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8/16/2019 Kendali Digital 5
1/13
Kendali Digital
Pertemuan 5
1
8/16/2019 Kendali Digital 5
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$.
:e per%ormanct!eo% Parameters
al#econstant ao#t tosettlesinp#t&!en t!eresponsestatesteadyo% si'et!edetermine&illsystemt!eo% gain*t!e+
responseentany transid#ringoscillatessystem%ast t!e!o&determine&ill%re,#encynat#ral#ndampedt!e
statesteadyto&arddecaysresponset!easoscillatessystemm#c! t!e!o&determine&illdampingt!e
:systemt!eo% Parameters
dc
n
•
•
•
ω
ζ
8/16/2019 Kendali Digital 5
3/13
Review of mapping s into z-planes (1)
-plane
!
ζ ω ω ζ
only t!e%#nctionais p!aset!e&!ileonly t!eo% %#nctionaismagnit#det!e plane-sn t!e ando% al#e by t!edeterminedis polest!eo% Position
c!ange.atoregardingsystemt!eo% e per%ormanct!edetermines p!ase)(magnit#de polesssystem/o% Position
n
n
•
•
•
plane-sonando% e%inition nω ζ
plane-sonando% 0ariation nω ζ
)%(P!ase
)%("agnit#de n
ζ
ω
=
=
8/16/2019 Kendali Digital 5
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0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2Step Response
Time (sec)
A m p l i t u d e
zeta=0.1
zeta=0.3
zeta=0.7
zeta=1.0
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
Step Response
Time (sec)
A m p l i t u d e
Wn=0.5 Wn=0.3
Wn=1.5
Wn=1
-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-1
-0.5
0
0.5
1Pole-Zero Map
Real Axis
I m a g i n a r y A x i s
zeta=0.7 zeta=0.3 zeta=0.1
zeta=1.0
S-Plane
-0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0-1.5
-1
-0.5
0
0.5
1
1.5
Pole-Zero Map
Real Axis
I m a g i n a r y A x i s Wn=0.3
Wn=0.5Wn=1.0
Wn=1.5
"
nando% meaningP!ysical ω ζ
22
2
2)(
)(
nn
ndc
s s
G
sr
s y
ω ζω
ω
++
=
8/16/2019 Kendali Digital 5
5/13
Review of mapping s into z-planes (2)
5
)($P!ase
)("agnit#de
: plane-'1n t!e
$:'ands bet&eenRelation
2n
2
T f T
T f e
j j se z
n
nT
nnd
sT
n
ω ζ ζ ω
ω ζ
ζ ω ζω ω σ
ζω
=−=
==
−+=+==
−
timeSampling:T
8/16/2019 Kendali Digital 5
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#
(2) plane-'onand nω ζ
8/16/2019 Kendali Digital 5
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$%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
$$3.5
2
==×−=−=
===
=
=→=××
=→=
××−−
ζ ω θ
ζ ω ζ
ω ω
ζω
θ
8/16/2019 Kendali Digital 5
8/13
n*uence of sampling time
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Pole-Zero Map
Real Axis
I m a g i n a r y A x i s
T=1s
T=0.1s
T=0.5s
T=1.5s
Z-Plane
0 2 4 6 8 10 12 140
0.2
0.4
0.6
0.8
1
1.2
1.4
T=1.5s
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
T=1.0s
0
0.2
0.4
0.6
0.8
1
1.2
1.4
T=0.5s
K
+
+
t,2 +,1"
t,2 +,2
t,2 +," .
)()($.5s
$+(s) 2 T f z G s
=⇒++
=
8/16/2019 Kendali Digital 5
9/13
/ury ta0ility est
/ury a0le'
nn
nn a z a z a z a z P ++++=−
−
$
$
$5 ...)(
:%ollo&sasexpressedise,#ationstic*!aracteri
$5
23
$5
2$
$5
$
2...32$5
$....32$5
+
−
+
−−−
+
−−
=
−==
−==
k
k k
k
k nnk
k
k nnk
p p
p pq
nk bb
bbc
nk aa
aab
52
52
5$
$$
$
5
...........
..........
4.
oddn%or5)(andeenn%or5)(.35)(.2
$.
:i% stable be&illsystem!e
cc
bb
z P z P z P
aa
n
n
z z
z
n
>
>
>
>
<
−
−
−=−=
=
8/16/2019 Kendali Digital 5
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.$$)$(2.
satis%ied$56.5$.
:conditionsstabilityt!e*!ecking
54
→>=−−++=−
→>=−++−=
→=
8/16/2019 Kendali Digital 5
11/13
$%ercise1
11
;#ry test bystability*!eck t!e
524.556.53.$)(
:systemao% e,#ationsticc!aracteri+ien t!e
23=+−−= z z z z P
8/16/2019 Kendali Digital 5
12/13
$%ercise 2
12
stableissystemt!e%or &!ic!
8/16/2019 Kendali Digital 5
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Rout3-4urwitz riterion
riterion of sta0ility' ystem is sta0le if all coe6cients in t3e 7rstcolumn 3ave t3e same sign8
1!
5...)($
$
)(...)(
5$
$
$
bilinear o% trans%orm-&$
$$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
−−
−
−−−
−
−−−
−
−−−
−=
−=
−=
−=