f.b. yeh & h.n. huang, dept. of mathematics, tunghai univ. 2004.nov.8 fang-bo yeh and huang-nan...

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ.

2004.Nov.8

Fang-Bo Yeh and Huang-Nan HuangFang-Bo Yeh and Huang-Nan Huang

Department of MathematicTunghai University

The 2 by 2 Spectral Nevanlinna Pick Controller Design Problem


2004.Nov.8 22

Outline

IntroductionIntroduction -- Analysis and SynthesisAnalysis and Synthesis Problem DescriptionProblem Description Spectral NP Interpolation Theory: 2 by 2 caseSpectral NP Interpolation Theory: 2 by 2 case Algorithm of Algorithm of -Synthesis via SNP Theory-Synthesis via SNP Theory Numerical ExamplesNumerical Examples ConclusionsConclusions


2004.Nov.8 33

Introduction -norm the-norm the structured singular value is a powerful is a powerful

tool in robust control..tool in robust control.. Spectral normSpectral norm is the lower bound of is the lower bound of -norm, and -norm, and

norm is its upper bound. Hnorm is its upper bound. H control is too conserva control is too conservative.tive.

No define theory for No define theory for -synthesis-synthesis.. SNP interpolation theory is developed with aims to SNP interpolation theory is developed with aims to

solve this problem.solve this problem. Formulate controller synthesis into SNP interpolatioFormulate controller synthesis into SNP interpolatio

n problem.n problem. Design Design --controller using SNP theory: 2 by 2 case.controller using SNP theory: 2 by 2 case.


2004.Nov.8 44

Robust Control Problem

Design K such that

is internally stable and track r under the influence:

1. perturbations in system model2. disturbance in actuator3. sensor noise

KA

uy +

r

S

Pe


2004.Nov.8 55

Type of Uncertainties

Real parametric uncertainty: e.g. a given plantReal parametric uncertainty: e.g. a given plant

Unstructured uncertainty: unmodeled dynamicsUnstructured uncertainty: unmodeled dynamics

2( ) , 0.8 1.2

2 1P s

s s

0( ) ( ) ( ), a aP s P s s

1. Additive type -

a

P0(s)

uy +


2004.Nov.8 66

Type of Uncertainties

Unstructured uncertainty: unmodeled dynamicsUnstructured uncertainty: unmodeled dynamics

2. Multiplicative type –

0( ) ( )( ( ))mP s P s I s 0( ) ( ( )) ( )mP s I s P s

m

m

P0(s)

uy +

m

P0(s)

uy +


2004.Nov.8 77

Robust Control Problem, Again

r: reference inputd: disturbancen: noise

KA

uy +

r

S

Pe

+

+

n

d

Design Philosophy: “Shaping” i.e. filtering W1, W2, W3

KA

uy +

r

S

Pe

+

+

n

d

W33z

W22z

W11z


2004.Nov.8 88

Structured Uncertainty

M

w

z

K

G

uy

w

z

G K M

0 1 2 3

1

2

3

,

P A S W W W K G

r z

d w z z

n z

Robust stability: (w=0,z=0) M+ is stable

Robust performance: Design K such that (i) M+ is stable (ii) 2

2

sup 1z

w


2004.Nov.8 99

IntroductionIntroduction -Analysis and Synthesis-Analysis and Synthesis Problem DescriptionProblem Description Spectral NP Interpolation Theory: 2 by 2 caseSpectral NP Interpolation Theory: 2 by 2 case Algorithm of Algorithm of -Synthesis via SNP Theory-Synthesis via SNP Theory Numerical ExamplesNumerical Examples ConclusionsConclusions

Outline


2004.Nov.8 1010

-Analysis and Synthesis

Definition of

Consider a matrix MCnn (the plant) and Cn n the structured uncertainty set.

