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Analysis and Design of Linear Control System –Part2-
Instructor: Prof. Masayuki Fujita
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3rd Lecture
11.4 Feedback Design via Loop Shaping
Loop ShapingBode’s Relations
Keyword :
11 Frequency Domain Design
(9.4 Bode’s Relations and Minimum Phase Systems)
11.5 Fundamental LimitationsRight Half-Plane Poles and ZerosGain
Crossover Frequency Inequality
Keyword :
(pp.326 to 331)
(pp.283 to 285)
(pp.331 to 340)
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11.4 Feedback Design via Loop Shaping
Loop transfer function
Sensitivity Complementary Sensitivity
• Load disturbance
• Tracking
• Robust stability
• Measurement noise(11.8) (12.11)
(12.15)
(12.6)
(12.13)
)()()( sCsPsL =
LLsT+
=1
)(L
sS+
=1
1)(
PSGyd = PdPS
GdG
yd
yd = /1 T
-
improve not only stability (Nyquist) but also performanceand
robustness
Loop shapingChoosing a compensator that gives a loop transfer
function with a desired shape
)( ωjC)( ωjL
11.4 Feedback Design via Loop Shaping
Loop transfer function)()()( sCsPsL =
Fig. 11.1
ur yη)(sC )(sPe ν ∑∑∑
1−
d n
-
(a) Frequency response ( ) (b) Frequency esponse ( )Fig.
11.8
• Load disturbances will be attenuated by a factor of 100
At low frequencies
Loop Gain Feedback Performance
1001
)(11)( <
+=
ωω
jLjS
<
−≤
+ 1001
11
11
LL101)( >ωjL
)( ωjL )( ωjS
Load DisturbanceAttenuation
)( ωjL)( ωjS
Loop Shaping
-
Loop Shaping
At high frequencies
(a) Frequency response ( ) (b) Frequency response ( )
High-frequency Measurement Noise
Loop Gain01.0
991
)(1)()( ≈<
+=
ωωωjL
jLjT
<
−≤
+ 991
11 LL
LL
01.0)(
-
)( ωjL
)( ωjL∠mϕ
gcngcω
Bode’s Relations (§9.4)
the phase is uniquely given by
at gain crossover frequency
(9.8)[rad]
ωωπω
log)(log
2)(arg 0 d
jGdjG ≈
gcm n2πϕπ =+− gcn : slope of the gain curve at
gain crossover frequency
(minimum phase systems)the shape of the gain curve
gcω
gcω
Loop Shaping
Fig. 11.8
-
)90(2/1 °=→−= πϕmgcn)0(02 °=→−= mgcn ϕ
)( ωjL
)( ωjL∠mϕ
gcngcω
mg
: slope of the gain curve at gain crossover
: phase margin
the slope of the gain curve at gain crossover cannot be too
steep
(11.11)
gcm n2πϕπ =+−
πϕm
gcn22+−=
Bode’s Relations (§9.4)
mϕ
gcngcω
Fig. 11.8
gcω
( )°−°= 9030mϕ
−≤≤− 1
35
gcn: gain marginmg( )52−=mg
pcω
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Bode’s Relations
High-frequency Measurement Noise
Load DisturbanceAttenuation
gcω)( ωjL
(a) Frequency response ( ))(sL
Loop Shaping
Fig. 11.8
)(/1 pcm iLg ω=
)(arg gcm iL ωπϕ +=
• Gain Margin
• Phase Margin
• Stability Margin
)52( −
)6030( °−°
)8.05.0( −Sm Ms /1=
-
Sensitivity Function )( ωjS
A
bSω
2
-
ImportantRelations
2
-
[Ex. 11.9] Balance system (§6.3)
Fig. 6.2 (a) Segway (b) Cart-pendulum system
Equations of motion( ) FmlpcmlpmM +−−=−+ 2sincos θθθθ ( ) θθγθθ
sincos2 mglpmlmlJ +−=−+
11.5 Fundamental Limitations
(6.4)
-
))(( 22222
ttt
tpF mglMslmJMs
mglsJH+−−
+−=
from to
*
poles:
RHP poleRHP zero
Fig. 6.2 (b)
from to
[Ex. 11.9] Balance system (§6.3)
zeros:
tttF mglMslmJM
mlH+−−
= 222 )(θ
mMM t +=2mlJJt +=
