Analysis and Design of Linear Control System –Part2-
Instructor: Prof. Masayuki Fujita
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)
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
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
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.
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 =+−≤−
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 2スライド番号 3スライド番号 4スライド番号 5スライド番号 6スライド番号 7スライド番号 8スライド番号 9スライド番号 10スライド番号 11スライド番号 12スライド番号 13スライド番号 14スライド番号 15スライド番号 16スライド番号 17スライド番号 18スライド番号 19スライド番号 20スライド番号 21スライド番号 22