Kendali Digital 5

8/16/2019 Kendali Digital 5

1/13

Kendali Digital

Pertemuan 5

1


2/13

Review of time response of a controlsystem

2

22

2

2)(

)( : byexpressedissystemorder-secondA

nn

ndc

s s

G

sr

s y

ω ζω

ω

++

=

error state-Steady4. timeSettling3.

timeRise3.

Peak time2.

oers!oot"axim#m$.

:e per%ormanct!eo% Parameters

al#econstant ao#t tosettlesinp#t&!en t!eresponsestatesteadyo% si'et!edetermine&illsystemt!eo% gain*t!e+

responseentany transid#ringoscillatessystem%ast t!e!o&determine&ill%re,#encynat#ral#ndampedt!e

statesteadyto&arddecaysresponset!easoscillatessystemm#c! t!e!o&determine&illdampingt!e

:systemt!eo% Parameters

dc

n

•

•

•

ω

ζ


3/13

Review of mapping s into z-planes (1)

-plane

!

ζ ω ω ζ

only t!e%#nctionais p!aset!e&!ileonly t!eo% %#nctionaismagnit#det!e plane-sn t!e ando% al#e by t!edeterminedis polest!eo% Position

c!ange.atoregardingsystemt!eo% e per%ormanct!edetermines p!ase)(magnit#de polesssystem/o% Position

n

n

•

•

•

plane-sonando% e%inition nω ζ

plane-sonando% 0ariation nω ζ

)%(P!ase

)%("agnit#de n

ζ

ω

=

=


4/13

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2Step Response

Time (sec)

A m p l i t u d e

zeta=0.1

zeta=0.3

zeta=0.7

zeta=1.0

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

A m p l i t u d e

Wn=0.5 Wn=0.3

Wn=1.5

Wn=1

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-1

-0.5

0

0.5

1Pole-Zero Map

Real Axis

I m a g i n a r y A x i s

zeta=0.7 zeta=0.3 zeta=0.1

zeta=1.0

S-Plane

-0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0-1.5

-1

-0.5

0

0.5

1

1.5

Pole-Zero Map

Real Axis

I m a g i n a r y A x i s Wn=0.3

Wn=0.5Wn=1.0

Wn=1.5

"

nando% meaningP!ysical ω ζ

22

2

2)(

)(

nn

ndc

s s

G

sr

s y

ω ζω

ω

++

=


5/13

Review of mapping s into z-planes (2)

5

)($P!ase

)("agnit#de

: plane-'1n t!e

$:'ands bet&eenRelation

2n

2

T f T

T f e

j j se z

n

nT

nnd

sT

n

ω ζ ζ ω

ω ζ

ζ ω ζω ω σ

ζω

=−=

==

−+=+==

−

timeSampling:T


6/13

#

(2) plane-'onand nω ζ


7/13

$%ample

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1Pole-Zero Map

Real Axis

I m a g i n a r y A

x i s

zoh method

tustin method

&sing some

appro%imation'

$s&it! plane-'onanditsS!o&

$.5s

$+(s):%#nctions%er&it! transystema+ien

n

2

=

++

=

ω ζ

s

on

T

j

n

nn

T

eer

re z

n

6.7487.5$3.5$$$

94.5)("agnit#de

3.5.52

$$

:lyAnalytical

:Ans&er

22

$$3.5

2

==×−=−=

===

=

=→=××

=→=

××−−

ζ ω θ

ζ ω ζ

ω ω

ζω

θ


8/13

n*uence of sampling time

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Pole-Zero Map

Real Axis

I m a g i n a r y A x i s

T=1s

T=0.1s

T=0.5s

T=1.5s

Z-Plane

0 2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

1.2

1.4

T=1.5s

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

T=1.0s

0

0.2

0.4

0.6

0.8

1

1.2

1.4

T=0.5s

K

+

+

t,2 +,1"

t,2 +,2

t,2 +," .

)()($.5s

$+(s) 2 T f z G s

=⇒++

=


9/13

/ury ta0ility est

/ury a0le'

nn

nn a z a z a z a z P ++++=−

−

$

$

$5 ...)(

:%ollo&sasexpressedise,#ationstic*!aracteri

$5

23

$5

2$

$5

$

2...32$5

$....32$5

+

−

+

−−−

+

−−

=

−==

−==

k

k k

k

k nnk

k

k nnk

p p

p pq

nk bb

bbc

nk aa

aab

52

52

5$

$$

$

5

...........

..........

4.

oddn%or5)(andeenn%or5)(.35)(.2

$.

:i% stable be&illsystem!e

qq

cc

bb

z P z P z P

aa

n

n

z z

z

n

>

>

>

>

<

−

−

−=−=

=


10/13

$%ample

1

56.53.559.52.$$!#s

556.53.559.52.$)(:e,#ationsticc!aracteri+ien t!e

432$5

234

−===−==

=−++−=

aaaaa

z z z z z P

satis%ied568.$56.53.559.52.$$)$(3.

satis%ied558.556.53.559.52.$$)$(2.

satis%ied$56.5$.

:conditionsstabilityt!e*!ecking

54

→>=−−++=−

→>=−++−=

→=


11/13

$%ercise1

11

;#ry test bystability*!eck t!e

524.556.53.$)(

:systemao% e,#ationsticc!aracteri+ien t!e

23=+−−= z z z z P


12/13

$%ercise 2

12

stableissystemt!e%or &!ic!


13/13

Rout3-4urwitz riterion

riterion of sta0ility' ystem is sta0le if all coe6cients in t3e 7rstcolumn 3ave t3e same sign8

1!

5...)($

$

)(...)(

5$

$

$

bilinear o% trans%orm-&$

$$5

=++++=→

−

+=

→ ++++=

−

−

−

−

bwbwbwbw P w

w z

w P a z a z a z a z P

n

n

n

n

nnnn

:array>#r&it'-Ro#t!

....

$

2$3$$

$

9$3

$

74$2

$

32$$

c

cbbcd

b

bbbbc

b

bbbbc

b

bbbbc

nn

n

nnnn

n

nnnn

n

nnnn

−−

−

−−−

−

−−−

−

−−−

−=

−=

−=

−=

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Kendali Digital 5