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Model Reductionwithin Control Systems

Design

Pepijn WortelboerPHILIPS OPTICAL STORAGE

January 2001


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Introduction

Control problem formulation Need for order reduction algorithms Introduction of CD-player mechanism Standard reduction techniques: modal and balanced Closed-loop reduction Iterative low-order control design for CD-player Concluding remarks


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Control problem formulation

Linear time-invariant model based standard plant formulation tuning by weighting function design

Goal:

simplest K achieving small z

G

K

w K M G z ),,(I=

z w


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Need for order reduction algorithms

High-speed high-accuracy servo-mechanisms implyhigh-order models

Highest frequency modes in model are not reliable

High-order model is not suited for optimal control Controller order exceeds model orderBut

order reduction not goal in itself

Part of design


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Rotating compact disc swing arm for radial tracking with optical pick-up unit

(vertically suspended lens)for focussing

mounting plate, disc, armnot infinitely stiff

2x2 servo controldesign problem

Introduction of CD-player mechanism


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Radial control loop

K G

distw

track w e i

No weights on input signals output weights

Ws to penalize low-frequenterror e

Wt to penalize high-frequentcurrent i


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Standard reduction techniques:

Modal A diagonal+ invariant for input & output+ direct+ no stability constraint

balanced Gramians P & Q diag+ input-output significant

Both suitable for truncation and singular perturbation Not optimal, how about closed-loop?


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Transform

model state controller state

to get closed-loop systemGramians with

closed-loop G-HSV closed-loop K-HSV

Closed-loop balancing

K

M

G

T

I

T

(

(

K

G

G

K

M

G

K

M


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Closed-loop balanced reduction

Solve P, Q from Lyapunov equations of I (G,M,K)

extract G -part: P G & Q G find balancing transformation:

T * Q G T = T-1 P G T

-* = diag( HSV G )

apply T to G : G bal =(T -1 AG T, T -1 B G , C G T ,D G ) truncate G

bal


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Reduction in twin feedback configuration

),,( bal K M GG hmm R I=(

G

K

M

Tuning freedom:

selection of order

controller or model


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Separate weighting in the form of power spectra

wh z mm K M GG R =

),,( bal I

(

( )

piecewise quadratic interpolationG

K

z w


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Iterative low-order control design

Available: High order model G Weights in M First guess controller K

Model reduction

Controller reduction

),,(

),,(),,(

K M G

K M G K M G

h

mh

I

II =

),,( r K M GI=

Optimal control


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CD-player example

Model G120 Ws order 1, Wt order 2 First guess

order-3 controller:


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Closed-loop evaluation

10

2

10

3

10

4-80

-60

-40

-20

0

20

40

60

open-loop -G b

*K b

[ d B ]

10

2

10

3

10

4-1 5

-1 0

-5

0

5

10

15

20

controller K b

[ d B ]

102

103

104

-50

-40

-30

-20

-10

0

10

20sensitivity

Frequency [Hz]

[ d B ]

102

103

104

-8 0

-6 0

-4 0

-2 0

0

20complementary sensitivity

Frequency [Hz]

[ d B ]

ComplementarySensitivity

Sensitivity

bb K G b K

76.91323 = 77.91317 =

No visual difference

Model reduction OK

K23 K17


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HSV and performance measures

10 20 30 4010

-6

10-5

10-4

10-3 s tep aa

d e

l t a

& e

s t i m a

t e s

reduced order model4 6 8 10 1 2 14 1 6 1 8 20 2 2

0

500

1000

1500

2000

2500s t ep cc

g a m m a

& e

s t i m a

t e s

reduced order controller

5 10 15 2010

-6

10-4

10-2

100 step dd

d e

l t a

& e

s t i m a

t e s

reduced order model4 6 8 10 12 1 4 16

0

500

1000

1500

2000

2500step ff

g a m m a

& e

s t i m a

t e s

reduced order controller

HSV(G)

HSV(G)

HSV(K)

HSV(K)


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102

103

104

-120

-100

-8 0

-6 0

-4 0

-2 0

0

20

40

G120G120-G20G20-G14

G120

G20-G14

G120-G20

Models


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102

103

104

- 20

- 15

- 10

- 5

0

5

1 0

1 5

2 0K23K9K3

Controllers

K3

K23K9


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102

1 03

104-4 0

-3 0

-2 0

-1 0

0

10

20

30

40

max. sv of G23, G13, and difference

Direct controller reductionwith manually shaped

frequency weights

102

103

104-15

-10

-5

0

5

10

15

20

102

103

104-4 0

-3 0

-2 0

-1 0

0

10

20

30

40

max. sv of G23, G11, and difference

Plain balanced reduction

23

12

Interval weightedbalanced reduction23 to 11

Interval weightedbalanced reduction23 to 13

K23Kreduced

error

weighting


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Recommendations from Experience: See order reduction as a tool for control design Use robust and fast algorithms Exploit specific structure of model Apply iterative scheme (with frequency tuning) Evaluate results in more than one measure (graphics) Choose odd or even order

Do not rely on a priori HSV-based error bounds Customize and automate in later stage

Further reading:

P.M.R. Wortelboer, M. Steinbuch and O.H. BosgraIterative Model and Controller Reduction using Closed-loop Balancing,

with application to a Compact Disc Mechanism Int. J. Robust Nonlinear Control 9, 123-142 (1999)

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