1 1

1

: repeated scalar block,

: full complex block.j j

r

S rS

F

i ri i

m mj

I

I

I

Uncertainty

M

1 1

,

( ) 1

S F

i ji j

r m n

B

Δ

Δ

= the smallest that causes M “instability”


2004.Nov.8 1111

(M) (M) (M)

maxU U

(UM) (M)

maxU U

(UM) (M) infDD

(DMD 1)

- When (S=1,F=0,r1=n) , (S=0,F=1,m1=n), the equality hold.

1 1 1*

*

diag , , , , :{ : },

, 0, , 0i i

S m k mk

n r ri i i j j

D D d I d Iu U UU I D

D D D d d

Bounds on

* Lower bound always holds, but the set of (UM) is not convex,

* Upper bound holds when 2S+F≤3.

1 ( ) = inf ( )D

M DM D

D

-


2004.Nov.8 1212

Linear Fractional Transformation(LFT)Let M be a complex matrix of the form

M M11 M12

M21 M22

C( p1 p2 )(q1 q2 )

M

w

z

M

w

z

Define the lower LFT Fl as

111 12 22 21

( , ) , wit

( (

h

, ) )

l

l

F M M M I M

M w

M

z F

122 21 11 12

( , ) ,

(

with

, ) ( )u

uz F M w

F M M M I M M

Define the upper LFT Fu as


2004.Nov.8 1313

: sup ( ( )), : sup ( ( )), : sup ( ( ))s

s

M M j M M j M M j

M M M

- Norm

{ : ( ) for all Re 0, }s s s S S C

Robust Stability using -Synthesis

-Let S denote the set of real-rational, proper, stable transfer matrices. Let

Robust Stability The loop shown is well-posed and internally stable for all S with ||||<1 if and only if

|| || : sup ( ( )) 1M M j

M


2004.Nov.8 1414

|| || : sup ( ( )) 1RP

M M j

Robust Performance

Robust Performance For all S with ||||<1, the loop shown is well-posed, internally stable, and || Fu(M, ) || <1 if and only if

Mwz

Mwz

F

RP

M

0: : ,

0w zn n

RP FF

S


2004.Nov.8 1515


Outline


2004.Nov.8 1616

Problem DescriptionFind K such that where

|| || : sup ( ( )) 1M M j

111 12 22 21( , ) ( )lM F G K G G K I G K G

G is chosen, respectively, as

• nominal performance (=0):

• robust stability only:

• robust performance:

11 12 13

21 22 23

31 32 33

P P P

G P P P P

P P P

K

Gwz

M

11 13

31 33

P PG

P P

22 23

32 33

P PG

P P


2004.Nov.8 1717

where

111 12 22 21 1 2 3( ) 1M G G K I G K G T T QT

1 122 22( ) or ( )Q K I G K K I QG Q

By using lower bound on 1

D( ) max ( ) ( ) inf ( ) ( )

DU uM UM M DMD M

we arrive at new problem: Find Q such that

1 2 3( ) 1, T T QT

Spectral Model Matching Problem

Q Parameterization

Q

* *2 2 3 3T T T T I


2004.Nov.8 1818

1 2 3( ) ( ) 1T T QT F

Spectral NP Interpolation Problem

Interpolation Problem

Let pi, i=1,2,…,n be the RHP poles of T2(G12) , T3(G21); zj, j=1,2,…,mbe the RHP zeros of T2(G12) , T3(G21).

The problem becomes find analytic function F on RHP satisfying the interpolation conditions:

1

1

( ) ( ), ( ) 0

( ) ( )i i i

j j

F p T p Q p

F z T z

F Q K Solve


2004.Nov.8 1919

Remark for QOnce F is solved, the Q is computed as following:

• T2, T3 are square and invertible,

• T2 is left invertible, T3 is right invertible, hence there exists such that

and then

2 3,T T

1 12 1 3( )Q T F T T

*

3 * *22 2 3 3*

32

TTT T T T I

TT

31 2 3 2 2

3

** *2

1 3 3*2

0

0 0

0( )

