第六讲主自由度的选择和传感器布置

1Dalian

University

of

Technology主自由度的选择、传感器布置方法

和评价准则

(Choose

of

master

DOFs

in

Model

reduction

/

sensor

placement

methods

in

SHM)1.2.3.4.5.6.7.

OutlineReview

of

existing

sensor

placement

methodsProblem

formulation

for

sensor

placementConnection

between

MKE

and

EIFast

computation

of

Effective

Independencethrough

QR

downdatingReview

of

evaluation

criteria

for

sensorplacementExtended

Minmac

algorithm

and

its

connectionwith

EIAnovel

Load

Dependent

Sensor

placementmethod

2SHM

and

HumanStructureinstalled

withsensorsControl

center

–Human

brainSub-structure-armSensors-nervesStructural

Health

Monitoring

SystemInterrogation

System

Data

Storage

&Processing

UnitControl

CenterAdministrationRemote

MonitoringStructures

SensorsdeployedWindows

FieldComputerWindows

RemoteComputerContents

of

SHMHealth:

or

NotLoad

/

Environment

Parameters

Structures1.

Existence

of

Damage

2.

Location

of

Damage

3.

Severity

of

Damage

4.

Remaining

lifeMonitoring

•Acceleration•

Displacement

•

Pressure

•

Temperature

StructuralcharacteristicsSensors:

Embeddible

andAttachable

传感器布置方法的必要性1，结构理论模型与试验模型验证(模型缩聚)的要求；2，结构自由度众多和传感器测试设备有限性的矛盾；问题：1，主自由度选择不当缩聚不成功、不收敛于第一阶频率(作业)；2，美国

MIR

–

SHUTTLE

空间站因传感器位置布置不当损失数百万美元；Ref:

Zimmerman,

D.C.,

"Model

Validation

and

Verification

of

Large

and

Complex

SpaceStructures,"

Inverse

Problems

in

Engineering,2000:8(2):93-118.

67Mir

Structural

Dynamics

Experiment(MiSDE)

--

Boeing

North

America1.

Vibration

Testing

in

Space2.

Model

Correlation/Updating3.

Part

of

Risk

Mitigation

forInternational

Space

Station8Importance

of

sensor

placement

(SP)Bad

sensor

placement

yields

inadequate

spatial

resolution

of

modes.Boeing

engineers

spent

6

months

to

“tune”

model,

with

little

success.Lost:

3

Mio.

U.S.

$

because

of

bad

sensor

placement.Acknowledgement:

Prof.

D.C.

Zimmerman

(Uni.

of

Houston)

Model

validation

and

verification

of

largeand

complex

space

structures.

Inverse

Problems

In

Science

And

Engineering

(2000),

8(2):

93-118.910

Motivation

for

sensor

placement

researchThe

number

of

sensors

deployed

in

large

bridges.1.

TingKau

Bridge

(67)2.

Anhui

Tongling

Bridge

(116)3.

Qsingma

Bridge

(more

than

600

sensors)11

Motivation

for

sensor

placement

researchNecessity

for

sensor

placement

research

in

SHM.1.

Many

sensors

and

huge

amount

of

data2.

Measuring

under

operating

condition,

difficult

to

relocated3.

Quality

of

measured

data

are

crucial

to

the

success

of

SHMExamples

of

instrumented

bridges:

evolution

of

the

number

of

sensors

with

time

Ref:

New

Trends

in

Vibration

Based

Structural

Health

Monitoring,A.

Deraemaeker

and

K.

Worden

(Eds.),Springer,

Wien,

2010.

q

i

M

Ciq

i

M

Kiqi

M

Bou,

y

Problem

formulation

1

1

1

T

i

i

i

y

q

1,

what

is

the

least

number

of

sensors?

s

(no.of

sensor)

>=

m

(no.

of

modes)

实质是组合优化问题2,

where

should

these

sensors

be

placed

among

n

positions?Where

should

additional

sensors

be

placed,

in

other

differentpositions

or

as

redundant

sensors?3,

How

to

evaluate

the

effectiveness

of

these

methods?

12

y1

2

y

y3

yn

m

,

Q

tm

s

AE

QR13Connections

of

existing

SP

methods1.

Visual

Inspection

/

Modal

Kinetic

Energy

(MKE)2.

Drive

Point

Residue

(DPR)3.

Eigenvector

Component

Product

(ECP)4.

