自控课件-自动控制chapter

Chapter

4Best

estimateindexBasic

knowledge

of

random

systemBasic

method

of

estimateKalman

filter

the

waves

in

basic

discreKalman

filter

the

waves

in

normal

discreinear

systeminear

systeminear

system

under5.Kalman

filter

the

waves

in

discrecolorful

yawp.The

stability

of

Kalman

filter

the

waves

and

error

yze.The

relationship

of

Kalman

filter

the

waves

with

best

control.北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

）；咨询Best

estimateintroductionThe

systems

we yzed

before

are

determinate

systems.

In

those

systems,theinitially

states

are

determinate

known,

the

input

is

a

determinate

knowntimefunction

or

a

linear

state

function,

and

the

output

is

a

linear

state

function

and

nosurvey

error.But

in

fact,

the

systems

we

often

met

are

the

other

type

systems.

In

those

system,theinitially

states

are

random

vector,

and

we

don’t

know

its

determinate

value

but

knowits

mathematical

expectation

and

variance.

Those

system

are

effected

not

only

by

thedeterminate

input,

but

also

by

some

random

interfere.

So

the

states

of

those

systemare

not

determinate

functions

but

random

processes.

How

to

estimate

the

systemstates

form

the

random

interfere

and

to

best

control

is

the

problems

of

best

estimate.The

problems

of

best

estimate

include

two

types,

one

is

parameter

estimate,

the

otheris

state

estimate.

The

main

method

to

solving

the

second

type

problem

is

Kalmanfilter

the

wave,

which

is

the

emphases

of

this

chapter.北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

）；咨询Basic

knowledge

of

randomsystem

(1)F

()

p{x

}0

F

()

1The

derivative(导数)of

the

distributionfunction

is

probability

density

function

p().dF

()dp()

F

()

p(x)dx

p(x)dx

1一.The

Statistic

speciality

of

random

variable

and

random

vector.1.The

Statistic

speciality

of

random

variable.随

量的统计特性.Consider

X

is

a

random

variable, is

its

possible

value, can

becontinuous

or

discrete,

and

X

is

called

respectively

continuous

randomvariable

or

discrete

random

variable.As

a

continuous

random

variable

X,

F

()

is

its

distribution

function.Basic

knowledge

of

randomsystem

(2)Some

other

mathematical

expression

of

random

variable.(1)mathematical

expectation

数学期望E[x]x

p

i

i

xE[x]

p(

)d

i1E[x]

E[C]

CC

is

a

constantmathematical

expectation

ofcontinuous

random

variable:mathematical

expectation

ofdiscrete

random

variable:mathematical

expectation

havefollow

theorem:E[

X

Y

]

E[

X

]

E[Y

]X,Y

are

random

variablen

nE[i

Xi

]

i

E[

Xi

]i1

i1i

are

constantBasic

knowledge

of

randomsystem

(3)(2)Variance

方差Var(x)xVar(

X

)

E[(

X

E(

X

) ]

E[(

X

)

]2

(

)

p(

)dx2

2IfX

is

in

normal

distribution(正态分布),its

probability

density

is:12

xxe2

2(

x

)2p()

It

can

be

simply

note2x

xX

~

N

(

,

)Variance

havefollow

theorem:Var(C)

0Var(CX

)

C

2

Var(

X

)n

naia

j

E[(Xi

xi

)(

X

j

xj

)]i1

i1

j

1nVar[

ai

Xi

]

Basic

knowledge

of

randomsystem

(4)

E[(

X1

x

)(

X

2

x1

2

E[(

X1

X

2

)]

x

x1

2Correlation

coefficient

相关系数)](3)Covariance

协方差COV(X1,X

2)COV

(

X1

,

X

2

)

E[(

X1

EX1

)(

X

2

EX

2

)]

x

x1

21

2COV

(

X1

,

X

2

)

x

xVar(

X

)

