2 modelovanje i identifikacija sistema

7/23/2019 2 Modelovanje i Identifikacija Sistema

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..

--3,3, . 2014.. 2014.


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(Laplace)


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()

.

.


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(1)(1)

, ,, , ,

. ,

( ) .


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(, ) (, ).

, .

() (, ,

, ) .

()

.


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7/47


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( ) .

,

:- ,- - .


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, ,

. . :

, ,

,

. () .


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.

. .

w.

.


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,

, :- ()

;- () .

,

q . ( ) ( )

.


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{x1,x2,...,xn} , t0 ,

y(t) tt0

u(t), tt0.


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x

, (2.1)

, x(k0)-dato (2.2)

u , y .

)),(),(()(

)),(),((

ttutxgty

ttutxfdt

dx

=

=

)),(),(()()),(),(()1(

kkukxgkykkukxfkx

d

d

=

=+

datotx )( 0


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.

x01 x02 h01(t) h02(t),, u1(t) u2(t)

h11(t) h12(t),,

u(t) =u1(t) + u2(t)

x0=x01 +x02

y(t)= h01(t) +h02(t) + h11(t) +h12(t) .


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() ,

u(t) y(t),

u1(t)=u(t+) y1(t)=y(t+).


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,

, x(t0)-dato (2.3)

A, B, CD

.

)()()(

)()(

tDutCxty

tButAxdt

dx

+=

+=


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. -

l .

0))(,...,)(

),(,...,)(

( =tu

dt

tudty

dt

tydl

m

m

n

n


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RLC


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(1)(1) RLC


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(2)(2)

.

:

!


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,

.

(2.14)

.

= )()),(),(()(

0txtutxfdt

tdx

))(),(()( tutxgty =

{ }Rttytutx eee );(),(),(


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(1)(1)

.

= )()),(),(()(

0txtutxfdt

tdxeee

e

))(),(()( tutxgty eee =Rttytutx eee ),(),(),(

eee yux ,,

0=dtdxe

0),( =ee uxf


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(2)(2)

(x,u). Taylor-ov ,

(2.16)

(x,u

).

Rttytutx ),(),(),(

))(())((),()(

))(())((),()(

eeeeee

eeeeee

utuu

gxtx

x

guxgty

utuu

fxtx

x

fuxf

dt

tdx

+

+

+

+


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(( ))


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(())

(3)(3)


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(3)(3)

ueu

gx

x

guxgF

uu

fx

x

fuxfE

u

gD

x

gC

u

fB

x

fA

eeeee

eeeeee

eeee

=

=

=

=

=

=

),(

),(

,,,

FtDutCxty

EtButAx

dt

tdx

++=

++=

)()()(

)()()(

(4)(4)


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(4)(4)

A, B, C, D, E,F . ,

.

,

,

(2.19)

)()()( txtxtx e= eututu = )()(

)()()(

)()()(

tuDtxCty

tuBtxAdt

txd

+=

+=


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--

-

(2.20)

Taylor- (y0,u0)

(2.21) (2.20)

:

0))(,...,

)(

),(,...,

)(

( =tudt

tud

tydt

tyd

l m

m

n

n

0),...,0,,...,0( 00 =uyl

)(...)(

)(...)()(

001

1

1 tubdt

tudbtya

dt

tyda

dt

tydm

m

mn

n

nn

n

++=+++

),...,0(),1,...0( mibnka ik ==

,...,...,,, 010001

00

=

=

=

=

u

l

bu

l

by

l

ay

l

a &&

(2.22)


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--

-

(2.20)

Taylor- (y0,u0)

(2.21) (2.20)

:

0))(,...,

)(

),(,...,

)(

( =tudt

tud

tydt

tyd

l m

m

n

n

0),...,0,,...,0( 00 =uyl

)(...)(

)(...)()(

001

1

1 tubdt

tudbtya

dt

tyda

dt

tydm

m

mn

n

nn

n

++=+++

),...,0(),1,...0( mibnka ik ==

,...,...,,, 010001

00

=

=

=

=

u

l

bu

l

by

l

ay

l

a &&

(2.22)

::


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::

-- :

x


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::


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::

--(2)(2) xe

/maxmax ..

/maxmax

xe

- 20, /

maxmax

0.950.95..

--

maxmax


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((LaplaceLaplace))


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((LaplaceLaplace))

(2.23)

s .

(2.24)

((2.23) (2.24))

,

,0),(


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(1)(1)

.

, :

1. :

2.

3. ..

)Re(s

)()(

11

sFatfal

i

ii

l

i

ii ==

=

l

0

11

1 )()(

)(=

=

=

t

k

ii

iikk

k

k

dt

tydssYs

dt

tydl

)(1

)(0

sYs

dy

t

=

l


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(2.22)

n-

f(s,x0)

:

),()(...)()()(...)()( 001

10

1

1 xsfsUbsUsbsUsbsYasYsasYs m

m

m

m

n

n

n +++=+++

)()()( sUsGsY =

)(

)(

)( sA

sB

sG =

0

1

1

0

1

1

...)(

...)(

bsbsbsB

asassA

m

m

m

m

n

n

n

+++=

+++=


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,

.

.

)()()()()()0()(

sDUsCXsYsBUsAXxssX

+=+=

)0()()())(()(

)()()0()()(

11

11

xAsICsUDBAsICsY

sBUAsIxAsIsX

++=

+=

...)(3

2

2

1 +++= s

A

s

A

s

IAsI

{ } AtetAAtIAsI =+++= ...!2)(

22

11l

Ate


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u(t), d(t).

/

: u(t)=u0, d(t)=d0 y(t)=y0

0))(,...,)(),(,...,)(),(,...,)(( =tddt

tddtudt

tudtydt

tydlk

k

m

m

n

n

)(...)(

)(...)(

)(...)()(

00

01

1

1

tdc

dt

tddctub

dt

tudb

tyadt

tydadt

tyda

k

k

km

m

m

n

n

nn

n

n

++++

=+++


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(1)(1)

D(s)= L(d(t))

:

),()(...)(

)(...)()(

)(...)()(

00

01

1

011

xsfsDcsDsc

sUbsUsbsUsb

sYasYsasYsa

k

k

mm

mm

nnnn

+++

+++=

+++

01

1

011

01

1

...)(

...)(

...)(

cscscsC

bsbsbsB

asasasA

kk

kk

mmmm

nn

nn

+++=

+++=

+++=


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(2)(2)

:

- u(t) y(t)

- d(t) y(t)

)(

),(

)()(

)(

)()(

)(

)( 0

sA

xsf

sDsA

sC

sUsA

sB

sY ++=

)(

)()(

sA

sBsG =

)(

)()(

sA

sCsGd =


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((33)) a

G(s)Gd(s) .

G(s)U(s)

Gd(s)D(s)

Y(s)+

+


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. :

- x0= x01+ x02+ ...,

- u(t)=u1(t)+u2(t)+...- d(t)= d1(t) +d2(t)+....,

y(t)= h01

(t)+h02

(t)+...+ yu1

(t)+yu2

(t)+... yd1

(t)+ yd2

(t)+....,

h01(t),h02(t),..; yu1(t),yu2(t),..; yd1(t), yd2(t),... : x0, u(t) d(t),.

2 modelovanje i identifikacija sistema

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