Guyan

Reduction

(Penny,

1994)5.

Effective

Independence

(EI)

(Kammer,

1991)6.

Flexibility

(Flanigan,

1992)7.

Genetic

Algorithm

(GA)

(Worden,

2001)8.

Modal

Norm

Index

(Gawronski,

1998)9.

Singular

Value

Decomposition

(SVD)10.

MinMAC

algorithm

(Carne,

1995)11.

QR

decomposition

(Link,

1996)Ref:

Li

D.S.

Fritzen

C.P.

and

Li

H.N.,

Extended

MinMAC

algorithm

and

comparison

of

sensorplacement

methods,

in

IMAC-XXVI,

Florida,

U.S.A,

February

4-7,

2008.

Paper

No.

78.

MKE

tm

M

tmDPR

tm

tm

tm

ECP

tm

tm

Ri

KGii

/MGii

s

USV

TT

Kmm

Kms

I

Ksm

Kmm

0

依次排除Q中最小的分量，每个分量对应一个DOF柔度14

Modal

Kinetic

Energy

method

(MKE)

The

method

ranks

all

candidate

sensor

positions

by

their

MKE

indices

as

follows,

n

s

1The

MKE

provides

a

rough

measure

of

the

dynamic

contribution

of

each

candidate

sensor

to

the

target

mode

shapes.

The

reason

to

adopt

MKE

resides

in

that

it

tells

which

DOFs

capture

most

of

the

relevant

dynamic

features

of

the

structure.

MKE

helps

to

select

those

sensor

positions

with

possible

large

amplitudes,

and

to

increase

the

signal

to

noise

ratio,

which

is

critical

in

harsh

and

noisy

circumstances.

H

H

Q

[0

]1

s

The

Effective

Independence

MethodObjectives:

FEM

validation

1:

the

measured

frequencies

comparable

with

FEM,

2:

the

measured

mode

shapes

independent.Observation

vector：Estimation

Covariance：Cramer-Rao

Covariance

lower

bound：ˆ

ˆP

E[(q

q)(q

q)T

]

Q

1ATT222

1

0

1

0

q

q

y(t)

H(q)

N

sq(t)

N

（1）（2）（3）i

n

n

A

Φ

s

i

s

i

1

i

1随着每个自由度的增加或删除，信息阵的信息也响应的增

加或者减少

15FE

s[

s

s]

1

s16The

EI

Computation

StepsA

–

positive

symmetric

matrix[A

I]

0TT

xxxxxxxx

n

j

1

x

*

(Bj)

x

（4）（5）（6）（9）

TA

(

s

)T(

s

)

[

s

]

[

s

]

G

(Bj)(Bj)T

jFE

[

s

]

[

s

]T

1（7）（8）1.

Row

summation.2.

Projector

diagonal.EI

的物理意义Ref:

Kammer

DC,

Sensor

placement

for

on-orbit

modal

identification

and

correlation

of

large

space

structures.

J

of

Guidance

and

Control

Dynamics,

1991,

(14):

251–259.

1718

Effective

Independence

method

(EI)

The

method

ranks

all

candidate

sensor

positions

by

their

EI

indices

as

follows,

TThe

EI

is

to

select

measurement

positions

that

make

the

mode

shapes

of

interest

as

linearly

independent

as

possible

while

containing

sufficient

information

about

the

target

modal

responses

in

the

measurements.

The

method

originates

from

estimation

theory

by

sensitivity

analysis

of

the

parameters

to

be

estimated,

and

then

it

arrives

at

the

maximization

of

the

Fisher

information

matrix.Φ

Φ

1MKE

diag(Φ1Φ1

)

Connection

between

MKE

and

EIViewpoint

of

Mode

Reduction:

First

iteration

of

EI,

y1

1q1

1)1

11TTED1

diag(Φ1

ΦTWhen

the

reduced

mode

shapes

are

re-orthonormalized,

T

T

T

T

19

diag(Φ

Φ

Φ

1Φ

)EDi

i

/(

i

)

i20Connection

between

MKE

and

EI

(cont.1)QR

decompostion:

QR

1

MKE

diag(ΦΦT)When

the

reduced

mode

shapes

are

re-orthonormalized,

QAspecial

case,

only

one

mode

shape

is

examined2MKEi

i22

2

ni

1MKE

=

EITTED21Connection

between

MKE

and

EI

(cont.2)

EI

requires

iteration

computations,

but

MKE

not.

