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1
Basic Control Theory and Its Application in AMB Systems
Zongli Lin
University of Virginia
2
Representation of a Plant
Example 1:
my F u
uy
m
2
1 2
2
1
1
Y(s) 1Transfer function:
U(s)
1State space:
00 1where , , 1 0
0 0 m
ms
x x
x um
y x
x Ax Bu
y Cx
A B C
3
1 11 12 1
2 21 22 2
4 2
2
Example 2:
Y ( ) G ( ) G ( ) U ( )
Y ( ) G ( ) G ( ) U ( )
, or
s s s s
s s s s
x Ax Bu x u
y Cx y
1 11 12 1
2 21 22 2
2 2
2
Example 3:
Y ( ) G ( ) G ( ) U ( )
Y ( ) G ( ) G ( ) U ( )
, or
n
s s s s
s s s s
x Ax Bu x u
y Cx y
Representation of a Plant
4
Representation of a Plant
1 1
2 2
1 1
2 2
11 14
41 44
Y ( ) U ( )G ( , ) G ( , )
Y ( ) U ( )
Y ( ) U ( )G ( , ) G ( , )
Y ( ) U ( )
x x
x x
y y
y y
s ss s
s s
s ss s
s s
or
ω
ω
x
y
uA G B 0x x
uG A 0 B
C 0y x
0 C
Example 4
5
Principles of Feedback
Tracking Requirement
R
Y( ) K( )G( )T ( )
R( ) 1 K( )G( )
s s ss
s s s
( )r t )K(s G( )s ( )y t
R
K( )G( )T ( ) 1, 0,
1 K( )G( ) b
j jj
j j
K( )G( ) 1
Closed-loop Stability
j j
6
( )r t )(G2 s)K(s )(G1 s ( )y t
( )d t
Disturbance Rejection
2D
2D
G ( )Y( )T ( )
D( ) 1 K( )G( )
G ( )T ( )
1 K( )G( )
sss
s s s
jj
j j
K( )G( ) 1
Closed-loop Stability
j j
Disturbance Rejection Requirement
1 2G( ) G ( )G ( )s s s(change in load, aero or mechanical forces, etc)
7
Sensitivity
( )r t )K(s G( )s ( )y t
Sensitivity to plant uncertainties
R
R R RR
R
T Ra
R
T /T TK( )G( )T ( ) , Sensitivity:
1 K( )G( ) T
TS ( : natural frequency, damping ratio, unbalance, etc)
T
s s as
s s a/a a
aa
a
K( )G( ) 1
Closed-loop Stability
j j
R
R
TG
TG
1S ( )
1 K( )G( )
1S ( )
1 K( )G( )
ss s
jj j
8
High Gain Instability
High-Gain Causes Instability
2
1
( 4) 16s s
( )r t k ( )y t
R 3 2T ( )
8 32
ks
s s s k
k=256
9
Limitations of Constant Feedback
Constant feedback is often insufficient
2
1
ms
( )r t k ( )y t
T
R 2
/T ( )
/
k ms
s k m
1,2
/ , 0
/ , 0
j k m ks
k m k
10
Integral Action
Use of integral action for zero steady state error (r(t)=1(t), e.g., raise rotor vertical position)
0 0
1Y( )
1 ( 1)
y ( ) lim Y( ) lim1
1, as 1
ss s s
k ks
s k s s s k
kt s s
s kk
kk
1
1s
( )r t K( )s ( )y t
K(s) = k K(s) = k/s
2
0
20
1Y( )
y ( ) lim Y( )
lim
y ( ) 1, 0
ss s
s
ss
ks
ss s kt s s
k
s s kt k
Observation: Large k causes actuator saturation
11
Differential Action
Use of differential action for closed loop stability
2
1
s
( )r t K( )s ( )y t
2Y( )
Unstable for any
ks
s kk
2
Y( )
Stable for any 0, 0
P D
D P
D P
k k ss
s k s k
k k
1K( ) I Ds kP k k s
s
•K(s)=k •K(s)=kP+kDs
In general: PID control
12
Example
Decentralized PI/PD position control of active magnetic bearings
Above: cross section of the studied active magnetic bearing system
Right: cross section of the studied radial magnetic bearing
Reference: B. Polajzer, J. Ritonja, G. Stumberger, D. Dolinar, and J. P. Lecointe, “Decentralized PI/PD position control for active magnetic bearings”, Electrical Engineering, vol. 89, pp. 53-59, 2006.
13
Stabilization
Stabilization: PD control
2G( )
y i y
i
y
my K y K i
Ks
ms K
'PD '
1G ( ) , 1
1d
d dd
sTs K T
sT
'Routh-Hurwitz Table: , y
di
kTd T Kd
k
14
Steady State Error
Steady state error reduction: PI Control (e.g., to avoid mechanical contact)
PI/PD control configuration
PI
1G ( ) i
ii
sTs K
sT
15
PD/PD vs PID Control
PD: Choice of Kd and Td is ad hoc.
