Control Systems - Lehigh Universityinconsy/lab/css/ME389/lectures/lecture08_… · Rule 2: The...
Transcript of Control Systems - Lehigh Universityinconsy/lab/css/ME389/lectures/lecture08_… · Rule 2: The...
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Classical Control – Prof. Eugenio Schuster – Lehigh University 1
Control Systems
Lecture 8 Root Locus
Classical Control – Prof. Eugenio Schuster – Lehigh University 2
Root Locus Controller
)(sC-
+ )(sR )(sY)(sG)(sE )(sU
)(1)()(
)()()(1)()(
)()(
sKLsGsC
sHsGsCsGsC
sRsY
+=
+=
)(sH
Plant
Sensor
Writing the loop gain as KL(s) we are interested in tracking the closed-loop poles as �gain��K varies
⇒= )()( sKDsC
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus Characteristic Equation:
0)(1 =+ sKL
The roots (zeros) of the characteristic equation are the closed-loop poles of the feedback system!!!
The closed-loop poles are a function of the �gain��K
Writing the loop gain as
nnnn
mmmm
asasasbsbsbs
sasbsL
++++++++
==−
−−
−
11
1
11
1
)()()(
The closed loop poles are given indistinctly by the solution of:
KsLsKbsa
sasbKsKL 1)(,0)()(,0)()(1,0)(1 −==+=+=+
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus
{ } { })()()(1 LKLsKL numdenrootszerosRL +=+=when K varies from 0 to ∞ (positive Root Locus) or
from 0 to -∞ (negative Root Locus)
!"
!#$
=∠
=⇔−=>
180)(
1)(1)(:0sL
KsL
KsLK
Phase condition
Magnitude condition
!"
!#$
=∠
−=⇔−=<
0)(
1)(1)(:0sL
KsL
KsLK
Phase condition
Magnitude condition
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus by Characteristic Equation Solution
Example: K-
+ )(sR )(sY( )( )110
1++ ss
)(sE )(sU
)10(11)()(
2 KssK
sRsY
+++=
Closed-loop poles: 0)10(110)(1 2 =+++⇔=+ KsssL
24815.5 Ks −
±−=
28145.5
5.524815.5
10,1
−±−=
−=
−±−=
−−=
Kis
s
Ks
s
0481048104810
<−
=−
>−
=
KKKK
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Classical Control – Prof. Eugenio Schuster – Lehigh University
We need a systematic approach to plot the closed-loop poles as function of the gain K → ROOT LOCUS
Root Locus by Characteristic Equation Solution
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Phase and Magnitude of a Transfer Function
011
1
011
1)(asasasbsbsbsbsG n
nn
mm
mm
++++++++
= −−
−−
The factors K, (s-zj) and (s-pk) are complex numbers:
)())(()())(()(
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21
n
m
pspspszszszsKsG
−−−−−−
=
K
pk
zj
i
ipkk
izjj
eKK
pkerps
mjerzs
φ
φ
φ
=
==−
==−
…
1,)(
1,)(
pn
pp
zm
zzK
ipn
ipip
izm
izizi
erererererereKsG
φφφ
φφφφ
21
21
21
21)( =
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Phase and Magnitude of a Transfer Function
Now it is easy to give the phase and magnitude of the transfer function:
( )
( )
( ) ( )[ ]pnppzm
zzK
pn
pp
zm
zzK
pn
pp
zm
zzK
ipn
pp
zm
zz
ipn
pp
izm
zzi
ipn
ipip
izm
izizi
errrrrrK
errrerrreK
erererererereKsG
φφφφφφφ
φφφ
φφφφ
φφφ
φφφφ
+++−++++
+++
+++
=
=
=
2121
21
21
21
21
21
21
21
21
21
21)(
( ) ( )pnppzm
zzK
pn
pp
zm
zz
sG
rrrrrrKsG
φφφφφφφ +++−++++=∠
=
2121
21
21
)(
,)(
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus by Phase Condition
Example: K-
+ )(sR )(sY( )( )845
12 ++++
sssss)(sE )(sU
( )( )
( )( )( )isissssssss
ssL
222251845
1)( 2
−+++++
=
++++
=
iso 31+−=
belongs to the locus?