EI

is

an

iterated

version

of

MKE

with

re-orthonormalized

mode

shapes.In

the

following

iterations

of

EI,

it

redistributesthe

modal

kinetic

energy

into

the

retaining

DOFsand

recomputed

their

MKE

index

for

the

reducedsystem

using

re-orthonormalized

mode

shapes.ED

Φs[Φs

Φs]

1ΦsTB

2m

n

2m

mnComparison

of

Method

A,

B

and

C22

33

TC

2m2n

2

m3

2mnConclusions:1.

Obviously

TA

>

TC2.

TB

>

TC

when

n

>

2

which

is

naturally

satisfied.22T

TED

diag(QQT)MethodA:

Row

summationMethod

B:

Projector

diagonalMethod

C:

QR

decomposition

TA

2m2n

8m3

2mnED

[ΦsΨ]

[ΦsΨ]T

λ

1Method

C

is

optimaln:

row

dimensionm:

column

dimension

QR

U

U

New

Method

D

through

QR

downdating

zT

Q1

zT

1

zT

zT

1

Q1

T

R

1

0

vT

vT

h

0

0

Q1

R1

Q1R1

R

h

0

1

Q1R1Mathematical

tricksHouseholder

transformation

+

Gram-Schmidt

processBasic

ideas:

first

row

transformed

into

a

unit

vector,

its

first

column

also

turned

into

a

unit

vector.Ref:

K.

Yoo,

and

H.

Park,

Accurate

downdating

of

a

modified

Gram-Schmidt

QR

decomposition.

BIT

Numerical

Mathematics,

36

(1996)

166-181.23

QR...

...Q1R1

Method

D:Steps:

1.

QR

downdating

algorithm:

needs

2m2+

10mn

flops

2.

Compute

the

norms

of

the

row

vectors:

2mn

flopsTotal

computation

flops

needed:

2

The

QR

downdating

is,

in

fact,

a

rank

one

modification

to

the

previous

QR

decomposition

with

fewer

operations

Much

faster

!New

EI

computation

method

D

zT

1

24253

Comparison

of

four

MethodsMethod

C:

QR

decomposition

TC

2m2n

2

m3

2mnMethod

D:

QR

downdating

TD

2m2

12mn26Connection

between

EI

and

QRDQRD

operates

on

a

transposed

mode

shape

matrix

and

finds

the

mostindependent

rows.

Since

the

diagonal

elements

of

the

upper

triangularmatrix

in

QR

decomposition

is

arranged

in

descending

order

of

theirabsolute

values,

the

first

m

linearly

independent

rows

selected

are

alsothose

rows

with

large

row

norms.QRD

selects

sensor

positions

according

to

their

row

norms

and

linearindependency

in

the

row

space,

whereas

EI

computes

QRdecomposition

in

the

column

space

of

the

modal

matrix.ED

diag(QQT)

[

q1

2,

q2

2,

,

qn

2]TEI:

QRQRD:

ΦT

AE

QR27Sub-conclusions

Computation

of

EI

with

QR

decomposition

is

proposed;Computation

of

EI

with

QR

downdating

is

developed;Comparison

of

four

methods

to

compute

EI

and

foundthat

Methods

C

and

D

with

QR

(/

downdating)

areconvenient

and

faster;Amatlab

program

with

QR

downdating

is

appended.MKEpq

pq

M

ps

sq28Current

evaluation

criteria1.

ModalAssurance

Criterion2.

Singular

Value

Decomposition

ratio3.

Measured

energy

per

mode4.

Fisher

Information

Matrix5.

Visulization

of

the

mode

shapes

Tj

k(

Tj

j)(

Tk

k)MAC

jk

σ1σmSVDratio

n

s

1FIM

ΦTΦ

Current

evaluation

criteria

for

SP

methods1,

Modal

Assurance

Criterion,Requires

that

the

measured

mode

shape

vectors

to

be

aslinearly

independent

as

possible,

the

off-diagonal

terms

ofthe

MAC

matrix

provides

such

a

measure.