{ }tJmgl /±68.2=p09.2=z
×Pole ○Zero
Im
Re
F θ
F p
( ){ }22/,0,0 lmJMmglM ttt −±
:pFH
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Effect of RHP Poles
Zero（○）: 0
Pole（×） : 1 ,1−
sssC 1)( −=
0)1)(1()(
11)( y
ssssr
ssy
−++
+=
11
)()()(
−==
ssusysP
Unstable0y : initial value
Im
Re01− 1
))()()(()( sysrsCsu −=
1 2 3 4 5 6
5.1
5.0
2
0
1
t0
)(ty
Step response
yr )(sC )(sP−
du
00 =y
01.00 =y
-
t
1.0=a
0 2 4 86 10
0
5.0
1
5.1
Effect of RHP Zeros
)12)(1(1)(++
+=
ssassG
Zero（○）:a1
−
Pole（×） : ,1− 5.0−
：Small No Effect
：Large Overshoot
0
-
(11.13)
: minimum phase part: all-pass system
Factor the process transfer function as
Ex. )
minimum phase part all-pass system
*
RHP poles, zeros and time delay
Gain Crossover Frequency Inequality
(nonminimum phase part s.t. : negative)
)()()( sPsPsP apmp=
mpP
apP1)( =ωjPap apParg
ss
sss
ssssP
+−
⋅++
+=
++−
=11
11
11)( 22
11
)(111)(
22
22
=+
−+=
+−
=ω
ωωωω
jjjPap
mpP apP
-
Ex. )
minimum phase part all-pass system
Q??=
-270
ss
sss
ssssP
+−
⋅++
+=
++−
=11
11
11)( 22
apP
mpP=P
mpP apP
mpParg
apPargParg
-
(11.14)
*
*
Phase of
Slope of at
Derivation of the Gain Crossover Frequency Inequality
)( gcjL ω CPPPCL apmp==
m
gcgcmpgcap
gc
jCjPjPjL
ϕπ
ωωω
ω
+−≥
++= )(arg)(arg)(arg
)(arg
)( gcjL ω
mϕ
)( ωjL
)(arg ωjL
gcngcω
gc
gc
djCjPd
djLd
n
mp
gc
ωω
ωω
ωωω
ωω
=
=
=
=
log)()(log
log)(log
1)( =ωjPap
gcω
-
mϕ
)( ωjL
)(arg ωjL
gcngcω
Derivation of the Gain Crossover Frequency Inequality
(11.15)
Bode’s Relations
holds for minimum phase systems
Combining it with (11.14)(11.14)
ωωπω
log)(log
2)(arg 0 d
jGdjG ≈
( )
2
log)()(log
2
)()(arg
π
ωωωπ
ωω
gc
gc
gcgcmp
gcgcmp
n
djCjPd
jCjP
=
≈
(9.8)
mgcgcmpgcap jCjPjP ϕπωωω +−≥++ )(arg)(arg)(arg( )
2 )()(arg πωω gcgcgcmp njCjP ≈
lgcmgcap njP ϕπϕπω :2
)(arg =+−≤−
-
mpParg
apPargParg
Gain Crossover Frequency Inequality
(11.15)
Gain Crossover Frequency Inequality
lgcmgcap njP ϕπϕπω :2
)(arg =+−≤−
: slope at
: gain crossover freq.
: required phasemargin
)( ωjL
)(arg ωjLmϕ
gcn
gcωgcω
• The phase lag of the nonminimum phase component must not betoo
large at the crossover frequency.• Nonminimum phase components
imposes severe restrictions on possible crossover frequencies.
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Gain Crossover Frequency Inequality
slope:
• for high robustness required phase margin :
slope :
• for lower robustness required phase margin :
30
90
allowable phase lag of at : °° −= 6030mϕapP gcω lϕ
gcω
°= 60mϕ1−=gcn
°= 30lϕ
°= 45mϕ2/1−=gcn
°= 90lϕ
apParg
mpParg
Parggcω
Gain Crossover Frequency Inequality
(11.15)lgcmgcap njP ϕπϕπω :2
)(arg =+−≤−
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3rd Lecture
11.4 Feedback Design via Loop Shaping
Loop ShapingBode’s Relations
Keyword :
11 Frequency Domain Design
(9.4 Bode’s Relations and Minimum Phase Systems)
11.5 Fundamental LimitationsRight Half-Plane Poles and ZerosGain
Crossover Frequency Inequality
Keyword :
(pp.326 to 331)
(pp.283 to 285)
(pp.331 to 340)
Analysis and Design of Linear Control System –Part2-スライド番号
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