analysis and design of linear control system part2-...bode diagram fig. 12.6: pi controller ( ) 1 (...
TRANSCRIPT
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Analysis and Design of Linear
Control System –Part2-
Instructor: Assoc. Prof. Takeshi Hatanaka (S5-204B)
Spring, 2015
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12.1 Modeling Uncertainty
9.3 Stability Margins
2nd Lecture
Modeling Uncertainty, Stability Margin
Robust Stability
Keyword :
12 Robust Performance
(12.3 Performance in the Presence of Uncertainty)
Complementary Sensitivity Function
Small Gain Theorem
Keyword :
12.2 Stability in the Presence of Uncertainty
(11.5 Fundamental Limitation)
(9.2 The Nyquist Criterion)
12.2 Stability in the Presence of Uncertainty
(pp. 347--352)
(pp. 278--282)
(pp. 352--358)
(pp. 358--361)
(pp. 331--340)
(pp. 270--278)
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12.1 Modeling Uncertainty
• Parametric uncertainty
parameters describing the system are unknown
Mass of a car changes with the number of passengers
Fig. 3.3Disturbance response
(3rd) (4th) (5th)
mass
gear ratio
20001600 m
14 ,12 ,10 speed 4010 ev
3
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Modeling Uncertainty
• Parametric uncertainty
parameters describing the system are unknown
• Unmodeled dynamics
neglected mechanisms such that the simple
model does not include.
• detailed model of the engine dynamics
• slight delays that can occur in electronically
controlled engines
The design based on a simple nominal
model will give satisfactory control. Ex. 12.1
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4
122 2
2.05.0~
i iii ss
s
sP
Unmodeled Dynamics
Vibration mode
Frequency response
Nominal System
(Rigid body mode)
vibration modeUnmodeled dynamics
http://mobile.jaxa.jp/gallery_list/
index.php?category=iss
P~
P
P
sP
5.0
P
02.0,10
2,5.0,2.0
4
321
i
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Unmodeled Dynamics
Additive perturbations Multiplicative perturbations
: nominal model
: unmodeled
dynamics
: actual model
PP~
P
PP )1(~
P
P/:
PP~
,
Frequency response
P~
P
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[Ex. 12.2] Similar in Open Loop but Large Differences in Closed Loop
(a) Step response (open loop) (b) Step response (closed loop)
open loop
Fig. 12.3(a)
(12.1)
[When Are Two Systems Similar ?]
1)(1
s
ksP
22 )1)(1()(
sTs
ksP 025.0T
100kur y
)(sC )(sPe
1
r y)(sP 1)( sC
closed loop
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[Ex. 12.3] Different in Open Loop but Similar in Closed Loop
(a) Step response (open loop) (b) Step response (closed loop)
open loop
Fig. 12.3(b)
(12.2)
[When Are Two Systems Similar ?]
closed loop
1)(1
s
ksP 100k
)1()(2
s
ksP
r y)(sP
ur y)(sC )(sP
e
1
1)( sC
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Nyquist Criterion (§9.2)
Re
1
O
Im
)( jL
Fig. 11.1
ur y)(sC )(sP
e
1
d n
PCL
Loop Transfer Function
Nyquist’s Stability Theorem
Theorem 9.1 and 9.2
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)( jL
Stability Margin (§9.3)
Re
1
OP
G
Im
m
Fig. 9.9 (a)
mg/1
0 dB
180
gc
pcPhase
Gain
mg1 02 0 lo g
m
Fig. 9.9 (b)
)(/1 pcm iLg
Gain Margin
(9.5) 52mg
Phase Margin
)(a r g g cm iL (9.6) 6 03 0m
Stability Margin ms
8.05.0 mssm Ms /1
( : maximum sensitivity)ms sM /1
( 6-14 dB)
Fig. 11.6(b)
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Robust Stability Using Nyquist’s Criterion
Additive Uncertainty
: stable perturbations
Perturbed Nyquist curve
doesn’t reach -1 (i.e. perturbed
closed loop is stable) when
Fig. 12.5(b)
Nyquist plot (additive uncertainty)
PP
PCL
CPCCP )(
LC 1C
Im
ReO
Loop Transfer Function
CL
P
C
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Fig. 12.5(b)
C
Robust Stability Using Nyquist’s Criterion
Perturbed Nyquist curve doesn’t reach -1
when
LC 1C
PC
1
CS
1
PCS
1
1
TPC
PC
P
11
PC
PCT
1
P/:
Im
ReO
Additive Uncertainty
Multiplicative Uncertainty
P
C
P
C
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Robust Stability Using Nyquist’s Criterion
small big is allowed
big small is allowed
)( jT
)( jT
)( j
)( j
Sufficient condition for robust stability
(12.6)0
)(
1)(
jTj
Uncertainty
P
C
-
HW
un
P
T
PC
CGun
1 P
dPT
G
dG
un
un (12.13)
Performance in the Presence of Uncertainty (§12.3)
Fig. 11.1
ur y)(sC )(sP
e
1
d n
Measurement noise typically has high frequencies
dP
noise
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Bode’s Integral Formula (§11.5)
Complementary sensitivity function:
where the summation is over all right
half-plane zeros.