Var(

X

)1

21

20

x

x

1is

independence.1

2

=0,x

x1

2X

,

Xx1x2

=1,

X1

,

X

2iscomple y

linearcorrelation.Basic

knowledge

of

randomsystem

(5)(4)high-order

moment

高阶距E(

X

n

)

n

p(

)dnE[(

X

)]

n(

)

p()dxn

order

origin

momentn阶原点距n

order

central

momentn阶中心距2.The

Statistic

speciality

of

random

vector.Consider

vector

X,

which

component

X1

,

X

2

,variable,

that

vector

is

called

random

vector.Xn

are

random讲解！北理工《自动控制理论》考研

，

、考点、典型题、命题规律独家详见：网学天地（

）；咨询Basic

knowledge

of

randomsystem

(6)Union

distribution

function

of

random

vector

F

(),

n

)

p{X1

1,

X

2

2

,,

Xn

,

xn

)dx1dx2,

xnn

}F

()

F

(1,2

,F

()

F

(1,2

,d

x

n1

2

n,

n

)

p(x1,

x2

,p(x1,

x2

,Union

distribution

density

function

isOrp(x1,x2,

,

xn

))Some

qualities

of:p(x1,

x2

,

,

xn

)

0

p(x1

,

x2

,dxn

1,

xn

)dx1dx2p{(1,2

,d

x

np(x1,

x2

,

n

)

S}

sp(x1,

x2

,

,

xn

)dx1dx2

,

xk

)

p(x1

,

x2

,

,

xn)dxk

1

dxn

Basic

knowledge

of

randomsystem

(7)There

is

also

average

value

matrix

andvariance

matrix

of

random

vector

X,

andcovariance

matrix

of

X,Y.

xn

n

E(

X

)E(

X

)

2E(

X

)

x2

E(

X1

)

x1

nxnn

xnnxn

R

1

nCOV

(

X

,

X

)Var(

X

)COV

(

X

,

X

))(

X)]Var(

X

)

E{[

X

E(

X

)][X

E(

X

)]T

}

n

1COV

(

X

,

X

)2

n22

1Var(

X1)2

x2222)

]n

xn

2

x2x21

x12x221

x11

x1

nVar(

X

)COV

(

Xn

,

X

2

)

COV

(

X1

,

X

2

)COV

(

X1,

Xn

)E[(

X

)2

])]

E[(

X

1E[(

X

)(X

x1E[(

X

)(

X

)]1

E[(

Xx

n

)(

Xx

E[(

X

)(

X

)]

)]xE[(

X

)2

])]E[(

X

E[(

X

)(

XVar(X)

is

a

symmetrymatrixBasic

knowledge

of

randomsystem

(8)n

2

n

m

2

1

2

2

2

mCOV

(

X

,Y

)COV

(

Xn

,Y1

)

COV

(

X

,Y

)COV

(

X

,Y

)

E{[

X

E(

X

)][Y

E(Y

)]T

}

XC1,OYVm

)(X

,Y

)

COV

(

X

,Y

)

COV

(

X

,Y

)COV

(

X1

,Y1)

COV

(

X1

,Y2

)

COV

(

COV

(

X

,Y

)

[COV

(Y,

X

)]TSoIf

COV(X,Y)=0,

random

vectors

X

and

Y

are

independence.If

random

vector

X

is

in

normal

distribution,

which

is

called

Gaussrandom

vector,

and

its

union

distribution

density

function

is11(2

)2

R

22exp{

1

(x

)T

R1

(x

)}np(x)

p(x1,

x2

,

,

xn

)讲解！北理工《自动控制理论》考研

，

、考点、典型题、命题规律独家详见：网学天地（

）；咨询Basic

knowledge

of

randomsystem

(9)For

example,

X

is

2

dimensions

vector.