0 0

TQF T T QT T T

T

Q TF T T T

T


2004.Nov.8 2020


Outline


2004.Nov.8 2121

Spectral NP Interpolation Problem

(1) ( ( )) 1, D,

(2) ( ) , 1, 2, , .i i

F

F W i n

Given distinct points …ninside open unit di

sk D and WW…WnCmm

find an analytic mm matrix function F such that

( ) 1m mm W W Define

( ) , D.mF then


2004.Nov.8 2222

Existence of the function F

(Bercovici, Foias & Tannenbaum,1989) Such a functio

n F exists if and only if there exists invertible mm m

atrices Mi, i=1,…,n such that

Difficulty: there are mmn unknowns in Mi, i=1,…,n.

1 1 *

, 1

( )0

1

n

i i i j j j

i j i j

I M W M M W M

Pick Matrix for NP problem: Choose Mi=I.


2004.Nov.8 2323

Existence of F (m=2)

(Agler & Young, 2001) Such a function F exists if and

only if there exist b1,…,bn,c1,…,cn such that

Note: there are only 2n unknowns instead of 2 2

n.

* 1122

1122

, 1

01

n

j ji i

j ji i

i j

i j

s bs bI

c sc s

where 2

trace , det , , 14

jj j j j j j j

ss W p W b c p j n


2004.Nov.8 2424

SNP Interpolation Problem: n=m=2 case

2 22 : ( ) 1 ,

a bW W W

c d

22

1 2 1 2 1 2

: {( , ) : 0,| | 1}

{( , ) :| | 1,| | 1}

s p s p

1 0 1= , trace( ), det( )RWR s W p W

p s

four complex unknowns a, b, c, and d.

If exists R such that

with two unknowns s and p.

Define the symmetrized bidisc

2 2(trace( ),det( ))W W W


2004.Nov.8 2525

Problem Transformation

1W

2W

2

D2

1

2

2 2( , )s p

1 1( , )s p

F

R


2004.Nov.8 2626

Modified SNP Interpolation Problem

2

1 1 2 2

(1) ( ) , ,

(2) ( ) , ( ) .

F

F W F W

D

2

1 1 1 2 2 2

(1) ( ) , ,

(2) ( ) ( , ), ( ) ( , ).s p s p

D

Agler & Young, 2000


2004.Nov.8 2727

Solution of Modified SNP Problem

2 22

0 20 0

( )( ) , ( ) 2

a b c pp s

c b a

21 2 1 2 0 0 0 0 0, , (4 2 ), 2 ( ), t s s q s s a t q b t q c q

Alger-Yeh-Young Theorem 2003 :

Suppose 0 is defined by

where

0 1 21

arg min 2 s s

then the solution () =(s(), p()) with

1 1 1

2 2 2

( ) ( , )

( ) ( , )

s p

s p

Given 1, 2D, (s1,p1),(s1,p2)2 find analytic function , ()2, 2Dsuch that

and i, si satisfy0

0

, 1, 22

ii

i

si

s


2004.Nov.8 2828

Symmetrized Bidisc

22

1 2 1 2 1 2

: {( , ) : 0,| | 1}

{( , ) :| | 1,| | 1}

s p s p


2004.Nov.8 2929

Geometry of

2

2with ,s p


2004.Nov.8 3030

Geometry of 222( , ) 2 4 4

Im 0

s p s sp s p s

p


2004.Nov.8 3131

Geometry of 222( , ) 2 4 4

Im 0

s p s sp s p s

s


2004.Nov.8 3232

Spectral InterpolationFind Analytic function Find Analytic function

such thatsuch that

2:

( ( ), ( ))

D

s p

D2

1

2

2 2( , )s p

1 1( , )s p

1 1 1

2 2 2

( ) ( , )

( ) ( , )

s p

s p


2004.Nov.8 3333

Main Idea

r0. .