PI: Choice of Ki and Ti is ad hoc. PID: Choice of 3 parameters even harder
16
Experimental Results
17
Experimental Results
18
Lag and Lead Compensation
Compensation objectives:
Increased PM
Increased PM
Increased steady state accuracy
19
Lag and Lead Compensation
Phase lag compensator
00 0
1
( ) ,1
p
p
ss
C s a s sss
0 0
One can exactly determine the values
of , and to achive a pre-specified
amount of improvement in phase margin
and steady state accuracy.
pa s s
20
Lag and Lead Compensation
Phase lead compensator
00 0
1
( ) ,1
p
p
ss
C s a s sss
0 0
One can exactly determine the values
of , and to achive a pre-specified
amount of improvement in phase margin.
pa s s
21
Two mass symmetric model of the rotor in an LP centrifugal compressor
sy by
, ,s s um k e
,b rm k
by
,b rm k
An LP Centrifugal Compressor (ISO 14839)
22
Mathematical model
2
2
cos
2 ( 2 ) 2
sin
2 ( 2 ) 2
s s s s s b a s s u
b b s s s r b i x
s s s s s b a s s u
b b s s s r b i y
m x k x k x q y m e t
m x k x k k x k i
m y k y k y q x m e t
m y k y k k y k i
7
6
where:
lumped shaft mass 560kg
lumped bearing mass 110kg
shaft bending stiffness
6.5 10 N/m
position stiffness 2 10 N/m
s
b
s
r
m
m
k
k
6
actuator gain 110N/A
aero cross coupling stiffness
2.5 10 N/m
unbalance eccentricity
rotor rotational speed
i
a
u
k
q
e
An LP Centrifugal Compressor (ISO 14839)
23
4 2 2
10 6
5 4 10 2 14 2 7
2( ) ( ) ( )
2 2 2 2
1.474 10 2.5 10 ( ) ( )
1.232 10 4.846 10 2.6 10 560 6.5 10
s i as x s
b s s s s r b s s r s s
x s
k k qX s I s Y s
m m s m k m k m k s k k m s k
I s Y ss s s
Transfer functions:
4 2 2
10 6
5 4 10 2 14 2 7
2( ) ( ) ( )
2 2 2 2
1.474 10 2.5 10 ( ) ( )
1.232 10 4.846 10 2.6 10 560 6.5 10
s i as y s
b s s s s r b s s r s s
y s
k k qY s I s X s
m m s m k m k m k s k k m s k
I s X ss s s
An LP Centrifugal Compressor (ISO 14839)
24
Decentralized control design
Stabilization requires a large increase in the phase
An LP Centrifugal Compressor (ISO 14839)
Open-loop poles:
1.4155e 014 6.3138e 002i
1.4155e 014 6.3138
7.2760e 00
e 002i
1
7.2760e 001
25
3( ) 600C s s
Three PD controllers (to approximate 3 lead compensators)
An LP Centrifugal Compressor (ISO 14839)
26
Compensated bode plots
An LP Centrifugal Compressor (ISO 14839)
Closed-loop poles: 117820.98, 769.35 526.27 90.11 , 526.27 90.11j j
27
An LP Centrifugal Compressor (ISO 14839)
Closed-loop poles in the presence of aero cross coupling:
72.76
116071.43
0 631.38
0 631.38
0 631.38
0 631.38
72.76
72.76
7
2.76
j
j
j
j
0.012
28
Challenges in Control of AMB Systems
PID Control and lead/lag compensators Choice of parameters Coupling between channels Flexible rotor leads to higher order plant model
State Space Representation and
Robust Control
29
A More Realistic Design Example
Axial Sensor
Backup Bearings
Radial AMB
Motor
Radial AMB
Middle Disk
Radial Sensors
Gyroscopic Disk
Thurst Bearing
Radial Sensors
Backup Bearing
Exciter
Speed Sensor
Radial Sensors
30
A More Realistic Design Example
Parameter varying (gyroscopic effects):
0
0x x xr r
y y yr r
q q uA G Bdq q uG A Bdt
Potential approach: LPV (Linear Parameter Varying) Approach
– Based on gain scheduling
– Conservative in performance
31
A More Realistic Design Example
Piecewise Design several
controllers at different speeds;
Each controllers robust in a speed range;
Switch between controllers as the speed varies;
Bumpless switching is the key
0 5000 10000 15000 20000 25000 30000 0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4Robust Performance
Speed (rpm)
Perf
orm
ance (
Mu)
Controller1Controller2
32
A More Realistic Design Example
Control Switching
Conditions for a Bumpless Transfer:
)()( 12 ss tutu
2 1( ) ( )s s
k k
t t t tk k
d u t d u t
dt dt
,2,1k
33
A More Realistic Design Example
Bumpless TransferBuild an observer that estimates the off line controller state from the on line controller output
Use the estimated state as the initial state at time of switching
As a result,
2 1( ) ( )s s
k k
t t t tk k
d u t d u t
dt dt
2 1( ) ( )s su t u t
34
A More Realistic Design Example
Piecewise controller design
• Controller 1 at nominal speed 10,000 rpm
• Controller 2 at nominal speed 15,000 rpm
• Each covers +/- 4000 rpm
• 48th order controllers
0 5000 10000 15000 20000 25000 30000 0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4Robust Performance
Speed (rpm)
Per
form
ance
(Mu)
Controller1Controller2
35
A More Realistic Design Example
Transfer at 12,000 rpm (upper bearing, x direction)
36
Transfer at 12,000 rpm (upper bearing, y direction)
A More Realistic Design Example
37
Nonlinearity of the AMB input-output characteristics
Constrained Control Theory
More Challenges in Control of AMB Systems