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![Page 1: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/1.jpg)
3-5 steady-state error calculation The steady-state performance is an important ch
aracteristic of control system , it represses the ability to follow input signal and resist interference 。
一 .error and steady-state error
1.definition:
(1) according to output
e(t): system error,Cr(t): requested output, c(t) :practical output 。
)()()( tctCte rc
![Page 2: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/2.jpg)
)(sG
)(sH
)(sC
-
)(sR
Steady-state error:
The steady-state error relies on the system's construction ,still the form of the input signal.But stability depends only on system’s construction.
)]()([lim)(lim tctCtee rt
ct
css
0 t
C(t)
essCr(t)
![Page 3: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/3.jpg)
(2) according to input
)()()( tbtrte
)]()([lim)(lim tbtrteett
ss
)(sG
)(sH
)(sC
-
)(sR
)(sB
)(sE
![Page 4: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/4.jpg)
2.calculation
(1).final theorem
when input is , final theor
em can be used.but when input is sine or co
sine signal,cannot used.
(2).from the definition
a.solve error TF
)]()([lim)(lim)(lim00
sBsRssEsteesst
ss
2
2
1,),(1),( tttt
)(
)(
sR
sE
![Page 5: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/5.jpg)
b.solve
c. solve
d. Solve limit that is steady-state error.
If the system exists at the same time
input and interference,method is as
follows :
)()(
)()( sR
sR
sEsE
)(te)(lim te
t
R(s)
N(s)
E(s)+
+
)(ser
)(sen
![Page 6: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/6.jpg)
TF between error and input,
TF between error and interference 。
so:
)(ser
)]()()()([lim
)(lim
0
0
sNssRss
ssEe
eners
sss
)(sen
![Page 7: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/7.jpg)
2
n(t)=-1(t)r(t)=t E(s)
-C(s)
12.0
1
s )1(
2
ss
exam:given a block diag. As follows.when input
r(t)=t and interference n(t)=-1(t),ess=?
![Page 8: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/8.jpg)
to interference N(s),we hope it produce output 0,that is ,output is independent of the interference N(s) 。
when input and interference acts on a system,output is respectively:
G1(s)
N(s)R(s) E(s)
-
C(s)
H(s)
G2(s)
![Page 9: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/9.jpg)
error produced by R(s) and N(s) :
)()()(1
1)(
)()()(
21
1
sHsGsGs
sRssE
er
er
)()()()()()()(
)()()(1
)()()(
)()()(
21
21
2
2
sNssRssEsEsE
sHsGsG
sHsGs
sNssE
ener
en
en
![Page 10: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/10.jpg)
in this exam.,stability must be assured first,or else,steady-state error cannot exist. 。
Characteristic is:
02)1(
2
12.0
11
0)()()(1 21
sss
sHsGsG
![Page 11: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/11.jpg)
That is
System is stable 。from:
0321
23
8.02.1
042.12.0
aaaa
sss
,即
)1(2
12.01
1
2)1(
2
)(
2)1(
212.0
11
1)(
sss
sss
sss
s
en
er
![Page 12: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/12.jpg)
The steady-state error relies on the system's construction ,still the form of the input signal.
4
7)]()()()([lim
)(lim
1)()(1)(
1)()(
0
0
2
sNssRss
ssEes
sNttn
ssRttr
eners
sss
![Page 13: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/13.jpg)
二 . steady-state error analysis when input acts on a system
)()()(1
1)( sR
sHsGsE
1)()(,0
)()()()(
00
00
sHsGswhen
sHsGs
ksHsG
v---open-loop TF
![Page 14: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/14.jpg)
whenγ=0,1,2 called respectively 0 type system,Ⅰtype system,Ⅱ type system (in generalγ<2)
so
)12()1(
)12()1()()(
2222
21
2122
2100
sTsTsT
ssssHsG
)()()(1
1lim)(lim
)()()(1
1)(
00
00
00
sRsHsG
sk
sssEe
sRsHsG
sk
sE
ssss
![Page 15: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/15.jpg)
Define kp,kv,ka error coefficient 。
Step input ,represented by kp
position error coefficient 。Speed input, represented by kv
velocity error coefficient 。
Acceleration input, represented by ka
Acceleration error coefficient 。
,lim)()(lim0
000 s
ksHsG
s
kk
ssp
,lim)()(lim10
0010 s
ksHsG
s
kk
ss
,lim)()(lim20
0020 s
ksHsG
s
kk
ssa
![Page 16: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/16.jpg)
20
10
01
1
lim
)()(1
1lim
1
)()(1
1lim
/1)(
)()()(1
1lim)(lim
)()()(1
1)(
0
00
0
00
0
00
00
00
k
ks
s
sHsGskssHsG
sk
se
ssRwhen
sRsHsG
sk
sssEe
sRsHsG
sk
sE
s
ssss
ssss
![Page 17: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/17.jpg)
20
1/1
0
lim
)]()(1[
1lim
1
)()(1
1lim
/1)(
1
0
00
0
2
00
0
2
kks
s
sHsGsk
s
ssHsGsk
se
ssRwhen
s
s
sss
![Page 18: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/18.jpg)
30
2/11
0
lim
)]()(1[
1lim
1
)()(1
1lim
/1)(
2
0
0020
3
00
0
3
kks
s
sHsGsk
s
ssHsGsk
se
ssRwhen
s
s
sss
![Page 19: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/19.jpg)
system
Stationary error coefficient
Steady state error
type
0
Ⅰ
Ⅱ
)(1)( 0 tRtr tVtr 0)( 2)( 20tAtr
k
R
10
k
V0
k
A0
0
00
0
00k
k
k
pk vk ak
sse
![Page 20: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/20.jpg)
to increase system type and open-loop gain can increase system's accuracy, but would lower stability, and must consider completely.