Then

:

2

1

2

2

1

2

21222

)]

x21

x12

x2x21x1

1

x12E[(X

)

]E[(X

)(X

)]

E[(X

)

]

E[(X

)(XR

222121Var(X

)Var(X

)Thereinto:

1

2is

the

correlation

coefficient

of

X

,

XThen

:

2

2

21

221 2

212

211

1(1

)(1

)

))

(1

(1

R

1

12

2

2

)

R

(1Basic

knowledge

of

randomsystem

(10)So11)11exp112

22

x223

212

2

21

21

212

21

2222121

21

22

2

x2

1

21

x1

2x1T

2

(x

)(

x

)

]}

(x

)

[(x

)

)2

(1

exp{

11

2x1

}

x

x2

x2

1 2

2

)

(1

(1

(1

)

(1

)1

x1

2

x2

x

2

1

x

{1

2p(x)

p(x

,

x

)

Union

distribution

densityfunction

of

2

dimensions

randomvector

express

by

figure

is:If

0

then

p(x1,

x2

)

p(x1

)

p(x2

)1

2p(x

,

x

)x1x2x1x2Basic

knowledge

of

randomsystem

(11)There

are

some

theorems

about

normal

random

vector.Theorem

1:

If

normal

random

variables

(vectors)X,

Yarenoncorrelation, then

they

are

independent

.Theorem

2:

If

random

variables

(vectors)

are

unian

normaldistribution

,then

their

variable

(vector)

are

also

normaldistribution.Theorem

3:

Linear

transform

and

linear

assemble

for

normalvariable

(vector),

also

are

normal

variable

(vector).p(x,

y)

p(x

|

y)

p2

(

y)

p(

y

|

x)

p1(x)p(,

)

p(

|

)

p2

(

)

p(

|

)

p1()Bayes’s

rule:orBasic

knowledge

of

randomsystem

(12)E[x|y]

is

when

y=

,the

averagevalue

of random

vector

X,calledconditional

average

value.E[

X

|

Y

y]

E[

X

|

Y

]

xp(x

|

y)dxE[

X

|

Y

]

E[

X

|

]

p(

|

)dTheorem

4:X,Y,and

Z

are

union

distribute

variables

or

vectorsx,yand

zare

the

possible

value

.a,bare

constants,

g(.)

is

scale

functionand

E[X],

E[Z],

E[g(Y)X]

are

existE[

X

|

y]

E[

X

]E(

X

)

E{E[

X

|

y]}E[g(Y

)

X

|

y]

g(Y

)

E[

X

|

y]E[g(Y

)x]

E{g(Y

)

E[

X

|

y]}E[a

|

y]

aE[g(Y

)

|

y]

g(

y)E[aX

bZ

|

y]

a

E[

X

|

y]

b

E[Z

|

y]北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

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knowledge

of

randomsystem

(13)Theorem

5:Assume

random

variabls

(vectors)

X,Y

are

union

normaldistribution

,when

given

Y

condition

distribution

density

of

X

is

alsonormal

distribution

and

conditio

age

and

condition

variance

are:E[

X

|

Y

]

E[

X

]

COV

(

X

,Y

)[Var(Y

)]1(

y

EY

)Var(

X

|

Y

)

Var(

X

)

COV

(

X

,Y

)[Var(Y

)]1COV

(Y

,

X

)北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

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knowledge

of

randomsystem

(14)二.Basic

knowledge

of

random

process.1.Concept

and

distribution

function

of

random

process.If

random

variable

X

follo rameter

t

to

change,

then

thecollectivity

of

X

{X(t),tD}

is

called

random

process.

Thereintoparameter

t

can

be

time

or

space

coordinate,

D

is

the

valuecollection

of

parameter

t.

If

collection

D

is

all

or

half

of

numberaxis,

then

{X(t)}

is

continuous

random

process.

And

if

collectionD

is

positive

integer,

then

{X(t)}

is

discrete

random

process,random

sequence

or

time

sequence.To

random

process

{X(t),t

D},

when

t

equal

N

values,

it

can

get

Nrandom

variables

(

X

(t1

)),

(

X

(t2

)

)

, (

X

(tN

)).