(11.20)
(11.19)
Re
Im
Zero
slow fast
izd
jT 1)(log
0 2
kpdjS
0 )(log
0||log T1|| T
T
1|| T
0||log T
RHP zeros fast (big): better
slow (small): worse
Sensitivity function: (1st lecture)
Re
Im
××Pole
slow fast
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Complementary Sensitivity Function
)()(1
)()()(
sCsP
sCsPsT
1
11
1
PC
PC
PCTS
: Complementary Sensitivity
Bandwidth Frequency bT
])dB[3( 2
1)( jT
])dB[2( 25.1TM
)( jT
]dB[3bT
: Maximum Peak
Magnitude of
)(max
jTMT
])dB[ 2( 25.1TM
)( jTTM
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Robust Stability
(sufficient condition)
Fig. 12.6Bode diagram
: PI controller
)(
1)(
jTj
0142.0
38.1)(
ssP
ssC
18.072.0)(
[Ex. 12.5] Cruise Control
40
20
0
-20
-40 210 110 010 110 210
should be smaller than
most sensitive
T
13.1TM
T/1
T/135.0mt
21.1bT
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around the gain crossover frequencies
small is required
Fig. 12.6 ’Bode diagram
A simple model that describes
the process dynamics well
around the crossover frequency
is often sufficient for design
Robust Stability (sufficient condition)
1
T
)( j
T
T/1
[Ex. 12.5] Cruise Control
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Robust Stability Using Small Gain Theorem
sufficient condition for robust stability
(12.6)
another interpretation by using small gain theorem
Theorem 9.4 Small Gain Theorem (§9.5)
Consider the closed loop system shown in Fig. 9.15,
where and are stable systems and the signal
spaces are properly defined. Let the gains of the
systems and be and . Then the closed
loop system is input/output stable if .
Fig. 9.15
1H 2H
1H 2H 1 2
121
2H
1H
0 )(
1)(
jT
j
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Robust Stability Using Small Gain Theorem
sufficient condition for robust stability
Small Gain Theorem
Closed loop system is input/output stable
Fig. 12.7
P
C
1
T
1 T
T
PC
PCT
1
-
for small
for large
: with integral action
Load Sensitivity Function* (§11.3)
(11.8, 11.9)C
TPS
PC
PGyd
1
)(sCs
ki
Fig. 11.1
ur y)(sC )(sP
e
1
d n
C
1
1
1
sP
P
iyd k
sCC
TG 1 )1( T
PPSGyd )1( S
10-2 10-1 100 101 102
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for small
for large
Noise Sensitivity Function* (§11.3)
(11.10)
PID + 2nd-order
noise filter
Roll off
P
TCS
PC
CGun
1
Fig. 11.1
ur y)(sC )(sP
e
1
d n
C
P
1
PPTGun
1 )1( T
CCSGun )1( S
3)1(1
s
P
12/1
22
sTsTsk
s
kkC
ff
di
p
10-1 100 101 102
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12.1 Modeling Uncertainty
9.3 Stability Margins
2nd Lecture
Modeling Uncertainty, Stability Margin
Robust Stability
Keyword :
12 Robust Performance
(12.3 Performance in the Presence of Uncertainty)
Complementary Sensitivity Function
Small Gain Theorem
Keyword :
12.2 Stability in the Presence of Uncertainty
(11.5 Fundamental Limitation)
(9.2 The Nyquist Criterion)
12.2 Stability in the Presence of Uncertainty
(pp. 347--352)
(pp. 278--282)
(pp. 352--358)
(pp. 358--361)
(pp. 331--340)
(pp. 270--278)
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Reading Assignment: 3rd Lecture
11.4 Feedback Design via Loop Shaping
Loop Shaping
Bode’s Relations
Keyword :
11 Frequency Domain Design
(9.4 Bode’s Relations and Minimum Phase Systems)
11.5 Fundamental Limitations
Right Half-Plane Poles and Zeros
Gain Crossover Frequency Inequality
Keyword :
(pp. 326--331)
(pp. 283--285)
(pp. 331--340)