1-s2.0-S095915241300111X-main资源-CSDN文库

稳健估计

需积分: 15 195 浏览量 2014-05-22 19:16:31 上传评论收藏 808KB PDF 举报

资源推荐

资源详情

资源评论

Journal

Process

Control

23 (2013) 1555–

1561

Contents

lists

available

ScienceDirect

Journal

Process

Control

nal

homep

age:

www.elsevier.com/locate/jprocont

Short

communication

Robust

derivative-free

Kalman

ﬁlter

based

Huber’s

M-estimation

methodology

Lubin

Chang

a,∗

Baiqing

Guobin

Chang

a,b

Department

Navigation

Engineering,

Naval

University

Engineering,

Wuhan,

People’s

Republic

China

Tianjin

Institute

Hydrographic

Surveying

and

Charting,

Tianjin

300000,

People’s

Republic

China

Article

history:

Received

November

2012

Received

revised

form

April

2013

Accepted

May

2013

Available online 28 June 2013

Keywords:

Huber’s

M-estimation

Nonlinear

ﬁltering

Robust

regression

Unscented

Kalman

ﬁlter

this

study,

discrete-time

robust

nonlinear

ﬁltering

algorithm

proposed

deal

with

the

contami-

nated

Gaussian

noise

the

measurement,

which

based

robust

modiﬁcation

the

derivative-free

Kalman

ﬁlter.

interpreting

the

Kalman

type

ﬁlter

(KTF)

the

recursive

Bayesian

approximation,

the

innovation

reformulated

capitalizing

the

Huber’s

M-estimation

methodology.

The

proposed

algo-

rithm

achieves

not

only

the

robustness

the

M-estimation

but

also

the

accuracy

and

ﬂexibility

the

derivative-free

Kalman

ﬁlter

for

the

nonlinear

problems.

The

reliability

and

accuracy

the

proposed

algorithm

are

tested

the

Univariate

Nonstationary

Growth

Model.

Introduction

The

Kalman

ﬁlter

(KF)

[1]

concerned

with

estimation

the

dynamic

state

from

noisy

measurements

the

class

estima-

tion

problems

where

the

dynamic

and

measurement

processes

can

approximated

linear

Gaussian

state

space

models.

Unfor-

tunately,

the

performance

the

can

severely

degraded

when

the

distribution

the

true

noise

deviates

from

the

assumed

Gaussian

model,

often

being

characterized

heavier

tails

and

gen-

erating

high-intensity

noise

realizations,

named

outliers

[2].

The

contaminated

Gaussian

noise

and

outliers

arise

naturally

many

areas

engineering.

Heavy

tailed

densities

occur

applications

glint

noise,

air

turbulence,

and

asset

returns

[3].

Outliers

can

appear

due

clutter,

glint

noise,

and

model

mismatch

caused

e.g.,

linearization

[4].

Therefore,

there

appears

consider-

able

motivation

for

considering

ﬁlters

that

are

robust

perform

fairly

well

non-Gaussian

environments.

handle

the

non-Gaussian

noise,

many

methods

have

been

proposed.

minimizing

the

worst-case

estimation

error

averaged

over

all

samples,

the

∞

based

can

used

treat

model-

ing

errors

and

uncertainties

unknown-but-bounded

noise

[5,6].

However,

breaks

down

the

presence

randomly

occurring

outliers

since

the

design

matrices

the

∞

ﬁlter,

just

the

covariance

matrix

the

classical

KF,

cannot

accommodate

well

the

outliers

induced

the

thick

tails

noise

distribution

[13].

∗

Corresponding

author.

E-mail

address:

changlubin@163.com

(L.

Chang).

The

Gaussian

sum

methodology

can

also

yield

robust

estimate,

but

extreme

cost

computation.

Speciﬁcally,

the

number

terms

kept

the

Gaussian

sum

grows

exponentially

with

time

[7].

general

and

versatile

method

the

particle

ﬁltering

(PF)

which

approximates

the

minimum

variance

estimate

using

stochastic

simulation

methods.

However,

the

also

very

compu-

tationally

intensive

since

requires

Monte

Carlo

integration

[8,9].

addition,

typically

requires

delicate

tuning

proposal

den-

sities

improve

its

convergence

rate.

The

extension

the

concept

Huber’s

M-estimation

method-

ology

the

problem

robust

Kalman

ﬁltering

has

been

widely

researched

and

emphasized

over

the

other

ones,

since

moti-

vated

the

maximum

likelihood

estimation,

which

makes

natural

and

rather

easy

[10–12].