And

its

uniondistribution

function

is

called

random

process

N

dimensions

distribute.FN

(x1,

x2

,

,

xN

;t1,

t2

,

,

tN

)

p{X

(t1)

x1,

X

(t2

)

x2

,

,

X

(tN

)xN

}Basic

knowledge

of

randomsystem

(15),

xN

;t1,

t2

,If

function

p(x1,

x2

, ,

tN

)

exist,

to

makex1

x2

xN,

tN

)

pN

(1,2

,FN

(x1

,

x2

, ,

xN

;t1

,

t2

,,

N

;t1

,

t2

,,

tN

)d1,d2,,

dNright.

,

xN

;t1,

t2

, ,

tN

)Then

called

function

p(x1,

x2

,

is

Ndimension probability

density

function

of

thisrandom

process.When

t

equal

N

other

values,

it

can

getN

other

union

distributionfunction

and

N

dimensions

probability

density

function.

Itsensemble

distribution

function

is

called

random

process

limiteddimension

distribution

group.随机过程的有限维分布族。{FN

(x1,

x2

,

,

xN

;t1,

t2

,

,

tN

),t1,

t2

,

,

tN

D,

N

Basic

knowledge

of

randomsystem

(16)Characteristic

function

of

random

process.随机过程特征函数。Some

characteristic

functions

followed.Average

value

function

and

Average

value

functionvector.均值函数与均值函数向量。Covariance

function

and

Covariance

function

matrix.协方差函数与协方差函数阵。Auto-correlation

function

andAuto-correlation

function

matrix.自相关函数与自相关函数阵。Cross

covariance

function

and

Cross

correlation

function

.互协方差函数与互相关函数北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

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knowledge

of

randomsystem

(17)N3.independence,

correlation

and

stationary

of

random

process.独立性、相关性和平稳性。If

for

a

random

process,the

follow

equation

is

true,

then

therandom

process

is

independence.FN

(x1,

x2

,

,

xN

;t1,

t2

,

,

tN

)

F1(xi

,

ti

)i1If

for

two

random

processes,

the

follow

equation

is

true,then

the

two

random

processes

is

cross

independence.FN

(x1

,

x2

,,xN

;

y1,

y2

,,yN

;t1,

t2

,,

tN

)

FN

(x1

,

x2

,, ,tN

),

xN

;t1,

t2

,,tN

)

FN

(

y1,y2

,,yN

;t1,

t2北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

）；咨询Basic

knowledge

of

randomsystem

(18)2If

a

random

vector

process: (t

,

t

)

E[

X

(t

)

X

T

(t

)]

E[

X

(t

)]

E[

X

T

(t

)],t

tx

1

2

1

2

1

2

1Or:

R

(t

,

t

)

(t

,

t

)

(t

)

T

(t

)

0x

1

2

x

1

2

x

1

x

2Then

the

random

process

is

non-correlation

process.If

n

dimensions

random

vector

process

{X(t),t D}

and

m

dimensionsrandom

vector

process

{Y(t),t D}

: (t

,t

)

E[

X

(t

)

YT

(t

)]

E[

X

(t

)]

E[YT

(t

)],t

,

t

,

Dxy

1

2

1

2

1

2

1

2Txy

1

2

1

x

1

2

y

2R

(t

,

t

)

E{[

X

(t

)

(t

)][Y

(t

)

(t

)] }

0Or:Then

the

two

random

processes

are

cross

non-correlation

processes.北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

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knowledge

of

randomsystem

(19)FN

(x1

,

x2

,

,

xN

;t1,

t2

,

,

tN

)

FN

(x1

,

x2

,If

a

random

process

is

a

independence

process

then

it

is

a

non-correlation

process

without

fail,

but

a

non-correlation

process

isnot

always

a

independence

process.