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus by Phase Condition
43.10887.36
45
70.78
90
[ ] 18070.784587.3643.10890 −≈+++− iso 31+−=⇒ belongs to the locus!
Note: Check code rlocus_phasecondition.m 10
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus by Phase Condition
We need a systematic approach to plot the closed-loop poles as function of the gain K → ROOT LOCUS
iso 31+−=
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus
Basic Properties: • Number of branches = number of open-loop poles • RL begins at open-loop poles • RL ends at open-loop zeros or asymptotes • RL symmetrical about Re-axis
{ } { })()()(1 LKLsKL numdenrootszerosRL +=+=
0)(0 =⇒= saK
0)()(1)(0)(1 =+⇔−=⇔=+ sKbsaK
sLsKL
!"#
>−∞→
=⇔=⇒∞=
)0(0)(
0)(mns
sbsLK
when K varies from 0 to ∞ (positive Root Locus) or from 0 to -∞ (negative Root Locus)
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus
Rule 1: The n branches of the locus start at the poles of L(s) and m of these branches end on the zeros of L(s). n: order of the denominator of L(s) m: order of the numerator of L(s)
Rule 2: The locus is on the real axis to the left of and odd number of poles and zeros. In other words, an interval on the real axis belongs to the root locus if the total number of poles and zeros to the right is odd. This rule comes from the phase condition!!!
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus
Rule 3: As K→∞, m of the closed-loop poles approach the open-loop zeros, and n-m of them approach n-m asymptotes with angles
1,,1,0,)12( −−=−
+= mnlmn
ll … πφ
and centered at
1,,1,0,11 −−=−
−=
−−
= ∑∑ mnlmnmn
ab…
zerospolesα
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus
Rule 4: The locus crosses the jω axis (looses stability) where the Routh criterion shows a transition from roots in the left half-plane to roots in the right-half plane.
Example:
)54(5)( 2 ++
+=
sssssG
5,20 jsK ±==
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus
Example: 4673
1)( 234 +++++
=ssss
ssG
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus
Design dangers revealed by the Root Locus: • High relative degree: For n-m≥3 we have closed loop instability due to asymptotes. • Nonminimum phase zeros: They attract closed loop poles into the RHP
46731)( 234 ++++
+=
ssssssG
11)( 2 ++
−=
ssssG
Note: Check code rootlocus.m 17
Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus
Viete�s formula: When the relative degree n-m≥2, the sum of the closed loop poles is constant
∑−= poles loop closed1a
nnnn
mmmm
asasasbsbsbs
sasbsL
++++++++
==−
−−
−
11
1
11
1
)()()(
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Phase and Magnitude of a Transfer Function Example:
( )( ) )20(51)735.6()(+++
+=
sssssG
p1φ
p2φ
p3φ
z1φ
pr3zr1 pr2
pr1 ( )pppz
ppp
z
sG
rrrrsG
3211
321
1
)(
)(
φφφφ ++−=∠
=iss o 57 +−==
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus- Magnitude and Phase Conditions
{ } { })()()(1 LKLsKL numdenrootszerosRL +=+=when K varies from 0 to ∞ (positive Root Locus) or
from 0 to -∞ (negative Root Locus)
⇔−=>K
sLK 1)(:0
⇔−=<K
sLK 1)(:0
( ) ( )[ ]pnppzm
zzpKipn
pp
zm
zz
pn
mp e
rrrrrrK
pspspszszszsKsL φφφφφφφ +++−++++=
−−−−−−
=
2121
21
21
21
21
)())(()())(()(
( ) ( )
180)(
1)(
2121
21
21
=+++−++++=∠
==
pn
ppzm
zzK
pn
pp
zm
zz
p
psL
KrrrrrrKsL
φφφφφφφ
( ) ( )
0)(
1)(
2121
21
21
=+++−++++=∠
−==
pn
ppzm
zzK
pn
pp
zm
zz
p
psL
KrrrrrrKsL
φφφφφφφ
20
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus
Selecting K for desired closed loop poles on Root Locus: If so belongs to the root locus, it must satisfies the characteristic equation for some value of K Then we can obtain K as
KsL o
1)( −=
)(1)(
1
o
o
sLK
sLK
=
−=
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus
Example: ( )( )511)()(++
==ss
sGsL
( ) ( ) 204242
54314351)(
143
2222 =++−=
++−++−=++==⇒+−=
iisssL
Kis ooo
o
sys=tf(1,poly([-1 -5])) so=-3+4i [K,POLES]=rlocfind(sys,so)
Using MATLAB:
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus Example: ( )( )51
1)()(++
==ss
sGsL
06.42)(
1
57
==
⇓
+−=
o
o
sLK
is
57 iso +−=
43 iso +−=
When we use the absolute value formula we are assuming that the point belongs to the Root Locus!