The

Huber’s

M-estimation

methodology

essentially

based

modifying

the

quadratic

cost

function

the

robust

convex

Huber

-function,

which

exhibits

-norm

properties

for

small

residuals

and

-norm

prop-

erties

for

large

residuals.

Possessing

moderate

breakdown

point

(BP),

high

efﬁciency,

and

manageable

computational

complexity,

the

Huber’s

M-estimation

methodology

still

subject

consider-

able

research

interest

recent

years

[13–17].

However,

most

the

work

limited

the

linear

case.

Although

these

methods

can

directly

extended

the

nonlinear

problems

making

use

the

extended

Kalman

ﬁlter

(EKF),

the

crude

approximation

and

the

cumbersome

derivation

and

evaluation

Jacobian

matrices

the

EKF

may

degrade

their

performance.

recent

years,

there

has

been

renewed

interest

the

topic

nonlinear

ﬁltering

with

the

discovery

the

“sigma-point”

Kalman

ﬁlters

(SPKF)

[18].

The

various

sigma

point

algorithms

differ

from

0959-1524/$

–

see

front

matter ©

http://dx.doi.org/10.1016/j.jprocont.2013.05.004

1556 L.

Chang

al.

Journal

Process

Control

23 (2013) 1555–

1561

each

other

their

choice

the

sigma

points,

for

instance

the

unscented

Kalman

ﬁlter

(UKF)

[19–22],

the

divided

difference

ﬁl-

ter

(DDF)

[23]

and

on.

Eliminating

the

cumbersome

derivation

and

evaluation

Jacobian

matrices,

these

ﬁlters

are

much

easier

implement

and

perform

better

than

the

EKF.

Combining

the

M-

estimation

methodology

with

these

derivative-free

ﬁlters

has

also

been

researched

[16,17].

However,

these

methods

are

all

based

the

concept

statistical

linearization

[24]

and

haven’t

made

full

use

the

advantages

these

derivative-free

ﬁlters.

this

paper,

shall

focus

deriving

the

robust

derivative-free

using

Huber’s

M-estimation

methodology

without

linearization

statistical

linearization.

Basing

the

recursive

Bayesian

interpre-

tation

the

KF,

point

out

that

the

M-estimation

methodology

used

modify

the

quadratic

cost

function

only

affects

the

innovation.

this

respect,

any

nonlinear

Kalman

type

ﬁlter

(KTF)

can

robustiﬁed

modifying

the

innovation

using

M-estimation

methodology.

Then

the

UKF,

taken

representation

the

derivative-free

ﬁlters,

robustiﬁed

using

the

proposed

methodol-

ogy.

The

proposed

robust

derivative-free

algorithm:

(i)

can

handle

the

contaminated

Gaussian

noise

and

outliers

the

measurement;

(ii)

exhibits

the

accuracy

and

ﬂexibility

the

UKF

for

the

nonlinear

problems;

(iii)

performs

well

under

nominal

conditions

(i.e.,

with

contaminated

Gaussian

noise

and

outliers

present);

(iv)

out-

performs

the

conventional

linear

statistical

linear

based

robust

KTF.

The

rest

the

paper

organized

follows.

Section

presents

the

Bayesian

recursive

interpretation

KF.

The

method

embedding

the

Huber’s

M-estimation

methodology

into

the

discussed

Section

Combining

the

Huber’s

M-estimation

methodology

with

the

UKF

the

subject

Section

Simulations

are

presented

Section

and

Section

gives

some

concluding

remarks.

Bayesian

regression

The

best

known

solution

the

optimal

ﬁltering

problem

when

the

following

linear

state-space

model

subject

Gaussian

noise

considered:

k−1

(1)

where

∈

and

∈

denote

the

state

and

measurement

vec-

tors

time

step

and

are

mutually

independent,

zero-mean

nominal

noise

vectors,

each

independent

across

time,

with

respec-

tive

covariance

matrices

and

k−1

the

dynamic

model

function

and

the

measurement

model

function.

Let

k|l

−

denote

the

linear

least

squares

estimate

given

≤

l},

and

let

k|l

denote

the

corresponding

error

covariance

matrix.