A

random

vector

process

isthesame.If

a

random

process

:FN

(x1

,

x2

,

,

xN

;t1

,

t2

,

,

tN

)

FN

(x1

,

x2

,

,

xN

;t1

,

t2

, ,

tN

)Then

it

is

a

strict

stationary

random

process.(严格平稳随机过程)then,

xN

;0,t2

t1As

a

strict

stationary

random

process

:

t1,

tN

t1

)北理工《自动控制理论》考研,，、考点、典型题、命题规律独家讲解！详见：网学天地（

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knowledge

of

randomsystem

(20)As

a

strict

stationary

random

process

:p1(x,

t)

p(x,0)

p(x)x

(t)

E[

X

(t)]

xp1(x)dx

xp2

(x1,

x2

;t1,

t2

)

p2

(x1,

x2

;0,

t2Rx

(t1,

t2

)

COV

[

X

(t1

),

X

(t2

)]

1

dimension

distribution

densityaverage

value2

dimensions

distribution

density2Tx

1

2

t1

)dx1dx2

]

p(

x

,

x

;0,

t[x1

x

][x2

const

t1)

Covariancefunction

matrixx

xT

p(x

,

x

;0,t

t

)dx

dx

(

)R(0,

t2

t1

)

R(t2

t1

)

R(

)

x

(t1,

t2

)

COR[

X

(t1

),

X

(t2

)]

1

2

1

2

2

1

1

2

xAuto-correlation

functionmatrixBasic

knowledge

of

randomsystem

(21)Stationary

random

process

of

broad

sense.(广义平稳随机过程)Average

value

and

Covariance

function

:111TTTx][x(t

)

]

dt

(

)xT

xTTCOV

[

X

(t),

X

(t

)]

lin

[x(t)

x(t)dt

2TE[

X

(t)]

linT

To

be

a

stationary

random

process

of

broad

sense

must:

lin

(

)

0

lin

(

)

0Or北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

）；咨询Basic

knowledge

of

randomsystem

(22)Some

specifically

random

process.几个特定的随机过程。white

noise

process

and

white

noise

sequence.白噪声过程和白噪声序列。normal

random

process.正态随机过程。(3)Markov

Process.过程北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

）；咨询Basic

knowledge

of

randomsystem

(+1)(1)

Average

value

function

and

Average

value

function

vector.均值函数与均值函数向量。To

random

variable

process,

its

average

value

function

is

:E[

X

(t)]

x

(t)

x

p1

(x,

t)dxThereinto,

p(x,t)

is

1

dimension

density

function.x

(t)

is

non-random

time

function.To

random

vector

process,

average

value

function

is

a

vector.x

(t)

[x

(t),

x

(

t

), ,

x(t)]T1

2

nx

(t)

E[

Xi

(t)]iAndReturnBasic

knowledge

of

randomsystem

(+2)2x

x

xVar[

X

(t)]

E{[

X

(t)

(t)][

X

(t)

(t)]}

(t)

x

(t)

is

average

Variance

of

random

process.Covariance

function

matrix

of

random

vector

process.(2)

Covariance

function

and

Covariance

function

matrix.协方差函数和协方差函数阵。Covariance

function

of

random

process.COV

[

X

(t1),

X

(t2

)]

E{[

X

(t1

)

x

(t1

)][

X

(t2

)

x

(t2

)]}

x

(t1,

t2

)All

appearance,

COV

[

X

(t1

),

X

(t2

)]

is

the

function

oft1,

t2,when

t1

t2

t

it

is

Variance

function

of

random

process.北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

）；咨询Basic

knowledge

of

randomsystem

(+3)COV[

X

(t

),

X

(t

)]COV[

Xn

(t1),

X1

(t2

)]

COV

[

Xn

(t1),

X

2

(t2

)]

COV

[

Xn

(t1),

Xn

(t2

)]Tx

2

x

1

2COV[

X

(t1),

X

(t2

)]

E{[

X

(t1)

x

(t1

)][X

(t2

)

COV

[

X1

(t1),

X1

(t2

)]

COV

[

X1

(t1),

X

2

(t2

)]2

1

n

2COV

[

X

(t

),

X

(t

)]

COV

[

X

(t

),

X

(t

)]2

1

1

22

1

2

2

COV

[

X1

(t1),

Xn

(t2

)](t

)]

}

R

(t

,

t

)

All

appearance,

COV

[

X

(t1),

X

(t2

)]

COV

[

X

(t

2),

X

(t1)]TWhen

t1

t2

t

,Var[

X

(t)]