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus - Compensators
Example: ( )( )511)()(++
==ss
sGsL
Can we place the closed loop pole at so=-7+i5 only varying K? NO. We need a COMPENSATOR.
( )( )( )51
110)()()(++
+==ss
ssGsDsL( )( )51
1)()(++
==ss
sGsL
The zero attracts the locus!!! 24
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus – Phase lead compensator
Pure derivative control is not normally practical because of the amplification of the noise due to the differentiation and must be approximated:
( )( )zp
sspszssGsDsL >
++++
== ,51
1)()()(
How do we choose z and p to place the closed loop pole at so=-7+i5?
zppszssD >
++
= ,)( Phase lead COMPENSATOR
When we study frequency response we will understand why we call �Phase Lead� to this compensator.
Classical Control – Prof. Eugenio Schuster – Lehigh University
19.14080.11104.21
?1zφ
Root Locus – Phase lead compensator Example:
Phase lead COMPENSATOR
( )( )zp
sspszssGsDsL >
+++
+== ,
511)()()(
03.9303.45304.2180.11119.1401801 ==+++=zφ 735.6−=⇒ z( ) ( ) 180)( 2121 =+++−++++=∠ p
nppz
mzzK psL φφφφφφφ
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Let us choose p=20
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus – Phase lead compensator Example:
Phase lead COMPENSATOR
( )( )511
20735.6)()()(
++++
==sss
ssGsDsL
11757
=
+−=
Kiso
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus – Phase lead compensator
Selecting z and p is a trial an error procedure. In general:
• The zero is placed in the neighborhood of the closed-loop natural frequency, as determined by rise-time or settling time requirements. • The poles is placed at a distance 5 to 20 times the value of the zero location. The pole is fast enough to avoid modifying the dominant pole behavior.
The exact position of the pole p is a compromise between:
• Noise suppression (we want a small value for p) • Compensation effectiveness (we want large value for p)
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zppszs
sD > ,++
=)( Phase lead COMPENSATOR
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus – Phase lag compensator Example:
Phase lag COMPENSATOR
( )( )511
20735.6)()()(
++++
==sss
ssGsDsL
( )( )2
00010735.6
511
20735.6lim)()(lim)(lim −
→→→×=
++++
===sss
ssGsDsLKsssp
What can we do to increase Kp? Suppose we want Kp=10.
( )( )zp
sssspszssGsDsL <
++++
++
== ,51
120735.6)()()(
We choose: 48.14810735.61 3 =×=
pz
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus – Phase lag compensator Example:
( )( )511
20735.6
001.014848.0)()()(
++++
++
==sss
ssssGsDsL
31.1803.594.6
=
+−=
Kiso
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Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus – Phase lag compensator
Selecting z and p is a trial an error procedure. In general:
• The ratio zero/pole is chosen based on the error constant specification. • We pick z and p small to avoid affecting the dominant dynamic of the system (to avoid modifying the part of the locus representing the dominant dynamics) • Slow transient due to the small p is almost cancelled by an small z. The ratio zero/pole cannot be very big.
The exact position of z and p is a compromise between:
• Steady state error (we want a large value for z/p) • The transient response (we want the pole p placed far from the origin)
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zppszs
sD < ,++
=)( Phase lag COMPENSATOR
Classical Control – Prof. Eugenio Schuster – Lehigh University
Root Locus - Compensators
Phase lead compensator: pzpszssD <
++
= ,)(
pzpszssD >
++
= ,)(Phase lag compensator:
We will see why we call �phase lead� and �phase lag� to these compensators when we study frequency response
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