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