Rx

(t)which

is

Variance

function

matrix

of

random

processReturn北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

）；咨询Basic

knowledge

of

randomsystem

(+4)(3)Auto-correlation

function

and

Auto-correlation

function

matrix.自相关函数与自相关函数阵。Auto-correlation

function

of

random

process:COR[X

(t1),

X

(t2

)]

E{X

(t1)

X

(t2

)}

x

(t1,

t2

)The

relation

of

auto-correlation

function,

covariance

functionand

averagevalue

function.

x

(t1,

t2

)

x

(t1,t2

)

x

(t1)

(t2

)For

random

process,

it

is

follow

equations:COR[

X

(t

),

X

(t

)]

E{X

(t

)

X

T

(t

)}

(t

,

t

)1

2

1

2

x

1

2R

(t

,

t

)

(t

,

t

)

(t

)

T

(t

)x

1

2

x

1

2

x

1

x

2Basic

knowledge

of

randomsystem

(+5)(4)Cross

covariance

function

and

Cross

correlation

function

.互协方差函数与互相关函数Cross

covariance

function

of

two

random

process

{X(t),tD}

and{Y(t),tD}

in

two

time

t1,

t2

.

And

of

vectorprocess.COV

[

X

(t1

),Y

(t2

)]

E{[

X

(t1

)

x

(t1

)][Y

(t2

)

y

(t2

)]}

xy

(t1,

t2

)Ty

2

xy

1

2(t

)]

}

R

(t

,

t

)COV

[

X

(t1

),Y

(t2

)]

E{[

X

(t1)

x

(t1

)][Y

(t2

)

Cross

correlation

function

of

two

random

process

and

vector

process.COR[

X

(t1

),Y

(t2

)]

E{X

(t1)

Y

(t2

)}

xy

(t1,

t2

)(t

,

t

)1

2

1

2

xy

1

2TCOR[X

(t

),Y

(t

)]

E{X

(t

)

Y

(t

)}

Return北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

）；咨询Basic

knowledge

of

randomsystem

(+6)The

relation

of

Cross

covariance

function, Cross

correlationfunction

and

Average

value

function.

xy(t1,

t2

)

xy

(t1,

t2

)

x

(t1)

y

(t2

)Rxy

(t1,

t2

)

xy

(t1,

t2

)

x

(t1

)

T

(t

)y

2The

Characteristic

functions

we

studied

hereinbefore

is

forcontinuous

random

process,

and

discrete

random

process

havecorresponding

functions

but

the

parameter

is

discrete.Return北理工《自动控制理论》考研，、考点、典型题、命题规律独家讲解！详见：网学天地（

）；咨询Basic

knowledge

of

randomsystem

(+7)(1)white

noise

process

and

white

noise

sequence.白噪声过程和白噪声序列。

(t

)dt

1,

t

(t

)

0,t

Dirac

functionk

j(k

)

COR[

X

(k),

X

(

j)]

CThen

it

is

white

noise

sequence.

Andk

j0,

k

j

1,

k

jIf

a

random

vector

process

which

auto-correlationfunction

matrix

can

be

transform

to:

x

(t,

)

COR[X

(t),

X

(

)]

K

(t)

(t

)Then

it

is

white

noise

process.

And

()

C

()To

discrete

time

sequence,

if

it

is

non-correlation,and

its

auto-correlation

function

can

be

transformto:

Kroneckerfunctionk

j

x

(k)

COR[

X

(k

),

X

(

j)]

RkBasic

knowledge

of

randomsystem

(+8)(2)normal

random

process.正态随机过程。For

n

dimensions

vector

process,

if

its

union

probability

density

functioncan

be

transform

to

the

follow

form,

then

it

is

normal

random

process.21

11R

2x

xexp{ (x

)

R

(x

)}(2

)p(x)

T

1m

n2Thereinto

:

xm

x

m

x

x2

E[

X

(t

)]

m

x

,

2

E[

X

(t

)]

E[

X

(t1

)]

x

1

x

2

x1

They