Smallest Smallest

(0,1)r

2

2 2( , )s p

1 1( , )s p

1 1 2 2, & ,s p s p2is a Complex Geodesic of throughis a Complex Geodesic of through

1 1

2 2

(0) ( , )

( ) ( , )

s p

r s p

D


2004.Nov.8 3434

12

12

,1

p ss p

s

D

21

2

2 2( , )s p

1 1( , )s p

D

21

Totally geodesic disc

C

1 1 21 2

1 2

, tanh1

d

IsometryIsometry

( , ) ( ( ), ) D Dd C


2004.Nov.8 3535

Caratheodory Distance

: Caratheodory distance: Caratheodory distance C

1 1 2 2

1 1 2 2

( , ) , ( , )

sup d ( , ) , ( , )G

C s p s p

G s p G s p

2where : analyticG D

2

2 2( , )s p

1 1( , )s pD

21

G

0

0 01 1 2 2 = d ( , ) , ( , )s p s p

0

10 2

102

,1

p ss p

s

C


2004.Nov.8 3636

Conclusion :

2

2, , ( , ) 1

2

p ss p s p

s

D

2

2 2( , )s p

1 1( , )s p D

21

: Caratheorary Extremal

1 1 2 2 1 1 2 2( , ), ( , ) d ( , ), ( , )C s p s p s p s p


2004.Nov.8 3737

Kobayashi Distance

: Kobayashi distance: Kobayashi distance

is a corresponding extremal

2over all f : analytic D

1 1 2 2

1 2

( , ) , ( , )

inf d ,

s p s p

1 1 1

2 2 2

f( ) = ( , )

f ( ) = ( , )

s p

s p

D

21

2

2 2( , )s p

1 1( , )s p


2004.Nov.8 3838

known ：1. Schwarz Lemma ：

2. Lempert’ Lemma ：

If is convex ,

then

C

C

If C

D

21

2

2 2( , )s p

1 1( , )s p

D21

00

Did

1 2 1 2Then ( ( ), ( )) ( , ) C d


2004.Nov.8 3939

Complex Geodesic of

complex geodesic of :complex geodesic of :

or or

wherewhere

2

D

21

2

2 2( , )s p

1 1( , )s p

D21

00

Did

2

2

0

20 0

,

1 , maximizes

Re 2

t q

t q

( ) , , 1

0

0

2 20 0 0

2 20 0 0

( )( ) 2

1

4 2 2 ( )

4 2

ps

t q t q qp

t q t q q

( ) ( ), ( ) , s p


2004.Nov.8 4040

IntroductionIntroduction -Analysis and Synthesis-Analysis and Synthesis Problem DescriptionProblem Description Spectral NP Interpolation Theory: 2 by 2 caseSpectral NP Interpolation Theory: 2 by 2 case Algorithm of Algorithm of Synthesis via SNP TheorySynthesis via SNP Theory Numerical ExamplesNumerical Examples ConclusionsConclusions

Outline


2004.Nov.8 4141

Algorithm of -Synthesis via SNP Theory

First transform the robust performance First transform the robust performance

problem toproblem to thethe model matching formmodel matching form

1 2 3sup ( )( ) 1p

RT T QT j

( , ) sup ( ( , ))( ) 1

pl lR

F G K F G K j

1 2 3( ) ( ) 1T T QT F ( ( ) max ( ) ( ) )U U

M UM M

K

G

uy

w

z

122( ) K I QG Q


2004.Nov.8 4242

Algorithm of -Synthesis via SNPT (cont’d)

Modify the problem to the Modify the problem to the situation such that we can use situation such that we can use the solution of SNP problem.the solution of SNP problem.

Solve the SNP problem for the Solve the SNP problem for the function function FF . .

Find the controller Find the controller KK..

Iterate for the desired Iterate for the desired KK..

