy

yn

t OTV t linearparametrizationwhere Oi bn 1 bn z bi bo an i an 2 a ao

V ft win Ct n'Cti ult

yin HI g y Ct yetT

Before O CA b c Now D bn i bo.am ao

k n't Zn ke Zn

y'm o.TV V Cain ufu yin 3 y yJTenlarged input

y output

Recall ylthoultfor vital math g Ctk Late

g Out

Let's take Us sntXn isn tt n d stdoa stable polynomial in S all roots have

neg real parts

s Y s OTV slls Ns

Vs Smuts sU s UlsSn Cs sycsl yes T

Define2 s Sn yes

167 7

s Is Uls SVG bcslls lls

n l

sus t s z Y's 41st T

Take inverse Laplace transform2 t OT fit

where both 2 and f are signals that can beobtained from u and y by passing them throughstable linear filters

tics Ies s s g

Goes bn Isn t r t bistboU 7 Gold y Smtane Sn It ta stao

y y O bn 1 b bo tlinear filters n l a a o

v vZ f where 2 atf

Now if 0 is unknown we can use the derivedsignals 2 It Iott to estimate 0

Jt I Ect Ect 2 Elt _hotties E off ft fitted o

Jeff fit f It T rank 1 matrix

Now can estimate Oct using gradient descentit P 00210 Ctl

where f rt 20 fixed adaptation gain matrixm ft 0 to give Dithnle C Lo

2 IE Ot fitf It lotti Elek att

mlt s

Ertle Otflt E C L EEL

Ie E _Lz Ect Ect E It ITIL E 0515ft Ice T 18 o

r 05 18 8 2,00 18

AnalysisMTA _It 16h12

I I 0 parameter estimation error

I i I IIt I IOT II G T IITE

Ftl 8 Lz Fct Ift 72Iz f e TICE Ittf B o

T Itf Ift Ito

Fecit f It o exercise

Ict

e t f It

Gradient law I rife 18REI

Preview o I C La e CNLCSlow adaptationno accumulations oferrors

if I is PE to be def'd thenI o exponentially fast

Key tools I Later IPE UCO property of a certain

LTV systemIoannou Sun

Recall regret minimization

Lz Bcs Icsi Ils Is

iff f i'Ics ElstTds