Tj

k(

Tj

j)(

Tk

k)MAC

jk

Comments:1,

Mode

shapes

are

only

strictly

orthogonal

with

respect

to

massmatrix;2,

MAC

without

mass

weighting

is,

in

fact,

to

compare

the

direction

oftwo

vectors.3,

The

off-diagonal

terms

of

the

MAC

matrix

ranges

between

0

and

1.Related

SP

method:

MinMAC

algorithm;

29

Current

evaluation

criteria

for

SP

methods2,

Singular

value

decomposition

(SVD)

ratio,Uses

the

ratio

of

the

largest

to

the

smallest

singular

value

ofthe

reduced

mode

shape

matrix,

the

smaller,

the

better.Comments:1,

SVD

ratio

is

similar

to

the

MAC

in

terms

of

modal

orthogonality;2,

Generalized

inverse,

condition

of

modal

expansion

for

FEMupdating;3,

The

numerical

rank

of

the

observability

matrix.Related

SP

method:

to

minimize

the

SVD

ratio;

30

1

mSVDratio

sq

ps

pq

pq

MKE

M31Current

evaluation

criteria

for

SP

methods3,

Measured

energy

per

mode,Provides

a

rough

measure

of

the

dynamic

contribution

of

eachcandidate

sensor

to

the

target

mode

shapes.Comments:1,

Measurable

modes

capture

a

larger

part

of

the

total

kineticenergy

of

the

structure;2,

Energy

contained

in

the

measured

dofs

for

each

mode

shouldbe

a

significant

portion

of

that

mode

so

that

the

mode

to

besufficiently

identified

in

the

noise

environment;3,

Signal

to

noise

ratio

consideration.Related

SP

method:

Modal

Kinetic

Energy

methodns

1E

q

q

q

q

1

y

1

1

1

y

0

2

A

132

Current

evaluation

criteria

for

SP

methods4,

Fisher

Information

Matrix,Minimize

the

covariance

matrix

of

the

estimate

error

for

anefficient

unbiased

estimator

.Comments:2,

Use

MKE

as

a

first

step;

Related

SP

method:

Effective

Independence

(EI)

method;

Currently,

the

most

influential

method.

T

T

q

q

0

T1,

MaximizingA

will

result

in

the

best

state

estimate

ofq

;33Current

evaluation

criteria

for

SP

methods5,

Visulization

of

the

mode

shapes

criterionTest

engineers

have

to

first

visualize

the

mode

shape

vectorsidentified

from

modal

experiments

to

have

a

first

impression

ofthe

overall

motion

of

the

structure

under

consideration.

Thiscriterion

has

no

concrete

mathematical

formulations

as

theaforementioned

four

criteria.

It

depends

on

the

structure

andusually

the

points

in

the

frame

corner

or

middle

are

picked

up.Comments:1,

Effectively

avoid

the

clustering

of

sensor

positions;2,

Could

not

evitable

been

kept

from

subjective

bias;Related

SP

method:

Visual

Inspection

(MKE)

;First

step

of

MinMAC

algorithm.34

MinMAC

MaxSVMaxTRACE

Comp.

of

9

sensor

placement

methods

MKE

ECPMSSP

DPR

EI

QRD

Ladder

structure

and

its8th

mode

shapeAcknowledgement:Prof.

T.G.

Carne(Sandia

Lab.,

USA)MKE78,77,120,35,79,119,76,36,80,8,118,9ECP78,79,120,77,119,80,35,118,76,36,102,53MSSP78,120,77,79,35,119,76,36,80,118,75,37DPR27,26,25,28,24,29,23,22,30,54,55,53EI120,115,101,95,88,42,35,34,30,22,18,11QRD78,120,77,35,8,27,56,100,94,22,109,31MinMAC120,104,101,100,99,98,96,78,77,35,9,6MaxSV4,9,25,26,31,35,76,77,78,79,119,120MaxTRACE8,9,35,36,76,77,78,79,80,118,119,12035Sensor

positions

selected

by

9

methodsMACSVDratioAver.EnergyFIMVisualizationMKE0.97742320580.283583.4029+++ECP0.968053320.90.271193.2543+MSSP0.988994.6e50.280233.3627++DPR0.940391.1e60.106561.2787++EI0.8276425.8920.132731.5928+QRD0.8676835.3140.203762.4451--MinMAC0.74995623.50.211312.5357+/-MaxSV0.94943101.610.246842.9621-MaxTRACE0.92618152.570.372514.4701++36Comp.

of

evaluation

criteria37MinMAC

algorithm1,

Objectives

of

the

MinMAC

algorithmProposed

by

Carne

and

Dohrmann

aims

to

ensure

modalcorrespondence

between

analytical

and

experimental

mode

shapesthrough

minimizing

the

off-diagonal

elements

of

the

MAC

matrix.Use

an

intuition

set

of

dofs

to

adequately

covers

the

structure

andareas

of

special

interest

to

ensure

modal

visualization

besidescorrespondence.2,

Computation

stepsA.