Now

consider

the

posterior

mean

estimate

KF.

k|k



p(x



1:k

)dx

(2)

where

p(x

1:k

)

denotes

the

conditional

density

with

measure-

ments

time

Making

use

Bayes

rule

and

the

fact

that

the

observation

sequence

conditionally

independent

given

the

current,

the

posterior

distribution

density

can

factored

[18]

p(x



1:k

)

p(y



)p(x



1:k−1

)

p(y



1:k−1

)

(3)

deﬁning

p(x



1:k−1

)

k−1

)

and

substituting

(3)

into

(2),

(2)

can

rewritten

after

some

algebra

k|k

k|k−1

p(y



1:k−1

)

k|k−1



p(y

)(P

−1

k|k−1

−

k|k−1

))p(

k|k−1

)dx

(4)

the

prior

state

estimate

(before

the

new

observation

received)

assumed

distributed

according

Gaussian

probability

density

function,

i.e.

k|k−1

)

N(x

;

k|k−1

)

(5)

Carrying

out

the

differentiation

(5)

results

∂

∂x

k|k−1

)

−P

−1

k|k−1

−

k|k−1

)p(

k|k−1

)

(6)

Substituting

(6)

into

(4)

results

k|k

k|k−1

−

p(y



1:k−1

)

k|k−1



p(y



)

∂

∂x

k|k−1

)dx

(7)

the

observation

considered

independent

and

iden-

tically

distributed,

the

likelihood

density

function

reduces

function

only

the

residuals

between

the

estimate

and

the

obser-

vation,

i.e.

p(y



)

p(y

−

)

(8)

Substituting

(8)

into

(7)

and

integrating

parts

results

k|k

k|k−1

p(y



1:k−1

)

k|k−1



k−1

)

∂

∂x

p(y

−

)dx

(9)

since

∂

∂x

p(y

−

)

−H

∂

∂y

p(y

−

)

(10)

Eq.

(9)

can

rewritten

k|k

k|k−1

−

p(y



1:k−1

)

k|k−1



k|k−1

)

∂

∂y

p(y

−

)dx

(11)

Noticing

that

the

observation

considered

independent

and

identically,

∂

∂y

k|k−1

)

∂

∂y

p(x



1:k−1

)

(12)

Combining

(11)

and

(12)

gives

k|k

k|k−1

−

p(y



1:k−1

)

k|k−1



∂

∂y

(p(y

−

)p(

k|k−1

))dx

(13)

Changing

the

order

differentiation

and

integration

yields

k|k

k|k−1

−

p(y



1:k−1

)

k|k−1

∂

∂y

p(y



1:k−1

)

(14)

which

can

rewritten

k|k

k|k−1



−

∂

∂y

p(y



1:k−1

)



(15)

Denote

the

innovation

−

k|k−1

with

its

corresponding

covariance

Denote

the

transformed

innovation



−1/2

1/2



k|k−1

(16)

Combining

(15)

and

(16)

results

k|k

k|k−1

−1/2



−

∂

∂

p(



1:k−1

)



(17)

Deﬁne

the

inﬂuence

function

(

)

the

negative

likelihood

score

the

transformed

innovation

density

(

)

−

∂

∂

p(



1:k−1

)

(18)

剩余6页未读，继续阅读

评论收藏

内容反馈

掌鈊化樰

粉丝: 0
资源: 1

1-s2.0-S095915241300111X-main

1-s2.0-S0967066117300515-main_FAULTDETECTION_

1-s2.0-S1877050913006984-main.pdf

1-s2.0-S0301421516300763-main_businessmodalss_

1-s2.0-S0008884620315647-main.pdf

1-s2.0-S156849461200573X-main.pdf

1-s2.0-S026322411300482X-main.pdf

1-s2.0-S030142151630060X-main_businessmodals_

1-s2.0-S026130691400990X-main_FSW_

1-s2.0-S093336572100052X-main(科研通-ablesci.com)1

1-s2.0-S0007850610000041-main_frictionstir_

1-s2.0-S1570870516303092-main.pdf

1-s2.0-S0921509306012391-main_FSW_

1-s2.0-S0925231220309978-main.pdf

1-s2.0-0012365X9190142O-main-1 (2).pdf

1-s2.0-S0968090X20306598-main.pdf

1-s2.0-S016502701100522X-main_2.rar_Multiple sclerosis _magnetic

1-s2.0-S153751101630318X-main.zip_Precision Farming_WSN_farming

X-AnyLabeling的yolox-s-onnx自动标注模型

NASA's Coutributions Aeronautics.pdf

1-s2.0-S0022169420308404-main.pdf

1-s2.0-S1084804514002124-main.pdf

1-s2.0-S1568494615007784-main.pdf

1_1-s2.0-S1361920923000512-main.pdf

时间序列异常检测精选论文-2020.zip

r40_tinav2.1_最终验证通过_使用CB-S来验证SPI2.0成功_20171114_0945没有外层目录.7z

vue移动端轻量级的轮播组件实现代码

romfs.2.0.x.gz

最新资源