exam.: given a system.its block diag. is
when H(s)=1and H(s)=0.5,solve ess 。
12
102 ss
)(sH
)(sC
-
)(15)( ttr
![Page 21: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/21.jpg)
When H(s)=1,open-loop TF
When H(s)=0.5,
0,10,12
10)()(
2
k
sssHsG
11
5
101
5
10
k
Ress
0,5,12
5.010)()(
2
kss
sHsG
6
5
51
5
10
k
Ress
![Page 22: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/22.jpg)
if in exam. above, H(s)=1,ess=0.2,solve k=?
when , solve ess i
n exam. above.
Because 0 type system ess = ∞ under the speed input and acceleration input, from addition rule,ess=∞
2412.0
51,
100
ssss e
Rkso
k
Re
1)(,2
1)(1)( 2 sHttttr
![Page 23: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/23.jpg)
三 .steady state error analysis under interference
We hope ess=0 under interference,but it is impossible 。
if input R(s)=0, under interference N(s), ess :
)()()()(1
)()(lim
)()()()(1
)()()(
21
2
0
21
2
sNsHsGsG
sHsGse
sNsHsGsG
sHsGsE
sss
![Page 24: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/24.jpg)
G1(s)
N(s)R(s)
E(s)
-
C(s)
H(s)
G2(s)
)()()()(1
)()(lim
)()()()(1
)()()(
21
2
0
21
2
sNsHsGsG
sHsGse
sNsHsGsG
sHsGsE
sss
![Page 25: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/25.jpg)
under interference N(s),steady state output changes, it is connected with open-loop TF
G(s)=G1(s)G2(s)H(s) ,interference N(s) and its action position.
r(t)=0
-C(t)
12
Ts
k
)(1)( 0 tMtn
0k s
k1
(a)
![Page 26: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/26.jpg)
r(t)=0
-
C(t)1
2
Ts
k
)(1)( 0 tMtn
0k s
k1
(b)
0
00
210
21
0
0
210
2
0
)1(1
)1(lim:)(
0
)1(1
1lim:)(
k
M
s
M
Tsskkk
Tsskk
seb
s
M
Tsskkk
Tsk
sea
sss
sss
![Page 27: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/27.jpg)
The action position is different, the steady state error is different too.
in front of action position,if increase an integer( proportion) to replace
)1
1(0
0 sTk 0k
r(t)=0
-
C(t)1
2
Ts
k
)(1)( 0 tMtn
)1
1(0
0 sTk
s
k1
(b)
![Page 28: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/28.jpg)
improving integer No.in front of interference action point can decrease or cancel steady state error under interference , but lower the system's stability.
0
)1()
11(1
)1(lim 0
21
00
21
0
s
M
Tsskk
sTk
Tsskk
ses
ss
![Page 29: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/29.jpg)
in a word, to decrease steady state error under input, we can increase open-loop TF integer No. and gain.
in order to decrease interference error,we must improve integer No.and gain in front of interference action point.this can lower the system's stability,and even transient response goes bad.
Following is a good method.
![Page 30: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/30.jpg)
四 . method of decreasing or canceling steady state error
1. compensation according to interference If interference can be measured ,at the same
time,the influence to system is clear and definite, compensation can then improve the steady state accuracy.
G2(s)
Gn(s)
G1(s)
C(s)
R(s) E(s)
N(s)
-
+
![Page 31: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/31.jpg)
Under interference,output:
completely cancel the influence to system output.
adding compensating equips makes the system's steady state output independent of interference, that is to say, steady state error 0.
,0)(,)(1)(
)()()(1
)()()()()(
1
21
212
sCthensGsGif
sNsGsG
sGsGsGsGsC
n
n
![Page 32: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/32.jpg)
Exam.:
System output:
)1( sTs
k
m
m
1
1
1 sT1k
)(sGn
-
R(s)=0
N(s)
C(s)
Compensation equip.
amplifier filter
)(
)1)(1(1
]1
)(1[)1(
)(
1
1
1
1
sN
sTsTs
kksTk
sGsTs
k
sC
m
m
nm
m
n
![Page 33: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/33.jpg)
if select the system's output
independent of interference,physically too difficult to
realize, because the order of denominator of a TF is higher
than numerator.
if select when steady state,
This is completely steady state compensated,and easily realized.