Select

an

intuition

sensor

set.B.

Add

available

candidate

sensors

one

by

one,

and

selects

onethat

minimizes

the

maximum

off-diagonal

element

of

theMAC

matrix

at

each

step.C.

Repeats

the

second

step

by

adding

one

sensor

at

a

time

until

arequired

number

of

sensors

are

selected.38Extended

MinMAC

algorithm1,

Motivation

and

problem

of

traditional

MinMACThe

maximum

off-diagonal

term

is

not

monotonically

decreasingwith

number

of

sensors.Adirection

to

alleviate

the

problem:Combining

forward

and

backward

MinMAC

approaches.2,

Computation

stepsA...,

B…

the

same

as

original.

(Forward

approach

FS-MinMAC

)C.

Repeats

the

second

step

by

adding

one

sensor

at

a

time

until

a

a

preset

number

of

sensors

(s1

larger

than

required)

are

selected.D.

Exclude

one

sensor

a

time

from

(s1)

until

the

required

number

ofsensors

(s)

is

reached.

This

is

the

backward

sequential

MinMACapproach

(BS-MinMAC).E.

Compare

forward

and

backward

curves,

and

select

the

curve

with

a

smaller

value

at

point

s,

to

minimize

the

maximum

off-

diagonal

terms

of

the

MAC

matrix...Example

of

Extended

MinMAC

algorithmThe

I-40

Bridge

was

located

over

the

Rio

Grande

inAlbuquerque,

NewMexico.

There

are

in

total

13

accelerometers

used

along

the

length

of

thebridge,

for

a

total

of

26

responses.

The

dofs

in

horizontal

axis

ofFigs(following

slide)

are

numbered

as

1,

2,y,

13

for

candidate

sensorpositions

S1,

S2,y,

S13

and

14,

15,y,

26

for

N1,

N2,y,

N13,

respectively.

3940Extended

MinMAC

algorithm

A1TA1

A1

A2

A1TA1

a1Ta1

A1

A2

a1

a2

A2

A1

a2

a1

A2

A2

a2

a2

41Relationship

between

MinMAC

and

EIDalian

University

of

TechnologyFirst

a

special

case

:

1,

Only

two

mode

shapes

are

of

interest

.

A1A2

a1

a2

TMAC1

T

T

A2

A1

A2

A2

1

0

0

1

T

T

T

TT

TT

2,

Mode

shapes

are

already

orthonormalized.

a1,a2

is

the

row

with

the

smallest

Frobenius

norm,

and

is

to

be

deleted

according

to

EI

in

the

first

iterationFor

the

reduced

mode

shape

after

one

EI

iteration:

T

TBy

MinMAC

algorithm,

a1Ta2

is

to

be

minimized,

which

isequivalent

of

EI

to

minimize

the

row

Frobenius

norm,

a12

a22

MinMAC

=

EI

(for

this

case)University

Siegenthe

MinMAC

algorithm.MinMAC

≈

EIEI

is

equivalent

to

MinMAC

algorithm

in

the

global

sense.

42

Relationship

between

MinMAC

and

EI

(cont.1)

When

more

than

two

mode

shapes

are

considered:

2

m

m

2

p

1

p

1,q

1

p

qThe

first

term

on

the

right

is

the

jth

or

kth

MKE

(EI)

indices.

Under

thisformulation,

EI

tries

to

maximize

the

first

term

on

the

right

hand

side,whereas

MinMAC

aims

to

minimize

the

second

term.

Both

EI

andMinMAC

will

lead

to

similar

results.From

the

perspective

of

matrix

theory:

EI

tries

to

maximize

the

traceof

the

information

matrix:

T

max(tr(

T))

max(tr(

T

))

T

is

just

the

MAC

matrix.

Maximization

of

the

diagonal

terms

of

the

information

matrix

by

EI

leads

naturally

to

the

minimization

of

the

off-diagonal

terms

of

the

MAC

matrix

byMACE

q

q

q

q

1

y

1

1

1

y

0

2

A

143Implicit

assumptions

in

FIM

criterion1.