Di

D

iD

i ( ,0)is2

h

i M

0


2004.Nov.8 4343


Outline


2004.Nov.8 4444

Numerical Examples

Real parameter uncertaintyReal parameter uncertainty

Dynamical uncertainty: SISO caseDynamical uncertainty: SISO case

Dynamical uncertainty: 2 Input 2 output caseDynamical uncertainty: 2 Input 2 output case


2004.Nov.8 4545

10.375 0.95 0.375 ( ) ( , ) (1 )

0.95 0.95 0.95 0.95

0.375 0.3751 (1 )

0.95 0.95(1 ) 0.95

l

sM s F G K K K

s s s s

sKsQ

s s K s

1M

11

1

1 ( ) , [ 0.325,0.425].

1P s

s

Plant:

Structured uncertainty notation:

1

1 0.05 0.375 ( ) ( , )

(1 0.05 0.375)

0.95 0.375 0.3751

0.95 0.95 0.95 0.95

uP s F Gs

s

s s s s

G wz

Closed loop system:

0.375 0.375( ( )) ( (1 )) (1 ) 1

0.95 0.95M s sQ sQ

s s

1 2 1 T T Q

Solve max ( ) ( )sU

UM UM M M

Usince and

Real Parameter Uncertainty


2004.Nov.8 4646

Block Diagram:

Closed loop system:

K

p

Wu

+ PWp +yz

u

w

1 1

1 1

(1 ) (1 )( , )

(1 ) (1 )u u

lp p

W K PK P W K PKM F G K

W PK P W PK

Dynamic Uncertainty

Standard Notation:

K

G

uy

w

z

0 0 u

p p p

W

G W P W W P

P I P

diag( , )p F


2004.Nov.8 4747

Q parameterization:

0 0

(1 ) (1 )u u u

p p p p p

W QP W Q WM Q P I

W PQ P W PQ W P W W P

Dynamic Uncertainty (cont’d)1( )K I QP Q

Model Matching:

0 0

( ) ( )u

p p p

WM Q P I

W P W W P

Interpolation conditions: let zi, pj be the zeros and poles of P(s), then

0 ( ) ( )( ) ,

0 ( )u i i

ip i

W z Q zM z

W z

( ) 0( )

( ) / ( ) 0u j

jp j j

W pM p

W p K p

Restriction: M(s) must be 22 matrices and the total number of i+j, must be 2.


2004.Nov.8 4848

1 0.25 0.6 4 8( ) , , .

1 0.006 32p u

s sP s W W

s s s

Plant:

Closed loop system:

Dynamic Uncertainty: SISO case

1

4( 2) 4( 2) ( 1)

( 32)( 1 ) ( 32)( 1 )( , )

0.25( 2.4) 0.25( 2.4)( 1)

( 0.006)( 1 ) ( 0.006)( 1 )

0 1

, 4( 2) 0.25( 2.4)( 1)0

( 32)( 1 ) ( 0.006)( 1 )

l

s K s K s

s s K s s KM F G K

s s s

s s K s s K

R Rs K s s

s s K s s K

det( ) 0M

0

0p

F

Interpolation Condition:2

32( ) , , such that (0) ( ,0)

33 D

Remark: for all SISO system det( ( )) 0M s Choose bj,cj16/33, the existence of is guaranteed.


2004.Nov.8 4949

3 2

4 3 2

2103 5983 28711 28896

6500 33539 3799 24

s s sK

s s s

( )M


2004.Nov.8 5050

( )M

2.8 1sK

s


2004.Nov.8 5151

Algorithm:Step 1: transform RHP into unit disk via s=(1+)/(1) and zeros of T2 are

Step 2: 1=1,the interpolation conditions are

1. Solve

2

2

1 1 10 1 0

4(2 1)(3 1) 2(3 1) (3 1)( ) ( 3)( 1) , , 3 1

01 2 30 2(3 1)

8(2 1) (3 1)(5 3) 2(3 1)

p u

s

s s s sP s s s W W s

ss

s s s s

Plant:

Model matching problem:

Dynamic Uncertainty: 2x2 case

1 2( ) ( ) 1F T T Q

00

0

0 F

1 2, p p uT W T W P PW

1 20, 1/ 2

1 11 1

1 0 00 0 0 101 1 1

(0) (0) (0) , ( ) ( ) ( )5 11 12 2 200 0

0 0 54 4

F R R F R R

1 1 2 2

1 1 1(0) ( , ) ( ,0), ( ) ( , ) ( ,0)

4 2 5s p s p


2004.Nov.8 5252

2. Compute

Dynamic Uncertainty: 2x2 case (cont’d)

00 1 0 2 1 2

10

22 min | 2 |,

2i i

ii

p ss s s s

s

0 1 2

1 11, ,

7 11 leads to

3. 2 2

1 2 2

74 1 77 4 1( ) ( 2 , )

4 77 4 77

2

1

36 612 169 18 (2 1)( ) ( ( )) ( , ), 0

4(18 169) 18 169M g

4. The Möbius transform formation is 2 2

2 2

(182 103 6) (42 97 26)( ) , 1

(26 97 42) (6 103 182)

gM g

g

5.

4 3 2 3 2

1 3 2 3 2

130 38 1012.9 38 130 520 46 119 6( ) ( ( )) ( , ), 1

6 119 46 520 6 119 46 520M g


2004.Nov.8 5353

Dynamic Uncertainty: 2x2 case (cont’d)12 1

2( )

12

R

11

0 1( ) ( ) ( )

( ) ( )F R R

p s

and choose6. Let ( ) ( ), ( )s p

we have

7. Replace with (s-1)/(s+1). For example when g=0,

11 121 2

21 22

3 211

212

3 221

3 244

( ) ( )1( ) ,

( ) ( )(3 1)(151 187)( 1)

1( ) ( 1)(271 1351 161 521)

21

( ) ( 1)( 3)(377 106 449)41

( ) ( 1)(31 373 889 147)21

( ) ( 3)(679 707 13934

F s F sF s

F s F ss s s

F s s s s s

F s s s s s

F s s s s s

F s s s s s

75)


2004.Nov.8 5454


2004.Nov.8 5555

Step 3: 1=1/2,repeat again.


2004.Nov.8 5656

Step 4: 1=1/3,repeat again.


2004.Nov.8 5757

IntroductionIntroduction - Analysis and Synthesis- Analysis and Synthesis Problem DescriptionProblem Description Spectral NP Interpolation Theory: 2 by 2 caseSpectral NP Interpolation Theory: 2 by 2 case Algorithm of Algorithm of -Synthesis via SNP Theory-Synthesis via SNP Theory Numerical ExamplesNumerical Examples ConclusionsConclusions

Outline


2004.Nov.8 5858

Conclusions

J.Agler, F.B.Yeh, N.J.Young J.Agler, F.B.Yeh, N.J.Young Realization of Functions Into the Symmetrised Bidisc,Realization of Functions Into the Symmetrised Bidisc,

Operator Theory: Advances and Applications,Vol.143,p1-37, 2003.Operator Theory: Advances and Applications,Vol.143,p1-37, 2003.Web: http://www.math.thu.edu.tw/~fbyehWeb: http://www.math.thu.edu.tw/~fbyeh

The transformation from the The transformation from the --synthesis problem to a synthesis problem to a spectral model matching problem is given.spectral model matching problem is given.

Propose a algorithm using the SNP theory to solve Mu-Propose a algorithm using the SNP theory to solve Mu-synthesis.synthesis.

With the development of SNP theory, this method could With the development of SNP theory, this method could be used more practically.be used more practically.


2004.Nov.8 5959

Comments and Questions.

Thanks for your attention.Thanks for your attention.

f.b. yeh & h.n. huang, dept. of mathematics, tunghai univ. 2004.nov.8 fang-bo yeh and huang-nan...

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