),1(1
)( 11
sTk
sGn
,1
)(1k
sGn
0)(
)1)(1(1
]1
)1
(1[)1(
lim
)(lim)()(lim
1
1
1
1
1
0
0
sN
sTsTskk
sTk
ksTsk
s
scsctc
m
m
m
m
s
ns
nnt
![Page 34: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/34.jpg)
2. Compensation according to given input.
If completely compensated,
R(s) G1(s) G2(s)
Gr(s)
-
C(s)
Compensation equipment
![Page 35: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/35.jpg)
Similarly, complete compensation is too difficult to realize. usually adopt the steady-state compensation to cancel the steady-state error to some inputs.
)(
1)()()(
0)()()(
)()()(1
)()]()([)(
2
21
21
sGsGsCsR
sCsRsE
sRsGsG
sGsGsGsC
r
r
![Page 36: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/36.jpg)
No compensation , open-loop TF
typeⅠ,input is a ramp,there are
some steady-state error.
tVtrsTs
ksG
sT
ksGif 0
2
22
1
11 )(,
)1()(,
1)(
)1)(1()()()(
21
2121
sTsTs
kksGsGsG
21
0
kk
Vess
![Page 37: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/37.jpg)
Introduce compensation Gr(s) to input
If select so
but is difficult to realize physically 。
)()()(1
)()(1
)()()(1
)()]()([)()(
21
2
21
21
sRsGsG
sGsG
sRsGsG
sGsGsGsRsE
r
r
2
2
2
)1(
)(
1)(
k
sTs
sGsGr
0)( sE
2
2 )1()(
k
sTssGr
![Page 38: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/38.jpg)
If take ,so
Thus steady-state compensated 。
2
)(k
ssGr
)(lim
)()(
)(
)1)(1(1
)1(1
)()()(1
)(1
)(
0
20
0
21
21
2
2
2
21
22
ssEes
VsRtVtr
sR
sTsTskksTsk
ks
sRsGsG
sGks
sE
sss
![Page 39: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/39.jpg)
3-8 using Matlab analyze in time domain
3.8.1 stability analysisPzmap :plot zeros and poles diag.Tf2zp:transform transfer function to zero-pole typeRoots:solve roots of a polynominal
27243
64523)(
.
235
23
4
4
sssss
sssssG
systemfollowingofstabilitytheupcheck
![Page 40: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/40.jpg)
>> num=[3 2 5 4 6]
num =
3 2 5 4 6
>> den=[1 3 4 2 7 2]
den =
1 3 4 2 7 2
>> [z,p,k]=tf2zp(num,den)
z =
0.4019 + 1.1965i
0.4019 - 1.1965i
-0.7352 + 0.8455i
-0.7352 - 0.8455i
p =
-1.7680 + 1.2673i
-1.7680 - 1.2673i
0.4176 + 1.1130i
0.4176 - 1.1130i
-0.2991
k =
3
>> pzmap(num,den)
![Page 41: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/41.jpg)
-2 -1.5 -1 -0.5 0 0.5-1.5
-1
-0.5
0
0.5
1
1.5Pole-Zero Map
Real Axis
Imag
inar
y A
xis
![Page 42: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/42.jpg)
roots(den)
ans =
-1.7680 + 1.2673i
-1.7680 - 1.2673i
0.4176 + 1.1130i
0.4176 - 1.1130i
-0.2991
>> roots(num)
ans =
0.4019 + 1.1965i
0.4019 - 1.1965i
-0.7352 + 0.8455i
-0.7352 - 0.8455i
To illustrate the meaning of the following command.
![Page 43: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/43.jpg)
3.8.2 transient characteristic analysis
Step:solve unit step response
num=[25]
num =
25
>> den=[1 4 25]
den =
1 4 25
>> step(num,den)
>> grid0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (sec)
Am
plitu
de
![Page 44: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/44.jpg)
[num,den]=zp2tf([-2],[-1,-0.2+10*i,-0.2-10*i],100)
num =
0 0 100 200
den =
1.0000 1.4000 100.4400 100.0400
>> [y,x,t]=step(num,den)
subplot(2,1,1),plot(t,y)
[yi,xi,t]=impulse(num,den);
>> subplot(2,1,2),plot(t,yi)
![Page 45: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/45.jpg)
0 5 10 15 20 250
1
2
3
0 5 10 15 20 25 30-10
-5
0
5
10
15
![Page 46: 3-5 steady-state error calculation The steady-state performance is an important characteristic of control system, it represses the ability to follow input.](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649e495503460f94b3ca82/html5/thumbnails/46.jpg)
homework:
3-15
3-18