The

rationale:

smallest

variance

gives

best

solution2.

An

underlying

implicit

premise:

equally

unbiased

estimatorsThis

is

not

always

valid!

T

T

q

q

0

TGroup

1

with

5

positions:

yG1

[y1,y3,y5,y7,y9]qG1

G1

yG1Group

2

with

5

positions:

yG2

[y2,y4,y6,y8,y10]qG2

G2

yG2

A

simple

example10

candidate

sensor

positions

in

total

are

available,

5

among

of

10

will

be

finally

selected.y

[y1,y2,...,y10]TTT10

measurements:

q0

y

qG1

qG2

q0Questionable

to

compare

the

variances

of

different

least

squaresestimates

with

different

groups

of

data

in

sensor

placement

problem.

4445Solution

to

the

problemMean

squared

error

(MSE)

can

be

alternatively

used

as

ameasure

as

followsMSE

consists

of

two

parts,

one

is

the

variance

of

the

estimator,

andthe

other

is

biased

distance

squared.ˆ

ˆ

ˆ

ˆMSE(q)

E(q

q)2

Var(q)

(E(q)

q)2ˆ

ˆ

ˆ

ˆJgu

(qs

qOLS)T(qs

qOLS)ˆ

s

sˆ

qS

(

T

s)

1

TysqOLS

(

T

)

1

TyThe

estimation

method

with

the

objective

as

above

defined

is

namedRepresentative

Least

Squares

(RLS).46Demonstration

example

of

RLS

ri

(yi

y

ˆi)

(yi

Xi

β)

247ββS

βS

β

T

β

βˆy

X

e,

S

(XSTXS)

1XSTyS,

ˆ

OLS

(XTX)

1XTyTβ

ni

1

ni

12

ni

12JOLS(

ˆ)

RLS

compared

with

OLSRepresentative

Least

Squares

(RLS)

method:

JRLS(

ˆ

)

(

ˆ

ˆ

OLS)

W

(

ˆ

S

ˆ

OLS)Ordinary

Least

Squares

(OLS)

method:

(y

ˆS,i

y

ˆOLS,i)

r

ˆ

i48ˆ

β

β

β

T

β

β

β

β

T

β

β

ni

12

ni

12minimize

partial

response

residualsCase

2:

W

=

I

(identity

matrix)Euclidean

distance22β

β

β

T

β

β

β

βˆ

β

βJRLS(

ˆ

S)

(

ˆ

S

ˆ

OLS)(

ˆ

S

ˆ

OLS)

ˆ

S

ˆ

OLSy

y

X

(

ˆ

S

ˆ

OLS)

minimize

all

response

residualsβS

βS

β

T

β

β

Physical

meaning

of

RLS

JRLS(

ˆ

)

(

ˆ

ˆ

OLS)

W

(

ˆ

S

ˆ

OLS)Case

1:

W

=

var(

OLS)

2(XTX)

1

Mahalanobis

distance

JRLS(

ˆ

S)

(

ˆ

S

ˆ

OLS)

XTX(

ˆ

S

ˆ

OLS)

(Xˆ

S

Xˆ

OLS)(Xˆ

S

Xˆ

OLS)qA

2

F

2

A

2

4922

O(

2)

1f

2y

2

Aq(

)

q

2

Analysis

of

the

RLS

criterionMatrix

sensitivity

analysis

from

Golub

and

Van

LoanThe

change

of

q(

)comes

from

two

contribution

1.

one

is

the

change

in

the

design

matrix

2.

the

other

in

the

responses

y

1determining

the

estimate

with

a

subgroup

data

by

RLS

criterion.A

Q

A1

Q1

0

A2

A

A1

A

A

z*zR

R1

RTR

z*zT50

Connection

between

FIM

and

RLSQR

decomposition

A1

0

zT

R

z2T

R1

T

T

T1

T1RLS

criterion.1.

FIM:

R1

is

the

minimum

perturbation

among

other

alternatives,

and

its

condition

number

as

well.2.

RLS:

implemented

in

a

sub-optimal

sense,

i.e.,

to

seek

for

the

minimum

change

of

the

condition

number

of

the

design

matrix,

(

)

the

criterion

of

the

RLS

agrees

exactly

with

criterion

of

the

FIM.Conclusion:

the

FIM

criterion

can

be

regarded

as

a

special

case

of

theC

(n

s)!s!C100

1.6e2751ˆ

ˆ